Reynolds Stress Anisotropy Research Articles

On the minimal representation of non-linear eddy-viscosity models

This study attempts to provide a minimal explicit non-linear relationship between the Reynolds stresses and the mean strain rate and vorticity. Such relationship is also referred to as the nonlinear eddy-viscosity model (EVM) of the Reynolds stresses. Based on the observation of the independent terms in the Reynolds stress anisotropy tensor, it is proposed here that the explicit Reynolds stress closure can be formulated in a compact vorticity tensor to form a minimal representation. With the application of the generalized Cayley–Hamilton theory, it is shown that the explicit algebraic stress model with 10 integrity bases can be transformed into a five-term model expression. Thus, the present work provides a framework for the development of the explicit Reynolds stress closure in the compact form. Indeed, a cubic EVM is developed here that is capable of revealing the quadratic behaviour of the tangential velocity with respect to the radial distance in the fully developed turbulent rotating pipe flow. This model, hence, removed some major defects in the lower-order EVMs, while the model complexity is kept at minimum.

Numerical modelling of the carrier gas phase in a laboratory-scale coal classifier model

The pneumatic transport of fine ideally combustible coal dust to the burner furnace is an important process in coal fired power plants. The strongly swirling air phase responsible for the particle separation and transport in a coal pulverising mill was characterised experimentally and numerically. Measurements of the swirl velocity component were taken in a scaled laboratory model of the device and compared to CFD model. In particular, an evaluation of the turbulence models used to describe the flow was performed. The modified isotropic k-epsilon turbulence models (RNG k-ε and Realizable k-ε) were compared to the anisotropic Reynolds stress model (RSM) and their ability to predict the bulk flow structure present in the classifier was assessed. The experiments showed that the swirling flow structure, responsible for coarse-fine particle classification, has several flow regimes which are governed by the areas it is bounded by. The numerical model predictions generally corroborate the results. However, a distinction in performance between the three models can be made based on accuracy, solution generation time and numerical stability. The RSM model predicted both the trends and magnitude the most accurately when compared to the isotropic models. However, the Realizable k-epsilon model, with its relatively low solution generation time, shows potential when using CFD as a classifier design optimization tool. The investigation has given some insight on single phase classifier flow and suggests a design improvement based on the results.

A priori evaluation of the Pantano and Sarkar model in compressible homogeneous shear flows

In this study, a Reynolds stress closure, including the Pantano and Sarkar model of the mean part of the pressure–strain correlation is used for the computation of compressible homogeneous at high-speed shear flow. Several studies concerning the compressible homogeneous shear flow show that the changes of the turbulence structures are principally due to the structural compressibility effects which significantly affect the pressure field and then the pressure–strain correlation. Eventually, this term appears as the main term responsible for the changes in the magnitude of the Reynolds stress anisotropies. The structure of the gradient Mach number is similar to that of turbulence, therefore this parameter may be appropriate to study the changes in turbulence structures that arise from structural compressibility effects. Thus, the incompressible model of the pressure strain correlation and its corrected form by using the turbulent Mach turbulent only, fail to correctly evaluate the compressibility effects at high shear flow. An extension of the widely used incompressible Launder, Reece and Rodi model on compressible homogeneous shear flow is the major aim of the present work. From this extension, the standard coefficients C i become a function of the extra compressibility parameters (the turbulent Mach number M t and the gradient Mach number M g ) through the Pantano and Sarkar model. Application of the model on compressible homogeneous shear flow by considering various initial conditions shows reasonable agreement with the DNS results of Simone et al. and Sarkar. The observed trend of the dramatic increase in the normal Reynolds stress anisotropies, the significant decrease in the Reynolds shear stress anisotropy and the increase of the turbulent kinetic energy amplification rate with increasing the gradient Mach number are well predicted by the model. The ability of the model to predict the equilibrium states for the flow in cases A 1 to A 4 from DNS results of Sarkar is examined, the results appear to be very encouraging. Thus, both parameters M t and M g should be used to model significant structural compressibility effects at high-speed shear flow.

