Session GO: Turbulence Modeling II

Towards an improved interface treatment in Hybrid RANS-LES methods

Almost all hybrid RANS-LES methods are confronted with the problem of having a proper interface treatment. In the case of DES, in which the same eddy-viscosity transport equation is solved throughout the domain, the RANS-LES interface has pronounced effects on the overall result. Since the interface is usually determined by the computational grid, this places much emphasis on the grid used. Several investigators have explored different interface treatments. Examples include adding a forcing term, specifying a better boundary condition at the interface, or changing the length scale such that the interface moves as the flow evolves. Our own research effort initially focused on moving the interface in such a way that the production and destruction of eddy viscosity always balances at the interface. In another approach, we used a narrow buffer near the interface allowing a smooth transition of eddy viscosity from high RANS values to low LES values. Recently, Germano proposed a hybrid filter which when applied to the Navier-Stokes equation allows smooth blending of RANS and LES models without the need of an interface treatment. Numerical results from some of these approaches will be presented.   

Analysis of 3-D ``poor man's Navier--Stokes equation'' for application to LES SGS models

We briefly outline derivation of a discrete dynamical system (DDS) that we demonstrate is capable of producing temporal behaviors associated with incompressible Navier-Stokes (N.--S.) equation flows. We compare results from the 3-D system with those of an analogous 2-D DDS previously studied by McDonough and Huang (Int. J. Numer. Meth. Fluids 45, 545, 2004) and show that the same wide range of states is achieved in 3D, and that there are no further ones. But we show that details of results from the 2-D and 3-D equations differ, as occurs for the N.--S. equations, themselves. In particular, the 3-D results appear to be generically more anisotropic, as might be intuitively expected. We present some selected comparisons between this 3-D poor man's Navier--Stokes (PMNS) equation and experimental data and then provide a brief mathematical analysis suggesting its relationship to a symbol and pseudodifferential operator of the N.--S. equation. We argue that this relationship serves to explain the ability of the PMNS equation to faithfully reproduce physical turbulence over a wide range of conditions, thus justifying its use as part of a subgrid-scale model for LES.   

Three regularization models as large-eddy simulations

We test three regularizations, the $\alpha-$model, {Leray$-\alpha$}, and {Clark$-\alpha$}, as sub-grid models for LES by comparison with a $1024^3$ direction numerical simulation (DNS), $R_\lambda \approx 800$, with a Taylor-Green forcing. Both the $\alpha-$model and {{Clark$-\alpha$} }are able to reproduce the large-scale anisotropy of the flow as well as the time scale of developing turbulence. {{Leray$-\alpha$} }fails in both these regards. We study intermittency corrections through pdfs and the anomalous scaling of the velocity increment structure functions. {{Leray$-\alpha$} }is somewhat less intermittent than the DNS and produces an energy spectrum that is too shallow in the inertial range, while {{Clark$-\alpha$} }produces a broad $k^{-5/3}$ spectrum and stronger intermittency corrections. Finally, the agreement of the DNS and $\alpha-$model spectra, in disparity with results for lower Reynolds number simulations, is worse than in the Clark$-\alpha$ model. We conjecture that this enhanced intermittency in the $\alpha$ model is related to the steeper than $k^{-5/3}$ spectrum now reported for the very highest Reynolds number simulations and atmospheric observations.   

Large eddy simulation with energy and helicity flux constraint

The energy and helicity flux constraints are imposed on the dynamic procedure of LES. The coefficients in the sub-grid scale stress model are determined by minimizing the error $E=\left<(L_{ij}-(T^{mod}_{ij}-\overline{\tau^{mod}_{ij}}))^2 \right>$ in satisfying the Germano identity under the energy and helicity flux constraints $ \left<\tau^{mod}_{ij}\tilde S_{ij}\right>=\Pi_E$, and $\left<\tau^{mod}_{ij}\tilde R_{ij}\right>=\Pi_H$. Both a \emph{priori} test and a \emph{posteriori} test of the present SGS model are performed. Given proper energy and helicity flux functions $\Pi_E$ and $\Pi_H$, this constrained dynamic SGS model not only achieves a good r.m.s approximation of real SGS stress but also preserves the correct energy and helicity flux.   

A Self-Adapting Turbulence Model for Hybrid RANS/LES

A self-adapting turbulence model is discussed which automatically adapts to the mesh provided so that as much turbulence as possible is resolved, and only the subgrid scale turbulence is modeled. The model automatically performs RANS, LES, or DNS depending on the mesh provided to it. Perturbation of the initial conditions does not affect the model. A smooth RANS initial condition on an LES mesh will eventually result in an LES solution. Both two-equation and Reynolds stress transport equation versions of the model are presented. Results are shown for low and high Reynolds number decaying turbulence.   

