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In fluid dynamics, potential flow, also know as
irrotational flow (of incompressible
fluids) is steady flow defined by the equations
- (zero rotation = no
viscosity)
- (zero divergence =
volume conservation)
Equivalently,
-
where:
- v is the vector fluid velocity
- Φ is the fluid flow potential, scalar
- "
×" is curl
- "
·" is divergence.
The equations above imply , or
Laplace's equation, holds.
Together with the Navier-Stokes equations and the
Euler equations, these equations can be used to calculate solutions
to many practical flow situations. In two dimensions, potential flow reduces to a very simple system that is analysed using
complex numbers (see potential flow in 2d))
Potential flow does not include all the characteristics of flows that are encountered in the real world. For example,
potential flow excludes turbulence, which is commonly encountered in nature.
Richard Feynman considered potential flow to be so unphysical that the only fluid to
obey the assumptions was "dry water".
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.
More precisely, potential flow cannot account for the behaviour of flows that include a boundary layer.
Nevertheless, understanding potential flow is important in many branches of fluid mechanics. In particular, simple potential
flows such as the free vortex and
the point source possess ready analytical solutions. These flows correspond
closely to real-life flows over the whole of fluid mechanics; in addition, many valuable insights arise when considering the
deviation (often slight) between an observed flow and the corresponding potential flow.
Potential flow finds many applications in fields such as aircraft design. For instance, in computational fluid dynamics, one technique is to
couple a potential flow solution outside the boundary layer to a
solution of the boundary layer equations inside the boundary layer.
Analysis
Potential flow in two dimensions is simple to analyse using complex
numbers, viewed for convenience on the Argand diagram.
The basic idea is to define a holomorphic function f. If we write
- f(x + iy) = φ + iψ
then the Cauchy-Riemann equations show that
-
(it is conventional to regard all symbols as real numbers; and to write z = x +
iy and w = φ + iψ).
The velocity field , specified
by
-
then satisfies the requirements for potential flow:
-
and
-
Lines of constant ψ are known as streamlines and lines of constant φ are known as equipotential lines (see
equipotential surface).
The two sets of curves intersect at right angles, for
-
Examples: general considerations
Any differentiable function may be used for f. The examples that follow use a variety of
elementary functions; special functions may also be used.
Note that multi-valued functions such as the natural logarithm may be used, but attention must be confined to a single
Riemann surface.
Examples: Power laws
If
- w = Azn
then, writing x + iy = reiθ, we
have
- φ = Arncosnθ
and
- ψ = Arnsinnθ
n = 1">
Power law with n = 1
If w = Az1, that is, a power law with n = 1, the streamlines (ie lines of constant ψ) are a system of
straight lines parallel to the x-axis. This is easiest to see by writing in terms of real and imaginary components:
-
thus giving φ = Ax and ψ =
Ay.
n = 2">
Power law with n = 2
If n = 2, then w = Az2 and
the streamline corresponding to a particular value of ψ are those points satisfying
- ψ = Ar2sin2θ
which is a system of rectangular hyperbolae. This may be seen by again rewriting in terms of real and imaginary components.
Noting that and
rewriting sinθ = y / r and cosθ = x /
r it is seen (on simplifying) that the streamlines are given by
- ψ = 2Axy.
The velocity field is given by , or
-
In fluid dynamics, this corresponds to a stagnation point. Note that the fluid velocity is zero at the origin (this follows on differentiation of
f(z) = z2 at z = 0).
The ψ = 0 streamline is particularly interesting: it has two (or four) branches, following
the coordinate axes, ie x = 0 and y = 0.
As no fluid flows across the x-axis, it (the x-axis) may be treated as a solid boundary (remember that the physical system to
which this analysis corresponds is an inviscid (ie zero viscosity) fluid; there
are thus no boundary layers to worry about). It is thus possible to
ignore the flow in the lower half-plane where y < 0 and to focus on the flow in the upper
half-plane.
With this interpretation, the flow is that of a vertically directed jet impinging on a horizontal flat plate.
The flow may also be interpreted as flow into a 90 degree corner if the regions specified by (say) x < 0 and y < 0 are ignored.
n = 3">
Power law with n = 3
If n = 3 the resulting flow is a sort of hexagonal version of the n = 2 case considered above. Streamlines are given by x2y
- y3 = ψ.
n = - 1">
Power law with n = - 1
if , the streamlines are given by
-
This is more easily interpreted in terms of real and imaginary components:
-
-
-
Thus the streamlines are circles that are tangent to the x-axis at the origin. The
velocity field is given by
-
The circles in the upper half-plane thus flow clockwise, those in the lower half-plane flow anticlockwise. Note that speeds go
as r - 2; and the speed at the origin is infinite.
Power law with n equals minus 2
{this section is to be completed}
See also: Laplacian field, conformal mapping.
External links
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