# Numerical Simulation of the Field Velocities and Local Disturbances of a Long Gravity Wave Passing above an Immersed Vertical Barrier

Authors: Laouar Abdelhamid; Guerziz Allaoua
Journal: Differential Equations and Nonlinear Mechanics
 (2008)

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## Abstract

This work is interested in the study of the passage of a long gravity wave above an immersed vertical barrier. The latter is placed at a right angle in the middle of the occupied fluid domain which is limited vertically by both a free surface and an impermeable horizontal bottom. We want to determine the field velocity and the local disturbances in the vicinity of the barrier. For this, we use the generalized theory of shallow water and complex variables method. For illustration, we consider a solitary wave as an emitted long wave.


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## Body

## 1. Introduction

The problem of the passage of gravity  waves above an immersed obstacle was studied by several authors (cf. [1–5]). Thus we mention some results
already obtained in this field. Dean's work [2] is classified in the theory of
the short waves; it concerns particularly the determination of the coefficients
of reflection and transmission of the waves passing a completely immersed
obstacle (e.g., a barrier). Gulli [4] has studied a same type of obstacle but
in the case of the long waves and he has concluded, at the first order of
approximation, that there is not a reflected wave by the obstacle. The studies
of Seabra-Santos et al. [5] concern the deformation and the dephasing of the free surface in
shallow water due to the solitary mutual interactions or with an isolated
obstacle. The theoretical and experimental contributions of Barthélemy et al.
[1] concern the phenomena resulting from the internal long waves. In [6], the
study of a passage of a long wave over vertical barrier, Germain's shallow
water theory associated with the complex variable method has been used to
determine the flow.Our objective in the present paper is to apply the
shallow water theory and complex variable method in order to determine the
local disturbances at the vicinity of the obstacle and simulate the velocity
field. So, we would like to know particularly with accuracy the kinematic of
the flow at the vicinity of the obstacle while the passage of the long gravity
wave above the thin immersed vertical barrier. For illustration, we consider a
solitary wave generated at the upstream by a piston wave maker.The plan of this paper is as follows. Section2
contains two parts: the first part describes the phenomenon, the second part
gives general equations and mathematical model. Section 3 gives a determination
of the system of equations and the technique of resolution. The last section
presents an application and a numerical simulation.


## 2. Position of the Problem

### 2.1. Description of the Phenomenon

We consider a fixedOxy reference system, where the axis Oy is vertically ascendant and the axis Ox coincides with the initial free surface. The
position of the fluid particle at the moment t, t>0,
is denoted by (x,y) and their coordinates at the initial position
by (a,b), where a, b, and t are the variables of Lagrange. Now we
introduce new components X and Y as follows:
(2.1)X(a,b,t)=x(a,b,t)−a,Y(a,b,t)=y(a,b,t)−b.
The assumption of the shallow water theory (see [3]) introduces
distortion space and temporal variables,
translating the difference in scale between the horizontal and
vertical sizes. This distortion will be characterized by the small parameter which is dependent on the relative long wave amplitude. Thus
(2.2)α=εa,β=b,τ=εgh⋅t,
where gh represents the critical celerity of the
propagated long waves (h and g are depth of fluid at rest and gravity, resp.).The description
of the phenomenon is as follows: we consider the domainD occupied by fluid of an infinite horizontal
band which is limited vertically by a free surface β=0 and an impermeable horizontal bottom β=−h.
A vertical, thin, and impermeable barrier is placed
in the middle of the occupied fluid at right angle to the bottom. The top of
the barrier is defined by (α=0; β=−l).
A data long wave Xe(α−τ) emitted upstream passes above the
obstacle by creating local disturbances in its vicinity (see Figure 1).Figure 1
Local disturbances at vicinity of the barrier.


