Abstract: The potential of frequency-tuned surfaces as a passive control strategy for reducing drag in wall-bounded turbulent flows is investigated using resolvent analysis. These surfaces are considered to have geometries with impedances that permit transpiration and/or slip at the wall in response to wall pressure and/or shear and are tuned to target the dynamically important structures of wall turbulence. It is shown that wall impedance can suppress the modes resembling the near-wall cycle and the very-large-scale motions and the Reynolds stress contribution of these modes. Suppression of the near-wall cycle requires a more reactive impedance. In addition to these dynamically important modes, the effect of wall impedance across the spectral space is analysed by considering varying mode speeds and wavelengths. It is shown that the materials designed for suppression of the near-wall modes lead to gain reduction over a wide range across the spectral space. Furthermore, a wall with only shear-driven impedance is found to suppress turbulent structures over a wider range in spectral space, leading to an overall turbulent drag reduction. Most importantly, the present analysis shows that the drag-reducing impedance is non-unique and the control performance is not sensitive to variations of the surface impedance within a favourable range. This implies that specific frequency bandwidths can be targeted with periodic material design.

Publication: Journal of Fluid Mechanics Vol.: 959ISSN: 0022-1120

ID: CaltechAUTHORS:20230412-109103400.2

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Abstract: In this work, we demonstrate how physical principles—such as symmetries, invariances and conservation laws—can be integrated into the dynamic mode decomposition (DMD). DMD is a widely used data analysis technique that extracts low-rank modal structures and dynamics from high-dimensional measurements. However, DMD can produce models that are sensitive to noise, fail to generalize outside the training data and violate basic physical laws. Our physics-informed DMD (piDMD) optimization, which may be formulated as a Procrustes problem, restricts the family of admissible models to a matrix manifold that respects the physical structure of the system. We focus on five fundamental physical principles—conservation, self-adjointness, localization, causality and shift-equivariance—and derive several closed-form solutions and efficient algorithms for the corresponding piDMD optimizations. With fewer degrees of freedom, piDMD models are less prone to overfitting, require less training data, and are often less computationally expensive to build than standard DMD models. We demonstrate piDMD on a range of problems, including energy-preserving fluid flow, the Schrödinger equation, solute advection-diffusion and three-dimensional transitional channel flow. In each case, piDMD outperforms standard DMD algorithms in metrics such as spectral identification, state prediction and estimation of optimal forcings and responses.

Publication: Proceedings of the Royal Society A: Mathematical, physical, and engineering sciences Vol.: 479 No.: 2271 ISSN: 1364-5021

ID: CaltechAUTHORS:20230321-821389800.34

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Abstract: The probability density function (PDF) of the instantaneous streamwise velocity has consistently been used to extract information on the formation of uniform momentum zones (UMZs) in wall-bounded flows. Its temporal evolution has previously revealed patterns associated with the geometry and amplitude of the underlying velocity fluctuations [Laskari and McKeon, J. Fluid Mech. 913, A6 (2021)]. In this paper, we examine the robustness of these patterns in a variety of data sets including experiments and wall-bounded flow models. Experimental data sets spanning a range of Reynolds numbers, with very long temporal and spatial domains, suggest that the rate of the observed temporal variations scales in inner units. The use of a convection velocity, uniform across heights, to transform space into time has a marginal effect on these features. Similarly, negligible effects are observed between internal and external geometries. Synthetic databases generated following the resolvent framework and the attached eddy model are employed to draw comparisons to the experimental databases. Our findings highlight the distinctive strengths of each: The broadband frequency input of the attached eddy model allows for a better statistical description as opposed to a narrow frequency input in the resolvent data sets; instantaneously, however, representative eddies are seen to lack some structural details leading to the observed temporal behavior, which is better replicated by resolvent modes. Overall, given the considerable variety of the input data tested, the agreement between the data sets highlights the robustness of the spatiotemporal characteristics of the examined UMZs. It also underpins the need for their proper inclusion in UMZ modeling from a statistical as well as an instantaneous viewpoint; the current analysis accentuates important performance indicators for both.

Publication: Physical Review Fluids Vol.: 7 No.: 10 ISSN: 2469-990X

ID: CaltechAUTHORS:20221108-874083900.5

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Abstract: Research in modern data-driven dynamical systems is typically focused on the three key challenges of high dimensionality, unknown dynamics and nonlinearity. The dynamic mode decomposition (DMD) has emerged as a cornerstone for modelling high-dimensional systems from data. However, the quality of the linear DMD model is known to be fragile with respect to strong nonlinearity, which contaminates the model estimate. By contrast, sparse identification of nonlinear dynamics learns fully nonlinear models, disambiguating the linear and nonlinear effects, but is restricted to low-dimensional systems. In this work, we present a kernel method that learns interpretable data-driven models for high-dimensional, nonlinear systems. Our method performs kernel regression on a sparse dictionary of samples that appreciably contribute to the dynamics. We show that this kernel method efficiently handles high-dimensional data and is flexible enough to incorporate partial knowledge of system physics. It is possible to recover the linear model contribution with this approach, thus separating the effects of the implicitly defined nonlinear terms. We demonstrate our approach on data from a range of nonlinear ordinary and partial differential equations. This framework can be used for many practical engineering tasks such as model order reduction, diagnostics, prediction, control and discovery of governing laws.

Publication: Proceedings of the Royal Society A: Mathematical, physical, and engineering sciences Vol.: 478 No.: 2260 ISSN: 1364-5021

ID: CaltechAUTHORS:20220414-26938000

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Abstract: This paper discusses the modeling activity of the NATO-STO Research Task Group AVT-349. The aim of this group is to improve the understanding and modeling of boundary layers in the complex flow around water vehicles. As such, the focus is on incompressible, high-Reynolds-number flows that can be subject to non-equilibrium conditions such as strong pressure gradients, three-dimensionality, and surface roughness and heterogeneity. The Task Group has identified a reduced number of simpler problems in which the above conditions can be studied separately and in controlled environments. These include two-dimensional rough-wall boundary layers under both zero and non-zero pressure gradients, two-dimensional smooth-wall boundary layers subject to pressure gradients, and boundary layers around smooth bodies of revolution and three-dimensional obstacles. An experimental and computational data set is being assembled for further analysis and insight into the flow mechanisms involved, as well as the shortcomings of state-of-the-art models. This paper gives an outlook of the modeling effort within the Task Group, as well its different objectives. These include predicting the effect of roughness in equilibrium conditions; assessing the applicability and/or extension of equilibrium models and predictions to non-equilibrium conditions, in particular when outer-layer similarity is lost; the development of near-wall models based on a reduced-order resolvent framework; and the use of machine-aided methods in closure models.

ID: CaltechAUTHORS:20220210-928380000

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Abstract: We consider a linearized Navier–Stokes based model for compressible laminar boundary layers and study the response of these equations to stochastic white-in-time forcing. In particular we look at the different components of the forcing and the response of this linear model with the aim of understanding how the different mechanisms captured by the model change with increasing compressibility effects. We therefore analyze the response of the linear operator to individual components of forcing, i.e. the forcing to each of the momentum equations, the continuity and the energy equations of the linear operator. We also analyze the response obtained in the three components of velocity, in density and in temperature individually. For a fixed Reynolds number of Re=400, we consider Mach numbers ranging between Ma=0.05 and Ma=10 and different wall-cooling ratios. We find that, for all the Mach numbers considered here, the most amplified structures are the streamwise streaks forced by streamwise vortices. Previous studies have shown that these modes are highly amplified in the incompressible regime as well. However, as the Mach number increases, the contribution of the streamwise velocity to these streaks decrease, and the contribution of density and temperature to the streaks increase. Finally, we briefly look at the resolvent operator of the flow, and find that all the components of the forcing are important for the amplification of the Mach waves of the flow, and these modes are not captured by the stochastically forced model.

ID: CaltechAUTHORS:20220210-928407000

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Abstract: The spatial organization of small scales around large-scale coherent structures in a flat plate turbulent boundary layer is studied using a conditional-averaging technique applied to experimental and computational data. The technique averages the small-scale velocity conditioned on the projection coefficient between the instantaneous streamwise velocity field and a model for large-scale velocity structures in the wake and logarithmic regions. Two distinct scenarios are identified for the organization of the small scales: amplitude variation, in which at a given wall-normal location the small-scale intensity varies in amplitude across the streamwise extent of the large-scale structure, and height variation, in which the small-scale velocity intensity remains nearly constant along a curve that changed its wall-normal location across the streamwise extent of the large-scale structure. Small scales that are energetic at the wall-normal location where the large-scale structure is centered primarily show evidence of height variation, while small scales that are energetic at wall-normal locations far from the center of the large-scale structure primarily show evidence of amplitude variation. Connections can be drawn between the statistical observations characterized by the amplitude modulation statistic and the structural picture associated with vortex clusters.

Publication: Physical Review Fluids Vol.: 7 No.: 1 ISSN: 2469-990X

ID: CaltechAUTHORS:20220121-870773000

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Abstract: The conceptual picture underlying resolvent analysis is that the nonlinear term in the Navier-Stokes equations acts as an intrinsic forcing to the linear dynamics, a description inspired by control theory. The inverse of the linear operator, defined as the resolvent, is interpreted as a transfer function between the forcing and the velocity response. From a theoretical point of view this is an attractive approach since it allows for the vast mathematical machinery of control theory to be brought to bear on the problem. However, from a practical point of view, this is not always advantageous. The inversion of the linear operator inherent in the control theoretic definition obscures the physical interpretation of the governing equations and is prohibitive to analytical manipulation, and for large systems it leads to significant computational cost and memory requirements. In this work we suggest an alternative, inverse-free, definition of the resolvent basis based on an extension of the Courant–Fischer–Weyl min-max principle in which resolvent modes are defined as stationary points of a constrained variational problem. This definition leads to a straightforward approach to approximate the resolvent (response) modes of complex flows as expansions in any arbitrary basis. The proposed method avoids matrix inversions and requires only the spectral decomposition of a matrix of significantly reduced size as compared to the original system. To illustrate this method and the advantages of the variational formulation we present three examples. First, we consider streamwise constant fluctuations in turbulent channel flow where an asymptotic analysis allows us to derive closed form expressions for the optimal resolvent mode. Second, to illustrate the cost-saving potential and investigate the limits of the proposed method, we apply our method to both a two-dimensional, three-component equilibrium solution in Couette flow and, finally, to a streamwise developing turbulent boundary layer. For these larger systems we achieve a model reduction of up to two orders of magnitude. Such savings have the potential to open up RA to the investigation of larger domains and more complex flow configurations.

Publication: Physical Review Fluids Vol.: 7 No.: 1 ISSN: 2469-990X

ID: CaltechAUTHORS:20220207-90383000

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Abstract: We present an optimization-based method to efficiently calculate accurate nonlinear models of Taylor vortex flow. Through the resolvent formulation of McKeon & Sharma (J. Fluid Mech., vol. 658, 2010, pp. 336–382), we model these Taylor vortex solutions by treating the nonlinearity not as an inherent part of the governing equations but rather as a triadic constraint which must be satisfied by the model solution. We exploit the low-rank linear dynamics to calculate an efficient basis, the coefficients of which are calculated through an optimization problem to minimize the triadic consistency of the solution with itself as well as the input mean flow. Our approach constitutes, what is to our knowledge, the first fully nonlinear and self-sustaining, resolvent-based model in the literature. Our results compare favourably with the direct numerical simulation (DNS) of Taylor–Couette flow at up to five times the critical Reynolds number. Additionally, we find that as the Reynolds number increases, the flow undergoes a fundamental transition from a classical weakly nonlinear regime, where the forcing cascade is strictly down scale, to a fully nonlinear regime characterized by the emergence of an inverse (up scale) forcing cascade. Triadic contributions from the inverse and traditional cascade destructively interfere implying that the accurate modelling of a certain Fourier mode requires knowledge of its immediate harmonic and sub-harmonic. We show analytically that this is a direct consequence of the structure of the quadratic nonlinearity of the Navier–Stokes equations formulated in Fourier space. Finally, using our model as an initial condition to a higher Reynolds number DNS significantly reduces the time to convergence.

Publication: Journal of Fluid Mechanics Vol.: 924ISSN: 0022-1120

ID: CaltechAUTHORS:20210315-104125027

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Abstract: Direct simulations of two-dimensional channel flow of a viscoelastic fluid have revealed the existence of a family of Tollmien-Schlichting (TS) attractors that is nonlinearly self-sustained by viscoelasticity [Shekar et al., J. Fluid Mech. 897, A3 (2020)]. Here, we describe the evolution of this branch in parameter space and its connections to the Newtonian TS attractor and to elastoinertial turbulence (EIT). At Reynolds number Re=3000, there is a solution branch with TS-wave structure but which is not connected to the Newtonian solution branch. At fixed Weissenberg number, Wi, and increasing Reynolds number from 3000 to 10 000, this attractor goes from displaying a striation of weak polymer stretch localized at the critical layer to an extended sheet of very large polymer stretch. We show that this transition is directly tied to the strength of the TS critical layer fluctuations and can be attributed to a coil-stretch transition when the local Weissenberg number at the hyperbolic stagnation point of the Kelvin cat's eye structure of the TS wave exceeds 1/2. At Re=10000, unlike 3000, the Newtonian TS attractor evolves continuously into the EIT state as Wi is increased from zero to about 13. We describe how the structure of the flow and stress fields changes, highlighting in particular a “sheet-shedding” process by which the individual sheets associated with the critical layer structure break up to form the layered multisheet structure characteristic of EIT.

Publication: Physical Review Fluids Vol.: 6 No.: 9 ISSN: 2469-990X

ID: CaltechAUTHORS:20210504-093917792

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Abstract: The interaction between shear-driven turbulence and stratification is a key process in a wide array of geophysical flows with spatiotemporal scales that span many orders of magnitude. A quick numerical model prediction based on external parameters of stratified boundary layers could greatly benefit the understanding of the interaction between velocity and scalar flux at varying scales. For these reasons, here we use the resolvent framework [McKeon and Sharma, J. Fluid Mech., 658 (2010)] to investigate the effects of an active scalar on incompressible wall-bounded turbulence. We obtain the state of the flow system by applying the linear resolvent operator to the nonlinear terms in the governing Navier-Stokes equations with the Boussinesq approximation. This extends the formulation to include the scalar advection equation with the scalar component acting in the wall-normal direction in the momentum equations [Dawson, Saxton-Fox and McKeon, AIAA Fluid Dyn. Conf. 4042 (2018)]. We use the mean velocity profiles from a direct numerical simulation (DNS) of a stably stratified turbulent channel flow at varying friction Richardson number Ri_τ. The results obtained from the resolvent analysis are compared to the premultiplied energy spectra, autocorrelation coefficient, and the energy budget terms obtained from the DNS. It is shown that despite using only a very limited range of representative scales, the resolvent model is able to reproduce the balance of energy budget terms as well as provide meaningful insight into coherent structures occurring in the flow. Computation of the leading resolvent models, despite considering a limited range of scales, reproduces the balance of energy budget terms, provides meaningful predictions of coherent structures in the flow, and is more cost-effective than performing full-scale simulations. This quick model can provide a further understanding of stratified flows with only information about the mean profile and prior knowledge of energetic scales of motion in the neutrally buoyant boundary layers.

Publication: Physical Review Fluids Vol.: 6 No.: 8 ISSN: 2469-990X

ID: CaltechAUTHORS:20210223-153658783

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Abstract: Resolvent analysis identifies the most responsive forcings and most receptive states of a dynamical system, in an input–output sense, based on its governing equations. Interest in the method has continued to grow during the past decade due to its potential to reveal structures in turbulent flows, to guide sensor/actuator placement and for flow control applications. However, resolvent analysis requires access to high-fidelity numerical solvers to produce the linearized dynamics operator. In this work, we develop a purely data-driven algorithm to perform resolvent analysis to obtain the leading forcing and response modes, without recourse to the governing equations, but instead based on snapshots of the transient evolution of linearly stable flows. The formulation of our method follows from two established facts: (i) dynamic mode decomposition can approximate eigenvalues and eigenvectors of the underlying operator governing the evolution of a system from measurement data, and (ii) a projection of the resolvent operator onto an invariant subspace can be built from this learned eigendecomposition. We demonstrate the method on numerical data of the linearized complex Ginzburg–Landau equation and of three-dimensional transitional channel flow, and discuss data requirements. Presently, the method is suitable for the analysis of laminar equilibria, and its application to turbulent flows would require disambiguation between the linear and nonlinear dynamics driving the flow. The ability to perform resolvent analysis in a completely equation-free and adjoint-free manner will play a significant role in lowering the barrier of entry to resolvent research and applications.

Publication: Journal of Fluid Mechanics Vol.: 918ISSN: 0022-1120

ID: CaltechAUTHORS:20210315-104726854

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Abstract: The flow around a cylinder oscillating in the streamwise direction with a frequency, f_f, much lower than the shedding frequency, f_s, has been relatively less studied than the case when these frequencies have the same order of magnitude or the transverse oscillation configuration. In this study, particle image velocimetry and the dynamic mode decomposition (DMD) are used to investigate the streamwise-oscillating cylinder wake for forcing frequencies f_f/f_s ∼0.04–0.2 and mean Reynolds number, Re₀ = 900. The amplitude of oscillation is such that the instantaneous Reynolds number remains above the critical value for vortex shedding at all times. Characterization of the wake reveals a range of phenomena associated with the interaction of the two frequencies, including modulation of both the amplitude and frequency of the wake structure by the forcing. DMD reveals a frequency spreading of dynamic modes. A scaling parameter and associated transformation are developed to relate the unsteady, or forced, dynamics of a system to that of an unforced system. For the streamwise-oscillating cylinder, it is shown that this transformation leads to a dynamic mode decomposition similar to that of the unforced system.

Publication: Physical Review Fluids Vol.: 6 No.: 7 ISSN: 2469-990X

ID: CaltechAUTHORS:20200811-092143234

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Abstract: We present a framework for predicting the interactions between motion at a single scale and the underlying stress fluctuations in wall turbulence, derived from approximations to the Navier–Stokes equations. The dynamical equations for an isolated scale and stress fluctuations at the same scale are obtained from a decomposition of the governing equations and formulated in terms of a transfer function between them. This transfer function is closely related to the direct correlation coefficient of Duvvuri & McKeon (J. Fluid Mech., vol. 767, 2015, R4), and approximately to the amplitude modulation coefficient described in Mathis et al. (J. Fluid Mech., vol. 628, 2009, pp. 311–337), by consideration of interactions between triadically consistent scales. In light of the agreement between analysis and observations, the modelling approach is extended to make predictions concerning the relationship between very-large motions and small-scale stress in the logarithmic region of the mean velocity. Consistent with experiments, the model predicts that the zero-crossing height of the amplitude modulation statistic coincides with the wall-normal location of the very large-scale peak in the one-dimensional premultiplied spectrum of streamwise velocity fluctuations, the critical layer location for the very large-scale motion. Implications of fixed phase relationships between small-scale stresses and larger isolated scales for closure schemes are briefly discussed.

Publication: Journal of Fluid Mechanics Vol.: 914ISSN: 0022-1120

ID: CaltechAUTHORS:20210312-151430797

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Abstract: The nonlinear mechanism in the self-sustaining process (SSP) of wall-bounded turbulence is investigated. Resolvent analysis is used to identify the principal forcing mode that produces the maximum amplification of the velocities in numerical simulations of the minimal channel for the buffer layer and a modified logarithmic (log) layer. The wavenumbers targeted in this study are those of the fundamental mode, which is infinitely long in the streamwise direction and once-periodic in the spanwise direction. The identified mode is then projected out from the nonlinear term of the Navier–Stokes equations at each time step from the simulation of the corresponding minimal channel. The results show that the removal of the principal forcing mode of the fundamental wavenumber can inhibit turbulence in both the buffer and log layer, with the effect being greater in the buffer layer. Removing other modes instead of the principal mode of the fundamental wavenumber only marginally affects the flow. Closer inspection of the dyadic interactions in the nonlinear term shows that contributions to the principal forcing mode come from a limited set of wavenumber interactions. Using conditional averaging, the flow structures that are responsible for generating the nonlinear interaction to self-sustain turbulence are identified as spanwise rolls interacting with oblique streaks. This method, based on the equations of motion, validates the similarities in the SSP of the buffer and log layer, and characterises the underlying quadratic interactions in the SSP of the minimal channel.

