Session EC: Turbulence Modeling II

Improving the near-wall behavior of multiscale models for LES

The multiscale models (in fact the ``multiscale approaches'' applied to the Smagorinsky model) have gained a growing interest in the LES community because of the appealing properties resulting from the involved high-pass filtering (HPF). One could hope that this filtering is sufficient for wall-resolved LES; hence no need for further near-wall damping (explicit or using a dynamic procedure). Unfortunately, the dissipation profile of such models does not tend to zero at the wall, even though the dissipation is indeed reduced compared to the corresponding unfiltered model. This leads to relatively poor results and, quite importantly, to severe and unpractical time-step restrictions because of stability issues. This unsatisfactory behavior is basically due to the fact the HPF velocity field has the same near-wall scaling as the unfiltered field. Hence, the SGS viscosity scalings previously developed to provide the proper $y^3$ near-wall dissipation behavior when computed using the unfiltered field, also provide the proper behavior when computed using the HPF field. In this study, several new and classical scalings, used in the multiscale approach, are investigated in turbulent channel flow LES at $Re_\tau = 395$ using a fourth order finite difference solver.   

A mixed LES model based on the residual-based variational multiscale formulation

In the residual-based variational multiscale (VMS) formulation of large eddy simulation (LES) a projection operator is used to separate the solution of the Navier-Stokes equations into coarse and fine scales. The coarse scale equations are solved numerically while the fine scale equations are solved analytically. In particular, an algebraic approximation for the fine scale velocities is derived wherein they are expressed in terms of the residual of the Navier Stokes operator applied to the coarse scale solution. In this talk we analyze the residual-based VMS model in wavenumber space and conclude that while it accurately models the cross-stress term, it under-estimates the contribution from the Reynolds-stress term. To remedy this we add to it a Smagorinsky eddy viscosity which provides a good approximation to the Reynolds-stress term. This leads us to a mixed model capable of accurately modeling all components of the subgrid stress. We implement the mixed model in a Fourier-spectral method and use it to predict the decay of homogeneous isotropic turbulence. We determine the two unknown parameters in this model dynamically using the variational counterpart of the Germano identity. We note that the mixed model yields better agreement with direct numerical simulation than either of its components: the dynamic Smagorinsky model and the dynamic version of the residual-based VMS model.   

Sequential approximation of velocity fields using episodic POD

The problem of approximating velocity fields at future and past times based on information available at the current time is addressed. A novel method called ``episodic POD'' is described that enables us to achieve our objective. Application of episodic POD to an ensemble of flow data results in a set of spatio-temporal eigenfunctions and a set of coefficients associated with the eigenfunctions. From these eigenfunctions, we develop two models called the ``forward model'' and ``inverse model'' that enable us to approximate the velocity fields at future and past times based on information provided at the current time. A second set of models, the forward and inverse sequential models are also developed that enable the dynamic update of approximated velocity fields when new information is made available, making these models more adept at on-line estimation. Different examples are used to validate and highlight the proposed method. It is also shown that episodic POD outperforms the linear Kalman filters in the presence of noise.   

Radial basis function approach to modeling dynamical systems.

We are interested in developing dynamical systems models that are based on discrete multivariate time series information only, with application to fluid flow phenomena. A method that uses radial basis functions and linear multi-step methods is developed to construct continuous nonlinear models that approximate the original dynamical system. Information, such as the structure of the original system, is incorporated into the models through weak constraints. The formulation of the model and its advantages associated with modeling are described. Different examples are presented that highlight the various characteristics of the model and its effectiveness in dealing with various problems encountered in fluid flow.   

