Potential flow

Potential flow is called as the vector field of the velocity is mathematically a nature such that it has a potential of the flow of a fluid ( liquid or gas). Such a flow is present in a homogeneous fluid, if it is irrotational ( irrotational ) and no viscous forces (friction forces) occur or are negligibly small. Each starting from rest flow of a homogeneous, inviscid fluid has such potential. Potential flow is a special case of an ideal fluid hydrodynamics described by the Euler 's equations of motion.

Force potential ( accelerating potential )

The mathematical term potential has its origin in the physics of conservative force fields. There, the potential is the ability of a field to do work per unit mass or charge (potential energy per mass or charge ). The potential therefore describes there the effect of the force field on masses or charges, regardless of the size of these masses or charges.

Hereinafter, the respective potential is ( are different ), the Greek letter (phi) respectively.

The force field = ( Fx, Fy, Fz), a three-dimensional vector field is obtained by the partial derivative of the potential on the place (position), so the spatial variation of the potential:

Where the operator grad ( gradient) or ( the nabla operator ) in Cartesian coordinates

Means. here is the position vector indicating the position in space (position). The derivative with respect to the formation of the gradient of a discharge line symbolizes all components (coordinates) of. In the one-dimensional case, the gradient is the ordinary derivative with respect to the path towards this dimension.

Velocity potential

Different from the potential of a force field, the velocity potential or streaming potential is the ability (potential energy) of a field to do work. The potential of the velocity field describes a special inner order of this field, the (formal) is mathematically the structure of the conservative force fields, but physically represents something else, namely Rotationslosigkeit ( no vorticity ):

Since the velocity potential does not include potential energy, the sign is not fixed and is usually defined as positive:

In the potential flow the Bernoulli equation applies not only along a streamline, but also " transverse" to the adjacent stream lines, ie for the whole field of potential flow. Euler's equations of motion are the motion equations of hydrodynamics of the ideal fluids, which includes the potential flow. These equations of motion are also seen for homogeneous flows, inviscid fluids with rotation. However, if the resistance has to be considered, such as in each boundary layer ( hydrodynamic boundary layer) and in the center of a vortex, however, to be expected with the Navier -Stokes equations and the flow is not a potential flow.

The contrast to the potential flow is not only the vortex flow with vorticity (there are also circling potential flows without vorticity ), but above all do not frictionless flow in which occur with shear movements viscous forces.

Potential flow with circulation ( potential vortex )

The potential vortex or "free vortex " is a circular, yet irrotational potential flow, ie in which the rotation of the velocity field

Is and therefore there is a real potential flow which ( such as the air space in a hall with center column ) has a circulation in a topologically doubly connected domain.

Free vortex in real fluids, these potential flow approximation with the exception of a core topic in its center, where viscous forces lead to a quasi- rigid rotation and the velocity field has vorticity, so is not a potential flow. In the free vortex all fluid particles move in concentric circular orbits with velocities v, which (except in the core area ) the inverse square law with constant c and axis distance r

Correspond. Thus, a completely different velocity and pressure distribution than the quasi- rigid vortex is given that has no potential and its speed is proportional to the axial distance r.

A special potential vortex can be observed at a free water surface when the pressure in the center is so small that the surface noticeably einsenkt and a swirl cup ( whirlpool ) forms. Where the funnel unlimited far in depth, prevails throughout the liquid potential flow field, but not in the air-filled core.