Potential flow
Such a description typically arises in the limit of vanishing viscosity, i.e., for an inviscid fluid and with no vorticity present in the flow.
As a result, a potential flow is characterized by an irrotational velocity field, which is a valid approximation for several applications.
The irrotationality of a potential flow is due to the curl of the gradient of a scalar always being equal to zero.
The velocity potential is not uniquely defined since one can add to it an arbitrary function of time, say
The non-uniqueness is usually removed by suitably selecting appropriate initial or boundary conditions satisfied by
In case of an incompressible flow — for instance of a liquid, or a gas at low Mach numbers; but not for sound waves — the velocity v has zero divergence:[3]
As evident, in the incompressible case, the velocity field is determined completely from its kinematics: the assumptions of irrotationality and zero divergence of flow.
Dynamics in connection with the momentum equations, only have to be applied afterwards, if one is interested in computing pressure field: for instance for flow around airfoils through the use of Bernoulli's principle.
If the potential flow satisfies the necessary conditions, then it is the required solution of the incompressible Navier–Stokes equations.
(stream function), incompressible potential flow reduces to a very simple system that is analyzed using complex analysis (see below).
where the last equation follows from the fact that entropy is constant for a fluid particle and that square of the sound speed is
Since in front of the leading shock wave, we have a potential flow, Bernoulli's equation shows that
Shock waves at the pointed leading edge of two-dimensional wedge or three-dimensional cone (Taylor–Maccoll flow) has constant intensity.
characterizes the small departure from the uniform flow and satisfies the linearized version of the full equation.
Note that also the oscillatory parts of the pressure p and density ρ each individually satisfy the wave equation, in this approximation.
Richard Feynman considered potential flow to be so unphysical that the only fluid to obey the assumptions was "dry water" (quoting John von Neumann).
[9] Incompressible potential flow also makes a number of invalid predictions, such as d'Alembert's paradox, which states that the drag on any object moving through an infinite fluid otherwise at rest is zero.
[1] Nevertheless, understanding potential flow is important in many branches of fluid mechanics.
These solutions can be superposed to create more complex flows satisfying a variety of boundary conditions.
Potential flow finds many applications in fields such as aircraft design.
[dubious – discuss] Potential flow in two dimensions is simple to analyze using conformal mapping, by the use of transformations of the complex plane.
However, use of complex numbers is not required, as for example in the classical analysis of fluid flow past a cylinder.
While x, y, φ and ψ are all real valued, it is convenient to define the complex quantities
So φ can be identified as the velocity potential and ψ is called the stream function.
Note that multi-valued functions such as the natural logarithm may be used, but attention must be confined to a single Riemann surface.
In case the following power-law conformal map is applied, from z = x + iy to w = φ + iψ:[12]
With this interpretation, the flow is that of a vertically directed jet impinging on a horizontal flat plate.
This flow pattern is usually referred to as a doublet, or dipole, and can be interpreted as the combination of a source-sink pair of infinite strength kept an infinitesimally small distance apart.
in fact is the volume flux per unit length across a surface enclosing the source or sink.
in fact is the volume flux across a closed surface enclosing the source or sink.
