The damping of viscous gravity waves

The damping of viscous gravity waves

M. Antuono a, A. Colagrossi a,b,∗,aCNR-INSEAN, The Italian Ship Model Basin, Rome, Italy

bCESOS: Centre of Excellence for Ship and Ocean Structures, NTNU,Trondheim, Norway

Abstract

Using a solution of the linearized Navier-Stokes equations, an approximate formulahas been derived for the damping rate of gravity waves in viscous fluids. Theproposed solution extends the results found by Lamb [1] for waves propagatingin deep-water conditions for large Reynolds numbers and those derived by Biesel [2]under more general hypotheses. Specifically, comparisons with the Lamb solutionhighlight large differences in intermediate and shallow depths and/or for moderateReynolds numbers while significant discrepancies are observed with the Bieselsolution in deep-water conditions. For these reasons, the proposed solution is of greatimportance for the estimate of the viscous dissipations during the wave motion andrepresents a useful benchmark for the validation of numerical solvers. With respectto this, the theoretical findings have been compared with numerical simulationsobtained by means of a well-known Smoothed Particle Hydrodynamics solver.

Key words: gravity waves, free-surface flows, wave damping

1 Introduction

The propagation of gravity waves is an important subject of research becauseof its applications to various problems and phenomena connected to humanworks (e.g. coastal and environmental engineering, naval activities, renewableenergies etc.). The greatest part of these phenomena is characterized by highReynolds numbers (Re = H∗0

√g∗H∗0/ν

∗ where H∗0 is the fluid depth, g∗ thegravity acceleration, ν∗ is the kinematic viscosity and stars give dimensional

∗ Corresponding author: Tel.: +39 06 50 299 343; Fax: +39 06 50 70 619.Email addresses: [email protected] (M. Antuono),

[email protected] (A. Colagrossi).

Preprint submitted to Elsevier Science 11 June 2015

variables) and, consequently, it is often described through the potential theory,that is, assuming that the fluid is inviscid and irrotational. Despite this,dissipations may play an important role when the propagation occurs overlong times or when the Reynolds number is not so high. Then, a correctestimate of the dissipative effects turns out to be of fundamental importancefor a proper description of the gravity wave evolution.

Many numerical works have been devoted to this subject (e.g. Harlow & Welch,[3]; Haddon & Riley, [4]; Raval et al., [5]; just to cite a few). Conversely,a further theoretical inspection is still needed, as the largest part of theanalytical results just applies to gravity waves in deep-water conditions andwith very low viscosity. The first attempts to estimate dissipative effects dateback to the works of Basset [6], Lamb [1] (first edition 1879, second edition1895) and Boussinesq [7]. It is not clear who was the first to find out theattenuation law for gravity waves but, nowadays, this solution is generallyreferred to Lamb’s work. Lamb [1] derived a damping coefficient for thewave amplitude using the linearized Navier-Stokes equations. His approachwas based on the assumption that the water depth was infinite and that theviscosity was small. The same result was found by Basset [6] who also triedan extension in finite depths. In this way, he was able to recover the exactlinear dispersion relation for the wave celerity but, due to the assumption oflow viscosity, he exactly recovered the damping coefficient of Lamb.

Over the years, the same assumptions (that is, deep-water conditions andlow viscosity) have been used in several works that confirmed the validity ofLamb’s result through different techniques/approaches. For example, Longuet-Higgins [8,9] found Lamb’s coefficient by using a boundary layer model andinspecting the equivalence of this model with the theory on weakly-dampedwaves by Ruvinsky & Freiman [10]-[11]. Lighthill [12] used the deep-waterinviscid solution to approximate the dissipative terms inside the kineticenergy equation for a viscous fluid. In this way, he obtained a dampingcoefficient for the kinetic energy evolution that can be straightforwardlyrelated to the Lamb’s one. A more rigorous approach was adopted by Wu etal. [13] to study the fluid motion inside a two-dimensional rectangular tank.Specifically, they solved an initial condition problem for the linearized Navier-Stokes equations using a no-slip condition along the bottom and a free-slipcondition along the tank walls. The latter assumption was forced by sometheoretical and numerical difficulties in imposing a no-slip condition at theintersection between the free surface and the walls. Despite this, the dampingrate predicted by Wu et al. [13] coincides with that predicted by Lamb. Morerecently, a linear solution of the Navier-Stokes equation has been used byDias et al. [14] under the hypotheses of infinite depth. They also providedsome insight in the action of nonlinearities, showing that the damping ratepredicted by Lamb can be also applied to weakly-nonlinear waves.

2

A significant improvement to Lamb’s findings was obtained by Biesel [2] who,thanks to an asymptotic expansion for high Reynolds numbers, derived higher-order terms for the damping rate. Surprisingly, such a work is not muchrefereed to in the current scientific literature. The work of Biesel shows severalsimilarities with the present analysis but, for deep water conditions, it maystill lead to an inaccurate prediction of the damping rate.

