Jiang Changbo1, 2 Bai yuchuan1 Zhao Zidan1
[1]Institute of Sediment on River and Coast Engineering
Tianjin University, Tianjin 300072,China
Liu Xiaopin2
[2]Dept. of River and Coast Engineering
Changsha Communications University Changsha 410076, China
Tel 86-22-27401747, E-mail: jcb36@263.net, ychbai@tju.edu.cn
Abstract:
Vortex ripple is widely formed in the coastal region, and the dynamic of vortex
is quite important because it is responsible for sediment transport. The flow
structure around the vortex ripples can be modeled as qusi-2D flow due to the
geometry of the flow boundaries. In this paper, 2D Large-Eddy-Simulation method
(LES) was used to predict the flow structure and the dynamic of vortex, the
numerical simulation results show a completely process of vortex formation,
evolvement and disappearance.
Keywords: oscillatory boundary layer flow, large eddy simulation, vortex ripple
As waves propagate into shallow water region, various shapes of sand ripple are formed in the sea bottom. In generally, two common types of sand ripple are rolling ripple and vortex ripple, the rolling ripple always can be found in the first stage of sand ripple formation, with the developing of flow conditions, the vortex ripple is widely formed in the coastal region. The vortex ripple enhanced the separation of flow and production of turbulence, it plays a vital role in various coastal process, such as sediment transport, the wave attenuation and the mass transport due to wave. Nevertheless, the flow structure and sediment movement above vortex ripple has not been investigated owing to the complexity of the phenomena.
Du Toit & Sleath (1981) used the
Laser-Doppler to measure the flow over a ripple bed, the similar work was
conducted by Sato S.J.et
al.
(1986), very detailed measurements of velocity closed to the bed over fixed
smooth and rough bed are made by using LAD. In resent years, most of the effort
has been focused on obtaining a numerical solution of oscillatory flow over
ripple, the first work was described by Lounguet-Higgins (1981), who was used
the discrete vortex model. Blondeaux & Vittorri (1991) who treat a laminar
flow situation by solving the vorticity transportation equations. Sato S.J.et
al.
(1986), Aydin (1987) and Tsujimoto et al (1991) used a two equation turbulent
model
to study the flow around the ripple and the suspended sediment concentration
distribution. Recently, Fredsoe,J et al. Have conducted experiments and
numerical simulation by
model to study the flow structure
around the vortex ripples under wave action. Where the turbulence model only
captures the averaged velocities, the more advanced numerical model need to be
studied the dynamic of vortex.
In the present study, a numerical simulation system based on LES method is developed for analyzing flow structure and dynamic of vortex, the sub-grid-scale turbulent stress is evaluated by the Smagorinsky model (1963). LES techniques originated from the global weather simulation with the Smargorinsks' work, since Deardoff (1970) made his first pioneer contribution on the problem of engineering, and then it has been developed by Schumann (1975), Madabushi & Vanka (1991), Yuan.M and Song,C.C.S (1989,1990,1994) and others, The simplify marker and cell method (SMAC) is used to solve the basic equations in the the curvilinear coordinater system, and a detailed of vortex dynamics is discussed.
In this study, the flow is governed by
incompressible Naverier-Storkes and continuity equations. By
applying the grid filter to continuity and Navier-Storkers equations one obtain
the filtered governing equations.
(1)
(2)
Here,
correspond to
,respectively; x is the steam-wise direction, y is the direction normal to the
walls, z is the span-wise direction;
is the velocity filed,
is the pressure,
is the fluid density.
Considering a case of the turbulent flow passing
over ripple, a body-fitted coordinater are introduced. the equationas are
transformed from physical space to the curvilinear coordinater computational
space with transformation:
(3)
In this paper, two dimensional flows are
considered, and a bodyfitted coordinates are intruduced according to :
(4)
Using this transformation, the two dimensional
(
) filter continuity equation and momentun equations are:
(5)
(6)
In the LES computations, the effect of small
scales appears in the the subgrid-scale (SGS) stress
, which must be modeled. In this paper, the eddy-viscosity concept model was
used as following:
(7)
where
is the fluid viscosity,
is the sub-grid-scale eddy viscosity,
is the model parameter,
is the length scale related to the filter width (see below), and the
is the magnitude of the resolved-scale-rate tensor:
(8)
The
parameter
can be either constants or functions of the time and space. Such as they can be
evaluated using Smargorinsky model and dynamic eddy-viscosity model. In this
paper, as the first step in modeling wave boundary layers using LES, only the
constant parameter values (
) of Smargrionsky model are considered.
The governing equations have been solved by means of a SMAC method. The forward differencing scheme is used for the time dependence, and the centered differencing in the pace for the diffusive terms. The differencing of the convective terms used the donor-cell (second-order upstream) method. The more details about the numerical method can be found in the paper of H. Miyata et. al(1984).
The computational domain is shown in the Fig.1, the periodic conditions are used on the sides of the domain, the symmetric condition is used at the top, i.e.
and
at
the boundary conditions on the bottom is no-slip conditions, it can be described as following:
and
at
As the flow is periodic in the stream-wise, an oscillatory volume force
was added to drive the flow all over the domain. Fig. 2 shows an orthogonal
curvilinear gird system.
