Chapter 6 Potential flow

Contours of constant stream function are streamlines of the flow. To see this, consider the directional derivative of the stream function in the direction of the streamline, i.e. in the direction of velocity

$\mathbf{\nabla }\psi \cdot 𝒖={\displaystyle \frac{\partial \psi }{\partial x}}u+{\displaystyle \frac{\partial \psi }{\partial y}}v=-vu+uv=0.$

(6.11)

Thus, irrespective of location in the flow, $\psi$ does not vary in the direction of streamlines. Therefore, $\psi$ is a constant along a streamline. In other words, streamlines are contours of constant stream function.

Contours of constant stream function and those of constant velocity potential meet orthogonally. To see this, consider the dot product of $\mathbf{\nabla }\varphi$ and $\mathbf{\nabla }\psi$, the normals of the contours at a given point:

$\mathbf{\nabla }\varphi \cdot \mathbf{\nabla }\psi ={\displaystyle \frac{\partial \psi }{\partial x}}{\displaystyle \frac{\partial \varphi }{\partial x}}+{\displaystyle \frac{\partial \psi }{\partial y}}{\displaystyle \frac{\partial \varphi }{\partial y}}=0.$

(6.12)

Since the normals are orthogonal, the tangents must be too, and therefore the curves must be too.

Consider the complex potential

$w(z)=U{e}^{-i\alpha }z⟹\varphi =U(x\cos \alpha +y\sin \alpha ),\psi =U(y\cos \alpha -x\sin \alpha ).$

(6.13)

The flow velocity corresponding to this potential is

$u=U\cos \alpha ,v=U\sin \alpha ,$

(6.14)

i.e., a uniform flow at an angle $\alpha$ to the $x$-axis. An example of this flow is shown in Figure 6.1(a).

Consider the complex potential

$w(z)={\displaystyle \frac{Q}{2\pi }}\log z⟹\varphi ={\displaystyle \frac{Q}{2\pi }}\log r,\psi ={\displaystyle \frac{Q\theta }{2\pi }},$

(6.15)

where $z=r^{i\theta }$ is the polar representation of the complex variable $z$. The flow velocity corresponding to this potential is

$𝒖={\displaystyle \frac{Q}{2\pi r}}{\widehat{𝒆}}_r,$

(6.16)

i.e., a radially outward flow decaying inversely proportional with distance. An example of this flow is shown in Figure 6.1(b). For what follows, let $𝒖\cdot {\widehat{𝒆}}_r=u_r$ and $𝒖\cdot {\widehat{𝒆}}_{\theta }=u_{\theta }$.

To understand why the flow corresponding to this potential is termed as the point source, consider the divergence and curl of the velocity field:

	$\mathbf{\nabla }\cdot 𝒖={\displaystyle \frac{1}{r}}\left({\displaystyle \frac{\partial (r{u}_r)}{\partial r}}+{\displaystyle \frac{\partial u_{\theta }}{\partial \theta }}\right)=0,$		(6.17)
	$\mathbf{\nabla }\times 𝒖={\displaystyle \frac{1}{r}}\left({\displaystyle \frac{\partial (r{u}_{\theta })}{\partial r}}-{\displaystyle \frac{\partial u_r}{\partial \theta }}\right){\widehat{𝒆}}_z=\mathrm{𝟎}$		(6.18)

for $r>0$, but this calculation says nothing about $r=0$. To deduce that, consider the line integrals along the circle of radius $r$

	${\displaystyle {\int }_C}𝒖\cdot \widehat{𝒏}𝑑s={\displaystyle {\int }_0^{2\pi }}{\displaystyle \frac{Q}{2\pi r}}{\widehat{𝒆}}_r\cdot {\widehat{𝒆}}_{r}r𝑑\theta =Q,$		(6.19)
	${\displaystyle {\int }_C}𝒖\cdot \widehat{𝒕}𝑑s={\displaystyle {\int }_0^{2\pi }}{\displaystyle \frac{Q}{2\pi r}}{\widehat{𝒆}}_r\cdot {\widehat{𝒆}}_{\theta }r𝑑\theta =0,$		(6.20)

where $\widehat{𝒏}$ and $\widehat{𝒕}$ are the unit normal and tangential vectors to the curve $C$. Given that, due to the divergence and Stokes theorems, respectively,