Publication: Journal of Fluid Mechanics Vol.: 914ISSN: 0022-1120

ID: CaltechAUTHORS:20191223-155902555

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Abstract: The response of a compliant surface in a turbulent boundary layer forced by a dynamic roughness is studied using experiments and resolvent analysis. Water tunnel experiments are carried out at a friction Reynolds number of Re_τ ≈ 410, with flow and surface measurements taken with 2D particle image velocimetry (PIV) and stereo digital image correlation (DIC). The narrow band dynamic roughness forcing enables analysis of the flow and surface responses coherent with the forcing frequency, and the corresponding Fourier modes are extracted and compared with resolvent modes. The resolvent modes capture the structures of the experimental Fourier modes and the resolvent with eddy viscosity improves the matching. The comparison of smooth and compliant wall resolvent modes predicts a virtual wall feature in the wall normal velocity of the compliant wall case. The virtual wall is revealed in experimental data using a conditional average informed by the resolvent prediction. Finally, the change to the resolvent modes due to the influence of wall compliance is studied by modeling the compliant wall boundary condition as a deterministic forcing to the smooth wall resolvent framework.

Publication: Fluids Vol.: 6 No.: 5 ISSN: 2311-5521

ID: CaltechAUTHORS:20210429-144554642

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Abstract: The probability density function (p.d.f.) of the streamwise velocity has been shown to indicate the presence of uniform momentum zones in wall-bounded turbulent flows. Most studies on the topic have focused on the instantaneous characteristics of this p.d.f. In this work, we show how the use of time-resolved particle image velocimetry data highlights robust features in the temporal behaviour of the p.d.f. and how these patterns are associated with the change of the number of zones present in the flow over time. The use of a limited resolvent model provides a clear link between this experimentally observed behaviour and the underlying velocity structures and their phase characteristics. This link is further supported by an extended resolvent model consisting of self-similar hierarchies centred in the logarithmic region, with triadically consistent members, yielding much more complex patterns in the p.d.f. Results indicate that the geometric similarity of these members instantaneously, as well as their relative evolution in time (dictated by their wall-normal varying wave speed), both inherent to the model, can reproduce many experimentally identified features.

Publication: Journal of Fluid Mechanics Vol.: 913ISSN: 0022-1120

ID: CaltechAUTHORS:20210311-134751536

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Abstract: Reduced-order models are often sought to efficiently represent key dynamical phenomena present among the broad range of temporal and spatial scales associated with unsteady and turbulent flow problems. Linear ‘input–output’ approaches and resolvent analyses reveal that important information about the most dangerous (most amplified) disturbances and the corresponding fluctuation response can be found with knowledge only of the base flow, or the turbulent mean field. In the work by Padovan et al. (J. Fluid Mech., vol. 900, 2020, A14), an important advance is made with regards to flows which have a periodically time-varying base flow, for example during unsteady vortex shedding from a body. By forming a harmonic resolvent relative to this base flow, limitations associated with the traditional linear resolvent are overcome to determine efficient bases for modelling of limit cycle flows and reveal novel information about key triadic (resonant) interactions.

Publication: Journal of Fluid Mechanics Vol.: 904ISSN: 0022-1120

ID: CaltechAUTHORS:20201022-112713087

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Abstract: Small spatial perturbations grow into fingers along the unstable interface of a fluid displacing a more viscous fluid in a porous medium or a Hele-Shaw cell. Mitigating this Saffman-Taylor instability increases the efficiency of fluid displacement applications (e.g., oil recovery), whereas amplifying these perturbations is desirable in, e.g., mixing applications. In this work, we investigate the Saffman-Taylor instability through analysis and experiments in which air injected with an oscillatory flow rate outwardly displaces silicone oil in a radial Hele-Shaw cell. A solution for linear instability growth that shows the competing effects of radial growth and surface tension, including wetting effects, is defined given an arbitrary reference condition. We use this solution to define a condition for stability relative to the constant flow rate case and make initial numerical predictions of instability growth by wave number for a variety of oscillations. These solutions are then modified by incorporating reference conditions from experimental data. The morphological evolution of the interface is tracked as the air bubble expands and displaces oil between the plates. Using the resulting images, we analyze and compare the linear growth of perturbations about the mean interfacial radius for constant injection rates with and without superimposed oscillations. Three distinct types of flow rate oscillations are found to modulate experimental linear growth over a constant phase-averaged rate of fluid displacement. In particular, instability growth at the interface is mildly mitigated by adding to the base flow rate provided by a peristaltic pump a second flow with low-frequency oscillations of small magnitude and, to a lesser extent, high-frequency oscillations of large amplitude. In both cases, the increased stability results from the selective suppression of the growth of large wave numbers in the linear regime. Contrarily, intermediate oscillations consistently destabilize the interface and significantly amplify the growth of the most unstable wave numbers of the constant flow rate case. Numerical predictions of low-frequency oscillations of opposite sign (initially decreasing) show promise of even greater mitigation of linear instability growth than that observed in this investigation. Looking forward, proper characterization of the unsteady, wetting, and nonlinear dynamics of instability growth will give further insight into the efficacy of oscillatory injection rates.

Publication: Physical Review Fluids Vol.: 5 No.: 12 ISSN: 2469-990X

ID: CaltechAUTHORS:20201217-110801490

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Abstract: This study investigates the physics underlying the drag increase in a low Reynolds number turbulent channel flow due to varying-phase opposition control by means of direct numerical simulation and modal analysis. The drag increase occurs for an extended region of the parameter space and we consider a controller with a positive phase shift in Fourier domain between sensor measurement and actuator response as a representative example for this regime. Analyses of instantaneous flow fields as well as spatial power spectra show that the structure of drag-increased flows is remarkably different from that of drag-reduced and canonical flows. In particular, the near-wall region is dominated by structures of short streamwise and large spanwise extent. Isolation of a representative control scale shows that these energetic structures can be characterized as spanwise rollers, which induce strong ejection and sweep motions and lead to drag increase. The presence of rollers, and therefore drag increase, in the full nonlinear system correlates well with the presence of an amplified eigenvalue in the eigenspectrum of the linearized Navier–Stokes operator. It is further shown that the scales responsible for drag increase at positive phase shifts are inactive at negative phase shifts and do not contribute to drag reduction. These scales can therefore be excluded from a controller aimed at drag reduction, which relaxes the spatial resolution requirements on the control hardware. The eigenspectrum may be used as a computationally cheap tool to identify such detrimental scales during an early design stage.

Publication: International Journal of Heat and Fluid Flow Vol.: 85ISSN: 0142-727X

ID: CaltechAUTHORS:20200805-102610802

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Abstract: This work applies resolvent analysis to compressible zero-pressure-gradient turbulent boundary layers with freestream Mach numbers between 2 and 4, focusing exclusively on large scale motions in the outer region of the boundary layer. We investigate the effects of Mach number on predicted flow structures, and in particular, look at how such effects may be attributed to changes in mean properties. By leveraging the similarity between the compressible and incompressible resolvent operators, we show that the shape of the streamwise velocity and temperature components of resolvent response modes in the compressible regime can be approximated by applying ideas from wavepacket pseudospectral theory to a simple scalar operator. This gives a means of predicting the shape of resolvent mode components for compressible flows without requiring the singular value decompositions of discretized operators. At a Mach number of 2, we find that accurate results are obtained from this approximation when using the compressible mean velocity profile. At Mach numbers of 3 and 4, the quantitative accuracy of these predictions is improved by also considering a local effective Reynolds number based on the local mean density and viscosity.

Publication: International Journal of Heat and Fluid Flow Vol.: 85ISSN: 0142-727X

ID: CaltechAUTHORS:20201009-091931261

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Abstract: Direct simulations of two-dimensional plane channel flow of a viscoelastic fluid at Reynolds number Re = 3000 reveal the existence of a family of attractors whose structure closely resembles the linear Tollmien–Schlichting (TS) mode, and in particular exhibits strongly localized stress fluctuations at the critical layer position of the TS mode. At the parameter values chosen, this solution branch is not connected to the nonlinear TS solution branch found for Newtonian flow, and thus represents a solution family that is nonlinearly self-sustained by viscoelasticity. The ratio between stress and velocity fluctuations is in quantitative agreement for the attractor and the linear TS mode, and increases strongly with Weissenberg number, Wi. For the latter, there is a transition in the scaling of this ratio as Wi increases, and the Wi at which the nonlinear solution family comes into existence is just above this transition. Finally, evidence indicates that this branch is connected through an unstable solution branch to two-dimensional elastoinertial turbulence (EIT). These results suggest that, in the parameter range considered here, the bypass transition leading to EIT is mediated by nonlinear amplification and self-sustenance of perturbations that excite the TS mode.

Publication: Journal of Fluid Mechanics Vol.: 897ISSN: 0022-1120

ID: CaltechAUTHORS:20200128-134130309

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Abstract: A resolvent-based reduced-order representation is used to capture time-averaged second-order statistics in turbulent channel flow. The recently proposed decomposition of the resolvent operator into two distinct families related to the Orr-Sommerfeld and Squire operators [Rosenberg and McKeon, Efficient representation of exact coherent states of the Navier-Stokes equations using resolvent analysis, Fluid Dyn. Res. 51, 011401 (2019)] results in dramatic improvement in the ability to match all components of the energy spectra and the uv cospectrum. The success of the new representation relies on the ability of the Squire modes to compete with the vorticity generated by Orr-Sommerfeld modes, which is demonstrated by decomposing the statistics into contributions from each family. It is then shown that this competition can be used to infer a phase relationship between the two sets of modes. Additionally, the relative Reynolds number scalings for the two families of resolvent weights are derived for the universal classes of resolvent modes presented by Moarref et al. [Moarref, Sharma, Tropp, and McKeon, Model-based scaling of the streamwise energy density in high-Reynolds-number turbulent channels, J. Fluid Mech. 734, 275 (2013)]. These developments can be viewed as a starting point for further modeling efforts to quantify nonlinear interactions in wall-bounded turbulence.

Publication: Physical Review Fluids Vol.: 5 No.: 8 ISSN: 2469-990X

ID: CaltechAUTHORS:20200128-133412087

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Abstract: The regeneration mechanism of streamwise vortical structures in the self-sustaining process of wall-bounded turbulence is investigated. Resolvent analysis [1] is used to identify the principal forcing mode which produces the maximum amplification of the response modes in the minimal channel for the buffer [2] and logarithmic layer [3]. The identified mode is then projected out from the nonlinear term of the Navier-Stokes equations at each time step from the direct numerical simulations (DNS) of the corresponding minimal channel. The results show that the removal of the principal forcing mode is able to significantly inhibit turbulence for the buffer and logarithmic layer while removing the subsequent modes instead of the principal one only marginally affects the flow. Analysis of the dyadic interactions in the nonlinear term shows that the contributions toward the principal forcing mode come from a limited number of wavenumber interactions. Using conditional averaging, the flow structures that are responsible for generating the principal forcing mode, and thus the nonlinear interaction to self-sustain turbulence, are identified to be spanwise rolls interacting with meandering streaks.

Publication: Journal of Physics Conference Series Vol.: 1522ISSN: 1742-6588

ID: CaltechAUTHORS:20200701-133438084

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Abstract: A series of experiments have been performed to study the response of a compliant (gelatin) surface to a turbulent boundary layer that has been forced by dynamic roughness. Through the synthetic flow structure generated by dynamic roughness, this work reduces the complexity of the fluid–structural problem in order to develop a fundamental framework with which to study flow control mechanisms. Flow velocity and surface deformations are measured using 2D particle image velocimetry and stereo digital image correlation, respectively. Both measurement methods are phase-locked to the roughness motion to allow the phase-averaged velocity and deformation fields to be isolated and correlated. The surface response of the non-dynamic roughness forced system is analyzed, and several spectral features are characterized. This analysis is used as context for the roughness forced deformations, subsequently confirming the response of the compliant surface to the synthetic flow mode. This demonstrates the potential of dynamic roughness experiments for studying flow control schemes and sets the stage for more detailed investigations of the effect of the compliant surface on the synthetic flow mode.

Publication: Experiments in Fluids Vol.: 61 No.: 4 ISSN: 0723-4864

ID: CaltechAUTHORS:20200316-150528520

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Abstract: The spatio-temporal response of a smooth-wall turbulent boundary layer to two-dimensional dynamic roughness forcing was investigated using flow structure-resolved particle image velocimetry. While such a flow has been extensively studied through pointwise temporal measurements, we characterize the streamwise-spatial nature of the streamwise and wall-normal velocity components of the synthetic mode excited by the roughness, which had not been previously fully characterized. Investigation of the spanwise variation of the response confirms a broad extent of two-dimensional characteristics in the streamwise and spanwise directions. Spatial amplitude modulation is observed in the synthetic structure and investigated directly through the spatial spectra. A parametric study of the influences of forcing frequency (and roughness height) leads to an empirical relationship between input frequency and resultant streamwise spatial lengthscale, which is proposed to extend the usefulness of dynamic roughness as a well-characterized input by which to probe the structure and dynamics of wall turbulence. In turn, dynamic roughness forcing may be used to study the response of complex fluid systems to these deterministic structures and explore the efficacy of possible flow control strategies in a directed manner.

Publication: Flow, Turbulence and Combustion Vol.: 104 No.: 2-3 ISSN: 1386-6184

ID: CaltechAUTHORS:20190910-154544575

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Abstract: Velocity structures associated with large streamwise density gradients in an incompressible turbulent boundary layer (with air as the working fluid, Pr = 0.71) are analysed experimentally using planar image velocimetry and aero-optic measurements. The resulting flow topologies for the velocity fluctuations associated with large negative and positive density gradients are in excellent agreement with results for coolings and heatings in time, respectively (Antonia & Fulachier, J. Fluid Mech., vol. 198, 1989, pp. 429–451). The current results are complimentary to those from Saxton-Fox et al. (AIAA J., vol. 57 (7), 2019, pp. 2828–2839), on the signature of the vertical velocity structures associated with large density gradients. In the present work, these structures are shown to exhibit a sign change, consistent with the scalar gradient, and are localised in the wall-normal direction with an average height of approximately 0.1δ, almost constant for increasing distance from the wall. The corresponding small-scale streamwise fluctuations also exhibit a consistent sign change, which is found to originate, on average, from upstream leaning structures. The emerging picture for the velocity field is then that of a multiscale phenomenon, where small-scale structures, responsible for large optical aberrations, are superimposed on the back of large-scale bulge-like structures that are known to populate the outer layers. The proposed conceptual model is consistent with early ideas of ‘typical’ eddies (Falco, Phys. Fluids, vol. 20 (10), 1977, pp. S124–S132), the hairpin vortex model and associated shear layers (Adrian et al., J. Fluid Mech., vol. 422, 2000, pp.1–54), as well as with notions of multiscale velocity organisation in shear layers (Klewicki & Hirschi, Phys. Fluids, vol. 16 (11), 2004, pp. 4163–4176; Saxton-Fox et al. 2019), and it provides new insight into the geometry of the small-scale velocity structures.

Publication: Journal of Fluid Mechanics Vol.: 885ISSN: 0022-1120

ID: CaltechAUTHORS:20200116-083832946

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Abstract: A methodology is presented that exploits both data-assimilation techniques and resolvent analysis for reconstructing turbulent flows, containing organized structures, with an efficient set of measurements. The mean (time-averaged) flow is obtained using variational data-assimilation that minimizes the discrepancy between a limited set of flow measurements, generally from an experiment, and a numerical simulation of the Navier–Stokes equations. The fluctuations are educed from resolvent analysis and time-resolved data at a single point in the flow. Resolvent analysis also guides where measurements of the mean and fluctuating quantities are needed for efficient reconstruction of a simple example case study: flow around a circular cylinder at a Reynolds number of Re=100. For this flow, resolvent analysis reveals that the leading singular value, most amplified modes, and the mean profile for 47

ID: CaltechAUTHORS:20190820-132839798

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Abstract: The resolvent formulation of McKeon & Sharma (J. Fluid Mech., vol. 658, 2010, pp. 336–382) is applied to supersonic turbulent boundary layers to study the validity of Morkovin’s hypothesis, which postulates that high-speed turbulence structures in zero-pressure-gradient turbulent boundary layers remain largely the same as their incompressible counterparts. Supersonic zero-pressure-gradient turbulent boundary layers with adiabatic wall boundary conditions at Mach numbers ranging from 2 to 4 are considered. Resolvent analysis highlights two distinct regions of the supersonic turbulent boundary layer in the wave parameter space: the relatively supersonic region and the relatively subsonic region. In the relatively supersonic region, where the flow is supersonic relative to the free-stream, resolvent modes display structures consistent with Mach wave radiation that are absent in the incompressible regime. In the relatively subsonic region, we show that the low-rank approximation of the resolvent operator is an effective approximation of the full system and that the response modes predicted by the model exhibit universal and geometrically self-similar behaviour via a transformation given by the semi-local scaling. Moreover, with the semi-local scaling, we show that the resolvent modes follow the same scaling law as their incompressible counterparts in this region, which has implications for modelling and the prediction of turbulent high-speed wall-bounded flows. We also show that the thermodynamic variables exhibit similar mode shapes to the streamwise velocity modes, supporting the strong Reynolds analogy. Finally, we demonstrate that the principal resolvent modes can be used to capture the energy distribution between momentum and thermodynamic fluctuations.

Publication: Journal of Fluid Mechanics Vol.: 883ISSN: 0022-1120

ID: CaltechAUTHORS:20200128-124558577

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Abstract: Analysis of the resolvent operator is used to study the properties of high-speed turbulent boundary layers for cooled walls. Previous study [1] shows that the resolvent response modes in the relatively subsonic region of high-speed turbulent boundary layers with adiabatic wall boundary conditions follow the same scaling law as those of the incompressible boundary layer case, validating Morkovin’s hypothesis on a mode-by-mode basis. Here, we study the effect of the cooled-wall boundary condition on the individual resolvent response modes to understand the underlying mechanisms that cause the failure of Morkovin’s hypothesis and velocity transformations for increasingly non-adiabatic walls. In particular, we show that the density and temperature resolvent mode shapes for the cooled-wall case exhibit a secondary peak in the inner and logarithmic layer, which is a result of the non-monotonic mean temperature profile that is absent in adiabatic cases. We also show that the secondary peak becomes more prominent with decreasing surface temperature ratio. The deviation of the mean velocity profiles is attributed to the change in the response modes in the near-wall region, the effect of which is propagated further away from the wall through nonlinear interactions.

ID: CaltechAUTHORS:20200113-074130443

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Abstract: The flows around a NACA 0018 airfoil at a chord-based Reynolds number of R_e = 10250 and angles of attack of α=0° and α=10° are modelled using resolvent analysis and limited experimental measurements obtained from particle image velocimetry. The experimental mean velocity fields are data assimilated so that they are solutions of the incompressible Reynolds-averaged Navier–Stokes equations forced by Reynolds stress terms which are derived from experimental data. Resolvent analysis of the data-assimilated mean velocity fields reveals low-rank behaviour only in the vicinity of the shedding frequency for α=0° and none of its harmonics. The resolvent operator for the α=10° case, on the other hand, identifies two linear mechanisms whose frequencies are a close match with those identified by spectral proper orthogonal decomposition. It is also shown that the second linear mechanism, corresponding to the Kelvin–Helmholtz instability in the shear layer, cannot be identified just by considering the time-averaged experimental measurements as an input for resolvent analysis due to missing data near the leading edge. For both cases, resolvent modes resemble those from spectral proper orthogonal decomposition when the resolvent operator is low rank. The α=0° case is classified as an oscillator and its harmonics, where the resolvent operator is not low rank, are modelled using parasitic modes as opposed to classical resolvent modes which are the most amplified. The α=10° case behaves more like an amplifier and its nonlinear forcing is far less structured. The two cases suggest that resolvent-based modelling can be achieved for more complex flows with limited experimental measurements.

