Session M08: Boundary Layers: Turbulent Boundary Layers II

Reynolds stress scaling in the near-wall region of wall-bounded flows

A new scaling is derived that yields a Reynolds number independent profile for all components of the Reynolds stress in the near-wall region of wall bounded flows, including channel, pipe and boundary layer flows.  The scaling demonstrates the important role played by the wall shear stress fluctuations and how the large eddies determine the Reynolds number dependence of the near-wall turbulence behavior.   

Tracking streaks in the buffer layer of wall-bounded turbulence

Among the many organized structures observed in near-wall turbulent flows, streaks, defined as regions of slowly moving fluid elongated in the direction of the mean flow, are considered to be of major importance for their role in the regeneration of turbulent energy. Here, we identify and track individual streaks in time using time-resolved direct numerical simulation data of a low Reynolds number channel flow. The analysis of the streaks shows that there is a clear distinction between wall-attached and detached streaks and that the former can be further categorized into streaks that are contained in the buffer layer and the ones that reach the outer region. The results reveal that streaks are born in the buffer layer, coalescing with each other to create larger streaks that are still attached to the wall. These tall-attached streaks eventually split into wall-attached and wall-detached components. The wall-detached component not only has a larger wall-normal velocity compared to its wall-attached counterpart, but it also has a larger (less negative) streamwise velocity, reminiscent of ejections or burst events.   

Wall-pressure fluctuations in a turbulent boundary layer: a comparison of two LES models

The fluctuating wall-pressure on a zero-pressure-gradient flat-plate turbulent boundary layer (BL) was numerically investigated using a large eddy simulation (LES). The pressure fluctuations generate vibrations and noise, which are of interest in many applications involving fluid-structure interactions. In this work, the pressure frequency-wavenumber spectrum generated by two LES subgrid-scale (SGS) closures, namely: the constant-coefficient Smagorinsky-Lilly model and the stretched-vortex model, were compared to empirical, experimental, and simulated data.

The two LES closure models were first validated with results from literature. The stretched-vortex model showed good agreement in the first and second ordered statistics compared to existing data. The constant-coefficient Smagorinsky model followed the same trend when compared to data from literature. The wall-pressure fluctuations were assessed in terms of the frequency-wavenumber spectrum. Both models showed a clear convective region, which compared well with the empirical models. However, in the subconvective region, the constant-coefficient Smagorinsky model diverged from the empirical models, while the stretched-vortex model captured the low wavenumber region.   

Near-wall patch wall-model for large eddy simulation

Wall-modeled large-eddy simulation (WMLES) requires predicting the near-wall stress, which is imposed as a boundary condition at the wall. Most wall models employ simplified formulations of the Reynolds-averaged Navier-Stokes equations (RANS) near the solid boundary, further supplemented with empirical closures. RANS-based wall models rely on averaged flow quantities without utilizing the available information about the structure of wall-bounded turbulent flows. We present a wall model based on a near-wall patch simulation that accurately resolves the near-wall flow structure. The patch’s domain size scales in inner units such that its cost is Reynolds number independent. The patch’s top boundary condition is derived from statistical constraints extracted from the LES and structural rescaling of its velocity field. The resulting wall-shear stress predicted by the patch is applied as the boundary condition to the LES, closing the system without utilizing a-priori fitted coefficients. Additionally, the patch provides predictions beyond the wall-shear stress and the subgrid mean velocity profile, such as velocity and pressure fluctuations below the first LES grid cell. The new wall model system is tested in various flow environments.   

A Lagrangian relaxation towards equilibrium wall model for large eddy simulation

A large eddy simulation wall model is developed based on a formal interpretation of quasi-equilibrium that governs the momentum balance integrated in the wall-normal direction. The model substitutes the law-of-the-wall velocity profile for a smooth surface into the wall-normal integrated momentum balance, leading to a Lagrangian relaxation towards equilibrium (LaRTE) transport equation for the friction velocity vector. This PDE includes a relaxation timescale governing the rate at which the wall stress can respond to imposed fluctuations due to the inertia of the fluid layer from the wall to the wall-model height. A-priori tests based on channel flow direct numerical simulation (DNS) data show that the identified relaxation timescale ensures self-consistency with assumed quasi-equilibrium conditions. The new approach enables us to formally distinguish quasi-equilibrium from additional, non-equilibrium contributions to the wall stress. For the latter, an additional model is derived motivated by laminar Stokes layer dynamics in the viscous sublayer. The wall model is first tested in standard equilibrium channel flow to document its properties and then is tested for various unsteady non-equilibrium flows.   

