Week 2 - Flow over a Cylinder.

Problem Statement:

1. Simulate flow over a cylinder for different Reynolds number by varying the velocity of the flow using steady solver and calculate the drag and lift coefficient.
2. Simulate the flow over a cylinder for steady and unsteady solver and calculate the Strouhal Number for Re 100.
3. Study and discuss the effect of Reynold number on the coefficient of drag.
4. explain the effect of Von Karman Vortex street.

Introduction

Flow over a blunt body such as circular cylinder experiences boundary layer separation and flow oscillations in the wake region behind the body. When Reynolds number exceeds a threshold value flow motion developed in the wake region behind the body is a result of boundary layer separation from both the sides of the body. This pattern of separation is called Von Karman Vortex Street. A Von Karman vortex street is repeating patterns of swirling vertices that are arising in the wake of flows around the blunt objects. The frequency of these oscillations is related to the Reynolds number of the flow which in this case depends on the velocity of the inlet flow.

Vortex Shedding:

Vortex shedding is an oscillating flow that takes place when a fluid flows past a blunt body at certain velocity. In this flow the boundary layer separated from the surface forms a shear layer which roll into vortex and detach from the surface whereas in Von Karman Vortex Street shedding occurs from both the sides of the body which interact with each other. They shed alternatively from the cylinder and generates a regular vortex pattern called Karman Vortex Street.

Animation of vortex street created by a cylindrical object; the flow on opposite sides of the object is given different colours, showing that the vortices are shed from alternating sides of the object

visualization of the vortex street behind a circular cylinder in the air; the flow is made visible through the release of glycerol vapour in the air near the cylinder

Strauhal Number:

Strauhal number is the non dimensional number describing the oscillating flow mechanism.

`Sr = (f*L)/v`

where Sr is Strauhal Number

f - frequency of oscillation in Hz

L - characteristic length in m.

In this case L = D, where D is the diameter of the cylinder in m.

v - velocity in m/s

Large Strauhal number indicates that the viscosity dominates the flow whereas lower Strauhal number indicates high speed fluid.

Drag Coefficient.

Drag Coefficient is a dimensionless quantity that is used to quantify the resistance of an object. It is used in the drag equation which lower drag coefficient indicates less aerodynamic or hydrodynamic drag.

Formula for Coefficient of drag (`c_d`):

`C_d = (2.F_d)/(rho.V^2.A)`

where `F_d`is drag force in N.

`rho` - density of the material in `(kg)/m^3`

V - velocity of the inlet flow in m/s

A - area of the object in `m^2`, here A is area of cylinder with diameter 2m.

Lift Coefficient.

Lift coefficient is non dimensionaless quantity that relates to the lifting force to the density of the fluid and velocity of the fluid and associated area of the object.

Formula for Coefficient of lift(`C_l`):-

`C_l = (2.F_l)/(rho.V^2.A)`

Where `F_l` is the lift force in N.

`rho`- density of the material in Kg/`m^3`

V - velocity of the inlet flow in m/s

A - area of the object in `m^2`, here A is area of cylinder with diameter 2m.

Procedure:

Geometry:

An infinitely long circular cylinder of diameter D=2.0 m is placed in an otherwise undisturbed uniform crossflow. The lateral boundary and the exit boundary in the far wake are placed at 10 Dand 20D from the centre of the circular cylinder, respectively. The cylinder is used to be drawn in SpaceClaim.

Dimensions of the Geometry:

Total length = 60m.

Total Width = 20m

Diameter of the cylinder = 2m.

Distance between inlet and center of the cylinder = 10D = 20m.

Distance between outlet and center of the cylinder = 20D = 40m.

Converting the geometry into the surface by using command pull and select. Also, delete the inside the circle to carry out the simulation.

Then they select the construction line under the curves.

Mesh:

First of all, the baseline mesh autogenerated quad mesh was created. After that, it will change according to simulation requirements, and it was finely tuned with the help of triangular mesh generated. The model meshed with an element size of 0.23m.

Edge sizing:

Body fitted mesh is generated.

Inflation layer:

To capture the better results near the cylinder wall inflation layer set up is used.

Named Selection of the created geometry.

Setting up physics, boundary conditions and output:

Solver: Unsteady State

Steady State

Viscous Model:

Material:

Reference Values:

Solving method ( Unsteady state ):

For velocity-pressure coupling, the SIMPLEC algorithm was used. When the SIMPLEC algorithm is used, the pressure under-relaxation is automatically increased to 1.0. The setting for pressure under-relaxation is 0.3 (when the default SIMPLE algorithm is used). Here the momentum under-relaxation has also been increased from 0.7 to 0.9 to yield better convergence.

