Session G33: Turbulent Boundary Layers: Walls and Modeling

Extending the restricted nonlinear model for wall-turbulence to high Reynolds numbers

The restricted nonlinear (RNL) model for wall-turbulence is motivated by the long-observed streamwise-coherent structures that play an important role in these flows. The RNL equations, derived by restricting the convective term in the Navier-Stokes equations, provide a computationally efficient approach due to fewer degrees of freedom in the underlying dynamics. Recent simulations of the RNL system have been conducted for turbulent channel flows at low Reynolds numbers (Re), yielding insights into the dynamical mechanisms and statistics of wall-turbulence. Despite the computational advantages of the RNL system, simulations at high Re remain out-of-reach. We present a new Large Eddy Simulation (LES) framework for the RNL system, enabling its use in engineering applications at high Re such as turbulent flows through wind farms. Initial results demonstrate that, as observed at moderate Re, restricting the range of streamwise varying structures present in the simulation (i.e., limiting the band of $x$ Fourier components or $k_x$ modes) significantly affects the accuracy of the statistics. Our results show that only a few well-chosen $k_x$ modes lead to RNL turbulence with accurate statistics, including the mean profile and the well-known inner and outer peaks in energy spectra.   

A Nonlinear System Model of Wall Turbulence Generation Under Active Suppression and Enhancement of Streak Transient Growth Instability

Schoppa and Hussain (1998, 2002) demonstrated streak transient growth (STG) as the dominant streamwise coherent structure generation mechanism required for wall turbulence production. A novel, flush surface-mounted pulsed-DC plasma actuator was recently developed at the University of Notre Dame to actively intervene in STG. In recent high Reynolds number, zero pressure gradient turbulent boundary layer experiments, drag reduction of up to 68{\%} was achieved. This is due to a plasma-induced near-wall, spanwise mean flow sufficient in magnitude to prevent the lift-up of low-speed streaks. This limits their flanking wall-normal component vorticity-a critical parameter in STG. Experiments also show that sufficiently large plasma-induced spanwise flow can exacerbate STG and increase drag by 80{\%}. The ability to significantly increase or decrease drag by near-wall actuation provides an unprecedented new tool for clarifying the open questions regarding the interaction between near-wall coherent structures and those in the logarithmic region. In the reported experiments this interaction is experimentally characterized by a second-order Volterra nonlinear system model under both active suppression and enhancement of STG.   

Comparison two different LES closure models of the transitional boundary layer flow

The goal of the present research is to measure the velocity profile in the thin boundary layer of a flat plate at zero angle of attack. We consider a flow over a flat plate with a uniform velocity profile. The uniform velocity fluid hits the leading edge of the flat plate, and a laminar boundary layer begins to develop. The near-wall, subgrid-scale (SGS) model is used to perform Large Eddy Simulation (LES) of the incompressible developing, smooth--wall, flat--plate turbulent boundary layer. In this model, the stretched--vortex, SGS closure is utilized in conjunction with a tailored, near--wall model designed to incorporate anisotropic vorticity scales in the presence of the wall. The composite SGS-wall model is presently incorporated into a computer code suitable for the LES of developing flat-plate boundary layers. Presently this model is extended to the LES of the zero--pressure gradient, flat-plate turbulent boundary layer. LES solver using Smagorinsky and the One-equation LES turbulence models. Results show that the normalized mean velocity profile is in good agreement with the universal law-of-the-wall and previous published data. In order to ensure the quality of the numerical results a convergence study was performed.   

Application of Wall-modeled LES to Turbulent Separated Flows

Resolved Large-Eddy Simulations (LES) and Direct Numerical Simulations (DNS) are unaffordable for very high Reynolds number ($Re$) wall-bounded flows. While the Reynolds Averaged Navier-Stokes (RANS) based methods predict high $Re$ attached flows accurately with little cost, their fidelity is degraded significantly in flows involving separation. A popular compromise between cost and accuracy is to use a Wall-modeled LES (WMLES) approach. In WMLES, the outer portion of the boundary layer is resolved with LES while the inner portion is modeled. In order to assess the performance of the widely used wall-stress models in separated flows, we perform WMLES simulations using an unstructured, compressible finite volume LES solver. The equilibrium and non-equilibrium wall models that require the solution of the simplified/full RANS on a separate near-wall domain are employed. Two configurations are studied: the shock-induced separation in a transonic flow over an axisymmetric bump placed on a cylinder, and a low-Mach flow past a NACA 4412 airfoil at a near-stall condition. Detailed comparisons will be made with available experimental data to comment on the applicability of WMLES in predicting complex turbulent flows involving separation.   

