A Close-form Model for Dispersion in Shallow Waves over...

Adv. Theor. Appl. Mech., Vol. 3, 2010, no. 1, 1 - 29

A Close-form Model for Dispersion in Shallow Waves

over Varying Shear Current

S. Patil

Department of Biological & Agricultural Engineering Texas A & M University, 2117 TAMU

College Station, Texas 77843, USA [email protected]

V. P. Singh

Department of Biological & Agricultural Engineering

Texas A & M University, 2117 TAMU College Station, Texas 77843, USA

[email protected]

Abstract

The longitudinal dispersion coefficient of pollutants in a wave-steady current flow field (WCLDC) is theoretically quantified by deriving a stream function for the combined flow. The steady current profile is simulated using Blasius power law. The first-order wave component influenced by this current is derived using a linear wave theory and is empirically extended to second order, based on experiments reported in the literature. The combined stream function is period-averaged in Elder’s (1958) longitudinal dispersion coefficient to obtain a closed form solution of WCLDC. The magnitude of WCLDC shows a finite increase over the no-wave case and further increases with second-order wave effect for higher amplitudes. Keywords: Longitudinal dispersion coefficient, waves, steady shear

2 S. Patil and V. P. Singh 1 Introduction

The longitudinal transport of pollutants in the flow direction is governed by the interaction between differential advection and cross-sectional mixing. The spread of pollutants reflects the change in cross-sectionally averaged concentration due to the effect of turbulent diffusion and differential advection and is called longitudinal dispersion. The longitudinal dispersion of pollutants can be described by one dimensional non-conservative mass conservation equation given as

⎟⎠⎞

⎜⎝⎛

∂∂

∂∂

=∂∂

+∂∂

xCAE

xAxCU

tC

x1 (1)

where C is the cross-sectional averaged concentration of a substance, U is the cross-sectional averaged velocity, A is the cross-sectional area of the channel, Ex is the longitudinal dispersion coefficient, t is time, x is the longitudinal distance along the direction of flow.

Studies on the solution of Eq. 1 are mainly focused on an accurate estimate of Ex. Taylor (1953) pioneered the derivation of an analytical expression of Ex for laminar flow between parallel plates as

∫ ∫ ∫−=h y y

x dydydyyUD

yUh

E0 0 0

)('1)('1 (2)

where h is the flow depth, y is the dimension along h; UyUyU −= )()(' [ )(yU is the parabolic velocity profile along h, and U is the mean velocity over h]; and D is the molecular diffusivity. He further extended Eq. 2 for turbulent flows in pipes (Taylor, 1954) using a parabolic-logarithmic velocity profile and turbulent diffusivity and showed that the magnitude of Ex in Eq. 2 is governed by an appropriate velocity profile and diffusivity. As a result, Elder (1958) used a perfect logarithmic velocity profile for channel flows and vertical turbulent diffusivity and proposed a new form of Eq. 2 for wide prismatic open channels as

∫ ∫ ∫−=h y y

yx dydydyyU

eyU

hE

0 0 0

)('1)('1 (3)

where ye = the vertical turbulent diffusivity, and )(yU is now the depth-wise vertical logarithmic velocity profile. For this current profile, he derived an expression for ye as

Close-form model for dispersion in shallow waves 3

6

*huke v

y = (4)

and solving Eq. 3, Elder derived a close form expression for the longitudinal dispersion coefficient for wide open streams as

huEx *93.5= (5)

where bghSu =* is the shear velocity, h is the flow depth, bS is the bed slope, and g = 9.81m/sec2. Along the same line, Eq. 3 had been further modified by Fischer (1979) for natural streams in which a transverse velocity profile and a transverse turbulent diffusion were used. Apart from the above theoretical investigations, several empirical dispersion coefficients for steady current flow have been proposed, as for example, by McQuivey and Keefer (1974) for subcritical flows, Liu (1977) for lateral velocity gradient, and Seo and Cheong (1998), and Deng et al. (2002) for natural streams. Comparisons between such formulae have also been reported (Seo and Cheong, 1998; Rowinski et al., 2005; Ayyoubzadeh et al., 2004).

The longitudinal dispersion described above has mainly been investigated in steady uniform flow. However, apart from the advection and diffusion processes, surface waves due to the ubiquity of wind shear also contribute to the mixing of pollutants. In several situations, free-surface flows are characterized by the wave-current interaction. Huang and Mei (2003) observed that a turbulent current in the direction of wave propagation decreases the strength of current near the surface but slightly increases near the bottom. In turn, wave patterns can be significantly modified by currents as well (Umeyama, 2005). The wave-induced drift velocity provides an additional steady component in the flow direction and increases the dispersion of pollutants (Law, 2000). Jets and streams can be highly affected by waves (Koole and Swan, 1994), whereas diffraction of wave can be affected in the presence of both wave and current (Teng et al., 2001). The quantification of this wave-current effect mainly depends on the current profile. Dean and Dalrymple (1991) assumed a depth-wise uniform velocity under a sinusoidal wave to derive wave dispersion equation which has been extended by Badour and Song (1998) for depth-wise linear varying current profile. Attempts have also been made to simulate the depth-wise non-linear current by using several linear components (Thompson, 1949), trigonometric functions (Yih, 1972; Dalrymple, 1973), WKB approximation (Margaretha, 2005), and power series (Umeyama, 2005). A comprehensive review of waves over different current profiles is given in Badour and Song (1998), and Margaretha (2005).

4 S. Patil and V. P. Singh This paper considers a Blausius law profile for depth-wise current profile, a type of nonlinear current profile. Then a combined stream function for wave over the Blausius power law (Fig. 1a for power =7) is formulated and the resultant velocity field is used to derive an expression for the longitudinal dispersion coefficient in the combined wave-steady shear current flow field (hereto denoted as wave-steady current longitudinal dispersion coefficient, WCLDC). Examples of waves over steady current are uniform flow in open channels with ubiquity of wind waves or long-period tides in costal zones.

