Transactions on the Built Environment vol 58 © 2001 WIT Press, www.witpress.com, ISSN 1743-3509
On the splitting of the sediment fluxes balance:
a new formulation for the sand waves equation
F. Saint-Cast1,J.P. Caltagironel, P. Bonneton2
' ~ a b o r a t o r yMASTER-ENSCPB, University of Bordeaux, France
Department of Geology and Oceanography,
2
University of Bordeaux, France
Abstract
Computing of coastal morphological evolution requires high numerical
resolution for the sediment conservation equation. The splitting method
exploits the relationship between the sediment discharge formula and the
hydrodynamics variables. After the sedimentary hydrodynamics model has
been defined, the sediment fluxes balance is splitted in order to obtain a new
formulation for the sand waves equation. Therefore, several terms of great
physical and mathematical interest are obtained. The physical mechanism of
the non-linear sand waves propagation is found to vary with respect to the
water depth dependence of the sediment fluxes and it is weighted by the flow
regime. The current streamline divergence effect induces a lateral expansion
of sand waves. Moreover, the competition between wave-induced radiation
stresses and bottom friction introduces local erosion or accretion. Finally.
this sand waves equation can be computed with high order non-oscillating
numerical schemes.
1 Sand bed evolution
The physical knowledge and mathematical modelling of hydrodynamics
field in two horizontal dimensions (2DH) have advanced so far. that
quantitatively reliable predictions of 2DH waves and currents patterns can
be obtained from numerical models. Together with increasing knowledge of
sediment transport due to waves and currents, this has opened the way to
compound numerical models of morphological evolutions in coastal zone,
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C o a m ! Engineering V: Computer Modelling of Seas and Coastal Regions
De Vriend [l].
The sediment transport rate is defined as the amount of sediment per
unit time passing through a vertical plane of unit width perpendicular to the
flow direction. The transport formulas class that predicts the total sediment
discharge controlled by currents have such mathematical properties that it
is often difficult to directly analyse the physical sand waves behaviour, and
it introduces stiff problems for the computation of bottom evolution.
The rate of net accretion or erosion of an area of sea bed depends on
the difference in the rates at which sand is entering and leaving the area.
The modelling of bed evolution required to make a prediction of the pattern
of erosion and accretion in the study area:
where Zf is the bottom-level, @, is the volumetric sediment flux, t is
the time and ac' = ( X ,y) are used for the two horizontal coordinates. V
stands for the divergence operator
The hydrodynamics induced by a variety of currents (wind waves, tides,
storm surges, river currents, ...) needs to be known in order to describe the
fluid effects on the sediment. Among these currents, a prominent role is
played by the breaking-wave-induced current. The radiation stress is defined as the excess momentum flux induced by the wave motion. This
concept introduced by Longuet-Higgins [5] has been widely applied to predict and analyse the nearshore current system. Simple expressions for the
conservation of mass and momentum in the nearshore current can be obtained by integrating Navier-Stokes equations over depth and then taking
its time average over the wave period, Mei [6]. The unsteady shallow water
equations are obtained:
2 + &.
where Q = (q,, q y ) is the horizontal depth averaged water discharge, h
is the total water depth, g is the gravitational acceleration, h + Zf = 7 is
the mean sea level, h 9 is the forcing terms due to the wave field (related
to the radiation stresses divergence), 6 is the bed shear-stress and p is the
density of water.
In order to obtain the medium-term morphological evolution, we need
to know the residual circulation of currents at the morphodynamical timescale.
That defines the sedimentary hydrodynamics. The distinct timescales between bedform evolution and current dynamics allows to drop time derivatives in the unsteady shallow water equations (2) and (3). This quasi-steady
Transactions on the Built Environment vol 58 © 2001 WIT Press, www.witpress.com, ISSN 1743-3509
Coastal Engineering V: Computer Modelling of Seas and Coastal Regions
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hypothesis means that fluid adjusts instantaneously to the bottom configuration all the time. The sedimentary hydrodynamics equations are given
by:
Total sediment transport is a very complicated matter, especially in
the surf zone. However, where current plays a dominant role, the general
description can be written as follows:
2 A new formulation for sand waves evolution
A global morphodynamical model is obtained using Exner's equation for
sediment mass conservation ( l ) , the formulation of sediment discharge (7)
and the sedimentary hydrodynamics equations (4), (5) and (6).
