Reynolds Stress Tensor Research Articles

An Explicit Algebraic Closure for Passive Scalar-Flux: Applications in Channel Flows at a Wide Range of Reynolds Numbers

In this paper, we propose an algebraic model for turbulent scalar-flux vector that stems from tensor representation theory. The resulting closure contains direct dependence on mean velocity gradients and quadratic products of the Reynolds stress tensor. Model coefficients are determined from Direct Numerical Simulations (DNS) data of homogeneous shear flows subjected to arbitrary mean scalar gradient orientations, while a correction function was applied at one model coefficient based on a turbulent channel flow case. Model performance is evaluated in Poiseuille and Couette flows at several Reynolds numbers for Pr=0.7, along with a case at a higher Prandtl number (Pr=7.0) that typically occurs in water–boundary interaction applications. Overall, the proposed model provides promising results for wide near-wall interaction applications. To put the performance of the proposed model into context, we compare with Younis algebraic model, which is known to provide reasonable predictions for several engineering flows.

Quantifying structural uncertainties in Reynolds-averaged Navier–Stokes simulations of wind turbine wakes

Reynolds-averaged Navier Stokes (RANS) based modeling is considered the mainstream computational fluid dynamics (CFD) approach for wind energy applications. Considering the inherent shortcomings associated with RANS models, quantification of uncertainties is of obvious importance if these models are to be used for design and optimization of wind farms. In the present work, structural uncertainties of RANS closure are quantified for simulations of wake flow behind a stand-alone wind turbine. The uncertainty is modeled by introducing perturbations to the Reynolds stress tensor. We specifically focus on perturbations in the eigenvalues of the tensor. The k-ω SST model and large-eddy simulation (LES) data are used as the baseline RANS model and reference data, respectively. Initially we compare the unperturbed RANS simulation against LES data, and show that the k-ω SST model generally tends to predict higher levels of isotropy in the turbulent wake. A comparison between LES data and differently perturbed k-ω SST simulations is made for the evolution of velocity deficit and turbulence intensity behind the turbine. A satisfactory coverage of LES profiles is observed when the amount of introduced perturbation is adjusted based on a priori comparison of the baseline RANS and LES results with the exception of the turbulent intensity immediately behind the turbine.

Speed–direction description of turbulent flows

In this note, we introduce speed and direction variables to describe the motion of incompressible viscous fluid. Fluid velocity u is decomposed into u = ur, with u = |u| and r = u/|u|. We consider a directional split of the Navier–Stokes equations into a coupled system of equations for u and for r. The equation for u is particularly simple but solely maintains the energy balance of the system. Under the assumption of a weak correlation between fluctuations in speed and direction in a developed turbulent flow, we further illustrate the application of u–r variables to describe mean statistics of a shear turbulence. The standard (full) Reynolds stress tensor does not appear in a resulting equation for the mean flow profile.

Turbulence anisotropy with higher-order moments in flow through passive grid under rigid boundary influence

The investigation presents the estimate of the degree of deviation from the isotropic turbulence in terms of Reynolds stress tensor for grid generated turbulence under the influence of bottom boundary. The turbulence triangle, Eigen values, and the invariant functions are presented at near and far field regions of the grids with different solidity ratio. In addition, the work also deals with the analysis based on third-order moments of the velocity fluctuations and the ratio of momentum flux to the turbulent kinetic energy in the frequency domain. The Reynolds stress anisotropy exposes that the anisotropic invariant maps possess a closed looping trend in the near field region and an open looping trend in the far-field region of the grid. Further, to describe the physical behaviour of the velocity time-series of random fluctuating components in the stream-wise directions, the probability distribution function are estimated and interpreted.

Characterization of anisotropic turbulence behavior in pulsatile blood flow

Turbulent-like hemodynamics with prominent cycle-to-cycle flow variations have received increased attention as a potential stimulus for cardiovascular diseases. These turbulent conditions are typically evaluated in a statistical sense from single scalars extracted from ensemble-averaged tensors (such as the Reynolds stress tensor), limiting the amount of information that can be used for physical interpretations and quality assessments of numerical models. In this study, barycentric anisotropy invariant mapping was used to demonstrate an efficient and comprehensive approach to characterize turbulence-related tensor fields in patient-specific cardiovascular flows, obtained from scale-resolving large eddy simulations. These techniques were also used to analyze some common modeling compromises as well as MRI turbulence measurements through an idealized constriction. The proposed method found explicit sites of elevated turbulence anisotropy, including a broad but time-varying spectrum of characteristics over the flow deceleration phase, which was different for both the steady inflow and Reynolds-averaged Navier–Stokes modeling assumptions. Qualitatively, the MRI results showed overall expected post-stenotic turbulence characteristics, however, also with apparent regions of unrealizable or conceivably physically unrealistic conditions, including the highest turbulence intensity ranges. These findings suggest that more detailed studies of MRI-measured turbulence fields are needed, which hopefully can be assisted by more comprehensive evaluation tools such as the once described herein.

