Compressible Flow

Question 1

Define internal energy

Answer 1

dE = δQ - δW

Difference in heat added to the system, with work done by the system on environment.

Question 2

Define a ‘specific’ quantity

Answer 2

Quantity divided by mass

Question 3

Define enthalpy (full and differential)

Answer 3

H = E + pV

dH = VdP

Question 4

Define baratropic fluid

Answer 4

A fluid whose pressure depends on density.

Question 5

Why is kinetic energy no longer conserved in an compressible fluid (vs incompressible)?

Answer 5

For a compressible fluid, kinetic energy can be converted into internal energy by forcing the fixed mass into a smaller volume.

Question 6

Define a polytropic gas

Answer 6

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.

Question 7

What condition do we impose on internal energy (give new formula)

Answer 7

Adiabatic: no heat exchange. dQ =0

dE = -PdV

Question 8

How do enthalpy and pressure relate?

Answer 8

dh = dp/ρ

∇h =1/ρ ∇P

Question 9

What KE conditions do the unforced, baratropic Euler eqs satisfy

Answer 9

dE/dt = 0

Check notes for other

Question 10

How do we set up analysis of sound waves

Answer 10

Disturbances around ρ0, p0 = P(ρ0), and zero velocity.

Sub in and linearise.

Question 11

How do we find the three-dimensional wave equation?

Answer 11

Differentiate the linearised continuity and combine with pressure

Question 12

State the three-dimensional wave equation

Answer 12

∂^2p/∂t^2 = a∆p

a = sqrt P’(ρ0)

Question 13

What assumptions do we make about u for sound waves and what boundary condition does this give?

Answer 13

Irrot

u = ∇φ

Gives equiv wave equation in φ

Question 14

What procedure do we follow to solve wave eq?

Answer 14

solve for φ, then compute u = ∇φ and p = −ρ0 ∂φ/∂t

Question 15

What is the general 1d solution for wave-eq?

Answer 15

φ(x, t) = X(x − c0t)

Substitute
u(x, t) = x − c0t
v(x, t) = x + c0t

φ(x, t) = F(u) + G(v)

Question 16

What is the general 1d standing wave solution for wave-eq?

Answer 16

φ(x, t) = X(x) sin(ωt)

Question 17

What is the general 3d solution for wave-eq?

Answer 17

Superposition of plane waves in diff directions with form
φ(x, t) = F(k · x − ωt)

Question 18

Define a characteristic

Answer 18

A curve which allows a pde to be written as an ode (using reverse chain rule?)

Question 19

Describe how to find a incomp characteristic

Answer 19

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)

Question 20

How can we eliminate ρ from the polytropic Euler eqs to find the characteristics?

Answer 20

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.

Question 21

State the Riemann invariants for a polytropic gas and the lines they are constant along

Answer 21

F_+- = u +- 2c/γ-1 for dx+-/dt = u +- c