The current profile is vertically non-linear (Kemp and Simons, 1982, 1983;

Klopman, 1994). The longitudinal dispersion of pollutants in such a flow field is specific and is different from that in wave-only flow (Dalrymple, 1976; Fischer, 1979; Yasuda, 1984; Putrevu and Svendsen, 1992; Karambas, 1996; Pearson et al. 2002) or in current-only flow (Liu, 1977; Seo and Cheong, 1998; Deng et al., 2002). The necessary shear in the current-only flow is provided by the cross-sectional variation in velocity (Fischer, 1979) caused by the shear stress generated at the fluid-solid boundaries, whereas in wave-only flow it is generated by the cross-sectional variation in the amplitude of oscillations (Law, 2000). Moreover, the non-linearity in the surface waves (second-order effect) is drift wielding (Stoke, 1847) which should further enhance the shear effect in Eq. 3 relative to diffusivity.

In this paper, the derived stream function considers all these three

contributions and is used to calculate the resultant shear from Eq. 3 and by assuming small amplitude waves, ye for the current-only case is assumed to obtain a closed form expression for WCLDC. The finding of the investigation improves the accuracy in the prediction of pollutant concentration inland and in coastal hydrodynamics, where wave over shear current is ubiquitous. The paper is organized as follows. Giving the introduction of the topic with a brief literature review in section 1, the 1-D dispersion equation is integrated over a wave period to derive WCLDC in section 2. In section 3, a combined stream function for wave-steady current flow using the Blausius law is derived and is substituted in Elder’s longitudinal dispersion coefficient (1958) to obtain a closed form solution of WCLDC in section 4. The results and conclusions are discussed in section 5. 2. Period-averaged dispersion equation 2.1 Analysis of period-averaging of dispersion equation

The 2-D convection-diffusion equation for unidirectional flow can be written

as


⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+⎟⎠⎞

⎜⎝⎛

∂∂

=∂

∂+

∂∂

yCD

yxCD

xxUC

tC

yx ∂∂

∂∂ (6)

where all the quantities are instantaneous: C is the concentration of pollutant; xD and

yD are the diffusivities in x and y, directions, respectively; U is the unidirectional velocity in the longitudinal (x) direction; and y is the vertical direction. As advection in the flow direction is greater than molecular diffusion in the same direction, the molecular term can be neglected. Averaging Eq. 1 over a wave period,

yCD

yxUC

tC

y ∂∂

=∂

∂+

∂

∂

∂∂ and using the Reynolds analogy, the advection and

dispersion terms can be rewritten as UC = U C + ′ U ′ C and

yCD

yCD

yCD yyy ∂

′∂′+∂∂

=∂∂ , where primes indicate fluctuations over period-

averaged terms. As these fluctuating terms are sinusoidal, their period-averaged values are zero. Thus, neglecting the second term on the right side, one obtains

yCD

yxC

UtC

y ∂∂

=∂

∂+

∂

∂

∂∂ (7a)

For brevity, we will neglect the brackets (that denote period-averaged quantities) henceforth. The next operation will be averaging over the flow depth. Thus, in Eq. 7a, C and U are divided into depth-averaged quantities and fluctuating quantities over the depth-averaged quantities. Thus, Eq. 7 is written as

( ) ( ) ( )⎟⎟⎠

⎞⎜⎜⎝

⎛∂

′∂=

∂′+∂′++

∂′+∂

2

2

yCD

xCCUU

tCC

y . Changing the time and space

coordinate into moving co-ordinate system (τ ,ξ ), it is

2

2

yCDCUCUCC

y ∂′∂

=∂

′∂′+∂∂′+

∂′∂

+∂∂

ξξττ. Averaging this over depth is

∂C ∂τ

+ ′ U ∂ ′ C ∂ξ

= 0 which on subtracting from itself provides

2

2

yCDCUCUCUC

y ∂′∂

=∂

′∂′−∂

′∂′+∂∂′+

∂′∂

ξξξτ (7b)

In Eq. 7b, the third and fourth terms balance each other and are much smaller than the second term. Thus, Eq. 7b is

6 S. Patil and V. P. Singh

2

2

yCDCUC

y ∂′∂

=∂∂′+

∂′∂

ξτ (8)

After a long time, the solution of Eq. 8 is its steady state form, i.e.,

2

2

yCDCU y ∂

′∂=

∂∂′

ξ with hyat

yC ,00 ==∂

′∂ ( 9a)

Solution of Eq. 9a is

( ) ( )01

0 0

CdydyUxC

DyC

y y

y

′+′∂∂

=′ ∫ ∫ (9b)

The rate of mass transport now can be written as

( )dyCUdydydyUUxC

DdyCUM

hy yh

y

h

01

00 000

′′+′′∂∂

=′′= ∫∫ ∫∫∫ (10a)

In Eq. 10a, ′ U 0

h

∫ ′ C 0( )dy = 0 because ′ U 0

h

∫ dy = 0. The rate of mass transport can also

be given as xChKM wc ∂

∂−= . Thus equating this with Eq. 10a, we get

∫ ∫∫ ′′−=y yh

ywc dydydyUU

DhK

0 00

1 (10b)

As the quantities on the right side term are period-averaged already, the above expression can be rewritten as

∫ ∫ ∫ ∫ ′′−==T h y y

yxwx dtdydydyyUyU

hTeEE

0 0 0 0

)()(11 (11)

where yy eD = , vertical turbulent diffusivity in y direction can be replaced for turbulent flow. Note that the effective depth of flow is employed for integration over depth. If the velocity is time dependent, the dispersion coefficient will also be time dependent.