Incidentally, we highlight that a further subject of research relies on howto include dissipation in models which, usually, do not account for viscouseffects. For example, Liu & Orfila [15] derived a system of integro-differentialdepth-averaged equations that includes dissipative effects due to the bottomboundary layer. Similarly, a viscous-potential formulation has been proposedin Dutykh & Dias [16] and further inspected in Dutykh [17] to account forthose effects in standard potential theory.

The aim of the present work is to provide a further contribution to the studyof viscous gravity waves, that is, waves whose evolution is mainly driven bygravity. Specifically, our analysis is based on a solution of the linearized Navier-Stokes equations over finite depths with a no-slip condition along the bottom.The main result is the derivation of approximate expressions for the wavecelerity and the damping rate. Indeed, these also hold true for gravity wavespropagating in intermediate- and shallow-water conditions and for Reynoldsnumbers that are not very large (e.g. Re ≥ 50). The latter point makes theseexpressions also suitable for fluids different from water.

For waves propagating in deep-water conditions with large Reynolds numbers,the leading order of the proposed solution reduces to that derived by Lamb.In this case, the higher-order terms provide corrections which are generallysmall but not negligible. Conversely, when the Reynolds number is not so highand/or the motion takes place in conditions that are far from deep water, thedamping rate displays significant differences with respect to that predictedby Lamb. As a consequence, the proposed solution may be important for theestimation of the viscous dissipations during the wave motion. Further, itmay be properly used as a benchmark for the validation of numerical solvers(see, for example, Carrica et al. [18]). With respect to this latter topic, wepropose some applications in the final part of the present work where theanalytical results are compared with the numerical outputs of a SmoothedParticle Hydrodynamics scheme (SPH hereinafter).

The paper is organized as follows: the analytical solution and the dampingcoefficient are described in Section §2 while Section §3 contains applicationsand comparisons with the Lamb’s and Biesel’s solutions.

3

x

y λ

H

y = ε η(x, t)

Fig. 1. Sketch of the geometry (dimensionless variables): λ is the wave length, His the still water depth and η(x, t) is the free surface. The dotted line indicates theundisturbed free surface.

2 Approximate analytical solution for viscous gravity waves

Let us consider the evolution of viscous gravity waves propagating over finitedepths. Figure 1 displays a sketch of the problem and of the Cartesian frameof reference, whose origin is at the undisturbed free surface with the y-axispointing upward. Hereinafter, the unstarred variables denote dimensionlessquantities. We introduce the nonlinearity parameter ε = 2A∗0/H

∗0 where H∗0

is the reference depth in still-water conditions and A∗0 is the wave amplitude.Accordingly, we use the following scaling:

x∗ = H∗0 x, y∗ = H∗0 y, t∗ =

√H∗0g∗

t, η∗ = 2A∗0 η,

u∗ = ε√g∗H∗0 u, v∗ = ε

√g∗H∗0 v, H∗ = H∗0 H, p∗ = 2 ρ∗0 g

∗A∗0 p .

(1)

Here, (u, v) are respectively the horizontal and vertical components of thevelocity field, η is the free surface, p is the pressure field, ρ∗0 is the fluid density(assumed constant) and g∗ is the gravity acceleration. Similarly to Wu et al.[13], we define the Reynolds number as Re = H∗0

√g∗H∗0/ν

∗ where ν∗ is thekinematic viscosity. Then, the dimensionless Navier-Stokes equations read:

4

ux + vy = 0 ,

ut + ε (uux + v uy ) = − px +1

Re(uxx + uyy ) ,

vt + ε (u vx + v vy ) = − py +1

Re( vxx + vyy ) ,

(2)

where p = y/ε + p is the dynamic pressure field. The kinematic boundaryconditions at the free surface y = ε η(x, t) and at the bottom y = −H(x, t)are:

ηt + ε u∣∣∣εηηx − v

∣∣∣εη

= 0Ht

ε+ u

∣∣∣−H

Hx + v∣∣∣−H

= 0 . (3)

Hereinafter we assume H to be constant. Because of the scaling in (1), it comesH = 1. In any case, we prefer to explicitly use symbol H instead of the value1 for the sake of clarity. Then, (3) becomes:

ηt + ε u∣∣∣εηηx − v

∣∣∣εη

= 0 v∣∣∣−H

= 0 , (4)

and the last equality is consistent with the no-slip boundary conditions alongthe bottom:

u∣∣∣−H

= 0 v∣∣∣−H

= 0 . (5)

By definition, a free surface is an interface along which the stress (that is,the surface force t) is zero. This is a common assumption for gravity wavesand corresponds to neglect both the action of the gaseous phase on the fluidand the effects due to the interface curvatures. This implies t = T · n = 0at y = ε η(x, t). Here, T is the stress tensor for a Newtonian fluid, that isT = − p1 + 2D/Re, D is the strain-rate tensor and n is the unit vectornormal to the free surface. Then, the following dynamic boundary conditionshold true:

p∣∣∣εη− η =

2

Re(n ·D · n )

∣∣∣εη

( τ ·D · n )∣∣∣εη

= 0 , (6)

where τ is the unit vector orthogonal to n (that is, the unit vector tangentialto the free surface).

