Week 2 - Flow over a Cylinder.

AIM :-

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

PART-I

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

PART-II

1. 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)
2. 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.
4. Mention the referred material (Reference Paper or Fluid Mechanics Textbook) at the end of the challenge submission page.

CONTENTS

Chapter 1 - Theory of External Flow.

Chapter 2 - Theory of Reynolds Number and its effects.

Chapter 3 - Explaination of Karman Vortex Sheet.

Chapter 4 - Part 1

a. Simulation of flow with steady case for Re = 100.

b. Simulation of flow with unsteady case for Re = 100.

c. Strouhal Number and its calculation for Re = 100.

Chapter 4 - Part 2

1. a. Cacluate the Coefficient of drag and Lift over a cylinder by setting the Re = 10.

b. Cacluate the Coefficient of drag and Lift over a cylinder by setting the Re = 100.

c. Cacluate the Coefficient of drag and Lift over a cylinder by setting the Re = 1000.

d. Cacluate the Coefficient of drag and Lift over a cylinder by setting the Re = 10000.

e. Cacluate the Coefficient of drag and Lift over a cylinder by setting the Re = 100000.

Chapter 5 - Results Discussion.

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Chapter 1

Theory of External Flow

Here we are going to discuss about

1. What is External flow.
2. Difference Between Internal flow and External Flow.
3. Reynolds Number and Geomtry Effects.
4. Definition Laminar flows, Transitional Flows, Turbulent Flows.
5. Drag and Lift.
6. Experimental External Flows.

1. What is external Flow.

This chapter is devoted to "Eternal Flow" around the bodies which means flow past immersed bodies. These types of flow will have Viscous (Shear and No slip) Affects near the body surface and in its wake, but will typicall be inviscid far from the bodies.therefore these are called Unconfined boundary layer flows.

immersed-body flows are commonly encountered in engineering studies:

• Aerodynamics (airplanes, rockets, projectiles),
• Hydrodynamics (ships, submarines, torpedos),
• Transportation (automobiles, trucks, cycles),
• Wind engineering (buildings, bridges, water towers, wind turbines),
• Ocean engineering (buoys,breakwaters, pilings, cables, moored instruments).

2. Difference between Internal Flow and External flow.

4. Definition of Laminar, Transition and Turbulent flow.

Laminar Flow

• Some flows are smooth and orderly while others are rather chaotic. The highly ordered fluid motion characterized by smooth layers of fluid is called laminar.

Transition Flow

• A flow that alternates between being laminar and turbulent is called transitional.

Turbulent Flow

• The highly disordered fluid motion that typically occurs at high velocities and is characterized by velocity fluctuations is called turbulent.

5. Drag and Lift.

Flow over bodies can also be classified as

• Incompressible flows (e.g., flows over automobiles, submarines, and buildings)
• Compressible flows (e.g., flows over high-speed aircraft, rockets, and missiles).

Compressibility effects are negligible at low velocities (flows with Ma ≲ 0.3), and such flows can be treated as incompressible with little loss in accuracy.

DRAG - A fluid may exert forces and moments on a body in and about various directions. The force a flowing fluid exerts on a body in the flow direction is called drag. Drag is usually an undesirable effect, like friction, and we do our best to minimize it. Reduction of drag is closely associated with the reduction of fuel consumption in automobiles, submarines, and aircraft;

Drag Force

When Integrated over the body

When done with differential area

Drag Coefficient

Where

Fd = Drag Force

P = Pressure Force

`tau` = Shear force

V = Velocity or free stream Velocity

A = Area

`rho` = Density

LIFT - A stationary fluid exerts only normal pressure forces on the surface of a body immersed in it. A moving fluid, however, also exerts tangential shear forces on the surface because of the no-slip condition caused by viscous effects. Both of these forces, in general, have components in the direction of flow, and thus the drag force is due to the combined effects of pressure and wall shear forces in the flow direction. The components of the pressure and wall shear forces in the direction normal to the flow tend to move the body in that direction, and their sum is called lift.

Lift Force

When Integrated over the body

When done with differential area

Lift Co-efficient

Where

Fl = Lift Force

P = Pressure Force

`tau` = Shear force

V = Velocity or free stream Velocity

A = Area

`rho` = Density

The average drag and lift coefficients for the entire surface are determined by integration from.

The drag and lift forces depend on the density 𝜌 of the fluid, the upstream velocity V, and the size, shape, and orientation of the body, among other things, and it is not practical to list these forces for a variety of situations. Instead, it is more convenient to work with appropriate dimensionless numbers that represent the drag and lift characteristics of the body. These numbers are the drag coefficient CD, and the lift coefficient CL.

where A is ordinarily the frontal area (the area projected on a plane normal to the direction of flow) of the body. In other words, A is the area seen by a person looking at the body from the direction of the approaching fluid. The frontal area of a cylinder of diameter D and length L, for example, is A =LD. In lift and drag calculations of some thin bodies, such as airfoils, A is taken to be the planform area, which is the area seen by a person looking at the body from above in a direction normal to the body. The drag and lift coefficients are primarily functions of the shape of the body, but in some cases they also depend on the Reynolds number and the surface roughness.