	${\displaystyle {\int }_C}𝒖\cdot \widehat{𝒏}𝑑s={\displaystyle {\int }_A}\mathbf{\nabla }\cdot 𝒖𝑑A,\text{and}$		(6.21)
	${\displaystyle {\int }_C}𝒖\cdot \widehat{𝒏}ds={\displaystyle {\int }_A}(\mathbf{\nabla }\times \cdot 𝒖)\cdot {\widehat{𝒆}}_{z}dA,$		(6.22)

the inesacapable conclusion is that for this flow $\mathbf{\nabla }\cdot 𝒖=Q\delta (𝒙)$, where $\delta$ is the Dirac-delta function and $\mathbf{\nabla }\times 𝒖=\mathrm{𝟎}$. Thus, this flow is irrotational everywhere, but incompressible only for $r>0$. Volume is not conserved at $r=0$, but instead is generated there. Hence the terminology to describe this flow as a “point source”.

Consider the complex potential

$w(z)=-{\displaystyle \frac{i\mathrm{\Gamma }}{2\pi }}\log z⟹\varphi ={\displaystyle \frac{\mathrm{\Gamma }\theta }{2\pi }},\psi =-{\displaystyle \frac{\mathrm{\Gamma }}{2\pi }}\log r.$

(6.23)

The flow velocity corresponding to this potential is

$𝒖={\displaystyle \frac{Q}{2\pi r}}{\widehat{𝒆}}_{\theta },$

(6.24)

i.e., an azimuthal flow decaying inversely proportional with distance. An example of this flow is shown in Figure 6.1(c).

We again examine $\mathbf{\nabla }\cdot 𝒖$ and $\mathbf{\nabla }\times 𝒖$ to understand the terminology associated with this flow. Just as for the point source,

	$\mathbf{\nabla }\cdot 𝒖={\displaystyle \frac{1}{r}}\left({\displaystyle \frac{\partial (r{u}_r)}{\partial r}}+{\displaystyle \frac{\partial u_{\theta }}{\partial \theta }}\right)=0,$		(6.25)
	$\mathbf{\nabla }\times 𝒖={\displaystyle \frac{1}{r}}\left({\displaystyle \frac{\partial (r{u}_{\theta })}{\partial r}}-{\displaystyle \frac{\partial u_r}{\partial \theta }}\right){\widehat{𝒆}}_z=\mathrm{𝟎}$		(6.26)

for $r>0$, but which says nothing about $r=0$. The line integrals along the circle of radius $r$

	${\displaystyle {\int }_C}𝒖\cdot \widehat{𝒏}𝑑s={\displaystyle {\int }_0^{2\pi }}{\displaystyle \frac{\mathrm{\Gamma }}{2\pi r}}{\widehat{𝒆}}_{\theta }\cdot {\widehat{𝒆}}_{r}r𝑑\theta =0,$		(6.27)
	${\displaystyle {\int }_C}𝒖\cdot \widehat{𝒕}𝑑s={\displaystyle {\int }_0^{2\pi }}{\displaystyle \frac{\mathrm{\Gamma }}{2\pi r}}{\widehat{𝒆}}_{\theta }\cdot {\widehat{𝒆}}_{\theta }r𝑑\theta =\mathrm{\Gamma },$		(6.28)

which, in view of (6.21-6.22) lead to the inescable conclusion that the flow is incompressible everywhere but irrotational only for $r>0$. The vorticity is, in fact, $\mathbf{\nabla }\times 𝒖=\mathrm{\Gamma }\delta (𝒙){\widehat{𝒆}}_z$. Hence the flow due to this potential is termed as the “point vortex”.

The flow constructed by the linear superposition of a point source and an equal point sink (i.e., a source with negative strength) located vanishingly small distances apart constitutes a point dipole. The complex potential is constructed as

$w(z)=\underset{ϵ\to 0}{lim}{\displaystyle \frac{Q}{2\pi }}\left(\log (z+ϵ)-\log (z-ϵ)\right).$

(6.29)

Of course, in the limit $ϵ\to 0$, the potential would vanish, unless the source-sink pair also strengthened in the process. In particular, let $Q=D/(2ϵ)$, so that the source-sink pair strength is inversely proportional to the distance between them. Here $D$ is the strength of the point dipole. With this assumption,

$w(z)=\underset{ϵ\to 0}{lim}{\displaystyle \frac{D}{2\pi }}\left({\displaystyle \frac{\log (z+ϵ)-\log (z-ϵ)}{2ϵ}}\right)={\displaystyle \frac{D}{2\pi z}}.$

(6.30)

The flow due to the point dipole is depicted in Figure 6.1(d). The streamlines and equipotential lines are both sets of circles that are tangential to the $x$ and $y$ axes, respectively, at the origin.

