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We present a scheme for coupling a 2D lattice Boltzmann free-surface solver with a 1D lattice Boltzmann shallow water solver, allowing to save computational effort and efficiently realize multiscale systems. The accuracy of the coupling is validated with two tests. First, we compare the numerical and analytical solutions in a setup with fixed inflow current and outflow water level in a canal. Secondly, the physics of wave propagation and reflection in a domain is investigated in a coupled simulation, and compared to the solutions obtained in both a pure free-surface and a pure shallow-water simulation. Finally, a performance test is carried out to demonstrate that the overhead of the coupling is negligible. A quantitative validation of this type of coupling for the lattice Boltzmann method is novel, and opens the door to a range of large-scale simulations of canals and other hydrodynamic systems.
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Coupling of lattice Boltzmann shallow water model with
lattice Boltzmann free-surface model
Yann Thorimbert1,
Abstract
We present a scheme for coupling a 2D lattice Boltzmann free-surface solver with
a 1D lattice Boltzmann shallow water solver, allowing to save computational ef-
fort and efficiently realize multiscale systems. The accuracy of the coupling is
validated with two tests. First, we compare the numerical and analytical so-
lutions in a setup with fixed inflow current and outflow water level in a canal.
Secondly, the physics of wave propagation and reflection in a domain is investi-
gated in a coupled simulation, and compared to the solutions obtained in both
a pure free-surface and a pure shallow-water simulation. Finally, a performance
test is carried out to demonstrate that the overhead of the coupling is negligi-
ble. A quantitative validation of this type of coupling for the lattice Boltzmann
method is novel, and opens the door to a range of large-scale simulations of
canals and other hydrodynamic systems.
Keywords: lattice Boltzmann, free-surface, shallow water, two-phase flows,
validation
1. Introduction
The lattice Boltzmann (LB) method is now recognized as a successful numer-
ical solver for the incompressible Navier-Stokes equations (NSE). The method
is based on the Boltzmann equation (see [1, 2] for more details on the method),
and it is therefore sufficiently general to simulate a wide set of physical systems,
including multi-phase and free-surface flows. Free-surface and shallow water
models are commonly used to simulate, among others, rivers and coastal flows.
The work of [3] demonstrates the ability of shallow water models to simulate
erosion with high performance, while [4] use it to study sediment transport. An
application of a shallow water model to marine flow in coastal areas can also
be found in [5]. Free-surface flows can be computed with good efficiency in the
LB method using a Volume-of-Fluid (VOF) approach [6]. Such an approach is
for instance used in [7] for the simulation of sediment transport in rivers. This
paper highlights the limits of the VOF approach from a computational perspec-
tive, as it shows the inability to simulate a river segment significantly longer
than 10 km even using a parallel computer with several hundred CPU cores.
A VOF free-surface model was used in [8] for simulating an Oscillating Water
Column wave energy conversion device, and a similar approach using the LB
Preprint submitted to Elsevier August 7, 2018
method was studied in our previous work [9]. While a free-surface model allows
us to fully simulate a 3D flow, an approach based on the shallow water equation
saves computational cost, allowing to simulate much larger domains. Shallow
water models may also be considered as a solution to overcome spurious energy
dissipation (see [9]) occurring over long distances in underresolved LB VOF
simulations. As a downside, the shallow water equation is based on more limiting
physical assumptions (see Section 2.1 below) than the free-surface approach.
A coupling between a LB shallow water model and a LB free-surface model
therefore offers a way to obtain the best of both worlds, using a free-surface
model to simulate the fluid on small portions of the domain requiring high
accuracy and with complex flow patterns, while pure wave or flow propagation
and reflection phenomena are simulated on larger portions with a shallow water
model.
The purpose of this article is to present a scheme for coupling LB free-surface
model with LB shallow water model. To illustrate this coupling and assess its
quality, we consider here the 2D-1D case. The coupling accuracy is studied
here through two different tests, the former focusing on flow rate transmission
between coupled lattices, while the latter focuses on wave propagation. The aim
of such a coupling is to save computational time thanks to the shallow water
model while preserving accuracy allowed by the free-surface model. As a result
of the lower dimensionality of the shallow water model, its computational cost
(see section 5) is one order of magnitude lower than the one of the free-surface
model; this particular point is crucial since one can expect this computational
scale difference to be reflected in the time and space scale of the corresponding
simulated domains, giving the opportunity to simulate multiscale systems.