Publication: Journal of Fluid Mechanics Vol.: 881ISSN: 0022-1120

ID: CaltechAUTHORS:20200124-083217262

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Abstract: This work develops a methodology for approximating the shape of leading resolvent modes for incompressible, quasi-parallel, shear-driven turbulent flows using prescribed analytic functions. We demonstrate that these functions, which arise from the consideration of wavepacket pseudoeigenmodes of simplified linear operators (Trefethen, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 461, 2005, pp. 3099–3122. The Royal Society), give an accurate approximation for the energetically dominant wall-normal vorticity component of a class of nominally wall-detached modes that are centred about the critical layer. We validate our method on a model operator related to the Squire equation, and show for this simplified case how wavepacket pseudomodes relate to truncated asymptotic expansions of Airy functions. Following the framework developed in McKeon & Sharma (J. Fluid Mech., vol. 658, 2010, pp. 336–382), we next apply a sequence of simplifications to the resolvent formulation of the Navier–Stokes equations to arrive at a scalar differential operator that is amenable to such analysis. The first simplification decomposes the resolvent operator into Orr–Sommerfeld and Squire suboperators, following Rosenberg & McKeon (Fluid Dyn. Res., vol. 51, 2019, 011401). The second simplification relates the leading resolvent response modes of the Orr–Sommerfeld suboperator to those of a simplified scalar differential operator – which is the Squire operator equipped with a non-standard inner product. This characterisation provides a mathematical framework for understanding the origin of leading resolvent mode shapes for the incompressible Navier–Stokes resolvent operator, and allows for rapid estimation of dominant resolvent mode characteristics without the need for operator discretisation or large numerical computations. We explore regions of validity for this method, and show that it can predict resolvent response mode shape (though not necessary the corresponding resolvent gain) over a wide range of spatial wavenumbers and temporal frequencies. In particular, we find that our method remains relatively accurate even when the modes have some amount of ‘attachment’ to the wall, and that that the region of validity contains the regions in parameter space where large-scale and very-large-scale motions typically reside. We relate these findings to classical lift-up and Orr amplification mechanisms in shear-driven flows.

Publication: Journal of Fluid Mechanics Vol.: 877ISSN: 0022-1120

ID: CaltechAUTHORS:20190429-080541243

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Abstract: Experimental studies of the changes in turbulence characteristics inside a boundary layer due to external forcing were performed using hot-wire anemometry. The forcing was created by a periodically forced shear layer that was external to a compressible subsonic turbulent boundary layer. The convecting coherent structures in the shear layer create a concomitant unsteady pressure and velocity field and provide an external disturbance for the turbulent boundary layer on the wall of the tunnel close to the forced shear layer. Both the pressure and velocity fluctuations inside the boundary layer were simultaneously measured along with the forcing signal, and a phase-locked analysis was performed. Regions of amplified turbulence inside the boundary layer were observed. Near the wall, the region of amplified turbulence was slightly upstream or lagging of the external forcing and away from the wall it was downstream or leading the forcing signal. Analysis of the convective speeds in the region of amplified turbulence supported the existence of the critical layer inside the wake region of the boundary layer, and the critical layer is believed to be responsible for the amplified levels of the turbulence in the wake region. Various modulation and amplification correlation coefficients were computed and analyzed, and the results also indicated the presence of the critical layer. Examination of the phase-locked turbulence revealed similarities between the turbulence amplitude amplification results due to these externally forced experiments and modulation response of an internally forced, subsonic boundary layer in the literature.

Publication: AIAA Journal Vol.: 57 No.: 9 ISSN: 0001-1452

ID: CaltechAUTHORS:20190806-084439876

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Abstract: We present an iterative method to compute traveling wave exact coherent states (ECS) in Couette and Poiseuille flows starting from an initial laminar profile. The approach utilizes the resolvent operator for a two-dimensional, three-component streamwise-averaged mean and exploits the underlying physics of the self-sustaining process. A singular value decomposition of the resolvent operator is used to obtain the mode shape for a single streamwise-varying Fourier mode. The self-interaction of the single mode is computed and used to generate an updated mean velocity input to the resolvent operator. The process is repeated until a nearly neutrally stable mean flow that self-sustains is obtained, as defined by suitable convergence criteria; the results are further verified with direct numerical simulation. The approach requires the specification of only two unknown parameters: the wave speed and amplitude of the mode. It is demonstrated that within as few as three iterations, the initial one-dimensional laminar field can be transformed into three-dimensional ECS.

Publication: Physical Review E Vol.: 100 No.: 2 ISSN: 2470-0045

ID: CaltechAUTHORS:20190911-153849331

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Abstract: It is demonstrated that time-evolving coherent structures with features consistent with Townsend's attached eddies and the developments associated with the reconstruction of flow statistics using the attached eddy hypothesis can be obtained from analysis of the (linear) resolvent operator associated with the Navier-Stokes equations. A discrete number of members of a single self-similar hierarchy, chosen by assuming a constant ratio of critical layer heights between neighboring members with overlapping wall-normal footprints, produces a geometrically self-similar, fractal-like structure of the spatial distribution of the velocity field and an associated signature of uniform momentum zones. The range of convection velocities associated with the hierarchy gives rise to time evolution of the velocity. The scaling of the streamwise wave number on a hierarchy can be reconciled with Townsend's distance-from-the-wall scaling for self-similar, attached eddies, at least conceptually, by considering coherent spatial structures in the velocity field and thus enforcing an exact equivalence between compared features. It has been shown previously that both the linear and nonlinear terms in the resolvent framework have self-similar representations in the logarithmic overlap region of the turbulent mean velocity profile. The conditions under which self-sustaining self-similar behavior may be obtained from self-similar hierarchical members in this region are elaborated and some similarities with other results and theories associated with self-similarity in this region are identified.

Publication: Physical Review Fluids Vol.: 4 No.: 8 ISSN: 2469-990X

ID: CaltechAUTHORS:20190826-092240301

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Abstract: These notes are intended to provide a description of some aspects of applications of resolvent analysis in fluid mechanics. They are targeted at students beginning research in this or related area, with the goal of providing a bridge between the generic linear algebra of textbooks and the archival journal articles implementing these techniques. With regards to mathematical techniques, a working understanding of fundamental concepts of linear algebra, and in particular the singular value decomposition, as well a familiarity with the goals underlying modal analysis are assumed. Requiring more background in fluid mechanics, the reader will already be familiar with the formulation of resolvent analysis from the Navier-Stokes equations. These notes outline approaches to gaining insight into the characteristics of fluid system by viewing the governing equations in terms of linear dynamics driven by endogenous (nonlinear) or exogenous forcing. Such systems may exhibit laminar or turbulent behavior, may be forced or unforced, and may perhaps be under the influence of control actuation. Physical interpretation of resolvent modes, the importance of mode weights and techniques for data reconstruction, and the incorporation of control into the analysis are covered.

ID: CaltechAUTHORS:20230224-222303039

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Abstract: The deflection of a small-aperture laser beam was studied as it passed through an incompressible turbulent boundary layer that was heated at the wall. The heating at the wall was sufficiently mild that the temperature and density fields acted as passive scalars with a Prandtl number of 0.71. Simultaneous particle image velocimetry and Malley probe laser deflection measurements were performed in overlapping regions of the boundary layer to identify correlations between coherent velocity structures, passive scalar transport, and optical beam deflection. Streamwise gradients in the streamwise and wall-normal velocity fields were observed to be correlated to the deflection of the optical beam and to streamwise density gradients. The passage of a large-scale motion through the beam path was shown to affect the statistics of the optical beam deflection as well as the local distribution of small-scale velocity features. The wall-normal small-scale velocity features were consistently correlated to the beam deflection, throughout different phases of the large-scale motion convection. The observations motivated a hypothesis that views the large scales as heat carriers, whereas the small scales modify the local sense of a velocity and density gradient toward a streamwise gradient that directly affects the optical beam deflection.

Publication: AIAA Journal Vol.: 57 No.: 7 ISSN: 0001-1452

ID: CaltechAUTHORS:20190806-081816859

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Abstract: The present study evaluates the capabilities of a low-order flow model based on the resolvent analysis of McKeon and Sharma [B. J. McKeon and A. S. Sharma, J. Fluid Mech. 658, 336 (2010)] for the purpose of controller design for drag reduction in wall-bounded turbulent flows. To this end, we first show that the model is able to approximate the change in mean wall shear stress, which is commonly used as measure for drag reduction. We also derive an analytical expression that decomposes the drag reduction in internal flows into terms that can be predicted directly by the model and terms that allow for quantification of model error if high-fidelity data are available. We then show by example of varying-phase opposition control in a low-Reynolds-number turbulent channel flow that the drag reduction predicted by the resolvent model captures the trend observed in direct numerical simulation (DNS) over a wide range of controller parameters. The DNS results confirm the resolvent model prediction that the attainable drag reduction strongly depends on the relative phase between sensor measurement and actuator response, which raises interesting flow physics questions for future studies. The good agreement between the resolvent model and DNS further reveals that resolvent analysis, which at its heart is a linear technique, is able to approximate the response of the full nonlinear system to control. We also show that in order to make accurate predictions the model only needs to resolve a small subset of the DNS wave numbers and that the controlled resolvent modes obey the Reynolds-number scaling laws of the uncontrolled resolvent operator derived by Moarref et al. [R. Moarref et al., J. Fluid Mech. 734, 275 (2013)]. As a consequence, our results suggest that resolvent analysis can provide a suitable flow model to design feedback flow control schemes for the purpose of drag reduction in incompressible wall-bounded turbulent flows even at technologically relevant Reynolds numbers.

Publication: Physical Review Fluids Vol.: 4 No.: 7 ISSN: 2469-990X

ID: CaltechAUTHORS:20190731-130426261

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Abstract: We use the feedback formulation of McKeon and Sharma [J. Fluid Mech. 658, 336 (2010)], where the nonlinear term in the Navier-Stokes equations is treated as an intrinsic forcing of the linear resolvent operator, to educe the structure of fluctuations in the range of scales (wave numbers) where linear mechanisms are not active. In this region, the absence of dominant linear mechanisms is reflected in the lack of low-rank characteristics of the resolvent and in the disagreement between the structure of resolvent modes and actual flow features. To demonstrate the procedure, we choose low Reynolds number cylinder flow and the Couette equilibrium solution EQ1, which are representative of very low-rank flows dominated by one linear mechanism. The former is evolving in time, allowing us to compare resolvent modes with dynamic mode decomposition (DMD) modes at the first and second harmonics of the shedding frequency. There is a match between the modes at the first harmonic but not at the second harmonic where there is no separation of the resolvent operator's singular values. We compute the self-interaction of the resolvent mode at the shedding frequency and illustrate its similarity to the nonlinear forcing of the second harmonic. When it is run through the resolvent operator, the “forced” resolvent mode shows better agreement with the DMD mode. A similar phenomenon is observed for the fundamental streamwise wave number of the EQ1 solution and its second harmonic. The importance of parasitic modes, labeled as such since they are driven by the amplified frequencies, is their contribution to the nonlinear forcing of the main amplification mechanisms as shown for the shedding mode, which has subtle discrepancies with its DMD counterpart.

Publication: Physical Review Fluids Vol.: 4 No.: 5 ISSN: 2469-990X

ID: CaltechAUTHORS:20190501-124814530

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Abstract: Simulations of elastoinertial turbulence (EIT) of a polymer solution at low Reynolds number are shown to display localized polymer stretch fluctuations. These are very similar to structures arising from linear stability (Tollmien-Schlichting modes) and resolvent analyses, i.e., critical-layer structures localized where the mean fluid velocity equals the wave speed. Computations of self-sustained nonlinear Tollmien-Schlichting waves reveal that the critical layer exhibits stagnation points that generate sheets of large polymer stretch. These kinematics may be the genesis of similar structures in EIT.

Publication: Physical Review Letters Vol.: 122 No.: 12 ISSN: 0031-9007

ID: CaltechAUTHORS:20190329-083647606

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Abstract: We describe the experimental generation of isolated vortical gusts and the interaction between these gusts and a downstream airfoil at a Reynolds number of 20,000. A standard method of generating a vortical gust has been to rapidly pitch an airfoil. A different approach is presented here: heaving a plate across a tunnel and changing direction rapidly to release a vortex. This method is motivated by the desire to limit a test article’s exposure to the wake of the gust generator by moving it to the side of the tunnel. Two suites of experiments were performed to characterize the performance of the gust generators and to measure the forces on and flow around the downstream airfoil. The novel mechanism allowed for measurement of the resumption of vortex shedding from the downstream airfoil, which was impossible with the pitching generator.

Publication: AIAA Journal Vol.: 57 No.: 3 ISSN: 0001-1452

ID: CaltechAUTHORS:20190314-082614909

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Abstract: A resolvent analysis of exact coherent states (ECS) of the Navier–Stokes equations (NSE) in a low Reynolds number channel is performed. The resolvent framework recasts the NSE into an input/output form in which the nonlinear term is treated as an internal forcing that drives the linear dynamics of the system. The framework has previously shown promise with regards to producing low-dimensional representations of ECS; here, we show that a componentwise analysis of the resolvent operator along with a Helmholtz decomposition of the nonlinear term reveals a simplified input/output form that clearly identifies and isolates the contributions of particular solenoidal forcing components to velocity/vorticity outputs. This new approach leads to an improved method for compact representations of ECS for both forcing and response fields, and establishes interesting connections to Orr–Sommerfeld/Squire modes in a nonlinear context.

Publication: Fluid Dynamics Research Vol.: 51 No.: 1 ISSN: 1873-7005

ID: CaltechAUTHORS:20190122-161813137

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Abstract: Analysis of the resolvent operator is used to study the properties of compressible planar Couette flow. In particular, we study how changing the Mach number affects the shape and amplitude of responses to optimal disturbances across a range of spatial and temporal frequencies. We consider Mach numbers up to 5, and show that the dependence of the resolvent norm(leading singular value) on streamwise and spanwise wavenumbers follows similar trends to the incompressible case, with the amplitude of the resolvent norm typically decreasing with increasing Mach number. An exception to this occurs when acoustic eigenmodes (which are not present in the incompressible regime) have eigenvalues sufficiently close to the temporal frequency ω such that modal resonance with this mode is the dominant contributor to the resolvent gain. This occurs, for example, for streamwise-constant disturbances for sufficiently low spanwise wavenumber. In addition, the resolvent formulation of the governing equations allows us to study independently the effects due to an altered mean/equilibrium profile due to compressibility, and the effects due to the changing linearized Navier-Stokes equations. This approach provides a framework for the study of compressible turbulent wall-bounded flows.

ID: CaltechAUTHORS:20190805-134837522

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Abstract: This work develops a framework for studying the behavior of a passive scalar field in incompressible wall-bounded turbulence using ·the resolvent operator. This approach expresses the stale of the system as the result of applying a linear (resolvent) operator to the nonlinear terms in the governing Navier-Stokes equations. By augmenting the system with a passive scalar equation, this formulation is used to study the relationship between velocity and scalar fluctuations. Additional insight into the mechanisms responsible for driving scalar fluctuations is attained by considering the resolvent form of the passive scalar equation in isolation from the momentum equations. We demonstrate that the passive scalar resolvent operator admits rescaling properties that relates the behavior or scalar fields with different diffusivities, and investigate the ability of this modeling framework to predict statistical properties of the fluctuating scalar field.

ID: CaltechAUTHORS:20190816-111358078

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Abstract: The use of dynamic roughness or small, oscillating, wall roughness elements to manipulate the structure of turbulent boundary layers is reviewed with a view to imposing true active control techniques. Linear and nonlinear responses to simple oscillations are detailed, such that the modifications of the overall structure of the turbulence and interactions between scales can be inferred. Although the work considered here uses mechanical actuation, the approach is considered sufficiently general to be implemented by a range of techniques, including engineered microelectromechanical system devices and metamaterial actuators.

Publication: AIAA Journal Vol.: 56 No.: 6 ISSN: 0001-1452

ID: CaltechAUTHORS:20180627-120113196

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Abstract: The structure of a turbulent boundary layer over a singly-periodic roughness of large wavelength is shown to give insight into the physics of rough-wall boundary layers. To this end, a roughness consisting of a single spanwise-varying mode and a single streamwise-varying mode was 3D printed with wavelengths on the order of the boundary layer thickness. The large length scale introduced by such a roughness creates spatial inhomogeneity of the mean velocity field throughout the entire boundary layer. A hot-wire probe was used to take time series of streamwise velocity at a grid of points in the x,y, and z directions, covering the volume over a single period of roughness, and allowing Fourier transforms of field variables to isolate the spatial variations correlating to the periodic geometry. The pre-multiplied Taylor-transformed wavelength power spectrum of streamwise velocity λTΦ(y, λT, x, z) can be Fourier-transformed in space to reveal that the portion of the power spectrum which varies most strongly in the streamwise direction is the portion with Taylor-transformed wavelength λT equal to the roughness wavelength λx. The spatial variation of the power spectrum at this wavelength exhibits a systematic change in phase across the boundary layer, which can be correlated to the phase of the spatially-varying time-averaged velocity field to reveal amplitude modulation of particular wavelengths by a roughness-induced synthetic scale. In a canonical smooth-wall boundary layer, the spatial variation of the mean velocity and the power spectrum would be identically zero due to translational symmetry. The introduction of a periodic roughness introduces the spatial variation in the power spectrum, but not directly. The roughness creates a stationary time-averaged velocity mode, but this mode does not appear in the power spectrum as it does not convect. The connection to the power spectrum must therefore be through non-linear interactions. It is shown that the correlation between the mean velocity and the power spectrum can be interpreted exactly as a measure of phase organization between pairs of convecting velocity modes which are triadically consistent with the stationary roughness velocity mode, analogously to amplitude modulation in canonical flows. Implications for real-world roughness are discussed.

Publication: International Journal of Heat and Fluid Flow Vol.: 71ISSN: 0142-727X

ID: CaltechAUTHORS:20180507-091749221

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Abstract: Eigenspectra and pseudospectra of the mean-linearized Navier-Stokes operator are used to characterize amplification mechanisms in laminar and turbulent flows in which linear mechanisms are important. Success of mean flow (linear) stability analysis for a particular frequency is shown to depend on whether two scalar measures of non-normality agree: (1) the product between the resolvent norm and the distance from the imaginary axis to the closest eigenvalue and (2) the inverse of the inner product between the most amplified resolvent forcing and response modes. If they agree, the resolvent operator can be rewritten in its dyadic representation to reveal that the adjoint and forward stability modes are proportional to the forcing and response resolvent modes at that frequency. Hence the real parts of the eigenvalues are important since they are responsible for resonant amplification and the resolvent operator is low rank when the eigenvalues are sufficiently separated in the spectrum. If the amplification is pseudoresonant, then resolvent analysis is more suitable to understand the origin of observed flow structures. Two test cases are studied: low Reynolds number cylinder flow and turbulent channel flow. The first deals mainly with resonant mechanisms, hence the success of both classical and mean stability analysis with respect to predicting the critical Reynolds number and global frequency of the saturated flow. Both scalar measures of non-normality agree for the base and mean flows, and the region where the forcing and response modes overlap scales with the length of the recirculation bubble. In the case of turbulent channel flow, structures result from both resonant and pseudoresonant mechanisms, suggesting that both are necessary elements to sustain turbulence. Mean shear is exploited most efficiently by stationary disturbances while bounds on the pseudospectra illustrate how pseudoresonance is responsible for the most amplified disturbances at spatial wavenumbers and temporal frequencies corresponding to well-known turbulent structures. Some implications for flow control are discussed.

Publication: Physical Review Fluids Vol.: 3 No.: 5 ISSN: 2469-990X

ID: CaltechAUTHORS:20180517-082629521

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Abstract: Simple aerodynamic configurations under even modest conditions can exhibit complex flows with a wide range of temporal and spatial features. It has become common practice in the analysis of these flows to look for and extract physically important features, or modes, as a first step in the analysis. This step typically starts with a modal decomposition of an experimental or numerical dataset of the flowfield, or of an operator relevant to the system. We describe herein some of the dominant techniques for accomplishing these modal decompositions and analyses that have seen a surge of activity in recent decades [1–8]. For a nonexpert, keeping track of recent developments can be daunting, and the intent of this document is to provide an introduction to modal analysis that is accessible to the larger fluid dynamics community. In particular, we present a brief overview of several of the well-established techniques and clearly lay the framework of these methods using familiar linear algebra. The modal analysis techniques covered in this paper include the proper orthogonal decomposition (POD), balanced proper orthogonal decomposition (balanced POD), dynamic mode decomposition (DMD), Koopman analysis, global linear stability analysis, and resolvent analysis.