2.1 Sediment fluxes splitting method
The sediment fluxes divergence in equation (1) is splitted. This splitting
aims at applying the spatial derivatives of the divergence operator to the
hydrodynamics variables, in spite of applying it to the sediment fluxes. The
relationship between the hydrodynamics variables and the sediment fluxes
are deduced from the sediment transport formula (7). Therefore, taking
into account the water mass conservation (4), it follows:
The partial derivative of the sediment flux with respect to the water
and the partial derivative of the sediment flux with respect to the
depth
flow discharge
are shown to be important parameters and they depend
on the transport formula used.
To go further, the sedimentary hydrodynamics equations (4), (5) and
(6), are used to express both hydrodynamics terms B . ~ ( Q and
)
G.a(h)
with respect to Zf .
%
Equation (9) can be related to d.?(Zf) as it can be extracted from
the sum of the weighted momentum equations qx(5)+ qy (6). Moreover, the
Transactions on the Built Environment vol 58 © 2001 WIT Press, www.witpress.com, ISSN 1743-3509
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Coastal Engineering V: Computer M o d e l l f ~ ~ofgSeas and Coastal Regions
inertial terms are extended in order to separate the water discharge effect
from the water depth effect. It follows that:
c.v(?d)
+cYv($4)
Q--
Q2
''
= S;Q.V(Q) - ~ Q . v ( h )
The sedimentary hydrodynamics model yields to:
(1 - F,') G . f ( h ) = -
with F,' =
G.f(zf)
S.Froude number F, stands for the flow regime.
2.2 Sand waves equation
A new formulation for the sand waves equation is obtained from (8) and
( l l ) ,if F, # 1 is assumed:
which becomes the fundamental equation for bottom evolution. Formulation (12) allows to analyse the effects of sedimentary hydrodynamics
on bottom evolution.
F, < 1 is obtained for a wide range of environmental flows. Subcritical flow regime is being considered herein. On the one hand, if the Froude
number is close to zero, equation (11) shows that d.?(h) is closely related
to -G.?(Z~). On the other hand, in the one dimensional (lD) case and for
any Froude number, B.?(Q) = 0 because of the mass balance equation (4).
It follows that Q . ~ ( Qis) independent of Q . f ( z f ) . Therefore, if the Froude
number is weak and if the flow is hardly ID, three classes of mechanisms
are identified in the sand waves equation (12).
2.2.1 Sand waves equation
The convection effect class represents the internal sand wave evolution mechanism. This transport dynamics of the interface Zf is obtained by the
hyperbolic equation:
Transactions on the Built Environment vol 58 © 2001 WIT Press, www.witpress.com, ISSN 1743-3509
Coastal Engineering V: Computer Modelling of Seas and Coa~talRegions
where the non linear propagation celerity
7
ezfof the bed is obtained:
The bottom elevation is constant along characteristic curves given by:
The non linear sand wave will propagate and will be deformed, but
it will preserve its height until the shock discontinuity is formed. After
the shock has been reached, dissipation occurs and the sand wave with a
steep face keeps moving but its height decays, Lighthill [4]. This wave-type
behaviour of the bottom changes appears to be typical for the class of sediment transport models (7) considered herein. It will occur for any transport
formula expressing in terms of Q and h.
2.2.2 Streamline divergence effect
The streamline divergence effect class shows the consequences of the water flow deviation induced by the shape of the bedform on the sediment
circulation. In contrast to convection effect, that sediment transport mechanism makes the bedform shape changed without horizontal motion. This
mechanism is modelled by the equation:
As earlier described by De Vriend [l],this specific flow of sediment
around the bedforms is provided by the streamline divergence self-induced
by the bottom perturbation.