Relaminarized and recovered turbulence under nonuniform body forces

We show that wall-bounded turbulence can be relaminarized and reorganized by adding nonuniform streamwise body forces. A nonuniform body force can distort the parabolic mean velocity profile, thus altering turbulence production due to mean shear. In the quasi-laminar state, all Reynolds stress tensor components are fairly weak except for streamwise fluctuations far from the wall, indicating a collapse of the near-wall turbulence self-sustaining cycle. In the recovered turbulence regime, the Reynolds shear stress has a negative range in the bulk connected with a positive range near the wall corresponding to the M-shaped velocity profile.

Turbulence modelling analysis in a corner separation flow

This paper studies RANS turbulence modelling for a linear compressor cascade corner separation flow, using large-eddy simulation results as reference. The Boussinesq and the quadratic constitutive relations are investigated with two versions of Wilcox’s k−ω turbulence model through a priori and a posteriori analyses. In the a priori analysis the quadratic constitutive relation shows improvement on the alignment between the rate-of-strain tensor and the Reynolds stress tensor for the inlet region, compared to the Boussinesq constitutive relation. But the improvement is less significant in the highly vortical region. Using the turbulent kinetic energy and the specific dissipation rate provides a fairly good estimate of the turbulent viscosity. The turbulent kinetic energy budget is also investigated. Large-eddy simulation results present non-equilibrium turbulence, i.e. the production and dissipation are not balanced, whereas the RANS models predicts equilibrium turbulence.

Reynolds stress anisotropy in flow over two-dimensional rigid dunes

Characteristics of turbulence anisotropy in flow over two-dimensional rigid dunes are analysed. The Reynolds stress anisotropy is envisaged from the perspective of the stress ellipsoid shape. The spatial evolutions of the anisotropic invariant map (AIM), anisotropic invariant function, eigenvalues of the scaled Reynolds stress tensor and eccentricities of the stress ellipsoid are investigated at various streamwise distances along the vertical. The data plots reveal that the oblate spheroid axisymmetric turbulence appears near the top of the crest, whereas the prolate spheroid axisymmetric turbulence dominates near the free surface. At the dune trough, the axisymmetric contraction to the oblate spheroid diminishes, as the vertical distance below the crest increases. At the reattachment point and one-third of the stoss-side, the oblate spheroid axisymmetric turbulence formed below the crest appears to be more contracted, as the vertical distance increases. The AIMs suggest that the turbulence anisotropy up to edge of the boundary layer follows a looping pattern. As the streamwise distance increases, the turbulence anisotropy at the edge of the boundary layer approaches the plane-strain limit up to two-thirds of the stoss-side, intersecting the plane-strain limit at the top of the crest and thereafter moving towards the oblate spheroid axisymmetric turbulence.

Analysis of the effect of freestream turbulence on dynamic stall of wind turbine blades

The aerodynamics of a wind turbine blade at various fixed and dynamically changing angles of attack for laminar and turbulent freestream is investigated by large-eddy simulations. The effects of freestream turbulence on the aerodynamic coefficients are analyzed for dynamic stall conditions being generated by prescribing gusts, i.e., periodically changing incoming velocity fields, onto a ClarkY airfoil at a chord based Reynolds number of Re=150,000. A synthetic turbulence generation (STG) method is extended and applied to produce the turbulent gusts in a time varying mean flow with predefined turbulence statistics, such as mean velocity, Reynolds stress tensor components, and velocity spectra. The length scales in the turbulent freestream are smaller than the blade chord length. The reduced frequency of the incoming laminar and turbulent gusts is kred=0.2 based on the freestream velocity, chord length, and gust frequency. Dynamic stall is analyzed for an angle of attack ranging from 10° to 25°. It is shown how much the freestream turbulence level delays stall and increases lift at high angle of attack. The effect of the turbulence is most evident on the aerodynamic forces during the phase of decreasing angle of attack of the dynamic stall process, in which the reattachment of the boundary layer is enhanced by the freestream turbulence. The negative damping from the moment coefficient is also significantly reduced due to the incoming turbulence. The results show that the freestream turbulence in a turbulent flow at a turbulence level of 5% improves the aerodynamic performance of the wind turbine blade.

