Computational analysis of steady and unsteady flow of air over cylinder

Objective :- To simulate the flow over a cylinder and to understand the Von Karman vortex street phenomena.

PART-I

• Simulate the flow with the steady and unsteady case and calculate the Strouhal Number for Re= 100

Introduction :- The flow of fluids over any object has always been fascinating in terms of the patterns it produce and the physics it contains. One such phenomena is the formation of Von karman vortex street which has been observed in the nature as shown in figure.

Figure 1:- NASA\'s terra satellite image of the von karman phenomena between two islands.

(Image courtesy :- NASA)

• In order to understand the Von karman vortex street it is necessary to know the dynamics of the fluid. The vortex is simply a rotating flow of fluid around an axis which results in to lower pressure region at the centre.
• Due to formation of vortex, there are two velocity vectors which can be seen at the extremeties, one in the direction of the fluid flow and the other in the direction opposite to it. Due to this, the magnitude of the velocity vector in the direction to the fluid is more than the velocity of fluid in the direction opposite to it.
• This give rise to the LIFT of the vortex in the direction normal to the fluid flow. Similar phenomena occurs in the bottom vortex which is formed opposite to the initial vortex.
• Now according to the bernouli\'s principle the higher the velocity results into having lower pressure region for the vortex formed in the direction of fluid while the quite opposite in the other. This results into an oscillating motion of the vortices formed continously at that region.
• This oscillating motion gives rise to the formation of von karman vortex street. The frequency of this oscillation is of quite importance since it should not match the natural frequency of the object being placed or else it can result into resonance and thus failure of the structure.
• One such pheneomena was seen in the classic Tacoma bridge collapse which is shown below.

https://www.youtube.com/watch?v=XggxeuFDaDU

• Hence the book definition of the Von Karman vortex street is the repeating pattern of the swirling of the vortices produced at the wake of the cylinder.
• During the flow of fluid over a cylinder at high velocities there exist a point where the boundary layer shears i.e it sperates itself at a point which gives rise to the pressure gradient and hence there is instability in the forces which gives rise to continous forming of the vortices at the back of the cylinder. This is a dangerous phenomena and can lead to unwanted noise and failure of the structure due to resonance.

CASE I:- Steady and Unsteady state with Re=100 and it\'s Strouhal number.

The process was done in 4 steps starting with geometry, processing and postprocessing.

• The geometry parameters are given below

.

• After geometry creation the meshing was performed with the below specified parameters.
• A Multizone method with all tri mesh parameter was used to generate tri mesh on the domain. An edge sizing (local meshing) was done around the cylinder edge in order to smoothly shaped circle.
• Since the output result is going to be affected by the mesh near the cylinder edge (because of the seperation of the boundary layer from the surface), an inflation layer was provided in order to body fit the mesh.

Meshing Parameter	Mesh Size	Mesh quality (min)	Mesh quality (avg)	No. of nodes	No. of elements
Multi Zone (all tri)/ Inflation/ Edge sizing	0.25 m	5.14e-2	0.993	22705	44518

Step 3:- The input parameters were given as mentioned below.

Condition 1:- Steady State Analysis

Fluid	Density (kg/m3)	Kinematic viscosity (kg-m/s)	Inlet velocity (m/s)	Outlet Gauge pressure (Pa)	Reynolds Number
Air	1	0.05	2.5	0	100

• Air as fluid was chosen and in steady state condition SIMPLE( Semi Implicit Method forr Pressure Linked Equaltions). Since from trhe literatures it was shown that between 40<Re<100 or less than it ,the flow is said to be laminar hence a laminar model was used to calculate the Strouhal number and coefficeint of lift and drag.
• In order to see the effect of vortex shedding (here it is happening because Re number is just above the critical Re no.) a point was created at a distance of 10 m from the center of the cylinder.
• Using it as a point to monitor, the velocity was monitored at that point in order to see the effect of vortex shedding.

• As it can be seen for the residual plot that the values of residuals were found to be converging after approximately 300 iterations. The fluctuations in the velocity values can be seen after 175 approx equations indicating phenomena of vortex shedding. This can also be seen from the velocity contour plot.
• Residual plot can be seen in order to predict the convergence of the flow. In this case it is approximately after 300 iterations.