The term

`1/2(rho*V^2)` is the dynamic pressure

Chapter 2

Theory of Reynolds Number and its effects

Reynolds number is a property of the flow, not of the fluid. Reynolds number is always a function of the flow velocity and some characteristic length; the fluid properties of importance are its viscosity and density. It is a measure of how turbulent a flow is.

• Reynolds number is the ratio of characteristic time scales for diffusive and advective transport of momentum in the fluid.
• If is the characteristic flow velocity and L">L is the characteristic length scale and `nu` is the kinematic viscosity of the fluid we have:
• Characteristic time scale for diffusion:

• Characteristic time scale for advection:

• Rynolds number:

So, when diffusive transport of momentum is fast compared to advection,

we have and so, in this case, we have Re≪1">Re≪1

When advective transport of momentum is faster than diffusion,

we have and so Re≫1">Re≫

Example: Flow in a pipe. In this case, u0">u0 is the average flow velocity (flow rate over pipe cross-section) and L">L is the pipe diameter.

To change the Reynolds number of a flow (not fluid!), you must change the ratio between the diffusive and advective momentum transfer. If you use a fluid which is more viscous, this will make diffusive transport of momentum more effective, and thus lower the Reynolds number. If you increase the flow velocity, this will make advection transport of momentum more effective and thus the Reynolds number will increase.

Chapter 3

Explaination of Karman Vortex Sheet.

Vortex shedding — What is it and why does it occur?

In fluid Dynamics, vortex shedding is an oscillating flow that takes place when a fluid such as air or water flows past a bluff (as opposed to streamlined) body at certain velocities, depending on the size and shape of the body. In this flow, vortices are created at the back of the body and detach periodically from either side of the body forming a Von Karman Vortex street. The fluid flow past the object creates alternating low-pressure vortices on the downstream side of the object. The object will tend to move toward the low-pressure zone.

There was no body involved in von Karman’s stability analysis, nor did he consider viscous effects or turbulence, as potential theory was applied. Nevertheless, the observed wake behind a bluff body has later been linked to his work as a Karman vortex street. After the seminal work of these early researchers, a wide and deep ocean of research has evolved in the field of vortex shedding and vortex-induced response of cylinders.

Many different angles of attack have been followed to approach the problem, and a large variety of completeness have been achieved. A very crude classification of the different research areas can be to distinguish between the fluid problem and the structural problem. The fluid problem. Those who study the vortex shedding process behind a bluff body, a study of the flow.

Figure 1 - Source - Google

In studying the response problem, it is of great importance to understand how the forces acting on the body surface are generated due to vortex shedding. The attention is thus drawn towards the flow in the wake close behind the cylinder, and to how the vortices are formed in this vortex formation region. The vortex formation region discussed in the following, may be thought of as the distance from the rear of the cylinder to the position of the centre of the vortex when it is shed. The position of maximum strength of the vorticity marks the end of the formation region, in other words, the vortex is shed at the moment it reaches its maximum vorticity.

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 (s^-1), is the diameter of the cylinder (m), and is the flow velocity (m s^-1)

The Strouhal number depends on the Reynolds number

Figure - 2 - Vortex shedding over a circular cylinder

Reynolds number and its effect

The technique of boundary layer (BL) analysis can be used to compute viscous effects near solid walls and to “patch” these onto the outer inviscid motion. This patching is more successful as the body Reynolds number becomes larger. The characteristic length for a circular cylinder or sphere is taken to be the external diameter D. Thus, the Reynolds number is defined as Re=VD/𝜈 where V is the uniform velocity of the fluid as it approaches the cylinder or sphere. The critical Reynolds number for flow across a circular cylinder or sphere is about Recr ≅ 2× 105. That is, the boundary layer remains laminar for about Re ≲ 2 × 105, is transitional for 2 × 105 ≲ Re ≲ 2 × 106 , and becomes fully turbulent for Re ≳ 2 × 106.

At very low upstream velocities (Re ≲ 1), the fluid completely wraps around the cylinder and the two arms of the fluid meet on the rear side of the cylinder in an orderly manner. Thus, the fluid follows the curvature of the cylinder. At higher velocities, the fluid still hugs the cylinder on the frontal side, but it is too fast to remain attached to the surface as it approaches the top (or bottom) of the cylinder. As a result, the boundary layer detaches from the surface, forming a separation region behind the cylinder. Flow in the wake region is characterized by periodic vortex formation and pressures much lower than the stagnation point pressure. The nature of the flow across a cylinder or sphere strongly affects the total drag coefficient CD. Both the friction drag and the pressure drag can be significant. The high pressure in the vicinity of the stagnation point and the low pressure on the opposite side in the wake produce a net force on the body in the direction of flow. The drag force is primarily due to friction drag at low Reynolds numbers (Re ≲ 10) and to pressure drag at high Reynolds numbers (Re ≳ 5000). Both effects are significant at intermediate Reynolds numbers.

• For Re ≲ 1, we have creeping flow, and the drag coefficient decreases with increasing Reynolds number. For a sphere, it is CD=24/Re. There is no flow separation in this regime
• At about Re ≅ 10, separation starts occurring on the rear of the body with vortex shedding starting at about Re ≅ 90. The region of separation increases with increasing Reynolds number up to about Re ≅ 103. At this point, the drag is mostly (about 95 percent) due to pressure drag. The drag coefficient continues to decrease with increasing Reynolds number in this range of 10 ≲ Re ≲ 103. (A decrease in the drag coefficient does not necessarily indicate a decrease in drag. The drag force is proportional to the square of the velocity, and the increase in velocity at higher Reynolds numbers usually more than offsets the decrease in the drag coefficient.)
• Some useful empirical relations for drag coefficient over a sphere in this
  range of Reynolds numbers are:
  `Cd = (24/(Re))(1+0.0916*Re)` for 0.1 < Re < 5
  `Cd = (24/(Re))(1+0.0916*(Re^(2/3)))` for 5 < Re < 1000

Average drag coefficient for cross-flow over a smooth circular cylinder and a smooth sphere.

• In the moderate range of 103 ≲ Re ≲ 105, the drag coefficient remains relatively constant. This behavior is characteristic of bluff bodies. The flow in the boundary layer is laminar in this range, but the flow in the separated region past the cylinder or sphere is highly turbulent with a wide turbulent wake.
• There is a sudden drop in the drag coefficient somewhere in the range of 105 ≲ Re ≲ 106 (usually, at about 2 × 105). This large reduction in CD is due to the flow in the boundary layer becoming turbulent, which moves the separation point further on the rear of the body, reducing the size of the wake and thus the magnitude of the pressure drag. This is in contrast to streamlined bodies, which experience an increase in the drag coefficient (mostly due to friction drag) when the boundary layer becomes turbulent.
• There is a “transitional” regime for 2 × 105 ≲ Re ≲ 2 × 106 , in which CD dips to a minimum value and then slowly rises to its final turbulent value.

Flow visualization of flow over a smooth sphere at Re = 15,000,

Flow visualization of flow over a sphere at Re = 30,000

Chapter 4

Simulation

PART-I

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

Case 1 :- Simulation of Flow over a cylinder (Steady State Simulation) (Re = 100)

Details

• Steady State Simulation
• Solver - Pressure Based
• Viscous Model - Laminar Flow
• Fluid - Air
• Re Number = 100
• Density of fluid - 1 Kg/m3
• Viscosity = 0.02 Kg/m-s
• Velocity = 1m/s
• Total Number of Iterations = 1200
• Solution Method - Simple Based Method
• Inflation Layer = 9
• Wall = No slip Condition

Figure 1 - Mesh Method	Figure 2 - Mesh Sizning	Figure - 3 - Solution Method
Figure - 4 - Reference Values

Solution / Results

Re = 100 = Steady State Simulation

Figure - 4 - Velocity Vector Magnitude	Figure - 5 - Drag Coefficient
Figure - 6 - Lift Cofficient	Figure - 7 - Vertex Average Vlocity
Figure - 8 - Scaled Residuals

Case 2 :- Simulation of Flow over a cylinder (UnSteady State Simulation) (Re = 100)

Details

• UnSteady State Simulation
• Solver - Pressure Based
• Viscous Model - Laminar Flow
• Fluid - Air
• Re Number = 100
• Density of fluid - 1 Kg/m3
• Viscosity = 0.02 Kg/m-s
• Velocity = 1m/s
• Here Time steps are taken
• Solution Method - Simple Based Method
• Inflation Layer = 9
• Wall = No slip Condition

PART-II

1. 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)
2. 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 %]

1. 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)

Re = 10:

Re = 100:

Re = 1000:

Re = 10000:

Re = 100000:

CHAPTER - 5

Results and Conclusion

• As the Reynolds number increases, vortex shedding at the extreme cylinder surface becomes more pronounced.
• As the inertia for viscous forces increases, the drag coefficient (Cd) decreases.
• The rate of decrease of Cd from Reynolds number 10 to 100 is much more drasatic than after that
• As the value of Reynolds number increases, so does the wake region behind the cylinder.
• Vortex shedding occurs for Reynolds numbers greater than 100.