One-equation RG hybrid RANS/LES computation of a turbulent impinging jet

Abstract

A one-equation variant of a previously developed two-equation, renormalization group based, hybrid RANS/LES model is presented. The model consists of a transport equation for the mean dissipation rate and an algebraically prescribed length scale. The length scale is proportional to the wall-distance in RANS regions of the flow and to the filter width in LES regions. A very simple, but efficient, near-wall model is also presented, in the manner of Yakhot and Orszag's RNG modeling. The only free parameter entering the low-Reynolds formulation has been calibrated against channel flow, and the value corresponds well with experimental values for the same parameter obtained for homogeneous isotropic turbulence. The model is then validated for a turbulent impinging jet heat transfer problem. Results are satisfactory and compare very favorably against DES and dynamic LES for the same computation.

Keywords:

• turbulence modeling
• subgrid modeling
• hybrid RANS/LES modeling
• renormalization group

Acknowledgment

The first author works as a Postdoctoral Fellow for the Bijzonder Onderzoeksfonds (BOF) of the UGent. The authors acknowledge the support from the Instituut voor de Aanmoediging van Innovatie door Wetenschap en Technologie in Vlaanderen (IWT, contract MuTEch SBO 040092).

Notes

1. Apart from some models that were developed with a specific application in mind, such as the Spalart–Allmaras model for aerodynamic applications.

2. Universal hybrid RANS/LES models [6] use the same model in the whole domain. The model has a RANS limit when the grid spacing becomes too coarse to perform LES. In practice RANS only occurs near-walls. Zonal hybrid RANS/LES models will not always allow such simple length scale modeling.

3. The stochastic nature of the hybrid RANS/LES flow field necessitates longer time averages than for 3D unsteady RANS, and even when the time steps in both computations are taken equal a hybrid RANS/LES computation generally needs more time steps to obtain statistical convergence than unsteady RANS [7].

4. In [7] we showed, for the case of a swirling jet, that the RG model is not very sensitive to the exact value of its model constants (as long as consistent sets are used), while, for some statistics, the DES results showed quite some sensitivity to its adjustable constant C _DES .

5. As can be seen from the mean velocity magnitude profiles below, the results for the fine grid, without refinement at the pipe exit, already show that the development of the shear layer is quite well modeled (with the RG hybrid model). Finer grid resolution just shows more LES activity, and less energy in the modeled part, without too much influence on the mean velocity (and Nusselt number) profiles.

6. For every computation we performed, we noticed irregular behavior of the mean Nusselt profiles in the stagnation zone, even after long time averaging. This is a consequence of numerical instabilities of the central differencing scheme, and was also noted in the LES and hybrid RANS/LES studies in [14]. The combined effect of the rapid radial acceleration of the flow in that region, the zero turbulent viscosity and the unstructured mesh in the zone where r/D < 0.5 lead to these observed unphysical oscillations in the mean Nusselt number profiles. Even with the bounded central differencing scheme (which was developed to suppress such wiggles), some numerical instabilities remained visible. In [14] these wiggles were removed by relying upon the QUICK upwind scheme in a region close to the stagnation point (in Fluent, however, one cannot use different schemes in different regions of the computational domain, and therefore the results still contain the effects of the numerical instabilities). Taking the uncertainties due to these wiggles near the stagnation zone into account, the computations still clearly show the general tendency of the Nusselt profile there.

7. Although the better prediction compared to DES of the asymptotic behavior of the Nusselt profile is mostly due to the RANS limit of the RG model, it should be mentioned that this does not mean that a full RANS computation with the one-equation RG ϵ−L model of the impinging jet would also lead to good results for the Nusselt profile in this asymptotic region, since the outer part of the boundary layer and the incoming velocity field from the jet would be wrongly predicted with such a simple RANS length scale.

8. The second maximum in the Nusselt profile is quite peculiar, and only occurs for small nozzle-to-plate distances. For larger distances (see below), no secondary peak is seen.

9. The hybrid RANS/LES model investigated in [14], which also uses a linear constitutive relationship and a grid with about the same resolution as our fine grid, does not show the second maximum neither, very probably for the same reasons explained above.

10. The influence of the radial extension was investigated with a RANS computation, and 12D was found to be sufficient to avoid influence of the outflow boundary condition.

11. Some other experiments provided Nusselt profiles that lie quite far apart, with the experiment of Baughn et al. and Colucci and Viskanta lying in between.