On the performance of linear and nonlinear [formula omitted] turbulence models in various jet flow applications

In this paper, a thorough numerical investigation of the performance of several linear and nonlinear k – ε turbulence model variants in various jet flow applications is carried out. Three k – ε based turbulence models are considered, namely the standard k – ε model, the υ 2 – f model, and the nonlinear k – ε model. The selected turbulence models are applied for the prediction of simple as well as complex jet flow applications to underpin knowledge about the accuracy obtained from the two-equation turbulence models. The numerical code developed by the present authors solves the unsteady RANS equations by using the control volume approach on a non-staggered grid system. Three jet flow applications are selected, namely a turbulent free jet, a turbulent jet impinging on a flat plate, and a turbulent wall jet. In order to validate the numerical results obtained and to investigate the performance of the different turbulence models considered, different experimental measurements from the literature are used. The present work is primarily motivated by the desire to provide a rational way for deciding how complex the turbulence model is required to be for a given application and to find out how the accuracy changes with model complexity. Due to the superior predictive performance of modern turbulence models in a wide range of complex industrial and engineering applications, it was believed that a ‘universal’ turbulence model might exist. In general, that is not true. Simple flows can be analysed using standard two-equation models. The present numerical investigation showed that the linear turbulence model could give good results in simple (non-impinging) jet flows. However, in complicated flows, such as impinging jet problems or wall jet flows, a more elaborate level of modeling is required. In such contexts, nonlinear models are appropriate for predicting the turbulent viscosity structure, namely the inhomogeneous near-wall flow region and the anisotropic Reynolds stresses, which is a vital part of turbulent jet flow prediction.

An alternative assessment of weak-equilibrium conditions in turbulent closure modeling

In formulating polynomial representations of turbulent stress anisotropy tensors, it is necessary to impose equilibrium conditions on the turbulent field in order to reduce the governing transport equations to a set of implicit algebraic equations. Imposing these conditions on the anisotropy tensor rather than the turbulent stress tensor directly results in a set of weak-equilibrium conditions on the evolution of the anisotropy tensor. A second kind of assumption is a requirement that the turbulent transport and viscous diffusion be related to the anisotropy tensor in a manner consistent with both the dynamics and related tensorial properties. In the study reported here, an alternative analysis of the weak-equilibrium hypothesis is performed in order to improve the physical interpretation of this assumption. One important result of the analysis is that previous turbulent weak-equilibrium hypothesis available in the literature impose that the flow has reached a Motion With Constant Relative Principle Anisotropic -Reynolds-Stress History, which means that the eigenvalues of the anisotropic Reynolds stress tensor do not change in time. Since virtually every geometry used to test this kind of hypothesis do not have a change on the cross sectional area and lead to viscometric flows, is natural that at equilibrium conditions such motion occurs. However, the eigenvalues of the Reynolds stress can change in time on turbulent extensional flows due to action of the mean flow in a different process from a turbulent relaxation. In this case, the extended weak-equilibrium assumption has to be related to an objective time derivative which is not purely co-rotational.

Decay of magnetohydrodynamic turbulence at low magnetic Reynolds number

We report a detailed numerical investigation of homogeneous decaying turbulence in an electrically conducting fluid in the presence of a uniform constant magnetic field. The asymptotic limit of low magnetic Reynolds number is assumed. Large-eddy simulations with the dynamic Smagorinsky model are performed in a computational box sufficiently large to minimize the effect of periodic boundary conditions. The initial microscale Reynolds number is about 170 and the magnetic interaction parameter N varies between 0 and 50. We find that except for a short period of time when N = 50, the flow evolution is strongly influenced by nonlinearity and cannot be adequately described by any of the existing theoretical models. One particularly noteworthy result is the near equipartition between the rates of Joule and viscous dissipations of the kinetic energy observed at all values of N during the late stages of the decay. Further, the velocity components parallel and perpendicular to the magnetic field decay at different rates, whose value depends on the strength of the magnetic field and the stage of the decay. This leads to a complex evolution of the Reynolds stress anisotropy ellipsoid, which goes from being rod-shaped, through spherical to disc-shaped. We also discuss the possibility of the power-law decay, the comparison between computed, experimental and theoretical decay exponents, the anisotropy of small-scale fluctuations, and the evolution of the spectral energy distributions.

