Fluid Mechanics Y2

Question 1

Material Derivative

Answer 1

D()/Dt = δ()/δt + grad(*)·u>

Question 2

Grad(x)

Answer 2

∇x

Question 3

Div(x)

Answer 3

∇·x

Question 4

Curl(x)

Answer 4

∇×x

Question 5

Continuity Equation

Answer 5

0 = δρ/δt + div(ρu>)

Question 6

Euler Equations

Answer 6

ρ Du>/Dt = -grad>(p) + f_body>

Question 7

Speed of sound in a perfect gas

Answer 7

c = sqrt( γ R T )

Question 8

Isentropic Pressure relation (fb)

Answer 8

p₀/p = ( 1 + (γ-1)/2 * M² ) ^{γ / (γ-1)}

Question 9

Isentropic Density Relation (fb)

Answer 9

ρ₀/ρ = ( 1 + (γ-1)/2 * M² ) ^{1 / (γ-1)}

Question 10

Isentropic Temperature Relation (fb)

Answer 10

T₀/T = 1 + (γ-1)/2 * M²

Question 11

Area-Mach Relation In converging-diverging nozzles (fb)

Answer 11

A/A* = 1/M * { 2/(γ+1) * [1 + (γ-1)/2 * Μ²] } ^{(γ+1)/2(γ-1)}

Question 12

Maximum nozzle flow rate (fb)

Answer 12

mdot_max = p₀ A* sqrt( γ/RT ) (2/γ+1) ^{(γ+1)/2(γ-1)}

Question 13

Area-Velocity Relation for 1D compressible isentropic flow

Answer 13

dA/A = dU/U ( M² - 1 )

Question 14

Cauchy equations

Answer 14

δu>/δt + grad(u>) * u> = -1/ρ grad(p) + 1/ρ div(τ) + f_body/ρ

Question 15

Navier Stokes equations for incompressible flow

Answer 15

div(u>) = 0

δu>/δt + grad(u>) * u> = -1/ρ grad(p) + μ/ρ Lap(u) + f_body/ρ

Question 16

Boundary Layer Equations

Answer 16

δu/δx + δv/δy = 0

u δu/δx + v δu/δy = -1/ρ δp/δx + v δ^2u/δy^2

δp/δy = 0

Question 17

All equations assume

Answer 17

Continuum

Question 18

Euler Eqs Assume

Answer 18

Inviscid fluid
Continuum

Question 19

Isentropic Flow Relations Assume

Answer 19

• Steady
• No body forces
• Inviscid
• Perfect Gas
• Isentropic flow (duh)
• Flow along streamline
• Continuum

Question 20

Max flow rate eqns assume

Answer 20

Same as isentropic flow

Question 21

Cauchy Eqns Assume

Answer 21

Continuum only

Question 22

Navier-Stokes eqns assume

Answer 22

• Incompressible Flow
• Newtonian Fluid
• Continuum

Question 23

Boundary Layer Eqns assume

Answer 23

• Steady Flow
• Incompressible Flow
• No Body Forces
• Newtonian Fluid

Question 24

Newtonian Viscosity Constitutive Relation

Answer 24

τ = 2 μ D + λ div(u>) I

Question 25

What is D?

Answer 25

D = 1/2 ( grad(u>) + grad(u>)^T )

Question 26

Displacement thickness for boundary layer

Answer 26

δ1(x) = int₀^∞( 1 - u/U0 )dy

Question 27

Momentum thickness for boundary layer

Answer 27

δ2(x) = int₀^∞( u* (1 - u*) )dy

Question 28

Von Karman integral for friction coefficient

Answer 28

C_f = 2 dδ2/dx

Question 29

Drag coefficient

Answer 29

C_D = 2 δ2 / L

Question 30

Blasius Solution validity

Answer 30

Self Similarity Laminar Flow