Relationship between $\phi$ and $\psi$

We notice that velocity potential $\phi$ and stream function $\psi$ are connected with velocity components. It is necessary to bring out the similarities and differences between them.

1

Stream function is defined in order that it satisfies the continuity equation readily (eqn. 4.20 see Stream function). We do not know yet if it satisfies the irrotationality condition. So we test out below. Recall that the velocity components are given by

$$u = {\partial \psi \over {\partial y}} ,    v = -{\partial \psi \over {\partial x}}$$

Substituting these in the irrotationality condition, we have

$${\partial v\over {\partial x}} - {\partial u \over {\partial y}}... ...artial^2 \psi} \over {\partial x^2}} - {{\partial^2 \psi} \over {\partial y^2}}$$ (4.61)

Which leads to the condition that $\nabla^2 \psi = 0$ for irrotationality.

Thus we see that the velocity potential $\phi$ automatically complies with the irrtotatioanlity condition, but to satisfy the continuity equation it has to obey that $\nabla^2 \phi = 0$. On the other hand the stream function readily satisfies the continuity condition, but to meet with the irrotationality condition it has to obey $\nabla^2 \psi = 0$.

Thus we see that the streamlines too follow the Laplace Equation. So it is possible to solve for a potential flow in terms of stream function.

 

Property	$\psi$	$\phi$
Continuity Equation	Automatically Satisfied	satisfied if $\nabla^2 \phi$=0
Irrotationality Condition	satisfied if $\nabla^2 \psi$=0	Automatically Satisfied

                                                                    Table 4.1: Properties of stream function and velocity potential

2

Streamlines and equipotential lines are orthogonal to each other. We have seen that the velocity components of the flow are given in terms of velocity potential and stream function by the equations,

$$u = {{\partial \phi} \over {\partial x}} = {{\partial \psi} \over {\partial y}}$$

$$v = {{\partial \phi} \over {\partial y}} = -{{\partial \psi} \over {\partial x}}$$ (4.62)

Those familiar with Complex Variables theory will recognise that these are the Cauchy-Riemann equations and that $\phi = C$ and $\psi = D$ are orthogonal and that both $\phi$ and $\psi$ obey Laplace Equation. However, we will prove the orthogonality condition by other means.

Figure 4.13: Orthogonality of Stream lines and Equi-potential lines

 

Since $\phi = C$, it follows that

$$d\phi = {{\partial \phi} \over {\partial x}}dx + {{\partial \phi} \over {\partial y}} dy$$

$$=0$$

$$=u dx + v dy$$ (4.63)

The gradient of the equipotential line is hence given by

$$\left( {dy} \over {dx} \right)_{\phi = C} = -{u \over v}$$ (4.64)

On the other hand the gradient of a stream line is given by

$$\left( {dy} \over {dx} \right)_{\psi = D} = {v \over u}$$ (4.65)

Thus we find that

$$\left( {dy} \over {dx} \right)_{\phi = C}\left( {dy} \over {dx} \right)_{\psi = D} = -1$$ (4.66)

showing that equipotential lines and streamlines are orthogonal to each other. This enables one to calculate the stream function when the velocity potential is given and vice versa.

Fig. 4.14 shows the flow through a bend where the streamlines and the equipotential lines have been plotted. The two form an orthogonal network.

Figure 4.14: Stream lines and Equi-potential lines for flow through a bend

(c) Aerospace, Mechanical & Mechatronic Engg. 2005
University of Sydney