Simulation of Vortex Shedding

Flow over a Cylinder – Von Karman Vortex Street

Introduction:

This study is to analyze the flow over a cylinder with laminar flow (assumed), but our main interest is to look for the formation of von karman vortex street in steady and unsteady flow nature using ansys-fluent.

Flow over a Cylinder:

The real flow over a cylinder is unlike an ideal inviscid flow over the cylinder having net zero drag by d’alembert’s paradox. It will have a boundary layer adjacent to surface of the bluff body of study and also the boundary layer separation occurs which will tend to form a wake region downstream of the flow which will be a low pressure region resulting in drag on that body in downstream direction.

The resultant flow is an oscillated flow by having an periodical formation of low pressure regions of same pattern in an oscillated manner in the downstream. The low pressure regions are in the form fluid circulating region where at the center of the region there will be a void space due to this flow circulation happens but this region disappears on spatial marching in downstream flow direction.

This is called as vortices and this vortices follows a regular pattern based on the flow velocity, shape and size of the body. This pattern of formation of vortices repeatedly in an oscillating manner is called as vortex shedding and the downstream region of bluff body is called as Vortex Street.

The frequency of the oscillated flow ( vortex shedding ) is used to determine the effect of the vortices formed, that frequency is related to the strouhal number which is a dimensionless number.

St = f*D/V

St – strouhal number of the vortex shedding.

f – Frequency of vortex shedding

D – Characteristic length of the bluff body

V – Free stream velocity

Our motive is to find out the strouhal number for the flow over our cylinder of specified dimensions in steady as well as unsteady state.

Model:

The below shown 2D model having the dimensions is considered for analysis. This 2D drawing is just for a referral. Modelling is done in space claim of ansys for analysis.

All dimensions are in mm.

Note: diameter of the cylinder is not 4000 mm it is 2000 mm. this is modeled in solid works , only used for reference purpose.

Input Parameter:

Density of fluid: 1 kg/m³

Viscosity of fluid: 0.02 kg/m.s              As we fixed Re=100.

Inlet velocity of fluid: 1 m/s

Boundary conditions:

• Left and right edges are assigned as velocity inlet and pressure outlet respectively.
• Top and bottom edges were assigned symmetrical boundary condition.
• Circular Cylinder (2D) is considered as wall.

Meshing: (both cases)

• Element size = 0.25m.
• Mesh type: Triangular mesh.
• Area near Cylinder wall edge is finely meshed by using edge sizing option, giving an input of 36 no of division on cylinder wall edge.
• Inflation method is used to generate inflation layers to capture the physics around the surface of cylinder.
• No of inflation layer: 6.
• Growth rate: 1.2.
• No of elements generated: 40,502.

Results of Analysis:

• Transient
• Analysis were done in transient state having time step of 0.1 second with 10 iterations per time step.
• Plots monitored at the run time of the analysis were MPV (monitor point velocity), C_L (coefficient of lift), SR (scaled residuals).

Convergence of solution:

• In the transient case of flow over a cylinder for Re=100 the formation of vortex street occurs.
• But for convergence of the solution it is very difficult to do it because the solution is varying after certain time step periodically but following a steady pattern.
• On referring fig 1 & 2 we can clearly notice that the monitor point velocity and coefficient of lift is varying in sinusoidal manner after crossing the time step of 125 seconds, following a steady pattern which doesn’t show any variation that pattern.
• Which indicates that the flow reaches its steady state. Thus we conclude that the physics has been converged.

Fig 1: Flow-Time Vs Monitor point velocity

                The fig 1 shows the graph of monitor point velocity variation with respect to flow time, the graph gives useful information like the velocity variation is sinusoidal but it follows a regular pattern of variation which says that the flow is a oscillatory flow.

Fig 2: Flow-Time Vs Coefficient of lift.

                The above fig 2 shows the graph of variation of C_L with respect to flow time. On observing the graph this also clearly shows the flow is a oscillatory flow by having positive and negative lifts in a sinusoidal manner also having a regular pattern at equal interval of time.

Fig 3: Velocity contour of flow over cylinder (2D)

                The above fig 2 shows the velocity contour plot of our study, this plot shows the fully developed oscillating flow which can be also known as von karman Vortex Street. At the beginning of the downstream of the cylinder you can see the vortex formation, when that vortex moves further in the direction of downstream dissipation of vortex is clearly seen.

Fig 4: Plot of Strouhal number-FFT on on CL	Fig 5: Log plot of Strouhal number-FFT on CL

                            The above shown fig 4 and fig 5 showing the plot and log plot of Strouhal number generated using FFT on C_L.

Fig 6: Plot of Strouhal number-FFT on monitor point velocity	Fig 7: Log plot of Strouhal number-FFT on monitor point velocity

                            The above shown fig 6 and fig 7 showing the plot and log plot of Strouhal number generated using FFT on monitor point velocity.

