TURBULENT SHEAR FLOW OVER STEEP STOKES WAVES

Advances and Applications in Fluid Mechanics © 2015 Pushpa Publishing House, Allahabad, India Published Online: September 2015 http://dx.doi.org/10.17654/AAFMOct2015_245_276 Volume 18, Number 2, 2015, Pages 245-276 ISSN: 0973-4686

Received: April 24, 2015; Accepted: June 8, 2015 2010 Mathematics Subject Classification: 35K06, 65M99. Keywords and phrases: air-sea interactions, turbulence model, third-order stokes waves.

Communicated by K. K. Azad

TURBULENT SHEAR FLOW OVER STEEP STOKES WAVES

S. G. Sajjadi1,2 and H. Khanal2 1Trinity College University of Cambridge U. K.

2Department of Mathematics Embry-Riddle Aeronautical University Florida, U. S. A.

Abstract

The problem of water wave generation and attenuation by wind is proposed by numerically calculating the turbulent air flow over a third-order Stokes wave train. Most existing works [10, 28] have used a turbulent flow closure model of one equation type and assumed the flow to be aerodynamically rough to avoid the difficulties arising from the viscous sublayer. However, air flows over water waves are known to be aerodynamically transitional [26]. Thus, in this investigation we have adopted Sajjadi’s [24] roughness parameter to circumvent this difficulty. The turbulence model adopted for this investigation is based on the two equation closer scheme proposed by Saffman and Wilcox [21], and is used to simulate turbulent flow within and outside the viscous sublayer over steep non-linear surface waves. Thus, the present contribution is an extension to the earlier investigation of Al- Zanaidi and Hui [1]. The linearized turbulent flow equations for small (yet finite) wave slope are solved numerically up, and including, the

S. G. Sajjadi and H. Khanal 246

third-order in wave steepness, taking into account the dynamical and kinematical boundary conditions at wave free surface. The resulting partial differential equations are first decomposed into a system of first order non-linear ordinary differential equations and solved numerically using the multiple shooting method [2]. The main aim of the present investigation is to calculate the vertical structure of wind field, the perturbation pressure as well as the fractional rate of energy input from wind to non-linear surface waves and hence calculate the energy transfer as well as the growth rate to a third-order Stokes wave due to turbulent shear flow flowing over it. The results show good agreement with computations of Conte and Miles [8] and also support the recent theoretical investigation of Sajjadi [24] for very high-Reynolds number (almost inviscid flow). The formation of Kelvin cat’s-eyes over a third-order Stokes waves is also calculated and their variations with the wave age show remarkable similarity with the work of Sullivan et al. [27].

1. Introduction

A semi-empirical study of turbulent flow over a steep Stokes waves is carried out, with the objective of evaluating normal and tangential stresses on the surface of the wave and thence estimating the growth of non-linear water waves due to a turbulent wind which is blowing over it. Let us first provide a basic review of the problem which has led to much research over the last sixty years, but not much contribution has been made to non-linear or growing waves (in which the complex celerity of the wave is not zero).

In 1956, Ursell [29] provided a critical summary of the theories of generation of waves by wind, and stated that waves grow by mechanism which is yet to be understood. It was due to this review that gave the necessary impetus for the following sixty years of intensive research by various researchers.

Invigorated by Ursell’s review, the following year, Miles [16] and Phillips [11] both formulated a problem for the theory of water wave generation by wind. Miles theory (commonly known as the critical layer mechanism) with inclusion of turbulence in wind and extension to steep

Turbulent Shear Flow Over Steep Stokes Waves 247

water waves is essentially what we will be concerned here. Thus, we consider the primary transfer of energy from turbulent wind to water, and ignore the secondary transfers of energy between water waves models.

Miles original formulation is purely two-dimensional and shows that an exponential buildup of water waves of velocity rc would occur if the mean

airflow velocity U in the direction of wave propagation vanishes with height y above the water surface in such a way that ( ) ,0<yU for more detailed

physical explanation, see Lighthill [14]. Also, in Miles method turbulence in the airflow was essentially neglected, other than in creating a ‘logarithmic’ wind mean velocity profile for which the condition is strongly satisfied. A substantial energy transfer to water then occurs provided that the ‘critical’ layer at ,cyy = where ( ) rc cyU = is small enough compared to the

wavelength, λ, of the water. This condition must be satisfied in order that the factor ( ) ( ),2exp4exp cc kyy −=λπ− where k is the wave number, from

being too small. Miles’ inviscid theory was later extended to include the viscous forces by Benjamin [6] who adopted a curvilinear orthogonal coordinates.

Moreover, Miles-Benjamin theory both assumed that wave-induced velocities and pressure are small, and this justified linearization of the equations of motion. In both theories, it was assumed the water is inviscid incompressible and irrotational with the slope of the displaced surface, ak, to be small. Note that, only the component of the aerodynamic force in phase with the wave slope is of primary importance for wave generation. The viscous drag forces in the air may be neglected compared to the normal pressure in their on the surface wave.

Some ten years later, Phillips [12] made an attempt to include wave-induced turbulence in Miles instability theory. He did this by inclusion in the momentum flux equation, an integral over the entire layer of air due to contribution made by perturbation eddy viscosity. This integral contained an arbitrary multiplier A whose magnitude was estimated by experimental results from flow over fixed wavy surfaces. Phillips estimated the uncertainty


in A is about ± 50%, but he argued that even with this degree of uncertainty the results do provide a framework in which experimental measurements may be correctly interpreted.

However, Miles [18] remarked that experimental results indicate that a theoretical model based on quasi-laminar flow may only be adequate in the laboratory but not on an oceanographic scale. It is now well known that the inviscid, quasi-laminar model underestimates the energy transfer from wind to water waves over at least a significant portion of the spectrum for an open sea. He further remarked that this suggests the significance of wave-induced perturbations in the turbulent Reynolds stresses will increase the momentum and energy transfer from wind to waves.

Despite the large amount of research conducted over the last sixty years, still the mechanism by which ocean waves are generated by wind is not fully understood. For example, when energy is transferred by wind to the ocean, it is still not known what percentage of energy and momentum go into waves and what percentage go into currents.

In recent years a number of mechanisms have been proposed for how an airflow over waves, with small steepness ,1ak or over a horizontal body

of liquid that produces waves on its surface [23]. Most of these mechanisms have been linear and therefore can be applied to any spectrum of waves. However, the mechanisms and models based on them are regularly applied when the surface disturbances significantly affect the gas and liquid flows, so that the mechanisms are non-linear, and the waves are growing which, in general, are not monochromatic. Typically the waves move in groups, which affect how the wind flows over the waves, how the waves break and thence how droplets form. This weakly non-linear interaction of mechanisms significantly influence the average momentum, heat and mass transfer associated with waves.

