Pressure-strain Correlation Research Articles

Assessment of RANS-based turbulence models for isothermal confined swirling flow in a realistic can-type gas turbine combustor application

The present study assesses RANS-based turbulence models to simulate the isothermal confined swirling flow in a combustor representing a constituent can combustor of the can-annular configuration used in jet engines. The two-equation models (standard k−ϵ, realizable k−ϵ, standard k−ω, SST k−ω), and seven-equation model (linear pressure strain-Reynolds stress model, LPS-RSM), are assessed by comparing their predictions of mean axial velocity, mean transverse velocity, turbulent kinetic energy, and shear stress with the experimental data at two different positions (i.e., the primary and dilution hole planes) in the combustor. While the two-equation models generally have failed to predict the confined swirling flow at both positions accurately, the SST k−ω model yielded the most accurate, followed by standard k−ω and realizable k−ϵ models. The discrepancies between the computational and experimental results could be attributed to the isotropic turbulence assumptions, which, however, are invalid for confined swirling flows. Further, the two-equation model formulations cannot capture the intricacies of vortex flow and its interaction with the surroundings in confined swirling flows. The LPS-RSM, which considers turbulence anisotropy, showed some promise, although overpredicted results follow the trend with experimental values at the primary holes plane. However, at the dilution holes plane, the model overpredicted the velocity field, and underestimated the turbulence field, including turbulent kinetic energy and shear stress. These observed discrepancies can be ascribed to the pressure-strain correlation in the LPS-RSM, which assumes the pressure is a linear function of the strain-rate tensor. However, this linear assumption is quite simplistic for complex flows. Further, the influence of discretization (SOU and third-order MUSCL) schemes of convective terms is also assessed, and the differences in predictions resulted from MUSCL scheme having lower diffusion and superior ability to capture sharper gradients, however, did not translate into improving the solution accuracy. Hence, this study suggests that more advanced high-fidelity turbulence models (e.g., hybrid RANS-LES, LES, DNS) are needed to accurately predict the confined swirling flow in combustors.

Axial friction coefficient of turbulent spiral Poiseuille flows

Direct numerical simulations of spiral Poiseuille flows in a narrow gap geometry are performed with the aim of identifying the mechanisms governing the dynamics of the axial friction coefficient. The investigation has explored a small portion of the Reynolds number–Taylor number phase space ( $600 \leq Re \leq 5766$ and $1500 \leq Ta \leq 5000$ ), for which reference experimental results are available. The study is focused on the mechanism leading to the enhancement of the axial friction coefficient with the Taylor number when the Reynolds number is kept constant. The analysis of the spatial distribution of the Reynolds stress tensor and of the turbulent energy budget has evidenced the key role of the pressure–strain correlation in the energy transfer from the azimuthal to the axial component. The latter eventually determines the increase of the axial friction coefficient through the enhanced radial mixing of axial momentum. Data have also shown that the flow dynamics is heavily dependent on the $Ta/Re$ ratio, and different regimes develop (ranging from laminar to turbulent), each with peculiar behaviours.

Spectra scaling of velocity components and pressure–strain correlations for hypersonic boundary layers at a range of wall-to-recovery temperatures

Flow similarity is one of the most desirable features for turbulence under different conditions. For hypersonic boundary layers, different wall temperatures are known to change the turbulence evolution by influencing the intercomponent energy transfer, while the flow similarity has received less attention. Based on direct numerical simulations, this work investigates the spectral distribution of velocity components and pressure–strain correlations for hypersonic boundary layers at Mach 6 and wall-to-recovery temperature ratios equal to 0.3, 0.5, 0.76, and 1. The wall-normal evolution of turbulent structures is compared by the spectra peak scale λSP, wall-normal location ySP, and the vertical profiles of peak scale λP. For all cases, the peak scale profiles exhibit linear increases with a height of 0.1≤y/δ≤0.25. The linear growth of the turbulence scale in the logarithmic region indicates the existence of self-similar structures, and the agreement of peak scales in the outer scaling suggests the flow similarity under different wall temperatures. Likewise, the streamwise pressure–strain term has the peak scale profile linearly dependent on the height for the same vertical range 0.1≤y/δ≤0.25 with velocity components, while the wall-normal and spanwise terms exhibit both linear and constant vertical dependence depending on the wall temperatures. With the increase in wall temperature, the peak scale exhibits broader constant vertical dependence and narrower linear vertical dependence. The cospectral analysis further confirms that the pressure–strain correlations are maximized at different spanwise-vertical aspect ratios of turbulence structures.