Fig. 1 the computational domain Fig. 2 A grid system around the vortex ripple
A comparison between the measurement of the flow over plane bed (Jesen 1989) and ripples bed (Jorgen et al. 1999) with numerical results are conducted, it shows the simulation results are good, more details can be found in (Jiang 2001).
The most important characteristic of the flow
around vortex ripples are the dynamic of the vortices, it can be visualized from
the velocity field. In generally, the velocity field obtained from the
turbulence model (
,
model) is the averaged velocity field. However, using LES method can obtain the
instantaneous velocity, so that it can easily reveal the dynamic of the vortices
over the vortex ripple.
Fig.3 clearly shows the formation, evolvement
and disappearance of the separation bubble during one wave period. A very strong
rotation is produced after the flow over the ripple crest, the separation bubble
grows quickly from the ripple crest with the increasing of the free stream
velocity, and it reaches its maximum value shortly after the free stream
velocity begin to decelerate, the phase is around
. Even the outer flow velocity begin to decelerate, the separation bubble still
expend and transport to the next ripple. At the same time, the bubble is curling
up into the main flow, but it is still maintains the strong rotation. As the
outer flow reverses, the separation bubble lifts over the ripple crest and eject
into the outer flow, became the free vortex, the phase is around
. Then the next separation bubble begin to form at lee back of the ripple, at
the same time, the former ejected vortex breaks up gradually in the outer flow.
Fig.
3 The flow velocity field and vorticity field at different phase
The
characteristic quality of vortex is very important, because the vortex is
responsible for sediment transport. The numerical experiments with different
value
, the
is the sand ripple length and the
represents the amplitude of the motion of fluid. Fig. 4 illustrates the
calculated voracity with different value
at same phase. For the shorter ripper or strong outer flow, the vortex is big
and reaches to the crest of next ripple, a lot of sediment will be eroded.
However, if the ripple is long and the outer flow is weak, the vortex does not
extend to the next ripple, there is no strong interaction between the neighbor
ripples.
Fig.
4 The vorticity field with different
(phase
)
Numerical model based on 2D-LES was presented to simulate the vortex dynamics in the oscillatory boundary layer flow over ripple beds. The results reveal the vortex characteristics over one period, the main factors influenced the vortex are discussed. It shows that the flow can be simplified to 2D flow structure at such circumstance; in addition, the 2D LES method is more efficient to simulate the flow structure around complex geometry boundary.
In generally, all flows are unstable
three-dimensional in the nature. However, there exist some situations which the
flow can be treated as quasi-2D turbulent flows, these types flow are widely
found in geophysics and engineering. There are two major factors that may cause
a flow to become quasi-2D: geometry of the flow boundaries and/or certain body
forces. Although modeling of quasi-2D flows has important practical applications,
particularly in the river and oceanic sciences, it has not received as much
attention in the literature as the modeling of the 3D flows. Thus there exists a
hope that in the near future, practically important quasi-2D problems can be
solved using LES or DNS method.
This paper shows that LES model proves to be a
good tool to predict the large scales of turbulent flows as well as period
unsteady flows. The number of adjustable constants is significantly reduced as
compared with the
and
models. However, the present model has a major drawback, since fine meshes are
necessary to accurately predict small turbulent scales. The latter point is
particularly crucial near the walls. Very fine grids, accurate numerical schemes
as well as large computation times are required. Despite those drawbacks, the
LES model proves to have many potentialities. Further studies are expected on
the vortex dynamics and sediment transport under the action of combine wave and
current over ripple bed.
Acknowledgements
This is research was supported by the National Natural Science Foundation of China under contracts no. 59809006 and No. 59890200, and The state key laboratory of Estuarine & coastal Research under contract No.98001.
References
[1] Aydin I., et al., Coastal Eng. In Japan., 30: 11-24 (1988).
[2] Blondeaux, P. and Vittori,G. J. Fluid Mech. 226, 257 (1991).
[3] Deardorff J W, J. Fluid Mech., 1970, 41: 453-480.
[4] Du Toit & Sleath. J. Fluid Mech. 112, 71-96, (1981)..
[5] J. Fredsoe, K. H. Andersen and M. Sumer, Coastal Engineering 38, 177-21 (1999).
[6] Jiang ,C.B. Phd. Thesis, Tianjin University, 2001
[7] Jensen,B.L. et al . J. Fluid Mech. 206, 265-297, (1989).
[8] Lounguet-Higgins,M.S, J. Fluid Mech. 107, 1-35, (1981).
[9] Miyata H, Nishimura S, J Comp Phys, 1985, 60:391-436
[10] Sato, S., et al., Coastal Eng. In Japan., 29: 65-78 (1986.
[11] Schumann,U, S. J., Comp. Phys., 1975,18: 376-404.
[12] Song C. C. S., Yuan M, ASME, J Fluid Eng, 1988,110:441-445
[13] Song C. C. S., Yuan M,. Proc. of the international symposium on large hydraulic machinery and associated equipment, Beijin, China,1989.
[14] Smagorinsky J,, Monthly Weather Review, 1963,19:99-165.
[15] Tsujimoto,G et al. Coastal Eng. In Japan., 34: 178-189 (1991).
[16] Vittori, G. and Blondeaux,P.. J. Fluid Mech. 239, 23, (1991).
[17] Yuan,M Song C C S, ASME,J. Fluid Eng., 1990,112:415-444.