Note that, due to its construction, the potential due to a point dipole is the derivative of the potential due to a point source. This can be extended to higher orders called “multipoles”. For example, a point quadrupole is a point dipole of point dipoles with a complex potential proportional to $1/z^2$. And octupole is a point dipole of quadrupoles with potential $1/z^3$. And so on.

In this manner, the general complex potential with the farfield approaching uniform flow may be written as

$w(z)=Uz+{\displaystyle \frac{Q-i\mathrm{\Gamma }}{2\pi }}\log z+{\displaystyle \frac{a_1}{z}}+{\displaystyle \frac{a_2}{z^2}}+{\displaystyle \frac{a_3}{z^3}}+\mathrm{\dots }.$

(6.31)

Here, the coefficients $U$, $Q$, $\mathrm{\Gamma }$, $a_1$, $a_2$, etc, may be functions of time.

The linear superposition of a uniform flow and a point source may be used to represent flow past a semi-infinite body, called the Rankine half-body. The complex potential is

$w(z)=Uz+{\displaystyle \frac{Q}{2\pi }}\log z.$

(6.32)

The flow streamlines and equipotential lines are shown in Figure 6.2(a).

Any region of the plane on one side of a streamline may be replaced by a solid body and the complex potential in the remaining region describs potential flow past this solid body. For example, consider the streamline given by $\psi =Q/2$, where

$\psi =\text{Im}(w)=Ur\sin \theta +{\displaystyle \frac{Q\theta }{2\pi }}\text{so the streamline is}r={\displaystyle \frac{Q(\pi -\theta )}{2\pi U\sin \theta }}.$

(6.33)

The streamline is shown in Figure 6.2(b) and the area inside the streamline, shaded blue in the figure, is the Rankine half-body. The complex potential $w(z)$ in (6.32) describes the flow past this Rankine half-body.

The linear superposition of a uniform flow, a point source and a point sink may be used to represent flow past a finite body, called the Rankine oval. The complex potential is

$w(z)=Uz+{\displaystyle \frac{Q}{2\pi }}\left(\log (z-1)-\log (z+1)\right).$

(6.34)

The flow streamlines and equipotential lines are shown in Figure 6.3(a).

Consider the streamline given by $\psi =0$, where

$\psi =\text{Im}(w)=Ur\sin \theta +{\displaystyle \frac{Q({\theta }_1-{\theta }_2)}{2\pi }}\text{so the streamline is given by}Ur\sin \theta +{\displaystyle \frac{Q({\theta }_1-{\theta }_2)}{2\pi }}=0,$

(6.35)

where $\tan {\theta }_1=r\sin \theta /(r\cos \theta +1)$ and $\tan {\theta }_2=r\sin \theta /(r\cos \theta -1)$. The streamline is shown in Figure 6.3(b) and the area inside the streamline, shaded blue in the figure, is the Rankine oval. The complex potential $w(z)$ in (6.34) describes the flow past this Rankine oval.

The Rankine oval approaches a circle as the source and the sink approach each other to form a dipole. The complex potential in this case is

$w(z)=U\left(z+{\displaystyle \frac{a^2}{z}}\right).$

(6.36)

The flow streamlines and equipotential lines are shown in Figure 6.4(a).

Consider the streamline given by $\psi =0$, where

$\psi =\text{Im}(w)=U\sin \theta \left(r-{\displaystyle \frac{a^2}{r}}\right),\text{ so that }\psi =0\text{ on}r=a.$

(6.37)

The streamline is shown in Figure 6.4(b) and the area inside the streamline, shaded blue in the figure, is a circle of radius $a$. The complex potential $w(z)$ in (6.36) describes the flow past this circle.

An arbitrary point vortex centered at the center of the circle may be added to the mix without violating the boundary condition on the circle. The most general potential flow around a circle is given by

$w(z)=U\left(z+{\displaystyle \frac{a^2}{z}}\right)-{\displaystyle \frac{i\mathrm{\Gamma }}{2\pi }}\log z.$

(6.38)

Here $\mathrm{\Gamma }$ is the circulation around the circle, perhaps due to the spinning of the circle. The flow due to this potential is shown in Figure 6.4(c).

