Shallow Water & Energy Equations

Question 1

What is the flow like for shallow water waves?

Answer 1

The flow is not steady but can be made quasi-static by choosing the frame moving with the wave.

Question 2

Problem: waves with amplitude Ψ/2 propagatae along a channel width Ac at speed c(w). What is c(w): what is the first step to solve this?

Answer 2

-Mass conservation, assuming uniform flow in h: v1Ac(h+Ψ/2) = v2Ac(h-Ψ/2) -> (v1+v2/2)Ψ = (v2-v1)h

Question 3

What is the second step to solve the shallow water problem before?

Answer 3

• Apply Bernoulli’s equation between points 1 and 2 (maxima and minima of waveform)
• P0 + ρg(h+Ψ/2) + ρv1^2/2 = P0 + ρg(h-Ψ/2) + ρv2^2/2 -> simplify, get gΨ = (v2^2+v1^2)/2 = 1/2 (v1+v2)(v1-v2) = 1/h ((v1+v2)/2)Ψ((v1+v2)/2)
• gh = ((v1+v2)/2)^2, and c(w) = sqrt(gh)

Question 4

Using the problem just discussed, when does the wave break?

Answer 4

When wave speed at h1 (at peak of waveform) is greater than speed at h2 (at trough of waveform) so wave peak catches trough.

Question 5

What is the first step in finding the energy in incompressible flows?

Answer 5

Curl of first term: ∇(ρdv/dt) = ρd/dt(∇v) = ρdw/dt, where w = ∇v, the vorticity

Question 6

What is the second step in finding the energy in incompressible flows?

Answer 6

Curl of second term: ∇ ∇Ф = 0 for any scalar Ф: pressure and gravity drop out

Question 7

What is the third step in finding the energy in incompressible flows?

Answer 7

Curl of third term: ∇(μ∇^2 v) = μ∇^2 w

Question 8

What is the fourth step in finding the energy in incompressible flows?

Answer 8

Curl of fourth term: -∇((v.∇)v) = (v.∇)w - (w.∇)v + w(∇.v) + v(∇.w) -> last 2 terms = 0, so D/Dt w = d/dt w + (v.∇)w = μ∇^2 w + (w.∇)v
-v.∇ = 0, so D/Dt w = (w.∇)v + μ∇^2 w

Question 9

Which equation would we use if we wanted the pressure for steady flow (ignoring gravity)?

Answer 9

ρ(v.∇)v = -∇P + μ∇^2 v, but μ∇^2 v = 0, so ∇^2 p = -ρ∇.[(v.∇)v]

Question 10

What is the equation for the circulation around any closed loop?

Answer 10

K = closed integral of v.dl = closed double integral of (∇v).ds -> ∇v is called the vorticity, if it =0, then K = 0

Question 11

What is Kelvins circulation theorem?

Answer 11

The circulation around any loop (Γ) moving with the fluid is constant for inviscid flow.

Question 12

What does dK/dt equal?

Answer 12

Curve Γ is moving with the fluid, so dK/dt = closed integral over Γ of dv/dt .dl + closed integral over Γ of v.d/dt(dl)

Question 13

What is another way of writing closed integral over Γ of v.d/dt(dl)?

Answer 13

= closed integral of ∇(v^2/2).dl

Question 14

What is the invisicid NS equation?

Answer 14

ρ dv/dt = -∇(P+ρgz)

Question 15

How can we ue the invisicid NS equation for Kelvins circulation theorem?

Answer 15

Rearrange for dv/dt and sub in:

dK/dt = closed integral over Γ of ∇(v^2/2 - P/ρ - gz) .dl

Question 16

What does the final version of dK/dt equal?

Answer 16

We know that closed integral of ∇(Ф).dl = 0 for any scalar Ф, so dK/dt = 0 for inviscid flow.

Question 17

How can we express v in cylindrical coordinates?

Answer 17

v = v r(hat) + v θ(hat) + v z(hat)