We present the LB, free-surface LB VOF and LB shallow water methods in
the second section. In the third section, we describe the scheme used to couple
shallow water and free-surface systems. The coupling model is tested in the
fourth section, where different benchmarks are used. Performance considera-
tions are given in the last section.
2. Lattice Boltzmann method
The fundamental principles of the LB method (see [2, 10, 11] for more details)
originate from statistical mechanics. The Boltzmann equation describes the
statistical kinetics of gas molecules [12]. As a consequence, the quantity solved
for in this method is the particle distribution function, and the macroscopic
variables such as velocity or pressure are derived quantities, defined as velocity
moments of the distribution function. The method is Cartesian mesh based,
allowing fast mesh generation, and the method is inherently parallelizable and
scalable [13] thanks to spatial locality of its interaction pattern.
The lattice Boltzmann equation is obtained by truncating the continuum
Boltzmann equation in velocity space [14, 15]. A number of qparticle distribu-
tions, also named populations, is attached to each lattice site. These populations
can be seen as the discrete counterpart of the particle distribution function in
the continuous Boltzmann equation. The qvelocities ciof the lattice correspond
2
to the Gaussian quadrature points used to resolve integrals in the discrete ve-
locity space. The population representing the statistics of particles entering the
lattice site rat time twith velocity ciis denoted by fi(r, t). The number of
discrete velocities used in the numerical scheme determines the connectivity of
lattice nodes with their neighbors. One uses the notation “DdQq”, where dis
the number of dimensions and qthe number of velocities, to denominate the
chosen scheme. For instance, a D1Q3 lattice has been used in the present study
to run the shallow water simulations.
The collision term of the Boltzmann equation is often approximated by the
Bhatnagar-Gross-Krook (BGK) collision model [16], which uses a single relax-
ation time. In this model, the time evolution is expressed through the relation
fi(r+ ∆tci, t + ∆t)fi(r, t) = 1
τ(fi(r, t)feq
i(r, t)) ,(1)
where feq
iare the equilibrium populations, xand tare the discrete space
and time steps respectively, and τis the relaxation parameter. The equilibrium
populations feq
iare obtained as a finite expansion of the Maxwell-Boltzmann
equilibrium distribution, up to second order with respect to the Mach number.
The macroscopic density and velocity can be computed from the populations
as:
ρ=Xfi, ρu=Xcifi.(2)
A Chapman-Enskog analysis (see [15] for instance) demonstrates that the
above scheme is asymptotically equivalent to solutions of the incompressible
Navier-Stokes equations, in the limit of small Mach number.
In the present study, 2D VOF simulations are run using an approach of large-
eddy simulation (LES), based on a Smagorinsky subgrid-scale model( [17, 11,
10]). Although the flow is expected to be laminar in coupling interface regions,
it can become turbulent in any other part of the domain.
2.1. LB Shallow Water model
An early attempt to solve the shallow water equations using the LB method
can be found in [18]. The LB shallow water model used in this study follows
the line of [19]. As in [20], we consider a 1D flow in a rectangular channel with
slope Iand width B. The shallow water equations, also named Saint-Venant
equations in 1D, are derived from mass and momentum Navier-Stokes equations,
assuming that the water depth is very small as compared to the characteristic
length of flow structures. Further assumptions include fluid incompressibility, a
hydrostatic pressure profile, negligible viscosity effects, and a small value of the
slope I. Under these assumptions, the Navier-Stokes equations are reduced to
the following relations:
∂h
∂t +(hu)
∂x = 0,(3)
3
(hu)
∂t +(gh2/2 + hu2)
∂x =gh(IJ),(4)
where hdenotes the water height from the bottom to the surface, uis the
depth-averaged horizontal velocity, gis the gravitational acceleration, and Ja
term accounting for the friction force due to channel bottom. The latter term
is modeled as [21]:
J=n2u22h+B
hB 4/3
,(5)
where nis the Manning coefficient of the canal.
