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GRAVITY WAVES WITH EFFECT OF SURFACE TENSION AND FLUID VISCOSITY CHEN Xiao-Bo1, DUAN Wen-Yang2 & LU Dong-Qiang3 1

Research Department, BV, Paris (France) - e-mail: [email protected] College of Shipbuilding Engineering, HEU, Harbin (China) 3 Shanghai Institute of Applied Mathematics and Mechanics, SHU, Shanghai (China) 2

ABSTRACT: The potential flow in a viscous fluid due to a point impulsive source is considered within the framework of linear Stokes equations. The combined effect of fluid viscosity and surface tension on the potential function below the water surface is studied. Dependent on the wavenumbers associated with the level of the effect due to surface tension, the oscillations can be grouped as gravity-dominant waves and capillary-dominant waves. It is shown that the wave form of gravity-dominant oscillations is largely modified by the surface tension while the wave amplitude of capillary-dominant oscillations is mostly reduced by the fluid viscosity. 1. INTRODUCTION The solution to the classical Cauchy-Poisson problem of water waves generated by an impulsive disturbance under the pure-gravity effect (Lamb 1932) presents a perplexing peculiarity - the surface elevation in a region approaching to the impulsive disturbance (an initial elevation concentrated along a line of surface in 2D case) is found to diminish continuously in length and to increase continuously in height without limit. It was interpreted that the region in the immediate neighborhood of the point disturbance can be regarded as some kind of source emitting on each side an endless succession of waves of continually increasing amplitude and frequency, and that the mathematical infinity is hoped to disappear if a finite distribution of line sources - integration over the breadth of the band - is considered. In 3D case, we consider the transient velocity potential generated by an impulsive source approaching to the free surface. It is confirmed in Cl´ement (1998) that the transient waves at a given instant oscillate with increasing amplitude and decreasing wavelength when we approach to the source point. Furthermore, the amplitude of transient waves in

a given radial distance from the disturbance grow linearly with time while the wavelength decreases at the same rate. This peculiar property of puregravity waves hinders the numerical development to solve the boundary-value problem associated with a floating body in which the space integral over body’s surface as well as the time-convolution integral are difficult to be accurate. The same situation is present in the frequency domain. The work on the wave pattern due to a steady-moving concentrated pressure on the free surface by Ursell (1960) showed that the wave elevation near the track of pressure point from the linearized pure-gravity theory oscillates with indefinitely increasing amplitude and indefinitely decreasing wavelength. For the more general case of a point source both pulsating and advancing at a uniform speed, the same behavior of the generated potential flow is revealed by Chen & Wu (2001) and described in an analytical way. 1 surface tension is commonly ignored in deThe scribing water waves around large floating bodies, since its effect is considered to be significant only for rather short waves such as ripples. However, the theory of gravity waves may yield waves of very short length which cannot be ignored and cause substantial difficulties in modelling them as described above. These singular and highly oscillatory properties being manifestly non-physical, it is expected that the surface tension plays an important role in modelling surface waves. In fact, the capillarygravity waves have been studied since Kelvin (1871) as summarized in Wehausen & Laitone (1960). A classical analysis of asymptotic behavior of gravity waves including the effect of surface tension was given in Crapper (1964). The wave patterns of capillary-gravity waves in deep water were studied in Yih & Zhu (1989). Surface waves affected by surfactants (which induce a variation of surface

