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April 2016
Abstract

The original investigation of Lamb (1932, §349) for the effect of viscosity on monochromatic surface waves is extended to account for second-order Stokes surface waves on deep water in the presence of surface tension. This extension is used to evaluate interfacial impedance for Stokes waves under the assumption that the waves are growing and hence the surface waves are unsteady. Thus, the previous investigation of Sajjadi et al. (2014) is further explored in that (i) the surface wave is unsteady and nonlinear, and (ii) the effect of the water viscosity, which affects surface stresses, is taken into account. The determination of energy-transfer parameter, from wind to waves, are calculated through a turbulence closure model but it is shown the contribution due to turbulent shear flow is some 20% lower than that obtained previously. A derivation leading to an expression for the closed streamlines (Kelvin cat-eyes), which arise in the vicinity of the critical height, is found for unsteady surface waves. From this expression it is deduced that as waves grow or decay, the cats-eye are no longer symmetrical. Also investigated is the energy transfer from wind to short Stokes waves through the viscous Reynolds stresses in the immediate neighborhood of the water surface. It is shown that the resonance between the Tollmien-Schlichting waves for a given turbulent wind velocity profile and the free-surface Stokes waves give rise to an additional contribution to the growth of nonlinear surface waves.

Growth of Stokes Waves by Wind on a Viscous Liquid] Growth of Stokes Waves Induced by Wind on a Viscous Liquid of Infinite Depth S. G. Sajjadi]SHAHRDADG.SAJJADI

1 Introduction

The energy exchange from wind to waves crucially depends on accurate determination of stresses on the water surface. The energy-transfer parameter (as is commonly known in the literature) is determined from the complex part of the interfacial impedance (as is termed by Miles), see (2.0) below. John Miles made several analytical attempts to improve upon the energy-transfer parameter, beginning with his pioneering work in 1957 and his final contribution in 1996. In all his contributions he assumed the initial surface is composed of a monochromatic surface wave of small steepness. Miles (2004) remarked ”… it will be interesting to see the extension of my 1996 contribution to Stokes waves, and its comparison with some numerical studies.” Although it was not explicitly mentioned by Miles, we assume he was referring to numerical contributions, for example, by Al-Zanaidi & Hui (1984) and Mastenbroek et al. (1996). However, he had recognized a major obstacle for this task, and further commented ”… the proper determination of the interfacial impedance for an a priori assumed nonlinear waves is by no means straightforward…”. In this note we offer a way to resolve this anomaly.

In a recent study by Sajjadi, Hunt, and Druillion (2014), it was shown that the growth rate, , where is the wave number and is the wave complex part of the wave phase speed, for growing waves critically depend on the energy-transfer parameter . Moreover, Sajjadi and Hunt (2003) (SHD therein) have suggested that wave steepness (for nonlinear waves, such as those observed in the sea) are also a contributing factor for the momentum transfer from wind to surface waves, see also Sajjadi (2015). Thus, one goal of the present study is the accurate determination of , and this requires the calculation of the complex amplitude of the wave-induced pressure at the surface.

To achieve our objective we must extend the effect of viscosity on monochromatic waves (Lamb 1932, §349) to a nonlinear surface wave, here we shall consider the simplest case, i.e. that of the Stokes wave. Thus, we shall adopt the bicrohomic assumption for the mean motion which admits the representation of the form (see section 4)

 (σ,χ,τ)=Re{(P1,Ξ1,T1)eik(x−ct)+ka(P2,Ξ2,T2)e2ik(x−ct)}

With , representing complex amplitude of stresses, and where ( is the wave speed) and is the wave steepness.

For an unsteady monochromatic surface wave SHD showed that the total energy transfer comprises the sum of two components ,

 βc=−π(U′′c/kU′c)(¯¯¯¯¯¯¯¯W2c/U21h20x),()′≡ddz (1.0)

is the contribution associated with the singularity at the critical layer Miles (1957), but due to the unsteadiness of the surface wave is additionally a function of (see SHD). In (1.1), the overbar signifies an average over is the kinematic friction velocity, is the wave-induced vertical velocity, and the subscript denotes evaluation at the critical point , where . The second component, is the rate of energy transfer to the surface due to the turbulent shear flow blowing over it.