### 2.2. General Equations and Mathematical Model

General equations and mathematical model are listed below:(i)
the kinematic condition expresses the incompressibility
of fluid(2.3)∂Y∂β+ε[∂X∂α+∂X∂α∂Y∂β−∂X∂β∂Y∂α]=0;(ii)
the dynamic condition for an
irrotational movement(2.4)∂2X∂β∂τ+ε[∂X∂α∂2X∂β∂τ−∂X∂β∂2X∂α∂τ+∂Y∂α∂2Y∂β∂τ−(1+∂Y∂β)∂2Y∂α∂τ]=0;(iii)
the impermeability boundary
conditions(2.5)∂Y∂α+εh∂2X∂τ2+ε2h[∂X∂α∂2X∂τ2+∂Y∂α∂2Y∂τ2]=0atthefreesurface(β=0),Y(α,β,τ)=0,atthebottom(β=−h),X(α,β,τ)=0,ontheobstacle;(iv)
the initial
conditions(2.6)X(α,β,−∞)=0,Y(α,β,−∞)=0(atrest).
The resolution of (2.3)–(2.6) requires that the solutions take into account the interaction fluid-obstacle.
According to the shallow water theory (see Germain [3]), the solution will be
calculated under the entire series in ε:
(2.7)X(α,β,τ)=∑n=1∞εn[Xn,0(α,β,τ)+∑m=1∞Xn,m(α,β,τ)exp(−mλαε)],Y(α,β,τ)=∑n=1∞εn[Yn,0(α,β,τ)+∑m=1∞Yn,m(α,β,τ)exp(−mλαε)],where λ is a determining constant.The double sum
(m≠0) in the formula (2.7) characterizes the local
disturbances whose amplitude decreases exponentially with the distance. The
technique of resolution consists to inject these series in the general
equations and to write that they are satisfied with the order (n,m) desired. For example, at the first order of
approximation (i.e., n=1), the components Xn,m and Yn,m for m=0 are given as
(2.8)X1,0(α,β,τ)=X1,0(α,τ),Y1,0(α,β,τ)=0,
and for any m≠0,(2.9)X1,m(α,β,τ)=A1,m(τ)cosmλβ,Y1,m(α,β,τ)=A1,m(τ)sinmλβ.


## 2.1. Description of the Phenomenon

We consider a fixedOxy reference system, where the axis Oy is vertically ascendant and the axis Ox coincides with the initial free surface. The
position of the fluid particle at the moment t, t>0,
is denoted by (x,y) and their coordinates at the initial position
by (a,b), where a, b, and t are the variables of Lagrange. Now we
introduce new components X and Y as follows:
(2.1)X(a,b,t)=x(a,b,t)−a,Y(a,b,t)=y(a,b,t)−b.
The assumption of the shallow water theory (see [3]) introduces
distortion space and temporal variables,
translating the difference in scale between the horizontal and
vertical sizes. This distortion will be characterized by the small parameter which is dependent on the relative long wave amplitude. Thus
(2.2)α=εa,β=b,τ=εgh⋅t,
where gh represents the critical celerity of the
propagated long waves (h and g are depth of fluid at rest and gravity, resp.).The description
of the phenomenon is as follows: we consider the domainD occupied by fluid of an infinite horizontal
band which is limited vertically by a free surface β=0 and an impermeable horizontal bottom β=−h.
A vertical, thin, and impermeable barrier is placed
in the middle of the occupied fluid at right angle to the bottom. The top of
the barrier is defined by (α=0; β=−l).
A data long wave Xe(α−τ) emitted upstream passes above the
obstacle by creating local disturbances in its vicinity (see Figure 1).Figure 1
Local disturbances at vicinity of the barrier.


## 2.2. General Equations and Mathematical Model

General equations and mathematical model are listed below:(i)
the kinematic condition expresses the incompressibility
of fluid(2.3)∂Y∂β+ε[∂X∂α+∂X∂α∂Y∂β−∂X∂β∂Y∂α]=0;(ii)
the dynamic condition for an
irrotational movement(2.4)∂2X∂β∂τ+ε[∂X∂α∂2X∂β∂τ−∂X∂β∂2X∂α∂τ+∂Y∂α∂2Y∂β∂τ−(1+∂Y∂β)∂2Y∂α∂τ]=0;(iii)
the impermeability boundary
conditions(2.5)∂Y∂α+εh∂2X∂τ2+ε2h[∂X∂α∂2X∂τ2+∂Y∂α∂2Y∂τ2]=0atthefreesurface(β=0),Y(α,β,τ)=0,atthebottom(β=−h),X(α,β,τ)=0,ontheobstacle;(iv)
the initial
conditions(2.6)X(α,β,−∞)=0,Y(α,β,−∞)=0(atrest).
The resolution of (2.3)–(2.6) requires that the solutions take into account the interaction fluid-obstacle.
According to the shallow water theory (see Germain [3]), the solution will be
calculated under the entire series in ε:
(2.7)X(α,β,τ)=∑n=1∞εn[Xn,0(α,β,τ)+∑m=1∞Xn,m(α,β,τ)exp(−mλαε)],Y(α,β,τ)=∑n=1∞εn[Yn,0(α,β,τ)+∑m=1∞Yn,m(α,β,τ)exp(−mλαε)],where λ is a determining constant.The double sum
(m≠0) in the formula (2.7) characterizes the local
disturbances whose amplitude decreases exponentially with the distance. The
technique of resolution consists to inject these series in the general
equations and to write that they are satisfied with the order (n,m) desired. For example, at the first order of
approximation (i.e., n=1), the components Xn,m and Yn,m for m=0 are given as
(2.8)X1,0(α,β,τ)=X1,0(α,τ),Y1,0(α,β,τ)=0,
and for any m≠0,(2.9)X1,m(α,β,τ)=A1,m(τ)cosmλβ,Y1,m(α,β,τ)=A1,m(τ)sinmλβ.


