High-order Large Eddy Simulations of Confined Rotor-Stator Flows

In many engineering and industrial applications, the investigation of rotating turbulent flow is of great interest. In rotor-stator cavities, the centrifugal and Coriolis forces have a strong influence on the turbulence by producing a secondary flow in the meridian plane composed of two thin boundary layers along the disks separated by a non-viscous geostrophic core. Most numerical simulations have been performed using RANS and URANS modelling, and very few investigations have been performed using LES. This paper reports on quantitative comparisons of two high-order LES methods to predict a turbulent rotor-stator flow at the rotational Reynolds number Re(=?Ωb ²/ν)?=4 × 10⁵. The classical dynamic Smagorinsky model for the subgrid-scale stress (Germano et al., Phys Fluids A 3(7):1760–1765, 1991) is compared to a spectral vanishing viscosity technique (Séverac & Serre, J Comp Phys 226(2):1234–1255, 2007). Numerical results include both instantaneous data and post-processed statistics. The results show that both LES methods are able to accurately describe the unsteady flow structures and to satisfactorily predict mean velocities as well as Reynolds stress tensor components. A slight advantage is given to the spectral SVV approach in terms of accuracy and CPU cost. The strong improvements obtained in the present results with respect to RANS results confirm that LES is the appropriate level of modelling for flows in which fully turbulent and transition regimes are involved.     

Nusselt number and friction factor in thermally stratified turbulent channel flow under Non-Oberbeck–Boussinesq conditions

In stably stratified turbulence, computations under Oberbeck–Boussinesq (OB) hypothesis of temperature-independent fluid properties may lead to inaccurate representation of the flow field and to wrong estimates of momentum/heat transfer coefficients. This is clearly assessed here comparing direct numerical simulations of stratified turbulence under OB conditions to simulations under NOB (Non-Oberbeck–Boussinesq) conditions of temperature-dependent fluid viscosity and thermal expansion coefficient. Compared to the OB case, NOB conditions may induce local flow relaminarization with significant variations (up to 30%) of heat and momentum transfer coefficients. Together with DNS results, we propose a phenomenological model (based on turbulent bursts) for heat transfer prediction in stratified turbulence under OB and NOB conditions. Implications of NOB assumptions on mixing efficiency (i.e. flux Richardson number Ri_f) and turbulent Prandtl number (Pr_t) are also discussed. These results are of specific importance in RANS modelling, where the condition Pr_t = 1 is usually assumed (Reynolds analogy). Although this assumption is valid in some situations (i.e. boundary layer, pipe flow) there is uncertainty about its validity for stably-stratified turbulence. We demonstrate that this assumption is inaccurate when NOB effects become significant.     

A stress transport equation model for simulating turbulence at any mesh resolution

Prior work has demonstrated the effectiveness of using two-equation closures as the basis for universal, self-adapting turbulence models that are effective at any mesh resolution (Perot and Gadebusch in Phys. Fluids 19:115105, 2007). In order to demonstrate the broad applicability of the fundamental approach, the same behavior is now demonstrated for a second-moment closure (SMC). The SMC has the advantage over the earlier two-equation universal closure of being more accurate in the coarse mesh limit and of having a natural mechanism for backscattering energy from the modeled to the resolved turbulent fluctuations. The mathematical explanation for why Reynolds averaged (RANS) transport equation closures are applicable at any mesh resolution, including the large eddy simulation (LES) regime, is reviewed. It is demonstrated that for the problem of isotropic decaying turbulence, the SMC model produces good predictions at any mesh resolution and with arbitrary initial conditions. In addition, it is shown that the proposed model automatically adapts to the mesh resolution provided. The self-adaptive nature of the method is clearly observed when different initial conditions are used. It is shown that classic RANS models (often thought to produce steady and smooth solutions) can produce three-dimensional, unsteady, and chaotic solutions when generalized correctly and when provided with sufficient mesh resolution. The implications of these observations on the fundamental theories of RANS and LES turbulence modeling are discussed.      