Thus, we determine the energy-transfer parameter through calculating the pressure , and the shear stress , at the surface. As mentioned above, this requires generalization of the interfaced impedance by extending the monochromatic viscous theory of water waves (Lamb 1932 §349) to account for the effect of viscosity for Stokes waves. Then, the extended Lambâs solution to Stokes wave in a viscous liquid, with prescribed stresses at the surface can be adopted to evaluate expressions for and in the limit as , as outlined in section 3 below.

Hence, following the procedure adopted by SHD, for unsteady monochromatic waves, which led through evaluation of and , to the following expressions

 βc=πξ3cL40[1+(4−13π2+10^c2i)Λ2+O(Λ2)] (1.0) βT=5κ2L0+O(Λ) (1.0)

is obtained for an unsteady Stokes wave in sections 4 and 5. In equations (1.2) and (1.3) , and is Eulerâs constant.

In section 6, we derive an expression for closed loop streamlines, namely Kelvinâs cat-eye, and the significance of which is explored and explained. Finally, the results and discussion is given in section 7.

2 Interfacial impedance

Miles-Sajjadi (Miles 1996, Sajjadi 1998, hereafter M96 and S98 respectively) theory of surface wave generation considers the role of wave-induced Reynolds stresses in the transfer of energy from a turbulent shear flow to gravity waves on deep water. In their theories the Reynolds-averaged equations for turbulent flow over a deep-water sinusoidal gravity wave, (M96) and fully nonlinear surface gravity wave (S98), are formulated. Their formulations uses the wave-following coordinates , where and is exponentially small for . The turbulent Reynolds stress equations are closed by a viscoelastic constituent equation-a mixing-length model with relaxation (M96) and by the rapid-distortion theory (S98). Both derive their evolution equation on the assumption that: (i) the basic velocity profile is logarithmic in , where is a roughness length; (ii) the lateral transport of turbulent energy in the perturbed flow is negligible; and (iii) the dissipation length is proportional to . In both theories an inhomogeneous counterpart of the Orr-Sommerfeld equation is derived for the complex amplitude of the perturbation streamfunction and then used to construct a quadratic functional for the energy transfer to the wave. A corresponding Galerkin approximation that is based on independent variational approximation for outer (quasi-laminar) and inner (shear-stress) domains yields the interfacial impedance (defined by Miles 1957) in the limit . The calculation of the interfacial impedance requires the solution of the linearized equations of motion of water bounded above by a monochromatic surface wave (Lamb 1932, §349).

However, for Stokes surface wave the extension of Lamb’s solution is not immediately obvious (and Lamb, as well as other researchers to date, did not address this problem). However, in the case of a shear flow over a sinusoidal wave (for the application to air-sea interactions), we may consider the solution of the linearized Navier-Stokes equations in the semi-infinite body of water bounded above by the surface wave

 z=aeik(x−ct)≡h0(x,t)(ka≪1) (2.0)

in a fixed frame of reference gives, after renaming variables, that is after letting , (where the letters on the left-hand side denote those used by Lamb), and neglecting surface tension therein; following Lamb and adopting complex dependent variables, we obtain

 uw=[k+C(k−m)]ch0 (2.0)
 τw=(τ13)w=(2νwkc+iCc2)kh0 (2.0)

and

 (2.0)

for the tangential velocity, tangential stress, and normal stress, respectively, at the surface. The subscript refers to water, , and being the ratio of air to water density.

Invoking continuity of the perturbation velocity and and , the tangential and normal stresses, eliminating , and letting , we obtain the interfacial conditionsWe emphasize that, is perturbation to tangential wind velocity and is the complex phase speed induced by the presence of the wind.

 u−i(kcνw)−\tiny12sτ=kch0 (2.0)
 s(σ+iτ)=(c2−c2w)kh0 (2.0)

where

 cw≡(g/k)\tiny12−2ikνw(|kνw/c|≪1) (2.0)

is the complex phase speed in the absence of the air comprehends (through its imaginary part, which may be replaced by an empirical equivalent) the dissipation in water. The ratio of the second term to the first term on the left-hand side of (2.0) is typically smaller than ; accordingly (2.0) may be approximated by . However, we note that (2.0) does not reduce to in the limit of air inviscid liquid .