## 3. Local Disturbances (Case the Obstacle is a Barrier)

### 3.1. System of Equations

At the first
order of approximation (n=1), we will calculate the coefficient of the local
disturbances. for this, we process as follows. We divide the domain D into two parts D−={−h≤β≤0;α<0} and D+={−h≤β≤0;α>0} (see Figure 2). On each domain D− or D+,
according to the point of view of Lagrange, the horizontal and vertical
components of the displacement of particle (X1−, Y1−) and (X1+, Y1+), respectively, can be written as
follows:(3.1)D−{X1−(α,β,τ)=Xe(α−τ)+∑m=1∞A1,m−(τ)cosmλ−β⋅exp(−mλ−αε),Y1−(α,β,τ)=∑m=1∞A1,m−(τ)sinmλ−β⋅exp(−mλ−αε),(3.2)D+{X1+(α,β,τ)=Xe(α−τ)+∑m=1∞A1,m+(τ)cosmλ+β⋅exp(−mλ+αε),Y1+(α,β,τ)=∑m=1∞A1,m+(τ)sinmλ+β⋅exp(−mλ+αε).A1,m− and A1,m+ are unknown functions depending on the
variable τ,
characterizing the amplitude of local disturbances in the domains D− and D+,
respectively, and λ− and λ+ are given coefficients (we take here λ+=−λ−=λ=π/h). We note that the approximation equations of
the phenomenon have been written at the first order of approximation (1,m).Figure 2
DomainsD+ and D−.Both the continuity conditions of the flow at the border of the two domains and the
impermeability condition of the barrier imply that(3.3a)X1−(0,β,τ)=X1+(0,β,τ),α=0,−h≤β≤0,(3.3b)Y1−(0,β,τ)=Y1+(0,β,τ),α=0,−l≤β≤0,(3.3c)X1+(0,β,τ)=0,α=0,−h≤β≤−l.
The condition (3.3a) applied to (3.1) and (3.2) gives the equality
(3.4)A1,m−(τ)=A1,m+(τ)=A1,m(τ).
The latter expresses the symmetry of the local disturbances at the vicinity of the barrier.The continuity
conditions (3.3a) and (3.3b) of the flow on the segment [α=0;−l≤β≤0] give the following equations:(3.5)∑m=1∞A1,m(τ)sinmπβh=0,α=0,−l≤β≤0,(3.6)Xe(−τ)+∑m=1∞A1,m(τ)cosmπβh=0,α=0,−l≤β≤0. By considering
the linearity of (3.6), if Xe(−τ)≠0,
we have(3.7)1+∑m=1∞a1,mcosmπβh=0,(3.8)wherea1,m=A1,m(τ)Xe(−τ)isaconstant.The problem leads us to solve
the following system:(3.9)1+∑m=1∞a1,mcosmπβh=0,α=0,−h≤β≤−l,∑m=1∞a1,msinmπβh=0,α=0,−l≤β≤0.Thus the coefficients a1,m are obtained starting from these two
time-independent equations.


