Session KO: Turbulence Modeling III

Statistics of SGS stress states in homogeneous isotropic turbulence

Two parameters are introduced that uniquely characterize the state of a third order symmetric tensor. We show that the proposed parameters are induced by the uniform metric in the matrix space, thus the joint PDF of these parameters can be used to determine the geometrical statistics of any third order symmetric tensor. We use the joint PDF of these parameters to describe the states of the subgrid-scale stress, which is of central interest in Large Eddy Simulation. Direct numerical simulation of forced isotropic turbulence is used in our {\it a priori} tests. With the proposed parameterization we can assess the most probable flow configuration on the scales of motion just above Kolmogorov scale. We also test four different subgrid-scale models on the subject of how well do they predict the structure, or state, of the subgrid-scale stress. It is found that models based on truncated Taylor series do not produce adequate distribution of state, even if augmented by turbulent viscosity term. On the other hand, models based on the scale-similarity assumption predict the distribution of states that is close to the actual.   

Extension of Boussinesq turbulence constitutive relation for partially-averaged velocity fields

Boussinesq hypothesis is still widely used to model Reynolds stresses in turbulent flows. Recently, there have been many attempts to extend the Boussinesq constitutive relation for modeling the sub-grid scale stresses (SGS) for filtered or partially-averaged velocity fields, for example, limited numerical scales method (LNS), flow simulation methodology (FSM), unsteady Reynolds averaged Navier-Stokes (URANS), very large eddy simulation (VLES) and partially-averaged Navier-Stokes (PANS). The major proposals can be classified into two categories. In VLES, FSM, and LNS the effective SGS viscosity is given by: $\nu_T =fC_{\mu} \frac{k^2}{\varepsilon}$ where $f$ is determined by the filter size. In PANS, the SGS viscosity is given by $\nu_{Tu} =C_{\mu} \frac{k_u^2}{\varepsilon_u}$ where $k_{u}$ and \textit{$\varepsilon$}$_{u}$ are the unresolved turbulent kinetic energy and dissipation rate respectively. The validity of these two proposals is investigated in two turbulent flow fields: (i) flow past a circular cylinder; (ii) flow past a backward facing step. We investigate the degree of eddy viscosity reduction obtained \textit{a posteriori} in the computations and compare them to the prescribed value. The effect of the two viscosity reduction proposals on the unsteadiness of the computed velocity fields is also examined.   

Shear Improved Smagorinsky Model

Shear Improved Smagorinsky Model (SISM) - a new eddy viscosity model for Large Eddy Simulations (LES) - has recently been introduced and tested in channel flow geometries (E. Leveque et al. nlin.CD/0605053). The formulation of the model is based on the following definition for the eddy viscousity $\nu_T = (C_s \Delta)^2 (|\overline S|-|\langle \overline S\rangle|)$, where $|\overline S|$ is the norm of the resolved strain tensor (as for the usual Smagorinsky model) and $|\langle \overline S\rangle|$ is the norm of the average shear. With respect to Smagorinsky model, the SISM has zero viscosity in laminar flow regions and a damped viscosity at approaching solid boundaries: these features allow to avoid the use of wall damping functions. Here we present tests benchmarking the model, in particular comparing it to the dynamical model, in channel flows and in a more complex flow geometry, namely the backward facing step.   

Dynamic high-pass filtered eddy-viscosity models for LES of turbulent boundary layers

Turbulent flow in a spatially developing boundary layer is simulated by large-eddy simulation (LES) employing a dynamic version of the high-pass filtered (HPF) eddy-viscosity approach. With these models, an a-priori spatial scale separation is performed via a filter with constant filter width acting in physical space. To close the governing equations, only the small- scale (i.e. high-pass filtered) fraction of the velocities are considered to compute the model terms, similar to the multiscale approach. Standard subgrid closures as e.g. the Smagorinsky model or the relaxation-term model are used. The dynamic determination of the model coefficient is based on a consistent formulation of the Germano/Lilly procedure. To this end it is necessary to introduce suitable high-pass filters for both the grid and the test-filter level. Simulations are performed based on spectral discretisation. Results are presented for zero-pressure gradient turbulent boundary-layer flow with $Re_\theta$ up to 1800. Additionally, computations with an adverse-pressure gradient, eventually leading to a separated boundary layer, are also shown. Comparisons are made to experimental data, direct numerical simulations and LES employing standard subgrid-scale closures.   

