CaltechAUTHORS: Article
https://feeds.library.caltech.edu/people/McKeon-B-J/article.rss
A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenThu, 13 Jun 2024 13:42:03 -0700Reynolds number dependence of streamwise velocity spectra in turbulent pipe flow
https://resolver.caltech.edu/CaltechAUTHORS:MORprl02
Year: 2002
DOI: 10.1103/PhysRevLett.88.214501
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.https://resolver.caltech.edu/CaltechAUTHORS:MORprl02Static pressure correction in high Reynolds number fully developed turbulent pipe flow
https://resolver.caltech.edu/CaltechAUTHORS:MCKmst02
Year: 2002
DOI: 10.1088/0957-0233/13/10/314
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.https://resolver.caltech.edu/CaltechAUTHORS:MCKmst02Pitot probe corrections in fully developed turbulent pipe flow
https://resolver.caltech.edu/CaltechAUTHORS:MCKmst03
Year: 2003
DOI: 10.1088/0957-0233/14/8/334
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.https://resolver.caltech.edu/CaltechAUTHORS:MCKmst03Further observations on the mean velocity distribution in fully developed pipe flow
https://resolver.caltech.edu/CaltechAUTHORS:MCKjfm04b
Year: 2004
DOI: 10.1017/S0022112003007304
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 600https://resolver.caltech.edu/CaltechAUTHORS:MCKjfm04bThe response of hot wires in high Reynolds-number turbulent pipe flow
https://resolver.caltech.edu/CaltechAUTHORS:LIJmst04
Year: 2004
DOI: 10.1088/0957-0233/15/5/003
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.https://resolver.caltech.edu/CaltechAUTHORS:LIJmst04Scaling of the streamwise velocity component in turbulent pipe flow
https://resolver.caltech.edu/CaltechAUTHORS:MORjfm04
Year: 2004
DOI: 10.1017/S0022112004008985
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.https://resolver.caltech.edu/CaltechAUTHORS:MORjfm04Friction factors for smooth pipe flow
https://resolver.caltech.edu/CaltechAUTHORS:MCKjfm04a
Year: 2004
DOI: 10.1017/S0022112004009796
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.https://resolver.caltech.edu/CaltechAUTHORS:MCKjfm04aA new friction factor relationship for fully developed pipe flow
https://resolver.caltech.edu/CaltechAUTHORS:MCKjfm05
Year: 2005
DOI: 10.1017/S0022112005005501
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%.https://resolver.caltech.edu/CaltechAUTHORS:MCKjfm05Manufacture of micro-sensors and actuators for flow control
https://resolver.caltech.edu/CaltechAUTHORS:20141008-165129047
Year: 2006
DOI: 10.1016/j.mee.2006.01.171
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.https://resolver.caltech.edu/CaltechAUTHORS:20141008-165129047Introduction: scaling and structure in high Reynolds number wall-bounded flows
https://resolver.caltech.edu/CaltechAUTHORS:20141006-135721326
Year: 2007
DOI: 10.1098/rsta.2006.1952
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).https://resolver.caltech.edu/CaltechAUTHORS:20141006-135721326Asymptotic scaling in turbulent pipe flow
https://resolver.caltech.edu/CaltechAUTHORS:20141008-160556305
Year: 2007
DOI: 10.1098/rsta.2006.1945
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.https://resolver.caltech.edu/CaltechAUTHORS:20141008-160556305The near-neutral atmospheric surface layer: turbulence and non-stationarity
https://resolver.caltech.edu/CaltechAUTHORS:20141008-163610413
Year: 2007
DOI: 10.1098/rsta.2006.1946
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.https://resolver.caltech.edu/CaltechAUTHORS:20141008-163610413A streamwise constant model of turbulence in plane Couette flow
https://resolver.caltech.edu/CaltechAUTHORS:20110131-095901289
Year: 2010
DOI: 10.1017/S0022112010003861
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.https://resolver.caltech.edu/CaltechAUTHORS:20110131-095901289Controlling Turbulence
https://resolver.caltech.edu/CaltechAUTHORS:20100409-112124108
Year: 2010
DOI: 10.1126/science.1187607
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.https://resolver.caltech.edu/CaltechAUTHORS:20100409-112124108Wall-bounded turbulent flows at high Reynolds numbers: Recent advances and key issues
https://resolver.caltech.edu/CaltechAUTHORS:20100810-131138266
Year: 2010
DOI: 10.1063/1.3453711
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.https://resolver.caltech.edu/CaltechAUTHORS:20100810-131138266Intermittency in the atmospheric surface layer: Unresolved or slowly varying?
