16thInternational Conference on Finite Elements in Flow Problems (FEF 2011)
W.A. Wall and V. Gravemeier (Eds) Munich, Germany, March 23–25, 2011
Application of the discontinuous Galerkin method to 3D compressible
RANS simulations of a high lift cascade flow
Marcus Drosson1, Bastien Gorissen and Koen Hillewaert2 1 University of Li`ege 2 Cenaero
Chemin des Chevreuils, 1, Bˆat. B52/3 Rue des fr`eres Wright, 29, Bˆat. Eole 4000 Li`ege, BELGIUM 6041 Gosselies, BELGIUM
m.drosson@ulg.ac.be koen.hillewaert@cenaero.be 0.1 1 10 100 1000 104 0 5 10 15 20 25 30 y+=uΤyΝ u += U u Τ y+=128 y+=96 y+=64 y+=32 y+=16 y+=8 y+=4 Wieghardt æ æ æ æ æ æ æ æ ææ à à à à à à à à ì ì ì ì ì ì ì ì ò ò ò ò ò ò ò ô ô ô ô ô ô ç ç ç ç ç ç á á á á á í í í í 0.1 1 10 100 1000 104 0 5 10 15 20 25 30 y+=uΤyΝ u += U u Τ í y+=128 áy+=96 çy+=64 ô y+=32 ò y+=16 ìy+=8 ày+=4 æWieghardt
(a) Velocity profile
0 2 4 6 8 10 2.0 2.5 3.0 3.5 4.0 4.5 5.0 106Rex 10 -3C f y+=128 y+=96 y+=64 y+=32 y+=16 y+=8 y+=4 Wieghardt æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ à à à à ì ì ì ò ò ò ô ô ô ç ç ç á á á í í í 0 2 4 6 8 10 2.0 2.5 3.0 3.5 4.0 4.5 5.0 106Rex 10 -3C f í y+=128 á y+=96 ç y+=64 ô y+=32 ò y+=16 ì y+=8 à y+=4 æ Wieghardt (b) Friction coefficient
Figure 1: Flat plate computations (M=0.2 and Re=5 × 106) with 4th order
polynomials on structured quadrangular grids.
A 3D high-order RANS solver in conservative variables has been developed, based on a dis-continuous Galerkin/Symmetric Interior Penalty discretisation. The turbulence model is the well-known one-equation Spalart-Allmaras model. It is shown that in order to stabilise the discretization scheme, it is necessary to adapt the transpose penalty term, which introduces an explicit dependency of the continuity equation on the turbulent viscosity.
The grid convergence of the method is illustrated by computations for a turbulent flat plate (M=0.2 and Re=5 × 106). Fig. 1 shows the velocity profile and the skin friction distribution for
different off-wall spacings y+of the first element in the case of 4storder polynomials and
struc-tured quadrangular grids. We observe that results stay in excellent agreement with experimental measurements, even for meshes which are much coarser (y+ = 64) and have much more severe stretchings (here 1.6) than the corresponding finite volume grids.
A second application demonstrates the performance of the presented method for complex 3D cases. Thereto the turbulent flow in a 3D compressor cascade (M=0.1 and Re=8.4 × 105),
featuring secondary flow and hub stall is computed, and detailed comparison with state of the art finite volume solvers are presented.
