Boundary conditions effect on linearized mud-flow shallow model

Czasopismo : Acta Geophysica
Tytuł artykułu : Boundary conditions effect on linearized mud-flow shallow model

Autorzy :
Nowożyński, K.
Institute of Geophysics, Polish Academy of Sciences, Warszawa, Poland, kn@igf.edu.pl,
Ślęzak, K.
Institute of Geophysics, Polish Academy of Sciences, Warszawa, Poland, katarzyna.slezak@igf.edu.pl,
Kądziałko-Hofmokl, M.
Institute of Geophysics, Polish Academy of Sciences, Warszawa, Poland, magdahof@igf.edu.pl,
Szczepański, J.
Institute of Geological Sciences, University of Wrocław, Wrocław, Poland, jacek.szczepanski@ing.uni.wroc.pl,
Werner, T.
Institute of Geophysics, Polish Academy of Sciences, Warszawa, Poland, twerner@igf.edu.pl,
Jeleńska, M.
Institute of Geophysics, Polish Academy of Sciences, Warszawa, Poland, bogna@igf.edu.p,
Nejbert, K.
Institute of Geochemistry, Mineralogy and Petrology, Warsaw University, Warszawa, Poland, knejbert@uw.edu.pl,
Shireesha, M.
National Geophysical Research Institute, Council of Scientific and Industrial Research, Hyderabad, India, shireeshageo.m@gmail.com,
Harinarayana, T.
National Geophysical Research Institute, Council of Scientific and Industrial Research, Hyderabad, India, thari54@yahoo.com,
Romashkova, L.
Institute of Earthquake Prediction Theory and Mathematical Geophysics, Russian Academy of Sciences, Moscow, Russia, lina@mitp.ru,
Peresan, A.
The Abdus Salam International Centre for Theoretical Physics, SAND Group, Trieste, Italy,
Arosio, D.
Department of Structural Engineering, Politecnico di Milano, Milan, Italy, diego.arosio@polimi.it,
Longoni, L.
Department of Environmental, Hydraulic, Infrastructures and Surveying Engineering, Politecnico di Milano, Milan, Italy, laura.longoni@polimi.it,
Papini, M.
Department of Environmental, Hydraulic, Infrastructures and Surveying Engineering, Politecnico di Milano, Milan, Italy, monica.papini@polimi.it,
Zanzi, L.
Department of Structural Engineering, Politecnico di Milano, Milan, Italy, luigi.zanzi@polimi.it,
Kostecki, A.
Oil and Gas Institute, Kraków, Poland, kostecki@inig.pl,
Półchłopek, A.
Oil and Gas Institute, Kraków, Poland, polchlopek@inig.pl,
Abbaszadeh, M.
Department of Surveying and Geomatics Engineering, Faculty of Civil Engineering, Babol Noushirvani University of Technology, Babol, Iran, m.abbaszadeh@nit.ac.ir,
Sharifi, M.
Department of Surveying and Geomatics Engineering, College of Engineering, University of Tehran, Tehran, Iran, sharifi@ut.ac.ir,
Nikkhoo, M.
Faculty of Geodesy and Geomatics, K.N. Toosi University of Technology, Tehran, Iran, Mehdi_nikkhoo@yahoo.com,
Di Cristo, C.
1Dipartimento di Ingegneria Civile e Meccanica, Università degli Studi di Cassino e del Lazio Meridionale, Cassino, Italy, dicristo@unicas.it,
Iervolino, M.
Dipartimento di Ingegneria Civile, Design, Edilizia e Ambiente, Seconda Università di Napoli, Aversa, Italy, michele.iervolino@unina2.it,
Vacca, A.
