# 1. Fluid Equations and Linear Wave Theory Flashcards

1
Q

Density Definition

A

-let Q denote the amount of a quantity in a volume V
-the density of Q, q, is defined as:
Q = ∫ q(x_,t) dV
-this also applies to vectors

2
Q

Mass Density

A
```Q = mass
q = ρ```
3
Q

Kinetic Energy Density

A
```Q = KE
q = 1/2 ρu²```
4
Q

Volume Density

A
```Q = V
q = 1```
5
Q

Linear Momentum Density

A
```Q_ = linear momentum
q = ρu_```
6
Q

Flux Definition

A

-flux of a quantity Q through a surface S is the rate at which Q passes through S
flux = ∫ q (u_ . n_) dS

7
Q

Mass Flux

A

mass flux = ∫ ρ (u_ . n_) dS

8
Q

Momentum Flux

A

flux = ∫ (ρu_) (u_ . n_) dS

9
Q

Conservation Laws

A

-the evolution of any quantity Q in a volume V is governed by a conservation law of universal form
dQ/dt = d/dt ∫qdV
= ∫qu_.(-n_)dS + ∫K.(-n_)dS + ∫adV
-first term is flux through S, second term includes any other surface sources of Q and the third term is for body sources

10
Q

Mass Conservation

A

-substitute q=ρ into the universal conservation law
-physics tells us that the second and third terms are 0
=>
d/dt ∫ρ dV = ∫ρu_.(-n_) dS
-put under single integarl using divergence theorem
∫∂ρ/∂t + ∇.(ρu_) dV = 0
-this is true for any V, thus:
∂ρ/∂t + ∇.(ρu_) = 0

11
Q

Momentum Conservation

A

q_ = ρu_
-sub into universal conservation law:
d/dt ∫ρ dV = ∫ρu_.(-n_) dS + ∫ρu_(u_.(-n_)) dS + ∫F_ dV
-put into suffix notation to put all terms under one volume integral, set equal to 0
=>
ρ Du_/Dt = -∇p + F_
-this is essentially Newton’s 2nd Law

12
Q

Equations of State

A
• physics => p = f(ρ,T)

- where f is material dependent

13
Q

Ideal Gas Law

A

p = RρT

• doubling the density will double the frequency of particle collisions so pressure would be twice as great
• doubling the temperature doubles the energy so you would get twice the momentum transfer per collision and thus double the pressure
14
Q

Linear Approximation to the Ideal Gas Law

A
```-when we make a small perturbation from the base state we can use a linear approximation, a Taylor expansion of (3) about the base state (ρ,T,p) = (ρo,To,po):
p = po + Aρ (ρ-ρo) + At (T-To)
-where:
Aρ = ∂f/∂ρ|ρo,To
At =  ∂f/∂T|ρo,To
-this is commonly written as:
ρ = ρo [1 + αp(p-po) + αt(T-To)]
-where αp is the coefficient of compressibility and αt is the thermal expansion coefficient```
15
Q

Fluid Mechanical Equations for a Compressible or Incompressible Fluid

A

∂ρ/∂t + ∇.(ρu_) = 0
ρ Du_/Dt = -∇p + F_
p = F(ρ), assuming T is constant

16
Q

Boundary Conditions

Moving Rigid Boundary

A

u_.n_ = U_.n_

-where u_ is the fluid velocity and U_ is the boundary velocity

17
Q

Boundary Conditions

Fluid-Fluid Interface

A
```u1_.n_ = u2.n_
-also need stress to be continuous
-for an invicid fluid this implies:
p1=p2 on S
-generally:
σ1_.n_ = σ2_.n_```
18
Q

Incompressibility

A

-many situations are incompressible, where αp~0
-if αp->0, we lose the relation between ρ and p, and:
Dρ/Dt = 0
-then mass conservation =>
∇.u_ = 0

19
Q

Fluid Mechanical Equations for an Incompressible Fluid

A

Dρ/Dt = 0
ρ Du_/Dt = -∇p + F_
∇.u_ = 0

20
Q

αp in the Atmosphere

A

αp~10^(-2) /Pa
=> αp(p-p0) ~ o(1)
-of order 1, this is significant so we would not model the atmosphere as incompressible

21
Q

αp in the Ocean

A
```αp~10^(-11) /Pa
-in the deep ocean p-p0~10^8Pa = 1000bar
-so:
αp(p-p0) ~ 10^(-3) << 1
-so this term is negligible and we can model the ocean as incompressible```
22
Q

Does incompressibiltiy mean that ρ is constant?

A
• NO

- only that Dρ/Dt=0

23
Q

Static States and Hydrostatic Pressure

u_=0

A
```F_ = -ρg ez^
0_ = -∇p + F_
=>
∂p/∂x = ∂p/∂y = 0
∂p/∂z = -ρg```
24
Q

Static States and Hydrostatic Pressure

u_=0, incompressible, ρ const.