2.2.3 Residdual source effect
The residual source effect class represents the sediment distribution due to
the balance between the fundamental forces applied to the water flow. The
residual hydrodynamics forcing is based on the competition between the
external forcing due to waves and the bed-shear stress resistance due to the
friction of the flow on the rough bed:
Sediment will accrete (or respectively erode) in areas where residual
forces are positive (respectively negative). The model (17) can be related to
Transactions on the Built Environment vol 58 © 2001 WIT Press, www.witpress.com, ISSN 1743-3509
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Coastill Eqineering V: Conputer Modelling of Seas and Coastal Regions
the mechanism for bed-flow instabilities reported by Falques [2]. The waveinduced forcing term is highly correlated with the bottom shape, whereas a
phase lag can appears between the bed friction resistance and the bedforms.
This unbalanced residual forces can becomes sufficient to provide local bed
instabilities.
3 Applications
Two morphodynamical cases are computed to illustrate these three previous
mechanisms. Both cases are computed using the sand wave equation (12)
coupled with the sedimentary hydrodynamics (4), (5) and (6) on a cartesian
grid. Periodic boundary conditions are applied in the current direction.
For both cases, the Engelund-Hansen sediment transport formulae is
used, and the sediment fluxes dependencies on hydrodynamics variables are
given:
where cr is a positive constant dependent on the sediment property and
the bed roughness. The partial derivatives of the sediment flux with respect
to the hydrodynamics variables $$= = -5% < 0 and aQ = 5% > 0 are
deduced.
A high order non oscillating numerical method is used to compute the
convection part of the sand waves equation. This total variation diminishing (TVD) scheme is based on the interface advection problem, Vincent 181.
For that reason, the good characteristic bed propagation velocity defined in
by equation (14) is exploited.
3.1 2D dune propagation in a straight shallow channel
The first case has been described by De Vriend [l]. An hydraulic straight
shallow channel is being considered. The incoming current is uniform (g, =
5 m2.s-l) and it flows over an horizontal sandy bottom (h = 10 m) perturbed by a sinusoidal hump. This yields to a very weak Froude number
2.10-~).As periodic boundary conditions are considered, basic bed
friction resistance has to be balanced with a basic external forcing term.
The initial state of the dune is shown in Figure 1.
To begin with, the non-linear wave propagation is illustrated in the De
Vriend's case. The mathematical model (13) is best explained by referring
to Figure 2. After several morphodynamical time steps, a horizontal movement of the dune is clearly observed. The dune travels upstream by erosion
at the upstream face (stoss side) and deposition at the downstream face (lee
side). The steep lee side formation is well computed. Sand wave distortion
is obtained without any spurious numerical oscillations.
,
Transactions on the Built Environment vol 58 © 2001 WIT Press, www.witpress.com, ISSN 1743-3509
Coastal Engineering V: Computer Modelling of Seas and Coastal Regio.1~
Figure 1: Initial bed topography of De Vriend's dune migration.
Figure 2: Convection of the dune according to equation (13).
Next, the streamline divergence effect according to the model (16) is
shown in Figure 3. After several morphodynamical time steps, the initially
circular bedform shown in Figure 1 has not moved downstream, but its
shape has been strongly modified. A two-dimensional star-shape expansion
of the bottom perturbation is observed. This mechanism is clearly described
in equation (16). The streamline divergence defined by Q . ~ ( Qis) weighted
by a positive term which depends on the transport formula. It follows that
the divergence of the streamlines produces a local accretion area, whereas
its convergence yields to the bottom erosion.
As an example, Figure 4 illustrates the whole sand waves model (12)
applied to De Vriend's case. Non-linear advection mechanism and streamline divergence mechanism take place simultaneously (residual forcing effect
Transactions on the Built Environment vol 58 © 2001 WIT Press, www.witpress.com, ISSN 1743-3509
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Coastal Engineering V: Computer Modelling of Seas and Coastal Regions
are found to be very weak). As a result, typical features such as advection,
the steep lee side formation and the lateral-shape expansion are coupled.
