Abstract: By employing dimensional analysis, we scale the equations of the mesoscopic model of finite temperature superfluid hydrodynamics. Based on this scaling, we set up three problems that depict the effects of kinematic, normal-fluid strain fields on superfluid vortex loops, and characterize small-scale processes in fully developed turbulence. We also develop a formula for the computation of energy spectra corresponding to superfluid vortex tangles in unbounded domains. Employing this formula, we compute energy spectra of superfluid vortex patterns induced by uniaxial, equibiaxial, and simple-shear normal-fluid flows. By comparing the steady-state superfluid spectra and vortex structures, we conclude that normal-flow strain fields do not play an important role in explaining the phenomenology of fully developed superfluid turbulence. This is in sharp contrast with the role of vortical normal-flow fields in offering plausible, structural explanations of superfluid vortex patterns and spectra entailed in numerical turbulent solutions of the mesoscopic model.

Publication: Physical Review Fluids Vol.: 6 No.: 4 ISSN: 2469-990X

ID: CaltechAUTHORS:20210423-084247606

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Abstract: We employ reconnection-capable, vortex filament methods and finite-volume, Navier–Stokes flow solvers to investigate the topological and helicity dynamics of vortex links for medium and high Reynolds numbers. Our vortex-dynamical model is based on discretization of vortex tubes into bundles of numerical analogues of vortex lines. Due to their nearly singular nature, the numerical vortex lines have topological writhe but not twist. By means of our reconnecting vortex tube model, it is shown that the helicity of a vortex link is conserved during the unknotting process. The dynamics of linking and writhe topological measures indicate that most of the initial linking becomes writhe during the post-reconnection evolution. The helicity spectra of the vortex link present alternating-sign helicity fluctuations at all (potential flow) scales up to the vortex core. At pre-reconnection times, these fluctuations are damped by Biot–Savart vortex stretching and helicity becomes single signed. The post-reconnection spectra indicate an inverse helicity cascade corresponding to the creation of a homogenized vortex blob, a process reminiscent of coherent structure formation in turbulence. An accompanying Navier–Stokes calculation of vortex link dynamics at Reynolds numbers Re=1500 confirms the post-reconnection transformation of linking into different topological measures, the pre-reconnection damping of helicity spectra fluctuations and the spectral shift to low wavenumbers at post-reconnection times. Due to viscous dissipation action, however, this shift is accompanied by progressive reduction of helicity peak values.

Publication: Journal of Fluid Mechanics Vol.: 911ISSN: 0022-1120

ID: CaltechAUTHORS:20210218-170937308

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Abstract: [no abstract]

Publication: Journal of Fluids and Structures Vol.: 89ISSN: 0889-9746

ID: CaltechAUTHORS:20200102-082032300

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Abstract: The technique of large-eddy simulation (LES) continues to play an important role in the numerical simulation of fluid dynamic processes in engineering and scientific applications. This paper will review and discuss a few of the various schemes for LES applied to the incompressible Navier-Stokes equations and to the scalar advection-diffusion equation. In particular, subgrid models based on deconvolution will be discussed. An interesting connection between the tensor diffusivity model, a particular version of a deconvolution model, and Lagrangian particle methods for the vorticity transport equation and for the scalar convection-diffusion equation will be explored. In addition, the possibility of using super-resolution, i.e. recovering fine-scale information knowing only coarse-scale information, in LES will be investigated.

ID: CaltechAUTHORS:20160930-083031310

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Abstract: We introduce an iterative Brinkman penalization method for the enforcement of the no-slip boundary condition in remeshed vortex methods. In the proposed method, the Brinkman penalization is applied iteratively only in the neighborhood of the body. This allows for using significantly larger time steps, than what is customary in the Brinkman penalization, thus reducing its computational cost while maintaining the capability of the method to handle complex geometries. We demonstrate the accuracy of our method by considering challenging benchmark problems such as flow past an impulsively started cylinder and normal to an impulsively started and accelerated flat plate. We find that the present method enhances significantly the accuracy of the Brinkman penalization technique for the simulations of highly unsteady flows past complex geometries.

Publication: Journal of Computational Physics Vol.: 280ISSN: 0021-9991

ID: CaltechAUTHORS:20150105-132839934

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Abstract: A low-order model for the arbitrary motion of a thin airfoil with trailing edge fluidic control is derived from basic fluid mechanics principles. The model consist of solving a single ordinary differential equation with a special treatment of a vortex shedding criteria. The model is compared with experimental and high-order numerical simulations and the results give a reasonable means of predicting the lift and moment on a thin airfoil. Furthermore, the model is extended to account for the actuation and control due to the synthetic jet actuation near the trailing edge. The model response is compared to experimental results.

Publication: Journal of Fluids and Structures Vol.: 27 No.: 5-6 ISSN: 0889-9746

ID: CaltechAUTHORS:20110906-075122465

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Abstract: We present a validation study for the hybrid particle-mesh vortex method against a pseudo-spectral method for the Taylor–Green vortex at Re_Γ = 1600 as well as in the collision of two antiparallel vortex tubes at Re_Γ = 10,000. In this study we present diagnostics such as energy spectra and enstrophy as computed by both methods as well as point-wise comparisons of the vorticity field. Using a fourth order accurate kernel for interpolation between the particles and the mesh, the results of the hybrid vortex method and of the pseudo-spectral method agree well in both flow cases. For the Taylor–Green vortex, the vorticity contours computed by both methods around the time of the energy dissipation peak overlap. The energy spectrum shows that only the smallest length scales in the flow are not captured by the vortex method. In the second flow case, where we compute the collision of two anti-parallel vortex tubes at Reynolds number 10,000, the vortex method results and the pseudo-spectral method results are in very good agreement up to and including the first reconnection of the tubes. The maximum error in the effective viscosity is about 2.5% for the vortex method and about 1% for the pseudo-spectral method. At later times the flows computed with the different methods show the same qualitative features, but the quantitative agreement on vortical structures is lost.

Publication: Journal of Computational Physics Vol.: 230 No.: 8 ISSN: 0021-9991

ID: CaltechAUTHORS:20110421-100240833

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Abstract: The interaction of two arbitrary bodies immersed in a two-dimensional inviscid fluid is investigated. Given the linear and angular velocities of the bodies, the solution of the potential flow problem with zero circulation around both bodies is reduced to the determination of a suitable Laurent series in a conformally mapped domain that satisfies the boundary conditions. The potential flow solution is then used to determine the force and moment acting on each body by using generalized Blasius formulas. The current formulation is applied to two examples. First, the case of two rigid circular cylinders interacting in an unbounded domain is investigated. The forces on two cylinders with prescribed motion forced-forced is determined and compared to previous results for validation purposes. We then study the response of a single “free” cylinder due to the prescribed motion of the other cylinder forced-free. This forced-free situation is used to justify the hydrodynamic benefits of drafting in aquatic locomotion. In the case of two neutrally buoyant circular cylinders, the aft cylinder is capable of attaining a substantial propulsive force that is the same order of magnitude of its inertial forces. Additionally, the coupled interaction of two cylinders given an arbitrary initial condition free-free is studied to show the differences of perfect collisions with and without the presence of an inviscid fluid. For a certain range of collision parameters, the fluid acts to deflect the cylinder paths just enough before the collision to drastically affect the long time trajectories of the bodies. In the second example, the flapping of two plates is explored. It is seen that the interactions between each plate can cause a net force and torque at certain instants in time, but for idealized sinusoidal motions in irrotational potential flow, there is no net force and torque acting at the system center.

Publication: Physics of Fluids Vol.: 22 No.: 10 ISSN: 1070-6631

ID: CaltechAUTHORS:20101130-152036162

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Abstract: The motion of a tube of vorticity with a cross sectional radius that is everywhere small compared to local radius of curvature of the tube is considered. In particular, we determine the inviscid motion of the 3D space curve that traces the centerline of the tube for an arbitrary distribution of axial vorticity within the core.

