The structure of the solution obtained with Reynolds-stress-transport models at the free-stream edges of turbulent flows

The behavior of Reynolds-stress-transport models at the free-stream edges of turbulent flows is investigated. Current turbulent-diffusion models are found to produce propagative (possibly weak) solutions of the same type as those reported earlier by Cazalbou, Spalart, and Bradshaw [Phys. Fluids 6, 1797 (1994)] for two-equation models. As in the latter study, an analysis is presented that provides qualitative information on the flow structure predicted near the edge if a condition on the values of the diffusion constants is satisfied. In this case, the solution appears to be fairly insensitive to the residual free-stream turbulence levels needed with conventional numerical methods. The main specific result is that, depending on the diffusion model, the propagative solution can force turbulence toward definite and rather extreme anisotropy states at the edge (one - or two-component limit). This is not the case with the model of Daly and Harlow [Phys. Fluids 13, 2634 (1970)]; it may be one of the reasons why this "old" scheme is still the most widely used, even in recent Reynolds-stress-transport models. In addition, the analysis helps us to interpret some difficulties encountered in computing even very simple flows with Lumley's pressure-diffusion model [Adv. Appl. Mech. 18, 123 (1978)]. A new realizability condition, according to which the diffusion model should not globally become "anti-diffusive", is introduced, and a recalibration of Lumley's model satisfying this condition is performed using information drawn from the analysis

New results on the model problem of the diffusion of turbulence from a plane source

The problem of the diffusion of turbulence from a plane source is addressed in the context of two-equation eddy-viscosity models and Reynolds-stress-transport models. In the steady state, full analytic solutions are given. At second order, they provide the equilibrium value of the anisotropy level obtained with different combinations of return-to-isotropy and turbulent-diffusion schemes and confirm the results obtained by Straatman et al. [AIAA J. 36, 929 (1998)] in an approximate analysis. In addition, all the characteristics of the turbulence decrease can be determined and it is shown that a special constraint on the value of the modeling constants should hold if turbulence fills the whole surrounding space. In a second step, precise results can be given for the unsteady model problem at the first-order-closure level. The evolution cannot be described with a single set of characteristic scales and one has to distinguish the cases of short and large times. In the short-time regime, the flow is governed by the characteristic scales of turbulence at the source and contamination of the flow proceeds as t^1/2. At large times, the flow is governed by time-dependent characteristic scales that correspond to the solution of the steady problem at the instantaneous location of the front. Contamination of the flow proceeds as a power of time that can be related to the value of the modeling constants. The role of a combination of these constants is emphasized whose value can be specified to produce a solution that matches simultaneously the experimental data for the decrease of turbulent kinetic energy in the steady state and the exponent of the propagation velocity in the transient regime

Large scale simulation of turbulence using a hybrid spectral/finite difference solver

Performing Direct Numerical Simulation (DNS) of turbulence on large-scale systems (offering more than 1024 cores) has become a challenge in high performance computing. The computer power increase allows now to solve flow problems on large grids (with close to 10^9 nodes). Moreover these large scale simulations can be performed on non-homogeneous turbulent flows. A reasonable amount of time is needed to converge statistics if the large grid size is combined with a large number of cores. To this end we developed a Navier-Stokes solver, dedicated to situations where only one direction is heterogeneous, and particularly suitable for massive parallel architecture. Based on an hybrid approach spectral/finite-difference, we use a volumetric decomposition of the domain to extend the FFTs computation to a large number of cores. Scalability tests using up to 32K cores as well as preliminary results of a full simulation are presented

Direct numerical simulation of unsheared turbulence diffusing toward a free-slip or no-slip surface