Close-form model for dispersion in shallow waves 7 2.2 Shear for combined wave-steady current flow

In the presence of short-period waves and steady current, the unsteady

velocity can be written as

),,(),,(),(),,( tyxUtyxUyxUtyxU dwc ++= (12) where Uc= the steady current velocity; Uw= the first-order wave induced velocity; and Ud= the second-order Stokes drift velocity and it is due to the fact that when a water particle is moving forward under a wave crest, it is at a higher level in the water column and moves faster, and when it is moving backward under the trough it is at a lower level and moves more slowly. Thus, a net forward movement results. The depth- and period-averaged velocity is given by

( )∫ ∫ ++=T h

dwc dydttyxUtyxUyxUxhT

xU0 00

0 ),,(),,(),()(

1)( (13)

The above velocity should be the Lagrangian velocity of particles, since the

dispersion phenomenon is considered. Subtracting Eq. 13 from Eq. 12, the fluctuating velocity component can be expressed as

),,(),,(U),(),(' 0

'0

'wL0

'0 tyxUtyxyxUyxU dc ++= (14)

where '

cU =the lateral deviation of the steady current velocity from the depth-averaged steady current velocity; '

wLU and 'dU = the lateral deviation of the wave

induced velocities from their time-averaged mean velocities. Assuming constant vertical diffusivity, the final form of the wave-current induced dispersion coefficient (WILDC) is given by

∫ ∫ ∫ ∫ ++++−=T h y y

dcdcy

wx dydydydttyUtyyUtyUtyyUehT

E0 0 0 0

''w

'''w

' )),(),(U)(()),(),(U)((1

(15)

yeI

−= (15a)


where T is the wave period and I is the shear term which is calculated by considering a combined rotational wave-current flow field and deriving a combined stream function as follows.

3. Stream function for wave-steady current flow and WCLDC 3.1 Theory of wave-steady current flow using power law

In this section, a combined stream function for the wave-shear current will

be derived. The Blausius power law is used to simulate shear current and its effect on the surface periodic wave is investigated. The result is in the form of a combined stream function ψ (x, y, t) for 2-D, incompressible steady flow with a surface periodic wave, where x, y are the flow and vertical directions, respectively, and t is the time. Thus, the shear current for the turbulent flow can be written using the Blausius law as

m

mc hyUU ⎟⎟

⎠

⎞⎜⎜⎝

⎛=

2 (16)

where m is the power and its value depends on Reynolds’ number, γ

hURe4

= (γ =

kinematic viscosity = 10-6 m2/s). This relationship is shown in Fig. 1b. The usual substitution of pipe radius = twice the flow depth is used to represent the depth-wise vertical velocity profile in open channels. Um is the maximum velocity, U is the mean flow velocity, and h is the flow depth. Changing the magnitude of m, Eq. 16 can be used to describe various shear velocity distributions over the flow depth of uniform, steady open-channel flow (See Fig. 1c). The shear in Eq. 16 provides vorticity and with a surface wave, it forms a 2-D rotational periodic flow. In the absence of shear (constant-velocity case), 0=m should be substituted in Eq. 16 which provides mc UU = , i.e., a depth-wise constant velocity. In this case, U (mean flow velocity) can be tentatively substituted in place of mU to get the mean flow velocity in the absence of shear and the flow becomes 2-D irrotational periodic flow.

To derive the combined stream function, the stream function of Eq. 16

(current part), i.e., 3

1

1 121 B

my

hUdyU

mm

mc +⎟⎟⎠

⎞⎜⎜⎝

⎛+

⎟⎠⎞

⎜⎝⎛==

+

∫ψ is considered apriori (B3 is

a constant and is zero for horizontal bottom) and the stream function for the wave components influenced by the current part are derived using the wave theory. The approach is inverse of that given by Baddour and Song (1998), who assumed wave component apriori and derived current component which, however, is a depth-wise linear current profile. To derive the combined stream function for wave-log current,

Close-form model for dispersion in shallow waves 9 the equations of motion neglecting the viscous effect are considered which can be written as

xp

tuv

xuu

tu

∂∂

−=⎟⎠⎞

⎜⎝⎛

∂∂

+∂∂

+∂∂ρ (17a)

gyp

yvv

xvu

tv ρρ −

∂∂

−=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+∂∂ (17b)

where ρ is the constant density, p is the pressure in the fluid, g is the acceleration due to gravity, t is the time, x is the flow direction, and y is the vertical direction

measured from the channel bottom. Also y

u∂∂

=ψ and

xv

∂∂

−=ψ are the velocity

components in the x and y directions, respectively, where ψ is the stream function. Eqs. 17a and 17b are integrated over x and y, respectively. Using the continuity

equation 0=∂∂

+∂∂

yv

xu & conservation of vorticity and eliminating pressure terms,

the resultant 2-D equation of motion for an incompressible, inviscid fluid is written as (Baddour and Song, 1998):

0.Vt

2 =ψ∇⎟⎠⎞

⎜⎝⎛ ∇+

∂∂ r

(18)

where t is the time, and x and y are the flow direction and vertical direction measured

from the channel bottom, respectively. jy

ix

rr

∂∂

+∂∂

=∇ is the 2-D gradient operator,

2

2

2

22

yx ∂∂

+∂∂

=∇ is the 2-D Laplacian operator, and jx

iy

Vrrr

∂∂

−∂∂

=ψψ is the velocity

vector, where ψ is the stream function. The vorticity in flow is generated by the logarithmic shear current and the complete flow is treated as rotational. The coordinate axis is fixed at the bottom so that at the bottom, y = zero and on the free surface is y = η (x, t). The no-flow condition perpendicular to the bottom can be given as

