Residual‐based variational multiscale methods for laminar,transitional and turbulent variable‐density flow at low Mach number

In the present study, residual‐based variational multiscale methods are developed for and applied to variable‐density flow at low Mach number. In particular, two different formulations are considered in this study: a standard stabilized formulation featuring SUPG/PSG/grad‐div terms and a complete residual‐based variational multiscale formulation additionally containing cross‐ and Reynolds‐stress terms as well as subgrid‐scale velocity terms in the energy‐conservation equation. The proposed methods are tested for various laminar flow test cases as well as a test case at laminar via transitional to turbulent flow stages. Stable and accurate results are obtained for all numerical examples. Substantial differences in the results between the two approaches do not become notable until a high temperature gradient is applied and the flow reaches a turbulent flow stage. The more pronounced influence of adding subgrid‐scale velocity terms to the energy‐conservation equation on the results than adding analogous terms to the momentum‐conservation equation in this situation appears particularly noteworthy. Copyright © 2009 John Wiley & Sons, Ltd.     

入口速度比对射流拟序结构的影响

为了深入了解湍流流动机理以及湍流拟序结构发现过程的影响因素，本文采用大涡模拟方法对不同入口射流伴流速度比的平面湍射流流动进行了数值模拟。采用分步投影法求解动量方程，亚格子项采用标准Smagorinsky亚格子模式模拟，压力泊松方程采用修正的循环消去法快速求解，空间方程采用二阶精度的差分格式，在时间方向上采用二阶精度的显式差分格式。模拟结果给出了平面射流中湍流拟序结构的瞬态发展演变过程，分析了入口速度比对射流拟序结构发展演化过程及宏观流场形态的影响。为进一步研究射流拟序结构及其在湍流流动中的作用提供了基础。     

Direct numerical simulation of supersonic pipe flow at moderate Reynolds number

We study compressible turbulent flow in a circular pipe at computationally high Reynolds number. Classical related issues are addressed and discussed in light of the DNS data, including validity of compressibility transformations, velocity/temperature relations, passive scalar statistics, and size of turbulent eddies. Regarding velocity statistics, we find that Huang’s transformation yields excellent universality of the scaled Reynolds stresses distributions, whereas the transformation proposed by Trettel and Larsson (2016) yields better representation of the effects of strong variation of density and viscosity occurring in the buffer layer on the mean velocity distribution. A clear logarithmic layer is recovered in terms of transformed velocity and wall distance coordinates at the higher Reynolds number under scrutiny (Re_τ ≈ 1000), whereas the core part of the flow is found to be characterized by a universal parabolic velocity profile. Based on formal similarity between the streamwise velocity and the passive scalar transport equations, we further propose an extension of the above compressibility transformations to also achieve universality of passive scalar statistics. Analysis of the velocity/temperature relationship provides evidence for quadratic dependence which is very well approximated by the thermal analogy proposed by Zhang et al. (2014). The azimuthal velocity and scalar spectra show an organization very similar to canonical incompressible flow, with a bump-shaped distribution across the flow scales, whose peak increases with the wall distance. We find that the size growth effect is well accounted for through an effective length scale accounting for the local friction velocity and for the local mean shear.     

Couplage d'un modèle stochastique lagrangien sous-maille avec une simulation grandes échelles

A large eddy simulation (LES) is used to compute passive scalar dispersion in a turbulent flow. Instead of resolving the passive scalar transport equation, fluid particles are tracked in a Lagrangian way and a Langevin stochastic modelling is used for the small scale component of the velocity of fluid particles at a subgrid-scale level. The stochastic model is written in terms of subgrid-scale statistics. The specificity of this study resides in the coupling of a LES with a stochastic model using the filtered subgrid-scale statistics in inhomogeneous turbulence. The results are compared with the wind tunnel experiments of Fackrell and Robins [J. Fluid Mech. 117 (1982) 1–26]. To cite this article: I. Vinkovic et al., C. R. Mecanique 333 (2005).     