Wall-modeled large-eddy simulation of transonic airfoil buffet at high Reynolds number

In this study, we conduct the wall-modeled large-eddy simulation (LES) of transonic buffet phenomena over the OAT15A supercritical airfoil at high Reynolds number. The transonic airfoil buffet involves shock-turbulent boundary layer interactions and shock vibration associated with the flow separation downstream of the shock wave. The wall-modeled LES developed by Kawai and Larsson PoF (2012) is tuned on the K supercomputer for high-fidelity simulation. We first show the capability of the present wall-modeled LES on the transonic airfoil buffet phenomena and then investigate the detailed flow physics of unsteadiness of shock waves and separated boundary layer interaction phenomena. We also focus on the sustaining mechanism of the buffet phenomena, including the source of the pressure waves propagated from the trailing edge and the interactions between the shock wave and the generated sound waves.   

Prediction of wall shear-stress fluctuations in wall-modeled large-eddy simulation

Wall-modeled large-eddy simulation (WMLES) is emerging as a viable and affordable tool for predicting mean flow statistics in high Reynolds number turbulent boundary layers. Recently, we examined the performance of two RANS-based wall models in prediction of wall pressure and shear stress fluctuations which are important in flow/structure interaction problems. Whereas the pressure statistics were predicted with reasonable accuracy, the magnitude of wall shear stress fluctuations was severely underestimated (Park \& Moin, \emph{Phys. Rev. Fluids} \textbf{1}, 024404 (2016)). The present study expands on this finding to characterize in more detail the capabilities of wall models for predicting $\tau_w'$. Predictions of several wall models in high Reynolds number channel flows ($Re_\tau$ = 2000) will be presented. Additionally, a recent empirical inner-outer model for $\tau_w'$ (Mathis \emph{et al.}, \emph{J. Fluid Mech.} \textbf{715}:163--180 (2013)) is reconstructed using channel flow DNS database , and it is coupled to WMLES to assess its performance as a predictive model in LES.   

A unified description of spatial and spectral distribution of fluctuation intensities in wall turbulence

The streamwise turbulent intensity in wall turbulence (pipe and boundary layer) presents non-uniform distribution in both physical and wave number space. The well-known Townsend-Perry attached eddy hypothesis divides the energy spectrum into three distinct ranges: a constant range at small wavenumbers k\textless kc, a k-1 law in the "attached eddy" range k\textless ki and the Kolmogorov form k-5/3 in the inertial range k\textless kd. However, the latest boundary layer experiment (Vallikivi et al., J. Fluid Mech., vol. 771, 2015, pp. 303-326) indicates that a more precise spectral model is needed. We present here a unified analytical expression, based on a generalized dilation-invariant ansatz. It will be shown that analytic description of a stress length ell giving rise to accurate description of the mean velocity profile yields equally accurate prediction of the integral scale wavenumber ki, and the predicted dissipation gives rise of good prediction of the Kolmorogov dissipation wavenumber kd. Finally, the large-scale characteristic wavenumber kc follows a simple scaling law in terms of the stress length ell. Furthermore, we find that the Princeton data reveals possible anomalous scaling in the k-1 and k-5/3 range. The spectral curves based on our generalized dilation-invariant ansatz agree very well with the experimental spectrum, and the kinetic energy profile is also accurately reproduced. We have thus achieved, for the first time, a unified description of spatial and spectral distribution of fluctuation intensity from a recently developed symmetry approach.   

Low-complexity stochastic modeling of spatially evolving flows

Low-complexity approximations of the Navier-Stokes (NS) equations are commonly used for analysis and control of turbulent flows. In particular, stochastically-forced linearized models have been successfully employed to capture structural and statistical features observed in experiments and direct simulations. In this work, we utilize stochastically-forced linearized NS equations and their parabolized equivalents to study the dynamics of flow fluctuations in transitional and turbulent boundary layers. We exploit the streamwise causality of the parabolized model to efficiently propagate statistics of stochastic disturbances into statistics of velocity fluctuations. Our study provides insight into interactions of slowly-varying base flow with streamwise streaks, oblique modes, and Tollmien--Schlichting waves. It also offers a systematic, computationally efficient framework for quantifying the influence of stochastic excitation sources (e.g., free-stream turbulence and surface roughness) on velocity fluctuations in weakly non-parallel flows.   

A simple model of inertial layer dynamics in turbulent boundary layers

Observations (e.g. Meinhart \& Adrian, Phys. Fluids., {\bf 7}, 694 (1995)) indicate that the inertial region of turbulent wall-flows consists of uniform momentum zones segregated by narrow vortical fissures. Multiscale analysis similarly reveals that the mean momentum equation admits a scaling layer hierarchy across the inertial region. Here, each layer increases in width with wall-normal distance, but the inner-normalized velocity increment remains fixed. The talk reports on a simple model that captures the essential elements of these observations and the theoretical scalings. In this model, the number of fissures is specified to satisfy the average total velocity increment across the inertial layer, while the average wall-normal locations of the fissures and their widths are informed by the theory. Ensembles of statistically independent instantaneous velocity profiles are then created by simply allowing the fissures to randomly displace in the wall normal direction. Results indicate that the model identically recovers a logarithmic mean profile, produces a logarithmic decay in the streamwise velocity variance, and generates sub-Gaussian behaviours in its skewness and kurtosis profiles on the inertial domain. These findings along with possible refinements are also discussed.