Boundary Conditions for 2D Boussinesq-type Wave-Current Interaction Equations

This research focuses on the development of a set of two-dimensional boundary conditions for specific governing equations. The governing equations are existing Boussinesqtype equations which is capable of simulating wave-current interaction. The present boundary conditions consist of for waves only case and for currents only case. To simulate wave-current interaction, the two kinds of the present boundary conditions are then combined. A numerical model based on both the existing governing equations and the present boundary conditions is applied to simulation of currents only and of wave-current interaction propagating over a basin with a submerged shoal. The results of the numerical model show that the present boundary conditions go well with the existing Boussinesq-type wave-current interaction equations.


Introduction
Boussinesq [1] developed the original formulation of the governing equations for a free surface flow, which included the effects of surface waves but in which the vertical dimension was eliminated.The formulation was in terms of the bottom velocity and was restricted to simulating waves moving over bathymetry with a flat bottom.The governing equations consist of one continuity equation and two momentum equations (in x and y directions).The governing equations were then called as Boussinesq equations.
Peregrine [2] developed two new formulations in two horizontal dimensions for the case of varying depth in terms of (i) the depth-averaged velocity vector and (ii) the velocity vector at still water level.The first formulation became known as the standard form of Boussinesq-type equations.
In 1993, Nwogu [3] developed a new approach in the derivation of a novel set of Boussinesq-type equations.The resulting equations are capable of simulating wave propagating over arbitrary bathymetry in terms of the horizontal velocity at an arbitrary level (Z = Z1) below still water level (-h < Z1 < 0) in which h is the local still water depth.The Boussinesq-type wave equations by Nwogu were solved numerically in one dimension by Nwogu and in two dimensions by Wei and Kirby [4].
Derivation of two-dimensional (2D) boundary conditions for the governing equations of Nwogu was also discussed by Wei and Kirby.
In 1998, Chen et al. [5] extended the Boussinesq-type wave equations of Nwogu by incorporating a steady ambient current.As a result, the corresponding equations were capable of simulating wave-current interaction.The continuity equation can be expressed as and the momentum equation is where g = gravitational acceleration η = free surface elevation The one dimensional (1D) form of these equations had been solved numerically by Chen et al. [5].However, the 2D one has not been solved yet.To do so, the suitable 2D boundary conditions need to be derived first.This is the objectives of the present study.Discussion regarding the numerical solution algorithm is not presented here, but can be found in Mera [6].

Boundary conditions for waves only case
The set of 2D boundary conditions for waves only case is discussed first.The other cases of currents only and waves plus currents will be considered in subsequent sub-section.

Incoming wave boundary conditions
The free surface elevation η at the incoming wave boundary can be varied sinusoidally as In which Hi, is the incoming wave height, k, the wave number vector, x, the horizontal spatial vector, ω, the angular frequency = 2π/T, T, the wave period, and t is time.
For a locally constant depth, the continuity equation (Eq. 1) reduces to where A periodic, small amplitude wave can be defined as where: ηa = amplitude of the water surface elevation; ua = amplitude of the horizontal velocity vector; i = 1 θi = incoming wave angle.
Substituting Equation 5into Equation 4gives the horizontal velocity vector at the incoming wave boundary, i.e.

Outgoing wave boundary conditions
The boundary condition for η is considered first.At the outgoing wave boundary, the Sommerfeld radiation condition [3] is used to allow the passage and egress of the wave energy, that is where the wave celerity C = |C|cos θi + |C|sin θi.
The boundary condition for the velocity components is considered next.The depth-integrated continuity equation can be expressed in terms of the depthaveraged horizontal velocity vector u as To eliminate ηt, substitute Equation 8 into the Sommerfeld radiation condition (Eq.7) to give For a locally constant depth, Equation 9 may be integrated over the fluid domain to obtain the horizontal velocity vector Having solved for u in Equation 10 is used to determined u.

Reflecting wave boundary conditions
The kinematic boundary condition at an impermeable wall can be stated as where n is an outward normal vector, Ω is the fluid domain, ∂Ω is the boundary and x is a position in the boundary.Consider, for example, the case of an impermeable wall being parallel to the x-axis.Equation 11 is a boundary condition and can be written as The slope and the curvature of v normal to the impermeable wall is assumed to be zero and expressed respectively as or can be written symply as vy = 0 and vyy = 0 in which the subscript y denotes partial differentiation with respect to the y direction The continuity equation (Eq. 1) can be expressed in terms of the volume flux vector Q as where The kinematic boundary condition in terms of the volume flux vector at an impermeable wall as For the case of the impermeable wall being parallel to the x-axis, the volume flux in the y-direction at the boundary becomes zero or where the subscripts x and y denote partial differentiation with respect to the x and y directions respectively.Substituting Equation 18 into Equation 1gives a reflecting wave boundary condition for calculating the free surface elevation at the boundary wall as set out below where For a locally constant depth, the horizontal velocity in the x-direction may be obtained by substituting Equations 12 and 13 into Equation 18giving For a boundary being parallel to the x-axis, the boundary conditions are Equations 19, 20 and 12 for η, u and v respectively.The derivation of the present boundary conditions for waves only case follows Mera [7], who derived a set of boundary conditions for the Boussinesq-type wave equations of Nwogu.

