A dynamic two-equation Sub Grid Scale model

Continuum Mechanics and Thermodynamics, 2000

All dynamic Smagorinsky-type Sub-Grid Scale (SGS) models deduce the eddy viscosity expression by assuming that a local balance exists between the SGS turbulent kinetic energy production and its dissipation. It is shown that the local balance assumption is not correctly formulated in these models. A new dynamic model is proposed in which the eddy viscosity is defined as a symmetric second rank tensor, proportional to the product of a turbulent length scale with an ellipsoid of turbulent velocity scales. The employed definition of the eddy viscosity allows to remove the local balance assumption of the SGS turbulent kinetic energy formulated in all the dynamic Smagorinsky-type SGS models. Furthermore, because of the tensorial structure of the eddy viscosity the alignment assumption between the principal axes of the SGS turbulent stress tensor and the resolved strain-rate tensor is equally removed, an assumption which is employed in the scalar eddy viscosity SGS models. The proposed model is tested for a turbulent channel flow. Comparison with the results obtained with other dynamic SGS models (Dynamic Smagorinsky Model, Dynamic Mixed Model and Dynamic K -equation Model) shows that the tensorial definition of the eddy viscosity and the removal of the local balance assumption of the SGS turbulent kinetic energy considerably improves the agreement between results obtained with Large Eddy simulation (LES) and Direct Numerical Simulations (DNS), respectevely.

A new LES model is proposed. The proposed closure relation for the generalized SGS turbulent stress tensor: complies with the principle of turbulent frame indifference; takes into account both the anisotropy of the turbulence velocity scales and turbulence length scales; removes any balance assumption between the production and dissipation of SGS turbulent kinetic energy. In the proposed model the generalized SGS turbulent stress tensor is related exclusively to the generalized SGS turbulent kinetic energy (which is calculated by means of its balance equation) and the modified Leonard tensor. The filtered momentum equations are solved by using a staggered fourth order finite difference scheme. The proposed model is tested for a turbulent channel flow at Reynolds numbers (based on friction velocity and channel half-width) ranging from 395 to 2340.

1994

The properties of turbulence subgrid-scale stresses are studied using experimental data in the far field of a round jet, at a Reynolds number of R, z 310. Measurements are performed using two-dimensional particle displacement velocimetry. Three elements of the subgrid-scale stress tensor are calculated using planar filtering of the data. Using a priori testing, eddy-viscosity closures are shown to display very little correlation with the real stresses, in accord with earlier findings based on direct numerical simulations at lower Reynolds numbers. Detailed analysis of subgrid energy fluxes and of the velocity field decomposed into logarithmic bands leads to a new similarity subgridscale model. It is based on the 'resolved stress' tensor L,,, which is obtained by filtering products of resolved velocities at a scale equal to twice the grid scale. The correlation coefficient of this model with the real stress is shown to be substantially higher than that of the eddy-viscosity closures. It is shown that mixed models display similar levels of correlation. During the a priori test, care is taken to only employ resolved data in a fashion that is consistent with the information that would be available during largeeddy simulation. The influence of the filter shape on the correlation is documented in detail, and the model is compared to the original similarity model of Bardina et al. (1980). A relationship between L,, and a nonlinear subgrid-scale model is established. In order to control the amount of kinetic energy backscatter, which could potentially lead to numerical instability, an ad hoc weighting function that depends on the alignment between Lii and the strain-rate tensor, is introduced. A 'dynamic' version of the model is shown, based on the data, to allow a self-consistent determination of the coefficient. In addition, all tensor elements of the model are shown to display the correct scaling with normal distance near a solid boundary.

Physical Review E, 2009

Two approaches for closing the turbulence subgrid-scale stress tensor in terms of matrix exponentials are introduced and compared. The first approach is based on a formal solution of the stress transport equation in which the production terms can be integrated exactly in terms of matrix exponentials. This formal solution of the subgrid-scale stress transport equation is shown to be useful to explore special cases, such as the response to constant velocity gradient, but neglecting pressure-strain correlations and diffusion effects. The second approach is based on an Eulerian-Lagrangian change of variables, combined with the assumption of isotropy for the conditionally averaged Lagrangian velocity gradient tensor and with the 'Recent Fluid Deformation' (RFD) approximation. It is shown that both approaches lead to the same basic closure in which the stress tensor is expressed as the product of the matrix exponential of the resolved velocity gradient tensor multiplied by its transpose. Short-time expansions of the matrix exponentials are shown to provide an eddy-viscosity term and particular quadratic terms, and thus allow a reinterpretation of traditional eddy-viscosity and nonlinear stress closures. The basic feasibility of the matrix-exponential closure is illustrated by implementing it successfully in Large Eddy Simulation of forced isotropic turbulence. The matrix-exponential closure employs the drastic approximation of entirely omitting the pressure-strain correlation and other 'nonlinear scrambling' terms. But unlike eddy-viscosity closures, the matrix exponential approach provides a simple and local closure that can be derived directly from the stress transport equation with the production term, and using physically motivated assumptions about Lagrangian decorrelation and upstream isotropy.

Journal of Fluid Mechanics, 1999

The response, evolution, and modelling of subgrid-scale (SGS) stresses during rapid straining of turbulence is studied experimentally. Nearly isotropic turbulence with low mean velocity and Rλ˜290 is generated in a water tank by means of spinning grids. Rapid straining (axisymmetric expansion) is achieved with two disks pushed towards each other at rates that for a while generate a constant strain rate. Time-resolved, two-dimensional velocity measurements are performed using cinematic PIV. The SGS stress is subdivided to a stress due to the mean distortion, a cross-term (the interaction between the mean and turbulence), and the turbulent SGS stress τ(T)ij. Analysis of the time evolution of τ(T)ij at various filter scales shows that all scales are more isotropic than the prediction of rapid distortion theory, with increasing isotropy as scales decrease. A priori tests show that rapid straining does not affect the high correlation and low square-error exhibited by the similarity model...

