Nicholas K.-R. Kevlahan

Non-Gaussianity and coherent vortex simulation for two-dimensional turbulence using an adaptive orthogonal wavelet basis

We decompose turbulent flows into two orthogonal parts: a coherent, inhomogeneous, non-Gaussian component and an incoherent, homogeneous, Gaussian component. The two components have different probability distributions and different correlations, hence different scaling laws. This separation into coherent vortices and incoherent background flow is done for each flow realization before averaging the results and calculating the next time step. To perform this decomposition we have developed a nonlinear scheme based on an objective threshold defined in terms of the wavelet coefficients of the vorticity. Results illustrate the efficiency of this coherent vortex extraction algorithm. As an example we show that in a 256 2 computation 0.7% of the modes correspond to the coherent vortices responsible for 99.2% of the energy and 94% of the enstrophy. We also present a detailed analysis of the nonlinear term, split into coherent and incoherent components, and compare it with the classical separation, e.g., used for large eddy simulation, into large scale and small scale components. We then propose a new method, called coherent vortex simulation ~CVS!, designed to compute and model two-dimensional turbulent flows using the previous wavelet decomposition at each time step. This method combines both deterministic and statistical approaches: ~i! Since the coherent vortices are out of statistical equilibrium, they are computed deterministically in a wavelet basis which is remapped at each time step in order to follow their nonlinear motions. ~ii! Since the incoherent background flow is homogeneous and in statistical equilibrium, the classical theory of homogeneous turbulence is valid there and we model statistically the effect of the incoherent background on the coherent vortices. To illustrate the CVS method we apply it to compute a two-dimensional turbulent mixing layer.

Wavelets and turbulence

We have used wavelet transform techniques to analyze, model, and compute turbulent flows. The theory and open questions encountered in turbulence are presented. The wavelet-based techniques that we have applied to turbulence problems are explained and the main results obtained are summarized.

Computation of turbulent flow past an array of cylinders using a spectral method with Brinkman penalization

Abstract A major difficulty in computing engineering flows at high Reynolds number is the need for non-uniform grids adapted to solid boundaries that may be moving or changing shape. These non-uniform grids are expensive to calculate and cannot be used with the most accurate or efficient numerical schemes. We present one solution to this problem: a Brinkman (volume) penalization of the obstacle which allows an efficient pseudo-spectral method to be used to solve the Navier–Stokes equations on a Cartesian grid. Although this is the most severe test of the penalization (due to the global support of the Fourier basis), it is shown that the method still yields reasonable results. We also present an analytical solution of Stokes flow calculated using the penalization which illustrates the error and continuity properties of the approach. Work is currently underway to implement the penalization approach in a wavelet basis.

Vorticity filaments in two-dimensional turbulence : creation, stability and effect

Vorticity lamentsy are characteristic structures of two-dimensional turbulence. The formation, persistence and eect of vorticity laments are examined using a highresolution direct numerical simulation (DNS) of the merging of two positive Gaussian vortices pushed together by a weaker negative vortex. Many intense spiral vorticity laments are created during this interaction and it is shown using a wavelet packet decomposition that, as has been suggested, the coherent vortex stabilizes the laments. This result is conrmed by a linear stability analysis at the edge of the vortex and by a calculation of the straining induced by the spiral structure of the lament in the vortex core. The time-averaged energy spectra for simulations using hyper-viscosity and Newtonian viscosity have slopes of 3 and 4 respectively. Apart from a much higher eective Reynolds number (which accounts for the dierence in energy spectra), the hyper-viscous simulation has the same dynamics as the Newtonian viscosity simulation. A wavelet packet decomposition of the hyper-viscous simulation reveals that after the merger the energy spectra of the lamentary and coherent parts of the vorticity eld have slopes of 2 and 6 respectively. An asymptotic analysis and DNS for weak external strain shows that a circular lament at a distance R from the vortex centre always reduces the deformation of a Lamb’s (Gaussian) vortex in the region r > R. In the region r< Rthe deformation is also reduced provided the lament is intense and is in the vortex core, otherwise the lament may slightly increase the deformation. The results presented here should be useful for modelling the coherent and incoherent parts of two-dimensional turbulent flows.

