BAIL 2006 Book of Abstracts - Institut für Numerische und ...
BAIL 2006 Book of Abstracts - Institut für Numerische und ... BAIL 2006 Book of Abstracts - Institut für Numerische und ...
TH. ALRUTZ, T. KNOPP: Near-wall grid adaptation for wall functions ✬ ✫ 0.006 exp. low-Re y + (1) = 1 y + (1) = 20 0.004 c f 0.002 0 y + (1) = 40 y + (1) = 60 c f = 0 0 0.25 0.5 0.75 1 x/c 0.006 exp. low-Re y + (1) = 1 y + (1) = 20 0.004 c f 0.002 0 y + (1) = 40 y + (1) = 60 c f = 0 0 0.25 0.5 0.75 1 x/c Figure 2: RAE2822 case 10 [5]: Prediction for skin friction coefficient�for Spalart-Allmaras-Edwards model [3] without adaptation (left) and withÝ -adaptation of the prismatic near-wall grid (right). c p -4.2 -3.8 -3.4 -3 exp. low-Re y + (1) = 12 y + (1) = 24 y + (1) = 40 y + (1) = 80 0 0.05 0.1 0.15 x/c Ý ÆÝ Ü×�Ô� Ü×�Ô� SA-Edwards Menter SST low-Re 33 0.771 0.866 1 33 0.770 0.866 4 28 0.755 0.863 7 26 0.759 0.868 12 24 0.761 0.861 24 21 0.787 0.861 (0.864) 50 19 (0.771) 0.881 (0.873) 70 17 (0.788) 0.903 (0.867) Figure 3: A-airfoil [6]: Detail of pressure coefficientÔfor SST model [4] on adapted grid (left). Right: Prediction of the separation point without adaptation and withÝ -adaptation (values in brackets). [2] Th. Alrutz, “Hybrid grid adaptation in TAU”, In: MEGAFLOW - Numerical flow simulation for aircraft design, Notes on Numerical Fluid Mechanics and Multidisciplinary Design. N. Kroll and J.K. Fassbender, Eds., (2005). [3] J.R. Edwards and S. Chandra, “Comparison of eddy viscosity-transport turbulence models for threedimensional, shock separated flowfields”, AIAA Journal, 34, 756–763 (1996). [4] F.R. Menter, “Zonal two equation�/�turbulence models for aerodynamic flows”, AIAA Paper 1993-2906, (1993). [5] P. H. Cook, M. A. McDonald and M.C.P. Firmin, “Aerofoil RAE 2822 - Pressure distributions and boundary layer and wake measurements”, AGARD Advisory Report AR-138, A6.1-A6.77 (1979). [6] Ch. Gleyzes, “Opération décrochage - Résultats de la 2ème campagne d’essais à F2 – Mesures de pression et vélocimétrie laser”, RT-DERAT 55/5004 DN, ONERA, (1989). Speaker: ALRUTZ, TH 70 BAIL 2006 2 ✩ ✪
TH. APEL, G. MATTHIES: A family of non-conforming finite elements of arbitrary order for the Stokes problem on anisotropic quadrilateral meshes ✬ ✫ A family of non-conforming finite elements of arbitrary order for the Stokes problem on anisotropic quadrilateral meshes Thomas Apel 1 and Gunar Matthies 2 1 Institut für Mathematik und Bauinformatik, Universität der Bundeswehr München 2 Fakultät für Mathematik, Ruhr-Universität Bochum, The solution of the Stokes problem in polygonal or polyhedral domains shows in general a singular behaviour near corners and edges of the domain. Both edge singularities and layers are anisotropic phenomena since the solution changes only slightly in one direction while the derivatives in the perpendicular direction(s) are large. These anisotropic behaviour can be well approximated on anisotropic triangulations. Let Ω ⊂ R 2 be a bounded polygonal domain. We consider the Stokes problem −△u + ∇p = f in Ω, div u = 0 in Ω, u = 0 on ∂Ω, where u and p are the velocity and the pressure, respectively, while f is a given force. We solve this problem by using finite element methods on triangulations with special properties. Let the domain Ω be partitioned by an admissible triangulation which consists of shape regular and isotropic macro-cells. Each macro-cell is further refined by applying admissible patches which are adapted to boundary layers and corner singularities, respectively. For a detailed description of such meshes, we refer to [1]. We are interested in solving the Stokes problem by non-conforming finite element methods of higher order. To this end, we use the families with finite element pairs of arbitrary order which were given recently in [2]. Each pair consists of a non-conforming space of order r for approximating the velocity and a discontinuous, piecewise polynomial pressure