Question

Q7. The flow field is defined by the complex potential function, f(z)- Uz+mlnz) a) Define the stream and potential functions, b) Define the types of the potential flows superpositioned c) Using the values, U-8 m/s and m-3 m'/s determine the pressure distribution and obtain the location op the stagnation point or points in the flow field.

Answer 1

a) The idea of introducing stream function works only if the continuity equation is reduced to two terms. There are 4-terms in the continuity equation that one can get by expanding the vector equation

dV ρ- dt Momentum : ρ--VD+Vr

61 CZ

For a steady, incompressible, plane, two-dimensional flow, this equation reduces to

Cu v -+-=0 схсу

Here, the striking idea of stream function works that will eliminate two velocity components u and vinto a single variable So, the stream function relates to the velocity components in such a way that continuity equation is satisfied.

ay oy cy テ @w": cu, or,V =

Velocity Potential

An irrotational flow is defined as the flow where the vorticity is zero at every point. It gives rise to a scalar function which is similar and complementary to the stream function . Let us consider the equations of irrortional flow and scalar function . In an irrotational flow, there is no vorticity

Now, take the vector identity of the scalar function

i.e. a vector with zero curl must be the gradient of a scalar function or, curl of the gradient of a scalar function is identically zero. Comparing, Eqs.

b)

Velocity components Stream Function Velocity Potential Description of Flow Field u», cos θ y-,(ycose-xsin θ)| | φ», (xcos θ + y sin θ)| Uniform flow at an angle θ V-V, sin θ with x-axis Source (A> 0) or Sink(A <0) Doublet -In 2π 2TTY 2π cos θ ksin d Kcos θ Ksin Free Vortex Counter-clockwise(T < 0) Clockwise(T > 0) 2π 2π Tr