The vorticity jump across a shock in a non-uniform flow

The vorticity jump across an unsteady curved shock propagating into a two-dimensional non-uniform flow is considered in detail. The exact general expression for the vorticity jump across a shock is derived from the gasdynamics equations. This general expression is then simplied by writing it entirely in terms of the Mach number of the shock MS and the local Mach number of the flow ahead of the shock MU. The vorticity jump is very large at places where the curvature of the shock is very large, even in the case of weak shocks. Vortex sheets form behind shock-shocks (associated with kinks in the shock front). The ratio of vorticity production by shock curvature to vorticity production by

Simultaneous space-time adaptive wavelet solution of nonlinear parabolic differential equations

Dynamically adaptive numerical methods have been developed to efficiently solve differential equations whose solutions are intermittent in both space and time. These methods combine an adjustable time step with a spatial grid that adapts to spatial intermittency at a fixed time. The same time step is used for all spatial locations and all scales: this approach clearly does not fully exploit space-time intermittency. We propose an adaptive wavelet collocation method for solving highly intermittent problems (e.g. turbulence) on a simultaneous space-time computational domain which naturally adapts both the space and time resolution to match the solution. Besides generating a near optimal grid for the full space-time solution, this approach also allows the global time integration error to be controlled. The efficiency and accuracy of the method is demonstrated by applying it to several highly intermittent (1D+t)-dimensional and (2D+t)-dimensional test problems. In particular, we found that the space-time method uses roughly 18 times fewer space-time grid points and is roughly 4 times faster than a dynamically adaptive explicit time marching method, while achieving similar global accuracy.

An adaptive wavelet collocation method for the solution of partial differential equations on the sphere

A dynamic adaptive numerical method for solving partial differential equations on the sphere is developed. The method is based on second generation spherical wavelets on almost uniform nested spherical triangular grids, and is an extension of the adaptive wavelet collocation method to curved manifolds. Wavelet decomposition is used for grid adaption and interpolation. An O(N) hierarchical finite difference scheme based on the wavelet multilevel decomposition is used to approximate Laplace-Beltrami, Jacobian and flux-divergence operators. The accuracy and efficiency of the method is demonstrated using linear and nonlinear examples relevant to geophysical flows. Although the present paper considers only the sphere, the strength of this new method is that it can be extended easily to other curved manifolds by considering appropriate coarse approximations to the desired manifold (here we used the icosahedral approximation to the sphere at the coarsest level).

Nonlinear interactions in turbulence with strong irrotational straining

The rate of growth of the nonlinear terms in the vorticity equation are analysed for a turbulent flow with r.m.s. velocity u 0 and integral length scale L subjected to a strong uniform irrotational plane strain S, where (u 0/L)/S = e ≪ 1. The Rapid Distortion Theory (RDT) solution is the zeroth order term of the perturbation series solution in terms of e. We use the asymptotic form of the convolution integrals for the zeroth order nonlinear terms when exp(St) ≫ 1 to determine when (in wavenumber k and time t) the perturbation series in e fails, and hence estimate precisely the domain of validity of inviscid and viscous RDT.

WKB theory for rapid distortion of inhomogeneous turbulence

A WKB method is used to extend RDT (rapid distortion theory) to initially inhomogeneous turbulence and unsteady mean flows. The WKB equations describe turbulence wavepackets which are transported by the mean velocity and have wavenumbers which evolve due to the mean strain. The turbulence also modies the mean flow and generates large-scale vorticity via the averaged Reynolds stress tensor. The theory is applied to Taylor’s four-roller flow in order to explain the experimentally observed reduction in the mean strain. The strain reduction occurs due to the formation of a large-scale vortex quadrupole structure from the turbulent spot conned by the four rollers. Both turbulence inhomogeneity and three-dimensionality are shown to be important for this eect. If the initially isotropic turbulence is either homogeneous in space or two-dimensional, it has no eect on the large-scale strain. Furthermore, the turbulent kinetic energy is conserved in the two-dimensional case, which has important consequences for the theory of two-dimensional turbulence. The analytical and numerical results presented here are in good qualitative agreement with experiment.

Nonlinear RDT theory of near-wall turbulence

A WKB method was recently used to extend rapid distortion theory (RDT) to initially inhomogeneous turbulence strained by irrotational mean flows [S.V. Nazarenko, N. Kevlahan, B. Dubrulle, J. Fluid Mech. 390 (1999) 325]. This theory takes into account the feedback of turbulence on the mean flow, and it was used by Nazarenko et al. to explain the effect of strain reduction caused by turbulence observed by Andreotti et al. [B. Andreotti, S. Douady,Y. Couder, in: O. Boratav, A. Eden, A. Erzan (Eds.), Turbulence Modeling and Vortex Dynamics, Proceedings of a Workshop held at Istanbul, Turkey, 2‐6 September 1996, pp. 92‐108]. In this paper, we develop a similar WKB RDT approach for shear flows. We restrict ourselves to problems where the turbulence is small-scale with respect to the mean flow length-scale and turbulence vorticity is weak compared to the mean shear. We show that the celebrated log-law of the wall exists as an exact analytical solution in our model if the initial turbulence vorticity (debris of the near-wall vortices penetrating into the outer regions) is statistically homogeneous in space and shortly correlated in time. We demonstrate that the main contribution to the shear stress comes from very small turbulent scales which are close to the viscous cut-off and which are elongated in the stream-wise direction (streaks). We also find that anisotropy of the initial turbulent vorticity changes the scaling of the shear stress, but leaves the log-law essentially unchanged. ©2000 Elsevier Science B.V. All rights reserved.