The physics involved in the interaction between statistically steady, shearless turbulence and a blocking surface is investigated with the aid of direct numerical simulation. The original conguration introduced by Campagne et al. [ECCOMAS CFD 2006] serves as the basis for comparing cases in which the blocking surface can be either a free-slip surface or a no-slip wall. It is shown that in both cases, the evolutions of the anisotropy state are the same throughout the surface-influenced layer (down to the surface), despite the essentially different natures of the inner layers. The extent of the blocking effect can thereby be measured through a local (surface) quantity identically defined in the two cases. Examination of the evolution and content of the pressure-strain correlation brings information on the mechanisms by which energy is exchanged between the normal and tangential directions: In agreement with an earlier analysis by Perot and Moin [J. Fluid Mech. 295 (1995)], it appears that the level of the pressure strain correlation is governed by a splat/antisplat disequilibrium which is larger in the case of the solid wall due to viscous effects. However, in contradiction with the latter, the pressure-strain correlation remains as a signicant contributor to both Reynolds-stress budgets; it is argued that the net level of the splat/antisplat disequilibrium is set, in the first place, by the normal-velocity skewness of the interacting turbulent field. The influence of viscous friction on the intercomponent energy transfer at the solid wall only comes in the second place and part of it can also be measured by the skewness. The remainder seems to originate from interactions between the strain field and ring-like vortices in the vicinity of the splats

Two-equation modeling of turbulent rotating flows

The possibility to take into account the effects of the Coriolis acceleration on turbulence is examined in the framework of two-equation eddy-viscosity models. General results on the physical consistency of such turbulence models are derived from a dynamical-system approach to situations of time-evolving homogeneous turbulence in a rotating frame. Application of this analysis to a (k,epsilon) model fitted with an existing Coriolis correction [J. H. G. Howard, S. V. Patankar, and R. M. Bordynuik, "Flow prediction in rotating ducts using Coriolis-modified turbulence models", ASME Trans. J. Fluids Eng. 102, (1980)] is performed. Full analytical solutions are given for the flow predicted with this model in the situation of homogeneously sheared turbulence subject to rotation. The existence of an unphysical phenomenon of blowup at finite time is demonstrated in some range of the rotation-to-shear ratio. A direct connection is made between the slope of the mean-velocity profile in the plane-channel flow with spanwise rotation, and a particular fixed point of the dynamical system in homogeneously sheared turbulence subject to rotation. The general analysis, and the understanding of typical inaccuracies and misbehavior observed with the existing model, are then used to design a new model which is free from the phenomenon of blowup at finite time and able to account for both of the main influences of rotation on turbulence: the inhibition of the spectral transfer to high wave numbers and the shear/Coriolis instability

The structure of a statistically steady turbulent boundary layer near a free-slip surface

The interaction between a free-slip surface with unsheared but sustained turbulence is investigated in a series of direct numerical simulations. By changing (i) the distance between the (plane) source of turbulence and the surface, and (ii) the value of the viscosity, a set of five different data sets has been obtained in which the value of the Reynolds-number varies by a factor of 4. The observed structure of the interaction layer is in agreement with current knowledge, being made of three embedded sublayers: a blockage layer, a slip layer, and a Kolmogorov layer. Practical measures of the different thicknesses are proposed that lead to a new Reynolds-number scaling based on easy-to-evaluate surface quantities. This scaling is consistent with previous proposals but makes easier the comparison between free-surface flows when they differ by the characteristics of the distant turbulent field. Its use will be straightforward in a turbulence-modeling framework

Deflection, drift and advective growth in variable-density, laminar mixing layers

Specific features of the variable-density mixing layers without gravity effects are studied using self-similar solutions to the laminar and time-evolving variant of this flow. Density variations come from either mass or temperature mixing, accounting for, in the latter case, the effect of the Mach number. The transverse profiles of the flow quantities, as well as the time evolutions of the global characteristic scales of the mixing layer, are given for a wide range of density ratio and Mach-number values. When compared to the constant-density case, it appears that most of the specificity of these flows comes from the emergence of a nonzero transverse component of the velocity. First, it produces a deflection of the flow that can be either confined in the core of the layer or global, the whole layer being tilted at an angle from the initial flow direction. In most cases, this deflection is such that some part of the higher-density fluid is "entrained" in the direction of the lower-density fluid, leaving no possibility to define a dividing streamline. Second, it leads to a shift between the density profile and the profiles of the other flow quantities. This shift scales on the time-increasing mixing-layer thickness and therefore appears as a time drift. When global deflection is present, the tilting of the layer can be shown to be equivalent to a global drift of the mixing/shear layer toward the light-fluid side of the flow. Third, transport by the transverse velocity component affects the spreading of the mixing layer, giving rise to an additional effect referred to as advective growth. Examination of the density-ratio and Mach-number effects leads to surprising results: While the momentum thickness is always observed to decrease when increasing these parameters, conventional thicknesses based on the profiles of the different variables can show opposite behaviors depending on the form of the diffusion model for the considered variable

