Stretched-vortex Subgrid-scale Model Research Articles

Assessing Stretched-Vortex Subgrid-Scale Models in Finite Volume Methods for Unbounded Turbulent Flows

Simulations of complex, compressible, high-Reynolds-number flows require high-fidelity physics and turbulence models to be appropriately coupled with strong numerical regularization methods. Obtaining grid-independent and scheme-independent solutions of these flows when using both explicit turbulence models and additional numerical regularization is especially important for further testing and development of accurate physics models. To this end, the current study investigates the interaction between the stretched-vortex subgrid-scale model and both the fourth-order piecewise parabolic limiter and a fifth-order upwinding interpolation (or hyperviscosity). It is demonstrated that computing the subgrid-scale kinetic energy estimate for the stretched-vortex model at a coarser resolution than the base mesh provides results which are independent of the use of numerical regularization techniques. This is shown to be the case for a temporally-evolving shear-layer, the inviscid Taylor–Green vortex problem, and a decaying, homogeneous turbulent flow.

Large-eddy simulation and modelling of Taylor–Couette flow

Wall-resolved large-eddy simulations (LES) of the incompressible Navier–Stokes equations together with empirical modelling for turbulent Taylor–Couette (TC) flow are presented. LES were performed with the inner cylinder rotating at angular velocity Ω_i and the outer cylinder stationary. With R_i, R_₀ the inner and outer radii respectively, the radius ratio is η = 0.909 . The subgrid-scale stresses are represented using the stretched-vortex subgrid-scale model while the flow is resolved close to the wall. LES is implemented in the range Re_i = 10⁵-10⁶ where Re_i = Ω_iR_id/v and d = R₀-R_i is the cylinder gap. It is shown that the LES can capture the salient features of the flow, including the quantitative behaviour of spanwise Taylor rolls, the log variation in the inner-cylinder mean-velocity profile and the angular momentum redistribution due to the presence of Taylor rolls. A simple empirical model is developed for the turbulent, TC flow for both a stationary outer cylinder and also for co-rotating cylinders. This consists of near-wall, log-like turbulent wall layers separated by an annulus of constant angular momentum. Model results include the Nusselt number Nu (torque required to maintain the flow) and measures of the wall-layer thickness as functions of both the Taylor number Ta and η. These are compared with results from measurement, direct numerical simulation and the LES. A model extension to rough-wall turbulent flow is described. This shows an asymptotic, fully rough-wall state where the torque is independent of Re_i/Ta, and where Nu ~ Ta^(1/2).

Wall-modelled large-eddy simulation of turbulent flow past airfoils

We present large-eddy simulation (LES) of flow past different airfoils with $Re_{c}$, based on the free-stream velocity and airfoil chord length, ranging from $10^{4}$ to $2.1\times 10^{6}$. To avoid the challenging resolution requirements of the near-wall region, we develop a virtual wall model in generalized curvilinear coordinates and incorporate the non-equilibrium effects via proper treatment of the momentum equations. It is demonstrated that the wall model dynamically captures the instantaneous skin-friction vector field on arbitrary curved surfaces at the resolved scale. By combining the present wall model with the stretched-vortex subgrid-scale model, we apply the wall-modelled LES approach to three different airfoil cases, spanning different geometrical parameters, different attack angles and low to high $Re_{c}$. The numerical results are verified with direct numerical simulation (DNS) at low $Re_{c}$, and validated with experiment data at higher $Re_{c}$, including typical aerodynamic properties such as pressure coefficient distributions, velocity components and also more challenging measurements such as skin-friction coefficient and Reynolds stresses. All comparisons show reasonable agreement, providing a measure of validity that enables us to further probe simulation results into aspects of flow physics that are not available from experiments. Two techniques to quantify hitherto unexplored physics of flows past airfoils are employed: one is the construction of the anisotropy invariant map, and the second is skin-friction portraits with emphasis on flow transition and unsteady separation along the airfoil surface. The anisotropy maps for all three $Re_{c}$ cases, show clearly that a portion of the flow field is aligned along the axisymmetric expansion line, corresponding to the turbulent boundary layer log-law behaviour and the appearance of turbulent transition. The instantaneous skin-friction portraits reveal a monotonic shrinking of the near wall structure scale. At $Re_{c}=10^{4}$, the interaction between the primary separation bubble and the secondary separation bubble contributes to turbulent transition, similar to the case of flow past a cylinder. At higher $Re_{c}=10^{5}$, the primary separation breaks into several small separation bubbles. At even higher $Re_{c}=2.1\times 10^{6}$, near the turbulent separation, the skin-friction lines show small-scale reversal flows that are similar to those observed in DNS of the flat plate turbulent separation. A notable feature of turbulent separation in flow past an airfoil is the appearance of turbulence structures and small-scale reversal flows in the spanwise direction due to the vortex shedding behaviour.