In the LB model for simulating shallow water equations, the water height his
assimilated to the first-order velocity moment ρ(“the density”), and is computed
as the sum of the populations. In this study we use the D1Q3 model presented
in [20]. One can show that, for Froude number F r =u/gH close to 1, spurious
viscous contribution appears in this model [20], thus constraining the type of
regime simulated.
2.2. LB free-surface VOF model
The free-surface model considers two-phase fluid compounds with a large
density ratio, akin to water-air systems in normal atmospheric conditions. It
simulates the physics of the heavy fluid only, and replaces the effect of the light
fluid onto the heavy one through a condition of zero parallel shear stress along
two-phase interface. In the LB VOF approach, additionally to the usual fluid
variables, every cell keeps trace of the local mass mand volume fraction V F .
While V F = 1 on bulk cells of the heavy fluid, it is related to the fluid density
on interface cells: V F =m/ρ. Thus, the cells of the fluid mesh are split into
three categories: “fluid cells” (cells completely occupied by the heavy fluid) in
which V F = 1, “empty cells” (cells completely occupied by the light fluid) with
V F = 0, and “interface cells” on which the volume fraction is comprised between
0 and 1. The interface layer between fluid and empty cells is always closed, so
that an empty cell is never in the neighboring of a fluid cell.
At each iteration, the progression of the two-phase interface is simulated
through a mass advection process, and the cell types are updated accordingly.
For an interface cell, the population fout
Ithat is streamed from an empty cell in
the direction iis given by fout
I=feq
i+feq
Ifi, where Idenotes the direction
opposite to i. The equilibrium populations here are computed using a constant
reference air density, set to 1 in our model.
Classical bounce-back conditions [22] are applied to wall cells, in the same
way as in standard single-phase simulations. A more detailed presentation of
VOF free-surface models and their LB implementation can be found in [23, 24].
Note that even if the solution of the flow field is obtained via a LB approach,
the advection equation for the VOF fill level can be solved in different ways, as
for instance in [25], where it is discretized with a finite volume method.
4
2.3. Other works on SW-FS coupling
To the best of the authors’ knowledge, there exist only few studies regarding
the coupling of shallow water and free-surface (SW-FS) models. A coupling
between a 3D free-surface solver and a 1D shallow water solver, both based on
finite differences method, is studied in [26]. The numerical scheme is validated
through the study of the flow in a curved channel, with an oscillatory inflow
velocity condition and imposed outflow. The study is however limited, as it
does not represent effects of friction along the channel bed, and the water depth
is constant along the channel. At the coupling interface, the water level in the
1D simulation is imposed as the average of the water level of the 3D simulation,
while the velocity in the 1D simulation is used as the value to impose in all
the corresponding 3D sites. Within this setup, a good match for the time
evolution of the free-surface height is found between the coupled system and
a single 3D free-surface simulation. A coupling between a 3D LB VOF free-
surface solver and a 2D LB shallow water solver is also presented in [27]. Since
this publication is targeted at a computer visualization community, it does not
include any quantitative validation of the coupling.
In [28, 29], a distinction is made between coupling strategies involving sys-
tems that are dimensionally homogeneous and physically heterogeneous, and
systems that are dimensionally heterogeneous and physically homogenous. Ap-
plications with different coupled shallow water models are depicted. The present
work, however, involves both dimensionally heterogeneous and physically het-
erogeneous systems. A study of different coupling types such as via source terms,
boundary conditions or state variables, is also found in [30] and is applied to
shallow water systems.
3. Coupling scheme
In a coupled simulation, we use free-surface and shallow water lattices to
represent different parts of the physical domain. At the connection between the
two domains, an overlap region is used, in which the two lattices are superim-
posed. At both ends of this overlap region, a coupling between the lattices is
performed. In this way, the coupling is not performed at the same location on
both lattices, in order to avoid amplification of errors and resulting numerical
instabilities. Figure 1 depicts the configuration. Note that, in a similar way as
done in [27] for the 3D-2D case, the coupling interfaces at coordinates x=B
and x=Care vertical columns with a thickness of 1 cell in the xdirection.
The overlap zone has a length of CBcells.
In the coupling scheme presented below, lattices are synchronously updated.
This implies that, at the coupling interface only, data from one of the lattices
has to be saved in temporary variables before the coupling scheme is applied.
