Session LC: Turbulence Modelling II

Evaluation of Turbulence Closures from a DNS Database of Shock Boundary Layer Interaction

Reynolds averaged closures have limited predictive capabilities when applied to the problem of shock boundary layer interaction. Several modifications to RANS models have been proposed in the literature, including compressibility corrections, limiters, and alternative forms of the turbulence production terms. Our objective is to characterize the errors introduced by the various approximations used in typical two-equation models, such as turbulence isotropy, linear stress-strain relationship, dissipation rate, etc. by isolating each contribution separately. We first solve the mean flow equations using the turbulence statistics obtained from a DNS database (Wu \& Martin, 2008); we alter the Reynolds stresses by perturbing the turbulent time scale and the stress anisotropy to investigate the effect of potential modeling errors on the resulting skin friction and wall pressure. As a second step, we use the same methodology to study the effect of modeling assumptions in the turbulent transport equations. We also discuss the use of the invariant map to introduce Reynolds stress perturbations that remain realizable.   

A priori comparison of RANS scalar flux models using DNS data of a Mach 5 boundary layer

In order to investigate the applicability of Reynolds-averaged scalar flux models (SFM) to scalar dispersion in high speed turbulent flows, a priori comparisons have been performed utilizing the results of direct numerical simulations (DNS) of a Mach 5 boundary layer. At a small patch on the solid surface boundary, a scalar was introduced into the flow at a rate depending upon the local surface temperature. This configuration mimics surface ablation in hypersonic flows. In different simulations, the scalar injection rate was varied, and the scalar was treated as both passive, not affecting the flow field, and active, affecting the flow field due to having different molecular properties than the bulk flow and having an injection velocity. Statistics of the simulated scalar fields have been calculated and compared a priori with terms from SFMs. Comparisons from the passive scalar case show that the scalar flux terms in the standard gradient diffusion model fail to predict even the trend of the DNS values. The generalized gradient diffusion models, while an improvement for the streamwise component of scalar flux, nevertheless fail to predict the wall normal and spanwise fluxes. Additionally, production and dissipation models for the scalar variance equation are evaluated.   

The Influence of Spanwise Rotation on the Redistribution of Turbulent Kinetic Energy in Fully-Developed Channel Flows

A recently developed universal, realizable, anisotropic prestress (URAPS-) closure for the normalized Reynolds (NR-) stress is used to predict the influence of spanwise rotation on the components of the NR-stress in fully-developed channel flows. Direct numerical simulation (DNS-) results are used to determine the relative time scales needed to solve the non-linear URAPS-equation. The new closure, which predicts the existence of a region of zero intrinsic vorticity on the high pressure side of the channel, provides an explanation of how rotation redistributes turbulent kinetic energy among the three components of the fluctuating velocity. The non-linear algebraic URAPS-equation is formulated as a mapping of the NR-stress into itself; therefore, fixed points of the URAPS-equation are \textit{realizable} for all turbulent flows, regardless of the benchmark flows used to calibrate the model parameters. The URAPS-closure together with closed transport equations for the turbulent kinetic energy and the turbulent dissipation provide a low-order closure for the RANS-equation.   

Uncertainty Quantification for RANS Turbulence Model Predictions

We examine the application of a Bayesian uncertainty quantification (UQ) framework to Reynolds-averaged Navier-Stokes (RANS) turbulence model predictions. The UQ framework involves the solution of a statistical inverse problem, where probability density functions (pdfs) for the parameters of a chosen stochastic model are calibrated, and a statistical forward problem, where the uncertainty represented by the calibrated pdfs is propagated through the model to the quantity of interest (QoI). In the context of RANS models, which are deterministic, a significant challenge is the development of an appropriate stochastic extension of the deterministic physical model. This stochastic extension is required to enable the model structural uncertainty---i.e., the uncertainty in the form of the physical model---to be represented in the inverse problem and properly propagated in the forward problem. Multiple possible formulations of a stochastic extension of popular RANS models will be discussed, and preliminary results in modeling turbulent boundary layers will be shown.   

