Vorticity Equation
Vorticity Equation: Theoretical Foundations
What is Vorticity?
Professor, the name "vorticity equation" sounds difficult, but first of all, what is "vorticity"?
Vorticity $\boldsymbol{\omega}$ is a vector quantity representing the local rotation of a fluid, defined as the curl of the velocity field.
In two dimensions, it becomes a scalar: $\omega = \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y}$. It quantifies how much a fluid element is rotating.
I see, so the entire fluid doesn't need to be swirling; if there's local rotation, there is vorticity?
Exactly. For example, even in a linear shear flow $u = ky$, $v = 0$, we have $\omega = -k \neq 0$, so there is vorticity. In other words, vorticity is a concept separate from the existence of a "vortex".
Derivation of the Vorticity Equation
So, how do you derive the equation that describes the time evolution of vorticity?
Applying $\nabla \times$ to both sides of the Navier-Stokes equations yields the vorticity equation. For an incompressible fluid, it becomes:
The left side is the material derivative (Lagrangian derivative) of vorticity, representing the rate of change of vorticity seen from a moving fluid particle.
What is the meaning of each of the two terms on the right side?
The first term $(\boldsymbol{\omega} \cdot \nabla)\mathbf{u}$ is the vortex stretching/tilting term. In 3D flow, when a vortex tube is stretched, its cross-section becomes thinner, and due to angular momentum conservation, the vorticity increases. The effect where a tornado or a bathtub vortex spins faster as it becomes thinner is due to this.
The second term $\nu \nabla^2 \boldsymbol{\omega}$ is the viscous diffusion term, representing the effect of vorticity diffusing due to molecular viscosity. The larger the kinematic viscosity coefficient $\nu$, the faster vorticity diffuses and dissipates.
In 2D, the vortex stretching term disappears, right?
Good observation. In 2D, $\boldsymbol{\omega}$ is only in the $z$ direction, so $(\boldsymbol{\omega} \cdot \nabla)\mathbf{u} = 0$, and the vorticity equation becomes a simple advection-diffusion equation.
This means vorticity generation can only occur at boundary surfaces.
Vorticity Generation Mechanism
Doesn't vorticity just appear spontaneously?
In incompressible, barotropic fluids, vorticity is not generated within the fluid interior. The main sources of vorticity are as follows.
- Solid walls: The no-slip condition creates velocity gradients at the wall, generating vorticity. The wall vorticity flux shown by Lighthill (1963) is expressed as $\nu \frac{\partial \omega}{\partial n}\big|_{wall}$.
- Baroclinic effect: When density gradients and pressure gradients are not parallel, vorticity is generated from the term $\frac{1}{\rho^2}(\nabla \rho \times \nabla p)$. Ocean thermohaline circulation is a prime example.
- Non-uniform distribution of body forces: Such as the Coriolis force in a rotating coordinate system.
In CFD, how can we accurately capture vorticity at walls?
Mesh resolution near the wall is critically important. The choice is either to make the $y^+$ of the first cell near the wall less than 1, or to use wall functions to approximate the vorticity flux from the wall shear stress.
Relationship with Kelvin's Circulation Theorem
How are the vorticity equation and Kelvin's circulation theorem connected?
For an inviscid, barotropic fluid with only potential body forces acting, the circulation $\Gamma = \oint \mathbf{u} \cdot d\mathbf{l}$ along a closed curve is conserved. By Stokes' theorem, $\Gamma = \int_S \boldsymbol{\omega} \cdot d\mathbf{S}$, so this corresponds to the vorticity equation when the viscous and baroclinic terms are zero.
In actual CFD, viscosity exists, so circulation is not conserved. Is the vorticity equation like a generalization of the circulation theorem?
That understanding is correct. The vorticity equation is a more general description that includes all viscous and non-barotropic effects.
History of the Vorticity Equation—Helmholtz's Vortex Theorems (1858) and Vortex Tube Conservation
The concept of vorticity and the "Helmholtz vortex theorems" governing the behavior of vortex tubes were published by Hermann von Helmholtz in 1858. The essence of these theorems: ① A vortex tube moves as a line of fluid particles (vortex strength is conserved as it advects), ② A vortex tube cannot end in the fluid—it must form a closed loop or terminate at a fluid boundary. This also explains why a tornado cannot detach from the ground and why vortices are "sucked up" from the sea surface. Lord Kelvin developed this further to derive "Kelvin's circulation theorem (for a barotropic perfect fluid, the circulation Γ around a material curve is constant)". This theorem is the fundamental principle behind lift generation mechanisms (bound vortex) for propellers and wings, and forms the mathematical foundation for modern CFD codes like the vortex lattice method (VLM) and vortex panel methods.
Computational Methods for Vorticity Equation
Vorticity-Stream Function Method
When solving the vorticity equation in CFD, what is the most basic method?
For 2D incompressible flow, the vorticity-stream function method is a classical approach. It couples the vorticity equation with the Poisson equation for the stream function.
Velocity is recovered as $u = \frac{\partial \psi}{\partial y}$, $v = -\frac{\partial \psi}{\partial x}$. The benefit of this method is that pressure disappears from the unknowns.
What advantages does it have compared to solving the N-S equations directly?
In 2D, there are only two unknowns, $\omega$ and $\psi$, avoiding the pressure-velocity coupling problem (no need for iterations like the SIMPLE method). However, extension to 3D is complex because vorticity becomes a 3-component vector, so it's not very practical.
Spatial Discretization
How do you discretize in space?
Let's look at a typical discretization using the finite difference method (FDM). Using central differences with uniform grid spacing $h$, the Poisson equation becomes:
For the advection term, central differences can cause wiggles when the grid Reynolds number $Re_h = |u|h/\nu > 2$. To avoid this, upwind differencing or the QUICK method is often used.
How does it change with the finite volume method (FVM)?
In FVM, the vorticity flux at cell faces is evaluated. For advective flux, second-order upwind schemes (Second Order Upwind) or TVD schemes are used; for diffusive flux, central differences are used. OpenFOAM's scalarTransportFoam solver is a good reference for FVM implementation solving vorticity advection-diffusion.
Time Integration
How do you advance in time?
Let's summarize typical time integration schemes.
| Scheme | Accuracy | Stability | Features |
|---|---|---|---|
| Forward Euler | 1st order | Conditionally stable | Constraint: $\Delta t < h^2/(4\nu)$ |
| Crank-Nicolson | 2nd order | Unconditionally stable | Implicit method; solve tridiagonal matrix each step |
| RK3 (Runge-Kutta 3rd) | 3rd order | Conditionally stable | Used in turbulence simulations; low memory overhead |
| RK4 (Runge-Kutta 4th) | 4th order | Conditionally stable | Classical method; computationally expensive |
Which method should I choose?
For vorticity-based simulations with high Reynolds numbers, RK3 is popular because it offers good accuracy without excessive computational cost. For stability at large time steps and when using implicit solvers, Crank-Nicolson is effective. Always check the CFL condition: $\mathrm{CFL} = |u|\Delta t / h \leq 1$ for explicit schemes.