Irrotational Flow

Flow is said to be irrotational when the vorticity $\omega$ has the magnitude zero everywhere. It immediately follows, from Equation (4.77), that the circulation around any arbitrary loop in an irrotational flow pattern is zero (provided that the loop can be spanned by a surface that lies entirely within the fluid). Hence, from Kelvin's circulation theorem, if an inviscid fluid is initially irrotational then it remains irrotational at all subsequent times. This can be seen more directly from the equation of motion of an inviscid incompressible fluid which, according to Equations (1.39) and (1.79), takes the form

$$\frac{\partial{\bf v}}{\partial t} + ({\bf v}\cdot \nabla)\,{\bf v} = - \nabla\left(\frac{p}{\rho}+{\mit\Psi}\right),$$ (4.82)

because $\rho$ is a constant. However, from Equation (A.171),

$$({\bf v}\cdot\nabla)\,{\bf v} =\nabla(v^{\,2}/2) -{\bf v}\times \mbox{\boldmath$\omega$} .$$ (4.83)

Thus, we obtain

$$\frac{\partial {\bf v}}{\partial t} = -\nabla\left(\frac{p}{\rho}+\frac{1}{2}\,v^{\,2}+{\mit\Psi}\right)+ {\bf v}\times \mbox{\boldmath$\omega$} .$$ (4.84)

Taking the curl of this equation, and making use of the vector identities $\nabla\times \nabla\phi\equiv 0$ [see Equation (A.176)], $\nabla\cdot\nabla\times {\bf A}\equiv {\bf0}$ [see Equation (A.173)], as well as the identity (A.179), and the fact that $\nabla\cdot{\bf v}=0$ in an incompressible fluid, we obtain the vorticity evolution equation

$$\frac{D{\mbox{\boldmath$\omega$}}}{Dt} = (\mbox{\boldmath$\omega$}\cdot\nabla)\,{\bf v}.$$ (4.85)

Thus, if $\omega$ $={\bf0}$ , initially, then $D$$\omega$ $/Dt={\bf0}$ , and, consequently, $\omega$ $={\bf0}$ at all subsequent times.

Suppose that $O$ is a fixed point, and $P$ an arbitrary movable point, in an irrotational fluid. Let $O$ and $P$ be joined by two different paths, $OAP$ and $OBP$ (say). It follows that $OAPBO$ is a closed curve. Because the circulation around such a curve in an irrotational fluid is zero, we can write

$$\int_{OAP} {\bf v}\cdot d{\bf r} + \int_{PBO} {\bf v}\cdot d{\bf r} = 0,$$ (4.86)

which implies that

$$\int_{OAP} {\bf v}\cdot d{\bf r} =\int_{OBP} {\bf v}\cdot d{\bf r} = -\phi_P$$ (4.87)

(say). It is clear that $\phi_P$ is a scalar function whose value depends on the position of $P$ (and the fixed point $O$ ), but not on the path taken between $O$ and $P$ . Thus, if $O$ is the origin of our coordinate system, and $P$ an arbitrary point whose position vector is ${\bf r}$ , then we have effectively defined a scalar field $\phi({\bf r})=-\int_O^P {\bf v}\cdot d{\bf r}$ .

Consider a point $Q$ that is sufficiently close to $P$ that the velocity ${\bf v}$ is constant along $PQ$ . Let $\eta$ be the position vector of $Q$ relative to $P$ . It then follows that (see Section A.18)

$$- \mbox{\boldmath$\eta$} \cdot\nabla\phi = -\phi_Q+\phi_P = \int_{P}^Q {\bf v}\cdot d{\bf r} \simeq {\bf v}\cdot \mbox{\boldmath$\eta$} .$$ (4.88)

The previous equation becomes exact in the limit that $\vert$$\eta$ $\vert\rightarrow 0$ . Because $Q$ is arbitrary (provided that it is sufficiently close to $P$ ), the direction of the vector $\eta$ is also arbitrary, which implies that

$${\bf v} = -\nabla\phi.$$ (4.89)

We, thus, conclude that if the motion of a fluid is irrotational then the associated velocity field can always be expressed as minus the gradient of a scalar function of position, $\phi({\bf r})$ . This scalar function is called the velocity potential, and flow which is derived from such a potential is known as potential flow. Note that the velocity potential is undefined to an arbitrary additive constant.

We have demonstrated that a velocity potential necessarily exists in a fluid whose velocity field is irrotational. Conversely, when a velocity potential exists the flow is necessarily irrotational. This follows because [see Equation (A.176)]

$$\mbox{\boldmath$\omega$} =\nabla\times {\bf v} = -\nabla\times\nabla\phi= {\bf0}.$$ (4.90)

Incidentally, the fluid velocity at any given point in an irrotational fluid is normal to the constant-$\phi$ surface that passes through that point.

If a flow pattern is both irrotational and incompressible then we have

$${\bf v} = -\nabla\phi$$ (4.91)

and

$$\nabla\cdot{\bf v} = 0.$$ (4.92)

These two expressions can be combined to give (see Section A.21)

$$\nabla^{\,2}\phi = 0.$$ (4.93)

In other words, the velocity potential in an incompressible irrotational fluid satisfies Laplace's equation.

According to Equation (4.84), if the flow pattern in an incompressible inviscid fluid is also irrotational, so that $\omega$ $={\bf0}$ and ${\bf v}=-\nabla\phi$ , then we can write

$$\nabla\left(\frac{p}{\rho} + \frac{1}{2}\,v^{\,2} + {\mit\Psi} - \frac{\partial\phi}{\partial t}\right)=0,$$ (4.94)

which implies that

$$\frac{p}{\rho} + \frac{1}{2}\,v^{\,2} + {\mit\Psi} - \frac{\partial\phi}{\partial t} = C(t),$$ (4.95)

where $C(t)$ is uniform in space, but can vary in time. In fact, the time variation of $C(t)$ can be eliminated by adding the appropriate function of time (but not of space) to the velocity potential, $\phi$ . Note that such a procedure does not modify the instantaneous velocity field ${\bf v}$ derived from $\phi$ . Thus, the previous equation can be rewritten

$$\frac{p}{\rho} + \frac{1}{2}\,v^{\,2} + {\mit\Psi} - \frac{\partial\phi}{\partial t} = C,$$ (4.96)

where $C$ is constant in both space and time. Expression (4.96) is a generalization of Bernoulli's theorem (see Section 4.3) that takes non-steady flow into account. However, this generalization is only valid for irrotational flow. For the special case of steady flow, we get

$$\frac{p}{\rho} + \frac{1}{2}\,v^{\,2} + {\mit\Psi} = C,$$ (4.97)

which demonstrates that for steady irrotational flow the constant in Bernoulli's theorem is the same on all streamlines. (See Section 4.3.)