## Abstract

The depth-integrated continuity and momentum equations developed by Liu & Orfila are extended to include the effects of turbulent bottom boundary layer. The eddy viscosity model is employed in the boundary layer, in which the eddy viscosity is assumed to be a power function of the vertical elevation from the bottom. The leading-order effects of the turbulent boundary layer appear as a convolution integral in the depth-integrated continuity equation because of the boundary-layer displacement. The bottom stress is also expressed as a convolution integral of the depth-averaged horizontal velocity. For simple harmonic progressive waves, the analytical expression for the phase shift between the bottom stress and the depth-averaged velocity is obtained. The analytical solutions for the solitary wave damping rate due to a turbulent boundary layer are also derived. Prandtl's one-seventh power law is described in detail as an example.

## 1. Introduction

While studying many shallow water wave propagation problems, such as dam-break generated waves (Hogg & Pritchard 2004) and the run-up of solitary waves on a sloping beach (Carrier *et al*. 2003), the effects of bottom stress can become significant. Furthermore, the bottom stress has always been viewed as a driving force of sediment transport in shallow water (e.g. Pritchard & Hogg 2003). In most of the existing studies, the bottom stress is usually modelled empirically as a function of the free-stream velocity, in which the phase shift between the bottom shear stress and the free-stream velocity is ignored. Consequently, by using such a bottom shear stress model, the existing morphology models still cannot predict the evolution of seafloor, such as onshore sandbar migration (e.g. Gallagher *et al*. 1998).

To formulate more rigorously the effects of bottom boundary layer on the long-wave propagation, Liu & Orfila (2004) (hereafter referred as LO) adopted the following perturbation expansions for the dimensionless horizontal velocity vector, , and the vertical velocity component, *w*:(1.1)(1.2)where the small parameters are defined as(1.3)with being a typical wave amplitude, the water depth, a horizontal length-scale, which is related to the magnitude of wavelength, and the time-scale. The velocity potential has been introduced for the irrotational flow in the core region. The flow motions are essentially irrotational except in the bottom boundary layer, in which the leading order of magnitude of the horizontal rotational velocity component, , is of order so as to satisfy the no-slip boundary condition on the bed. From the continuity equation, a vertical velocity component of is generated inside the bottom boundary layer. Therefore, the irrotational flow in the core region must be corrected at order to fulfil the no-flux condition on the bed.

In terms of the dimensionless depth-averaged velocity, , and the total water depth, , the dimensionless continuity equation and momentum equation can be expressed as(1.4)(1.5)In LO, *ν* is considered as either the kinematic viscosity of the fluid or a constant eddy viscosity. In the continuity equation (1.4), represents the leading-order vertical rotational velocity on the bottom, . Thus, the leading-order boundary-layer effects are associated with the velocity deficit in the streamwise direction inside the boundary layer. The assumption was adopted. Specifically, in LO when the viscosity is a constant, is expressed as(1.6)We note that the theory developed in LO has been confirmed by laboratory measurements of viscous damping of solitary waves propagating in a wave tank (Liu *et al*. 2006).

In sediment transport modelling, the constant eddy viscosity assumption is usually employed for ripple bed condition. However, the eddy viscosity is usually better approximated as a function of the vertical distance from the bottom, using the mixing length concept, for flat bed (sheet flow) conditions. For the steady-state turbulence boundary layer over a flat plate, Prandtl (1952) has shown that the one-seventh power law is a good approximation for the velocity profile up to the Reynolds number, , based on the distance from the tip of the plate. Thus, the streamwise velocity, , is proportional to the seventh root of the distance from the plate, ,(1.7)in which is the frictional velocity. For greater values of the Reynolds number, the streamwise velocity can be approximately proportional to the eighth, ninth and tenth roots of . These formulae are approximations to the logarithmic law. Assuming that shear stress is approximately constant in the boundary layer and adopting the eddy viscosity model,(1.8)one can express the eddy viscosity in terms of as(1.9)Therefore, in a steady-state turbulent boundary layer if the streamwise velocity is approximately proportional to the seventh, eighth, ninth and tenth roots of , then the corresponding eddy viscosity is proportional to six-seventh, seven-eighth, eight-ninth and nine-tenth roots of , respectively. We remark here that the formulation of the eddy viscosity shown above is based on a smooth bed and the viscous wall scale, , is employed. However, this formulation can be extended to rough bed conditions, in which an empirical roughness height, , is used as the proper length-scale in (1.5).