Enhancing turbulence modeling: Machine learning for pressure-strain correlation and uncertainty quantification in the Reynolds stress model

This research study delves into the realm of enhancing the Reynolds stress model through the remarkable capabilities of machine learning. Focusing on two key aspects, we investigate the modeling of fluctuating pressure-rate-of-strain tensor and uncertainty quantification in turbulent fluid flows. In the first part of our investigation, a neural network is employed to predict the pressure-strain correlation term for the frictional Reynolds number, even in cases where it was not trained. Through the use of high-fidelity datasets, we surpass the limitations of traditional numerical methods, achieving more accurate results with reduced computational costs. Going beyond conventional analysis, we explore the isotropic and anisotropic components of the pressure-strain correlation term. Intriguingly, our findings unveil the significant influence of production and redistribution terms on the anisotropic component by 50% and viscous dissipation impacts the isotropic part by 65% in wall-bounded turbulent flows. The second part introduces a novel approach for uncertainty quantification in the Reynolds stress model. Leveraging a binary classification model, we identify error-prone points within the simulation domain with an impressive accuracy of up to 95%. This critical step paves the way for a deeper understanding of uncertainties in turbulent flows for more reliable predictions. By harnessing the power of machine learning, our study provides two compelling ways to enhance the Reynolds stress model, contributing to a more efficient and dependable prediction of turbulent fluid dynamics.

Large-eddy simulation of free-surface turbulent channel flow over square bars

The results of large-eddy simulations of free-surface turbulent channel flow over spanwise-aligned square bars are used to investigate the effects of bed roughness and water surface deformations on the root-mean-squared velocity fluctuations, dispersive shear stress, double-averaged Reynolds shear stress, wake kinetic energy and double-averaged turbulent kinetic energy. Two bar spacings, corresponding to transitional and k-type roughness, at similar Reynolds and Froude numbers are considered. The main peak of all statistical quantities occurs at the bar crest height. The effects of a standing wave at the water surface in flow over k-type roughness is marked by a local peak under the water surface for all statistical quantities considered here except wall-normal and spanwise velocity fluctuations. Quadrant analysis shows that sweeps and ejections are the strongest events contributing to both dispersive and double-averaged Reynolds shear stress but their contributions are different for the two bar spacings. Examining the budgets of dispersive and double-averaged Reynolds shear stress reveals that the dominant terms of these stresses are pressure–strain correlation and pressure transport and the contribution of wake production is similar for both of these stresses but with opposite sign. In addition to the main role of the bars in consuming or producing wake kinetic energy through production and transport and convection, the standing wave at the water surface in flow over k-type roughness induces large convection in the bulk flow too. The dominant terms in the double-averaged turbulent kinetic energy budget are similarly production, transport and convection. Large shear production renders large temporal fluctuations than spatial fluctuations of flow variables. The interaction of bars, bed and water surface is seen in the convection term in flow over k-type roughness.

A turbulence modeling study of the precessing vortex core in a pipe flow

The phenomenon of precessing vortex core is observed experimentally when swirl is imparted on an axial flow in a pipe. It manifests as a coherent structure in the form of a helical vortex of regular wavelength whose axis is coincident with the pipe’s axis. The most striking consequence of this pattern of flow is the generation of periodic fluctuations in the streamwise distribution of the wall static pressure and skin friction. While the prediction of the precessing vortex has proved possible with large-eddy simulations, there is no record of this phenomenon being captured in great detail by Reynolds-averaged Navier-Stokes methods utilizing turbulence models to close the time-averaged equations. The purpose of the research reported here was to determine whether the precessing vortex core and its impact on conditions at the wall can be captured using this approach. The turbulence model used was of the Reynolds-stress transport type which involves the solution of a differential transport equation for each of the six non-zero components of the Reynolds stress tensor. Previous studies in which such models were used in the prediction of rotating and swirling flows have shown that their performance is largely determined by the way in which the difficult fluctuating pressure-strain correlations that appear in these equations are modeled. To this end, four very different alternative models for these correlations were assessed by comparisons with experimental data for a swirling pipe flow at relatively high swirl number. It was found that the models do indeed capture the precessing vortex revealing it to be in the form of an exceptionally well-defined double vortex. It was also found that the expected periodic fluctuations in static pressure and wall skin friction, not previously obtained in turbulence model studies, are captured by the present closures.