https://resolver.caltech.edu/CaltechAUTHORS:20100809-092327681
Year: 2010
DOI: 10.1016/j.physd.2009.10.010
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.https://resolver.caltech.edu/CaltechAUTHORS:20100809-092327681Scaling the characteristic time of the bursting process in the turbulent boundary layer
https://resolver.caltech.edu/CaltechAUTHORS:20100809-153959601
Year: 2010
DOI: 10.1016/j.physd.2009.09.004
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.https://resolver.caltech.edu/CaltechAUTHORS:20100809-153959601A critical-layer framework for turbulent pipe flow
https://resolver.caltech.edu/CaltechAUTHORS:20101011-153852682
Year: 2010
DOI: 10.1017/S002211201000176X
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.https://resolver.caltech.edu/CaltechAUTHORS:20101011-153852682Large-eddy simulation of large-scale structures in long channel flow
https://resolver.caltech.edu/CaltechAUTHORS:20101206-152355868
Year: 2010
DOI: 10.1017/S0022112010002995
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.https://resolver.caltech.edu/CaltechAUTHORS:20101206-152355868High–Reynolds Number Wall Turbulence
https://resolver.caltech.edu/CaltechAUTHORS:20110524-094826972
Year: 2011
DOI: 10.1146/annurev-fluid-122109-160753
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.https://resolver.caltech.edu/CaltechAUTHORS:20110524-094826972Interactions within the turbulent boundary layer at high Reynolds number
https://resolver.caltech.edu/CaltechAUTHORS:20110307-143116884
Year: 2011
DOI: 10.1017/S0022112010004544
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.https://resolver.caltech.edu/CaltechAUTHORS:20110307-143116884The effect of small-amplitude time-dependent changes to the surface morphology of a sphere
https://resolver.caltech.edu/CaltechAUTHORS:20110527-113052456
Year: 2011
DOI: 10.1017/S0022112011000164
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.https://resolver.caltech.edu/CaltechAUTHORS:20110527-113052456New perspectives on the impulsive roughness-perturbation of a
turbulent boundary layer
https://resolver.caltech.edu/CaltechAUTHORS:20110711-100700826
Year: 2011
DOI: 10.1017/jfm.2011.75
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.https://resolver.caltech.edu/CaltechAUTHORS:20110711-100700826Amplification and nonlinear mechanisms in plane Couette flow
https://resolver.caltech.edu/CaltechAUTHORS:20110722-112408053
Year: 2011
DOI: 10.1063/1.3599701
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.https://resolver.caltech.edu/CaltechAUTHORS:20110722-112408053A streamwise-constant model of turbulent pipe flow
https://resolver.caltech.edu/CaltechAUTHORS:20111115-094643184
Year: 2011
DOI: 10.1063/1.3640081
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.https://resolver.caltech.edu/CaltechAUTHORS:20111115-094643184The effect of a small isolated roughness element on the forces on a sphere in uniform flow
https://resolver.caltech.edu/CaltechAUTHORS:20111021-085953294
Year: 2011
DOI: 10.1007/s00348-011-1126-y
The effect of an isolated roughness element on the forces on a sphere was examined for a Reynolds number range of 5×10^4https://resolver.caltech.edu/CaltechAUTHORS:20111021-085953294A study of the three-dimensional spectral energy distribution in a zero pressure gradient turbulent boundary layer
https://resolver.caltech.edu/CaltechAUTHORS:20111025-073430962
Year: 2011
DOI: 10.1007/s00348-011-1117-z
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.https://resolver.caltech.edu/CaltechAUTHORS:20111025-073430962Unsteady force measurements in sphere flow from subcritical to supercritical Reynolds numbers
https://resolver.caltech.edu/CaltechAUTHORS:20111216-150038357
Year: 2011
DOI: 10.1007/s00348-011-1161-8
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.https://resolver.caltech.edu/CaltechAUTHORS:20111216-150038357Dynamic roughness perturbation of a turbulent boundary layer
https://resolver.caltech.edu/CaltechAUTHORS:20120120-115846312
Year: 2011
DOI: 10.1017/jfm.2011.375
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.https://resolver.caltech.edu/CaltechAUTHORS:20120120-115846312Relaminarisation of Re_τ=100 channel flow with globally stabilising linear feedback control
