Reynolds Shear Stress Profiles Research Articles

Spectral proper orthogonal decomposition analysis of a spatially developing underexpanded round jet at Re = 45 000

This study conducts a comprehensive analysis of the coherent structures within a spatially developing underexpanded axisymmetric jet at a Reynolds number of 45 000, utilizing high-fidelity implicit large eddy simulations (iLES) in conjunction with spectral proper orthogonal decomposition (SPOD). In the frequency–wavenumber space, the global SPOD analysis identifies three distinct coherent structures, corresponding to three different mechanisms, namely, Kelvin–Helmholtz (KH), Orr, and lift-up. Their salient characteristics are discussed in detail. Local SPOD analysis further explores the streamwise evolution of these coherent structures, revealing that the influence of KH mechanism is confined to the near field, while the lift-up mechanism persists and dominates the energy content beyond the potential core, with the streaks of azimuthal wavenumbers one and two being the most energetic. The reconstruction of turbulent kinetic energy (TKE) and Reynolds shear stress from SPOD modes is assessed, and the first few azimuthal modes with low wavenumbers and frequencies are found crucial for capturing the dominant features of the flow. It is found that only the m = 0 and m = 1 modes contribute to the TKE at the centerline. The Reynolds shear stress reconstruction quality is comparable to TKE, but with a negligible contribution from the m = 0 mode. The azimuthal mode m = 1 captures the slope of the actual Reynolds shear stress profile in the vicinity of centerline, while m = 2 and higher modes capture the peak location of the actual profile.

Improving modeling of submerged canopy flows with a vortex-based Spalart–Allmaras model

Accurately modeling the hydrodynamics of submerged canopies is crucial for predicting sediment dynamics and geomorphodynamics. This paper introduces vortex-based Spalart–Allmaras (VBSA) models, considering the spatial structure of canopy-scale vortices and stem wakes. The VBSA models are innovative in incorporating the physics of canopy-overflow interaction. They were validated using experimental data across a wide range of velocities and canopy densities, and their performance was compared with other turbulence models (i.e., previous SA models and k-ε model). Despite lacking high-order numerical schemes, the VBSA models outperform other turbulence models, exhibiting less numerical dissipation and better replication of mean velocity and Reynolds shear stress profiles in the vortex penetration space and below the surface. A sensitivity analysis was conducted to identify optimal values for new model parameters. Analyses suggest that the porous canopy layer suppresses the mixing efficiency by reducing the turbulence length scale. The analysis of eddy viscosity flux balance reveals that the eddy viscosity is transferred to the canopy layer from the overflow by the canopy-scale vortices and is destructed in the canopy, with production being less important, particularly in large submergence scenarios.

Turbulent Channel Flow: Direct Numerical Simulation-Data-Driven Modeling

Turbulent Channel Flow: Direct Numerical Simulation-Data-Driven Modeling

Patterns of turbulent flow structures over dune shaped bed features under the influence of downward seepage

The research presents an experimental investigation on change in streamwise flow velocity and turbulent characteristics over a fixed dune shaped bed feature, in the presence and absence of downward seepage. Streamwise velocity, Reynolds Shear Stress (RSS), turbulent intensities, and third order correlation profiles are presented along the flow depth at eleven locations over the dune. Amplification in the values of streamwise velocities, RSS, and turbulent intensities has been observed at the initial sections on the stoss side as well as at the sections on the lee side of the dune under the influence of downward seepage. At the initial and lee side sections of the dune, average values of the streamwise velocity, RSS, and streamwise turbulent intensity along the depth of flow increase by ∼0.2 %−31 %, ∼9 %−50 %, and ∼2 %−30 % respectively, for the 10 % seepage condition as compared to the no seepage condition. For the 15 % seepage condition, there has been found an increase in these value values by ∼3 %−67 %, ∼52 %−150 %, and ∼35 %−43 % respectively, at the considered sections over the dune. Further, it has also been observed that under the influence of downward seepage, bed shear stress increases at the initialand lee side sectionsof the dune. Through the third order correlation studies, occurrence of rushing flow due to seepage was confirmed at the lee side and initial sections of the dune. Further analysis at the wake zone revealed a reduction in the values of second and third-order moments demonstrating greater streamwise flow of Reynolds stress and diffusion of vertical Reynolds stress in streamwise direction as a result of increased velocity owing to downward seepage. Results indicate that the introduction of seepage leads to the creation of large angled dunes with wake zone stretched in the flow direction as well as the same wake zone was found squeezed in the vertical direction.