A compressible turbulence model for the dissipation rate

In this work, the ability of a Reynolds stress model to compute turbulent homogeneous shear flow with significant compressibility effects is discussed. Several studies of compressible turbulent flows carried out in the past years have shown that the pressure strain correlation is mainly responsible for the strong changes in the magnitude of the Reynolds stress anisotropies. Two recent compressible models of this term are considered in conjunction with the standard model of the dissipation rate of the turbulent kinetic energy to predict compressible homogeneous flow highly sheared are tested. It is found that deficiencies appear in the calculations even if the pressure strain model is improved by compressibility corrections. Consistent with earlier studies, this deficiency is attributed to the use of the incompressible model for turbulent dissipation. However, a compressibility correction of this equation model uncovers the main focus of the present study. This correction makes the standard coefficients of this equation depend on the turbulent gradient Mach and Mach numbers. The proposed model is tested for low and strong compressibility cases from the DNS results of Sarkar. A comparison of the proposed model predictions with the DNS results shows good qualitative agreement. Therefore, compressibility correction for the incompressible model of the turbulent dissipation rate is found to be an important issue for the compressible homogeneous turbulent shear flow.

Prandtl number effects on the hydrodynamic stability of compressible boundary layers: flow–thermodynamics interactions

Hydrodynamic stability of compressible boundary layers is strongly influenced by the Mach number ( $M$ ), Prandtl number ( $Pr$ ) and thermal wall boundary condition. These effects manifest on the flow stability via the flow–thermodynamics interactions. Comprehensive understanding of stability flow physics is of fundamental interest and important for developing predictive tools and closure models for integrated transition-to-turbulence computations. The flow–thermodynamics interactions are examined using linear analysis and direct numerical simulations in the following parameter regime: $0.5 \leq M \leq 8$ ; and $0.5 \leq Pr \leq 1.3$ . For the adiabatic wall boundary condition, increasing Prandtl number has a destabilizing effect. In this work, we characterize the behaviour of production, pressure–strain correlation and pressure dilatation as functions of the Mach and Prandtl numbers. First and second instability modes exhibit similar stability trends but the underlying flow physics is shown to be diametrically opposite. The Prandtl number influence on instability is explicated in terms of the base flow profile with respect to the different perturbation mode shapes.

On the Non-monotonic Variation of the Entrainment Buoyancy Flux with Wind Shear

The magnitude of the entrainment buoyancy flux, and hence the growth rate of the convective boundary layer, does not increase monotonically with wind shear. Explanations for this have previously been based on wind-shear effects on the turbulence kinetic energy. By distinguishing between turbulent and non-turbulent regions, we provide an alternative explanation based on two competing wind-shear effects: the initial decrease in the correlation between buoyancy and vertical velocity fluctuations, and the increase in the turbulent area fraction. The former is determined by the change in the dominant forcing; without wind shear, buoyancy fluctuations drive vertical velocity fluctuations and the two are thus highly correlated; with wind shear, vertical velocity fluctuations are partly determined by horizontal velocity fluctuations via the transfer of kinetic energy through the pressure–strain correlation, thus reducing their correlation with the buoyancy field. The increasing turbulent area fraction, on the other hand, is determined by the increasing shear production of turbulence kinetic energy inside the entrainment zone. We also show that the dependence of these conditional statistics on the boundary-layer depth and on the magnitude of the wind shear can be captured by a single non-dimensional variable, which can be interpreted as an entrainment-zone Froude number.

Evaluation of machine learning algorithms for predictive Reynolds stress transport modeling

The application of machine learning (ML) algorithms to turbulence modeling has shown promise over the last few years, but their application has been restricted to eddy viscosity based closure approaches. In this article, we discuss the rationale for the application of machine learning with high-fidelity turbulence data to develop models at the level of Reynolds stress transport modeling. Based on these rationales, we compare different machine learning algorithms to determine their efficacy and robustness at modeling the different transport processes in the Reynolds stress transport equations. Those data-driven algorithms include Random forests, gradient boosted trees, and neural networks. The direct numerical simulation (DNS) data for flow in channels are used both as training and testing of the ML models. The optimal hyper-parameters of the ML algorithms are determined using Bayesian optimization. The efficacy of the above-mentioned algorithms is assessed in the modeling and prediction of the terms in the Reynolds stress transport equations. It was observed that all three algorithms predict the turbulence parameters with an acceptable level of accuracy. These ML models are then applied for the prediction of the pressure strain correlation of flow cases that are different from the flows used for training, to assess their robustness and generalizability. This explores the assertion that ML-based data-driven turbulence models can overcome the modeling limitations associated with the traditional turbulence models and ML models trained with large amounts of data with different classes of flows can predict flow field with reasonable accuracy for unknown flows with similar flow physics. In addition to this verification, we carry out validation for the final ML models by assessing the importance of different input features for prediction.