Large-eddy simulation of flow over a grooved cylinder up to transcritical Reynolds numbers

We report wall-resolved large-eddy simulation (LES) of flow over a grooved cylinder up to the transcritical regime. The stretched-vortex subgrid-scale model is embedded in a general fourth-order finite-difference code discretization on a curvilinear mesh. In the present study $32$ grooves are equally distributed around the circumference of the cylinder, each of sinusoidal shape with height $\unicode[STIX]{x1D716}$, invariant in the spanwise direction. Based on the two parameters, $\unicode[STIX]{x1D716}/D$ and the Reynolds number $Re_{D}=U_{\infty }D/\unicode[STIX]{x1D708}$ where $U_{\infty }$ is the free-stream velocity, $D$ the diameter of the cylinder and $\unicode[STIX]{x1D708}$ the kinematic viscosity, two main sets of simulations are described. The first set varies $\unicode[STIX]{x1D716}/D$ from $0$ to $1/32$ while fixing $Re_{D}=3.9\times 10^{3}$. We study the flow deviation from the smooth-cylinder case, with emphasis on several important statistics such as the length of the mean-flow recirculation bubble $L_{B}$, the pressure coefficient $C_{p}$, the skin-friction coefficient $C_{f\unicode[STIX]{x1D703}}$ and the non-dimensional pressure gradient parameter $\unicode[STIX]{x1D6FD}$. It is found that, with increasing $\unicode[STIX]{x1D716}/D$ at fixed $Re_{D}$, some properties of the mean flow behave somewhat similarly to changes in the smooth-cylinder flow when $Re_{D}$ is increased. This includes shrinking $L_{B}$ and nearly constant minimum pressure coefficient. In contrast, while the non-dimensional pressure gradient parameter $\unicode[STIX]{x1D6FD}$ remains nearly constant for the front part of the smooth cylinder flow, $\unicode[STIX]{x1D6FD}$ shows an oscillatory variation for the grooved-cylinder case. The second main set of LES varies $Re_{D}$ from $3.9\times 10^{3}$ to $6\times 10^{4}$ with fixed $\unicode[STIX]{x1D716}/D=1/32$. It is found that this $Re_{D}$ range spans the subcritical and supercritical regimes and reaches the beginning of the transcritical flow regime. Mean-flow properties are diagnosed and compared with available experimental data including $C_{p}$ and the drag coefficient $C_{D}$. The timewise variation of the lift and drag coefficients are also studied to elucidate the transition among three regimes. Instantaneous images of the surface, skin-friction vector field and also of the three-dimensional Q-criterion field are utilized to further understand the dynamics of the near-surface flow structures and vortex shedding. Comparison of the grooved-cylinder flow with the equivalent flow over a smooth-wall cylinder shows structural similarities but significant differences. Both flows exhibit a clear common signature, which is the formation of mean-flow secondary separation bubbles that transform to other local flow features upstream of the main separation region (prior separation bubbles) as $Re_{D}$ is increased through the respective drag crises. Based on these similarities it is hypothesized that the drag crises known to occur for flow past a cylinder with different surface topographies is the result of a change in the global flow state generated by an interaction of primary flow separation with secondary flow recirculating motions that manifest as a mean-flow secondary bubble. For the smooth-wall flow this is accompanied by local boundary-layer flow transition to turbulence and a strong drag crisis, while for the grooved-cylinder case the flow remains laminar but unsteady through its drag crisis and into the early transcritical flow range.

