Potential flow

The term potential flow designates (apart from isolated singularities) a vortex-free and source-free flow. Its velocity field ^➔v fulfils the condition of being irrotational rot(^➔v) = ^➔o and can be derived in accordance with ^➔v = grad(Φ) from a velocity potential (Φ).

In the case of incompressible media, the velocity potential Φ satisfies the potential equation ΔΦ = 0, where, Δ is the Laplace operator. The potential equation can verify the condition of tangential flow direction at solid walls, but not no-slip conditions. In a potential flow, a closed body surrounded by flow on all sides does not experience drag, only lift.

Simple examples of potential flow are parallel, source or sink flow and the potential vortex. In a source or sink flow, the radial component (v_r) (which is the only flow component present) varies inversely with the radial distance (r) from the centre; in a potential vortex, the same applies for the tangential component (v_u) of the velocity (again, the only component present). Both flows feature a singularity at the centre where r = 0, as here the velocity becomes infinite.

Strictly speaking, potential flows can only be flows of frictionless fluids. However, flows of real fluids subject to friction can be treated as approximating potential flows at sufficiently large distances from solid walls, while friction forces need only be taken into account in a thin boundary layer close to the wall.

If the body contour is increased in the calculation by the displacement thickness of the boundary layer and the Reynolds number is high (i.e. for thin boundary layers), the flow around bodies – in particular the flow and pressure distribution in a vane cascade – can be calculated reasonably accurately as a potential flow.

Important methods of calculating a potential flow (e.g. aerofoil theory) include the conformal representation, or conformal mapping, function (by which a flow field is transformed from one complex plane into another preserving angles) or the singularity method (by which the flow field is represented by superimposing separate or continuous singularities such as sources, sinks, vortices on the body surrounded by the flow).

Numerical flow simulation methods also allow a potential flow to be directly calculated on the basis of the differential equations describing the flow field (Euler's equations). In contrast to Navier-Stokes equations, these equations do not include the friction term. When considering the potential flow in a rotating vane cascade, it must be taken into account that although the absolute flow (^➔v) is irrotational, the rigid body reference system prevents this condition for the relative flow (^➔w), meaning that rot(^➔w) does not equal zero. As a result, a so-called relative vortex rotating in the opposite direction to the rotation of the impeller develops in the vane channel of an impeller through which a frictionless flow passes.

Note: All arrows (^➔) mark vector quantities. For technical reasons, it is not possible to display them correctly above the letters.