Week 2 - Flow over a Cylinder.

AIM : To simulate a flow over a cylinder and explain the phenomenon of Karman vortex street .

OBJECTIVE :

1. To simulate the flow over a cylinder with steady and unsteady cases and calculate the Strouhal number for Re = 100.
2. To simulate the steady case for various Reynolds number and calculate the lift and drag coefficients.
3. To discuss the effect of Reynolds number on the drag coefficients.

INTRODUCTION: 

   Karman vortex street is a phenomenon of repeating the pattern of swirling vortices that is caused by the process of vortex shedding , which in turn gives rise to unsteady  seperation of flow of fluid over a blunt body . When a single vortex is shed it forms asymmetrical flow pattern is formed around the body and the pressure distribution changes . This will cause alternate sheddign of vortices to form in a periodic manner and forces acting due to the periodic fluctuation causes vibration and even lead to catastrophical damages.

 

Solution and Modelling approach:

Part 1 : we will be simulating the steady and unsteady state for Re =  100 

Part 2 : we will be simulating the steady state cases for Re = 10,100,1000,10000,100000

 

Part 1 - Steady State : 

here we will be simulating the steady state case of flow ovwe a cylinder with Re = 100 and observe the plots for the lift and drag coefficients , the vertex average velocity plot for the moniter point and also the pressure and velocity contours and finally calculate the strouhal number.

For the geometry we will be considering a 2D case , so the cylinder in the 3D will turn into a circle in our 2D calculations. we will be having a cylinder of diameter 2m and the flow region will be about 10 times the diameter before the cylinder and the 20 times the diameter after the body and height of theflow region is kep to be 10 times the diameter symmetrical distributed along the flow axis.

We create our geometry in SpaceClaim as shown below

 

After creating this geometry we move to meshing .

while meshing the geometry we wil be adding method as triangular

after this we add edge sizing at the cylinder wall to the number of edges as 36 so the we get a good circular profile .

 

after this we add inflation  to capture the boundary layer at the cylinder wall

we finally add a global element sizing of 0.25 m as the element size

Our final mesh is shown in the following image

this mesh is quite good and if we see the mesh metrics of element quality

most of the elements are triangular and have a quality close to 1.

also the statics of mesh shows that there are more than 40000 elements which is good for our case

Finally adding named selections to the mesh geometry

 

Now we will load this mesh in to fluent to do our simulation of the flow over a cylinder in steady state

Moving to fluent we first check the mesh 

as there are no error or warnings so now we setup physics

changing the material property to match our Re number we keep the density as 1 kg/m^3 and viscosity as 0.02 kg/m_s

next setting the inlet velocity as 1 m/s

all the other boundary conditions are set to default , the outlet is pressure outlet , wall at the cylinder wall , and symmetry boundary condition at the symmetry 

next is setting the reference values , compute from the inlet and then change the area and length to 2 m^2 and 2 m respectively

Now we move to setting up the solution

first changing the solution methods to SIMPLE

then uncheck the convergence for the residuals

adding report definition 

1. vertex average velocity at a moniter point at a distance of 4D from the centre of the cylinder on the downstream

for this we will first create the moniter point 

and we can see this moniter point in the geometry as shown below

now we setup the report definition for the vertex average velocity

2. drag coefficient at the cylinder wall

3. lift coefficeint at the cylinder wall

after this we initialize the simulation using hybrid initialization and then run the simulation for about  700 iterations

 

after the calculation is complete we look at the residuals plot 

our residuals seems to have stabilized at about 400 iterations

now chack for the plots of the vertex average valocity at moniter point

form this plot it looks like the vertex average velocity is fluctuating between a stable range after about 500 iterations

now we wil be looking at the lift and drag coefficients plots

both this plot are fluctuating in a stable range after about 600 iterations

so we will say that out solution has converged at abotu 600 iterations

Now to validate out result for the drag coefficient we will us the reference paper by Rajani, Kandasamy and Majumdar given in the reference section

form this paper we see that our drag coefficient is close to the values for the paper.

 

as our result is validated we will now be calculating the Strouhal number 

using the FFT plot we will be ploting strouhal number vs magnitude using the lift coefficients values and the maximum values is obtained at strouhal number of 0.0429

 

Finally we will lok at the pressure and velocity contour to see the karman vortex street phenomenon

velocity contour

pressure contour

form this contour we are able to see the vortex sheddng and this vortices are forming alternately around the axis of flow

Now we will be simulating the same flow in an unsteady case and check the differences

Part 1 - Unsteady State

In the undteady state we will be using the same geoometry, same mesh and simulate the flow over a cylinder using a transient case settings

after loading the mesh in the fluent setup we will be setting the case to transient

Now adding the same boundary condition and material property as earlier we will jump to setting up the solution

there also we wiil be using solution methods as SIMPLE scheme and adding the same report definitions as in the steady state 

Using hybrid initialization and usign the time step value as 0.1 and number of time steps as 1800 we will calculate the solution for our simulation

After the calculation is complete we will be looking at all the plots 

Residuals Plot:

we see that our residual values for continuity are below and 1e-3 and quite stable and the residuals for the velocity in x and y are very quite negligible

Vertex average velocity plot for the moniter point:

 

the fluctuation in the vertex average velocity at the moniter point are in a stablised range.