0=∂∂

xψ at y = 0 (19)

whereas on the free surface, the pressure is zero. The free surface kinematic boundary condition is


0. =⎟⎠⎞

⎜⎝⎛ ∇+

∂∂ SVt

r on y = η (x, t) (20)

and the periodic and permanent type free surface is expressed as

( ) 0),(,, =−= txytyxS η (21)

where ∑∞

=

−=0

)(cos),(n

n tkxnatx ση (22)

in which an are the Fourier coefficients, k=2π/L is the wave number, and L is the wavelength. As η (Eq. 21) is periodic, ψ must also be a periodic function, i.e.,

),(),,( ytkxtyx σψψ −= for the horizontal bed and can be written as

∑∞

=

−=0

)(cos)(),,(n

n tkxnyPtyx σψ (23)

where )(yPn are the Fourier coefficients and are determined later. Taylor series expansion of Eq. 22 and Eq. 23 to the second order expansion can be written as )(cos)(cos),( 21 tkxatkxahtx σση −+−+= (24) and )(2cos)()(cos)()(),,( 210 tkxyAtkxyAyAtyx σσψ −+−+= (25) where h is the depth of flow, 1a is the wave amplitude ( a ), and 2a is the second order coefficient. The stream function 0A (y) in Eq. 25 corresponds to the log-current velocity given by Eq. 16. The corresponding stream function A0 (y) is now known a priori and can be given as

3

1

0 121

2)( B

my

hUdy

hyUyA

mm

m

m

m +⎟⎟⎠

⎞⎜⎜⎝

⎛+

⎟⎠⎞

⎜⎝⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛=

+

∫ (26)

First of all, an expression for 1A in Eq. 25 to the first order of wave

amplitude is found by substituting Eq. 12 in Eq. 18 as


( ) ( )

{ } ( )0

22sin

sin121

22

1111

12

113

12

1

=−′′′+′′′−+

−⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−⎟⎠⎞

⎜⎝⎛−′′⎟

⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛+−′′ −

tkxAAkAAk

tkxAkymmh

AkhyAk

hyAkA m

mmm

σ

σσσ (27)

where the prime denotes differentiation with respect to y. Eq. 27 suggests that for all )( tkx σ− , a set of differential equations is:

( ) 0121

22 12

113

12

1 =−⎟⎠⎞

⎜⎝⎛−′′⎟

⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛+−′′ − Akymm

hAk

hyAk

hyAkA m

mmm

σσ (28)

and 01111 =′′′+′′′− AAkAAk (29)

Eq. 29 is the differential equation for wave-only flow. From Eq. 19, 0=ψ for 0=y . For this, the periodic part of Eq. 25 suggests 1A (y) = 0 at y = 0, for which Eq. 29 suggests

)(sinh)(sinh)( 111 kymyyA ββ == (30) where 1β is a constant to be determined, and m = k. For m > k and m < k, trigonometric current profiles result (Badour and Song, 1998) and are out of scope of the present study. Substitution of Eq. 30 in Eq. 28

yields ( ) ( ) 0sinh121

12 =+⎟

⎠⎞

⎜⎝⎛− − kykymm

dU m

m

m β . For the shallow water waves

( 05.0/ ⟨Lh ; 314.0<kh ) considered in this paper, this expression satisfies the condition of 0)sinh( →kh (replaced y = h). Thus Eq. 28 satisfies Eq. 16 and therefore validates the apriori assumed Blausius velocity profile.

Along the same line, an expression for the second order Fourier coefficient

can be written as )(2sinh)( 22 kyyA β= (31) where 1β is a constant to be determined. Substituting 0A , 1A and 2A in Eq. 25, the stream function for the combined wave-current rotational flow field is

)(2cos)(2sinh)(cos)(sinh12

1),,(

21

3

1

tkxkytkxky

Bmy

dUtyx

mm

m

σβσβ

ψ

−+−+

+⎟⎟⎠

⎞⎜⎜⎝

⎛+

⎟⎠⎞

⎜⎝⎛=

+

(32)

12 S. Patil and V. P. Singh where the first three terms on the right side are the steady current components, the fourth term is the first order wave motion and the fifth term is the second order expansion which reveals the drift velocity. Constant 3B = 0 for the no slip condition at y = 0 and thus it is necessary that ψ (x, y, t) = 0 for all t. 3.2 Kinematic boundary condition

Coefficient 1β in Eq. 32a can now be determined by Taylor series expansion of the kinematic free surface boundary condition about y = h to the first order, i.e.,

using Eq. 20. Thus, substituting jy

ix

rr

∂∂

+∂∂

=∇ ; jx

iy

Vrrr

∂∂

−∂∂

=ψψ and Eq. 21, Eq. 20

is simplified as:

0=∂∂

+∂∂

∂∂

+∂∂

xxytψηψη (33)

For the first order, the third term in Eqs. 24 and 32a is neglected. The remaining part of these equations is substituted in Eq. 33 to get

( ) ( ) 0)sin(sinhcosh)(2sin2

)(sin)(sin30log)(sin

44

2

*

=−−−−

−−−−−

tkxkykBkytkxBak

tkxakUtkxk

yakkutkxa c

sv

σσ

σσσσ

The term consisting of )(2sin tkx σ− corresponds to the second order expansion and is hence neglected. From the remaining terms and using y = h, 1β can be expressed as

)(sinh

21 kd

dyUCa

m

m⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−

=β (34)

and kC σ= is determined. Coefficient 2β in Eq. 32a is associated with the second order expansion. Thus, the second order expansion of Eq. 33 can be derived as

02

2

2

=∂∂

∂+

∂∂

∂∂

+∂∂

+∂∂

∂∂

+∂∂

yxyxxxytψηψηηψηψη (35)

