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Mathematics of fluid dynamics University of Bath 25-26, MA32051/MA52129

 

Section 4.1 The velocity potential

In this chapter, we focus on 2D flows where the velocity vector is given by
\begin{equation*} \bu(x, y, t) = u(x, y, t)\bi + v(x, y, z)\bj = [u, v]. \end{equation*}
With the velocity given as above, the vorticity is then
\begin{equation} \bomega = \nabla \times \bu = \left(\pd{v}{x} - \pd{u}{y}\right)\bk.\tag{4.1.1} \end{equation}
If we assume that the flow is irrotational according to Definition 3.5.7, then \(\nabla \times \bu = 0\) and
\begin{equation} \pd{v}{x} - \pd{u}{y} = 0.\tag{4.1.2} \end{equation}
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Further, we know that if the flow is irrotational, then there exists a velocity potential, \(\phi\text{,}\) such that \(\bu = \nabla \phi\text{.}\) Thus the velocities are expressed as
\begin{equation} u = \pd{\phi}{x} \quad \textrm{and} \quad v = \pd{\phi}{y}.\tag{4.1.3} \end{equation}
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The above result about irrotational flows is a standard result in Vector Calculus, but we will re-state the result here for reference, and provide a review of its proof.
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Proof.

Define
\begin{equation*} \phi(\bx, t) \equiv \phi_0(t) + \int_C \bu \cdot \de{x}, \end{equation*}
where \(C\) is any contour connecting an arbitrary origin point to the point \(\bx\) (changing the origin point will change the "constant" of integration \(\phi_0(t)\)).
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We can verify, using the definition of differentiation, and the fundamental theorme of calculus, applied along each of the three coordinate directions, that \(\nabla \phi = \bu\) as desired.
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The key is to prove that the above definition is unique, regardless of the choice of contour \(C\text{.}\) To this end, consider two contours, \(C_1\) and \(C_2\text{,}\) both with the same origin point, \(O\text{,}\) and end point \(P\text{.}\) Then the contour \(C_1 - C_2\) is a close contour beginning and ending at \(O\text{.}\)
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Figure 4.1.2. Proof of the uniqueness of the potential
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By Stokes’ theorem,
\begin{equation*} \int_{C_1 - C_2} \bu \cdot \de{x} = \iint_S (\nabla \times \bu) \, \bn \, \de{S}, \end{equation*}
where \(S\) is any surface with bounding curve \(C_1 - C_2\text{,}\) and with unit normal \(\bn\) positively oriented with the bounding curve. However, the right hand-side is zero by irrotationality, and therefore
\begin{equation*} \int_{C_1} \bu \cdot \de{x} = \int_{C_2} \bu \cdot \de{x}, \end{equation*}
and our choice of curve in the definition of the potential is irrelevant.
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The converse direction of the theorem follows directly from the fact that "curl grad equals zero", i.e. \(\nabla \times \nabla \phi = 0\text{.}\)
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Let us return to discussing the setting of potential flow.
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In addition to being irrotational, we furthermore have assumed that the flow is incompressible. Therefore from Theorem 3.4.1,
\begin{equation} \nabla \cdot \bu = \pd{u}{x} + \pd{v}{y} = \pd{^2\phi}{x^2} + \pd{^2\phi}{y^2} = 0.\tag{4.1.4} \end{equation}
This is the crucial result, which is that in potential flows, we need only solve the Laplace equation:
\begin{equation} \nabla^2 \phi = 0,\tag{4.1.5} \end{equation}
within the flow region. This is effectively a single linear equation for the single unknown \(\phi\text{.}\) However, for different problems, the boundary conditions can render even this "simple" problem difficult.
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Once the velocity potential \(\phi\) has been solved, the velocities in the flow can be recovered from the relationship (4.1.3). The pressure in the flow also follows from Bernoulli’s equation. For the situation of a steady potential flow, following Theorem 3.5.9, it is
\begin{equation} \frac{p}{\rho} + \frac{1}{2} |\nabla \phi|^2 + \chi = \textrm{constant}.\tag{4.1.6} \end{equation}
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Subsection 4.1.1 Elementary flows

The next three examples will introduce you to the elementary flows consisting of uniform flow, stagnation point flow, and line source/sink flows. You will also investigate the notion of a source "strength".
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Example 4.1.3. Uniform flow.

Consider the potential given by the linear function
\begin{equation*} \phi(x, y) = Ux \cos\alpha + Uy \sin\alpha, \end{equation*}
with constants \(U\) and \(\alpha\text{.}\) Then by differentiation we have that the velicity is
\begin{equation*} \bu = U[\cos\alpha, \sin\alpha]. \end{equation*}
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The image shown below shows the streamlines of the flow.
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Figure 4.1.4. Streamlines (or velocity field) of uniform flow with \(U = 1\) and \(\alpha = \pi/4\text{.}\)
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Example 4.1.5. Stagnation point flow.