5

2.1 Dispersion relation

In the present section we summarize the procedure to find out a solution forthe linearized Navier-Stokes equations. The complete derivation is describedin appendix A.

First, we map the fluid domain (x, y) ∈ R × [−H, εη] in the following plane(x, ξ) ∈ R× [−H, 0]. The map is given by:

ξ =H

H + ε η( y − ε η ) ⇐⇒ y =

(H + ε η)

Hξ + ε η . (7)

The transformation described in (7) is a straightforward application of thestrained coordinate technique (see, for example, Lighthill [19]) and is awidespread technique in the scientific literature. In the present case, sucha transformation enables one to overcome the inconsistency of the assignmentof the boundary conditions at the free surface. Indeed, a straightforwardlinearization of the Navier-Stokes equations would lead to an assignment aty = 0 while the actual free surface is at y = εη. Conversely, in the (x, ξ)-planethe assignment is done at ξ = 0 and this line coincides with the free surface.

Assuming ε 1 (that is, small gravity waves), the following relations holdtrue between the derivatives of a generic function f(x, y, t) and its transformedform f(x, ξ, t):

fx = fx + O(ε) fy = fξ + O(ε) ft = ft + O(ε) . (8)

Consequently, the linearized Navier-Stokes equations in the (x, ξ)-plane are:

ux + vξ = 0 ,

ut = − px +1

Re( uxx + uξξ ) ,

vt = − pξ +1

Re( vxx + vξξ ) ,

(9)

while the boundary conditions in (4), (5) and (6) become:

6

ηt = v∣∣∣ξ=0

u∣∣∣ξ=−H

= 0 v∣∣∣ξ=−H

= 0 , (10)

p∣∣∣ξ=0− η =

2

Re

(n · D · n

) ∣∣∣ξ=0

(τ · D · n

) ∣∣∣ξ=0

= 0 , (11)

where n = (0, 1) and τ = (1, 0). Similarly to Lamb [1], we look for solutionswith the following structure:

u = U(ξ) eı θ + c.c. + O(ε) v = V (ξ) eı θ + c.c. + O(ε) (12)

p = P (ξ) eı θ + c.c. + O(ε) η = E eı θ + c.c. + O(ε) (13)

where θ = k x−σ t, k is the dimensionless wave number (i.e. k = 2π/λ whereλ is the dimensionless wave length) and ‘c.c.’ indicate the complex conjugate.Note that U , V , P , E and σ are complex numbers. Specifically, the real part ofσ represents the circular frequency while the imaginary part gives the dampingrate of the viscous gravity wave.

Here, we focus on the dispersion relation that is obtained by substitutingexpressions (12)-(13) in system (9) and in boundary conditions (10)-(11). Inparticular, the solution for U , V , P and E are given in appendix A while thedispersion relation is:

− ik4

Re− ik m3 tanh (kH)

Re tanh (mH)+ik3 tanh (kH)m

Re tanh (mH)+ik2m2

Re+

+4σm3k2

Re2 sinh (mH) cosh (kH)+σm4 k tanh (kH)

Re2 − σm5

Re2 tanh (mH)+

+4σ k4m

Re2 sinh (mH) cosh (kH)+σ k5 tanh (kH)

Re2 − 5σ k4m

Re2 tanh (mH)+

+6σ k3m2 tanh (kH)

Re2 − 2σm3k2

Re2 tanh (mH)= 0 . (14)

This has been derived by Basset [6] who also tried to find an explicit expressionfor the damping rate of the free surface, finding =(σ) = −2 k2/Re. This result,that coincides with that of Lamb [1], is a leading-order approximation forlow-viscosity waves propagating in deep-water conditions. Here, we give anextension for waves propagating in different regimes (from shallow- to deep-water) and for values of Re that are not very large. In the following section,we verify through comparisons with numerical simulations that the proposedsolution holds true up to Re = 50, at least. Specifically, the solution for σ hasbeen expressed in form of a perturbation expansion:

7

<(σ) =√k tanh(kH) +

√2 k5/4 (tanh(kH)2 − 1)

4 tanh(kH)3/4Re−1/2 +

−√

2 k11/4 (7 tanh(kH)6+147 tanh(kH)4−11 tanh(kH)2−15)

128 tanh(kH)13/4Re−3/2 +

+O(Re−2) , (15)

=(σ) =

√2 k5/4 (tanh(kH)2 − 1)

4 tanh(kH)3/4Re−1/2 +

+k2 (tanh(kH)4 − 8 tanh(kH)2 − 1)

4 tanh(kH)2Re−1 +

+

√2 k11/4 (7 tanh(kH)6+147 tanh(kH)4−11 tanh(kH)2−15)

128 tanh(kH)13/4Re−3/2 +

+O(Re−2) , (16)

As highlighted in appendix A, such a solution is well-posed for Re 1 andfor Re−1 kH Re2/3 (we recall that H = 1 because of the adopted scalingand, therefore, does not influence the results). The bounds above ensure thateach term inside the perturbation expansion is smaller than the preceding termand larger than the subsequent one for every value of k. The requirement forRe 1 is a mathematical one and, consequently, Reynolds numbers thatare generally regarded as moderate in the fluid dynamics literature actuallysatisfy such a constraint. This motivates the good agreement of the proposedsolution with the numerical results for Re = 50 (see section 3.2).

