Session AC: Turbulence Modeling I

A modified artificial viscosity approach for compressible turbulence simulations

Standard high-order methods give rise to spurious oscillations near shocks which can be controlled by using localized artificial viscosity (AV). Schemes which give a high wavenumber bias to the numerical dissipation around shocks are gaining popularity. Using simulations of compressible isotropic turbulence with optimized high-order schemes at different resolutions we investigated the range of scales where artificial dissipation is active. We observed that the impact of AV was not limited to high wavenumbers. This is especially true for moderately high Mach number isotropic turbulence which spontaneously forms shocklets, for which the AV method is found to excessively damp the dilatational motions. We propose a modified form using a dynamic coefficient which activates AV only in the regions of strong compression, such as shocks, turning it completely off for turbulence and expansion waves. This is found to give improved statistics for all quantities, not just dilatation. This formulation reverts back to the traditional one for strong shocks, so that its shock capturing capability is not compromised.   

Finite-Volume Optimal LES Formulations for Coarsely Resolved Channel Flows

In finite-volume formulations of optimal LES (OLES), the effects of unresolved turbulence are represented in terms of models for the momentum fluxes through the finite-volume cell boundaries. The models are quadratic in the velocities of the cell volumes, and given the finite-volume stencil to be used are optimized using stochastic estimation. The resulting model is effectively a finite-volume scheme that has been designed to represent the effects of subgrid turbulence. This approach has been found to be highly effective in isotropic turbulence, but in a wall-bounded flow, new complications arise. These include the treatment of the mean velocity, subgrid anisotropy and the momentum flux to the wall. In this study, correlations derived from DNS of turbulent channel flow are used to perform the stochastic estimation for a number of model dependencies (stencils), and the properties of the resulting OLES models are evaluated. Of particular interest are stability of the schemes, energy transfer to the small scales, and the {\it a posteriori} accuracy of the resulting LES. Schemes that correctly represent energy transfer can easily be constructed, but stable and accurate LES results are more sensitively dependent on the formulation of the schemes and their interaction with models for the wall shear stress. The properties of models that perform well {\it a posteriori} will be discussed.   

Large-eddy simulations based on the subgrid-scale kinetic energy transport equation

This presentation focusses on the development of SGS models based on transport equations for the SGS kinetic energy and SGS scalar variance. These models do not suffer from the limitations of the local equilibrium assumption that is used by the great majority of existing SGS models. In virtually all SGS models using transport equations, the diffusion terms are lumped together, and their joint effect is modeled using a ``gradient-diffusion'' model. It is shown that this is a poor approximation for inertial range filter sizes and high Reynolds numbers. The reason for this lies in a loss of local balance between the SGS turbulent diffusion and diffusion caused by GS/SGS interactions, and in the deficient modeling of the diffusion by SGS pressure-velocity interactions. In order to improve this situation, a new model, inspired by Clark's SGS model, is developed for this term. The new model shows very good agreement with the exact SGS pressure-velocity term in a priori tests and better results than the classical model in a posteriori LES tests. We assess several models currently used for the molecular/viscous SGS dissipation terms. The model used in hybrid RANS/LES tested here gives very poor results. The reason behind this is connected with the deficient spectral representation of the exact molecular SGS dissipation term.   

Energy transfers in shell models for MHD turbulence

A systematic procedure to derive shell models for MHD turbulence is proposed. It takes into account the conservation of ideal quadratic invariants such as the total energy, the cross-helicity and the magnetic helicity as well as the conservation of the magnetic energy by the advection term in the induction equation. This approach also leads to simple expressions for the energy exchanges as well as to unambiguous definitions for the energy fluxes. When applied to the existing shell models with nonlinear interactions limited to the nearest neighbour shells, this procedure reproduces well known models but suggests a reinterpretation of the energy fluxes. This formalism also yields general constraints on the shell models that are independent of the shell model expressions for the helicities. The final structure of the shell model requires however explicit definitions for both the vorticity and the magnetic potential vector in terms of the shell variables.   