tension) were recently studied by Zilman & Miloh (2001). More recently, Chen (2002) gave a updated analysis on the steady ship waves including the effect of surface tension. It is shown that the introduction of surface tension in the formulation of ship waves eliminates the singularity of ship waves in the region near the track of the source point at the free surface. Chen & Duan (2003) considered capillary-gravity waves due to an impulsive disturbance and performed interesting analysis on the transient waves due to an impulsive source, especially, at large time as well as in the region near the disturbance. It is confirmed that the gravitydominant waves represent faithfully the physical properties of free-surface waves. Very recently, the potential flow due to a point impulsive force applying at the free surface is formulated in Chen et al. (2006). Both the surface tension and fluid viscosity effects are included in the analysis. Further to this study, we consider here the potential flow induced by a point impulsive source with the framework of linear Stokes equation. The combined effect of fluid viscosity and surface tension on the potential function at the water surface is studied. Dependent on the wavenumbers associated with the level of the effect due to surface tension, the oscillations can be grouped as gravity-dominant waves and capillary-dominant waves. Same as the potential flow due to a point impulsive force, it is shown that the wave form of gravity-dominant oscillations is largely modified by the surface tension while the wave amplitude of capillary-dominant oscillations is mostly reduced by the fluid viscosity. 2. STOKES EQUATION We consider the lower half-space filled with water limited on the top by the water-air interface. A Cartesian coordinate system is defined by placing the (x, y)-plane coincided with the undisturbed free surface and the z-axis oriented positively upward. In this gravity-dominant fluid domain, the reference length L, the acceleration of gravity g and the water density ρ are used to define the nondimensional coordinates (x, y, z), the time t, the fluid velocity u = (u, v, w), the velocity potential Φ, the dynamic pressure p P and forces F with respect p √ to (L, g/L, gL, gL3 , ρgL, ρgL3 ), respectively. We study the flow in the fluid domain x = (a, b, c ≤ 0) due to a point source of unit strength located at x′ = (a′ , b′ , c′ ≤ 0). By assuming the incompressibility, the fluid flow is governed by the continuity equation :

∇ · u = δ(x−x′ )δ(t)

(1a)

and the momentum equation : ut = −∇P + ǫ∇2 u (1b) p where ǫ = µ/(ρ gL3 ) with µ the fluid viscosity and δ(·) is the Dirac delta function. On the free surface z = η(x, y, t), the boundary conditions are linearized by assuming small wave amplitudes and written on the undisturbed free surface z = 0 : ηt = w (2a) as the kinematic condition stating no fluid particles cross the free surface and ǫ(uz + wx ) = 0 = ǫ(vz + wy )

(2b)

η − σ 2 (ηxx + ηyy ) + 2ǫwz = P

(2c)

as the dynamic conditions representing the vanishing of shear stress in both x and y directions (2b) and the p equation of normal stress (2c). In (2c), σ = T /(ρgL2 ) with T is the surface tension of water-air interface. In addition, the initial values of the velocity, the hydrodynamic pressure and the free-surface elevation are taken as those of the quiescent fluid, i.e. u = P = η = 0 at t = 0

(3)

The equations (1-3) construct an initial-boundaryvalue problem. 3. INTEGRAL SOLUTIONS To solve the initial-boundary-value problem preceding defined (1-3), the unknowns (u, P ) are decomposed as the sum of an unbounded singular Stokes flow (uS , P S ) and the regular flow (uR , P R ) which represents the free-surface effect. Thus, we write : (u, P ) = (uS , P S ) + (uR , P R )

(4)

The singular Stokes flow (ΦS , uS ) satisfies ∇ · uS = δ(x−x′ )δ(t)

uSt

2 S

− ǫ∇ u = −∇P

S

(5a) (5b)

which gives the solution : uS = ∇ΦS with :

and P S = −ΦSt

ΦS = −δ(t)/(4π|x−x′ |)

(6a) (6b)

Using the integral representation of ΦS , we have uS written as : Z γ+i∞Z ∞ Z ∞ ′ −i ∇ ds dβ dα e−k|c−c |+i(αx+βy)+st /k uS = 16π 3 γ−i∞ −∞ −∞ (6c)

p with (x, y) = (a−a′ , b−b′), k = α2 +β 2 and γ is the Laplace convergence abscissa in the Bromwich integral for the inversion of Laplace transform.

which gives :

Furthermore, we write the regular Stokes flow (uR , P R ) as the sum of an irrotational and a solenoidal vectors :

with

uR = ∇Φ + uT

˜ 0 = 1/k + N/D Φ

(13)

N = −2(ω 2 − 4ǫ2 k 3 kǫ )/k

(14a)

2

(7)

where the scalar function Φ(x, t) represents the irrotational flow while uT the rotational flow. The Stokes flow (Φ, uT ) taking account of free-surface effect satisfies :

2 2

2 3

D = ω + (s + 2ǫk ) − 4ǫ k kǫ

(14b)

In (13), D is often called the dispersion function and ω is defined by p ω = k + σ2 k3 (14c)

The boundary conditions (2) can now be expressed in terms of (uS , P S , Φ, uT ) on the undisturbed free surface (z = 0) :