The influence of normal stress anisotropy in predicting scalar dispersion with the v–f model

A numerical study of scalar dispersion is presented to investigate the effectiveness of pairing the v²–f turbulence model with algebraic models for the scalar flux. This approach is contrasted with utilizing a full Second Moment Closure (SMC) as the flow field input to the scalar model. Predictions of scalar transport in a turbulent channel and over a wavy wall are compared to available DNS databases. The latter case includes a scalar release from a point source and therefore detailed comparisons of the three-component turbulent scalar flux are reported. It is found that the transported variable v², representing the near wall turbulent velocity fluctuation scale, can be used to increase the level of normal stress anisotropy provided to algebraic scalar models and thereby improve mean scalar prediction over that of the Standard Gradient Diffusion Hypothesis (SGDH). Improvement is most significant in the near wall region. Three specifications of the normal stresses, derived from v², are considered to provide the link from the v²–f model to the algebraic flux models used to close the scalar transport equation. Barycentric maps are used to examine the state of turbulence anisotropy in each case. As the anisotropy in the normal stress specification becomes more accurate, improvements are realized in the prediction of the spanwise flux as well as the mean concentration.     

Exact transport equation for eddy diffusivity in turbulent shear flow

Two-equation models that treat the transport equations for two variables are typical models for the Reynolds-averaged Navier–Stokes equation. Compared to the equation for the turbulent kinetic energy, the equation for the second variable such as the dissipation rate does not have a theoretical analogue. In this work, the exact transport equation for the eddy diffusivity was derived and examined for better understanding turbulence and improving two-equation models. A new length scale was first introduced, which involves the response function for the scalar fluctuation. It was shown that the eddy diffusivity can be expressed as the correlation between the velocity fluctuation and the new length scale. The transport equations for the eddy diffusivity and the length-scale variance were derived theoretically. Statistics such as terms in the transport equations were evaluated using the direct numerical simulation of turbulent channel flow. It was shown that the streamwise component of the eddy diffusivity is greater than the other two components in the whole region. In the transport equation for the eddy diffusivity, the production term due to the Reynolds stress is a main positive term, whereas the pressure–length-gradient correlation term plays a role of destruction. It is expected that the analysis of the transport equations is helpful in developing better turbulence models.     

Simulations of Turbulent Non-Premixed Counterflow Flames with First-Order Conditional Moment Closure

Simulations of turbulent CH₄-air counterflow flames are presented, obtained in terms of zero and two-dimensional first-order Conditional Moment Closure (CMC) to study the flame structure and extinction limits. The CMC equation with detailed chemistry is solved without the need for operator splitting, while the accompanying flow field is determined using a commercial CFD software employing a Reynolds stress turbulence model and additional transport equations for the turbulent scalar flux and for the mean scalar dissipation rate. Two detailed chemical mechanisms and different conditional scalar dissipation rate models have been examined and small differences were found.The first-order CMC captures the overall structure of the counterflow flame accurately for the unconditional averages. The calculated conditional averages behave as if the scalar dissipation rate were under-predicted, although a comparison with measurement of the conditional scalar dissipation rate is reasonable. The calculated extinction velocity is found to be much higher than the experimental value, but the trend of increasing extinction velocity with air dilution of the fuel stream is captured well. The discrepancies with the data are mostly attributed to the neglect of conditional fluctuations.     

Conditional Moment Closure for Large Eddy Simulations

A conditional moment closure (CMC) based combustion model for large-eddy simulations (LES) of turbulent reacting flow is proposed and evaluated. Transport equations for the conditionally filtered species are derived that are consistent with the LES formulation and closures are suggested for the modelling of the conditional velocity, conditional scalar dissipation and the fluctuations around the conditional mean. A conventional β-probability density distribution of the scalar is used together with dynamic modelling for the sub-grid fluxes. The model is validated by comparison of simulations with measurements of a piloted, turbulent methane-air jet diffusion flame.     

基于光流算法的粒子图像测速技术研究和验证

光流测量技术作为一种新的空气动力学实验技术,以其像素级分辨率的矢量场测量优势获得广泛的应用。光流测量技术使用光流约束方程,配合平滑限定条件,可以进行速度场测量,获得高分辨率的全局矢量场。本文首先通过研究积分最小化光流测速理论和算法,采用C++编写光流速度测量程序;然后通过三种典型的人工位移图像对光流计算程序进行了验证,并将结果和标准位移分布进行比对分析,以指导如何在实际应用中获得高精度光流速度场;最后进行小型风洞后向台阶实验,利用高速相机拍摄示踪粒子图像,使用光流计算程序获得速度矢量场,同采用互相关算法的粒子图像测速计算结果相比较,体现出光流计算方法像素级分辨率的矢量场测量优势。     

Large eddy simulation and a simple wall model for turbulent flow calculation by a particle method

