Vorticity

In Chapter 4 we will be deriving a surface-based energy whose minimum is attained when the water surface is physically valid. When we interpret this energy as a physical potential, we obtain equations of motion that are strikingly similar to equations of motion for vortex sheet, which are two-dimensional surfaces of concentrated vorticity. In Section 2.8.3 we will have a look at the vortex sheet equations, but before that let us try to understand vorticity a bit better.

In Section 2.4 we saw the definition of vorticity as the curl of velocity ξ=∇×u

We also saw that it is proportional to the (local) angular velocityωof the velocity.

Notice that vorticity (and angular velocity) is avectorwhose direction defines the axis of local rotation and whose magnitude specifies the speed of the rotation.

Another way of looking at this, is through the concept ofcirculation. If Cis a closed curve sitting in a velocity fielduinR³then the circulationΓ aroundC is defined to be

Γ := Z

C

u·dl

IfC=∂Sis the boundary of a surfaceS, then it follows from Stokes’ theorem that Γ =

which suggests that vorticity is circulation per unit area around an infinitesimal loop. Conversely, the flux of vorticity throughSis the circulation.

From Section 2.7 we know that divergence-free velocity fields correspond to incompressible fluid flow. Therefore, one might equivalently choose to work with vorticity instead of velocity, which practitioners actually do as we will see in Sec-tions 2.8.2 and 2.8.3. The question is, why would you? One of the main reasons is that while velocity is non-zero almost everywhere (at least for any interesting mo-tion) vorticity can be far more concentrated. This makes sense once you consider that vorticity is a spatial derivative of velocity, which means constant or slowly varying regions of velocity correspond to (almost) zero vorticity. As we shall see in Section 2.8.1, it is also instructive to think of electromagnetism where the current through a wire (vorticity) can generate a whole magnetic (velocity) field. This

has led to models for (zero-dimensional) vortex particles [Selle et al. 2005], (one-dimensional) vortex filaments [Angelidis and Neyret 2005] and (two-(one-dimensional) vortex sheets [Kim et al. 2009; Pfaff et al. 2012] as we will see in Section 2.8.3.

2.8.1 Velocity from vorticity

In Chapter 4 we will be deriving a surface-based energy to reduce the deviation of a liquid surface from a physically valid state. We interpret the negative gradient (the direction of steepest descent) of this energy as the (local) angular velocityω.

Eventually, we have to turn these local rotations into motion of the surface. Luckily, we saw in Section 2.4 that ωis proportional to the vorticityξ=∇×uof some velocity fieldu. This allows us to leverage existing techniques for turning vorticity into velocity in which we may then move the surface.

One way of relating vorticity to velocity is through Equation (2.34) which gives us a vector-Poisson equation for the so-calledstream functionΨ

∇²Ψ=−ξ. (2.37)

This equation can be solved subject to∇ ·Ψ=0using methods from Section 2.3.4.

The velocity can then be recovered by noticing that

−ξ=−∇×u=∇²Ψ=−∇×∇×Ψ+∇:0 (∇ ·Ψ) which suggests thatu=∇×Ψ.

Since Equation (2.37) is a vector-Poisson equation, it can also be solved using Green’s functions (cf.Section 2.3.2). Concretely, we can apply the Green’s function for the three-dimensional Laplacian for each of the three dimensions independently.

According to Equation (2.16) we have

u=∇×Ψ=∇× derivation we have additionally used the quotient rule and the identity∇kx−x⁰k=

(x−x⁰)/kx−x⁰k. Equation (2.38) is called theBiot-Savart law.

Since Equation (2.38) is singular (shoots to infinity) when evaluated whereξ is non-zero, it is common (this is also what we do) to instead use theregularized

version [Pfaff et al. 2012]

u=− 1 4π

Z

R³

(x−x⁰) kx−x⁰k²+"²³₂

×ξ(x⁰)dx⁰

where" > 0is a regularization parameter that effectively controls the minimal size of the vortices.

2.8.2 Vorticity-velocity form

Previously, in Section 2.7, we derived the equations of motion for an incompressible Newtonian fluid, the incompressible Navier-Stokes equations. These equations describe the time evolution ofvelocity. As a gently warm-up to the next section on vortex sheets (Section 2.8.3), we will look at the equations of motion forvorticity. As we discussed in the beginning of Section 2.8, working with vorticity is just as valid as working with velocity when considering incompressible fluids. Recall that vorticity is defined asξ= ∇×u. The equations of motion can be obtained by taking the curl of the momentum equation (Equation (2.30)), which gets us

∂ξ

∂t + (u· ∇)ξ+ (ξ· ∇)u−ν∇²ξ=∇×f. (2.39) The first thing we notice about Equation (2.39) is that the term involving the gradient of pressure has dropped out because the curl of a gradient is zero by vector calculus identity. This has the benefit that we no longer have to solve for the unknown pressurep, however, since velocity still appears in the equation, we have effectively traded solving a scalar Poisson equation for the pressurepfor solving a vector Poisson equation for the vector potentialΨ. Nevertheless, this may still pay off when vorticity is sparse as we will see in the next section. Other than this fact, Equation (2.39) closely resembles the momentum equation, except for the addition of the newvortex stretchingterm(ξ· ∇)ω.

2.8.3 Vortex sheets

In this next section we will see the equations of motion for avortex sheet, which is simply the concentration of vorticity on a two-dimensional surface M embedded in three-dimensional spaceR³. We will also see that these equations bear striking similarity to our surface-based dynamics in Chapter 4.

Physically, a vortex sheet arises when there is a discontinuity in the tangential velocity of two contacting inviscid immiscible fluids with densities ρ1 and ρ2

and constant surface tension γ. Letting (u₂−u₁) be the (tangential) velocity discontinuity between the two fluids andnˆbe the unit normal toM pointing into the exterior fluid labeled2, we can define thevortex strengthof the sheetMas

ζ^def=ˆn×(u₂−u₁)

Note that this suggests that ζ is tangential to M. The equations of motion are obtained as in Wu [1995] or Pozrikidis [2000] and are given by

Dζ

Dt −(ζ· ∇)u+ζP(∇)·u=2Aˆn×(a−g) + 4γ ρ1+ρ2

ˆ

n×∇H (2.40) whereA= ^ρ_ρ¹₁^−ρ_+ρ²₂ is the Atwood ratio, P=I₃−nˆ⊗ˆnis the tangential projection operator,I₃is the 3-by-3 identity matrix and a= ¹₂ ^Du_Dt¹ +^Du_Dt²

is the average of the accelerations on each side of the sheet. Compare this to our surface correction forces in Section 4.4, reproduced in the same notation as Equation (2.40) for convenience

Dζ

Dt =2ˆn×(−∇p) +4γˆn×∇H

where we have set β = 2 as discussed in Section 4.4. The primary difference to Equation (2.40) is that (a−g)has been replaced with the negative pressure gradient−∇p. Also, the vortex stretching−(ζ· ∇)uand dilationζP(∇)·uterms have been neglected.

Since a vortex sheet is simply the concentration of vorticity, we can compute the velocity induced by the vortex sheet in a similar manner to how we computed the velocity from vorticity in Section 2.8.1. In Chapter 4 we use the Biot-Savart law to recover velocity from the rotational velocities and accelerations in our surface correction algorithm.