Very small unsteady waves are generated by turbulence and/or growing Tollmien-Schlichting instabilities in the sheared air flow over the surface and Kelvin-Helmholtz coupled instability of the airflow over the liquid, discovered originally by Miles [17] and their significance was further


explored by Sajjadi [22] and recently revisited by Sajjadi [24]. When steady waves are generated artificially in an airflow, for example in a wind-wave tank, the linear mechanisms for the growth of the waves are the pressure drag caused by asymmetric slowing of airflow over the downwind slopes of the waves and turbulence stresses caused by the disturbed flow, and wind-induced variations of surface roughness disturbed surface [3, 4]. Both mechanisms are affected by the relative speed of the wave, ,rc to the friction

velocity, ,∗U of the airflow, and the disturbed flow changes at the critical

height, ,cy where the wave speed, ,rc is equal to the wind speed, ( ).cyU

Consider now those waves, with wavelength ,2 kπ=λ that begin to

grow at the rate .ikc If this is comparable with the frequency of wave

passing, i.e., ,kU∗ then the critical layer is above the inner shear layer near

the surface. Also the dynamics across the critical layer are determined by inertial forces as the flow accelerates and decelerates over the wave. But only if the wave is growing (or decaying), i.e., ,0≠ic is there a net force on the

wave caused by critical layer dynamics [5]. The triple deck analysis [25] agrees with that of Miles [16] different method of analysis for a growing wave. But, they do not agree with his, and many subsequent authors (e.g. [14]), conclusion is that there is a net inviscid force on monochromatic non-growing waves, (i.e., when .)0=ic

Wave shapes also affect the wave growth [23, 24]. Whether (as in the photographs in Jeffreys [9]) the wave groups are capillary waves on a Cambridge duck pond or breaking rollers in the Atlantic ocean, the wave shapes as well as their height vary in a group. Since their slopes tend to increase downwind, this is likely to amplify the growth rate. By considering the dynamics of steep non-linear waves, it becomes possible to estimate rationally how air flow affects the non-linear interactions between waves, and compare how this relates to the wave-wave hydrodynamic interactions, that are assumed to dominate the distribution of ocean waves. Thus, variations of wave shapes could also affect the net wave growth, while violent erratic winds can prevent the formation of wave groups, so that wave


growth may be reduced (but spray from waves is increased) as is observed near the centre of hurricanes [24].

At higher wave speeds, another mechanism is also significant, namely the displacement of the critical layer outside the surface shear layer ( )..,i.e ∗> Ucr This acts to reduce the sheltering mechanism, by contrast

with Belcher and Hunt’s [3] analysis ( )∗< Ucrwhen which showed how

the critical layer within the shear layer increases the sheltering mechanism [7]. Thus, the decrease of the growth rate as ∗Ucr increases is compensated

by the increase in growth rate as waves form into groups at higher wind speeds (which also needs to be modelled). The decrease in the local sheltering mechanism as cy increases over the downwind part of a wave

group further affects the dynamical effect of the critical layer. We note that the existence of a critical layer over a monochromatic with a significant role on the boundary layer dynamics still does not mean that the Miles inviscid mechanism is operative (cf. [27]).

Earlier theories [16], in which the interaction of the atmospheric turbulence and wave fields was neglected, have provided some explanation of the physics of the air flow over water waves. Recent works [3, 22, 24] have provided a more detailed physical mechanism for this processes, in particular, they have shown that the surface pressure is extremely sensitive to the turbulence closure schemes adopted. However, the equations of motion which govern the turbulent air flow over water waves are not amenable to analytical solutions and therefore must be solved numerically.

After recent discovery by Sajjadi et al. [25] (SHD therein), we construct a numerical model for turbulent air flow blowing over steep water waves. The water wave is represented by a reasonably steep (third-order non-growing) Stokes wave in a frame of reference moving with the wave, and turbulence model adopted for the air flow over such a wave is taken from the two equation model originally proposed by Saffman and Wilcox [21].

Since the boundary layer above water waves is not large compared to the wave amplitude, the boundary conditions cannot be prescribed on the mean


undisturbed surface. Thus, orthogonal curvilinear coordinates are adopted and the equations of motion and the boundary conditions are transformed into this coordinate system. The resulting equations and the boundary conditions, in the new coordinates, are then expressed in perturbation expansions with respect to the wave slope, up to and including the third-order in wave steepness.

The perturbation equations consist of a set of coupled ordinary differential equations, with respect to the vertical coordinate, which are solved numerically using the multiple shooting method as described by Ascher et al. [2].

2. Formulation of the Problem

We consider motion of an incompressible air flow over water waves, and refer the equations to Cartesian coordinates ( )zyx ,, in which the y-axis

is measured vertically upwards from the undisturbed water surface.

If ( )wvuui~,~,~~ = are the local components of flow velocity in Eulerian

frame of reference at a point in Cartesian coordinates ( ),,, zyx then the

Navier-Stokes equations may be cast in the form

,~~~~~

~,0

~

⎭⎬⎫

⎩⎨⎧

⎟⎠

⎞⎜⎝

⎛∂∂

+∂∂

μ∂∂+

∂∂−=⎟

⎠

⎞⎜⎝

⎛∂∂

+∂∂

ρ=∂∂

i

j

ji

jiji

jj

ji

xu

xu

xxp

xuut

uxu (2.1)

where p~,ρ and μ are, respectively, the air density, the pressure and the

dynamic viscosity. We will consider a fully developed turbulent flow over a third-order Stokes wave

( )34232 3cos832cos2

1cos kakxkakxkakxays O+++= (2.2)

in a frame of reference moving with the wave with a speed ( )0, == ir ccc

in the positive x-direction, where λπ= 2k is the wave number, λ the

wavelength and a the amplitude.


Decomposing the instantaneous velocity fields and the pressure into mean and fluctuating components according to iii uUu +=~ and ,~ pPp +=

then upon substitution into (2.1), followed by time averaging we obtain the Reynolds-averaged Navier-Stokes equations

.,0⎭⎬⎫

⎩⎨⎧

ρ−⎟⎠

⎞⎜⎝

⎛∂∂

+∂∂

μ∂∂+

∂∂−=

∂∂

ρ=∂∂

jii

j

ji

jiji

jii uux

UxU

xxP

xUUx

U (2.3)

In equation (2.3), jiuuρ are the unknown Reynolds stresses which must

be provided through a closure scheme. Here, we follow the closure model suggested by Saffman and Wilcox [21] and express the Reynolds stresses as

,21,2

1,322 ji

i

j

ji

ijijijji uuExU

xUSESuu =⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

=δρ−ρε=ρ− (2.4)

where ε is a scalar eddy viscosity, ijS is the mean rate of strain tensor, and E

is the specific turbulent kinetic energy.