Similar results for (˜ η0 , u˜T0 , v˜0T , w ˜0T ) with the same dispersion function D in denominator can be obtained. The wave elevation η has been considered in a number of studies as in Miles (1957) and in Lu & Chwang (2004). Since the amplitude functions of (˜ uT0 , v˜0T , w ˜0T ) are of order (ǫ1/2 , ǫ1/2 , ǫ), respectively, we are interested here to the potential function Φ which is obtained by taking the inverse integral transform :

ηt − (Φz + wT ) = wS

(10a)

Φ = δ(t)/(4π|x−x′1 |) + ΦF

(10b)

with x′1 = (a′ , b′ , −c′ ) is the mirror point of x′ = (a′ , b′ , c′ ) with respect to the mean free surface z = 0, and ΦF is the wave potential function : Z γ+i∞Z ∞ Z ∞ 1 N F Φ =− ds dβ dα ekz+i(αx+βy)+st 16π 3 i γ−i∞ −∞ −∞ D (16) in which (x, y, z) = (a − a′ , b − b′, c + c′ ) while N and D are given in (14). The potential function ΦF representing waves due to free-surface effect is analyzed below.

∇2 Φ = 0 = ∇ · uT uTt = ǫ∇2 uT

(8a) (8b)

The dynamic pressure P R is defined by : P R = −Φt

Φt + η

2Φzx + uTz + wxT = −(uSz + wxS ) 2Φzy + vzT + wyT = −(vzS + wyS ) − σ 2 (ηxx + ηyy ) + 2ǫ(Φzz + wzT )

= −ΦSt − 2ǫwzS

(9)

(10c) (10d)

Now we introduce the joint integral transform for (η, Φ, uT ) as :     Z ∞Z ∞ Z ∞ η˜ η ˜  = dt db da  Φ0 ekc  e−i(αa+βb)−st Φ 0 −∞ −∞ uT0 ekǫ c ˜T u (11) in which, we have used the notations : p kǫ = k 2 + s/ǫ Φ0 = Φ(a, b, c = 0, t)

uT0 = uT (a, b, c = 0, t) Taking the joint integral transform (11) over the left hand side of (10) as well as ∇·uT = 0, and introducing the integral form of (ΦS , uS ) on the right hand side of (10), a system of linear equations is obtained for the five unknowns :    η˜ s −k 0 0 −1 ˜   0 2ikα kǫ 0 iα   Φ   0 0 2ikβ 0 kǫ iβ     u˜T0  1 + σ 2 k 2 s + 2ǫk 2 0 0 2ǫk   v˜T  ǫ 0 0 0 iα iβ kǫ w ˜0T   1   ′ −2iα   kc −i(αa′ +βb′ ) −2iβ = (12) e (s + 2ǫk 2 )/k  0

(15)

4. WAVE POTENTIAL FUNCTION To evaluate the wave potential function ΦF (x, t), we examine the equation D = 0 which gives two poles of the Laplace integral : s± = ±iω − 2ǫk 2 + O(ǫ3/2 )

(17)

By taking a contour integration in the complex s plane using the Cauchy residue theorem, we obtain : ΦF = −

1 4π 2

Z

∞Z ∞ 2 dβ dα e−2ǫk t (ω/k) sin(ωt)

−∞ −∞

ekz+i(αx+βy) + O(ǫ)

(18)

which can be further written by introducing the Bessel function J0 (·) of the first kind : Z ∞ 2 1 F Φ =− dk e−2ǫk t ω sin(ωt)ekz J0 (kh) + O(ǫ) 2π 0 (19) p in which h = x2 + y 2 .