Dipartimento di Ingegneria Civile, Design, Edilizia e Ambiente, Seconda Università di Napoli, Aversa, Italy, vacca@unina.it,
Abstrakty : The occurrence of roll-waves in mud-flows is investigated based on the formulation of the marginal stability threshold of a linearized onedimensional viscoplastic (shear-thinning) flow model. Since for this kind of non-Newtonian rheological models this threshold may occur in a hypocritical flow, the downstream boundary condition may have a nonnegligible effect on the spatial growth/decay of the perturbation. The paper presents the solution of the 1D linearized flow of a Herschel and Bulkley fluid in a channel of finite length, in the neighbourhood of a hypocritical base uniform flow. Both linearly stable and unstable conditions are considered. The analytical solution is found applying the Laplace transform method and obtaining the first-order analytical expressions of the upstream and downstream channel response functions in the time domain. The effects of both the yield stress and the rheological law exponent are discussed, recovering as particular cases both power-law and Bingham fluids. The theoretical achievements may be used to extend semi-empirical criteria commonly employed for predicting roll waves occurrence in clear water even to mud-flows.

Słowa kluczowe : boundary conditions, Herschel and Bulkley fluid, linear stability, roll-waves, shallow flow model,
Wydawnictwo : Instytut Geofizyki PAN
Rocznik : 2013
Numer : Vol. 61, no. 3
Strony : 649 – 667
Bibliografia : 1. Ancey, C., N. Andreini, and G. Epely-Chauvin (2012), Viscoplastic dam break waves: Review of simple computational approaches and comparison with experiments, Adv. Water Resour. 48, 79-91, DOI: 10.1016/j.advwatres. 2012.03.015.
2. Anderson, K., S. Sundaresan, and R. Jackson (1995), Instabilities and the formation of bubbles in fluidized beds, J. Fluid Mech. 303, 327-366, DOI: 10.1017/ S0022112095004290.
3. Anderson, T.B., and R. Jackson (1967), Fluid mechanical description of fluidized beds: equations of motion, Ind. Eng. Chem. Fundamen. 6, 4, 527-539, DOI: 10.1021/i160024a007.
4. Balmforth, N.J., and J.J. Liu (2004), Roll waves in mud, J. Fluid Mech. 519, 33-54, DOI: 10.1017/s0022112004000801.
5. Berlamont, J.F. (1976), Roll-waves in inclined rectangular open channels. In: Proc. Int. Symp. “Unsteady Flow in Open Channels”, BHRA Fluid Engineering, Newcastle, A2, 13-26.
6. Berlamont, J.F., and N. Vanderstappen (1981), Unstable turbulent flow in open channels, J. Hydraul. Div. 107, 4, 427-449.
7. Bialik, R.J., V.I. Nikora, and P.M. Rowiński (2012), 3D Lagrangian modelling of saltating particles diffusion in turbulent water flow, Acta Geophys. 60, 6, 1639-1660, DOI: 10.2478/s11600-012-0003-2.
8. Bird, R.B., G.C. Dai, and B.J. Yarusso (1983), The rheology and flow of viscoplastic materials, Rev. Chem. Eng. 1, 1-70.
9. Bose, S.K., and S. Dey (2013), Turbulent unsteady flow profiles over an adverse slope, Acta Geophys. 61, 1, 84-97, DOI: 10.2478/s11600-012-0080-2.
10. Brock, R.R. (1969), Development of roll-wave trains in open channels, J. Hydraul. Div. 95, 4, 1401-1427.
11. Coussot, P. (1994), Steady, laminar, flow of concentrated mud suspensions in open channel, J. Hydraul. Res. 32, 2, 535-559, DOI: 10.1080/00221686.1994. 9640151.
12. Di Cristo, C., and A. Vacca (2005), On the convective nature of roll waves instability, J. Appl. Math. 2005, 3, 259-271, DOI: 10.1155/JAM.2005.259.
13. Di Cristo, C., M. Iervolino, A. Vacca, and B. Zanuttigh (2008), Minimum channel length for roll-wave generation, J. Hydraul. Res. 46, 1, 73-79, DOI: 10.1080/00221686.2008.9521844.
14. Di Cristo, C., M. Iervolino, A. Vacca, and B. Zanuttigh (2009), Roll-waves prediction in dense granular flows, J. Hydrol. 377, 1-2, 50-58, DOI: 10.1016/ j.jhydrol.2009.08.008.
15. Di Cristo, C., M. Iervolino, A. Vacca, and B. Zanuttigh (2010), Influence of relative roughness and Reynolds number on the roll-waves spatial evolution, J. Hydraul. Eng. ASCE 136, 1, 24-33, DOI: 10.1061/(ASCE)HY.1943-7900.0000139.