In steady potential flow, only the term $\rho {|𝒖|}^2/2$ contributes to the net unbalanced force beyond buoyancy. In complex variables, let the components of the force $𝑭=F_x{\widehat{𝒆}}_x+F_y{\widehat{𝒆}}_y$ be written as $F_x+i{F}_y$. If the tangent to $C$ makes an angle $\theta$ to the $x$-axis, then $𝒖=|𝒖|e^{i\theta }$ because the flow is tangential to the streamline. Also, because the components of velocity are $w^{\prime }(z)=u-iv=|𝒖|e^{-i\theta }$, ${|𝒖|}^2=w^{\prime }{(z)}^2{e}^{2i\theta }$. The product of the unit normal and the infinitesimal line element is $-i{e}^{i\theta }ds$. The complex conjugate of the force is

$F_x-i{F}_y={\displaystyle {\int }_C}{\displaystyle \frac{1}{2}}\rho w^{\prime }{(z)}^2{e}^{2i\theta }(i{e}^{-i\theta }ds)={\displaystyle \frac{i\rho }{2}}{\displaystyle {\int }_C}{w}^{\prime }{(z)}^2𝑑z,$

(6.41)

where the complex integration infinitesimal $dz=e^{i\theta }ds$.

Using this expression for force, and substituting (6.31) in (6.41) yields

$F_x-i{F}_y=-\rho U(Q-i\mathrm{\Gamma }),$

(6.42)

where Cauchy’s residue theorem is employed for calculating the contour integral. Law of conservation of mass demands that for a body immersed in the flow $Q=0$, and therefore,

$F_x=0\text{and}{F}_y=-\rho U\mathrm{\Gamma }.$

(6.43)

The drag on an object is the force it experiences in the direction of the flow, while the lift is the force perpendicular to the flow. According to (6.43), the lift on an immersed body due to the potential flow is $\rho U\mathrm{\Gamma }$, which is the statement of the Kutta-Joukowsky (also spelled Joukowski, Zhukowski or Zhukovsky) theorem in aerodynamics. The lift on the spinning circle, due to this theorem, is $\rho U\mathrm{\Gamma }$.

The fact that a body immersed in a potential flow does not experience any drag is the subject of the D’Alembert paradox. Neither the Rankine oval, nor the circle, nor body of any other shape experiences any drag in potential flow. D’Alembert paradox exist because potential flow does not satisfy the no-slip condition. The friction of a viscous fluid with the solid boundary generates vorticity, which is transported by the fluid to regions away from the immersed body. In this manner, application of the no-slip boundary condition on viscous fluids invalidates the assumptions underlying potential flow, especially that the flow be irrotational. In fact, the shedding of vorticity into the flow is the origin of fluid dynamical drag (this we state without proof at this point).

Consider the contribution of the term $\rho {\displaystyle \frac{\partial \varphi }{\partial t}}$. The resulting force is

${\displaystyle {\int }_C}\rho {\displaystyle \frac{\partial \varphi }{\partial t}}\widehat{𝒏}𝑑s.$

(6.44)

Let us examine this expression for the case of flow around a circle, so that $w(z)$ is given by (6.36). This corresponds to

$\varphi =\text{Re}(w)=U\cos \theta \left(r+{\displaystyle \frac{a^2}{r}}\right),\text{which on }r=a\text{ is}\varphi =2Ua\cos \theta .$

(6.45)

The integral for force yields

$𝑭={\displaystyle {\int }_0^{2\pi }}2\rho {\displaystyle \frac{dU}{dt}}a\cos \theta (\cos \theta {\widehat{𝒆}}_x+\sin \theta {\widehat{𝒆}}_y)a𝑑\theta =\rho \pi a^2{\displaystyle \frac{dU}{dt}}{\widehat{𝒆}}_x.$

(6.46)

This result may be interpreted in the following way. Any acceleration of the circle implies an acceleration of the fluid around it. This acceleration needs an additional force proportional to the acceleration. The coeffient of proportionality between this additional force and the acceleration is the “added mass” the fluid presents to the motion of the circle. In this case, the added mass in $\rho \pi a^2$.