Publication: Theoretical and Computational Fluid Dynamics Vol.: 24 No.: 1-4 ISSN: 0935-4964

ID: CaltechAUTHORS:20100407-113730016

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Abstract: A simple low-order model is derived for developing flight control laws for controlling the longitudinal dynamics of an aircraft using synthetic jet type actuators. Bi-directional changes in the pitching moment over a range of angles of attack are effected by controllable, nominally-symmetric trapped vorticity concentrations on both the suction and pressure surfaces near the trailing edge. Actuation is applied on both surfaces by hybrid actuators that are each comprised of a miniature obstruction integrated with a synthetic jet actuator to manipulate and regulate the vorticity concentrations. In previous work, a simple model was derived from a reduced order vortex model that includes one explicit nonlinear state for fluid variables and can be easily coupled to the rigid body dynamics of an aircraft. This paper further simplifies this model for control design. The control design is based on an output feedback adaptive control methodology that illustrates the effectiveness of using the model for achieving flight control at a higher bandwidth than achievable with typical static actuator assumptions. A unique feature of the control design is that the control variable is a pseudo-control based on regulating a control vortex strength. Wind tunnel experiments on a unique dynamics traverse verify that tracking performance is indeed better than control designs employing standard actuator modeling assumptions.

ID: CaltechAUTHORS:20191008-111635395

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Abstract: Regions of high curvature of a material line as they evolve in a chaotic flow are considered. In such a region, the curvature as a function of arclength along the line is found to have a universal form with the peak curvature the only parameter involved. The alignment of the principal axes of the strain tensor with respect to the local tangent vector of the curve and the ratio of the two largest finite-time Lyapunov exponents play a key role. Numerical experiments with ABC flow demonstrate the result.

Publication: Journal of Fluid Mechanics Vol.: 622ISSN: 0022-1120

ID: CaltechAUTHORS:20090824-134558385

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Abstract: The method of contour dynamics, developed for two-dimensional vortex patches by Zabusky et al. [N.J. Zabusky, M.H. Hughes, K.V. Roberts, Contour dynamics for the Euler equations in two-dimensions, J. Comp. Phys. 30 (1979) 96-106] is extended to vortex rings in which the vorticity distribution varies linearly with normal distance from the symmetry axis. The method tracks the motion of the boundaries of the vorticity regions and hence reduces the dimensionality of the problem by one. We discuss the formulation and implementation of the scheme, verify its accuracy and convergence, and present illustrative examples.

Publication: Journal of Computational Physics Vol.: 227 No.: 21 ISSN: 0021-9991

ID: CaltechAUTHORS:SHAjcp08

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Abstract: A simple low-order model is derived to determine the flow forces and moments on an airfoil that arbitrarily pitches and plunges with the presence of synthetic jet actuation for the use in an adaptive closed-loop control scheme. The low-order model captures the attached flow response of an airfoil in the presence of synthetic jet actuators near the trailing edge. The model includes two explicit non-linear states for fluid variables and can be easily coupled to the rigid body dynamics of the system. The model is validated with high fidelity numerical simulations and experiments. The low-order model agreement with experiments is good for low reduced frequency pitching. The agreement to numerical simulations is also good for reduced frequencies that are an order of magnitude higher than those attainable in experiments.

ID: CaltechAUTHORS:20200324-125732644

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Abstract: We present direct numerical simulations of the turbulent decay of vortex rings with ReΓ = 7500. We analyse the vortex dynamics during the nonlinear stage of the instability along with the structure of the vortex wake during the turbulent stage. These simulations enable the quantification of vorticity dynamics and their correlation with structures from dye visualization and the observations of circulation decay that have been reported in related experimental works. Movies are available with the online version of the paper.

Publication: Journal of Fluid Mechanics Vol.: 581ISSN: 0022-1120

ID: CaltechAUTHORS:BERjfm07

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Abstract: The emergence of coherent vortical structures is a hallmark of the evolution of two-dimensional turbulence. Two fundamental processes of this evolution have been identified in vortex merging and vortex axisymmetrization. The question of whether axisymmetrization is a universal process has recently been answered in the negative. In the linear approximation, vortices indeed become axisymmetric, due to shear-enhanced diffusion. In the case of nonlinear interactions, other outcomes are possible; in the present work, we discuss a situation in which the flow reorganizes into a tripolar vortex. By performing an extensive numerical study, spanning the parameter space, we pursue the questions of what dictates if the flow will become axisymmetric or will develop into a quasisteady tripolar vortex, and what are the stages and the time scales of the flow evolution. The initial condition in this study consists of a Gaussian monopole with a quadrupolar perturbation. The amplitude of the perturbation and the Reynolds number determine the evolution. A tripole emerges for sufficiently large amplitude of the perturbation, and we seek to find a critical amplitude that varies with Reynolds number. We make several physical observations derived from visualizing and postprocessing numerous flow simulations: looking at the decay of the perturbation with respect to viscous or shear diffusion time scales; applying mixing theory; obtaining the first few azimuthal modes of the vorticity field; and describing the long-time evolution.

Publication: Physics of Fluids Vol.: 19 No.: 1 ISSN: 1070-6631

ID: CaltechAUTHORS:BARpof07

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Abstract: We have studied the effects of controlled damping on the amplitude and frequency response profiles of an elastically mounted cylinder in cross-flow. The dimensionless damping parameter, b^*=2b/ρLDU, which is closely related to the traditional “mass-damping” parameter, m^*ζ, was varied over a wide range of values through the use of a variable magnetic eddy current damping system. For low damping and sufficiently high Reynolds number we observe the previously described large-amplitude, three-branch (initial, upper, lower) response profile, and for high damping or low Reynolds number we observe the small-amplitude, two-branch (initial and lower) response profile. However we find that, because of the influence of Reynolds number, the traditional labels of “high mass-damping” and “low mass-damping” are incomplete with regard to predicting a large or small-amplitude response profile. In our experiments, as damping is systematically increased, we observe a transition between these two profiles characterized by a gradual “erosion” and eventual disappearance of the large-amplitude section (upper branch) and the scaling down of the lower branch region. We find that jumps from the upper to the initial branch originate on the 2S/2P boundary in the Williamson–Roshko plane. Another new finding is a hysteresis between the lower branch and the desynchronized region, which only appears at low Reynolds numbers. We also explore changes in the frequency response profile, which are connected with the changes in the amplitude profile, for our upper branch cases. We observe that analogous to the three amplitude branches, there are three distinct branches for the frequency response.

Publication: Journal of Fluids and Structures Vol.: 22 No.: 6-7 ISSN: 0889-9746

ID: CaltechAUTHORS:20110630-142700268

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Abstract: It is known that the stable and unstable manifolds of dynamical systems theory provide a powerful tool for understanding Lagrangian aspects of time-periodic flows. In this work we consider two time-periodic vortex ring flows. The first is a vortex ring with an elliptical core. The manifolds provide information about entrainment and detrainment of irrotational fluid into and out of the volume transported with the ring. The likeness of the manifolds with features observed in flow visualization experiments of turbulent vortex rings suggests that a similar process might be at play. However, what precise modes of unsteadiness are responsible for stirring in a turbulent vortex ring is left as an open question. The second situation is that of two leapfrogging rings. The unstable manifold shows striking agreement with even the fine features of smoke visualization photographs, suggesting that fluid elements in the vicinity of the manifold are drawn out along it and begin to reveal its structure. We suggest that interpretations of these photographs that argue for complex vorticity dynamics ought to be reconsidered. Recently, theoretical and computational tools have been developed to locate structures analogous to stable and unstable manifolds in aperiodic, or finite-time systems. The usefulness of these analogs is demonstrated, using vortex ring flows as an example, in the paper by Shadden, Dabiri, and Marsden [Phys. Fluids 18, 047105 (2006)].

Publication: Physics of Fluids Vol.: 18 No.: 4 ISSN: 1070-6631

ID: CaltechAUTHORS:SHApof06a

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Abstract: The evolution of initially weak structures of vorticity as they evolve in an incompressible turbulent flow is investigated. Such objects are candidates for important structures in the inertial range and in the dissipation range of scales. As these structures are strained by the flow, fine-scales of vorticity are produced along the direction of maximum compression with a consequent flow of energy to the high wavenumbers. It is shown that, under certain circumstances, the self-energy spectrum of such a structure may be time-averaged, producing a fractional power law. The exponent of the power law depends on the ratio of the first two Lyapunov exponents of the strain tensor.

