Session E09: Boundary Layers: Turbulent Boundary Layers Wall Modeling (3:10pm - 3:55pm CST)

A relaxation wall model for large eddy simulations of turbulent flows

The equilibrium wall model (EQWM) is the simplest and most widely adopted wall model in large eddy simulations (LES). The standard EQWM utilizes a known velocity profile, observed in a statistically stationary configuration, to relate the wall stress to the LES velocity. Even in non-equilibrium conditions the EQWM performs remarkably well since LES includes non-equilibrium effects outside the wall model layer. However, in rapidly changing conditions, near-wall changes are not captured by EQWM-LES. In this scenario, it appears conceptually important to be able to separate equilibrium and non-equilibrium effects to model each separately. We posit that the wall stress determined by the assumed velocity profile in EQWM can only respond within a relaxation timescale to changes in the LES velocity. From the integrated boundary layer equations, we derive an evolution equation for the wall stress which relaxes to the equilibrium value at the derived timescale. This ``relaxation wall model’’ (ReWM) accounts for the effectively time-filtered response of the wall stress due to the viscous layer between the LES velocity and the wall. The model thus enables us to properly separate the quasi-equilibrium response captured by EQWM and the non-equilibrium parts requiring more advanced modeling.   

Wall-modeled LES of a turbulent thermal boundary layer with a non-equilibrium behavior

The studied configuration features a turbulent channel flow (Re\textunderscore \textunderscore $=$ 395) that is fully established and adiabatic in the first part of the domain and then encounters an isothermal wall. The evolution of the thermal boundary layer has previously been studied with DNS: initially at equilibrium, it is perturbed by the abrupt change of boundary conditions, and a non-equilibrium transient phase is observed downstream. The present study aims at characterizing the ability of wall-modeled large-eddy simulations to predict the boundary layer spatial evolution and the associated wall heat flux. The modeling strategy first relies on describing the unresolved inner layer by 1D equilibrium equations that are solved numerically at each wall face. The method is first validated in equilibrium channel flows before being applied to the target configuration. Results are analyzed and compared to the DNS reference results. Finally, the wall model is extended by taking into account non-equilibrium terms identified by the DNS analysis to improve the predictions.   

Assessing the equilibrium wall model for Low-speed Flows with Heat Transfer

We report wall-modeled large-eddy simulation (WMLES) results of low-speed turbulent flows in plane channel and in ribbed ducts. We compare our WMLESs to Pirozzoli's direct numerical simulations (DNSs) of low-speed plane channel flow and our own DNSs of ribbed ducts with various pitch to height ratios. We consider Mach number effects below the often quoted low Mach number limit Ma$=$0.2. The results show that Mach number has significant effects on the normalized mean temperature profile even below the often quoted low Mach number limit Ma $=$ 0.2. In addition, we compare the first-point implementation (FGI) and the third-point implementation (TGI) of the equilibrium wall model. The results show that, for WMLES with its typical resolution, TGI is likely to miss the thermal field. The objective of this study is to systematically assess WMLES in terms of its ability to predict heat transfer for low-speed flows. For the flows considered here, i.e., plane channel and ribbed duct, we show that WMLES with FGI is able to accurately model heat transfer at a much reduced cost than WRLES and DNS.   

A new ODE-based wall model for boundary layers accounting for pressure gradient and Re effects

In wall-modeled large-eddy simulations (WMLES), the near wall model plays a significant role in predicting the skin friction, although the majority of the boundary layer is resolved by LES. In this work, we propose a new ODE-based wall model for boundary layers accounting for pressure gradient and Reynolds number effects. The new model sensitizes the law of the wall to the boundary layer shape factor. By consulting the local outer profile, the inner wall model can incorporate the non-equilibrium effects captured by the outer LES solver as well as the non-universality of the law of the wall at low Reynolds number. As a result, the proposed wall model greatly extends the predictive capability of WMLES for flows with strong pressure gradients and a wide range of Reynolds numbers . Specifically, the wall model improves the skin friction prediction in simulations of canonical favorable pressure gradient flows (pipes and channels), zero and adverse pressure gradient flat plates, and the flow over a NACA 4412 airfoil at an angle of attack of 5 degrees and Re\textunderscore c in the range of 100,000 to 1,000,000. The suite of test cases exhibits a range of friction Reynolds numbers between 130 and 8000 and Clauser pressure gradient parameters in the range of -0.3 and 4.   