Turbulence in a wall-wake flow downstream of two horizontal cylinders

Experimental investigations of flow past two (one above other) horizontal cylinders have been presented in this paper. The experimental data captured over a rough-bed with double cylinders having three different diameters, were used for the analysis. Firstly, general characteristics such as streamwise velocity, Reynolds shear stress and turbulence intensity profiles have been elucidated at different downstream locations from cylinder position; then some advanced analysis such as length scales, turbulent kinetic energy (TKE) fluxes and budget have been investigated to exhibit their variations. Length scale profiles exhibit higher values in near bed; indicating comparatively larger eddies’ downstream of both cylinders. TKE budget profiles indicate highly negative pressure energy diffusion rate together with a sufficient TKE production rate, stabilized by the amplified TKE diffusion and dissipation rates. To determine one of the most prime behaviors of turbulence characteristics, namely TKE dissipation rate more accurately, the concept of structure function has been employed. Primarily velocity gradient method elucidates TKE dissipation rate, then it has been verified by Kolmogorov’s two-thirds law using second order structure function and Kolmogorov’s − 5/3 law using the method of power spectra.

Velocity spectra and scale decomposition of adverse pressure gradient turbulent boundary layers considering history effects

Experiments at relatively large Reynolds number were conducted to better understand the influence of an adverse pressure gradient on the turbulence in two-dimensional boundary layer flows. One focus is on documenting the scale contributions to the Reynolds normal and shear stress profiles under the influence of a positive streamwise pressure gradient. Toward these aims, velocity spectra at friction Reynolds numbers (7100≤Reτ≤8600) are presented, analyzed, and compared to a zero pressure gradient case from the same facility at nominally matching Reynolds number. Distance-from-the-wall scaling is central to numerous theoretical and modeling considerations of the canonical flat plate flow. Consistently, the present spectral measurements reveal clear evidence for its preservation on the inertial sublayer under modest adverse pressure gradients. Increases are seen in the viscous-scaled premultiplied spectra in the outer region as the pressure gradient increases — commensurate with the increases observed in the outer peaks of the streamwise and wall-normal velocity variances, as well as the Reynolds shear stress. Effects of varying Reτ on the streamwise and wall-normal velocity variances and the Reynolds shear stress are studied by comparing the present data at varying Clauser pressure gradient parameter, β, with previous experiments having similarly varying β at significantly lower Reτ (≈1900). This allows to study the effects of changing Reynolds number at matched β. Cases at similar β and Reynolds number, but with varying flow history are also examined, wherein the same β value is arrived either by starting from a smaller or larger initial β. The effects of β>0 on the large and small scale motions are investigated by segregating the signal contributions according to a cut-off wavelength established in previous zero pressure gradient flows. Under a normalization that uses a hybrid velocity scale this analysis reveals similarity between the outer regions of the spectrally-decomposed variances and Reynolds shear stress. These analyses also reveal a more dramatic reduction in the near-wall large scale contribution to the Reynolds shear stress gradient in the β>0 flow when compared to the β=0 flow.

