Steady Vs Unsteady flow over a cylinder
STEADY & UNSTEADY FLOW OVER CYLINDER
1. PROBLEM STATEMENT
- Simulate the flow with the steady and unsteady case and calculate the Strouhal Number for Re= 100.
- Calculate the coefficient of drag and lift over a cylinder by setting the Reynolds number to 10,100,1000,10000 & 10000
2. INTODUCTION & THEORY
Flow is a very important area in the field of fluid mechanics. There might be generation of instabilities during flow over a cylinder. Whenever viscous flow passes over a cylinder (say it blunt body) at low Reynolds number, it makes them unstable. These instabilities result in generation of wake, separated shear layer and boundary layer. The first wake instability was observed at a Reynold number of approximately 50. Although the flow was laminar but it became asymmetric & unstable.
In order to see the presence of von Karman vortex we must maintain the value of Reynold number at 50. As we increase the value of Reynold number the vortex shedding pattern changes and results in generation of 3-D instabilities such as formation of vortex loops, deformation of primary vortices and stream wise and span wise vortices, appear in wake. 3-D instabilities make the flow turbulent. Beyond a certain critical Re, the shear layer separating from the upper and lower surface of the cylinder, starts becoming unstable. The transition point, beyond which the separated layer becomes unstable, moves upstream with increase in Reynolds number. At Re = 200000, the boundary layer on the cylinder surface undergoes a transition from laminar to turbulent. This transition leads to a delay of the separation of flow from the cylinder surface causing a substantial reduction in the drag force that the cylinder experiences.
3. AIM
The focus of project is to observe the vortex shedding phenomena at a Reynolds number of 100 & compare their behaviour in steady as well as transient state. The effect on drag coefficient has been observed by increasing the Reynolds number from 10 to 100000.
4. GEOMETRY & MESHING
4.1 GEOMETRY
2-D flow region has been taken in consideration to save the computational time. Rectangular flow domain has been taken in consideration. The dimension of rectangular domain was 60x15 meter. The diameter of cylinder was 2 meters.
The following figure shows the geometry of flow domain
4.2 MESHING
Meshing was done with all triangle methods. Different meshing was performed for transient & steady state simulation.
4.2.1 STEADY STATE
The total number of nodes & elements are 4294 &8036 respectively. The element size is 500 mm.
This was applied to the region away from the cylindrical. As vortex generation & all other instabilities phenomenon starts from near cylindrical wall region so mesh has been refined carefully to get better results.
The circular cylindrical edge has been divided into 36 number of divisions. The helps in maintaining the circular curvature of cylinder. Additionally, inflammation layer has been added on circular edge.
The growth rate has been maintained at 1.2. The maximum number of layers was 6 & the height of first layer thickness was 5mm. Following figure shows the meshing near wall in details.
4.2.2 TRANSIENT STATE
The total number of nodes & elements are 29825 & 43678 respectively. The element size is 250 mm.
This was applied to the region away from the cylindrical. As vortex generation & all other instabilities phenomenon starts from near cylindrical wall region so mesh has been refined carefully to get better results.
The circular cylindrical edge size was maintained at 25 mm. This helps in maintaining the circular curvature of cylinder. Additionally, inflammation layer has been added on circular edge.
The growth rate has been maintained at 1.5. The maximum number of layers was 60 & the height of first layer thickness was 5mm. Following figure shows the meshing near wall in details.
5. BOUNDARY CONDITIONS & FLUID PROPERTIES
The inlet velocity of fluid was taken as 50 m/s as to maintain a Reynold number of 100. The density of fluid was taken as 1Kg/m^3. The viscosity of fluid was also 1 kg-m/s.
In order to maintain different Reynold number from 10 to 100000 only velocity of fluid was changed. The fluid property was kept constant throughout the study.
| CASE |
|
DENSITY Kg/m3 | VELOCITY (m/s) | Reynolds Number | |
| 1 | 1 | 1 | 5 | 10 | |
| 2 | 1 | 1 | 50 | 100 | |
| 3 | 1 | 1 | 500 | 1000 | |
| 4 | 1 | 1 | 5000 | 10000 | |
| 5 | 1 | 1 | 50000 | 100000 |
Left edge of fluid domain was given as inlet WALL & right edge was given outlet wall. The top & bottom edge was given as symmetric boundary condition. Circular cylinder was also considered as cylindrical wall conditions.
6. SOLUTION METHOEDS
SIMPLE scheme was chosen for solving pressure velocity coupling. SECOND ORDER spatial discretisation was implemented for Pressure. Momentum equation was discretised with the help of SECOND ORDER UPWIND SCHEME.
Transient simulation was also done with the help of SECOND ORDER UPWIND formulation at a Reynold number of 100.