The complex part =(σ) is strictly negative and gives the dissipation rate of aviscous gravity wave propagating with velocity <(σ)/k. As expectable, at theleading order <(σ) coincides with the dispersion relation of the potential flowtheory while at the higher-orders strictly negative terms appear. This meansthat the viscosity tends to reduce the circular frequency and, consequently,the phase velocity of the wave.

A bit more complex is the analysis of the dissipative effects which,consequently, is reported in a dedicated section.

2.1.1 Damping rate

Let us focus on the damping rate of the kinetic energy. In this case, thedamping coefficient predicted by the linear solution is β = − 2=(σ). In deepwater conditions (that is, for kH 1), tanh(kH) ' 1 and, using equation(16), we obtain:

8

Fig. 2. Damping rate: relative error between the Lamb’s solution [1] and the presentone (β = −2=(σ)) for different Reynolds numbers: εrel = βL/β − 1

β = 4 k2Re−1 − 2√

2 k11/4Re−3/2 + O(Re−2) . (17)

As expectable, the leading-order term coincides with the damping coefficientpredicted by Lamb, that is, βL = 4 k2Re−1 (this comes from the damping rateof the wave elevation, i.e. 2 k2Re−1). However, the higher-order term is strictlynegative and this implies that βL, generally, overestimates the dissipations,especially when the Reynolds number is not very large. Conversely, inintermediate- and shallow-water conditions the leading term of β is of orderO(Re−1/2), that is, at least one order of magnitude larger than βL [that isof order O(Re−1)]. This leads to a larger damping rate with respect to thatpredicted by Lamb’s solution, even when the Reynolds number is high (see, forexample, figure 2). This behavior is mainly due to the fact that Lamb’s solutionrelies on the assumption that the fluid depth is infinite and, consequently, doesnot account for the dissipations caused by the boundary layer at the bottom.The presence of terms of order O(Re−1/2) due to the bottom boundary layerhas been also pointed out by Keuleghan [20], Lighthill [12] and Dutykh & Dias[16].

Regarding the solution of Biesel [2], this coincides with the solution given in(15)-(16) up to O(Re−1) while terms of order Re−3/2 are neglected. Biesel’ssolution has been derived by using a perturbation expansion similar to thatdescribed in the present work (see Appendix A) and gives good estimates of thedamping rate for intermediate and shallow depths. Conversely, for deep-waterconditions it exactly reduces to Lamb’s solution. In this case, the absence of theterm of order Re−3/2 [see equation (17)] generally leads to an overestimate ofthe damping rate. This is shown in figure (3), where the relative error betweenthe damping rate predicted by using the Biesel’s solution (βB) and the presentone (β) is displayed. Specifically, high Reynolds numbers are considered,namely Re = 100, 000 and Re = 1, 000, 000. As expectable, the error ispractically negligible for small and moderate values of kH and decreases as theReynolds number increases. On the contrary, it is of some percent for kH 1(deep-water conditions). These are enough to cause significant differences inthe wave evolution. Indeed, waves propagating at large Reynolds numbersevolve for very long times before they are completely damped. Consequently,small relative errors may lead to sensibly wrong estimates of the dampingphenomenon. As shown in the following section, for smaller Reynolds numbers

9

Fig. 3. Damping rate: relative error between the Biesel’s solution [2] and the presentone (β = −2=(σ)) for different Reynolds numbers: εrel = βB/β − 1

this behavior becomes pronounced even for moderate values of kH.

3 Applications

In the present section we show some numerical applications. The proposedsolution may be regarded as a reference solution for the validation of numericalsolvers (see, for example, Carrica et al. [18]). At the same time, since theanalytical results are not exact but are obtained from a linearized systemthrough a perturbation expansion, comparisons with numerical solutionsprovide a further foundation to the present analysis.

Specifically, we compare the analytical solution with the results obtainedthrough a Smoothed Particle Hydrodynamics model (SPH hereinafter). Sucha scheme proved to be robust and accurate in describing free-surface flows(see, for example, Antuono et al. [21]) and, as a major advantage, does notneed any explicit enforcement of the free-surface boundary conditions, sincethese are implicitly satisfied (for details, see Colagrossi et al. [22]).

To reduce the computational domain and, therefore, the computational costs,we consider a viscous standing wave.