A Bayesian approach to flow in a channel with small but arbitrarily directional system rotation

We show dimensional analysis that when a boundary-layer flow is subjected to small system rotation, the constant stress layer survives, and the mean flow $U^+$ is a function only of $y^+$, $\Omega_{x}^+$, $\Omega_{y}^+$, and $\Omega_{z}^+$, where $U$ is the mean flow, $y$ is the distance from the wall, $\Omega_i$ is the system rotation speed in the $i$th direction. Determining the mean flow behavior $U^+(y^+, \Omega_{x}^+, \Omega_{x}^+,\Omega_{z}^+)$ is non-trivial, and taking an analytical approach incurs large errors. Here, we pursue a Bayesian approach, where we survey the three dimensional parameter space of $\Omega_x^+$, $\Omega_y^+$, $\Omega_z^+$ via direct numerical simulation. Surveying a parameter space for knowledge of a flow quantity is conventionally considered to be a ``brutal force'' approach. However, because a Bayesian surrogate gives not only a prediction of the objective function but also an estimate of the prediction's uncertainty level, we are able to very efficiently sample the parameter space and quickly obtain an accurate surrogate of $U^+$. Four independent surveys are conducted with 146 DNSs in total. Validating a surrogate with the data in other surveys, we show that the the Bayesian approach yields accurate estimates of $U^+$.   

Wall-modeled large-eddy simulation of turbulent boundary layers with mean-flow three-dimensionality

We examine the performance of wall-modeled LES (WMLES) to predict turbulent boundary layers (TBLs) with mean-flow three-dimensionality. The analysis is performed for an ordinary-differential-equation-based equilibrium wall model (EQWM) due to its widespread use and ease of implementation. Two test cases are considered: a spatially-developing TBL in a square duct with a 30-degree bend, following the experiment of Schwarz & Bradshaw (JFM, 1994), and the flow behind a wall-mounted skewed bump with a 3-D separation bubble, following the experimental study by Ching, Elkins & Eaton (Exp. in Fluids, 2018). In the duct simulation, WMLES predicts mean velocity profiles and crossflow angles in the outer region of the flow to within 1--5\% accuracy using 10 points per boundary layer thickness. The largest disagreement is observed in the crossflow angles in the bend region, where 3-D effects are the most significant. In the bump simulation, the EQWM with a grid resolution of 40 points across the 3-D separation region predicts mean velocity profiles, separation location to within 1--3\% accuracy. The bubble size and vortex structures in the wake are also well predicted. It is demonstrated that capturing the shear layer at the apex of the bump is key for achieving accurate results.   

An efficient implementation of the ODE equilibrium wall model using Gauss quadrature method

Owing to its ease of implementation and reasonable accuracy at a moderate cost, the ODE equilibrium wall model has been popular for the computation of flows with complex wall geometries. The model implementation typically employs a finite-volume discretization, which entails the solution of a tridiagonal system on each wall face at each time step. Frequent inversion of these linear systems is the most expensive part of this wall-modeling approach. To this end, we develop a low-cost grid-free implementation for the ODE wall model based on Gauss-quadrature. The method is based on the integral form of velocity profile obtained from the constant-stress layer statement. The wall stress is then found iteratively using the shooting method and the spectral evaluation of the velocity integral using the Gauss-Lobatto-Legendre quadrature method. A priori validation of the model has been conducted using available data for the turbulent channel, pipe and boundary layer flows, for Reynolds number up to $Re_{\tau}\sim10^{5}$. The costs of the finite-volume and the Gauss-quadrature approach will be contrasted. Additionally, the Reynolds number dependence of the wall-modeling cost and the number of quadrature points required for a fixed accuracy of the predicted wall-stress will be investigated.   

