GATE 2023 AE – Question 58
Enstrophy is squared vorticity magnitude. For $\vec V=(4x-1.5y+2.5z)\hat i+(1.5x-1.5y)\hat j+0.7xy\hat k$, enstrophy at (1,1,1), to two decimal places, is _____.
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Correct answer: 12.67 to 12.79
Explanation
**Enstrophy** is the squared magnitude of the vorticity: $\Omega=|\vec\omega|^2$, with $\vec\omega=\nabla\times\vec V$.
For $\vec V=(u,v,w)$ with
$$u=4x-1.5y+2.5z,\quad v=1.5x-1.5y,\quad w=0.7xy,$$
**Components of the curl:**
- $\omega_x=\dfrac{\partial w}{\partial y}-\dfrac{\partial v}{\partial z}=0.7x-0=0.7x$
- $\omega_y=\dfrac{\partial u}{\partial z}-\dfrac{\partial w}{\partial x}=2.5-0.7y$
- $\omega_z=\dfrac{\partial v}{\partial x}-\dfrac{\partial u}{\partial y}=1.5-(-1.5)=3$
**At $(1,1,1)$:** $\vec\omega=(0.7,\ 1.8,\ 3.0)$.
$$|\vec\omega|^2=0.49+3.24+9=\mathbf{12.73}.$$