Direct numerical simulations of turbulence subjected to a straining and destraining cycle

In many turbulent flows, significant interactions between fluctuations and mean velocity gradients occur in nonequilibrium conditions, i.e., the turbulence does not have sufficient time to adjust to changes in the velocity gradients applied by the large scales. The simplest flow that retains such physics is the time dependent homogeneous strain flow. A detailed experimental study of initially isotropic turbulence subjected to a straining and destraining cycle was reported by Chen et al. [“Scale interactions of turbulence subjected to a straining-relaxation-destraining cycle,” J. Fluid Mech. 562, 123 (2006)]. Direct numerical simulation (DNS) of the experiment of Chen et al. [“Scale interactions of turbulence subjected to a straining-relaxation-destraining cycle,” J. Fluid Mech. 562, 123 (2006)] is undertaken, applying the measured straining and destraining cycle in the DNS. By necessity, the Reynolds number in the DNS is lower. The DNS study provides a complement to the experimental one including time evolution of small-scale gradients and pressure terms that could not be measured in the experiments. The turbulence response is characterized in terms of velocity variances, and similarities and differences between the experimental data and the DNS results are discussed. Most of the differences can be attributed to the response of the largest eddies, which, even if are subjected to the same straining cycle, evolve under different conditions in the simulations and experiment. To explore this issue, the time evolution of different initial conditions parametrized in terms of the integral scale is analyzed in computational domains with different aspect ratios. This systematic analysis is necessary to minimize artifacts due to unphysical confinement effects of the flow. The evolution of turbulent kinetic energy production predicted by DNS, in agreement with experimental data, provides a significant backscatter of kinetic energy during the destraining phase. This behavior is explained in terms of Reynolds stress anisotropy and nonequilibrium conditions. From the DNS, a substantial persistency of anisotropy is observed up to small scales, i.e., at the level of velocity gradients. Due to the time dependent deformation, we find that the major contribution in the Reynolds stresses budget is provided by the production term and by the pressure/strain correlation, resulting in large time variation of velocity intensities. The DNS data are compared with predictions from the classical Launder–Reece–Rodi isoptropic production [B. E. Launder et al., “Progress in the development of a Reynolds stress turbulence closure,” J. Fluid Mech. 68, 537 (1975)] Reynolds stress model, showing good agreement with some differences for the redistribution term.

A methodology to quantify the nonlinearity of the Reynolds stress tensor

Turbulent models provide closure equations that relate the Reynolds stress with kinematic tensors. In this study, we present a methodology to quantify the dependence of the Reynolds stress tensor on mean kinematic tensor basis. The methodology is based upon tensor decomposition theorems which allow to extract from the anisotropic Reynolds stress tensor the part that is linear or nonlinear in the strain rate tensor D, and the parts that are in-phase (sharing the same eigenvectors) and out-of-phase with the strain rate. The study was conducted using direct numerical simulation (DNS) data for turbulent plane channel (from Reτ=180 to Reτ=1000) and square duct flows (Reτ=160). The results have shown that the tensorial form of the linear Boussinesq hypothesis is not a good assumption even in the region where production and dissipation are in equilibrium. It is then shown that the set of tensor basis composed by D, D2 and the persistence-of-straining tensor D·(W-ΩD)-(W-ΩD)·D, where W is the vorticity tensor and ΩD is the rate of rotation of the eigenvectors of D, is able to totally reproduce the anisotropic Reynolds stress. With the proposed methodology, the scalar coefficients of nonlinear algebraic turbulent models can be determined, and the adequacy of the tensorial dependence of the Reynolds stress can be quantified with the aid of scaled correlation coefficients.

Turbulence structures over irregular rough surfaces

Turbulent flow in a channel with irregular two-dimensional rough surfaces is analysed through wall-resolving large eddy simulation (LES). Both walls of the channel are roughened through the superimposition of sinusoidal functions having random amplitude and four different wavelengths. The downward shift of the velocity profile in the log region due to the roughness, known as roughness function, is well captured in the simulations. The spanwise and wall-normal turbulence intensities are found to increase with the roughness height, while the streamwise component decreases. The analysis of the Reynolds stress anisotropy tensor highlights a tendency towards isotropisation, confirmed by the vorticity rms. The analysis of the statistics shows that the effects of the roughness on the turbulent flow are greatly related to the increase of the height of the maximum peaks of the corrugations. Although the inner layer is dramatically affected by the wall irregularities, the outer layer appears not affected by the specific wall shape and a collapse of the turbulence statistics in smooth- and rough-wall conditions is observed. Much of the present results are consistent with the observations made over regular roughness, nevertheless the turbulence structures and the turbulent kinetic energy production analysis shows that the turbulence dynamics are directly affected by the local shape of the irregularities. The spatial inhomogeneities of the height of the roughness peaks and of the cavity region locally modifies the structures introducing variable length-scales. Overall the irregular roughness destroys the coherent pattern of the flow selectively, depending on the spatial distribution of the higher roughness peaks.

