Russian Academy of Science Institute for Problem in Mechanics  

 

Roman N. Bardakov 

Internal wave generation problem exact analytical and numerical solution

 

gvv

νzP

t 00

00v

zt0div v

pressuredensity,velocity,)zv,yv,x(v Pv

Basic set of equations

.0v

,22

v

0

0

zz

zxxUt

aUt

axU

Boundary conditions

Boundary conditions

02

2

2

2

2

2

22

2

2

2

2

2

2

zxtxN

zxt

.0

,22

0

0

z

z

x

xUta

Uta

xUz

Navier-Stokes equation for stream function

3

222 4

112

,Nkii

kkkw

3

222 4

112

,Nkii

kkki

dkkkUkkkUkee

eka

kiU

tzxiw

zkkUiikzkkUwikUtxik

,,2sin

1,,

,,

Exact solution for stream function

Dispersion equation

022222222

zkkikNzkk

Velocity absolute value

L = 1 cm, plate moving speed U = 0.25 cm/s, buoyancy

period Tb = 14 s. (Fr = U/LN = 0.55, Re =UL/ = 25,  

= UTb = 3.5 cm).

Vertical component of velocity L = 1 cm, plate moving speed U = 0.25 cm/s,

buoyancy period Tb = 14 s. (Fr = U/LN = 0.55, Re =UL/n = 25,   l = UTb = 3.5 cm).

Stream lines (N = 0.45 s-1, U=0.25 cm/s =UTb=3.5 cm, L=4 cm, Fr =

0.14)

Absolute value (left) and horizontal component (right) of velocity

boundary layer (U = 1 cm/s, L = 4 cm, Fr = 0.56, Re = 400, N = 0.45 s-

1,   = UTb = 14 cm).

Vertical component of velocity boundary layer

(U = 1 cm/s, L = 4 cm, Fr = 0.56, Re = 400, N = 0.45 s-

1,   = UTb = 14 cm).

0.2 0.4 0.6 0.80

1

2

3

2.5 см 5 см 10 см

Z/(a)-2

X

0.2 0.4 0.6 0.80.0

0.1

0.2

0.3

0.4

0.5

0.6

2.5 см 5 см 10 см

Z

X

=7.5 с, =0.11 см, =20 см, =2.6

0.11 cmN N 7.5 c,bT 20 cm,bUT 3 2.6,L g N

Vertical component of velocity (N = 0.45 s-1, U=0.25 cm/s, =UTb= 3.5 cm, Fr = 0.014,

Re = 1000)

Absolute value and vertical component of velocity. (N = 1 s-1, Tb = 6 s, U=0.01 cm/s, =UTb=0.06 cm, L=1

cm, Fr = 0.01, Re = 1)

Comparing with experimental results

Comparing with experimental results