3.1. Coupling from shallow water to free-surface
At the coordinate x=C, the coupling acts as a boundary condition for
the free-surface lattice, which imposes a water height and velocity determined
5
Figure 1: Configuration of the lattices and their coupling. The area below the curved line in
the free-surface lattice represents the VF value of interface cells, while the height simulated
in the shallow water lattice is represented by a discrete water column.
by the shallow water flow, incoming from the right. The following steps are
performed each iteration:
First step.. Update the free-surface cell types according to the shallow water
height. In order to perform this step, the volume fraction V F of the free-
surface lattice is computed according to the water height hof the shallow water
lattice on the same interface cell. The volume fraction of a corresponding,
superimposed interface free-surface cell is h bhc. The ycoordinate of this cell
is bh·xc.
Second step.. The horizontal velocity is obtained from the shallow water lattice
at the same interface location x=C. This value is used as a basis for the
velocity to impose at each cell of the fluid column of the free-surface lattice.
Here, missing information for the horizontal velocity along depth u(y)has to
be completed. Three types of distribution have been tested: a constant velocity
profile, a quadratic velocity profile and a logarithmic velocity profile. In the
first case, one imposes the same velocity to each cell along the depth. For
the quadratic case, the imposed velocity profile corresponds to a steady-state
solution of a laminar free-surface flow, with a constant current Qand a constant
fluid height H. Figure 2 displays an example of a quadratic profile. As indicated
in [31] for instance, the expected velocity profile is parabolic in a theoretical
laminar free-surface flow. The parabolic profile is characterized by the following
three parameters: the mean velocity ¯uQ/S, where Sis the cross section of the
domain, and the velocity at the bottom and at the level of the free surface, u(0)
and u(H)respectively. These values depend on the properties of the walls. The
latter two conditions, combined with the flow condition 1/H RH
0u(y)dy = ¯u,
6
yield a unique velocity profile. In our case, we apply a no slip condition at
the bottom, u(0) = 0. Since the free-surface model assumes zero parallel shear
stresses at the interface, the maximum theoretical velocity is reached at the
surface y=H, and the theoretical velocity profile reads:
u(y) = uy
H1y
2H.(6)
The maximum velocity value u/2is in theory obtained at y=H. However, real
laminar flows in a canal show a slight velocity decrease near the water surface,
due to the so-called dip effect [32] that arises in canals with finite width due to
friction forces, as well as non-zero parallel shear stress at the interface. These
phenomena are not implemented in our study. Also, our free-surface model is
two-dimensional and we assume there are no lateral walls. An example of the
velocity profile for laminar flow in the free-surface model is shown in Figure 2.
In summary, as we couple the velocity from the shallow water to the free-
surface lattice at x=C, we apply the theoretical velocity profile to the water
column, as provided by Equation 6, with mean velocity ¯u=uSW and a water
height H=hSW . These values are imposed to free-surface cells as Zou-He
velocity boundary conditions [33], even for the last, interface cell of the column
(both the free-surface completion scheme and Zou-He completion scheme are
involved at the top cell, the latter using the value from the former if needed).
When the water height changes, the populations of the new interface cell are
initialized at equilibrium.
In turbulent flows, the velocity profile is known to take a logarithmic form
[34]. This profile has been tested and we show in Section 4.2.1 that, for the
benchmarks used, the difference in the results is not significantly impacted by
the choice of the velocity profile. In the canal test cases (see Section 4.1), the use
of either quadratic or logarithmic velocity profile improved the overall accuracy
of the result by approximately 5%, while in the wave propagation tests (see
Section 4.2), the improvement of accuracy amounted to approximately 2.5%. In
this article, the quadratic profile was used unless otherwise specified.
We would like to point out that, if the main phenomenon of interest is the
propagation of surface gravity waves, an alternative velocity profile could be
considered. In this case, if the properties (e.g. period and amplitude) of the
waves created during the simulation are known, one can deduce the pressure and
velocity distributions under the water surface. The profile for these quantities
is expressed in terms of the amplitude, the period, and the water depth [34],
and could be directly imposed at the coupling interface. However, the deter-
mination of wave properties, and in particular their period, is non trivial, and
an additional numerical effort would be required to compute their time-varying
properties.