Explicit algebraic subgrid stress models with application to rotating channel flow

New explicit subgrid stress models are proposed involving the strain rate and rotation rate tensor, which can account for rotation in a natural way. The new models are based on the same methodology that leads to the explicit algebraic Reynolds stress model formulation for Reynolds-averaged Navier–Stokes simulations. One dynamic model and one non-dynamic model are proposed. The non-dynamic model represents a computationally efficient subgrid scale (SGS) stress model, whereas the dynamic model is the most accurate. The models are validated through large eddy simulations (LESs) of spanwise and streamwise rotating channel flow and are compared with the standard and dynamic Smagorinsky models. The proposed explicit dependence on the system rotation improves the description of the mean velocity profiles and the turbulent kinetic energy at high rotation rates. Comparison with the dynamic Smagorinsky model shows that not using the eddy-viscosity assumption improves the description of both the Reynolds stress anisotropy and the SGS stress anisotropy. LESs of rotating channel flow at Reτ = 950 have been carried out as well. These reveal some significant Reynolds number influences on the turbulence statistics. LESs of non-rotating turbulent channel flow at Reτ = 950 show that the new explicit model especially at coarse resolutions significantly better predicts the mean velocity, wall shear and Reynolds stresses than the dynamic Smagorinsky model, which is probably the result of a better prediction of the anisotropy of the subgrid dissipation.

Evaluation of linear and nonlinear convection schemes on multidimensional non‐orthogonal grids with applications to KVLCC2 tanker

AbstractAlmost all evaluations of convection schemes reported in the literature are conducted using simple problems on uniform orthogonal grids; thus, having limited contribution when solving industrial computational fluid dynamics (CFD), where the grids are usually non‐orthogonal with distortions. Herein, several convection schemes are assessed in uniform and distorted non‐orthogonal grids with emphasis on industrial applications. Linear and nonlinear (TVD) convection schemes are assessed on analytical benchmarks in both uniform and distorted grids. To evaluate the performance of the schemes, four error metrics are used: dissipation, phase and L1 errors, and the schemes' effective order of accuracy. Qualitative and quantitative deterioration of these error metrics as a function of the grid distortion metrics are investigated, and rigorous verifications are performed. Recommendations for effective use of the convection schemes based on the range of grid aspect ratio (AR), expansion ratio (ER) and skewness (Q) are included. A ship hydrodynamics case is studied, involving a Reynolds averaged Navier–Stokes simulation of a bare‐hull KVLCC2 tanker using linear and nonlinear convection schemes coupled with isotropic and anisotropic Reynolds‐stress (ARS) turbulence models using CFDShip‐Iowa v4. Predictions of local velocities and turbulent quantities from the midships to the nominal wake plane are compared with experimental fluid dynamics (EFD), and rigorous verification and validation analyses for integral forces and moments are performed for 0° and 12° drift angles. Best predictions are observed when coupling a second‐order TVD scheme with the anisotropic turbulence model. Further improvements are observed in terms of prediction of the vortical structures for 30° drift when using TVD2S‐ARS coupled with DES. Copyright © 2009 John Wiley & Sons, Ltd.

