Rapid Pressure-strain Correlation Research Articles

A review of pressure strain correlation modeling for Reynolds stress models

Most investigations of turbulent flows in academic studies and industrial applications use turbulence models. Out of the different turbulence modeling approaches Reynolds stress models have the highest potential to replicate complex turbulent flow phenomena at a reasonable computational expense. The Reynolds stress modeling framework is constituted by individual closures that approximate the effects of separate turbulence processes like dissipation, turbulent transport, pressure strain correlation, etc. Owing to its complexity and importance in flow evolution the modeling of the pressure strain correlation mechanism is considered the crucial challenge for the Reynolds stress modeling framework. In the present work, the modeling of the pressure strain correlation for homogeneous turbulent flows is reviewed. The importance of the pressure strain correlation and its effects on flow evolution via energy transfer are established. The fundamental challenges in pressure stain correlation modeling are analyzed and discussed. Starting from the governing equations we outline the theory behind models for both the slow and rapid pressure strain correlation. Established models for both these are introduced and their successes and shortcomings are illustrated using theoretical analysis, computational fluid dynamics simulations, and comparisons against experimental and numerical studies. Recent advances and developments in this field are presented and discussed. The application of machine learning algorithms such as Deep Neural Networks, Random Forests, and Gradient Boosted Regression Trees is summarized and examined. We report fundamental problems in the application of machine learning algorithms for pressure strain correlation modeling. Finally, challenges and hurdles for pressure strain correlation modeling are outlined and explained in detail to guide future investigations.

Linear analysis of non-local physics in homogeneous turbulent flows

Characterization of non-local physical mechanisms is one of the important challenges toward deeper understanding of turbulent flows. In this investigation, we study the role of pressure in the evolution of three-dimensional incompressible homogeneous turbulent flows. We find that the evolution of turbulence, the nature of inertial physics, and pressure action are dependent on the mean flow invariants. We identify and explain the intercomponent energy transfer induced by the rapid pressure strain correlation for different regimes of flow. Additionally, the structuring effect of pressure on turbulent fluctuations of different alignments is determined and explicated. We exhibit that in regimes of flow where the action of the pressure strain correlation is significant, there is a switching of the flow evolution between diametric stages of evolution. This phenomenon is explained, and its lack of amenability to single-point turbulence modeling is detailed.

A Representation Theory-Based Model for the Rapid Pressure Strain Correlation of Turbulence

In the presence of mean strain or rotation, the anisotropy of turbulence increases due to the rapid pressure strain term. In this paper, we consider the modeling of the rapid pressure strain correlation of turbulence. The anisotropy of turbulence in the presence of mean strain is studied and a new model is formulated by calibrating the model constants at the rapid distortion limit. This model is tested for a range of plane strain and elliptic flows and compared to direct numerical simulation (DNS) results. The present model shows agreement with DNS and improvements over the earlier models like those by Launder et al. (1975, “Progress in the Development of a Reynolds-Stress Turbulence Closure,” J. Fluid Mech., 68(3), pp. 537–566.) and Speziale et al. (1991, “Modelling the Pressure–Strain Correlation of Turbulence: An Invariant Dynamical Systems Approach,” J. Fluid Mech., 227(1), pp. 245–272.) that have been reported to give satisfactory performance for hyperbolic flows but not satisfactory for elliptic flows.

Reynolds stress models applied to canonical shock-turbulence interaction

ABSTRACTShock waves in high-speed flows can drastically alter the nature of Reynolds stresses in a turbulent flow. We study the canonical interaction of homogeneous isotropic turbulence passing through a normal shock, where the shock wave generates significant anisotropy of Reynolds stresses. Existing Reynolds stress models are applied to this canonical problem to predict the amplification of the stream-wise and transverse normal Reynolds stresses across the shock wave. In particular, the efficacy of the different models for the rapid pressure–strain correlation is evaluated by comparing the results with available direct numerical simulation (DNS) data. The model predictions are found to be grossly inaccurate, especially at high-Mach numbers. We propose physics-based improvement to the Reynolds stress-transport equation in the form of shock-unsteadiness effect and enstrophy amplification for turbulent dissipation rate . The resulting model is found to capture the essential physics of Reynolds stress amplification, and match DNS data for a range of Mach numbers. Numerical error encountered at shock waves are also analysed and the model equations are cast in conservative form to obtain physically consistent results with successive grid refinement. Finally, the proposed model for canonical shock-turbulence interaction is generalised to multi-dimensional flows with shock of arbitrary orientation.