3.1 The SPH model

The SPH scheme is a meshless Lagrangian method in which the fluid domain isdiscretized in a finite number of particles representing elementary fluid volumesV ∗, each one with its own local mass m∗ and other physical properties. In thiscontext a generic field f ∗ at the position r∗i of the i-th particle is approximatedthrough the convolution sum

〈f ∗〉(r∗i ) =∑j

f ∗j W∗(r∗i − r∗j , h∗)V ∗j (18)

10

where f ∗j is the value of f ∗ associated to the generic particle j, V ∗j is its volumeand, finally, W ∗(r∗i − r∗j , h∗) is a weight function that, in the SPH literature,is generally called kernel function. The kernel function has a compact supportwhose radius is proportional to h∗ (also known as smoothing length) andtends to a delta Dirac “function” when h∗ goes to zero. The integration of thekernel function on its support is imposed to be equal to one. For the ease ofnotation, hereinafter we denote W ∗(r∗i − r∗j , h∗) simply through W ∗

ij and usethe Wendand kernel ψ3,1 (see Wendland [23]).

In the present work we consider a weakly-compressible SPH scheme (see, formore details, Monaghan [24]). In this framework, the fluid is assumed to beweakly-compressible and barotropic. Similarly to the work of Antuono et al.[21], the state equation of the fluid is linearized around a reference value,that, in the present case, is the density value at the free surface in still waterconditions. The (dimensionless) equations of the SPH scheme considered hereare:

DρiDt

= − ε ρi∑j

(uj − ui) · ∇iWij Vj ,

ρiDuiDt

= −∑j

(pj + pi)∇iWij Vj + ρie3

ε+

1

Re

∑j

πij∇iWij Vj

pi =c2

0

ε(ρi − 1) ,

DriDt

= ui ,

(19)

where ρi, pi and ui are the density, the pressure and the velocity ofthe i-th particle and ∇i indicates the differentiation with respect to ri.Symbol e3 denotes the vertical unit vector (pointing upwards) and c0 is the(dimensionless) sound velocity. The scaling factors are:

V ∗i = V ∗0 Vi =

(m∗

ρ∗0

)Vi W ∗

ij =Wij

V ∗0, c∗0 =

√g H∗0 c0 ,

where ρ∗0 is the reference density (that is, the density along the free surface).The argument of viscous terms is:

πij = K(uj − ui) · (rj − ri)

|rj − ri|2.

11

Fig. 4. Dynamic pressure field (left) and vorticity field (right) at t = 0 forRe = 50, k = π and ε = 0.1.

where K = 8 in two dimensions.

To reduce the computational effort, it is a common practice in the weakly-compressible SPH solvers to use a sound velocity much smaller than thephysical one. In the specific case of gravity waves, Antuono et al. [21] provedc∗0 = 10

√g∗H∗0 to be a reliable choice.

A no-slip condition has been implemented along the bottom using the fixedghost particles described in Marrone et al. [25]. In all the numerical simulationsthe initial conditions for the SPH scheme have been obtained by using thelinear viscous solution. The spatial resolution of the SPH scheme is indicatedthrough the parameter H∗/∆x∗ where ∆x∗ is the initial particle distance. Thenumber of particles inside the kernel domain is proportional to h∗/∆x∗. In allthe numerical simulations the ratios H∗/∆x∗ and h∗/∆x∗ has been chosen toensure attainment of a converged solution.

3.2 Results

Figure 4 displays the dynamic pressure field and the vorticity at t = 0 forRe = 50, k = π and ε = 0.1. The initial free surface displacement has been setto zero to simplify the assignment of the initial conditions for the SPH scheme.This implies that at t = 0 the dynamic pressure field coincides with the actualpressure along the free surface. Here, the pressure is sensibly different from zerobecause of the dynamic boundary condition in (6). The vorticity field (rightpanel of the same figure) is large along the free surface while it is much smallerat the bottom. This was expectable since k = π corresponds to deep waterconditions, with large gradients close to the free surface and weak dynamicsalong the bottom. The pattern of the pressure field is shifted of a quarter ofwave length with respect to the vorticity field.

Figures 5 and 6 display some snapshots of the subsequent evolution

12

Fig. 5. Vorticity field at t = 0.5 as predicted by the linear solution (left panel) andby the SPH scheme (right panel, H∗/∆x∗ = 400) (Re = 50, k = π and ε = 0.1).

Fig. 6. Vorticity field at t = 1.0 as predicted by the linear solution (left panel) andby the SPH scheme (right panel, H∗/∆x∗ = 400) (Re = 50, k = π and ε = 0.1).

(specifically, at t = 0.5 and t = 1.0) comparing the vorticity field predicted bythe linear solution with that obtained through the SPH scheme. Apart fromsome small discrepancies due to nonlinearities, the agreement between thesesolutions is good and confirms a satisfactory qualitative description of the flowdynamics. The evolution of the flow field in agreement with the free surfaceis possible thanks to the nonlinear transformation defined in (7).

3.2.1 Kinetic energy, enstrophy and damping rate of the standing wave

For a more quantitative analysis, we compare the evolution of the kineticenergy and enstrophy of the linear viscous solution with the outputs of theSPH scheme. For the sake of simplicity, we briefly recall the definition ofenstrophy (i.e. rotational energy):

Eω =∫

Ω

‖ω‖2

2dV . (20)

The analysis that follows is divided in three parts. The first one is devoted tocomparisons at different Reynolds numbers for a fixed ε and k in the range

13

of deep-water conditions. In the second part k varies to describe motions inintermediate and shallow depths (while Re and ε are constant) and, finally,to inspect the influence of nonlinearities on the wave motion, the last partcontains cases at different ε (with fixed values of Re and k).

