Potential flow

 ( Fluid Dynamics ) 

In fluid dynamics, potential flow or irrotational flow refers to a description of a fluid flow with no vorticity in it. Such a description typically arises in the limit of vanishing viscosity, i.e., for an inviscid fluid and with no vorticity present in the flow.

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 . 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 airfoil, water waves, electroosmotic flow, and groundwater flow. 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 , potential flow theory is not able to provide reasonable predictions of the flow.Batchelor (1973) pp. 378–380. 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, , and electroosmotic flow.

$$\backslash boldsymbol\backslash omega\; \backslash equiv\; \backslash nabla\backslash times\backslash mathbf\; v=0,$$

where $\backslash mathbf\; v(\backslash mathbf\; x,t)$ is the velocity field and $\backslash boldsymbol\backslash omega(\backslash mathbf\; x,t)$ is the vorticity field. Like any vector field having zero curl, the velocity field can be expressed as the gradient of certain scalar, say $\backslash varphi(\backslash mathbf\; x,t)$ which is called the velocity potential, since the curl of the gradient is always zero. We therefore haveBatchelor (1973) pp. 99–101.

$$\backslash mathbf\{v\}\; =\; \backslash nabla\; \backslash varphi.$$

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

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

$$\backslash Gamma\; \backslash equiv\; \backslash oint\_C\; \backslash mathbf\; v\backslash cdot\; d\backslash mathbf\; l\; =\; \backslash int\; \backslash boldsymbol\backslash omega\backslash cdot\; d\backslash mathbf\; f=0$$

where $d\backslash mathbf\; l$ is the line element on the contour and $d\backslash mathbf\; f$ 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 $\backslash Gamma$ need not be zero. In the former case, Stokes theorem cannot be applied and in the later case, $\backslash boldsymbol\backslash omega$ is non-zero within the region bounded by the contour. Around a contour encircling an infinitely long solid cylinder with which the contour loops $N$ times, we have

$$\backslash Gamma\; =\; N\; \backslash kappa$$

where $\backslash kappa$ is a cyclic constant. This example belongs to a doubly-connected space. In an $n$-tuply connected space, there are $n-1$ such cyclic constants, namely, $\backslash kappa\_1,\backslash kappa\_2,\backslash dots,\backslash kappa\_\{n-1\}.$

$$(c^2-\backslash varphi\_x^2)\backslash varphi\_\{xx\}+(c^2-\backslash varphi\_y^2)\backslash varphi\_\{yy\}+(c^2-\backslash varphi\_z^2)\backslash varphi\_\{zz\}-2(\backslash varphi\_x\backslash varphi\_y\backslash varphi\_\{xy\}+\backslash varphi\_y\backslash varphi\_z\backslash varphi\_\{yz\}+\backslash varphi\_z\backslash varphi\_x\backslash phi\_\{zx\})=0$$

where $c=c(v)$ is expressed as a function of the velocity magnitude $v^2=(\backslash nabla\backslash phi)^2$. For a polytropic gas, $c^2\; =\; (\backslash gamma-1)(h\_0-v^2/2)$, where $\backslash gamma$ is the specific heat ratio and $h\_0$ is the stagnation enthalpy. In two dimensions, the equation simplifies to

$$(c^2-\backslash varphi\_x^2)\backslash varphi\_\{xx\}+(c^2-\backslash varphi\_y^2)\backslash varphi\_\{yy\}-2\backslash varphi\_x\backslash varphi\_y\backslash varphi\_\{xy\}=0.$$

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

$$\backslash nabla\; (h+v^2/2)\; -\; \backslash mathbf\; v\backslash times\backslash boldsymbol\backslash omega\; =\; T\; \backslash nabla\; s$$

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

$$\backslash mathbf\; v\backslash times\backslash boldsymbol\backslash omega\; =\; -T\; \backslash nabla\; s$$

