Pratik Vadher - Fluid Mechanics

A short guide to Velocity Potential & Potential Flow in Fluid Mechanics

1. Shantilal Shah Engineering College Production Engineering Sem.- 4 th. Group- 22

2. Velocity Potential & Potential Flow, Relation between Stream function & Velocity Potential by: P ratik Vadher 130430125113 Guided by : Prof . Vinay A Parikh Prof. Prashant V Sartanpara Prof. Nital P Nirmal

3. DIFFERENCES BETWEEN f and y Flow field variables are found by: Differentiating f in the same direction as velocities Differentiating y in direction normal to velocities Potential function f applies for irrotational flow only Stream function y applies for rotational or irrotational flows 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 )] Stream function y applies for 2D y ( x,y ) or y (r, q ) flows only 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

4. 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

5. 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 U r = Velocity component in radial direction & U q = Velocity component in tangential direction

6. The continuity equation for an incompressible steady flow is Substituting the values of u, v & w from above equation, we get

7. 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..

8. Substituting the values of u, v and w from equation in the above rotational components, we get

9. 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 : If velocity potential (f) exits, the flow should be irrotational . If velocity potential (f) satisfies the Laplace equ . It represents the possible steady incompressible irrotational flow.

10. Stream Function It 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

11. The velocity components in cylindrical polar co-ordinates in terms of stream function are given as Where U r = radial velocity and U q = tangential velocity. The continuity equation for two dimensional flow is

12. 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 W z is Given by

13. Substituting the values of u and v from equation in the above rotational component, we get For irrotational flow, W z = 0. Hence above equation becomes as Which is Laplace equation for y.

14. The Properties of Stream Function (y) are : If stream function ( y) exists, it is possible case of fluid flow which may be rotational or irrotational . If stream function ( y) satisfies the L aplace equation, it is a possible case of an irrotational flow.

15. Relation between Stream Function & Velocity Potential Functions We have, From stream function equation we have,

16. Thus, we have Hence