Following the work by Chen & Duan (2003) in which a potential function similar to (19) but in an inviscid fluid was analyzed, there are two saddle points kg and kT associated with the phase function ψ = ω − kv of the oscillatory part of the integrand in (19) : kg = 1/(4v 2 ) + O(σ/v 2 ) 2

2

(20a)

2

kT = 4v /(9σ ) + O(σ/v )

(20b)

√ for v ≫ σ, where v = h/t is the wave velocity. √ When v is of the same order as σ, the wavenumbers kg and kT become close and in particular, √ kg = kT = k0 ≈ 0.393/σ for v = v0 ≈ 1.086 σ. When v < v0 , the wavenumbers kg and kT are complex. This analysis shows that there are two waves propagating at the same speed v : one is gravity-dominant wave with a lower wavenumber kg (20a) and another capillary-dominant wave with a much larger wavenumber kT (20a). There is also a minimum speed v0 below which two waves become evanescent. By using the method of steepest descent, we obtain the asymptotic behaviors of ΦF (h, z → 0, t) at above three different cases with respect to the parameter v. Firstly, F

Φ ≈ +

s

s

1+σ 2 kg2 2 sin(ωg t−kg h) ekg z−2ǫkg t −ωg′′ ht

2 1+σ 2 kT2 cos(ωT t−kT h) ekT z−2ǫkT t ωT′′ ht

(21a)

with ωg,T = ω(kg,T ) defined by (14c) for v ≫ v0 . Secondly, p  ′  (ω0 − v)τ 2/3 2π(1 + σ 2 k02 ) F √ Ai Φ ≈ (ω0′′′ /2)1/3 (τ ω0′′′ /2)1/3 h

2

· sin(ω0 t − k0 h + π/4)ek0 z−2ǫk0 t

(21b)

for v ≈ v0 in which Ai(·) is the Airy function defined in Abramowitz & Stegun (1967) and {ω0 , ω0′ , ω0′′ , ω0′′′ } = {ω(k0 ), ω ′ (k0 ), ω ′′ (k0 ), ω ′′′ (k0 )}. And finally for v ≪ v0 :

ΦF ≈ −4/t3 + 5!(3h2 + 4σ)/t7 + O(t−11 )

(21c)

which is associated with the contribution from the end point of (16), i.e. at the origin k = 0 since that from the complex saddle points is exponentially small. The above equation (21a) indicates that the amplitude of ΦF (h, z = 0, t) is of order 2 O(e−2ǫkg,T t /t1/2 ) as t → ∞ for h/t = v > v0 . If the ratio v = h/t keeps constant, the amplitude of 2 ΦF (h, z = 0, t) is of order O(e−2ǫkg,T t /t) for t → ∞.

There are two wave systems associated with kg and kT corresponding to the first and second terms in (21a), respectively. At σ → 0, the wave system associated with kT - the second term - disappears while the first term associated with kg is reduced to that for the pure-gravity waves. The kg - and kT waves are thus called as the gravity-dominant and capillary-dominant wave systems, respectively. The two wave systems are merged into one as expressed by (21b) at v = v0 . For v < v0 , the amplitude of wave systems decreases exponentially and the leading term is no-wavy and decreases at the time rate −4/t3 everywhere as indicated by the first term in (21c). Physically, v = h/t is the propagation velocity of capillary-gravity waves which has a minimum limit v0 . In other words, we should find a calm region near the disturbance whose radius h0 = v0 t increases with time. Unlike the inviscid potential function in Chen & 2 Duan (2003), the factor e−2ǫk t is present in the integrand of (19). This exponentially-decreasing function reduces the amplitude of ΦF at large time and for waves of large wavenumbers. The capillarydominant waves are then heavy damped by the viscous effect while gravity-dominant waves are much less affected at small values of time. To confirm above analysis, we have performed the numerical computation of (19) in the complex k-plane by using the Romberg’s algorithm along the steepest descent contour. The figures on the last page of the present paper illustrate ΦF for z = −1/1000 at a fixed t = 10 and h varying from 0 to 8/5. 5. DISCUSSION AND CONCLUSION The potential function ΦF defined by (19) with the fluid viscosity but without taking account of the effect of surface tension is shown on Figure 1. Large waves with small wavelength are present at a region of small distance from the singularity. Due to the viscous effect and the immersion z = −1/1000, the wave amplitude of larger wavenumbers close to the singularity is reduced to zero. This is consistent with that the transient pure-gravity waves on the free surface at a given instant oscillate with increasing amplitude and decreasing wavelength when we approach to the impulsive source point, as stated in Lamb (1932). Furthermore, the amplitude of pure-gravity waves increases linearly with time in a rate of order O(t/h2 ) and wavenumber increases in an order of O[t/(4h2 )]. This peculiar property of pure-gravity potential hinders the numerical development to solve the boundary-value problem associated with a floating body in which the space integral over body’s surface as well as the time-convolution