Numerical error evaluation for tip clearance flow calculations in a centrifugal compressor

Since globally mesh independent solution are still beyond available computer resources for industrial cases, a method to quantify locally the numerical error is proposed. The design of experiments method helps selecting mesh parameters that influence the tip clearance solution, so that additional meshes are computed to evaluate the numerical error on the shroud friction coefficient. In the field of CFD applied to turbomachinery, this study results from a partnership between ENSICA, Liebherr-Aerospace Toulouse and Numeca International. This paper focuses on numerical error evaluation for RANS simulations, applied to centrifugal compressor flow field calculations. CFD is now commonly used for centrifugal compressor design optimization, but, as Hutton and Casey develop in [1], there is an urging demand for improved quality and trust in industrial CFD. Indeed, this stresses the need for comprehensive and thorough numerical error evaluation, namely the process of verification, as defined for example by Oberkampf and Trucano in [2]. Unfortunately, 3D turbulent calculations for turbomachinery components are still very demanding in computational resources and, to the knowledge of the author, there is no published result concerning comprehensive verification of the entire flow field in centrifugal compressors. As a first step on the way to achieve that, this paper presents a method aiming at the obtention of a numerical solution that can be regarded as locally mesh-independent. In other words, the objective is to compute the flow field on a grid such that the solution obtained has a specific region where the numerical error is negligible. It has long been recognized that the tip clearance of a centrifugal compressor is of paramount importance for aerodynamic performances, which means that accurately predicting the flow field in this region is crucial for accurate prediction of performances by means of CFD codes. Numerous studies have been published that compare numerical and experimental results in the tip region. However, in these studies, numerical error still remains an issue; for instance Basson and Lakshminarayana [3] show excellent comparisons with experiments, but they attribute the remaining discrepancies to insufficient grid resolution. Indeed, accurate predictions of global effects, such as efficiency, require a fine description of flow details. Therefore, friction at the shroud endwall is the concern of the study, since it is a very sensitive indicator of the quality of the velocity profile’s prediction at the wall

Compressibility effects on shock-turbulence interaction

Direct numerical simulation (DNS) and inviscid linear analysis (LIA) are used to study the interaction of a normal Mach 1.5 shock wave and isotropic turbulence. The influence of the nature of the incoming turbulence on the interaction is emphasized. The presence of upstream entropy fluctuations enhance the amplification of the turbulent kinetic energy and transverse vorticity variance across the shock compared to the solenoidal (pure vorticity) case. More reduction of the transverse Taylor microscale is also observed in the vorticity-entropy case while no influence can be seen on the longitudinal microscale. When acoustic and vortical fluctuations are associated upstream, less amplification of the turbulent kinetic energy, less reduction of the transverse microscale and more amplification of the transverse vorticity variance are observed through the discontinuity. All the DNS results are in good qualitative agreement with LIA

Direct numerical simulation of the interaction between a shock wave and various types of isotropic turbulence

Direct Numerical Simulation (DNS) is used to study the interaction between normal shock waves of moderate strength (M1=1.2 and M1=1.5) and isotropic turbulence. A complete description of the turbulence behaviour across the shock is provided and the influence of the nature of the incoming turbulence on the interaction is investigated. The presence of upstream entropy fluctuations satisfying the Strong Reynolds Analogy enhances the amplification of the turbulent kinetic energy and transverse vorticity variances across the shock compared to the solenoidal (pure vorticity) case. Budgets for the fluctuating-vorticity variances are computed, showing that the baroclinic torque is responsible for this additional production of transverse vorticity. More reduction of the transverse Taylor microscale and integral scale is also observed in the vorticity-entropy case while no influence can be seen on the longitudinal Taylor microscale. When the upstream turbulence is dominated by acoustic and vortical fluctuations, less amplification of the kinetic energy (for Mach numbers between 1.25 and 1.8), less reduction of the transverse microscale and more amplification of the transverse vorticity variance are observed through the shock compared to the solenoidal case. In all cases, the classic estimation of Batchelor relating the dissipation rate and the integral scale of the flow proves to be invalid. These results are obtained with the same numerical tool and similar flow parameters, and they are in good agreement with Linear Interaction Analysis (LIA)