Substituting Eqs. 24 and 32a in Eq. 35 and applying y = h, the first order terms (i.e., )(sin tkx σ− ) provides Eq. 34 as a solution for 1β , whereas the collection of coefficients of )(2sin tkx σ− provides an expression for 2β as


( )

)2(sinh

coth244

12 2

1

221

221

2 kh

khdyUCk

aha

hU

dyUC

aa

am

mm

m

m⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−−⎟⎟

⎠

⎞⎜⎜⎝

⎛−−

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−

=β (36)

3.3 Experimental determination of 2β

The only unknown in Eq. 36 is 2a . From the second-order expansion of the dynamic free surface boundary condition, 2a as a function of 1a can be obtained, however, the analysis becomes difficult for wave-log current flow. As such, 2β has been obtained empirically using experimental data. First, a functional relationship for 2β based on Eq. 36 is formed as

( ) ⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

m

m dyUC

khaf

2,

2sinh

21

2β (37)

The two independent variables on the right hand side of Eq. 37 include all the wave and current parameters. Eq. 37 shows a nonlinear relationship between 2β and independent variables. However, from Eq. 34, it can be surmised that the

variable ( )kha

2sinh

21 would have been maintained, had the analytical solution of 2β

been obtained. This can be confirmed from the same expression as obtained by Baddour and Song (1998) for wave-linear current interaction. As such, 2β as a nonlinear function of the independent variables is expressed as

( )

φ

β⎟⎟

⎠

⎞

⎜⎜

⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−=

m

m dyUC

kha

22sinh

21

2 (38)

where φ is the regression coefficient to be determined from experimental data.

The term )( tkx σ− in Eq. 32a suggests that the experimental data should be

within a wave period such as wave spectrum. Therefore, a theoretical expression of wave spectrum within a wave period at hy = is derived. Substituting Eqs. 38 and 34 in Eq. 32a, the vertical velocity is derived from Eq. 32a as

xtW

∂∂

−=∂∂

=ψη


)(2sin)2sinh(2)2sinh(

)2(

)(sin)sinh(2)sinh(

2

tkxkhdyUC

khka

tkxkhdyUC

khak

m

m

m

m

σ

σ

φ

−⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−+

−⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−=

(39)

Integrating Eq. 39 with respect to t, the surface wave spectrum η is derived as

)(2cos2

)(cos2

2

tkxdyUC

Catkx

dyUC

Ca

m

m

m

m σση

φ

−⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−+−

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−= (40)

Experiments by Kemp & Simons (1982, 1983), Klopman (1994), and

Umeyama (2005) have all been within an intermediate wave range of =Lh 0.1 to 0.2. The data of Kemp & Simons and Klopman maintained a constant depth and wave period for all waves. However, Umeyama conducted four experimental runs with a variable wave period which results in four wave spectra for different wave periods. Therefore, his data (Table 2) were used to find φ from Eq. 40 by the least square method.

In Umeyama’s experiments, the uniform velocity = 0.12m/sec corresponds

to the no-wave cases, whereas the wave data is from his four wave-only cases. He passed these waves over the uniform velocity and measured the change in the wave spectrum due to the wave-current interaction. This resultant spectrum of wave-current interaction is reproduced in Fig. 2 along with the theoretical wave spectrum by Eq. 40. A phase shift of 2π is provided to the cosine spectrum of Eq. 40 to get in phase matching with the observed data which is in the sine spectrum (Umeyama, 2005; Fig. 10). In the experiments, the first and third runs are producing wave spectra close to the linear sinusoidal wave which should not be the case as non-linear waves have a higher crest than trough, whereas the second and forth runs satisfy this requirement. The reason behind this experimental error is not known, however, a streaming effect might have been present in the experimental runs. Also, the theoretical velocity profile used by Umeyama (2005) is a power series that deviates from Eq. 16. However, Eq. 40 derived herein accurately simulates the trend of higher and sharper crest and shallower and flatter trough qualitatively. While matching Eq. 40 with experimental data, attention is provided to capture the crest and the crossing point, for which the range of φ has been found to be between 10 and 20. More

Close-form model for dispersion in shallow waves 15 experiments on the combined flow of wave-log current are needed for a better estimation ofφ .

The flow structure for the steady shear-only and wave-steady shear flow is

plotted in Figs. 3a and 3b. Fig 3b shows the periodicity induced in the steady shear by the presence of wave activity. Velocities to the first order wave are plotted, as the magnitude of φ is not certain. However, these figures suffice for comparison to distinguish between cases. The unidirectional flow for the steady shear is shown in Fig. 3a, in which the depth-wise change in the horizontal velocities can be seen. When the steady shear is superposed by wave (Fig. 3b), then the vector field depends on the relative magnitude of wave and current velocities. At L/2 along the x-axis, velocities of steady current and waves are opposite in direction and therefore cancel each other. The effect can be seen in Fig. 3b which shows negligible velocities at L/2.

3.4 WCLDC

The combined wave-log current horizontal velocity is

)(2cos)(2cosh2

)(cos)(cosh2

2

1

tkxykk

tkxkykdyU

yU

m

m

σβ

σβψ

−+

−+⎟⎟⎠

⎞⎜⎜⎝

⎛=

∂∂

= (41)

Averaging Eq. 41 over h, the depth-average velocity is

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−+

−+⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

+

)(2cos)(2sinh

)(cos)(sinh12

11

2

1

1

tkxhk

tkxkhmh

hU

hU

mm

m

σβ

σβ (42)

Thus, substituting UUyU −=′ )( in Eq. 15a, the shear term becomes

( ) ( ) dtdydydyUUUUUUIT h y y

dwcdwc∫ ∫ ∫ ∫ ′+′+′′+′+′=0 0 0 0

(43)

where cU ′ = the current only component, wU ′ = the first order wave component, and

dU ′ = the second order drift component of wave. Some of the terms in Eq. 43, such as cross co-variances of current with first and second order waves, for example wc UU ′′ and dc UU ′′ , will vanish under the time integral. Thus Eq. 43 becomes,