Figure 1. Schematic diagram showing a shear flow over various surface waves of different steepness. The figure also demonstrates the relationship between the Cartesian and the Curvilinear coordinates adopted.

Hence, the two-dimensional governing equations for the turbulent flow over Stokes wave may be cast in their final form as

{ ( ) } ,32

21,2,0 EPSxxx

UUxU

ijjij

ij

ii +=ε+ν

∂∂+

∂∂−=

∂∂

=∂∂

PP (2.5a)


where ρμ=ν represents the kinematic viscosity. The turbulent closure

(kinetic energy E and pseudo-vorticity ω) equations are

( ) ( ) ,2 321⎭⎬⎫

⎩⎨⎧

∂∂ε+ν

∂∂+ω−=

∂∂

jjijij

jj x

EbxbSSbExEU (2.5b)

( ) ,2

3542

2

⎭⎬⎫

⎩⎨⎧

∂ω∂ε+ν

∂∂+⎟⎟

⎠

⎞⎜⎜⎝

⎛ω−

∂∂

∂∂

ω=∂ω∂

jjji

ji

jj xbxbx

UxUbxU (2.5c)

where .ω=ε E

3. Equations in Curvilinear Coordinates

In the problem posed here, the boundary layer thickness is small compared with the wave amplitude. Thus, it is not satisfactory to apply the surface boundary conditions at the mean water level ( ).0=y Hence to

circumvent this difficulty a reference frame, moving with the wave, is chosen and use is made of a system of orthogonal curvilinear coordinates in which the wave surface is a coordinate line.

Therefore, we define

{ ( )[ ]} { ( )[ ]}η+ζη+ζ −η=−ζ= iikiik aeReyiaeRex ,

which to third-order in the wave steepness, ak, the coordinate 0=η

corresponds to the free surface ( )xhys 0≡ given by (2.2). The Jacobian of

the transformation may be expressed as

( )( ) { ( ) ( ) } 13322cos21,

, −η−η−η− ++ζ+=∂

ηζ∂= kkk eakeakkakeyxJ (3.1)

correct to the third-order.

Non-dimensionalizing of the governing equations is such that the velocities are scaled with respect to the friction velocity ∗U and the length is

scaled with respect to .∗ν U Thus, the equations of motion can be written in

( )ηζ, coordinates system as follows:


Continuity equation:

( ) ( ) ,021

21

=η∂∂+

ζ∂∂ −−

vJuJ (3.2)

where u and v are the mean velocities in the ζ and η directions, respectively.

u-momentum equation:

( ) ( ) ( )ζ∂∂+

ζ∂∂++

η∂∂+

ζ∂∂ −−−− pJJvuJuvJuuJ 1

22211

2

( ) ( )⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎥⎦

⎤

⎢⎢⎣

⎡

η∂∂+

ζ∂∂ε

ζ∂∂= − vJuJJ 2

121

1ˆ

( ) ( ) .ˆ 21

21

1

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎥⎦

⎤

⎢⎢⎣

⎡

ζ∂∂−

η∂∂ε

η∂∂− − vJuJJ (3.3)

Here ρμ=ν is the kinematic viscosity, ω=ε E is the turbulent

viscosity and .ˆ ε+ν=ε

v-momentum equation:

( ) ( ) ( )η∂∂+

ζ∂∂++

η∂∂+

ζ∂∂ −−−− pJJvuJvvJuvJ 1

22211

2

( ) ( )⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎥⎦

⎤

⎢⎢⎣

⎡

η∂∂+

ζ∂∂ε

ζ∂∂= − vJuJJ 2

121

1ˆ

( ) ( ) .ˆ 21

21

1

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎥⎦

⎤

⎢⎢⎣

⎡

η∂∂−

ζ∂∂ε

η∂∂+ − vJuJJ (3.4)

Turbulent kinetic energy equation:

{ }ω−−η∂∂+

ζ∂∂ −−−

2112

121

2 bSSbEJEvJEuJ ijij

( ) ( ) .0ˆˆ 33 =⎭⎬⎫

⎩⎨⎧

η∂∂ε+ν

η∂∂−

⎭⎬⎫

⎩⎨⎧

ζ∂∂ε+ν

ζ∂∂− EbEb (3.5)


Pseudo-vorticity equation:

{ }ω−ω−ζ∂ω∂+

η∂ω∂

ω−−−

5421

2212

21

bSbJvJuJ

( ) ( ) .0112

16

21

6 =⎭⎬⎫

⎩⎨⎧

η∂ω∂ω+

η∂∂−

⎭⎬⎫

⎩⎨⎧

η∂ω∂ω+

ζ∂∂− −− EbEb (3.6)

The stress terms in (3.5) and (3.6) are given by

( ) ( )2

21

21

222⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

η∂∂−

ζ∂∂=+= vJuJSS ntijij RR

( ) ( ) ,2

21

21

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

ζ∂∂+

η∂∂+ vJuJ (3.7a)

where

( ) ( ) ,21

21

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

η∂∂−

ζ∂∂ε=−=ε vJuJuvtR (3.7b)

( ) ( )⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

η∂∂−

ζ∂∂ε=−=+−=ε

−−vJuJEvEun 2

121

2232

32

R (3.7c)

are the tangential ( )tRε and the normal ( )nRε components of Reynolds

stresses in the orthogonal curvilinear coordinates, and

( ) ( ) .21

221

21

2

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

η∂∂−

ζ∂∂+=

−−ω vJuJJSSS ijij (3.7d)

The constants 1b to 6b in the transport equations (3.5) and (3.6) are

taken from Saffman and Wilcox [21] and their values are

,4,,3.0,21 2

1

6

2

514

212163

⎭⎬⎫

⎩⎨⎧

κ−=====bb

bbbbbbbbb and ,23

525 << b

b

where 41.0=κ is the von Kármán’s constant.