Large-eddy simulation of the flow in a lid-driven cubical cavity using dynamic approximate deconvolution models

LES of the flow in a lid-driven cubical cavity by the spectral element method using dynamic approximate deconvolution models (ADM) are considered. Explicit filtering is based on an invertible modal filter. Results for $\textrm{Re}=12'000$ are showing very good agreement with other experimental and DNS results. An under-resolved DNS has also been carried-out, but on a limited time range just to prove that the under-resolution is effective and the essential need for taking into account the subfilter scales (SFS) and the subgrid scales (SGS). Despite its simple geometry, this 3D unsteady flow at $\textrm{Re}=12'000$ is very challenging for SFS and SGS modeling. Indeed, maintaining the energy balance among scales in such a confined fluid domain is a difficult task allowing to track any under- or over-dissipative character of the SFS and SGS dynamic ADM. Furthermore, the transitional nature of this flow makes intense turbulent zones coexisting with laminar regions therefore requiring a proper activation of the SFS and SGS models.   

Dynamic Optimal Finite-Volume LES and its Application to a Temporally Evolving Plane Mixing Layer

A simple dynamic optimal finite-volume LES model has been developed and applied to a temporally evolving free shear layer. Unlike previous optimal models, the model used here does not depend on DNS data. The necessary velocity correlations for the stochastic estimation procedure, which yields the optimal model, are obtained by assuming isotropy of the turbulence at the filter scale, allowing the use of Kolmogorov's expressions for the third-order, two-point velocity structure functions. The model reduces to a second-order dissipation term whose strength is determined from the consistent kinetic energy dissipation rate and the average kinetic energy dissipation (anti-dissipation) of the numerical treatment of the nonlinear terms. Modifications to the modeling procedure due to the existence of a mean velocity profile and a direction of inhomogeneity are discussed. Results for the dynamic optimal finite-models are compared to the DNS simulations of Rogers and Moser (1994).   

A-posteriori tests of one-equation subgrid models for the LES of rotating turbulence

In rotating turbulence, a successful subgrid-scale (SGS) model should be able to capture partial transfer of energy from small to large scales and the formation of cyclonic vortical columns, both of which are observed in experiments and direct numerical simulation (DNS). The challenge is to simultaneously reflect the (anisotropic) 3D nature of the small scales and the primarily 2D nature of larger scales. To this end, we develop two consistent one-equation models using a dynamic procedure and an equation for the subgrid-scale kinetic energy. These models satisfy the constraint of material frame indifference and they do not assume that the principal axes of the strain rate tensor are aligned with those of the SGS tensor. At the a-posteriori test level, we assess various subgrid models using three flow configurations: (i) homogeneous decaying turbulence subjected to uniform background rotation; (ii) rotating turbulence forced at large scales; (iii) rotating turbulence forced at small scales. Compared to other existing models, the new models are better able to capture essential features of rotational flows.   

A new a posteriori test for the subgrid-scale stress models

Traditional \textit{a posteriori} tests of SGS models often compare large eddy simulation (LES) profiles of statistics with measurements. In this study a new \textit{a posteriori} test is employed to study SGS model performance. We compare the conditional means of the LES-generated SGS stress and the conditional stress production rate conditional on the resolvable-scale velocity, which must be reproduced by the SGS model for large eddy simulation (LES) to correctly predict the one-point resolvable-scale velocity joint probability density function, with measurements. The results for convective atmospheric boundary layers show that the level of the anisotropy of the SGS stress is underpredicted by both the Smagorinsky model and the Kosovic nonlinear model. The magnitudes of the conditional means are also underpredicted by both models whereas the trends are generally better predicted by the Kosovic nonlinear model. However, the alignment between the conditional SGS stress and its production rate are over-predicted by both models. The model strength and deficiencies observed here were also identified in our previous statistical \textit{a priori} tests analyzing these conditional statistics. The remarkable consistency between the two types of tests suggests that statistical tests analyzing the conditional SGS stress and its production rate are a highly capable approach for identifying specific model deficiencies and for evaluating SGS model performance in simulations. Supported by NSF.   

There Is No Smagorinsky Constant

There is significant discussion in the literature regarding the ``correct'' value for the Smagorinsky coefficient, $C_S$, which appears in the classical Smagorinksy subgrid model as a multiplier for the characteristic grid length scale. Careful consideration of the equations, however, shows that $C_S$ is not a physical quantity at all, and certainly is not a constant, even for a specific flow and flow solver. Rather, the coefficient includes both numerical and physical effects in that its value controls the trade-off between the numerical resolution and the range of turbulent scales which are explicitly resolved. Consequently, as the numerical resolution is improved the model should become insensitive to the particular value of the coefficient. In the current work, a simple model is developed which shows the qualitative behavior of the total simulation error as $C_S$ and the numerical resolution are varied. Simulations are presented to support this model. This has important ramifications for both model development and validation. It is further argued that the current results generalize to any model which includes a coefficient multiplying an implicit filter length scale.