The importance of turbulent equilibrium for Reynolds-stress modeling

Turbulence equilibrium state is analyzed for the modeled Reynolds-stress transport equation, assuming the most general formulation of pressure–strain correlation. In a two-dimensional mean flow at a high-Reynolds number, an algebraic equation system is obtained, providing Reynolds-stress anisotropies as functions of pressure–strain model coefficients. Conversely, the equations provide calibration conditions for the model coefficients to predict specified equilibrium anisotropies. The predicted von-Kármán constant depends on the predicted equilibrium anisotropies and, hence, the pressure–strain model coefficients. Identical equilibrium anisotropies can be obtained with different sets of model coefficients. Identical equilibrium values of invariants of the Reynolds-stress anisotropy tensor can be achieved, despite the differing anisotropy components. Numerical simulations with the Speziale–Sarkar–Gatski (SSG) model, using different sets of model coefficients, confirm the results of the theoretical analysis. They show that the predicted equilibrium value of the Reynolds-shear stress anisotropy determines the predicted skin friction of a boundary layer as well as the predicted spreading rate of a plane mixing layer. However, different values and, hence, different sets of model coefficients are required for achieving good agreement with experimental data for both flows. Therefore, for general improvement of turbulence models, the set of model coefficients probably needs to be adapted to the local type of flow. The required classification is supposed to be suitable for machine learning methods.

A Compressible Turbulence Model for Pressure—Strain

This work focuses on the performance and validation of compressible turbulence models for the pressure-strain correlation. Considering the Launder Reece and Rodi (LRR) incompressible model for the pressure-strain correlation, Adumitroaie et al., Huang et al., and Marzougui et al., used different modeling approaches to develop turbulence models, taking into account compressibility effects for this term. Two numerical coefficients are dependent on the turbulent Mach number, and all of the remaining coefficients conserve the same values as in the original LRR model. The models do not correctly predict the compressible turbulence at a high-speed shear flow. So, the revision of these models is the major aim of this study. In the present work, the compressible model for the pressure-strain correlation developed by Khlifi−Lili, involving the turbulent Mach number, the gradient, and the convective Mach numbers, is used to modify the linear mean shear strain and the slow terms of the previous models. The models are tested in two compressible turbulent flows: homogeneous shear flow and the newly developed plane mixing layers. The predicted results of the proposed modifications of the Adumitroaie et al., Huang et al., and Marzougui et al., models and of its universal versions are compared with direct numerical simulation (DNS) and experiment data. The results show that the important parameters of compressibility in homogeneous shear flow and in the mixing layers are well predicted by the proposal models.

A Comparative Study for Compressible Turbulence Models

The incompressible models for the pressure-strain correlation are unable to correctly predict the turbulence flows evolving with significant compressibility. Huang and Fu use a damping function of the turbulent Mach number to modify two numerical coefficients of the incompressible model for the pressure strain developed by Launder, Reece and Rodi. This model predicts the spreading rate and the shear stress behavior in compressible turbulent mixing well. However, the model does not show the well-known compressibility effects on the compressible homogenous shear flow. In the present work, the model of Huang-Fu is revised, all resulting model coefficients become dependent on the turbulent Mach number, the gradient Mach number and the convective Mach number. The proposed model is tested in different compressible turbulent homogeneous shear flow and mixing layers cases. In general, the predicted results from the proposed model are in an acceptable agreement with DNS and experiment data.

Experimental observation cycles of the generation and breakdown for turbulent stripes in plane Couette flow

In this study, visualization experiments of plane Couette flow were conducted to investigate the spatio-temporal variation of turbulent stripes, the spanwise wavelength, and the process of their generation and breakdown. The experiments were performed with three different film half-widths (δ = 2.90 mm, 3.39 mm, and 4.30 mm). Two cycles of generation and breakdown of turbulent stripe were successfully captured. We defined long-life and short-life cycles based on the order in their appearance, respectively. For each film half-width, the survival time of turbulent stripes, that appeared in the long-life cycle, was t* = 1620, 1960, and 556, respectively, and the number of turbulent stripes was confirmed as 2, 1, and 1. The elapsed time t* since the observation started was non-dimensionalized by the film velocity and half-width. Similarly, the survival time of turbulent stripes, that appeared in short-life cycle, was t* = 970, 1240, and 0, respectively, and the number of bands was 5, 1, and 0. In the case of a film half-width of 2.90 mm, five turbulent stripes were visualized, the largest number in this study. Based on the results, it was found that the number of turbulent stripes decreases and the spanwise wavelength increases as the film half-width increases. This tendency is not considered to be organized on an outer scale, given the standard deviations of the film half-widths. Considering the distribution ratio of the pressure-strain correlation term or friction velocity in turbulent stripes of numerical simulations in previous studies, the tendency for differences in the number of turbulent stripes and spanwise wavelengths at each film half-width to be organized on a viscous scale is considered reasonable.