A Bypass Transition Model Based on the Intermittency Function

An intermittency model that is formulated in local variables is proposed for representing bypass transition in Reynolds-Averaged Navier-Stokes (RANS) computations. No external data correlation is used to fix transition. Transition is initiated by diffusion, and a source term carries it to completion. A sink term is created to predict the laminar region before transition, then it vanishes in the turbulent region. Both the source and sink are functions of a wall-distance Reynolds number and turbulence scale. A modification is introduced to predict transition in separated boundary layers. The transition model is incorporated with the k?ω RANS model. The present model is implemented into a general purpose, computational fluid dynamics (CFD) code. The model is validated with several test cases. Decent agreement with the available data is observed in a range of flows.     

Hybrid RANS/LES computations of plane impinging jets with DES and PANS models

The qualities of a DES (Detached Eddy Simulation) and a PANS (Partially-Averaged Navier–Stokes) hybrid RANS/LES model, both based on the k–ω RANS turbulence model of Wilcox (2008, “Formulation of the k–ω turbulence model revisited” AIAA J., 46: 2823–2838), are analysed for simulation of plane impinging jets at a high nozzle-plate distance (H/B = 10, Re = 13,500; H is nozzle-plate distance, B is slot width; Reynolds number based on slot width and maximum velocity at nozzle exit) and a low nozzle-plate distance (H/B = 4, Re = 20,000). The mean velocity field, fluctuating velocity components, Reynolds stresses and skin friction at the impingement plate are compared with experimental data and LES (Large Eddy Simulation) results. The k–ω DES model is a double substitution type, following Davidson and Peng (2003, “Hybrid LES–RANS modelling: a one-equation SGS model combined with a k–ω model for predicting recirculating flows” Int. J. Numer. Meth. Fluids, 43: 1003–1018). This means that the turbulent length scale is replaced by the grid size in the destruction term of the k-equation and in the eddy viscosity formula. The k–ω PANS model is derived following Girimaji (2006, “Partially-Averaged Navier–Stokes model for turbulence: a Reynolds-Averaged Navier–Stokes to Direct Numerical Simulation bridging method” J. Appl. Mech., 73: 413–421). The turbulent length scale in the PANS model is constructed from the total turbulent kinetic energy and the sub-filter dissipation rate. Both hybrid models change between RANS (Reynolds-Averaged Navier–Stokes) and LES based on the cube root of the cell volume. The hybrid techniques, in contrast to RANS, are able to reproduce the turbulent flow dynamics in the shear layers of the impacting jet. The change from RANS to LES is much slower however for the PANS model than for the DES model on fine enough grids. This delays the break-up process of the vortices generated in the shear layers with as a consequence that the DES model produces better results than the PANS model.     

Joint RANS/LES Approach to Premixed Flame Modelling in the Context of the TFC Combustion Model

We present an original timesaving joint RANS/LES approach to simulate turbulent premixed combustion. It is intended mainly for industrial applications where LES may not be practical. It is based on successive RANS/LES numerical modelling, where turbulent characteristics determined from RANS simulations are used in LES equations for estimation of the subgrid chemical source and viscosity. This approach has been developed using our TFC premixed combustion model, which is based on a generalization of the Kolmogorov’s ideas. We assume existence of small-scale statistically equilibrium structures not only of turbulence but also of the reaction zones. At the same time, non-equilibrium large-scale structures of reaction sheets and turbulent eddies are described statistically by model combustion and turbulence equations in RANS simulations or follow directly without modelling in LES. Assumption of small-scale equilibrium gives an opportunity to express the mean combustion rate (controlled by small-scale coupling of turbulence and chemistry) in the RANS and LES sub-problems in terms of integral or subgrid parameters of turbulence and the chemical time, i.e. the definition of the reaction rate is similar to that of the mean dissipation rate in turbulence models where it is expressed in terms of integral or subgrid turbulent parameters. Our approach therefore renders compatible the combustion and turbulent parts of the RANS and LES sub-problems and yields reasonable agreement between the RANS and averaged LES results. Combining RANS simulations of averaged fields with LES method (and especially coupled and acoustic codes) for simulation of corresponding nonstationary process (and unsteady combustion regimes) is a promising strategy for industrial applications. In this work we present results of simulations carried out employing the joint RANS/LES approach for three examples: High velocity premixed combustion in a channel, combustion in the shear flow behind an obstacle and the impinging flame (a premixed flame attached to an obstacle).     

