Lecture Slides

All the things that were used in class or assigned as homework. Sent by email and hard copies scanned and uploaded.

Lectures 1 - 11: Mathematics of Fluid Mechanics

Lecture 1: A Topic in Classical Physics

7 pages

Fluid Definition: A structure (gas or liquid) where intermolecular forces are moderate to weak. This contrasts with solids where intermolecular forces are strong. Additionally, there is significant molecular thermal motion and disorder in arrangement. The counter example are crystals, where there is periodicity.

Continuum Hypothesis: A fluid exhibits no structure, no matter how small it is divided. Where $\lambda$ is the mean free path between molecules. Let $L$ be the characteristic length of interest where

Time-Space Scale Diagram: On a power log graph of a bivariate equation

\log t = \cancel{k}^{1} \log x + \cancel{\log a}^0

Where $x$ and $t$ correspond to SI length and time scales; the constant $k$ is unity, there distinct regions of length scales separating global, fluids, molecular, and atomic interactions. Which line every 3 orders of magnitude.

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$

Lecture 2: Fluid Properties

10 pages

Density: Total mass per unit volume is calculated by summation on mass of every molecule. When the increment in volume goes below the cube of the mean free path, increment in mass also loses its identity.

Pressure: Measured in Pascals [Pa], measure of the force per unit area. Derived by conservation of linear momentum along each direction along the 3 identical right triangular faces on the unit tetrahedron.

From geometry, the ratio of areas of the cosine of the angles between vectors describing each face to the large equiliateral triangular face.
From Newton's Law for a fluid at rest.
Pressure

Temperature: a measure of the molecular average of the kinetic energy of the particles in a fluid (typically gas). Related by Boltzmann's Constant.

Specific Version of an Extensive Quantity: is the density normalized version of some measure which otherwise would scale with volume of a fluid.

Specific Internal Energy: (how do I even explain this)

du = c_VdT - \left[T\left.\left(\frac{\partial P}{\partial T}\right)\right|_{\rho_{const}}-p\right]\frac{d\rho}{\rho^2}

Specific Enthalpy:

Energy of creation is one way to think of it

h = u + \frac{P}{\rho} \quad dh = du+ d\left(\frac{P}{\rho}\right)

Specific Entropy

Measure of disorder. Or measure of deviation from "perfect" order. Second definition comes from Gibbs Eqn.

TdS = c_VdT - \left.\left(\frac{\partial P}{\partial T}\right)\right|_{\rho_{const}}\frac{d \rho}{\rho^2} = dh - \frac{d\rho}{\rho}

Simple Compressible Fluid

Where 2 variables define the rest, so that a surface is described in R^3 (P,\rho,T).

Phase Diagram

Three regions of Solid, Liquid, and Vapor meet at one location: the critical point. This is the inflection point with respect to the fluid system energy. At critical point:

\left.\left(\frac{\partial P}{\partial \rho}\right)\right|_{T} = \left.\left(\frac{\partial^2 P}{\partial \rho^2}\right)\right|_{T} = 0.

These state first, that the energy surface is continuous, and then that it is smooth; this is only true when both are identically zero, or homogenous.

Reducible Values

Akin to the specific versions of extensive properties, normalize one of energy surface's quantities by its critical value. Can be done on pressure, density (or its' inverse, specific volume), or temperature. It's also how we have dimensionless versions of working quantities.

Vander Waals Equation of State

P = \frac{\rho R T}{1-b\rho}-a\rho^2

Redlich Kwong Equation of State

P=\frac{R T}{V_{m}-b}-\frac{a}{\sqrt{T} V_{m}\left(V_{m}+b\right)}

In the limit as the reduced pressure approaches 0, then we see equation of state of a thermodynamically perfect equation of an ideal gas.

P = \rho RT

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

tab 3 as

Lecture 3: Vector and Tensor Math

9 pages

Scalar is a property with no preferred direction. These include density, average pressure, temperature, viscosity, enthalpy, and entropy:

(\rho, P, T, \mu,h,s).

Vector properties have preferred directions. List them in order, typically (x,y,z).

Tensor properties have preferences in 2 directions simultaneously. Weird notation used in class but the point comes across.