https://resolver.caltech.edu/CaltechAUTHORS:20120221-073318177
Year: 2011
DOI: 10.1063/1.3662449
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.https://resolver.caltech.edu/CaltechAUTHORS:20120221-073318177Obtaining accurate mean velocity measurements in high Reynolds number turbulent boundary layers using Pitot tubes
https://resolver.caltech.edu/CaltechAUTHORS:20130226-135510873
Year: 2013
DOI: 10.1017/jfm.2012.538
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.https://resolver.caltech.edu/CaltechAUTHORS:20130226-135510873Experimental manipulation of wall turbulence: A systems approach
https://resolver.caltech.edu/CaltechAUTHORS:20130606-132057755
Year: 2013
DOI: 10.1063/1.4793444
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.https://resolver.caltech.edu/CaltechAUTHORS:20130606-132057755Natural logarithms
https://resolver.caltech.edu/CaltechAUTHORS:20130308-085213459
Year: 2013
DOI: 10.1017/jfm.2012.608
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.https://resolver.caltech.edu/CaltechAUTHORS:20130308-085213459Phase relationships between large and small scales in the turbulent boundary layer
https://resolver.caltech.edu/CaltechAUTHORS:20130605-100354415
Year: 2013
DOI: 10.1007/s00348-013-1481-y
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.https://resolver.caltech.edu/CaltechAUTHORS:20130605-100354415Time-resolved measurements of coherent structures in the turbulent boundary layer
https://resolver.caltech.edu/CaltechAUTHORS:20130605-101847312
Year: 2013
DOI: 10.1007/s00348-013-1508-4
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.https://resolver.caltech.edu/CaltechAUTHORS:20130605-101847312On coherent structure in wall turbulence
https://resolver.caltech.edu/CaltechAUTHORS:20130820-101633052
Year: 2013
DOI: 10.1017/jfm.2013.286
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.https://resolver.caltech.edu/CaltechAUTHORS:20130820-101633052Model-based scaling of the streamwise energy density in high-Reynolds-number turbulent channels
https://resolver.caltech.edu/CaltechAUTHORS:20131121-132351952
Year: 2013
DOI: 10.1017/jfm.2013.457
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.https://resolver.caltech.edu/CaltechAUTHORS:20131121-132351952Compact representation of wall-bounded turbulence using compressive sampling
https://resolver.caltech.edu/CaltechAUTHORS:20140320-104900351
Year: 2014
DOI: 10.1063/1.4862303
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.https://resolver.caltech.edu/CaltechAUTHORS:20140320-104900351Influence of a local change of depth on the behavior of walking oil drops
https://resolver.caltech.edu/CaltechAUTHORS:20140502-074953981
Year: 2014
DOI: 10.1016/j.expthermflusci.2013.12.023
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.https://resolver.caltech.edu/CaltechAUTHORS:20140502-074953981A low-order decomposition of turbulent channel flow via resolvent analysis and convex optimization
https://resolver.caltech.edu/CaltechAUTHORS:20140519-153243814
Year: 2014
DOI: 10.1063/1.4876195
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.https://resolver.caltech.edu/CaltechAUTHORS:20140519-153243814Opposition control within the resolvent analysis framework
https://resolver.caltech.edu/CaltechAUTHORS:20140725-153829494
Year: 2014
DOI: 10.1017/jfm.2014.209
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.https://resolver.caltech.edu/CaltechAUTHORS:20140725-153829494On the structure and origin of pressure fluctuations in wall turbulence: predictions based on the resolvent analysis
https://resolver.caltech.edu/CaltechAUTHORS:20140725-141354609
Year: 2014
DOI: 10.1017/jfm.2014.283
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.https://resolver.caltech.edu/CaltechAUTHORS:20140725-141354609Experimental control of natural perturbations in channel flow
https://resolver.caltech.edu/CaltechAUTHORS:20140815-113320134
Year: 2014
DOI: 10.1017/jfm.2014.317
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.https://resolver.caltech.edu/CaltechAUTHORS:20140815-113320134On the origin of frequency sparsity in direct numerical simulations of turbulent pipe flow
