Potential flow

thumb|300px|right|Potential-flow [[Streamlines, streaklines, and pathlines|streamlines around a NACA 0012 airfoil at 11° angle of attack, with upper and lower streamtubes identified. The flow is two-dimensional and the airfoil has infinite span.]]

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:

The flow of an inviscid fluid
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 aerofoils, 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 wakes 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

thumb|A potential flow is constructed by adding simple [[elementary flows and observing the result.]]

thumb|right|[[Streamlines, streaklines, and pathlines|Streamlines for the incompressible potential flow around a circular cylinder in a uniform onflow.]]

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

<math display="block">\boldsymbol\omega \equiv \nabla\times\mathbf v=0,</math>

where <math>\mathbf v(\mathbf x,t)</math> is the velocity field and <math>\boldsymbol\omega(\mathbf x,t)</math> is the vorticity field. Like any vector field having zero curl, the velocity field can be expressed as the gradient of certain scalar, say <math>\varphi(\mathbf x,t)</math> which is called the velocity potential, since the curl of the gradient is always zero. We therefore have

<math display="block"> \mathbf{v} = \nabla \varphi.</math>

The velocity potential is not uniquely defined since one can add to it an arbitrary function of time, say <math>f(t)</math>, without affecting the relevant physical quantity which is <math>\mathbf v</math>. The non-uniqueness is usually removed by suitably selecting appropriate initial or boundary conditions satisfied by <math>\varphi</math> and as such the procedure may vary from one problem to another.

In potential flow, the circulation <math>\Gamma</math> around any simply-connected contour <math>C</math> is zero. This can be shown using the Stokes theorem,

<math display="block">\Gamma \equiv \oint_C \mathbf v\cdot d\mathbf l = \int \boldsymbol\omega\cdot d\mathbf f=0</math>

where <math>d\mathbf l</math> is the line element on the contour and <math>d\mathbf f</math> 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 or point vortices, or in smoke rings), the circulation <math>\Gamma</math> need not be zero. In the former case, Stokes theorem cannot be applied and in the later case, <math>\boldsymbol\omega</math> is non-zero within the region bounded by the contour. Around a contour encircling an infinitely long solid cylinder with which the contour loops <math>N</math> times, we have

<math display="block">\Gamma = N \kappa</math>

where <math>\kappa</math> is a cyclic constant. This example belongs to a doubly-connected space. In an <math>n</math>-tuply connected space, there are <math>n-1</math> such cyclic constants, namely, <math>\kappa_1,\kappa_2,\dots,\kappa_{n-1}.</math>

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:

<math display="block">(c^2-\varphi_x^2)\varphi_{xx}+(c^2-\varphi_y^2)\varphi_{yy}+(c^2-\varphi_z^2)\varphi_{zz}-2(\varphi_x\varphi_y\varphi_{xy}+\varphi_y\varphi_z\varphi_{yz}+\varphi_z\varphi_x\phi_{zx})=0</math>

where <math>c=c(v)</math> is expressed as a function of the velocity magnitude <math>v^2=(\nabla\phi)^2</math>. For a polytropic gas, <math>c^2 = (\gamma-1)(h_0-v^2/2)</math>, where <math>\gamma</math> is the specific heat ratio and <math>h_0</math> is the stagnation enthalpy. In two dimensions, the equation simplifies to

<math display="block">(c^2-\varphi_x^2)\varphi_{xx}+(c^2-\varphi_y^2)\varphi_{yy}-2\varphi_x\varphi_y\varphi_{xy}=0.</math>

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

<math display="block">\nabla (h+v^2/2) - \mathbf v\times\boldsymbol\omega = T \nabla s</math>

where <math>h</math> is the specific enthalpy, <math>\boldsymbol\omega</math> is the vorticity field, <math>T</math> is the temperature and <math>s</math> is the specific entropy. Since in front of the leading shock wave, we have a potential flow, Bernoulli's equation shows that <math>h+v^2/2</math> is constant, which is also constant across the shock wave (Rankine–Hugoniot conditions) and therefore we can write

<math display="block">\mathbf v\times\boldsymbol\omega = -T \nabla s</math>

1) When the shock wave is of constant intensity, the entropy discontinuity across the shock wave is also constant i.e., <math>\nabla s=0</math> and therefore vorticity production is zero. Shock waves at the pointed leading edge of two-dimensional wedge or three-dimensional cone (Taylor–Maccoll flow) has constant intensity. 2) For weak shock waves, the entropy jump across the shock wave is a third-order quantity in terms of shock wave strength and therefore <math>\nabla s</math> can be neglected. Shock waves in slender bodies lies nearly parallel to the body and they are weak.

Nearly parallel flows: When the flow is predominantly unidirectional with small deviations such as in flow past slender bodies, the full equation can be further simplified. Let <math>U\mathbf{e}_x</math> be the mainstream and consider small deviations from this velocity field. The corresponding velocity potential can be written as <math>\varphi = x U + \phi</math> where <math>\phi</math> characterizes the small departure from the uniform flow and satisfies the linearized version of the full equation. This is given by

<math display="block">(1-M^2) \frac{\partial^2\phi}{\partial x^2} + \frac{\partial^2\phi}{\partial y^2} + \frac{\partial^2\phi}{\partial z^2} =0</math>

where <math>M=U/c_\infty</math> is the constant Mach number corresponding to the uniform flow. This equation is valid provided <math>M</math> is not close to unity. When <math>|M-1|</math> is small (transonic flow), we have the following nonlinear equation