Finally by replacing and with their complex amplitudes and (defined as in Miles 1957) the interfacial impedance is obtained which may be expressed in the form

 (2.0)

where is the kinematic shear stress, is von Kármán’s constant, and the suffix zero indicates evaluation on , which to is the same as evaluation at .

Sajjadi (1998) followed Miles (1996) and calculated the interfacial impedance for every harmonic of the fully nonlinear surface wave. We remark that although M96 and S98 formulations are basically different for the turbulent flow over a surface wave, nevertheless the form of the interfacial impedance adopted are the same, provided Sajjadi’s series, for the representation of a fully nonlinear surface wave, is truncated after the first harmonic. Moreover, in S98, the inclusion of surface tension leads to an ambiguous results, even for the second-order Stokes wave, when his series is truncated after the second harmonic.

This ambiguity can readily be seen from equation (3.0) below in which

 (ϕ,ζ,w)=∑s(Φs,Ξs,Ws)

where

 Φs=Aseksz,Ξs=(Aseksz−iCsemsz),Ws=−ksΞs,

then substituting into (3.0) then for the -harmonic we have

 [LHS (???)]=−1ns{[n2s+ks(g+\@fontswitchTk2)+2νwk2sns]As−iks[2νwmsns+g+\@fontswitchTk2]Cs}

Here distinction has to be made between , the wavenumber of the entire wave, and , the wavenumber associated with the -harmonic. One may make the seemingly obvious assumption that , however, this will lead to a result which appears to be wrong. Note incidently, this ambiguity can be circumvented if surface tension is neglected as in S98.

The purpose of this note is to extend Lamb’s original investigation, for monochromatic waves, to Stokes waves in the presence of surface tension but under the assumption that the wave steepness .

3 Stokes waves on a viscous liquid

We consider the effect of viscosity on Stokes waves on deep water whose profile is given by

 z=aeik(x−ct)+12ka2e2ik(x−ct)≡h0+kah1=h(x,t),(ka≪1) (3.0)

see figure 1.

If we take the -axis to be vertically upwards, and if we assume a two-dimensional motion with velocities and being confined to the -coordinates and pressure , then ignoring the inertia terms, the equations of motion may be cast as

 ∂u∂t=−1ρw∂p∂x+νw∇2u,∂w∂t=−1ρw∂p∂z−g+νw∇2w, (3.0)

together with the continuity equation

 ∂u∂x+∂w∂z=0 (3.0)

where and are the water density, the kinematic viscosity of water and acceleration due to gravity, respectively.

Equations (3.0) and (3.0) are satisfied by

 u=−∂ϕ∂x−∂ψ∂z,w=−∂ϕ∂z+∂ψ∂x, (3.0)

and the linearized dynamic condition

 pρw=∂ϕ∂t−gz (3.0)

provided

 ∇21ϕ=0,∂ψ∂t=νw∇21ψ, (3.0)

where

 ∇21≡∂2∂x2+∂2∂z2.

We consider the solutions in normal mode by assuming that they are periodic in with a prescribed wavelength . Thus, assuming transient factors and spacial factors , for first and second harmonics, respectively. The solution of (3.0) may therefore be expressed in the following form:

 ϕ=(A1ekz+B1e−kz)eikx+nt+14ka(A2e2kz+B2e−2kz)e2(ikx+nt)ψ=(C1emz+D1e−mz)eikx+nt+14ka(C2e2mz+D2e−2mz)e2(ikx+nt)⎫⎪ ⎪⎬⎪ ⎪⎭ (3.0)

with

 m2=k2+n/νw (3.0)

The boundary conditions will provide equations which are sufficient to determine the nature of the various modes, and the corresponding values of .

In the case of infinite depth one of these conditions takes the form that the motion must be finite at . Excluding for the present case where is purely imaginary, this requires that for provided denote the root of equation (3.0) with . Hence

 (3.0)

Since denotes the elevation at the free surface, then the linearized kinematic condition is . Taking the origin of at the undisturbed level, this condition gives

 h=−kn(A1−iC1)eikx+nt−k2a4n(A2−iC2)e2(ikx+nt). (3.0)

Let be the surface tension, then the stress conditions at the free surface are given by

 p(zz)=T∂2h∂x2,p(xz)=0 (3.0)

to the first order, since we have assumed the inclination of the surface to the horizontal is sufficiently small, so that .