A Gas-Kinetic Scheme for Turbulent Flow

RANS simulations may not provide accurate results for all flow conditions. The interaction between a shock wave and a turbulent boundary layer is an example which may still be difficult to simulate accurately. Beside the inability to reproduce physical phenomena such as shock unsteadiness, the argument is put forward that the conventional numerical schemes, based on the Navier-Stokes equations, may be unable to generate a physically consistent turbulent stress tensor in the presence of large unresolved scales of motion. A large ratio between unresolved and resolved scales of motion, a sort of Knudsen number based on turbulent fluctuations, might introduce inaccuracies for which the turbulence model is not accountable. In order to improve the accuracy of RANS simulations, researchers have suggested various ad-hoc modifications to standard turbulence models which limit eddy viscosity or the turbulent stress tensor in the presence of strong gradients. Gas-kinetic schemes might be able to improve RANS predictions in shocklayers by removing or limiting the errors caused by the large scales ratio. These schemes are a class of their own; in the framework of a finite-volume or finite-elements discretizations, they model the numerical fluxes on the basis of the Boltzmann equation instead of the Navier-Stokes equations as is conventionally done. In practical terms, these schemes provide a higher accuracy and, more importantly, an in-built “multiscalar” mechanism, i.e. the ability to adjust to the size of unresolved scales of motion. This property makes them suitable for shock-capturing and rarefied flow. Gas-kinetic scheme may be coupled to a conventional RANS turbulence model; it is shown that the turbulent stress tensor is naturally adjusted as a function of the unresolved-to-resolved scales ratios and achieves a higher physical consistency than conventional schemes. The simulations shown - well-known benchmark cases with strong shock-boundary layer interactions - have been obtained with a standard two-equation turbulence model (k- ω). It is shown that the gas-kinetic scheme provides good quality predictions, where conventional schemes with the same turbulence model are known to fail.     

Assessment of Reynolds stresses tensor reconstruction methods for synthetic turbulent inflow conditions. Application to hybrid RANS/LES methods

Hybrid or zonal RANS/LES approaches are recognized as the most promising way to accurately simulate complex unsteady flows under current computational limitations. One still open issue concerns the transition from a RANS to a LES or WMLES resolution in the stream-wise direction, when near wall turbulence is involved. Turbulence content has then to be prescribed at the transition to prevent from turbulence decay leading to possible flow relaminarization. The present paper aims to propose an efficient way to generate this switch, within the flow, based on a synthetic turbulence inflow condition, named Synthetic Eddy Method (SEM). As the knowledge of the whole Reynolds stresses is often missing, the scope of this paper is focused on generating the quantities required at the SEM inlet from a RANS calculation, namely the first and second order statistics of the aerodynamic field. Three different methods based on two different approaches are presented and their capability to accurately generate the needed aerodynamic values is investigated. Then, the ability of the combination SEM + Reconstruction method to manufacture well-behaved turbulence is demonstrated through spatially developing flat plate turbulent boundary layers. In the mean time, important intrinsic features of the Synthetic Eddy method are pointed out. The necessity of introducing, within the SEM, accurate data, with regards to the outer part of the boundary layer, is illustrated. Finally, user’s guidelines are given depending on the Reynolds number based on the momentum thickness, since one method is suitable for low Reynolds number while the second is dedicated to high ones with a transition located around Re_θ = 3000.     

基于基因表达式编程的数据驱动湍流建模

计算流体动力学是湍流研究的重要手段, 其中雷诺平均模拟在航空航天等实际工程中得到了广泛应用. 雷诺平均模拟的结果很大程度上依赖于湍流模型的预测精度, 而实际工程应用中常用的模型往往精度有限. 近年来, 数据驱动的湍流建模方法得到越来越多的关注. 本文介绍了基于基因表达式编程 (gene-expression programming, GEP) 方法的湍流建模相关进展. 本文首先讨论基因表达式编程应用于湍流建模的具体方法, 包括基本算法、显式代数应力模型和湍流传热两种建模框架、模型测试方法以及损失函数设置等. 在此基础上, 基因表达式编程方法被应用于涡轮叶栅尾流混合、竖直平板间自然对流、三维横向流中的射流等问题. 结果表明, GEP可以有效提升常用模型对于尾流混合损失、壁面热通量等关键参数的预测精度. 基因表达式编程方法可以显式给出模型方程, 因此模型具有可解释性强等特点. 基于双向耦合方法得到的模型还被证明具有较好的后验测试精度和鲁棒性. 基因表达式编程方法还被初步应用于大涡模拟亚格子应力和边界层转捩等问题的建模, 在不同湍流建模领域表现出很大的潜力.      