Furthermore, the tangential and normal stresses, due to the presence of turbulence, are given, respectively, by

( ) ( ) ( ) ,1 21

21

⎥⎥⎦

⎤

⎢⎢⎣

⎡

ζ∂∂+

η∂∂

⎟⎠⎞⎜

⎝⎛

ω+=τ ζη vJuJE

( ) ( ) ( ) .1 21

21

⎥⎥⎦

⎤

⎢⎢⎣

⎡

η∂∂−

ζ∂∂

⎟⎠⎞⎜

⎝⎛

ω+−−=σ ηη vJuJEP

Boundary conditions:

At ,∞η=η which we typically take to be one wave length above the

water surface, are

( )ηκ

=ω===−+η+κ

=11

1,1,0,1ln1bbEPvcBu (3.8)

whilst on the water surface ( )0=η the boundary conditions are given by

( ) ( ) .0,0,0 1102

1bJUQEvcJu −∗−

⎟⎠⎞⎜

⎝⎛

νη

=ω==−= (3.9)

The constant B in (3.9) depends on the nature of the surface. 0η is the

roughness length and ∗U is the friction velocity. Here, Q is the universal

function defined by

( )

( )⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎥⎦⎤

⎢⎣⎡ +⎟

⎠⎞

⎜⎝⎛

νη

⎥⎦⎤

⎢⎣⎡ +⎟

⎠⎞

⎜⎝⎛

νη

=⎟⎠⎞

⎜⎝⎛

νη

∗

∗

∗

.Rough,68.0ln

44.1

,altransitionandSmooth,38.2ln

26.6

20

20

0

U

UUQ (3.10)

3.1. Linearized perturbation equations

We expand u, v, P, E, ω and J in order of wave steepness ak as


( ) ( ) ( ) ( ) ζζ +η+η=ηζ iKiK eUakeakUUu 22

210,

( ) ( ) ,433

3 akeUak iK O++ ζ

( ) ( ) ( ) ( ) ( ) ,, 433

322

21 akeVakeVakeakVv iKiKiK O+++η=ηζ ζζζ

( ) ( ) ( ) ( ) ( ) ,, 433

322

21 akePakePakeakPP iKiKiK O+++η=ηζ ζζζ

( ) ( ) ( ) ( ) ζζ +η+η=ηζ iKiK eEakeakEEE 22

210,

( ) ( ) ,433

3 akeEak iK O++ ζ

( ) ( ) ( ) ( ) ζζ Ω+ηΩ+ηΩ=ηζω iKiK eakeak 22

210,

( ) ( ) ,433

3 akeak iK O+Ω+ ζ

( ) ( ) ( ) ( ) ( ) .1, 433

322

21 akeJakeJakeakJJ iKiKiK O+++η+=ηζ ζζζ

Here ( ) ( ) η−η−∗ =η=ην= KK eJeJUkK 2

21 2,2, and ( ) .2 33

η−=η KeJ Substituting these expansions into the partial differential equations (3.2)-(3.6), and the boundary conditions and equating the coefficients of ( ),1O

( ),akO ( )2akO terms, and ( )3akO we obtain a system of 54 first order non-linear ordinary differential equations, see the Appendix.

3.2. Numerical methods

We transform the independent variable η to ξ using the substitution =ξ

( ).1ln η+ We then separate the real and imaginary parts and put the system

in a normal form. The resulting boundary value problem, that consists of 54 first order non-linear ordinary differential equations, are then solved numerically using the multiple shooting method as described by Ascher et al. [2].

4. Results

4.1. Energy transfer from wind to waves

The wave-perturbation pressure p is proportional to the wave steepness ak and may be expressed as


( ) ( ) ( )⎢⎣

⎡β+α

σ+β+α

σρ= ζζ iKiK

w eiakeiakcp 222

2

211

12

0

( ) ( ) ,333

3

3⎥⎦

⎤β+α

σ+ ζiKeiak (4.1)

where [ ( ) ]21 akkgc += is the wave phase velocity, ( )3,2,1=σ nn are

the frequencies of the first, second and third harmonics, respectively, and wρ

is the water density. The total energy transfer parameter from the wind to the wave is given by

( ) 32

21 β+β+β=β akak (4.2)

which is related to fractional growth of wave per radian .aς The energy

transfer parameter, β, is related to the total energy of the water wave, ,E by

( ) ,2

11 ⎟⎠⎞⎜

⎝⎛β=

∂∂−

cUstkc EE (4.3)

where awas ρρρ= ,1 being the air density, ∗U the wind friction

velocity, and .1 κ= ∗UU The present turbulent closure model adopted here

automatically produces a logarithmic mean velocity for the wind, which agrees exactly with the following analytical profile:

00

1 ,ln ηη⎟⎠⎞⎜

⎝⎛ηη= UU (4.4)

with 0η is given by Sajjadi [24].

We generalize Miles [16] formulation for rate of energy transfer from wind to a monochromatic surface wave ( ),cos0 kxah =

,20

21

2

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛⎟⎠⎞⎜

⎝⎛

′′′

π−=βx

cc

c

hUUkU V


where V is the vertical component of the wave-induced velocity, to a third-order Stokes wave. Thus, following the generalization proposed by Sajjadi [24], we use the following formulation:

( )

22

321 ⎟

⎠⎞

⎜⎝⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛

′

′′π−=β ∫

∞

η−

ηc

dzweUUUk w

c

cnn (4.5)

for the energy transfer rate to each harmonic of the surface wave. In (4.5),

( ) ( )cn

ncn zknkkUUw =η=η= ,,1 and the suffix c indicates evaluation at

the critical height at which .cU = Hence, after calculating ( ) ( )η=ηζ Uu f ,

at a fixed value of ,ζ namely ,fζ=ζ we substitute the result in (4.5) and

evaluate the integral numerically using Gauss-Laguerre quadrature. In (4.5), the critical layer height, ,cη is given by

,12

1 Ucc ec

U⎟⎠⎞⎜

⎝⎛Ω=η

where 210 Ugη=Ω is the dimensionless Charnock’s constant, and g is the

acceleration due to gravity.

The energy growth rate per radian due to wind may thus be expressed as

.2

1 β⎟⎠⎞⎜

⎝⎛=

ςc

Usa (4.6)

Since the damping ratio associated with viscous action in the water is given by ([13], Section 348)

ckww ν−=ς 4 (4.7)

hence the total growth rate per radian may be cast in the following form:

( ) ,142

2221

ckag

cUs w +ν

−⎟⎠⎞⎜

⎝⎛β=ς (4.8)

where wν is the kinematic viscosity of the water.


In comparing our results with the numerical simulations of Conte and Miles [8], we have considered the transitional flow, for which νη∗ 0U

25.0= for three values of 23 10,103 −−×=Ω and .102 2−× Figure 2

shows the plot of β against 1Uc for .103 3−×=Ω As can be seen from the

figure the result of the present computation agrees well with Conte and Miles in the narrow range ,96 1 ≤≤ Uc where Miles [16] formulation is expected

to become important Phillips [12, Section 4.3]. For 61 <Uc however, the

present calculated result is smaller compared with Conte and Miles’ values. Similarly, for 91 >Uc the result is larger than theirs. The reason for this

may be attributed to the fact that Miles’ critical layer is moving faster (for smaller values) and closer (for the larger values) to the wave surface as 1Uc

decreases (for smaller values) and increases (for larger values) from the narrow range. Also, Conte and Miles’ model is an inviscid laminar model and hence does not account for the effect of turbulence, particularly as 1U

increases and thus 1Uc decreases. In Figure 3, we consider the transitional

air flow for which .10 2−=Ω As can be seen from this figure the agreement between the present computations and Conte and Miles’ is in excellent agreement for 31 ≤Uc and also in a narrow region .76 1 ≤≤ Uc However,

for a fully rough flow, ,102 2−×=Ω where the effect of turbulence becomes

more important, we observe the result of the present computation is higher, over the entire range of ,1Uc compared with that of Conte and Miles, as

shown in Figure 4. This is, of course, to be expected as Miles’ model neglects the interaction of turbulent air flow with surface waves.