A turbulent channel flow and the flow around a cubic obstacle are calculated by the moving particle semi‐implicit method with the subparticle‐scale turbulent model and a wall model, which is based on the zero equation RANS (Reynolds Averaged Navier‐Stokes). The wall model is useful in practical problems that often involve high Reynolds numbers and wall turbulence, because it is difficult to keep high resolution in the near‐wall region in particle simulation. A turbulent channel flow is calculated by the present method to validate our wall model. The mean velocity distribution agrees with the log‐law velocity profile near the wall. Statistical values are also the same order and tendency as experimental results with emulating viscous layer by the wall model. We also investigated the influence of numerical oscillations on turbulence analysis in using the moving particle semi‐implicit method. Finally, the turbulent flow around a cubic obstacle is calculated by the present method to demonstrate capability of calculating practical turbulent flows. Three characteristic eddies appear in front of, over, and in the back of the cube both in our calculation and the experimental result that was obtained by Martinuzzi and Tropea. Mean velocity and turbulent intensity profiles are predicted in the same order and have similar tendency as the experimental result. Copyright © 2012 John Wiley & Sons, Ltd.     

Multi-scale analysis of subgrid stress and energy dissipation in turbulent channel flow

In present study, the subgrid scale （SGS） stress and dissipation for multiscale formulation of large eddy simulation are analyzed using the data of turbulent channel flow at Ret = 180 obtained by direct numerical simulation. It is found that the small scale SGS stress is much smaller than the large scale SGS stress for all the stress components. The dominant contributor to large scale SGS stress is the cross stress between small scale and subgrid scale motions, while the cross stress between large scale and subgrid scale motions make major contributions to small scale SGS stress. The energy transfer from resolved large scales to subgrid scales is mainly caused by SGS Reynolds stress, while that between resolved small scales and subgrid scales are mainly due to the cross stress. The multiscale formulation of SGS models are evaluated a priori, and it is found that the small- small model is superior to other variants in terms of SGS dissipation.     

Analysis of high-order velocity moments in a strained channel flow

In the current study, model expressions for fifth-order velocity moments obtained from the truncated Gram-Charlier series expansions model for a turbulent flow field probability density function are validated using data from direct numerical simulation (DNS) of a planar turbulent flow in a strained channel. Simplicity of the model expressions, the lack of unknown coefficients, and their applicability to non-Gaussian turbulent flows make this approach attractive to use for closing turbulent models based on the Reynolds-averaged Navier-Stokes equations. The study confirms validity of the model expressions. It also shows that the imposed flow deformation improves an agreement between the model and DNS profiles for the fifth-order moments in the flow buffer zone including when the flow reverses its direction. The study reveals sensitivity of particularly odd velocity moments to the grid resolution. A new length scale is proposed as a criterion for the grid generation near the wall and in the other flow areas dominated by high mean velocity gradients when higher-order statistics have to be collected from DNS.     

Large eddy simulation (2D) of spatially developing mixing layer using vortex‐in‐cell for flow field and filtered probability density function for scalar field

A large eddy simulation based on filtered vorticity transport equation has been coupled with filtered probability density function transport equation for scalar field, to predict the velocity and passive scalar fields. The filtered vorticity transport has been formulated using diffusion‐velocity method and then solved using the vortex method. The methodology has been tested on a spatially growing mixing layer using the two‐dimensional vortex‐in‐cell method in conjunction with both Smagorinsky and dynamic eddy viscosity subgrid scale models for an anisotropic flow. The transport equation for filtered probability density function is solved using the Lagrangian Monte‐Carlo method. The unresolved subgrid scale convective term in filtered density function transport is modelled using the gradient diffusion model. The unresolved subgrid scale mixing term is modelled using the modified Curl model. The effects of subgrid scale models on the vorticity contours, mean streamwise velocity profiles, root‐mean‐square velocity and vorticity fluctuations profiles and negative cross‐stream correlations are discussed. Also the characteristics of the passive scalar, i.e. mean concentration profiles, root‐mean‐square concentration fluctuations profiles and filtered probability density function are presented and compared with previous experimental and numerical works. The sensitivity of the results to the Schmidt number, constant in mixing frequency and inflow boundary conditions are discussed. Copyright © 2005 John Wiley & Sons, Ltd.     