Publication: AIAA Journal Vol.: 55 No.: 12 ISSN: 0001-1452

ID: CaltechAUTHORS:20171213-075938896

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Abstract: Large-scale motions (LSMs) in wall-bounded turbulent flows have well-characterised instantaneous structural features (Kovasznay et al., J. Fluid Mech., vol. 41 (2), 1970, pp. 283–325; Meinhart & Adrian, Phys. Fluids, vol. 7 (4), 1995, pp. 694–696) and a known spectral signature (Monty et al., J. Fluid Mech., vol. 632, 2009, pp. 431–442). This work aims to connect these previous observations through the development and analysis of a representative model for LSMs. The model is constructed to be consistent with the streamwise energy spectrum (Monty et al. 2009) and amplification characteristics of the Navier–Stokes equations (McKeon & Sharma, J. Fluid Mech., vol. 658, 2010, pp. 336–382), and is found to naturally recreate characteristics of instantaneous turbulent structures, including a bulge shape (Kovasznay et al. 1970) and the presence of uniform momentum zones (Meinhart & Adrian 1995) in the streamwise velocity field. The observed structural similarity between the LSM representative model and instantaneous experimental data supports the use of travelling wave models to connect statistical and instantaneous descriptions of coherent structures, and clarifies a simple general equivalency between symmetry in a Reynolds-decomposed velocity field and asymmetry in the laboratory frame.

Publication: Journal of Fluid Mechanics Vol.: 826ISSN: 0022-1120

ID: CaltechAUTHORS:20170911-104649363

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Abstract: The mixture-averaged thermal diffusion model originally proposed by Chapman and Cowling is validated using multiple flame configurations. Simulations using detailed hydrogen chemistry are done on one-, two-, and three-dimensional flames. The analysis spans flat and stretched, steady and unsteady, and laminar and turbulent flames. Quantitative and qualitative results using the thermal diffusion model compare very well with the more complex multicomponent diffusion model. Comparisons are made using flame speeds, surface areas, species profiles, and chemical source terms. Once validated, this model is applied to three-dimensional laminar and turbulent flames. For these cases, thermal diffusion causes an increase in the propagation speed of the flames as well as increased product chemical source terms in regions of high positive curvature. The results illustrate the necessity for including thermal diffusion, and the accuracy and computational efficiency of the mixture-averaged thermal diffusion model.

Publication: Journal of Turbulence Vol.: 18 No.: 12 ISSN: 1468-5248

ID: CaltechAUTHORS:20171211-080203757

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Abstract: The resolvent analysis of McKeon & Sharma (2010) recasts the Navier-Stokes equations into an input/output form in which the nonlinear term is treated as a forcing that acts upon the linear dynamics to yield a velocity response. The framework has shown promise with regards to producing low-dimensional representations of exact coherent states. Previous work has focused on a primitive variable output; here we show a velocity-vorticity formulation of the governing equations along with a Helmholtz decomposition of the nonlinear forcing term reveals a simplified input/output form in the resolvent analysis. This approach leads to an improved method for compact representations of exact coherent states for both forcing and response fields, with a significant reduction in degrees of freedom in comparison to the primitive variable approach.

ID: CaltechAUTHORS:20170731-084031416

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Abstract: Data assimilation can be used to combine experimental and numerical realizations of the same flow to produce hybrid flow fields. These have the advantages of less noise contamination and higher resolution while simultaneously reproducing the main physical features of the measured flow. This study investigates data assimilation of the mean flow around an idealized airfoil (Re = 13,500) obtained from time-averaged two-dimensional particle image velocimetry (PIV) data. The experimental data, which constitute a low-dimensional representation of the full flow field due to resolution and field-of-view limitations, are incorporated into a simulation governed by the two-dimensional, incompressible Reynolds-averaged Navier–Stokes (RANS) equations with an unknown momentum forcing. This forcing, which corresponds to the divergence of the Reynolds stress tensor, is calculated from a direct-adjoint optimization procedure to match the experimental and numerical mean velocity fields. The simulation is projected onto the low-dimensional subspace of the experiment to calculate the discrepancy and a smoothing procedure is used to recover adjoint solutions on the higher dimensional subspace of the simulation. The study quantifies how well data assimilation can reconstruct the mean flow and the minimum experimental measurements needed by altering the resolution and domain size of the time-averaged PIV.

Publication: Experiments in Fluids Vol.: 58 No.: 5 ISSN: 0723-4864

ID: CaltechAUTHORS:20170426-063702442

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Abstract: Known structures and self-sustaining mechanisms of wall turbulence are reviewed and explored in the context of the scale interactions implied by the nonlinear advective term in the Navier–Stokes equations. The viewpoint is shaped by the systems approach provided by the resolvent framework for wall turbulence proposed by McKeon & Sharma (J. Fluid Mech., vol. 658, 2010, pp. 336–382), in which the nonlinearity is interpreted as providing the forcing to the linear Navier–Stokes operator (the resolvent). Elements of the structure of wall turbulence that can be uncovered as the treatment of the nonlinearity ranges from data-informed approximation to analysis of exact solutions of the Navier–Stokes equations (so-called exact coherent states) are discussed. The article concludes with an outline of the feasibility of extending this kind of approach to high-Reynolds-number wall turbulence in canonical flows and beyond.

Publication: Journal of Fluid Mechanics Vol.: 817ISSN: 0022-1120

ID: CaltechAUTHORS:20170428-132759772

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Abstract: Phase relations between specific scales in a turbulent boundary layer are studied here by highlighting the associated nonlinear scale interactions in the flow. This is achieved through an experimental technique that allows for targeted forcing of the flow through the use of a dynamic wall perturbation. Two distinct large-scale modes with well-defined spatial and temporal wavenumbers were simultaneously forced in the boundary layer, and the resulting nonlinear response from their direct interactions was isolated from the turbulence signal for the study. This approach advances the traditional studies of large- and small-scale interactions in wall turbulence by focusing on the direct interactions between scales with triadic wavenumber consistency. The results are discussed in the context of modelling high Reynolds number wall turbulence.

Publication: Philosophical Transactions A: Mathematical, Physical and Engineering Sciences Vol.: 375 No.: 2089 ISSN: 1364-503X

ID: CaltechAUTHORS:20170213-124117937

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Abstract: Previous work has established the usefulness of the resolvent operator that maps the terms nonlinear in the turbulent fluctuations to the fluctuations themselves. Further work has described the self-similarity of the resolvent arising from that of the mean velocity profile. The orthogonal modes provided by the resolvent analysis describe the wall-normal coherence of the motions and inherit that self-similarity. In this contribution, we present the implications of this similarity for the nonlinear interaction between modes with different scales and wall-normal locations. By considering the nonlinear interactions between modes, it is shown that much of the turbulence scaling behaviour in the logarithmic region can be determined from a single arbitrarily chosen reference plane. Thus, the geometric scaling of the modes is impressed upon the nonlinear interaction between modes. Implications of these observations on the self-sustaining mechanisms of wall turbulence, modelling and simulation are outlined.

Publication: Philosophical Transactions A: Mathematical, Physical and Engineering Sciences Vol.: 375 No.: 2089 ISSN: 1364-503X

ID: CaltechAUTHORS:20170213-124117603

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Abstract: Time-resolved velocity fields around a pitching and surging NACA 0018 airfoil were analyzed to investigate the influence of three independent timescales associated with the unsteady flowfield. The first of these timescales, the period of the pitch/surge motion, is directly linked to the development of dynamic stall. A simplified model of the flow using only a time constant mode and the first two harmonics of the pitch surge frequency has been shown to accurately model the flow. Full stall and leading-edge flow separation, however, were found to take place before the maximum angle of attack, indicating that a different timescale was associated with leading-edge vortex formation. This second, leading-edge vortex, timescale was found to depend on the airfoil convection time and compare well with the universal vortex formation time. Finally, instantaneous non-phase-averaged measurements were investigated to identify behavior not directly coupled to the airfoil motion. From this analysis, a third timescale associated with quasi-periodic Strouhal vortex shedding was found before flow separation. The interplay between these three timescales is discussed in detail, particularly as they relate to the periodic velocity and angle-of-attack change apparent to the blades of a vertical axis wind turbine.

Publication: AIAA Journal Vol.: 54 No.: 11 ISSN: 0001-1452

ID: CaltechAUTHORS:20161202-090243214

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Abstract: This paper employs the resolvent framework to consider the design of compliant walls for turbulent skin friction reduction. Specifically, the effects of simple spring–damper walls are contrasted with the effects of more complex walls incorporating tension, stiffness and anisotropy. In addition, varying mass ratios are tested to provide insight into differences between aerodynamic and hydrodynamic applications. Despite the differing physical responses, all the walls tested exhibit some important common features. First, the effect of the walls (positive or negative) is the greatest at conditions close to resonance, with sharp transitions in performance across the resonant frequency or phase speed. Second, compliant walls are predicted to have a more pronounced effect on slower moving structures because such structures generally have larger wall-pressure signatures. Third, two-dimensional (spanwise constant) structures are particularly susceptible to further amplification. These features are consistent with many previous experiments and simulations, suggesting that mitigating the rise of such two-dimensional structures is essential to designing performance-improving walls. For instance, it is shown that further amplification of such large-scale two-dimensional structures explains why the optimal anisotropic walls identified in previous direct numerical simulations only led to drag reduction in very small domains. The above observations are used to develop design and methodology guidelines for future research on compliant walls.

Publication: Journal of Turbulence Vol.: 17 No.: 8 ISSN: 1468-5248

ID: CaltechAUTHORS:20160901-123400889

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Abstract: Vertical axis wind turbine blades undergo dynamic stall due to the large angle of attack variation they experience during a turbine rotation. Particle image velocimetry on a pitching and surging airfoil was used to perform time resolved measurements at blade Reynolds numbers near turbine operating conditions of 10^5. These experiments were compared to simulations performed in the rotating turbine frame as well as the linear, experimental, frame at a Reynolds number of 10^3 to investigate rotational and Reynolds number effects. The flow was shown to develop similarly prior to separation, but the kinematics of vortices shed post separation were reference frame dependent.

No.: 185 ISSN: 0930-8989

ID: CaltechAUTHORS:20161202-082406384

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Abstract: A novel reduced-order model for time-varying nonlinear flows arising from a resolvent decomposition based on the time-mean flow is proposed. The inputs required for the model are the mean-flow field and a small set of velocity time-series data obtained at isolated measurement points, which are used to fix relevant frequencies, amplitudes and phases of a limited number of resolvent modes that, together with the mean flow, constitute the reduced-order model. The technique is applied to derive a model for the unsteady three-dimensional flow in a lid-driven cavity at a Reynolds number of 1200 that is based on the two-dimensional mean flow, three resolvent modes selected at the most active spanwise wavenumber, and either one or two velocity probe signals. The least-squares full-field error of the reconstructed velocity obtained using the model and two point velocity probes is of the order of 5 % of the lid velocity, and the dynamical behaviour of the reconstructed flow is qualitatively similar to that of the complete flow.

Publication: Journal of Fluid Mechanics Vol.: 798ISSN: 0022-1120

ID: CaltechAUTHORS:20160630-111131742

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Abstract: The relationship between Koopman mode decomposition, resolvent mode decomposition, and exact invariant solutions of the Navier-Stokes equations is clarified. The correspondence rests upon the invariance of the system operators under symmetry operations such as spatial translation. The usual interpretation of the Koopman operator is generalized to permit combinations of such operations, in addition to translation in time. This invariance is related to the spectrum of a spatiotemporal Koopman operator, which has a traveling-wave interpretation. The relationship leads to a generalization of dynamic mode decomposition, in which symmetry operations are applied to restrict the dynamic modes to span a subspace subject to those symmetries. The resolvent is interpreted as the mapping between the Koopman modes of the Reynolds stress divergence and the velocity field. It is shown that the singular vectors of the resolvent (the resolvent modes) are the optimal basis in which to express the velocity field Koopman modes where the latter are not a priori known.

Publication: Physical Review Fluids Vol.: 1 No.: 3 ISSN: 2469-990X

ID: CaltechAUTHORS:20160729-123350653

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Abstract: An experimental investigation of nonlinear scale interactions in a forced turbulent boundary layer is presented here. A dynamic wall perturbation mechanism was used to externally force two distinct large-scale synthetic modes with well-defined spatial and temporal wave numbers in a fully turbulent flow. The focus is on characterizing the nonlinear flow response at triadically consistent wave numbers that arises from the direct interactions of the two synthetic modes. These experimental results isolate triadic scale interactions in wall turbulence in a unique fashion, and provide the ability to explore the dynamics of scale coupling in a systematic and detailed manner. The ideas advanced here are intended to contribute towards modeling efforts of high-Reynolds-number wall turbulence.

Publication: Physical Review Fluids Vol.: 1 No.: 3 ISSN: 2469-990X

ID: CaltechAUTHORS:20160729-122826684

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Abstract: As part of an effort to examine the impact of vortical gusts on airfoils, a simple gust generator has been built and investigated. This consists of a heaving at plate capable of following a specifed transverse trajectory across a water tunnel. The relationship between the trajectory and the properties of the gusts that are shed downstream is characterized for non-periodic heaving motion described by Eldredge's smooth motion equation. PIV experiments show that the circulation of the vortical gust is proportional to the heaving speed of the plate. Tests with a downstream NACA 0018 airfoil demonstrate repeatable forces in response to the produced gusts.

Publication: AIAA JournalISSN: 0001-1452

ID: CaltechAUTHORS:20161128-141948592

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Abstract: A large scale spatio-temporally periodic disturbance was excited in a turbulent boundary layer via a wall-actuated dynamic roughness. Streamwise velocity, wall pressure, and direct wall shear stress measurements were made with a hot wire, pressure microphone, and a micro-scale differential capacitive sensor, respectively. Phase-averaged fields for the three quantities were calculated and analyzed. A phase calibration between the various sensors was performed with an acoustic plane wave tube over a range of operating conditions to validate a direct phase comparison between the respective quantities. Results suggest encouraging agreement between the phase of the wall shear stress and velocity near the wall; however, more refined velocity measurements are needed to make quantitative comparisons to the wall pressure. Overall, this work highlights the potential for wall-based control with applications towards reducing turbulent drag.

ID: CaltechAUTHORS:20161128-143308353

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Abstract: The effect of streamwise-varying steady transpiration on turbulent pipe flow is examined using direct numerical simulation at fixed friction Reynolds number Re_τ=314. The streamwise momentum equation reveals three physical mechanisms caused by transpiration acting in the flow: modification of Reynolds shear stress, steady streaming and generation of non-zero mean streamwise gradients. The influence of these mechanisms has been examined by means of a parameter sweep involving transpiration amplitude and wavelength. The observed trends have permitted identification of wall transpiration configurations able to reduce or increase the overall flow rate −36.1% and 19.3%, respectively. Energetics associated with these modifications are presented. A novel resolvent formulation has been developed to investigate the dynamics of pipe flows with a constant cross-section but with time-mean spatial periodicity induced by changes in boundary conditions. This formulation, based on a triple decomposition, paves the way for understanding turbulence in such flows using only the mean velocity profile. Resolvent analysis based on the time-mean flow and dynamic mode decomposition based on simulation data snapshots have both been used to obtain a description of the reorganization of the flow structures caused by the transpiration. We show that the pipe flows dynamics are dominated by a critical-layer mechanism and the waviness induced in the flow structures plays a role on the streamwise momentum balance by generating additional terms.

Publication: Journal of Fluid Mechanics Vol.: 796ISSN: 0022-1120

ID: CaltechAUTHORS:20160610-084044942

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Abstract: A structure configured to modify its surface morphology between a smooth state and a rough state in response to an applied stress. In demonstrated examples, a soft (PDMS) substrate is produced, and is pre-strained. A relatively stiff overlayer of a metal, such as chromium and gold, is applied to the substrate. When the pre-strained substrate is allowed to relax, the free surface of the stiff overlayer is forced to become distorted, yielding a free surface having a roughness of less than 1 millimeter. Repeated application and removal of the applied stress has been shown to yield reproducible changes in the morphology of the free surface. An application of such morphing free surface is to control a boundary layer transition of an aerodynamic fluid flowing over the surface.

ID: CaltechAUTHORS:20200224-133536093

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Abstract: The present contribution explores the relationship between response and forcing via amplification mechanisms in the Navier–Stokes equations applied to a turbulent pipe flow. A novel numerical method coupling direct numerical simulation with the resolvent model [J. Fluid Mech. 658, 336-382 (2010)] is developed in order to reveal the exact distribution of the nonlinear forcing terms, originally unknown in the model. The obtained results highlight the major role of the nonlinear terms in the energy spectra.

No.: 165 ISSN: 0930-8989

ID: CaltechAUTHORS:20161111-092925356

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Abstract: We report that many exact invariant solutions of the Navier-Stokes equations for both pipe and channel flows are well represented by just a few modes of the model of McKeon and Sharma [J. Fluid Mech. 658, 336 (2010)]. This model provides modes that act as a basis to decompose the velocity field, ordered by their amplitude of response to forcing arising from the interaction between scales. The model was originally derived from the Navier-Stokes equations to represent turbulent flows and has been used to explain coherent structure and to predict turbulent statistics. This establishes a surprising new link between the two distinct approaches to understanding turbulence.

Publication: Physical Review E Vol.: 93 No.: 2 ISSN: 2470-0045

ID: CaltechAUTHORS:20160315-160338397

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Abstract: The Extreme Flows Workshop was held in Princeton in May 2014 to honor the contributions by Professor Lex Smits to the field of fluid dynamics. His work was represented through a mix of oral and poster presentations, and discussions covered the state of the art and challenges associated with fundamental research over a range of Mach and Reynolds numbers, energy harvesting, biomimetic flows, vortex dynamics and the development of instrumentation and analysis techniques. The workshop captured Lex’s holistic approach to research, the breadth of his contributions and the synergy he has achieved between scientific advances and the many developments in facilities, instrumentation and techniques.

Publication: Experiments in Fluids Vol.: 57 No.: 1 ISSN: 0723-4864

ID: CaltechAUTHORS:20160218-122950356

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Abstract: This paper describes an experiment in which a forced shear layer external to a turbulent boundary layer is used to impart external large-scale forcing to the boundary layer. The shear-layer forcing creates periodic coherent vortical structures in the shear layer that convect at the shear-layer convective velocity. The convecting coherent structures create a concomitant unsteady pressure field that provides a disturbance for the turbulent boundary layer on the upper wall of the tunnel above the forced shear layer. The unsteady pressure field in turn creates a variation of the effective freestream velocity experienced by the boundary layer, and both the pressure disturbance and the concomitant velocity fluctuations are reported. The character of the turbulence in the boundary layer due to the external forcing is studied through hot-wire anemometry. Thorough examination of the turbulence results in similarities between the turbulence amplitude modulation results due to these externally-forced experiments and modulation response of an internally-forced boundary layer done by others.

ID: CaltechAUTHORS:20160212-132731202

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Abstract: Vertical axis wind turbine blades undergo dynamic stall due to the large angle of attack variation they experience during a turbine rotation. The flow over a single blade was modeled using a sinusoidally pitching and surging airfoil in a non-rotating frame with a constant freestream flow at a mean chord Reynolds number of 10^5. Two-dimensional, time-resolved velocity fields were acquired using particle image velocimetry. Vorticity contours were used to visualize shear layer and vortex activity. A low-order model of dynamic stall was developed using dynamic mode decomposition, from which primary and secondary dynamic separation modes were identified. The interaction between these two modes was able to capture the physics of dynamic stall and as such can be extended to other turbine configurations and problems in unsteady aerodynamics. Results from the linear pitch/surge frame are extrapolated to the rotating VAWT frame to investigate the behavior of identified flow structures.

Publication: Experiments in Fluids Vol.: 56 No.: 8 ISSN: 0723-4864

ID: CaltechAUTHORS:20150908-141014592

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Abstract: Vertical axis wind turbine blades undergo a large angle of attack and velocity variation during every turbine rotation. The flow over a NACA 0018 airfoil undergoing sinusoidal pitch, surge and combined motions at similar frequency, amplitude and Reynolds number as a commercial turbine was measured to investigate the effect of the different motions on the flow field. Two-dimensional, time resolved velocity fields were acquired using particle image velocimetry. Vorticity contours of the phase-averaged flow were used to visualize the separation process and shear layer development. The pitch-only ease is seen to separate earlier and at a lower angle of attack than the combined case. Results for surging airfoils at two angles of attack are presented, one fully separated, and one partially separated. Surge forward is shown to move the separation point toward the leading edge, while surge back moved it aft. The dynamics of the leading edge vortex in the combined pitch/surge and pitch motions are discussed.

