Session L05: Boundary Layers: Pressure Gradients

Structure of the turbulent boundary layer over a grooved bump

Coherent structures in a turbulent boundary layer over a smooth (SW) transverse bump and also with fine-scale longitudinal square grooves (GW) are studied using DNS. The spanwise coherence of the rollers in the shear layer above the separation bubble downstream of the bump peak was explored via a phase-aligned ensemble average of 250 realizations for both SW and GW. In GW, counter-rotating streamwise swirling jets emanating at the same x downstream of the bump peak impinge on the shear layer and increase the spanwise coherence of the downstream shear layer rollup – also decreasing turbulence intensity, Reynolds shear stress and production with respect to SW. Interestingly, wall-attached minibubbles occur within the separation bubble with opposite circulation, adding further to the form-drag of the bump – by 20% over SW. In contrast, in SW, the minibubbles are intermittent, smaller, and occur less frequently, presumably contributing less to the SW form-drag. The minibubbles caused by the separation of the upstream wall flow move downstream due to induction by the image vortices. Zones of time-averaged negative production are explained in terms of the local dynamics. Negative production within the grooves is due to counter-gradient (positive) Reynolds shear stress, while outside the grooves, it is due to Reynolds normal stress and flow acceleration. Our study should explain various flows over rough bumps in nature and technology, including flows over mountains with trees, sand dunes, and subaqueous dunes.    

Data-driven turbulence model for flow separation over the Boeing Gaussian Bump

Existing RANS models have struggled to predict the smooth-body separation on the Boeing Gaussian Bump. We create a new RANS model that is based on a transport equation for the RANS eddy viscosity. We use a neural network to learn the eddy viscosity source terms using training data from DNS and experimental data published in the literature. We further discuss the applicability of this effort to develop models that generalize to predicting flow separation for airfoils at angles of attack near the stall condition.   

Direct numerical simulation of the turbulent boundary layer over the Boeing bump at ReL=2M

The turbulent boundary layer over a two-dimensional Gaussian-shaped bump is computed by direct numerical simulation (DNS) of the incompressible Navier–Stokes equations. At the inflow, the momentum thickness Reynolds number is approximately 1900 and the boundary layer thickness is 1/9 of the bump height. At this Reynolds number, the flow does not experience significant re-laminarization and the resulting flow experiences a large separation soon after the bump peak where the pressure gradient switches from favorable to adverse. A review of the strong pressure gradient flow physics from past presentations will be included, however this talk will focus on the strong adverse pressure gradient and separated regions on the downstream side of the bump. Owing to the slower time scales and the demand for a wider domain width to avoid confinement issues, convergence of statistical quantities of interest has required a substantial increase in computational resources.   

Wall-modeled large-eddy simulations of flow over a Gaussian-shaped bump with a novel sensor-based blended approach

A novel sensor-based blended wall-modelling approach is proposed to investigate the flow over a Gaussian-shaped bump geometry at three different Reynolds numbers: Re_L = 10⁶, 2·10⁶, and 4·10⁶, with a freestream Mach number of M = 0.2. This flow configuration poses challenges for traditional wall-modeling approaches that assume the boundary layer to be fully turbulent, overestimating friction and momentum losses in quasi-laminar flow regions. To address this limitation, we propose several sensors capable of identifying relaminarization regions, taking into account a hysteresis time scale along fluid pathlines, and a blending strategy that combines an equilibrium wall model and a no-slip boundary condition based on the local sensor value. To evaluate the performance, we compare spanwise periodic simulation results with experimental data from Williams et al. (2020, 2021) and Gray et al. (2022), as well as DNS results from Uzun & Malik (2020-2022). Previously developed laminar-to-turbulent transition sensors by Bodart & Larsson (2012) and Mettu & Subbareddy (2018) are also assessed. The predictive capabilities and robustness of the newly proposed approach are evaluated for all three Reynolds numbers. A posteriori analyses suggest that the sensor correctly identifies relaminarization present in the Re_L = 10⁶ case, improving flow predictions.   