No.: 71
ID: CaltechAUTHORS:20200304-094728369

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Abstract: The response of a freely oscillating circular cylinder (“free vibration”) in cross-flow has been studied experimentally using controlled magnetic eddy current to provide variable damping. In general, the nondimensional response amplitude, A*, and dominant frequency, ω*, depend on the Reynolds number, Re, and the nondimensional mass, m*, damping, b*, and elasticity, k*, of the system. The main objective of this study is to characterize the maximum amplitude that is achieved for a given system as cross-flow velocity is varied. We find that this maximum amplitude, A*_(max), occurs within a small range of values of k*_(eff) = ω*^2m* + k*. For values of Reynolds number in the range 525

ID: CaltechAUTHORS:20141201-122718180

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Abstract: A vortex method has been developed where spatial adaption of the Lagrangian vortex particles is provided by the technique of radial basis function interpolation. In this way, the meshless formulation of the vortex method is preserved throughout. Viscous effects are provided by the core spreading method, where core size control is accomplished in the spatial adaption, thus ensuring convergence. Numerical experiments demonstrate considerable increase in accuracy, in comparison with standard remeshing schemes used with vortex methods. Proof-of-concept is achieved successfully on a problem of quasi-steady tripole vortex flow, and parallel implementation of the method has permitted high-accuracy computations of vortex interactions at high Reynolds number.

Publication: International Journal for Numerical Methods in Fluids Vol.: 47 No.: 8-9 ISSN: 0271-2091

ID: CaltechAUTHORS:20150115-110003024

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Abstract: Vortex methods have a history as old as finite differences. They have since faced difficulties stemming from the numerical complexity of the Biot–Savart law, the inconvenience of adding viscous effects in a Lagrangian formulation, and the loss of accuracy due to Lagrangian distortion of the computational elements. The first two issues have been successfully addressed, respectively, by the application of the fast multipole method, and by a variety of viscous schemes which will be briefly reviewed in this article. The standard method to deal with the third problem is the use of remeshing schemes consisting of tensor product interpolation with high-order kernels. In this work, a numerical study of the errors due to remeshing has been performed, as well as of the errors implied in the discretization itself using vortex blobs. In addition, an alternative method of controlling Lagrangian distortion is proposed, based on ideas of radial basis function (RBF) interpolation (briefly reviewed here). This alternative is formulated grid-free, and is shown to be more accurate than standard remeshing. In addition to high-accuracy, RBF interpolation allows core size control, either for correcting the core spreading viscous scheme or for providing a variable resolution in the physical domain. This formulation will allow in theory the application of error estimates to produce a truly adaptive spatial refinement technique. Proof-of-concept is provided by calculations of the relaxation of a perturbed monopole to a tripole attractor.

Publication: International Journal for Numerical Methods in Fluids Vol.: 47 No.: 5 ISSN: 0271-2091

ID: CaltechAUTHORS:20150114-160021492

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Abstract: Recent developments of the 3-D Lagrangian vortex element method for bluff body flows are presented. In this approach attached boundary layer regions are modelled using infinitely thin vortex sheets while Lagrangian vortex elements are used for the separation regions and the wake. Preliminary results for the flow past a simplified generic truck geometry are presented. Further developments, aimed at the development of a hybrid Eulerian-Lagrangian solver, are briefly introduced.

No.: 19 ISSN: 1613-7736

ID: CaltechAUTHORS:20200205-150626934

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Abstract: We elaborate the physics of systems of unconstrained, reconnecting vortex filaments with dynamic finite cores of uniform ("quantized") circulation interacting via Biot-Savart and viscous forces. The phenomenology of this purely structured turbulent system includes an inertial range with Kolmogorov's k^–5/3 scaling for the energy spectrum, as well as Kolmogorov's linear in r scaling for the third order longitudinal structure function.

Publication: Physical Review Letters Vol.: 90 No.: 23 ISSN: 0031-9007

ID: CaltechAUTHORS:KIVprl03

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Abstract: We investigate numerically the Navier-Stokes dynamics of reconnecting vortex rings at small Reynolds number for a variety of configurations. We find that reconnections are dissipative due to the smoothing of vorticity gradients at reconnection kinks and to the formation of secondary structures of stretched antiparallel vorticity which transfer kinetic energy to small scales where it is subsequently dissipated efficiently. In addition, the relaxation of the reconnection kinks excites Kelvin waves which due to strong damping are of low wave number and affect directly only large scale properties of the flow.

Publication: Physical Review Letters Vol.: 90 No.: 5 ISSN: 0031-9007

ID: CaltechAUTHORS:CHAprl03

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Abstract: The evolution of initially weak structures of vorticity as they evolve in an incompressible turbulent flow is investigated. Such objects are candidates for being important structures in the inertial range and in the dissipation range of scales. Initially, these structures evolve passively as a result of the induced velocity field of the large-scale vorticity field. This field is three dimensional and time dependent, so these objects are subjected to straining apropos of Lagrangian chaos, characterized by a distribution of finite-time Lyapunov exponents.

Publication: Journal of Turbulence Vol.: 3ISSN: 1468-5248

ID: CaltechAUTHORS:LEOjot02

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Abstract: In vortex particle methods one is concerned with the problem of clustering and depletion of particles in different regions of the flow. The overlap of the vortex blobs is indeed of primary importance for the convergence of the method. In this paper we consider face-centred cubic (FCC) lattices for particle redistribution in three dimensions. This lattice is in fact the most natural way to pack spheres (the FCC is also known as a closest-sphere packing lattice). As a consequence, a point has 12 equidistant close neighbours rather than six for the cubic lattice. The FCC lattice thus offers some symmetry properties that should prove useful for a number of reasons, e.g., the core overlap issue. A few results for this scheme are presented. The problem of two colliding vortex rings at Re = 250 and 500 is studied with both the FCC and cubic lattice schemes. This problem subjects the vortex tubes to a quite strong stretching field and can amply test the quality of the lattice and the remeshing.

Publication: Journal of Turbulence Vol.: 2002 No.: 3 ISSN: 1468-5248

ID: CaltechAUTHORS:CHAjot02

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Abstract: Vortex methods have become useful tools for the computation of incompressible fluid flow. In this work, a vortex particle method for the simulation of unsteady two-dimensional compressible flow is developed. By decomposing the velocity into irrotational and solenoidal parts, and using particles that are able to change volume and that carry vorticity, dilatation, enthalpy, entropy and density, the equations of motion are satisfied. Spatial derivatives are treated using the method of particle strength exchange with high-order-accurate, non-dissipative kernels. The new vortex method is applied to co-rotating and leapfrogging vortices in compressible flow, with the far acoustic field computed using a two-dimensional Kirchhoff surface.

Publication: Journal of Turbulence Vol.: 2002 No.: 3 ISSN: 1468-5248

ID: CaltechAUTHORS:ELDjot02

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Abstract: A unified approach to approximating spatial derivatives in particle methods using integral operators is presented. The approach is an extension of particle strength exchange, originally developed for treating the Laplacian in advection-diffusion problems. Kernels of high order of accuracy are constructed that can be used to approximate derivatives of any degree. A new treatment for computing derivatives near the edge of particle coverage is introduced, using "one-sided" integrals that only look for information where it is available. The use of these integral approximations in wave propagation applications is considered and their error is analyzed in this context using Fourier methods. Finally, simple tests are performed to demonstrate the characteristics of the treatment, including an assessment of the effects of particle dispersion, and their results are discussed.

Publication: Journal of Computational Physics Vol.: 180 No.: 2 ISSN: 0021-9991

ID: CaltechAUTHORS:20190708-164305261

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Abstract: A vortex particle method is developed for simulating two-dimensional, unsteady compressible (low. The method uses the Helmholtz decomposition of the velocity field to separately treat the irrotational and solenoidal portions of the Now, and the particles are allowed to change volume to conserve mass. In addition to having vorticity and dilatation properties, the particles also carry density, enthalpy, and entropy. The resulting evolution equations contain terms that are computed with techniques used in some incompressible methods. Truncation of unbounded domains via a nonreflecting boundary condition is also considered. The fast multipole method is adapted to compressible particles in order to make the method computationally efficient. The new method is applied to several problems, including sound generation by corotating vortices and generation of vorticity by baroclinic torque.

