Hybrid model based on integral length as parameter switching between LES and URANS

16.3 Hybrid model based on integral length as parameter switching between LES and URANS

A simple hybrid model proposed in [29] is based on the observation that the basic transport equations have the same form in LES and RANS

but the interpretation of the overline differs. In LES it means filtering, but in RANS it stands for the Reynolds, or ensemble, averaging. Here we used the standard notation of p

ij for the

ij

F igure 16.4: The division of the computational domain into the URANS (dark) and LES (light) regions at one time instant for hybrid calculation of tanker.

laminar and turbulent stresses respectively. Note that the turbulent stresses are calculated in different ways in LES and URANS regions. The computational domain in our model is dynamically (i.e. at each time step) divided into the LES and URANS regions. A cell of the mesh belongs to one or the other region depending on the relation between the integral

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The integral length scale is calculated from the known formula of Kolmogorov and Prandtl with the correction factor 0:168 taken from [12]

where k is the turbulent kinetic energy and " is the dissipation rate. The constant C is C 0:7 and C the boundary layer thickness. L varies from one time step to another, which results in varying decomposition of the computational domain into the LES and URANS regions. The extended LES filter is computed as

(16.5) where d max is the maximal length of the cell edges d max

d max 2 Cı 2 ;

D max.d x ;d y ;d z / and ı D .the cell volume/ 1=3 is the common filter width used in LES. This

choice ensures that very flat cells in the boundary layer (for which ı but d max >0 precomputed only once before the main computation.

As a sample LES and URANS regions are shown in Fig. 16.4 for flow around

a tanker. The URANS region is located close to the ship surface and plays the role of a dynamic wall function. In areas of bilge vortices formation, the boundary layer is shedding from the hull and penetrates into the outer flow part. Since the boundary layer is a fine scale flow the procedure (16.3) recog- nizes the bilge vortex formation zones as URANS ones. There is a technical issue concerning the cells which are far from the ship hull and where both k and " are small, so large numerical errors are introduced into the integral length scale computed according to Eq. (16.4). To avoid an irregular dis- tribution of URANS and LES zones, the general rule (16.3) of the domain decomposition is corrected in such a way that the LES region is switched to URANS one if k is getting less than some threshold. This procedure has no influence on the ship flow parameters since it is used far from the area of the primary interest.

We have performed several calculations with different combinations of LES and URANS models to find the most efficient one for the problem under consideration. Among the models we used in our computations are the linear

and nonlinear k-", k-! SST and k"v 2 f URANS models combined with the simple and dynamic Smagorinsky as well as with the dynamic mixed LES closure models. The experience shows that the most satisfactory results are

obtained using the URANS approach based on the k"v 2 f turbulent model of [30] and LES approach based on the Smagorinsky dynamic model. The

ij are calculated from the Boussinesq approximation using the concept of the turbulent viscosity. The only difference between LES

and URANS is the definition of the kinematic viscosity. Within LES it is considered as the subgrid viscosity and calculated according to the dynamic model of Smagorinsky:

D min 2 0:09 ; 0:22v T t ;

where v 2 is the wall normal component of the stresses and T t is the turbulent time T t D max.k="; 6