Stochastic Coherent Adaptive Large Eddy Simulation of forced isotropic turbulence with variable thresholding

In this talk we discuss the progress in the development of the novel methodology for the numerical simulation of turbulent flows, called Stochastic Coherent Adaptive Large Eddy Simulation (SCALES). SCALES is an extension of the Large Eddy Simulation approach that uses a wavelet filter-based dynamic grid adaptation strategy to solve for the most energetic coherent structures in a turbulent flow field, while modelling the effect of the less energetic background flow with a local dynamic subgrid-scale (SGS) model. In contrast to previous formulations that used a global relative wavelet threshold, in this study we explore the {\em spatially variable} wavelet thresholding strategy to ensure the adequate resolution of local flow characteristics. For example, the local wavelet thresholding level can be adjusted by ensuring a prescribed level of SGS dissipation with respect to the resolved viscous dissipation. A number of numerical experiments for linearly forced homogeneous turbulence are presented and the results are compared with pseudo-spectral reference solutions. The agreement holds not only in terms of global statistical quantities but also in terms of spectral distribution of energy and, more importantly, enstrophy all the way down to the dissipative scales.   

Helical vortex-based model of deterministic stresses for Large-Eddy-Simulation of a wind turbine wake

When representing a wind turbine in LES using a drag disk (e.g. A. Jimenez et al. 2007), the periodic effects due to the turbine's rotating elements remain unresolved. The periodic effects on the mean flow can be represented in a simulation using deterministic stresses in the wake. In this work, based on the Biot-Savart law with a helical vortex street and various simplifications, we develop an analytical expression for the deterministic, periodic velocity fluctuations in the wake. Then, the deterministic stress tensor is obtained by the product of the approximated fluctuating components of velocity, and integration over a helical period. The resulting model is implemented within a Large Eddy Simulation of an array of wind turbines, using the scale-dependent Lagrangian dynamic model (Bou-Zeid et al. 2005). The importance of the deterministic stresses on the computed wake structure is examined by varying the strength of the helical vortices.   

A dynamical systems approach to modeling the rapid pressure strain correlation

Models for the rapid pressure strain correlation, under the auspices of the classical Reynolds stress closure schemes, represent dynamical systems in the state space composed of the Reynolds stress tensor components. For 2 dimensional mean flows, these are single parameter systems, dependent on the mean flow gradient. A classification of the topology and behavior of the same would provide invaluable guidelines for developing improved models. In line with the maxim of understanding before prediction, the authors aim to classify the dynamical behavior of this hypothetical system. With the objective of isolating the effects of pressure strain correlation, the behavior of the Navier Stokes equations is contrasted against its pressure released analogue, the Burgers equations, in the rapid distortion limit. The authors carry out numerical simulations, in addition to the analytical modeling, for both the systems. The corresponding invariant set topologies are classified and a concomitant bifurcation analysis is conducted. Some salient issues addressed in this study include the requisite nature of the model, viz. a linear or nonlinear structure; and the inability of models to capture elliptic flows.   

Truncated Navier-Stokes Equations with the Automatic Filtering Criterion

Truncated Navier-Stokes (TNS) methodology is a LES technique in which Navier Stokes equations are solved in the DNS mode on a coarse LES mesh. Because no explicit SGS model is used, such simulations would quickly depart from DNS results obtained on a full DNS mesh, with departures observed first in the small scales. In TNS this process is controlled by periodically filtering out the small scales of the numerical solution and replacing them by new, estimated scales. In previous work the filtering time interval was normally fixed through trial and error. We report details of a modified TNS procedure where the filtering interval is determined automatically during the course of the simulations. The procedure employs the criterion that prevents the energy buildup in the small scales beyond limits allowed by the inertial and dissipation range dynamics. The procedure is tested in a sequence of TNS simulations for turbulent channel flow and several Reynolds numbers for which detailed DNS data are available for comparison, up to $Re_{\tau}=2000$.   

A kinematic sub-grid scale model for large-eddy simulation of turbulence-generated sound

In the hybrid simulation of large-eddy simulation (LES) and Lighthill's acoustic analogy for turbulence-generated sound, an LES is used to solve the Navier-Stokes equations and the turbulence-generated sound at far fields is calculated from Lighthill's acoustic analogy. As only the filtered velocity fields are available from the LES, the Lighthill stress tensor, serving as a source term in Lighthill's acoustic equation, has to be evaluated from the resolved velocity fields and thus, the unresolved velocity fields are missing in the conventional LES. The sound of missing scales has been shown to be important and hence need to be modeled. The present study proposes a kinematic sub-grid model which recasts the unresolved velocity fields into Lighthill's stress tensors. A kinematic simulation is used to construct the unresolved velocity fields with an imposed temporal statistics, which is consistent with the random sweeping hypothesis. The model is used to calculate sound power spectra from isotropic turbulence with an improved result: the missing portion of the sound power spectra is approximately recovered in this calculation.