Publication: Journal of Computational Physics Vol.: 179 No.: 2 ISSN: 0021-9991

ID: CaltechAUTHORS:20190708-164305134

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Abstract: This work describes an unsplit, second-order accurate algorithm for multidimensional systems of hyperbolic conservation laws with source terms, such as the compressible Euler equations for reacting flows. It is a MUSCL-type, shock-capturing scheme that integrates all terms of the governing equations simultaneously, in a single time-step, thus avoiding dimensional or time-splitting. Appropriate families of space-time manifolds are introduced, along which the conservation equations decouple to the characteristic equations of the corresponding 1-D homogeneous system. The local geometry of these manifolds depends on the source terms and the spatial derivatives of the flow variables. Numerical integration of the characteristic equations is performed along these manifolds in the upwinding part of the algorithm. Numerical simulations of two-dimensional detonations with simplified kinetics are performed to test the accuracy and robustness of the algorithm. These flows are unstable for a wide range of parameters and may exhibit chaotic behavior. Grid-convergence studies and comparisons with earlier results, obtained with traditional schemes, are presented.

Publication: Computers and Mathematics with Applications Vol.: 44 No.: 1-2 ISSN: 0898-1221

ID: CaltechAUTHORS:20160602-163756768

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Abstract: A vortex particle method for the simulation of two-dimensional compressible flows is developed. The computational elements are Lagrangian particles that carry vorticity,dilatation,enthalpy,entropy and density. The velocity field is decomposed into irrotational and solenoidal parts,which allows its calculation in terms of the particles' vorticity and dilatation. The particle coverage is truncated and incident acoustic waves are absorbed using a suitable boundary treatment. A Kirchhoff surface formulation is developed for computing the far-field sound. The method is applied to a co-rotating vortex pair and the results are discussed.

ID: CaltechAUTHORS:20190726-104729692

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Abstract: Phenomena associated with flow-induced transverse oscillation of an elastically mounted body are considered. The use of a recently introduced parameter that combines the effect of mass and elasticity—effective elasticity—is exploited to demonstrate the predictive value of the new approach and to provide insights into solution branching, the maximum amplitude of vibration, and modeling.

Publication: Journal of Fluids and Structures Vol.: 15 No.: 3-4 ISSN: 0889-9746

ID: CaltechAUTHORS:20141201-122214519

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Abstract: Drag reduction in two-dimensional flow over a circular cylinder, achieved using rotary oscillation, was investigated with computational simulations. In the experiments of Tokumaru & Dimotakis (1991), this mechanism was observed to yield up to 80% drag reduction at Re = 15 000 for certain ranges of frequency and amplitude of sinusoidal rotary oscillation. Simulations with a high-resolution viscous vortex method were carried out over a range of Reynolds numbers (150–15 000) to explore the effects of oscillatory rotational forcing. Significant drag reduction was observed for a rotational forcing which had been very effective in the experiments. The impact of the forcing is strongly Reynolds number dependent. The cylinder oscillation appears to trigger a distinctive shedding pattern which is related to the Reynolds number dependence of the drag reduction. It appears that the source of this unusual shedding pattern and associated drag reduction is vortex dynamics in the boundary layer initiated by the oscillatory cylinder rotation. The practical efficiency of the drag reduction procedure is also discussed.

Publication: Journal of Fluid Mechanics Vol.: 431ISSN: 0022-1120

ID: CaltechAUTHORS:SHIjfm01

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Abstract: Transverse oscillation of a dynamically supported circular cylinder in a flow at Re=100 has been numerically simulated using a high-resolution viscous-vortex method, for a range of dynamical parameters. At the limiting case with zero values of mass, damping and elastic force, the cylinder oscillates sinusoidally at amplitude A/D=0·47 and frequency fD/U_∞=0·156. For zero damping, the effects of mass and elasticity are combined into a new, “effective” dynamic parameter, which is different from the classic “reduced velocity”. Over a range of this parameter, the response exhibits oscillations at amplitudes up to 0·6 and frequencies between 0·15 and 0·2. From this response function, the classic response in terms of reduced velocity can be obtained for fixed values of the cylinder/fluid ratio m*. It displays “lock-in” at very high values of m*.

Publication: Journal of Fluids and Structures Vol.: 15 No.: 1 ISSN: 0889-9746

ID: CaltechAUTHORS:20141201-121315237

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Abstract: An experimental technique is introduced for deducing the unsteady fluid forces, F(t), on elastically vibrating structures using their oscillation trace y(t) , and the free stream velocity U. This new technique involves an accurate modeling of the elastic structure as an ordinary differential equation and employing time-dependent oscillation traces, y(t) and/or the acceleration a(t), to extract the side force on the structure. Filtering the oscillation signals and modeling the non-linear damping terms in the structural equation turn out to be the challenges of this technique. This technique may be applied in force studies relating vortex patterns and the the unsteady forces as well as methods for predicting or modeling fluid forces and amplitudes. Although the results are not compared with any independent technique, the unsteady lift coefficient traces show high values of above three as observed by other researchers.

ID: CaltechAUTHORS:20130930-085939898

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Abstract: Using a time-periodic perturbation of a two-dimensional steady separation bubble on a plane no-slip boundary to generate chaotic particle trajectories in a localized region of an unbounded boundary layer flow, we study the impact of various geometrical structures that arise naturally in chaotic advection fields on the transport of a passive scalar from a local 'hot spot' on the no-slip boundary. We consider here the full advection-diffusion problem, though attention is restricted to the case of small scalar diffusion, or large Peclet number. In this regime, a certain one-dimensional unstable manifold is shown to be the dominant organizing structure in the distribution of the passive scalar. In general, it is found that the chaotic structures in the flow strongly influence the scalar distribution while, in contrast, the flux of passive scalar from the localized active no-slip surface is, to dominant order, independent of the overlying chaotic advection. Increasing the intensity of the chaotic advection by perturbing the velocity held further away from integrability results in more non-uniform scalar distributions, unlike the case in bounded flows where the chaotic advection leads to rapid homogenization of diffusive tracer. In the region of chaotic particle motion the scalar distribution attains an asymptotic state which is time-periodic, with the period the same as that of the time-dependent advection field. Some of these results are understood by using the shadowing property from dynamical systems theory. The shadowing property allows us to relate the advection-diffusion solution at large Peclet numbers to a fictitious zero-diffusivity or frozen-field solution, corresponding to infinitely large Peclet number. The zero-diffusivity solution is an unphysical quantity, but is found to be a powerful heuristic tool in understanding the role of small scalar diffusion. A novel feature in this problem is that the chaotic advection field is adjacent to a no-slip boundary. The interaction between the necessarily non-hyperbolic particle dynamics in a thin near-wall region and the strongly hyperbolic dynamics in the overlying chaotic advection field is found to have important consequences on the scalar distribution; that this is indeed the case is shown using shadowing. Comparisons are made throughout with the flux and the distributions of the passive scalar for the advection-diffusion problem corresponding to the steady, unperturbed, integrable advection field.

Publication: Journal of Fluid Mechanics Vol.: 372ISSN: 0022-1120

ID: CaltechAUTHORS:GHOjfm98

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Abstract: The flow around two leap-frogging vortex rings is studied using the dynamical systems approach with a view to estimating the mixing taking place. The controlling parameter is the ratio of the initial radii of the rings to the axial distance between them. Small initial distances correspond to very little mixing, with interesting dynamical features in the region close to the rings. As the initial distance increases, the mixing region increases rapidly upto some point, and for larger distances the level of mixing remains approximately the same.

No.: 515
ID: CaltechAUTHORS:20200715-091033504

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Abstract: The present work is concerned with an application of the theory of characteristics to conservation laws with source terms in one space dimension, such as the Euler equations for reacting flows. Space-time paths are introduced on which the flow/chemistry equations decouple to a characteristic set of ODE's for the corresponding homogeneous laws, thus allowing the introduction of functions analogous to the Riemann invariants in classical theory. The geometry of these paths depends on the spatial gradients of the solution. This particular decomposition can be used in the design of efficient unsplit algorithms for the numerical integration of the equations. As a first step, these ideas are implemented for the case of a scalar conservation law with a nonlinear source term. The resulting algorithm belongs to the class of MUSCL-type, shock-capturing schemes. Its accuracy and robustness are checked through a series of tests. The stiffness of the source term is also studied. Then, the algorithm is generalized for a system of hyperbolic equations, namely the Euler equations for reacting flows. A numerical study of unstable detonations is performed.