<math display="block">2\alpha_*\frac{\partial\phi}{\partial x} \frac{\partial^2\phi}{\partial x^2} = \frac{\partial^2\phi}{\partial y^2} + \frac{\partial^2\phi}{\partial z^2}</math>

where <math>\alpha_*</math> is the critical value of Landau derivative <math>\alpha = (c^4/2\upsilon^3)(\partial^2 \upsilon/\partial p^2)_s</math> and <math>\upsilon=1/\rho</math> is the specific volume. The transonic flow is completely characterized by the single parameter <math>\alpha_*</math>, which for polytropic gas takes the value <math>\alpha_*=\alpha=(\gamma+1)/2</math>. 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

<math display="block">\frac{\partial\rho}{\partial t} + \rho \nabla\cdot\mathbf v + \mathbf v\cdot\nabla \rho = 0, \quad \frac{\partial\mathbf v}{\partial t}+ (\mathbf v \cdot\nabla)\mathbf v= -\frac{1}{\rho}\nabla p =-\frac{c^2}{\rho}\nabla \rho=-\nabla h.</math>

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

<math display="block">\frac{\partial\varphi}{\partial t} + \frac{v^2}{2} + h = f(t), \quad \Rightarrow \quad \frac{\partial h}{\partial t} = -\frac{\partial^2\varphi}{\partial t^2} - \frac{1}{2}\frac{\partial v^2}{\partial t} + \frac{df}{dt}</math>

where <math>f(t)</math> is an arbitrary function. Without loss of generality, we can set <math>f(t)=0</math> since <math>\varphi</math> is not uniquely defined. Combining these equations, we obtain

<math display="block">\frac{\partial^2\varphi}{\partial t^2} + \frac{\partial v^2}{\partial t}=c^2\nabla\cdot\mathbf v - \mathbf v\cdot (\mathbf v \cdot \nabla)\mathbf v.</math>

Substituting here <math>\mathbf v=\nabla\varphi</math> results in

<math display="block">\varphi_{tt} + (\varphi_x^2+ \varphi_y^2+ \varphi_z^2)_t= (c^2-\varphi_x^2)\varphi_{xx}+(c^2-\varphi_y^2)\varphi_{yy}+(c^2-\varphi_z^2)\varphi_{zz}-2(\varphi_x\varphi_y\varphi_{xy}+\varphi_y\varphi_z\varphi_{yz}+\varphi_z\varphi_x\phi_{zx}).</math>

Nearly parallel flows: As in before, for nearly parallel flows, we can write (after introducing a recaled time <math>\tau=c_\infty t</math>)

<math display="block">\frac{\partial^2\phi}{\partial \tau^2} + 2M \frac{\partial^2\phi}{\partial x\partial\tau}= (1-M^2) \frac{\partial^2\phi}{\partial x^2} + \frac{\partial^2\phi}{\partial y^2} + \frac{\partial^2\phi}{\partial z^2}</math>

provided the constant Mach number <math>M</math> is not close to unity. When <math>|M-1|</math> is small (transonic flow), we have the following nonlinear equation

<math display="block">\frac{\partial^2\phi}{\partial \tau^2} + 2\frac{\partial^2\phi}{\partial x\partial\tau} = -2\alpha_*\frac{\partial\phi}{\partial x} \frac{\partial^2\phi}{\partial x^2} + \frac{\partial^2\phi}{\partial y^2} + \frac{\partial^2\phi}{\partial z^2}.</math>

Sound waves: In sound waves, the velocity magnitude <math>v</math> (or the Mach number) is very small, although the unsteady term is now comparable to the other leading terms in the equation. Thus neglecting all quadratic and higher-order terms and noting that in the same approximation, <math>c</math> is a constant (for example, in polytropic gas <math>c^2=(\gamma-1)h_0</math>), we have

<math display="block">\frac{\partial^2 \varphi}{\partial t^2} = c^2 \nabla^2 \varphi,</math>

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, 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.

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

<math display="block">\begin{align}

z &= x + iy \,, \text{ and } &

w &= \varphi + i\psi \,.

\end{align}</math>

Now, if we write the mapping as

then, writing in polar coordinates as , we have

Line source and sink

A line source or sink of strength <math>Q</math> (<math>Q>0</math> for source and <math>Q<0</math> for sink) is given by the potential

where <math>Q</math> in fact is the volume flux per unit length across a surface enclosing the source or sink. The velocity field in polar coordinates are

<math display="block">u_r = \frac{Q}{2\pi r},\quad u_\theta=0</math>

i.e., a purely radial flow.

Line vortex

A line vortex of strength <math>\Gamma</math> is given by

<math display="block">w=\frac{\Gamma}{2\pi i}\ln z</math>

where <math>\Gamma</math> is the circulation around any simple closed contour enclosing the vortex. The velocity field in polar coordinates are

<math display="block">u_r = 0,\quad u_\theta=\frac{\Gamma}{2\pi r}</math>

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

The velocity potential of a point source or sink of strength <math>Q</math> (<math>Q>0</math> for source and <math>Q<0</math> for sink) in spherical polar coordinates is given by

where <math>Q</math> in fact is the volume flux across a closed surface enclosing the source or sink. The velocity field in spherical polar coordinates are

<math display="block">u_r = \frac{Q}{4\pi r^2}, \quad u_\theta=0, \quad u_\phi = 0.</math>

Notes

References

External links

— Java applets for exploring conformal maps
Potential Flow Visualizations - Interactive WebApps