Numerical computations of solitary waves in a two-layer fluid

Transcription

1 Numerical computations of solitary waves in a two-layer fluid Emilian Pǎrǎu, Hugh Woolfenden School of Mathematics University of East Anglia

2 Outline Background Internal/interfacial waves (observation and experiments) Formulation Two-layer / Free-surface / Surface tension Results Gravity waves Gravity-Capillary waves Surface tension only Gravity-Capillary waves Tension on both surfaces Gravity-Capillary waves Interfacial tension only 3D solutions Summary

3 Brief history of internal waves Internal waves Salinity or temperature gradients Where do they occur? Images Global Ocean Associates ( European Space Agency - ESA ( NASA ( Dead water Observation Tacitus, 5 AD Observation Nansen, 893 Experiments Ekman, 94 Recent experiments Mercier et al., 2

4 Internal waves in the strait of Gibraltar Figure: Spain at top, Morocco below. Wavelength 2 km. c European Space Agency - ESA Amplitude: 5 m at creation Ref: D.M. Farmer & L. Armi Prog. Oceanogr. 2

5 Experiments on interfacial waves with a free surface Figure: Experimental setup tank dimensions are 3x5x.5 cm. Ref: - M.J. Mercier, R. Vasseur & T. Dauxois. 2. Resurrecting dead-water phenomenon. Nonlin. Processes Geophys. - L. Walker Interfacial solitary waves in a two fluid medium, Phys. Fluids. - H. Michallet & E. Barthelemy Experimental study of interfacial solitary waves, JFM. The surface and/or interfacial tension may be important for the waves in these experiments, as the scales are small (e.g. h = cm)

6 Two-layer fluid system U y y = h 2 + ζ(x) y = h 2 Fluid 2 (ρ 2) y = η(x) x y = Fluid (ρ ) Figure: Physical domain y = h

7 Assumptions, scales and parameters Assumptions: Irrotational and incompressible flow, inviscid fluids Velocity potentials: Φ and Φ 2 Satisfying Laplace s equation Look for: Steady, symmetrical solitary waves Scales: U and h 2 Parameters: H = h 2, R = ρ 2, F 2 = U2, τ I = σ I h ρ g h 2 ρ g h2 2, andτ F = σ F ρ 2 g h2 2 (E.g. R.99 for salt-water, σ I.5 dyne/cm, σ F 73 dyne/cm R.78 for petrol-water, σ I 23 dyne/cm, σ F 25 dyne/cm) Profiles: Free-surface is y = +ζ(x) and interface is y = η(x)

8 Boundary conditions Kinematic (bed, interface and free-surface): Φ y =, Bernoulli (free-surface): Bernoulli (interface): Φ j x η = Φ j y and 2 F2( Φ 2 2 ) ζ +ζ τ F ( +(ζ ) 2)3 2 Φ 2 x ζ = Φ 2 y +ǫ F p D = 2 F2( Φ 2 R Φ 2 2 ( R) ) η +( R)η τ I ( Φj Radiation: u j = x, Φ ) j y (+(η ) 2 ) 3 2 (,) as x. ) Pressure forcing: p D (x) = exp when x < and zero otherwise ( x 2 R ǫ I p D =

9 Two-layer fluid system Inverse plane Ψ j Ψ 2 = Φ j Ψ = Ψ 2 = Ψ = /H Figure: Inverse plane Φ = Φ 2 = at x = w j (z) = Φ j (x,y)+i Ψ j (x,y), x = x(φ j,ψ j ), y = y(φ j,ψ j ) Define Φ 2 = g(φ ) and let X(Φ ) = x(φ 2,), Y(Φ ) = y(φ 2,)

10 Dynamic conditions Inverse plane Free-surface: ( 2 F2 (g ) 2 )+(Y (X ) 2 +(Y ) 2 X Y X Y ) τ F ( (X ) 2 +(Y ) 2)3 2 +ǫ F p D = Interface: ( R(g ) 2 )+( R)η 2 F2 (x ) 2 +(η ) 2 ( R) x η x η τ I ( (x ) 2 +(η ) 2)3 2 Rǫ I p D =

11 Integral equations In lower fluid ( x (Φ ) = 2Ĥ(x ) (Φ Φ )η ) η dφ dφ, π (Φ Φ ) 2 +4Ĥ2 Φ Φ where Ĥ = /H and Φ is the evaluation point on Φ. In upper fluid x (Φ ) g (Φ ) = g (Φ ) π X (Φ ) g (Φ ) = g (Φ ) π where g = g(φ ). ( ( (X g ) (g g )Y (g g ) 2 + (x g )+(g g )η (g g ) 2 + η ) dφ dφ, g g Y ) dφ dφ g g