Publication: Journal of Computational Physics Vol.: 134 No.: 1 ISSN: 0021-9991

ID: CaltechAUTHORS:20160602-181443806

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Abstract: Significant progress was made in the third year of an interdisciplinary experimental, numerical and theoretical program to extend the state of knowledge and understanding of the effects of chemical reactions in hypervelocity flows. The program addressed the key problems in aerothermochemistry that arise from.the interaction between the three strongly nonlinear effects: Compressibility; vorticity; and chemistry. Important new results included: • New data on transition in hypervelocity carbon dioxide flows • New method of free-piston shock tunnel operation for lower enthalpy • Accurate new method for computation of self-similar flows • New experimental data on flap-induced separation at high enthalpy • Insight into mechanisms active in reacting shear layers from comparison of experiment and computation • Extensive new data from Rayleigh scattering diagnostics of supersonic shear layer • Comparison of new experiments and computation of hypervelocity double-wedge flow yielded important differences • Further first-principles computations of electron collision cross-sections of CO, N_2 and NO • Good agreement between EFMO computation and experiment of flow over a cone at high incidence • Extension of LITA diagnostics to high temperature.

ID: CaltechAUTHORS:20141111-111211793

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Abstract: The probability density functions (PDFs) of the velocity and the velocity difference field induced by a distribution of a large number of discrete vortex elements are investigated numerically and analytically. Tails of PDFs of the velocity and velocity difference induced by a single vortex element are found. Treating velocities induced by different vortex elements as independent random variables, PDFs of the velocity and velocity difference induced by all vortex elements are found using limit distribution theorems for stable distributions. Our results generalize and extend the analysis by Takayasu [Prog. Theor. Phys. 72, 471 (1984)]. In particular, we are able to treat general distributions of vorticity, and obtain results for velocity differences and velocity derivatives of arbitrary order. The PDF for velocity differences of a system of singular vortex elements is shown to be Cauchy in the case of small separation r, both in 2 and 3 dimensions. A similar type of analysis is also applied to non-singular vortex blobs. We perform numerical simulations of the system of vortex elements in two dimensions, and find that the results compare favorably with the theory based on the independence assumption. These results are related to the experimental and numerical measurements of velocity and velocity difference statistics in the literature. In particular, the appearance of the Cauchy distribution for the velocity difference can be used to explain the experimental observations of Tong and Goldburg [Phys. Lett. A 127, 147 (1988); Phys. Rev. A 37, 2125, (1988); Phys. Fluids 31, 2841 (1988)] for turbulent flows. In addition, for intermediate values of the separation distance, near exponential tails are found.

Publication: Physics of Fluids Vol.: 8 No.: 5 ISSN: 1070-6631

ID: CaltechAUTHORS:MINpof96

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Abstract: The present work is concerned with the extension of the theory of characteristics to conservation laws with source terms in one space dimension, such as the Euler equations for reacting flows. New spacetime curves are introduced on which the equations decouple to the characteristic set of O.D.E's for the corresponding homogeneous laws, thus allowing the introduction of functions analogous to the Riemann Invariants. The geometry of these curves depends on the spatial gradients for the solution. This particular decomposition can be used in the design of efficient unsplit algorithms for the numerical integration of the equations. As a first step, these ideas are implemented for the case of a scalar conservation law with a nonlinear source term. The resulting algorithm belongs to the class of MUSCL-type, shock-capturing schemes. Its accuracy and robustness are checked through a series of tests. The aspect of the stiffness of the source term is also studied. Then, the algorithm is generalized for a system of hyperbolic equations, namely the Euler equations for reacting flows. An extensive numerical study of unstable detonations is performed.

ID: CaltechAUTHORS:20141111-103448502

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Abstract: Velocity autocorrelations and the mean-square displacements of fluid particles are obtained for decaying, isotropic homogeneous turbulence by numerical simulation of the flow field, using 1283 and 2563 grids, and tracking several tens of thousands of fluid particles, using a third-order interpolation scheme. A self-preserving Lagrangian velocity autocorrelation coefficient is found in terms of a dimensionless time variable s, defined by ds=dt/[script T]s(t), under the observation of a power-law energy decay and the assumption that [script T]s(t) is proportional to the Lagrangian integral timescale [script T]_{[script L]}. This timescale is in turn assumed to be proportional to the length scale of the energy-containing eddies [script L]e~K3/2/epsilon divided by the turbulent velocity u[prime], where K=3/2u[prime]2 is turbulent energy and epsilon is the energy dissipation rate.

Publication: Physics of Fluids Vol.: 7 No.: 10 ISSN: 1070-6631

ID: CaltechAUTHORS:HUApof95

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Abstract: The development of a two-dimensional viscous incompressible flow generated from a circular cylinder impulsively started into rectilinear motion is studied computationally. An adaptative numerical scheme, based on vortex methods, is used to integrate the vorticity/velocity formulation of the Navier–Stokes equations for a wide range of Reynolds numbers (Re = 40 to 9500). A novel technique is implemented to resolve diffusion effects and enforce the no-slip boundary condition. The Biot–Savart law is employed to compute the velocities, thus eliminating the need for imposing the far-field boundary conditions. An efficient fast summation algorithm was implemented that allows a large number of computational elements, thus producing unprecedented high-resolution simulations. Results are compared to those from other theoretical, experimental and computational works and the relation between the unsteady vorticity field and the forces experienced by the body is discussed.

Publication: Journal of Fluid Mechanics Vol.: 296ISSN: 0022-1120

ID: CaltechAUTHORS:20120217-081614132

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Abstract: This is the final report of our program on "Chemical Reactions in Turbulent Mixing Flows," supported under the AFOSR Grant No. F49620-92-J-0290, which was granted a no-cost extension to permit the completion of the Supersonic Shear Layer Facility upgrade that extended the operating envelope to higher Mach-number flows. As part of this upgrade, a variety of new diagnostic and safety features were also implemented in this unique facility. The purpose of this program has been to conduct fundamental investigations of turbulent mixing, chemical reaction and combustion processes in turbulent, subsonic and supersonic flows. Scientific progress in these areas was documented in our most recent Annual Report (Dimotakis & Leonard 1994).

ID: CaltechAUTHORS:20141111-095158441

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Abstract: The decay of nominally isotropic, homogeneous incompressible turbulence is studied by direct numerical simulations for Re-lambda in the range (5-50) with 256(3) spectral coefficients. A power-law decay of the turbulent energy is observed with exponents approximately equal to 1.5 and 1.25, apparently dependent on Re-lambda. A new complete similarity form for the double and triple velocity correlation functions, f(r,t) and k(r,t), is proposed for low to intermediate Re-lambda that is consistent with the Karmia-Howarth equation and the results of the numerical experiments. The results are also consistent with Saffman's proposed asymptotic behavior of f(r,t) for large separation r for runs with a decay exponent of 1.5. The so-called final period of decay is not observed.

