13. Vorticity equation

A differential equation governing vorticity can be derived starting from the momentum equation. Starting from the convective form of the momentum equation

\[\begin{split}\begin{aligned} \partial_t \mathbf{u} + \left( \mathbf{u} \cdot \nabla \right) \mathbf{u} & = \mathbf{g} + \dfrac{1}{\rho} \nabla \cdot \left[ - p \mathbb{I} + 2 \mu \mathbb{D} + \lambda \left( \nabla \cdot \mathbf{u} \right) \mathbb{I} \right] = \\ & = \mathbf{g} - \dfrac{1}{\rho} \nabla p + \frac{1}{\rho} \left[ 2 \mu \nabla^2 \mathbf{u} + \left( \mu + \lambda \right) \nabla \left( \nabla \cdot \mathbf{u} \right) \right] \ . \end{aligned}\end{split}\]

with

\[\begin{split}\begin{aligned} 2 \nabla \cdot \mathbb{D} & = \partial_i \left( \partial_i u_j + \partial_j u_i \right) = u_{j/ji} + u_{i/jj} = \nabla \left( \nabla \cdot \mathbf{u} \right) + \nabla^2 \mathbf{u} \\ \nabla \cdot \left( \left( \nabla \cdot \mathbf{u} \right) \mathbb{I} \right) & = \partial_{k} \left( u_{l/l} \delta_{ki} \right) = u_{l/li} = \nabla \left( \nabla \cdot \mathbf{u} \right) \ . \end{aligned}\end{split}\]

Taking the curl of the momentum equation above, vorticity equation follows

\[\begin{split}\begin{aligned} \partial_t \boldsymbol\omega + \nabla \times \left( \boldsymbol\omega \times \mathbf{u} \right) & = - \nabla \times \frac{\nabla p}{\rho} + \dots \\ \partial_t \boldsymbol\omega + \mathbf{u} \cdot \nabla \boldsymbol\omega - \boldsymbol\omega \cdot \nabla \mathbf{u} + \boldsymbol\omega \, \nabla \cdot \mathbf{u} & = \frac{1}{\rho^2} \nabla \rho \times \nabla p + \dots \ , \end{aligned}\end{split}\]

or, highlighting the material derivative,

\[ D_t \boldsymbol\omega = \boldsymbol\omega \cdot \nabla \mathbf{u} - \boldsymbol\omega \, \nabla \cdot \mathbf{u} + \frac{1}{\rho^2} \nabla \rho \times \nabla p + \dots \ . \]

Vector caluclus identities

\[\nabla \left( \mathbf{a} \cdot \mathbf{b} \right) = a_{i/j} b_i + a_i b_{i/j}\]

\[\begin{split}\begin{aligned} \left(\nabla \times \mathbf{a} \right) \times \mathbf{b} & = \varepsilon_{ijk} \varepsilon_{jlm} \partial_l a_m b_k = \\ & = \left( \delta_{kl} \delta_{im} - \delta_{km} \delta_{il} \right) \partial_l a_m b_k = \\ & = a_{i/k} b_k - a_{k/i} b_k = \mathbf{b} \cdot \nabla \mathbf{a} - \nabla \mathbf{a} \cdot \mathbf{b} \end{aligned}\end{split}\]

Thus, \(\left(\nabla \times \mathbf{u}\right) \times \mathbf{u}\) can be written as

\[\left(\nabla \times \mathbf{u}\right) \times \mathbf{u} = \mathbf{u} \cdot \nabla \mathbf{u} - \nabla \mathbf{u} \cdot \mathbf{u} = \mathbf{u} \cdot \nabla \mathbf{u} - \nabla \frac{|\mathbf{u}|^2}{2} \ ,\]

so that the advective term can be recast as \(\mathbf{u} \cdot \nabla \mathbf{u} = \boldsymbol\omega \times \mathbf{u} - \nabla \frac{|\mathbf{u}|^2}{2} \ ,\)

being \(\boldsymbol\omega = \nabla \times \mathbf{u}\) the vorticity field. Using vector calculus identity