Figure 2. Variation of β with 1Uc for .103 3−×=Ω ,−◊− Computation of

Conte and Miles [8]; •, Present computation.

Figure 3. Variation of β with 1Uc for .10 2−=Ω ,−◊− Computation of


Figure 4. Variation of β with 1Uc for .102 2−×=Ω ,−◊− Computation of



Figure 5. Variation of sς with cU1 for 23 10,103 −− =Ω×=Ω and

.102 2−×=Ω Symbols, computation of Conte and Miles [8], solid lines, present computation.

In Figure 5, the fractional growth of the wave per unit radian, ,sς is

plotted against cU1 for all values of Charnock’s constant considered. Also,

for comparison, the results of computation by Conte and Miles are also plotted. As can be seen from this figure there is a very good agreement between the present computations and that of Conte and Miles [8]. We remark that the results indicate that there is very little sensitivity to the choice of Charnock’s constant, Ω, used, particularly in the region where turbulence is dominant. This is because in the presence of turbulence, the critical layer moves much closer to the surface and hence there will be insignificant contribution to the wave growth, provided that .0=ic

Finally, Figure 6 shows comparisons of the theoretical prediction of the growth rate due to energy transfer from wind to the surface wave, β, which is normalized by the wave frequency, ,1 Tf = with the oceanic data collected

by Plant [20]. In this figure, 0ηλ has the value of 510 which is

representative of ocean waves. The present calculations, using (4.5), shows good agreement with the experimental data through a wide range of conditions. The solid line is the prediction made by Miles [19], drawn here for comparison.


Figure 6. The growth rate of the energy transfer rate to the wave β due to the effect of the asymmetric pressure, made non-dimensional with the wave frequency f, as a function of .cU∗ The theoretical curves, ◊ are the present

calculations, and the solid line from Miles [19] prediction. The data collected by Plant [20] from oceanic experiments.

4.2. Effect of the critical layer

The important dynamics of the critical layer for flows over water waves has long been recognized. We remark that the concept of the critical layer has been the central ingredient of Miles [16] inviscid theory. The physical interpretation of this phenomenon and how it affects the wave growth was given by Lighthill [14]. Basically, the consequence of the critical layer is that a closed streamlines, commonly known as the ‘cat’s eye’, will exist if the mean flow ( )ηU has periodic variation along the wave. Lighthill [14]

expressed these streamlines in the form

( ) ( ) ( ) ( )( ) .coshsin

21 02

cacc U

kkpUηρ

ηζ+η−ηη=ψ


It is interesting to note that Townsend [28], using a linearized theory, stated that in the presence of turbulence the critical layer is not as important of an equilibrium layer. This, of course, has been disputed by later work of Miles [19] and others. Furthermore, in a numerical study carried out by Mastenbroek et al. [15] they commented that no computational studies, to date, even using second-moment closure scheme, over slow growing waves show any dynamical effects of a critical layer. Mastenbroek’s comment is not in fact true.∗

In fact apart from a second-moment closure model, certain other turbulence models, such as the one adopted in the present study, have shown the importance of the critical layer not only for flows over water waves but also other flows such as gravity wave turbulence interaction. In Figures 7(a)-(d) our numerical simulations show the appearance of cat’s-eye over a wide range the wave age .∗Uc As can be seen from these figures, the cat’s-eye

pattern is centered about the critical height, and no separation or re-attachment occur at the surface of the waves. However, as will be shown in subsequent papers, turbulent fluctuations will lead to small transient separated regions, particularly for very steep breaking waves.

Our simulations are carried out over steep waves of steepness .33.0≈ak From Figures 7(a)-(d), we see that when ∗Uc is small the mean cat’s-eye

surrounds the lower boundary whose center is located just upwind of the wave trough which extends over the entire wavelength (cf. [27]). However, just above the center of the cat’s eye pattern, the flow is displaced vertically and does not attach to the profile of the surface wave. This clearly shows that the critical layer dynamically alters the mean flow patterns above the waves. As the wave speed increases the cat’s-eye pattern moves closer to the crest and extends vertically higher from the surface of the wave. Further increase in ∗Uc shows that the cat’s-eye lifts well above the surface. These findings

∗A very detailed study beyond the work of SHD will soon be published in this journal which demonstrates an intriguing result which will demonstrate the comments made by Mastenbroek are in fact false.


are in complete agreement with the direct numerical simulation of Sullivan et al. [27].

In Figure 7(e), we plot the pressure contour over a third-order Stokes wave for 8.7=∗Uc and .33.0≈ak We can see that the pressure field has a

great impact on the velocity field. The pressure contours show clearly a tilt in the downstream direction below the critical height cη and then bend back

above .cη It can also be observed the minimum (maximum) pressure is

almost centered over the wave crest (trough). This, of course, yields a small negative form drag, which will affect the energy transfer parameter β.

Figure 7. The streamlines pattern, (a-d), and pressure contours over moving waves with .33.0≈ak (a) ,0.3=∗Uc (b) ,9.3=∗Uc (c) ,8.7=∗Uc (d)

,5.11=∗Uc (e) Pressure contours over the wave with the same wave steepness for .8.7=∗Uc

5. Conclusions

A computational of turbulent flow over a third-order Stokes wave of steepness 33.0≈ak is carried out which is an extension to the earlier work of Al-Zanaidi and Hui [1]. The turbulence model adopted is based on that of Saffman and Wilcox [21] expressed in a curvilinear orthogonal coordinates.

A perturbation scheme is used up and including ( )3akO which led (after


separating the equations in their real and imaginary parts) to 54 coupled non-linear ordinary differential equations. These equations were solved using the numerical scheme described by Ascher et al. [2].

The computation of airflow was performed in the limit of ∞→Re (where Re is the Reynolds number), thereby the flow field could be assured to be nearly inviscid.