1) When the shock wave is of constant intensity, the entropy discontinuity across the shock wave is also constant i.e., $\backslash nabla\; s=0$ 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 $\backslash nabla\; s$ 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 $U\backslash mathbf\{e\}\_x$ be the mainstream and consider small deviations from this velocity field. The corresponding velocity potential can be written as $\backslash varphi\; =\; x\; U\; +\; \backslash phi$ where $\backslash phi$ characterizes the small departure from the uniform flow and satisfies the linearized version of the full equation. This is given by

$$(1-M^2)\; \backslash frac\{\backslash partial^2\backslash phi\}\{\backslash partial\; x^2\}\; +\; \backslash frac\{\backslash partial^2\backslash phi\}\{\backslash partial\; y^2\}\; +\; \backslash frac\{\backslash partial^2\backslash phi\}\{\backslash partial\; z^2\}\; =0$$

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

$$2\backslash alpha\_*\backslash frac\{\backslash partial\backslash phi\}\{\backslash partial\; x\}\; \backslash frac\{\backslash partial^2\backslash phi\}\{\backslash partial\; x^2\}\; =\; \backslash frac\{\backslash partial^2\backslash phi\}\{\backslash partial\; y^2\}\; +\; \backslash frac\{\backslash partial^2\backslash phi\}\{\backslash partial\; z^2\}$$

where $\backslash alpha\_*$ is the critical value of Landau derivative $\backslash alpha\; =\; (c^4/2\backslash upsilon^3)(\backslash partial^2\; \backslash upsilon/\backslash partial\; p^2)\_s$1942, Landau, L.D. "On shock waves" J. Phys. USSR 6 229-230Thompson, P. A. (1971). A fundamental derivative in gasdynamics. The Physics of Fluids, 14(9), 1843-1849. and $\backslash upsilon=1/\backslash rho$ is the specific volume. The transonic flow is completely characterized by the single parameter $\backslash alpha\_*$, which for polytropic gas takes the value $\backslash alpha\_*=\backslash alpha=(\backslash gamma+1)/2$. Under hodograph transformation, the transonic equation in two-dimensions becomes the Euler–Tricomi equation.

$$\backslash frac\{\backslash partial\backslash rho\}\{\backslash partial\; t\}\; +\; \backslash rho\; \backslash nabla\backslash cdot\backslash mathbf\; v\; +\; \backslash mathbf\; v\backslash cdot\backslash nabla\; \backslash rho\; =\; 0,\; \backslash quad\; \backslash frac\{\backslash partial\backslash mathbf\; v\}\{\backslash partial\; t\}+\; (\backslash mathbf\; v\; \backslash cdot\backslash nabla)\backslash mathbf\; v=\; -\backslash frac\{1\}\{\backslash rho\}\backslash nabla\; p\; =-\backslash frac\{c^2\}\{\backslash rho\}\backslash nabla\; \backslash rho=-\backslash nabla\; h.$$

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

$$\backslash frac\{\backslash partial\backslash varphi\}\{\backslash partial\; t\}\; +\; \backslash frac\{v^2\}\{2\}\; +\; h\; =\; f(t),\; \backslash quad\; \backslash Rightarrow\; \backslash quad\; \backslash frac\{\backslash partial\; h\}\{\backslash partial\; t\}\; =\; -\backslash frac\{\backslash partial^2\backslash varphi\}\{\backslash partial\; t^2\}\; -\; \backslash frac\{1\}\{2\}\backslash frac\{\backslash partial\; v^2\}\{\backslash partial\; t\}\; +\; \backslash frac\{df\}\{dt\}$$

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

$$\backslash frac\{\backslash partial^2\backslash varphi\}\{\backslash partial\; t^2\}\; +\; \backslash frac\{\backslash partial\; v^2\}\{\backslash partial\; t\}=c^2\backslash nabla\backslash cdot\backslash mathbf\; v\; -\; \backslash mathbf\; v\backslash cdot\; (\backslash mathbf\; v\; \backslash cdot\; \backslash nabla)\backslash mathbf\; v.$$