Inflow Boundary Conditions
At the upstream end, the depth-averaged velocity is specified but the boundary condition also needs to involve η.Inflow currents are bound to flow in the xdirection.One way of linking u and η at the upstream end is to combine the Sommerfeld radiation condition (Eq.7) and the continuity equation (Eq.8) to give

Outflow Boundary Conditions
At the downstream end, the free surface elevation can be predicted using Equation 7 in the x direction or η t + Cη x = 0 (23) Then, the horizontal velocities are predicted using Equation 16 in the x direction or and in the y direction v = 0 (25)

No-flow Boundary Conditions
At the boundaries, which are paralel to the flow, the boundary conditions are equivalent to the reflecting wave boundary conditions.

Boundary conditions for wave-current interaction cases
The governing equations considered here were derived based on a steady ambient current.In the model tests, the following procedure is adopted.
(1) Model is run with current only.
(2) The results from the model settle down to a steady state.(3) After the steady state is reached, a sinusoidally varying surface elevation is imposed at the inflow or outflow boundary.This results in a wave train propagating into the computational domain.
Please note that breaking waves are not included in the existing governing equations and in the present boundary conditions considered in this paper.But, Purwanto [8] discussed a Quasi-Equilibrium Turbulent Energy (QETE) model with boundary conditions for breaking and non-breaking waves.
The QETE model was intended for simulation of (density) current only case.The density current is a type of current that occurs when fluid flow enters a fluid body of different density.

Numerical set-up
The numerical set-up consists of a basin with a submerged shoal.The basin is 18 m long and 18.2 m wide (Figure 1).Side walls are at y = 0 and 18.2 m.The centre of the shoal was located at (x,y) = (13,9.22)m with the perimeter given by (x -13)

Current only case
In the first test, a flat water surface (initial value of η = 0) and a steady velocity of 0.10 m/s is imposed at the Southern boundary (x = 0 m).The computation is carried out with a mesh of ∆x = ∆y = 0.1 m and ∆t = 0.02 s.The imposed current flows from x = 0 m to x = 18 m, and reaches a steady state condition after about t = 65 s.Figures 2 shows a perspective view of the free surface elevation (upside down) at t = 65 s predicted by a numerical model based on the existing governing equations of Chen et al. [5] and the present boundary conditions.This figure indicates that the present boundary conditions go well with the existing governing equations.In the second test, a flat water surface (initial value of η = 0) and a constant inflowing velocity of 0.10 m/s is imposed in the opposite direction to that in first test, i.e. at the Northern boundary (x = 18 m) instead of at the Southern boundary.This leads to a steady current flowing from x = 18 m to x = 0 m of the basin (not presented here).

Waves and opposing current
Once the currents in the basin reach a steady state (after about t = 65 s) in the first test, the free surface elevation at the southern boundary (x = 18 m) is sinusoidally varied with time to generate an incident wave.The incoming wave specifications and the grid resolution remain the same as is used in the previous tests i.e. ∆x = ∆y = 0.1 m and ∆t = 0.02 s.A wave train with a period of 1.0 s and a wave height of 0.0118 m comes prependicularly to the fluid domain.At the incoming wave boundary (x = 18 m), the ambient current is allowed to pass through, leaving the flow domain.A perspective view of the predicted surface elevation at t = 20 s and at t = 40 s are shown in Figures 3 and 4 respectively.From these figures can be seen that the present boundary conditions are still suitable to the existing governing equations.

Waves and current in same direction
On top of the steady current field (second test), a sinusoidal wave train with a period of 1.0 s and a wave height of 0.0118 m is imposed at x = 18 m.The incoming wave period and wave height are same as is used in the last test.Perspective views of the predicted free surface elevation at t = 20 s and at t = 40 s are shown in Figures 5 and 6 respectively in which the present boundary conditions are still going well with the existing governing equations.

Conclusions
A set of two-dimensional boundary conditions for the existing Boussinesq-type wave-current interaction equations is developed.The boundary conditions consist of for waves only case and for currents only case.Two kinds of the present boundary conditions are then combined in simulation of wave-current interaction.
A numerical set-up consists of a basin of 18 m long and 18.2 m wide with a submerged shoal.The numerical model runs stably at least for 40 s in simulation time as shown by the numerical results.This indicates that the present boundary conditions are suitable to the existing governing equations.

Figure 1 .
Figure 1.A perspective view of numerical set-up.

Figure 3 .
Figure 3. Waves propagating against a steady, opposing current at t = 20 s.

Figure 5 .
Figure 5. Waves propagating with a co-flowing steady current at t = 20 s.

Figure 6 .
Figure 6.Waves propagating with a co-flowing steady current at t = 40 s.