16. Di Cristo, C., M. Iervolino, and A. Vacca (2012a), Discussion of “Analysis of dynamic wave model for unsteady flow in an open channel” by Maurizio Venutelli, J. Hydraul. Eng. ASCE 138, 10, 915-917, DOI: 10.1061/(ASCE) HY.1943-7900.0000538.
17. Di Cristo, C., M. Iervolino, and A. Vacca (2012b), Green’s function of the linearized Saint–Venant equations in laminar and turbulent flows, Acta Geophys. 60, 1, 173-190, DOI: 10.2478/s11600-011-0039-8.
18. Di Cristo, C., M. Iervolino, and A. Vacca (2013a), Waves dynamics in a linearized mud-flow shallow model, Appl. Math. Sci. 7, 8, 377-393.
19. Di Cristo, C., M. Iervolino, and A. Vacca (2013b), Gravity-driven flow of a shearthinning power-law fluid over a permeable plane, Appl. Math. Sci. 7, 33, 1623-1641.
20. Di Nucci, C., and A. Russo Spena (2011), Discussion of “Energy and momentum under critical flow conditions” by O. Castro-Orgaz, J.V. Giráldez and J.L. Ayuso, J. Hydraul. Res. 49, 1, 127-130, DOI: 10.1080/00221686.2010. 538573.
21. Di Nucci, C., A. Russo Spena, and M.T. Todisco (2007), On the non-linear unsteady water flow in open channels, Il Nuovo Cimento B 122, 3, 237-255, DOI: 10.1393/ncb/i2006-10174-x.
22. Dooge, J.C.I., and J.J. Napiórkowski (1984), Effect of downstream control in diffusion routing, Acta Geophys. Pol. 32, 4, 363-373.
23. Dooge, J.C.I., J.J. Napiórkowski, and G. Strupczewski (1987), The linear downstream response of a generalized uniform channel, Acta Geophys. Pol. 35, 3, 277-291.
24. Drew, D.A. (1983), Mathematical modeling of two-phase flow, Ann. Rev. Fluid Mech. 15, 261-291, DOI: 10.1146/annurev.fl.15.010183.001401.
25. Dyke, P.P.G. (1999), An Introduction to Laplace Transforms and Fourier Series, Springer Undergraduate Mathematics Series, Springer, London, 248 pp.
26. Gradshteyn, I.S., and I.M. Ryzhik (1963), Table of Integrals, Series, and Products, 5th ed., Academic Press, London.
27. Greco, M., M. Iervolino, A. Leopardi, and A. Vacca (2012), A two-phase model for fast geomorphic shallow flows, Int. J. Sediment Res. 27, 4, 409-425, DOI: 10.1016/S1001-6279(13)60001-3.
28. Huang, X., and M.H. García (1997), A perturbation solution for Bingham-plastic mudflows, J. Hydraul. Eng. ASCE 123, 11, 986-994, DOI: 10.1061/(ASCE) 0733-9429(1997)123:11(986).
29. Huang, X., and M.H. García (1998), A Herschel–Bulkley model for mud flow down a slope, J. Fluid Mech. 374, 305-333, DOI: 10.1017/S0022112098002845.
30. Hwang, C.-C., J.-L. Chen, J.-S. Wang, and J.-S. Lin (1994), Linear stability of power law liquid film flows down an inclined plane, J. Phys. D: Appl. Phys. 27, 11, 2297-2301, DOI: 10.1088/0022-3727/27/11/008.
31. Iverson, R.M. (1997), The physics of debris flows, Rev. Geophys. 35, 3, 245-296, DOI: 10.1029/97RG00426.
32. Jackson, R. (2000), The Dynamics of Fluidized Particles, Cambridge Monographs on Mechanics, Cambridge University Press, Cambridge.
33. Johnson, A.M. (1970), Physical Processes in Geology, Freeman, Cooper and Co., San Francisco, 577 pp.
34. Liggett, J.A. (1975), Basic equations of unsteady flow. In: K. Mahmood and V. Yevjevich (eds.), Unsteady Flow in Open Channels, Vol. 1, Water Resources Publications, Fort Collins, 29-62.
