Stream Function Analysis

Let us consider a two-dimensional incompressible flow parallel to the x - y plane in a rectangular cartesian coordinate system. The flow field in this case is defined by

u = u(x, y, t)

v = v(x, y, t)

w = 0             

The equation of continuity is
 


If a function ψ(x, y, t) is defined in the manner



so that it automatically satisfies the equation of continuity (Eq. (1)), then the function is known as stream function.

Note that for a steady flow, ψ is a function of two variables x and y only.

Constancy of ψ on a Streamline

Since ψ is a point function, it has a value at every point in the flow field. Thus a change in the stream function ψ can be written as

     
The equation of a streamline is given by

     
It follows that dψ = 0 on a streamline. This implies the value of ψ is constant along a streamline. Therefore, the equation of a streamline can be expressed in terms of stream function as

ψ(x, y) = constant         (3)

Once the function ψ is known, streamline can be drawn by joining the same values of ψ in the flow field.
 
Stream function for an irrotational flow

In case of a two-dimensional irrotational flow

     
Conclusion drawn: For an irrotational flow, stream function satisfies the Laplace’s equation
 
Physical Significance of Stream Funtion ψ

Figure 1 illustrates a two dimensional flow.
 


Let A be a fixed point, whereas P be any point in the plane of the flow. The points A and P are joined by the arbitrary lines ABP and ACP. For an incompressible steady flow, the volume flow rate across ABP into the space ABPCA (considering a unit width in a direction perpendicular to the plane of the flow) must be equal to that across ACP. A number of different paths connecting A and P (ADP, AEP...) may be imagined but the volume flow rate across all the paths would be the same.
 
This implies that the rate of flow across any curve between A and P depends only on the end points A and P.

Since A is fixed, the rate of flow across ABP, ACP, ADP, AEP (any path connecting A and P) is a function only of the position P. This function is known as the stream function ψ.

The value of ψ at P represents the volume flow rate across any line joining P to A.

The value of ψ at A is made arbitrarily zero. If a point P’ is considered (Fig. 1b), PP’ being along a streamline, then the rate of flow across the curve joining A to P’ must be the same as across AP, since, by the definition of a streamline, there is no flow across PP'

The value of ψ thus remains same at P’ and P. Since P’ was taken as any point on the streamline through P, it follows that ψ is constant along a streamline. Thus the flow may be represented by a series of streamlines at equal increments of ψ.

In fig (1c) moving from A to B net flow going past the curve AB is

     
The stream function, in a polar coordinate system is defined as

     
The expressions for V_r and V_θ in terms of the stream function automatically satisfy the equation of continuity given by

     
Stream Function in Three Dimensional Flow

In case of a three dimensional flow, it is not possible to draw a streamline with a single stream function.

An axially symmetric three dimensional flow is similar to the two-dimensional case in a sense that the flow field is the same in every plane containing the axis of symmetry.

The equation of continuity in the cylindrical polar coordinate system for an incompressible flow is given by the following equation

     
For an axially symmetric flow (the axis r = 0 being the axis of symmetry),  the term  ,and simplified equation is satisfied by  functions defined as



The function ψ , defined by the Eq.(4) in case of a three dimensional flow with an axial symmetry, is called the stokes stream function.

Stream Function in Compressible Flow

For compressible flow, stream function is related to mass flow rate instead of volume flow rate because of the extra density term in the continuity equation (unlike incompressible flow)

The continuity equation for a steady two-dimensional compressible flow is given by

     
Hence a stream function ψ is defined which will satisfy the above equation of continuity as
     

 
ρ₀ is used  to retain the unit of ψ same as that in the case of an incompressible flow. Physically, the difference in stream function between any two streamlines multiplied by the reference density ρ₀ will give the mass flow rate through the passage of unit width formed by the streamlines

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