We can verify that the velocity potential
\begin{equation*} \phi = \frac{1}{2} (x^2 - y^2), \end{equation*}
satisfies Laplace’s equation. The corresponding velocity field is given by
\begin{equation*} [u, v] = [x, -y]. \end{equation*}
and corresponds to stagnation point flow.
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The streamlines (or velocity field) is shown below.
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Figure 4.1.6. Streamlines (or velocity field) of stagnation point flow.
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Example 4.1.7. Line source.

We aim to derive the potential and velocity for a line source, imagined as the flow consisting of a point source or point sink that ejects/drains fluid from a point in space. Since it would be expected for the potential to be axisymmetric, we attempt to solve \(\nabla^2 \phi = 0\) in plane polar coordinates. This is given by
\begin{equation*} \nabla^2 \phi = \frac{1}{r} \pd{}{r}\left(r \pd{\phi}{r}\right) + \frac{1}{r^2} \pd{^2 \phi}{\theta^2} = 0. \end{equation*}
We assume that the potential takes the form \(\phi = \phi(r)\text{.}\) Then direct integration gives
\begin{equation*} \phi = \frac{Q}{2\pi} \log r, \end{equation*}
where we have set an additional constant of integration to zero without loss of generality. The leading constant has been set to \(Q/(2\pi)\) so that \(Q\) can be later identified with a physical quantity.
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The velocity then follows from consideration of the gradient in polar form,
\begin{equation*} \bu = \nabla \phi = \pd{\phi}{r} \be_r + \frac{1}{r}\pd{\phi}{\theta} \be_{\theta} = \frac{Q}{2\pi r} \be_r, \end{equation*}
where the unit vectors written in the Cartesian basis are \(\be_r = [\cos\theta, \sin\theta]\) and \(\be_{\theta} = [-\sin\theta, \cos\theta]\text{.}\) Thus we can write the velocity as
\begin{equation*} \bu = \frac{Q}{2\pi r^2} r[\cos\theta, \sin\theta] = \frac{Q}{2\pi r^2} [x, y]. \end{equation*}
The above corresponds to a velocity field directed radially outwards from the origin. The flow is a called a line source because fluid is ejected from the origin (a source). It refers to a "line" because in \((x, y, z)\text{,}\) the source runs parallel to the \(z\)-axis.
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The streamlines (or velocity field) are shown below.
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Figure 4.1.8. Streamlines (or velocity field) of line source flow.
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Let us also identify the strength of this line source. Consider a closed contour \(C\) around the origin. Then the quantity
\begin{equation*} \int_C \bu \cdot \bn \, \de{s}, \end{equation*}
is the flux (the flow per unit time) of fluid crossing the contour, with \(\bn\) denoting the unit normal along \(C\text{.}\)
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For simplicity, let us take the contour \(C\) to be a circle of constant radius \(r = a\text{.}\) Then since the unit normal is precisely \(\be_r\text{,}\) we have that
\begin{equation*} \int_C \bu \cdot \bn \, \de{s} = \int_0^{2\pi} \frac{Q}{2\pi a} \be_r \cdot \be_r \, (a \, \de\theta) = Q. \end{equation*}
In computing the above integral, remember that the conversion following the polar Jacobian is \(\de{s} = r \de{\theta}\) where \(r = a\text{.}\)
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Therefore, \(Q\) is the rate at which fluid is produced from the line source. If \(Q \lt 0\text{,}\) we refer to the flow as a line sink.
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Crucially, because the governing fluid mechanical equation is only Laplace’s equation: this is a linear partial differential equation, and therefore the summation of elementary flows also produces an admissible flow.
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Example 4.1.9. Line source in a uniform flow.

For instance, we may combine a uniform flow in the \(x\)-direction with a line source:
\begin{equation*} \phi = Ux + \frac{Q}{2\pi} \log r = Ux + \frac{Q}{2\pi} \log \sqrt{x^2 + y^2}. \end{equation*}
We can then obtain the velocity field as
\begin{equation*} \bu = [U, 0] + \frac{Q}{2\pi(x^2 + y^2)} [x, y]. \end{equation*}
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The streamlines (or velocity field) is shown below.
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Figure 4.1.10. Streamlines (or velocity field) of a line source in a uniform flow with \(U = 1\) and \(Q = 1\text{.}\)
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Where do you think the stagnation point lies in this flow?
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