Condition 2:- Transient State Analysis

• The fluctuations in the values of velocity dives rise to the formation of the vortex creation. This frequency can help in calculating Strouhal number which represents the ratio of inertial forces due to variation or unsteadiness in the flow to the inertial forces due to change in the magnitude of velocity from one point to anothe point in the flow.
• It is a function of Reynold\'s number.

https://youtu.be/k7y8vgw8gOA

• For calculating Strouhal number the oscillation frequency needs to be calculated which can be found from the below graph of co efficient of lift. The Strouhal number is given by :-

St=ω.l/v , where ω = vortex shedding oscillating frequency

l= charateristic length

v= free stream velocity

ω= 2/10 ( FROM THE ABOVE GRAPH)

ω=0.2

• St = 0.2*2/2.5

St = 0.16

• Tudor Baracu et. al. [1] mentioned the empirical relation to find the Strouhal number. It is given for any fluid having Re < 2*10^5 by the following equation.

`St= 0.212(1-21.2 / Re)` , For Re= 100 it comes out to be St=0.167 which is similar to the result obtained form the analysis. This shows our analysis is correct.

• Transient analysis is only the correct approcah for such problems which are severly dependent on time. Hence eventhough both steady and transient analysis can predict the same resulr but this is more of time dependent phenomena before reaching steady state hence transient analysis is the correct method to solve such problems.

Cases	Coefficient of lift	Coefficient of drag	Velocity at vertex (m/s)	Pressure at monitor point (Pa)
Transient (Re=100)	-0.057	1.291	1.541	-0.39
Steady (Re=100)	-0.121	1.31	1.580	-0.310

PART-II

• Calculate the coefficient of drag and lift over a cylinder by setting the Reynolds number to 10, 100, 1000, 10000 & 100000. (Run with steady solver)
• Discuss the effect of Reynolds number on the coefficient of drag.[Results should be validated with any standard literature & error must be within 5%.]

This was done with the same parameters as the above part and only velocities were made variable for getting different Reynold\'s number.

• By varying the velocity the following results of the coefficient of lift and drag coefficient were found .

Cases	Velocity (m/s)	Coefficient of lift	Coefficient of drag
Steady (Re=10)	0.25	0.000422	3.26
Steady (Re=100)	2.5	-0.121	1.31
Steady (Re=1000)	25	-0.126 &0.025	0.73 &0.757
Steady (Re=10000)	250	-0.1177	0.617
Steady (Re=100000)	2500	-0.19	0.432

• B.N Rajani et. al [2] has mentioned the value of coefficient of lift and drag for Re= 100 for a 2D flow over a cylinder.

The computed values are in well agreement with the values of the literature, hence the results can be considered valid.

The convergence can also be seen from the residual plots indicating 650 iterations for Re=10, 350 iterations for Re= 100 and 250 to 300 iterations for the further cases.

Discussion :- The Reynold\'s number effects the velocity contour and coefficient of drag and lift.

For Re =10:- The formation of vortex shedding is not seen because the flow is laminar and the boundary layer seperation is not predominant for formation of vortex shedding.The flow is extremely slow anad thus the viscosity is not predominant and due to which the vortex is not formed.

Re =100 &1000:- As the velocity increases the vortex begins to form due to boundary layer seperation. This phenomena is caused when the fluid flows over the surface and at the downstream of the cylinder the boundary layer seperates which cause pressure gradient and thus gives rise to the lift force. The below figure is taken from the literature [1].

Re=10000 & 100000:- During this the velocity is very high and gives rise to large wake behind the cylinder. This gives rise to high turbulent vortices forming behind the cylinder. The decreasing value of drag wrt reynolds number can be seen from the graph.

References:-

[1] TUDOR BARACU, STEFAN GRIGORAS-BENESCU (n.d.) \'COMPUTATIONAL ANALYSIS OF THE FLOW AROUND A CYLINDER AND OF THE DRAG FORCE\', , (), pp. .

[2]B.N. Rajani a , A. Kandasamy b , Sekhar Majumdar a (2008) \'Numerical simulation of laminar flow past a circular cylinder\', Applied Mathematical Modelling, (), pp. .