（1）
Chunguang Chen2, Lingkan Yao and Qin Wang1
1School of Civil Eng.,Southwest Jiaotong University,
Chengdu 610031, China
2 Ph.D.candidate, associate professor, School of Civil Eng., Southwest Jiaotong University, Chengdu 610031, FAX:7524007.TEL.028-7602045
Abstract: In this paper,a 2-D governing equation with the tensor form is put forward first to describe the confluence of debris flow and main channel. Then,a simulat caculating model is established by means of the MAC method. Finally, some practical examples are preseted and the macroscopical domino offects of the interaction between debris flow and main channel are shown by the computer cartoons.
Keywords: debris flow, main channel , confluence, coupling analysis, MAC method
When the debris flow flows into a main river, not only the mass , density and particle distribution will vary greatly with both the space and the time ,but also the mechanical character between debris flow and main channel, especially in rheology. It is hard to calculat this complex problem by using common numerical sumulation method . In this paper,a 2-D governing equation with the tensor form is put forward first to describe the confluence of debris flow and main channel based on theoretics of both the classical fluid mechanics and non-Newtonian fluid mechanics. Then,a simulat caculating model is established by means of the Marker-Cell method and Particle-In-Cell method. And the rheologic equation and rheologic parameters to be needed can be obtained through the current theory of debris flow or sedimentation. Finally, some practical examples are preseted and the macroscopical domino offects of the interaction between debris flow and main channel in the confluence area are shown by the computer cartoons.
In the case of water field there are following equation of continuity (1) and dynamic equation (2) are derived :
（1）
（2）
where
is a shear stress tensor,
Debris flow belongs to non-Newtonian fluid, the governing equation for debris flow can be also derived from newtonian physics:
（3）
（4）
It is similar to Newtonian fluid that rheologic equation for non-Newtonian fluid can be written as 1),
（5）
where ,
is the first Rivlin-Erichsen tensor，
;
is a viscosity function for
non-Newtonian fluid;
.
Thus, in this analysis the basic equation for describing the confluent of debris flow and main channel can be written as a unified equation:
（6）
（7）
From the results of observation for the confluent of debris flow , we assume reasonally all of flow parameters in a vertical direction vary less than in horizontall direction, therefore, in this analysis we can omit term of an acceleration and a shear stress on the third direction in (7), hence
Fig.1 Diagram of water depth
（8）
Integrating this equation yields
（9）
where,
is free surface of fluid as shown in Fig. 1.
Thus, equation (3) can be manipulated into the form to be two equations in horizontal direction:
(10)
Now, we can integrate Eq.(6) over the
depth from x3=-h to x3=
A mass flow rate per unit width over the depth
is introduced
here. Moreover, notice that
as shown Fig.1. From which we get
two-dimensional equation of continuity:
（11）
Integrating Eq.(10) over the depth
from x3=-h to x3=
,then substituting Eq.(5) into it ,we have
（12）
where
is a shear force from the bed，it can be expressed as follows:
（13）
where
is a coefficient of friction on the
bed.
If
，and assuming that the change with the time for the dipth (H)
approximates to zero , further, substituting
and
below into Eq.(11) and (12), we
have
（14）
（15）
These are the two-dimesional flow equation to apply to flow yield in the confluence of debris flow and main channel.
The confluence of debris flow and main channel has some features such as : (1)The flow is unsteady ; (2)There is a strong interaction between main channel and debris flow ; (3) The changes in density , size distribution of sediments and rheology of flow caused by interraction are mainly in the confluent boundary of two flow fields first. Based on these features, we decided to sumulat this problem by using the marker-and-cell (MAC) method and particle-in-cell method(PIC)2) due to the following reasons: (1)MAC method is suitable for calculating unsteady flows ; (2)Combining MAC with PIC method, a coupling model of confluence can be established easily in each calculating step and the rheologic equation and rheologic parameters to be needed can be obtained through the current theory of debris flow or sedimentation ; (3)By changing particle distribution, different density distribution and size distribution ,which are definitude or random,can be produced for the more accurate coupling model.
First, integrating Eq.(15) from time
(t) to time (
)，in the case that time interval
is very little，yields:
（16）
where
Making a divergence to both sides of
Eq.(16), and considering the continuity Eq.(14) ,that is
.then Eq.(16) can be rewritten as
（17）
Discretization of Posson Eq.（17）is:
（18）
When calculate the flows in the confluence of debris flow and main channel by means of the MAC method, it is necessary to build up a relation of the density and rheology for the mixing fluid. First, initial particle distribution are given in each mesh through PIC method. A type of particle and its number in debris flow region usually can be decided to base on both computing time and memory. After particle number in each mesh is calculated , density of fluid in each mesh is given as
（19）
where n is
marker number of water; m is marker number of debris flow;
is density of water ;
is density of debris flow.