12 Numerical method Apply symmetry conditions: x( φ) = x(φ), x ( φ) = x (φ), η( φ) = η(φ), η ( φ) = η (φ) g( φ) = g(φ), g ( φ) = g (φ) X( φ) = X(φ), X ( φ) = X (φ), Y( φ) = Y(φ), Y ( φ) = Y (φ) η () =, Y () = Method Truncate and discretise (typically Φ 4 with 64 points) Use Newton s method (construct Jacobian) 5N unknowns: g, x, η, X, Y Apply forcing Forced wave (free-stream) Amplitude continuation... Forced solitary wave Remove forcing Solitary wave

13 Dispersion relation For wave-number, k, F 2 ± (k) = b(k) ± b 2 (k) 4a(k)c(k) 2k a(k) where ) a(k) = +R tanh( k tanh(k) ( ) H b(k) = tanh k +tanh(k) +k 2( ( ) ( τ H I tanh k +τ H F (R tanh k )+tanh(k)) ) ) H c(k) = ( R +τ I k 2 )( +τ F k )tanh( 2 k tanh(k) H Two modes: F + is the fast and F slow Critical values (long-wave limit, k ): ( F± ) 2 (+H)± (+H) = 2 4H( R) 2H Theoretical works (gravity only): Peters & Stoker (96), Kakutani & Yamasaki (978), Craig, Guyenne & Kalisch (25) etc.

14 Gravity case Dispersion relation (a) R =.3 (b) R = F.8 F k k Figure: H =, τ I =, τ F =. Fast mode solid line. Slow mode dashed line. F ± as k.

15 Ref: J.N. Moni & A.C. King Guided and unguided interfacial solitary waves. Gravity solitary waves - F near (above) the fast critical value 2.5 y/h x/h 2 Figure: Gravity wave profiles for H =, R =.3 and F =.44. Bed is at y/h 2 =. The interfacial and free-surface amplitudes are.37h 2 and.562h 2 respectively. F is higher than the fast critical number F +.

16 Generalized gravity solitary waves - F near (above) the slow critical value Figure: Generalized solitary waves for F close to F slow were computed by Michallet & Dias (999), P. & Dias (2) (see also Akylas & Grimshaw (992), Rusas & Grue (22), Sun & Shen (993) etc.)

17 Gravity-Capillary waves Surface tension only Dispersion relation F Figure: H = 3, R =.9, τ I = and τ F. τ F =.6 solid line. τ F =.5 dashed line. F + as k. k Conditions for fast minimum? Small k analysis = critical τ F (τ F )

18 Froude number near (below) the fast critical value.8.6 y/h x/h 2 Figure: In-phase depression solitary waves for H = 3, R =.9, τ I =, τ F =.6(> τ F) and F =.857.

19 Froude vs. Amplitude near F + (a) Peak interface amplitude, η() (b) Peak free-surface amplitude, ζ() η() -.3 ζ() F F Figure: Froude number vs. amplitude for H = 3, R =.9, τ I = and τ F =.6(F + =.44). Forced waves : ǫ F =. dashed line.

20 Froude number near the fast critical value y/h x/h 2 Figure: In-phase depression solitary waves for H = 3, R =.9, τ I =, τ F =.6(< τ F) and F =.857.

21 Froude number near (above) slow critical value (high H).8.6 y/h x/h 2 Figure: Out-of-phase solitary waves for H = 3, R =.9, τ I =, τ F =. and F =.83.

22 Froude number near slow critical value (high H) η() ζ() Figure: Froude number vs. amplitude for H = 3, R =.9, τ I = and τ F =.. Forced waves: ǫ I =. dashed lines. F

23 Froude number near slow critical value (high H) y/h x/h 2 2 Figure: Solitary wave broadening for H = 3, R =.9, τ I =, τ F =. with F =.83 (solid line), F =.8358 (dashed line) and F = (dotted line). Bed lies at y/h 2 = /3. Limiting case when F near F front (see e.g. F. Dias & A. Il ichev (2), K.G. Lamb (2))

24 Froude near slow critical value (low H) Froude number vs. amplitude Similar story, signs reversed y/h x/h 2 Figure: Solitary waves for H =.3, R =.9, τ I =, τ F =.5 and F =.334. Solitary waves exist when depth is infinite Ref: M. Barrandon & G. Iooss. 25. Chaos: An Interdisciplinary Journal of Nonlinear Science 5

25 Froude near slow critical value (low H).4.2 y/h x/h 2 Figure: Solitary waves for H =.3, R =.9, τ I =, τ F =.5, F =.334 (solid line), F = (dashed line) and F = (dotted line).

