Student Notes

My notes

Attached are the handwritten notes, collected during lectures by Dr. Rusak. They are mostly an exercise in wrote just as brainless is and attacked for completeness.

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MANE 6529: Lecture Notes

First half

Lecture 1

Fluid mechanics is a topic in classical physics of fluid is a substance gas or liquid where intermolecular forces are medium to week and there's a significant random movement of molecules and disorder in the arrangement

Continuum hypothesis a fluid is modeled as a Continuum or material that is exhibiting no structure however small is divided

$\lambda$ is the mean free path

the mean free path is a distance between molecular collisions 10-7 M for air at STP greater than a meter at the edge of the atmosphere 10 -10 m in water liquid

Knudsen number

$\text{Kn} = \frac{\lambda}{L} << 1$

but L characteristic length of Interest much greater than Lambda Kn = lambda/L <<< 1 Collision between thermal equilibrium tau is less than 10 to the -8 second and air at STP free continuing Behavior how is much less than the time of Interest which we measure

Is important to recognize that for very small distances approximately within an order of magnitude of the mean free path of the situation at hand

density as a function control volume $\rho = \lim_{\Delta V\rightarrow min} \frac{\Delta m}{\Delta Vol}$

specific volume $ v = 1/\rho$

Time Space Scale Diagram

Lecture 2

To find the pressure of a hydrostatic fluid look at the unit tetrahedra each of the minor faces has a force orthogonal to eachother. The normal force balancing has to be the balaning them allso that the pressure is the force pre unit area of some arbitrary are

$$p_n = \stackrel{\text{lim}}{\text{min}} \frac{F_n}{A_n} = p + O(\mu)$$

Inreality there are viscous effects but are very small for most slightly viscous fluids so the viscous component can usually be neglected

Temperature of a Fluid Element

Definition

$$E_{kinetic} = \frac{1}{N} \sum\frac{1}{2} m_nV_v^2 = \frac{3}{2}kT$$

Average kinetic energy

k = 1.38\timex 10^{-26}J/K

Specific properties

• internal energy :$du = c_vdT - [T(\frac{\partial p}{\partial T}-p)]\frac{d\rho}{\rho^2}$

• enthalpy: $h = u + \frac{P}{\rho}$ or $dh=du+d(p/\rho)$

• entropy: $Tds = cVdT - (\frac{\partial P}{\partial T}){cost. \rho}\frac{d\rho}{\rho^2}$ leads to $Tds = dh - \frac{d\rho}{\rho}$ or the gibbs eqn

simple compressible fluid: any two properties define the rest $P = f(\rho,T)$ but that is not the only choice. Just need to link the fluid to the critical point

Phase diagram: solid liquid vapor regions

Critical point $(P_c ,\rho_c, T_c)$ is the location where first and second partials of pressure with respect to density for constant temperature is 0

$$\left(\frac{\partial P}{\partial \rho }\right)_T = \left(\frac{\partial^2 P}{\partial \rho^2 }\right)_T = 0$$

• Liquid: high first derivative so we can usually hold \rho constant rate of temperature is not necessarily high tho

• Gas vapor is moderate so that small changes in density create comporable changes in presssure

Reduction of Gasses wrt Critical Point

Carried out on the variable $(\cdot)$ by dividing by the corresponding $(\cdot)_c$ so that $(\cdot)_R = (\cdot)/(\cdot)_c$. for $P,\rho = 1/\mathcal{V},T$ and the compressibility factor $Z = \frac { P } { \rho RT}$ where $R = \mathcal{R}/M$ the universal gas constant div the molar weight Z is a function of $P_R,T_R$

For low pressure and density, we have the conditions for a thermodynamically perfect ideal gas $P = \rho RT$

Vander Waals Eqn of State

def $\left( P + a \frac { 1 } { V { m } ^ { 2 } } \right) \left( V { m } - b \right) = R T$ or as he preferes $P = \rho RT/(1-b\rho)-a\rho^2$

Redlich-Kwong Eqn. of State

def: $p = \frac { R T } { V { m } - b } - \frac { a } { \sqrt { T } V { m } \left( V _ { m } + b \right) }$

Lecture 3

Scalars and Tensors

Scalars are properties without a preferred direction $(\rho,P,T,\mu,h,s)$ so that the normal operations are defined on

Vectors are a properties with a prefered direction so that ${\bf V} = u{\bf e}_x + v {\bf e}_y + w{\bf e}_z = V_x{\bf e}_x + V_y{\bf e}_y + V_z{\bf e}_z$