Rheology of fluid in mixing region may be clasified in many ways such as : a.water (Newton body), b. sediment laden flow (Newton body or Bingham body), c. debris flow (Newton body or Bingham body).
for Bingham body, a viscosity function is given as :
（20）
where
is the coefficient of rigidity;
is the yielding shear.
A factor affected rheology of fluid maily have solid concentration, size of particle, size-grading of particle, physical and chemical property of solid, shape of particle , flow condition of fluid and so on. At present the rheologic equation and parameter for expressing debris flow and sediment laden flow are very much . 3), 4). Theoretically,any one of these model or parameter have been investigated through the field experiments. Even though each model or parameter have a certain range, but all of those are good firsthand data. Furthermore, because it is a general discussion in this paper, in order convenience to discuss, we can use the model in article 4):
（21）
（22）
where
is a relative viscosity
coefficient; Svm is limit concentration of thick liquid; SVF
is solid concentration in thick liquid; SVC is concentration of solid
that size is over 0.1～0.3mm; S’Vm is limit concentration of
mixing fluid.
Because large stone in debris flow
affect yielding stress(
) little; we can substitute yielding stress(
) of thick liquid eliminated large stone for yielding stress(
) of debris flow . Like this, substitute yielding stress(
) of mixing fluid in the confluence can be replaced by yielding stress(
) of thick liquid that is corresponding to mixing fluid, so4)
（23）
where B is a
coefficient;
.
Limit concentration(SVm) of
thick liquid have a concern with size-grading of particle in thick liquid, and
it is constant in certain case. Also, limit concentration(S’Vm)
of mixing fluid concern the quantity contained and size-grading of particle.
Evidently, SVC and S’Vm depend upon state of
flow in the confluence. therefore its value should be a important point studied
in simulating. But this problem need to experiment even more. Tentatively , in
this paper values of SVC and S’Vm are
estimated to base on existing equation, to be function of concentration of
mixing flow. Hence, first decide a ratio of SVF and SVT
from the results to study previously, let
, thus
（24）
volume-concentration of solid in mixing region is:
（25）
where
is the density of debris flow;
is the density of soil .
The critical concentration from Newtonain fluid
to Bingham body in the mixing region is4)
（26）
Through the above analyzing, we are aware of that using MAC method is a good useful way to provide enough information about density and particle distribution as well as size-grading in mixing region, from this a accurate couple model can be made.
In order to show this simulation method , we calculated the behavior of flow for the case of that debris flow ditch is perpendicular to river. The simulating results were shown by the computer cartoons. Because the space of a printed page is limited, so only pattern of flow at some time are shown in this paper. Fig. 2 shows Water depth pattern of confluence, Fig. 3 shows Particles distribution of confluence. Calculating condition are : the width of main river B=4.0m, the width of debris flow ditch b=2.0m, the velocity of river is 1.0m/s, the velocity of debris flow is (a)1.0m/s and, (b)2.0m/s. Another parameter and constant are given in the table 1. In order to compare , the case such as water flow along branch are considered , and the results are shown in Figure 2 (c) and Figure 3 (c).
Fig.2 Water depth pattern of confluence
Table1 Parameter and constant
|
|
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Fig.3 Particles distribution of confluence
The following conclusions can be drawn from the example: (1) The initial velocity of debris flow will affect the flow of main channel greatly. (2)When the debris flow flows into main river,the rise in water will be more obvies(Fig.2-b) but the sucking ability weaker(Fig.3-b) as compared with the confluence of water flow(Fig.2-c and Fig.3-c). (3) Comparing Fig.3-b with Fig.3-c,the debris flow could pass through a farther distance in the same time.
[1] CHEN, W. F. , 1984,Non-Newtonain fluid mechanics (In Chinese), peking, Sciece Press.
[2] LI, D. Y. ,XU, G. R. and SHUI, H. S. , 1997, Numerical method for tow dimension unsteady flow fluid mechanics (In Chinese), peking, Sciece Press.
[3] Wu, J. S., et al. 1993, Debris flow and its comprehensive control (In Chinese), peking, Sciece Press.
[4] QIAN, N., 1989, sediment laden flow (In Chinese), peking, Tsinghua Unversity Press.