Substituting here $\backslash mathbf\; v=\backslash nabla\backslash varphi$ results in

$$\backslash varphi\_\{tt\}\; +\; (\backslash varphi\_x^2+\; \backslash varphi\_y^2+\; \backslash varphi\_z^2)\_t=\; (c^2-\backslash varphi\_x^2)\backslash varphi\_\{xx\}+(c^2-\backslash varphi\_y^2)\backslash varphi\_\{yy\}+(c^2-\backslash varphi\_z^2)\backslash varphi\_\{zz\}-2(\backslash varphi\_x\backslash varphi\_y\backslash varphi\_\{xy\}+\backslash varphi\_y\backslash varphi\_z\backslash varphi\_\{yz\}+\backslash varphi\_z\backslash varphi\_x\backslash phi\_\{zx\}).$$

Nearly parallel flows: As in before, for nearly parallel flows, we can write (after introducing a recaled time $\backslash tau=c\_\backslash infty\; t$)

$$\backslash frac\{\backslash partial^2\backslash phi\}\{\backslash partial\; \backslash tau^2\}\; +\; 2M\; \backslash frac\{\backslash partial^2\backslash phi\}\{\backslash partial\; x\backslash partial\backslash tau\}=\; (1-M^2)\; \backslash frac\{\backslash partial^2\backslash phi\}\{\backslash partial\; x^2\}\; +\; \backslash frac\{\backslash partial^2\backslash phi\}\{\backslash partial\; y^2\}\; +\; \backslash frac\{\backslash partial^2\backslash phi\}\{\backslash partial\; z^2\}$$

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

$$\backslash frac\{\backslash partial^2\backslash phi\}\{\backslash partial\; \backslash tau^2\}\; +\; 2\backslash frac\{\backslash partial^2\backslash phi\}\{\backslash partial\; x\backslash partial\backslash tau\}\; =\; -2\backslash alpha\_*\backslash frac\{\backslash partial\backslash phi\}\{\backslash partial\; x\}\; \backslash frac\{\backslash partial^2\backslash phi\}\{\backslash partial\; x^2\}\; +\; \backslash frac\{\backslash partial^2\backslash phi\}\{\backslash partial\; y^2\}\; +\; \backslash frac\{\backslash partial^2\backslash phi\}\{\backslash partial\; z^2\}.$$

Sound waves: In sound waves, the velocity magnitude $v$ (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, $c$ is a constant (for example, in polytropic gas $c^2=(\backslash gamma-1)h\_0$), we haveLamb (1994) §287, pp. 492–495.

$$\backslash frac\{\backslash partial^2\; \backslash varphi\}\{\backslash partial\; t^2\}\; =\; c^2\; \backslash nabla^2\; \backslash varphi,$$

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.

 z &= x + iy \,, \text{ and } &
 w &= \varphi + i\psi \,.
     

 f(x + iy) &= \varphi + i\psi \,, \text{ or } &
      f(z) &= w \,.
     

 \frac{\partial\varphi}{\partial x} &=  \frac{\partial\psi}{\partial y} \,, &
 \frac{\partial\varphi}{\partial y} &= -\frac{\partial\psi}{\partial x} \,.
     

 u &= \frac{\partial\varphi}{\partial x} = \frac{\partial\psi}{\partial y}, &
 v &= \frac{\partial\varphi}{\partial y} = -\frac{\partial\psi}{\partial x} \,.
     

 \Delta\varphi &= \frac{\partial^2\varphi}{\partial x^2} + \frac{\partial^2\varphi}{\partial y^2} = 0 \,,\text{ and } &
    \Delta\psi &= \frac{\partial^2\psi}{\partial x^2} + \frac{\partial^2\psi}{\partial y^2} = 0 \,.
     

 \nabla \varphi \cdot \nabla \psi =
 \frac{\partial\varphi}{\partial x} \frac{\partial\psi}{\partial x} + \frac{\partial\varphi}{\partial y} \frac{\partial\psi}{\partial y} =
 \frac{\partial \psi}{\partial y} \frac{\partial \psi}{\partial x} - \frac{\partial \psi}{\partial x} \frac{\partial \psi}{\partial y} =
 0 \,.
     

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

                \psi = \frac{-A y}{r^2} &= \frac{-A y}{x^2 + y^2} \,, \\
           x^2 + y^2 + \frac{A y}{\psi} &= 0 \,, \\
 x^2 + \left(y+\frac{A}{2\psi}\right)^2 &= \left(\frac{A}{2\psi}\right)^2 \,.