PAMM · Proc. Appl. Math. Mech. 4, 458–459 (2004) / DOI 10.1002/pamm.200410210
Investigations of shear free turbulent diffusion in a rotating frame Guenther, S.∗1 , Oberlack1 , Brethouver, G2 , and Johansson, A.V.2 1 2
Hydromechanics and Hydraulics Group, T U Darmstadt Petersenstraße 13, 64287 Darmstadt, Germany. Department of Mechanics, KTH, SE-100 44 Stockholm, Sweden.
We reconsider the problem of shear free turbulent diffusion in a rotating frame, rotating about x 1 . Shear free turbulence is generated at a vibrating grid in the x2 − x3 plane and diffuses away from the grid in x1 direction. An important property of this flow case is that there is no mean flow-velocity. With the help of Lie-group methods Reynolds-stress transport models can be analyzed for this kind of flow in a rotating frame. From the analysis it can be found, that the turbulent diffusion only influences a finite domain. Implicating this solution in the model equations shows that even fully nonlinear Reynolds-stress transport models (non-linear in the Reynolds-stresses for the pressure-strain model) are insensitive to rotation for this type of flow. © 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
1 Symmetries and invariant solutions of the Reynolds stress equations For the subsequent analysis we employ the Reynolds stress equations in a rotating frame. The rotation rate Ω 1 , enters these equations through the Coriolis term. The problem is one-dimensional in the sense that there is only one independent variable x1 , which defines the direction of inhomogeneity for this case. From two successive Lie group analysis we receive the final form of the invariant solutions (see also [3]): tΩ x1 2 x ˜1 = + xo e− to , ui uj = Ω2 (x1 + γ(Ω)x0 ) ug x1 ), i uj (˜ γ(Ω) (1) 1 2 2 g 2 3 K = Ω (x1 + γ(Ω)x0 ) ui ui (˜ x1 ), = Ω (x1 + γ(Ω)x0 ) e (˜ x1 ) . 2
The surprising result for this solution is that even for t → ∞ the turbulent diffusion only influences a finite domain due to the quadratic behavior of the large-scale turbulence quantities. The magnitude of the finite domain depends thereby through the function γ(Ω) on the rotation rate. The point at which the turbulent kinetic energy becomes zero, which corresponds to the coordinate γ(Ω)x0 is in the following called fix point.
2 Large eddy simulation To obtain a better understanding of the given flow case, a large-eddy simulations (LES) in a rotating frame at constant angular velocity Ω1 about the x1 axis have been performed. For the LES a standard pseudo-spectral method is used with periodic boundary conditions in all three directions. In order to simulate shear free turbulence, the flow field with zero mean velocity is forced in a limited part of the domain. Outside this region the flow field is not forced. A method introduced by Alvelius [1] is used to generate a random but statistically stationary forcing at the largest velocity scales. This random forcing in a limited part of the computational domain provides conditions that are similar to a vibrating grid in an experiment. Turbulent fluctuations develop in the forcing region, which then diffuse in x1 -direction. The subgrid scales of the velocity field are modelled by means of a simple eddy-viscosity model due to Smagorinsky with C s = 0.115 and a low Reynolds number correction. We observed that the subgrid scales have only a significant contribution inside and close to the part of the domain where the forcing is applied. Further away from the forcing region the subgrid contribution is small because of the lower turbulent intensity. The simulations have been running using a 1024 × 48 × 48 grid in a 40π × 2π × 2π box at various rotation rates. Figure 1 shows the decay of the turbulent kinetic energy depending on the rotation rate if anisotropic forcing is considered. From (1) one can easily see that the spatial decay of the turbulent kinetic energy is faster for small rotation rates than for larger ones. The same results have already been obtained for the temporal decay of isotropic turbulence in various former numerical simulations and experiments (for a list of relevant numerical simulations and experiments see [2]). The slower decay of kinetic energy is due to the fact that the rotation inhibits the transfer of energy from large to small scales and therefore results in a decrease of the dissipation rate. ∗
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© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
Section 12
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Fig. 1: Decay of turbulent kinetic energy obtained by large eddy simulation for various rotation rates: − − − Ω = 0.5, − · − Ω = 1, · · · Ω = 2.
3 Approaches for turbulence modelling
Only a few turbulence models seem to be capable to give a dependence of the Reynolds stresses respectively the kinetic energy on the rotation rate. The two models which are tested in the following are the model from Sj o¨ gren and Johansson [5] and Shimomura [4]. In the approach of Sjo¨ gren and Johansson a term which is quadratic in the rotation tensor Wij = ∂ui ∂xj
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To account for the rotation Shimomura presented a new equation by augmenting the classical equation with the term KWij Wji ω = −C3 2 . 1 + C4 Wij Wji K2
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The Launder Reece and Rodi (LRR) model augmented by the Nij term and the K − model augmented by the ω term have c S. Wallin, been investigated numerically for the given flow geometry with the help of a numerical tool, called 1D solver ( FOI). From figures 2 and 3 it can be seen that both models give a finite domain diffusion due to rotation. The position of the 0
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Fig. 3: Decay of turbulent kinetic energy using the K − Fig. 2: Decay of turbulent kinetic energy using the LRR model augmented by ω : − − − Ω = 0.5, − · − Ω = 1, model augmented by Nij : − − − Ω = 0.5, − · − Ω = 1, · · · Ω = 2. · · · Ω = 2. fix point depends thereby on the rotation rate Ω1 as predicted in section 1. Considering the dependence of the decay of kinetic energy on the rotation rate, it is found that the Shimomura model gives a correct prediction while the model from Sj o¨ gren and Johansson gives a faster decay for higher rotation rates. Taking a closer look at the terms it is further detected that in contrast to the ω term the Nij term vanishes for isotropic turbulence. Therefore the model becomes insensitive to rotation in isotropic turbulence what is in disagreement with the studies mentioned in [2].
References [1] Alvelius, K., 1999, Random Forcing of Three-Dimensional Homogeneous Turbulence, Pys.Fluids, 11, (7), 1880 [2] Jacquin, L., Leuchter, O., Cambon, C., Mathieu, J., 1990, Homogeneous turbulence in the presence of rotation, J. Fluid Mech., 220, 1-52 [3] Oberlack, M., Guenther, S., 2003, Shear-Free Turbulent Diffusion- Classical and New Scaling Laws, Fluid Dyn. Res., 33, (5-6), 453-476 [4] Shimomura, Y. 1993, Near-Wall Turbulent Flows, Proceedings of the International Conference on Near-Wall Turbulent Flows, Tempe, Arizona, U.S.A. [5] Sj¨ogren, T., Johansson, A.V., 2000, Development and calibration of algebraic nonlinear models for terms in the Reynolds stress transport equation, Phys. Fluids, 12, (6), 1554-1572 © 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