Statistics of a turbulent boundary layer with sequential favorable and adverse pressure gradients

A turbulent boundary layer is studied under an imposed favorable and adverse pressure gradient in sequence. A deformable ceiling panel (Parthasarathy & Saxton-Fox, 2022) is used to impose multiple pressure gradients on the boundary layer. This talk will focus on the turbulent statistics and structures in the non-equilibrium statically held pressure gradients. Planar particle image velocimetry and preliminary stereo particle image velocimetry data will be discussed. In previous studies (Parthasarathy & Saxton-Fox, 2023), the formation of an internal boundary layer has been observed under various pressure gradients of this type (FAPGs), however, the process of formation of the internal layer has not been studied. This talk focuses on observing the statistics in the region where the internal layer is formed as well as the preceding FPG region. This talk will also look at the statistics in the spanwise-wall normal plane at the point of neutral pressure gradient between the FPG and the APG. Experiments are being performed at multiple Re_Tau, in order to investigate any Reynolds number dependence, however, this talk will only focus on Re_Tau=1000.   

Large-eddy simulation and analysis of non-equilibrium turbulent boundary layers

Evolution of a turbulent boundary layer (TBL) under streamwise varying pressure gradient conditions is studied using wall-resolved large-eddy simulation (LES). The simulations are based on the experiments of Volino (Journal of Fluid Mechanics 897, A2, 2020), who performed profile measurements at several streamwise stations in an initially zero pressure gradient TBL evolving through a region of favorable pressure gradient, followed by a zero pressure gradient recovery and a subsequent adverse pressure gradient region. The pressure gradient quantified by the acceleration parameter was held constant in each of these three regions. The simulation setup and the computational grid are carefully designed to mimic the experimental conditions without requiring a long development region for the incoming zero pressure gradient TBL. The computational grid is able to resolve the rapid thinning and thickening of the TBL in the favorable and adverse pressure gradient regions respectively. The LES results show good agreement with the experimental data at several stations in the domain. The results are analyzed to understand the behavior of non-equilibrium TBL, which are common in many engineering applications.   

Direct numerical simulation of non-zero pressure gradient turbulent boundary layers

Turbulent boundary layers (TBL) subjected to pressure gradients are of great interest in wall modeling and aerodynamic design considerations. In the present work, direct numerical simulation (DNS) of TBLs with various favorable and adverse pressure gradients at low Reynolds numbers will be carried out. The objective of the study is to analyze and model the non-equilibrium effects of pressure gradients as well as create a comprehensive high-fidelity data set of non-zero pressure gradient TBLs.   

Extension of Thwaites Method to turbulent boundary layers

Thwaites developed an approximate method for predicting the momentum thickness (θ) in laminar boundary layers using a first-order expansion of the Von Karman integral equation. Such a model is lacking for turbulent flows; where the pressure gradient affects the inner and outer flow scales differently. In this work, the Thwaites method is extended for determining the momentum thickness in non-equilibrium turbulent boundary layers using boundary layer edge variables. It is shown that the method provides good estimates of the momentum thickness in comparison to recent high-fidelity simulation datasets for attached non-equilibrium boundary layers and also for boundary layers on the verge of separation in both subsonic and transonic regimes. The results indicate that the proposed linear model can capture the “history effects” associated with pressure gradient on the growth of the boundary layer.

We perform adjoint analysis using the ODE model to study the sensitivity of turbulent flow separation to upstream perturbations. Our analysis also reveals that the proposed model agrees fairly well with Alber’s method (9th Aerospace Sciences Meeting, 1971) for predicting the point of separation. Finally, using the continuity equation, and a proposed fit, the growth rate of the displacement thickness is evaluated which may be of use in practical potential flow solvers   

Angular momentum integral equation on turbulent boundary layers with pressure gradient