Evaluation of the closure hypothesis in the Launder–Reece–Rodi rapid pressure–strain model

ABSTRACTThe derivation of the Launder–Reece–Rodi (LRR) stress transport model assumes that the rapid pressure–strain correlation is a function of the Reynolds stresses. This model is compared to a model derived from an approximate solution of the equation satisfied by the two-point correlation tensor. The comparison suggests that the assumption of functional dependence used to derive the LRR model lacks fundamental justification.

Algebraic Reynolds stress modeling of turbulence subject to rapid homogeneous and non-homogeneous compression or expansion

A recently developed explicit algebraic Reynolds stress model (EARSM) by Grigoriev et al. [“A realizable explicit algebraic Reynolds stress model for compressible turbulent flow with significant mean dilatation,” Phys. Fluids 25(10), 105112 (2013)] and the related differential Reynolds stress model (DRSM) are used to investigate the influence of homogeneous shear and compression on the evolution of turbulence in the limit of rapid distortion theory (RDT). The DRSM predictions of the turbulence kinetic energy evolution are in reasonable agreement with RDT while the evolution of diagonal components of anisotropy correctly captures the essential features, which is not the case for standard compressible extensions of DRSMs. The EARSM is shown to give a realizable anisotropy tensor and a correct trend of the growth of turbulence kinetic energy K, which saturates at a power law growth versus compression ratio, as well as retaining a normalized strain in the RDT regime. In contrast, an eddy-viscosity model results in a rapid exponential growth of K and excludes both realizability and high magnitude of the strain rate. We illustrate the importance of using a proper algebraic treatment of EARSM in systems with high values of dilatation and vorticity but low shear. A homogeneously compressed and rotating gas cloud with cylindrical symmetry, related to astrophysical flows and swirling supercritical flows, was investigated too. We also outline the extension of DRSM and EARSM to include the effect of non-homogeneous density coupled with “local mean acceleration” which can be important for, e.g., stratified flows or flows with heat release. A fixed-point analysis of direct numerical simulation data of combustion in a wall-jet flow demonstrates that our model gives quantitatively correct predictions of both streamwise and cross-stream components of turbulent density flux as well as their influence on the anisotropies. In summary, we believe that our approach, based on a proper formulation of the rapid pressure-strain correlation and accounting for the coupling with turbulent density flux, can be an important element in CFD tools for compressible flows.

Assessment of turbulence model performance: Large streamline curvature and integral length scales

For the flow over curved surfaces, an extra wall-normal pressure gradient is imposed to the flow through excessive surface pressure, such that the flow turns in alignment with the surface. In turn, turbulent fluctuations are suppressed over the convex surface; whereas, they are enhanced over the concave. Recently, the direct numerical simulation (DNS) of turbulent channel flow experiencing a 60 degree circular bend shows highly complex flow phenomena. Particularly, the mean flow properties are directly related to the channel geometry; in the impulse response of the mean flow to the step change of streamline curvature, sudden changes in mean strain rate and extra rates of strain emerge. This mean flow process is prior to the response of the turbulence structures. Due to the large streamline curvature, the underlying turbulence lagging mechanism and the stress strain misalignment are difficult to model. For this, the new DNS data for the wall bounded flow with high streamline curvature and large integral length scales is used to explore RANS performance. For eddy-viscosity models, this leads to the Boussinesq approximation being questionable. Also, for a Reynolds-stress model (RSM) with closure approximations applicable to homogeneous turbulent flows that are nearly in equilibrium, the current case can result in substantial predictive error. This is because of, for example, the linear approximation for the rapid pressure–strain correlation. To help move towards better turbulence modelling, Reynolds-averaged Navier–Stokes (RANS) predictions are compared for the same flow configuration as the DNS, using some popular turbulent models. These models include the second-order closure with the stress-ω formulation, the standard k−ω and the Menter’s shear-stress transport (SST) models, the standard Spalart–Allmaras (S-A) model with and without the corresponding strain–vorticity correction. As expected, overall, the RSM provides closer predictions to the DNS data than the selected eddy-viscosity models, even though the predictive accuracy needs to be further improved. Potentially, a non-linear constitutive relation or second-order closure, incorporating a relaxation approximation for the lagging mechanism, may lead to a remedy for the current non-equilibrium flow. Moreover, all models would also benefit from sensitisation to the impact of the large integral length scales.

