Stream Functions, Incompressibility and Fluid accelerations

This week we are looking further into flow visualisation and looking at Stream Functions.

We are now concentrating on incompressible 2D flows (which are flows in which the velocity is in the form (u,v,0). The continuity equation in these flows is:

1

Suppose that we can find a scalar function si such that:

2

This scalar function is called the stream function for the flow. Once we know the velocity field, we can find the stream function and also the other way round!

If we say that the streamline is parametrised by writing x=x(s) and y=y(s) for some parameter s, then dx/ds is the tangent vector to the streamline and is therefore parallel to the fluid velocity at x. Therefore the vector cross product between u and dx/ds is zero. I.e:

3

We can simplify this down to:

4

Therefore this becomes dsi/dsk=0. Therefore dsi/ds=0 along a streamline and hence si is constant along a streamline. We can then solve the equation of the stream function, and this gives us another method for producing streamlines!

We have also seen the Principle of Superposition which stated if we add two different velocity fields together it has the effect of combining the two relative stream functions together.

Please find below a few examples from example sheet 1:

IMG_0027IMG_0028

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