7. RESULT & DISSCUSION
7.1 STEADY STATE V/S TRANSIENT SIMULATION AT A Re of 100
7.1.1 STEADY STATE
The fluid parameter was constant as mentioned in above section & vortex was observed at a Reynold number of 100. The velocity of monitor point was observed for declaring the convergence of solution.
Above figure shows the monitor point velocity. The point was located at 8 meters away along x axis from overall reference (at the centre of cylinder). The simulation was run for 350 iteration. The monitor point velocity was repeating its value after 250 iteration.
Oscillatory values of velocity have been observed at monitor point. This confirms the presence of sufficient vortex near the cylinder. Figure below shows the vortex structure formulation after 350 iteration of calculation.
Coefficient of lift & drag has been observed carefully. The plot below shows the same values. First figure shows the plot of drag along x-direction & second plot shows the lift coefficient.
Below is the plot for coefficioent of lift
Following figure shows the output for all parameter like output velocity, coefficient for lift & drag.
Monitor point velocity - 25.5139 m/s
Coefficient of drag - 1.67
Coefficient of lift - 0.14152
Result are observed at the end of 350iteration. The Coefficient of drag achieved almost constant value whereas Coefficient of lift was oscillating between -0.15 to 0.15 & monitor point velocity was also oscillating between 22m/s to 33m/s.
7.1.2 TRANSIENT STATE
The fluid parameter was constant as mentioned in above section & vortex was observed at a Reynold number of 100. The velocity of monitor point was observed for declaring the convergence of solution.
Time step size was 0.01 & simulation was run for 500 time step.
Above figure shows the monitor point velocity. The point was located at 8 meters away along x axis from overall reference (at the centre of cylinder). The simulation was run for 500 time step. The monitor point velocity was repeating its value after 2.75 second.
Oscillatory values of velocity have been observed at monitor point. This confirms the presence of sufficient vortex near the cylinder. Figure below shows the vortex structure formulation after 5 second of calculation.
Coefficient of lift & drag has been observed carefully. The plot below shows the same values. First figure shows the plot of drag along x-direction & second plot shows the lift coefficient.
Below is the plot for coefficioent of lift
Following figure shows the output for all parameter like output velocity, coefficient for lift & drag
Monitor point velocity- 37.771 m/s
Coefficient of drag- 1.4859
Coefficient lift- 0.5180
Result are observed at the end of 5 second of simulation. The Coefficient of drag achieved almost constant value whereas Coefficient of lift was oscillating between -0.275 to 0.275 & monitor point velocity was also oscillating between 37.5 m/s to 47.5 m/s.
7.1.3 STROUHAL NUMBER-STEADY V/S UNSTEADY STATE SIMULATION
STROUHAL NUMBER FOR STEADY STATE SIMULATION BASED ON LIFT COEFFICIENT AT A REYNOLD NUMBER OF 100.
STROUHAL NUMBER FOR TRANSIENT STATE SIMULATION BASED ON LIFT COEFFICIENT AT A REYNOLD NUMBER OF 100.
7.2 COMPARISION OF COEFFICIENT OF LIFT & DRAG FOR REYNOLD NUMBER RANGING FROM 10 TO 100000 AT STEADY STATE SIMULATION
7.2.1 REYNOLD NUMBER =10
FOLLOWING PLOT SHOWS THE VARATION OF DRAG CO-EFFICIENT .
This parameter achieves alomost a constant value of 2.1288 after 500 iteration.
FOLLOWING PLOT SHOWS THE VARATION OF LIFT CO-EFFICIENT .
This parameter achieves alomost a constant value of 0.000636 after 500 iteration.
MONITER POINT VELOCITY AT A REYNOLD NUMBER OF 10 HAS BEEN SHOWN IN FOLLOWING PLOT
Above plot of average vortex velocity achieve a constant value of 5.6952 m/s near about 500 iterations.
Following figure shows the value of above 3 parameters after 500 iterations
Monitor point velocity- 5.6592 m/s
Coefficient of Drag- 2.1288
Coefficient of Lift- 0.000636
7.2.2 REYNOLD NUMBER =100
All details regarding Reynold number 100 has been disscused in section 7.1.1
For reference purpose following data has been attached
7.2.3 REYNOLD NUMBER =1000
FOLLOWING PLOT SHOWS THE VARATION OF DRAG CO-EFFICIENT .
This parameter achieves alomost a constant value of 0.7601 after 225 iteration.
FOLLOWING PLOT SHOWS THE VARATION OF LIFT CO-EFFICIENT .
This parameter achieves alomost a constant value of -0.0861 after 225 iteration.
MONITER POINT VELOCITY AT A REYNOLD NUMBER OF 1000 HAS BEEN SHOWN IN FOLLOWING PLOT
Above plot of average vortex velocity achieve a constant value of 262.9 m/s near about 225 iterations.
