Introduction

When a fluid flows over a cylindrical surface different flow pattern appears in the laminar flow depending upon the Reynolds number. Different flow patterns show in the image below.

Re<1		The flow is streamlined and symmetrical about the horizontal and vertical axis. The flow covers the entire cylinder and there is no wake region.
Re<4		Flow gradually gets disturbed and the flow symmetry disappears. Inertia force starts dominating the viscous force. boundary layer will have a tendency to get separated from the surface.
4<Re<40		Flow circulates behind the cylinder and the circulation of flow oscillates with the increase in Reynolds number.
40<Re<400		Von carmen vortex street is visible in this range. The vortex disappears beyond this range. Vortices move away from the cylinder and eventually break.

(Hmf, 2019)

When the inertia force dominates the viscous force the boundary layer close to the surface may not have the energy to stick to the surface and forms a free shear layer and is highly unstable. This unstable layer forms a discrete vortex and moves away from the surface. If the unstable shear layer formed from the top and bottom surface, a regular vortex flow is generated in the wake region called Karmen vortex. The frequency of vortex shedding is depended upon the Reynolds number. The dimensionless frequency number called Strouhal number can be specified to identify the nature of the vortex formation.

In this project, Ansys fluent software tool is used to study the von Karmen vortex street due to fluid flow over a cylinder. The flow of an imaginary fluid with viscosity 0.02 Pa.s is used over a cylinder of diameter 2m. Since cross-section is constant along the axial direction, a 2D computational domain is created using Solidworks to make the simulation is easier. The computational domain is shown below.

The computational domain is selected in such a way that the effect of the boundary would not affect the simulation result. The cylinder is located at 10D from the inlet and 20D from the exit. The height of the domain is 10D. Further increasing the domain has an only mere effect. (Nidhul, 2014)

Discretization And Boundary Conditions

Ansys meshing is performed by creating unstructured grids. After many trials, Karmen vortex street appears with the number of nodes and elements is 20645 and 40356 respectively. Meshing near the wall boundary made with inflation layers to capture good accuracy. The wall is divided into 50 parts so that the circular effects is captured. The mesh structure is triangular everywhere. The final grid is shown in the image below.

The extreme left end is treated as the inlet through which the fluid is entering the computational domain with the velocity of 1m/s. The fluid viscosity is 0.02Pa s. The extreme right side is the outlet with the boundary condition of 1atm pressure. The top and bottom are treated as symmetrical about the horizontal axis and the inner circular edge is the wall and no-slip conditions are applied.

Results

Steady flow

Steady flow is carried out in the pressure-based solver with the flow model is applied as laminar. A monitor point is created 8 meters behind the cylinder to check the velocity convergence. The location must be after the cylinder to check the velocity variation so that the Karmen vortex is created. Velocity measured to check the convergence at the monitor point. The iterations are performed 3000 times to make sure Karmen vortex street is developed. Velocity at the monitor point is studied along with the residuals. The results show that the residuals are constant after 300 iterations and there is no convergence obtained for the velocity at the monitor point. But the velocity fluctuation started after 100 iterations. It is seen from the plot that the velocity obtained at the monitor point fluctuates from 0.4m/s to 0.7m/s and the fluctuation continues till the iteration ends. It means that the Karmen vortex street is will be unchanged unless there is no change in the boundary conditions. Residuals in the continuity reach one during the vortex formation and the residuals of x moment and y momentum are less than 1e-3. To find the Strouhal number in the steady flow is impossible because there is no evidence of time taken for each vortex fluctuations.

The velocity and pressure contours are shown in the images shown below. Karmen vortex street is formed and the fluctuations are clearly visible in the velocity profile. The velocity close to the wall is zero and the boundary layer formation identified in the velocity contour. The boundary layer thickness increases from the front end to the midsection of the cylinder. The boundary layer separation developed from the mid-section of the cylinder. The maximum velocity is visible at the midsection of the cylinder. Just after the boundary layer separation, there is a considerable velocity gradient in the vertical direction. Velocity is zero at the front end of the cylinder and the pressure is maximum (stagnation point). The pressure is maximum at the front and minimum at the rear side of the cylinder. The pressure gradient exists along the direction of flow and thereby pressure drag is acting over the cylinder. The vortex streets are formed at the top and bottom alternatively and move away from the cylinder. The vertices disappear in the upstream.

Unsteady flow

In the case of unsteady flow, the convergence criteria at each time step have to be satisfied. The best time step for the unsteady simulation for this case is calculated based on the Courant number.

`(U Deltat)/(Deltax)<1`

From the mesh result, the smallest area is 9.978904e-04m2 assume it as a square of one side length 0.031 m. For velocity of U = 1m/s, the suitable time step for this simulation will be

`∆t< (1*0.031)/1 = 0.031`

To satisfy the convergence criteria the time step used in the simulation is 0.03 seconds. A total of 8000-time steps is performed which means the total time is 8000*0.03 = 240 seconds. Based on a few trials after each iteration, the convergence criteria of continuity and momentum were satisfied in two iterations for every time step. The total number of iterations performed is 8000*2=16000

From the unsteady results, the velocity at the monitor point is fluctuating after25 seconds. The amplitude of fluctuation increases gradually to till1 50 seconds. After 150 seconds the fluctuations are repeated. In each Von Karmen vortex cycle, there are two peaks. One peak is due to the shear layer separation at the top side and the second peak is because of the bottom shear layer separation. From the repeated cycles the time take for two cycles is found out to be 13.23 seconds. The frequency is calculated as 2/13.23 = 0.151. The Strouhal number obtained is

`St = (f L)/U = (0.151* 2)/1 =0.302`

The velocity and pressure contours are shown below

Conclusion

Steady flow simulations may not appear in any meshing sizes. The more precise close mesh gives more accurate results. From the unsteady state analysis, it takes a few seconds to initiate the Von Karmen vortex street. The frequency of vortex formation is 0.151 Hz. The unstable wake regions form close to the cylinder surface. Further upstream side the wake regions are disappeared. Since the frequency of vortex formation depends upon the Reynolds number, Srouhal number varies with Reynolds number. Convergence criteria in the vortex formation have to be checked on the continuity and momentum equations.

References

Hmf.enseeiht.fr. (2019). Strouhal Instability - Von Karman Vortex Street. [online] Available at: http://hmf.enseeiht.fr/travaux/CD0102/travaux/optmfn/gpfmho/01-02/grp1/phy_know.htm [Accessed 12 Nov. 2019].

Nidhul, K., Sunil, A, S. and Benphil, C. (2014). Effect of Domain Size and Grid Spacing on Flow Past A Circular Cylinder At Low Reynolds Number. International Journal of Engineering Research & Technology (IJERT, [online] 3(8), pp.1365-1368. Available at: https://www.ijert.org/research/effect-of-domain-size-and-grid-spacing-on-flow-past-a-circular-cylinder-at-low-reynolds-number-IJERTV3IS081045.pdf [Accessed 12 Nov. 2019].