Convective Term

Gradient of the velocity vector field produces a tensor field, typically referred to as the velocity gradient which can act on the velocity vector. This is seen in the convective term of Navier-Stokes, or Reynold's Transport Theorem applied to Conservation of Linear Momentum.

\frac{D}{Dt} = \frac{\partial}{\partial t} + {\bf v}\cdot \nabla

Trace of Velocity Gradient or Divergence of Velocity

This is a measure of the compressibility of the fluid and typically zero.

Vorticity Vector, or application of the differential cross product

Also known as the rotor of a generic vector quantity.

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 7: Intensive and Extensive Properties

13 pages

Equations of Motion

Forces

Body
Surface

Navier Stokes Equation

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

Lectures 12 - 22: Physical Flows and Turbulence

Lecture 12: Vorticity Transport Equation

6 pages

From the equations of motion and the vector identity

Balance or Transport Equation of Vorticity

tilting and the stretching/cromressing of vorticity by velocity gradient
measure of the compressibility of of vorticity
Baroclinic Effect
Viscous Diffusion of Vorticity

Fluid Statics

Body Force Due to Gravitation
Due to Atmospheric Pressure

Focus is on incompressible flow or constant density fluid.

Lecture 12

I missed this lecture, mom had a bad day and I had a bad call. RIP

Vorticity Transport Equation

From 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 {\text{traction}}}

We find using the vector identity ${\bf V}\cdot\nabla{\bf V} = \nabla \left(\frac{2}\right) -{\bf V}\times{\underline{\omega}}$ then applying $\nabla \times (\cdot)$ of the expression.

for a newtonian fluid the balance of vorticity $\underline{\omega}$. Effect to change in the flow are each of the terms:$\textcolor{red}{\text{screen caps would be nice}}$

$\underline{\omega}\cdot(\nabla{\bf V})$ is the streching
$\underline{\omega} \left(\nabla \cdot{\bf V}\right)$ measure of compressibility and stretching of rotation
$\nabla T\times \nabla s$ boroclinic effect since $s(T, p), \quad \vec{\nabla} s=(\ldots) \vec{\nabla} T+( \ldots) \vec{v} p$
the rest is viscous diffusion

Fluid statics

If the fluid is not in motion, pressure becomes the dominant effect this is $\bf V=\underline{0}$ and $\nabla {\bf V} = \underline{\underline{0}}$. Equations of motion become $\nabla p = \rho {\bf B}$ the fluid static equation.

Assuming that ${\bf B } = g {\bf e}z$ and $\rho$ is a constant for most liquids. so $\frac{d p}{dz} = \rho g$ and $p|{z=0} = p_\text{atm}$. So integrate from the surface atmospheric for the hydrostatic equation $p(z) = \rho g z+p_a $ for $z \geq 0$.

F = \int_{z_1}^{z_2} \left(p(z)-p_a\right)dz\\ M = \int_{z_1}^{z_2} \left(p(z)-p_a\right)(z-z_1)dz

Force and moment but idk if this is right… don’t trust the units...

Atmospheric pressure

Use the expression $\frac{dp}{dz} = - \rho g$, assuming a perfect gas $p = \rho RT$ creates the aerostatics equation

\frac{dp}{dz} = \frac{-p}{RT}g

Using a given T(z)$\textcolor{red}{\text{he corrected but I still can’t read his correction….}}$

\int_{p_\text{ref}}^{p} \frac{dp}{p} = \int_{z_\text{ref}}^{z} \frac{-g}{R\space T(z)}dz \rightarrow p(z)

Generally T is const $T(z) = \text{const}= 216.6 \text{ K.}$

Stratosphere $1.1\times 10^{4}<z<2.5\times 10^{4}$ where $z\text{bot} = 1.1\times 10^{4}$, $\ln (\frac{p}{p\text{bot}})=\frac{-g}{RT}(z-z_\text{bot})$ is solved straightforward

p=p_{11,000} \exp \left(\frac{-g}{R T}(z-z_\text{bot})\right)

Focus on liquids where incompressible have $\rho = \text{const}$. for low $\text{Ma}$. of gases and most liquids. Use integral equation of property balance for fixed volume