ID: CaltechAUTHORS:20190828-110109774

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Abstract: This paper extends the resolvent formulation proposed by McKeon & Sharma (J. Fluid Mech., vol. 658, 2010, pp. 336–382) to consider turbulence–compliant wall interactions. Under this formulation, the turbulent velocity field is expressed as a linear superposition of propagating modes, identified via a gain-based decomposition of the Navier–Stokes equations. Compliant surfaces, modelled as a complex wall admittance linking pressure and velocity, affect the gain and structure of these modes. With minimal computation, this framework accurately predicts the emergence of the quasi-two-dimensional propagating waves observed in recent direct numerical simulations. Further, the analysis also enables the rational design of compliant surfaces, with properties optimized to suppress flow structures energetic in wall turbulence. It is shown that walls with unphysical negative damping are required to interact favourably with modes resembling the energetic near-wall cycle, which could explain why previous studies have met with limited success. Positive-damping walls are effective for modes resembling the so-called very-large-scale motions, indicating that compliant surfaces may be better suited for application at higher Reynolds number. Unfortunately, walls that suppress structures energetic in natural turbulence are also predicted to have detrimental effects elsewhere in spectral space. Consistent with previous experiments and simulations, slow-moving spanwise-constant structures are particularly susceptible to further amplification. Mitigating these adverse effects will be central to the development of compliant coatings that have a net positive influence on the flow.

Publication: Journal of Fluid Mechanics Vol.: 768ISSN: 0022-1120

ID: CaltechAUTHORS:20150420-130124855

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Abstract: A formal relationship between the skewness and the correlation coefficient of large and small scales, termed the amplitude modulation coefficient, is established for a general statistically stationary signal and is analysed in the context of a turbulent velocity signal. Both the quantities are seen to be measures of phase in triadically consistent interactions between scales of turbulence. The naturally existing phase relationships between large and small scales in a turbulent boundary layer are then manipulated by exciting a synthetic large-scale motion in the flow using a spatially impulsive dynamic wall roughness perturbation. The synthetic scale is seen to alter the phase relationships, or the degree of modulation, in a quasi-deterministic manner by exhibiting a phase-organizing influence on the small scales. The results presented provide encouragement for the development of a practical framework for favourable manipulation of energetic small-scale turbulence through large-scale inputs in a wall-bounded turbulent flow.

Publication: Journal of Fluid Mechanics Vol.: 767ISSN: 0022-1120

ID: CaltechAUTHORS:20150227-022811215

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Abstract: The possibility of creating reduced-order models for canonical wall-bounded turbulent flows based on exploiting energy sparsity in frequency domain, as proposed by Bourguignon et al. [Phys. Fluids26, 015109 (2014)], is examined. The present letter explains the origins of energetically sparse dominant frequencies and provides fundamental information for the design of such reduced-order models. The resolvent decomposition of a pipe flow is employed to consider the influence of finite domain length on the flow dynamics, which acts as a restriction on the possible wavespeeds in the flow. A forcing-to-fluctuation gain analysis in the frequency domain reveals that large sparse peaks in amplification occur when one of the possible wavespeeds matches the local wavespeed via the critical layer mechanism. A link between amplification and energy is provided through the similar characteristics exhibited by the most energetically relevant flow structures, arising from a dynamic mode decomposition of direct numerical simulation data, and the resolvent modes associated with the most amplified sparse frequencies. These results support the feasibility of reduced-order models based on the selection of the most amplified modes emerging from the resolvent model, leading to a novel computationally efficient method of representing turbulent flows.

Publication: Physics of Fluids Vol.: 26 No.: 10 ISSN: 1070-6631

ID: CaltechAUTHORS:20141211-082426965

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Abstract: An analytical framework for studying the logarithmic region of turbulent channels is formulated. We build on recent findings (Moarref et al., J. Fluid Mech., 734, 2013) that the velocity fluctuations in the logarithmic region can be decomposed into a weighted sum of geometrically self-similar resolvent modes. The resolvent modes and the weights represent the linear amplification mechanisms and the scaling influence of the nonlinear interactions in the Navier-Stokes equations (NSE), respectively (McKeon & Sharma, J. Fluid Mech., 658, 2010). Originating from the NSE, this framework provides an analytical support for Townsend’s attached-eddy model. Our main result is that self-similarity enables order reduction in modeling the logarithmic region by establishing a quantitative link between the self-similar structures and the velocity spectra. Specifically, the energy intensities, the Reynolds stresses, and the energy budget are expressed in terms of the resolvent modes with speeds corresponding to the top of the logarithmic region. The weights of the triad modes -the modes that directly interact via the quadratic nonlinearity in the NSE- are coupled via the interaction coefficients that depend solely on the resolvent modes (McKeon et al., Phys. Fluids, 25, 2013). We use the hierarchies of self-similar modes in the logarithmic region to extend the notion of triad modes to triad hierarchies. It is shown that the interaction coefficients for the triad modes that belong to a triad hierarchy follow an exponential function. The combination of these findings can be used to better understand the dynamics and interaction of flow structures in the logarithmic region. The compatibility of the proposed model with theoretical and experimental results is further discussed.

Publication: arXiv
ID: CaltechAUTHORS:20180831-112157832

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Abstract: A combined approach using system identification and feed-forward control design has been applied to experimental laminar channel flow in an effort to reduce the naturally occurring disturbance level. A simple blowing/suction strategy was capable of reducing the standard deviation of the measured sensor signal by 45 %, which markedly exceeds previously obtained results under comparable conditions. A comparable reduction could be verified over a significant streamwise extent, implying an improvement over previous, more localized disturbance control. The technique is effective, flexible, and robust, and the obtained results encourage further explorations of experimental control of convection-dominated flows.

Publication: Journal of Fluid Mechanics Vol.: 752ISSN: 0022-1120

ID: CaltechAUTHORS:20140815-113320134

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Abstract: We generate predictions for the fluctuating pressure field in turbulent pipe flow by reformulating the resolvent analysis of McKeon and Sharma (J. Fluid Mech., vol. 658, 2010, pp. 336–382) in terms of the so-called primitive variables. Under this analysis, the nonlinear convective terms in the Fourier-transformed Navier–Stokes equations (NSE) are treated as a forcing that is mapped to a velocity and pressure response by the resolvent of the linearized Navier–Stokes operator. At each wavenumber–frequency combination, the turbulent velocity and pressure field are represented by the most-amplified (rank-1) response modes, identified via a singular value decomposition of the resolvent. We show that these rank-1 response modes reconcile many of the key relationships among the velocity field, coherent structure (i.e. hairpin vortices), and the high-amplitude wall-pressure events observed in previous experiments and direct numerical simulations (DNS). A Green’s function representation shows that the pressure fields obtained under this analysis correspond primarily to the fast pressure contribution arising from the linear interaction between the mean shear and the turbulent wall-normal velocity. Recovering the slow pressure requires an explicit treatment of the nonlinear interactions between the Fourier response modes. By considering the velocity and pressure fields associated with the triadically consistent mode combination studied by Sharma and McKeon (J. Fluid Mech., vol. 728, 2013, pp. 196–238), we identify the possibility of an apparent amplitude modulation effect in the pressure field, similar to that observed for the streamwise velocity field. However, unlike the streamwise velocity, for which the large scales of the flow are in phase with the envelope of the small-scale activity close to the wall, we expect there to be a π/2 phase difference between the large-scale wall-pressure and the envelope of the small-scale activity. Finally, we generate spectral predictions based on a rank-1 model assuming broadband forcing across all wavenumber–frequency combinations. Despite the significant simplifying assumptions, this approach reproduces trends observed in previous DNS for the wavenumber spectra of velocity and pressure, and for the scale-dependence of wall-pressure propagation speed.

Publication: Journal of Fluid Mechanics Vol.: 751ISSN: 0022-1120

ID: CaltechAUTHORS:20140725-141354609

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Abstract: Vertical axis wind turbine (VAWT) blades undergo dynamic separation due to the large angle of attack variation they experience during a turbine rotation. The flow over a single blade was modeled using a sinusoidally pitching and surging airfoil in a constant free stream flow at a mean chord Reynolds number of 10^5. Two-dimensional, time resolved velocity fields were acquired using particle image velocimetry (PIV). Vorticity contours were used to visualize shear layer and vortex activity. A low order model of dynamic separation was developed using Dynamic Mode Decomposition (DMD). A primary and secondary dynamic separation mode were identified as the critical drivers for the unsteady flow field.

ID: CaltechAUTHORS:20150210-091011617

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Abstract: This paper extends the resolvent analysis of McKeon & Sharma (J. Fluid Mech., vol. 658, 2010, pp. 336–382) to consider flow control techniques that employ linear control laws, focusing on opposition control (Choi, Moin & Kim, J. Fluid Mech., vol. 262, 1994, pp. 75–110) as an example. Under this formulation, the velocity field for turbulent pipe flow is decomposed into a series of highly amplified (rank-1) response modes, identified from a gain analysis of the Fourier-transformed Navier– Stokes equations. These rank-1 velocity responses represent propagating structures of given streamwise/spanwise wavelength and temporal frequency, whose wall-normal footprint depends on the phase speed of the mode. Opposition control, introduced via the boundary condition on wall-normal velocity, affects the amplification characteristics (and wall-normal structure) of these response modes; a decrease in gain indicates mode suppression, which leads to a decrease in the drag contribution from that mode. With basic assumptions, this rank-1 model reproduces trends observed in previous direct numerical simulation and large eddy simulation, without requiring high-performance computing facilities. Further, a wavenumber–frequency breakdown of control explains the deterioration of opposition control performance with increasing sensor elevation and Reynolds number. It is shown that slower-moving modes localized near the wall (i.e. attached modes) are suppressed by opposition control. Faster-moving detached modes, which are more energetic at higher Reynolds number and more likely to be detected by sensors far from the wall, are further amplified. These faster-moving modes require a phase lag between sensor and actuator velocity for suppression. Thus, the effectiveness of opposition control is determined by a trade-off between the modes detected by the sensor. However, it may be possible to develop control strategies optimized for individual modes. A brief exploration of such mode-optimized control suggests the potential for significant performance improvement.

Publication: Journal of Fluid Mechanics Vol.: 749ISSN: 0022-1120

ID: CaltechAUTHORS:20140725-153829494

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Abstract: A synthetic spanwise-constant spatio-temporal mode is excited in a flat plate turbulent boundary layer through a spatially impulsive patch of dynamic wall-roughness. The streamwise wavelength of the synthetic mode approximately corresponds to the very large-scale motions present in high Reynolds number wall turbulence. Hot wire anemometer measurements made downstream of the roughness forcing reveal the nature of the two dimensional synthetic large-scale and its influence on the small-scale turbulence. A clear phase organizing effect on the small-scales is noticed in presence of the synthetic large-scale. A thorough understanding of these phase relations lays the foundation for a framework that allows for practical manipulation of energetic small-scale turbulence through large-scale inputs by utilizing the inherent non-linear coupling present in the governing Navier-Stokes equations.

ID: CaltechAUTHORS:20140713-153917772

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Abstract: Aero-optic measurements of turbulent boundary layers were performed in wind tunnels at the University of Notre Dame and California Institute of Technology for heated walls at a range of Reynolds numbers. Temporally resolved measurements of wavefronts were collected at a range of Mach numbers between 0.03 and 0.4 and the range of Re_θ between 1,700 and 20,000. Wavefront spectra for both heated and un-heated walls were extracted and compared to demonstrate that wall heating does not noticeably alter the shape of wavefront spectra in the boundary layer. The effect of Reynolds number on the normalized spectra was also presented, and an empirical spectral model was modified to account for Reynolds number dependence. Measurements of OPD_(rms) for heated walls were shown to be consistent with results from prior experiments, and a method of estimating OPD_(rms) and other boundary layer statistics from wavefront measurements of heated-wall boundary layers was demonstrated and discussed.

ID: CaltechAUTHORS:20141107-115405464

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Abstract: We combine resolvent-mode decomposition with techniques from convex optimization to optimally approximate velocity spectra in a turbulent channel. The velocity is expressed as a weighted sum of resolvent modes that are dynamically significant, non-empirical, and scalable with Reynolds number. To optimally represent direct numerical simulations (DNS) data at friction Reynolds number 2003, we determine the weights of resolvent modes as the solution of a convex optimization problem. Using only 12 modes per wall-parallel wavenumber pair and temporal frequency, we obtain close agreement with DNS-spectra, reducing the wall-normal and temporal resolutions used in the simulation by three orders of magnitude.

Publication: Physics of Fluids Vol.: 26 No.: 5 ISSN: 1070-6631

ID: CaltechAUTHORS:20140519-153243814

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Abstract: Bouncing liquid silicone oil drops on a vertically oscillating bath of the same liquid were studied experimentally. The different bouncing regimes previously described in the literature were observed, with transitions depending mainly on the droplet size and the forcing acceleration of the oil bath. In a particular range of forcing amplitudes, just below the Faraday instability threshold where standing waves appear on the free surface, walking drops traveling at a constant velocity over the surface were observed, consistent with previous studies. The influence of a local change of depth on this walking behavior was studied by submerging an obstacle in the oil bath. Notably different to the study of Eddi et al. (2009) [1], the depth change was such that walking was still observed over the obstacle. Previously unobserved drop trajectories, including trapping of a walking drop over the obstacle, crossing for non-normal drop approach to the obstacle, and reflection from the rear face of the obstacle were observed and explained in light of recent results and models in the literature.

Publication: Experimental Thermal and Fluid Science Vol.: 54ISSN: 0894-1777

ID: CaltechAUTHORS:20140502-074953981

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Abstract: Compressive sampling is well-known to be a useful tool used to resolve the energetic content of signals that admit a sparse representation. The broadband temporal spectrum acquired from point measurements in wall-bounded turbulence has precluded the prior use of compressive sampling in this kind of flow, however it is shown here that the frequency content of flow fields that have been Fourier transformed in the homogeneous spatial (wall-parallel) directions is approximately sparse, giving rise to a compact representation of the velocity field. As such, compressive sampling is an ideal tool for reducing the amount of information required to approximate the velocity field. Further, success of the compressive sampling approach provides strong evidence that this representation is both physically meaningful and indicative of special properties of wall turbulence. Another advantage of compressive sampling over periodic sampling becomes evident at high Reynolds numbers, since the number of samples required to resolve a given bandwidth with compressive sampling scales as the logarithm of the dynamically significant bandwidth instead of linearly for periodic sampling. The combination of the Fourier decomposition in the wall-parallel directions, the approximate sparsity in frequency, and empirical bounds on the convection velocity leads to a compact representation of an otherwise broadband distribution of energy in the space defined by streamwise and spanwise wavenumber, frequency, and wall-normal location. The data storage requirements for reconstruction of the full field using compressive sampling are shown to be significantly less than for periodic sampling, in which the Nyquist criterion limits the maximum frequency that can be resolved. Conversely, compressive sampling maximizes the frequency range that can be recovered if the number of samples is limited, resolving frequencies up to several times higher than the mean sampling rate. It is proposed that the approximate sparsity in frequency and the corresponding structure in the spatial domain can be exploited to design simulation schemes for canonical wall turbulence with significantly reduced computational expense compared with current techniques.

Publication: Physics of Fluids Vol.: 26 No.: 1 ISSN: 1070-6631

ID: CaltechAUTHORS:20140320-104900351

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Abstract: The effects of introducing a two- or three-dimensional streamwise traveling wave type body forcing in a turbulent channel flow at Re_τ = 180 are investigated using direct numerical simulations (DNS). The optimal forcing shape (that is, the forcing that leads to the most amplified or highest-gain velocity response) is obtained from the resolvent analysis of McKeon & Sharma (2010) for wave-numbers and wave-speed representative of the near-wall cycle. The velocity response due to imposed forcing obtained from DNS is found to agree well with resolvent analysis predictions at small forcing amplitude. The changes in mean velocity, shear stress and kinetic energy are characterized for various amplitudes of forcing.

ID: CaltechAUTHORS:20141107-145723375

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Abstract: We study the Reynolds-number scaling and the geometric self-similarity of a gainbased, low-rank approximation to turbulent channel flows, determined by the resolvent formulation of McKeon & Sharma (J. Fluid Mech., vol. 658, 2010, pp. 336–382), in order to obtain a description of the streamwise turbulence intensity from direct consideration of the Navier–Stokes equations. Under this formulation, the velocity field is decomposed into propagating waves (with single streamwise and spanwise wavelengths and wave speed) whose wall-normal shapes are determined from the principal singular function of the corresponding resolvent operator. Using the accepted scalings of the mean velocity in wall-bounded turbulent flows, we establish that the resolvent operator admits three classes of wave parameters that induce universal behaviour with Reynolds number in the low-rank model, and which are consistent with scalings proposed throughout the wall turbulence literature. In addition, it is shown that a necessary condition for geometrically self-similar resolvent modes is the presence of a logarithmic turbulent mean velocity. Under the practical assumption that the mean velocity consists of a logarithmic region, we identify the scalings that constitute hierarchies of self-similar modes that are parameterized by the critical wall-normal location where the speed of the mode equals the local turbulent mean velocity. For the rank-1 model subject to broadband forcing, the integrated streamwise energy density takes a universal form which is consistent with the dominant near-wall turbulent motions. When the shape of the forcing is optimized to enforce matching with results from direct numerical simulations at low turbulent Reynolds numbers, further similarity appears. Representation of these weight functions using similarity laws enables prediction of the Reynolds number and wall-normal variations of the streamwise energy intensity at high Reynolds numbers (Re_τ ≈ 10^3–10^(10)). Results from this low-rank model of the Navier–Stokes equations compare favourably with experimental results in the literature.

Publication: Journal of Fluid Mechanics Vol.: 734ISSN: 0022-1120

ID: CaltechAUTHORS:20131121-132351952

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Abstract: This article accompanies a fluid dynamics video entered into the Gallery of Fluid Motion of the 66th Annual Meeting of the APS Division of Fluid Dynamics.

Publication: arXiv
ID: CaltechAUTHORS:20180831-112154414

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Abstract: The work of Couder et al [1] (see also Bush et al [3, 4] inspired consideration of the impact of a submerged obstacle, providing a local change of depth, on the behavior of oil drops in the bouncing regime. In the linked videos, we recreate some of their results for a drop bouncing on a uniform depth bath of the same liquid undergoing vertical oscillations just below the conditions for Faraday instability, and show a range of new behaviors associated with change of depth. This article accompanies a fluid dynamics video entered into the Gallery of Fluid Motion of the 66th Annual Meeting of the APS Division of Fluid Dynamics.

Publication: arXiv
ID: CaltechAUTHORS:20180831-112151003

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Abstract: Recent experimental investigations of the canonical turbulent boundary layer at high Reynolds numbers has provided evidence on the active role of very large‐scale structures, extending in the streamwise direction for several boundary layer heights, and modulating near‐wall turbulence from the energy containing eddy scales down to the dissipative scales. However, the physical mechanisms governing such interactions are not completely clear yet, and the reason may be related to the fact that the structure of wall turbulence at high Reynolds numbers still deserves further investigation. In this contribution we present recent results on the structural population in wall turbulence. We compare statistical trends obtained in two very different Reynolds number experiments, one in the atmospheric surface layer at SLTEST and one in a flat plate turbulent boundary layer. While the very large‐scale structures of turbulence and the near‐wall turbulent streaks are observed to have a well‐defined location in physical space and in the energetic domain, based on the frequency or wave number spectra, the intermediate scale motions that manifest as ramplike structures still seem to suffer from Reynolds number effects. Results suggest that outer scaling may not be appropriate, implying that ramplike structures are more likely confined to the near‐wall region in very high Reynolds‐number flows such as the atmospheric surface layer. Spatially resolved measurements at high Reynolds numbers are needed to univocally define the correct scaling of ramplike structures and to assess Reynolds number effects in the structural description of zero pressure gradient turbulent boundary layers.