Turbulent boundary layers (TBL) encountering pressure gradients have been the subject of several works because of their engineering application. A fundamental understanding of their physics is essential for modeling and prediction of engineering quantities such as skin friction coefficient. In this study, we apply the angular momentum integral (AMI) equation and re-formulate it to determine the contribution of different flow phenomena such as pressure gradient and Reynolds shear stress to skin friction. These flow phenomena are represented as torques about a length scale, l, that is obtained from zero-pressure-gradient (ZPG) Blasius laminar solution. Such a choice isolates the skin friction of a ZPG laminar flow at the same Reynolds number and evaluates the other flow phenomena relative to the laminar friction. The AMI equation is applied to a set of DNS and LES TBL simulations undergoing various adverse pressure gradients. Although the contribution of the edge pressure gradient and mean fluxes to skin friction respond substantially according to the strength of the pressure gradient, the explicit turbulent contribution is less sensitive to the edge pressure gradient. The study of the TBL history on the statistics using the AMI budget yields a Clauser parameter based on l, β_l. We observe a more robust collapse of the mean velocity and Reynolds stress profiles at matching β_l and Reynolds numbers for the TBL dataset compared with the previous studies.   

Structure of an Axisymmetric Turbulent Boundary Layer with Pressure Gradients

The axisymmetric turbulent boundary layer of a body of revolution was measured in a wind tunnel. The model studied was the hull of a generic submarine, the DARPA SUBOFF, without its appendages or sail. High dynamic spatial range (in excess of 1,000) particle image velocimetry (PIV) was performed along the model's entire tail region (70 to 95% of body length), where the boundary layer experienced strong pressure gradients and surface curvatures. Surface static pressure measurements and flow statistics from PIV showed agreement with historical data. Further, comparisons of the flow statistics and structures were made between this axisymmetric boundary layer and the canonical flat plate boundary layer. This analysis reveals the combined effects of pressure gradient and surface curvatures on boundary layer development.   

Influence of time-varying freestream conditions on the unsteady separation of a turbulent boundary layer

The Large-Eddy Simulation technique is used to carry out numerical simulations of unsteady separation in a turbulent boundary layer. The unsteadiness is generated by an oscillating freestream vertical velocity profile V_∞(x, t). In previous meetings we described the effects of the oscillation frequency on the dynamics of unsteady flow separation. Here we fix the reduced frequency at k = 1, and we describe the effects of the freestream condition on the flow physics. Three cases are analyzed: while in case A, corresponding to k = 1 in Ambrogi et al. [J. Fluid Mech. 245, A10, 2022], V_∞ changes from suction-blowing to blowing-suction in a complete cycle, case B and C are both suction-blowing only, and the strength of the adverse pressure gradient is modulated in time. Moreover, the boundary layer in case B never approaches a zero pressure gradient condition. A closed separation bubble is formed in all cases, and for case B its dimensions match the steady calculation. The time evolution of turbulent kinetic energy (TKE) reveals that an advection mechanism is present in all cases. Case A and C are characterized by a similar advection process in which the high TKE region generated in the separated shear layer is washed out of the domain as a rigid body. In case B, on the other hand, the separation bubble never disappears and the separated shear layer is characterized by a breathing pattern.   

Wall-Resolved LES Turbulence Investigations of BoLT-II Hypersonic Vehicle

Wall-Resolved Large-Eddy Simulations (WRLES) of the flow over the BoLT-II hypersonic flight vehicle are performed at a Mach 6 condition along the flight trajectory. The vehicle has highly swept leading edges, spanwise surface concavity and significant bow-shock curvature. The flow over the vehicle is highly three-dimensional with multiple instabilities that modify the boundary layer (BL) energy balance. We demonstrate the successful generation of grids that support the stability of WRLES and present the detailed flow data with particular interest in the analysis of the 3D turbulent BL, which show departure from canonical 2D BL theory, energy injection by the inviscid vortical structures, and scale interactions between the inviscid flow and the turbulent BL. The large scale in the BoLT2 flow environment is described. The hypersonic turbulent boundary layer flow is characterized using the spectrally resolved high-fidelity simulation data.