Pressure-strain Correlation Model Research Articles

Modeling the pressure strain correlation in turbulent flows using deep neural networks

The pressure strain correlation plays a critical role in the Reynolds stress transport modeling. Accurate modeling of the pressure strain correlation leads to the proper prediction of turbulence stresses and subsequently the other terms of engineering interest. However, classical pressure strain correlation models are often unreliable for complex turbulent flows. Machine learning–based models have shown promise in turbulence modeling, but their application has been largely restricted to eddy viscosity–based models. In this article, we outline a rationale for the preferential application of machine learning and turbulence data to develop models at the level of Reynolds stress modeling. As an illustration, we develop data-driven models for the pressure strain correlation for turbulent channel flow using neural networks. The input features of the neural networks are chosen using physics-based rationale. The networks are trained with the high-resolution DNS data of turbulent channel flow at different friction Reynolds numbers (Reλ). The testing of the models is performed for unknown flow statistics at other Reλ and also for turbulent plane Couette flows. Based on the results presented in this article, the proposed machine learning framework exhibits considerable promise and may be utilized for the development of accurate Reynolds stress models for flow prediction.

Assessment of turbulence models using DNS data of compressible plane free shear layer flow

The present paper uses the detailed flow data produced by direct numerical simulation (DNS) of a three-dimensional, spatially developing plane free shear layer to assess several commonly used turbulence models in compressible flows. The free shear layer is generated by two parallel streams separated by a splitter plate, with a naturally developing inflow condition. The DNS is conducted using a high-order discontinuous spectral element method (DSEM) for various convective Mach numbers. The DNS results are employed to provide insights into turbulence modelling. The analyses show that with the knowledge of the Reynolds velocity fluctuations and averages, the considered strong Reynolds analogy models can accurately predict temperature fluctuations and Favre velocity averages, while the extended strong Reynolds analogy models can correctly estimate the Favre velocity fluctuations and the Favre shear stress. The pressure–dilatation correlation and dilatational dissipation models overestimate the corresponding DNS results, especially with high compressibility. The pressure–strain correlation models perform excellently for most pressure–strain correlation components, while the compressibility modification model gives poor predictions. The results of an a priori test for subgrid-scale (SGS) models are also reported. The scale similarity and gradient models, which are non-eddy viscosity models, can accurately reproduce SGS stresses in terms of structure and magnitude. The dynamic Smagorinsky model, an eddy viscosity model but based on the scale similarity concept, shows acceptable correlation coefficients between the DNS and modelled SGS stresses. Finally, the Smagorinsky model, a purely dissipative model, yields low correlation coefficients and unacceptable accumulated errors.

A Reliable Pressure Strain Correlation Model for Complex Turbulent Flows

A Reliable Pressure Strain Correlation Model for Complex Turbulent Flows

Experimental and numerical analysis of grid generated turbulence with and without mean strain

This paper presents experimental and numerical analysis of grid generated turbulence with and without the effects of applied mean strain. We conduct a series of experiments on decaying grid generated turbulence and grid turbulence with mean strain. Experimental data of turbulence statistics including Reynolds stress anisotropies is collected, analyzed and then compared to the predictions of Reynolds Stress Models to assess their accuracy. The experimental data is used to evaluate the variability in the coefficients of the rate of dissipation model and the pressure strain correlation models used in Reynolds Stress Modeling. For both models we recommend optimal values of coefficients that should be used for experimental studies of grid generated turbulence.

Toward approximating non-local dynamics in single-point pressure–strain correlation closures

A key hurdle in turbulence modelling is the closure for the pressure–strain correlation. Herein, the challenge stems from the fact that the non-local dynamics due to pressure cannot be comprehensively incorporated in a single-point closure expression. In this article, we analyse different aspects of the dynamics due to pressure for their amenability with the single-point modelling framework. Based on this, an approach is proposed that incorporates a set of pragmatic compromises in the form and the scope of the model to augment the degree of non-local dynamics that may be approximated by a single-point pressure strain correlation model. Thence, this framework is utilized to formulate an illustrative model. The predictions of this model are compared to numerical and experimental data and contrasted against other established closures over a range of homogeneous flows, under diverse conditions. Finally, the regions of validity in anisotropy space for this illustrative model are delineated using the process realizability criteria for different flows.

