Week 2 - Flow over a Cylinder.

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

OBJECTIVE:

PART-I

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

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 and error should be within 5 %]

Expected results:-

1. In both approaches, show that the flow has converged. That is, what quantities or plots need to be looked at to determine that the flow has converged?
2. Plots of the coefficient of drag and lift.
3. Plots to show vortex shedding behind the cylinder.

Theory:

In spite of extensive experimental and numerical studies almost over a century, flow around a circular cylinder still remains a challenging problem in fluid mechanics, where intensive investigations are continued even today to understand the complex unsteady dynamics of the cylinder wake flow. Cross-flow normal to the axis of a stationary circular cylinder and the associated problems of heat and mass transport are encountered in a wide variety of engineering applications. Both experimental measurements and numerical computations have confirmed the onset of instability of the wake flow behind a cylinder beyond a critical Reynolds number, leading finally to a kind of periodic flow identified by definite frequencies, well-known in the literature as the Von Karman vortex street. In case of laminar flow past cylinders with regular polygonal cross-section, the flow usually separates at one or more sharp corners of the cross-section geometry itself, forming a system of vortices in the wake on either side of the mid symmetry plane. On the other hand, for a circular cylinder, where the point of flow separation is decided by the nature of the upstream boundary layer, the physics of the flow is much more complex than what its relatively simple shape might suggest.

Strouhal number:

In dimensional analysis, the Strouhal number is a dimensionless number describing oscillating flow mechanisms.

For large Strouhal numbers (order of 1), viscosity dominates fluid flow, resulting in a collective oscillating movement of the fluid "plug". For low Strouhal numbers (order of 10−4 and below), the high-speed, quasi-steady-state portion of the movement dominates the oscillation. Oscillation at intermediate Strouhal numbers is characterized by the buildup and rapidly subsequent shedding of vortices.

The Strouhal number is often given as

S_t=fD/U

Where,

f= frequency of vortex shedding

D= characteristic diameter of the cylinder

U= fluid flow velocity

Co-efficient of drag:

In fluid dynamics, the drag coefficient (Cd) is a dimensionless quantity that is used to quantify the drag or resistance of an object in a fluid environment, such as air or water. It is used in the drag equation in which a lower drag coefficient indicates the object will have a less aerodynamic or hydrodynamic drag. The drag coefficient is always associated with a particular surface area. The drag coefficient of any object comprises the effects of the two basic contributors to fluid dynamic drag: skin friction and form drag. The drag coefficient of a lifting airfoil or hydrofoil also includes the effects of lift-induced drag. The drag coefficient of a complete structure such as an aircraft also includes the effects of interference drag. The drag coefficient Cd is defined as

Cd=2.Fd/ρu²A

where:

Fd is the drag force, which is by definition the force component in the direction of the flow velocity,

ρ is the mass density of the fluid,

u is the flow speed of the object relative to the fluid,

A is the reference area.

Co-efficient of lift:

The lift coefficient (CL) is a dimensionless coefficient that relates the lift generated by a lifting body to the fluid density around the body, the fluid velocity, and an associated reference area. A lifting body is a foil or a complete foil-bearing body such as a fixed-wing aircraft. CL is a function of the angle of the body to the flow, its Reynolds number, and its Mach number. The section lift coefficient cl refers to the dynamic lift characteristics of a two-dimensional foil section, with the reference area replaced by the foil chord.

C_L=L/qS=L/0.5* ρu²S= 2L/ ρu²S

where

L, is the lift force,

S, is the relevant surface area, and

q is the fluid dynamic pressure, in turn, linked to the fluid density, and to the flow speed.

RESULTS AND SOLUTION:

There are 2 cases to solve.

CASE 1:

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

Where we calculate Reynolds number by using

Re_L=UL/ υ ₀

U = the free stream flow speed

L = a characteristic length parameter of the body or channel

υ = the free stream kinematic viscosity

We have 2 parts to this case,

PART 1 :

STEADY-STATE:

A brief explanation to solve the flow over a cylinder

1. Geometry:

Creating the geometry by sketching using the line and circle command, circle diameter of 2m and the dimensions having 20X60m and the circle is placed 20m from the inlet.