( ) ( )( )( )

( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( )

( )

∫ ∫

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎭

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎨

⎧

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+−

−+

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛+

−−

−+

−

+

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+−

−+

⎥⎦

⎤⎢⎣

⎡−

++⎟⎟⎠

⎞⎜⎜⎝

⎛

=

+

T h

xtxt

xtxt

xtxt

xtxtxtxt

xtxtxtxt

xtxtxtxt

xtxt

xtxt

mm

mom

dydt

OkhhyOkykh

kh

OkykhhkyOky

OOkhkhhy

OOkhkykh

OOkykhkh

OOkhkyhkyOOkyky

OOkykhhkyOOkyky

OkhhyOkykh

kh

OkykhhkyOky

yhmm

yh

u

ThI

0 0

222

22

22

22

21

2

2

2

2

21

222

22

22

22

21

22

)2(cos)2(sinh2

)2(cos)2cosh()2sinh(21

)2(cos)2cosh()2sinh()2(cos2cosh

)cos()2cos(sinh2sinh2

2

)cos()2cos(sinh2cosh21)cos()2cos(2cosh2sinh1

)cos()2cos(sinh2cosh)cos()2cos(cosh2cosh2

)cos()2cos(cosh2sinh2

)cos()2cos(cosh2cosh21

)(cos)(sinh2

)(cos)cosh()sinh(1

)(cos)cosh()sinh(2

)(coscosh

2212

1

β

ββ

β

(44) where ( )tkxOxt σ−= . Note that all the cross co-variances between first and second order terms are in the bracket associated with 21ββ and will be zero in the period averaging, because the integral over a period of )cos()2cos( xtxt OO is zero, as shown in appendix B. This suggests that there is no association of trigonometric terms of first order with second order. As time and depth are independent variables, the time integral is first solved:

( ) ( )( )( )

( ) ( )

( ) ( )

∫

⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

⎥⎦

⎤⎢⎣

⎡+−−+

⎥⎦

⎤⎢⎣

⎡+−−+

⎥⎦

⎤⎢⎣

⎡−

++⎟⎟⎠

⎞⎜⎜⎝

⎛

=

+

h

mm

mm

dy

yh

khkykh

khkyykhhkky

yh

khkykh

khkyykhh

kky

yhmm

yh

U

TTh

I0

22

2222

2

22

2222

1

22

2)2(sinh)2cosh(

2)2sinh()2cosh()2sinh(2cosh

2)(sinh)cosh()sinh()cosh()sinh(

2cosh

2212

21

β

β (45)

The period integrals used are included in appendix B. Arranging terms, we get


( ) ( )( )( )( )

( )( )

( )( )( )( )

⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ +−++

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ +−++

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+

+++−

+−

+++−

=

+++

khhk

hkkh

kh

khhk

hkkh

kh

dhmmm

dhm

dhmmm

hh

Uh

I

mmmmmmm

2sinh12

3216

4sinh341

sinh6

68

2sinh341

621332213221

222

2222

222

222

1

323332

2

2

β

β (46)

In each bracket of 2

1β and 22β , the first two terms stems from the depth-wise velocity

profile (Eq. 41) and the third term stems from the depth-averaged velocity profile (Eq. 42). The second and third bracket should be negative to get the positive dispersion coefficient. This can be checked as follows: For first-order oscillatory flow,

0sinh6

68

2sinh341 2

22

22

⟨⎟⎟⎠

⎞⎜⎜⎝

⎛ +−+ kh

hkhk

khkh is the necessary condition for the bracket

to be negative and for which, khhk

hkkh

kh 222

22

sinh6

68

2sinh341

⎟⎟⎠

⎞⎜⎜⎝

⎛ +⟨+ . For small

amplitude waves, khkh ≈sinh . This gives ( ) ( )222

22

66

823

41 kh

hkhk

khkh

⎟⎟⎠

⎞⎜⎜⎝

⎛ +⟨+ . This

provides ( ) 02 ⟩kh i.e. 0⟩kh which is true and the term is negative. The third bracket governs second-order oscillatory flow, which, to be

negative, should have 02sinh12

3216

4sinh341 2

22

22

⟨⎟⎟⎠

⎞⎜⎜⎝

⎛ +−+ kh

hkhk

khkh . This gives,

khhk

hkkh

kh 2sinh12

3216

4sinh341 2

22

22

⎟⎟⎠

⎞⎜⎜⎝

⎛ +⟨+ and for khkh ≈sinh , will be 0⟩kh . This is

true and thus the third bracket will be negative. Assuming small amplitude waves, the vertical diffusivity, ye for depth-wise

steady flow given by Eq. 4 is used. Moreover, the steady shear term in Eq. 46 is approximately calculated as steady dispersion coefficient (Eq. 5) multiplied by the corresponding vertical diffusivity (Eq. 4) which provides 22

*4.0 hu= . The resultant of Eq. 46 with Eq. 4 is substituted in Eq. 15a to get a closed form solution for WCLDC as


( ) ( )( )( )( )

( )( )

( )( )( )( )

⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ +−++

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ +−++

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+

+++−

+−

+++−

−=

+++

khhk

hkkh

kh

khhk

hkkh

kh

dhmmm

dhm

dhmmm

hh

Uh

hukE

mmmmmmm

vwx

2sinh12

3216

4sinh341

sinh6

68

2sinh341

621332213221

6

222

2222

222

222

1

323332

2

2

*

β

β (47)