A priori tests of eddy viscosity models in square duct flow

We carry out a priori tests of linear and nonlinear eddy viscosity models using direct numerical simulation (DNS) data of square duct flow up to friction Reynolds number {text {Re}}_tau =1055. We focus on the ability of eddy viscosity models to reproduce the anisotropic Reynolds stress tensor components a_{ij} responsible for turbulent secondary flows, namely the normal stress a_{22} and the secondary shear stress a_{23}. A priori tests on constitutive relations for a_{ij} are performed using the tensor polynomial expansion of Pope (J Fluid Mech 72:331–340, 1975), whereby one tensor base corresponds to the linear eddy viscosity hypothesis and five bases return exact representation of a_{ij}. We show that the bases subset has an important effect on the accuracy of the stresses and the best results are obtained when using tensor bases which contain both the strain rate and the rotation rate. Models performance is quantified using the mean correlation coefficient with respect to DNS data {widetilde{C}}_{ij}, which shows that the linear eddy viscosity hypothesis always returns very accurate values of the primary shear stress a_{12} ({widetilde{C}}_{12}>0.99), whereas two bases are sufficient to achieve good accuracy of the normal stress and secondary shear stress ({widetilde{C}}_{22}=0.911, {widetilde{C}}_{23}=0.743). Unfortunately, RANS models rely on additional assumptions and a priori analysis carried out on popular models, including k–varepsilon and v^2–f, reveals that none of them achieves ideal accuracy. The only model based on Pope’s expansion which approaches ideal performance is the quadratic correction of Spalart (Int J Heat Fluid Flow 21:252–263, 2000), which has similar accuracy to models using four or more tensor bases. Nevertheless, the best results are obtained when using the linear correction to the v^2–f model developed by Pecnik and Iaccarino (AIAA Paper 2008-3852, 2008), although this is not built on the canonical tensor polynomial as the other models.

Characterization of the vertical evolution of the three-dimensional turbulence for fatigue design of tidal turbines.

A system of two coupled four-beam acoustic Doppler current profilers was used to collect turbulence measurements over a 36-h period at a highly energetic tidal energy site in Alderney Race. This system enables the evaluation of the six components of the Reynolds stress tensor throughout a large proportion of the water column. The present study provides mean vertical profiles of the velocity, the turbulence intensity and the integral lengthscale along the streamwise, spanwise and vertical direction of the tidal current. Based on our results and considering a tidal-stream energy convertor (TEC) aligned with the current main direction, the main elements of turbulence prone to affect the structure (material fatigue) and to alter power generation would likely be: (i) the streamwise turbulence intensity (Ix), (ii) the shear stress, [Formula: see text], (iii) the normal stress, [Formula: see text] and (iv) the vertical integral lengthscale (Lz). The streamwise turbulence intensity, (Ix), was found to be higher than that estimated at other tidal energy sites across the world for similar height above bottom. Along the vertical direction, the length (Lz) of the large-scale turbulence eddies was found to be equivalent to the rotor diameter of the TEC Sabella D10. It is considered that the turbulence metrics presented in this paper will be valuable for TECs designers, helping them optimize their designs as well as improve loading prediction through the lifetime of the machines. This article is part of the theme issue 'New insights on tidal dynamics and tidal energy harvesting in the Alderney Race'.

Assessing the turbulent kinetic energy budget in an energetic tidal flow from measurements of coupled ADCPs.

Two coupled four-beam acoustic Doppler current profilers were used to provide simultaneous and independent measurements of the turbulent kinetic energy (TKE) dissipation rate ε and the TKE production rate [Formula: see text] over a 36 h long period at a highly energetic tidal energy site in the Alderney Race. The eight-beam arrangement enabled the evaluation of the six components of the Reynolds stress tensor which allows for an improved estimation of the TKE production rate. Depth-time series of ε, [Formula: see text] and the Reynolds stresses are provided. The comparison between ε and [Formula: see text] was performed by calculating individual ratios of ε corresponding to [Formula: see text]. The depth-averaged ratio [Formula: see text] averaged over whole flood and ebb tide were found to be 2.2 and 2.8 respectively, indicating that TKE dissipation exceeds TKE production. It is shown that the term of diffusive transport of TKE is significant. As a result, non-local transport is important to the TKE budget and the common assumption of a local balance, i.e. a balance between production and dissipation, is not valid at the measurement site. This article is part of the theme issue 'New insights on tidal dynamics and tidal energy harvesting in the Alderney Race'.