Experiments on rapidly rotating turbulent flows

A novel laboratory experiment for investigating statistically steady rotating turbulence is presented. Turbulence is produced nonintrusively by means of electromagnetic forcing. Depending on the rotation rate the Taylor-based Reynolds number is found to be in the range of 90≲Reλ≲240. Relevant properties of the turbulence, both with and without rotation, have been quantified with stereoscopic particle image velocimetry (SPIV). This method enables instantaneous measurement of all three velocity components in horizontal planes at a distance H from the bottom. The root-mean-square turbulent velocity decreases inversely proportional to H in the nonrotating experiments and is approximately constant when background rotation is applied. The integral length scale shows a weak H-dependence in the nonrotating experiments which is presumably due to the spatial extent of the forcing. Based on the behavior of the principal invariants of the Reynolds stress anisotropy tensor, the rotating turbulence has been characterized as a three-dimensional two-component flow. Furthermore, these SPIV measurements provide supporting evidence for (i) reduction of the dissipation rate, (ii) suppression of the vertical velocity as compared to the horizontal velocity, and (iii) increased spatial and temporal correlation of the horizontal velocity components, with the temporal correlation growing ever stronger as the rotation rate is increased. A less commonly known feature of rotating turbulence, quantified here for the first time in a laboratory setting, is the reverse dependence on the rotation rate of the spatial horizontal velocity correlation functions. Another interesting result concerns the linear (anomalous) scaling of the longitudinal spatial structure function exponents in the presence of rotation, consistent with a study by Baroud et al. [Phys. Rev. Lett. 88, 114501 (2002)].

Reduction of the Germano-identity error in the dynamic Smagorinsky model

We revisit the Germano-identity error in the dynamic modeling procedure in the sense that the current modeling procedure to obtain the dynamic coefficient may not truly minimize the error in the mean and global sense. A “corrector step” to the conventional dynamic Smagorinsky model is proposed to obtain a corrected eddy viscosity which further reduces the error. The change in resolved velocity due to the coefficient variation as well as nonlocal nature of the filter and flow unsteadiness is accounted for by a simplified suboptimal control formalism without resorting to the adjoint equations. The objective function chosen is the Germano-identity error integrated over the entire computational volume and pathline. In order to determine corrected eddy viscosity, the Fréchet derivative of the objective function is directly evaluated by a finite-differencing formula in an efficient predictor-corrector-type framework. The proposed model is applied to decaying isotropic turbulence and turbulent channel flow at various Reynolds numbers and resolutions to obtain noticeable reduction in the Germano-identity error and significantly improved flow statistics. From channel flow large-eddy simulation, it is shown that conventional dynamic model underestimates subgrid scale eddy viscosity when the resolution gets coarse, and this underestimation is responsible for increased anisotropy of predicted Reynolds stress. The proposed model raises both the overall and near-wall subgrid scale eddy viscosity to reduce exaggerated Reynolds stress anisotropy and yield significantly improved flow statistics.

Combined effects of flow curvature and rotation on uniformly sheared turbulence

This paper describes an experiment in which a uniformly sheared turbulence was subjected to simultaneous streamwise flow curvature and rotation about the streamwise axis. The distortion of the turbulence is complex but well defined and may serve as a test case for turbulence model development. The uniformly sheared turbulence was developed in a straight wind tunnel and then passed into a curved tunnel section. At the start of the curved section the plane of the mean shear was normal to the plane of curvature so as to create a three-dimensional or ‘out of plane’ curvature configuration. On entering the curved tunnel, the flow developed a streamwise mean vorticity that rotated the mean shear about the tunnel centreline through approximately 70°, so that the shear was nearly in the plane of curvature and oriented so as to have a stabilizing effect on the turbulence. Hot wire measurements of the mean velocity, mean vorticity, mean rate of strain and Reynolds stress anisotropy development along the wind tunnel centreline are reported. The observed effect of the mean shear rotation on the turbulence was to diminish the shear stress in the plane normal to the plane of curvature while generating non-zero values of the shear stress in the plane of curvature. A rotating frame was identified for which the measured mean velocity field took the form of a simple shear flow. The turbulence anisotropy was transformed to this frame to estimate the effects of frame rotation on the structure of sheared turbulence.