26 Parameters in experiments Two-layer experiments of Mercier et al. 2 H =.357 (h = 4 cm, h 2 = 5 cm) R =.976 (ρ =.227gcm 3, ρ 2 =.998gcm 3 ) σ F = 73 dyne/m = τ F =.3 σ I = Depression solitary wave with amplitude 3 cm in experiment and numerical model (similar Froude numbers).

27 Generalised gravity-capillary solitary waves y 7.8 x x 4 x y x Figure: Generalised gravity-capillary solitary waves for F near slow critical value (low H) and weak surface tension

28 Gravity-Capillary case Tension on both surfaces Dispersion relation F Figure: R =.9, H =.3, τ I and τ F =.. τ I =. dashed line. τ I =. dotted line. F ± as k. [In experiments: Walker (973): τ F = , τ I = Michallet and Barthelemy (998): τ F = , τ I = ] k

29 Froude number near(below) slow critical value (low H) y/h x/h 2 Figure: Out-of-phase solitary waves for H =.3, R =.9, τ F =., τ I =.2(> τ I ) and F =.26. Bed lies at y/h 2 = /3.

30 Froude number near slow critical value (low H) y/h x/h 2 Figure: Damped oscillations in solitary wave profiles for H =.3, R =.9, τ F =., τ I =.6(< τ I ) and F =.26.

31 Froude number near critical value (high H) y/h x/h 2 Figure: Out-of-phase solitary wave profiles for H = 3, R =.9, τ I =.2, τ F =. and F =.5. Bed lies at y/h 2 = /3

32 Solitary wave coexistence.8.6 y/h x/h 2 Figure: Coexisting solitary waves for F =.8, H =.5, R =.9, τ I =.2, τ F =.. Solid line from low H wave. Dashed line from high H wave. Bed lies at y/h 2 = 2/3. Ref: O. Laget & F. Dias Numerical computation of capillary gravity interfacial solitary waves. J. Fluid Mech. 349

33 Gap solitary waves Interfacial tension only Dispersion relation 2.2 F.5.5 F k k Figure: Dispersion relation curves (solid lines) for τ =.5, H =.4, R =.5 (left) and for τ =., H = 3, R =.5 (right). The Froude number corresponding to the rigid-lid approximation is also shown (dotted line). The upper curve F + has a minimum and the lower branch F has a maximum for finite values of k close to each other. By decreasing τ I, the gap between these two extrema may become very small (see also Grimshaw, R. & Christodoulides, P. (28)).

34 Figure: Amplitudes of in-phase and out-of-phase solitary waves for H =.4, R =.5, τ = y y x x Figure: Profiles of out-of-phase solitary waves for F =.72 (left) and in-phase solitary waves for F =.9 (right) η() and ζ() F

35 3D solitary waves When free surface is replaced by a rigid lid, 3D interfacial solitary waves are predicted (see Kim & Akylas 25 (weakly nonlinear), P. et al 27) When the free surface is present we compute fully-localised gravity-capillary solitary waves using a numerical method based on boundary integral methods (Green functions) Both in-phase 3D solitary waves and out-of-phase solitary waves are found

36 3D numerical results.7 Free surface/interface amplitude, 2D( )/3D( ) η() and ζ() x 4 2 y F Figure: 3d fully-localised solitary waves (left). Amplitudes of solitary waves in 2D/3D (right). H =.3, R =.9, τ I = τ F =.

37 Summary Numerical method for 2-layer free-surface flow with surface and interfacial tension (fully nonlinear) Wave orientations, broadening and coexistence Gravity waves: In-phase elevation Gravity-Capillary waves: τ I =, τ F Near F+ : In-phase depression, damped oscillations (τ F < τ ) F Near F : Out-of-phase orientation depends on H Broadening at a critical Froude number Gravity-Capillary waves: τ I, τ F Near F : Out-of-phase, damped oscillations (τ I < τ I ), coexistence Gravity-Capillary waves: τ I, τ F = ( gap solitary waves) 3D solitary waves Further work... Time-dependent/Internal waves under an ice sheet / Three-dimensions