Here with tensors or order 1 or greater, operations are not commuitive, directions matter

Examples

• component extraction ${\bf c} = k{\bf a} = ka_i{\bf e}_i\rightarrow c_i = ka_i $

• addition subtraction

  ${\bf c} = {\bf a} \pm {\bf n} = (a_i + b_i){\bf e}_i$

• multiplication is a measure of orthogonality of two vectors

  • dot inner product : c is now a scalar $c = {\bf a\cdot b} =a_i \space b_i$

  • cross outer product: c is of the same rank $\bf c = a\times b$ or $ci = \epsilon{ijk}a_jb_k$

Tensors of Rank 2 and up

Rusak’s notation is kinda fucked up. I’m going to copy one verbatim and describe my qualms

$\stackrel{T}{\underline{\underline{}}} = \stackrel{\rightarrow}{a}\stackrel{\rightarrow}{b}$ and does not define the operation which takes place between. Obviously it’s an outer dyatic product but notationally it does not make sense unless indexes are applied

$\textcolor{red}{fucked up }$

Lecture 4

Fluid Element Kinematics

Pathline

what happens to a single element

Streakline

all the points which have passed through a point from some $t_0$

Streamline

hold time constant and show tangent line of the velocity field or path element would take if time was constant so that $d{\bf l} \times {\bf V} = 0 $.

$$\frac{dy}{dx} = \left.\frac{V_y}{V_x}\right|_{t=\text{const}}\quad \frac{dz}{dx} = \left.\frac{V_z}{V_x}\right|_{t=\text{const}}$$

Local Global Approaches

Eulerian

Construct field so that

Lagrangian

Follow the element in time so that every variable has a definite derivative wrt time.

Lecture 5

Let the quantity of interest be $\bf V$.

${\bf a}_e = \frac{D{\bf V}_e}{Dt} = \frac{d{\bf V}_e}{dt} = \left(\frac{\partial{\bf V}_e}{\partial t} \right)_e + {\bf V}_e(t) \cdot \left(\nabla {\bf V}\right)_e $

Decompose the velocity gradient into even and odd components

$$\nabla {\bf V} = \overbrace{\frac{1}{2} \left(\nabla {\bf V}+ \left(\nabla {\bf V}\right)^T\right)}^{\bf D} + \overbrace{\frac{1}{2} \left(\nabla {\bf V}- \left(\nabla {\bf V}\right)^T\right)}^{\bf \Omega}$$

$\bf D$ deformation tensor

$\bf \Omega$ spin tensor

Then the velocity vector is described in a Taylor series approximation for fixed time $\textcolor{red}{\texttt{I think he fucked up in the notes here}}$ we have comments of the velocity field in all directions not just ${\bf e}_x$

$${\bf V}(x_o+dx,y_o+dy,z_o+dz,t) = \\ \left.{\bf V}\right|_{(x_0,y_0,z_0,t)} + \left.\frac{\partial {\bf V}_x}{\partial x}\right|_{(x_0,y_0,z_0,t)}dx{\bf e}_x + \left.\frac{\partial {\bf V}_y}{\partial y}\right|_{(x_0,y_0,z_0,t)}dy{\bf e}_y + \left.\frac{\partial {\bf V}_z}{\partial z}\right|_{(x_0,y_0,z_0,t)}dz{\bf e}_z + \text{h.o.t.}\ldots \\ = {\bf V}(x_o,y_o,z_o,t) + (\nabla{\bf V})|_{(x_0,y_0,z_0)}\cdot d{\bf l} + \ldots$$

Note that $d{\bf l}= dx{\bf e}_x + dy{\bf e}_y + dz{\bf e}_z$. In the limit as the differential segment length approaches zero

Fluid FLow Equations

When Fi is an extensive property (volume dependent), the we normalize not by volume but by mass to get a specific quantity : total to specific

As the fluid element evolves in time, we get reynolds transport theorem

$$\frac{d}{d t}(F_{II})=\dot{\mathcal{F}}_{\text {inlet}}-\dot{\mathcal{F}}_{\text {outlet}}+\dot{\mathcal{Q}}_{F}$$

Lecture 6

Reynolds Transport theory

let $F{II} = \int\text{vol} \rho f d\text{vol}$ where $f$ is the normalized $F$ per unit mass

• $f = 1 $, then $F_{II} \rightarrow \text{mass in volume at }t $

• $f = {\bf V} $, then $F_{II} \rightarrow \text{momentum in volume at }t $

• $f = e = u + \frac{1}{2}{\bf V}\cdot {\bf V}$, then $F_{II} \rightarrow \text{total energy in volume at }t $

