Vorticity Equation
(Derivation from Karamcheti (1980), section 9.4.)
We start with the vector form of the Momentum conservation equation (for inviscid compressible flow) with body forces \(f\)
\[ \Dt{\boldsymbol{U}} = -\frac{1}{\rho} \nabla p + \mathbf{f} \]
where \(\Dt{}\) is the material derivative, \(\boldsymbol{U}\) is the velocity field, \(p\) is the pressure field, \(\rho\) is the density and \(\mathbf{f}\) is the body force per unit mass.
Now cosider the left hand side of above equation. By definition,
\[ \Dt{\boldsymbol{U}} = \frac{\partial \boldsymbol{U}}{\partial t} + \boldsymbol{U}\cdot \grad{\boldsymbol{U}} \]
where,
\[ \grad{\boldsymbol{U}} = [\nabla u ^T, \nabla v ^T, \nabla w ^T]^T \]
which is a tensor of rank 2. The components of this tensor are the gradients of the components of the velocity field given by
\[ \grad{\boldsymbol{U}} = \begin{bmatrix} \frac{\partial u}{\partial x} \frac{\partial u}{\partial y} \frac{\partial u}{\partial z} \\ \frac{\partial v}{\partial x} \frac{\partial v}{\partial y} \frac{\partial v}{\partial z} \\ \frac{\partial w}{\partial x} \frac{\partial w}{\partial y} \frac{\partial w}{\partial z} \\ \end{bmatrix} \]
Now, it can shown that
\[ \boldsymbol{U}\cdot\grad{\boldsymbol{A}} = \frac{1}{2} \left( \grad(\boldsymbol{U}\cdot \boldsymbol{A}) - \boldsymbol{U}\times (\nabla \times \boldsymbol{A}) - \boldsymbol{A}\times (\nabla \times \boldsymbol{U}) - \curl{(\boldsymbol{U}\times \boldsymbol{A})} + \boldsymbol{U}(\nabla \cdot \boldsymbol{A}) - \boldsymbol{A}(\nabla \cdot \boldsymbol{U}) \right) \]
where \(\boldsymbol{A}\) is any vector field. Putting \(\boldsymbol{A}= \boldsymbol{U}\) in above equation, we get
\[ \boldsymbol{U}\cdot\grad{\boldsymbol{U}} = \frac{1}{2} \left( \grad(\boldsymbol{U}\cdot \boldsymbol{U}) - \boldsymbol{U}\times (\nabla \times \boldsymbol{U}) - \boldsymbol{U}\times (\nabla \times \boldsymbol{U}) - \curl{(\boldsymbol{U}\times \boldsymbol{U})} + \boldsymbol{U}(\nabla \cdot \boldsymbol{U}) - \boldsymbol{U}(\nabla \cdot \boldsymbol{U}) \right) \]
The fourth term of the rhs is zero as \(\boldsymbol{U}\times \boldsymbol{U}= 0\). The fifth and sixth terms are zero as \(\nabla \cdot \boldsymbol{U}= 0\) for incompressible flow. Therefore, the above equation simplifies to
\[ \boldsymbol{U}\cdot\grad{\boldsymbol{U}} = \frac{1}{2} \grad(\boldsymbol{U}\cdot \boldsymbol{U}) - \boldsymbol{U}\times (\nabla \times \boldsymbol{U}) \]
Noting that \(\nabla \times \boldsymbol{U}= \boldsymbol{\xi}\), where \(\boldsymbol{\xi}\) is the vorticity vector, we can write
\[ \boldsymbol{U}\cdot\grad{\boldsymbol{U}} = \frac{1}{2} \grad(\boldsymbol{U}\cdot \boldsymbol{U}) - \boldsymbol{U}\times \boldsymbol{\xi} \]
Therefore the momentum equation can be written as
\[ \frac{\partial \boldsymbol{U}}{\partial t} + \frac{1}{2} \grad(\boldsymbol{U}\cdot \boldsymbol{U}) - \boldsymbol{U}\times \boldsymbol{\xi}= -\frac{1}{\rho} \nabla p + \mathbf{f} \]
Taking curl of the above equation, we get
\[ \frac{\partial \boldsymbol{\xi}}{\partial t} + \nabla \times \frac{1}{2} \grad(\boldsymbol{U}\cdot \boldsymbol{U}) - \curl{(\boldsymbol{U}\times \boldsymbol{\xi})} = -\curl{\frac{1}{\rho} \nabla p} + \curl{\mathbf{f}} \]
Clearly, second term on lhs is zero as curl of a gradient is zero. The third term on lhs can be simplified as
\[ \curl{(\boldsymbol{U}\times \boldsymbol{\xi})} = \boldsymbol{U}(\diver{(\boldsymbol{\xi})}) - \boldsymbol{U}\cdot \grad{\boldsymbol{\xi}} - \boldsymbol{\xi}(\nabla \cdot \boldsymbol{U}) + \boldsymbol{\xi}\cdot \grad{\boldsymbol{U}} \]
The first term on rhs is zero as \(\diver{\boldsymbol{\xi}} = \diver{\curl{\boldsymbol{U}}} = 0\) for incompressible flow. The third term on rhs is zero as \(\nabla \cdot \boldsymbol{U}= 0\). Therefore, the above equation simplifies to
\[ \curl{(\boldsymbol{U}\times \boldsymbol{\xi})} = - \boldsymbol{U}\cdot \grad{\boldsymbol{\xi}} + \boldsymbol{\xi}\cdot \grad{\boldsymbol{U}} \]
Putting this in the vorticity equation, we get
\[ \frac{\partial \boldsymbol{\xi}}{\partial t} - \boldsymbol{U}\cdot \grad{\boldsymbol{\xi}} + \boldsymbol{\xi}\cdot \grad{\boldsymbol{U}} = -\curl{\frac{1}{\rho} \nabla p} + \curl{\mathbf{f}} \]
Now we observe that the first term on the rhs is zero as curl of a gradient is zero. Therefore, the vorticity equation can be written as
\[ \frac{\partial \boldsymbol{\xi}}{\partial t} - \boldsymbol{U}\cdot \grad{\boldsymbol{\xi}} + \boldsymbol{\xi}\cdot \grad{\boldsymbol{U}} = \curl{\mathbf{f}} \]
Noting the definition of the total derivative of vorticity, above equation can be written as
\[ \Dt{\boldsymbol{\xi}} - \boldsymbol{\xi}\cdot \grad{\boldsymbol{U}} = \curl{\mathbf{f}} \]