4. Evaluation of WCLDC

To observe the performance of Eq. 47, some step-wise sample values of kh

have been chosen within shallow wave range (h/L < 0.05) as h/L= 0.05, 0.04, 0.03, 0.02 and 0.01. The corresponding values of kh = 0.31, 0.26, 0.19, 0.13 and 0.09, respectively. For the assumed flow depth h = 0.2m and Um unity, φ =10 in β2 is assumed to calculate the magnitude of dispersion coefficient for four values of wave amplitudes for which the range of a/L = 0.0007-0.01 is well within the assumed small amplitude waves. Fig. 4a shows dispersion coefficients calculated from Eq. 47 under no-wave and first and second order waves for the power 7. It is seen that the contribution from the first-order wave is distinctly different from the no-wave case. The higher wave amplitude causes more churning of particles which enhances the dispersion coefficient. Higher kh also has the same effect, however, the increase in the wave amplitude has a higher effect on dispersion than kh and it is because the increase in the wave effect is higher for amplitude. The contribution from the second-order wave starts at relatively higher wave amplitudes.

For higher values ofφ , the difference between the contribution from the first-

order wave and the second-order wave will increase. Figure 4a also shows that as kh increases, the dispersion coefficient increases. This is because the increase in kh decreases the wavelength and therefore the period of wave activity for a fixed flow depth. The resultant waves have a short period that increases the churning action and thus the mixing of pollutants. Moreover, as mentioned earlier, the shear in the steady current can be changed by changing the magnitude of m (Fig. 1c). The effect of change in shear on longitudinal dispersion coefficient can be seen in Fig. 4a and 4b in which the magnitude of m is decreases from 7 to 2. The vertical diffusivity for both the cases is assumed same. This causes an increase in the gradient of velocity profile which increases the shear dispersion due to steady current. The effect of wave-induced dispersion is an addition over the shear dispersion as can be seen in Fig. 6b.

Close-form model for dispersion in shallow waves 19 5. Summary and Conclusions The paper describes the longitudinal dispersion in a combined wave-steady shear current flow field. The wave-current stream function, derived by Baddour and Song (1998) for depth-wise flow profile, has been modified by using the Blausius power law for current profile and the wave components to second order influenced by the Blausius law have been derived. Thus, the resultant combined stream function has three components: steady current, first order wave and its second order empirical expansion. The depth-wise combined velocity has been derived from the stream function and substituted in Elder's dispersion coefficient to obtain a closed form solution of WCLDC for small amplitude waves. The vertical diffusivity for the current-only case is used for the assumption of small amplitude waves. The derived WCLDC has three components corresponding to those described above for the stream function. The contribution from the first-order wave is important as it is distinctly different from the no-wave case, whereas the second-order wave contributes to relatively higher values of the wave amplitude. The increase in wave amplitude as well as kh enhances churning of particles which increases the dispersion coefficient. The effect of wave amplitude has been found to be higher than kh. The individual contribution to the dispersion can be assessed by equating the coefficients of other two components to zero. For example, the dispersion coefficient for the current-only case can be obtained by substituting the wave amplitude = 0, i.e., by equating the coefficients of first and second order wave profile, i.e., β1 and β2 to zero.

The derived WCLDC has significance as the mixing due to waves and

currents greatly enhances the transfer of chemical and biological elements due to the change in the strength of current by surface wave. Moreover, the wave-current flow in coastal flows affects long-term evolution of morphological patterns, vegetation growth and sedimentation. In view of the ubiquitous wave-current flow existing in real environment, the outcome of the present work improves, quantitatively and qualitatively, the mechanism of pollutants dispersion in the combined wave-steady current flow. References [1] A. Ayyoubzadeh, M. Faramarz and K. Mohammadi, “Estimating longitudinal dispersion coefficient in rivers, Regional Characteristics and Water Problems, Conference of Asia Oceania Geosciences Society, (2004), 1-8. [2] R. E. Baddour and S. W. Song, The rotational flow of finite amplitude periodic water waves on shear current, Applied Ocean Research, 20(3) (1998), 163–171.

20 S. Patil and V. P. Singh [3] R. G. Dean and R. A. Dalrymple, Water Wave Mechanics for Engineers and Scientists, Prentice Hall, World Scientific, Singapore, 1991, 353 pp. [4] R. A. Dalrymple, Water wave models and wave forces with shear currents, Coastal Ocean Eng. Lab. Tech. Report No 20., Univ. of Florida, Gainesville, pp. 173, 1973. [5] R. A. Dalrymple, Wave induced mass transport in water waves, Journal of Waterways Harbour and Coastal Engineering Division, (1976), 255-264. [6] Z. Q. Deng, L. Bengtsson, Vijay P. Singh, and D. D. Adrian, Longitudinal dispersion coefficient in single-channel streams, J. Hydraul. Engg., 128(10) (2002), 901–916. [7] J. W. Elder, The dispersion of marked fluid in turbulent shear flow, J. Fluid Mech., (1958), 544-560 [8] H. B. Fischer, J. E. List, C. R. Koh, J. Imberger and N. H. Brooks, Mixing in Inland and Coastal Waters, Acad. Press, Inc., NY, 1979. [9] T. V. Karambas, On the pollutant dispersion subjected to near-shore circulation, Proc. 3rd Int. Conf. on Env. Pollution, Univ. of Thessaloniki, Greece. 1996. [10] Z. Huang and C. C. Mei, Effects of surface waves on turbulent current, J. Fluid Mech., 497(24) (2003), 253–287. [11] P. H. Kemp and R. R. Simons, The interaction between waves and a turbulent current: waves propagating with the current, J. Fluid Mech., 116 (1982), 227–250. [12] P. H. Kemp and R. R. Simons, The interaction between waves and a turbulent current: waves propagating against the current, J. Fluid Mech, 130(1983), 73–89. [13] G. Klopman, Vertical structure of the flow due to waves and currents, Delft Hydraulics, Progress Report No. H840.30, Part II, prepared for Rijkwaterstaat, Tidal WaterDivision and Commission of the European Communities, Directorate General XII , Delft, The Netherlands, 1994. [14] R. Koole and C. Swan, Dispersion of pollutant in a wave environment.” Proc., 24th Int. Conf. on Coastal Engineering, Japan, ASCE, NY, (1994), 3017-3085. [15] H. Liu, Predicting dispersion coefficient of stream, J. Envir. Engg. Div., ASCE, 103(EE1) (1977), 59-69.