1. Mesh: After the geometry is loaded we should mesh it. But in the meshing nodes, the shape is more quadrilateral than the triangle so we should this by changing the method to the triangle in the insert option

After changing the node's shape into triangles we should improve the mesh around the cylinder by sizing in the mesh which is used to control precisely the mesh. Then we should add inflation which is used to create body-fitted mesh

3.Setting up Ansys fluent :

Before opening the set up we should update the mesh. Then we should change the viscous flow to laminar.

Open Materials and create a fluid that is to be used for the simulation for this case, it is air and also inputs the density and kinematic viscosity values.

Open Zones and input the inlet velocity as per the reynolds number and check if the fluid is updated. Update the reference values which is located inside set up at the right-hand side.

Procedures followed in fluent set-up-

1. laminar model
2. fluid: Air
3. Method of discretization: Simple
4. Density/kinematic viscosity: 1/0.05
5. Creating a monitor point behind the cylinder by selecting point under surfaces in the results bar.
6. After creating the monitor point, create reports/plots for drag, lift, velocity wrt cylinder.
7. Create plots for the velocity at the monitor point to analyze the vortex shedding.
8. Use hybrid initialization for steady-state and standard for transient state simulations.

Now, Simulate the flow with the steady case and calculate the Strouhal number for Re=100 and velocity 2.5m/s.

CaseI:

Part I: Steady state (Re=100)

For Re=100, velocity=2.5 m/s

Residual plot:

Vertex Average of Velocity Magnitude Plot:

Coefficient of Drag Plot:

Coefficient of Lift Plot:

Velocity Contour Plot:

Pressure Contour Plot:

Strouhal no:

The strouhal number for steady case is 0.016

PartII: Unsteady state (Re=100)

Residual plot:

Vertex Average of Velocity Magnitude Plot:

Coefficient of Drag Plot:

Coefficient of Lift Plot:

Velocity Contour Plot:

Pressure Contour Plot:

Strouhal no:

The strouhal number for unsteady case is 0.128

Case II: Steady state (Re=10)

For Re=10, velocity=0.25 m/s

Residual plot:

Vertex Average of Velocity Magnitude Plot:

Coefficient of Drag Plot:

Coefficient of Lift Plot:

Velocity Contour Plot:

Pressure Contour Plot:

(Re=1000)

For Re=1000, velocity=25 m/s

Residual plot:

Vertex Average of Velocity Magnitude Plot:

Coefficient of Drag Plot:

Coefficient of Lift Plot:

Velocity Contour Plot:

Pressure Contour Plot:

(Re=10000)

For Re=10000, velocity=250 m/s

Residual plot:

Vertex Average of Velocity Magnitude Plot:

Coefficient of Drag Plot:

Coefficient of Lift Plot:

Velocity Contour Plot:

Pressure Contour Plot:

(Re=100000)

For Re=100000, velocity=2500 m/s

Residual plot:

Vertex Average of Velocity Magnitude Plot:

Coefficient of Drag Plot:

Coefficient of Lift Plot:

Velocity Contour Plot:

Pressure Contour Plot:

Drag and Lift coefficients for different Re no.

Re	Cd	CL
10	3.32	0.0175
100	1.35	-0.026
1000	1.066	-0.3118
10000	1.174	-0.49
100000	1.39	-0.164

Error between theoretical and numerical value of Cd

Reynolds no	Theoretical	Numerical	Error %
10	3.5	3.34	2.8
100	1.4	1.35	3.5

Conclusions:

In the first case,

1. The flow over a cylinder is simulated for both Steady and unsteady cases with Re=100, and with the results generated the phenomena of Von Karman Vortex Street are also observed.

2. Necessary plots are plotted in order to observe that the flow is converged and the vortex shedding is happening.

3. Contours are generated in order to observe the flow over a cylinder. By plotting Power Spectral Density vs Frequency, the Strouhal number is calculated.

In the second case,

six different simulations i.e (Re = 10,100,1000,10000,100000,1000000) are simulated to observe the behavior of the drag and lift coefficients by changing the Reynolds number.

1. In order to observe the behavior of the drag coefficients for the simulations, the velocity of the flow is varied to change the Reynolds number and the rest of the parameters are kept constant.

2. For Re=10, it is observed that there is no formation of vortex shedding and in the rest of the cases, it is clearly observed.

3. As per the basic conditions and parameters, the results are validated from the literature.
As we refine the mesh we need to run more iterations.

4. From the above table, it can be concluded that the error percentage is less than 5%.