Potential flow

From Academic Kids

fluid dynamics, potential flow, also known as irrotational flow (of incompressible fluids) is steady flow defined by the equations

<math> \nabla \times \mathbf{v} = 0<math> (zero rotation = no viscosity)
<math> \nabla \cdot \mathbf{v} = 0<math> (zero divergence = volume conservation)

Equivalently,

<math> \mathbf{v} = \nabla \Phi \; , <math>

where:

  • v is the vector fluid velocity
  • Φ is the fluid flow potential, scalar
  • "Missing image
    Del.gif
    Image:del.gif

    ×" is curl
  • "Missing image
    Del.gif
    Image:del.gif

    ·" is divergence.

The equations above imply <math>\nabla^2 \Phi=0 <math>, 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 (called elemental flows) such as the free vortex and the point source possess ready analytical solutions. These solutions can be superposed to create more complex flows satisfying a variety of boundary conditions. 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.

Since the flow is inviscid and free of shear forces, this means that any streamline can be replaced with a solid boundary with no change in the flow field, a technique used in many aerodynamic design approaches. Another technique would be the use of Riabouchinsky solids.

Contents

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 <math>f<math>. If we write

<math>

f(x+iy)=\phi+i\psi <math>

then the Cauchy-Riemann equations show that

<math>

\frac{\partial\phi}{\partial x}=\frac{\partial\psi}{\partial y}, \qquad \frac{\partial\phi}{\partial y}=-\frac{\partial\psi}{\partial x}. <math> (it is conventional to regard all symbols as real numbers; and to write <math>z=x+iy<math> and <math>w=\phi+i\psi<math>).

The velocity field <math>\underline{u}=(u,v)<math>, specified by

<math>

u=\frac{\partial\phi}{\partial x},\qquad v=\frac{\partial\phi}{\partial y} <math> then satisfies the requirements for potential flow:

<math>

\nabla\cdot\underline{u}= \nabla^2\phi= \frac{\partial^2\phi}{\partial x^2}+\frac{\partial^2\phi}{\partial y^2}= {\partial \over \partial x} {\partial \psi \over \partial y} + {\partial \over \partial y} \left( - {\partial \psi \over \partial x} \right) = 0 <math> and

<math>

\left|\nabla\times\underline{u}\right|= \frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}= \frac{\partial^2\phi}{\partial x\partial y}- \frac{\partial^2\phi}{\partial y\partial x}=0. <math>

<math>\psi<math> is defined as the stream function. Lines of constant <math>\psi<math> are known as streamlines and lines of constant <math>\phi<math> are known as equipotential lines (see equipotential surface).

The two sets of curves intersect at right angles, for

<math>

\nabla \phi \cdot \nabla \psi = \frac{\partial\phi}{\partial x}\frac{\partial\psi}{\partial x}+ \frac{\partial\phi}{\partial y}\frac{\partial\psi}{\partial y}= {\partial \psi \over \partial y} {\partial \psi \over \partial x} - {\partial \psi \over \partial x} {\partial \psi \over \partial y} = 0. <math>

Examples: general considerations

Any differentiable function may be used for <math>f<math>. 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

<math>

w=Az^n<math>

then, writing <math>x+iy=re^{i\theta}<math>, we have

<math>

\phi=Ar^n\cos n\theta <math> and

<math>

\psi=Ar^n\sin n\theta <math>

Power law with <math>n=1<math>

If <math>w=Az^1<math>, that is, a power law with <math>n=1<math>, the streamlines (ie lines of constant <math>\psi<math>) 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:

<math>

f(x+iy)=A\times(x+iy)=Ax+i\cdot Ay <math> thus giving <math>\phi=Ax<math> and <math>\psi=Ay<math>.

Power law with <math>n=2<math>

If <math>n=2<math>, then <math>w=Az^2<math> and the streamline corresponding to a particular value of <math>\psi<math> are those points satisfying

<math>

\psi=Ar^2\sin 2\theta <math>

which is a system of rectangular hyperbolae. This may be seen by again rewriting in terms of real and imaginary components. Noting that <math>\sin 2\theta=2\sin\theta\,\cos\theta<math> and rewriting <math>\sin\theta=y/r<math> and <math>\cos\theta=x/r<math> it is seen (on simplifying) that the streamlines are given by

<math>

\psi=2Axy. <math>

The velocity field is given by <math>\nabla\phi<math>, or

<math>

(u,v)= \left( {\partial \phi \over \partial x}, {\partial \phi \over \partial y} \right) = \left( {\partial \psi \over \partial y}, - {\partial \psi \over \partial x} \right) = \left(2Ax,-2Ay\right) <math>

In fluid dynamics, the flowfield near the origin corresponds to a stagnation point. Note that the fluid at the origin is at rest (this follows on differentiation of <math>f(z)=z^2<math> at <math>z=0<math>).

The <math>\psi=0<math> streamline is particularly interesting: it has two (or four) branches, following the coordinate axes, ie <math>x=0<math> and <math>y=0<math>.

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 <math>y<0<math> 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) <math>x<0<math> and <math>y<0<math> are ignored.

Power law with <math>n=3<math>

If <math>n=3<math> the resulting flow is a sort of hexagonal version of the <math>n=2<math> case considered above. Streamlines are given by <math>x^2y-y^3=\psi<math>.

Power law with <math>n=-1<math>

if <math>n=\,-1<math>, the streamlines are given by

<math>

\psi=-\frac{A}{r}\sin\theta. <math>

This is more easily interpreted in terms of real and imaginary components:

<math> \psi = {-A y \over r^2} = {-A y \over x^2 + y^2}, <math>
<math> x^2 + y^2 + {A y \over \psi} = 0, <math>
<math>

x^2+\left(y+\frac{A}{2\psi}\right)^2=\left(\frac{A}{2\psi}\right)^2. <math>

Thus the streamlines are circles that are tangent to the x-axis at the origin. The velocity field is given by

<math>

(u,v)=\left( {\partial \psi \over \partial y}, - {\partial \psi \over \partial x} \right) = \left(A\frac{y^2-x^2}{(x^2+y^2)^2},-A\frac{2xy}{(x^2+y^2)^2}\right) <math>

The circles in the upper half-plane thus flow clockwise, those in the lower half-plane flow anticlockwise. Note that speeds go as <math>r^{-2}<math>; 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.

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