Now, if denotes the dynamic viscosity of the water,

 p(zz)=−p+2μw∂2w∂z2,p(xz)=μw(∂w∂x+∂u∂z) (3.0)

whence, by (3.0) and (3.0) we find, at the surface

 p(zz)ρw−\@fontswitchT∂2ζ∂x2=−∂ϕ∂t+(g+\@fontswitchTk2)ζ+2νw∂w∂z (3.0)

where . Next, writing , , and , then from (3.0), (3.0), (3.0) and (3.0) we have, after equating coefficients for and , respectively,

 p(zz)1ρw−\@fontswitchT∂2h0∂x2=−1n{[n2+k(g+\@fontswitchTk2)+2νwnk2]A1−ik[2νwmn+g+\@fontswitchTk2]C1} p(zz)2ρw−\@fontswitchT∂2h1∂x2=−1n{[n22+k4(g+\@fontswitchTk2)+2νwnk2]A2−ik[2νwmn+14(g+\@fontswitchTk2)]C2}

Similarly, writing and , we obtain from (3.0) and (3.0)

 p(xz)1ρw=−{2iνwk2A1+(n+2νwk2)C1} (3.0) p(xz)2ρw=−{2iνwk2A2+(n+2νwk2)C2} (3.0)

Substituting (3.0) into the first equation of (3.0), and eliminating the ratio , we see to that

 (n+2νwk2)2+gk+\@fontswitchTk3=4ν2wk2m (3.0)

Similarly, substitution of (3.0) into the second equation of (3.0), and eliminating the ratio , yields, to

 (n+4νwk2)(n+2νwk2)+gk+\@fontswitchTk3=16ν2wk2m (3.0)

Eliminating between (3.0) and (3.0) then gives

 n=−2νwk2±√4ν2wk4−6σ2

and by virtue of the fact that , we obtain

 n=−2νwk2±iσ∗ (3.0)

where and .

The condition shows that

 C1A1=C2A2=−2iνwk2n+2νwk2=∓2νwk2σ∗ (3.0)

which is, under the same circumstances, very small. Hence the motion is approximately irrotational, with a velocity potential

 ϕ=A1e−2νwk2t+kz+i(kx±σ∗t)+14kaA2e−4νwk2t+2kz+2i(kx±σ∗t) (3.0)

If we put

 A1=A2=∓iσ∗ak

the equation (3.0) of the free surface becomes, on taking the real parts

 h=ae−2νwk2tcos(kx±σ∗t)+14a2ke−4νwk2tcos2(kx±σ∗t) (3.0)

Since the motion is nearly (but not exactly) irrotational, there is vorticity present whose magnitude is given by

 ω=∂w∂x−∂u∂z≡∇21ψ

Thus, from (3.0) and (3.0), we have approximately

 m=(1±i)bwhereb=(σ∗/2νw)1/2

Hence, with the same notation as before, we find

 ω= ∓ 2σ∗kae−2νwk2t+bzcos{kx±(σ∗t+bz)} (3.0) ∓ 2σ∗k2a2e−4νwk2t+2bzcos2{kx±(σ∗t+bz)}

From equation (3.0) it can be seen that the vorticity diminishes rapidly from the surface downwards. Moreover, since the motion has an oscillatory character, the sign of the vorticity which is being diffused inwards from the surface is continually reversing, such that (paraphrasing Lamb) ‘beyond a stratum’ of thickness of the effect diminishes.

The above analysis gives results for the first two components of the normal modes of the prescribed wavelength. For a fully nonlinear Stokes wave, there are an infinitely more of these modes exist and they correspond to pure-imaginary values of , which are less persistent in character.