ID: CaltechAUTHORS:20140826-104525799

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Abstract: A new theory of coherent structure in wall turbulence is presented. The theory is the first to predict packets of hairpin vortices and other structure in turbulence, and their dynamics, based on an analysis of the Navier–Stokes equations, under an assumption of a turbulent mean profile. The assumption of the turbulent mean acts as a restriction on the class of possible structures. It is shown that the coherent structure is a manifestation of essentially low-dimensional flow dynamics, arising from a critical-layer mechanism. Using the decomposition presented in McKeon & Sharma (J. Fluid Mech., vol. 658, 2010, pp. 336–382), complex coherent structure is recreated from minimal superpositions of response modes predicted by the analysis, which take the form of radially varying travelling waves. The leading modes effectively constitute a low-dimensional description of the turbulent flow, which is optimal in the sense of describing the resonant effects around the critical layer and which minimally predicts all types of structure. The approach is general for the full range of scales. By way of example, simple combinations of these modes are offered that predict hairpins and modulated hairpin packets. The example combinations are chosen to represent observed structure, consistent with the nonlinear triadic interaction for wavenumbers that is required for self-interaction of structures. The combination of the three leading response modes at streamwise wavenumbers 6; 1; 7 and spanwise wavenumbers ±6; ±6; ±12, respectively, with phase velocity 2/3, is understood to represent a turbulence ‘kernel’, which, it is proposed, constitutes a self-exciting process analogous to the near-wall cycle. Together, these interactions explain how the mode combinations may self-organize and self-sustain to produce experimentally observed structure. The phase interaction also leads to insight into skewness and correlation results known in the literature. It is also shown that the very large-scale motions act to organize hairpin-like structures such that they co-locate with areas of low streamwise momentum, by a mechanism of locally altering the shear profile. These energetic streamwise structures arise naturally from the resolvent analysis, rather than by a summation of hairpin packets. In addition, these packets are modulated through a ‘beat’ effect. The relationship between Taylor’s hypothesis and coherence is discussed, and both are shown to be the consequence of the localization of the response modes around the critical layer. A pleasing link is made to the classical laminar inviscid theory, whereby the essential mechanism underlying the hairpin vortex is captured by two obliquely interacting Kelvin–Stuart (cat’s eye) vortices. Evidence for the theory is presented based on comparison with observations of structure in turbulent flow reported in the experimental and numerical simulation literature and with exact solutions reported in the transitional literature.

Publication: Journal of Fluid Mechanics Vol.: 728ISSN: 0022-1120

ID: CaltechAUTHORS:20130820-101633052

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Abstract: Despite being one of the earliest - and most studied - active control techniques proposed for wall-bounded turbulent flows, the opposition control method of Choi et al., [J.Fluid Mech., Vol. 262, 1994, pp. 75-110] remains to be fully understood. In this paper, we develop a simple model for opposition control by extending the forcing-response analysis presented in McKeon and Sharma [J. Fluid Mech., Vol. 658, 2010, pp. 336-382]. Based on a gain analysis of the Navier-Stokes equations, the velocity field in turbulent pipe flow is decomposed into a series of highly-amplified response modes (i.e., propagating helical waves). Opposition control, introduced via the boundary condition on wall-normal velocity, alters the amplification characteristics and structure of these response modes, whereby a reduction in gain (mode suppression) leads to a reduction in drag. With simple assumptions, and minimal computation, our model reproduces the leading-order integrated effects of opposition control predicted by DNS. By breaking down opposition control into modal subsystems, our analysis provides new physical insight into the deterioration of control performance with increasing sensor elevation and Reynolds number. We show that opposition control is only effective for specific wavenumber-frequency combinations; others require the introduction of a phase lag between sensed and actuated velocity. Moving forward, this mode-by-mode approach can enable the design and evaluation of targeted control techniques, as well as the definition of a theoretical limit for controller performance.

ID: CaltechAUTHORS:20150211-141351820

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Abstract: The resolvent-based analysis of wall turbulence (McKeon & Sharma 2010) is extended with an explicit treatment of the non-linearity in the Navier-Stokes equations. The equivalent of triadic interaction in the wall-normal direction is described, and it is found that the resulting forcing has a phase shift of π/2 in terms associated with symmetric spatial directions. Explicit forms for the nonlinearity in fully-developed pipe and channel flows, i.e. in cylindrical and Cartesian coordinate systems.

ID: CaltechAUTHORS:20150211-102550604

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Abstract: We evaluate the efficacy of a gain-based rank-1 model, developed by McKeon & Sharma (J. Fluid Mech., 2010), for representing the energy spectra and the streamwise/wall-normal co-spectrum in a turbulent channel. This is motivated by our previous observation that the streamwise turbulent energy intensity is well approximated by the rank-1 model subject to a broadband forcing in the wall-parallel directions and a properly selected temporal intensity. In the present study, the evaluation is based on finding the optimal forcing spectrum that minimizes the deviation between the two-dimensional velocity spectra at different wall-normal locations obtained from direct numerical simulations at friction Reynolds number 2003 (Hoyas & Jiminénez, Phys. Fluids, 2006) and from the rank-1 model at equal Reynolds number. It is shown that the optimally forced rank-1 model captures the streamwise energy spectrum for streamwise wavelengths smaller than approximately 1000 viscous units throughout the channel. For larger wavelengths, the streamwise spectrum is matched in the outer region of the channel, i.e. wall-normal distances larger than approximately 0.15 times the channel half-height, and the mismatch close to the wall results in less than 5 percent error in the inner-scaled peak of the streamwise energy intensity. In addition, we show that the rank-1 model with optimal forcing captures the essential features of the wall-normal and spanwise spectra and the streamwise/wall-normal co-spectrum. We observe that the predicted magnitudes of the latter three spectra are smaller in the rank-1 model compared to the simulation results suggesting that a higher-order or different rank-1 model may be necessary for accurate representation of these spectra.

ID: CaltechAUTHORS:20150218-094025300

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Abstract: Systems and methods for providing dynamic control to a vehicle in a dynamic fluid. The systems and methods of the invention relate to one or more morphable surfaces that can be controlled by a controller and an actuator in an active manner to provide asperities that interact with a fluid moving across the morphable surfaces. By controlling the size, shape and location of the asperities, one can exert control authority over the motion of the vehicle relative to the fluid, including a speed, a direction and an attitude of the vehicle. Examples of materials that provide suitable morphable surfaces include ionic polymer metal composites and shape memory polymers, both of which types of material are commercially available. Useful morphable surface systems have been examined and are described.

ID: CaltechAUTHORS:20160223-153919542

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Abstract: Time-resolved particle image velocimetry was used to examine the structure and evolution of swirling coherent structure (SCS), one interpretation of which is a marker for a three-dimensional coherent vortex structure, in wall-parallel planes of a turbulent boundary layer with a large field of view, 4.3δ × 2.2δ. Measurements were taken at four different wall-normal locations ranging from y/δ = 0.08–0.48 at a friction Reynolds number, Re_τ = 410. The data set yielded statistically converged results over a larger field of view than typically observed in the literature. The method for identifying and tracking swirling coherent structure is discussed, and the resulting trajectories, convection velocities, and lifespan of these structures are analyzed at each wall-normal location. The ability of a model in which the entirety of an individual SCS travels at a single convection velocity, consistent with the attached eddy hypothesis of Townsend (The structure of turbulent shear flows. Cambridge University Press, Cambridge, 1976), to describe the data is investigated. A methodology for determining whether such structures are “attached” or “detached” from the wall is also proposed and used to measure the lifespan and convection velocity distributions of these different structures. SCS were found to persist for longer periods of time further from the wall, particularly those inferred to be “detached” from the wall, which could be tracked for longer than 5 eddy turnover times.

Publication: Experiments in Fluids Vol.: 54 No.: 4 ISSN: 0723-4864

ID: CaltechAUTHORS:20130605-101847312

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Abstract: We review recent progress, based on the approach introduced by McKeon and Sharma [J. Fluid Mech. 658, 336–382 (2010)10.1017/S002211201000176X], in understanding and controlling wall turbulence. The origins of this analysis partly lie in nonlinear robust control theory, but a differentiating feature is the connection with, and prediction of, state-of-the-art understanding of velocity statistics and coherent structures observed in real, high Reynolds number flows. A key component of this line of work is an experimental demonstration of the excitation of velocity response modes predicted by the theory using non-ideal, but practical, actuation at the wall. Limitations of the approach and promising directions for future development are outlined.

Publication: Physics of Fluids Vol.: 25 No.: 3 ISSN: 1070-6631

ID: CaltechAUTHORS:20130606-132057755

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Abstract: Marusic et al. (J. Fluid Mech., vol. 716, 2013, R3) show the first clear evidence of universal logarithmic scaling emerging naturally (and simultaneously) in the mean velocity and the intensity of the streamwise velocity fluctuations about that mean in canonical turbulent flows near walls. These observations represent a significant advance in understanding of the behaviour of wall turbulence at high Reynolds number, but perhaps the most exciting implication of the experimental results lies in the agreement with the predictions of such scaling from a model introduced by Townsend (J. Fluid Mech., vol. 11, 1961, pp. 97–120), commonly termed the attached eddy hypothesis. The elegantly simple, yet powerful, study by Marusic et al. should spark further investigation of the behaviour of all fluctuating velocity components at high Reynolds numbers and the outstanding predictions of the attached eddy hypothesis.

Publication: Journal of Fluid Mechanics Vol.: 718ISSN: 0022-1120

ID: CaltechAUTHORS:20130308-085213459

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Abstract: The apparent amplitude modulation effect between large- and small-scale motions in the turbulent boundary layer, including both streamwise and wall-normal velocity components, is explored by cross-correlation techniques. Single-point hotwire and planar PIV measurements are employed to consider the envelopes of small-scale fluctuations in both directions and their correlation with the fluctuations of large-scale motions in the streamwise direction. The degree of correlation is interpreted as a measure of phase lag between the different scale motions, and these phase measurements are used to demonstrate that the fluctuations in the envelope of small-scale motions in both directions tend to lead corresponding fluctuations in the large scales in the streamwise direction. The cospectral density of the cross-correlation between the different scales is used to identify the particular large-scale motions dominant in the modulation effect, and it is shown that the dominant interacting (or ‘modulating’) scale corresponds in size to the very large-scale motions observed in internal flows but not normally observed in the outer region of the boundary layer.

Publication: Experiments in Fluids Vol.: 54 No.: 3 ISSN: 0723-4864

ID: CaltechAUTHORS:20130605-100354415

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Abstract: This article reports on one component of a larger study on measurement of the zero-pressure-gradient turbulent flat plate boundary layer, in which a detailed investigation was conducted of the suite of corrections required for mean velocity measurements performed using Pitot tubes. In particular, the corrections for velocity shear across the tube and for blockage effects which occur when the tube is in close proximity to the wall were investigated using measurements from Pitot tubes of five different diameters, in two different facilities, and at five different Reynolds numbers ranging from Re_θ = 11 100 to 67 000. Only small differences were found amongst commonly used corrections for velocity shear, but improvements were found for existing near-wall proximity corrections. Corrections for the nonlinear averaging of the velocity fluctuations were also investigated, and the results compared to hot-wire data taken as part of the same measurement campaign. The streamwise turbulence-intensity correction was found to be of comparable magnitude to that of the shear correction, and found to bring the hot-wire and Pitot results into closer agreement when applied to the data, along with the other corrections discussed and refined here.

Publication: Journal of Fluid Mechanics Vol.: 715ISSN: 0022-1120

ID: CaltechAUTHORS:20130226-135510873

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Abstract: Systems and methods for providing dynamic control to a surface immersed in a dynamic fluid. The systems and methods of the invention relate to one or more morphable surfaces that can be control in an active manner to provide asperities that interact with a fluid moving across the morphable surfaces. By controlling the size, shape and location of the asperities, one can exert control authority over the motion of the surface relative to the fluid. Examples of materials that provide suitable morphable surfaces include ionic polymer metal composites and shape memory polymers, both of which types of material are commercially available. Useful morphable surface systems have been examined and are described.

ID: CaltechAUTHORS:20160212-094145157

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Abstract: Systems and methods for providing dynamic control to a vehicle in a dynamic fluid. The systems and methods of the invention relate to one or more morphable surfaces that can be controlled by a controller and an actuator in an active manner to provide asperities that interact with a fluid moving across the morphable surfaces. By controlling the size, shape and location of the asperities, one can exert control authority over the motion of the vehicle relative to the fluid, including a speed, a direction and an attitude of the vehicle. Examples of materials that provide suitable morphable surfaces include ionic polymer metal composites and shape memory polymers, both of which types of material are commercially available. Useful morphable surface systems have been examined and are described.

ID: CaltechAUTHORS:20160212-100724553

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Abstract: In shear coaxial injectors, commonly used for cryogenic liquid rocket engines, propellants traveling at different velocities are separated by the inner jet post before they come into contact with each other, mix, and combust. Knowing how the fluids mix and how susceptible they are to hydrodynamic instabilities is paramount for a successful liquid rocket engine. In this study, the wake behind a blunt trailing edge of a long plate, similar to an unwrapped coaxial injector, was studied in a water tunnel. Two fluid streams of different velocities were introduced on opposite sides of the plate. PIV was used to visualize and determine the influence of the velocity ratio of the split stream on the wake behavior. Measurements of the vortex shedding frequency were taken at various velocity ratios and compared with well characterized cases with a uniform free stream. Operating conditions ranged from Reynolds number 6,000 to 22,000 and velocity ratios 0.30 to 1.00.

ID: CaltechAUTHORS:20190826-092412052

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Abstract: The laminar separation over a symmetric, idealized airfoil was altered by employing a leading edge roughness element ahead of the separation point. Experimental tests were performed with a dynamic roughness, with a time-dependent amplitude, in order to determine the range of control authority associated with this type of leading edge manipulation. The dynamic roughness tests demonstrated the sensitivity of the separation to frequency perturbations at the leading edge induced by the roughness element. Two low dimensional analysis techniques, Proper Orthogonal Decomposition and Dynamic Mode Decomposition, were employed to examine the coupling between the wall motion and the flow, with a view to investigating the practicality of using leading edge roughness perturbations to in future closed loop control configurations.

ID: CaltechAUTHORS:20190826-092412147

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Abstract: The zero-pressure-gradient turbulent boundary layer over a flat plate was perturbed by a temporally oscillating, spatial impulse of roughness, and the downstream response of the flow field was interrogated by hot-wire anemometry and particle-image velocimetry. The key features common to impulsively perturbed boundary layers, as identified in Jacobi & McKeon (J. Fluid Mech., 2011), were investigated, and the unique contributions of the dynamic perturbation were isolated by contrast with an appropriately matched static impulse of roughness. In addition, the dynamic perturbation was decomposed into separable large-scale and small-scale structural effects, which in turn were associated with the organized wave and roughness impulse aspects of the perturbation. A phase-locked velocity decomposition of the entire downstream flow field revealed strongly coherent modes of fluctuating velocity, with distinct mode shapes for the streamwise and wall-normal velocity components. Following the analysis of McKeon & Sharma (J. Fluid Mech., vol. 658, 2010, pp. 336–382), the roughness perturbation was treated as a forcing of the Navier–Stokes equation and a linearized analysis employing a modified Orr–Sommerfeld operator was performed. The experimentally ascertained wavespeed of the input disturbance was used to solve for the most amplified singular mode of the Orr–Sommerfeld resolvent. These calculated modes were then compared with the streamwise and wall-normal velocity fluctuations. The discrepancies between the calculated Orr–Sommerfeld resolvent modes and those experimentally observed by phase-locked averaging of the velocity field were postulated to result from the violation of the parallel flow assumption of Orr–Sommerfeld analysis, as well as certain non-equilibrium effects of the roughness. Additionally, some difficulties previously observed using a quasi-laminar eigenmode analysis were also observed under the resolvent approach; however, the resolvent analysis was shown to provide reasonably accurate predictions of velocity fluctuations for the forced Orr–Sommerfeld problem over a portion of the boundary layer, with potential applications to designing efficient flow control strategies. The combined experimental and analytical effort provides a new opportunity to examine the non-equilibrium and forcing effects in a dynamically perturbed flow.

Publication: Journal of Fluid Mechanics Vol.: 688ISSN: 0022-1120

ID: CaltechAUTHORS:20120120-115846312

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Abstract: The problems of nonlinearity and high dimension have so far prevented a complete solution of the control of turbulent flow. Addressing the problem of nonlinearity, we propose a flow control strategy which ensures that the energy of any perturbation to the target profile decays monotonically. The controller’s estimate of the flow state is similarly guaranteed to converge to the true value. We present a one-time off-line synthesis procedure, which generalises to accommodate more restrictive actuation and sensing arrangements, with conditions for existence for the controller given in this case. The control is tested in turbulent channel flow (Re_τ = 100) using full-domain sensing and actuation on the wall-normal velocity. Concentrated at the point of maximum inflection in the mean profile, the control directly counters the supply of turbulence energy arising from the interaction of the wall-normal perturbations with the flow shear. It is found that the control is only required for the larger-scale motions, specifically those above the scale of the mean streak spacing. Minimal control effort is required once laminar flow is achieved. The response of the near-wall flow is examined in detail, with particular emphasis on the pressure and wall-normal velocity fields, in the context of Landahl’s theory of sheared turbulence.

Publication: Physics of Fluids Vol.: 23 No.: 12 ISSN: 1070-6631

ID: CaltechAUTHORS:20120221-073318177

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Abstract: The flow over a smooth sphere is examined in the Reynolds number range of 5.0 × 10^4 < Re < 5.0 × 10^5 via measurements of the fluctuating forces and particle image velocimetry measurements in a planar cut of the velocity field. Comprehensive studies of the statistics and spectra of the forces are presented for a range of subcritical and supercritical Reynolds numbers. While the subcritical lateral force spectra are dominated by activity corresponding to the large-scale vortex shedding frequency at a Strouhal number of approximately 0.18, there is no such peak apparent in the supercritical spectra, although resolution effects may become important in this region. Nor does the large-scale vortex shedding appear to have a significant effect on the drag force fluctuations at either sub- or super-critical Reynolds numbers. A simple double spring model is shown to capture the main features of the lateral force spectra. The low-frequency force fluctuations observed in earlier computational studies are shown to have important implications for statistical convergence, and in particular, the apparent mean side force observed in earlier studies. At least one thousand dimensionless time units are required for reasonable estimates of the second and higher moments below the critical Reynolds number and even more for supercritical flow, stringent conditions for computational studies. Lastly, investigation of the relationship between the motion of the instantaneous wake shape, defined via the local position where the streamwise velocity is equal to half the freestream value, and the in-plane lateral force for subcritical flow reveals a significant negative correlation throughout the near wake, which is shown to be related to a structure inferred to arise from the large-scale vortex shedding convecting downstream at 61% of the freestream velocity. In addition to its utility in understanding basic sphere flow, the apparatus is also a testbed that will be used in future studies, examining the effect of both static and dynamic changes to the surface morphology.

Publication: Experiments in Fluids Vol.: 51 No.: 5 ISSN: 0723-4864

ID: CaltechAUTHORS:20111216-150038357

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Abstract: Time-resolved particle image velocimetry (PIV) measurements performed in wall parallel planes at three wall normal locations, y^+ = 34, 108, and 278, in a zero pressure gradient turbulent boundary layer at Re_τ = 470 are used to illuminate the distribution of streamwise velocity fluctuations in a three-dimensional energy spectrum (2D in space and 1D in time) over streamwise, spanwise, and temporal wavelengths. Two high-speed cameras placed side by side in the streamwise direction give a 10δ × 5δ streamwise by spanwise field of view with a vector spacing of Δx^+ = Δz^+ ≈ 37 and a time step of Δt^+ = 0.5. Although 3D wavenumber-frequency spectra have been calculated in acoustics studies, to the authors’ knowledge this is the first time they has been calculated and presented for a turbulent boundary layer. The calculation and normalization of this spectrum, its relation to 2D and 1D spectra, and the effects of the PIV algorithm on its shape are carefully analyzed and outlined.

Publication: Experiments in Fluids Vol.: 51 No.: 4 ISSN: 0723-4864

ID: CaltechAUTHORS:20111025-073430962

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Abstract: The effect of an isolated roughness element on the forces on a sphere was examined for a Reynolds number range of 5×10^4

ID: CaltechAUTHORS:20111021-085953294

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Abstract: A streamwise-constant model is presented to investigate the basic mechanisms responsible for the change in mean flow occuring during pipe flow transition. The model is subject to two different types of forcing: a simple forcing of the axial momentum equation via a deterministic form for the streamfunction and a stochastic forcing of the streamfunction equation. Using a single forced momentum balance equation, we show that the shape of the velocity profile is robust to changes in the forcing profile and that both linear non-normal and nonlinear effects are required to capture the change in mean flow associated with transition to turbulence. The particularly simple form of the model allows for the study of the momentum transfer directly by inspection of the equations. The distribution of the high- and low-speed streaks over the cross-section of the pipe produced by our model is remarkably similar to one observed in the velocity field near the trailing edge of the puff structures present in pipe flow transition. Under stochastic forcing, the model exhibits a quasi-periodic self-sustaining cycle characterized by the creation and subsequent decay of “streamwise-constant puffs,” so-called due to the good agreement between the temporal evolution of their velocity field and the projection of the velocity field associated with three-dimensional puffs in a frame of reference moving at the bulk velocity. We establish that the flow dynamics are relatively insensitive to the regeneration mechanisms invoked to produce near-wall streamwise vortices, such that using small, unstructured background disturbances to regenerate the streamwise vortices in place of the natural feedback from the flow is sufficient to capture the formation of the high- and low-speed streaks and their segregation leading to the blunting of the velocity profile characteristic of turbulent pipe flow. We propose a “quasi self-sustaining process” to describe these mechanisms.