Capturing turbulent density flux effects in variable density flow by an explicit algebraic model

The explicit algebraic Reynolds stress model of Grigoriev et al. [“A realizable explicit algebraic Reynolds stress model for compressible turbulent flow with significant mean dilatation,” Phys. Fluids 25, 105112 (2013)] is extended to account for the turbulent density flux in variable density flows. The influence of the mean dilatation and the variation of mean density on the rapid pressure-strain correlation are properly accounted for introducing terms balancing a so-called “baroclinic” production in the Reynolds stress tensor equation. Applying the weak-equilibrium assumption leads to a self-consistent formulation of the model. The model together with a K − ω model is applied to a quasi-one-dimensional plane nozzle flow transcending from subsonic to supersonic regimes. The model remains realizable with constraints put on the model parameters. When density fluxes are taken into account, the model is less likely to become unrealizable. The density variance coupled with a “local mean acceleration” also can influence the model acting to increase anisotropy. The general trends of the behaviour of the anisotropy and production components under the variation of model parameters are assessed. We show how the explicit model can be applied to two- and three-dimensional mean flows without previous knowledge of a tensor basis to obtain the general solution. Approaches are proposed in order to achieve an approximate solution to the consistency equation in cases when analytic solution is missing. In summary, the proposed model has the potential to significantly improve simulations of variable-density flows.

On the realizability of pressure–strain closures

AbstractThe realizability condition for statistical models of turbulence is augmented to ensure that not only is the Reynolds stress tensor positive semi-definite, but the process of its evolution is physically attainable as well. The mathematical constraints due to this process realizability requirement on the rapid pressure strain correlation are derived. The resulting constraints reveal important limits on the inter-component energy transfer and the consequent flow stability characteristics, as a function of the mean flow. For planar mean flows, the realizability constraints are most stringent for the case of purely sheared flows rather than elliptic flows. The relationship between the constraints and flow stability is explained. Process realizability leads to closure model guidance not only at the two-component (2C) limit of turbulence (as in the classical realizability approach) but throughout the anisotropy space. Consequently, the domain of validity and applicability of current models can be clearly identified for different mean flows. A simple framework for incorporating these process realizability constraints in model formulation is outlined.

A realizable explicit algebraic Reynolds stress model for compressible turbulent flow with significant mean dilatation

The explicit algebraic Reynolds stress model of Wallin and Johansson [J. Fluid Mech. 403, 89 (2000)] is extended to compressible and variable-density turbulent flows. This is achieved by correctly taking into account the influence of the mean dilatation on the rapid pressure-strain correlation. The resulting model is formally identical to the original model in the limit of constant density. For two-dimensional mean flows the model is analyzed and the physical root of the resulting quartic equation is identified. Using a fixed-point analysis of homogeneously sheared and strained compressible flows, we show that the new model is realizable, unlike the previous model. Application of the model together with a K − ω model to quasi one-dimensional plane nozzle flow, transcending from subsonic to supersonic regime, also demonstrates realizability. Negative “dilatational” production of turbulence kinetic energy competes with positive “incompressible” production, eventually making the total production negative during the spatial evolution of the nozzle flow. Finally, an approach to include the baroclinic effect into the dissipation equation is proposed and an algebraic model for density-velocity correlations is outlined to estimate the corrections associated with density fluctuations. All in all, the new model can become a significant tool for CFD (computational fluid dynamics) of compressible flows.

