数理解析研究所講究録 1413 巻 2005 年 1-8

Numerical Study of the Formation of Star Dunes With Three and Four arms

鳥取大学乾燥地研究センター

張汝岩(Ruyan ZHANG)

Arid Land Research Center, Tottori University

お茶の水女子大学人間文化研究科 河村哲也 (Tetuya KAWAMURA)

Graduate School of Humanities and Sciences, Ochanomizu University

Two kinds of Star dunes are simulated numerically in order to make clear themechanism of their formation. The flow above the sand dune has been
investigated by using Large-Eddy Simulation (LES) method. The numericalmethod employed in this study can be divided in three parts: (i) Calculation of the
air flow above the sand dune by using standard MAC method with ageneralized
coordinate system; (ii) Estimation of the sand transfer caused by the flow throughthe friction; (iii) Determination of the shape of the sand surface. Since thecomputational domain is changed due to step (iii), these procedures arerepeated until typical shape of the sand dune is formed. Two cases of dunes aredirections, the sandsimulated. the first case, when the winds blow from thastar with threeoftheshapebecomes
andthreeatdirections
dune ki ends$\ln$arms.

$\ln$$\mathrm{r}\mathrm{e}\mathrm{e}$the second case, when the winds blow from three pairs of opposing

directions, the sand dune extends in four directions, becomes the shape of astar

with four arms.

1. INTRODUCTION(Arrows show the wind directions)Fig.l Typical sandin the desert. Most dunes can be classified intofondbecandunes
sandofshapes
Various. Barchan dunes and transverse
barchan dunes, transverse dunes, linear dunes and stardunes are formed when wind blows from one direction (Fig 1aand Fig. 1 ). The difference betwee$\mathrm{d}\mathrm{u}\mathrm{n}\mathrm{e}\mathrm{s}^{2)}$$\mathrm{d}\mathrm{u}\mathrm{n}\mathrm{e}\mathrm{s}^{1)}$$\mathrm{b}$them is in the sand supply. When sand supply is abundant, transverse dunes are formed, otherwise,barchan dunes are formed. Linear dunes are formed from barchan dunes by two directional winds
and extends at the converging direction (Fig.l ). Star dunes are formed by winds blow from several
directions. They commonly have a high central peak and three or more arms extending radially. The$\mathrm{c}$.

of transverse dunes and linear dunes in $2003^{4)}$ .
this study, two types of the star dunes are
simulated in order to make clear the mechanism of theof them. If the flow feld over thesand dune is computed, the friction of the wind can be determined. Using the formula between thesurface friction and the sand transfer derived by Bagnold, the mass of the sand transfer can be
estimated quantitatively. Therefore it is possible to compute the change of the shape and to predictthe movement of the sand dunes caused by the wind.
$\ln$$\mathrm{f}\mathrm{o}\ulcorner \mathrm{m}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$2. NUMERICAL METHODSThe numerical method employed in this study consists of the following three parts.
Calculation of the air flow above the sand dunes.
(\"u) Estimation of the sand transfer caused by the air flow through the friction between the air flow
and the sand suffice.
(\"ui) Determination of the shape of the sand dunes.
Because the shape of the sand suffice is changed due to step (i\"u), steps-(\"ui) are repeateduntil typical shapes of the sand dunes are formed. We will explain the procedures below.
$(\mathrm{i})$$(\mathrm{i} )$2.1 Calculation of the Air Flow

Because the strength of the wind in the saltation layer which can transfer the sand is more than, the Reynolds number of the air flow over the sand sufface is high enough that the flow is inturbulence regime. Therefore, we use LES method to compute turbulent flow over the complex
geometry. Standard MAC method is employed to solve 3-dimensional Navier-Stokes equations.
The shape of the sand dune is rather complex and is changing with time.
order to impose
boundary condition accurately along the sand sufface, the time dependent body fitted$0.02\mathrm{m}/\mathrm{s}$$\ln$coordinate system$\xi=\xi(x,y,z,t)$ ,$\eta=\eta(x,y,z,t)$ ,$\sigma$$=\sigma(x,y,z,t)$is used in this study. Then the computation can be done on the time independent rectangular.
2.2 Estimation of the Sand Transfer, there are three types of sand transfer. They are surface
According to the study ofcreep, saltation and suspension. These processes are depending on the radius of the sandgranularity and the strength of the wind. Observation of the sand transfer shows that saltation isdominant. In this study, we assume that the sand is transferred only by saltation. the processof saltation, the relation between the sand transfer and the friction velocity
is given$\mathrm{g}\dot{\mathrm{n}}\mathrm{d}\mathrm{s}^{5)}$$\mathrm{B}\mathrm{a}\mathrm{g}\mathrm{n}\mathrm{o}1\mathrm{d}^{6)}$$\ln$$\mathrm{b}\mathrm{y}^{6\rangle}$$u_{\mathrm{r}}$$q=cu\underline{\rho}.3$

$g$

(1)

3
is the density of the air,is the friction velocity,whereis experimental constant. The frlction velocity is given by$u_{*}$$q$$\rho$|s the mass transfer of the sand.