approximation of order r − 1. For the stability of finite element methods for solving the Stokes problem and its relatives, it is necessary that the discrete spaces for the velocity and the pressure fulfil an inf-sup condition. In order to get error estimates with constants which are independent of the aspect ratio of the underlying mesh, it is important that on the one hand the inf-sup constant is independent of the aspect ratio and that on the other hand the approximation error and the consistency error can be bounded by expressions whose constants don’t depend on the aspect ratio. Considering the families given in [2], we will show that we obtain for one family optimal error estimates on anisotropic meshes, i.e., the constant are independent of the aspect ratio. The other families give only error estimates with constant which depend on the aspect ratio, i.e., the constants blow up with increasing aspect ratio. We will show by means of numerical results how the inf-sup constant behaves on special families of triangulation with increasing aspect ratio. References [1] Th. Apel, S. Nicaise, The inf-sup condition for low order elements on anisotropic meshes, CALCOLO, 41, 89–113 (2004). [2] G. Matthies, Inf-sup stable nonconforming finite elements of higher order on quadrilaterals and hexahedra. Bericht Nr. 373, Fakultät für Mathematik, Ruhr-Universität Bochum (2005). Speaker: MATTHIES, G. 71 BAIL 2006 1 ✩ ✪
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TH. ALRUTZ, T. KNOPP: Near-wall grid adaptation for wall functions<br />
✬<br />
✫<br />
0.006 exp.<br />
low-Re<br />
y + (1) = 1<br />
y + (1) = 20<br />
0.004<br />
c f<br />
0.002<br />
0<br />
y + (1) = 40<br />
y + (1) = 60<br />
c f = 0<br />
0 0.25 0.5 0.75 1<br />
x/c<br />
0.006 exp.<br />
low-Re<br />
y + (1) = 1<br />
y + (1) = 20<br />
0.004<br />
c f<br />
0.002<br />
0<br />
y + (1) = 40<br />
y + (1) = 60<br />
c f = 0<br />
0 0.25 0.5 0.75 1<br />
x/c<br />
Figure 2: RAE2822 case 10 [5]: Prediction for skin friction coefficient�for Spalart-Allmaras-Edwards<br />
model [3] without adaptation (left) and withÝ -adaptation <strong>of</strong> the prismatic near-wall grid (right).<br />
c p<br />
-4.2<br />
-3.8<br />
-3.4<br />
-3<br />
exp.<br />
low-Re<br />
y + (1) = 12<br />
y + (1) = 24<br />
y + (1) = 40<br />
y + (1) = 80<br />
0 0.05 0.1 0.15<br />
x/c<br />
Ý ÆÝ Ü×�Ô� Ü×�Ô�<br />
SA-Edwards Menter SST<br />
low-Re 33 0.771 0.866<br />
1 33 0.770 0.866<br />
4 28 0.755 0.863<br />
7 26 0.759 0.868<br />
12 24 0.761 0.861<br />
24 21 0.787 0.861 (0.864)<br />
50 19 (0.771) 0.881 (0.873)<br />
70 17 (0.788) 0.903 (0.867)<br />
Figure 3: A-airfoil [6]: Detail <strong>of</strong> pressure coefficientÔfor SST model [4] on adapted grid (left). Right:<br />
Prediction <strong>of</strong> the separation point without adaptation and withÝ -adaptation (values in brackets).<br />
[2] Th. Alrutz, “Hybrid grid adaptation in TAU”, In: MEGAFLOW - Numerical flow simulation for<br />
aircraft design, Notes on Numerical Fluid Mechanics and Multidisciplinary Design. N. Kroll and<br />
J.K. Fassbender, Eds., (2005).<br />
[3] J.R. Edwards and S. Chandra, “Comparison <strong>of</strong> eddy viscosity-transport turbulence models for threedimensional,<br />
shock separated flowfields”, AIAA Journal, 34, 756–763 (1996).<br />
[4] F.R. Menter, “Zonal two equation�/�turbulence models for aerodynamic flows”, AIAA Paper<br />
1993-2906, (1993).<br />
[5] P. H. Cook, M. A. McDonald and M.C.P. Firmin, “Aer<strong>of</strong>oil RAE 2822 - Pressure distributions and<br />
bo<strong>und</strong>ary layer and wake measurements”, AGARD Advisory Report AR-138, A6.1-A6.77 (1979).<br />
[6] Ch. Gleyzes, “Opération décrochage - Résultats de la 2ème campagne d’essais à F2 – Mesures de<br />
pression et vélocimétrie laser”, RT-DERAT 55/5004 DN, ONERA, (1989).<br />
Speaker: ALRUTZ, TH 70 <strong>BAIL</strong> <strong>2006</strong><br />
2<br />
✩<br />
✪