Progress in the extension of a second-moment closure for turbulent environmental flows

An advanced second-moment closure for the double-averaged turbulence equations of porous medium and vegetation flows is proposed. It treats three kinds of second moments which appear in the double-averaged momentum equation. They are the dispersive covariance, the volume averaged (total) Reynolds stress and the micro-scale Reynolds stress. The two-component-limit pressure–strain correlation model is applied to model the total Reynolds stress equation whilst a novel scale-similarity non-linear k–ε two-equation eddy viscosity model is employed for the micro-scale turbulence. For the dispersive covariance, an algebraic relation is applied. Model validation in several fully developed homogeneous porous medium flows, porous channel flows and aquatic vegetation canopy flows is performed with satisfactory agreement with the data.

The Effect of a Pressure-Containing Correlation Model on Near-Wall Flow Simulations with RST Models.

It is accustomed to think that turbulence models based on solving the Reynolds-Averaged Navier-Stokes equations require empirical functions to accurately reproduce the behavior of flow characteristics of interest, particularly near a wall. The current paper analyzes how choosing a model for pressure-strain correlations in second-order closures affects the need for introducing empirical functions in model equations to reproduce the flow behavior near a wall correctly. An axially-rotating pipe flow is used as a test flow for the analysis. Results of simulations demonstrate that by using more physics-based models to represent pressure-strain correlations, one can eliminate wall functions associated with such models. The higher the Reynolds number or the strength of imposed rotation on a flow, the less need there is for empirical functions regardless of the choice of a pressure-strain correlation model.

Explicit algebraic Reynolds stress model (EARSM) for compressible shear flows

We develop an explicit algebraic Reynolds stress model (EARSM) for high-speed compressible shear flows and validate the model with direct numerical simulation (DNS) data of homogeneous shear flow and experimental data of high-speed mixing-layers. Starting from a pressure–strain correlation model that incorporates compressibility effects, the weak-equilibrium assumption is invoked to derive the EARSM closure expression. The resulting closure is fully explicit and physically realizable and is a function of mean flow strain rate, rotation rate, turbulent kinetic energy, dissipation rate, and gradient Mach number. Homogeneous shear flow calculations show that the model captures the asymptotic behavior of DNS quite well. Linear EARSM calculations of a plane supersonic mixing-layer are performed, and comparison with experimental data shows good agreement. Salient results are agreement of streamwise velocity similarity profiles, mixing-layer spreading rates, and capturing the Langley curve trend.

A compressibility correction of the pressure strain correlation model in turbulent flow

This paper is devoted to the second-order closure for compressible turbulent flows with special attention paid to modeling the pressure–strain correlation appearing in the Reynolds stress equation. This term appears as the main one responsible for the changes of the turbulence structures that arise from structural compressibility effects. From the analysis and DNS results of Simone et al. and Sarkar, the compressibility effects on the homogeneous turbulence shear flow are parameterized by the gradient Mach number. Several experiment and DNS results suggest that the convective Mach number is appropriate to study the compressibility effects on the mixing layers. The extension of the LRR model recently proposed by Marzougui, Khlifi and Lili for the pressure–strain correlation gives results that are in disagreement with the DNS results of Sarkar for high-speed shear flows. This extension is revised to derive a turbulence model for the pressure–strain correlation in which the compressibility is included in the turbulent Mach number, the gradient Mach number and then the convective Mach number. The behavior of the proposed model is compared to the compressible model of Adumitroiae et al. for the pressure–strain correlation in two turbulent compressible flows: homogeneous shear flow and mixing layers. In compressible homogeneous shear flows, the predicted results are compared with the DNS data of Simone et al. and those of Sarkar. For low compressibility, the two compressible models are similar, but they become substantially different at high compressibilities. The proposed model shows good agreement with all cases of DNS results. Those of Adumitroiae et al. do not reflect any effect of a change in the initial value of the gradient Mach number on the Reynolds stress anisotropy. The models are used to simulate compressible mixing layers. Comparison of our predictions with those of Adumitroiae et al. and with the experimental results of Goebel et al. shows good qualitative agreement.