Pressure–Strain Correlation Modeling: Towards Achieving Consistency with Rapid Distortion Theory

In turbulence closure modeling, it is widely accepted that the rapid pressure-strain correlation (RPSC) model be consistent with the rapid distortion theory (RDT). It is desirable to achieve this consistency with a closure model that is computationally tractable and satisfies the requisite mathematical constraints of realizability and linearity in the appropriate variables. In this investigation, starting from a detailed modal analysis of two-dimensional mean flows, we identify important flow features to be incorporated into the model. However, the dynamical system analysis shows that the suggested physics cannot be embodied in a model with all desired computational and mathematical attributes. To resolve this conflict, we propose a slight compromise in the physical requirement and ease one of the linearity constraints leading to a "best possible" tractable model. Overall, the present work provides important insight into RPSC closure modeling challenges—arising from the interplay among physical fidelity, computational viability and mathematical constraints—and proposes avenues for future improvement.

Nonlocal form of the rapid pressure-strain correlation in turbulent flows

A new fundamentally based formulation of nonlocal effects in the rapid pressure-strain correlation in turbulent flows has been obtained. The resulting explicit form for the rapid pressure-strain correlation accounts for nonlocal effects produced by spatial variations in the mean-flow velocity gradients and is derived through Taylor expansion of the mean velocity gradients appearing in the exact integral relation for the rapid pressure-strain correlation. The integrals in the resulting series expansion are solved for high- and low-Reynolds number forms of the longitudinal correlation function f(r), and the resulting nonlocal rapid pressure-strain correlation is expressed as an infinite series in terms of Laplacians of the mean strain rate tensor. This formulation is used to obtain a nonlocal transport equation for the turbulence anisotropy that is expected to provide improved predictions of the anisotropy in strongly inhomogeneous flows.

Research of the rapid pressure-strain correlation model in the rapid distortion limit

Even though a number of rapid pressure-strain models have been suggested and successfully tested for different flow situations by various authors, the model proposals still exhibit some apparent deficiencies when subjected to the flows with rapid distortion. From Mansour’s relatively straightforward rapid distortion analysis, if an initially anisotropic flow undergoes a purely rapid rotation, the anisotropy measures will exhibit the behavior of the damped oscillations. Within the current framework of modeling the rapid pressure-strain correlation, i.e., the models based on the assumption that the M-tensor for the rapid pressure-strain term is expandable in the Reynolds-stress anisotropy tensor alone, all the model predictions fail to give the damped oscillations in the turbulence anisotropy. In the case of initially isotropic turbulence subjected to rapid distortion, Sjögren and Johansson showed that all the existing rapid pressure-strain models would deliver the identical path in the anisotropy-invariant map for both homogeneous plane strain and shear flows. The rapid distortion analysis shows two distinct curves reflecting different flow physics. In this work, we try to present a possible way to create a system that can overcome these deficiencies with the aid of the rapid distortion theory (RDT).

Analyzing the influence of compressibility on the rapid pressure–strain rate correlation in turbulent shear flows

The influence of compressibility on the rapid pressure–strain rate tensor is investigated using the Green’s function for the wave equation governing pressure fluctuations in compressible homogeneous shear flow. The solution for the Green’s function is obtained as a combination of parabolic cylinder functions; it is oscillatory with monotonically increasing frequency and decreasing amplitude at large times, and anisotropic in wave-vector space. The Green’s function depends explicitly on the turbulent Mach number M t , given by the root mean square turbulent velocity fluctuations divided by the speed of sound, and the gradient Mach number M g , which is the mean shear rate times the transverse integral scale of the turbulence divided by the speed of sound. Assuming a form for the temporal decorrelation of velocity fluctuations brought about by the turbulence, the rapid pressure–strain rate tensor is expressed exactly in terms of the energy (or Reynolds stress) spectrum tensor and the time integral of the Green’s function times a decaying exponential. A model for the energy spectrum tensor linear in Reynolds stress anisotropies and in mean shear is assumed for closure. The expression for the rapid pressure–strain correlation is evaluated using parameters applicable to a mixing layer and a boundary layer. It is found that for the same range of M t there is a large reduction of the pressure–strain correlation in the mixing layer but not in the boundary layer. Implications for compressible turbulence modeling are also explored.

