Timothy T. Clark

A numerical study of the statistics of a two-dimensional Rayleigh–Taylor mixing layer

A two-dimensional Rayleigh–Taylor mixing layer was numerically simulated to determine the growth rate of the mixing layer for an ensemble of initial conditions. The numerical algorithm used is a recently developed lattice Boltzmann method for multiphase flow. Small variations in the initial conditions of the fluid interface are observed to yield large variations in the growth rate of the mixing-layer width. Consequently, an ensemble of simulations has been generated to provide a statistical basis for determining the mixing-layer growth rate. The results for this ensemble indicate that the mixing-layer attains an approximate state of self-similarity and yields a distribution of mixing-layer growth rates.

A spectral model applied to homogeneous turbulence

Because a spectral model describes distributions of turbulent energy and stress in wave‐number space or, equivalently, in terms of a distribution of length scales, it can account for the variation of evolution rates with length scale. A spectral turbulence model adapted from a model introduced by Besnard, Rauenzahn, Harlow, and Zemach is applied here to homogeneous turbulent flows driven by constant mean‐flow gradients and to free decay of such flows. To the extent permitted by the experimental data, initial turbulent spectra are inferred, and their evolutions in time are computed to obtain detailed quantitative predictions of the spectra, relaxation times to self‐similarity, self‐similar spectrum shapes, growth rates, and power‐law time dependence of turbulent energies and dominant‐eddy sizes, and integral data, such as the components of the Reynolds stress tensor and the Reynolds stress anisotropy tensor. The match to experimental data, within the limits of experimental uncertainties, is good. Some qual...

Symmetries and the approach to statistical equilibrium in isotropic turbulence

The relaxation in time of an arbitrary isotropic turbulent state to a state of statistical equilibrium is identified as a transition to a state which is invariant under a symmetry group. We deduce the allowed self-similar forms and time-decay laws for equilibrium states by applying Lie-group methods (a) to a family of scaling symmetries, for the limit of high Reynolds number, as well as (b) to a unique scaling symmetry, for nonzero viscosity or nonzero hyperviscosity. This explains why a diverse collection of turbulence models, going back half a century, arrived at the same time-decay laws, either through derivations embedded in the mechanics of a particular model, or through numerical computation. Because the models treat the same dynamical variables having the same physical dimensions, they are subject to the same scaling invariances and hence to the same time-decay laws, independent of the eccentricities of their different formulations. We show in turn, by physical argument, by an explicitly solvable a...

Small scale response and modeling of periodically forced turbulence

The response of the small scales of isotropic turbulence to periodic large scale forcing is studied using two-point closures. The frequency response of the turbulent kinetic energy and dissipation rate, and the phase shifts among production, energy, and dissipation are determined as functions of the Reynolds number. It is observed that the amplitude and phase of the dissipation exhibit nontrivial frequency and Reynolds number dependence that reveals a filtering effect of the energy cascade. Perturbation analysis is applied to understand this behavior which is shown to depend on distant interactions between widely separated scales of motion. Finally, the extent to which finite dimensional models (standard two-equation models and various generalizations) can reproduce the observed behavior is discussed.

Self-similar turbulence evolution and the dissipation rate transport equation

The dissipation rate transport equation is analyzed in the setting of time-dependent isotropic turbulence driven by a statistically unsteady force. In the limit of slow spectral variation, the balance between vortex stretching and enstrophy destruction postulated by Tennekes and Lumley [A First Course in Turbulence (MIT Press, Cambridge, MA, 1972)] is verified and spectral closure is used to identify the O(Re0) difference between them; however, no definite formulation of an ϵ-equation results. The ϵ equation is usually calibrated to predict self-similar unit flows such as decaying turbulence and homogeneous shear flow. The limitations of this approach are shown by constructing classes of self-similar states of forced isotropic turbulence. Any choice of constants in the ϵ equation yields a model that is consistent with some self-similar states, but not with all possible states: two-equation models of the standard form select a class of admissible self-similar states and rule out the others.

Time-dependent isotropic turbulence*

Homogeneous isotropic turbulence subject to linearly increasing forcing is investigated as a unit problem for statistically unsteady turbulence. The transient spectral dynamics is analysed using a closure theory. A long time asymptotic state is found with k −7/3 corrections to the energy spectrum as proposed by Yoshizawa. Although the cancellation of O(Re 1/2) terms underlying the standard dissipation rate equation is confirmed in this asymptotic state, it is found that this equation cannot predict the transient dynamics accurately. The discrepancies are explained in terms of the basic mechanisms of vortex stretching and enstrophy destruction responsible for the evolution of the dissipation rate. This paper was chosen from Selected Proceedings of the Third International Symposium on Turbulence and Shear Flow Phenomena (Sendai, Japan, 24–27 June 2003).