Drag coefficeint plot

the value of the drag coefficient also have a stable fluctuation and the value is close to the values from the research paper

Lift coefficient plot:

we can see that the lift coefficeient also has a stable fluctuation and also the values of the lift coefficeint are close to the values from the research paper. So we can say that our validation is fine and now we can calculate the strouhal number and check for the contours for the pressure and velocity 

The plot for the strouhal number vs the magnitude is shown below

from this plot we see that at strouhal number of 0.16657 we have highest magnitude .

Next we will see the at contours 

velocity contour

Pressure contour

from the contour plots the von karman vortex street is clearly visible.

 

Now we will be moving on to the part 2 of our objectives where we will be doing the steady state simulation for various Reynolds and check for the variations of the drag coefficents of the same.

 

Part 2 - Steady State with varying Reynolds Number

For this part we will be using the Reynold number = 10 , 100, 1000, 10000, 100000 

So to find the inlet velocity for each case we wil follow the table

Reynolds number(`Re`)	viscosity(`mu`)	density(`rho`)	diameter(`D`)	velocity(`V= (Re * mu)/(rho *D)`)
10	0.02	1	2	0.1
100	0.02	1	2	1
1000	0.02	1	2	10
10000	0.02	1	2	100
100000	0.02	1	2	1000

 

we will be comparing the results for various Reynolds numbers

Residuals plot

1. Re = 10

2. Re = 100

 

3. Re = 1000

 

4. Re = 10000

 

5. Re = 100000

We can see that the residulas keeps on fluctuating for all the values of Re except 10.

Drag coefficeint plot

1. Re = 10

 

2. Re = 100

 

3. Re = 1000

 

4, Re = 10000

 

5. Re = 100000

 

the drag coefficient is steady for Re =10  and it is fluctuating for other values of Re. Also the drag coefficeint decreases as the reynolds number increases.

 

Lift ceofficient plot

1. Re = 10

2. Re = 100

 

3. Re = 1000

 

4. Re  = 10000

 

5. Re  = 100000 

 

the lift coefficient is zero for Re = 10 and for the rest the lift ocsillates between a ceratain value from positive to negative maxima . Also as the Reynolds number increases the maxima of lift coefficeint increases.

 

Vertex average velocity at the moniter point

1. Re = 10

 

2. Re = 100

 

3. Re = 1000

 

4. Re  = 10000

 

5. Re  = 100000 

The vertex average velocity for the moniter point keep fluctuating at about 80 % of the flow velocity for all the values of Reynold number except 10 where the vertex average velocity becomes almost 40 % of the flow velocity and remains constant

 

Now we will look at the contour plots

Velocity Contour

1. Re = 10

2. Re = 100

 

3. Re = 1000

 

4. Re  = 10000

 

5. Re  = 100000 

Pressure Contour

1. Re = 10

2. Re = 100

 

3. Re = 1000

 

4. Re  = 10000

 

5. Re  = 100000 

we can see that as our Reynolds number = 10, as it is lower than the critical value for vortex shedding , we see that the are no vortices formed in the contour plots and the flow is symmetrical along the flow axis. for rest of the values of reynolds number as it is greater than the critical value we are able to see the vortex shedding.

Strouhal Number Plots

1. Re = 10

2. Re = 100

 

3. Re = 1000

 

4. Re  = 10000

 

5. Re  = 100000 

As the Reynolds number increases the Strouhal number decreases . for Re = 10 we see that the straohal number is almost zero and the maxima is small as compared to other values this is because there is no von karman vortex phenomenon at such low reynolds .

 

In the following table we will be comparing the drag coefficient and the strouhal number 

Reynolds number	drag coefficeint
10	3.25
100	1.33
1000	0.95
10000	0.95
100000	0.9

form this table we can say that as the darg coefficient decreases as the reynolds number increases 

 

CONCLUSION :

Part 1 

1. The steady state and transient state for the flow over a cylinder in 2D case is simulated using FLUENT.
2. Form the results we can see that the residuals for convergence are constant in steady state and the residuals are fluctuating for the transient case.
3. the vertex average velocity is fluctuating in both the steady and unsteady case. 
4. the drag coefficeint is fluctuating around 1.33 in both the cases 
5. the lift coefficient is fluctuating in the range of -1.2 to +1.2 for the steady state and for the transient state the value oscilates between -2.5 to +2.5.
6. the velocity contour shows vortex shedding for the part of flow after the cylinder , this phenomenon is known as von karman vortex shedding  as seen in the theory.
7. the pressure contour alos shows the different pressure region that causes the vortex formations
8. the strouhal number 0.0429 is calculated in the steady state and for the unsteady state we get the strouhal number value as 0.16

 

Part 2

1. the steady state simulation for flow over a cylinder with various reynolds number is simulated
2. From the results we see that the residuals are converged for the Re = 10 and for rest of the values of Reynolds number the residuals fluctuates
3. the Drag and lift ceofficent for Re= 10 is converged and for the rest of the values we get fluctuating results
4. the drag coefficient decreases with increase in the reynolds number .
5. the range for which lift coefficient fluctuates increases with increase in the reynolds number
6. the vertex average velocity is almost the same  ratio to that of the inlet velocity for all the cases except for Re = 10.
7. the velocity contour shows vortex shedding for all the cases except for Re = 10.
8. the pressure contour shows fluctuating pressure regions except for Re = 10.
9. Strouhal number decreses with increase in the reynolds number. 

 

REFERENCE :

1. B.N.Rajani, A. Kandasamy, S. Majumdar - Numerical simulation of laminar flow past a circular cylinder , Applied Mathematical Mdelling 33 (2009) 1228-1247
2. Kármán vortex street - Wikipedia