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Velocity Potential & Potential Flow, Relation between Stream
function & Velocity Potential
by: Pratik Vadher 130430125113
Guided by : Prof. Vinay A Parikh
Prof. Prashant V Sartanpara Prof. Nital P Nirmal
DIFFERENCES BETWEEN f and y
1. Flow field variables are found by: Differentiating f in the same direction as velocities Differentiating y in direction normal to velocities
2. Potential function f applies for irrotational flow only3. Stream function y applies for rotational or irrotational flows
4. Potential function f applies for 2D flows [f(x,y) or f(r,q)] and 3D flows [f(x,y,z) or f(r,q, f)]
5. Stream function y applies for 2D y(x,y) or y(r, q) flows only
6. Stream lines (y =constant) and equipotential lines (f =constant) are mutually perpendicular Slope of a line with y =constant is the negative reciprocal of the
slope of a line with f =constant
Velocity Potential Function
It is defined as a scalar function of space & time such that its negative derivative w.r.t any direction gives the fluid velocity in that direction. It is defined by f (Phi). Mathematically, the velocity, potential is defined as f = f (x,y,z) for steady flow such that
Where u, v and w are the components of velocity in x, y & z directions respectively.
The velocity components in cylindrical polar co-ordinates in terms of velocity potential function are given by
Where Ur = Velocity component in radial direction& Uq = Velocity component in tangential direction
The continuity equation for an incompressible steady flow is
Substituting the values of u, v & w from above equation, we get
For two-dimensional case, above equation reduces to
If any value of (f Phi) that satisfies the Laplace equation, will correspond to some case of fluid flow.Properties of the Potential function. The rotational components are given by
Conti..
If f is a continuous function,
Then equ.
Therefore
When rotational components are zero, the flow is called irrotational. Hence the properties of the potential function are :
1. If velocity potential ( ) f exits, the flow should be irrotational.
2. If velocity potential ( ) f satisfies the Laplace equ. It represents the
possible steady incompressible irrotational flow.
Stream FunctionIt is defined as the scalar function of space &
time, such that it’s partial derivative w.r.t any direction gives the velocity component at right angles to that direction. It is denoted as y (Psi) and defined only for two-dimensional flow. Mathematically, for steady flow it is defined as y = f(x,y) such that
And
The velocity components in cylindrical polar co-ordinates in terms of stream function are given as
Where Ur = radial velocity and Uq= tangential velocity.
The continuity equation for two dimensional flow is
Substituting the values of u and v from above equation, we get
Hence existence of y means a possible case of fluid flow. The flow may be rotational or irrotational.The rotational component Wz is
Given by
Substituting the values of u and v from equation in the above rotational component, we get
For irrotational flow, Wz = 0. Hence above equation becomes as
Which is Laplace equation for .y
The Properties of Stream Function ( )y are :
1. If stream function ( ) y exists, it is possible case of fluid flow which may be rotational or irrotational.
2. If stream function ( ) y satisfies the Laplace equation, it is a possible case of an irrotational flow.
Relation between Stream Function & Velocity Potential Functions
We have,
From stream function equation we have,