• Nie Znaleziono Wyników

ρ1) in gravitational field separated by a sharp, plane interface as shown in the figure

N/A
N/A
Protected

Academic year: 2021

Share "ρ1) in gravitational field separated by a sharp, plane interface as shown in the figure"

Copied!
2
0
0

Pełen tekst

(1)

University of Warsaw Advanced Hydrodynamics

Faculty of Physics Selected Topics in Fluid Mechanics

Summer Semester 2019/20

Homework 10 Due June 5, 2020

Solutions should be sent togustavo.abade@fuw.edu.pl

1. Consider two ideal, imiscible fluids of densityρ1andρ2(withρ2 > ρ1) in gravitational field separated by a sharp, plane interface as shown in the figure. Assume additionaly an existing stationary flow of the form

u =¯  U ˆx for y > 0,

0 fory ≤ 0. (1)

y

x ρ1

ρ2

g y = ξ(x, t)

U

Figure 1: Idealized flow configuration with sharp discontinuities in density and velocity.

We then introduce a small perturbation of the interface described byy = ξ(x, t) with

ξ(x, t) = a ei(kx−ωt), (2)

the real part being understood. If U = 0 then this perturbation will induce gravity waves that propagate without either growth or decay (provided ρ2 > ρ1), as seen in the Exercise sheet 10. Otherwise, ifU > 0 the perturbation may grow (under certain conditions) at the expense of the energy contained in the existing flow current.

Follow the procedures described in the Exercise sheet 10 to perform the following tasks.

(a) What should be a convenient stationary base state (¯u, ¯p) for this problem?

(2)

(b) Show that the frequency equation for the dispersion relationω(k) is

122− 2U kρ1ω + ρ1U2k2− (ρ2− ρ1)gk = 0, (3) yielding

ω(k) = ρ1

ρ12

U k ±p

∆(k), (4)

with

∆(k) = ρ2 − ρ1

ρ12

gk − ρ1ρ2

12)2U2k2. (5)

(c) Show that if surface tension is accounted for thenω(k) has the form (4) with ∆(k) including an extra term,

∆(k) = ρ2 − ρ1

ρ12

gk − ρ1ρ2

12)2U2k2+ γ ρ12

k3, (6)

whereγ is the surface tension coefficient.

(d) Instability. Consider the possibility of a complex-valued frequency,

ω = ω0+iω00. (7)

Show that infinitesimal sinusoidal perturbations (2) of wavelengthλ = 2π/k will grow exponentially if

ω00= Im{ω(k)} > 0. (8)

(e) Assume the system sketched in Fig. 1 is air blowing with velocity U over wa- ter surface. For known values of the physical constants (ρ1, ρ2, γ, g) plot the wave-number dependence of ∆(k) for different values of U . Show that there is a threshold valueUc, so that

i. ∆(k) > 0 for all k, if U < Uc;

ii. ∆(k) < 0 for k1 < k < k2, ifU > Uc.

The frequencyω is complex-valued if ∆(k) < 0. Then for U > Uc, wind gener- ates water waves with wave-numbersk in the interval k1 < k < k2.

Estimate the numerical value of Uc(and its associated critical wave number kc) for the air-water system.

(f) Show that gravity (providedρ2 > ρ1) and surface tension are stabilizing mecha- nisms. Surface tension forces suppress the growth of small-wavelength perturba- tions, while gravity limits those at long wavelengths.

2

Cytaty

Powiązane dokumenty

4. Einsteinowska religią kosmiczna jest emocjonalno-psychicznym prze- życiem tajemnicy wyrażającym się w wierze w racjonalną strukturę świata. Geniusze, choć

Przyczyn powstawania licznych fundacji nie można dopatrywać się jedynie w wyrzutach sumienia bogatych kupców6 , lecz ra­ czej w duchowości ludzi tamtych

– dr Aneta Maria Abramowicz podczas Międzynarodo- wej Konferencji Naukowej online International Scientific Congress Trnava Days of Law – Legal Politics and Legislation,

Ponadto, skoro wymienia się kolejne pozycje biblio­ graficzne, podając miejsce i rok wydania, to ta zasada powinna dotyczyć wszyst­ kich prac, tymczasem w jednym miejscu (s. Do

Results of railpads stiffness variation (v01 and v03): (a) vertical contact force of wing rail and crossing nose, (b) maximum VM stress distribution on

Van enkele andere serviceflats zijn bewoners uitgekocht door een belegger die het complex vervol- gens sloopte en er een ander type huisvesting voor in de plaats bouwde, wat

We therefore hypothesize that the effect of the imaginary part on the JND in the real part (i.e., the effect of a system’s damping on the JND in that system’s stiffness and

Ubóstwo baka- łarzy i brak uposażenia niektórych szkół niekiedy stawały się nawet przyczyną ich likwidacji (np.. Bieliczna