Steady Vs Unsteady flow over a cylinder
Project Aim: To simulate the 2 D laminar flow over a cylinder and study vortex shedding
Project Objectives:
PART-I
- Simulate the flow with the steady and unsteady case and calculate the Strouhal Number for Re= 100.
PART-II
- Calculate the coefficient of drag and lift over a cylinder by setting the Reynolds number to 10,100,1000,10000 & 100000. (Run with steady solver)
- Discuss the effect of Reynolds number on the coefficient of drag. [ Results should be validated with any standard literature and error should be within 5 %]
Expected results:
- In both approaches, show that the flow has converged. That is, what quantities or plots need to be looked at to determine that the flow has converged?
- Plots of the coefficient of drag and lift.
- Plots to show vortex shedding behind the cylinder.
- Mention the preferred material at the end of the project.
Procedure:
The figure below shows the stepwise procedure followed to get the simulation
Fig1. Procedure followed
Part 1: Simulation with steady and unsteady case with calculation of the Strouhal Number for Re=100
Step 1. Creating the 2D geometry of cylinder and the flow field
Fig2. 2-D Geometry and boundary
The geometry and the boundary conditions used are as shown in the figure. The same geometry will be used for part 1 and part 2 as well. But in case of part 2 the velocity will be changed to get the results at different values of Reynold's number.
Fig3. Actual geometry
Step 2. Meshing and creating named selections
First of all the base line autogenerated quad mesh was created and the simulation was run. Once the simulation ran with the baseline mesh, the mesh was fine tuned and tri elements were used to generate the mesh and also 6 inflation layers were added near the cylinder walls to get better approximation of the geometry. Figure below shows the mesh.
Fig4. a) Triangular Mesh b)Mesh Quality and c) Inflation Layers
a)
b)
c)
After the mesh was generated, named selections were created 1) Inlet 2) Outlet 3) Walls 4) Symmetry as shown in figure 1.
Step 3. Setting up physics, boundary conditions and outputs
Solver: Pressure-based
Material: user-defined material with following properties (density = 1 `(kg)/m^3` and dynamic viscosity = 0.02 `(kg)/(m.s)`)
Inlet Velocity: Adjusted to get the proper value of Reynold's Number for every case of simulation
Outlet pressure: 0 Pa (gauge)
Boundary Conditions: Cylinder (Wall) and end of domain (Symmetry)
Initailization: hybrid
Outputs (observed at monitor point at in the expected wake region):
[For part 1]
1) Residuals 2) Velocity Magnitude
[For part 2]
1) Residuals 2) Velocity Magnitude 3) Drag and lift coefficients (near cylinder walls)
Results: 1) Velocity contour 2) Pressure contours 3) Strouhal number (can be calculated for transient case only)
Step 4. Solution and Results
Part 1: Simulating the flow with
case1) Steady State (Re = 100)
`Re = (Uxxrho xxD)/mu`
`implies100 = (Uxx1 xx2)/0.02`
`implies U = 1m/s`
For the Reynold's number of 100 the following results were obtained for residuals and the velocity at a point in the wake region
Fig5. Residuals for the steady state case for 500 iterations
Fig6. Velocity fluctuations at the monitor point (due to vortex shedding) at Reynold's number 100
Fig7. Velocity contour (for visualizing vortex shedding) throughout the domain
Fig8. Pressure contour for visualizing the magnitude of pressure difference at up and downstream side of the cylinder
For this case Strouhal Number cannot be calculated. Strouhal Number can be calculate if the simulation is transient. The next case is the transient case where it is possible to calculate Strouhal Number.
case2) Unsteady State (Re = 100)
The inlet velocity (1 m/s) remains the same because the calculation is to be done at Re=100. Only the solution method is changed to transient.
Fig9. Residuals for the transient state simulation
From the above plot it can be rightly said that vortex shedding being a periodically oscillating phenomenon will never settle down or converge. So, the residuals never die down but keep on oscillating throughout the 12000 iterations.
