Compressible Flow Flashcards
Define internal energy
dE = δQ - δW
Difference in heat added to the system, with work done by the system on environment.
Define a ‘specific’ quantity
Quantity divided by mass
Define enthalpy (full and differential)
H = E + pV
dH = VdP
Define baratropic fluid
A fluid whose pressure depends on density.
Why is kinetic energy no longer conserved in an compressible fluid (vs incompressible)?
For a compressible fluid, kinetic energy can be converted into internal energy by forcing the fixed mass into a smaller volume.
Define a polytropic gas
A fluid with pressure defined as P = k rho^(gamma)
Where gamma is the polytropic index and depends on the degrees of freedom of the gas.
What condition do we impose on internal energy (give new formula)
Adiabatic: no heat exchange. dQ =0
dE = -PdV
How do enthalpy and pressure relate?
dh = dp/ρ
∇h =1/ρ ∇P
What KE conditions do the unforced, baratropic Euler eqs satisfy
dE/dt = 0
Check notes for other
How do we set up analysis of sound waves
Disturbances around ρ0, p0 = P(ρ0), and zero velocity.
Sub in and linearise.
How do we find the three-dimensional wave equation?
Differentiate the linearised continuity and combine with pressure
State the three-dimensional wave equation
∂^2p/∂t^2 = a∆p
a = sqrt P’(ρ0)
What assumptions do we make about u for sound waves and what boundary condition does this give?
Irrot
u = ∇φ
Gives equiv wave equation in φ
What procedure do we follow to solve wave eq?
solve for φ, then compute u = ∇φ and p = −ρ0 ∂φ/∂t
What is the general 1d solution for wave-eq?
φ(x, t) = X(x − c0t)
Substitute
u(x, t) = x − c0t
v(x, t) = x + c0t
φ(x, t) = F(u) + G(v)
What is the general 1d standing wave solution for wave-eq?
φ(x, t) = X(x) sin(ωt)
What is the general 3d solution for wave-eq?
Superposition of plane waves in diff directions with form
φ(x, t) = F(k · x − ωt)
Define a characteristic
A curve which allows a pde to be written as an ode (using reverse chain rule?)
Describe how to find a incomp characteristic
State the incomp Euler equations and notice that u(x,t) = dx/dt gives our characteristic, where u is constant for a choice of t and x. Make these choices based on initial condition and evaluate.
Invert to find x0 and mult by u(0,0) to find u(x,t)
How can we eliminate ρ from the polytropic Euler eqs to find the characteristics?
Consider sound speed c(ρ) = sqrt(P’(ρ)) and differentiate wrt to x and t. Substitute in to euler to get two equations. Add and subtract to get the expressions.
State the Riemann invariants for a polytropic gas and the lines they are constant along
F_+- = u +- 2c/γ-1 for dx+-/dt = u +- c
