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Chapter 7
Potential flow theory
Flows past immersed bodies generate boundary layers within which the presence of the body is felt through viscous effects. Outside these boundary layers, the flow is typically inviscid and irrotational. Such flows are generally zeros of a Laplace operator and are called potential flows.
In cases where the Reynolds number is large enough, we can use techniques based on the fact that the boundary layer is asymptotically thin and we can patch it onto the outer potential flow (Chapter 6). While the viscous flow has to be solved for experimentally or numerically, simplifying theories for the outer flow exist.
Last, potential flow theory is not only limited to applications to the outer region in boundary layer flows. Very low Reynolds number flows, such as electroosmotic flows or creeping flows, the momentum equations can be reduced to Laplace equations and potential flow theory applies.
7.1 Basic concepts
7.1.1 Velocity potential
For an irrotational flow (∇ × u ≡ 0 ), we can define a velocity potential φ such that
u = ∇φ, (7.1)
or u = ∂xφ, v = ∂y φ, w = ∂z φ. (7.2)
The definition of the velocity through the velocity potential ensures the irrotationality of the flow: ∇ × u = ∇ × ∇φ = 0. (7.3)
The Euler equations for an inviscid flow writes:
ρ (∂tu + (u · ∇)u) = −∇p − ρg, (7.4)
for which we have kept the gravitational acceleration term for the sake of exhaustivity. These equations are nothing but the Navier–Stokes equations for which the viscous term has been dropped. Using vectorial calculus, we can express the advective term in a different way:
ρ
∂tu +
∇(u · u) + (∇ × u) × u
= −∇p − ρg. (7.5)
As the flow is irrotational, ∇ × u ≡ 0 and equation (7.5) becomes
ρ
∂tu +
∇(u · u)
which can be written with the velocity potential as follows:
ρ
∂t∇φ +
∇(u · u)
- ∇p + ∇ (ρg · x) = 0, (7.7)
where x = (x, y, z) is the position vector. We now assume g = gˆz and integrate the equation once to get the time-dependent Bernoulli equation:
ρ∂tφ +
ρ|u|^2 2
where |u|^2 = u · u = |∇φ|^2.
The incompressibility constraint, expressed with the velocity potential, writes
∇ · u = 0, (7.9) ∇ · ∇φ = 0, (7.10) ∇^2 φ = 0, (7.11)
which is the Laplace equation.
When one solves for a potential flow, the outer boundary conditions are written for the velocity (gradient of the velocity potential) while at the surface of the immersed structure, the boundary condition is a vanishing normal derivative of the velocity potential: ∂nφ = 0. This translates the fact that the immersed structure is an impenetrable boundary. All these boundary conditions turn out to involve derivatives. As a result, the solution to the Laplace equation is not unique and φ is defined to a constant. A common way to fix this ill-posedness is to set the average of φ to zero. Such a resetting does not impact the equations as the velocity potential only appears through its derivatives.
7.1.2 Streamfunction
For a two-dimensional incompressible flow, we can define a streamfunction ψ such that
u = ∂y ψ, v = −∂xψ. (7.12)
The same calculation can be carried out for the streamfunction with the line element defined as (dx′, dy′):
dψ = ∂xψdx′^ + ∂yψdy′, (7.23) ⇒ 0 = −vdx′^ + udy′, (7.24)
⇒ dy′^ =
vdx′ u
Taking the dot product between the potential line element and the streamline element, we find:
dx dx′^ + dy dy′^ = dx dx′^ −
udx v
vdx u
= dx dx′^ − dx dx′, (7.27) = 0, (7.28)
showing that potential lines and streamlines are orthogonal.
7.2 Elementary solutions
We illustrate the use of the velocity potential and of the streamfunction through a few examples of elementary solutions in the absence of a structure. In this section, the flow is two-dimensional, irrotational, incompressible and inviscid and obeys equations (7.11) and (7.16). As we shall see in Chapter 8, these equations (and associated boundary conditions) are linear and we can therefore add two solutions together to create a third solution. The solutions introduced here represent building blocks of potential flow theory.
7.2.1 Uniform streams
A uniform stream u = U ˆx is both incompressible (∇· u = 0) and irrotational (∇×u = 0). This flow therefore possesses both a velocity potential and a streamfunction.
Definition (7.2) yields U = ∂xφ, 0 = ∂y φ, (7.29)
indicating the velocity potential does not depend on y and that upon integrating the first equation we get φ = Ux, (7.30)
where we have set the constant of integration to zero.
In the same spirit, definition (7.12) yields
U = ∂y ψ, 0 = ∂xψ, (7.31)
which shows that the streamfunction is independent of x. It follows, upon integration of the first equation: ψ = Uy. (7.32)
No immersed structure is present to set the constant of integration, so we set it to zero.