As a final comment, the y-component of the velocity cannot be directly
obtained from the shallow water lattice. However, we deduce it, at a first-order
approximation, by keeping track of the hvalues on interface cells from the
previous iterations. The value (htht1)/tis then taken as an approximation
of the vertical component of the velocity, and is uniformly distributed over the
7
Figure 2: Velocity profile along the water height yfrom the bottom y= 0 to the free surface
y= 0.4 m, obtained from a free-surface simulation with constant inlet and outlet flows. In
this example, x= 8 ·103mand t= 8 ·105s.
8
depth. Higher order schemes may be considered, but we have not investigated
them presently, and this study includes only the above one.
3.2. Coupling from free-surface to shallow water
At the position x=B, the coupling acts as a boundary condition for the
cell of the shallow water lattice, using flow properties of the free-surface lattice
arriving from the left.
The water height of a free-surface column at coordinate xcan be computed
as:
hF S = ∆x
H
X
y=0
VF(x, y).(7)
The mean horizontal velocity uFS of the free-surface lattice is similarly com-
puted as an average over a water column. These values are imposed to the shal-
low water lattice on interface cell, by setting the populations to fi=feq
i(hF S , uF S )+
fneq
i(hSW , uSW ). The non-equilibrium part is preserved as it does not change
the evaluation of macroscopic height and velocity. The value of the non-equilibrium
part can be computed as indicated in [20] :
fneq
0=τt1gh
v23u2
v2x(hu)+2u2
v2gh
v2u∂xh,(8)
and fneq
1=fneq
2=fneq
0/2. The spatial derivatives of hand uare evaluated
through a second order centered scheme.
3.3. Coupling parameters
The numerical parameters that can be freely chosen in our model are the
length of the overlap zone and the time and space steps tand x. In this
paper, the latter two parameters are equal in the free-surface and the shallow
water simulation. Although it is in principle wasteful to apply the same time step
in both simulations whereas the free-surface model often requires a finer time
step, this choice remains appropriate in our case, in which the computational
effort is largely dominated by the free-surface simulations (see Section 5), while
the computations of the shallow water lattice are negligible.
The stability of the simulations is affected by the size of the overlaping zone.
For the benchmarks studied, an overlap zone of 0.2 m was found to be a good
value in terms of stability. One can see the overlap zone as a buffer area allowing
to make a relaxation of the coupling variables going from one lattice to the other
one. When the size is too small, the spurious quantities are increased by the
feedback between lattices. This behaviour was expected, since the physics of
each lattice are of different nature, in addition to the inhomogeneity of lattice
topology. Hence, the coupling here is substantially different from that involved
in a grid refinement scheme for instance, which allows for smaller interaction
stencils.
Another parameter of the simulation is the relaxation time τ, which relates
to the fluid viscosity. We have kept this parameter free, and its influence on the
accuracy is presented in the results below.
9
It should be pointed out that, for our coupled system to produce meaningful
results, one should make sure that physical phenomena occurring near the cou-
pling interface are properly captured by both the free-surface and the shallow
water model. It is reminded that the assumptions of the shallow water model are
valid only if the horizontal characteristic scale (e.g. the wavelength of a wave)
is much larger than the vertical scale (e.g. water depth). In the case of surface
wave propagation in a canal of depth H, waves in the shallow water system
propagate with velocity v=gH, while in the free-surface system waves travel
with velocity v=ptanh(2πH/λ)/(2π), assuming their behavior obey linear
wave theory (see [34] for instance), with λthe wavelength. Since tanh(x)x
for small values of x, these expressions tend to be equivalent for small values of
H/λ.
We conclude this section by remarking that height, velocity, or any physical
quantity, are defined twice in overlapping areas, by both the free-surface and
the shallow water lattices. Since there is no a priori reason to choose the
value from one lattice rather than from the other one, we choose here to define
any quantity qof the fluid in the overlap area through linear interpolation as
q(x) = α·qSW (x) + (1 α)·qF S (x), where α= (xB)/(CB). While this
choice has an impact on the visualization of the flow quantities, it does not
affect the results of either the free-surface or the shallow water simulation, as
the physical variables of the overlap zone are not explicitly used in the coupling
scheme.