Close-form model for dispersion in shallow waves 21 [16] A. W. K. Law, Taylor dispersion of contaminant due to surface waves, Journal of hydraulic research, 38 (2000), 41-48. [17] R. S. McQuivey and T. N. Keefer, Simple method for predicting dispersion in stream, J. Envir. Engg. Div., 100(EE4) (1974), 997-1011. [18] H. Margaretha, Mathematical modelling of wave-current interaction in a hydrodynamic laboratory basin, PhD Thesis, University of Twente, Netherlands, 2005. [19] I. Nezu and H. Nakagawa, Turbulence in open-channel flows. Balkema Publishers, Rotterdam, Netherlands, pp. 281, 1993. [20] J. M. Pearson, I. Guymer, J. R. West and L. E. Coates, Effect of wave height on cross-shore solute mixing, J. Waterway, Port, Coastal and Ocean Engineering, ASCE, 128 (2002), 10-20. [21] U. Putrevu and I. A. Svendsen, A mixing mechanism in the near-shore region, Proc. 23rd Int. Conf. on Coastal Engineering, Italy, ASCE, New York, (1992), pp. 161-172. [22] P. M. Rowiński, A. Piotrowski and J. J. Napiórkowski, Are artificial neural network techniques relevant for the estimation of longitudinal dispersion coefficient in rivers?, Hydrological sciences–journal–des sciences, Hydrologiques, 50(1) (2005), 175-187. [23] G. G. Stokes, On the theory of oscillatory waves, Proceedings of the Cambridge Philosophical Society, 8 (1847), 441-455. [24] Seo Won II. and T. S. Cheong, Prediction of longitudinal dispersion coefficient in natural stream, J. Hydraulic Engg., ASCE, 124 (1998), 25-31. [25] B. Teng, M. Zhao and W. Bai, Wave diffraction in a current over a local shoal, Coastal Engineering, 42(22) (2001), 163–172. [26] P. D. Thompson, The propagation of small surface disturbance through rotational flow, Ann. N. Y. Acad. Sci., 51(3) (1949), 463–474. [27] G. I. Taylor, Proc. of Roy. Soc., London, England, 219A (1953), 186; see also 225, 473. [28] G. I. Taylor, The dispersion of matter in Turbulent flow through a pipe, Proceedings of the Royal Society, London, England, 223A (1954), 446-468.

22 S. Patil and V. P. Singh [29] M. Umeyama, Reynolds stresses and velocity distributions in a wave–current coexisting environment, J. Waterway, Port, Coastal, and Ocean Engineering, 131(5) (2005), 203–212. [30] H. Yasuda, Longitudinal dispersion of matter due to the shear effect of steady and oscillatory currents, J. Fluid Mech., 48 (1984), 383-403. [31] C. S. Yih, Surface waves in flowing water, J. Fluid Mech., 51(2) (1972), 209–220.

h

Blausius power law for steady current

Progressive surface wave Flow direction

Fig. 1a: Combined wave-steady shear flow field

a


0

0.05

0.1

0.15

0.2

1.0E+03 1.0E+04 1.0E+05 1.0E+06 1.0E+07Re

m

Fig 1b: Reynolds number v/s power (m)


Fig. 1c: Shear velocity profiles according to magnitude of m

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1 1.2

m=0.3 m=0.5

m=1

m=4m=7

m=15

m=50

N

orm

aliz

ed fl

ow d

epth

(y/d

)

Flow velocity for Um = 1


-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0 90 180 270 360

Fig. 2: Wave spectrum within a wave period. Lines with dots (Eq. 32); Only dots (experiments by Umeyama (2005)). (▲Run 1; ■ Run 2; ▬ Run 3; ● Run 4)

Phase Angle in degrees

Wav

e am

plitu

de in

met

ers


1 0.5 0

Non

-dim

ensi

onal

flow

dep

th

0 L/2 L Wavelength

Fig 3a: Vector diagram of current-only flow for m=7

0

2

4

6

8

10

12

14

16

18

20

Fig 3a: Vector diagram of current-only flow for m = 7


0

2

4

6

8

10

12

14

16

18

20 1 0.5 0

Non

-dim

ensi

onal

flow

dep

th

0 L/2 L Wavelength

Fig 3b: Flow structure of wave-sheady shear (m = 7) case for a = 0.04m, T = 2.85s, Kd = 0.315 and current as in Fig 1.


Fig 4a: LDC (♦) and WCVLDC (1st order, ■; 2nd order, ▲) v/s kh (m = 7)

0

0.12

0.24

0.07 0.14 0.21 0.28 0.35

a = 4cm

a = 3cm

a = 2cm

a = 1cm

no wave

kh

Dis

pers

ion

coef

ficie

nt (m

2 /sec

)

Close-form model for dispersion in shallow waves 29 Received: November, 2008

0

0.12

0.24

0.36

0.07 0.14 0.21 0.28 0.35

a = 4cm

a = 3cm

a = 2cm

a = 1cm

no wave

Fig 4b: LDC (♦) and WCVLDC (1st order, ■; 2nd order, ▲) v/s kh (m = 2)

kh

Dis

pers

ion

coef

ficie

nt (m

2 /sec

)

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