Fixed cylinder Re 100

Modeling of flow field around a circular cylinder is a classical benchmark problem that is used to assess the accuracy of numerical methods that are based on transient Navier–Stokes equations. The physical problem consists of a cylinder that is fixed in space and is surrounded by flowing fluid. As the flow develops, the boundary layer detaches from the cylinder and gives rise to two dominant vortices behind the cylinder. As the Reynolds number gets sufficiently high, usually considered Re ≥ 40−45, these vortices are convected down-
stream of the cylinder and new vortices are formed behind the cylinder in a periodic fashion. Typically, the numerical accuracy of the formulation for simulations of this problem is checked by means of the Strouhal number (inverse of the period in which vortices are detached from the cylinder).

Figure below shows a schematic diagram of this problem.

For comparison purposes, we have used the benchmark by Calderer & Masud (2010), which has the same problem description, dimension, and spatial discretization as the one used in Hauke and Hughes (1998). The viscosity of the fluid is 0.01. The velocity field on the inflow boundary is set equal to 1. No-slip boundary condition is prescribed on the surface of the cylinder to account for the viscous adhesion, and traction-free conditions along with zero pressure gauge are imposed on the outflow boundary. Zero normal velocity and zero shear stress conditions are prescribed on the remaining boundaries of the computational domain. The flow is initially in uniform flow with the inflow velocity. Reynolds number based on the inflow velocity and the diameter of the cylinder is 100. The computational domain is discretized into 136482 triangular elements with size 0.01 m around the cylinder and 0.1 m along the boundary of the domain. The time step t is set equal to 0.01.

The 2007 tau diagonal incompressible is employed.

Mesh is defined in the mesh.geo file, properties of the fluid are set in the props.txt file, initial and boundary conditions are assigned in the fluid_ic_bc.hpp file, while the problem setup is stated in setup.txt file.

The video shows velocity and pressure fields during the development of the vortex shedding.

Flow field generates viscous and pressure forces that act on the surface of the cylinder. We define the lift coefficient as C_L = F_y /0.5ρv_x² D², where F_y is the force acting on the cylinder in the transverse direction, ρ = 1 is the density, v_x = 1 is the input velocity, D = 1 is the diameter of the cylinder. Likewise, we define the drag coefficient as C_D = F_x /0.5ρv_x² D², where F_x is the force acting on the cylinder along the mean-stream direction. The evolution of the lift and drag coefficients are shown in Figures below, with the comparison with the reference solution. The differences in values of the coefficients is due to the slightly different definition of the coefficients themselves.

The period of vortex shedding T can be calculated from the time-plot of the lift coefficient. The computed period is 5.7, in agreement with the findings by Calderer & Masud (2010). The corresponding Strouhal number (St = 1/T ) is 0.175. Similar results for this problem have been reported in Hauke and Hughes [33], wherein using time step t = 0.025 they obtained St = 0.172.

References

Ramon Calderer & Arif Masud “A multiscale stabilized ALE formulation for incompressible flows” (2010) Comput. Mech.
with moving boundaries