Cylinder Flow

 For this test of the hybrid incompressible code we consider two-dimensional flow past a circular cylinder. The cylinder has unit diameter and the domain surrounding the cylinder is a rectangle $[-22,69]\times[-22,22]$. Uniform upwind boundary conditions are employed at the box. No slip and wall temperature are imposed on the cylinder. Figure 5.7 shows the domain, mesh and boundary conditions. This is the same mesh that was used for the incompressible cylinder flow tests.

The mesh has 332 quadrilaterals and 448 triangles. It was designed to direct resolution around the cylinder, have regular resolution in the wake behind the cylinder, and block out with large elements to push the farfield boundaries away from the cylinder. We ran the simulation at N=8 for Re = 100,150,200 and 250. A summary of the simulation parameters is given in table 7.1. In table 7.2 we show that the Strouhal number is only changed by about three percent by the compressibility at Mach 0.5.

 

Parameter Value

Dimension 2d

Re 100, 150, 200 and 250

Mach 0.5 and 0.7

$$\Delta t$$ 0.05 to 0.0002

N-Range 8 to 10

K_Tri 332

K_Quad 448

Method Discontinuous Galerkin

 

Re Incompressible Compressible (Mach 0.5)

100 0.1662 0.1611

150 0.1853 0.1812

200 0.1972 0.1934

250 0.2054 0.2020

We also ran the cimulation at Mach 0.7 in order to compare with the results of Lomtev [73] who used Discontinuous Galerkin with spectral element triangles and Beskok [81] who used Galerkin on quadrilaterals. We show good agreement in table 7.3 for the Strouhal frequency, however the drag (C_d) and lift (C_l) coefficients are different by about three percent. This difference can be accounted for by the different dimensions of the meshes used. For the run the outflow was 69 diameters from the cylinder, whereas for the other methods the outflow was only 35 diameters from the cylinder. When we ran with the smaller mesh it was obvious that there was a higher level of noise induced when the von Karman street exited the domain. The comparision with Lomtev shows that using quadrilaterals has not adversely affected the polymorphic implementation of the Discontinuous Galerkin method. The comparison with Beskok shows that the Discontinuous Galerkin method on unstructured polymorphic grids compares well with the more traditional Galerkin methods on semi-unstructured quadrilateral meshes using collocation.

 

$$(\frac{x}{D})$$ Code Method St C_d C_l

dgL [73] Discontinuous Galerkin 0.158 1.846 0.246 35

$$\mu$$ Explicit Galerkin 0.159 1.843 0.248 35

Discontinuous Galerkin 0.157 1.802 0.255 69

In figure 7.3 we show instantaneous iso-contours for compressible flow past a cylinder at Mach=0.5. Also in figure 7.4 we show vorticity which is quite similar to the incompressible case shown in figure 5.9.

  

Figure 7.3: Instantaneous iso-contours for the simulation of compressible flow past a cylinder (Re=100, M=0.5) From the top: (1) density, (2) pressure field, (3) x component of the velocity field, (4) y component of the velocity field

  

Figure 7.4: Instantaneous iso-contours of vorticity for the simulation of compressible flow past a cylinder (Re=100, M=0.5)