In today’s lesson, we are going to be discussing the following topics:
1. Stream Function
2. Example using Stream Function in Cartesian Coordinates
3. Steady Plane Compressible Flow
4. Incompressible Axisymmetric Flow
5. Example using Stream Function in Polar Coordinates
video coming soon
Stream Function
The stream function is denoted by the following greek letter psi (Ψ). It is usually used with incompressible flow in the xy (2D) plane. It helps us to simplify horizontal steady fluid flow with the single variable Ψ(x,y) (it is a function of both x and y) since it is 2D. Let’s write out the conservation of mass equation in 2D:
Now let us define a function Ψ such that:
If we want to rewrite the components of velocity, they can be rewritten with Ψ as such. We have now taken out u and v, and the velocity equation is only with respect to one variable Ψ (the derivatives are taken with respect to x and y in the denomiator, but those do not introduce new variables because it’s just telling us how to take the derivative):
Now if we substitute these Ψ equations for u and v into the Mass Conservation Equation above, we get the following. This shows that the terms end up cancelling out, and the Mass Conservation Equation is satisfied!:
There is another equation that can be found through stream functions. By taking the change in the stream function (Ψ2 – Ψ1), we find the volume flow (aka flow rate) Q!
We can use these stream function definitions in further examples below.
Example using Stream Function in Cartesian Coordinates
(a) Define any assumptions. Here, we can assume incompressible and 2D flow (since the z-component, w, is 0).
(b) Make sure that this velocity field satisfies the continuity equation. This ensures that a stream function even exists for this velocity field:
(c) So we can see that the continuity equation is satisfied and a stream function exists. Now let’s use the definition of Ψ to find the stream function, substituting in the equation we got for u. You can choose to do this with either u or v, and you should get the same answer. You will need to integrate both sides to get rid of the derivative, like I have done below to solve for the stream function Ψ. Remember you need to add a +C at the end of the integral because we didn’t have specific bounds to integrate between (this is a general rule from integral calculus):
(d) Now let’s plot and interpret what this means. Let’s say C = 0, to make it easier. To graph this, we need to define values for both Ψ and a at certain points. We can set Ψ = a, Ψ = -a, Ψ = -2a, etc to eliminate extra variables. Plugging some of these Ψ and a values into an online calculator, this is what the stream function looks like on an x-y coordinate system:
We should notice that the origin of the graph (the origin of the flow) is a stagnation point (velocity is zero).
Note that when we have lines of constant Ψ, for example where Ψ = -2a or Ψ = a along that entire line from the graph above, this is called a streamline. These are lines of constant Ψ. These are also known as lines in the flow field that are tangent to the direction of flow. Since they are tangent to the flow (to the velocity vectors), there is no flow across a streamline (dΨ = 0). Plotting the stream function gives us the streamlines!
Also notice the arrows on the graph from the textbook. I have highlighted the major lines in green. All the flow falls in between these lines, so based on where it falls, the arrows are placed accordingly. I did not have arrows to show the direction of flow in my graph above, but you would need to add them in. So this would be the final graph of the stream function (showing the streamlines).
However, the actual method of determining the direction of flow is to look at the upper and lower streamlines around the one you are trying to determine the arrow for. Take a look at the following image, where U and L stand for upper and lower, respectively:
Steady Plane Compressible Flow
We usually deal with incompressible flow when using the stream function. If we have compressible flow, we need to change up the stream function definition to include density.
The change in the stream function is now mass flow rate, not volume flow rate:
Incompressible Axisymmetric Flow
We usually deal with 2D flow when using the stream function. If we have 3D flow, and the circumference of the pipe which the fluid is going through is constant (no circumferential variation), this is called axisymmetric flow. The only equation that changes is the stream function, which then becomes the following:
Example using Stream Function in Polar Coordinates
(a) In this case, r and theta are the x- and y- coordinates. This means that we need to select values for U, R and Ψ. Let us rearrange the equation to put Ψ, R and U on the same side, since they are all constant. We can factor one R out from the right side, to have as many constants on the left as possible:
(b) Now let us choose the left side Ψ/UR = 1, and choose values for r/R. Then we can find values of theta (which will be along the x-axis). I will choose values of 1, 2, 3, etc for r/R, then solve for theta:
(c) Plotting these in a graphing calculator gives the following – you can then connect the points to get the streamline for Ψ/UR = 1.
(d) Then repeat the same calculations but instead with Ψ/UR = 2 on the left side of the equation and choosing values of 1, 2, 3, 4, etc for r/R, and plot, then connect the dots. You can see the pink curves that are shown for different values of Ψ/UR from the textbook graph:
