Potential flow

In fluid dynamics, a potential flow is described by means of a velocity potential φ, being a function of space and time. The flow velocity v is a vector field equal to the gradient, ∇, of the velocity potential φ:[1]

${\displaystyle \mathbf {v} =\nabla \varphi .}$

Sometimes, also the definition v = −∇φ, with a minus sign, is used. But here we will use the definition above, without the minus sign. From vector calculus it is known that the curl of a gradient is equal to zero:[1]

${\displaystyle \nabla \times \nabla \varphi =\mathbf {0} \,,}$

and consequently the vorticity, the curl of the velocity field v, is zero:[1]

${\displaystyle \nabla \times \mathbf {v} =\mathbf {0} \,.}$

This implies that a potential flow is an irrotational flow. This has direct consequences for the applicability of potential flow. 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.[2] Fortunately, there are often large regions of a flow where the assumption of irrotationality is valid which is why potential flow is used for various applications. For instance in: flow around aircraft, groundwater flow, acoustics, water waves, and electroosmotic flow.[3]

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:[1]

${\displaystyle \nabla \cdot \mathbf {v} =0\,,}$

with the dot denoting the inner product. As a result, the velocity potential φ has to satisfy Laplace's equation[1]

${\displaystyle \nabla ^{2}\varphi =0\,,}$

where ∇² = ∇ ⋅ ∇ is the Laplace operator (sometimes also written Δ). In this case the flow can be determined completely from its kinematics: the assumptions of irrotationality and zero divergence of flow. Dynamics only have to be applied afterwards, if one is interested in computing pressures: for instance for flow around airfoils through the use of Bernoulli's principle.

In two dimensions, potential flow reduces to a very simple system that is analyzed using complex analysis (see below).

Potential flow theory can also be used to model irrotational compressible flow. The full potential equation, describing a steady flow, is given by:[4]

${\displaystyle \left(1-M_{x}^{2}\right){\frac {\partial ^{2}\Phi }{\partial x^{2}}}+\left(1-M_{y}^{2}\right){\frac {\partial ^{2}\Phi }{\partial y^{2}}}+\left(1-M_{z}^{2}\right){\frac {\partial ^{2}\Phi }{\partial z^{2}}}-2M_{x}M_{y}{\frac {\partial ^{2}\Phi }{\partial x\,\partial y}}-2M_{y}M_{z}{\frac {\partial ^{2}\Phi }{\partial y\,\partial z}}-2M_{z}M_{x}{\frac {\partial ^{2}\Phi }{\partial z\,\partial x}}=0\,,}$

with Mach number components

${\displaystyle {\begin{aligned}M_{x}&={\frac {1}{a}}{\frac {\partial \Phi }{\partial x}}\,,&M_{y}&={\frac {1}{a}}{\frac {\partial \Phi }{\partial y}}\,,{\text{ and }}&M_{z}&={\frac {1}{a}}{\frac {\partial \Phi }{\partial z}}\,,\end{aligned}}}$

where a is the local speed of sound. The flow velocity v is again equal to ∇Φ, with Φ the velocity potential. The full potential equation is valid for sub-, trans- and supersonic flow at arbitrary angle of attack, as long as the assumption of irrotationality is applicable.[4]

In case of either subsonic or supersonic (but not transonic or hypersonic) flow, at small angles of attack and thin bodies, an additional assumption can be made: the velocity potential is split into an undisturbed onflow velocity V_∞ in the x-direction, and a small perturbation velocity ∇φ thereof. So:[4]

${\displaystyle \nabla \Phi =V_{\infty }x+\nabla \varphi \,.}$

In that case, the linearized small-perturbation potential equation — an approximation to the full potential equation — can be used:[4]

${\displaystyle \left(1-M_{\infty }^{2}\right){\frac {\partial ^{2}\varphi }{\partial x^{2}}}+{\frac {\partial ^{2}\varphi }{\partial y^{2}}}+{\frac {\partial ^{2}\varphi }{\partial z^{2}}}=0,}$

with M_∞ = V_∞/a_∞ the Mach number of the incoming free stream. This linear equation is much easier to solve than the full potential equation: it may be recast into Laplace's equation by a simple coordinate stretching in the x-direction.

Potential flow theory can also be used to model unsteady irrotational compressible flow. The full potential equation, describing a unsteady flow, is given by:[4]

${\displaystyle {\begin{aligned}0=-{\frac {1}{a^{2}}}\left[{\frac {\partial }{\partial t}}(\nabla \Phi \cdot \nabla \Phi )+{\frac {\partial ^{2}\Phi }{\partial t^{2}}}\right]&+\left(1-M_{x}^{2}\right){\frac {\partial ^{2}\Phi }{\partial x^{2}}}+\left(1-M_{y}^{2}\right){\frac {\partial ^{2}\Phi }{\partial y^{2}}}+\left(1-M_{z}^{2}\right){\frac {\partial ^{2}\Phi }{\partial z^{2}}}\\[3pt]&-2M_{x}M_{y}{\frac {\partial ^{2}\Phi }{\partial x\,\partial y}}-2M_{y}M_{z}{\frac {\partial ^{2}\Phi }{\partial y\,\partial z}}-2M_{z}M_{x}{\frac {\partial ^{2}\Phi }{\partial z\,\partial x}}\end{aligned}}}$

with Mach number components

${\displaystyle {\begin{aligned}M_{x}&={\frac {1}{a}}{\frac {\partial \Phi }{\partial x}}\,,&M_{y}&={\frac {1}{a}}{\frac {\partial \Phi }{\partial y}}\,,{\text{ and }}&M_{z}&={\frac {1}{a}}{\frac {\partial \Phi }{\partial z}}\,,\end{aligned}}}$

where a is the local speed of sound. The flow velocity v is again equal to ∇Φ, with Φ the velocity potential. The full potential equation is valid for sub-, trans- and supersonic flow at arbitrary angle of attack, as long as the assumption of irrotationality is applicable.[4]

In case of either subsonic or supersonic (but not transonic or hypersonic) flow, at small angles of attack and thin bodies, an additional assumption can be made: the velocity potential is split into an undisturbed onflow velocity V_∞ in the x-direction, and a small perturbation velocity ∇φ thereof. So:[4]

${\displaystyle \nabla \Phi =V_{\infty }x+\nabla \varphi \,.}$

In that case, the linearized small-perturbation potential equation — an approximation to the full potential equation — can be used:[4]

${\displaystyle -{\frac {1}{a^{2}}}\left[2V_{\infty }{\frac {\partial ^{2}\varphi }{\partial x\partial t}}+{\frac {\partial ^{2}\varphi }{\partial t^{2}}}\right]+\left(1-M_{\infty }^{2}\right){\frac {\partial ^{2}\varphi }{\partial x^{2}}}+{\frac {\partial ^{2}\varphi }{\partial y^{2}}}+{\frac {\partial ^{2}\varphi }{\partial z^{2}}}=0,}$

with M_∞ = V_∞/a_∞ the Mach number of the incoming free stream.

Small-amplitude sound waves can be approximated with the following potential-flow model:[7]

${\displaystyle {\frac {\partial ^{2}\varphi }{\partial t^{2}}}={\overline {a}}^{2}\Delta \varphi ,}$

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

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,[2] 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).[8] 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.[9] More precisely, potential flow cannot account for the behaviour of flows that include a boundary layer.[2] 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.

Examples of conformal maps for the power law w = Azⁿ

Examples of conformal maps for the power law w = Azⁿ, for different values of the power n. Shown is the z-plane, showing lines of constant potential φ and streamfunction ψ, while w = φ + iψ.

In case the following power-law conformal map is applied, from z = x + iy to w = φ + iψ:[11]

${\displaystyle w=Az^{n}\,,}$

then, writing z in polar coordinates as z = x + iy = re^iθ, we have[11]

${\displaystyle \varphi =Ar^{n}\cos n\theta \qquad {\text{and}}\qquad \psi =Ar^{n}\sin n\theta \,.}$

In the figures to the right examples are given for several values of n. 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 n are:[11]

• n = 1/2: this corresponds with flow around a semi-infinite plate,
• n = 2/3: flow around a right corner,
• n = 1: a trivial case of uniform flow,
• n = 2: flow through a corner, or near a stagnation point, and
• n = −1: flow due to a source doublet

The constant A is a scaling parameter: its absolute value |A| determines the scale, while its argument arg(A) introduces a rotation (if non-zero).

If w = Az¹, that is, a power law with n = 1, the streamlines (i.e. 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:

${\displaystyle f(x+iy)=A\,(x+iy)=Ax+iAy}$

thus giving φ = Ax and ψ = Ay. This flow may be interpreted as uniform flow parallel to the x-axis.