### 3.2. Technique of Resolution

In the domainD+,
we consider two relations (3.2) of the components X1+ and Y1+ and (3.8). We can then
construct an analytic complex variable function z=(α/ε−iβ) such that(3.10)f+(αε−iβ)=1+∑m=1∞a1,mexp(−mπh(αε−iβ)).This function bounded in the
domain D+ except at the singular points satisfies the
impermeability condition at the free surface β=0 and the bottom β=−h.
Therefore,
(3.11)Y+(α,β)=Imf+(αε−iβ)=0,ontheboundariesβ=0,β=−h,
and the impermeability condition of the barrier leads to
(3.12)X+(α,β)=Ref+(αε−iβ)=0,onthesegment(α=0;−h≤β≤−l).
In the same way, we build in D− a second analytical complex variable function z=α/ε−iβ as follows:(3.13)f−(αε−iβ)=1+∑m=1∞a1,mexp(mπh(αε−iβ)).The continuity conditions at the
border of the two domains and by the analytical extension allow the following
equality:(3.14)f+(αε−iβ)=f−(αε−iβ)=f(αε−iβ).The determination of local
disturbances a1,m leads to find a function f,
analytical in the domain {−∞<α<+∞;−h≤β≤0} without the segment {α=0;−h≤β≤−l}.
This function is bounded in this domain except at the singularity point.
Furthermore, it verifies the following restrictions:(3.15)Imf(αε−iβ)=0,ontheboundariesβ=0,β=−h,Ref(αε−iβ)=0,on[α=0;−h≤β≤−l],f(αε−iβ)equaltounityatinfinity. According to [6],
we can obtain a conform transformation of this domain on the superior
half-plane (see Figure 3) and seek an analytic function satisfying the
conditions on the boundaries in the later. We find, in particular, the value of δ=(cos(πl/2h))−1 (see [6]).Transformation of
the plane(z) on the half plane (ζ).
(a)(b)Figure 4
The field velocityh=20cm, l=10cm, and ε=0.6.Figure 5
The local
disturbancesh=20cm, l=10cm, and ε=0.6.By using the
formula of Schwarz-Christofel [6], the conform transformation is written
as(3.16)z=−hπln(ζ2−1−tan(πl/2h)ζ2−1+tan(πl/2h)),and its inverse transformation
is(3.17)ζ=1+tan2πl2hcoth2πz2h.The function H(ζ)=f(z) is holomorphic in the superior half-plane and
also bounded except at the point A3′.
On the real axis, this function verifies the following
conditions:(3.18)Im[H(ζ)]=0,onthesegment[A6′,A1′];[A1′,A2′],Re[H(ζ)]=0,onthesegment[A2′,A3′];[A3′,A4′],Im[H(ζ)]=0,onthesegment[A4′,A5′];[A5′,A6′].By applying the formula of
Signorini [6], we find the function(3.19)H(ζ)=1sin(πl/2h)(ζ2−1ζ),and returning to the z-plane using (3.17), we finally
find(3.20)f(αε−iβ)={sinπl2h⋅1+[tanh(π/2h)(α/ε−iβ)tan(πl/2h)]2}−1.This function describes the full
flow in particular the field of the disturbances in the vicinity of the
barrier. The coefficients a1,m can be calculated easily. In fact on the
vertical α=0,
the function (3.20) is(3.21)f(iβ)=2cos(πβ/2h)cos(πβ/h)−cos(πl/h). Furthermore the
relations (3.10) to (3.14) permit the function(3.22)f(iβ)=1+∑m=1∞a1,m(cosmπβh+isinmπβh),where the coefficients a1,m are obtained by using the Fourier series
expansion on the interval [−h,0].It follows
that(3.23)a1,m=2h∫0−hf(iβ)cosmπβhdβ.Thus(3.24)a1,m=Pm(cosπlh)+Pm−1(cosπlh), where Pm and Pm−1 are the Legendre polynomials of degree m and m−1,
respectively, in the third integral representation.ThePm formula is given below:(3.25)Pm(cosπlh)=21.3.5⋯(2m−1)2.4.6⋯2m{cosmπlh+1⋅m1⋅(2m−1)cos(m−2)πlh+1.3⋅m(m−1)1.2⋅(2m−1)(2m−3)cos(m−4)πlh+⋯}. Using (3.1), (3.2), (3.4), (3.8), and
(3.14), the horizontal and vertical components of the displacement of the flow,
in all domain D,
can be written as follows:(3.26)X1(α,β,τ)=Xe(α−τ)+Xe(−τ)∑m=1∞a1,mcosmπβhexp(−mπ|α|εh),Y1(α,β,τ)=Xe(−τ)∑m=1∞a1,msinmπβhexp(−mπ|α|εh),where X1 and Y1 are, respectively, the horizontal and vertical
components of the displacement of the particle.Remark 3.1.
According to (3.1), (3.2), and (3.26), one notices that displacements X1(α,β,τ) and Y1(α,β,τ) are the independent contributions of the parts
“wave” and “local disturbances” of the flow. On the first
order of approximation, the part “wave” of the flow imposes a
uniform distribution of displacements and speeds on a vertical. The part “disturbances local” permits to take into account the presence of the
obstacle.