Publication: Physics of Fluids Vol.: 23 No.: 9 ISSN: 1070-6631

ID: CaltechAUTHORS:20111115-094643184

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Abstract: We study the input-output response of a streamwise constant projection of the Navier-Stokes equations for plane Couette flow, the so-called 2D/3C model. Study of a streamwise constant model is motivated by numerical and experimental observations that suggest the prevalence and importance of streamwise and quasi-streamwise elongated structures. Periodic spanwise/wall-normal (z–y) plane stream functions are used as input to develop a forced 2D/3C streamwise velocity field that is qualitatively similar to a fully turbulent spatial field of direct numerical simulation data. The input-output response associated with the 2D/3C nonlinear coupling is used to estimate the energy optimal spanwise wavelength over a range of Reynolds numbers. The results of the input-output analysis agree with previous studies of the linearized Navier-Stokes equations. The optimal energy corresponds to minimal nonlinear coupling. On the other hand, the nature of the forced 2D/3C streamwise velocity field provides evidence that the nonlinear coupling in the 2D/3C model is responsible for creating the well known characteristic “S” shaped turbulent velocity profile. This indicates that there is an important tradeoff between energy amplification, which is primarily linear, and the seemingly nonlinear momentum transfer mechanism that produces a turbulent-like mean profile.

Publication: Physics of Fluids Vol.: 23 No.: 6 ISSN: 1070-6631

ID: CaltechAUTHORS:20110722-112408053

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Abstract: The zero-pressure-gradient turbulent boundary layer over a flat plate was perturbed by a short strip of two-dimensional roughness elements, and the downstream response of the flow field was interrogated by hot-wire anemometry and particle image velocimetry. Two internal layers, marking the two transitions between rough and smooth boundary conditions, are shown to represent the edges of a ‘stress bore’ in the flow field. New scalings, based on the mean velocity gradient and the third moment of the streamwise fluctuating velocity component, are used to identify this ‘stress bore’ as the region of influence of the roughness impulse. Spectral composite maps reveal the redistribution of spectral energy by the impulsive perturbation – in particular, the region of the near-wall peak was reached by use of a single hot wire in order to identify the significant changes to the near-wall cycle. In addition, analysis of the distribution of vortex cores shows a distinct structural change in the flow associated with the perturbation. A short spatially impulsive patch of roughness is shown to provide a vehicle for modifying a large portion of the downstream flow field in a controlled and persistent way.

Publication: Journal of Fluid Mechanics Vol.: 677ISSN: 0022-1120

ID: CaltechAUTHORS:20110711-100700826

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Abstract: Typical approaches to manipulation of flow separation employ passive means or active techniques such as blowing and suction or plasma acceleration. Here it is demonstrated that the flow can be significantly altered by making small changes to the shape of the surface. A proof of concept experiment is performed using a very simple time-dependent perturbation to the surface of a sphere: a roughness element of 1% of the sphere diameter is moved azimuthally around a sphere surface upstream of the uncontrolled laminar separation point, with a rotational frequency as large as the vortex shedding frequency. A key finding is that the non-dimensional time to observe a large effect on the lateral force due to the perturbation produced in the sphere boundary layers as the roughness moves along the surface is ˆt =tU_(∞)/D ≈4. This slow development allows the moving element to produce a tripped boundary layer over an extended region. It is shown that a lateral force can be produced that is as large as the drag. In addition, simultaneous particle image velocimetry and force measurements reveal that a pair of counter-rotating helical vortices are produced in the wake, which have a significant effect on the forces and greatly increase the Reynolds stresses in the wake. The relatively large perturbation to the flow-field produced by the small surface disturbance permits the construction of a phase-averaged, three-dimensional (two-velocity component) wake structure from measurements in the streamwise/radial plane. The vortical structure arising due to the roughness element has implications for flow over a sphere with a nominally smooth surface or distributed roughness. In addition, it is shown that oscillating the roughness element, or shaping its trajectory, can produce a mean lateral force.

Publication: Journal of Fluid Mechanics Vol.: 675ISSN: 0022-1120

ID: CaltechAUTHORS:20110527-113052456

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Abstract: Simultaneous streamwise velocity measurements across the vertical direction obtained in the atmospheric surface layer (Re_τ ≃ 5 × 10^5) under near thermally neutral conditions are used to outline and quantify interactions between the scales of turbulence, from the very-large-scale motions to the dissipative scales. Results from conditioned spectra, joint probability density functions and conditional averages show that the signature of very-large-scale oscillations can be found across the whole wall region and that these scales interact with the near-wall turbulence from the energy-containing eddies to the dissipative scales, most strongly in a layer close to the wall, z^+ ≲ 10^3. The scale separation achievable in the atmospheric surface layer appears to be a key difference from the low-Reynolds-number picture, in which structures attached to the wall are known to extend through the full wall-normal extent of the boundary layer. A phenomenological picture of very-large-scale motions coexisting and interacting with structures from the hairpin paradigm is provided here for the high-Reynolds-number case. In particular, it is inferred that the hairpin-packet conceptual model may not be exhaustively representative of the whole wall region, but only of a near-wall layer of z^+ = O(10^3), where scale interactions are mostly confined.

Publication: Journal of Fluid Mechanics Vol.: 666ISSN: 0022-1120

ID: CaltechAUTHORS:20110307-143116884

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Abstract: We review wall-bounded turbulent flows, particularly high–Reynolds number, zero–pressure gradient boundary layers, and fully developed pipe and channel flows. It is apparent that the approach to an asymptotically high–Reynolds number state is slow, but at a sufficiently high Reynolds number the log law remains a fundamental part of the mean flow description. With regard to the coherent motions, very-large-scale motions or superstructures exist at all Reynolds numbers, but they become increasingly important with Reynolds number in terms of their energy content and their interaction with the smaller scales near the wall. There is accumulating evidence that certain features are flow specific, such as the constants in the log law and the behavior of the very large scales and their interaction with the large scales (consisting of vortex packets). Moreover, the refined attached-eddy hypothesis continues to provide an important theoretical framework for the structure of wall-bounded turbulent flows.

Publication: Annual Review of Fluid Mechanics Vol.: 43ISSN: 0066-4189

ID: CaltechAUTHORS:20110524-094826972

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Abstract: The majority of practical flows, particularly those flows in applications of importance to transport, distribution and climate, are turbulent and as a result experience complex three-dimensional motion with increased drag compared with the smoother, laminar condition. In this study, we describe the development of a simple model that predicts important structural and scaling features of wall turbulence. We show that a simple linear superposition of modes derived from a forcing-response analysis of the Navier-Stokes equations can be used to reconcile certain key statistical and structural descriptions of wall turbulence. The computationally cheap approach explains and predicts vortical structures and velocity statistics of turbulent flows that have previously been identified only in experiments or by direct numerical simulation. In particular, we propose an economical explanation for the meandering appearance of very large scale motions observed in turbulent pipe flow, and likewise demonstrate that hairpin vortices are predicted by the model. This new capability has clear implications for modeling, simulation and control of a ubiquitous class of wall flows.

Publication: arXiv
ID: CaltechAUTHORS:20180831-112140731

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Abstract: We investigate statistics of large-scale structures from large-eddy simulation (LES) of turbulent channel flow at friction Reynolds numbers Re_τ = 2K and 200K (where K denotes 1000). In order to capture the behaviour of large-scale structures properly, the channel length is chosen to be 96 times the channel half-height. In agreement with experiments, these large-scale structures are found to give rise to an apparent amplitude modulation of the underlying small-scale fluctuations. This effect is explained in terms of the phase relationship between the large- and small-scale activity. The shape of the dominant large-scale structure is investigated by conditional averages based on the large-scale velocity, determined using a filter width equal to the channel half-height. The conditioned field demonstrates coherence on a scale of several times the filter width, and the small-scale–large-scale relative phase difference increases away from the wall, passing through π/2 in the overlap region of the mean velocity before approaching π further from the wall. We also found that, near the wall, the convection velocity of the large scales departs slightly, but unequivocally, from the mean velocity.

Publication: Journal of Fluid Mechanics Vol.: 661ISSN: 0022-1120

ID: CaltechAUTHORS:20101206-152355868

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Abstract: A model-based description of the scaling and radial location of turbulent fluctuations in turbulent pipe flow is presented and used to illuminate the scaling behaviour of the very large scale motions. The model is derived by treating the nonlinearity in the perturbation equation (involving the Reynolds stress) as an unknown forcing, yielding a linear relationship between the velocity field response and this nonlinearity. We do not assume small perturbations. We examine propagating helical velocity response modes that are harmonic in the wall-parallel directions and in time, permitting comparison of our results to experimental data. The steady component of the velocity field that varies only in the wall-normal direction is identified as the turbulent mean profile. A singular value decomposition of the resolvent identifies the forcing shape that will lead to the largest velocity response at a given wavenumber–frequency combination. The hypothesis that these forcing shapes lead to response modes that will be dominant in turbulent pipe flow is tested by using physical arguments to constrain the range of wavenumbers and frequencies to those actually observed in experiments. An investigation of the most amplified velocity response at a given wavenumber–frequency combination reveals critical-layer-like behaviour reminiscent of the neutrally stable solutions of the Orr–Sommerfeld equation in linearly unstable flow. Two distinct regions in the flow where the influence of viscosity becomes important can be identified, namely wall layers that scale with R^(+1/2) and critical layers where the propagation velocity is equal to the local mean velocity, one of which scales with R^(+2/3) in pipe flow. This framework appears to be consistent with several scaling results in wall turbulence and reveals a mechanism by which the effects of viscosity can extend well beyond the immediate vicinity of the wall. The model reproduces inner scaling of the small scales near the wall and an approach to outer scaling in the flow interior. We use our analysis to make a first prediction that the appropriate scaling velocity for the very large scale motions is the centreline velocity, and show that this is in agreement with experimental results. Lastly, we interpret the wall modes as the motion required to meet the wall boundary condition, identifying the interaction between the critical and wall modes as a potential origin for an interaction between the large and small scales that has been observed in recent literature as an amplitude modulation of the near-wall turbulence by the very large scales.

Publication: Journal of Fluid Mechanics Vol.: 658ISSN: 0022-1120

ID: CaltechAUTHORS:20101011-153852682

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Abstract: We present streamwise velocity structure functions <δv_L(τ)>=<|v(t+τ)−v(t)|^p> (with p=1:5) obtained in the near neutral atmospheric surface layer at the Utah SLTEST site at the highest terrestrial Reynolds number Re_τ=O(10^6). We show that the occurrence of very large scale coherent oscillations in the streamwise velocity throughout the wall region, interpreted as genuine structural features of the canonical turbulent boundary layer, affects the scaling exponents of the p>3 order structure functions. This results in a slight alteration of the intermittent behavior of the velocity field. It was found that for positive (fast) large scale oscillation of the low-pass filtered velocity signal, deviations from the Kolmogorov K41 prediction (absence of multiscaling) are more marked, as compared to negative (slow) excursion. The results are discussed in terms of convergence of statistics from atmospheric boundary layer measurements.

Publication: Physica D Vol.: 239 No.: 14 ISSN: 0167-2789

ID: CaltechAUTHORS:20100809-092327681

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Abstract: Characteristic time scales associated with bursting events in the turbulent boundary layer were examined over a very large range of Reynolds numbers based on momentum thickness of 1010, 2870, 4850, and 5×10^6. Well-resolved hot-wire measurements were obtained at the lowest three Reynolds numbers in a low-speed wind tunnel with a long development length and compared to hot-wire and sonic anemometer measurements in the near-neutral atmospheric surface layer over the salt flats of Utah’s western desert. Bursting events were detected using the modified U-level threshold-crossing algorithm outlined by Luchik and Tiederman (1987) [1]. The same procedure and codes were used to analyze all time series records from both the wind tunnel and field experiments. The time between events, T_e, and the event duration, ΔT, were calculated and normalized using four different types of scalings: inner, outer, mixed, and Taylor microscales. It was found that both Reynolds number and wall-normal trends in the mode of T_e were eliminated when scaled by the Taylor microscale. Furthermore, constant (Reynolds number independent) values of the nondimensional mean T_e and ΔT were found in a narrow wall-normal region near the top of the buffer layer when the data were normalized by the Taylor microscale.

Publication: Physica D Vol.: 239 No.: 14 ISSN: 0167-2789

ID: CaltechAUTHORS:20100809-153959601

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Abstract: Time-resolved DPIV measurements performed in wall parallel planes at several wall normal locations in a turbulent boundary layer (TBL) are used to illuminate the distribution of wall parallel velocities in a three-dimensional energy spectrum over streamwise, spanwise, and temporal wavelengths. To our knowledge, this is the first time this type of spectral distribution has been reported. Slices of the 3D spectrum can give insight into the propagation of different scales in the ow as well as the streamwise and spanwise extent of dominant scales. Measurements were performed at three wall normal locations, y^+ = 34; 108; and 278, in a zero pressure gradient TBL at Re_τ = 470 . Two high speed cameras placed side-by-side in the streamwise direction give a 10δ streamwise field of view with a time step of Δt^+ = 0:5 between consecutive fields. Far from the wall the convection velocities of all scales are very close to the local mean velocity in agreement with the work of Dennis and Nickels, while at y^+ = 34 it was found that all measured scales in the flow convect faster than the local mean in agreement with Krogstad et. al. The variation of the convection velocity with scale and distance from the wall will be discussed.

ID: CaltechAUTHORS:20150226-100222113

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Abstract: Statistical and spectral analyses of the manipulation of a canonical zero pressure gradient turbulent boundary layer using static roughness and low-frequency dynamic roughness patches are presented. A shift of spectral energy away from the wall downstream of the roughness patch is observed. The dynamic roughness is shown to disrupt the structure of the boundary layer, while embedding its periodic signature in an extensive stretch of the downstream flow field.

ID: CaltechAUTHORS:20150226-094906075

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Abstract: Wall-bounded turbulent flows at high Reynolds numbers have become an increasingly active area of research in recent years. Many challenges remain in theory, scaling, physical understanding, experimental techniques, and numerical simulations. In this paper we distill the salient advances of recent origin, particularly those that challenge textbook orthodoxy. Some of the outstanding questions, such as the extent of the logarithmic overlap layer, the universality or otherwise of the principal model parameters such as the von Kármán “constant,” the parametrization of roughness effects, and the scaling of mean flow and Reynolds stresses, are highlighted. Research avenues that may provide answers to these questions, notably the improvement of measuring techniques and the construction of new facilities, are identified. We also highlight aspects where differences of opinion persist, with the expectation that this discussion might mark the beginning of their resolution.

Publication: Physics of Fluids Vol.: 22 No.: 6 ISSN: 1070-6631

ID: CaltechAUTHORS:20100810-131138266

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Abstract: Pipes feature strongly in the infrastructure of everyday life, from domestic water pipes to oil and natural gas conduits. A primary consequence of the onset of turbulence in the fluid flowing through the pipes is the dramatically increased power required to pump stuff at the same rate. Thus, the incentives to understand and control the transition process are strong. However, more than 100 years after Osborne Reynolds's seminal experiments on the transition of flow through a pipe from a laminar (smooth) to a turbulent state, the exact physical mechanism that drives this phenomenon still vexes the fluid mechanics community. On page 1491 of this issue, Hof et al. (1) describe a mechanism that feeds energy into a turbulent flow system, allowing the onset of the transition to be manipulated and even the suppression of the turbulence.

Publication: Science Vol.: 327 No.: 5972 ISSN: 0036-8075

ID: CaltechAUTHORS:20100409-112124108

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Abstract: An experimental study was undertaken to determine the effects of step excrescences on boundary layer transition using a unique ground test facility in which the test model was propelled though still air. The models used were designed to have a nominally constant pressure gradient so that the results would be relevant to laminar flow aircraft whose wings often have long runs of mildly favorable pressure gradient. The models had an integrated continuously adjustable two-dimensional step, which could be adjusted to be forward-facing or aft-facing. The large model was used to increase the Reynolds numbers examined so that the results are applicable to laminar flow flight vehicles. Multiple measurement methods, including Preston tubes, hot wires, accelerometers, a boundary layer traverse, and static pressure taps were used to provide comparison data, and to add to the physical understanding of the results. The propelled-model test approach required that the instrumentation be self-contained and ride along with the model as the carrier vehicle moved down the test track. Due to the relatively short times available for data-taking (approximately 15-30 seconds per run), the initialization and data analysis techniques had to be tailored for this application.

ID: CaltechAUTHORS:20150303-112035324

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Abstract: We describe a method to investigate the mode shapes in turbulent pipe flow at a given wavenumber pair that are most responsive to harmonic forcing in the sense that the they correspond to the largest singular value in a Schmidt decomposition of the linear Navier-Stokes operator using the turbulent mean profile as the base flow. The ideas follow logically from the work of Sharma & McKeon (2009), who considered a similar approach for laminar pipe flow.

ID: CaltechAUTHORS:20150303-105749150

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Abstract: An examination of the effects of surface step excrescences on boundary layer transition was performed, using a unique experimental facility. The objective of the work was to characterize the variation of transition Reynolds numbers with measurable step size and boundary layer parameters, with the specific goal of specifying new tolerance criteria for laminar flow airfoils, alongside a fundamental investigation of boundary layer transition mechanisms. This paper focuses on interpretation of hot-wire measurements, including supporting stability calculations, undertaken as part of the study. The results for both forward and aft-facing steps indicated a substantial stabilizing effect of favorable pressure gradient on excrescence-induced boundary layer transition. These findings suggest that manufacturing tolerances for laminar flow aircraft could be loosened in areas where even mild favorable pressure gradients exist.

ID: CaltechAUTHORS:20150303-111551041

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Abstract: Manufacturing tolerances for laminar flow wings can be significantly tighter than those of conventional aircraft. The tighter tolerances can significantly affect the assessment of the practicality of designing for laminar flow. However, existing data on the effects of excrescences typical of manufacturing process are limited. Further, information on the effects—often beneficial—of pressure gradient present on the laminar flow wings is not generally available. To address these concerns, a series of experiments has been undertaken to examine the effects of surface steps in the presence of pressure gradients. The step geometries were selected to represent those that result from actual aircraft manufacturing processes. The range of pressure gradients correspond to those typical of laminar flow wings. Initial experiments were conducted in a low-speed wind tunnel. Later experiments used a novel propelled-model test facility. The results of these studies show that the allowable sizes of surface excrescences for laminar flow wings may be significantly greater than has conventionally been assumed. This could significantly influence the more widespread use of laminar flow for drag reduction, resulting in more efficient aircraft.

ID: CaltechAUTHORS:20150303-110715680

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Abstract: Streamwise and quasi-streamwise elongated structures have been shown to play a significant role in turbulent shear flows. We model the mean behaviour of fully turbulent plane Couette flow using a streamwise constant projection of the Navier–Stokes equations. This results in a two-dimensional three-velocity-component (2D/3C) model. We first use a steady-state version of the model to demonstrate that its nonlinear coupling provides the mathematical mechanism that shapes the turbulent velocity profile. Simulations of the 2D/3C model under small-amplitude Gaussian forcing of the cross-stream components are compared to direct numerical simulation (DNS) data. The results indicate that a streamwise constant projection of the Navier–Stokes equations captures salient features of fully turbulent plane Couette flow at low Reynolds numbers. A systems-theoretic approach is used to demonstrate the presence of large input–output amplification through the forced 2D/3C model. It is this amplification coupled with the appropriate nonlinearity that enables the 2D/3C model to generate turbulent behaviour under the small-amplitude forcing employed in this study.