integral are difficult to be accurate. Taking into account of the effect of surface tension, the wave form changes, in particular, there is not wave at all at small distance when h/t < √ v0 ≈ 1.086 σ. On the other side, at larger distance from the impulsive point, the capillary-dominant waves have very large amplitudes with wavenumbers proportional to 4h2 /(9t2 σ 2 ) as shown on Figure 2. These large and short waves of capillarity are fortunately heavy damped by the viscous effect as shown on Figure 3. In practice, the most interesting are the gravitydominant waves. Their properties of the potential function with the combine effect of surface tension and fluid viscosity are welcome and believed to be much useful in the numerical solution of wave-body problems. Especially, the use of this fundamental solution (potential function due to a point source) will remove difficulties associated with the highly oscillatory and singular property of pure-gravity waves in the evaluation of the waterline and timeconvolution integrals.

References [1] Abramowitz M. & Stegun I.A. (1967) Handbook of mathematical functions. Dover Publications. [2] Chen X.B. & Wu G.X. (2001) On singular and highly oscillatory properties of the Green function for ship motions. J. Fluid Mech. 445, 77-91. [3] Chen X.B. (2002) Role of surface tension in modelling ship waves. Proc. 17th Intl Workshop on Water Waves and Floating Bodies, Cambridge (UK), 25-28. [4] Chen X.B. & Duan W.Y. (2003) Capillarygarvity waves due to an impulsive disturbance. Proc. 18th Intl Workshop on Water Waves and Floating Bodies, Carry-Le-Rouet (France). [5] Chen X.B., Lu D.Q., Duan W.Y. & Chwang A.T. (2006) Potential flow below the capillary surface of a viscous fluid. Proc. 21st Intl Workshop on Water Waves and Floating Bodies, Loughborough (UK). [6] Cl´ ement A.H. (1998) An ordinary differential equation for the Green function of timedomain free-surface hydrodynamics. J. Eng. Math. 33(2), 201-217. [7] Crapper G.D. (1964) Surface waves generated by a travelling pressure point. Proc. Royal Soc. London A, 282, 547-558. [8] Lamb H. (1932) Hydrodynamics. 6th Ed. Dover Publications, New York.

[9] Lu D.Q. & Chwang A.T. (2004) Free-surface waves due to an unsteady stokeslet in a viscous fluid of infinite depth. Proc. 6th ICHD, 611-17. [10] Miles J.W. (1968) The Cauchy-Poisson problem for a viscous liquid. J. Fluid Mech. 34, 35970. [11] Ursell F. (1960) On Kelvin’s ship-wave pattern. J. Fluid Mech. 8, 418-431. [12] Wehausen J.V. & Laitone E.V. (1960) Surface waves. Handbuch des Physik SpringerVerlag. [13] Yih C.S. & Zhu S. (1989) Patterns of ship waves II: gravity-capillary waves. Q. Appl. Math. 47 35-44. [14] Zilman G. & Miloh T. (2001) Kelvin and Vlike ship waves affected by surfactants. J. Ship Res. 45, 2, 150-163.

60 40 20 0 -20 -40 -60 0

0.2

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0.8

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1.2

1.4

1.6

Figure 1: Wave potential function ΦF (h, z, t) at z = −1/1000 and t = 10 against h varying from 0 to 8/5, obtained by taking only account of the fluid viscosity (ǫ = 3.193e-7) but without the surface tension (σ = 0).

60 40 20 0 -20 -40 -60 0

0.2

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0.8

1

1.2

1.4

1.6

Figure 2: Gravity-dominant waves (solid line) and capillary-dominant waves (dashed line) of potential function ΦF (h, z, t) at z = −1/1000 and t = 10 against h varying from 0 to 8/5, obtained by taking account of surface tension (σ=2.713e-3) but without the fluid viscosity (ǫ = 0).

60 40 20 0 -20 -40 -60 0

0.2

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1

1.2

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1.6

Figure 3: Gravity-dominant waves (solid line) and capillary-dominant waves (dashed line) of potential function ΦF (h, z, t) at z = −1/1000 and t = 10 against h varying from 0 to 8/5, obtained by taking account of both the surface tension (σ=2.713e-3) and the fluid viscosity (ǫ=3.193e-7).