Inlet

$\dot{\mathcal{F}}_\text{inlet} = -\int\rho f({\bf V \cdot n})dS$

with outlet

$\dot{\mathcal{F}}_\text{outlet} = \int\rho f({\bf V \cdot n})dS$

then diff

$\dot{\mathcal{F}}\text{inlet} - \dot{\mathcal{F}}\text{outlet} = -\int\rho f({\bf V \cdot n})dS$

end with

$$\frac{d}{dt}\int_V\rho f d\text{vol} + \int_S\rho f({\bf V\cdot n})dS = \dot {\mathcal{Q}}_F$$

is the integral equation of balance on $F$ where $\dot{\mathcal{Q}}F = \int\text{Vol} \dot{q}_F d\text{vol}$

The Energy Equation

Let $f = e_t$ specific total energy , $e_t = u + \frac{1}{2} {\bf V \cdot V}$

Integral form: $\frac{d}{dt}\int_V \rho e_t d V+ \int_S \rho e_t({\bf V \cdot n}) = \dot{Q}_F$ power or energy per unit time

Conservative $\frac{d}{dt}(\rho e_t)+ \rho e_t({\bf V \cdot n}) = \dot{q}_F$

Regular $\rho\left[\frac{d}{dt}( e_t)+ e_t({\bf V \cdot n})\right] = \dot{q}_F$

a

Lecture 7

Extensive property $F = \int_V \rho f dV$ so it’s normalized by mass but volume dependent

Integral equation

$$\frac{d}{dt}\int_V \rho f dV = \int_S \rho f ({\bf V \cdot n})dA \textcolor{red}{\texttt{ ``he fucked up here too"}} = \dot{\mathcal{Q}}_F$$

Conservation form

$$\frac{\partial}{\partial t}(\rho f) + \nabla \cdot (\rho f{\bf V}) = \dot{q}_F$$

Note that $\dot{q}$ is $\dot{\mathcal{Q}}$ per unit volume

Regular Form through the domain

$$\rho \left( \frac{\partial f}{\partial t} + {\bf V}\cdot\nabla f \right) =\dot{q}_F$$

Lagrangian following a fluid element

$\rho_e(t)\left(\frac{df}{dt}\right)_e = \dot{q}_F$

Equations of Motion

The linear momentum equation where $f \rightarrow {\bf V}$ and $\dot{q}_F \neq0$

$\frac{\partial (\rho {\bf V})}{\partial t} + \nabla \cdot (\rho({\bf V\otimes V})) ={\bf \dot{q}}_F$ or $\rho(\frac{\partial ( {\bf V})}{\partial t} + \nabla \cdot ({\bf V\otimes V})) ={\bf \dot{q}}_F$

in Newton’s 2nd law form $\rho_e(t)(\frac{d{\bf V}}{dt})_e = ({\bf \dot{q}}_F)_e$

Forces Exerted on body

Body Force : $\int_V\rho {\bf B}dV$

Surface Force (traction) :

Then ${\bf \Theta }= -p{\bf n}+{\bf \tau}$

$$\int_A{\bf \Theta }dA= -\int_Ap{\bf n}dA+\int_A{\bf \tau}dA$$

More Regularly

$$\dot{\bf Q}_F = \int_V \rho{\bf B}dV -\int_S p{\bf n}dS + \int_S {\bf \tau}dS$$

Use Cauchy to show assuming symmetric stress tensor $\bf \tau = T \cdot n$.

Use divergence theorem $\int_S {\bf A\cdot n}dS=\int_V \nabla\cdot {\bf A}dV$

Use Green’s theorem $\int_S a{\bf n}dS = \int_V\nabla a\space dV$

we produce

$$\dot{\bf Q}_F = \int_V \rho {\bf B} dV - \int_V \nabla p dV - \int_V \nabla \cdot {\bf T}dV \\ = \int_V \left( \rho {\bf B} - \nabla p + \nabla\cdot{\bf T} \right) \rightarrow \dot{bf q}_F = \rho {\bf B} - \nabla p + \nabla\cdot{\bf T}$$

Integral

$$\frac{d}{dt}\int_V \rho {\bf V}dV + \rho{\bf V(V \cdot n)}dS = \int_V\rho {\bf B}dV - \int_S p{\bf n}dS + \int_S {\bf T \cdot n}dS$$