In this limit, energy transfer parameters as well as growth rate were calculated which broadly agree with Conte and Miles [8] numerical integration of the Rayleigh equation for various values of Charnocks constant Ω in the range ( )8~61 1 ≤≤ Uc depending on the value of Ω. For

aerodynamically smooth flow ( )3103 −×=Ω we see marked difference

between the present computation and Conte and Miles for .5<∗Uc This is

not surprising since this regime (i.e., slow moving waves) turbulence plays an important role. The agreement between the two models become very

much closer for a fully rough flow ( ).102 2−×=Ω

We remark even at very high Re, the flow in reality, is not inviscid and a turbulence model captures the main features of turbulent flows. We further remark, as was pointed out by Mastenbroek et al. [15], and we agree too, that the ε−k turbulence model yields to wrong results of flow fields for a turbulent flow over water waves. However, the present model based on the

ω−k developed by Saffman and Wilcox, captures the salient features of turbulent flow over water waves compared with a differential second-moment turbulence model. This is due to the fact that in the ω−k model the turbulent stresses are modelled in a manner which is similar to an algebraic stress turbulence model.

Finally, our simulations, for various values of ,∗Uc show the existence

of closed streamlines (the cat’s-eye pattern) which are centered about the critical layer. It is found that the cat’s-eye are dynamically important and alter the flow field above the water waves. We have also demonstrated that these cat’s-eye patterns have remarkable similarity with those calculated by Sullivan et al. [27]. Our results show that for small values of ∗Uc the center


of the cat’s-eye are just slightly upwind of the trough and close to the surface wave. However, as ∗Uc increases, the cat’s-eye patterns become thicker

and moves upstream of the wave trough. More strikingly, at ,5.11=∗Uc

the center of the cat’s-eye lifts up well above the surface of the wave and begins to influence the flow field. We remark that since the fluid moves more slowly within the cat’s-eye then the mean streamlines are deflected away from the moving surface, which is in agreement with the conclusion drawn by Sullivan et al. [27].

References

[1] M. A. Al-Zanaidi and W. H. Hui, Turbulent airflow over water waves - a numerical study, J. Fluid Mech. 148 (1984), 225-246.

[2] U. Ascher, R. Matheij and R. Russel, Numerical Solution of Boundary Value Problems for Ordinary Differential Equations, SIAM, Philadelphia, PA, 1995.

[3] S. E. Belcher and J. C. R. Hunt, Turbulent shear flow over slowly moving waves, J. Fluid Mech. 251 (1993), 109-148.

[4] S. E. Belcher and J. C. R. Hunt, Turbulent flow over hills and waves, Annu. Rev. Fluid Mech. 30 (1998), 507-538.

[5] S. E. Belcher, J. C. R. Hunt and Cohen, Turbulent flow over growing waves, In Wind over waves: Prospective and prospects, S. G. Sajjadi, N. H. Thomas and J. C. R. Hunt, eds., Oxford University Press, 1999.

[6] T. B. Benjamin, Shearing flow over a wavy boundary, J. Fluid Mech. 6 (1959), 161-205.

[7] Cohen and Belcher, Turbulent shear flow over fast-moving waves, J. Fluid Mech. 386 (1999), 345-371.

[8] S. Conte and J. W. Miles, On the numerical investigation of the Orr-Sommerfeld equation, J. Soc. Indust. Appl. Math. 7 (1959), 361-366.

[9] H. Jeffreys, On the formation of water waves by wind, Proc. R. Soc. Lond. A 107 (1925), 189-206.

[10] P. R. Gent and P. A. Taylor, A Numerical Model of Flow above Water Waves, J. Fluid Mech. 77 (1976), 105-128.

[11] O. M. Phillips, On the generation of waves by turbulent wind, J. Fluid Mech. 2 (1957), 417-445.


[12] O. M. Phillips, The Dynamics of the Upper Ocean, Cambridge University Press, 1966.

[13] H. Lamb Hydrodynamics, 6th ed., Cambridge University Press, 1932.

[14] M. J. Lighthill, Physical interpretation of the mathematical theory of wave generation by wind, J. Fluid Mech. 14 (1962), 385-398.

[15] C. Mastenbroek, V. K. Makin, M. H. Garat and J. P. Giovanangeli, Experimental evidence of the rapid distortion of turbulence in the air flow over water waves, J. Fluid Mech. 318 (1996), 273-302.

[16] J. W. Miles, On the generation of waves by shear flows, J. Fluid Mech. 3 (1957), 185-204.

[17] J. W. Miles, On the generation of waves by shear flows, Part 4, J. Fluid Mech. 13 (1962), 433-448.

[18] J. W. Miles, On the generation of waves by shear flows, Part 5, J. Fluid Mech. 30 (1967), 163-175.

[19] J. W. Miles, Surface-wave generation revisited, J. Fluid Mech. 256 (1993), 427-441.

[20] W. J. Plant, A relationship between wind stress and wave slope, J. Geophys. Res. 87 (1982), 1961-1967.

[21] P. G. Saffman and D. C. Wilcox, Turbulence model predictions for turbulent boundary layers, AIAA J. 12 (1974), 541-546.

[22] S. G. Sajjadi, On the growth of a fully non-linear Stokes wave by turbulent shear flow, Part 2, Rapid Distortion Theory, Math. Engng. Ind. 6 (1998), 247-260.

[23] S. G. Sajjadi, J. C. R. Hunt and N. H. Thomas, Wind over waves: Prospective and prospects, Oxford University Press, 1999.

[24] S. G. Sajjadi, Interaction of Turbulence due to Tropical Cyclones with Surface Waves, Adv. Appl. Fluid Mech. 1 (2007), 101-145.

[25] S. G. Sajjadi, J. C. R. Hunt and F. Drullion, Asymptotic multi-layer analysis of wind over unsteady monochromatic surface waves, J. Eng. Math. 84 (2014), 73-85.

[26] R. L. Snyder, F. W. Dobson, J. A. Elliot and R. B. Long, Array measurement of atmospheric pressure fluctuations above surface gravity waves, J. Fluid Mech. 102 (1974), 1-59.

[27] P. P. Sullivan, J. C. McWilliams and C.-H. Moeng, Simulation of turbulent flow over idealized water waves, J. Fluid Mech. 404 (2000), 47-85.


[28] A. A. Townsend, Flow in a deep turbulent boundary layer over a surface distorted by water waves, J. Fluid Mech. 55 (1972), 719-735.

[29] F. Ursell, Wave generation by wind, In Surveys in Mechanics, G. K. Batchelor, ed., Cambridge University Press, 1956.