Two-Point Closures and Statistical Equilibrium

A two-point, or spectral, turbulence transport model describes the evolution of the two-point velocity covariance tensor, or its Fourier transform, the spectral tensor. Such a model describes the turbulent dynamics as functions of length-scale or wave-number. This permits a more general description of turbulence than is available with a one-point closure. This greater generality is useful in understanding the behavior of turbulent flows that are undergoing rapid transients and that are therefore not in “equilibrium”. If the turbulent flow is in an “equilbrium” environment wherein the mean forces on the flow are relatively constant in time, the turbulent spectra tend toward self-similar forms. When applied to a specific spectral model (Besnard et al., 1996) (Clark and Zemach, 1995), these selfsimilar forms may be exploited to reduce the model to the more familiar R ij — E and K — E models. These one-point models have coefficients that are functions of the spectral distributions. We discuss the limits of validity of the two-point descriptions as well as the consequences of the equilibrium assumptions embedded in the one-point variants.

Reassessment of the classical closures for scalar turbulence

In deducing the consequences of the Direct Interaction Approximation, Kraichnan was sometimes led to consider the properties of special classes of nonlinear interactions in degenerate triads in which one wavevector is very small. Such interactions can be described by simplified models closely related to elementary closures for homogeneous isotropic turbulence such as the Heisenberg and Leith models. These connections can be exploited to derive considerably improved versions of the Heisenberg and Leith models that are only slightly more complicated analytically. This paper applies this approach to derive some new simplified closure models for passive scalar advection and investigates the consistency of these models with fundamental properties of scalar turbulence. Whereas some properties, such as the existence of the Kolmogorov–Obukhov range and the existence of thermal equilibrium ensembles, follow the velocity case closely, phenomena special to the scalar case arise when the diffusive and viscous effects become important at different scales of motion. These include the Batchelor and Batchelor–Howells–Townsend ranges pertaining, respectively, to high and low molecular Schmidt number. We also consider the spectrum in the diffusive range that follows the Batchelor range. We conclude that improved elementary models can be made consistent with many nontrivial properties of scalar turbulence, but that such models have unavoidable limitations.

Generation of anisotropy in turbulent flows subjected to rapid distortion

A computational tool for the anisotropic time-evolution of the spectral velocity correlation tensor is presented. We operate in the linear, rapid distortion limit of the mean-field-coupled equations. Each term of the equations is written in the form of an expansion to arbitrary order in the basis of irreducible representations of the SO(3) symmetry group. The computational algorithm for this calculation solves a system of coupled equations for the scalar weights of each generated anisotropic mode. The analysis demonstrates that rapid distortion rapidly but systematically generates higher-order anisotropic modes. To maintain a tractable computation, the maximum number of rotational modes to be used in a given calculation is specified a priori. The computed Reynolds stress converges to the theoretical result derived by Batchelor and Proudman [Quart. J. Mech. Appl. Math. 7, 83 (1954)QJMMAV0033-561410.1093/qjmam/7.1.83] if a sufficiently large maximum number of rotational modes is utilized; more modes are required to recover the solution at later times. The emergence and evolution of the underlying multidimensional space of functions is presented here using a 64-mode calculation. Alternative implications for modeling strategies are discussed.

On fully self-preserving solutions in homogeneous turbulence

Mathematical attributes of full self-preserving solutions (‘full’ meaning self-similar at all scales with non-zero viscosity) for isotropic decay as well as for homogeneous shear flow turbulence are examined from a fundamental theoretical standpoint. Fully self-preserving solutions are those wherein the two-point double and triple velocity correlations are self-similar at all scales. It is shown that the full self-preserving solutions for isotropic decay corresponds to a t −1 asymptotic power law decay, consistent with earlier studies. Fully self-preserving solutions for homogeneous shear flow correspond to a production-equals-dissipation equilibrium, with bounded turbulent kinetic energy and dissipation. It is then shown that the fully self-preserving solutions of isotropic decay and of homogeneous shear flow both require severe constraints on the behavior of the low-wavenumber energy spectra. These constraints render the full self-preserving solutions as mathematical consistent but having no physical re...