4. Results
4.1. Benchmark
Our benchmark case is similar to the one of [20]. It imposes an inflow current
Q0and an outflow water height hLon a canal of given length L, as depicted
in Figure 3, and in this way allows us to test the coupling in both ways (from
free-surface to shallow water and vice versa). This benchmark tests the response
Figure 3: Configuration of the benchmark test with a canal of length L. The inflow current
Q0and the outflow water height hLare imposed. The free-surface lattice is in the center of
the domain and is surrounded by longer, shallow water lattices.
of both height and velocity to perturbations, and is motivated by its similarity
with potential real-life applications to irrigation canals [20, 35]. The simulation
10
domain is decomposed into three parts: two shallow water parts and one free-
surface part in the center. A bounce-back condition is applied at the bottom
of the free-surface part. The inflow current and outflow height are imposed to
the shallow water lattices in the same way as in [20], by completing the unique
missing population. The shallow water differential equation corresponding to
the discussed condition reads:
∂h
∂x =gh(IJ)
gh u2,(9)
where u=Q0/(Bh), with the condition h(L) = hL, and
J=n2Q2
0
B2h2Bh
B+2h4/3.(10)
An approximated solution for this equation is obtained numerically using a
solver for ordinary differential equations based on Runge-Kutta method.
The results are validated by computing the mean relative error of the data
from the coupling model to the reference solution href of Equation (9). We
define the mean relative error as:
=1
N
N
X
i=0
|h(ix)href (ix)|
href (ix),(11)
where NL/xis the total number of cells along x-axis. All three do-
mains are included in the calculation of the error. The parameters used for the
simulations are x= 8 ·103m,t= 8 ·105s,H0= 0.4 m,τS W = 0.51,
L= 4 m,I= 2.6·103,B= 0.1 m,n= 0.0103 and g= 9.81 m/s2. The length
of the full free-surface domain is 0.8 m and the length of each overlap zone is
0.2 m.
For a given set of values for the quantities described above, the resulting
flow reads:
Qi=IBH
nBH
B+ 2H2/3
.(12)
The above value of Qiis used for the initial condition of the flow velocity
ui=Qi/(BH )everywhere in the domain. In an initial stage of the simulation,
the flow is linearly increased up to Q0= 1.5·Qi, as done in [20]. The Froude and
Reynolds number corresponding to the described setup are F r 0.5,Re 106
respectively. The water level is initialized at a value h(x) = H0everywhere in
the domain.
The proposed benchmark is highly demanding for our coupling model. In-
deed, the LB, VOF-based free-surface model suffers from energy dissipation over
long distances, and is therefore poorly adapted to a simulation of a long canal.
As pointed out in the introduction, this is precisely one of the reasons why
a SW-FS coupling model is needed in situations where both long-range wave
11
propagation and a precise representation of the flow structure are required in
different parts of the simulation domain.
Figure 4 shows the evolution of as a function of time for τFS = 0.55,
τF S = 0.53 and τF S = 0.501, and the solution from the shallow water model
alone. An installation wave and its reflections perturb the simulation at the
beginning, due to the initial flow increase from Qito Q0. At time t10 s,
a steady state is reached. Figure 6 shows that the mean relative error linearly
decreases with the lattice spacing x, for a given set of relaxation times for the
free-surface and shallow water lattices. The simulations show that the accuracy
of the results linearly decreases as the value of τFS departs from 0.5, as shown
in Figure 5. To explain this, we point out that the considered benchmark case
assumes an inviscid flow, and that the shallow water equation similarly solves
the non-viscous shallow water equations. Note that is has been shown in [20]
that even the shallow water model exhibits a non-null viscosity, that does not
correspond to the one of the free-surface model. Therefore, a residual error is
found, although the free-surface domain yields best results at low viscosity, when
τis close to 0.5. Figure 7 compares the elevation profile at time T= 29.7 s in
the theoretical case and in the coupled simulation, and shows that the coupling
scheme introduces discrepancies of the order of 102. As a result, the total mass
of fluid slightly differs from the initial one ; this effect is quantified in Section
4.3.
Figure 4: Evolution of mean relative error as a function of time, for three different values of
τF S with SW-FS model and shallow water model.
12
Figure 5: Mean relative error in the steady state as a function of lattice spacing xfor
constant τSW = 0.51 and τF S = 0.501. The dotted line corresponds to a linear fit.