## 3.1. System of Equations

At the first
order of approximation (n=1), we will calculate the coefficient of the local
disturbances. for this, we process as follows. We divide the domain D into two parts D−={−h≤β≤0;α<0} and D+={−h≤β≤0;α>0} (see Figure 2). On each domain D− or D+,
according to the point of view of Lagrange, the horizontal and vertical
components of the displacement of particle (X1−, Y1−) and (X1+, Y1+), respectively, can be written as
follows:(3.1)D−{X1−(α,β,τ)=Xe(α−τ)+∑m=1∞A1,m−(τ)cosmλ−β⋅exp(−mλ−αε),Y1−(α,β,τ)=∑m=1∞A1,m−(τ)sinmλ−β⋅exp(−mλ−αε),(3.2)D+{X1+(α,β,τ)=Xe(α−τ)+∑m=1∞A1,m+(τ)cosmλ+β⋅exp(−mλ+αε),Y1+(α,β,τ)=∑m=1∞A1,m+(τ)sinmλ+β⋅exp(−mλ+αε).A1,m− and A1,m+ are unknown functions depending on the
variable τ,
characterizing the amplitude of local disturbances in the domains D− and D+,
respectively, and λ− and λ+ are given coefficients (we take here λ+=−λ−=λ=π/h). We note that the approximation equations of
the phenomenon have been written at the first order of approximation (1,m).Figure 2
DomainsD+ and D−.Both the continuity conditions of the flow at the border of the two domains and the
impermeability condition of the barrier imply that(3.3a)X1−(0,β,τ)=X1+(0,β,τ),α=0,−h≤β≤0,(3.3b)Y1−(0,β,τ)=Y1+(0,β,τ),α=0,−l≤β≤0,(3.3c)X1+(0,β,τ)=0,α=0,−h≤β≤−l.
The condition (3.3a) applied to (3.1) and (3.2) gives the equality
(3.4)A1,m−(τ)=A1,m+(τ)=A1,m(τ).
The latter expresses the symmetry of the local disturbances at the vicinity of the barrier.The continuity
conditions (3.3a) and (3.3b) of the flow on the segment [α=0;−l≤β≤0] give the following equations:(3.5)∑m=1∞A1,m(τ)sinmπβh=0,α=0,−l≤β≤0,(3.6)Xe(−τ)+∑m=1∞A1,m(τ)cosmπβh=0,α=0,−l≤β≤0. By considering
the linearity of (3.6), if Xe(−τ)≠0,
we have(3.7)1+∑m=1∞a1,mcosmπβh=0,(3.8)wherea1,m=A1,m(τ)Xe(−τ)isaconstant.The problem leads us to solve
the following system:(3.9)1+∑m=1∞a1,mcosmπβh=0,α=0,−h≤β≤−l,∑m=1∞a1,msinmπβh=0,α=0,−l≤β≤0.Thus the coefficients a1,m are obtained starting from these two
time-independent equations.


