1.2. Vorticity formulation

An alternative formulation is with the vorticity. In the two-dimensional case, the vorticity is (essentially) a scalar, and the pressure disappears, making it easier to solve the system.

When the velocity field is u=(u,v)\mathbf{u} = (u, v) and the spatial variable is x=(x,y)\mathbf{x} = (x, y), the (scalar) vorticity is given by

ω=vx−uy, \omega = v_x - u_y,

where the subscripts denote the corresponding partial derivatives.

Alternatively, if u=(u,v,0)\mathbf{u} = (u, v, 0), then ω=∇×u=(0,0,ω)\mathbf{\omega} = \boldsymbol{\nabla} \times \mathbf{u} = (0, 0, \omega).

By taking the curl of the linear momentum equation, we obtain

∂t(∇×u)+∇×((u⋅∇u)⋅u)+∇×∇p=νΔ(∇×u)+∇×f. \partial_t (\boldsymbol{\nabla} \times \mathbf{u}) + \boldsymbol{\nabla} \times ((\mathbf{u} \cdot \boldsymbol{\nabla} \mathbf{u})\cdot \mathbf{u}) + \boldsymbol{\nabla} \times \boldsymbol{\nabla} p = \nu \Delta (\boldsymbol{\nabla} \times \mathbf{u}) + \boldsymbol{\nabla} \times \mathbf{f}.

Notice that

∇×∇p=(0,0,0),∂t(∇×u)=(0,0,ωt),νΔ(∇×u)=(0,0,νΔω),∇×f=(0,0,g), \begin{align*} \displaystyle \boldsymbol{\nabla} \times \boldsymbol{\nabla} p & = (0, 0, 0), \\ \displaystyle \partial_t (\boldsymbol{\nabla} \times \mathbf{u}) & = (0, 0, \omega_t), \\ \displaystyle \nu \Delta (\boldsymbol{\nabla} \times \mathbf{u}) & = (0, 0, \nu\Delta\omega), \\ \displaystyle \boldsymbol{\nabla} \times \mathbf{f} & = (0, 0, g), \end{align*}

where g=(f1)y−(f2)xg = (f_1)_y - (f_2)_x, where f1f_1 and f2f_2 are the two compenents of the force, f=(f1,f2)\mathbf{f} = (f_1, f_2).

As for the remaining term, we write

∇×((u⋅∇)u)=∇×(uux+vuyuvx+vvy0) \boldsymbol{\nabla} \times ((\mathbf{u}\cdot\boldsymbol{\nabla})\mathbf{u}) = \boldsymbol{\nabla} \times \left( \begin{matrix} uu_x + vu_y \\ uv_x + vv_y \\ 0 \end{matrix} \right)

The first two components of this curl vanish, while the last component is given by

(uvx+vvy)x−(uux+vuy)y=uxvx+uvxx+vxvy+vvyx−uyux−uuxy−vyuy−vuyy=u(vxx−uxy)+v(vyx−uyy)+ux(vx−uy)+vy(vx−uy)=uωx+vωy+uxω+vyω=(uω)x+(vω)y=∇⋅(ωu). \begin{align*} (uv_x + vv_y)_x - (uu_x + vu_y)_y & = u_xv_x + uv_{xx} + v_xv_y + vv_{yx} - u_yu_x - uu_{xy} - v_yu_y -vu_{yy} \\ & = u(v_{xx} -u_{xy}) + v(v_{yx}-u_{yy}) + u_x(v_x-u_y) + v_y(v_x-u_y) \\ & = u\omega_x + v\omega_y + u_x\omega + v_y\omega = (u\omega)_x + (v\omega)y = \boldsymbol{\nabla}\cdot(\omega\mathbf{u}). \end{align*}

Combining the expressions and considereing only the zz-component, we find the two-dimensional vorticity equation

ωt+∇⋅(ωu)=νΔω+g. \omega_t + \boldsymbol{\nabla}\cdot(\omega\mathbf{u}) = \nu\Delta \omega + g.

Due to the divergence-free condition on the velocity field, we have

∇⋅(ωu)=∇ω⋅u+ω∇⋅u=∇ω⋅u. \boldsymbol{\nabla}\cdot(\omega\mathbf{u}) = \boldsymbol{\nabla}\omega \cdot \mathbf{u} + \omega \boldsymbol{\nabla}\cdot \mathbf{u} = \boldsymbol{\nabla}\omega \cdot \mathbf{u}.

But we keep the form ∇⋅(ωu),\boldsymbol{\nabla}\cdot(\omega\mathbf{u}), in order to apply the Basdevant reduction, which requires less Fourier transforms; see 1.3. Basdevant reduction.



MIT License (Ricardo M. S. Rosa)
Last modified: March 21, 2022. Built with Franklin.jl, using the Book Template.