4.2. Wave transmission
In the next numerical test, we focus on the transmission of waves from the
shallow water to the free-surface simulation and vice versa, by tracking the
evolution of surface elevation along time. For this reason, we use a symmetric
test case in which the water elevation is initialized with the following profile:
h(x) = H0+Hdrop ·e(xX0)2/(2σ2).(13)
For the results presented below, the parameters are H0= 0.195 m,Hdrop =
0.1 m,X0= 2 m and σ= 0.15 m. The total length of the domain is 4 m while
the length of the central, free-surface domain is 1 m and overlap areas of 0.2 m.
The corresponding Froude and Reynolds numbers are F r 0.2,Re 105
respectively. We refer to this test under the term of “Gaussian drop”, due to
the similarity of its behaviour with the waves generated by a droplet falling into
water. The symmetry of the system, the reflections of waves on boundary walls
and the resulting superimposition of waves makes it relevant for the study of
wave propagation. It allows us to verify wave properties over a long time while
keeping the simulation domain small.
13
Figure 6: Mean relative error as a function of τFS for constant τS W = 0.51, in the steady
state.
14
Figure 7: Water elevation along the channel for the simulation using the coupling scheme (red
dotted line) and in the theoretical case (green line), at time T= 29.7 s. Overlap areas are
delimited by vertical dotted lines.
15
4.2.1. Velocity profile in free-surface model
As discussed in Section 3.1, the shape of the velocity profile requires an ad-
hoc choice to be made, given that the data from the shallow water lattice is
one-dimensional. Figure 8 shows the difference of water elevation at different
time steps for the Gaussian drop, for results obtained from constant, quadratic
and logarithmic velocity profiles. In the Gaussian drop test case as well as in the
canal test, the mean relative difference between the results for different profiles
is less than 2.5%. Apart from the observation that, in our benchmark, the actual
velocity profile appears to be much less important than the total flow, another
possible reason for this small difference is that, even starting from a constant
velocity profile along depth, the profile adapts after on the x-axis, as shown in
Figure 9. Thus, a constant profile has a small impact on the mean relative error
for a steady flow; however, it has an influence on the capacity of the coupling
scheme to react to quick flow changes, as in the canal benchmark test. For this
reason, the quadratic profile was preferred.
Figure 8: Comparison of the solution obtained for the Gaussian drop test with constant,
quadratic and logarithmic velocity profiles at different time steps.
4.2.2. Comparison between models
Figure 10 compares the solutions obtained with the shallow water model,
the free-surface model, and the coupled system respectively for the Gaussian
drop.
16
Figure 9: Example of velocity profile along y-axis at different locations on the x-axis, after
0.5 sof simulation (5·103iterations). In this test a constant velocity (continuous line) is
imposed at x= 0 in the free-surface lattice, and there is no shallow water lattice. A parabolic
distribution is recovered for the velocity after a short distance.
17
Figure 11 shows the mean relative deviation of the shallow water solution, as
compared to the coupled and the pure free-surface solutions. It can be observed
that the difference between coupled system and pure shallow water method is
essentially equal to the difference between coupled system and pure free-surface.
Regardless if one chooses free-surface or shallow water as reference solutions,
the mean relative error remains smaller than approximately 3% throughout the
simulation, which is consistent with the mean relative error found in the previous
benchmark.
Figure 10: Solutions obtained from shallow water model only, free-surface model only, and
coupled model for the Gaussian drop. The physical time Tis indicated in each case. Vertical
dashed lines delimit overlap areas between lattices.
4.3. Mass conservation
As discussed above, information has to be reduced/augmented when passing
from one lattice to the other, because the two involved models are 1D or 2D re-
spectively. This transformed information impacts the macroscopic variables on
the two lattices through the boundary conditions on the coupling interfaces (see
Sections 3.1 and 3.2 above). For this reason, one can expect mass conservation
to be affected by the coupling scheme.
The left plot of figure 12 shows the fluid mass in the Gaussian drop simu-
lation. Small variations of mass can be seen locally (less than 1% of the total
fluid mass), but no systematic loss of mass occurs. We conclude that spurious
18
Figure 11: Mean relative deviation of the coupled (SW-FS) and free-surface model compared
to the shallow water solution, in the case of the Gaussian drop test.
