Cylinder Flow

Flow past a cylinder provides a good way to verify the code. For $Re \geq 40$ vortex shedding occurs at the cylinder and a von Karman street of vortices forms in the wake of the cylinder. This shedding process causes the forces on the cylinder to oscillate with a distinct frequency. This is known as the Strouhal frequency (St). The Strouhal frequency will be used as a measure to compare the results from the hybrid code and other codes. For Reynolds numbers up to approximately 190 the flow structures remains two-dimensional. Above this number three-dimensional instabilities occur, causing the two-dimensional approximation to be increasingly inaccurate for higher Re. Thus, we will consider a range of Re up to 250 in order to investigate how difficult it will be to resolve high gradient fields.

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 velocity boundaries are used at the inflow, upper and lower boundaries. Zero Neumann boundary conditions are used for velocity and the pressure is set to zero at the outflow. Figure 5.7 shows the domain, mesh and boundary conditions.

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=10 for Re = 50,100,150,200 and 250.

In the following table we show good agreement between Strouhal numbers obtained by Hybrid , [30] and Prism [40]. A summary of the simulation parameters is given in table 5.2.

 

Parameter Value

Dimension 2d

Re 100 to 250

$$\Delta t$$ 0.002

N-Range 10 and 12

K_Tri 448

K_Quad 332

Method Galerkin

 

Resolution Re=100 Re=150 Re=200 Re=250

Hybrid K=780, N=10 0.1659 0.1851

K=780, N=12 0.1662 0.1853 0.1972 0.2054

[30]) K=173, N=11 0.1667 NA 0.1978 NA

Prism [12] 14000 dof 0.1664 0.1854 0.1969 0.2051

 

Resolution Re=100 Re=150 Re=200 Re=250

Hybrid K=780, N=12 1.34468 1.33136 1.34473 1.36339

Prism [12] 14000 dof 1.35000 1.33336 1.34116 1.35769

In figures 5.9 and 5.10 we show iso-contours of vorticity of the wake region of the cylinder at Re=50,100,150,200 and 250. We can see that at Re=50, 100 and 150 the vorticity is smooth and the vortex structure is well resolved. At Re=200 and more so at Re=250 we see that the bulk shape of the vortices is correct but the features are a little unresolved. There are also streaks of spurious vorticity at Re=250. These effects will be reduced by increasing resolution.

 

Figure 5.6: Domain for the two-dimensional incompressible flow past a circular cylinder simulations.

  

Figure 5.7: Mesh for the two-dimensional incompressible flow past a circular cylinder simulations.

 

Figure 5.8: Instantaneous contours of vorticity for incompressible flow past a two-dimensional cylinder at Reynolds numbers Re=50,100 .

  

Figure 5.9: Instantaneous contours of vorticity for incompressible flow past a two-dimensional cylinder at Reynolds numbers Re= 150.

  

Figure 5.10: Instantaneous contours of vorticity for incompressible flow past a two-dimensional cylinder at Reynolds numbers Re=200 and 250.

has been taken to higher Reynolds number flows past a cylinder with the two-dimensional module and with the Fourier module I wrote based on the model in [40]. The Fourier module uses two-dimensional planes connected with a homogeneous third direction for which we use a Fourier expansion. The results from these simulations have been published by Ma Xia [45] and Evangelinos et al. [46]. The numerical method was adapted from [40] to include both triangles and quadrilaterals in the two-dimensional planes and allow variable expansion order from element to element. The Fourier module is also being adapted by Evangelinos [47] to include a time dependent spatial mapping in the third direction. That work is an adaption of the work by [48] where only nodal quadrilaterals were used in the two-dimensional planes.