1. Fluid Equations and Linear Wave Theory Flashcards
Density Definition
-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
Mass Density
Q = mass q = ρ
Kinetic Energy Density
Q = KE q = 1/2 ρu²
Volume Density
Q = V q = 1
Linear Momentum Density
Q_ = linear momentum q = ρu_
Flux Definition
-flux of a quantity Q through a surface S is the rate at which Q passes through S
flux = ∫ q (u_ . n_) dS
Mass Flux
mass flux = ∫ ρ (u_ . n_) dS
Momentum Flux
flux = ∫ (ρu_) (u_ . n_) dS
Conservation Laws
-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
Mass Conservation
-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
Momentum Conservation
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
Equations of State
- physics => p = f(ρ,T)
- where f is material dependent
Ideal Gas Law
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
Linear Approximation to the Ideal Gas Law
-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
Fluid Mechanical Equations for a Compressible or Incompressible Fluid
∂ρ/∂t + ∇.(ρu_) = 0
ρ Du_/Dt = -∇p + F_
p = F(ρ), assuming T is constant
Boundary Conditions
Moving Rigid Boundary
u_.n_ = U_.n_
-where u_ is the fluid velocity and U_ is the boundary velocity
Boundary Conditions
Fluid-Fluid Interface
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_
Incompressibility
-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
Fluid Mechanical Equations for an Incompressible Fluid
Dρ/Dt = 0
ρ Du_/Dt = -∇p + F_
∇.u_ = 0
αp in the Atmosphere
αp~10^(-2) /Pa
=> αp(p-p0) ~ o(1)
-of order 1, this is significant so we would not model the atmosphere as incompressible
αp in the Ocean
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
Does incompressibiltiy mean that ρ is constant?
- NO
- only that Dρ/Dt=0
Static States and Hydrostatic Pressure
u_=0
F_ = -ρg ez^ 0_ = -∇p + F_ => ∂p/∂x = ∂p/∂y = 0 ∂p/∂z = -ρg
Static States and Hydrostatic Pressure
u_=0, incompressible, ρ const.
p = po - ρgz
-pressure gets larger the deeper you go as there is a greater weight above e.g. the ocean