https://resolver.caltech.edu/CaltechAUTHORS:20141211-082426965
Year: 2014
DOI: 10.1063/1.4900768
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.https://resolver.caltech.edu/CaltechAUTHORS:20141211-082426965Triadic scale interactions in a turbulent boundary layer
https://resolver.caltech.edu/CaltechAUTHORS:20150227-022811215
Year: 2015
DOI: 10.1017/jfm.2015.79
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.https://resolver.caltech.edu/CaltechAUTHORS:20150227-022811215A framework for studying the effect of compliant surfaces on wall turbulence
https://resolver.caltech.edu/CaltechAUTHORS:20150420-130124855
Year: 2015
DOI: 10.1017/jfm.2015.85
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.https://resolver.caltech.edu/CaltechAUTHORS:20150420-130124855Dynamic stall on a pitching and surging airfoil
https://resolver.caltech.edu/CaltechAUTHORS:20150908-141014592
Year: 2015
DOI: 10.1007/s00348-015-2028-1
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.https://resolver.caltech.edu/CaltechAUTHORS:20150908-141014592Introduction to Topical Issue on Extreme Flows
https://resolver.caltech.edu/CaltechAUTHORS:20160218-122950356
Year: 2016
DOI: 10.1007/s00348-015-2094-4
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.https://resolver.caltech.edu/CaltechAUTHORS:20160218-122950356Low-dimensional representations of exact coherent states of the Navier-Stokes equations from the resolvent model of wall turbulence
https://resolver.caltech.edu/CaltechAUTHORS:20160315-160338397
Year: 2016
DOI: 10.1103/PhysRevE.93.021102
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.https://resolver.caltech.edu/CaltechAUTHORS:20160315-160338397Streamwise-varying steady transpiration control in turbulent pipe flow
https://resolver.caltech.edu/CaltechAUTHORS:20160610-084044942
Year: 2016
DOI: 10.1017/jfm.2016.279
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.https://resolver.caltech.edu/CaltechAUTHORS:20160610-084044942Correspondence between Koopman mode decomposition, resolvent mode decomposition, and invariant solutions of the Navier-Stokes equations
https://resolver.caltech.edu/CaltechAUTHORS:20160729-123350653
Year: 2016
DOI: 10.1103/PhysRevFluids.1.032402
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.https://resolver.caltech.edu/CaltechAUTHORS:20160729-123350653Nonlinear interactions isolated through scale synthesis in experimental wall turbulence
https://resolver.caltech.edu/CaltechAUTHORS:20160729-122826684
Year: 2016
DOI: 10.1103/PhysRevFluids.1.032401
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.https://resolver.caltech.edu/CaltechAUTHORS:20160729-122826684A reduced-order model of three-dimensional unsteady flow in a cavity based on the resolvent operator
https://resolver.caltech.edu/CaltechAUTHORS:20160630-111131742
Year: 2016
DOI: 10.1017/jfm.2016.339
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.https://resolver.caltech.edu/CaltechAUTHORS:20160630-111131742On the design of optimal compliant walls for turbulence control
https://resolver.caltech.edu/CaltechAUTHORS:20160901-123400889
Year: 2016
DOI: 10.1080/14685248.2016.1181267
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.https://resolver.caltech.edu/CaltechAUTHORS:20160901-123400889Analysis of Flow Timescales on a Periodically Pitching/Surging Airfoil
https://resolver.caltech.edu/CaltechAUTHORS:20161202-090243214
Year: 2016
DOI: 10.2514/1.J054784
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.https://resolver.caltech.edu/CaltechAUTHORS:20161202-090243214Phase relations in a forced turbulent boundary layer: implications for modelling of high Reynolds number wall turbulence
https://resolver.caltech.edu/CaltechAUTHORS:20170213-124117937
Year: 2017
DOI: 10.1098/rsta.2016.0080
PMCID: PMC5311448
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.https://resolver.caltech.edu/CaltechAUTHORS:20170213-124117937Scaling and interaction of self-similar modes in models of high Reynolds number wall turbulence
https://resolver.caltech.edu/CaltechAUTHORS:20170213-124117603
Year: 2017
DOI: 10.1098/rsta.2016.0089
PMCID: PMC5311453
