Steady Vs Unsteady flow over a cylinder
Theory:
Karman vortex street:
Karman vortex street or a Von Karman vortex street is a repeating pattern of swirling vortices, caused by a called vortex shedding(it is an oscillating flow that takes place when a fluid such as air or water flows past a bluff body at certain velocities, depending on the size and shape of the body). Vortices or vortex is a region in the fluid in which the flow revolves around the axis line, which may be straight or curved.
This was named after the engineer and the fluid dynamicist Theodore Von Karman. A vortex street will form only at a certain range of flow velocities, specified by a range of Reynolds number(Re), typically the Re value limit is above 90. The range of the Re values will vary with the size and shape of the body.
`Re_L = (UL)/v`
where U = free stream fluid flow speed
L = The characteristic length parameter of the body
v = kinematic viscosity of the fluid
Strouhal number(St or Sr):
The Strouhal number is a dimensionless number describing the oscillating flow mechanism.
`St = (fd)/U`
where f = frequency of vortex shedding
d = diameter of the cylinder
U = flow velocity
Actually the below formula is generally hold for the range 40< Re <150:
`Re = 0.198(1-(19.7/(Re_d)))`
Part - 1: Flow with the steady and unsteady case:
Steady-state simulation for flow over a cylinder:
a. Geometry:
b. Mesh:
The mesh grid size is 0.25. Here we can also increase the mesh count around the circle by using the option sizing < select edge of the circle < in type; select no. of divisions < and mention 36 divisions (for every 10deg there is a division).
Here we added a 6 number of layers around the cylinder are known as the inflation layer. This layer helps to create a body-fitted mesh.
c. Setup:
Click on a check and open setting on physics < Time - unsteady.
Reference values:
Area: 2
density: 1
length: 2
Depth: 1
viscosity : 0.02
the velocity depends upon the Reynolds number.
Unsteady state(Transient time):
For Reynolds number = 100, velocity = 1m/s.
Time step = 0.5, No. of iterations = 300.
Residual plots:
Drag coefficient:
Lift coefficient:
Velocity simulation:
Drag coefficient = 1.4314, Lift coefficient = 0.0579.
For Reynolds number equal to 100, the Strouhal number comes out to be = 0.1589
Part - 2:
Change the time to steady-state in setting in physics.
a) for Re = 10, the velocity = 0.1m/s.
Residual plots:
Lift coefficient:
Drag coefficient:
Velocity simulation:
b) for Re = 100, the velocity = 1m/s.
Residuals:
Drag coefficient:
Lift coefficient:
Velocity simulation:
c) for Re = 1000, the velocity = 10.
Residuals:
Drag coefficient:
Lift coefficient:
Velocity simulation:
d) for Re = 10000, the velocity = 100.
Residuals:
Drag coefficient:
Lift coefficient:
Velocity simulation:
e) for Re = 100000, the velocity = 1000.
Residuals:
Drag coefficient:
Lift coefficient:
Velocity simulation:
Result:
| S.No |
Reynolds number |
Drag coefficient Cd | Lift coefficient Cl |
| 1. | 10 | 3.3002 | 0.0304 |
| 2. | 100 | 1.3124 | 0.0194 |
| 3. | 1000 | 1.2099 | 0.3063 |
| 4. | 10000 | 1.1973 | 0.02178 |
| 5. | 100000 | 1.1679 | -0.4982 |
Conclusion:
- The vortex street does not occur for the Re number 10. It exists for remaining Re numbers.
- The drag coefficient decreases with the increase in the Reynolds number.
- The drag coefficient of the unsteady state is higher than the steady-state.