Figures 7, 8 and 9 display the evolution of the kinetic energy (left panels)and enstrophy (right panels) for Re = 50, 500, 2500, for k = π (deep-water

conditions) and ε = 0.1. Here, E (0)K indicates the initial kinetic energy of the

standing wave. In all cases, the agreement with the SPH solver is good, provingthat the proposed linear solution well describes both the damping rate and theglobal wave evolution. These figures also display the kinetic energy attenuationobtained by using the damping rate predicted by the Biesel [2], namely βB(dotted lines in figures 7, 8 and 9). We recall that in deep-water conditionsβB coincides with the damping rate obtained by Lamb, that is, βL = 4 k2/Re. As expected, Biesel’s solution tends to overestimates the damping rate indeep water conditions when the Reynolds number is not so high (figure 7).Conversely, a good match with the proposed analytical solution is obtained forhigh Reynolds numbers (figure 9). The accuracy of Biesel’s solution decreaseswhen k increases. This is shown in figure 10 where the case considered in figure8 is repeated by doubling the wave number (k = 2π).

This part of the analysis highlights the importance of a correct estimate ofthe damping rate. The cases described in figures 8 and 10 represent wavespropagating in deep-water conditions at a moderately high Reynolds number(Re = 500). If we had used Biesel’s (or Lamb’s) solution to check the validityof the SPH scheme, we would have concluded that the SPH scheme was notaccurate enough to describe long-time evolution of viscous gravity waves. Theinaccuracy of the Biesel’s solution at moderate Reynolds number is quite animportant matter. Indeed, because of computational reasons, numerical solversoften implement a viscosity larger than the physical one and, consequently,simulate high Reynolds number flows as moderate Reynolds number flows.Then, under these conditions, use of Biesel’s solution (as well as the Lamb’sone) may lead to wrong results.

Incidentally, we highlight that for high Reynolds numbers (e.g. Re ≥1000) some care has to be given to the numerical implementation of thelinear solution because of the presence of exponentially-large numbers [e.g.cosh(mH), sinh(mH)]. To avoid errors due to ill-conditioned computations, itis sufficient to properly rearrange some of the analytical expressions of section§2.1. Further, the spatial resolution of the SPH has to be properly increasedas well as the number of particles inside the kernel domain to reduce thenumerical errors during the smoothing procedure.

In figures 11 and 12 the same analysis is repeated for k = π/3 (intermediatewater) and k = π/12 (shallow water) at Re = 500 and ε = 0.1. The match

14

t t

EK

E (0)K

Eω

Fig. 7. Kinetic energy (left panel) and Enstrophy (right panel) as predicted by thelinear solution (solid lines) and by the SPH scheme (dashed lines, H∗/∆x∗ = 400)for Re = 50, k = π and ε = 0.1. The dotted line indicates the rate of dampingpredicted by Biesel [2].

t t

EK

E (0)K

Eω

Fig. 8. Kinetic energy (left panel) and Enstrophy (right panel) as predicted by thelinear solution (solid lines) and by the SPH scheme (dashed lines, H∗/∆x∗ = 400)for Re = 500, k = π and ε = 0.1. The dotted line indicates the rate of dampingpredicted by Biesel [2].

t t

EK

E (0)K

Eω

Fig. 9. Kinetic energy (left panel) and Enstrophy (right panel) as predicted by thelinear solution (solid lines) and by the SPH scheme (dashed lines, H∗/∆x∗ = 400)for Re = 2500, k = π and ε = 0.1. The dotted line indicates the rate of dampingpredicted by Biesel [2].

15

t t

EK

E (0)K

Eω

Fig. 10. Kinetic energy (left panel) and Enstrophy (right panel) as predicted by thelinear solution (solid lines) and by the SPH scheme (dashed lines, H∗/∆x∗ = 400)for Re = 500, k = 2π and ε = 0.1. The dotted line indicates the rate of dampingpredicted by Biesel [2].

between the linear solution and the SPH scheme is still good even if somediscrepancies for long times are observed in the shallow water case. For theseregimes of motion Biesel’s solution generally provides a good estimate of thedissipation while, on the contrary, Lamb’s solution strongly underestimatesthe damping rate of the standing wave (dotted lines in figures 11 and 12).

Finally, the influence of nonlinearities on the theoretical findings is inspectedby studying a test case with ε = 0.2 and Re = 500, k = π (see figure 13).Again, the agreement between the SPH and the linear solution is fairly good,the damping rate is correctly represented and only a small discrepancy on thephase velocity is observed. This means that the results found in section §2.1may be usefully extended to weakly-nonlinear waves.