Wall-stress Modeling for Laminar Boundary Layers

As described in NASA's CFD Vision 2030 report (Slotnick, et al. 2014), for external aerodynamic applications such as flow over an airfoil, the number of computational volumes in the laminar and transitional region can exceed that of the turbulent region by up to two orders of magnitude (100 times) in a wall-modeled large eddy simulation (WMLES). The associated high computational cost is a key bottleneck in the application of such reduced-order models. The goal of this study is to develop a wall model capable of treating the laminar region without the need to fully resolve it. We demonstrate that for a stagnation flow, the semi-analytical wall stress, derived from the Hiemenz flow similarity solution, can be used as a wall-model for coarse laminar simulations. The wall model is then extended to flows over a flat plate with a spatially varying edge pressure gradient by considering the family of Falkner-Skan similarity solutions. The local behavior of the pressure gradient is used to select the appropriate Falkner-Skan wall stress. The similarity-solution based wall model is applied using the unstructured charLES solver. The wall stress of both cases is well predicted using the new wall model.   

Mean flow and turbulence characteristics of a spatially-developing pressure-driven 3D turbulent boundary layer

Statistically three-dimensional turbulent boundary layers (TBLs) are found commonly in nature and engineering applications. We conduct WMLES of a thin 3DTBL developing on the floor of a bent square duct to study the mean flow and turbulence characteristics in the outer portion of pressure-driven 3DTBLs. The simulations agree reasonably well with the experiment by Schwarz \& Bradshow (\textit{J. Fluid Mech.} (1994), vol. 272, pp. 183–210). The inviscid skewing mechanism which generates the mean three dimensionality in the outer part of the boundary layer is discussed based on the vorticity equation and the Johnston triangular plot. These characteristics are shown to be not found in the shear-driven 3D channel flow (Lozano-Durán \textit{et al.} \textit{J. Fluid Mech.} (2020), vol. 883, pp. A20). The anisotropy of turbulence are discussed using the Lumley triangle. In contrast to canonical 2D wall turbulence, the 3DTBL has a non-monotonic increase of anisotropy in the log layer, which corresponds to a sharp corner in the Lumley triangle.   

Examination of Machine Learning for the Modeling of Hypersonic Boundary Layers

There is a great need for accurate, applicable, and grid-flexible wall models for hypersonic flows. This work examines the use of machine learning for constructing wall models that are targeted to produce a minimum loss of accuracy on coarse-grid simulations when compared to fine-grid calculations. Specifically, we use machine learning to reconstruct the velocity gradient on the wall in the wall-normal direction by introducing a velocity gradient correction factor. This compensates for the loss of accuracy when using finite volume discretization to find the velocity gradient in the highly non-linear region for coarse grids. The examination of machine learning is done in the Reynolds-averaged Navier-Stokes simulations (RANS) of hypersonic boundary layers. Fine-grid RANS simulations are conducted to generate training data for machine learning. The Random forest bagged algorithm is used to model the velocity gradient correction factor. Different choices of input parameters are examined. A priori assessment of the model is compared against fine-grid RANS results. It is found that with the machine learning model, the modeled velocity gradient correction factor can reproduce the velocity gradient on the wall improving the relative error from $50\%$ to within six percent.   

Mean-velocity scaling of compressible turbulent boundary layer flows under non-adiabatic wall conditions

The Van Driest transformation, commonly considered to be the state-of-the-art in compressible mean-velocity scaling, fails when applied to flows with wall heat transfer. Recent efforts rectified the failure in channel flows and for moderate mean flow Mach numbers. Yet, the performance in boundary layers and in the supersonic/hypersonic regime remains unsatisfactory. Particularly in the log layer, a sizable discrepancy is observed when using modern mapping techniques to scale the profiles. Using physical arguments and a generalized stress balance derivation, in this talk, we investigate the shortcomings of the existing transformations. We identify the buffer layer as the region where the failure arises and provide a possible means to resolve the dilemma. A comment is also given on the validity of Morkovin's hypothesis and Prandtl's mixing length theory under those extreme conditions.