Invariant-based calibration of coefficients in nonlinear eddy viscosity turbulence closure

Based on the standard quasi-linear pressure–strain correlation model a nonlinear eddy viscosity turbulence closure is developed. In two-dimensional mean-flows it is uniquely determined by three model coefficients. The model coefficients are allowed to adapt to flow conditions described by the invariants of mean-flow deformation and Reynolds stress anisotropy. The adaption is done in a straightforward and inexpensive manner. In the rapid distortion limit the coefficients are based on recent numerical and theoretical predictions for initially anisotropic turbulence. The model performance is furthermore calibrated against DNS data of canonical flows of engineering and geophysical interest. Those are the boundary layer of channel flow, homogeneous shear flow and Ekman flow. The resulting turbulence model is demonstrated to be superior to those based on well-known pressure–strain correlation models. An interpolation procedure is suggested to cover the entire Reynolds stress anisotropy-invariant map.

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).

A simplified explicit algebraic model for the Reynolds stresses

A simplified consistency condition for the production-to-dissipation ratio is developed to obtain an explicit algebraic Reynolds stress model. The stress expression is based on the well-known tensor representation derived from quasi-linear pressure–strain correlation models for two-dimensional mean flows in the weak-equilibrium limit. Results obtained for homogeneous turbulent flows, a backward-facing step flow, and a channel bend flow indicate that the model combines improved accuracy with computational efficiency.

Réalisabilité d'un modèle non-linéaire de la corrélation pression-température

In this work, we propose a non linear model for the return to isotropy term of the pressure-temperature correlation as a development of a fifth power of the Reynolds stress anisotropy . This model is an essay of extension of the recent model of Chung & Kim that is retained for the slow pressure-strain correlation. The study of the stability of the equilibrium states of a homogeneous turbulence, in the absence of mean gradients, leads to a simple condition on one of the coefficients of the proposed model. Moreover, a study of realisability has been developed. We demonstrate that a sufficient condition of realisability of the proposed model is obtained when its coefficients are expressed according to those of the pressure-strain correlation model. Finally, a numerical simulation on the basis of the direct simulation results of Iida & Kasagi, has shown that the proposed model ensures a better prediction of the thermal turbulence with respect to Rotta's classic model.

3D numerical simulation of compound open-channel flow with vegetated floodplains by reynolds stress model

This paper presents a Reynolds stress modeling of compound open-channel flows with vegetation on the floodplain. In the Reynolds stress model, we use the SSG model by Spezialeet al., for the pressure-strain correlation term, Mellor and Herring's model for the turbulent diffusion term, and Hanjalic and Launder's model for the dissipation term. In order to take into account the anisotropy of turbulence due to the free surface, the combination of Shir's model and Gibson and Launder's model is included in the pressure-strain correlation model. Model validations are carried out for the compound open-channel flows without vegetation. Then, the model is applied to the compound open-channel flows with vegetated floodplains. The mean flow and turbulence structures are simulated and the impact of vegetation on the floodplains is investigated.

Pressure-strain correlation in homogeneous anisotropic turbulence subject to rapid strain-dominated distortion

Rapid distortion calculations of initially anisotropic turbulence are performed to better understand the physics of the pressure-strain correlation in strain-dominated mean flows. Based on the results of simulations we infer important physical characteristics of the “rapid” pressure-strain correlation Φij(r) in such flows: (i) it vanishes when there is no production of anisotropy, (ii) in the proximity of two-componential state it tends to decrease Reynolds stress anisotropy, and (iii) its magnitude is generally smaller than that of production. The observed characteristics are proposed as criteria that pressure-strain correlation models may be required to satisfy. All of the current popular models violate the above criteria for a sizeable subset of anisotropic initial conditions. Reynolds stress transport model calculations show that unphysical and unrealizable model behavior can be directly attributed to these violations.