Publication: Physics of Fluids Vol.: 6 No.: 11 ISSN: 1070-6631

ID: CaltechAUTHORS:HUApof94

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Abstract: A new approach for studying wave propagation phenomena in an inviscid gas is presented. This approach can be viewed as the extension of the method of characteristics to the general case of unsteady multidimensional flow. The general case of the unsteady compressible Euler equations in several space dimensions is examined. A family of spacetime manifolds is found on which an equivalent one-dimensional problem holds. Their geometry depends on the spatial gradients of the flow, and they provide, locally, a convenient system of coordinate surfaces for spacetime. In the case of zero entropy gradients, functions analogous to the Riemann invariants of 1-D gas dynamics can be introduced. These generalized Riemann Invariants are constant on these manifolds and, thus, the manifolds are dubbed Riemann Invariant Manifolds (RIM). In this special case of zero entropy gradients, the equations of motion are integrable on these manifolds, and the problem of computing the solution becomes that of determining the manifold geometry in spacetime. This situation is completely to the traditional method of characteristics in one-dimensional flow. Explicit espressions for the local differential geometry of these manifolds can be found directly from the equations of motion. The local direction and speed of propagation of the waves that these manifolds represent, can be found as a function of the local spatial gradients of the flow. Their geometry is examined, and in particular, their relation to the characteristic surfaces. It turns out that they can be space-like or time-like, depending on the flow gradients. Wave propagation can be viewed as a superposition of these Riemann Invariant waves, whenever appropriate conditions of smoothness are met. This provides a means for decomposing the equations into a set of convective scalar fields in a way which is different and potentially more useful than the characteristic decomposition. The two decompositions become identical in the special case of one-dimellsional flow. This different approach can be used for computational purposes by discretizing the equivalent set of scalar equations. Such a computational application of this theory leads to the possibility of determining the solution at points in spacetime using information that propagates faster than the local characteristic speed, i.e., using information outside the domain of dependence. This possibility and its relation to the uniqueness theorems is discussed.

ID: CaltechAUTHORS:20141110-161414265

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Abstract: Active circulation control of the two-dimensional unsteady separated flow past a semiinfinite plate with transverse motion is considered. The rolling-up of the separated shear layer is modelled by a point vortex whose time-dependent circulation is predicted by an unsteady Kutta condition. A suitable vortex shedding mechanism introduced. A control strategy able to maintain constant circulation when a vortex is present is derived. An exact solution for the nonlinear controller is then obtained. Dynamical systems analysis is used to explore the performance of the controlled system. The control strategy is applied to a class of flows and the results are discussed. A procedure to determine the position and the circulation of the vortex, knowing the velocity signature on the plate, is derived. Finally, a physical explanation of the control mechanism is presented.

Publication: Journal of Fluid Mechanics Vol.: 260ISSN: 0022-1120

ID: CaltechAUTHORS:CORjfm94

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Abstract: Area and axial flow variations on rectilinear vortex tubes are considered. The state of the flow is characterized by two dependent variables, a core area, and an azimuthal circulation per unit length, which vary in time and in distance along the length of the tube. Nonlinear integrodifferential equations of motion for these variables are derived by taking certain integrals of the vorticity transport equation. The equations are argued to be valid for moderately short waves (on the order of a few core radii) as well as for long waves. Applications to vortex breakdown and other wave phenomena are considered.

Publication: Physics of Fluids Vol.: 6 No.: 2 ISSN: 1070-6631

ID: CaltechAUTHORS:LEOpof94

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Abstract: The purpose of this research is to conduct fundamental investigations of turbulent mixing, chemical reaction and combustion processes in turbulent, subsonic and supersonic flows. The program during this reporting period was comprised of several parts: a. an experimental effort, b. a numerical simulation effort, and c. an effort to develop instrumentation and diagnostics; flow and combustion facilities; and data-acquisition systems. The latter as dictated by the specific needs of the experimental part of the program. Our approach in this research has been to carry out a series of detailed theoretical and experimental studies of turbulent mixing in primarily in two, well-defined, fundamentally important flow fields: free-shear layers and axisymmetric jets. To elucidate molecular transport effects, experiments and theory concern themselves with both reacting and non-reacting flows of liquids and gases, in fully-developed turbulent flows, i.e., in moderate to high Reynolds number flows. A criterion for fully-developed turbulence was recently developed and will be presented below. The computational studies are, at present, focused at fundamental formulation and implementation issues pertaining to the computational simulation of both compressible and incompressible flows characterized by strong fronts, such as shock waves and flames. Our diagnostic development efforts have recently been focused on improving the signal-to-noise ratio of flow images, in both gas- and liquid-phase flows, as well as the continuing development of data-acquisition electronics to meet very high-speed, high-volume data requirements; the acquisition of single, or pairs, of two-dimensional images in rapid succession; and the acquisition of data from arrays of supersonic flow sensors.

ID: CaltechAUTHORS:20141021-161231721

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Abstract: A method for computing one-dimensional unsteady compressible flows. with and without chemical reactions is presented. This work has focused on the accurate computation of the discontinuous waves that arise in such flows. The main feature of the method is the use of an adaptive Lagrangian grid. This allows the computation of discontinuous waves and their interactions with the accuracy of front-tracking algorithms. This is done without the use of additional grid points representing shocks, in contrast to conventional front-tracking schemes. The Lagrangian character of the present scheme also allows contact discontinuities to be captured easily. The algorithm avoids interpolation across discontinuities in a natural and efficient way. The method has been used on a variety of reacting and non-reacting flows in order to test its ability to compute accurately and in a robust way complicated wave interactions.

Publication: Journal of Computational Physics Vol.: 104 No.: 2 ISSN: 0021-9991

ID: CaltechAUTHORS:20141021-162327176

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Abstract: We observe that high-stretch tails of finite-time Lyapunov exponent distributions associated with interfaces evolving under a class of nonturbulent chaotic flows can range from essentially Gaussian tails to nearly exponential tails, and show that the non-Gaussian deviations can have a significant effect on interfacial evolution. This observation motivates new insight into stretch processes under chaotic flows.

Publication: Physical Review Letters Vol.: 70 No.: 3 ISSN: 0031-9007

ID: CaltechAUTHORS:BEIprl93

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Abstract: The purpose of this research is to conduct fundamental investigations of turbulent mixing, chemical reaction and combustion processes in turbulent, subsonic and supersonic flows. Our program is comprised of several parts: a. an experimental effort, b. an analytical effort, c. a computational effort, d. a modeling effort, and e. a diagnostics development and data-acquisition effort, the latter as dictated by specific needs of the experimental part of the overall program. Our approach has been to carry out a series of detailed theoretical and experimental studies of turbulent mixing in primarily in two, well-defined, fundamentally important flow fields: free shear layers and axisymmetric jets. To elucidate molecular transport effects, experiments and theory concern themselves with both reacting and non-reacting flows of liquids and gases, in fully-developed turbulent flows, i.e., in moderate to high Reynolds number flows. The computational studies are, at present, focused at fundamental issues pertaining to the computational simulation of both compressible and incompressible flows. Modeling has been focused on both shear layers and turbulent jets, with an effort to include the physics of the molecular transport processes, as well as formulations of models that permit the full chemical kinetics of the combustion process to be incorporated. Our primary diagnostic development efforts are currently focused on data-acquisition electronics to meet very high-speed, high-volume data requirements, the acquisition of single, or a sequence, of two-dimensional images, and the acquisition of data from arrays of supersonic flow sensors. Progress has also been made in the development of a dual-beam laser interferometer/correlator to measure convection velocities of large scale structures in supersonic shear layers and in a new method to acquire velocity field data using pairs of scalar images closely spaced in time.

ID: CaltechAUTHORS:20141020-101605714

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Abstract: N/A

Publication: Annual Review of Fluid Mechanics Vol.: 24ISSN: 0066-4189

ID: CaltechAUTHORS:20120328-151405455

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Abstract: The authors generalize notions of transport in phase space associated with the classical Poincare map reduction of a periodically forced two-dimensional system to apply to a sequence of nonautonomous maps derived from a quasiperiodically forced two-dimensional system. They obtain a global picture of the dynamics in homoclinic and heteroclinic tangles using a sequence of time-dependent two-dimensional lobe structures derived from the invariant global stable and unstable manifolds of one or more normally hyperbolic invariant sets in a Poincare section of an associated autonomous system phase space. The invariant manifold geometry is studied via a generalized Melnikov function. Transport in phase space is specified in terms of two-dimensional lobes mapping from one to another within the sequence of lobe structures, which provides the framework for studying several features of the dynamics associated with chaotic tangles.

