Fundamentals of Fluid Mechanics B Assignment 9 — Jets and Wakes  
Instructions
Write your solutions in single column format, with one statement following another vertically. Write your solutions neatly so that they are easy to read and verify. Don't write one line with two equal signs. Highlight your answers using a box. Failure to do this will result in a lower score and fewer comments on my part.
04.28.20
Question #1
Recall what when deriving self-similar solutions for the jet or the wake with the jet or wake moving downstream along the $x$ axis, we assumed that the shear stresses $\tau_{xx}$ is negligible compared to other terms within the momentum equations. Using order of magnitude analysis, compare the terms involving the shear stress $\tau_{xx}$ to other terms and determine under which conditions such are negligible across a wake or jet.
Question #2
Starting from the momentum equation in conservative form: $$ \frac{\partial \rho u}{\partial t} + \frac{\partial \rho u^2}{\partial x} + \frac{\partial \rho u v}{\partial y} + \frac{\partial \rho u w}{\partial z} = -\frac{\partial P}{\partial x} + \frac{\partial \tau_{xx}}{\partial x} + \frac{\partial \tau_{yx}}{\partial y} + \frac{\partial \tau_{zx}}{\partial z} +B_x $$ And using the Reynolds transport theorem: $$ \frac{d}{dt} \int_V \vec{F} dV = \int_V \frac{\partial \vec{F}}{\partial t}dV +\int_S \left(\vec{v}_{\rm cv} \cdot \vec{n} \right) \vec{F} dS $$ Prove that the $i$th component of the viscous momentum equation in integral form corresponds to: $$ \frac{{\rm d}}{{\rm d} t} \int_V \rho \vec{v}_i \,{\rm d} V + \int_S \rho \vec{v}_i (\vec{v}-\vec{v}_{\rm cv}) \cdot \vec{n}\, {\rm d} S = -\int_S P \vec{n}_i \,{\rm d} S + \int_S \sum_{j=1}^3 \tau_{ji} \vec{n}_j \,{\rm d} S + \int_V \vec{B}_i \, {\rm d} V $$ where $\vec{v}$ is the velocity of the flow and $\vec{v}_{\rm cv}$ is the velocity of the control volume surfaces.
Question #3
Consider a jet of fluid interacting with an infinite fluid at rest. In class, we derived the following streamfunction for this problem: $$ \psi= \left(\frac{9 \nu M x}{2 \rho} \right)^\frac{1}{3} \tanh(\eta) $$ with $$ \eta=\frac{y}{ x^{\frac{2}{3}}}\left( \frac{M}{48 \nu^2 \rho} \right)^\frac{1}{3} $$ with $M$ the momentum integral at any $x$ station. Starting from the above streamfunction, calculate the mass flow rate going through one plane perpendicular to the $x$ axis. Is this mass flow rate constant along $x$? If so, explain where in the derivation of the above streamfunction the conservation of mass was imposed. If not, explain what would need to be changed in the derivation to impose conservation of mass.
Question #4
Recall that we derived in class the far wake velocity distribution for a non-lifting cylinder as: $$ u_{\rm w}=u_\infty - \frac{F_{\rm D}}{\rho L} \sqrt{\frac{1}{4\nu \pi u_\infty x}} \exp \left( -\eta^2\right) $$ with $$ \eta=y \sqrt{\frac{u_\infty}{4 \nu x}} $$ where $L$ is the length of the cylinder and $F_{\rm D}$ is the drag force on the cylinder. Starting from the latter, do the following:
(a)  List the various boundary conditions that $u_{\rm w}$ must satisfy.
(b)  Verify that the expression for $u_{\rm w}$ does satisfy the boundary conditions listed in (a).
(c)  Following similar steps as done in class for the jet case, outline an adequate definition of the wake height $\delta$ given the velocity profile $u_{\rm w}$.
(d)  Using the definition for $\delta$ outlined in (c), determine how $\delta$ varies with $x$.
Due on Tuesday May 5th at 11:00. Do all problems.
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