The potential lines correspond to x = cst and the streamlines to y = cst. They are therefore orthogonal as shown in figure 7.1. Finally, it is straighforward to show that both
Figure 7.1: Uniform flow u = U ˆx represented using potential lines (x = cst, dashed lines) and streamlines (y = cst, solid pink lines). After White, Fluid Mechanics (2011).
φ and ψ are harmonic functions, i.e., that they solve ∇^2 φ = 0 and ∇^2 ψ = 0.
It might be interesting to consider a flow forming an angle α with the x coordinate. Definitions (7.2) and (7.12) imply:
U cos(α) = ∂xφ = ∂y ψ, U sin(α) = ∂y φ = −∂xψ, (7.33)
which gives the following expressions for the velocity potential and streamfunction:
φ = U(x cos(α) + y sin(α)), ψ = U(y cos(α) − x sin(α)). (7.34)
7.2.2 Sources and sinks
We inject some fluid at the origin (x = 0, y = 0). This creates an axisymmetric outward flow in cylindrical coordinates with no azimuthal velocity. This special point is called source. If, on the contrary, we suck up fluid at this point, the resulting velocity field is pointing inwards and the point is called sink.
In polar coordinates, the velocity components associated with a source or sink are:
ur =
Q
2 πr
, uθ = 0, (7.35)
where ur represents the radial velocity, Q the flow rate and uθ the azimuthal velocity. Note that the flow rate across a section r = cst in polar coordinates write:
Q =
∫ (^2) π
0
urrdθ. (7.36)
The fact we look for a constant flow rate, independent of r, explains the writing of the radial velocity in definition (7.35).
Similarly, the angle θ can be expressed in Cartesian coordinates: θ = tan−^1 y/x. The streamfunction corresponding to an offset source or sink then writes:
ψ =
Q
2 π
tan−^1
y − b x − a
Last, it might be easier to think in terms of the source/sink strength m = Q/ 2 π rather than the flow rate Q. Both writings are standard and can be used interchangeably.
7.2.3 Free vortices
The last elementary potential flow solution we introduce in this lecture is the irrotational vortex. The free vortex we consider here has a vanishing radial component of the velocity and the azimuthal component of the velocity only varies with r. We can then write:
ur = 0, uθ = f (r), (7.44)
Note that this flow naturally satisfies the incompressibility constraint. As we want it to be irrotational, we impose
∇ × u = 0, (7.45) ⇒ (^1) r [∂r(ruθ) − ∂θur] = 0. (7.46)
The solution of equation (7.46), provided the solution form in relations (7.44) is
ur = 0, uθ =
K
r
where K represents the strength of the vortex.
The definition of the velocity potential and of the streamfunction in polar coordinates writes:
0 = ∂r φ =
r
∂θ ψ,
K
r
r
∂θφ = −∂r ψ, (7.48)
which yields: φ = Kθ, ψ = −K ln r, (7.49)
where once again, the constants of integration have been dropped.
The potential lines associated with this irrotational vortex are spokes: θ = cst; while the streamlines are circles: r = cst. They are both orthogonal and the resulting flow is shown in figure 7.3. A similar calculation to that done for the source/sink in equation (7.39) and (7.40) can be carried out here to demonstrate that both the velocity potential and the streamfunction are harmonic functions.
We can also express the velocity potential and streamfunction of an offset free vortex using the same method as for the source/sink. The result for a vortex located at (x, y) = (a, b) is:
φ = K tan−^1
y − b x − a
, ψ = −
K
ln
[
(x − a)^2 + (y − b)^2
]
Figure 7.3: Irrotational vortex u = K/rθˆ represented using potential lines (θ = cst, dashed lines) and streamlines (r = cst, solid pink lines). After White, Fluid Mechanics (2011).
Note that K > 0 defines a anti-clockwise vortex while K < 0 defines a clockwise vortex.
7.3 Wrap-up
Your potential cheat sheet...
We summarise the definitions and properties of the velocity potential and the stream- function in table 7.1. In the case of an incompressible two-dimensional flow, it is usual
Function Velocity potential Streamfunction
Definition u = ∇φ u = ∂y ψ; v = −∂xψ Dimension 3D 2D
∇ · u = 0 ∇^2 φ = 0 Automatically satisfied ∇ × u = 0 Automatically satisfied ∇^2 ψ = 0
Isoline (φ = cst) ⊥ u (ψ = cst) ‖ u
Table 7.1: Summary of the definition and properties of the velocity potential φ and streamfunction ψ.
to prefer working with the streamfunction rather than the velocity potential. When the streamfunction is not defined, (in the case of an either three-dimensional or compressible flow), the choice of working with the velocity potential is natural over the primitive variables. Note that the streamfunction we have introduced in this Chapter is two-