Fig10. Velocity magnitude (at the monitor point) as a function of time
Strouhal Number:
The Strouhal number represents the ratio of interial forces due to the local acceleration of the flow to the inertial forces due to the convective acceleration. Mathematically, it is given by
`S_(tr)=(fL)/U`
where
`f` is the frequency of oscillation in dimensions [`T^-1`]
`L` is the characteristic length (D in this case)
`U` is the inlet velocity
From the above figure the value of frequency of oscillation can be calculated by
`f = 1/(52.5-45)=0.133/sec`
Thus Stouhal number is given by
`S_(tr)=(0.133xx2)/1=0.266`
From standard literature available on vortex shedding, it is stated that at Strouhal number values of 0.2-0.3, the vortex shedding can be observed. Also the Strouhal number also varies with Reynold's number. The following figure gives the basic idea (experimental data)
Fig12. Velocity contour at the end of 60 s
Above figure shows the vortex street at the end of 60 sec. The transient simulation took a lot of time to simulate the time accurate flow. These periodic oscillations never settled and remained unsteady throughout.
Fig13. Pressure contour at the end of 60 s
Part 2: Simulating the flow with different values of Reynold's Number (using steady state solver) to calculate values of of coefficients of drag and lift
[For Reynold's number of 10]
Fig14. Reference values for Reynold's number 10
In above figure the frontal area `(pi/4xxD^2)` is used for the cylinder because the simulation is 2D. Length is also `D`. Depth is taken as 1 m though the geometry is 2D. Viscosity is user defined value and velocity is adjusted such that the Reynold's number should be 10. Without these reference values the values of drag and lift coefficients would be incorrect.
Fig15. Velocity magnitude at monitor point (in wake region)
Above figure shows that for Re = 10 , no vortex shedding is observed behind the cylinder and the velocity also reaches a steady state value of 0.0425 m/s.
Fig16. Coefficient of drag for Re=10
The coefficient of drag is close to 2-3 for Re=10.
Fig17. Coefficient of lift for Re=10
Coefficient of lift is approximately 0 for Re=10.
Fig18. Velocity contour for Re=10
No vortex shedding is observed behind the cylinder (sphere in 2D) for Re=10 which is in accordance with the experimental results.
[For Reynold's number of 100]
Fig19. Reference values and residuals for Reynold's number 100
Just the inlet velocity is adjusted to get the Re=100
Fig20. Velocity magnitude at monitor point (in wake region)
With the above plot it can be seen that the velocity at A point in the wake region is fluctuating and the vortex shedding can be seen at Re=100
Fig21. Coefficient of drag for Re=100
Coefficient of drag is still around 3-5, slightly higher than the previous case as it can be seen from the figure.
Fig22. Coefficient of lift for Re=100
Average value of coefficient of lift is zero for Re=100.
Fig23. Velocity contour for Re=100
Vortex shedding can be visualized for Re=100.
[For Reynold's number of 1000]
Fig24. Reference values and residuals for Reynold's number 1000
Fig25. Velocity magnitude at monitor point (in wake region)
Fig26. Coefficient of drag for Re=1000
Fig27. Coefficient of lift for Re=1000
Fig28. Velocity contour for Re=1000
[For Reynold's number of 10,000]
Fig29. Reference values and residuals for Reynold's number 10,000
Fig30. Velocity magnitude at monitor point (in wake region)
Fig31. Coefficient of drag for Re=10,000
Fig32. Coefficient of lift for Re=10,000
Fig33. Velocity contour for Re=10,000
[For Reynold's number of 100,000]
Fig34. Reference values and residuals for Reynold's number 100,000
Fig35. Velocity magnitude at monitor point (in wake region)
Fig36. Coefficient of drag for Re=100,000
Fig37. Coefficient of lift for Re=100,000
Fig38. Velocity contour for Re=100,000
Fig39. Tabulated results for part 2 (for user defined fluid)
Fig40. Experimental results plot for the drag coefficients of spherical and cylindrical geometries for air as a fluid (Taken from a std Fluid Mechanics textbook by Cengel)
Conclusion: For the given simulation the results differ slightly with the experimental data because the fluid chosen for the simulation is quite diefferent from the air. But this simulation gives acceptable results to visualize the Vortex shedding phenomena for the cylinder or spheres.