Conservative

$$\frac{\partial(\rho {\bf V})}{\partial t} + \nabla \cdot (\rho ({\bf V \otimes V})) = \rho{\bf B} -\nabla p + \nabla\cdot{\bf T}$$

REgular

$$\rho(\frac{\partial{\bf V}}{\partial t} + ({\bf V \cdot \nabla V})) = \rho{\bf B} -\nabla p + \nabla\cdot{\bf T}$$

Lagrangian

$$\rho_e(t)(\frac{d{\bf V}}{d t})_e = \rho_e{\bf B}_e -(\nabla p)_e + (\nabla\cdot{\bf T})_e$$

What is the stress tensor? Well Euler is inviscid so $\bf T = 0$.

Euler Equations of Motion

$$\frac{\partial (\rho {\bf V})}{\partial t} + \nabla \cdot \left(\rho ({\bf V\otimes V})\right) = \rho {\bf B}-\nabla p$$

Regular Form

$$\rho\left(\frac{\partial {\bf V}}{\partial t} + {\bf V}\cdot \nabla {\bf V}\right) = \rho {\bf B}-\nabla p$$

Navier Stokes Model : linear Relation

$${\bf T} = \mu (T,p) \left[ \nabla{\bf V} + (\nabla{\bf V})^T - \frac{2}{3} \left( \nabla {\bf V} \right) {\bf I} \right] + \mu_\mathcal{V} \left( \nabla {\bf V} \right) {\bf I}$$

2 parameter viscosity and bulk viscosity for a newtonian fluid

$$\rho \left( \frac{\partial{\bf V}}{\partial t} + {\bf V} \cdot \left( \nabla {\bf V} \right) \right) = \rho {\bf B} - \nabla p + \nabla \cdot \left\{ \mu \left[ \nabla {\bf V} + \left( \nabla {\bf V} \right)^T - \frac{2}{3} \left( \nabla \cdot {\bf V} \right){\bf I} \right] + \mu_\mathcal{V} \left( \nabla \cdot {\bf V} \right){\bf I} \right\}$$

Nonlinear is ${\bf T} = f\left(\nabla \cdot {\bf V}\right)$

Lecture 8

Summary of Equations

Continuity $\frac{\partial \rho}{\partial t} \nabla\cdot(\rho{\bf V}) $

Equations of Motion

$\rho\left(\frac{\partial {\bf V}}{\partial t} + {\bf V}\cdot(\nabla {\bf V})\right)=\rho{\bf B} - \nabla p + \nabla \cdot {\bf T}$

Energy Equation

$\rho \left( \frac{\partial e_t}{\partial t} + {\bf V} \cdot \nabla e_t\right) = \rho {\bf B\cdot V} - \nabla \cdot (p {\bf V}) + \nabla \cdot ({\bf T\cdot V})-\nabla \cdot {\bf q}$

Equation of State

$p = f(\rho , T) $ or hold $\rho$ as constant

Lagrangian Following a Fluid Element

$\frac{1}{\rho_e}\left(\frac{d\rho}{dt}\right)_e = - \nabla \cdot {\bf V}$

$\rho_e(t) \left(\frac{d{\bf V}}{dt}\right)_e = \rho_e{\bf B}_e - (\nabla p)_e + (\nabla \cdot {\bf T})_e$

$\rho_e(t) \left(\frac{d{e_t}}{dt}\right)_e = \rho_e{(\bf B\cdot V)}_e - (\nabla \cdot (p{\bf V}))_e + (\nabla \cdot {\bf (T\cdot V)})_e -(\nabla \cdot {q})_e$

Boundary Conditions

Inlet

Outlet

Wall Conditions

Rigid Bodies

Internal boundary condition where $\bf V\cdot n$ is some value.

Solid

Porous

Far Field

Lecture 9

Nondimensional equations

Using a characteristic set of reference properties, we can create a solution which describes a family of solutions, of geometrically similar problems: care only about the shape of the airfoil

Choose a selection of characteristic properties

$$\text{Geometric: }L_x,L_y,L_z \\ \text{Velocity: }V_x, V_y,V_z \\ \text{Property: }t_c, p_c, \rho_c$$

Lecture 10

NRG equation

$$\rho\left(\frac{\partial e}{\partial t} + {\bf V}\cdot \nabla e\right) = \rho {\bf B\cdot V} - \nabla\cdot(p{\bf V}) + \nabla\cdot({\bf T \cdot V}) - \nabla \cdot {\bf q}$$

Lecture 11

Consider

Second half

summary

Consider