Validation of SST-SCM correlation-based transition model by the simulation of wall-bounded flows

A modified correlation-based transition model coupled with SST turbulence model(k−ω−γ−R˜eθt) is proposed in this work. For better verification of intermittency, the constraint definition strictly based on local variables is cited to ensure physical solution. In order to obtain more details of laminar and transition boundary layer for separation-induced transition, a local correction method based on streamline curvature modification (SCM) has been developed to extend model ability in predicting adverse-pressure-gradient flows. The mean strain rate involved with mean rotation rate and shear rate is modified according to wall-bounded condition with rotation and non-rotation features to preserve the anisotropic characteristics encountered in large range flows. Furthermore, separation-induced intermittency is corrected by considering the effect of average strain rate to prevent outliers in local intermittency whenever the laminar boundary layer separates. Considering that results of test cases simulation with different empirical correlations vary huge from each other, all cases are matched with its suitable empirical correlations to ensure the stability and accuracy calculated by proposed model. Transitional flat-plate cases are calculated to validate the basic ability for near-wall flow simulation.Massive separated flow over two-dimensional circular cylinder has been studied to verify the predicted accuracy for non-equilibrium flows. Flow over multi-element airfoil 30P30N and the simulation of vortex system over three-dimensional Low Pressure Turbine(LPT) cascade T106A are given to examine the predictions of Reynolds shear stress profiles and pressure distribution with flow disturbance; reasonable results show a good match between experimental data and calculation by proposed model.

Second Moment Closure Modeling and Direct Numerical Simulation of Stratified Shear Layers

AbstractBuoyant shear layers encountered in many engineering and environmental applications have been studied by researchers for decades. Often, these flows have high Reynolds and Richardson numbers, which leads to significant/intractable space–time resolution requirements for direct numerical simulation (DNS) or large eddy simulation (LES). On the other hand, many of the important physical mechanisms, such as stress anisotropy, wake stabilization, and regime transition, inherently render eddy viscosity-based Reynolds-averaged Navier–Stokes (RANS) modeling inappropriate. Accordingly, we pursue second-moment closure (SMC), i.e., full Reynolds stress/flux/variance modeling, for moderate Reynolds number nonstratified, and stratified shear layers for which DNS is possible. A range of submodel complexity is pursued for the diffusion of stresses, density fluxes and variance, pressure strain and scrambling, and dissipation. These submodels are evaluated in terms of how well they are represented by DNS in comparison to the exact Reynolds-averaged terms, and how well they impact the accuracy of full RANS closure. For the nonstratified case, SMC model predicts the shear layer growth rate and Reynolds shear stress profiles accurately. Stress anisotropy and budgets are captured only qualitatively. Comparing DNS of exact and modeled terms, inconsistencies in model performance and assumptions are observed, including inaccurate prediction of individual statistics, non-negligible pressure diffusion, and dissipation anisotropy. For the stratified case, shear layer and gradient Richardson number growth rates, and stress, flux and variance decay rates, are captured with less accuracy than corresponding flow parameters in the nonstratified case. These studies lead to several recommendations for model improvement.

Layered structure of turbulent natural convection over a vertical flat plate

The layered structure of turbulent natural convection over a vertical flat plate (TNCVP) is first investigated qualitatively by dimensional analysis. The scaling patch approach is then applied to quantify the layered structure of TNCVP based on the characteristics of the mean momentum and energy equations. Fluid flow in TNCVP is divided into two layers: an inner layer and an outer layer, but heat transport is divided into four layers: a molecular diffusion sub-layer, a gradient balance layer, a meso layer and an outer layer. Scaling patch analysis identifies proper scales for the length, velocity, temperature, Reynolds shear stress, and turbulent heat flux in different layers, and develops a multi-scaling of the mean momentum and energy equations. Part of the novelty of the approach is that different regions incorporate unique scaling lengths, temperatures, turbulence temperature flux, and Reynolds stresses. The inner and outer scaled mean velocity, temperature, Reynolds shear stress and turbulent temperature flux profiles at four streamwise locations in the experiments of Tsuji and Nagano [1,2] merge onto a single curve in the inner and outer layers, respectively. This behavior supports the validity of the analysis. The layered structure of TNCVP is compared with three canonical turbulent flows: differentially heated turbulent flow in a vertical channel (DHVC), forced convection in turbulent boundary layer flows (TBL), and turbulent plumes. Striking similarities and distinctive differences among these turbulent flow and heat transport situations are elucidated. The inner layer of the mean momentum equation of TNCVP is similar to the inner layer of DHVC, but the outer layer of TNCVP is similar to that of a turbulent planar plume. The four layer structure for the mean energy equation of TNCVP is similar to that of pressure- or shear-driven turbulent boundary layers.