$\underline{\text{mass:}} \int{\text{vol}} \frac{\partial}{\partial t}d\text{Vol} + \int\text{surf} \rho {\bf V}\cdot {\bf n}dS=0$ where usually $\int_\text{surf} {\bf V}\cdot {\bf n}dS=0$

$\underline{momentum}$aaaaayyy $\int\text{vol}\frac{\partial}{\partial t}(\rho{\bf V})d\text{vol} + \int \rho {\bf V}({\bf V}\cdot {\bf n})dS = \int\text{vol} \rho {\bf B} d\text{vol} - \int\text{surf} p {\bf n} dS + \int\text{surf} \underline{\tau} dS $

$\cancelto{0}{\int\text{vol}\frac{\partial}{\partial t}(\rho{\bf V})d\text{vol} }+ \int \rho {\bf V}({\bf V}\cdot {\bf n})dS = \int\text{vol} \rho {\bf B} d\text{vol} - \int\text{surf} p {\bf n} dS + \int\text{surf} \underline{\tau} dS \\textcolor{red}{\text{i can’t deal with these notes… impossible to transcribe}}$

Ya know, I'm not mathematician here, but even I can see a common denominator ...
from u/Synli on this post

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Lecture 1Specific volume Density Mean free path Fluid property

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 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 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 Lecture 2

Lecture 3Lecture 4Lecture 5Lecture 6Lecture 7Lecture 8Lecture 9Lecture 10Lecture 11Lecture 12

Finite Elements

# Problem 1 What fluid flow and the requirements for the application of fluid continuum mechanicsThermodynamic identity to differential internal energ What is the expression for specific internal energy using thermodynamic identity Use joe’s post in fluid Knudsen number for STP Air 10E-7 m What is vorticity and rotor? What is its relation to spin vector # Problem 2 Critical point something? Not sure right now Find the vorticity of a given field Complete the operation for gradient of a scalar and left hand dot # Problem 3 # Problem 4Write the lagrangian description of the fluid momentum or the equation of motionDefine each component Write the equation for conservation of energy in differential form Write the equation for the viscous stress tensor used in NS equations Will help to know the derivation 10-10 on so lecture 11

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Last updated 5 years ago

Lectures 1 - 11: Mathematics of Fluid Mechanics

Lecture 1: A Topic in Classical Physics

Lecture 2: Fluid Properties

Temperature of a Fluid Element

Phase diagram: solid liquid vapor regions

Reduction of Gasses wrt Critical Point

Lecture 3: Vector and Tensor Math

Scalars and Tensors

Examples

Tensors of Rank 2 and up

Lecture 4: Fluid Element Kinematics

Fluid Element Kinematics

Pathline

Streakline

Streamline

Local Global Approaches

Eulerian

Lagrangian

Lecture 5: Fluid Element Spin and Vorticity

Lecture 6: Balance Equations

The Energy Equation

Lecture 7: Intensive and Extensive Properties

Equations of Motion

Forces Exerted on body

Integral

Conservative

REgular

Lagrangian

Euler Equations of Motion

Navier Stokes Model : linear Relation

Lecture 8: Fluid Domain Boundaries

Summary of Equations

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

Equations of Motion

Energy Equation

Equation of State

Lagrangian Following a Fluid Element

Boundary Conditions

Inlet

Outlet

Wall Conditions

Rigid Bodies

Far Field

Lecture 9: Selection of Length Scale for Nondimensionalization

Nondimensional equations

Lecture 10: Similarity Parameters and Dimensionless Numbers

Lectures 12 - 22: Physical Flows and Turbulence

Lecture 12: Vorticity Transport Equation

Lecture 12

Vorticity Transport Equation

Fluid statics

Atmospheric pressure

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Lecture 13: Hydraulic Jump Equation

Lecture 14: Couette and Poiseuille Flow

Lecture 15: Turbulent Flow

Lecture 16: Blasius Boundary Layer Stability

Lecture 17: Boundary Layer Growth due to Pressure Gradient

Lecture 18: 2D Flow Over Flat Plate

Lecture 19: Boundary Layer Flow Over Flat Plate

Lecture 20: Boundary Layer on a Smooth Surface with a Pressure Gradient

Lecture 21: Stability of the Blasius Boundary Layer Solution

Lecture 22: Turbulent Flows

Reynolds Averaged Navier Stokes (RANS) Equations

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