A

p = po - ρgz

-pressure gets larger the deeper you go as there is a greater weight above e.g. the ocean

25
Q

Static States and Hydrostatic Pressure

u_=0, incompressible, ρ variant

A

ρ = ρo(z)
=>
p = -g ∫ρdz ez^ + po

26
Q

Static States and Hydrostatic Pressure

u_=0, compressible, ρ(z) unknown

A

-use appropriate equation of state for ρ

27
Q

Static States and Hydrostatic Pressure

Atmosphere

A
```p = RρT
=>
RT ∂ρ/∂z = -ρg
=>
ρ = ρo e^(-g/RT z)```
28
Q

Archimedes Principle

A

-object immersed in hydrostatic pressure experiences a net pressure force
-this buoyancy force is equal to the weight of the displaced fluid:
∫pn_ dS = ∫ρgdV ez^

29
Q

Wave

Definition

A
• a propagating disturbance that leaves the medium unchanged
• needs a restoring force / mechanism
30
Q

Particle in 1D

A
```x'' = f(x)
-suppose x=xo is the neutral state such that f(xo)=0
-consider a small perturbation x~ from xo:
x = xo + x~
x~'' = f(xo + x~)
-Taylor Expansion
-remove higher order terms to linearise
x~'' = -α² x~
-simple harmonic motion
-try solution of form:
x~ = Ae^(-iωt) => ω=±α
-general solution:
x~ = Ae^(-iαt) + Be^(iαt)```
31
Q

Linear Wave Analysis

Steps

A

1) define the neutral / background state from which perturbations are made
2) linearisation
3) wave ansatz (a guess that works), for waves:
u~ = Re[ u^ e^(i(kx-ωt))]

32
Q

Dispersion Relation

Definition

A

-the relation between frequency, ω and wave vector, k

33
Q

Non-Dispersive

Definition

A

-if ω(k)∝k, then the wave is non-dispersive

34
Q

Completeness Principle

A
• the most general superposition of linear wave modes IS the general solution
• obtain this by integrating over all values of k
• or summing if there are restrictions on the values that k can take
35
Q

Complex Amplitude and Phase

Definitions

A
```-in the wave ansatz:
u~ = Re[ u^ e^(i(kx-ωt))]
-the complex amplitude is u^
-define phase as: θ = kx-ωt
-and let:
u^ = |u^| e^(iθo)
-where θo=arg(u^)
-then the wave ansatz becomes:
u~ = |u^| Re[e^(i(kx-ωt+θo))] = |u^|cos(kx-ωt+θo)
-here |u^| is the amplitude and θo is the phase shift```
36
Q

Phase Velocity

Definition

A

-factorise k out of the exponent in the ansatz:
u~ = Re[ u^ e^(ik(x-Cp(k)t))]
-then Cp(k)=ω(k)/k, the phase velocity
-note that the ansatz is only constant when the exponent is constant, x-Cp(k)t=const.
-therefore crest/troughs etc. all move at the phase velocity

37
Q

Superposition

A

-by setting A(k) equal to a delta function in the general solution you can recover the superposition of two wave solutions

38
Q

Conditions for Equations to Support Waves

A
• to support waves, an equation must have a solution that allows both k and ω to be real at the same time
• this is only the case if all derivatives have the same parity
• i.e. they are all odd or all even
39
Q

Travelling Wave Solutions

A

-if a wave is non-dispersive, then it preserves its shape
-can then use a faster method, transformation of coordinates, to solve it
e.g.
ζ = x-Ut
τ = t

40
Q

Sound Waves

A

-apply linear wave analysis to the full compressible fluid mechanical equations with no body forces and a background state of rest

41
Q

The Wave Equation

A

∂²ρ~/∂t² - c²∇²ρ~ = 0

-ρ~, p~ and u_~ all obey the wave equation, just with different phases etc.

42
Q

3D Wave Modes

A
```-introduce the wave vector:
k_ = (k,l,m)
-then the 3D ansatz is given by:
u~ = Re[ u^ e^(i(k_.x_-ωt))]
-this is now constant when k_.x_-ωt=const. corresponding to planes moving in the k_ direction
-the 3D phase velocity is then given by:
Cp_ = k_/|k_|² * ω```
43
Q

Standing Waves

A

-introduce a boundary, then to meet the boundary condition, u~=0 at x=0, the ansatz becomes a superposition of incident and reflected waves:
u~ = Re[ u^ e^(i(kx-ωt)) + u^ e^(-i(kx-ωt))]
-introducing two boundaries (with waves inbetween) restricts the k to a discrete range of possible values:
u~ = 2|u^|sin(kx)cos(ckt)
-for sound