Speed ​​potential

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 . ${\displaystyle \phi }$

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. ${\displaystyle \phi (x,y)={\text{const.}}}$

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. ${\displaystyle \psi }$ ${\displaystyle \psi (x,y)={\text{const.}}}$

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

Basics

For an eddy-free two-dimensional flow field , the rotation is equal to 0: ${\displaystyle {\vec {u}}(x,y)}$

${\displaystyle {\vec {\nabla }}\times {\vec {u}}(x,y)=0}$

Similar to the case of the electrostatic potential, the velocity potential is now introduced . The gradient of this potential is precisely the flow field: ${\displaystyle \phi (x,y)}$

${\displaystyle {\vec {u}}(x,y)={\vec {\nabla }}\phi (x,y)=\left({\frac {\partial \phi }{\partial x}},{\frac {\partial \phi }{\partial y}}\right)}$

Because of this , the flow field is automatically free of eddies. ${\displaystyle {\vec {\nabla }}\times {\vec {\nabla }}\phi (x,y)=0}$

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

${\displaystyle {\vec {\nabla }}\cdot {\vec {u}}(x,y)=0}$

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: ${\displaystyle \phi (x,y)}$

${\displaystyle {\vec {\nabla }}\cdot {\vec {u}}(x,y)={\vec {\nabla }}\cdot {\vec {\nabla }}\phi (x,y)=\Delta \phi (x,y)=0}$

The current function

→ Main article : Electricity 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. ${\displaystyle \phi (x,y)}$

Now we introduce the current function , which is defined by: ${\displaystyle \psi (x,y)}$

${\displaystyle {\vec {u}}=\left({\frac {\partial \psi }{\partial y}},-{\frac {\partial \psi }{\partial x}}\right)}$

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

${\displaystyle {\vec {\nabla }}\cdot {\vec {u}}={\frac {\partial ^{2}\psi }{\partial x\cdot \partial y}}-{\frac {\partial ^{2}\psi }{\partial y\cdot \partial x}}=0}$

However, the freedom of rotation must be explicitly required:

${\displaystyle {\frac {\partial u_{y}}{\partial x}}-{\frac {\partial u_{x}}{\partial y}}={\frac {\partial ^{2}\psi }{\partial x^{2}}}+{\frac {\partial ^{2}\psi }{\partial y^{2}}}=0}$

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: ${\displaystyle \phi }$ ${\displaystyle \psi }$

${\displaystyle u_{x}={\frac {\partial \phi }{\partial x}}={\frac {\partial \psi }{\partial y}}\quad \wedge \quad u_{y}={\frac {\partial \phi }{\partial y}}=-{\frac {\partial \psi }{\partial x}}}$

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: ${\displaystyle \phi }$ ${\displaystyle \psi }$ ${\displaystyle w(z)}$

${\displaystyle w(z)=\phi (z)+i\cdot \psi (z)\quad {\textrm {mit}}\quad z=x+i\cdot y}$

The complex velocity potential thus also fulfills the Laplace equation:

${\displaystyle \Delta w(z)=\Delta \phi (z)+i\cdot \Delta \psi (z)=0}$

literature

• Ralf Greve: Continuum Mechanics . Springer, 2003, ISBN 3-540-00760-1 .
• M. Bestehorn: hydrodynamics and structure formation . Springer, 2006, ISBN 978-3-540-33796-6 .