Differential Analysis of Fluid Flow_A

Fluid Element Kinematics

• We need to know inlet and outlet conditions  ⇋   Integral or CV Analysis or No flow properties.
• We need to follow fluid particles  ⇋  Flow domain or Differential analysis.

Particles undergo four types of motion considering mass remain same:

1. Translation
2. Rotation
3. Linear Deformation (have its volume changed)
4. Angular Deformation (shape change)

Ques: How fluid flows (i.e. all four types of motion)?

All these motion take place all at a time because of difference in velocities (evidence: velocity profile in a pipe is look like parabolic i.e. velocity change).

OR

Pressure due to the fluid or wall friction changes.

Linear Motion & Deformation

Angular Motion & Deformation

• For Pure rotation  ⇋  Ý = 0  &  Pure angular deformation  ⇋  ω = 0.

Rate of Translation and Rotation

Vorticity

It is defined as, "curl of velocity vector OR Twice the angular velocity of fluid".

• て = 0  ⇋  flow is Irrotational.
• て is not equals to 0  ⇋  flow is Rotational.

Circulation

Differential Form of Continuity Equation

• The continuity equation reflects the fact that mass is conserved in any non-nuclear continuum mechanics analysis. 
• The equation is developed by adding up the rate at which mass is flowing in and out of a control volume, and setting the net in-flow equal to the rate of change of mass within it.

Stream Function

• Change in the stream function is related to volume flowrate.
• If lines of constant stream function plotted with provided the family of streamlines ⇋ help in visualization of flow pattern.

Velocity Potential Function

• Velocity Potential Function  ⇋  consequence of irrotationality of flow field, 3D flow.
• Stream Function  ⇋  consequence of mass conservation, 2D flow.

Basic or Elementary or Plane Potential Flows

Plane potential flows is defined as, "the inviscid, incompressible, irrotational, 2D flows".

• For simplicity   →   only 2D plane flows will be considered.

A. Uniform Flow

It is defined as, "Flow in which streamlines are all straight and parallel, the magnitude of the velocity is constant".

B. Source Flow

It is defined as, "Fluid flows radially outward from a line (point) through origin perpendicular to x-y plane".

• Streamlines are straight lines directed radially outward from a point.
• Strength of source: m = 2πrVr.

C. Sink Flow

It is defined as, "Fluid flows radially inward from a point towards the origin (i.e. opposite to source flow".

• Same as Source flow but opposite sign.

D. Free Vortex Flow

It is defined as, "streamlines are in the form of concentric circles".

• Fluid particles do not rotate while revolving around the vortex center.
• Strength of Vortex: K' = 2πrVӨ.
• Tangential velocity varies inversely with the distance from the origin.

Method of Superposition

It is defined as, "the combination of basic velocity potential and stream functions to yield a streamline that corresponds to a particular body shape of interest which describe flow around the body".

• Potential flows are governed by Laplace's equation, which is a linear partial differential equation.

1. Source & Sink Pair

2. Doublet

It is formed by combining a source and sink in a special way infinitely close to each other.

• Uniform flow and source  →  flow past a half body.
• Uniform flow and a Source and Sink pair  →  flow past a Rankine Oval.

3. Uniform Flow and Doublet

A doublet combine with a uniform flow in positive X-direction can be used to represent a flow around a stationary circular cylinder.

Pressure Distribution & Resultant Force on the Cylinder Surface

Pressure distribution on the cylinder surface is obtained from the Bernoulli equation by neglecting elevation differences.

• These results indicate that both drag and lift are predicted by Potential Theory.
• For a fixed cylinder in a uniform flow   →   Fx and Fy = 0.
• From experience, there is a significant drag developed on a cylinder when it is placed in a moving fluid.
• This discrepancy is known as D' Almbert's Paradox.

4. Uniform Flow, Doublet and Free Vortex

An additional potential flow can be developed by adding free vortex to the flow around a cylinder  →  Flow past a rotating cylinder.

• This flow field type can be approximately created by placing a rotating cylinder in a uniform stream.
• Because of viscosity, the fluid in contact with the rotating cylinder would rotate with the same velocity as the cylinder.

Pressure Distribution & Resultant Force on the Cylinder Surface

Pressure distribution on the cylinder surface is obtained from the Bernoulli equation by neglecting elevation differences.

• For the rotating cylinder, no force in the direction of the uniform flow is developed  →   FX = 0.
• Lift is developed equal to the product of fluid density, upstream velocity and circulation  →  Lift Force.

References:

• Material from Class Lectures + Book named Fundamentals of Fluid Mechanics by Munson, Young & Okiishi's (8th Edition) + my knowledge. 
• Pics and GIF from Google Images.  
• Videos from YouTube (Engineering Sights).