Decay Of Homogeneous Isotropic Turbulence Research Articles

DNS study of decaying homogeneous isotropic turbulence with polymer additives

In order to investigate the turbulent drag reduction phenomenon and understand its mechanism, direct numerical simulation (DNS) was carried out on decaying homogeneous isotropic turbulence (DHIT) with and without polymer additives. We explored the polymer effect on DHIT from the energetic viewpoint, i.e. the decay of the total turbulent kinetic energy and energy distribution at each scale in Fourier space and from the phenomenological viewpoint, i.e. the alterations of vortex structures, the enstrophy and the strain. It was obtained that in DHIT with polymer additives the decay of the turbulent kinetic energy is faster than that in the Newtonian fluid case and a modification of the turbulent kinetic energy transfer process for the Newtonian fluid flow is observed due to the release of the polymer elastic energy into flow structures at certain small scales. Besides, we deduced the transport equations of the enstrophy and the strain, respectively, for DHIT with polymer additives. Based on the analyses of these transport equations, it was found that polymer additives depress both the enstrophy and the strain in DHIT as compared to the Newtonian fluid case, indicating the inhibition effect on small-scale vortex structures and turbulence intensity by polymers.

Effects of multitemperature nonequilibrium on compressible homogeneous turbulence

We study the effects of the rotational-translational energy exchange on the compressible decaying homogeneous isotropic turbulence (DHIT) in three dimensions through direct numerical simulations. We use the gas-kinetic scheme coupled with multitemperature nonequilibrium based on the Jeans-Landau-Teller model. We investigate the effects of the relaxation time of rotational temperature, ZR, and the initial ratio of the rotational and translational temperatures, TR0/TL0, on the dynamics of various turbulence statistics including the kinetic energy K(t), the dissipation rate epsilon(t), the energy spectrum E(k,t), the root mean square of the velocity divergence theta'(t), the skewness Su(t) and the flatness Fu(t) of the velocity derivatives, and the probability distribution functions of the local Mach number Ma and the shocklet strength chi. The larger the ZR is, the faster the compressibility decays after an initial time. Similarly, with a fixed TL0, the higher the initial energy ratio TR0/TL0, the weaker is the compressibility in the flow. It is also observed that the effect of TR0/TL0 is strong in all times in the decay, while the effect of ZR is severe only in the later times passing through the stage with strong nonlinearity. We also observe that the multitemperature model does not affect the self-similarities obeyed by the probability distribution functions of Ma and chi , which appear to be a robust feature of the compressible DHIT.

Gas-kinetic schemes for direct numerical simulations of compressible homogeneous turbulence

We apply the gas-kinetic scheme (GKS) for the direct numerical simulations (DNSs) of compressible decaying homogeneous isotropic turbulence (DHIT). We intend to study the accuracy, stability, and efficiency of the gas-kinetic scheme for DNS of compressible homogeneous turbulence depending on both flow conditions and numerics. In particular, we study the GKS with multidimensional, quasi-one-dimensional, dimensional-splitting, and smooth-flow approximations. We simulate the compressible DHIT with the Taylor microscale Reynolds number Re(lambda)=72.0 and the turbulence Mach number Ma(t) between 0.1 and 0.6. We compute the low-order statistical quantities including the total kinetic energy K(t), the dissipation rate epsilon(t), the skewness S(u)(t), and the flatness F(u)(t) of the velocity field u(x,t). We assess the effects on the turbulence statistics due to the approximations made in the treatment of fluxes, the flux limiter, the accuracy of the interpolation, and the bulk viscosity. Our results show that the GKS is adequate for DNS of compressible homogeneous turbulence as far as the low-order turbulence statistics are concerned.

Progress in subgrid-scale transport modelling for continuous hybrid non-zonal RANS/LES simulations