Following figure shows the value of above 3 parameters after 500 iterations
Monitor point velocity- 262.962 m/s
Coefficient of Drag- 0.7601
Coefficient of Lift- -0.086184
7.2.4 REYNOLD NUMBER =10000
FOLLOWING PLOT SHOWS THE VARATION OF DRAG CO-EFFICIENT .
This parameter achieves alomost a constant value of 0.73988 after 300 iteration.
FOLLOWING PLOT SHOWS THE VARATION OF LIFT CO-EFFICIENT .
This parameter achieves alomost a constant value of 0.12983 after 300 iteration.
MONITER POINT VELOCITY AT A REYNOLD NUMBER OF 10000 HAS BEEN SHOWN IN FOLLOWING PLOT
Above plot of average vortex velocity achieve a constant value of 3441.1309 m/s near about 300 iterations.
Following figure shows the value of above 3 parameters after 300 iterations
Monitor point velocity- 3441.1309 m/s
Coefficient of Drag- 0.73988
Coefficient of Lift- 0.12983
7.2.5 REYNOLD NUMBER =100000
FOLLOWING PLOT SHOWS THE VARATION OF DRAG CO-EFFICIENT .
This parameter achieves alomost a constant value of 0.73652 after 300 iteration.
FOLLOWING PLOT SHOWS THE VARATION OF LIFT CO-EFFICIENT .
This parameter achieves alomost a constant value of -0.2589 after 300 iteration.
MONITER POINT VELOCITY AT A REYNOLD NUMBER OF 1000 HAS BEEN SHOWN IN FOLLOWING PLOT
Above plot of average vortex velocity achieve a constant value of 28143.797 m/s near about 300 iterations.
Following figure shows the value of above 3 parameters after 300 iterations
Monitor point velocity- 28143.797 m/s
Coefficient of Drag- 0.73652
Coefficient of Lift- -0.2589
7.3 VELOCITY COUNTER - VORTEX SHEDDING
VORTEX at a Re = 1000 has been shown in following plot
VORTEX at a Re = 10000 has been shown in following plot
VORTEX at a Re = 10000 has been shown in following plot
VORTEX at a Re = 10 has been shown in following plot. NO VORTEX at a Re = 10 has been observed.
The critical value for vortex sheeding is equal to reynolds number =50.
8. CONCLUSION
8.1 CONVERGENCE
Instead of residual we have focused on velocity of moniter point. The location of monitor point was at a distance of 8 meter from centre of cirlcle along positive x-axis. In each case velocity value keep ossilating within a certain range except the case of reynold number 10.
For very low reynold number (Re = 10), velocity achieve almost a constant value of 5.69 m/s. In this case inlet velocity was 10 m/s.
In each of simulation , a ossilatory value of velocity within some particular range has been achived. So we can expect a converged solution of each case.
8.2 STEADY & TRANSIENT STATE
The simulation was done at a Reynold number of 100.
Following table shows the comparision between different parameters.
| CASE | REYNOLDS NUMBER | INLET VELOCITY (m/s) | CO-EFFICIENT OF DRAG | CO-EFFICIENT OF LIFT | MONITOR POINT VELOCITY |
| STEADY STATE | 100 | 50 | 1.67 | 0.14152 | 25.5139 |
| TRANSIENT STATE | 100 | 50 | 1.48 | 0.5180 | 37.771 |
Velocity of moniter point was a little bit higher for transient state. Also coefficient of lift was higher than steady state whereas coefficient of drag was higher for steady state simulation.
8.3 DRAG & LIFT
COMPAIRASION OF CO-EFFICIENT OF DRAG & LIFT VALUES HAS BEEN SHOWN IN FOLLOWING TABLE. COMPARISION HAS BEEN DONE AT STEADY STATE SIMULATION
| CASE | REYNOLDS NUMBER | CO-EFFICIENT OF LIFT | CO-EFFICIENT OF DRAG |
| 1 | 10 | 0.000636 | 2.128 |
| 2 | 100 | 0.14152 | 1.671 |
| 3 | 1000 | -0.086184 | 0.7601 |
| 4 | 10000 | 0.12983 | 0.73988 |
| 5 | 100000 | -0.2589 | 0.73652 |
As reynolds number increases, CO-EFFICIENT OF DRAG keeps on decreasing. At very high reynolds number, the value of CO-EFFICIENT OF DRAG does not vary much (difference is 0.00336). At the same time CO-EFFICIENT OF LIFT values kept ossilating between a larger range from positive side to negative side. The value of drag coefficient achieve almost a constant value after a large number of iteration.
8.4 MATTERIAL REQUIRED
Permeable matterial can be preffered for controlling instability of flow.
The reference for preferred matterial has been taken from Hanru Liu1 , Jinjia Wei2.
Permeability plays an important role and can determine the effectiveness of the porous treatment for controlling the vortex shedding, unsteady forces and the far-field noise. The non-dimensional permeability about Da= 6.4×10-3 is found to achieve the maximum noise reduction and deviation from this critical value causes the degradation of control ability.