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.https://resolver.caltech.edu/CaltechAUTHORS:20170213-124117603The engine behind (wall) turbulence: perspectives on scale interactions
https://resolver.caltech.edu/CaltechAUTHORS:20170428-132759772
Year: 2017
DOI: 10.1017/jfm.2017.115
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.https://resolver.caltech.edu/CaltechAUTHORS:20170428-132759772Data assimilation of mean velocity from 2D PIV measurements of flow over an idealized airfoil
https://resolver.caltech.edu/CaltechAUTHORS:20170426-063702442
Year: 2017
DOI: 10.1007/s00348-017-2336-8
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.https://resolver.caltech.edu/CaltechAUTHORS:20170426-063702442Phase-relationships between scales in the perturbed turbulent boundary layer
https://resolver.caltech.edu/CaltechAUTHORS:20171211-080203757
Year: 2017
DOI: 10.1080/14685248.2017.1361536
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.https://resolver.caltech.edu/CaltechAUTHORS:20171211-080203757Coherent structures, uniform momentum zones and the streamwise energy spectrum in wall-bounded turbulent flows
https://resolver.caltech.edu/CaltechAUTHORS:20170911-104649363
Year: 2017
DOI: 10.1017/jfm.2017.493
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.https://resolver.caltech.edu/CaltechAUTHORS:20170911-104649363Modal Analysis of Fluid Flows: An Overview
https://resolver.caltech.edu/CaltechAUTHORS:20171213-075938896
Year: 2017
DOI: 10.2514/1.J056060
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.https://resolver.caltech.edu/CaltechAUTHORS:20171213-075938896Non-normality and classification of amplification mechanisms in stability and resolvent analysis
https://resolver.caltech.edu/CaltechAUTHORS:20180517-082629521
Year: 2018
DOI: 10.1103/PhysRevFluids.3.053902
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.https://resolver.caltech.edu/CaltechAUTHORS:20180517-082629521Dynamic Roughness for Manipulation and Control of Turbulent Boundary Layers: An Overview
https://resolver.caltech.edu/CaltechAUTHORS:20180627-120113196
Year: 2018
DOI: 10.2514/1.J056764
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.https://resolver.caltech.edu/CaltechAUTHORS:20180627-120113196Relation between a singly-periodic roughness geometry and spatio-temporal turbulence characteristics
https://resolver.caltech.edu/CaltechAUTHORS:20180507-091749221
Year: 2018
DOI: 10.1016/j.ijheatfluidflow.2018.04.005
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.https://resolver.caltech.edu/CaltechAUTHORS:20180507-091749221Efficient representation of exact coherent states of the Navier–Stokes equations using resolvent analysis
https://resolver.caltech.edu/CaltechAUTHORS:20190122-161813137
Year: 2019
DOI: 10.1088/1873-7005/aab1ab
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.https://resolver.caltech.edu/CaltechAUTHORS:20190122-161813137Vortical Gusts: Experimental Generation and Interaction with Wing
https://resolver.caltech.edu/CaltechAUTHORS:20190314-082614909
Year: 2019
DOI: 10.2514/1.J056914
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.https://resolver.caltech.edu/CaltechAUTHORS:20190314-082614909Critical-Layer Structures and Mechanisms in Elastoinertial Turbulence
https://resolver.caltech.edu/CaltechAUTHORS:20190329-083647606
Year: 2019
DOI: 10.1103/physrevlett.122.124503
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.https://resolver.caltech.edu/CaltechAUTHORS:20190329-083647606Role of parasitic modes in nonlinear closure via the resolvent feedback loop
https://resolver.caltech.edu/CaltechAUTHORS:20190501-124814530
Year: 2019
DOI: 10.1103/PhysRevFluids.4.052601
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.https://resolver.caltech.edu/CaltechAUTHORS:20190501-124814530Predicting the response of turbulent channel flow to varying-phase opposition control: Resolvent analysis as a tool for flow control design
https://resolver.caltech.edu/CaltechAUTHORS:20190731-130426261
Year: 2019
DOI: 10.1103/PhysRevFluids.4.073905
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.https://resolver.caltech.edu/CaltechAUTHORS:20190731-130426261Effect of Coherent Structures on Aero-Optic Distortion in a Turbulent Boundary Layer