Vortex-In-Cell and Probability Density Function Approach for a Passive Scalar Field in a Mixing Layer

A Reynolds-averaged simulation based on the vortex-in-cell (VIC) and the transport equation for the probability density function (PDF) of a scalar has been developed to predict the passive scalar field in a two-dimensional spatially growing mixing layer. The VIC computes the instantaneous velocity and vorticity fields. Then the mean-flow properties, i.e. the mean velocity, the root-mean-square (rms) longitudinal and lateral velocity fluctuations, the Reynolds shear stress, and the rms vorticity fluctuations are computed and used as input to the PDF equation. The PDF transport equation is solved using the Monte Carlo technique. The convection term uses the mean velocities from the VIC. The turbulent diffusion term is modeled using the gradient transport model, in which the eddy diffusivity, computed via the Boussinesq's postulate, uses the Reynolds shear stress and gradients of mean velocities from the VIC. The molecular mixing term is closed by the modified Curl model.

The computational results were compared with two-dimensional experimental results for passive scalar. The predicted turbulent flow characteristics, i.e. mean velocity and rms longitudinal fluctuations in the self-preserving region, show good agreement with the experimental measurements. The profiles of the mean scalar and the rms scalar fluctuations are also in reasonable agreement with the experimental measurements. Comparison between the mean scalar and the mean velocity profiles shows that the scalar mixing region extends further into the free stream than does the momentum mixing region, indicating enhanced transport of scalar over momentum. The rms scalar profiles exhibit an asymmetry relative to the concentration centerline, and indicate that the fluid on the high-speed side mixes at a faster rate than the fluid on the low-speed side. The asymmetry is due to the asymmetry in the mixing frequency cross-stream profiles. Also, the PDFs have peaks biased toward the high-speed side.     

RANS Simulations of a Series of Turbulent V-Shaped Flames using Conditional Source-term Estimation

In the present study, Reynolds Averaged Navier Stokes (RANS) simulations are applied to a series of turbulent V-shaped flames. Two formulations of Conditional Source-term Estimation (CSE) are developed using singly and doubly conditioned averages for turbulent premixed and partially premixed flames, respectively. Detailed chemistry is included. Conditionally averaged chemical source terms are closed by conditional averaged scalars which are obtained by inverting an integral equation. The objectives are to study a turbulent premixed V-shaped flame using the premixed CSE approach and apply the Doubly Conditional CSE (DCSE) combustion model to a case of stratified combustion. The partially premixed implementation involves double conditioning on two variables, mixture fraction and progress variable. The present study represents the first application of DCSE for a series of turbulent stratified flames. First, CSE is analysed for fully premixed conditions. A sensitivity analysis on the number of CSE ensembles and different scalar dissipation model closures is performed. Good results are obtained in terms of velocity and progress variable profiles. Finally, the partially premixed formulation is applied to the stratified case at three different conditions, corresponding to two different turbulence grids and three different profiles of the equivalence ratio, providing promising results.     

Scale properties of turbulent transport and coherent structure in stably stratified flows

The empirical mode decomposition (EMD) is used to study the scale properties of turbulent transport and coherent structures based on velocity and temperature time series in stably stratified turbulence. The analysis is focused on the scale properties of intermittency and coherent structures in different modes and the contributions of energy-contained coherent structures to turbulent scalar counter-gradient transport (CGT). It is inferred that the velocity intermittency is scattered to more modes with the development of the stratified flow, and the intermittency is enhanced by the vertical stratification, especially in small scales. The anisotropy of the field is presented due to different time scales of coherent structures of streamwise and vertical velocities. There is global counter-gradient heat transport close to the turbulence-generated grid, and there is local counter-gradient heat transport at certain modes in different positions. Coherent structures play a principal role in the turbulent vertical transport of temperature.     

Dissipation rate correction methods

Dissipation rates of the turbulent kinetic energy and of the scalar variance are underestimated when the measurement resolution of the small scales of a turbulent flow field are insufficient. Results are presented of experiments conducted in a salt-stratified water tunnel (Schmidt number ∼700). Dissipation rates are determined to be underestimated, and thus correction techniques based on velocity structure functions and mixed-moment functions are proposed. Dissipation rates in laboratory experiments of shear-free, grid-generated turbulence are determined from balance calculations of the kinetic energy and scalar variance evolution equations. Comparisons between the structure function and balance estimates of dissipation show that the corrections are O(1) for the kinetic energy dissipation rate, and are O(100) for the scalar variance dissipation rate. This difference is due to the lack of resolution down to the Batchelor scales that is required for a high Schmidt number flow. Simple correction functions based on microscale Reynolds numbers are developed for both turbulent kinetic energy and scalar variance dissipation rates. Application of the technique to the results of laboratory experiments of density stratified turbulence, sheared turbulence, and sheared density stratified turbulence yields successful corrections. It is also demonstrated that the Karman–Howarth equality (and the analogous Yaglom equation) that relates second and third-order structure functions to dissipation rates is valid for both unstrained (decaying grid-generated turbulence) and density stratified and sheared turbulence at least up to the magnitudes of strains of the current experiments Nt∼10, St∼10, respectively. This is helpful for it allows the use of these equations in the analysis of turbulence even when the large scale background profiles of velocity and scalar are unknown.     