Reduced-order Modeling of Laminar Boundary Layers

In external aerodynamic applications such as flow over an airfoil, the number of computational volumes

in the laminar and transitional region can be greater than that of the turbulent region by up to two

orders of magnitude in a wall-modeled large-eddy simulation (WMLES) (Slotnick, et al., 2014). The high

computational cost of this region is a barrier to the application of WMLES to transitional flows of

engineering interest, such as the flow near the leading edge of wings. Prior work (Gonzalez, et al., 2020)

implemented a reduced order model for laminar boundary layers based on the Falkner-Skan similarity

solution. This model was then tested in stagnation flow and a spatially varying pressure gradient

boundary layer and the results were in good agreement with the Navier Stokes solutions. In this

investigation, we present a further assessment of the model when applied to a NACA0012 airfoil. We

evaluate the accuracy of engineering quantities of interest, such as wall-stress, lift, and drag. In addition,

the model is integrated with the parabolized stability equations in order to identify the onset of

transition.   

Implementation of Integral wall model for LES in an unstructured-grid finite-volume flow solver and its application to non-equilibrium turbulent boundary layer.

Integral wall model (IWM) for LES developed by Yang et al. (Phys. Fluids 27, 025112 (2015)) has been shown to incorporate more near-wall physics than equilibrium wall model (EQWM) while being algebraic. However, widespread adoption of IWM warrants two key extensions to the original study. First, while IWM has been used largely in structured-grid framework to date, implementation in flow solvers that can handle complex geometries and unstructured grids is lacking, restricting the model’s potential for practical problems. Second, the model needs to be evaluated in flows with strong nonequilibrium effects. In this talk, we first discuss challenges unique to IWM’s implementation in the unstructured-grid framework associated with data exchange and wall-tangential gradient calculations. Cost of wall modeling will be discussed in comparison to EQWM. For assessment in nonequilibrium flow, we consider two experiments at moderately high Re_θ (~3000-7000): a TBL successively subjected to varying pressure gradient (ZPG, FPG and APG) zones (J. Fluid Mech. (2020), vol. 897, A2.), and a spatially developing 3DTBL subjected to a 30^o bend (J. Fluid Mech. (1994), vol. 212, pp. 183-209). Mean flow statistics, boundary layer integral quantities and wall stress predictions from WMLES will be analyzed.   

Wall model for LES based on building-block flows

A wall model for large-eddy simulation is proposed by devising the flow as a collection of building blocks, whose information enables the prediction of the stress as the wall.  The core assumption of the model is that simple canonical flows (such as turbulent channel flows, boundary layers, pipes, ducts, speed bumps, etc) contain the essential flow physics to devise accurate models. Three types of building block units are used to train the model, namely, turbulent channel flows, turbulent ducts, and turbulent boundary layers with separation.  The approach is implemented using two interconnected artificial neural networks: a classifier, which identifies the contribution of each building block in the flow; and a predictor, which estimates the wall stress via non-linear combinations of building-block units.  The output of the model is accompanied by the confidence in the prediction. The latter aids the detection of areas where the model underperforms, such as flow regions that are not representative of the building blocks used to train the model. The model is validated in a realistic aircraft geometry from NASA Juncture Flow Experiment, which is representative of external aerodynamic applications with trailing-edge separation.   

Numerical experiments with slip wall boundary conditions in large-eddy simulation

Traditional wall models for large-eddy simulation (LES) use Reynolds-averaged closure models to provide wall-stress boundary conditions to the LES equations. The slip wall boundary condition is derived directly from the LES constitutive equations, and allows for the development of dynamic wall models free from RANS phenomenological modeling (Bose & Moin, PoF, 2014). Recent wall-modeled LES calculations with slip wall boundary conditions have reported sensitivity to mesh resolution, subgrid-scale model, and numerical discretization (Bae et al., JFM, 2019).

In this study numerical experiments are performed with slip wall models in high-Reynolds number channels and flow over a Gaussian bump with separation (Williams et al., 2020). We investigate effects of mesh resolution, transpiration control, and subgrid-scale model wall closure. The RANS-based equilibrium wall model provides reasonable predictions of separation at coarser resolutions, but fails to separate at finer resolutions until a wall-resolved LES limit is reached. In contrast, the slip boundary condition is shown to predict the separation across the mesh resolutions considered.   

Error scaling of wall-modeled large-eddy simulation of compressible wall turbulence

Error scaling properties of large-eddy simulation of compressible wall-bounded turbulent flows are characterized. The statistical quantities of interest investigated are the mean velocity profile, wall stress, and wall heat transfer. The errors scale as (Δ/L)^α Re_τ^γ M^β, where Δ is the characteristic grid resolution, Re_τ is the friction Reynolds number, L is the meaningful length-scale to normalize Δ in order to collapse the errors across the wall-normal distance, and M is the Mach number to account for compressibility effects in the channel flow. Different length-scales are used in determining L such that errors along the wall-normal distance are effectively collapsed. The exponents α, γ, and β are estimated using theoretical analysis and validated through numerical simulations.