Fundamentals of Fluid Mechanics B Assignment 8 — 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
Consider a jet of height $h$ with uniform velocity $u_0$ exiting a channel and interacting with a gas at rest. In class, we derived an expression for the velocity as follows: $$ u= \frac{A}{B} x^{-1/3} f_\eta $$ with $$ A=\left(\frac{9\nu M}{2\rho} \right)^\frac{1}{3}~,~~ B=\left(\frac{48 \nu^2 \rho}{M} \right)^\frac{1}{3}~,~~f(\eta)={\rm tanh}(\eta)~,~~M=\rho u_0^2 h $$ Starting from the latter, do the following:
(a)  Outline an adequate definition of the jet height $\delta$ given the velocity profile $u_w$;
(b)  Using the definition for $\delta$ outlined in (a), determine how $\delta$ varies with $x$.
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 the mass conserved? 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)  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 $x$.
Question #5
Recall that 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.
04.15.21
Question #6
Consider water flowing around a cylinder as follows:
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The water temperature is of 27$^\circ$C, its density 1000 kg/m$^3$, its viscosity $0.001$ kg/ms, the velocity far from the cylinder $u_\infty$ is of 0.5 m/s, and the cylinder diameter $D$ is of 0.01 m. It is decided to obtain a steady-state solution of the wake by neglecting pressure gradients and streamwise viscous stresses within the momentum equation. Thus, the $x$ momentum equation reduces to: $$ \rho u \frac{\partial u}{\partial x} + \rho v \frac{\partial u}{\partial y} = \mu \frac{\partial^2 u}{\partial y^2} $$ Do the following:
(a)  Use order of magnitude analysis and find the condition when the term $\mu \frac{\partial^2 u}{\partial x^2}$ can be ignored in the $x$ momentum equation. Explain your reasoning in detail.
(b)  Using the condition found in (a) and the freestream flow properties estimate the error that originates from neglecting the term $\mu \frac{\partial^2 u}{\partial x^2}$ in the $x$ momentum equation.
(c)  Starting from the exact solution to the wake in the far field derived in class: $$ \Psi=\frac{F_{\rm D} {\rm erf}(\eta)}{2 \rho L u_\infty} $$ with $$ \eta=y \sqrt{\frac{u_\infty}{4 \nu x}} $$ derive an expression for $\frac{\partial^2 u}{\partial x^2}$ at $y=0$ and another expression for $\frac{\partial^2 u}{\partial y^2}$ at $y=0$.
(d)  Using the expressions found in (c), calculate the error due to neglecting the term $\mu \frac{\partial^2 u}{\partial x^2}$ in the $x$ momentum equation at $y=0$ for $x=2D,~5D,~20D$. Is the error found in (b) similar to the one found in (c)? Explain why there is a discrepancy.
Answers
1.  $\delta\propto x ^\frac{2}{3}$
Due on Tuesday April 27th at 11:00. Do Questions #1, #3, and #6 only.
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