Pressure–strain correlation modelling of complex turbulent flows

A methodology for deriving a pressure–strain correlation model with variable coefficients is developed. The methodology is based on two important premises: (i) the extreme states of turbulence – the rapid distortion and equilibrium limits – are more amenable to mathematically rigorous modelling because of significant simplifications not possible at other states; and (ii) the models of the extreme states collectively contain all of the relevant physics so that models for any intermediate state can be obtained by suitable interpolation. A pressure–strain model of the standard form is considered and the coefficients are determined from linear analysis in the rapid distortion limit and from a fixed point analysis in the equilibrium limit. The model coefficients, which depend on the mean deformation and turbulence state, vary from flow to flow in a manner consistent with Navier–Stokes physics.The exact causal relationship between the model coefficients and the model's equilibrium behaviour is established by fixed point analysis performed using representation theory. Then, the equilibrium values of the model coefficients are chosen to yield the observed equilibrium behaviour. The values of the model coefficients in the rapid distortion limit are determined by enforcing consistency with the Crow constraint. The new variable-coefficient model reduces to the traditional constant-coefficient model in strain-dominated turbulent flows near equilibrium. The model performance in bench-mark turbulent flows, in which the traditional models have been calibrated extensively, is preserved intact. The new model is significantly different from the traditional one in mean vorticity-dominated and non-equilibrium turbulence. These two important classes of flows, in which traditional models fail, are successfully captured by the new model.

Modelling of pressure-strain correlations for Taylor-Proudman turbulence

The theoretical foundation for the modelling of the turbulent pressure-strain correlations in rotating reference frame is presented. Based on the recently developed materially-frame indifference principle it is observed that most of the existing second-moment closures violate the Taylor-Proudman theorem in the limit of rapid rotation. It is shown that the application of the materially-frame indifference principle gives rise to the formation of a new pressure-strain correlation model which satisfies the Taylor-Proudman theorem.

Numerical Calculation of the Fully Developed Turbulent Flow in an Axially Rotating Pipe With a Second-Moment Closure

The effect of axial tube rotation on the fully developed pipe flow is analyzed by a low-Reynolds-number turbulence closure and compared with experimental results. A flow which is initially turbulent is stabilized by the rotation leading to a laminarized mean axial velocity distribution. The applied second-moment closure reveals an encouraging ability to capture this phenomenon as well as other features of the mentioned flow configuration. The “rapid” part of the pressure-strain correlation model is found to have a significant influence on the numerical results and seems to be the key for further improvements concerning highly swirling flows.

Assessment of pressure-strain models in predicting compressible, turbulent ramp flows

The study presented has the objective of isolating the effect of the pressure-strain correlation term in the prediction of a compression ramp flow, where shock-boundary-layer interaction had a significant effect on turbulence amplification. Variable density extensions of three incompressible pressure-strain models were tested to assess their performance. The study showed that all of the models had difficulty in predicting the rapid response of the turbulence to the presence of the shock and showed improper distribution among stress components. These results imply that the variable-density extensions of incompressible pressure-strain correlation models are insufficient in themselves to properly account for the effects of turbulence amplification through a shock. 14 refs.

Comparison of pressure-strain correlation models for the flow behinda disk

Attention is given to the behavior of Reynolds stresses in the separated wake region behind a disk that is attached in a normal fashion to a long cylinder of small diameter. Computations of the turbulent flow were made in a region beyond a disk by using the second-order closure model of turbulence. It is found that the models of Naot et al. (1970) and Launder et al. (1975) yield similar results and are reliable; the energy distribution may nevertheless be improved for the case of reattaching shear flows by taking the effects of mean strain into account.