As a final example, figure 14 displays a comparison between the proposedanalytical solution (solid lines) and the boundary layer solutions by Liu &Davis [26] (dashed lines) for ε = 0.1, k = π,Re = 5000 at (t, x) = (0, L/4).Specifically, the left panel shows the free-surface boundary layer (the freesurface is at ξ = 0) while the right one displays the boundary layer at thebottom (ξ = −H = −1). The overall match is fairly good and the globalbehavior of the solutions is quite similar, even if the dynamics predicted bythe present solutions appear somehow weaker. This may be due to the factthat the present solution is obtained from linearized equations. In any case,the thickness of the boundary layers is well predicted.

16

t t

EK

E (0)K

Eω

Fig. 11. Kinetic energy (left panel) and Enstrophy (right panel) as predicted by thelinear solution (solid lines) and by the SPH scheme (dashed lines, H∗/∆x∗ = 200)for Re = 500, k = π/3 and ε = 0.1. The dotted line indicates the rate of dampingpredicted by Lamb [1].

t t

EK

E (0)K

Eω

Fig. 12. Kinetic energy (left panel) and Enstrophy (right panel) as predicted by thelinear solution (solid lines) and by the SPH scheme (dashed lines, H∗/∆x∗ = 200)for Re = 500, k = π/12 and ε = 0.1. The dotted line indicates the rate of dampingpredicted by Lamb [1].

Conclusions

A linear solution of the Navier-Stokes equations has been derived to studydissipative effects in viscous gravity waves. The most important theoreticalresult is the definition of proper perturbation expansions for the circularfrequency and the damping rate of the viscous wave. These highlight significantdifferences with the results obtained by Lamb [1] under the hypotheses of highReynolds numbers and infinite fluid depth and by Biesel [2] under more generalassumptions. In this perspective, the proposed solution may be regarded asan extension of Lamb’s and Biesel’s findings.

17

t t

EK

E (0)K

Eω

Fig. 13. Kinetic energy (left panel) and Enstrophy (right panel) as predicted by thelinear solution (solid lines) and by the SPH scheme (dashed lines, H∗/∆x∗ = 200)for Re = 500, k = π and ε = 0.2.

εu εu

ξ ξ

Fig. 14. Horizontal velocity field close to the free-surface (left) and the bottom(right) (ε = 0.1, k = π,Re = 5000 at t = 0, x = L/4). Comparison between theproposed analytical solution and the boundary layer solution by Liu & Davis [26].

The solution found in the present work is not only useful for the estimateof the wave damping during the evolution but also as reference solutionfor numerical solvers. With respect to this, a possible application has beenshown and the theoretical findings have been compared with the numericalsimulations obtained through a Smoothed Particle Hydrodynamics model.The good agreement between the analytical solution and the numerical schemestands as a cross-validation of the proposed results.

Finally, a comparison with solutions obtained from a boundary layer approachhas been provided, showing a fair overall match and a good prediction of theboundary layer thickness.

18

Acknowledgments

This work was supported by the Centre of Excellence for Ship and OceanStructures of NTNU Trondheim (Norway) within the “Violent Water-VesselInteractions and Related Structural Load”. The authors would like to thankProf. Maurizio Brocchini for his useful comments and suggestions.

A Solution of the linearized Navier-Stokes equations

Let us consider system (9) and use the following expressions:

u = U(ξ) eı θ + c.c. + O(ε) v = V (ξ) eı θ + c.c. + O(ε) (A.1)

p = P (ξ) eı θ + c.c. + O(ε) η = E eı θ + c.c. + O(ε) (A.2)

where θ = k x − σ t and k is the wave number. For the sake of the notation,the hat over the variables in the (x, ξ)-plane has been dropped. Substitutingthese expressions into (9) and neglecting all terms of order O(ε), we get:

ı k U + V = 0

− ı σ U = − ı k P +1

Re

(− k2 U + U

)

− ı σ V = − P +1

Re

(− k2 V + V

)(A.3)

Differentiating the second equation of (A.3) by y and summing it to the thirdequation multiplied by −ık, we obtain:

− ı σ Re(U − ıkV

)−[− k2

(U − ıkV

)+(...U − ıkV

) ]= 0 . (A.4)

Extracting U from the first equation of system (A.3) and substituting into(A.4), we find:

....V −

(k2 +m2

)V + k2m2 V = 0 , (A.5)

where m2 = (k2 − ı σ Re). The solution for V has the following generalstructure:

19

V (ξ) = b1 cosh [k(ξ +H)] + b2 cosh [m(ξ +H)] +

+ b3 sinh [k(ξ +H)] + b4 sinh [m(ξ +H)] . (A.6)

Then, using the first equation of system (A.3), we find:

U(ξ) = ı b1 sinh [k(ξ +H)] + ım

kb2 sinh [m(ξ +H)] +

+ ı b3 cosh [k(ξ +H)] + ım

kb4 cosh [m(ξ +H)] . (A.7)

The boundary conditions (10) and (11) become:

U(0) + ı k V (0) = 0 U(−H) = 0 V (−H) = 0 (A.8)

− ı σ E = V (0) − E + P (0) = 2 V (0)/Re . (A.9)