A modified STRUCT model for efficient engineering computations of turbulent flows in hydro-energy machinery

A modified STRUCT (MST) turbulence model for efficient engineering computations of turbulent flows in hydro-energy machinery is proposed in this paper. The MST model switches between URANS and LES-like modes using a new damping function to adjust the turbulent viscosity. Compared with the original STRUCT method, the modifications are as follows: (1) the BSL k-ω model with the Spalart-Shur correction is chosen as the new baseline to improve the sensitivity to rotation and curvature; (2) a new adaptive time-scale ratio is proposed to avoid the arbitrariness of geometric averaging operation in the original method; (3) the normalized helicity is introduced into the new damping function to detect the energy backscatter phenomenon. Five classical high Reynolds number flow cases are tested. The results show that the turbulent viscosity of the MST model is reasonably reduced in the massively separated regions and LES-like mode is activated, which captures more turbulent vortices and fluctuations on the URANS grids. With high efficiency and robustness, the MST model inherits the advantages of the original STRUCT method and improves the prediction accuracy of the turbulence with rotation and curvature, which enables efficient engineering computations of turbulent flows in hydro-energy machinery.     

Towards a Unified Turbulence Simulation Approach for Wall-Bounded Flows

A hybrid Reynolds-averaged Navier–Stokes/Large-Eddy Simulation (RANS/LES) methodology has received considerable attention in recent years, especially in its application to wall-bounded flows at high-Reynolds numbers. In the conventional zonal hybrid approach, eddy-viscosity-type RANS and subgrid scale models are applied in the RANS and LES zones, respectively. In contrast, the non-zonal hybrid approach uses only a generalized turbulence model, which provides a unified simulation approach that spans the continuous spectrum of modeling/simulation schemes from RANS to LES. A particular realization of the non-zonal approach, known as partially resolved numerical simulation (PRNS), uses a generalized turbulence model obtained from a rescaling of a conventional RANS model through the introduction of a resolution control function F _R , where F _R is used to characterize the degree of modeling required to represent the unresolved scales of turbulent motion. A new generalized functional form for F _R in PRNS is proposed in this study, and its performance is compared with unsteady RANS (URANS) and LES computations for attached and separated wall-bounded turbulent flows. It is demonstrated that PRNS behaves similarly to LES, but outperforms URANS in general.     

Turbulence properties in the cylinder wake at high Reynolds numbers

The present contribution analyses the turbulence properties in unsteady flows around bluff body wakes and provides a database for improvement and validation of turbulence models, concerning the present class of nonequilibrium flows. The flow around a circular cylinder with a low aspect ratio and a high blockage coefficient is investigated. This confined environment is used in order to allow direct comparisons with realisable 3-D Navier–Stokes computations avoiding ‘infinite’ conditions. The flow is investigated in the beginning of the critical regime at Reynolds number 140 000. The analysis is carried out by means of 2-D PIV, of 3-C PIV and of high-frequency 2-D PIV. The experimental analysis contributes to confirm the validity of advanced statistical turbulence modelling for unsteady flows around bodies.     

Application of the PITM Method Using Inlet Synthetic Turbulence Generation for the Simulation of the Turbulent Flow in a Small Axisymmetric Contraction