Appendix A. Perturbation Equations

A.1. ( )1O equations

To ( )1O the equations of motion are:


( ) ( ) .011 01002

02

100 =

η⎭⎬⎫

⎩⎨⎧ Ω+

η+

ηΩ+ −−

ddUEd

dd

UdE (A1a)

Turbulent kinetic energy:

( ) ( )η⎭⎬

⎫⎩⎨⎧ Ω+

η+

ηΩ+ −−

ddEEbd

dd

EdEb 010032

02

1003 11

.000

103 =⎟⎠⎞⎜

⎝⎛

η−Ω+ Ed

dUbb (A1b)


( ) ( )ηΩ

⎭⎬⎫

⎩⎨⎧ Ω+

η+

η

ΩΩ+ −−

ddEbd

dd

dEb201

0062

20

21

006 11

.020

0405 =Ω⎟

⎠⎞⎜

⎝⎛

η−Ω+ d

dUbb (A1c)

The boundary conditions are given by

( )∞

∞ η=Ω=−+η+

κ=

10

100

1,1,1ln1KbbEcBU

(typically few meters above the wave) and

( ) 11

10

000 0,0, bJUQEcU −∗ ⎟⎠⎞⎜

⎝⎛

νη

=Ω=−=

on the water surface ( ).0=η


A.2. ( )akO equations

Similarly the ( )akO equations are given by


.0212 101

1 =⎟⎠⎞⎜

⎝⎛ −+

ηJUUiKd

dV (A2a)


( ) ( )η⎭⎬

⎫⎩⎨⎧ Ω+

η+

ηΩ+ −−

ddUEd

dd

UdE 11002

12

100 11

{ ( ) } 101

002 1 UiKUEK +Ω+− −

( ) 101

001 1 VddUEd

diKiKP⎭⎬⎫

⎩⎨⎧

η−Ω+

η+− −

( )⎭⎬⎫

⎩⎨⎧

ηΩΩ−Ω

η+ −−

ddUEEd

d 01

200

100

( ) ( )η

Ω+η

−Ω+− −−d

dUEddJJUEK 01

001101

002 12

1121

( ) ( ) .012112

121

21

00020

21

001 =η

Ω++η

Ω+− −−

dJdEU

dUdEJ (A2b)


( ) ( )η⎭⎬

⎫⎩⎨⎧ Ω+

η+

ηΩ+ −−

ddVEd

dd

VdE 11002

12

100 121

{ ( ) } 101

002 1 ViKUEK +Ω+− −

( )12

001

01012

01

21 ΩΩ−Ω⎟

⎠⎞⎜

⎝⎛

η+

η−

η− −− EEd

dUiKddJUd

dP

( ) ( ) .011 0100

10001 =

⎭⎬⎫

⎩⎨⎧

ηΩ++Ω+

η− −−

ddUEEd

dUiKJ (A2c)



( ) ( )η⎭⎬

⎫⎩⎨⎧ Ω+

η+

ηΩ+ −−

ddEEbd

dd

EdEb 110032

12

1003 11

( ) 1020

101

002 1 Ebd

dUbiKUEK⎭⎬⎫

⎩⎨⎧ Ω+

η−+Ω+− −

( ) 20

21

200

1013

ηΩΩ−Ω+ −−

dEdEEb

( )η⎭⎬

⎫⎩⎨⎧ −ΩΩ−Ω

η+ −−

ddEVEEd

db 011

200

1013

021

21

01121020

111

011

1 =⎭⎬⎫

⎩⎨⎧ +Ω−Ω+

η−

η+

η+ EiKVbJbd

dUJbddJUbd

dUb (A2d)


( ) 21

21

0060 1η

ΩΩ+Ω −

ddEb

( ) ( )ηΩ

⎭⎬⎫

⎩⎨⎧

ηΩ

Ω++Ω+η

Ω+ −−d

dd

dEbEbdd 101

0061

0060 121

( )⎢⎣⎡ Ω−

ηΩ+ΩΩ+− − 2

050

0401

0062

231 bd

dUbEbK

( ) 101

00600 1 Ω⎥⎦⎤⎭⎬⎫

⎩⎨⎧

ηΩ

Ω+η

+Ω− −d

dEbddiKU

ηηΩ+

η

ΩΩ+ −

ddU

ddUb

ddEb 102

042

20

21

016

.021 0

12041

305

20

1 =η

Ω−Ω+ηΩ

− ddUJbJbd

dV (A2e)

The respective boundary conditions at ∞η=η are

011111 =Ω==== EPVU


and on 0=η are

( ) ( ) .0,0,021

110

11111 bJUQEVcJU ⎟⎠⎞⎜

⎝⎛

νη

=Ω=== ∗

A.3. ( )2akO equations


.083

2122

121 2

102021

11

12 =⎟

⎠⎞⎜

⎝⎛ +−+

η−

η−

ηJUJUUiKd

dVJddJVd

dV (A3a)


( ) ( )η⎭⎬

⎫⎩⎨⎧ Ω+

η+

ηΩ+ −−

ddUEd

dd

UdE 21002

22

100 11

( )⎩⎨⎧

⎜⎝⎛

η+

η+

η+

ηΩ+

η+ −

ddUJd

dJUddUJd

dJUEdd 0

22

01

11

11

00 21

21

21

211

⎭⎬⎫⎟⎠⎞⎜

⎝⎛ ++

η−

η− 112

021

110 2

1281

41 JVViKd

dUJddJJU

{ ( ) }11

0012

001

011012 12

12 JEEEJUUK −−− Ω+−ΩΩ−Ω⎟⎠⎞⎜

⎝⎛ +−

[ ( ) ]⎩⎨⎧

ηΩ+−ΩΩ−Ωκ− −−−

ddVJEEEi 1

11

0012

001

01 12

⎭⎬⎫+−−+ 2

11102021

20 2 UJUUJUJU

⎭⎬⎫

⎩⎨⎧ ⎟

⎠⎞⎜

⎝⎛ +−−

η+

η− 2

102021

11

10 83

2122

121 JUJUUikd

dVJddJVU

η+

η−

η−

η− d

dVJUddUVd

dVUddUV 1

101

11

10

2

( 10220

11

110 UUJUiKd

dUVddJVU +−

η+

η+

) .02 112120 =−+− JPPJU (A3b)