Large-Eddy Simulation of Stratified Turbulence. Part II: Application of the Stretched-Vortex Model to the Atmospheric Boundary Layer

Abstract The buoyancy-adjusted stretched-vortex subgrid-scale (SGS) model is assessed for a number of large-eddy simulations (LESs) corresponding to diverse atmospheric boundary layer conditions. The cases considered are free convection, a moderately stable boundary layer [first Global Energy and Water Exchanges (GEWEX) Atmospheric Boundary Layer Study (GABLS)] case, shallow cumulus [Barbados Oceanographic and Meteorological Experiment (BOMEX)], shallow precipitating cumulus [Rain in Cumulus over the Ocean (RICO)] and nocturnal stratocumulus [Second Dynamics and Chemistry of the Marine Stratocumulus (DYCOMS-II) field study RF01]. An identical LES setup, including advection discretization and SGS model parameters, is used for all cases, which is a stringent test on the ability of LES to accurately capture diverse conditions without any flow-adjustable parameters. The LES predictions agree well with observations and previously reported model results. A grid-resolution convergence study is carried out, and for all cases the mean profiles exhibit good grid-resolution independence, even for resolutions that are typically considered coarse. Second-order statistics, for example, variances, converge at finer resolutions compared to domain means. The simulations show that 90% of the turbulent kinetic energy (at each level) must be resolved to obtain sufficiently converged mean profiles. This empirical convergence criterion can be used as a guide in designing future LES runs.

Large-Eddy Simulation of Stratified Turbulence. Part I: A Vortex-Based Subgrid-Scale Model

Abstract The stretched-vortex subgrid-scale (SGS) model is extended to enable large-eddy simulation of buoyancy-stratified turbulence. Both stable and unstable stratifications are considered. The extended model retains the anisotropic form of the original stretched-vortex model, but the SGS kinetic energy and the characteristic SGS eddy size are modified by buoyancy subject to two constraints: first, the SGS kinetic energy dynamics is determined by stationary and homogeneous conditions, and second, the SGS eddy size obeys a scaling analogous to the Monin–Obukhov similarity theory. The SGS model construction, comprising an ensemble of subgrid stretched-vortical structures, naturally limits vertical mixing but allows horizontal mixing provided the alignment of the SGS vortex ensemble is favorable, even at high nominal gradient Richardson numbers. In very stable stratification, the model recovers the z-less limit, in which a vortex-based Obukhov length controls the SGS dynamics, while in very unstable stratification, the model recovers the free-convection limit, in which a vortex-based Deardorff velocity controls the SGS dynamics. The efficacy of the present SGS model is demonstrated by simulating the canonical stationary and homogeneous, stratified sheared turbulence at high Reynolds numbers and moderately high Richardson numbers. In the postprocessing, the SGS dynamics of the stretched-vortex model is further interrogated to yield predictions of buoyancy-adjusted one-dimensional SGS spectra and SGS root-mean-square velocity-derivative fluctuations.

