Steady Vs Unsteady flow over a cylinder

Aim: To create a simulation of flow over a cylinder and to explain the phenomenon of Karman Vortex effect

Theory:

VORTEX SHEDDING

Vortex shedding is the oscillating flow that fluid such as air or water flows past a bluff body at certain velocities depending upon size and shape of the body. If the bluff structure is not mounted rigidly and the frequency of vortex shedding matches the resonance frequency of the structure, then the structure can begin to resonate vibrating with harmonic oscillations driven by the energy of the flow.

The flow around two-dimensional bluff bodies is defined by flow separations that results a wake in downstream which in turn causes a vortex street. The location and the time dependent behavior of the vortices are highly dependent on the cross-section of the bluff bodies. The circular cylinder is the most primary example of bluff bodies, demonstrating rounded geometries that have the highest degree of symmetry.

The periodic vortex shedding behind bluff bodies exposed to uniform flow. The occurrence of this flow phenomenon is due to instabilities and depends on the geometry of the bluff body and the Reynolds number.

The frequency at which vortex shedding takes place for an infinite cylinder is related to the Strouhal number by the following equation:

Where is the dimensionless Strouhal number, is the vortex shedding frequency, is the diameter of the cylinder, and is the flow velocity.

Computataonal Domain and Boundary Conditions

Meshing

We have choosen the element size as 0.25m to generate the mesh. Number of nodes = 20679, Number of elements = 40466, Growth rate = 1.2

The Circle is not perfect and includes many sharp edges. These sharp corners affect the flow field. The geometries on top of which we are trying to simulate the flow must be captured perfectly. We use 'Sizing' option to preciesly control the mesh. We are creating the edge for every 10 degrees on the circle for smooth development of flow. We are using the inflation layer to take care of the region around the cylinder where the flow takes place. Inflaton layer helps us create a body fitted mesh. If the Mesh perfectly fits the geometry then it is called as "Body fitted Mesh". In this case I have added 6 inflation layers.

Results of "STEADY STATE" simulation for Reynold's number 100

Reynolds number is the ratio Inertia force to the Visous force. Its a dimensionless number used to decide the type of flow in pipe.

where ρ-----> Density,v--> velocity, d--->Pipe Diameter, μ---> Dynamic viscocity, R_e ---> Reynolds number

Residuals plot

At 450 iterations the solution start to converge

Vertex average of Velocity Magnitude

Drag plot

Lift plot

Velocity Contour

Pressure Contour

Results of "TRANSIENT STATE" simulation for Reynold's number 100

Residuals plot

Vertex average of Velocity Magnitude

Drag Plot

Lift plot

For calculation of frequency we are considering One and half completed cycle

Velocity Contour Pressure Contour

Strouhal number- It is a dimensionless number describing oscillating flow mechanism. It is an integral part of fundamentals of Fluid Mechanics

f----> Frequency of vortex shedding, L----->Characterstic length of the body(diameter of the cylinder), U-----> Flow velocity

Drag coefficient

Drag coefficient is the dimensionless quantity used to quantify the drag or resistence of an object in a fluid environment such as air or water. It is used in the drag equation in which lower drag coefficient indicates object will have less hydrodynamic drag. The drag coefficient is always associated with particular surface area.

C_d- drag Coefficient , F_d---> Drag force, ρ--->Mass Density, u--->Flow speed of the object , A---> Refrence area

Lift coefficient

It is the dimensionless coefficient that relates the lift generated by a lifting body to the fluid density around the body, the fluid velocity and an associated refrence area . The lift force acts always perpendicular to the drag force.

C_L- Lift Coefficient, ρ--->Mass Density, u--->Flow speed of the object, A--->Refrence area, L--->Lift Force

Effect of Drag and Lift with varying values of Reyolds number

For Reynolds number(R_e ) = 10

Drag force Lift force

Velocity contour Pressure contour

Residual plots

For Reynolds number(R_e ) = 100

Drag force Lift force

Velocity contour Pressure contour

Residual plots

For Reynolds number(R_e ) = 1000

Drag force Lift force

Velocity contour Pressure contour

Residual Plots

For Reynolds number(R_e ) = 10,000

Drag force Lift force

Velocity contour Pressure contour

Residual Plots

For Reynolds number(R_e ) = 1,00,000

Drag force Lift force

Velocity contour Pressure contour

Residual Plots

The results of all the above 5 cases are tabulated as below

Tabulation

Observations

1] For both Steady and Unsteady cases flow over the cylinder was successfully simulated with Reynolds number of 100 and the phenomenon of Von Karman Vortex effect is observed.

2] Vortex formation, Vortex propogation, Pressure rise from the front of the cylinder was easily observed with the help of contours

3] As the Reynolds number value increases, the value of Coefficient of Drag decreases

4] Strouhal number of 0.15 was observed at Reynolds number of 100