Optimal harmonic forcing and drag reduction in pipe flow

A simple (RANS) model is used in combination with and explicit expression for the eddy viscosity, to model the linear response of turbulent pipe flow to harmonic forcing. Although the response is not so large as for the laminar case, it is shown that the greatest response still occurs for forcing on the largest scale, i.e.\ on the scale of the size of the domain. As this scale is independent of the flow rate, any strategy for drag reduction on this scale should be practical over a large range of Reynolds numbers. Also, as any drag reduction technique must modify the flow, it is convenient that it the large-scale motion is the most efficiently induced. The only question remaining, therefore, is whether the large scale motion can lead to drag reduction in this geometry. In this work we show that forcing large scale motions can produce competitive drag reduction and at little energetic cost.   

A New Turbulence Model for Capturing Physics in Variable-Density Flows

Variable-density flows are ubiquitous in nature. They are encountered on a larger scale in astrophysical, geophysical, atmospheric and oceanic flows, as well as on a smaller level, for example, in inertial confinement fusion and engine combustion. Our goal was to develop a turbulence model that is universally applicable across a wide variety of variable-density configurations by including more of the physics important to these types of flows. As part of this effort, the BHR turbulence model, which we use to close the traditional RANS equations, has been extended to capture the effects of molecular mixing in miscible fluids, as well as changes in turbulence quantities that occur due to de-mixing. Here, we demonstrate these new model capabilities for a flow that evolves from an unstable to stable Rayleigh-Taylor state. The model results are validated against experimental data.   

Effects of streamline curvature on separation prediction

In this study, the effects of streamline curvature on prediction of flow separation are investigated. The geometry is a circulation control airfoil, a high-lift configuration that has been under extensive research for more than two decades. A tangential jet is blown over a thick, rounded trailing edge, using the Coanda effect to delay separation. An attempt is made to understand, through numerical simulations, the dynamics of turbulent separation and reattachment on the Coanda surface. Highly curved, attached recirculation regions are seen to form. A physics based curvature correction proposed by Pettersson-Reif et al. (1999) is used in conjunction with $\zeta-f$ turbulence model. The chord-based Reynolds number is $Re = 10^{6}$. Two jet momentum coefficients of $C_{\mu}=0.03 $ and $0.1$ are computed. In this paper, comparisons between the computed and experimental pressure distributions, velocity profiles and the position of flow detachment are presented. Comparisons with other closures such as Menter's SST model are also discussed.   

Turbulent Flow Past Spinning Cylinders

Flow past cylinders aligned along their axis where a base freely spins while attached to a non-spinning forebody is considered from a computational and experimental point of view. The time-averaged equations of motion and energy are solved using the modeled form of transport equations for the turbulence kinetic energy and the scalar form of turbulence dissipation with an efficient finite-volume algorithm. 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. A subsonic wind tunnel with a forward-sting mounted spinning cylinder is used for experiments. Experiments are performed for a range of spin rates and free stream flow conditions. The experimental results of Carlucci {\&} Thangam (2001) are used to benchmark flow over spinning cylinders. The data is extended to munitions spinning in the wake of other munitions. Applications involving the design of projectiles are discussed.   

Predicting Law-of-the-Wall with LES: Role of SFS and Surface Stress Models

In previous work we presented a framework in which large-eddy simulation (LES) can be designed to capture the law-of-the-wall (LOTW) in mean velocity gradient when the first grid level is in the inertial layer. A proper combination of grid aspect ratio ($A_{R})$, subfilter-scale (SFS) stress model constant ($C_{t})$, and vertical grid resolution is required to suppress a spurious frictional length scale that underlies deviations from LOTW scaling. This occurs when the LES exceeds critical values of: (1) ratio of mean resolved to SFS shear stress ($R)$ at the first grid level, (2) an ``LES Reynolds number'' (\textit{Re}$_{LES})$, and (3) vertical resolution---the ``High Accuracy Zone'' (HAZ) on a plot of $R$ vs. \textit{Re}$_{LES}$. Here we demonstrate this framework for 2 eddy viscosity and one non eddy viscosity SFS models, and we show that for the eddy viscosity models both $R$ and \textit{Re}$_{LES}$ are inversely proportional to $D_t =C_t^a A_R^b $, where $a$ and $b$ are model-dependent constants. Commonly applied surface shear stress models create a spurious sink in velocity variance and oscillations in mean velocity gradient at the surface. Correcting the spurious sink both reduces these oscillations and increases the predicted value of the Von K\'{a}rm\'{a}n constant (VKC). We also show that $C_t $ and $A_R $ must be chosen within certain bounds and that the VKC predicted by LES asymptotes to a value of about 0.37 within these bounds when the LES is within the HAZ.