Turbulent Events around an Intermediately Submerged Boulder under Wake Interference Flow Regime

In-stream boulders significantly affect local flow hydrodynamics, sediment transport, and aquatic habitats. This study experimentally investigates the Reynolds shear stress (RSS) profiles and turbulent events around an intermediately submerged boulder under a wake interference flow regime. The frequency of turbulent events and their contribution to the RSS are quantified at two specific submergence ratios of 1.56 and 1.90 around the boulder in the near-bed and top-boulder regions. Ejection and sweep events were generally dominant in the near-bed and top-boulder regions. The dominance of turbulent events around the boulder varied as the submergence level changed. Relationships are proposed to predict the contribution of the near-bed turbulent events to the RSS around the boulder.

Revisiting rough-wall turbulent boundary layers over sand-grain roughness

Abstract

Resolvent-based design and experimental testing of porous materials for passive turbulence control

An extended version of the resolvent formulation is used to evaluate the use of anisotropic porous materials as passive flow control devices for turbulent channel flow. The effect of these porous substrates is introduced into the governing equations via a generalized version of Darcy’s law. Model predictions show that materials with high streamwise permeability and low wall-normal permeability (ϕxy=kxx/kyy≫1) can suppress resolvent modes resembling the energetic near-wall cycle. Based on these predictions, two anisotropic porous substrates with ϕxy>1 and ϕxy<1 were designed and fabricated for experiments in a benchtop water channel experiment. Particle Image Velocimetry (PIV) measurements were used to compute mean turbulence statistics and to educe coherent structure via snapshot Proper Orthogonal Decomposition (POD). Friction velocity estimates based on the Reynolds shear stress profiles do not show evidence of discernible friction reduction (or increase) over the streamwise-preferential substrate with ϕxy>1 relative to a smooth wall flow at identical bulk Reynolds number. A significant increase in friction is observed over the substrate with ϕxy<1. This increase in friction is linked to the emergence of spanwise rollers resembling Kelvin–Helmholtz vortices. Coherent structures extracted via POD analysis show qualitative agreement with model predictions.

One-dimensional turbulence: Application to incompressible spatially developing turbulent boundary layers

A map-based stochastic approach, One-Dimensional Turbulence (ODT), is applied to analyze the incompressible spatially developing turbulent boundary layer (SBL). In the present study, the SBL is formed by a plane moving wall and a free stream at rest. The flow variables are resolved on all scales along a 1-D domain. A deterministic process represents the molecular diffusion and a stochastic process is modeling the effect of turbulent advection and pressure fluctuations. Due to the reduced dimensions in the model, it achieves major cost reductions as compared to the full 3-D simulations and is, thus, able to explore large parameter regimes. The simulations are presented for momentum thickness Reynolds numbers, Reθ, varied in the range Reθ≈1968-8000. We have investigated various features related to the SBL, such as mean, root mean square, Reynolds shear stress, skewness, flatness and pre-multiplied velocity profiles and skin friction coefficient and shape factor, all for two bulk velocities using ODT and compared our results to the available reference Direct Numerical Simulations (DNS) and large-eddy simulations (LES) results. We have further compared the results for spatial and temporal ODT formulations used to investigate the turbulent boundary layer with the reference data at matched Reθ. The comparison presented suggests that ODT is a reasonably accurate approach for the simulations of the spatially developing turbulent boundary layers which can be further improved to yield accurate statistics at high Reynolds number by implementing Fourier transformation of the kernels which in the present formulation of the model is not implemented.

