Question

Enstrophy is defined as square of magnitude of vorticity. For velocity field \[ \vec{V} = (4x - 1.5y + 2.5z)\hat{i} + (1.5x - 1.5y)\hat{j} + (0.7xy)\hat{k}, \] find enstrophy at $(1,1,1)$. (round off to two decimal places)

Hint:

Enstrophy = $|\nabla \times \vec{V}|^2$. Always compute curl carefully component-wise and substitute coordinates.

Answer 1

Step 1: Vorticity definition.
\[ \vec{\omega} = \nabla \times \vec{V} \]

Step 2: Compute curl.
\[ \omega_x = \frac{\partial V_z}{\partial y} - \frac{\partial V_y}{\partial z} = \frac{\partial (0.7xy)}{\partial y} - 0 = 0.7x \] \[ \omega_y = \frac{\partial V_x}{\partial z} - \frac{\partial V_z}{\partial x} = \frac{\partial (4x - 1.5y + 2.5z)}{\partial z} - \frac{\partial (0.7xy)}{\partial x} \] \[ = 2.5 - 0.7y \] \[ \omega_z = \frac{\partial V_y}{\partial x} - \frac{\partial V_x}{\partial y} = \frac{\partial (1.5x - 1.5y)}{\partial x} - \frac{\partial (4x - 1.5y + 2.5z)}{\partial y} \] \[ = 1.5 - (-1.5) = 3.0 \]

Step 3: At point (1,1,1).
\[ \omega_x = 0.7(1) = 0.7, \omega_y = 2.5 - 0.7(1) = 1.8, \omega_z = 3.0 \]

Step 4: Magnitude.
\[ |\vec{\omega}|^2 = (0.7)^2 + (1.8)^2 + (3.0)^2 = 0.49 + 3.24 + 9 = 12.73 \] So enstrophy = 12.73 (approx). But double-checking arithmetic: $0.49 + 3.24 + 9 = 12.73$. Correct. \[ \boxed{12.73} \]