Applied Scientific Research, 1995

Several versions of similarity subgrid-scale turbulence models are tested a-priori using high Reynolds number experimental data. Measurements are performed by two-dimensional Particle Image Velocimetry (PIV) in the far field of a turbulent round jet. It is first verified that the usual Smagorinsky model is poorly correlated with the real stress Tij. On the other hand, a similarity subgrid-scate model based on the 'resolved stress' tensor Lij, which is obtained by filtering products of resolved velocities at a scale equal to twice the grid scale, displays a much higher level of correlation. Several variants of this model are examined: the mixed model, and the global and local dynamic procedure. Model coefficients are measured, based on the condition that the subgrid models dissipate energy at the correct rate. The experimental data are employed to show that the dynamic procedure [4] yields appropriate model coefficients based only on the resolved portion of the velocity field. Some features of the dynamic procedure in its local formulation are also explored.

1975

High Reynolds number turbulent flows of incompressible fluids in plane channels and annuli are simulated using a finite difference procedure which integrates the Navier-Stokes equations in time and in three-dimensional space. This paper describes the finite difference procedure and the subgrid scale (SGS) motion model. The model differs from earlier ones in the following points. The finite difference equations are based on integral conservation equations for each grid volume. As a consequence the SGS stresses are defined as surface mean rather than grid volume mean values of the fluctuating velocity products. This allows us to identify and model the effects of anisotropic grids (especially unequally sided grid volumes) and anisotropic finite difference operators. In this model SGS stresses are split into two parts, one accounting for locally isotropic turbulence, the other for inhomogeneous effects. This results in a model which is meaningful even if the size of the grid volumes is very large. The SGS kinetic energy is calculated using a separate transport equation. The boundary conditions are formulated in a manner consistent with the SGS theory. The method may be used for plane channels and annuli as well, and has been used to simulate flows with up to 65,536 grid volumes. The results agree rather well with experimental values, even for a smaller number of grid volumes.

International Journal of Heat and Fluid Flow, 2017

Some types of mixed subgrid-scale (SGS) models combining an isotropic eddy-viscosity model and a scale-similarity model can be used to effectively improve the accuracy of large eddy simulation (LES) in predicting wall turbulence. Abe (2013) has recently proposed a stabilized mixed model that maintains its computational stability through a unique procedure that prevents the energy transfer between the grid-scale (GS) and SGS components induced by the scale-similarity term. At the same time, since this model can successfully predict the anisotropy of the SGS stress, the predictive performance, particularly at coarse grid resolutions, is remarkably improved in comparison with other mixed models. However, since the stabilized anisotropy-resolving SGS model includes a transport equation of the SGS turbulence energy, k SGS , containing a production term proportional to the square root of k SGS , its applicability to flows with both laminar and turbulent regions is not so high. This is because such a production term causes k SGS to self-reproduce. Consequently, the laminar-turbulent transition region predicted by this model depends on the inflow or initial condition of k SGS. To resolve these issues, in the present study, the mixed-timescale (MTS) SGS model proposed by Inagaki et al. (2005) is introduced into the stabilized mixed model as the isotropic eddy-viscosity part and the production term in the k SGS transport equation. In the MTS model, the SGS turbulence energy, k es , estimated by filtering the instantaneous flow field is used. Since the k es approaches zero by itself in the laminar flow region, the self-reproduction property brought about by using the conventional k SGS transport equation model is eliminated in this modified model. Therefore, this modification is expected to enhance the applicability of the model to flows with both laminar and turbulent regions. The model performance is tested in plane channel flows with different Reynolds numbers and in a backward-facing step flow. The results demonstrate that the proposed model successfully predicts a parabolic velocity profile under laminar flow conditions and reduces the dependence on the grid resolution to the same degree as the unmodified model by Abe (2013) for turbulent flow conditions. Moreover, it is shown that the present model is effective at transitional Reynolds numbers. Furthermore, the present model successfully provides accurate results for the backward-facing step flow with various grid resolutions. Thus, the proposed model is considered to be a refined anisotropy-resolving SGS model applicable to laminar, transitional, and turbulent flows.

International Journal of Heat and Fluid Flow, 2002

The study of unsteady turbulent flows can lead to non-equilibrium turbulence spectra and there is a need to develop new subgridscale models that can be efficient in such situations. This is particularly necessary if the grid is coarse and the cutoff lies before the inertial zone of the spectrum. The model should be able to behave like a standard LES when the filter is sharp and like RANS models when the filter is wide, in a progressive way. The present work is a first step in that direction made by using transport equations for the subgrid-scale turbulence. We have used a transposition of the statistical model of Hassid and Poreh. The results of the simulations show that the one-equation model can predict the lag effects in the turbulent field response in pulsed channel flow, while the usual Smagorinsky model fails in several aspects. Ó

Flow Turbulence and Combustion, 1999

A new subgrid scale model is proposed for Large Eddy Simulations in complex geometries. This model which is based on the square of the velocity gradient tensor accounts for the effects of both the strain and the rotation rate of the smallest resolved turbulent fluctuations. Moreover it recovers the proper y3 near-wall scaling for the eddy viscosity without requiring dynamic