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Wikipedia / Potential flow

In fluid dynamics, potential flow or irrotational flow refers to the idealised, frictionless flow of a fluid. Flows of two kinds are visualised in this way:

  1. The flow of an inviscid fluid

  2. The flow of a fluid of low viscosity, in regions that do not contain a boundary layer. See Prandtl hypothesis.

Potential flow describes the velocity field as the gradient of a scalar function: the velocity potential. 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.

In the case of an incompressible flow the velocity potential satisfies Laplace's equation, and potential theory is applicable. However, potential flows also have been used to describe compressible flows and Hele-Shaw flows. The potential flow approach occurs in the modeling of both stationary as well as nonstationary flows. Applications of potential flow include: the outer flow field for aerofoilss, water waves, electroosmotic flow, and groundwater flow.

A region containing friction forces (described as shear forces and viscous forces) can be described as a region containing vorticity. For flows (or parts thereof) with strong vorticity effects, the potential flow approximation is not applicable. In flow regions where vorticity is known to be important, such as wakess and boundary layers, potential flow theory is not able to provide reasonable predictions of the flow. However, there are often large regions of a flow in which the assumption of irrotationality is valid, allowing the use of potential flow for various applications; these include flow around aircraft, groundwater flow, acoustics, water waves, and electroosmotic flow.

Description and characteristics

In potential or irrotational flow, the vorticity vector field is zero, i.e.,

is the vorticity which is called the velocity potential, since the curl of the gradient is always zero. We therefore have

and as such the procedure may vary from one problem to another.

In potential flow, the circulation around any simply-connected contour is zero. This can be shown using the Stokes theorem,

is the area element of any surface bounded by the contour. In multiply-connected space (say, around a contour enclosing solid body in two dimensions or around a contour enclosing a torus in three-dimensions) or in the presence of concentrated vortices, (say, in the so-called irrotational vortices times, we have

Incompressible flow

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 has zero divergence:

satisfies the Laplace equation

where is the Laplace operator (sometimes also written ). Since solutions of the Laplace equation are harmonic functions, every harmonic function represents a potential flow solution. 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.

In incompressible flows, contrary to common misconception, the potential flow indeed satisfies the full Navier–Stokes equations, not just the Euler equations, because the viscous term

is identically zero. It is the inability of the potential flow to satisfy the required boundary conditions, especially near solid boundaries, makes it invalid in representing the required flow field. 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).

Compressible flow

===Steady flow=== from Eulers equation is quite straightforward. The continuity and the (potential flow) momentum equations for steady flows are given by

where the last equation follows from the fact that entropy is constant for a fluid particle and that square of the sound speed from the two governing equations results in

results in

. For a polytropic gas is the specific heat ratio is the stagnation enthalpy. In two dimensions, the equation simplifies to

Validity: As it stands, the equation is valid for any inviscid potential flows, irrespective of whether the flow is subsonic or supersonic (e.g. Prandtl–Meyer flow). However in supersonic and also in transonic flows, shock waves can occur which can introduce entropy and vorticity into the flow making the flow rotational. Nevertheless, there are two cases for which potential flow prevails even in the presence of shock waves, which are explained from the (not necessarily potential) momentum equation written in the following form

is the specific enthalpy is the vorticity is constant, which is also constant across the shock wave (Rankine–Hugoniot conditions) and therefore we can write

and therefore vorticity production is zero. Shock waves at the pointed leading edge of two-dimensional wedge or three-dimensional cone (Taylor–Maccoll flow can be neglected. Shock waves in slender bodies lies nearly parallel to the body and they are weak.

Nearly parallel flows: characterizes the small departure from the uniform flow and satisfies the linearized version of the full equation. This is given by

is the constant Mach number is small (transonic flow), we have the following nonlinear equation

is the critical value of Landau derivative is the specific volume. Under hodograph transformation, the transonic equation in two-dimensions becomes the Euler–Tricomi equation. === Unsteady flow === The continuity and the (potential flow) momentum equations for unsteady flows are given by

The first integral of the (potential flow) momentum equation is given by

is not uniquely defined. Combining these equations, we obtain

results in

Nearly parallel flows:)

is small (transonic flow), we have the following nonlinear equation

Sound waves:), we have

which is a linear wave equation for the velocity potential . Again the oscillatory part of the velocity vector is related to the velocity potential by , while as before is the Laplace operator, and is the average speed of sound in the homogeneous medium. Note that also the oscillatory parts of the pressure and density each individually satisfy the wave equation, in this approximation.

