Speed ​​potential

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The velocity potential is introduced for eddy-free , two- and three-dimensional flows of fluid dynamics . With it, the calculations are simplified and you gain a deeper mathematical-physical understanding. The velocity potential of fluid dynamics corresponds mathematically to the electrostatic or gravitational potential .

This article deals with the two-dimensional case - the three- dimensional case is presented in the article Potential flow.

Solving the equation gives the equipotential lines of the flow field.

In addition, the current function is introduced , the clear meaning of which is that the solutions of the equation represent the streamlines of the velocity potential.

The complex speed potential is formed from the speed potential and the current function .


For an eddy-free two-dimensional flow field , the rotation is equal to 0:

Similar to the case of the electrostatic potential, the velocity potential is now introduced . The gradient of this potential is precisely the flow field:

Because of this , the flow field is automatically free of eddies.

Furthermore, the continuity equation also applies to the velocity field in the case of an incompressible flow :

If you insert the definition of the velocity potential into it, you can see that the Laplace equation (as a special case of the Poisson equation ) satisfies:

The current function

The speed potential was introduced in such a way that freedom from eddies is automatically fulfilled. However, the fulfillment of the continuity equation or the Laplace equation had to be explicitly required.

Now we introduce the current function , which is defined by:

From this definition one can see that the continuity equation is automatically fulfilled:

However, the freedom of rotation must be explicitly required:

The current function also fulfills the Laplace equation in eddy-free flows.

Complex speed potential

With the definitions of speed potential and current function, we get:

This is exactly of the form of the Cauchy-Riemann differential equations for a holomorphic function , with a real part and an imaginary part . Thus one introduces the complex speed potential:

The complex velocity potential thus also fulfills the Laplace equation: