Strömungsfunktionsdarstellung

Strömungsfunktion, often translated as stream function, is a concept used in fluid dynamics to simplify the analysis of two-dimensional, incompressible, and irrotational fluid flow. It is a scalar function whose partial derivatives with respect to spatial coordinates yield the velocity components of the fluid. For a two-dimensional flow in the xy-plane, the stream function, denoted by psi (ψ), is defined such that the velocity component in the x-direction (u) is equal to the partial derivative of psi with respect to y (∂ψ/∂y), and the velocity component in the y-direction (v) is equal to the negative of the partial derivative of psi with respect to x (-∂ψ/∂x).

The primary advantage of using the stream function is that it automatically satisfies the continuity equation for incompressible flow. The continuity equation, which states that mass is conserved, requires that the divergence of the velocity field is zero. For a 2D incompressible flow, this is ∂u/∂x + ∂v/∂y = 0. Substituting the stream function definition, we get ∂(∂ψ/∂y)/∂x + ∂(-∂ψ/∂x)/∂y = ∂²ψ/∂x∂y - ∂²ψ/∂y∂x, which is always zero due to the equality of mixed partial derivatives. This means any function ψ will automatically describe a physically possible flow in terms of mass conservation.

Lines of constant stream function are called streamlines. These streamlines represent the instantaneous paths of fluid particles. The difference in stream function values between two streamlines is proportional to the volume flow rate between them. The concept of stream function is particularly useful in potential flow theory and in analyzing flows around airfoils and other aerodynamic shapes.