Applicability and limitations

Potential flow does not include all the characteristics of flows that are encountered in the real world. Potential flow theory cannot be applied for viscous internal flows, except for flows between closely spaced plates. Richard Feynman considered potential flow to be so unphysical that the only fluid to obey the assumptions was "dry water" (quoting John von Neumann). 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. 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 elementary 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. The absence of boundary layer effects means that any streamline can be replaced by 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.==Analysis for two-dimensional incompressible flow== Potential flow in two dimensions is simple to analyze using conformal mapping, by the use of transformationss 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. It is not possible to solve a potential flow using complex numbers in three dimensions.

The basic idea is to use a holomorphic (also called analytic) or meromorphic function , which maps the physical domain to the transformed domain . While , , and are all real valued, it is convenient to define the complex quantities

Now, if we write the mapping as

Then, because is a holomorphic or meromorphic function, it has to satisfy the Cauchy–Riemann equations

The velocity components , in the directions respectively, can be obtained directly from by differentiating with respect to . That is

So the velocity field is specified by

Both and then satisfy Laplace's equation:

So can be identified as the velocity potential and is called the stream function. Lines of constant are known as streamlines and lines of constant are known as equipotential lines (see equipotential surface).

Streamlines and equipotential lines are orthogonal to each other, since

Thus the flow occurs along the lines of constant and at right angles to the lines of constant .

is also satisfied, this relation being equivalent to . So the flow is irrotational. The automatic condition then gives the incompressibility constraint .

Examples of two-dimensional incompressible flows

Any differentiable function may be used for . 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.

Power laws

In case the following power-law conformal map is applied, from to :

then, writing in polar coordinates as , we have

In the figures to the right examples are given for several values of . The black line is the boundary of the flow, while the darker blue lines are streamlines, and the lighter blue lines are equi-potential lines. Some interesting powers are:

  • : this corresponds with flow around a semi-infinite plate,

  • : flow around a right corner,

  • : a trivial case of uniform flow,

  • : flow through a corner, or near a stagnation point, and

  • : flow due to a source doublet

The constant is a scaling parameter: its absolute value determines the scale, while its argument introduces a rotation (if non-zero). ==== Power laws with : uniform flow ==== If , that is, a power law with , the streamlines (i.e. lines of constant ) are a system of straight lines parallel to the -axis. This is easiest to see by writing in terms of real and imaginary components:

thus giving and . This flow may be interpreted as uniform flow parallel to the -axis.

Power laws with n 2

If , then and the streamline corresponding to a particular value of are those points satisfying

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 and it is seen (on simplifying) that the streamlines are given by

The velocity field is given by , or

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 at ). The streamline is particularly interesting: it has two (or four) branches, following the coordinate axes, i.e. and . As no fluid flows across the -axis, it (the -axis) may be treated as a solid boundary. It is thus possible to ignore the flow in the lower half-plane where and to focus on the flow in the upper halfplane. 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) are ignored.

Power laws with n 3

If , the resulting flow is a sort of hexagonal version of the case considered above. Streamlines are given by, and the flow in this case may be interpreted as flow into a 60° corner. ==== Power laws with : doublet ==== If , the streamlines are given by

Thus the streamlines are circles that are tangent to the x-axis at the origin. The circles in the upper half-plane thus flow clockwise, those in the lower half-plane flow anticlockwise. Note that the velocity components are proportional to ; and their values at the origin is infinite. 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. The velocity field is given by

or in polar coordinates:

Power laws with n −2: quadrupole

If , the streamlines are given by

This is the flow field associated with a quadrupole.

Line source and sink

for sink) is given by the potential

in fact is the volume flux per unit length across a surface enclosing the source or sink. The velocity field in polar coordinates are

i.e., a purely radial flow.

Line vortex

is given by

is the circulation around any simple closed contour enclosing the vortex. The velocity field in polar coordinates are

i.e., a purely azimuthal flow.

Analysis for three-dimensional incompressible flows

For three-dimensional flows, complex potential cannot be obtained.

Point source and sink

for sink) in spherical polar coordinates is given by

in fact is the volume flux across a closed surface enclosing the source or sink. The velocity field in spherical polar coordinates are

See also

Notes

References

Further reading