https://resolver.caltech.edu/CaltechAUTHORS:20190806-081816859
Year: 2019
DOI: 10.2514/1.J058088
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.https://resolver.caltech.edu/CaltechAUTHORS:20190806-081816859Computing exact coherent states in channels starting from the laminar profile: A resolvent-based approach
https://resolver.caltech.edu/CaltechAUTHORS:20190911-153849331
Year: 2019
DOI: 10.1103/physreve.100.021101
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.https://resolver.caltech.edu/CaltechAUTHORS:20190911-153849331Self-similar hierarchies and attached eddies
https://resolver.caltech.edu/CaltechAUTHORS:20190826-092240301
Year: 2019
DOI: 10.1103/PhysRevFluids.4.082601
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.https://resolver.caltech.edu/CaltechAUTHORS:20190826-092240301Turbulence Amplitude Amplification in an Externally Forced, Subsonic Turbulent Boundary Layer
https://resolver.caltech.edu/CaltechAUTHORS:20190806-084439876
Year: 2019
DOI: 10.2514/1.J057871
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.https://resolver.caltech.edu/CaltechAUTHORS:20190806-084439876On the shape of resolvent modes in wall-bounded turbulence
https://resolver.caltech.edu/CaltechAUTHORS:20190429-080541243
Year: 2019
DOI: 10.1017/jfm.2019.594
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.https://resolver.caltech.edu/CaltechAUTHORS:20190429-080541243A tale of two airfoils: resolvent-based modelling of an oscillator versus an amplifier from an experimental mean
https://resolver.caltech.edu/CaltechAUTHORS:20200124-083217262
Year: 2019
DOI: 10.1017/jfm.2019.747
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.https://resolver.caltech.edu/CaltechAUTHORS:20200124-083217262Resolvent-based study of compressibility effects on supersonic turbulent boundary layers
https://resolver.caltech.edu/CaltechAUTHORS:20200128-124558577
Year: 2020
DOI: 10.1017/jfm.2019.881
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.https://resolver.caltech.edu/CaltechAUTHORS:20200128-124558577Mean and Unsteady Flow Reconstruction Using Data-Assimilation and Resolvent Analysis
https://resolver.caltech.edu/CaltechAUTHORS:20190820-132839798
Year: 2020
DOI: 10.2514/1.J057889
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 47https://resolver.caltech.edu/CaltechAUTHORS:20190820-132839798Spatial organisation of velocity structures for large passive scalar gradients
https://resolver.caltech.edu/CaltechAUTHORS:20200116-083832946
Year: 2020
DOI: 10.1017/jfm.2019.977
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.https://resolver.caltech.edu/CaltechAUTHORS:20200116-083832946Characterization of the Spatio-Temporal Response of a Turbulent Boundary Layer to Dynamic Roughness
https://resolver.caltech.edu/CaltechAUTHORS:20190910-154544575
Year: 2020
DOI: 10.1007/s10494-019-00069-1
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.https://resolver.caltech.edu/CaltechAUTHORS:20190910-154544575Measurements of a turbulent boundary layer-compliant surface system in response to targeted, dynamic roughness forcing
https://resolver.caltech.edu/CaltechAUTHORS:20200316-150528520
Year: 2020
DOI: 10.1007/s00348-020-2933-9
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.https://resolver.caltech.edu/CaltechAUTHORS:20200316-150528520Characterization of vortex regeneration mechanism in the self-sustaining process of wall-bounded flows using resolvent analysis
https://resolver.caltech.edu/CaltechAUTHORS:20200701-133438084
Year: 2020
DOI: 10.1088/1742-6596/1522/1/012001
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.https://resolver.caltech.edu/CaltechAUTHORS:20200701-133438084Interaction of forced Orr-Sommerfeld and Squire modes in a low-order representation of turbulent channel flow
https://resolver.caltech.edu/CaltechAUTHORS:20200128-133412087
Year: 2020
DOI: 10.1103/PhysRevFluids.5.084607
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.https://resolver.caltech.edu/CaltechAUTHORS:20200128-133412087Self-sustained elastoinertial Tollmien-Schlichting waves
https://resolver.caltech.edu/CaltechAUTHORS:20200128-134130309
Year: 2020
DOI: 10.1017/jfm.2020.372
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.https://resolver.caltech.edu/CaltechAUTHORS:20200128-134130309Prediction of resolvent mode shapes in supersonic turbulent boundary layers
https://resolver.caltech.edu/CaltechAUTHORS:20201009-091931261
Year: 2020
DOI: 10.1016/j.ijheatfluidflow.2020.108677
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.https://resolver.caltech.edu/CaltechAUTHORS:20201009-091931261On the origin of drag increase in varying-phase opposition control
https://resolver.caltech.edu/CaltechAUTHORS:20200805-102610802
Year: 2020
DOI: 10.1016/j.ijheatfluidflow.2020.108651
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.https://resolver.caltech.edu/CaltechAUTHORS:20200805-102610802Control of instability by injection rate oscillations in a radial Hele-Shaw cell
https://resolver.caltech.edu/CaltechAUTHORS:20201217-110801490
Year: 2020
DOI: 10.1103/physrevfluids.5.123902
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.https://resolver.caltech.edu/CaltechAUTHORS:20201217-110801490A basis for flow modelling
https://resolver.caltech.edu/CaltechAUTHORS:20201022-112713087
Year: 2020
DOI: 10.1017/jfm.2020.728
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.https://resolver.caltech.edu/CaltechAUTHORS:20201022-112713087Temporal characteristics of the probability density function of velocity in wall-bounded turbulent flows
https://resolver.caltech.edu/CaltechAUTHORS:20210311-134751536
Year: 2021
DOI: 10.1017/jfm.2020.1163
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.https://resolver.caltech.edu/CaltechAUTHORS:20210311-134751536Experiments and Modeling of a Compliant Wall Response to a Turbulent Boundary Layer with Dynamic Roughness Forcing
https://resolver.caltech.edu/CaltechAUTHORS:20210429-144554642
Year: 2021
DOI: 10.3390/fluids6050173
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.https://resolver.caltech.edu/CaltechAUTHORS:20210429-144554642Nonlinear mechanism of the self-sustaining process in the buffer and logarithmic layer of wall-bounded flows
https://resolver.caltech.edu/CaltechAUTHORS:20191223-155902555
Year: 2021
DOI: 10.1017/jfm.2020.857
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.https://resolver.caltech.edu/CaltechAUTHORS:20191223-155902555Interactions between scales in wall turbulence: phase relationships, amplitude modulation and the importance of critical layers
https://resolver.caltech.edu/CaltechAUTHORS:20210312-151430797
Year: 2021
DOI: 10.1017/jfm.2020.770
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.https://resolver.caltech.edu/CaltechAUTHORS:20210312-151430797Unsteady dynamics in the streamwise-oscillating cylinder wake for forcing frequencies below lock-on
https://resolver.caltech.edu/CaltechAUTHORS:20200811-092143234
Year: 2021
DOI: 10.1103/PhysRevFluids.6.074702
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.https://resolver.caltech.edu/CaltechAUTHORS:20200811-092143234Data-driven resolvent analysis
https://resolver.caltech.edu/CaltechAUTHORS:20210315-104726854
Year: 2021
DOI: 10.1017/jfm.2021.337
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.https://resolver.caltech.edu/CaltechAUTHORS:20210315-104726854Resolvent analysis of stratification effects on wall-bounded shear flows
https://resolver.caltech.edu/CaltechAUTHORS:20210223-153658783
Year: 2021
DOI: 10.1103/PhysRevFluids.6.084804
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.https://resolver.caltech.edu/CaltechAUTHORS:20210223-153658783Tollmien-Schlichting route to elastoinertial turbulence in channel flow
https://resolver.caltech.edu/CaltechAUTHORS:20210504-093917792
Year: 2021
DOI: 10.1103/PhysRevFluids.6.093301
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.https://resolver.caltech.edu/CaltechAUTHORS:20210504-093917792Closing the loop: nonlinear Taylor vortex flow through the lens of resolvent analysis
https://resolver.caltech.edu/CaltechAUTHORS:20210315-104125027
Year: 2021
DOI: 10.1017/jfm.2021.623
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.https://resolver.caltech.edu/CaltechAUTHORS:20210315-104125027Amplitude and wall-normal distance variation of small scales in turbulent boundary layers