$c$

(2)

’
$\sqrt{v\frac{d|\mathrm{U}|}{d_{-}^{-}}},\mathrm{u}.//\mathrm{L}$$|\mathrm{u}.|\approx$

where$\mathrm{U}$is the velocity parallel to the sand sufface and$v$

is the turbulent eddy viscosity.

2.3 Determination of the Shape of the Sand Dune
The sand dunes are changing its shape by the sand transfer estimated by equation (1).
Considering the local coordinate system along the sand surface, continuity equation of sandbecomes$p$,$\frac{dh}{dt_{h}}\approx$$- \frac{dq_{1}}{dY}-\frac{dq_{2}}{dV}$(3).

is the normal distance from the base plane parallelis the density of the sand andcomponents of the vector . Axes $X$ andto the sand sufface, whereas , are theare determined by the base plane and the original x-z plane and y-z plane respectively (Fig.3).
time equals to the net influx of the sand into the small
Eq.3 shows that the increment ofregion. By discretizing Eq.3, the shape of the sand dunes are determined in every time step.
Because the time scale of the change of air flow is quite different from that of the change ofthe sand surface, the time increment to integrate Eq.3 is 2000 times greater than the one used. It means that we estimate the sandto integrate the Navier-Stokes equationtransfer every
..

Where

$h$

$\rho_{\backslash }$$\mathrm{Y}$$q_{1}$$X,\mathrm{Y}$$q_{2}$$q$$h\mathrm{w}.\mathrm{K}\mathrm{h}$$(\Delta t_{h}\fallingdotseq 2000\Delta t)$$2000\mathrm{A}/$, the height of the sand at
If the slope of the sand exceeds the maximum angle of about
value and to satisfy themaximum
keep
theartificiallybothtochanged
point
isthe gridare determined by using the relation
andconservation of the sand. Namely,$32^{0}$$\Delta h_{1}$$S_{AX_{1}JX_{1}^{1}}‘=S_{BX_{2}OX_{2}}$,$\Delta h_{2}$(Fig.4)–—–A
Fig.4 Sand avalanche

3. RESULT

before avalanche
after avalanche

3.1 Star dunes with three arms
The initial sand dune is shown on Fig.5. A hill with circular cross section parallel to x-y planeand parabolic cross section parallel to both x-z and y-z planes is considered |n the simulation. The number of grid
analysis. The height of the hill isand the radius of the base ispoints is 139 in -direction, 117 in -direction and 20 in z-direction.
(Fig.5). After 10
Initial uniform wind is applied in velocity of -direction with lntensity$30\mathrm{m}$$25\mathrm{m}$$\mathrm{x}$$\mathrm{y}$$|\mathrm{u}_{1}|=8\mathrm{m}/\mathrm{s}$$\mathrm{y}$hours the wind velocity is changed withwind direction is$60^{0}$$|\mathrm{u}_{2}|=10\mathrm{m}/\mathrm{s}$and the angle between negative -axis and$\mathrm{x}$. After another 10 hours, the wind veloclty is changed to$|\mathrm{u}_{3}|=10\mathrm{m}/\mathrm{s}$andThere after, identical set of winds is applied
the angle between -axis and wind direction isagain. It is assumed that the wind blow over the whole region.
$60^{0}$$\mathrm{x}$Fig.5 Computational domain (left) and the grid near the hill (right)

The flow feld over the fixed sand dune is calculated without movement during the first 1000
for the Naviersteps (20 seconds) in order to obtain the initial conditions. Time increment-(\"ui) are. By using these initial conditions, the steps
Stokes equation is set toshapecomputed.
isof the sand dune
repeated as mentioned in section 2 and the change of theAKhough the shape of the sand sulface changes with time, n0-slip condition is imposed$\Delta t$$0.02\mathrm{s}$$(\mathrm{i} )$because the sand moves very slowly.
Tme development of sand surface contours are presented on Fig.6. When wind is changed
every 10 hours, the simulated dune, which has circular cross section parallel to x-y plane andparabolic cross section parallel to both x-z and y-z planes, extends at three directions andbecomes the shape of star with three arm. ( means day in Fig. 6)$\mathrm{d}$Fig.6 Contours of sand surface