Hydrodynamics of flow over two-dimensional dunes

The turbulence characteristics in flow over and within the interface of two-dimensional dunes are investigated experimentally. Besides the spatial flow and turbulence quantities, their double-averaged profiles are also analyzed. The flow over dunes is recognized to be a wake-interference flow, where the decelerated flow at the immediate downstream of the crest causes the kolk-boil effect. The flow reattachment can be explained from the perspective of the Coandă effect. The inner boundary layer edge follows the locus of the inflection points of velocity profiles having a velocity defect. The Reynolds shear stress profiles attain their respective peaks along this locus. In addition, the dispersive shear stress initiates from the edge of the form-induced sublayer being negative, indicating a spatially decelerated flow. The third-order correlations reveal that an inrush of rapidly moving fluid streaks coupled with a downward-downstream Reynolds stress diffusion prevails within the interfacial sublayer, while an arrival of slowly moving fluid streaks coupled with an upward-upstream stress diffusion governs the flow zone above the crest. The turbulent kinetic energy (TKE) flux results corroborate the similar findings. Concerning the TKE budget, the dispersive kinetic energy diffusion is found to be substantial within the roughness sublayer. The budget terms exhibit their respective peaks near the crest. The production rate is greater than the dissipation rate. However, the TKE diffusion and pressure energy diffusion rates are negative in the interfacial sublayer. The bursting analysis endorses that the sweeps and ejections govern within the interfacial sublayer and the flow zone above the crest, respectively.

Prediction of Turbulent Flow Over a Flat Plate With a Step Change From a Smooth to a Rough Surface Using a Near-Wall RANS Model

Abstract The present paper reports a numerical study of fully developed turbulent flow over a flat plate with a step change from a smooth to a rough surface. The Reynolds number based on momentum thickness for the smooth flow was Reθ=5950. The focus of the study was to investigate the capability of the Reynolds-averaged Navier–Stokes (RANS) equations to predict the internal boundary layer (IBL) created by the flow configuration. The numerical solution used a two-layer k−ε model to implement the effects of surface roughness on the turbulence and mean flow fields via the use of a hydrodynamic roughness length y0. The prediction for the mean velocity field revealed a development zone immediately downstream of the step in which the mean velocity profile included a lower region affected by the surface roughness below and an upper region with the characteristics of the smooth-wall boundary layer above. In this zone, both the turbulence kinetic energy and Reynolds shear stress profiles were characterized by a significant reduction in magnitude in the outer region of the flow that is unaffected by the rough surface. The turbulence kinetic energy profile was used to estimate the thickness of the IBL, and the resulting growth rate closely matched the experimental results. As such, the IBL is a promising test case for assessing the ability of RANS models to predict the discrete roughness configurations often encountered in industrial and environmental applications.

Turbulence in Wall-Wake Flow Downstream of an Isolated Dunal Bedform

This study examines the turbulence in wall-wake flow downstream of an isolated dunal bedform. The streamwise flow velocity and Reynolds shear stress profiles at the upstream and various streamwise distances downstream of the dune were obtained. The results reveal that in the wall-wake flow, the third-order moments change their signs below the dune crest, whereas their signs remain unaltered above the crest. The near-wake flow is featured by sweep events, whereas the far-wake flow is controlled by the ejection events. Downstream of the dune, the turbulent kinetic energy production and dissipation rates, in the near-bed flow zone, are positive. However, they reduce as the vertical distance increases up to the lower-half of the dune height and beyond that, they increase with an increase in vertical distance, attaining their peaks at the crest. The turbulent kinetic energy diffusion and pressure energy diffusion rates, in the near-bed flow zone, are negative, whereas they attain their positive peaks at the crest. The anisotropy invariant maps indicate that the data plots in the wall-wake flow form a looping trend. Below the crest, the turbulence has an affinity to a two-dimensional isotropy, whereas above the crest, the anisotropy tends to reduce to a quasi-three-dimensional isotropy.