                            Using an excel sheet median for strouhal number plot was taken it gives the following value for it

Strouhal number based on C_L =0.2497.

Strouhal number based on MPV =0.25.

• Steady state
  • The flow over a cylinder (2D) for Re=100 is solved in steady state mode to know what happens to the flow, as we already know that the flow is oscillating one with respect to time.
  • So the input parameters were set as same as the transient model but only change is solution is solved using steady state condition.
  • Here we can analyze and compare the solution especially the strouhal number of the flow solved using steady and unsteady condition, to know the importance of the time dependence .

Convergence of solution:

• In the steady state analysis the solution is said to be converged when the residuals meets the convergence criteria.
• But in our case solution cannot converge as the flow is an oscillating flow.
• So we can check for the convergence of physics, here I say the convergence of physics is nothing but the flow pattern that we obtain after certain iterations would follow an regular pattern of flow as we proceed increasing the number of iterations.

Fig 8: iterations Vs Monitor point velocity – Steady

The fig 8 shows the graph of monitor point velocity variation with respect to flow iterations, the graph gives useful information like the velocity variation is sinusoidal but it follows a regular pattern of variation which says that the flow is an oscillatory flow which is similar as like transient but range variation is different when we compare it.

     Fig 9: iterations Vs Coefficient of lift - steady.

The above fig 9 shows the graph of variation of C_L with respect to number of iterations. On observing the graph this also clearly shows the flow is a oscillatory flow by having positive and negative lifts on the cylinder in a sinusoidal manner also having a regular pattern at equal no of iteration intervals. But when we compare it with the transient model the range of the sinusoidal pattern is different.

 Fig 10: Velocity contour of flow over cylinder (2D) - Steady

The above fig shows the velocity contour of the flow over cylinder in steady state mode. Here also we can notice the vortex street in the downstream of flow over the cylinder. As analysis is conducted in steady state mode, it tries to give an output of flow which reaches the steady condition. When we compare the transient model with steady state model it has variation in vortex street profile.

Fig 11: Plot of Strouhal number-FFT on on CL - steady	Fig 12: Log plot of Strouhal number-FFT on CL - steady

                            The above shown fig 11 and fig 12 showing the plot and log plot of Strouhal number generated using FFT on C_L.

Fig 13: Plot of Strouhal number-FFT on monitor point velocity - steady	Fig 14: Log plot of Strouhal number-FFT on monitor point velocity - steady

                            The above shown fig 13 and fig 14 showing the plot and log plot of Strouhal number generated using FFT on monitor point velocity.

Strouhal number based on C_L = 0.25.

Strouhal number based on MPV = 0.25.

Inference from both the results:

• From the results we got from transient and steady state model the strouhal number doesn’t show any appreciable difference, but when we examine the flow pattern in fig 3 & 10 it was not similar. The frequency of vortex generation may be the same but the vortex street thus formed is not similar.
• But we have to be clear about a thing that transient mode analysis will capture the real physics from the starting 0 time to the time we specified with increment of time step for N no of time step. Which will be very accurate to capture the change in the flow with respect to time.
• So we can conclude that the both the model will yield the similar strouhal number (frequency of vortex generation) but the dissipation of vortex varies that too transient model accurately captures it but steady state model fails to do it.

Fig 15: velocity contour of FOC model solved in transient state	Fig 16: velocity contour of FOC model solved in steady state

The above two figures shows the velocity profile of flow over cylinder (FOC) model analyzed in transient state and steady state. From the above fig we can clearly notice that the flow analyzed in transient model is realistic, because downstream of the cylinder is highly influenced by the vortices generated. Disturbance felt by vortices is well distributed in transient model than steady state model. But the frequency of generated vortices remains the same for both the flow.

Steps to calculate Strouhal number:

• In both the model when we come to know that the flow is converged, go for plot option under that select FFT (Fast Fourier Transform).
• A dialogue box will be opened which is named as Fourier transform. There you have a tab called “Load input file” select that and navigate through folders to load the output file of the solution definitions, either C_L Vs flow time/iterations or MPV Vs flow time/iterations file.
• Select X axis function as strouhal number and Y axis function as magnitude.
• After that plot the strouhal number Vs magnitude using plot FFT option which will give a plot on St Vs magnitude plot.
• After generating the plot select the check box of “write FFT to file “and select “write FFT” which will give the output values of the plot.
• You can import this data in an excel sheet and can calculate the median value for the strouhal number.

Fig 17: residuals plot for staedy state analysis.

Fig 18: residuals plot for transient state analysis.

Conclusion:

• On comparative analysis on flow over a cylinder in both transient and steady state model, we can conclude that both the model can capture the frequency of vortices generated.
• But steady flow could not capture the disturbance imparted by the vortices generated in the downstream of the flow.
• Transient model clearly captures the realistic flow behavior of the FOC (Re=100) and maps the Von karman vortex street.