If n = 2, then w = Az² and the streamline corresponding to a particular value of ψ are those points satisfying

${\displaystyle \psi =Ar^{2}\sin 2\theta \,,}$

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

${\displaystyle \psi =2Axy\,.}$

The velocity field is given by ∇φ, or

${\displaystyle {\begin{pmatrix}u\\v\end{pmatrix}}={\begin{pmatrix}{\frac {\partial \varphi }{\partial x}}\\[2px]{\frac {\partial \varphi }{\partial y}}\end{pmatrix}}={\begin{pmatrix}+{\partial \psi \over \partial y}\\[2px]-{\partial \psi \over \partial x}\end{pmatrix}}={\begin{pmatrix}+2Ax\\[2px]-2Ay\end{pmatrix}}\,.}$

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 f(z) = z² at z = 0). The ψ = 0 streamline is particularly interesting: it has two (or four) branches, following the coordinate axes, i.e. x = 0 and y = 0. As no fluid flows across the x-axis, it (the x-axis) may be treated as a solid boundary. 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 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) x, y < 0 are ignored.

If n = 3, the resulting flow is a sort of hexagonal version of the n = 2 case considered above. Streamlines are given by, ψ = 3x²y − y³ and the flow in this case may be interpreted as flow into a 60° corner.

If n = −1, the streamlines are given by

${\displaystyle \psi =-{\frac {A}{r}}\sin \theta .}$

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

${\displaystyle {\begin{aligned}\psi ={\frac {-Ay}{r^{2}}}&={\frac {-Ay}{x^{2}+y^{2}}}\,,\\x^{2}+y^{2}+{\frac {Ay}{\psi }}&=0\,,\\x^{2}+\left(y+{\frac {A}{2\psi }}\right)^{2}&=\left({\frac {A}{2\psi }}\right)^{2}\,.\end{aligned}}}$

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 r⁻²; 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

${\displaystyle (u,v)=\left({\frac {\partial \psi }{\partial y}},-{\frac {\partial \psi }{\partial x}}\right)=\left(A{\frac {y^{2}-x^{2}}{\left(x^{2}+y^{2}\right)^{2}}},-A{\frac {2xy}{\left(x^{2}+y^{2}\right)^{2}}}\right)\,.}$

or in polar coordinates:

${\displaystyle (u_{r},u_{\theta })=\left({\frac {1}{r}}{\frac {\partial \psi }{\partial \theta }},-{\frac {\partial \psi }{\partial r}}\right)=\left(-{\frac {A}{r^{2}}}\cos \theta ,-{\frac {A}{r^{2}}}\sin \theta \right)\,.}$

If n = −2, the streamlines are given by

${\displaystyle \psi =-{\frac {A}{r^{2}}}\sin 2\theta \,.}$

This is the flow field associated with a quadrupole.[12]

A line source or sink of strength ${\displaystyle Q}$ (${\displaystyle Q>0}$ for source and ${\displaystyle Q<0}$ for sink) is given by the potential

${\displaystyle w={\frac {Q}{2\pi }}\ln z}$

where ${\displaystyle Q}$ in fact is the volume flux per unit length across a surface enclosing the source or sink. The velocity field in polar coordinates are

${\displaystyle u_{r}={\frac {Q}{2\pi r}},\quad u_{\theta }=0}$

i.e., a purely radial flow.

A line vortex of strength ${\displaystyle \Gamma }$ is given by

${\displaystyle w={\frac {\Gamma }{2\pi i}}\ln z}$

where ${\displaystyle \Gamma }$ is the circulation around any simple closed contour enclosing the vortex. The velocity field in polar coordinates are

${\displaystyle u_{r}=0,\quad u_{\theta }={\frac {\Gamma }{2\pi r}}}$

i.e., a purely azimuthal flow.

The velocity potential of a point source or sink of strength ${\displaystyle Q}$ (${\displaystyle Q>0}$ for source and ${\displaystyle Q<0}$ for sink) in spherical polar coordinates is given by

${\displaystyle \phi =-{\frac {Q}{4\pi r}}}$

where ${\displaystyle Q}$ in fact is the volume flux across a closed surface enclosing the source or sink. The velocity field in spherical polar coordinates are

${\displaystyle u_{r}={\frac {Q}{4\pi r^{2}}},\quad u_{\theta }=0,\quad u_{\phi }=0.}$