Numerical simulation of the three-dimensional river antidunes
Toshiki IWASAKI (Hokkaido University), Takuya INOUE, and Hiroki YABE (Civil Engineering Research Institute for Cold Regions, Sapporo, Japan) Shinichiro ONDA (Kyoto Univeristy, Japan)

Results
Introduction
Understanding the dynamics of bedforms in rivers is ofgreat importance from the engineering point of view. In steep
rivers, the flow becomes supercritical during a flood, resulting
in formation of antidunes on the bed. The antidunes havesometimes important role in developing a triangle-shape water
surface wave (standing wave) as shown in Figure 1. Since thissurface wave may be a potential risk for the river management
works, it is important to predict the dynamics of this kind ofwaves and bedforms.
The water surface wave shown above has a strong threedimensional shape, so that the antidunes formed on the bed
may be also three dimensional. However, there is a lack ofunderstanding the formation and development of threedimensional antidunes. We proposed a model framework tocapture the formation of three-dimensional antidunes (Iwasakiet al., 2016). The linear stability analysis shows that theproposed model is in part able to reproduce the threedimensional river antidunes (Figure 2). In this study, we perform
numerical simulations to understand the model performance atthe nonlinear level.

h Uh Vh

0
t
x
y

qby
t



x
U
U
U h   1  p
UV g sin   g  dz


t
x
y

x
x
h
x
h
 2px2

u px qby
x

Dimensionless spanwise
wavenumberu py qbyy

Upstream-migrating antidunes

Upstream-migrating antidunes

Downstream-migrating free bars

2232322h 2  2  2 2hHH2  Bg  2  2  U 2UVV22 2 xxyy  3y  x

pypxepye1u pye qbe  u py qb ls

S g gd 3

u , u , u , u    U  V 
1   U  V  h    U  V   dz  h  B  3  x t  y t   3 U  x 2  xy   V  xy  y 2 3

2Figure 2. The instability diagrams of 3D antidunes.
The parameters are Fr=1.25, *0=0.16, Cf=10.6-2.

3/ 2y h   1  p
VVV UV  g  dz tx
yh y y  h y  

2Figure 4. Numerical results of the 3D antidune and water surface wave dynamics. Left) relatively narrow channel (Channel width/depth= 6.3, Fr = 1.04, * = 0.27), and right) relatively wide channel (Channel width/depth = 20, Fr = 1.3, * = 0.16)

Dimensionless streamwise
wavenumberqbe  4 *   *c Vb  ub  vbFree bar
instability

qbx u px qbx u py qbx 1
 u pxe qbe  u px qb tx
yls

Flow model: Boussinesq model

pAntidune
instability

Non-equilibrium bedload transport model

Model



Figure 1. A triangle-shape water surface wave
observed during a flood in Toyohira River, Japan.

1
qbx , qby , qbxe , qbye La 1   uvvuqbex  qbe  qben , qbey  qbe  qbenVVVVz
 ubn*cqben  qbe Vns k * b qbx qby
1    
02txy h  1  U 
 2 V 
 2  1  U h
 2 h V h  
ub  U  h  
U 2 V
U 2 Vx
x y
xy  6  x x
x
x yxy 
 2  x x
 1  U  2 V  2  1  U h
 2 h V h
 2 h vb  V  h  U V 2   U V 2 xy y yy  6  y xxy y yy  2  y xhyFigure 5. Time series of the bed elevation change from the initial bed near the right bank given from the simulated bed bathymetry (Figure 4).

x

Figure 3. The coordinate system.

Notations: t is the time, x, y, z are the streamwise, spanwise and vertical coordinate, U, V are the depth-averaged flow velocity in x, y directions, h is the water depth,  is the deviation from theaverage bed elevation,  is the longitudinal angle of average bed elevation, H is the water level, p is the pressure, x, y are the bed shear stress in x, y directions, ub, vb are the flow velocity near thebed in x, y directions, Cf is the bed friction coefficient, s, n are the local streamline coordinate qbx, qby are the bedload flux in x, y directions, upx, upy are the bedload particle velocity in x, y directions,
qbe is the equilibrium bedload flux, * is the Shields number, *c is the critical Shields number, ls is the step length, La is the active layer thickness, Sg is the specific weight of sediment in fluid, c isthe dynamic Coulomb friction coefficient, B is the model constant proposed by Madsen and Sorensen (1992) .

 The model simulates three-dimensional antidunes (Upper panel of Figure 4). The left figure of Fig. 4 shows anantidune like an alternate bar, but the wavelength is much shorter than the bar scale. Right figure of Fig. 4 shows
the formation of multiple antidunes in transverse direction. These antidunes mostly migrate upstream (Figure 5).
 The long time calculation somewhat changes the initially-formed antidunes as shown in bottom panel of Figure 4.
The simulated antidunes tend to be 2D antidunes (left figure of Fig. 4), and in relatively wide channel for which
the alternate bar can form, the upstream-migrating antidune coexists with downstream-migrating free bars.

Conclusion
 We proposed a depth-integrated model to capture three-dimensional antidunes. The model performance is investigated by fully-nonlinear numerical simulations.
 The model can reproduce the formation and development of three-dimensional antidunes, and also coexistence of antidune and free bars.
 The limitations of the model are 1) simulated 3D antidunes tend to lose their three-dimensionality, and 2) the model can not reproduce downstream-migrating antidunes.

References
Iwasaki et al., 2016, AGU2016.
Madsen and Sorensen, 1992, Coastal Engineering