In this paper, we will consider the effects of a transient turbulent boundary layer on long-wave propagation. In the turbulent boundary layer, we will employ an eddy viscosity model, in which the eddy viscosity is assumed to be a power function of the distance from the bottom, in the same way as in Prandtl's theory for a steady-state boundary layer. The power considered is always greater than zero and less than one. Using the method of Green' function, we will find the analytical solutions for the transient turbulent boundary layer in terms of the free-stream velocity. The leading-order vertical rotational velocity, , which is required in the depth-integrated continuity equation, (1.4), can also be obtained analytically by integrating the continuity equation. Thus, a new set of Boussinesq equations with the consideration of the turbulent boundary layer is obtained. For simple harmonic progressive waves, the phase shift between the bottom stress and the free-stream velocity is obtained. The analytical solutions for the solitary wave damping rate are also derived for a turbulent boundary layer.

## 2. Bottom turbulent boundary-layer analysis

The leading-order shear stress inside the bottom boundary layer is modelled in the dimensional form as follows:(2.1)where the dimensional eddy viscosity is expressed as(2.2)in which denotes the location of the seafloor, and are considered as known dimensional constants. We remark that the eddy viscosity models, (2.1) and (2.2), are equivalent to Prandtl's eddy viscosity model for the steady-state turbulent boundary layer. In the case of Prandtl's one-seventh power law, and . To be consistent with the scales and notations introduced in LO, we introduce the effective viscosity, , so that the small parameter, *α*, is redefined as(2.3)Consequently, the dimensionless eddy viscosity is expressed as(2.4)and(2.5)We remark that the frictional velocity, , depends on . This makes the boundary-layer problem nonlinear. We shall simplify the problem by assigning a representative constant to .

Following LO, the leading-order continuity equation for the rotational velocity in the bottom boundary layer becomes(2.6)while the leading-order momentum equations can be expressed as(2.7)The no-slip conditions in the bottom require that the rotational velocity should satisfy the following boundary conditions:(2.8)At the outer edge of the boundary layer, where , the horizontal rotational velocity components vanish,(2.9)We note that represents the velocity potential in the core flow region (see LO). The depth-averaged velocity, , is defined in terms of as(2.10)

### (a) Analytical solution for boundary-layer velocities and bottom stress

To find the boundary-layer solution for , we first introduce the transformation,(2.11)and convert the boundary-layer equation (2.7) into the following form:(2.12)The initial condition for is(2.13)and the boundary conditions remain in the same form as (2.8) and (2.9). Thus,(2.14)(2.15)Note that the approximation (2.10) has been employed.

The analytical solution for the two-point initial-boundary-value problem described above can be obtained by means of Green's function method. By replacing by in equations (2.12)–(2.15), we can construct Green's function of equation (2.12), *G*, which must satisfy the adjoint equation(2.16)with the causality condition(2.17)and homogeneous boundary conditions(2.18)

By multiplying (2.12) by *G* and (2.16) by , taking the differences of the two resulting equations and integrating the difference over from 0 to , and over from 0 to *t*, we obtain(2.19)

By applying Green's theorem to the first integral on the right-hand side of equation (2.19) and performing the second integral, we can simplify it to(2.20)Thus, once Green's function is known and its normal derivative at the bottom can be evaluated, (2.20) provides the analytical solution for the initial-boundary-value problem given in (2.12)–(2.15).

To construct Green's function, we first split equation (2.16) into two equations (Stakgold 1968),(2.21)and(2.22)By using the method of separation of variables we can find the basic solutions of (2.21) as (Sutton 1943)(2.23)where denotes the Bessel function of the first kind and of order *q*, with . denotes the separation constant and *K* is an arbitrary constant. Green's function is the linear superposition of (2.23), and and *K* are determined by the boundary conditions (2.18) and (2.22).

It is known (Watson 1922) that(2.24)in which is the modified Bessel function. Let(2.25)then the above identity becomes(2.26)

Thus, the following possible solution is obtained:(2.27)where refers to the plus and minus sign of the modified Bessel function and is the arbitrary constant to be determined.

Both *G*_{1} and *G*_{2} satisfy the homogeneous conditions at for *q*>0 and . However, as the argument of Green's function becomes insignificant, , the modified Bessel function becomes(2.28)Therefore, the homogeneous boundary condition, (2.18), at can only be satisfied by as shown in (2.27). The coefficient can be determined by substituting into (2.22) and integrating with respect to from 0 to . The result shows that must be the same as *q*; . Thus, Green's function, satisfying all the requirements, is

It is straightforward to show that(2.29)By substituting (2.29) into (2.20), we obtain(2.30)Note that we have replaced by *T* as a dummy variable.