Large Eddy Simulation of Scalar Mixing in a Coaxial Confined Jet

The highly turbulent flow occurring inside gas-turbine combustors requires accurate simulation of scalar mixing if CFD methods are to be used with confidence in design. This has motivated the present paper, which describes the implementation of a passive scalar transport equation into an LES code, including assessment/testing of alternative discretisation schemes to avoid over/undershoots and excessive smoothing. Both second order accurate TVD and higher order accurate DRP schemes are assessed. The best performance is displayed by a DRP method, but this is only true on fine meshes; it produces similar (or larger) errors to a TVD scheme on coarser meshes, and the TVD approach has been retained for LES applications. The unsteady scalar mixing performance of the LES code is validated against published DNS data for a slightly heated channel flow. Excellent agreement between the current LES predictions and DNS data is obtained, for both velocity and scalar statistics. Finally, the developed methodology is applied to scalar transport in a confined co-axial jet mixing flow, for which experimental data are available. Agreement with statistically averaged fields for both velocity and scalar, is demonstrated to be very good, and a considerable improvement over the standard eddy viscosity RANS approach. Illustrations are presented of predicted time-resolved information e.g. time histories, and scalar pdf predictions. The LES results are shown, even using a simple Smagorinsky SGS model, to predict (correctly) lower values of the turbulent Prandtl number in the free shear regions of the flow, compared to higher values in the wall-affected regions. The ability to predict turbulent Prandtl number variations (rather than input these as in combustor RANS CFD models) is an important and promising feature of the LES approach for combustor flow simulation since it is known to be important in determining combustor exit temperature traverse.     

Agglomeration of inertial particles in a random rotating symmetric straining flow

This paper is concerned with the development and validation of a simple Lagrangian model for particle agglomeration in a turbulent flow involving the collision of particles in a sequence of correlated straining and vortical structures which simulate the Kolmogorov small scales of motion of the turbulence responsible for particle pair dispersion and collision. In this particular study we consider the collision rate of monodisperse spherical particles in a symmetric (pure) straining flow which is randomly rotated to create an isotropic flow. The model is similar to the classical model of Saffman and Turner (S&T) (1956) for the collision (agglomeration) of tracer particles suspended in a turbulent flow. However unlike S&T, the straining flow is not frozen in time persisting only for timescales ∼Kolmogorov timescale. Furthermore, we consider the collision of inertial particles as well as tracer particles, and study their behavior not only at the collision boundary but also in its vicinity. In the simulation, particles are injected continuously at the boundaries of the straining flow, the size of the straining region being typical of the Kolmogorov length scale η_K of the turbulence. For steady state conditions, we calculate the flux of particles colliding with a test particle at the centre of the straining flow and consider its dependence on the inertia of the colliding particles (characterized by the particle Stokes number, St). The model replicates the segregation and accumulation observed in DNS and in particular the maximum segregation for St ∼ 1 (where St is the ratio of the particle response time to the Kolmogorov timescale). We also calculate the contributions of the various turbulent forces in the momentum balance equation for satellite particles and show for instance that for small Stokes number, there is a balance between turbulent diffusion and turbophoresis (gradient of kinetic stresses) which in turn is responsible for the build-up of concentration at the collision boundary. As found in previous studies, for the case of inertialess tracer particles, the collision rate turns out to be significantly smaller than the S&T prediction due to a lowering of the concentration at the collision boundary compared to the fully mixed value. The increase in collision rate for St ∼ 0.5 is shown to be a combination of particle segregation (build-up of concentration near the collision boundary) and the decorrelation of the relative velocity between the local fluid and a colliding particle. The difference from the S&T value for the agglomeration kernel is shown to be a consequence of the choice of perfectly absorbing boundary conditions at collision and the influence of the time scale of the turbulence (eddy lifetime). We draw the analogy between turbulent agglomeration and particle deposition in a fully developed turbulent boundary layer.