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Reduced Equations and Special Solutions for Geomorphic Dam-break Flows
BEM2014 2014/ 10/ 04
Hervé Capart Dept of Civil Engineering National Taiwan University
In celebration of Professor Young Der-Liang’s 70th birthday
SLIDE 2 Sources 2 Capart, H., and D.L. Young (1998) Formation of a jump by the dam- break wave over a granular bed. Journal of Fluid Mechanics 372, 165– 187. Fraccarollo, L., and H. Capart (2002) Riemann wave description of erosional dam-break flows. Journal of Fluid Mechanics 461, 183–228. Capart, H., M. Bellal, and D.L. Young (2007) Self-similar evolution of semi-infinite alluvial channels with moving boundaries. Journal of Sedimentary Research 77, 13-22. Hsu, J.P.C., and H. Capart (2008) Onset and growth of tributary-dammed
- lakes. Water Resources Research 44(11), W11201.
Spinewine, B., and Capart, H. (2013) Intense bed-load due to a sudden dam-break. Journal of Fluid Mechanics 731, 579-614. Capart, H. (2013) Analytical solutions for gradual dam breaching and downstream river flooding. Water Resources Research 49(4), 1968- 1987.
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Dam break: sudden failure of rigid dam Example: Balin check dam, September 2007 Sudden water release produces geomorphic change. Dam breach: Gradual failure of loose dam Example: Tangjiashan landslide dam, June 2008 Water outflow and geomorphic change drive each other.
NTU-MHRG Xinhua
Geomorphic dam break and dam breach flows 3
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Research strategy 4 How to make these problems mathematically tractable ? 1) Use lab and field observations to guide the mathematics 2) Cast the flows as m oving boundary problems. 3) Exploit similarity and quasi-linearity to find special solutions.
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Taipei dam break experiments (Capart and Young, 1998) 5
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Louvain dam break experiments (Spinewine and Capart, 2013) 6
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Evolving boundaries 7
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Vertical flow structure 8
Experimental Theoretical
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Semi-empirical closure relations 9
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~ x u h t z ) ( x cu h c h c z t x z h u h x t u h ) (
2
Depth-integrated governing equations (Spinewine and Capart, 2013) 10
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Quasi-linear hyperbolic equations 11
W W B W x t ) (
) ( ) ( i i i
K BK
Eigenstructure Homogeneous hyperbolic system
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Riemann wave solution (Fraccarollo and Capart, 2002) 12
t ) / ( ) , ( t x t x W W
) ( 3 3 ) ( 2 2 ) ( 1 1 i i i
K dW K dW K dW
i
t x
Self-similar expansion Integrate across simple wave
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Comparison with experiments 13
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Predicted velocity and concentration maps 14
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Detailed comparisons with experiments (Spinewine and Capart, 2013) 15
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Dam breach example: 2009 Namaxia debris dam 16
Forestry Bureau, Taiwan
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Alluvial diffusion theory (Capart, Bellal, Young, 2007; Hsu and Capart, 2008) 17
2 2
x z KQ t z
w s
h z z
s w
x zw )) ( ( x z h z z
w s w
Water-driven diffusion Complementary constraints on water surface profile
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Self-similar behavior (Capart, Bellal and Young, 2007) 18
) (
2 2
x z t KQ t z
s s
x t xC ) ( Qdt x f Qdt zs Qdt xC
Diffusion problem Self-similar solution
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Dam breach solution (Capart, 2013) 19
Qdt R t
D
) (
2 / 3 2 / 1 27 8
) ( ) ( t bg t Q ) ( ) ( t Q dt t dz A
L L
S.Y.J. Lai
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Reduction to a pair of autonomous ODEs 20
2
2 ) (
2 8 1 2 2
1 4 ) ( t t t
2 / 3
) ( dt t d
2 / 3 2 / 3
) ( dt t d
ODE pair Solution
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Solution plots 21
3 2 3
) ˆ 1 ( ˆ 8 ) ( t t Q t Q
P
2 2
ˆ 1 ˆ ) ( t t d t
B
Breach drop Discharge hydrograph
P
T t t ˆ
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GSC
Application to 1996 Lake Ha! Ha! dyke breach 22
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B C Xinhua / GSC
Comparison with more field events 23
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Flood wave routing to the downstream valley 24
8 2 3
2 / 1
x h h f gS t h
V
Kinematic wave equation S.Y.J. Lai
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Solution by the method of characteristics 25
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Wave profile evolution and discharge hydrographs at downstream stations 26
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Comparison with 2008 Tangjiashan landslide dam breach flood 27
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Conclusions 28 For both dam break and dam breach problems: 1) Reduced equations provide good approximations of real behavior. 2) Similarity and quasi-linearity can be exploited to find useful special solutions.
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Acknowledgements and thanks 29
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Acknowledgements and thanks 30