19
0 1 2 3 4 5 6 7
Time [s]
0.0
0.2
0.4
0.6
0.8
1.0
1.2
M/ M 0
cou pled SW
cou pled FS
SW-FS
8
0 1 2 3 4 5 6 7 8
Time [s]
0.1 9
0.2 0
0.2 1
0.2 2
0.2 3
Mass [LB]
FS
SW
sw-FS
Figure 12: (left) Total mass of fluid during a simulation for the Gaussian drop test. Continuous
line shows the total (SW + FS) fluid mass Mdivided by the initial mass M0. The dotted
and dashed lines show the repartition of fluid mass into shallow-water and free-surface parts
respectively. (right) Total mass of fluid contained inside the overlap zone for three distinct
simulations: pure SW, pure FS and SW-FS systems.
mass gain or loss introduced by the coupling scheme is negligible. Simulations
of pure shallow water, pure free-surface and coupled SW-FS systems have been
performed in order to compare the evolution of the mass of fluid contained inside
the overlap area (even though for pure systems no special treatment is applied
in this area). This way, mass evolution at the SW-FS transition zone can be
compared to pure systems. The right plot figure 12 shows that this difference
is comparable to the error previously found for the water level in the same
problem.
5. Performance
The free-surface model is computationally much more expensive than the
shallow water model, mainly because of the representation of a 2D domain, as
opposed to 1D in the shallow water case. Figure 13 illustrates the dependence
of the total execution time on the relative length of the free-surface domain,
as a fraction of the total domain. Note that our code has not been optimized
and simply follows the description of the coupling model given here. Thus,
the performance results depicted below constitute an initial investigation, as
the coupling overhead may reveal sensitive to the specific implementation used.
Consistently with our expectations for the coupled model, the execution time
is substantially reduced when a larger part of the domain is represented by the
shallow water system. It is important to stress here that for sake of implemen-
tation simplicity, the shallow water lattice is in fact used over the whole domain,
although its values outside the domain defined for shallow water simulations are
ignored and never involved in computations of physical quantities. As one can
20
deduce from this figure, it is not worthwhile using the coupled system if the do-
main fraction represented by the free-surface model is larger than about 75 %, as
in this case the overhead due to the coupling algorithm weighs out the reduction
of computational cost obtained by the shallow water scheme. This condition is
not expected to cause any problems in practice, since the free-surface simulation
is intended to be concentrated on areas of restricted size, with highly resolved
local flow structures, while the shallow water model should be applied to areas
of extended lengths (LSW >> λ), where wave or current propagation is the
central investigated phenomenon.
Figure 13: Ratio of execution time between coupled model pure free-surface, for different values
of the fraction LF S of the domain simulated by the free-surface model. As a rough observation,
when less than 75 % of the domain is simulated by free-surface, the coupling scheme becomes
worthwile in terms of performance. Note that in our software implementation, for simplicity,
the shallow-water lattice covers the whole domain, though its values are ignored at locations
where free-surface domain is defined.
6. Conclusion
The proposed coupling scheme allows us to significantly reduce the compu-
tational time compared to a pure free-surface approach. Its accuracy is found
to be satisfying both in the case of flows dominated by current transmission, as
well as for flows dominated by wave transmission, as long as the flow conditions
at the coupling interface satisfy the assumptions of the shallow water model.
21
Time interpolation should be considered in the future as a possible improve-
ment of the current model, as well as an implicit scheme to solve the coupling
interface and reduce the length of the overlap zone. Also, for the use in large
scale simulations, a parallel implementation of our coupling scheme and its soft-
ware implementation should be investigated. While both shallow water and
free-surface models are proven to be efficiently parallelizable, the overhead of
the coupling scheme could have a negative impact on the overall performance,
or ability to achieve efficient load balancing.
A limitation of the model is that one should carefully ensure that the shallow
water lattice always simulates flows with Froude number less than unity, since
it yields spurious viscous terms in regimes with F r close to 1 (see Section 2.1).
Thus, regions with large Froude number, including transitions to these regions,
must be simulated within the free-surface part of the domain.
Finally, a natural generalization of our numerical scheme will include its
adaptation to a coupling between 1D or 2D shallow water and 3D free-surface.
In this case, a proper velocity profile will need to be formulated, not only along
the fluid height, but also in the transversal axis. The main ideas as developed
here will apply.
22
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