LES of the adverse-pressure gradient turbulent boundary layer

We describe large-eddy simulations (LES) of the flat-plate turbulent boundary layer in the presence of an adverse pressure gradient. The stretched-vortex subgrid-scale model is used in the domain of the flow coupled to a wall model that explicitly accounts for the presence of a finite pressure gradient. The LES are designed to match recent experiments conducted at the University of Melbourne wind tunnel where a plate section with zero pressure gradient is followed by section with constant adverse pressure gradient. First, LES are described at Reynolds numbers based on the local free-stream velocity and the local momentum thickness in the range 6560–13,900 chosen to match the experimental conditions. This is followed by a discussion of further LES at Reynolds numbers at approximately 10 times and 100 times these values, which are well out of range of present day direct numerical simulation and wall-resolved LES. For the lower Reynolds number runs, mean velocity profiles, one-point turbulent statistics of the velocity fluctuations, skin friction and the Clauser and acceleration parameters along the streamwise, adverse pressure-gradient domain are compared to the experimental measurements. For the full range of LES, the relationship of the skin-friction coefficient, in the form of the ratio of the local free-stream velocity to the local friction velocity, to both Reynolds number and the Clauser parameter is explored. At large Reynolds numbers, a region of collapse is found that is well described by a simple log-like empirical relationship over two orders of magnitude. This is expected to be useful for constant adverse-pressure gradient flows. It is concluded that the present adverse pressure gradient boundary layers are far from an equilibrium state.

Assessment of stretched vortex subgrid-scale models for LES of incompressible inhomogeneous turbulent flow

The physical space version of the stretched vortex subgrid scale model [Phys. Fluids 12, 1810 (2000)] is tested in large eddy simulations (LES) of the turbulent lid driven cubic cavity flow. LES is carried out using a higher order finite-difference method [J. Comput. Phys. 229, 8802 (2010)]. The effects of different vortex orientation models and subgrid turbulence spectrums are assessed through comparisons of the LES predictions against direct numerical simulations (DNS) [Phys. Fluids 12, 1363 (2000)]. Three Reynolds numbers 12000, 18000, and 22000 are studied. Good agreement with the DNS data for the mean and fluctuating quantities is observed.

Direct numerical simulation and large-eddy simulation of stationary buoyancy-driven turbulence

We report direct numerical simulation (DNS) and large-eddy simulation (LES) of statistically stationary buoyancy-driven turbulent mixing of an active scalar. We use an adaptation of the fringe-region technique, which continually supplies the flow with unmixed fluids at two opposite faces of a triply periodic domain in the presence of gravity, effectively maintaining an unstably stratified, but statistically stationary flow. We also develop a new method to solve the governing equations, based on the Helmholtz–Hodge decomposition, that guarantees discrete mass conservation regardless of iteration errors. Whilst some statistics were found to be sensitive to the computational box size, we show, from inner-scaled planar spectra, that the small scales exhibit similarity independent of Reynolds number, density ratio and aspect ratio. We also perform LES of the present flow using the stretched-vortex subgrid-scale (SGS) model. The utility of an SGS scalar flux closure for passive scalars is demonstrated in the present active-scalar, stably stratified flow setting. The multi-scale character of the stretched-vortex SGS model is shown to enable extension of some second-order statistics to subgrid scales. Comparisons with DNS velocity spectra and velocity-density cospectra show that both the resolved-scale and SGS-extended components of the LES spectra accurately capture important features of the DNS spectra, including small-scale anisotropy and the shape of the viscous roll-off.

LES approach for high Reynolds number wall-bounded flows with application to turbulent channel flow

We describe a large-eddy simulation approach for turbulent channel flow using the stretched-vortex subgrid-scale model. The inner region of the turbulent boundary layer is not included in the modeling of this attached, wall-bounded flow. Appropriate boundary conditions and closure are derived using a combination of elements from asymptotic expansions, matching, and well-established wall-modeling approaches. The modeling approach for this application combines the stretched-vortex subgrid model with a localized wall-shear-stress treatment that relates the instantaneous wall-parallel velocity to the shear stress via the log-law, as appropriate for this (near-) zero pressure gradient flow. The impermeability boundary condition is built into the method such that only the outer-flow solution is simulated, obviating the need to impose the stiff no-slip condition at the wall. This formulation attempts to minimize numerical and modeling errors introduced by the boundary-condition treatment, while preserving the fundamental elements required to predict low-order statistics of these flows. We present simulation results for turbulent channel flow up to Reynolds number based on the wall-friction velocity of 10 6 . These compare favorably with results from large-scale DNS and experimental correlations.