3.2 Star dunes with four arms

Fig.7 Computatlonal domain (left) and the grid near the hill (right)
The initial sand dune is shown on Fig.7, a hill with circular cross section parallel to x-y plane andand theparabolic cross section parallel to both x-z and y-z planes. The height of the hill isand-direction,y-direction
in139
inis
137
pointsgridofThenumber.
isradius of the baseinz-direction.
20$20\mathrm{m}$$30\mathrm{m}$$\mathrm{x}$6this simulation, the wind is assumed blow from three pairs of opposing directions. The
lntensity of the wind |s shown on Table 1 and the direction of the wlnd is shown on Fig.8 (g).
$\ln$Time duration is 10 hours.
As in the prevlous case, the initial conditions are calculated for the first 1000 steps (20seconds). The time increment
for the Navier-Stokes equation is set to. By using these-(iii) as mentioned in sectlon 2 and compute the changelnitial conditions, we repeat stepsof the shape of the sand dune. As the same reason as mentioned in section 3.1, n0-s|ip
condition is imposed on the sand surface.
Fig.8 shows the time development of sand surface contours. When winds are blowing form
three pairs of opposing directions, the simulated dune, which has circular cross section parallelto x-y plane and parabolic cross section parallel to both x-z and y-z planes, extends at four
$0.02\mathrm{s}$$\Delta t$$(\mathrm{i} )$dlrections, with high central peak and four arms extending radlally.

7Fig.8 Contours of sand surface
When the sand supply is increased to two times as shown in Fig.9, another shape of star.
dunes–complex linear dunes are rmed. lt is explained in our another paper$\mathrm{c}1\mathrm{e}\mathrm{a}\mathrm{r}1\mathrm{y}^{7)}$$\mathrm{f}\mathrm{o}$$\ovalbox{\tt\small REJECT} \mathrm{o}\mathrm{o}$[$\mathrm{a})\mathrm{t}\underline{-}$od

Fig. 9 Contours of sand surface of complex linear dunes

4.CONCLUDING REMARKS
this study, the formation of star dunes are simulated and the flow above the sand dunes
are investigated. One hill is placed on the sand sufface as the initial condition.
$\ln$

When the winds blow from three directions, the simulated dune extends at three directions,becoming the shape of a high central peak and three arms extending radially. When the winds
blow from three pairs of opposing directions, the simulated dune extends at four directions,becoming the shape of a high central peak and four arms extending radially.
Further problem is to investigate the factors that affect the number of the arms of star dunes
four arms and the complex linear dunes.
and the relationship between the star dunes$\mathrm{w}.\mathrm{R}\mathrm{h}$REFERENCES
1). Wasson and R. Hyde, “Factors determining desert dune typ\"e, Nature$\mathrm{R}.\mathrm{A}$$\mathrm{V}\mathrm{o}\mathfrak{l}.304$

8, pp. 337-339.
D. Mckee, A study of global sand seas, Introduction to a study of global sand seas”
L1S Geological Survey Professional Paper 1052 (1979), pp.1-19.
3) M. Kan and T. Kawamura, “Numerical simulation of the formation of the barchan sand(1999), pp.349-354.
dun\"e, Theoretical and Applied Mechanics,
4) R. Zhang, Y. Sato, M. Kan and T. Kawamura Numerical study of the effect of flow fields(2003), pp.205on the shape of sand dune” Theoretical and Applied Mechanic,$(\mathrm{f}983)$

2)

11$\mathrm{E}$$\mathrm{V}\mathrm{o}\mathrm{l}.48$$\mathrm{f}4$$\mathrm{V}\mathrm{o}\mathrm{l}.52$

210.

. Thompson, Z.U.A Warsi, C.W Mastin, “Numerical grid generation foundations andapplications”, Elsevier Science Pubulishing Co. Inc. (1985).
6) R. A. Bagnold, “The movement of desert sand”, Proc. Roy. Soc. A157 (1963).
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Computational Mechanics in conjunction with the Second Asian-Pacifc Congress on
Computational Mechanics, (September 5-10, 2004), Beijing, China,.
5)$\mathrm{J}.\mathrm{F}$$\mathrm{U}\mathrm{R}\mathrm{A}$$\mathrm{C}\mathrm{D}$