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FLUID MECHANICS - THEORY

GREEK "Phi" WARNING

Greek Lower Case Letter
Phi can be written two ways
or   (each Internet Browser does it differently).

Also, sometimes Upper Case Phi, , and Upper Case Psi, , is used for Velocity Potential and Stream Function, respectively.

 

It was shown previously that two-dimensional incompressible, inviscid, and irrotational flow can be described by the velocity potential, φ, or stream function, ψ, using the Laplace's equation:

     

In the following sections, some simple plane potential flows (e.g., uniform flow, source and sink, vortex and doublet) will be introduced. Also, since Laplace's equation is linear, various solutions can be combined to form other solutions. Therefore, some of the real flow problems (e.g., half-body) are obtained by combining these simple plane potential flows using the method of superposition.

     
    Uniform Flow


Uniform Flow
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Uniform flow is the simplest form of potential flow. For flow in a specific direction, the velocity potential is

     φ = U (x cosα + y sinα)

while the stream function is

     ψ = U (y cosα - x sinα)

where α represents the angle between the flow direction and the x-axis (as shown in the figure). Recall that the velocity potential and stream function are related to the component velocity in the 2 dimensional flow field as follows:

Cartesian coordinates:

       and 

Cylindrical coordinates:

       and  

Note that the lines of the constant velocity potential (equipotential lines) are orthogonal to the lines of the stream function (streamlines).

   
  Source and Sink

 

When a fluid flows radially outward from a point source, the velocities are

     vr = m / (2πr)   and   vθ = 0

where m is the volume flow rate from the line source per unit length. The velocity potential and stream function can then be represented as

     

respectively. When m is negative, the flow is inward, and it represents a sink. The volume flow rate per unit depth, m, indicates the strength of the source or sink. Note that as r approaches zero, the radial velocity goes to infinity. Hence, the origin represents a singularity. As shown in the figure, the equipotential lines are given by the concentric circles while the streamlines are the radial lines.

   
    Vortex


Vortex
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Vortex Example: Tornado

 

 

 

A vortex can be obtained by reversing the velocity potential and stream functions for a point source such that

     φ = Kθ   and     ψ = -K ln(r)

where K is a constant indicating the strength of the vortex. Now, the equipotential lines are radial lines while the streamlines are given by the concentric circles. The velocities of a vortex are,

     vr = 0   and   vθ = K/r

The strength of a vortex can be described using the concept of circulation (Γ), which is defined as,

     

where the integral is taken around a closed arbitrary curve C. For irrotational flows (), the circulation becomes

     

However, when the closed curve consists of a singularity point (such as the case of a vortex), the circulation is non-zero,

     

     
    Doublet


Superposition of a Source and a Sink
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Streamlines for a Doublet

 

By combining a source and a sink of equal strength using the method of superposition, the stream function is given by

     ψ = ψsource + ψsink = -(m/2π) (θ1 - θ2)

Through considerable manipulation (i.e., geometric relationships and trigonometric identities), the above equation can be rewritten as

     

For small values of a gives,

     

A doublet is obtained by letting the distance between the source and sink approach zero (i.e., distance "a" tends to zero) which means r/(r2 - a2) -> 1/r. The stream function for a doublet then becomes

     ψ = -Ksinθ/r

where K is a constand equal to ma/π, and is called the strength of the doublet. The streamlines of a typical doublet are shown in the figure. The corresponding velocity potential is

     φ = Kcosθ/r

For simplicity, the details of the velocity potential derivation are not given here. Students are encouraged to go through the above derivation process themselves for practice.

     
    Flow around a Half-Body


Superposition of a Uniform Flow
and a Source
Click to view movie (16k)

 

Flow around a half-body can be obtained by the superposition of a uniform flow with a source. The combined stream function is given by

     ψ = ψuniform flow + ψsource = U r sinθ + (m/2π) θ

and the corresponding velocity potential is

     φ = φuniform flow + φsource = U r cosθ + (m/2π) ln(r)

The velocity components are given by

     

     

     

Flow Past a Oval-type Body
 

It is interesting to note, the streamlines of this combined function can be used to represent a oval-like shape in a flow stream. The stagnation point of the flow can be used to define the half-bodyshape. The location of the stagnation point can be determined by setting vr and vθ equal to zero, yielding

     θ = π   and   rstagnation = b = m/2πU

The streamline that passes through the stagnation point is then obtained as

     ψstagnation = m/2 = πbU

By replacing this streamline with a solid boundary, one can then clearly see that flow around a half-body can indeed be represented by the superposition of a uniform flow with a source. The magnitude of the resultant velocity, V, at any point of the flow field is given by

     

The shape of the top or bottom half-body is found by putting ψstagnation value back into the stream function where θ = π. This gives,

     πbU = Ursinθ = bUθ

or

     r = b( π - θ)/sinθ

     

Combination of Uniform Flow and Point Souce