Steady-state

Solution initialization :

Unsteady state:

Steady-state:

Hybrid Initialization

Residuals:

Boundary condition type: Velocity inlet, pressure outlet.

The outlet pressure is set to gauge pressure of 0 Pascals. The side walls of the domain are assigned symmetry boundary conditions due to which velocity normal to the symmetry plane is zero and ensure zero gradient to all variables normal to symmetry plane. The cylinder surface is assigned wall boundary conditions which is no slip wall boundary condition.

Plots obtained using steady solver.

Residual Plot

Velocity at monitor point.

Pressure Contour.

From the contour plot the stagnation point at the front of the cylinder and the flow separation happening on the surface of the cylinder can be seen

Velocity Contour

Plots obtained using Transient solver.

Residual Plot.

Velocity at monitor point.

Pressure Plot:

Velocity plot

Drag Coefficient:

Lift coefficient.

Calculation for Strouhal Number.

`St = (f.D)/v`

`f = 2.5/(30) = 0.0833`

`St = (0.83.2)/1 =0.166`

Inference:

From Residual plot after 250 iterations the residuals are constant and also from velocity at monitor point follows the same pattern hence we can conclude that the solution has reached a steady state thus there will be no change in the resulting solution even if there is increase in the number of the solution.

In the simulation using steady state solver, the results are not time accurate due to which we cannot calculate the Strauhal number whereas in the simulation using unsteady solver the accurate solutions can be obtained from Strauhal number can be calculated which is 0.166.

Part II:

In this part the simulation is done for the range of Reynolds number by varying the velocity of the flow at the inlet and simulated using steady state solver.

Case 1: Re = 10.

Here Velocity v = 0.1m/s.

Residual plot:

Velocity at the monitor point

Coefficient of Drag

Coefficient of lift

Velocity contour plot

Case 2: Re=100

Velocity v = 1m/s

Residual plot:

Velocity at the monitor point.

Coefficient of Drag

Coefficient of lift

Velocity contour plot

Case 3: Re=1000

Velocity v = 10m/s

Residual plot:

Velocity at the monitor point

Coefficient of Drag

Coefficient of lift

Velocity contour plot

Case 4: Re=10000

Velocity v = 100m/s

Residual plot:

Velocity at the monitor point

Coefficient of Drag

Coefficient of lift

Velocity contour plot

Case 5: Re=100000

Velocity v = 1000m/s

Residual plot:

Velocity at the monitor point

Coefficient of Drag

Coefficient of lift

Velocity contour plot

Table depicting the effect of reynolds number on drag coefficient and lift coefficient.

Velocity in m/s	Reynolds Number	Drag Coefficient
0.1	10	3.3
1	100	1.32
10	1000	0.758
100	10000	0.72
1000	100000	0.62

Plot Re vs Cd

Conclusion

• When the Reynolds number increases the coefficient of drag decreases. Moreover, when the reynolds number is large (Re>4000) flow will be turbulent( Chaotic and swirling vortices), in which laminar flow is very small and which leads to lower skin friction drag.
• Firstly , when the velocity increases coeffecient of drag decreases.
• Secondly, for the reynolds number 10 no vortices can be seen because of low velocity and only boundary layer seperation occured.Also, in low reynolds number coefficient of drag increases.
• For Reynolds number 100 ( Unsteady state) the vortices can be seen and the shedding frequency calculated.In addition, the value of 0.16 strouhal number is calculated.We have validated this result with experimental data (0.165) were the error appears to be less than 5%.
• In this case the boundary layer separation decreases there will be less recirculation flow at wake region behind the cylinder with the reduced drag.For all the simulation parameters are kept constant except for the velocity, so increasing the velocity will increase reynolds number.
• For the steady case the flow is not dependent on time step so strouhal number cannot be calculated for the steady state but the solution has reached the steady state after some iterations. We can seen vortices for higher reynolds number only.
• For better visualization we can use contour, vector and streamline to see the vortex generation when the flow past over a cylinder.
• In transient case lift coeffecient keeps on changing beacuse of alternate nature of the vortices it never settle down and keeps on repeating same pattern after some iteration( residual plot).When it comes to steady case the flow is not dependent time so the lift coeffecient becomes constant after some iterations.

Preferred material for instability:

In order to prevent the unwanted vibration of cylindrical bodies, a longitudinal fin can be fitted on the downstream side, which, provided it is longer than the diameter of the cylinder, will prevent the eddies from interacting, and consequently they remain attached. Obviously, for a tall building , the relative wind could come from any direction. For this reason, helical projections resembling large screw threads are sometimes placed at the top, thereby discouraging the alternate shedding of vortices. Another countermeasure with tall buildings is using variation in the diameter with height, such as tapering - that prevents the entire building being driven at the same frequency. For example, spiral added to the chimney to avoid vortex shredding