Revisiting the role of intermittent heat transport towards Reynolds stress anisotropy in convective turbulence

Abstract

Analysis of the different sources of stress acting in fully rough turbulent flows over geometrical roughness elements

The discrete element roughness method is considered in this article for the prediction of turbulent flows over rough walls. This approach is derived by ensemble- and volume-averaging the Navier–Stokes equations, providing double-averaged Navier–Stokes equations, and yielding three unknown terms in the momentum equation: the Reynolds stress and dispersive stress tensors and the average drag force acting on the roughness elements. This work aims at analyzing these different terms, quantifying their respective contributions to the near-wall momentum budget, and providing guidance for their modeling. For this purpose, relevant data of turbulent flows are required. Roughness-resolved Reynolds-averaged Navier–Stokes simulations of transitionally and fully rough channel flows over academic roughness configurations are performed at high friction Reynolds numbers ranging from 3500 to 8000. Comparisons with existing and new velocity measurements performed in rough-wall turbulent boundary layers provide support to the simulation results, a particular emphasis being given on the validity of the numerical results in the wake of the roughness elements. These numerical results highlight the influence of roughness elements geometry and density on the roughness drag coefficient and the dispersive stress. It is particularly suggested that the standard roughness drag closure model should be revisited for double-averaged flows. Furthermore, the dispersive stress is shown to mainly originate from the wake of the roughness elements, an observation that could be leveraged for its modeling. However, since this stress contributes only marginally to the global stress and to the skin friction coefficient, such a modeling may not be critical at first order.

Two-equation and multi-fluid turbulence models for Richtmyer–Meshkov mixing

This paper concerns an investigation of two different approaches in modeling the turbulent mixing induced by the Richtmyer–Meshkov instability (RMI): A two-equation K-L multi-component Reynolds-averaged Navier–Stokes model and a two-fluid model. We have improved the accuracy of the K-L model by implementing new modifications, including a realizability condition for the Reynolds stress tensor and a threshold in the production of the turbulence kinetic energy. We examine the models in the one-dimensional (1D) form in the (re)-shocked mixing of a double-planar air and sulfur-hexafluoride (SF6) interface of the Atwood number |At| ≃ 0.6853. Furthermore, we investigated the models’ accuracy to RMI-induced mixing of a (re)-shocked planar-inverse chevron air–SF6 interface. Relevant integral quantities in time, as well as instantaneous profiles and contour plots, are used to assess the models’ accuracy against high-resolution implicit large eddy simulations. The proposed modifications improve the efficiency of the K-L model. The model is designed as a simple model capable of capturing the self-similar growth of Rayleigh–Taylor and Richtmyer–Meshkov flows. The two-fluid model remains more accurate but is also computationally more expensive.

Quantifying turbulence model uncertainty in Reynolds-averaged Navier–Stokes simulations of a pin-fin array. Part 1: Flow field

The assumptions that form the basis for Reynolds stress closure models have been formulated by considering canonical flows. As a result, the accuracy of Reynolds-averaged Navier–Stokes simulations can deteriorate significantly when modeling complex flows, and engineering applications would benefit from methods that can quantify the corresponding uncertainty in the predictions. This paper analyzes the performance of a previously proposed turbulence model uncertainty quantification (UQ) framework for simulations of flow through a pin-fin array. The method is a physics-based, data-free, interval approach that perturbs the Reynolds stress tensor shape towards the three limiting realizable states of anisotropy. The performance of the method is evaluated by determining whether large-eddy simulation results for the quantities of interest are encompassed by the intervals predicted by the UQ method. The results demonstrate that perturbing the stress shapes towards the one-component or two-component limit generally enhances the momentum transport between the bulk flow and wake regions, whereas perturbations towards the three-component limit suppress this transport. For quantities of interest that depend on the mean velocity and pressure field this results in predictions for uncertainty intervals that encompass the reference solution. For the turbulence kinetic energy the method fails to predict an adequate upper bound in regions with larger-scale turbulent structures, and it predicts overly conservative bounds in the pin stagnation regions. Based on these results several suggestions for improving the framework are made.