We investigate the turbulence modeling of second moment closure used both in RANS and PITM methodologies from a fundamental point of view and its capacity to predict the flow in a low turbulence wind tunnel of small axisymmetric contraction designed by Uberoi and Wallis. This flow presents a complex phenomenon in physics of fluid turbulence. The anisotropy ratio of the turbulent stresses τ ₁₁/τ ₂₂ initially close to 1.4 returns to unity through the contraction, but surprisingly, this ratio gradually increases to its pre-contraction value in the uniform section downstream the contraction. This point constitutes the interesting paradox of the Uberoi and Wallis experiment. We perform numerical simulations of the turbulent flow in this wind tunnel using both a Reynolds stress model developed in RANS modeling and a subfilter scale stress model derived from the partially integrated transport modeling method. With the aim of reproducing the experimental grid turbulence resulting from the effects of the square-mesh biplane grid on the uniform wind tunnel stream, we develop a new analytical spectral method of generation of pseudo-random velocity fields in a cubic box. These velocity fields are then introduced in the channel using a matching numerical technique. Both RANS and PITM simulations are performed on several meshes to study the effects of the contraction on the mean velocity and turbulence. As a result, it is found that the RANS computation using the Reynolds stress model fails to reproduce the increase of anisotropy in the centerline of the channel after passing the contraction. In the contrary, the PITM simulation predicts fairly well this turbulent flow according to the experimental data, and especially, the “return to anisotropy” in the straight section of the channel downstream the contraction. This work shows that the PITM method used in conjunction with an analytical synthetic turbulence generation as inflow is well suited for simulating this flow, while allowing a drastic reduction of the computational resources.     

Application of a differential equation for turbulent viscosity to the analysis of plane non-self-similar flows

A differential equation of the kinetic-energy balance of turbulence is used in a number of papers to close the equations describing average motion in turbulent flows. On the basis of this relation, a differential equation for turbulent viscosity is obtained herein. Numerical computations are carried out for incompressible non-self-similar turbulent and transition flows in awake, a jet, and a boundary layer; universal constants in the equation for the viscosity are refined. The flow in a wake and boundary layer with high longitudinal pressure gradients is investigated by analytical and numerical methods. Dimensionless criteria determining the nature of the effect of the pressure gradient on the average flow and turbulent viscosity are obtained.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 114–127, September–October, 1971.The author is greateful to I. P. Smirnov, S. Yu. Krasheninnikov, and V. B. Kuz'mich for aid in compiling the program for the numerical computations and to L. L. Bychkov for processing the computational results and plotting the graphs.     

Numerical study of an oscillatory turbulent flow over a flat plate

Oscillatory turbulent flow over a flat plate is studied using large eddy simulation (LES) and Reynolds-average Navier-Stokes (RANS) methods. A dynamic subgrid-scale model is employed in LES and Saffman's turbulence model is used in RANS. The flow behaviors are discussed for the accelerating and decelerating phases during the oscillating cycle. The friction force on the wall and its phase shift from laminar to turbulent regime are also investigated for different Reynolds numbers. The project supported by the Youngster Funding of Academia Sinica and by the National Natural Science Foundation of China     

A low Reynolds number variant of partially-averaged Navier-Stokes model for turbulence

A low Reynolds number (LRN) formulation based on the Partially Averaged Navier-Stokes (PANS) modelling method is presented, which incorporates improved asymptotic representation in near-wall turbulence modelling. The effect of near-wall viscous damping can thus be better accounted for in simulations of wall-bounded turbulent flows. The proposed LRN PANS model uses an LRN k-ε model as the base model and introduces directly its model functions into the PANS formulation. As a result, the inappropriate wall-limiting behavior inherent in the original PANS model is corrected. An interesting feature of the PANS model is that the turbulent Prandtl numbers in the k and ε equations are modified compared to the base model. It is found that this modification has a significant effect on the modelled turbulence. The proposed LRN PANS model is scrutinized in computations of decaying grid turbulence, turbulent channel flow and periodic hill flow, of which the latter has been computed at two different Reynolds numbers of Re = 10,600 and 37,000. In comparison with available DNS, LES or experimental data, the LRN PANS model produces improved predictions over the standard PANS model, particularly in the near-wall region and for resolved turbulence statistics. Furthermore, the LRN PANS model gives similar or better results - at a reduced CPU time - as compared to the Dynamic Smagorinsky model.     