Substituting (A.6) and (A.7) into (A.8), we find:

(b2

b1

)= − 1

(b3

b1

)= µ

(b4

b1

)= − k

mµ , (A.10)

where

µ = −

2 cosh(kH)−

(1 +

m2

k2

)cosh(mH)

2 sinh(kH)− k

m

(1 +

m2

k2

)sinh(mH)

. (A.11)

Using the first condition in (A.9), we find the solution for the free surface:

E =ı b1

σ

[cosh(kH)− cosh(mH) + µ sinh(kH)− µ k

msinh(mH)

]. (A.12)

For consistency with the definition of ε, it has to be |E| = 1/4. Specifically, wechoose E = −ı/4 and obtain b1 by inverting equation (A.12). The equationfor P (ξ) is obtained by rearranging the second equation of system (A.3):

P (ξ) =ı

k Re

(m2 U − U

). (A.13)

Substituting the solution for U(ξ), it follows:

20

P (ξ) =ı σ

k b1 sinh [k(ξ +H)] + b3 cosh [k(ξ +H)] . (A.14)

Finally, the second condition in (A.9) gives the following linear dispersionrelation:

− ik4

Re− ik m3 tanh (kH)

Re tanh (mH)+ik3 tanh (kH)m

Re tanh (mH)+ik2m2

Re+

+4σm3k2

Re2 sinh (mH) cosh (kH)+σm4 k tanh (kH)

Re2 − σm5

Re2 tanh (mH)+

+4σ k4m

Re2 sinh (mH) cosh (kH)+σ k5 tanh (kH)

Re2 − 5σ k4m

Re2 tanh (mH)+

+6σ k3m2 tanh (kH)

Re2 − 2σm3k2

Re2 tanh (mH)= 0 (A.15)

The equation above cannot be solved explicitly and, therefore, the solution isfound in form of an asymptotic expansion for Re 1. Because of the presenceof exponential terms (that is, tanh(mH) and sinh(mH)), we need some furtherhypotheses to simplify expression (A.15). Specifically, we assume that Re 1implies <(mH) 1. The validity (and the limits) of such an assumption willbe checked “a posteriori” after the asymptotic solution for σ is achieved. Then,we can write:

tanh (mH) = 1 + O(e−<(mH)

)sinh (mH) = O

(e<(mH)

). (A.16)

These enable one to simplify the dispersion relation as follows:

− ik4

Re− ik m3 tanh (kH)

Re+ik3 tanh (kH)m

Re+ik2m2

Re+

+σm4 k tanh (kH)

Re2 − σm5

Re2 +σ k5 tanh (kH)

Re2 − 5σ k4m

Re2 +

+6σ k3m2 tanh (kH)

Re2 − 2σm3k2

Re2 = O(e−<(mH)

). (A.17)

Since <(mH) 1, the right-hand side of (A.17) is infinitesimally smaller thanthe left-hand side and, therefore, is taken as zero. Conversely, the left-handside is solved by means of a perturbation expansion in terms of

√Re. We find:

21

<(σ) =√k tanh(kH) +

√2 k5/4 (tanh(kH)2 − 1)

4 tanh(kH)3/4Re−1/2 +

−√

2 k11/4 (7 tanh(kH)6+147 tanh(kH)4−11 tanh(kH)2−15)

128 tanh(kH)13/4Re−3/2 +

+O(Re−2) , (A.18)

and

=(σ) =

√2 k5/4 (tanh(kH)2 − 1)

4 tanh(kH)3/4Re−1/2 +

+k2 (tanh(kH)4 − 8 tanh(kH)2 − 1)

4 tanh(kH)2Re−1 +

+

√2 k11/4 (7 tanh(kH)6+147 tanh(kH)4−11 tanh(kH)2−15)

128 tanh(kH)13/4Re−3/2 +

+O(Re−2) , (A.19)

The validity of the expansions above has to be proved for both large and smallvalues of k. In the former case, they can be rearranged as follows:

<(σ)√k

= A0(kH) + A1(kH)µ1/2+ A3(kH)µ3/2+ O(µ2),

=(σ)√k

= B1(kH)µ1/2+ B2(kH)µ + B3(kH)µ3/2+ O(µ2),

where µ = (kH)3/2/Re while B1(kH) = A1(kH), B3(kH) = −A3(kH) and allthe coefficients An(kH), Bn(kH) are of order O(1) for large values of k (H = 1because of the scaling in use). This implies that the expansions (A.18) and(A.19) are well-posed for µ 1, that is (kH)3/2 Re. Specifically, the latterbound ensures that each term inside (A.18) and (A.19) is smaller than thepreceding term and larger than the subsequent one for large values of k.

For small values of k, the same procedure described above is applied to <(σ)/kand =(σ)/k, implying kH Re−1. Summarizing, the expansions (A.18) and(A.19) hold true for:

Re−1 kH Re2/3 Re 1 . (A.20)

It is simple to prove that the bounds above ensure <(mH) 1 and, therefore,confirm the validity of the simplifications done to obtain equation (A.17).

22

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