It is interesting to note that, if we now, in place of (3.0), assume

 ϕ=A1ekzeikx+nt+14kaA2e2kze2(ikx+nt)ψ=(C1cosℓz+D1sinℓz)eikx+nt+14ka(C2cos2ℓz+D2sinℓz)e2(ikx+nt)⎫⎪ ⎪⎬⎪ ⎪⎭ (3.0)

and carrying out the previous analysis, we find to

 (n2+2νwk2n+gk+\@fontswitchTk3)A1−i(gk+\@fontswitchTk3)C1+2iνwkℓnD1=02ik2A1+(k2−ℓ2)C1=0⎫⎪⎬⎪⎭ (3.0)

and to

 (2n2+8νwk2n+gk+\@fontswitchTk3)A2−i(gk+\@fontswitchTk3)C2+8iνwkℓnD2=02ik2A2+(k2−ℓ2)C2=0⎫⎪⎬⎪⎭ (3.0)

We note that now any value of is admissible in these equations for determining the ratios ; and the corresponding value of is

 n=−νw(k2+ℓ2)

We remark the extension of the above analysis to third or higher order Stokes waves, (if at all analytically tractable) is by no means an easy task.

4 Energy transfer to unsteady Stokes waves

We consider a turbulent shear flow of air whose density is blowing over an unsteady second-order Stokes wave of the form (3.0) with a complex phase speed , where is the wave speed and is the growth () or decay () rate.

We shall neglect the molecular viscosity of the air by virtue of the fact that and thus the viscous forces in the airflow becomes negligible. Then, the governing Reynolds-averaged equations are given by S98

 ∂i⟨ui⟩=0,D⟨ui⟩=−∂i⟨p/ρa⟩−∂j⟨u′iu′j⟩,

where

 D=∂t+⟨uj⟩∂j.

Hence, the horizontal and vertical momentum equations may be expressed, respectively, as

 D⟨u⟩=−σx+χx+τz (4.0)

and

 D⟨w⟩=−σz+τx (4.0)

where

 σ≡⟨p/ρa+w′2⟩,χ≡⟨u′2−w′2⟩,τ≡−⟨u′w′⟩.

Here and are the mean normal and shear stresses.

Following Townsend’s scaling argument (Townsend 1972), we may further neglect the components and without affecting the solution significantly. Accordingly, equations (4.0) and (4.0) reduce to

 D⟨u⟩=−σx+τzD⟨w⟩=−σz.

The continuity of the air-water at the surface requires

 u=c∂h∂η,sτ=τwandsσ=σw

where . Note we have used the same transformations given in section 2. However, we note that if the wave steepness the by virtue of which .

Using the expression for the horizontal velocity, given by the first of equations (3.0), in the curvilinear coordinates, namely

 u=−(ikA1ekη+mC1emη)eikξ+nt−12ka(ikA2e2kη+mC2e2mη)e2(ikξ+nt),

and using the following transformation:

 (A1,A2,C1,C2,n)=[iac(1+C1),2iac(1+C2),acC1,2acC2,−ikc]

we obtain

 uw=c[k+C1(k−m)]h0+2kac[k+C2(k−m)]h1,

where the subscript refers to water.

Neglecting the surface tension, for Stokes waves the total mean normal stress at the surface is (see section 3)

Thus, referring to the previous section, we have

 p(ηη)1=ka{gk−c2−2ikνwc−C1[c2−2i(k−m)νwc]} p(ηη)2=12ka{gk−2c2−8ikνwc−2C2[c2−4i(k−m)νwc]}

and similarly,

 τw=p(ξη)=(2νwk+iC1c)ckh0+2ka(2νwk+iC2c)ckh1.

We next express the mean surface stresses in the bichromatic perturbation form

 (σw,τw)=(P1h0+kaP2h1,T1h0+kaT2h1)

where and represent the complex amplitudes of the normal and shear stresses, respectively. Invoking continuity of the perturbation velocity (), eliminating and letting (as ), we obtain the interfacial conditions:

 u1−is(kcνw)−\footnotesize12T1h0+12ka[2u2−is(kcνw)−% \footnotesize12T2h1]
 s[(P1+iT1)h0+ka(P2+iT2)h1]=(c2−c2w)kh0+ka(2c2−c2w)kh1 (4.0)

where is given by (2.0). Hence from (4.0) we see that

 c2(1+2ka)−c2w(1+ka)sU21 = (P1+iT1)+ka(P2+iT2)kaU21 (4.0) = (α1+iβ1)+ka(α2+iβ2).

