§3.6 Steady plane potential flows

3.6.1 Velocity potential function and Laplace equation

In plane potential flows, ux, uy are equation of x and y, and ωz=0 or

So

which is

whereφis velocity potential function.Definition:irrotational flow.

The characteristic of velocity potential function are:

(1)Three components of velocity equal to the partial derivative of velocity potential φat respective coordinates.Or the gradient of velocity potentialφequal to the velocity.

(2)Irrotational flow existφ, and velocity potentialφis irrotational flow.

Differential continuity equation for incompressible flow:

This formula is the Laplace equation.

3.6.2 Stream function

1．Stream function and laplace equation

Because the flow is constant plane flow, and they are only a function of x and y. Differential continuity equation for incompressible flow:

So

which is

whereψis velocity potential function.Definition:impressible flow.

The stream function is applied to the plane potential flow to obtain laplace equation.

Plane potential flow:

then

or

The applicable conditions of formula (3-42)are incompressible fluid, constant flow, plane flow, and potential flow.

2．Physical meaning of stream function

(1)The lines of constant stream functionψ(x, y)=C are streamlines.

Solution:

ψ(x, y)=C

dψ=-uydx+uxdy=0

Obtain plane stream function:.

(2)The discharge per unit width through the cross section AB is

In incompressible plane flow, the difference in the stream functions between two different streamlines is equal to the discharge per unit width between the two streamlines.

3．Inference-the relationship between stream function and potential function

The formula is Cauchy-Riemann.φ, ψ satisfy Laplace equation and Cauchy-Riemann.

For example, there are two flow.(a)ux=1, uy=2. (b)ux=4 x, uy=-4 y.

(1)Whether the flow of (a)is stream function?

(2)Whether the flow of (b)is potential function?

Solution:

So,the flow is irrotational.

3.6.3 Flow net

1．The nature of flow nets

(1)Other streaming line of plane irrotational flows are perpendicularly intersecting with equipotential line.

Prove:equipotential line is stream line, dψ=ux dy-uy dx=0.

Slope of flow line at any point is:

Slope of potential line at same point is

Because m1m2=-1, so flow lines are perpendicularly intersecting with potential line.

(2)The ratio of each mesh side length in flow net is equal to , mesh of flow nets are curve square.

2．The draw of flow net

There are two ways to draw flow nets.One is Graphic, another is electrical analogy method.

The principles of Graphic:

(1)The equipotential lines are orthogonal with boundary.

(2)If flow rate at the free surface perpendicular to a surface equal is zero, the free surface is stream line.

(3)Draw stream lines and potential line based on the proportion of pre-selected grid lines.

3．The application of flow net

The flow net is the only solution in border at super position.So the flow net can be applicated in all similar flow field.

The theory of flow net has been widely used over potential flow of ideal fluid.

3.6.4 The superposition principle of potential flow

If the Potential function satisfy the Laplace equation, results of their superposition also satisfy the Laplace equation.For example, ∇2φ1=0 and ∇2φ2=0, results of their superposition areφ=φ1+φ2.