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The learning goal of this exercise is to understand how we can switch between the Lagrangian and Eulerian descriptions of ideal fluids (e.g. no diffusion). To do so, we need full information about the velocity field \(\mathbf{v}(t, \mathbf{x})\) in the Eulerian case or the trajectories \(\mathbf{X}(t, \mathbf{x}_0\) for all starting positions \(\mathbf{x}_0\) in the Lagrangian case.
Given the trajectories \(\mathbf X(t; \mathbf x_0) = (X(t; \mathbf x_0), Y(t; \mathbf x_0), Z(t; \mathbf x_0))^T\)
with \(\mathbf x_0 = (x_0, y_0, z_0)^T\) indicating coordinates at \(t=0\), we now measure a scalar \(\phi\) on the trajectory of all particles \(\mathbf{X}(t; \mathbf{x}_0)\) with their respective starting positions \(\mathbf{x}_0\):
Sketch two trajectories for \(t \in (0, 2 \pi)\) as well as their respective field values \(\phi(t, x=\mathbf X(t; \mathbf{x}_0))\) over time.
partial solution
Use the online tool linked in the lecture notes to verify your trajectory.
b)
Do you know which physical field in fluid flow applications exhibits the same bahavior as \(\phi\)?
partial solution
The density of incompressible fluids.
c)
We now want to describe \(\phi(t, \mathbf x)\) in the Eulerian frame. The final expression should therefore only depend on the independent variables \((t, \mathbf{x})\).
partial solution
Write the velocity field as a matrix-vector product \(\mathbf{X} = \underline{\underline{R}} \, \mathbf{x_0}\)
We can ‘reverse time’ and find the the initial position \(\mathbf{x}_0\) for each trajectory \(\mathbf{X}(t; \mathbf{x}_0)\) at time \(t\).
Substitute this relation to obtain the scalar field \(\phi(t,\mathbf{x})\) in the Eulerian frame.
d)
Compute \(\frac{d\phi(t, \mathbf{X}(t; \mathbf{x}_0))}{dt}\) and \(\frac{D\phi(t, \mathbf{x})}{dt}\) using their definitions.
partial solution
\(\frac{d\phi(t, \mathbf{X}(t; \mathbf{x}_0)}{dt} = 0\) is trivially computed in the Lagrangian frame. \(\frac{D\phi(t, \mathbf{x})}{dt}\) needs to give the same result, as we are only changing our frame of reference! You are on the right track if your velocity field reads \(\mathbf{v} = (y, -x)^T\).
e)
Consider experiment where you measure a scalar field \(\psi\) on the trajectory of all particles\(\psi(t, \mathbf{x}=\mathbf{X}(t; \mathbf{x}_0)) = X^2 + Y^2\) with their respective starting positions \(\mathbf{x}_0\). Rewrite \(\psi\) in the Eulerian frame.
partial solution
We only require the fact that we currently restrict \(\mathbf{x} \stackrel{!}{=} \mathbf{X}(t; \mathbf{x_0})\) to the particle trajectory. As the trajectory has the same form for all particles, we can directly move to the Eulerian frame.
Streamlines/Streaklines/Pathlines
Consider the velocity field
\[\begin{align*}
\mathbf{v}(t,x,y) =
\begin{pmatrix}
u \\
v
\end{pmatrix}
=
\begin{pmatrix}
- k x + \alpha t \\
k y
\end{pmatrix}
\end{align*}\]