https://resolver.caltech.edu/CaltechAUTHORS:20220121-870773000
Year: 2022
DOI: 10.1103/physrevfluids.7.014606
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.https://resolver.caltech.edu/CaltechAUTHORS:20220121-870773000Variational formulation of resolvent analysis
https://resolver.caltech.edu/CaltechAUTHORS:20220207-90383000
Year: 2022
DOI: 10.1103/physrevfluids.7.013905
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.https://resolver.caltech.edu/CaltechAUTHORS:20220207-90383000Kernel learning for robust dynamic mode decomposition: linear and nonlinear disambiguation optimization
https://resolver.caltech.edu/CaltechAUTHORS:20220414-26938000
Year: 2022
DOI: 10.1098/rspa.2021.0830
PMCID: PMC9006118
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.https://resolver.caltech.edu/CaltechAUTHORS:20220414-26938000Spatiotemporal characteristics of uniform momentum zones: Experiments and modeling
https://resolver.caltech.edu/CaltechAUTHORS:20221108-874083900.5
Year: 2022
DOI: 10.1103/physrevfluids.7.104603
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.https://resolver.caltech.edu/CaltechAUTHORS:20221108-874083900.5Physics-informed dynamic mode decomposition
https://resolver.caltech.edu/CaltechAUTHORS:20230321-821389800.34
Year: 2023
DOI: 10.1098/rspa.2022.0576
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.https://resolver.caltech.edu/CaltechAUTHORS:20230321-821389800.34Frequency-tuned surfaces for passive control of wall-bounded turbulent flow – a resolvent analysis study
https://resolver.caltech.edu/CaltechAUTHORS:20230412-109103400.2
Year: 2023
DOI: 10.1017/jfm.2023.149
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.https://resolver.caltech.edu/CaltechAUTHORS:20230412-109103400.2Towards real-time reconstruction of velocity fluctuations in turbulent channel flow
https://authors.library.caltech.edu/records/qfh54-kgd88
Year: 2023
DOI: 10.1103/PhysRevFluids.8.064612
<p>We develop a framework for efficient streaming reconstructions of turbulent velocity fluctuations from limited sensor measurements with the goal of enabling real-time applications. The reconstruction process is simplified by computing linear estimators using flow statistics from an initial training period and evaluating their performance during a subsequent testing period with data obtained from direct numerical simulation. We address cases where (i) no, (ii) limited, and (iii) full-field training data are available using estimators based on (i) resolvent modes, (ii) resolvent-based estimation, and (iii) spectral proper orthogonal decomposition modes. During training, we introduce blockwise inversion to accurately and efficiently compute the resolvent operator in an interpretable manner. During testing, we enable efficient streaming reconstructions by using a temporal sliding discrete Fourier transform to recursively update Fourier coefficients using incoming measurements. We use this framework to reconstruct with minimal time delay the turbulent velocity fluctuations in a minimal channel at <span class="mjx-chtml MathJax_CHTML"><span class="mjx-math"><span class="mjx-mrow"><span class="mjx-mrow"><span class="mjx-msub"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R">Re_<span class="mjx-sub"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I"><em>τ</em> </span></span></span></span></span></span></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R">≈ </span></span><span class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R">186</span></span></span></span></span></span> from sparse planar measurements. We evaluate reconstruction accuracy in the context of the extent of data required and thereby identify potential use cases for each estimator. The reconstructions capture large portions of the dynamics from relatively few measurement planes when the linear estimators are computed with sufficient fidelity. We also evaluate the efficiency of our reconstructions and show that the present framework has the potential to help enable real-time reconstructions of turbulent velocity fluctuations in an analogous experimental setting.</p>https://authors.library.caltech.edu/records/qfh54-kgd88