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.

Analyses of wave-phase variation of Reynolds shear stress underneath surface wave using streamline coordinates

The dependence of Reynolds shear stress on wave phase is investigated for initially isotropic turbulence distorted by a progressive surface wave through direct numerical simulations. A wave-following streamline coordinate frame is used to analyse the turbulence dynamics such that the information of the varying direction of wave orbital motions is embedded into the coordinate system, which helps capture the effect of flow curvature on the turbulence and quantify momentum exchange between the near-surface and deep regions in the wavy domain accurately. It is found that the Reynolds shear stress is enhanced under the backward slope of the wave and can be scaled by the wave steepness and the streamwise velocity fluctuations. Analyses of the budget of Reynolds shear stress indicate that such a variation with the wave phase is caused by the variation in the production of the Reynolds shear stress and the effect of pressure fluctuations. Further investigation shows that the production of the Reynolds shear stress is closely associated with the wave surface curvature. A model that includes a correction term for the curvature effect for the pressure–strain correlation term is examined and is found to agree reasonably well with the simulation result. The correction term is found to make an appreciable contribution to the model, further supporting our finding that the wave curvature plays an important role in the turbulence dynamics near the surface.

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.

Drag reduction in turbulent channel flows by a spanwise traveling wave of wall blowing and suction

A spanwise traveling wave, generated by blowing and suction at walls, is applied in direct numerical simulations to reduce the drag in a fully developed turbulent channel flow with Reτ≈180. The wave traveling direction reverses periodically to induce a spanwise motion of the flow near the wall, similar to that from spanwise wall oscillations. Such a motion can be approximated by an infinite series of the Stokes layers in an asymptotic expansion. A relation linking the blowing and suction velocity and frequency for the traveling wave is derived to achieve a drag reduction similar to the case with spanwise wall oscillation. A drag reduction of 24% is achieved in the present study. From the analyses of the energy budget of Reynolds stresses, the drag reduction mechanism is found strongly related to the pressure–strain correlations. The reduced wall-normal strain rate and pressure–strain correlation are the main causes of the decrease in energy components of turbulence. The induced spanwise motion also weakens the streamwise strain rate through inclining the vortex structures periodically and modulates the spanwise strain rate through its production term.

Determining a priori a RANS model’s applicable range via global epistemic uncertainty quantification

Calibrating a Reynolds-averaged Navier–Stokes (RANS) model against data leads to an improvement. Determining a priori if such an improvement generalizes to flows outside the calibration data is an outstanding challenge. This work attempts to address this challenge via global epistemic Uncertainty Quantification (UQ). Unlike the available epistemic UQ methods that are local and tell us a model’s uncertainty at one specific flow condition, the global epistemic UQ method presented in this work tells us also whether a perturbation of the original model would generalize. Specifically, the global epistemic UQ method evaluates a potential improvement in terms of its “effectiveness” and “inconsistency”. Any improvement can be put in one of the following four quadrants: first, high effectiveness, low inconsistency; second, high effectiveness, high inconsistency; third, low effectiveness, low inconsistency; and fourth, low effectiveness, high inconsistency. An improvement would generalize if and only if it is in the high effectiveness, low inconsistency quadrant. To demonstrate the concept, we apply the global epistemic UQ to full Reynolds stress modeling of a stratified shear layer. The global epistemic UQ results point to a model coefficient in the pressure-strain correlation closure (among others) as effective and consistent for predicting the quantity of interest of shear layer’s growth. We calibrate the model coefficients such that our RANS matches direct numerical simulation data at one flow condition. We show that the calibrated model generalizes to several other test flow conditions. On the other hand, when calibrating a high inconsistency term, the model does not generalize beyond the calibration condition.

Transition from axisymmetric to three-dimensional turbulence

In purely axisymmetric turbulence sustained by a linear forcing mechanism, a stable, entirely poloidal flow is observed when the toroidal component of the forcing is below a threshold. We investigate using numerical simulations whether this state persists when toroidal variations of the flow are continuously reintroduced and the forcing is purely poloidal. It is shown how the pressure–strain correlation allows the redistribution of the energy towards the toroidal component. A simple statistical model allows to capture the main physical effects on the level of the global energy balance. This model is then used to investigate the stability of the poloidal state for various toroidal-to-poloidal forcing strengths and different degrees of axisymmetry.