Publication: Journal of Fluid Mechanics Vol.: 665ISSN: 0022-1120

ID: CaltechAUTHORS:20110131-095901289

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Abstract: Turbulent flow over a “dynamically rough” wall is modeled by considering time-dependent velocity perturbations in the streamwise and wall-normal directions imposed at the wall, a crude linearization. Analysis of the linearized Navier-Stokes operator is performed to select roughness parameters that are predicted to lead to a large disturbance amplification in the body of the flow. Direct numerical simulations of turbulent channel flow at Re_τ ~ 500 with three different roughness amplitudes, a^+, indicate that for a^+ = 10 the response of the flow approximates this predicted form, including the development of a significant span-wise velocity component. The turbulence characteristics, as a function of amplitude, are hypothesized to offer insight of relevance to the static roughness problem.

No.: 22
ID: CaltechAUTHORS:20110713-114512764

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Abstract: Though the effect of distributed roughness on flow over a sphere has been examined in detail, there have been few observations as to the effect of an isolated roughness element on the forces induced on a sphere that is in uniform flow. In this experimental study, we examine how the forces are altered due to both a stationary and dynamic three-dimensional roughness element in the Reynolds number range of 5 x 104 to 5 x 105. It is found that even a small change to the geometry of the sphere, by adding a cylindrical roughness element with a width and height of 1% the sphere diameter, dramatically alters the drag and lateral forces over a wide range of Reynolds numbers. Of particular interest is that the mean of the lateral force magnitude can be increased by a factor of about seven, compared with a stationary stud, by moving the isolated roughness at a constant angular velocity about the sphere. These results can be applied to tripping a laminar boundary layer, steering a bluff body, and increasing the mixing of two fluids, using a minimal amount of energy input. This research is a first step towards understanding the interaction between time dependent surface motion and the subsequent alteration of the location of the boundary layer separation line and wake development.

ID: CaltechAUTHORS:20150303-112614795

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Abstract: First experimental measurements of manipulation of the structure of a canonical zero pressure gradient turbulent boundary layer using a low frequency (compared to the viscous frequency) mechanical dynamic roughness are presented. “Dynamic” (or time-dependent) surface roughness is proposed as a method for both control and diagnosis of turbulent boundary layers.

ID: CaltechAUTHORS:20150303-113157017

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Abstract: The response of pipe flow to physically realistic, temporally and spatially continuous(periodic) forcing is investigated by decomposing the resolvent into orthogonal forcing and response pairs ranked according to their contribution to the resolvent 2-norm. Modelling the non-linear terms normally neglected by linearisation as unstructured forcing permits qualitative extrapolation of the resolvent norm results beyond infinitesimally small perturbations to the turbulent case. The concepts arising have a close relationship to input output transfer function analysis methods known in the control systems literature. The body forcings that yield highest disturbance energy gain are identified and ranked by the decomposition and a worst-case bound put on the energy gain integrated across the pipe cross-section. Analysis of the spectral variation of the corresponding response modes reveals interesting comparisons with recent observations of the behavior of the streamwise velocity in high Reynolds number (turbulent) pipe flow, including the importance of very long scales of the order of ten pipe radii, in the extraction of turbulent energy from the mean flow by the action of turbulent shear stress against the velocity gradient.

ID: CaltechAUTHORS:20141107-113657314

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Abstract: Vortex shedding and turbulent motion in the wake of a sphere that is supported using a streamwise-aligned cylindrical sting are investigated at a subcritical Reynolds number of Re=3800, using high speed particle image velocimetry. The mechanism by which the presence of a sting of increasing diameter relative to the diameter of the sphere influences the wake, in terms of both the small-scale shear instability and the larger wake instability, is explored and brie y compared with the two-dimensional analog of the splitter plate introduced into a cylinder wake. The difficulties associated with obtaining converged statistics, along with the effect of free stream turbulence and sphere vibrations are detailed. An understanding of the mechanism by which the blockage, or interference, arising from the presence of the sting influences cross-wake communication and downstream development is a necessary precursor to studies of active control of the wake using surface actuation on a sting-mounted sphere.

ID: CaltechAUTHORS:20150303-113517097

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Abstract: High Reynolds number pipe flow data are used to demonstrate the importance of several conditions related to scale separation that are either assumed in the classical theories or may be used in light of recent results in wall turbulence to infer a minimum Reynolds number condition above which scaling results may be suitable for extrapolation. Results from the Princeton Superpipe have suggested Re_τ > 5000 as the minimum Reynolds number for which key properties of pipe flow reach a “fully-developed” condition, based on observations of streamwise mean and turbulent velocity structure. Additional values related to finer constraints on the structural development are also discussed. A “skeleton” of wall turbulence is introduced, based on structural components identified as having a dominant role in the dynamics of near-wall turbulence in recent experiments by a variety of authors. Possible interaction mechanisms between these components are described alongside some outstanding questions concerning scale separation and interaction.

ID: CaltechAUTHORS:20141107-121005176

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Abstract: An experimental investigation of the transition process in boundary layers subjected to forward- or aft-facing two-dimensional step excrescences is described. The objective of the work was to characterize the variation of transition Reynolds numbers with measurable roughness and boundary layer parameters, with the specific goal of specifying new tolerance criteria for laminar flow airfoils, alongside a fundamental investigation of linear boundary layer stability mechanisms. Results from an ongoing program of increasing complexity on effects of pressure gradient on excrescence-induced transition are presented. Preliminary N-factor calculations are used to determine the effects of boundary layer stability and attempt to isolate the effect of the disturbance due to the excrescence.

ID: CaltechAUTHORS:20150303-121020776

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Abstract: A simple model for roughness elements with a time-varying height was used to investigate the effect of time-dependent, “dynamic” roughness on wall-bounded flow. Temporally varying wall velocities were specified in a turbulent channel flow simulation in order to model the effect of introducing a roughness time scale in addition to a distribution of roughness length scales.

ID: CaltechAUTHORS:20141024-120928169

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Abstract: The streamwise velocity component in turbulent pipe flow is assessed to determine whether it exhibits asymptotic behaviour that is indicative of high Reynolds numbers. The asymptotic behaviour of both the mean velocity (in the form of the log law) and that of the second moment of the streamwise component of velocity in the outer and overlap regions is consistent with the development of spectral regions which indicate inertial scaling. It is shown that an ‘inertial sublayer’ in physical space may be considered as a spatial analogue of the inertial subrange in the velocity spectrum and such behaviour only appears for Reynolds numbers R^+>5×10^3, approximately, much higher than was generally thought.

Publication: Philosophical Transactions A: Mathematical, Physical and Engineering Sciences Vol.: 365 No.: 1852 ISSN: 1364-503X

ID: CaltechAUTHORS:20141008-160556305

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Abstract: According to Lighthill (1995), Prandtl’s (1904) boundary layer has had the same transforming effect on fluid dynamics as Einstein’s 1905 discoveries had on other parts of physics, which, by the way, were celebrated in 2005 as the World Year of Physics. That the boundary layer becomes turbulent was formally known to Blasius (1908), though, of course, the origin of turbulence in a pipe was studied earlier by Reynolds (1883). The problem of the turbulent boundary layer has since been a paradigm in the field of turbulence. Its practical importance in flows over air and water vehicles as well as in geophysical fluid dynamics has been recognized for nearly a century now. Advances in our understanding of the boundary-layer scaling and structure can be expected to shed further light on the complex and multiscale flow dynamics, and also offer basic input to flow control strategies for practically relevant problems such as reducing large vehicle drag (and hence, by implication, emission levels).

Publication: Philosophical Transactions A: Mathematical, Physical and Engineering Sciences Vol.: 365 No.: 1852 ISSN: 1364-503X

ID: CaltechAUTHORS:20141006-135721326

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Abstract: The neutrally stable atmospheric surface layer is used as a physical model of a very high Reynolds number, canonical turbulent boundary layer. Challenges and limitations with this model are addressed in detail, including the inherent thermal stratification, surface roughness and non-stationarity of the atmosphere. Concurrent hot-wire and sonic anemometry data acquired in Utah’s western desert provide insight to Reynolds number trends in the axial velocity statistics and spectra.

Publication: Philosophical Transactions A: Mathematical, Physical and Engineering Sciences Vol.: 365 No.: 1852 ISSN: 1364-503X

ID: CaltechAUTHORS:20141008-163610413

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Abstract: Measurements of the steady pressure in a fluid flow may be required to determine other thermodynamic properties, to determine forces on a body due to the pressure distribution over it, or in order to determine the dynamic head and flow velocity (for further details on the latter see Sect. 5.1. Pressure is a scalar representation of molecular activity, a measure of the nondirectional molecular motions. Thus it must, by definition, be measured by a device at rest relative to the flow. Whilst the common practice in the fluid mechanics community is to denote the pressure as static (as opposed to the coordinate-dependent total pressure, Sect. 3.1), this terminology introduces a fundamental redundancy. In practice, pressure is commonly measured both at walls and in the freestream using the types of measurement device shown in Fig. 4.1 connected to a transducer of suitable sensitivity and range. The orifice of a small wall tapping represents a simple way to obtain the pressure impressed on the wall by the external flow. So-called static pressure tubes approximate the local fluid pressure in the freestream if the disturbance presented to the flow can either be accounted for or is not large to begin with. However this can only ever be strictly true for steady laminar flow due to the normal velocity component introduced when a flow becomes turbulent. Measurement of freestream pressure is one of the hardest challenges in fluid mechanics. Fig. 4.1 This chapter addresses measurement of pressure using wall tappings (Sect. 4.1) and static pressure tubes (Sect. 4.2), and especially errors due to the intrusive flow presence of real, finite-sized devices and calibrations to correct for these. Bryer and Pankhurst [4.1] and Chue [4.2] provided seminal monographs on the general topic of pressure probes in 1971 and 1975, respectively, which give detailed descriptions of measurement devices, coverage of the background to the various corrections and a survey of older data. The topic is covered here more concisely, with a view to practical use by the engineer, and with reference to modern literature. The reader is referred to Bryer and Pankhurst [4.1] and Chue [4.2] for further details on most sections. In more recent years a further method for obtaining pressure on the surface of a wind tunnel model has been developed, based on pressure sensitive paints (PSP). The introduction of PSP provides a method to measure the pressure on the surface of a model directly without the transducers and tubing associated with conventional means. A paint, the luminescence of which is dependent on air pressure, is applied to the surface of a wind tunnel model and the pressure distribution is obtained from the images produced by proper illumination. In Sect. 4.4 the basics of PSP are discussed and further subsections address in detail different paints, paint application procedures, imaging systems and image processing. In discussing the achievable accuracy of PSP techniques, both the spatial and temporal resolution is examined. The thermal sensitivity of the paint dye is introduced and this is closely linked to temperature-sensitive paints (TSP), as discussed in Chap. 7, Sect. 7.4.

ID: CaltechAUTHORS:20141024-122054736

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Abstract: A method for nonlinear global stabilisation of the incompressible Navier-Stokes equations is presented and used to eliminate transient growth in linearly stable Poiseuille flow for the case of full-field actuation and sensing. In the absence of complete velocity field sensing and full actuation the controller synthesis procedure gives a controller that minimises the the attainable perturbation energy over all disturbances and thus maximises the disturbance threshold for transition to occur. The control laws are found using the theory of positive real systems, originating in the control systems community. It is found that a control law making the linearised part of the perturbed Navier-Stokes equations positive real, provides nonlinear global stability. A state-space synthesis procedure is presented that results in two game-theoretic algebraic Riccati equations.

ID: CaltechAUTHORS:20150303-121446831

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Abstract: A fluid flow control device (210) including an active actuator surface may be used to control boundary layer separation by controllably deforming the actuator surface to create a depression in an exterior face of the surface. Boundary layer control may thus be achieved by producing a turbulent state which is more resistant to separation than the original laminar flow by energising the boundary layer and preventing it from separating. The surface may comprise an electroactive polymer membrane (100) supported on a substrate (200) and a set of electrodes (300, 310) adjacent to a cavity in the substrate. The electrodes may be used to control the deformation of the electroactive polymer membrane to deflect into the cavity and thus create a depression. A fluid flow control system, vehicle, vessel, aircraft or stationary structure employing a fluid flow control device as described above is also envisaged. The fluid flow control device in a method of controlling a fluid flow over surface, including forming one or more depressions in the surface and varying the depths of the depression or depressions as a function of time.

ID: CaltechAUTHORS:20160212-111128964

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Abstract: The design, manufacture and testing of micro-sensors and -actuators for aerodynamic drag reduction by flow control is described. Key factors in the designs are discussed, for example, the type and shape of the actuator electrodes affect the critical parameters of frequency response and deflection amplitude. Some aspects of future work are also covered.

Publication: Microelectronic Engineering Vol.: 83 No.: 4-9 ISSN: 0167-9317

ID: CaltechAUTHORS:20141008-165129047

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Abstract: The friction factor relationship for high-Reynolds-number fully developed turbulent pipe flow is investigated using two sets of data from the Princeton Superpipe in the range 31×10^3 ≤ ReD ≤ 35×10^6. The constants of Prandtl’s ‘universal’ friction factor relationship are shown to be accurate over only a limited Reynolds-number range and unsuitable for extrapolation to high Reynolds numbers. New constants, based on a logarithmic overlap in the mean velocity, are found to represent the high-Reynolds-number data to within 0.5%, and yield a value for the von Kármán constant that is consistent with the mean velocity profiles themselves. The use of a generalized logarithmic law in the mean velocity is also examined. A general friction factor relationship is proposed that predicts all the data to within 1.4% and agrees with the Blasius relationship for low Reynolds numbers to within 2.0%.

Publication: Journal of Fluid Mechanics Vol.: 538ISSN: 0022-1120

ID: CaltechAUTHORS:MCKjfm05

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Abstract: Friction factor data from two recent pipe flow experiments are combined to provide a comprehensive picture of the friction factor variation for Reynolds numbers from 10 to 36,000,000.

Publication: Journal of Fluid Mechanics Vol.: 511ISSN: 0022-1120

ID: CaltechAUTHORS:MCKjfm04a

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Abstract: Statistics of the streamwise velocity component in fully developed pipe flow are examined for Reynolds numbers in the range 5.5 x 10^4 ≤ ReD ≤ 5.7 x 10^6. Probability density functions and their moments (up to sixth order) are presented and their scaling with Reynolds number is assessed. The second moment exhibits two maxima: the one in the viscous sublayer is Reynolds-number dependent while the other, near the lower edge of the log region, follows approximately the peak in Reynolds shear stress. Its locus has an approximate (R^+)^{0.5} dependence. This peak shows no sign of ‘saturation’, increasing indefinitely with Reynolds number. Scalings of the moments with wall friction velocity and $(U_{cl}-\overline{U})$ are examined and the latter is shown to be a better velocity scale for the outer region, y/R > 0.35, but in two distinct Reynolds-number ranges, one when ReD < 6 x 10^4, the other when ReD > 7 x 10^4. Probability density functions do not show any universal behaviour, their higher moments showing small variations with distance from the wall outside the viscous sublayer. They are most nearly Gaussian in the overlap region. Their departures from Gaussian are assessed by examining the behaviour of the higher moments as functions of the lower ones. Spectra and the second moment are compared with empirical and theoretical scaling laws and some anomalies are apparent. In particular, even at the highest Reynolds number, the spectrum does not show a self-similar range of wavenumbers in which the spectral density is proportional to the inverse streamwise wavenumber. Thus such a range does not attract any special significance and does not involve a universal constant.

Publication: Journal of Fluid Mechanics Vol.: 508ISSN: 0022-1120

ID: CaltechAUTHORS:MORjfm04

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Abstract: Issues concerning the accuracy of hot-wire measurements in turbulent pipe flow are addressed for pipe Reynolds numbers up to 6 × 106 and hot-wire Reynolds numbers up to Rew ap 250. These include the optimization of spatial and temporal resolution and the associated feature of signal-to-noise ratio. Very high wire Reynolds numbers enable the use of wires with reduced length-to-diameter ratios compared to those typical of atmospheric conditions owing to increased wire Nusselt numbers. Simulation of the steady-state heat balance for the wire and the unetched portion of wire are used to assess static end-conduction effects: they are used to calculate wire Biot numbers, \sqrt{c_0}l , and fractional end-conduction losses, σ, which confirm the 'conduction-only' theory described by Corrsin. They show that, at Rew ap 250, the wire length-to-diameter ratio can be reduced to about 50, while keeping \sqrt{c_0}l\gt3 and σ < 7% in common with accepted limits at Rew ap 3. It is shown that these limits depend additionally on the choice of wire material and the length of unetched wire. The dynamic effects of end-cooling are also assessed using the conduction-only theory.

Publication: Measurement Science and Technology Vol.: 15 No.: 5 ISSN: 0957-0233

ID: CaltechAUTHORS:LIJmst04

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Abstract: The measurements by Zagarola & Smits (1998) of mean velocity profiles in fully developed turbulent pipe flow are repeated using a smaller Pitot probe to reduce the uncertainties due to velocity gradient corrections. A new static pressure correction (McKeon & Smits 2002) is used in analysing all data and leads to significant differences from the Zagarola & Smits conclusions. The results confirm the presence of a power-law region near the wall and, for Reynolds numbers greater than 230×10^3 (R+ >5×10^3), a logarithmic region further out, but the limits of these regions and some of the constants differ from those reported by Zagarola & Smits. In particular, the log law is found for 600

ID: CaltechAUTHORS:MCKjfm04b

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Abstract: Mean flow measurements taken in fully developed turbulent pipe flow over a wide Reynolds number range are used to evaluate current methods of correcting Pitot probe data. Based on this evaluation, a new form for the displacement correction is proposed which appears to be more accurate over a wider range of conditions than those currently available. The difficulty of obtaining the true near-wall velocity profile near the wall is explored.

Publication: Measurement Science and Technology Vol.: 14 No.: 8 ISSN: 0957-0233

ID: CaltechAUTHORS:MCKmst03

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Abstract: Measurements are reported of the error in wall static pressure reading due to the finite size of the pressure tapping. The experiments were performed in incompressible turbulent pipe flow over a wide range of Reynolds numbers, and the results indicate that the correction term (as a fraction of the wall stress) continues to increase as the hole Reynolds number $d^+=u_\tau d/\nu$ increases, contrary to previous studies. For small holes relative to the pipe diameter the results follow a single curve, but for larger holes the data diverge from this universal behaviour at a point that depends on the ratio of the hole diameter to the pipe diameter.

Publication: Measurement Science and Technology Vol.: 13 No.: 10 ISSN: 0957-0233

ID: CaltechAUTHORS:MCKmst02

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Abstract: Spectra of the streamwise velocity component in fully developed turbulent pipe flow are presented for Reynolds numbers up to 5.7×10^6. Even at the highest Reynolds number, streamwise velocity spectra exhibit incomplete similarity only: while spectra collapse with both classical inner and outer scaling for limited ranges of wave number, these ranges do not overlap. Thus similarity may not be described as complete, and a region varying with the inverse of the streamwise wave number, k1, is not expected, and any apparent k1-1 range does not attract any special significance and does not involve a universal constant. Reasons for this are suggested.

Publication: Physical Review Letters Vol.: 88 No.: 21 ISSN: 0031-9007

ID: CaltechAUTHORS:MORprl02

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Abstract: Statistics of the streamwise velocity component in fully-developed pipe flow are examined for Reynolds numbers in the range 5.5 x 10^4 < Re_D < 5.7 x 10^6. The second moment exhibits two maxima: one in the viscous sublayer is Reynolds-number dependent while the other, near the lower edge of the log region, is also Reynolds-number dependent and follows roughly the peak in Reynolds shear stress. The behaviour of both peaks is consistent with the concept of inactive motion which increases with increasing Reynolds number and decreasing distance from the wall. No simple scaling is apparent, and in particular, so-called "mixed" scaling is no better than wall scaling in the viscous sublayer and is actually worse than wall scaling in the outer region. The second moment is compared with empirical and theoretical scaling laws and some anomalies are apparent. The scaling of spectra using y, R and u_τ is examined. It appears that even at the highest Reynolds number, they exhibit incomplete similarity only: while spectra do collapse with either inner or outer scales for limited ranges of wave number, these ranges do not overlap. Thus similarity may not be described as complete and any apparent k_1^(-1) range does not attract any special significance and does not involve universal constants. It is suggested that this is because of the influence of inactive motion. Spectra also show the presence of very long structures close to the wall.

ID: CaltechAUTHORS:20150304-112212308

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