A new perspective on realizability of turbulence models

In the second-moment-closure (SMC) method of turbulence modelling, measures to ensure a realizable turbulence model are currently limited to constraining the Reynolds stress to physically plausible values. These constraints address neither the realizability of the other statistical moments (e.g. pressure–strain correlation) nor the underlying causes of unrealizable Reynolds stress. For achieving increased consistency with flow physics in SMC, we propose the additional requirement that the closure model for each of the unclosed statistical moments in the Reynolds stress equation be individually realizable. We then proceed to derive two realizability constraints on the rapid-pressure statistics: (i) the rapid pressure-gradient variance must be positive which leads to the requirement that the tensor must be positive semi-definite, and (ii) the rapid pressure–strain correlation closure must satisfy the Schwarz inequality. Calculations with currently popular models show that unrealizable rapid-pressure–strain correlation precedes unrealizable Reynolds stress. It is also demonstrated that when the Launder, Reece and Rodi (LRR) rapid-pressure–strain correlation model is modified (truncated) to satisfy the new constraints, Reynolds stress realizability is always preserved. These findings clearly indicate that an unrealizable closure model is the cause of Reynolds stress realizability violation and highlight the importance of the new constraints.

Turbulence modeling for the axially rotating pipe from the viewpoint of analytical closures

A single-point model eddy viscosity model of rotation effects on the turbulent flow in an axially rotating pipe is developed based on two-point closure theories. Rotation is known to impede energy transfer in turbulence; this fact is reflected in the present model through a reduced eddy viscosity, leading to laminarization of the mean velocity profile and return to a laminar friction law in the rapid rotation limit. This model is compared with other proposals including linear redistribution effects through the rapid pressure-strain correlation, Richardson number modification of the eddy viscosity in a model of non-rotating turbulence, and the reduction of turbulence through the suppression of near-wall production mechanisms.

Study on the rapid pressure–strain rate in the second-moment closure for turbulent flows undergoing strong compression and large distortion

To clarify applicability of the existing models for the rapid pressure–strain correlation in the Reynolds-stress closures, modifications accounting for the nonzero divergence of the mean velocity have been set into five turbulence models. These are two models of Launder et al. (LRR1 and LRR2), model of Speziale et al. (SSG), model of Johansson et al. and model of Lee. Computations have been carried out with these models for compression process in the cylinder of an internal combustion engine. The calculation results of the mean turbulent kinetic energy, the anisotropic tensor of Reynolds stress and rapid pressure–strain have been compared with the analytical solutions of rapid distortion theory in order to identify a suitable model for turbulent flows undergoing strong compressions as in the engines. The results demonstrate that the linear model of Lee which includes the history effect of the total strain can even give the same accurate predictions as those of the Johansson's nonlinear models up to fourth order in the Reynolds-stress-anisotropic tensor.

Modeling the rapid pressure-strain correlation in homogeneous flows

For homogeneous flows the rapid pressure-strain correlation tensor can be written in terms of the mean velocity gradient and an integral of the two-point correlation tensor. In the present study the two-point correlation tensor is modeled and analytically integrated to obtain the pressure-strain model coefficients, which can then be expressed in terms of closed integrals of the two-point correlation model coefficients. Therefore, deficiencies of the pressure-strain models can be detected and resolved on the two-point correlation level. As an example for the new approach, direct numerical simulations results for a homogeneous shear flow are analyzed in detail and the resulting coefficients are compared to existing model coefficients for linear and nonlinear pressure-strain models.