The partially integrated transport modelling (PITM) method can be viewed as a continuous approach for hybrid RANS/LES modelling allowing seamless coupling between the RANS and the LES regions. The subgrid turbulence quantities are thus calculated from spectral equations depending on the varying spectral cutoff location [Schiestel, R., Dejoan, A., 2005. Towards a new partially integrated transport model for coarse grid and unsteady turbulent flow simulations. Theoretical and Computational Fluid Dynamics 18, 443–468; Chaouat, B., Schiestel, R., 2005. A new partially integrated transport model for subgrid-scale stresses and dissipation rate for turbulent developing flows. Physics of Fluids, 17 (6)] The PITM method can be applied to almost all statistical models to derive its hybrid LES counterpart. In the present work, the PITM version based on the transport equations for the turbulent Reynolds stresses together with the dissipation transport rate equation is now developed in a general formulation based on a new accurate energy spectrum function E ( κ ) valid in both large and small eddy ranges that allows to calibrate more precisely the c sgs ϵ 2 function involved in the subgrid dissipation rate ϵ sgs transport equation. The model is also proposed here in an extended form which remains valid in low Reynolds number turbulent flows. This is achieved by considering a characteristic turbulence length-scale based on the total turbulent energy and the total dissipation rate taking into account the subgrid and resolved parts of the dissipation rate. These improvements allow to consider a large range of flows including various free flows as well as bounded flows. The present model is first tested on the decay of homogeneous isotropic turbulence by referring to the well known experiment of Comte-Bellot and Corrsin. Then, initial perturbed spectra E ( κ ) with a peak or a defect of energy are considered for analysing the model capabilities in strong non-equilibrium flow situations. The second test case is the classical fully turbulent channel flow that allows to assess the performance of the model in non-homogeneous flows characterised by important anisotropy effects. Different simulations are performed on coarse and refined meshes for checking the grid independence of solutions as well as the consistency of the subgrid-scale model when the filter width is changed. A special attention is devoted to the sharing out of the energy between the subgrid-scales and the resolved scales. Both the mean velocity and the turbulent stress computations are compared with data from direct numerical simulations.

Power law of decaying homogeneous isotropic turbulence at low Reynolds number

We focus on an estimate of the decay exponent (m) in the initial period of decay of homogeneous isotropic turbulence at low Taylor microscale Reynolds number R lambda (approximately equal to 20-50). Lattice Boltzmann simulations in a periodic box of 256(3) points are performed and compared with measurements in grid turbulence at similar R lambda. Good agreement is found between measured and calculated energy spectra. The exponent m is estimated in three different ways: from the decay of the turbulent kinetic energy, the decay of the mean energy dissipation rate, and the rate of growth of the Taylor microscale. Although all estimates are close, as prescribed by theory, that from the Taylor microscale has the largest variability. It is then suggested that the virtual origin for the decay rate be determined from the Taylor microscale, but the actual value of m be estimated from the decay rate of the kinetic energy. The dependence of m on R lambda(0) (the value of R lambda at the beginning of the simulation) is also analyzed, using the present data as well as data from the literature. The results confirmed that m approaches 1, as R lambda(0) increases.

Variational formulation of the Germano identity for the Navier–Stokes equations

The variational counterpart of the Germano identity for the large eddy simulation (LES) of turbulent flows is derived. The differences between the variational and the filter-based identities are explained on the basis of the role of an LES model in these two formulations. The variational Germano identity is then used to determine parameters in LES models. The performance of this identity in predicting the Smagorinsky eddy viscosity during the decay of homogeneous isotropic turbulence is evaluated. It is found that it yields results that are as accurate as its filter-based counterpart and that it is less sensitive to the exact approach used in evaluating the unknown parameters.

A dynamic approach for evaluating parameters in a numerical method

A new methodology for evaluating unknown parameters in a numerical method for solving a partial differential equation is developed. The main result is the identification of a functional form for the parameters which is derived by requiring the numerical method to yield ‘optimal’ solutions over a set of finite-dimensional function spaces. The functional depends upon the numerical solution, the forcing function, the set of function spaces, and the definition of the optimal solution. It does not require exact or approximate analytical solutions of the continuous problem, and is derived from an extension of the variational Germano identity. This methodology is applied to the one-dimensional, linear advection–diffusion problem to yield a non-linear dynamic diffusivity method. It is found that this method yields results that are commensurate to the SUPG method. The same methodology is then used to evaluate the Smagorinsky eddy viscosity for the large eddy simulation of the decay of homogeneous isotropic turbulence in three dimensions. In this case the resulting method is found to be more accurate than the constant-coefficient and the traditional dynamic versions of the Smagorinsky model. Copyright © 2004 John Wiley & Sons, Ltd.

The multiscale formulation of large eddy simulation: Decay of homogeneous isotropic turbulence

The variational multiscale method is applied to the large eddy simulation (LES) of homogeneous, isotropic flows and compared with the classical Smagorinsky model, the dynamic Smagorinsky model, and direct numerical simulation (DNS) data. Overall, the multiscale method is in better agreement with the DNS data than both the Smagorinsky model and the dynamic Smagorinsky model. The results are somewhat remarkable when one realizes that the multiscale method is almost identical to the Smagorinsky model (the least accurate model!) except for removal of the eddy viscosity from a very small percentage of the lowest modes.