Publication: Nonlinearity Vol.: 4 No.: 3 ISSN: 0951-7715

ID: CaltechAUTHORS:BEInonlin91

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Abstract: A global, finite-time study is made of stretching and diffusion in a class of chaotic tangles associated with fluids described by periodically forced two-dimensional dynamical systems. Invariant lobe structures formed by intersecting global stable and unstable manifolds of persisting invariant hyperbolic sets provide the geometrical framework for studying stretching of interfaces and diffusion of passive scalars across these interfaces. In particular, the present study focuses on the material curve that initially lies on the unstable manifold segment of the boundary of the entraining turnstile lobe.A knowledge of the stretch profile of a corresponding curve that evolves according to the unperturbed flow, coupled with an appreciation of a symbolic dynamics that applies to the entire original material curve in the perturbed flow, provides the framework for understanding the mechanism for, and topology of, enhanced stretching in chaotic tangles. Secondary intersection points (SIP's) of the stable and unstable manifolds are particularly relevant to the topology, and the perturbed stretch profile is understood in terms of the unperturbed stretch profile approximately repeating itself on smaller and smaller scales. For sufficiently thin diffusion zones, diffusion of passive scalars across interfaces can be treated as a one-dimensional process, and diffusion rates across interfaces are directly related to the stretch history of the interface.An understanding of interface stretching thus directly translates to an understanding of diffusion across interfaces. However, a notable exception to the thin diffusion zone approximation occurs when an interface folds on top of itself so that neighboring diffusion zones overlap. An analysis which takes into account the overlap of nearest neighbor diffusion zones is presented, which is sufficient to capture new phenomena relevant to efficiency of mixing. The analysis adds to the concentration profile a saturation term that depends on the distance between neighboring segments of the interface. Efficiency of diffusion thus depends not only on efficiency of stretching along the interface, but on how this stretching is distributed relative to the distance between neighboring segments of the interface.

Publication: Physics of Fluids A Vol.: 3 No.: 5 ISSN: 0899-8213

ID: CaltechAUTHORS:BEIpofa91

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Abstract: We examine the transport properties of a particular two-dimensional, inviscid incompressible flow using dynamical systems techniques. The velocity field is time periodic and consists of the field induced by a vortex pair plus an oscillating strainrate field. In the absence of the strain-rate field the vortex pair moves with a constant velocity and carries with it a constant body of fluid. When the strain-rate field is added the picture changes dramatically; fluid is entrained and detrained from the neighbourhood of the vortices and chaotic particle motion occurs. We investigate the mechanism for this phenomenon and study the transport and mixing of fluid in this flow. Our work consists of both numerical and analytical studies. The analytical studies include the interpretation of the invariant manifolds as the underlying structure which govern the transport. For small values of strain-rate amplitude we use Melnikov's technique to investigate the behaviour of the manifolds as the parameters of the problem change and to prove the existence of a horseshoe map and thus the existence of chaotic particle paths in the flow. Using the Melnikov technique once more we develop an analytical estimate of the flux rate into and out of the vortex neighbourhood. We then develop a technique for determining the residence time distribution for fluid particles near the vortices that is valid for arbitrary strainrate amplitudes. The technique involves an understanding of the geometry of the tangling of the stable and unstable manifolds and results in a dramatic reduction in computational effort required for the determination of the residence time distributions. Additionally, we investigate the total stretch of material elements while they are in the vicinity of the vortex pair, using this quantity as a measure of the effect of the horseshoes on trajectories passing through this region. The numerical work verifies the analytical predictions regarding the structure of the invariant manifolds, the mechanism for entrainment and detrainment and the flux rate.

Publication: Journal of Fluid Mechanics Vol.: 214ISSN: 0022-1120

ID: CaltechAUTHORS:20120509-134404773

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Abstract: I have a rather provocative question to ask the speakers. What I am wondering is: When you apply the proper orthogonal decomposition - I know this is a procedure you use and it is probably standard - you remove the mean flow and you only look at the perturbations. Why do not you include the mean flow, because if you did you would suppress the cubic terms, and also you would be getting something that would be closer to what people call coherent structures, such as hairpin vortices.

No.: 357
ID: CaltechAUTHORS:20141201-140640640

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Abstract: The contour dynamics method is extended to vortex rings with vorticity varying linearly from the symmetry axis. An elliptic core model is also developed to explain some of the basic physics. Passage and collisions of two identical rings are studied focusing on core deformation, sound generation and stirring of fluid elements. With respect to core deformation, not only the strain rate but how rapidly it varies is important and accounts for greater susceptibility to vortex tearing than in two dimensions. For slow strain, as a passage interaction is completed and the strain relaxes, the cores return to their original shape while permanent deformations remain for rapidly varying strain. For collisions, if the strain changes slowly the core shapes migrate through a known family of two-dimensional steady vortex pairs up to the limiting member of the family. Thereafter energy conservation does not allow the cores to maintain a constant shape. For rapidly varying strain, core deformation is severe and a head-tail structure in good agreement with experiments is formed. With respect to sound generation, good agreement with the measured acoustic signal for colliding rings is obtained and a feature previously thought to be due to viscous effects is shown to be an effect of inviscid core deformation alone. For passage interactions, a component of high frequency is present. Evidence for the importance of this noise source in jet noise spectra is provided. Finally, processes of fluid engulfment and rejection for an unsteady vortex ring are studied using the stable and unstable manifolds. The unstable manifold shows excellent agreement with flow visualization experiments for leapfrogging rings suggesting that it may be a good tool for numerical flow visualization in other time periodic flows.

ID: CaltechAUTHORS:SHAnasatm102257

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Abstract: The purpose of this research has been to conduct fundamental investigations of turbulent mixing, chemical reaction and combustion processes in turbulent, subsonic and supersonic flows. Progress in this effort thus far has uncovered important deficiencies in conventional modeling of these phenomena, and offered alternative suggestions and formulations to address some of these deficiencies. This program is comprised of an experimental effort, an analytical modeling effort, a computational effort, and a diagnostics development and data-acquisition effort, the latter as dictated by specific needs of our experiments. Our approach has been to carry out a series of detailed theoretical and experimental studies primarily in two, well-defined, fundamentally important flow fields: free shear layers and axisymmetric jets. To elucidate molecular transport effects, experiments and theory concern themselves with both liquids and gases. Modeling efforts have been focused on both shear layers and turbulent jets, with an effort to include the physics of the molecular transport processes, as well as formulations of models that permit the full chemical kinetics of the combustion process to be incorporated. The computational studies are, at present, focused at fundamental issues pertaining to the computational simulation of both compressible and incompressible flows. This report includes an outline discussion of work completed under the sponsorship of this Grant, with six papers, which have not previously been included in past reports, or transmitted in reprint form, appended.

ID: CaltechAUTHORS:20141020-095543611

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Abstract: The use of a modified scheme for the dynamics of vortex singularities is shown to lead to a weak solution of the three-dimensional inviscid incompressible vorticity equation.

Publication: Physics of Fluids Vol.: 31 No.: 7 ISSN: 0031-9171

ID: CaltechAUTHORS:WINpof88

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Abstract: Our task is to describe recent progress and present prospects for research in turbulence using numerical methods. This is quite a challenge, because we know that Liepmann has not been a strong advocate of computational turbulence, or "compulence," as it has sometimes been called -- the incompressible version presumably being "incompulence." We want to recognize several objectives in exploring turbulence by numerical methods, and to order these in priority as follows: 1. To gain understanding and insight into the physics of turbulence, so as to complement the insight obtained from experiments and analysis; for example, by helping the experimentalist to understand what it is that has been observed experimentally, and by helping the analyst to explore solution space in more detail. 2. To provide special "data" for guiding, evaluating, and calibrating simpler predictive models of turbulent flows. 3. To predict turbulent flows by numerical simulation.

No.: 320
ID: CaltechAUTHORS:20190605-093421211

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Abstract: Work is continuing primarily in gas phase turbulent mixing and chemical reactions. The liquid phase work to date is in its final stages of being analyzed and documented for dissemination in the form of archival publications. In the gas phase shear layer work, our investigations are concentrating on shear layer free stream density ratio effects, finite kinetic rate (Damköhler number) effects, and a design effort in support of the planned extension of the work to supersonic flows. In jet flows, progress has been made in the gas phase laser Rayleigh scattering techniques developed for conserved scalar measurements down to diffusion space and time scales. A new technique has been developed under joint support with the Gas Research Institute that permits the imaging of soot sheets in turbulent flames and is being used to describe the combustion flame sheets in methane flames. Theoretical work in progress is addressing the finite chemical rate problem as well as the diffusion-limited shear layer mixing problem. Advances in our data acquisition capabilities during the last year are permitting higher temporal resolution measurements to be taken with digital image arrays.