Numerical investigation of skewed spatially evolving mixing layers

The sensitivity of turbulent dynamics in spatially evolving mixing layers to small skew angles is investigated via direct numerical simulation. Angle is a measure of the lack of parallelism between the two asymptotic flows, whose interaction creates the turbulent mixing region. The analysis is performed considering a large range of values of the shear intensity parameter . This two-dimensional parameter space is explored using the results of a database of 18 direct numerical simulations. Instantaneous fields as well as time-averaged quantities are investigated, highlighting important mechanisms in the emergence of turbulence and its characteristics for this class of flows. In addition, a stochastic approach is used in which and are considered as random variables with a given probability distribution. The response surfaces of flow statistics in the parameter space are built through non-intrusive generalized polynomial chaos. It is found that variations of the parameter have a primary effect on the growth of the mixing region. A secondary effect associated with is observed as well. Higher values for the skew angle are responsible for a rapid increase in growth of the inlet structures, enhancing the development of the mixing region. The impact on the turbulence features and, in particular, on the Reynolds stress tensor is also significant. A modification of the normalized diagonal components of the Reynolds stress tensor due to is observed. In addition, the interaction between the parameters and is here the governing element.

Analysis and modelling of the relation between the shear rate and Reynolds stress tensors in transitional boundary layers

A non-linear eddy-viscosity transition model is presented, tuned by a large experimental data set describing transitional boundary layers. Data have been acquired by TR-PIV on a flat plate placed in a 2D converging-diverging channel with variable opening angle, allowing variation of the adverse pressure gradient, the free-stream turbulence intensity and the flow Reynolds number. Overall, 48 different combinations of these flow parameters encompass different modes of transition from bypass to separated-flow mechanisms, thus allowing fine tuning of the model, spanning significantly different conditions. The model is tuned locally as a function of the turbulent kinetic energy, a Reynolds number based on the wall distance and the ℓ2-norm of the shear rate tensor. A first correlation determines the rotation for alignment of the principal axes of the shear and stress tensors. By a second correlation, the eigenvalues of the stress tensor are obtained. The non-linear eddy-viscosity relation reproduces the anisotropy of the turbulence field observed for both bypass and separated-flow transitional cases. The relation has been applied to another experimental data set that did not participate to the fitting of the model and that is characterized by a different range of Reynolds number and turbulence intensity and a significantly stronger adverse pressure gradient with respect to the tuning dataset. Such application further strengthens the capability of the proposed correlations, that can easily be implemented in existing CFD solvers.

Transitional and turbulent flows in rectangular ducts: budgets and projection in principal mean strain axes

We carry out Direct Numerical Simulation (DNS) of flows in closed rectangular ducts with several aspect ratios. The Navier–Stokes equations are discretised through a second-order finite difference scheme, with non-uniform grids in two directions. The duct cross-sectional area is maintained constant as well as the flow rate, which allows to investigate which is the appropriate length scale in the Reynolds number for a good scaling in the laminar and in the fully turbulent regimes. We find that the Reynolds number based on the half length of the short side leads to a critical Reynolds number which is independent on the aspect ratio (), for ducts with . The mean and rms wall-normal velocity profiles are found to scale with the local value of the friction velocity. At high friction Reynolds numbers, the Reynolds number dependence is similar to that in turbulent plane channels, hence flows in rectangular ducts allow to investigate the Reynolds number dependency through a reduced number of simulations. At low Re, the profiles of the statistics differ from those in the two-dimensional channel due to the interaction of flow structures of different size. The projection of the velocity vector and of the Reynolds stress tensor along the eigenvectors of the strain-rate tensor yields reduced Reynolds stress anisotropy and simple turbulence kinetic energy budgets. We further show that the isotropic rate of dissipation is more difficult to model than the full dissipation rate, whose distribution does not largely differ from that of turbulence kinetic energy production. We expect that this information may be exploited for the development of advanced RANS models for complex flows.

Hybrid datasets: Incorporating experimental data into Lattice‐Boltzmann simulations

SummaryA novel method, which combines both fluid‐mechanical experimental and numerical data from magnetic resonance velocimetry and Lattice‐Boltzmann (LB) simulations is presented. The LB method offers a unique and simple way of integrating the experimental data into the simulation by means of its equilibrium term. The simulation is guided by the experimental data, while at the same time potential outliers or noisy data are physically smoothed. In addition, the simulation allows to increase the resolution and to obtain further physical quantities, which are not measurable in the experiment. For a benchmark case, temporally averaged velocity data is included into the simulation. The proposed model creates a hybrid dataset, which satisfies the Reynolds‐averaged Navier‐Stokes equations, including the correctly deduced contribution from the Reynolds stress tensor.