## 3.2. Technique of Resolution

In the domainD+,
we consider two relations (3.2) of the components X1+ and Y1+ and (3.8). We can then
construct an analytic complex variable function z=(α/ε−iβ) such that(3.10)f+(αε−iβ)=1+∑m=1∞a1,mexp(−mπh(αε−iβ)).This function bounded in the
domain D+ except at the singular points satisfies the
impermeability condition at the free surface β=0 and the bottom β=−h.
Therefore,
(3.11)Y+(α,β)=Imf+(αε−iβ)=0,ontheboundariesβ=0,β=−h,
and the impermeability condition of the barrier leads to
(3.12)X+(α,β)=Ref+(αε−iβ)=0,onthesegment(α=0;−h≤β≤−l).
In the same way, we build in D− a second analytical complex variable function z=α/ε−iβ as follows:(3.13)f−(αε−iβ)=1+∑m=1∞a1,mexp(mπh(αε−iβ)).The continuity conditions at the
border of the two domains and by the analytical extension allow the following
equality:(3.14)f+(αε−iβ)=f−(αε−iβ)=f(αε−iβ).The determination of local
disturbances a1,m leads to find a function f,
analytical in the domain {−∞<α<+∞;−h≤β≤0} without the segment {α=0;−h≤β≤−l}.
This function is bounded in this domain except at the singularity point.
Furthermore, it verifies the following restrictions:(3.15)Imf(αε−iβ)=0,ontheboundariesβ=0,β=−h,Ref(αε−iβ)=0,on[α=0;−h≤β≤−l],f(αε−iβ)equaltounityatinfinity. According to [6],
we can obtain a conform transformation of this domain on the superior
half-plane (see Figure 3) and seek an analytic function satisfying the
conditions on the boundaries in the later. We find, in particular, the value of δ=(cos(πl/2h))−1 (see [6]).Transformation of
the plane(z) on the half plane (ζ).
(a)(b)Figure 4
The field velocityh=20cm, l=10cm, and ε=0.6.Figure 5
The local
disturbancesh=20cm, l=10cm, and ε=0.6.By using the
formula of Schwarz-Christofel [6], the conform transformation is written
as(3.16)z=−hπln(ζ2−1−tan(πl/2h)ζ2−1+tan(πl/2h)),and its inverse transformation
is(3.17)ζ=1+tan2πl2hcoth2πz2h.The function H(ζ)=f(z) is holomorphic in the superior half-plane and
also bounded except at the point A3′.
On the real axis, this function verifies the following
conditions:(3.18)Im[H(ζ)]=0,onthesegment[A6′,A1′];[A1′,A2′],Re[H(ζ)]=0,onthesegment[A2′,A3′];[A3′,A4′],Im[H(ζ)]=0,onthesegment[A4′,A5′];[A5′,A6′].By applying the formula of
Signorini [6], we find the function(3.19)H(ζ)=1sin(πl/2h)(ζ2−1ζ),and returning to the z-plane using (3.17), we finally
find(3.20)f(αε−iβ)={sinπl2h⋅1+[tanh(π/2h)(α/ε−iβ)tan(πl/2h)]2}−1.This function describes the full
flow in particular the field of the disturbances in the vicinity of the
barrier. The coefficients a1,m can be calculated easily. In fact on the
vertical α=0,
the function (3.20) is(3.21)f(iβ)=2cos(πβ/2h)cos(πβ/h)−cos(πl/h). Furthermore the
relations (3.10) to (3.14) permit the function(3.22)f(iβ)=1+∑m=1∞a1,m(cosmπβh+isinmπβh),where the coefficients a1,m are obtained by using the Fourier series
expansion on the interval [−h,0].It follows
that(3.23)a1,m=2h∫0−hf(iβ)cosmπβhdβ.Thus(3.24)a1,m=Pm(cosπlh)+Pm−1(cosπlh), where Pm and Pm−1 are the Legendre polynomials of degree m and m−1,
respectively, in the third integral representation.ThePm formula is given below:(3.25)Pm(cosπlh)=21.3.5⋯(2m−1)2.4.6⋯2m{cosmπlh+1⋅m1⋅(2m−1)cos(m−2)πlh+1.3⋅m(m−1)1.2⋅(2m−1)(2m−3)cos(m−4)πlh+⋯}. Using (3.1), (3.2), (3.4), (3.8), and
(3.14), the horizontal and vertical components of the displacement of the flow,
in all domain D,
can be written as follows:(3.26)X1(α,β,τ)=Xe(α−τ)+Xe(−τ)∑m=1∞a1,mcosmπβhexp(−mπ|α|εh),Y1(α,β,τ)=Xe(−τ)∑m=1∞a1,msinmπβhexp(−mπ|α|εh),where X1 and Y1 are, respectively, the horizontal and vertical
components of the displacement of the particle.Remark 3.1.
According to (3.1), (3.2), and (3.26), one notices that displacements X1(α,β,τ) and Y1(α,β,τ) are the independent contributions of the parts
“wave” and “local disturbances” of the flow. On the first
order of approximation, the part “wave” of the flow imposes a
uniform distribution of displacements and speeds on a vertical. The part “disturbances local” permits to take into account the presence of the
obstacle.