Reynolds Stress Anisotropy in Open-Channel Flow

This paper reports the results of an experimental study characterizing turbulence and turbulence anisotropy in smooth and rough shallow open-channel flows. The rough bed consists of a train of two-dimensional transverse square ribs with a ratio of the roughness height (k) to the total depth of flow (d) equal to 0.10. Three rib separations (p/k) of 4.5, 9, and 18 were examined. Here, p is the pitch between consecutive roughness elements and was varied to reproduce the classical condition of d- and k-type roughness. For each case, two-component velocity measurements were obtained using a laser Doppler velocimetry system at two locations for p/k=4.5 and 9: on the top of the rib and above the cavity, and an additional location for p/k=18. The measurements allow examination of the local variations of the higher-order turbulent moments, stress ratios as well as turbulence anisotropy. Large variations of the turbulence intensities, Reynolds shear stress, turbulent kinetic energy and turbulence production are found for y1<3k. In this region, the flow is more directly influenced by the shear layers from the preceding ribs. The higher-order moments appear to be similar for all rough surfaces beyond y1≈7k. In the outer layer (y1>3k), all higher-order turbulent moments for the k-type roughness show a substantial increase due to the complex interactions between the roughness and the remnants overlying shear layers shed from succeeding ribs. Analysis of the components of the Reynolds stress anisotropy tensor shows that at p/k=18, the flow at y1<5k tends to be more isotropic which implies that for this particular case, the effect of the roughness density could also be important. On the smooth bed, at the shallower depths, the correlation coefficient near the free surface increases and turbulence tends to become less anisotropic.

Velocity and conformation statistics based on reduced Karhunen–Loeve projection data from DNS of viscoelastic turbulent channel flow

To investigate the effectiveness of the Karhunen–Loeve (K–L) method as a data reduction approach, we study here its effect on the velocity and conformation statistics in a drag reducing turbulent polymer flow. The K–L method has been used to construct a set of basis velocity eigenfunctions from a large number of independent realizations of the velocity. Those were obtained from direct numerical simulation (DNS) of a viscoelastic turbulent channel flow using the Giesekus model. A subset of the K–L eigenfunctions, large enough to contain more than 90% of the fluctuating kinetic energy of the flow on the average, has then been subsequently used to obtain time series of projection coefficients of the velocity fields generated further from DNS. In a post-processing step, velocity fields were reconstructed using selected subsets of the projection coefficients. Those reconstructed velocity fields were then used to evaluate turbulent statistics as well as to integrate the constitutive equation. The turbulent statistics (r.m.s. velocities, Reynolds stress etc.) thus constructed showed good agreement with the full results from DNS. The Reynolds stress anisotropy was also calculated in this work for the first time. It was found to increase with viscoelasticity that was well reproduced in the reduced K–L data except near the channel centerline where the K–L data showed some loss of anisotropy. The biggest differences however between the K–L reduced data and the full DNS results were seen in the conformation statistics. The average polymer conformation extracted from the K–L reduced data was significantly less than that corresponding to the full DNS results anywhere except in the shear-dominated wall region. A further comparison of the energy and dissipation spectra between the full DNS and the K–L reconstructed data illustrated the impact of the K–L process in resulting to a significant damping of small turbulent scales even those contributing to the maximum in turbulent dissipation. This may also be the principal reason behind the poor quality of the K–L reconstructed conformation data.

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.

Investigation of Rayleigh–Taylor turbulence and mixing using direct numerical simulation with experimentally measured initial conditions. II. Dynamics of transitional flow and mixing statistics

A 1152×760×1280 direct numerical simulation (DNS) using initial conditions, geometry, and physical parameters chosen to approximate those of a transitional, small Atwood number, nonreacting Rayleigh–Taylor mixing experiment was presented in Paper I [Mueschke and Schilling, Phys. Fluids 21, 014106 (2009)]. In addition, the DNS model of the experiment was validated by comparing quantities from the simulation to experimental measurements, including large-scale quantities, higher-order statistics, and vertical velocity and density variance spectra. In Paper II of this study, other quantities not measured in the experiment are obtained from the DNS and discussed, such as the integral- and Taylor-scale Reynolds numbers, Reynolds stress and dissipation anisotropy, two-dimensional density and velocity variance spectra, hypothetical chemical product formation measures (similar to those used in reacting shear flow experiments), other local and global mixing parameters, and the statistical composition of mixed fluid. The integral- and Taylor-scale Reynolds numbers, together with visualizations of vertical and center plane slices of the density and vorticity fields, are used to elucidate the various evolutionary stages of the flow. It is shown that the early-time evolution retains a primarily two-dimensional character until the flow begins to transition to a more three-dimensional state at later times, as also observed in the experiment. The evolution of the three diagonal components of the anisotropy tensors showed that anisotropy persists to the latest times in the simulation. Compensated spectra at the latest time in the DNS suggest very short k−5/3 and k−5/4 inertial subrange scalings of the vertical velocity and density variance spectra, respectively. By interpreting the mixing between the two fluids as a hypothetical, infinitely fast, reversible chemical reaction between the species, the local formation of chemical product, equivalent product thickness, and other standard measures of mixing used in shear-driven turbulence are obtained from the DNS and discussed. Other measures of molecular mixing are shown to be qualitatively similar to the molecular mixing parameter θ on the center plane. Finally, the statistical composition of the mixed fluid is examined using the probability distribution function of the heavy-fluid volume fraction and the averaged composition of mixed fluid. Thus, DNS modeled closely after a physical Rayleigh–Taylor instability and mixing experiment can provide additional insights into the flow physics complementary to the experiment.

