Study the phenomenon of Von-Karman vortex with the variation of Reynold's number and calculating Strouhal number for the flow over a cylinder under steady and unsteady simulation.
AIM: Study the phenomenon of Von-Karman vortex with the variation of Reynold's number and calculating Strouhal number for the flow over a cylinder under steady and unsteady simulation.
ABSTRACT:
This project is divided into two parts
1) Calculation of strouhal number and studying the formation of vortex behind the cylinder.
2) Studying the behavior of vortex formation with variation in Reynolds number using a steady solver.
In part 2 Reynold's number is varied as follows - 10, 100, 1000, 10000, 100000.
Reynolds number:
Reynolds number is a dimensionless number used to categorize the fluid systems by considering the effects of viscosity which is one of the important factors in controlling the fluid flow pattern. Reynolds number can be defined as the ratio of inertial force to viscous force. Below is the formula for calculating Reynolds number for flow over a cylinder.
`Re = (rho*U*D)/mu`
U = velocity (m/s)
`rho` = Density = 1 `(kg)/m^3` (This value is considered for air during the simulations)
D = Diameter of pipe = 2 m.
Reynolds number formula is similar for flow over an aerofoil, just the diameter is replaced by the length of an aerofoil.
Strouhal number:
It is a dimensionless number that describes an oscillating flow.
`Sr = (f*D)/U`
where,
f = frequency of peaks per seconds
D = Diameter of cylinder in meters
U = velocity at inlet in m/s
GEOMETRY:
MESHING:
Part-1:
Mesh size = 0.25 m
Number of elements = 40420
Number of nodes = 20656
Part-2:
Mesh size = 0.1 m
Number of elements = 253687
Number of nodes = 127864
Mesh size was kept finer to get better graphical outputs and get accurate results. During actual industrial simulations, proper mesh size should be selected depending on the preference of results ( i.e accuracy or performance/cost).
RESULTS:
PART1: Calculating Strouhal number for Re = 100
Using Steady solver:
Residual plot
Vertex average plot at monitor point 8 m behind the cylinder where the vortex is formed
Plot of lift coefficient
From the residual plot and the vertex average plot, we can observe that the pattern starts repeating and stops increasing and decreasing. Hence, we can say that the solution has converged.
From the plot of the lift coefficient, we can calculate the number of peaks against the number of iterations. This being a steady-state solution we cannot get the number of peaks with respect to real-time hence we cannot obtain frequency. Without frequency, we cannot calculate the Strouhal number. To calculate the Strouhal number transient solver is used and the results are shown below.
Using Transient solver:
Residual plot
Vertex average plot at monitor point 8 m behind the cylinder where the vortex is formed
Plot of lift coefficient
From the residual plot and the vertex average plot, we can observe that the pattern starts repeating and stops increasing and decreasing. Hence, we can say that the solution has converged.
From the plot of the lift coefficient, we can calculate the frequency (number of peaks in 60 seconds). In our case, it can be observed 5 peaks per minute which can be seen between 200 and 260 seconds.
f = frequency = `5/(60)`
By using Strouhal numbers formula,
Sr = `5/(60)*2/1`
Sr = 0.1667
where,
f = `5/(60)` , D = 2m , U = 1 m/s
This can be verified from the literature [Numerical simulation of laminar flow past a circular cylinder] Sr = 0.1569.
PART - 2: Studying the behavior of vortex formation with variation in Reynolds number using a steady solver
All simulations were simulated using a steady solver for 1000 iterations. The time-dependent solution is not needed since no time-based study is done in part 2. Also, the transient solver needs more computing power and time so a steady solver was used for part 2 study.
It can be seen from all the residual plots and vertex average plots that the solutions are converged and the pattern starts repeating.
Re = 10, U = 0.1 m/s, `mu = 0.02 (N.s)/m^2 , rho = 1 (kg)/m^3 , D = 2 m`
Residual plot
vertex average velocity plot at point 4*D (8 m) behind the cylinder
Drag coefficient plot
Drag coefficient = 3.305715
Lift coefficient plot
Lift coefficient = -0.0061410003
Vortex formation
Re = 100, U = 1 m/s, `mu = 0.02 (N.s)/m^2 , rho = 1 (kg)/m^3 , D = 2 m`
Residual plot
vertex average velocity plot at point 4*D (8 m) behind the cylinder
Drag coefficient plot
Drag coefficient = 1.325361
Lift coefficient plot
Lift coefficient = 0.12785328
Vortex formation
Re = 1000, U = 10 m/s, `mu = 0.02 (N.s)/m^2 , rho = 1 (kg)/m^3 , D = 2 m`
Residual plot
vertex average velocity plot at point 4*D (8 m) behind the cylinder
Drag coefficient plot
Drag coefficient = 1.0441872
Lift coefficient plot
Lift coefficient = -0.35039309
Vortex formation
Re = 10000, U = 100 m/s, `mu = 0.02 (N.s)/m^2 , rho = 1 (kg)/m^3 , D = 2 m`
Residual plot
vertex average velocity plot at point 4*D (8 m) behind the cylinder
Drag coefficient plot
Drag coefficient = 1.0097153
Lift coefficient plot
Lift coefficient = -0.47591292
Vortex formation
Re = 100000, U = 1000 m/s, `mu = 0.02 (N.s)/m^2 , rho = 1 (kg)/m^3 , D = 2 m`
Residual plot
vertex average velocity plot at point 4*D (8 m) behind the cylinder
Drag coefficient plot
Drag coefficient = 0.91384594
Lift coefficient plot
Lift coefficient = 0.39372982
Vortex formation
ANIMATIONS:
Re = 10 U = 1 m/s, `mu = 0.2 (N.s)/m^2 , rho = 1 (kg)/m^3 , D = 2 m`
Re = 100 U = 1 m/s, `mu = 0.02 (N.s)/m^2 , rho = 1 (kg)/m^3 , D = 2 m`
Re = 1000 U = 10 m/s, `mu = 0.02 (N.s)/m^2 , rho = 1 (kg)/m^3 , D = 2 m`
Re = 10000 U = 20 m/s, `mu = 0.004 (N.s)/m^2 , rho = 1 (kg)/m^3 , D = 2 m`
Re = 100000 U = 50 m/s, `mu = 0.0001 (N.s)/m^2 , rho = 1 (kg)/m^3 , D = 2 m`
CONCLUSION
| Reynold's number | Lift Co-efficient | Drag Co-efficient |
| 10 | -0.0061410003 | 3.305715 |
| 100 | 0.12785328 | 1.325361 |
| 1000 | -0.35039309 | 1.0441872 |
| 10000 | -0.47591292 | 1.0097153 |
| 100000 | 0.39372982 | 0.91384594 |
1) Drag and lift coefficient can be observed from the above plots. The Lift coefficient goes on increasing with the increase in Reynolds number while the Drag coefficient decreases with an increase in Reynolds number.
2) The frequency of peaks also increases with an increase in the Reynolds number.
3) Strauhal number remains the same for all the simulations irrespective of the Reynolds number and is below 10 % error.
REFERENCE RESEARCH PAPER:
Numerical simulation of laminar flow past a circular cylinder