## 4. Application

For illustration, we consider as an example a solitary
wave (see [5]) Xe(α−τ) which is emitted at the upstream at time τ=0 and arrives exactly above the barrier at time τ=τ0:
(4.1)Xe(α−τ)=εh{1−tanh[34h(α−(τ−τ0))]}. Now, we
calculate the velocity field and the local disturbances of the flow. We observe
that the horizontal displacement is independent of the variable β in accordance with the theory of long gravity
waves.The derivative
of the expression (3.26) with respect to t gives the horizontal and vertical components,
respectively, u1 and v1 of the velocity field(4.2)u1=34ε2gh{1−tanh2[34h(α−(τ−τ0))]}+34ε2gh×{1−tanh2[34h(τ−τ0)]}∑m=1∞a1,mcosmπβhexp(−mπ|α|εh),v1=34ε2gh{1−tanh2[34h(τ−τ0)]}∑m=1∞a1,msinmπβhexp(−mπ|α|εh). We observe that
the horizontal (vertical) components of the velocity is symmetric, that is, u1(α,β)=u1(−α,β) (resp., antisymmetric, i.e., v1(α,β)=−v1(−α,β)).From (4.2), we
deduce the horizontal and vertical components, respectively, u1∗ and v1∗ of the field local
disturbances(4.3)u1∗=34ε2gh{1−tanh2[34h(τ−τ0)]}∑m=1∞a1,mcosmπβhexp(−mπ|α|εh),v1∗=34ε2gh{1−tanh2[34h(τ−τ0)]}∑m=1∞a1,msinmπβhexp(−mπ|α|εh).The components u1∗ and v1∗ can be written as(4.4)u1∗=c(τ)⋅u1∗∗(α,β),v1∗=c(τ)⋅v1∗∗(α,β),where(4.5)c(τ)=34ε2gh{1−tanh2[34h(τ−τ0)]},u1∗∗(α,β)=∑m=1∞a1,mcosmπβhexp(−mπ|α|εh),v1∗∗(α,β)=∑m=1∞a1,msinmπβhexp(−mπ|α|εh).Equations (4.4) show that the
coefficient c(τ) intervenes only as a parameter of scale of the
local disturbances (i.e., the kinematic of the flow remains unchanged
qualitatively whatever the position of the wave relatively to the barrier).
Naturally the intensity of the local disturbances is maximum at the moment t=t0.


### 4.1. Numerical Simulation

The numerical
simulation is realized on the Apollo station by using the subprogram of the
Legendre polynomials. The data used areh=20cm, l=10cm,
and ε=0.6.
The maximum velocity of incident flow is approximately 38cm/s.
We note that the series (3.26) converge slowly, for this, we have used 5 103 terms. The criterion of convergence is the
impermeability condition of the barrier (i.e., the norm of the horizontal
component remains within a specified tolerance 10−4).The graphical
representation of the field of velocity and the local disturbances are given,
respectively.The shallow water theory at the first order of
approximation permits to determine in a simple way the velocity field and the
local disturbances when the interaction of a solitary wave occurs with an
isolated obstacle. The numerical simulation illustrates the flow at the
vicinity of the barrier and shows, in particular, the zone of the influence of
the local disturbances.In a future work, we intend to study the problem by
using the shallow water theory at the second order of approximation in order to
consider, further, the effects of the reflected wave on the kinematic flow
barrier.


## 4.1. Numerical Simulation

The numerical
simulation is realized on the Apollo station by using the subprogram of the
Legendre polynomials. The data used areh=20cm, l=10cm,
and ε=0.6.
The maximum velocity of incident flow is approximately 38cm/s.
We note that the series (3.26) converge slowly, for this, we have used 5 103 terms. The criterion of convergence is the
impermeability condition of the barrier (i.e., the norm of the horizontal
component remains within a specified tolerance 10−4).The graphical
representation of the field of velocity and the local disturbances are given,
respectively.The shallow water theory at the first order of
approximation permits to determine in a simple way the velocity field and the
local disturbances when the interaction of a solitary wave occurs with an
isolated obstacle. The numerical simulation illustrates the flow at the
vicinity of the barrier and shows, in particular, the zone of the influence of
the local disturbances.In a future work, we intend to study the problem by
using the shallow water theory at the second order of approximation in order to
consider, further, the effects of the reflected wave on the kinematic flow
barrier.



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*Source: 135982-2008-12-24.xml*