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## Instructions:

Fluids are generally considered to be those materials that have the ability to constantly change their shape by adapting to the container, which is why liquids, vapors and gases are considered to be fluids. Fluid mechanics consists of two main branches:

Fluid mechanics deals with fluids that are stationary in an inertial system, i.e. with constant velocity in time and homogeneity in space. Historically, it was the first step towards the study of mechanics.

Fluid dynamics or fluid mechanics (including specifically aerodynamics, hydrodynamics, and oil dynamics), deals with fluids in motion.

Fluids are characterized by having their own volume and a density very similar to that of solids, which means that at the microscopic level, the distances between molecules remain small and the interaction forces are high. This is a fundamental difference from gaseous substances, which have a low density and therefore low intermolecular interactions, allowing them to expand at any volume.

(a) Supersonic flow enters a straight pipe of constant cross-sectional area. Heat transfer is negligible, but the pipe wall is rough. Draw a labelled graph to show how the Mach number distribution along the pipe evolves as the skin-friction coefficient increases from zero. You may assume that the exit pressure is low enough to ensure that the inlet conditions are always the same.

(a) The graph below shows the variation of the Mach number with distance along the pipe as the skin friction coefficient increases from zero. As the skin friction coefficient increases, the velocity near the pipe wall decreases, causing the boundary layer to thicken. This results in a reduction in the effective cross-sectional area available for flow, which in turn reduces the mass flow rate and increases the Mach number. The Mach number gradually increases until it reaches the sonic condition at the throat of the pipe, after which it remains constant until the exit.

(b) Air flows in a pipe of length $5.9 \mathrm{~m}$ and inside diameter $0.2 \mathrm{~m}$. The inlet stagnation pressure is 2.7 bar, and the static pressure at the pipe exit is 1 bar. If the exit is choked, and there are no shocks in the pipe, find: (i) the two possible values of the Mach number at the inlet; (ii) the skin-friction coefficient $c_f$ corresponding to each.

(b) From the given data, we can use the choked flow condition to find the Mach number at the inlet. The choked flow condition occurs when the flow velocity at the throat of the pipe reaches the local speed of sound. At the throat, the Mach number is therefore 1.

Using the isentropic relations for a perfect gas, we can relate the Mach number to the pressure ratio across the throat:

$\frac{P_{02}}{P_{01}}=\left(\frac{1+\frac{\gamma-1}{2} M_1^2}{\frac{\gamma+1}{2}}\right)^{\frac{\gamma}{\gamma-1}}=\frac{P_{02}}{P_e}=\left(\frac{A_e}{A_{02}}\right)^2=1$,

where $P_{01}$ is the stagnation pressure at the inlet, $P_{02}$ is the pressure at the throat, $P_e$ is the static pressure at the exit, $A_{02}$ is the area of the throat, and $A_e$ is the area of the exit.

Solving for $M_1$ using the given values, we find that there are two possible values of the Mach number at the inlet:

$M_1=\sqrt{\frac{2}{\gamma-1}\left[\left(\frac{P_{02}}{P_{01}}\right)^{\frac{\gamma-1}{\gamma}}-1\right]}=0.747,2.11$.

Next, we can use the Prandtl-Meyer function to find the Mach number corresponding to a given skin-friction coefficient $c_f$. The Prandtl-Meyer function is a relation between the Mach number and the turning angle of a supersonic flow, and it depends only on the specific heat ratio $\gamma$ of the gas. The skin-friction coefficient $c_f$ can be related to the friction Reynolds number $Re_{\tau}$ using the law of the wall:

$c_f=\frac{\tau_w}{\frac{1}{2} \rho_1 V_1^2}=\frac{0.026}{R e_\tau^{0.2}}$,

where $\tau_w$ is the wall shear stress, $\rho_1$ is the density at the inlet, and $V_1$ is the velocity at the inlet.

For air at room temperature and pressure, $\gamma=1.4$. Using a table or a calculator, we can find that the Prandtl-Meyer function for $\gamma=1.4$ is approximately $\nu=30.5^{\circ}$ at $M=0.747$, and $\nu=66.1^{\circ}$ at $M=2.11$.

To find the turning angle corresponding to a given skin-friction coefficient, we can use the relation

$\theta=\frac{c_f}{2 \nu}$

Using the values of $c

(a) For each of the following equations or systems of equations, state whether they are hyperbolic, elliptic or parabolic. Also briefly discuss the implications for boundary/initial conditions and for numerical solution methods.

(i) $$ \frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=0 $$

(ii) $$ u \frac{\partial u}{\partial x}=\alpha \frac{\partial^2 u}{\partial y^2} $$ where $\alpha$ is a positive constant.

(iii) The Euler equations in one dimension, i.e. $$ \frac{\partial}{\partial t}\left[\begin{array}{c} \rho \\ \rho u \\ E \end{array}\right]+\frac{\partial}{\partial x}\left[\begin{array}{c} \rho u \\ \rho u^2+p \\ u(E+p) \end{array}\right]=0, $$ where the symbols have their usual meanings.

(i) The equation $\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=0$ is an example of the Laplace equation, which is an elliptic partial differential equation. The solutions to elliptic equations are smooth and have no characteristic curves, which implies that the boundary/initial conditions should be given on the whole boundary/initial surface. Numerical methods for solving elliptic equations include finite difference, finite element, and spectral methods.

(ii) The equation $u \frac{\partial u}{\partial x}=\alpha \frac{\partial^2 u}{\partial y^2}$ is an example of a convection-diffusion equation, which can be hyperbolic or parabolic depending on the values of $u$ and $\alpha$. To determine the type of equation, we need to compute the characteristic speeds. The characteristic speeds are given by $\lambda_1 = u$ and $\lambda_2 = \pm \sqrt{\alpha u}$, which are real and distinct when $u>0$ and $\alpha>0$. Therefore, the equation is hyperbolic when $u>0$ and $\alpha>0$. On the other hand, when $u<0$ or $\alpha<0$, the equation is parabolic. For hyperbolic equations, initial/boundary conditions must be prescribed along characteristic curves, while for parabolic equations, initial/boundary conditions must be given at an initial time/surface. Numerical methods for solving hyperbolic equations include finite difference, finite volume, and finite element methods, while numerical methods for parabolic equations include finite difference, finite element, and boundary element methods.

(iii) The Euler equations in one dimension are a system of nonlinear hyperbolic partial differential equations. The solutions to hyperbolic equations have characteristic curves, which are the curves along which information propagates. Boundary/initial conditions must be specified along characteristic curves, and numerical methods for solving hyperbolic equations must be designed to respect the properties of the characteristic curves. Some popular methods for solving hyperbolic equations include finite difference, finite volume, and finite element methods, as well as shock-capturing and high-resolution schemes.