In terms of the stretched boundary-layer coordinate, *η*, the above equation becomes(2.31)The corresponding vertical rotational velocity can be obtained by integrating the continuity equation, (2.6), subject to the boundary condition, (2.9). On the bottom, where *η*=0, the vertical rotational velocity becomes(2.32)By definition, the dimensionless shear stress inside the boundary layer can be expressed as(2.33)Therefore, from (2.31), the leading-order bed shear stress, at *η*=0, can be expressed as(2.34)The bottom shear stress is the time integration of local acceleration, which is weighted by the function for . The memory is longer for smaller *q* values and hence higher Reynolds number flows. Note that to obtain equation (2.34), the initial velocity, , has been assumed to be zero.

### (b) Special cases

In this section, the boundary-layer solutions for two special cases are presented. The first case corresponds to the constant viscosity case, i.e. *p*=0 and . From (2.31), the horizontal rotational velocity becomes(2.35)and the vertical rotational velocity at the bottom becomes the same as (1.6). Note that has been used. For the bottom shear stress, (2.34) becomes(2.36)Equations (2.35) and (2.36) are the same as those found by LO.

In the second case, we examine Prandtl's one-seventh power velocity profile, i.e. and . Again, from (2.31), the horizontal rotational velocity becomes(2.37)with the vertical rotational velocity at *η*=0 as(2.38)For the bottom shear stress, (2.34) becomes(2.39)It is clear that the bottom stress has a longer memory on (or is affected more by) the past history of the bottom acceleration for the turbulent boundary layer.

Finally, if we consider only the periodic flows in time, i.e. , the bottom stress, (2.34), can be integrated analytically as(2.40)in which is the phase shift between the free-stream velocity and the bottom stress. For the laminar viscosity or the constant eddy viscosity, the phase shift is , while the phase shift is smaller for turbulent boundary layer, e.g. for Prandtl's formulation, the phase shift becomes . These results agree very well with the experimental data reported in Fredsoe & Deigaard (1992), in which the phase shift is about for the turbulent case.

## 3. Boussinesq equations

By substituting the expression of the vertical rotational velocity, (2.32), into the depth-integrated continuity equation, (1.4), we obtain(3.1)The corresponding leading-order depth-integrated momentum equation remains the same as that shown in (1.5). Thus, (3.1) and (1.5) constitute the Boussinesq equations in terms of the depth-averaged velocity, , and the total water depth, *H*, when the effects of a turbulent bottom boundary layer are included.

### (a) Viscous damping of solitary wave

To examine the viscous damping of solitary waves, we shall consider one-dimensional problems in this section. Thus, the continuity and momentum equations become(3.2)

(3.3)

Following the approach outlined in LO, (3.2) and (3.3) can be combined in terms of a moving frame, , as(3.4)in which the slow time variable is defined as . Without the damping effect, i.e. , the solitary wave solution can be written as(3.5)Thus, with the viscous damping, we introduce the perturbation solution as follows:(3.6)By substituting (3.6) into (3.4), and by collecting the terms at different order, we obtain the following equations for the first two orders in *δ*:(3.7)(3.8)where and are adjoint operators of each other, i.e.(3.9)Clearly, the solution for the leading-order equation is just the solitary wave solution,(3.10)Equation (3.9) provides a solvability condition for (3.11)The items in the integrand of the above integral can be expressed explicitly asThus, by substituting the above equations and equation (3.10) into the solvability condition, (3.11), we find(3.12)with(3.13)The above equation can be integrated numerically for a given *q* value. *β* can be approximated as a linear function of *q* for as(3.14)We note that for the constant eddy viscosity case, and *β* becomes , and (3.12) can be integrated to get(3.15)On the other hand, for the case of Prandtl's one-seventh power law, and *β* is approximately 0.5698. Hence, (3.12) can be integrated to get(3.16)

The dimensional forms of the damping rate can be recovered by using the definitions of the dimensionless variables. Thus, for the case of constant viscosity we obtain(3.17)On the other hand, for the case of Prandtl's one-seventh power law, the dimensional damping formula can be written as(3.18)in which we have specified the frictional velocity .

We remark here that Jensen *et al*. (2003) conducted an experimental study of solitary run-up on a steep slope. They reported that the observed wave attenuation was significantly bigger than that estimated by the laminar viscous damping. In view of the new results presented here, further examination of the experimental data will be performed.

## 4. Concluding remarks

Analytical solutions for a transient turbulent boundary-layer flow are obtained. The eddy viscosity is assumed to be a power function of the distance from the bottom. The corresponding Boussinesq type equations are presented with the turbulent boundary-layer effects considered. The effects of the boundary layer appear in the continuity equation as a convolution integral. An numerical scheme can be developed to solve the depth-integrated equations as shown in (3.1) and (1.5) for practical applications.

## Acknowledgements

This work was supported by the National Science Foundation. The author would also like to thank Drs A. Orfila and G. Simarro for their able assistance in preparing this paper.

## Footnotes

- Received December 8, 2005.
- Accepted May 11, 2006.

- © 2006 The Royal Society