Highly Resolved LES and URANS of Turbulent Buoyancy-Driven Flow Within Inclined Differentially-Heated Enclosures

In the present study, both LES and unsteady RANS computations are presented, for turbulent natural convection of air inside differentially-heated rectangular tilted cavities using a finite volume code (Code_Saturne), for an aspect ratio of H/L?=?28.6 and Rayleigh number of 0.86×10⁶. Attention is focused on two angles of inclination: 15° to the horizontal with hot lower and cold upper wall, the 15° unstable case, and the mirror image of this case where the angle is the same but with a hot upper and cold lower wall, the 15° stable case. In accordance with recent experimental data, the LES computations for both the stable and unstable tilted cavities returned three-dimensional time-averaged flow fields. In the case of the unstably stratified enclosure, the flow is highly unsteady with coherent turbulent structures in the core of the enclosure. Time-averaged temperature, velocity and resolved turbulence intensities resulting from LES computations show close agreement to measured data. Subsequent comparisons of different URANS schemes with LES are used in order to explore to what extent these models are able to reproduce the large-scale unsteady flow structures. All URANS schemes have been found to be able to reproduce the 3-D unsteady flow features present in the 15° unstable cavity. However, the low-Reynolds-number model tested, as well as requiring a high resolution near-wall grid, also needed a finer grid in the core region than the high-Reynolds-number models, thus making it computationally very expensive. Flow within the 15° stable cavity also shows some 3-D features, although it is significantly less unsteady, and the URANS models tested here have been less successful in reproducing this flow pattern. The overall heat transfer is presented here for both differentially heated enclosures.     

Upstream monotonic interpolation for scalar transport with application to complex turbulent flows

A new monotonic scheme for the approximation of steady scalar transport is formulated and implemented within a collocated finite-volume/pressure-correction algorithm for general turbulent flows in complex geometries. The scheme is essentially a monotonic implementation of the quadratic QUICK interpolation and uses a continuous and compact limiter to secure monotonicity. The principal purpose is to allow an accurate and fully bounded, hence stable, approximation of turbulence convection in the context of two-equation eddy viscosity and Reynolds stress transport modelling of two- and three-dimensional flows, both subsonic and transonic. Among other benefits, this capability permits an assessment to be made of the adequacy of approximating turbulence convection with first-order upwind schemes in conjunction with higher-order formulations for mean-flow properties—a widespread practice. The performance characteristics of the bounded scheme are illustrated by reference to computations for scalar transport, for a transonic flow in a Laval nozzle, for one separated laminar flow and for two separated turbulent flows computed with a non-linear RNG model and full Reynolds stress closure.     

Progress in subgrid-scale transport modelling for continuous hybrid non-zonal RANS/LES simulations

The partially integrated transport modelling (PITM) method can be viewed as a continuous approach for hybrid RANS/LES modelling allowing seamless coupling between the RANS and the LES regions. The subgrid turbulence quantities are thus calculated from spectral equations depending on the varying spectral cutoff location [Schiestel, R., Dejoan, A., 2005. Towards a new partially integrated transport model for coarse grid and unsteady turbulent flow simulations. Theoretical and Computational Fluid Dynamics 18, 443–468; Chaouat, B., Schiestel, R., 2005. A new partially integrated transport model for subgrid-scale stresses and dissipation rate for turbulent developing flows. Physics of Fluids, 17 (6)] The PITM method can be applied to almost all statistical models to derive its hybrid LES counterpart. In the present work, the PITM version based on the transport equations for the turbulent Reynolds stresses together with the dissipation transport rate equation is now developed in a general formulation based on a new accurate energy spectrum function E(κ) valid in both large and small eddy ranges that allows to calibrate more precisely the c_sgs2 function involved in the subgrid dissipation rate _sgs transport equation. The model is also proposed here in an extended form which remains valid in low Reynolds number turbulent flows. This is achieved by considering a characteristic turbulence length-scale based on the total turbulent energy and the total dissipation rate taking into account the subgrid and resolved parts of the dissipation rate. These improvements allow to consider a large range of flows including various free flows as well as bounded flows. The present model is first tested on the decay of homogeneous isotropic turbulence by referring to the well known experiment of Comte-Bellot and Corrsin. Then, initial perturbed spectra E(κ) with a peak or a defect of energy are considered for analysing the model capabilities in strong non-equilibrium flow situations. The second test case is the classical fully turbulent channel flow that allows to assess the performance of the model in non-homogeneous flows characterised by important anisotropy effects. Different simulations are performed on coarse and refined meshes for checking the grid independence of solutions as well as the consistency of the subgrid-scale model when the filter width is changed. A special attention is devoted to the sharing out of the energy between the subgrid-scales and the resolved scales. Both the mean velocity and the turbulent stress computations are compared with data from direct numerical simulations.