Figure 3: Streamline divergence effect according to equation (16).
Figure 4: Morphological evolution of the dune according to equation (12).
3.2 Sand waves instabilities on beaches
The second case has been described by Falques [2]. A basic longshore current
is flowing over a sloping flat beach. The surf-zone lenght (Ly N 90 m) is
defined between the coastline and the breaking line close to the longshore
current velocity peak (1 m.s-l). This yields to a very weak Froude number
(F,' 2 0.1). The basic equilibriumstate is obtained by the balance between
the wave-induced forces and the bed friction resistance. This flat beach is
then perturbed and linearly instable modes are obtained by Falques.
Sand waves equation (12) has been used t o compute the numerical solu-
Transactions on the Built Environment vol 58 © 2001 WIT Press, www.witpress.com, ISSN 1743-3509
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tion of the coastal morphodynamical case. Figure 5 shows the perturbed topography and Figure 6 illustrates its non linear evolution using model (12).
The sand waves on the beach have been propagated downstream. Their
migration velocity is strongly correlated with the longshore current. In the
surf-zone, the sand waves lee side becomes steep. Outside the surf-zone,
cross-shore sand waves with a weakly steep lee side have been prolonged.
Close to the breaking line, it seems to appear transversal sand waves.
Figure 5: Initial beach topography of Falques'bed-flow instabilities.
Figure 6: Morphological evolution of the beach using the equation (12)
4 Conclusion
A sand waves model is obtained by the splitting of the sediment fluxes
balance. This new formulation for sand waves equation underlines the hyperbolic nature of the bedforms which characteristic velocity is weighted
Transactions on the Built Environment vol 58 © 2001 WIT Press, www.witpress.com, ISSN 1743-3509
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by the Froude number. The streamline divergence effects and the residual
forcing mechanism are directly linked to the sand waves behaviour. Both
effects can contribute to sand bed instabilites.
As the characteristic bed propagation velocity is obtained by the splitting method, a high order TVD numerical scheme is used to compute the
sand waves equation. However, this splitting analysis cannot be easily done
using complex parametric sediment transport formulas described by Van
Rijn [7]. That is why other numerical methods are needed. Non oscillatory
central schemes presented by Jiang [4] allows to solve globally the hyperbolic conservation laws whatever the fluxes are, despite a larger amount of
numerical diffusion.
References
[l] De Vriend, H. J., 2DH Mathematical Modelling of Morphological Evo-
lution~in Shallow Water, Coastal Engineering, Vol. 11, pp. 1-27, 1987.
[2] Falques, A., Montoto, A. & Iranzo, V., Bed-flow instability of the longshore current, Continental Shelf Research, Vol. 16, No. 15, pp 19271964, 1996.
131 Jiang, G. S. & Tadamor, E., Nonoscillatory Central Schemes for Multidimensional Hyperbolic Conservation Laws, SIAM Journal on Scientific Computing , Vol. 19, No. 6, pp 1892-1917.
[4] Lighthill, J., Waves in Fluids, Cambridge University Press, 1978.
[5] Longuet-Higgins, M. S., Longshore Currents Generated by Obliquely
Incident Sea Waves, 1 and 2, Journal of Geophysical Research, Vol.
75, pp 6778-6801, 1970.
[6] Mie, C. C., The Applied Dynamics of Ocean Surface Waves , World
scientific, 1989.
[7] Van Rijn, L. C., Handbook of Sediment Transport by Currents and
Waves, Delft Hydraulic, 1990.
[8] Vincent, S. & Caltagirone, J.P., Efficient Solving Method for Unsteady
Incompressible Interfacial Flow Problems, International Journal for
Numerical Methods in Fluids , Vol. 30, pp 795-811,1999.