A generalized Landau model for oscillatory to complex shear flows --- enablers for reduced, low and least-order Galerkin models

Landau's (1944) celebrated amplitude equation $dA/dt = \sigma A - \beta A^3$ for a supercritical Hopf bifurcation connects linear instability with a nonlinear amplitude saturation mechanism, thereby describing the transient and post-transient phase of oscillations. This model is significantly generalized for a much larger class of laminar to turbulent shear flows within the finite-time thermodynamics (FTT) formalism (Noack et al.\ 2008 JNET). In this talk, we highlight the critical role of FTT in deriving reduced to least-order Galerkin models for oscillatory to complex shear flows. This includes shift modes as well as a novel nonlinear subgrid turbulence representation. Intriguingly, both can lead to a similar, nonlinear damping term for fluctuation energy as described by Landau's model.   

A study of the sensitivity of the POD eigenvalues to the density of the resolved measurement grid.

A study of the sensitivity of the convergence of the POD eigenvalues to the discretization of the measurement / computational grid is presented using a number of different experimental data sets. The purpose is to determine the necessary conditions, base on an a prior understanding of the statistical properties of the turbulence field, for sufficiently obtaining a reduced order representation of the system. The analysis is important for two reasons. The first is that when the grid resolution is too coarse, the first few POD eigenvalues are overestimated followed by underestimates in the higher POD eigenvalues. Conversely, for grid resolutions that are too dense, superfluous information is carried. Where the latter is concerned, the performance of physics based control architectures that are designed around low-dimensional analysis tools can be seriously hindered.   

A priori study of SGS flux of a passive scalar in LES

The DNS data is used to explore the properties of subgrid-scale flux $\tau_\phi$ of a passive scalar $\phi$ in the framework of Large Eddy Simulation. Geometrical characteristics such as alignment trends between the flux and resolved and SGS structures are studied. It is shown that the direction of the flux is strongly coupled with the SGS stress axes rather than the resolved flow quantities such as strain $\bar{S}_{ij}$, vorticity $\bar{\omega}$ or scalar gradient $\nabla \bar{\phi}$. We derive an approximate transport equation for the subgrid-scale flux of a scalar and look at the relative importance of the terms in the transport equation. A particular form of LES tensor-viscosity model for the scalar flux is investigated, which includes the subgrid-scale stress: $\tau_{i\phi}=1/|\bar{S}| \tau_{ij} \bar{\phi}_{,j}$. Effect of different models for the subgrid-scale stress $\tau_{ij}$ on the model for the subgrid-scale flux $\tau_{\phi}$ is studied.   

Computational and Experimental Investigations of Turbulent Flow Past Projectiles

Experimental and computational investigations of turbulent flow past projectiles is modeled as axial flow past a cylinder with a free-spinning base. A subsonic wind tunnel with a forward-sting mounted spinning cylinder is used for experiments. In addition, a free-jet facility is used for benchmarking the experimental set up. Experiments are performed for a range of spin rates and free stream flow conditions. An anisotropic two-equation Reynolds-stress model that incorporates the effect of rotation-modified energy spectrum and swirl is used to perform computations for the flow past axially rotating cylinders. Both rigid cylinders as well as that of cylinders with free-spinning base are considered from a computational point of view. Applications involving the design of projectiles are discussed.   

Mixing in turbulent clouds

Turbulent clouds in the earth's atmosphere constantly entrain dry air from the surrounding environment, and this entrainment process influences the cloud microphysical properties, and therefore cloud optical properties relevant to climate. How cloud droplet size distributions respond to turbulent mixing is analogous to the problem of turbulent reactive flows, and therefore depends on relative time scales for mixing and for water phase changes (Damkoehler number). We have studied turbulent mixing in small cumulus clouds using a helicopter-borne instrument payload with high resolution measurements of the three-dimensional wind, temperature and humidity fields, and the droplet size distribution. Small Damkoehler numbers correlate uniform evaporation of droplets, while large Damkoehler numbers correlate with constant mean droplet size and a reduction in droplet number density due to complete evaporation of a subset of droplets. In some cases the latter, inhomogeneous mixing led to the formation of drops that are larger than in the unmixed adiabatic cloud core, which is of potential importance for precipitation formation in warm cumulus clouds.