\[\begin{split}\begin{aligned} \nabla \times \left( \mathbf{a} \times \mathbf{b} \right) & = \varepsilon_{ijk} \partial_j \varepsilon_{klm} \left( a_l b_m \right) = \\ & = \left( \delta_{il} \delta_{jm} - \delta_{im} \delta_{jl} \right) \partial_j \left( a_l b_m \right) = \\ & = a_{i/j} b_j + a_i b_{j/j} - a_{j/j} b_i - a_j b_{i/j} = \\ & = \mathbf{b} \cdot \nabla \mathbf{a} + \mathbf{a} \, \nabla \cdot \mathbf{b} - \mathbf{b} \nabla \cdot \mathbf{a} - \mathbf{a} \cdot \nabla \mathbf{b} \ . \end{aligned}\end{split}\]

and recalling that the curl of a gradient is identically zero, the curl of the advective term reads

\[\begin{split}\begin{aligned} \nabla \times \left[ \left( \mathbf{u} \cdot \nabla \right) \mathbf{u} \right] & = \nabla \times \left[ \boldsymbol\omega \times \mathbf{u} + \nabla \frac{|\mathbf{u}|^2}{2} \right] = \\ & = \mathbf{u} \cdot \nabla \boldsymbol\omega + \boldsymbol\omega \nabla \cdot \mathbf{u} - \mathbf{u} \underbrace{\nabla \cdot \boldsymbol\omega}_{ = \mathbf{0} \, \text{div curl}\equiv\mathbf{0}} - \boldsymbol\omega \cdot \nabla \mathbf{u} \ . \end{aligned}\end{split}\]

The curl of a Laplacian reads

\[\left\{ \nabla \times \nabla^2 \mathbf{u} \right\}_i = \varepsilon_{ijk} \partial_j \partial_{ll} u_k = \partial_{ll} \left( \varepsilon_{ijk} \partial_j u_k \right) = \nabla^2 \left( \nabla \times \mathbf{u} \right) \ .\]

The curl of the product of a scalar and vector field reads

\[\nabla \times \left( a \mathbf{v} \right) = \varepsilon_{ijk} \partial_j ( a v_k ) = \varepsilon_{ijk} \left( a_{/j} v_k + a v_{k/j} \right) = \nabla a \times \mathbf{v} + a \nabla \times \mathbf{v} \ .\]

13.1. Vorticity and entropy

Starting from the potentials, and \(dh = d \left( e + \frac{p}{\rho} \right) = de + \frac{d p}{\rho} - \frac{p}{\rho^2} d\rho\),

\[\begin{split}\begin{aligned} de & = T ds + \frac{p}{\rho^2} d \rho \\ dh & = T ds + \frac{d p}{\rho} \end{aligned}\end{split}\]

and taking the gradient,

\[\frac{\nabla p}{\rho} = \nabla h - T \nabla s \ .\]

Using the momentum equation to replace the first term,

\[\begin{split}\begin{aligned} \frac{\nabla p}{\rho} & = - D_t \mathbf{u} + \mathbf{g} + \frac{1}{\rho} \nabla \cdot \mathbb{S} = \\ & = - \partial_t \mathbf{u} - \boldsymbol\omega \times \mathbf{u} - \nabla \frac{|\mathbf{u}|^2}{2} - \nabla \chi + \frac{1}{\rho} \nabla \cdot \mathbb{S} \ , \end{aligned}\end{split}\]

it follows

\[\mathbf{0} = \partial_t \mathbf{u} + \boldsymbol\omega \times \mathbf{u} + \nabla \left[ h + \frac{|\mathbf{u}|^2}{2} + \chi \right] + \frac{1}{\rho} \nabla \cdot \mathbb{S} - T \nabla s\]

Steady condition, inviscid flows. If \(\partial_t \equiv 0\) and \(\mathbb{S} \equiv \mathbb{0}\), it follows

\[\mathbf{0} = \boldsymbol\omega \times \mathbf{u} + \nabla \left[ h + \frac{|\mathbf{u}|^2}{2} + \chi \right]- T \nabla s \ .\]

todo Rotational flow behind an oblique shock

Scalar multiplication by \(\rho \mathbf{u} \cdot\) and transforming back the gradient of the entropy field as a function of the gradient of the internal energy and density,

\[\begin{split}\begin{aligned} 0 & = \rho \partial_t \frac{|\mathbf{u}|^2}{2} + \rho \mathbf{u} \cdot \nabla \left[ h^t + \chi \right] + \mathbf{u} \cdot \nabla \cdot \mathbb{S} - T \rho \mathbf{u} \cdot \nabla s = \\ & = \dots \end{aligned}\end{split}\]

todo Going back to Bernoulli…