Investigation of Rayleigh–Taylor turbulence and mixing using direct numerical simulation with experimentally measured initial conditions. I. Comparison to experimental data

A 1152×760×1280 direct numerical simulation (DNS) using initial conditions, geometry, and physical parameters chosen to approximate those of a transitional, small Atwood number Rayleigh–Taylor mixing experiment [Mueschke et al., J. Fluid Mech. 567, 27 (2006)] is presented. In particular, the Atwood number is 7.5×10−4, and temperature diffusion is modeled by mass diffusion with an equivalent Schmidt number of 7. The density and velocity fluctuations measured just off of the splitter plate in this buoyantly unstable water channel experiment were parametrized to provide physically realistic, anisotropic initial conditions for the DNS. The methodology for parametrizing the measured data and numerically implementing the resulting perturbation spectra in the simulation is discussed in detail. The DNS is then validated by comparing quantities from the simulation to experimental measurements. In particular, large-scale quantities (such as the bubble front penetration hb and the mixing layer growth parameter αb), higher-order statistics (such as velocity variances and the molecular mixing parameter θ on the center plane), and vertical velocity and density variance spectra from the DNS are shown to be in favorable agreement with the experimental data. The DNS slightly underestimates the growth of the bubble front hb but predicts αb≈0.07 at the latest time, in excellent agreement with the experimental measurement. While the molecular mixing parameter θ is also slightly underestimated by the DNS during the nonlinear and weakly turbulent growth phases, the late-time value θ≈0.55 compares favorably with the value θ≈0.6 measured in the experiment. The one-dimensional density and vertical velocity variance spectra are in excellent agreement between the DNS and experimental measurements. Differences between the quantities obtained from the DNS and from experimental measurements are related to limitations in the dynamic range of scales resolved in the DNS and other idealizations of the simulation. Specifically, the statistical convergence of the DNS results and confidence interval bounds are discussed. This work demonstrates that a parametrization of experimentally measured initial conditions can yield simulation data that quantitatively agrees well with experimentally measured low- and higher-order statistics in a Rayleigh–Taylor mixing layer. This study also provides resolution and initial conditions implementation requirements needed to simulate a physical Rayleigh–Taylor mixing experiment. In Paper II [Mueschke and Schilling, Phys. Fluids 21, 014107 (2009)], other quantities not measured in the experiment are obtained from the DNS and discussed, such as the integral- and Taylor-scale Reynolds numbers, Reynolds stress and dissipation anisotropy, two-dimensional density and velocity variance spectra, hypothetical chemical product formation measures, other local and global mixing parameters, and the statistical composition of mixed fluid. These quantities are valuable for assessing the predictions of Reynolds-averaged Navier–Stokes and large-eddy simulation models of Rayleigh–Taylor turbulent mixing.

Reynolds stress closure for nonequilibrium effects in turbulent flows

From consideration of turbulence anisotropy dynamics due to spatial or temporal variations in the mean strain rate, a new Reynolds stress closure for nonequilibrium effects in turbulent flows has been developed. This closure, formally derived from the Reynolds stress anisotropy transport equation, results in an effective strain rate tensor that accounts for the strain rate history to which the turbulence has been subjected. In contrast to prior nonequilibrium models that have sought to address nonequilibrium effects via changes in the eddy viscosity, the present approach accounts for nonequilibrium effects in the fundamental relation between the anisotropy tensor and the strain rate tensor. The time-local form of the nonequilibrium closure can be readily implemented in place of the classical equilibrium Boussinesq closure on which most existing computational frameworks are currently based. This new closure is applied here to four substantially different classes of nonequilibrium test problems. Results show dramatically improved agreement with experimental and computational data, without the need to vary any model parameters, when compared with the standard equilibrium closure and with various prior nonequilibrium closures.