( ) ( )η⎭⎬

⎫⎩⎨⎧ Ω+

η+

ηΩ+ −−

ddVEd

dd

VdE 21002

22

100 11

[ ( ) ]⎭⎬⎫

⎩⎨⎧ Ω+−ΩΩ−Ω⎟

⎠⎞⎜

⎝⎛

ηη+− −−−

12

0012

001

011

20 12 JEEEddV

ddViKU

[ ( ) ]⎭⎬⎫

⎩⎨⎧ ⎟

⎠⎞⎜

⎝⎛ +Ω+−ΩΩ−Ω

η− −−−

10112

0012

001

01 211 JUUJEEEd

diK

( )⎢⎣⎡

⎩⎨⎧

η+

η+

η+

ηΩ++ −

ddJUd

dUJddJUd

dUEiK 20

11

11

2100 2

121

2112

⎭⎬⎫

η−

η−

η+ d

dUJddJJUd

dUJ 021

110

02 8

141

21

{( ) ( ) ( ) 112

001

0121

0021

100

0 11 JEEJEJEddU

ΩΩ−Ω−Ω+−Ω+η

+ −−−−

}22

0021

3000

201

102 ΩΩ−ΩΩ+ΩΩ−Ω+ −−−− EEEE

{ ( ) }11

0012

001

01 1 JEEE −−− Ω+−ΩΩ−Ω+

⎥⎦⎤⎭⎬⎫

⎩⎨⎧ +

η+

η+

η× 1

01

10

121

21 iKVd

dUJddJUd

dU

( )⎭⎬⎫

⎩⎨⎧ ⎟

⎠⎞⎜

⎝⎛

η+

ηΩ+

η+ −

ddVJd

dJVEdd 1

11

11

00121

( )⎭⎬⎫

⎩⎨⎧ ⎟

⎠⎞⎜

⎝⎛ −++Ω+

η− − 2

10201121

00 81

21

2112 JUJUJUUEd

dik

( )η

−η

−−+− ddJUd

dVVJVUVUVUiK 220

111101120 2

122

( ) .011

211

2010 =

η−

η−

η−− d

dPJddP

ddJJUUU (A3c)



( ) ( )η⎭⎬

⎫⎩⎨⎧ Ω+

η+

ηΩ+ −−

ddEEbd

dd

EdEb 210032

22

1003 11

( ) 200

10220

21

01

0032 214 EiKUd

dUbbd

EdEbK⎭⎬⎫

⎩⎨⎧

+η

−Ω+η

Ω−Ω+− −−

( )⎭⎬⎫

⎩⎨⎧

ηΩΩ−Ω

η+ −−

ddEEEd

db 11

200

1013

( )⎭⎬⎫

⎩⎨⎧

ηΩΩ+ΩΩ−ΩΩ

η− −−−

ddEEEEd

db 02

200

21

3001

2013

⎩⎨⎧

⎟⎠⎞⎜

⎝⎛ +−−⎥⎦

⎤⎢⎣⎡ ⎟

⎠⎞⎜

⎝⎛ +

η⎟⎠⎞⎜

⎝⎛

η+

− 2101

221

22

1011

01 2

121 JUUKVKJUUd

dd

dUb

η⎟⎠⎞⎜

⎝⎛ +−⎟

⎠⎞⎜

⎝⎛

η+ d

dVJUUiKddV 1

1012

1212

02

1101 21

21 EiKVJUUd

d

⎭⎬⎫

⎥⎦⎤

⎢⎣⎡ +⎟

⎠⎞⎜

⎝⎛ +

η−

η−⎟

⎠⎞⎜

⎝⎛ +− d

dEVEJUUiK 111101 2

1

( ) ( ) 1101211011011 21 Ω−−

⎭⎬⎫

⎩⎨⎧ +⎟

⎠⎞⎜

⎝⎛ +

η−+ JEEbiKVJUUd

dJEEb

( ) ( )210201102

01120

2101 JEJEJEbd

dUJEJEJEb −+Ω+η

−−+

( ) .02 112

001

0132 =ΩΩ−Ω− −− EEEbK (A3d)


( ) ( )⎩⎨⎧

Ω+η

Ω−ηΩ

Ω−η

ΩΩ+Ω −−− 10060

202

00622

21

0060 1212 Ebdd

ddEb

ddEb


( )ηΩ

⎭⎬⎫

ηΩ

Ω+− −d

dd

dEb 20100614

( ) ( )⎩⎨⎧

η

ΩΩ+−Ω+Ω+ΩΩ+− −−

20

21

006002050

1006

2 12418d

dEbiKUbEbK

( ) 20

22

00601

00612η

ΩΩ+

ηΩ

Ω+η

− −−

ddEbd

dEbdd

( ) ( ) 1012

001

012

62201

006 4 ΩΩΩΩ−Ω−Ω⎭⎬⎫

ηΩ

Ωη

+ −−− EEKbddEd

db

( )⎭⎬⎫

⎩⎨⎧

ηΩ

Ω+η

+ −d

dEbdd 2

12

10061

( ) ( )⎭⎬⎫

⎩⎨⎧ ΩΩ

ηΩΩ−Ω

η+ −−−

120

11

200

10162 d

dEEddb

( )⎭⎬⎫

⎩⎨⎧

ηΩ

ΩΩ+ΩΩ−Ωη

+−

−−−d

dEEEddb

202

13

0012

011

026

( ) 1010121

1006

221214 ΩΩ⎟

⎠⎞⎜

⎝⎛ −−ΩΩ+− − JUUiKEbK

( )ηΩ

⎟⎠⎞⎜

⎝⎛ −−ΩΩ

η−Ω− d

dJVVddViKU

20

112101210 2

122

⎢⎣⎡ ⎟

⎠⎞⎜

⎝⎛ −−

η+

η⎟⎠⎞⎜

⎝⎛ −+ 11011

02

21 2

121

41 iKVJUUd

dJddUJJ

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎠⎞

⎜⎝⎛

η+−

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

η

⎟⎠⎞⎜

⎝⎛ +

⎟⎠⎞

⎜⎝⎛

η+

− 212

12

21011

021

212

1

21

21

ddVVKd

JUUd

ddU


⎪⎪⎭

⎪⎪⎬

⎫

⎟⎠⎞⎜

⎝⎛ +

η−⎟

⎠⎞⎜

⎝⎛ +− 101

12

1012

2122

1 JUUddViKJUUK

.0281 2

0

2101

1 =Ω⎥⎥

⎦

⎤

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞⎜

⎝⎛

η⎟⎠⎞⎜

⎝⎛

η+−

−

ddU

ddVJ (A3e)

Boundary conditions:

022222 at0 η=η=Ω==== EPVU (A3f)

and on 0=η

( ) ( ) ,0,02108

32222

212 ===⎟

⎠⎞⎜

⎝⎛ −−= EPVJJcU

[ ( ) ( )] .00 1221

02 bJJUQ −⎟

⎠⎞⎜

⎝⎛

νη

=Ω ∗ (A3g)

A.4. ( )3akO equations


⎭⎬⎫

⎩⎨⎧ ⎟

⎠⎞⎜

⎝⎛ −

η−

η−

η 1212

11343

21

21 VJJd

dd

VdJddV

12123 4

3233 UJJiKiKU ⎟

⎠⎞⎜

⎝⎛ −−+

085

23

23

031213 =⎟⎠⎞⎜

⎝⎛ ++− UJJJJiK (A4)

together with similar equations for u- and v-momentum, turbulent kinetic energy and pseudo-vorticity. However, these equations are far too lengthy and thus are not presented here.

TURBULENT SHEAR FLOW OVER STEEP STOKES WAVES

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