Final Period of Decay for a Viscoelastic Fluid in Isotropic Turbulence

AbstractThe final period of decay of homogeneous isotropic turbulence of an Oldroyd type fluid is studied. Expressions are derived for the double correlation function and the decay of turbulent energy. In comparing the results with those of Batchelor and Townsend for a Newtonian fluid, it is found that the difference from the Newtonian counterpart is the presence of two additional terms involving speed and acceleration of evolution of turbulent energy at the initial instant. As in the Newtonian case the two point velocity correlation function is in normal form. The energy decays according to the (− 5/2) power laws but modified by elasticity effects.

On the law of decay of homogeneous isotropic turbulence and the theories of the equilibrium and similarity spectra

AbstractThe requirements of Kolmogoroff's theory of the equilibrium spectrum are satisfied only at very high Reynolds numbers, higher than any at which experiments have yet been done. In particular, when the theory holds, the rate of decay of the mean-square vorticity ω must be negligible compared with either its rate of increase due to the stretching of the vortex filaments or the rate of dissipation due to viscosity.An extended version of Kolmogoroff's hypothesis may be proposed, in which the statistical properties of the turbulence in a range of wave-numbers (range of eddy sizes) depend not only on the rate of dissipation ∈ per unit volume and the viscosity ν, but also on the time rate of change d∈/dt of ∈. The result is to introduce a dependence on the Reynolds number R of the turbulence into quantities and constants which, on Kolmogoroff's original hypothesis, were independent of R. The Reynolds number R is defined from the decay law; u2t, with an origin of time suitably chosen, is a function of t, finite when t = 0, and R is defined as (u2t)t=0/ν. Lin's decay law follows logically from the extended hypothesis, according to which the rates of change of ω−1, due to the causes mentioned above, are constant during the decay; Lin's decay law would also follow if this less general part only of the extended hypothesis be assumed. The same decay law is also obtained if the similarity spectrum of Heisenberg is taken to apply not to the whole of the energy-bearing eddies, but only to the energy-dissipating eddies. But it is suggested that further generalization of the theory of the similarity spectrum, and of the decay law, is necessary; that the similarity spectrum is probably only asymptotically correct for a range of large wave-numbers, the range depending on the initial conditions and decreasing as the decay proceeds; that the general decay law is u2t = μRd(t), where d(t) is an integral function of t, such that d(0) = 1, and with an asymptotic value for large t to give correctly the law of decay in the final period; and that d(t), and the number of constants needed to specify it approximately, depend on the initial conditions. An experiment is suggested to test the dependence of the law of decay on the initial conditions. It is also suggested that the recently observed constancy of u2t in the initial period in the turbulence behind a single grid is only approximate, and this approximate constancy still needs explanation. Remarks are also included on the range of application of the equilibrium spectrum, for which some formulae are given when there is a definite cut-off in the spectrum.

A Contribution to the Theory of Turbulence

S of the problem of Wind-Tunnel Turbulence have recently been published by Taylor, von Karman, and Dryden. These studies are concerned with the decay of homogeneous isotropic turbulence. The scale and intensity of the turbulence is assumed to be known at a given instant throughout the fluid and the decay with respect to time is asked for. In this note the authors investigate the turbulence produced at a constant rate in the boundary, and decaying in a direction normal to the surface of generation. This problem, so far as the authors are aware, has not been treated before. The problem becomes especially simple if one considers a fluid in a semiinfinite space bounded by a plane along which the turbulence is produced. The mean motion is assumed to be constant throughout and parallel to the plane. This investigation is not only of theoretical interest but should be useful in the study of the behavior of turbulent boundary layers, and might be of help in the study of the influence of surface roughness. The stated problem can be treated by means of the equation of the statistical theory of turbulence recently proposed by von Karman. It furthermore might serve as the basis for an experiment in which the consequences of the theory can be tested. The considered turbulent motion has statistically an isotropic character due to the absence of mean shear stresses, i.e., the correlations of the velocity components at a certain point in the space with respect to a set of orthogonal axes are independent of the rotation of this system. The turbulence, however, is not homogeneous, since the correlations are functions of the distance from the boundary. Consider a space with the orthogonal axes xiy x2} x3, in which the fluid is bounded by a (xix3) plane and has a, constant mean velocity U in the X direction. The