5 Determination of energy-transfer parameter

The energy-transfer parameter, defined as in (4.0), requires the calculation of the complex amplitudes of the wave-induced pressure and shear stress at the surface of Stokes waves. We may obtain these from the solution of the Orr-Sommerfeld equations (for details see S98 or SHD),

 [νe(Φ′′1+U′′H′′1)]′′=ik[U(Φ′′1−k2Φ1)−U′′Φ1]
 [νe(Φ′′2+U′′H′′2)]′′=2ik[U(Φ′′2−4k2Φ2)−U′′Φ2]

where , subject to boundary conditions at and . In the above equations is the eddy viscosity, , and and are, respectively, the complex amplitudes of and the perturbation stream function , given by

 Ψ=∫η0U(η)dη+Uh(ξ,η)+ϕ(ξ,η) (5.0)

with understanding that , etc.

Alternatively we can follow generalization of M96 for Stokes waves and evaluate by taking the real part of the quadratic functional

 β=(kaU1)−2∫∞0{νe[UZ′′21+2U′Z′1Z′′1+U′′(Z1−H1)Z′′1]+ +ikU2(Z′21+k2Z21)}dη +ka(2kaU1)−2∫∞0{νe[UZ′′22+2U′Z′2Z′′2+U′′(Z2−H2)Z′′2] +2ikU2(Z′22+4k2Z22)}dη, (5.0)

which provides a Galerkin approximation for suitable approximations to and for . We note that

 h(ξ,η)=h0(ξ)e−kη+kah1(ξ)e−2kη

with

 [h,ζ]=[H1,Z1]eik(ξ−ct)+ka[H2,Z2]e2ik(ξ−ct).

Note further, is obtained from the linear approximation to the kinematic surface condition , which in turn yields , or

 ζ1+kaζ2=−U−1(ϕ1+kaϕ2).

Following M96, S98 and SHD, and generalizing the former and the latter to the second-order Stokes waves, we arrive at

 (5.0)

Choosing the simplest trial function for the variational integral (5.0), namely

 Zn=ae−knη/bwith k1≡k and k2≡2k,

where is a free parameter, we obtain

 Pn0/knaU21=b2(b−2+1){π26+ln2(2knγηcb)−2i^ciln(2knγηcb)+^c2i}

where , and being the Euler’s number. Proceeding as in SHD, we may cast the above expression as

 βnc=πk3nη3cL4n0{1+(4−π23+10^c2i)Λ2n+O(Λ3n)}

with

 Λ−1n=Ln0=γ−ln(2knηc).

Hence the energy-transfer contribution due to the initial critial layer, , becomes

 βc = β1c+kaβ2c = πk3η3c{(L410+8kaL420)+(4−π33+10^c2i)(L210+8kaL220)}

Similarly the contribution of the energy-transfer parameter due to turbulence may be evaluated from the integral (5.0) with the extra contribution

 βT=−4ki∫∞0λ′WZ1Z′1dη−8k2ai∫∞0λ′WZ2Z′2dη

where , and . Evaluating the integral it can be shown that the result may be put in the formSHD obtained a very similar results for monochromatic waves, namely , using a different approach.

 βT=4κ2(L01+2kaL02).

We note taht the above expression (for a monochromatic wave) is some 20% lower than that given by SHD, (cf. equation (1.3) above).

6 Kelvin cats-eye

Closed streamlines, commonly known as Kelvin cats-eye, or simply cats-eye, occur in the neighbourhood of the critical point , where . The stream function , for the basic flow, there

 ψb≡∫η0U(η)dη→ψc+12U′c(η−ηc)2asη→ηc

has minimum when where

 ψc≡∫ηc0U(η)dη≃−U′cη2c

when . The stream function for the perturbed flow, (5.0), in the neighbourhood of has the following expansion (cf. Lighthill 1962 and Phillips 1977 §4.3)

 ψ=ψc+12U′c(η−ηc)2+U′c(η−ηc)hc(ξ)+ϕc(ξ),

wherein the subscript implies evaluation at , and an error factor of is implicit in (6.2).

Assuming ,

 ϕc=Re{Φceik(ξ−ct)}≡−14A2ekcitU′ccosk(ξ−c