Coherent Vortex Simulations of 3D isotropic turbulence

This is the first of three talks on the wavelet filter based dynamically adaptive eddy capturing computational methodology that unifies variable fidelity simulation approaches such as wavelet-based DNS, Coherent Vortex Simulation (CVS), and Stochastic Coherent Adaptive Large Eddy Simulation. The commonality of these approaches is their ability to identify and ``track" on an adaptive mesh energetic coherent vortical structures. In CVS the velocity field is decomposed into two orthogonal parts: a coherent, inhomogeneous, non-Gaussian component and an incoherent, homogeneous, Gaussian component. This separation of coherent and incoherent components is achieved by wavelet thresholding which can be viewed as a non-linear filter that depends on each flow realization. The essence of the CVS approach is to solve for the coherent non-Gaussian component of a turbulent flow field. It has been shown previously that second generation bi-orthogonal wavelet threshold filtering is able to decompose a turbulent velocity field such that the total resulting SGS dissipation is approximately zero. This physically allows a CVS simulation to recover low order statistics with no SGS model. In this work CVS simulations of decaying incompressible 3D isotropic turbulence are compared to DNS results. \vspace{-6pt}   

Lagrangian dynamic SGS model for Stochastic Coherent Adaptive

This is the first of two talks, which describe ongoing localized SGS model development for the Stochastic Coherent Adaptive Large Eddy Simulation (SCALES) methodology. The SCALES approach has the potential for significant improvement over regular grid LES methods with its ability to resolve and dynamically track the most energetic coherent structures in a turbulent flow through dynamic grid adaptation based on wavelet threshold filtering. In this talk we propose a new local Lagrangian path-line/tube dynamic model, as an extension of the original formulation by Meneveau {\em et al.} (J. Fluid Mech., 1996). The new procedure involves the definition of the following filtered averages over the trajectory of a fluid particle: \linebreak[5] $ \displaystyle \overline{\mathcal{I}}_{LM}({\mathbf{x}},t) = {1 \over T} \int_{-\infty}^{t} e^{\frac{\tau-t}{T}} \int \!\!\!\int \!\!\!\int_{D} G \left( {\mathbf{y}}-{\mathbf{x}}\left(\tau\right),\tau \right) L_{ij}M_{ij}\left({\mathbf{x}}\left(\tau\right),\tau\right) {d}{\mathbf{y}} {d}\tau $, and, analogously, $\overline{\mathcal{I}}_{MM}({\mathbf{x}},t)$, $G$ being the local low-pass filter moving together with the particle. Preliminary numerical experiments are conducted for a freely decaying homogeneous turbulent flow. Good results are obtained in terms of both grid compression and low order flow statistics.   

SGS kinetic energy based dynamic models for Stochastic Coherent Adaptive Large Eddy Simulation

This is the second of two talks, which describe ongoing localized SGS model development for the Stochastic Coherent Adaptive Large Eddy Simulation (SCALES) methodology. In this talk, new localized one-equation dynamic closure models based on the subgrid-scale turbulent kinetic energy are presented. One of the main advantages of the present formulation is that the equilibrium assumption between production and dissipation of SGS kinetic energy is no longer required, since an additional transport model equation for the SGS kinetic energy is solved, together with the wavelet-filtered Navier-Stokes equations, to ensure the energy budget between resolved and unresolved motions. Both localized eddy-viscosity and dynamic structure non eddy-viscosity models are considered. Preliminary numerical experiments are conducted for freely decaying homogeneous turbulence. Good results are obtained in terms of both grid compression, that is a fundamental parameter for wavelet-based numerical solutions of turbulence, and low order flow statistics.   

A Novel Uncertainty Propagation Model for the Burgers’ Equation

Uncertainty quantification is indispensable to numerical simulations of complex multi-physics problems with sophisticated models. The Monte Carlo approach is statistically sound but extremely expensive; and for this reason it usually requires the development of reduced models of the original system to obtain sufficient realizations. In such a situation, since the prediction quality will be affected by the decreased fidelity of the models, there is no need to quantify the uncertainty too accurately. This suggests the possibility of modeling the uncertainty propagation with the objective of reducing the number of realizations required to achieve the same level of prediction quality for a given computational cost. This work focuses on developing such an approach for the Burgers’ equation. We decompose the unknown variables into mean and fluctuating parts (in probability space), then derive the corresponding equations starting from the original PDE. General observations and comparisons to several other uncertainty quantification approaches, such as polynomial chaos, are made. Like in turbulence modeling, the non-linearity in the PDE introduces unclosed higher-moment terms and these require modeling. A number of possible closures are proposed and investigated in details.