Homework 4
- Your homework solution has to be handed in as a group solution via Moodle.
- Clearly state name and matriculation number of each student
1 Pipe Flow
Consider the incompressible flow of a Newtonian fluid along a pipe of constant radius \(a\) with fixed walls. A schematic is shown in the figure below. The velocity can be written as \(\vec{u}=u_r\vec{e}_{r}+u_{\theta}\vec{e}_{\theta}+u_z\vec{e}_{z}\).
\[ \begin{aligned} \nabla \cdot \mathbf u &= 0 \\ \partial_t \mathbf u + \mathbf u \cdot \nabla \mathbf u &= - \frac{1}{\rho_0}\nabla p + \nu \triangle \mathbf u . \end{aligned} \tag{1}\]
where \(\rho_0\) is the reference density.
Tasks
Task 1
How is the process model called?
Task 2
For a constant pressure gradient \(\frac{\partial p}{\partial z} = -P\), is the pressure greater at \(z_0\) or \(z_1\)?
Task 3
How do mass and momentum balance read in cylindrical coordinates?
Gradient and divergence in cylindrical coordinates are given by:
\[\begin{align*} \nabla f & = \frac{\partial f}{\partial r } \vec{e}_r + \frac{1}{r} \frac{\partial f}{\partial \theta} \vec{e}_\theta+ \frac{\partial f}{\partial z} \vec{e}_z, \\ \nabla \cdot \vec{A} & = \frac{1}{r}\frac{\partial}{\partial r }(r A_r) + \frac{1}{r} \frac{\partial A_\theta}{\partial \theta} + \frac{\partial A_z}{\partial z} \\ \nabla \times \vec{A} & = \left(\frac{1}{r}\frac{\partial A_z}{\partial \theta}-\frac{\partial A_\theta}{\partial z}\right)\vec{e}_r+ \left(\frac{\partial A_r}{\partial z}-\frac{\partial A_z}{\partial r}\right)\vec{e}_\theta+\frac{1}{r}\left(\frac{\partial}{\partial r}(r A_\theta)-\frac{\partial A_r}{\partial \theta}\right) \vec{e}_z \end{align*}\]
Task 4
Assuming constant pressure in the cross-sectional area \(\frac{\partial p}{\partial {\theta}} = \frac{\partial p}{\partial r } = 0\), how do mass and momentum balance reduce? What does this imply for the velocity field?
Task 5
Furthermore, there will be no change in the pipe’s flow field in time and no flow in radial direction or rotational direction. Hence \(u_z = u_z(r, \theta)\) and \(u_r = u_{\theta} = 0\). How does the remaining process model look like?
Task 6
Derive an explicit expression for the velocity profile. Also, derive an expression that yields the volume discharge?
Task 7
Assume that the pressure grdient drops by a factor of \(50\%\). How can you change the setting to assure the same volume discharge?
2 Oil Skimmer
An oil skimmer is a device typically mounted on a ship that is designed to remove oil floating on the water surface from oil spills. Let’s consider an oil skimmer ship with a moving belt that is \(5\textrm{m}\) wide, \(6\textrm{m}\) long, \(2\textrm{mm}\) above a fixed platform inclined by \(30^{\circ}\), traveling at \(3\textrm{m/s}\) and discharging into an open container on the ship. In reality the fluid is a mixture of oil and water, but let’s simplify the analysis by assuming that density and viscosity of the transported fluid correspond to the properties of crude oil (\(\rho = 860\textrm{kg}/{\textrm{m}}^{3}\), \(\mu=10^{-2}{\textrm{Pa} \textrm{s}}\)). Assume a gravitational acceleration of \(\left|\vec{g}\right|=9.806\textrm{m}/{\textrm{s}}^2\).
For laminar, steady, uniform flow of an incompressible, linear viscous fluid, the general Navier-Stokes equation reduces to \[ - \vec{\nabla} p + \mu \Delta\vec{v} + \rho \vec{g} = \vec{0}. \]
We now consider the specific case of flow between two infinite parallel plates, separated by a distance \(a\) and inclined to the horizontal by an angle \(\theta\). The upper plate is moving with constant non-zero velocity \(\vec{v}=\left(v_0,0,0\right)\) and the lower plate is at rest. Assume that the \(x\)-axis is the inclined axis parallel to the plates and the \(y\)-axis is perpendicular to the plates.
Tasks
Task 1
Draw a sketch of the situation. Indicate all mentioned variables and the coordinate system. Explain why the velocity in both y- and z-direction vanish.
Task 2
Determine an equation for the velocity profile in the fluid.
The volume flowing through a specific cross-section per unit time calculates as \[\begin{equation*} \dot{V} = \int_F \vec{v}\cdot d\vec{F}. \end{equation*}\] Here, with the width \(L\), this simplifies to \[\begin{equation*} \frac{\dot{V}}{L} = \int_0^a v_x dy. \end{equation*}\]
Task 3
Derive an equation for the volume flow rate per unit width.
Task 4
Calculate how much oil does the oil skimmer actually remove per unit time?
3 Karman vortex street
You can find the zip file with the notebook and data in Moodle.
If you have conda/anaconda/mamba installed on your system, the environment can for instance be installed by typing
conda env create -f environment.yml -n karman
and afterwards activated with
conda activate karmanThe kernel can finally be installed in order to be available in Jupyterlab with
python -m ipykernel install --user --name karmanIn Jupyterlab, you can then find the kernel named karman, which contains the necessary packages.
If you prefer pip, you can find the requirements in the environment.yml file in the under the pip section.
In case you have problems installing the environment, please reach out as soon as possible. Once the RWTHJupyterhub is up and running again with our environment, we will let you know.