Direct numerical simulations of second-order Stokes wave driven smooth-walled oscillatory channel: investigation of net current formation

ABSTRACTIn wall-bounded time-periodic flows, nonlinearity, associated with higher harmonic term(s) in velocity and/or acceleration outside the boundary layer, can significantly change the wall turbulence compared with that in the linear Stokes Boundary Layer. A significant feature of a nonlinear wall-bounded turbulent time-periodic flow is the formation of a net current which has not yet been mechanistically explained. This study investigates the effects of asymmetric velocity outside the boundary layer on wall turbulence and net current formation through Direct Numerical Simulations of a smooth-walled planar channel driven by the Second-order Stokes Wave. Simulation results suggest that net current characteristics depend on whether developed turbulence is present. When turbulence is developed, asymmetric viscous length scale is found to be the primary reason of the net current whereby a vertical offset between negative and positive Reynolds shear stress profiles, associated with forward and reverse flows, respectively, is created in a cycle. After averaging over a cycle, residual Reynolds shear stress, which drives the net current, is observed to be within the offset layer.

On the virtual origin determined from momentum equation analysis using experimental data within the roughness sublayer

The centroid of distributed drag acting on a roughness element, which is interpreted as the origin of the distance from the wall, is discussed for a wall-bounded turbulent flow over a rough surface with square ribs arranged with a roughness pitch ratio, PR, of 4. The first and second moments of fluctuating velocity components were measured by one-dimensional laser Doppler velocimetry to estimate spatially averaged quantities in the roughness sublayer. The spatially averaged profiles of the streamwise mean velocity and Reynolds shear stress can be scaled with the representative scales of the mixing layer observed behind a roughness element; these are $${u_{\text{p}}}=\sqrt {{D_{\text{p}}}/\left( {\rho b} \right)}$$ and b, where up is the velocity scale of the driven force for the flow in a cavity, Dp is the form drag acting on a roughness element, $$\rho$$ is the fluid density, and b is the cavity width. The analytical solutions of the spatially averaged profiles can be formulated in exponential form, and are in good agreement with available experimental data. The displacement height, which is the distance from the centroid of the distributed drag to the roughness crest, can be approximately estimated by the spatially averaged Reynolds shear stress profile. The prediction of the displacement height well represents that of the experimental and direct numerical simulation data for 4 ≤ PR ≤ 8. The spatially averaged Reynolds shear stress profiles in the cavity of turbulent boundary layers over square-ribbed rough surfaces can be scaled with the representative scales of the mixing layer behind a roughness element for 4 ≤ PR ≤ 8, where PR is the roughness pitch ratio. The displacement height, which corresponds to the origin of the law of the wall, can be predicted using the scaled spatially averaged Reynolds shear stress profile and increases with the PR as the mixing layer develops over the cavity.

Exact coherent states with hairpin-like vortex structure in channel flow

Hairpin vortices are widely studied as an important structural aspect of wall turbulence. The present work describes, for the first time, nonlinear travelling wave solutions to the Navier–Stokes equations in the channel flow geometry – exact coherent states (ECS) – that display hairpin-like vortex structure. This solution family comes into existence at a saddle-node bifurcation at Reynolds number $Re=666$. At the bifurcation, the solution has a highly symmetric quasi-streamwise vortex structure similar to that reported for previously studied ECS. With increasing distance from the bifurcation, however, both the upper and lower branch solutions develop a vortical structure characteristic of hairpins: a spanwise-oriented ‘head’ near the channel centreplane where the mean shear vanishes connected to counter-rotating quasi-streamwise ‘legs’ that extend toward the channel wall. At $Re=1800$, the upper branch solution has mean and Reynolds shear-stress profiles that closely resemble those of turbulent mean profiles in the same domain.

Determination of wall shear stress from mean velocity and Reynolds shear stress profiles

A method for determining the wall shear stress under arbitrary streamwise pressure gradients and wall roughness conditions using Reynolds shear stress and mean velocity profile data is presented. Data from very near the wall are not required. The method is tested with experimental and direct numerical simulation data.