本文目的在於探討土石流的濃度分佈和流速分佈對土石流運動行為的影
響，並透過質量和動量守恆式的深度積分獲得控制方程式，以此為基礎進行研
究。由於本研究的水砂混合密度是變動的且是隨著流場土砂濃度改變的物理量，
在深度積分的質量守恆式與動量守恆式中將會產生四個修正係數。而這些修正
係數對於土石流的運動行為模擬將有著顯著的影響。本文建構出底床傾角、顆
粒間磨擦角、流速、濃度、修正係數的關係，可供之後對於土石流的模擬參考。
在穩態均勻流的條件下，可利用剪力平衡和壓力平衡推導出流速式和濃度
式。影響土石流流況有幾個因素，顆粒間磨擦角、底床傾角、土砂顆粒粒徑、
底床土砂濃度、水砂密度比以及底床條件。從研究結果可歸納六項結論: (1) 經
由嚴謹的深度積分我們發現在質量守恆式和動量守恆式中需有四個修正係數
才可以完整描述土石流運動行為；(2) 相同顆粒間磨擦角，底床傾角越大，濃
度分佈在深度方向上會越趨均勻；(3) 相同顆粒間磨擦角，底床傾角小時，流
體呈現清水層和混合層，且隨著底床傾角減小，清水層比例漸增；(4) 不論顆
粒間磨擦角與底床傾角的條件為何，混合層的流速分佈趨勢皆會相似且接近線
性；(5) 在定床條件下，相同的顆粒間磨擦角與底床傾角條件，不同的底床土
砂濃度對濃度分佈與流速分佈有顯著影響； (6) 底床條件對於濃度分佈與流速
分佈有顯著的影響，相同顆粒間磨擦角和底床傾角下，動床和定床的濃度與流
速分佈有明顯差異。

The purpose of this thesis is to study the effects of concentration distribution
and velocity distribution to the behavior of debris flows. My governing equations is
integrated by conservation equations. The flow density in my research is not
constant and is vary with flow’s depth and sediment concentration. There are four
correction factors in the mass and momentum conservation due to the depth integration,
they are important to the simulation of debris flow. Building the relationship
between friction angle, bed slope, velocity, concentration and correction factors
is useful to the simulation of debris flow.
In the condition of steady state and uniform flows, we can use the stress and
pressure conservation to derive the velocity and concentration equations. There are
some factors to debris flow, such as friction angle, bed slope, diameter of sediment,
bed’s sediment concentration, density ratio and whether the bed is erodible or rigid.
After the research, we can obtain six point: (1) By depth integration, we have four
correction factors in mass and momentum conservation to describe debris flows; (2)
Under same friction angle, when the bed slope is larger, the concentration distribution
will be more uniform; (3) Under same friction angle, when the bed slope is
small, the flow field will separate into water layer and mixture layer. The proportion
of the water layer will increase while the bed slope decrease; (4) No matter the
friction angle and bed slope are, the mixture layer’s velocity distribution are very
similar and linear; (5) On the rigid bed, the bed’s sediment concentration will influence
the distribution of concentration and velocity; (6) Bed’s condition will influence
the distribution of velocity and sediment concentration. Under same friction
angle and bed’s slope, the concentration and velocity profile are different apparently
between erodible and rigid bed.

1. Ashida, K., Egashira, S., Yajima, H. “Mechanism of Debris Flow.” Disas. Prev.
Res. Inst., 31(B-2), 1988.
2. Ashida, K., Egashira, S., Liu, B. “Numerical Calculation of Flow and Bed
Evolution in Compound Channels by a Two-Layer Flow Model.” Disas. Prev.
Res. Inst., 35(B-2), 1992.
3. Bose Sujit, K. and Dey, Subhasish. “Suspended load in flows on erodible bed.”
International Journal of Sediment Research, 24, 315-324, 2009.
4. Bovis, M. J., Jakob, M. “The role of Debris Supply Condition in Predicting
Debris Flow Activity.” Earth Surface Processes and Landforms, 24,
1039-1054, 1999.
5. Cao, Z., Pender, G., Carling, P. “Shallow Water Hydrodynamic Models for
Hyperconcentrated Sediment-laden floods over Erodible Bed.” Advances in
Water Resources, 29, 546-557, 2006.
6. Capart, H. and Fraccarollo, L. “Transport layer structure in intense bed-load.”
Geophys. Res. Lett, 38, L20402, 2011.
7. Chen, H., Lee, C. F. “Numerical simulation of debris flows.” Canadian
Geotechnical Journal, 37, 146, 2000.
8. Chen, X., Li, Y., Niu, X., Li, M., Chen, D., Yu, X. “A general two-phase
turbulent flow applied to the study of sediment transport in open channels”
International Journal of Multiphase Flow, 37, 1099-1108, 2011.
9. Currie, I. G. “Fundamental Mechanics of Fluids”, McGraw-Hill, 2003.
10. D'Ambrosio, D., Iovine, G., Spataro, W., Miyamoto, H. “A macroscopic
collisional model for debris-flows simulation.” Environmental Modelling &
Software, 22, 1417-1436, 2007.
11. Drew, D. A. “Mathematical modeling of two phase flow.” Ann. Rev. Fluid
Mech., 15, 261-291, 1983.
