An extensive study on LES, RANS and hybrid RANS/LES simulation of a narrow-gap open Taylor-Couette flow

The present paper concerns a numerical benchmark of various turbulence modelings, from RANS to LES, applied to Taylor-Couette-Poiseuille flows in a narrow gap cavity for six different combinations of rotational and axial Reynolds numbers. Two sets of refined Large-Eddy Simulation results, using the WALE and the Dynamic Smagorinsky subgrid scale models available within an in-house code based on high-order compact schemes, hold for reference data. The efficiency of a RANS model, the Elliptic Blending Reynolds Stress Model (EB-RSM) [1], and a hybrid RANS/LES method, the so-called ”Equivalent DES” [2], both run with Code Saturne, is then questioned. Thin coherent structures appearing as negative (resp. positive) spiral rolls are obtained by the LES but also the hybrid RANS/LES along the rotor (resp. stator) sides. More quantitatively, the hybrid RANS/LES does not improve the predictions of the EBRSM for both the mean and turbulent fields, stressing the need for further theoretical development. NOMENCLATURE C, Cm model parameters. Cε1,Cε2 model parameters. δ t time step. h length of the cylinders. k total (resolved+modeled) turbulence kinetic energy. km modeled part of the turbulence kinetic energy. ∗Address all correspondence to this author. L length scale. N rotation parameter, N = ReΩ/ReQ. r radial coordinate. rk energy ratio rk = km/k. ReQ axial Reynolds number, ReQ =Wm(R2 −R1)/ν . ReΩ rotational Reynolds number, ReΩ = ΩR1(R2 −R1)/ν . R1, R2 radii of the inner and outer cylinders respectively. Ri j Reynolds stress tensor components. Si j Strain-rate tensor components. U Velocity. Vθ , Vz mean tangential and axial velocity components. Wm bulk inlet axial velocity. z axial coordinate. Subscripts and superscripts c cutoff. d deviatoric. i, j indices for tensors; (i, j) = (r,θ ,z). m modeled. s sweeping. .̃ filtered quantity. SFS subfilter-scale. t turbulent. Greek symbols α parameter of the EB-RSM. β0 model constant. Γ aspect ratio of the cavity, Γ = h/(R2 −R1). ∆ grid step. ε dissipation rate ε = εkk/2. 1 Copyright c © 2014 by ASME εi j dissipation rate tensor components. η radius ratio, η = R1/R2. ν fluid kinematic viscosity. νt turbulent eddy viscosity. φ∗ i j redistribution tensor components. θ azimuthal coordinate. ω pulsation. Ω rotation rate. Acronyms DES Detached-Eddy Simulation. EB-RSM Elliptic Blending Reynolds Stress Model. LES Large-Eddy Simulation. RANS Reynolds-Averaged Navier-Stokes. WALE Wall-Adapting Local Eddy Viscosity. INTRODUCTION Since Taylor’s first theoretical results obtained one century ago, huge research efforts have been done to better describe and understand the rich dynamical behavior exhibited by the flow induced by the differential rotation of two concentric cylinders. This kind of flow has found many applications in chemical engineering or in the turbomachinery industry for examples. In the present case, the main motivation arises from the cooling of electrical motors, which can be modeled quite faithfully by considering a narrow-gap Taylor-Couette system with an axial Poiseuille flow. A better knowledge of the hydrodynamic field is absolutely necessary before considering the heat transfer aspect as pointed out recently by Fénot et al. [3]. These last authors provided a very useful and exhaustive literature survey on Taylor-Couette flows with or without an axial Poiseuille flow and including or not heat transfer processes. They explained the difficulty to establish universal correlations for the heat transfer coefficients by the large numbers of parameters involved in the problem. Moreover, most of the previous studies were mainly concerned with temperature measurements without any idea of what the hydrodynamic flow was and especially what the inlet flow conditions were. In the present work, one proposes a numerical benchmark for various combinations of the flow parameters in the isothermal case. It is a step forward the turbulence modeling of the full problem with heat transfer. Nouri and Whitelaw [4] then Escudier and Gouldson [5] provided very useful experimental databases for middle-gap cavities (η ≃ 0.5) of large aspect ratios (Γ = 98 and 244 respectively). Until now, most numerical studies used turbulence modeling, which provided rather limited informations. It was shown by [6, 7] that increasing the rotation rate of the inner cylinder amplified the turbulence kinetic energy, resulting in the enhancement of heat transfer along the rotor. More recently, the development of computational methods (including Direct and Large Eddy Simulations) has led to an increase in numerical studies of rotating flows, but few works were concerned by turbulent Taylor-Couette-Poiseuille flows and especially the near-wall turbulent structures in such systems. As example, in the middlegap configuration, Chung and Sung [8] established the destabilization of the near-wall turbulent structures due to rotation of the inner wall giving rise to an increase of sweep and ejection events. In the same way, Hadziabdic et al. [9] studied by LES a fully-developed turbulent flow in a concentric annulus of radius ratio η = 0.5, with the outer wall rotating at a range of rotation rates N = ReΩ/ReQ from 0.5 up to 4. They focused their attention on the effect of the rotation parameter N on the turbulence statistics and coherent structures in the near-wall regions. To our knowledge, there is no reference numerical or experimental data for the narrow-gap case (η > 0.8). Thus, the present in-house code based on fourth-order compact schemes using the dynamic Smagorinsky subgrid scale model has been first validated by Oguic et al. [10] against the experimental data of Nouri and Whitelaw [4]. Its slightly improved the LES results of Chung and Sung [8] based on second-order numerical schemes highlighting the importance of the order of the spatial schemes. It will be considered here as providing the reference data to discuss the capability of RANS and hybrid RANS/LES of predicting the mean and turbulent flow fields in the narrow-gap case. The present work is an extension of Friess et al. [11] to more combinations of the rotational and axial Reynolds numbers and to more turbulence modelings. Its goal remains twofold: (i) providing some reference LES data using two subgrid scale modes available within an in-house high-order solver and (ii) questioning the capabilities of a hybrid RANS / LES method, as well as the underlying RANS model, in predicting this kind of flow. The paper is organized as follows: the flow configuration is first described. The numerical approaches are then presented. Afterwards, the results about the hydrodynamic fields are discussed in details in terms of the coherent structures and the mean and turbulent flow fields, before some final conclusions. FLOW CONFIGURATION AND PARAMETERS The fluid is confined between two concentric cylinders of radii R1 and R2 and height h (see Figure 1). The inner cylinder rotates at a constant rate Ω, while the outer cylinder is stationary. The cavity is characterized by two geometrical parameters : its aspect ratio Γ = h/(R2 −R1) and its radius ratio η = R1/R2. An axial throughflow is imposed within the gap at a constant bulk velocity Wm. The main flow parameters are the rotational Reynolds number ReΩ =ΩR1(R2−R1)/ν and the bulk Reynolds number ReQ =Wm(R2 −R1)/ν , ν being the fluid kinematic viscosity. The present study considers two values of the axial flow rate and three rotation rates of the inner cylinder, resulting in five distinct values of N = ReΩ/ReQ (see Table 1). 2 Copyright c © 2014 by ASME