Velocity Gradient Decomposition
Given a velocity vector field \(\mathbf{v}\) in \(\mathbb{R}^3\), we have at any point in space
\[
\nabla \mathbf{v} := \mathbf{L}
\] referred to as the velocity gradient tensor, which can be further decomposed into a symmetric tensor \[
\dfrac{1}{2} (\mathbf{L} + \mathbf{L}^{T}) := \mathbf{D}
\] and a skew symmetric tensor \[
\dfrac{1}{2} (\mathbf{L} - \mathbf{L}^{T}) := \mathbf{W}.
\]
Herein, \(\mathbf{D}\) is called the rate of deformation tensor and \(\mathbf{W}\) is called the spin tensor and we have
\[
\mathbf{L} = \mathbf{D} + \mathbf{W}
\]
Velocity Gradient Decomposition
The rate of deformation tensor \(\mathbf{D}\) can be further decomposed into a sperical part describing dilatation motion \[
\text{sph}(\mathbf{D}) = \dfrac{1}{3} \text{tr}(\mathbf{D}) \mathbf{I}
\]
and a deviatoric part describing shearing motion
\[
\text{dev}(\mathbf{D}) = \mathbf{D} - \text{sph}(\mathbf{D})
\]
Velocity Gradient Decomposition
Ex1
For the given velocity field \(\mathbf{v}\), compute the values of:
- Velocity Gradient Tensor \(\nabla \mathbf{v}\)
- Rate of Deformation Tensor \(\mathbf{D}\)
- Spin Tensor \(\mathbf{W}\)
- \(\text{sph}(\mathbf{D})\)
- \(\text{dev}(\mathbf{D})\)
Comment on the significance of the computed values by comparing it to the plot of velocity field.
Velocity Gradient Decomposition
Ex1.a:
\[
\mathbf{v} = \left(\begin{array} \text{a} \\ 0\end{array}\right)
\]
Velocity Gradient Decomposition
Ex1.b:
\[
\mathbf{v} = \left(\begin{array} \text{2y} + 10 \\ 0\end{array}\right)
\]
Velocity Gradient Decomposition
Ex1.c:
\[
\mathbf{v} = \left(\begin{array} \text{bx} \\ -by\end{array}\right)
\]
Velocity Gradient Decomposition
Ex1.d:
\[
\mathbf{v} = \left(\begin{array} \text{cx} \\ cy\end{array}\right)
\]
Velocity Gradient Decomposition
Ex1.e:
\[
\mathbf{v} = \left(\begin{array} \text{dy} \\ dx\end{array}\right)
\]
Velocity Gradient Decomposition
Ex1.f:
\[
\mathbf{v} = \left(\begin{array} \text{-}fy \\ fx\end{array}\right)
\]
Vorticity
Given a velocity vector fielf \(\mathbf{v}\), at any point in space we can calculate
\[
\mathbf{\omega} = \nabla \times \mathbf{v}
\] where \(\mathbf{\omega}\) is called the vorticity.
Ex2
For the given velocity field \(\mathbf{v}\), calculate the vorticity field. Interpret the connection of the vorticity field with:
- the plot of the velocity field
- the spin tensor \(\mathbf{W}\) calculated in the previous exercise
Vorticity
Ex2.a:
\[
\mathbf{v} = \left(\begin{array} \text{a} \\ 0\end{array}\right)
\]
Vorticity
Ex2.b:
\[
\mathbf{v} = \left(\begin{array} \text{2y} + 10 \\ 0\end{array}\right)
\]
Vorticity
Ex2.c:
\[
\mathbf{v} = \left(\begin{array} \text{bx} \\ -by\end{array}\right)
\]
Vorticity
Ex2.d:
\[
\mathbf{v} = \left(\begin{array} \text{cx} \\ cy\end{array}\right)
\]
Vorticity
Ex2.e:
\[
\mathbf{v} = \left(\begin{array} \text{dy} \\ dx\end{array}\right)
\]