12. Duan, J. G. and Nanda, S. K. “Two-dimensional depth-averaged model
simulation of suspended sediment concentration distribution in a groyne field.”
Journal of Hydrology, 327, 426-437, 2006.
13. Egashira, S., Miyamoto, K., Itoh, T. “Constitutive equations of debris flow and
their applicability.” Debris Flow Hazards Mitigation, 340-349, 1997.
14. Egashira, S., Honda, N., Itoh, T. “Experimental Study on the Entrainment of
Bed Material into Debris Flow.” Phys. Chem. Earth, 26(9), 645-650, 2000.
15. Egashira, S., Itoh, T., Takeuchi, H. “Transition Mechanism of Debris Flow
Over Rigid Bed to Over Erodible Bed.” Phys. Chem. Earth, 26(2), 169-174,
2001.
16. Egashira, S., Ashida, K., Takahama, J. I., Tanonaka, S. “Sediment Transport
Formula Derived from an Energy Dissipation Model of Solid-Fluid Mixture.”
Annuals, Disas. Prev. Res. Inst., 33(B-2), 1990.
17. Egashira, S., Ashida, K., Tanonaka, S., Takahashi, T. “Bed-Load Formula
Derived from Constitutive Equations of Solid Particle-Water Mixture.” Disas.
Prev. Res. Inst., 34(B-2), 1991.
18. Egashira, S., Sato, T., Chishiro, K. “Effect of Particles size on the Flow
Structure of Sand-Water Mixture.” Disas. Prev. Res. Inst., 37(B-2), 1994.
19. Egashira, S., Ashida, K., Yajima, H., Takahama, J. “Constitutive Equations of
Debris Flow.” Disas. Prev. Res. Inst., 32(B-2), 1989.
20. Einstein, H.A. and Chien, N. “Effect of heavy sediment concentration near the
bed on the velocity and sediment distribution.” Report, 8, University of
California, Berkeley, CA, 1955.
21. Fraccarollo, L., Papa, M. “Numerical simulation of real debris flow events.”
Phys. Chem. Earth, 25(9), 757-763, 2000.
22. Gongdan, Z. “The mechanisms of debris flow.” HKUST Library, 2010.
23. Hsu, T. J. “A two phase flow approach for sediment transport.” UMI, 2002.
24. Hunt, J. N. “The Turbulent Transport of Suspended Sediment in Open
Channels.” Mathematical, Physical and Engineering Sciences, 224(1158),
322-335, 1954.
25. Iverson, R. M. “The physics of debris flows.” Geophysics, 35(3), 245-296,
1997.
26. Liu, S. H., Zhao, S. L., Luo, Q. S. “Simulation of Low-concentration
Sediment-laden Flow based on two-phased Flow Theory.” Journal of
Hydrodynamics, 19(5), 653-660, 2007.
27. Mazumder, B. S. and Ghoshal, K. “Velocity and Concentration profiles in
uniform sediment-laden flow.” Mathematical Modelling, 30, 164-176, 2006.
28. Mazumder, B. S. “Grain size distribution in suspension from bed materials.”
Sedimentology, 41, 271-277, 1994.
29. Papa, M., Egashira, S., Itoh, T. “Critical conditions of bed sediment
entrainment due to debris flow.” Natural Hazards and Earth System Sciences, 4,
469-474, 2004.
30. Pouliquen, O. “Scaling laws in granular flows down rough inclined planes.”
Physics of Fluids, 11(3), 542, 1999.
31. Rijn, L. C. “Sediment Transport, Part II: Suspended Load Transport.” J.
Hydraul. Eng., 110, 1613-1641, 1984.
32. Shieh, C. L., Jan, C. D., Tsai, Y. F. “A numerical simulation of debris flow and
its application.” Natural Hazards, 13, 39-54, 1996.
33. Shrestha, B. B., Nakagawa, H., Kawaike, K., Baba, Y. “Numerical Simulation
on Debris-Flow Deposition and Erosion Processes Upstream of a Check Dam
with Experimental Verification.” Disas. Prev. Res. Inst., 51(B), 2008.
34. Spinewine, B. and Capart, H. “Intense bed-load due to a sudden dam-break.”
Journal of Fluid Mechanics., 731, 579-614, 2013.
35. Tai, Y. C., Noelle, S., Gray, J. M. N. T., Hutter, K. “Shock-Capturing and
Front-Tracking Methods for Granular Avalanches.” Journal of Computational
Physics, 175, 269-301, 2002.
36. Takahama, J. I., Fujita, Y., Kondo, Y., Hachiya, K. “A two layer simulation
model unifying debris flow and sediment sheet flow.” Matsumoto Congress
Publication, 1, 113-124, 2002.
37. Umeyama, M., Gerritsen, F. “Velocity Distribution in Uniform Sediment-laden
Flow” J. Hydraul. Eng., 118, 229-245, 1992.
38. Woo, H. S., Julien, P. Y. “Suspension of Large Concentrations of Sands.”
Journal of Hydraulic Engineering, 114(8), 1989.
39. 詹錢登. “土石流概論.” 科技圖書, 03001B, 2000.
40. 謝正倫. “土石流預警系統之現況與展望.” 環境教育季刊, 35, 29-35, 1998.