ID: CaltechAUTHORS:20141020-092935685

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Abstract: The deformation of a hairpin-shaped vortex filament under self-induction and in the presence of shear is studied numerically using the Biot–Savart law. It is shown that the tip region of an elongated hairpin vortex evolves into a vortex ring and that the presence of mean shear impedes the process. In addition, evolution of a finite-thickness vortex sheet under self-induction is investigated using the Navier–Stokes equations. The layer evolves into a hairpin vortex, which in turn produces a vortex ring of high Reynolds stress content. These results indicate a mechanism for the generation of ring vortices in turbulent shear flows, and a link between the experimental and numerical observation of hairpin vortices and the observation of Falco [Phys. Fluids 20, S124 (1977)] of ring vortices in the outer regions of turbulent boundary layers.

Publication: Physics of Fluids Vol.: 29 No.: 4 ISSN: 0031-9171

ID: CaltechAUTHORS:MOIpof86

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Abstract: N/A

Publication: Annual Review of Fluid Mechanics Vol.: 17ISSN: 0066-4189

ID: CaltechAUTHORS:20120629-092755487

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Abstract: The problem of electromagnetic wave propagation in a turbulent plasma is formulated in terms of the radiative transport equation. A singular eigenfunction solution is obtained for the case of isotropic plasma turbulence, and detailed numerical calculations are presented. The intensity distribution is studied as a function of the turbulent spectrum and relative strength of scattering attenuation to total attenuation. For a highly forward peaked scattering law characteristic of many physical situations it is found that the reflected backscatter intensity is relatively insensitive to the angle of incidence, except as grazing incidence is approached. The importance of multiple scatter is studied as a function of the properties of the medium.

Publication: Physics of Fluids Vol.: 15 No.: 9 ISSN: 0031-9171

ID: CaltechAUTHORS:FEIpof72

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Abstract: The effect of thermal radiation on the propagation of small disturbances generated by the harmonic oscillation of a planar wall, either in position or temperature or both, is considered. By separation of variables, it is shown that the governing linearized equations can be reduced to a homogeneous integral equation that admits both regular and singular solutions to form a complete set; thus, the problem can be solved by the application of singular eigenfunction expansions analogous to those used in neutron-transport theory. An exact closed form solution is obtained for disturbed quantities in the flow and radiation fields. The exact solution shows that the disturbances consist of a damped continuum mode with infinite wave speed in addition to two discrete modes with finite wave speed. One of the discrete modes represents the "modified classical" wave whereas the other discrete mode plus the continuum mode corresponds to the radiation-induced wave. For sufficiently small Bouguer numbers, the discrete mode of the radiation-induced wave disappears. The newly found continuum mode damps faster than the discrete mode. Consequently, only the discrete modes persist away from the boundary. The differential approximation is found to predict the modified classical wave accurately but predicts the radiation-induced wave only approximately. The implications as well as the accuracy of the differential approximation are discussed and compared.

Publication: Physics of Fluids Vol.: 14 No.: 5 ISSN: 0031-9171

ID: CaltechAUTHORS:CHEpof71

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Abstract: Methods derived from the theory of several complex variables are used as a means of analyzing a class of two‐dimensional transport problems in a scattering and absorbing quarter space (0 ≤ x_1, 0 ≤ x_2, −∞ ≤ x_3 ≤ ∞) described by a linear, one‐speed Boltzmann equation. Using Fourier transformation and the Bochner decomposition, the multivariable analog of the Wiener‐Hopf factorization, we find the Green's function in transform space, which solves all source problems having a solution bounded at infinity. The transform of the density asymptotically far from the corner (x_1 = x_2 = 0) is determined explicitly, while the remainder is given in terms of the solution to a pair of Fredholm equations.

Publication: Journal of Mathematical Physics Vol.: 12 No.: 5 ISSN: 0022-2488

ID: CaltechAUTHORS:20120810-145444423

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Abstract: The governing equations for the problem of linearized flow through a normal shock wave in an emitting, absorbing, and scattering grey gas are reduced to two linear coupled integro-differential equations. By separation of variables, these equations are further reduced to an integral equation similar to that which arises in neutron-transport theory. It is shown that this integral equation admits both regular (associated with discrete eigenfunctions) and singular (associated with continuum eigenfunctions) solutions to form a complete set. The exact closed-form solution is obtained by superposition of these eigen-functions. If the gas downstream of a strong shock is absorption–emission dominated, the discrete mode of the solution disappears downstream. The effects of isotropic scattering are discussed. Quantitative comparison between the numerical results based on the exact solution and on the differential approximation are presented.

Publication: Journal of Fluid Mechanics Vol.: 45 No.: 4 ISSN: 0022-1120

ID: CaltechAUTHORS:CHEjfm71

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Abstract: The integral form of the one-speed, steady-state Boltzmann transport equation is solved for a point source in a homogeneous, isotropically scattering slab. In addition, solutions are obtained for line sources and plane sources in the slab, both normal and parallel to the slab faces. Using Fuorier and Laplace transforms, the problem is reduced to that of solving a 1-dimensional integral equation with a difference kernel. This equation is transformed into a singular integral equation which is solved using standard methods. The Green's functions are subsequently obtained as generalized eigenfunction expansions over the spectrum of the 1-dimensional integral operator. This form yields a simple solution far from the source, and alternate expressions are obtained to facilitate evaluation near the source. In a thick slab the exact solutions are shown to reduce to simple closed expressions plus correction terms that decrease exponentially as the slab thickness increases. Most of the work previously done in multidimensional transport in slabs is shown to be easily reproduced using this theory in the thick-slab approximation. Also, virtually all other problems of this type can be solved using the theory presented here. In particular, the density from a pencil beam of particles normally incident to the slab is obtained.

Publication: Journal of Mathematical Physics Vol.: 11 No.: 2 ISSN: 0022-2488

ID: CaltechAUTHORS:GARjmp70

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Abstract: The fugacities of ideal Fermi and Bose gases are calculated explicity as a function of density and temperature, using the analytic properties of the integral appearing in the fugacity-density equation. The Hilbert problem of the theory of analytic functions is encountered.

Publication: Physical Review Vol.: 175 No.: 1 ISSN: 0031-899X

ID: CaltechAUTHORS:LEOpr68

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Abstract: A spectral analysis of the transport kernel for anisotropic scattering in finite slabs is achieved by first solving a type of generalized scattering problem for a subcritical slab. Initially, the scattering problem is stated as an inhomogeneous integral transport equation with a complex-valued source function. This is readily transformed to singular integral equations and linear constraints in which the space and angle variables enter as parameters. Dual singular equations appear in applications of Case's method to transport problems, but we cannot yet completely explain this duality. The singular equations are transformed to Fredholm equations by an extension of Muskhelishvili's standard method and by analytic continuation. It is shown that, for a wide class of scattering functions, this particular Fredholm reduction yields equations which converge rapidly under iteration for all neutron productions and slab thicknesses. The ultimate solution of the singular equations contains arbitrary constants which, when evaluated by the aforementioned linear constraints, display explicitly the Fredholm determinant and the eigenfunctions of the transport kernel. An immediate consequence of this result is the criticality condition and the associated neutron distribution. Specific applications to linear anisotropic and isotropic scattering in slab geometry are discussed. In addition, it is seen that the case of isotropic scattering in spheres can be treated with this method, and, in fact, the spectral analysis of the kernel for the slab problem immediately applies to the sphere kernel.

Publication: Journal of Mathematical Physics Vol.: 5 No.: 3 ISSN: 0022-2488

ID: CaltechAUTHORS:LEOjmp64

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Abstract: It is shown that, within the static approximation, the neutron diffraction pattern from a sample can be interpreted in terms of multiple particle correlation functions. In particular, the relation between double scattering and the three-particle correlation function is derived. Although the formula given cannot be formally inverted to give the three-particle correlation function, it may be useful for making corrections for multiple scattering in experiments.

Publication: Physical Review Vol.: 128 No.: 5 ISSN: 0031-899X

ID: CaltechAUTHORS:FERpr62

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