2018 Viscous Flow Final Exam  
Viscous Flow
Final Examination
Friday December 14, 2018
19:00 — 22:00

Question #1
Starting from the Navier-Stokes equations and the mass conservation transport equation, show that the constant-density constant-viscosity $y$-momentum equation corresponds to: $$ \rho \left(\frac{\partial v}{\partial t} + u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y} + w \frac{\partial v}{\partial z} \right) = -\frac{\partial P}{\partial y} + \mu \frac{\partial^2 v}{\partial x^2} + \mu \frac{\partial^2 v}{\partial y^2} + \mu \frac{\partial^2 v}{\partial z^2} $$
Question #2
Consider a journal bearing as follows:
Q2.png  ./download/file.php?id=4615&sid=aff4ecf5bfde075c9adf122ebae62d10  ./download/file.php?id=4615&t=1&sid=aff4ecf5bfde075c9adf122ebae62d10
Starting from the Navier-Stokes equations in cylindrical coordinates, derive an expression for $v_\theta$ as a function of $r$. It is known that $R_2-R_1=\frac{1}{4} R_1$. Indicate clearly the assumptions (including terms dropped) and explain why this is valid in this case.
Question #3
Consider the following two-dimensional channel with smooth walls:
duct.png  ./download/file.php?id=4614&sid=aff4ecf5bfde075c9adf122ebae62d10  ./download/file.php?id=4614&t=1&sid=aff4ecf5bfde075c9adf122ebae62d10
Using the finite volume method, derive the discretized equations for the boundary and inner nodes for the following governing equation: $$ \frac{\partial }{\partial y}\left((\mu+\mu_{\rm t})\frac{\partial u}{\partial y} \right)= \frac{\partial P}{\partial x} $$ Then, write down the C code for the functions init_y_u() and find_coefficients_and_rhs():
/* initialize y[], u[], k[], omega[] on all nodes given
   N -> number of nodes
   H -> height of half channel in m
void init_y_u(long N, double H, double dy1, double *y, double *u){
  long j;


/* finds the discretization coefficients a,b,c in kg/m3s and the RHS rhs in Pa/m for all nodes given
   N -> number of nodes
   H -> half height of the channel in m
   rho -> density in kg/m3
   mu -> viscosity in kg/ms
   dPdx -> pressure gradient in Pa/m
   u[] -> velocity on all nodes in m/s */
void find_coefficients_and_rhs(long N, double *y, double rho, double mu, double dPdx, double *u, double *a, double *b, double *c, double *rhs){

Your code must have the following characteristics:
(a)  It must take advantage of the symmetry of the problem: reduce the domain from $0\le y \le 2 H$ to $0 \le y \le H$ by imposing a symmetrical boundary condition at $y=H$.
(b)  It must implement variable grid spacing and cluster more nodes near the wall than near the axis of symmetry.
Recall: $$ \mu_t \approx \rho L_m^2 \left| \frac{\partial u}{\partial y}\right| $$ with $$ L_m = 0.435 y \left(1- \exp(-y^+/26) \right) $$ and with $$ y+\equiv \frac{y \sqrt{\rho \tau_w}}{\mu} $$ with $\tau_w$ the shear stress at the wall.
Note: you must write down only one code that has both the symmetry condition and the variable node spacing.
Question #4
Consider a fluid flowing at steady-state in a pipe of diameter $D$ and length $L$. The flow can be assumed to be fully developed from the entrance of the pipe to the exit. For a fixed mass flow rate $\dot{m}$, a fixed viscosity $\mu$, a fixed density $\rho$, and a fixed length $L$, do the following:
(a)  Show that $\Delta P \propto D^{-4}$ should the flow be laminar.
(b)  Show that $\Delta P \propto D^{-4.8}$ should the flow be turbulent and the pipe walls smooth.
(c)  Show that $\Delta P \propto D^{-5}$ should the flow be turbulent and the pipe walls rough with $e/D>0.01$ and ${\rm Re}_D>10^5$.
Notes: $\Delta P$ is the pressure difference between the entrance and the exit of the pipe (i.e., the pressure drop); you should start your proof from the balance of the forces acting on a fluid slice within the pipe.
Question #5
Consider the following pump:
pump.png  ./download/file.php?id=4612&sid=aff4ecf5bfde075c9adf122ebae62d10  ./download/file.php?id=4612&t=1&sid=aff4ecf5bfde075c9adf122ebae62d10
Water with a density of 1000 kg/m$^3$ and a viscosity of $0.001$ kg/ms flows through the pump at a rate of 1 kg/s. The radii of the pipes at station 1 and 2 are of $R_1=0.5$ m and $R_2=1$ m respectively. It is known that the flow properties at station 1 are all uniform and that the pressure at station 2 is also uniform. However, the specific internal energy at station 2 is not uniform but rather varies according to: $$ e_2=e_1 + a\left(1- \frac{r}{R_2} \right) $$ with $r$ the radial coordinate and $a$ a constant equal to 100 J/kg. Knowing that the pump power ${\cal P}=200$ W and that the pump loses heat to the environment as $\dot{Q}=50$ W, do the following:
(a)  Find the pressure increase through the pump $P_2-P_1$.
(b)  Find the pump efficiency.
Hint: the energy change due to gravity can be neglected.
Question #6
Consider the following node arrangement:
The governing equation is: $$ \frac{\partial G}{\partial y}=S $$ with $G=\eta\frac{\partial u}{\partial y}$ and with $S=k$ where $k$ is a constant. Recall that the finite volume method can be expressed as: $$ G_{j+1/2}-G_{j-1/2}=\widetilde{S}\Delta y_j $$ with $j$ the node index and $\widetilde{S}$ the average of $S$ over the finite volume. Knowing that $y_2=1$ m, $y_3=2$ m, and $y_4=4$ m, do the following:
(a)  Find the most accurate representation of $G_{j=3.5}$ using values of $\eta$, $y$, and $u$ on nodes 3 and 4 only. I.e., the expression for $G_{j=3.5}$ should be function of $\eta_3$, $\eta_4$, $y_3$, $y_4$, $u_4$, and $u_3$ only.
(b)  Find the most accurate representation of $G_{j=3.5}$ using values of $\eta$, $y$, and $u$ on nodes 2, 3 and 4 only.
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