35. Liu, K.F., and C.C. Mei (1989), Slow spreading of a sheet of Bingham fluid on an inclined plane, J. Fluid Mech. 207, 505-529, DOI: 10.1017/ S0022112089002685.
36. Montuori, C. (1963), Discussion of “Stability aspect of flow in open channels” by F.F. Escoffier and M.B. Boyd, J. Hydraul. Div. 89, 4, 264-273.
37. Moreno, P.A., and F.A. Bombardelli (2012), 3D numerical simulation of particleparticle collisions in saltation mode near stream beds, Acta Geophys. 60, 6, 1661-1688, DOI: 10.2478/s11600-012-0077-x.
38. Napiórkowski, J.J., and J.C.I. Dooge (1988), Analytical solution of channel flow model with downstream control, Hydrol. Sci. J. 33, 3, 269-287, DOI: 10.1080/02626668809491248.
39. Ng, C.-O., and C.C. Mei (1994), Roll waves on a shallow layer of mud modelled as a power-law fluid, J. Fluid Mech. 263, 151-184, DOI: 10.1017/ S0022112094004064.
40. O’Brien, J.S., and P.Y. Julien (1988), Laboratory analysis of mudflow properties, J. Hydraul. Eng. ASCE 114, 8, 877-887, DOI: 10.1061/(ASCE)0733-9429 (1988)114:8(877).
41. Pascal, J.P. (2006), Instability of power-law fluid flow down a porous incline, J. Non-Newton. Fluid Mech. 133, 2-3, 109-120, DOI: 10.1016/j.jnnfm. 2005.11.007.
42. Pascal, J.P., and S.J.D. D’Alessio (2007), Instability of power-law fluid flows down an incline subjected to wind stress, Appl. Math. Model. 31, 7, 1229-1248, DOI: 10.1016/j.apm.2006.04.002.
43. Pitman, E.B., and L. Le (2005), A two-fluid model for avalanche and debris flows, Phil. Trans. Roy. Soc. A 363, 1832, 1573-1601, DOI: 10.1098/rsta.2005. 1596.
44. Ponce, V.M., and D.B. Simons (1977), Shallow wave propagation in open channel flow, J. Hydraul. Div. 103, 12, 1461-1476.
45. Ridolfi, L., A. Porporato, and R. Revelli (2006), Green’s function of the linearized de Saint–Venant equations, J. Eng. Mech. 132, 2, 125-132, DOI: 10.1061/ (ASCE)0733-9399(2006)132:2(125).
46. Stoker, J.J. (1957), Water Waves, Interscience, New York.
47. Supino, G. (1960), Sopra le onde di traslazione nei canali, Rendiconti Lincei 29, 5-6, 543-552 (in Italian).
48. Takahashi, T. (1991), Debris Flow, IAHR/AIRH Monograph, Balkema, Rotterdam.
49. Trèves, F. (1966), Linear Partial Differential Equations with Constant Coefficients, Gordon and Breach Science Publishers, New York.
50. Trowbridge, J.H. (1987), Instability of concentrated free surface flows, J. Geophys. Res. 92, C9, 9523-9530, DOI: 10.1029/JC092iC09p09523.
51. Tsai, C.W-S., and B.C. Yen (2001), Linear analysis of shallow water wave propagation in open channels, J. Eng. Mech. 127, 5, 459-472, DOI: 10.1061/ (ASCE)0733-9399(2001)127:5(459).
52. Vedernikov, V.V. (1946), Characteristic features of a liquid flow in open channel, C. R. Acad. Sci. URSS 52, 3, 207-210.
53. Venutelli, M. (2011), Analysis of dynamic wave model for unsteady flow in an open channel, J. Hydraul. Eng. ASCE 137, 9, 1072-1078, DOI: 10.1061/(ASCE) HY.1943-7900.0000405.
54. Zanuttigh, B., and A. Lamberti (2007), Instability and surge development in debris flows, Rev. Geophys. 45, 3, RG3006, DOI: 10.1029/2005RG000175.
DOI :
Cytuj : Nowożyński, K. ,Ślęzak, K. ,Kądziałko-Hofmokl, M. ,Szczepański, J. ,Werner, T. ,Jeleńska, M. ,Nejbert, K. ,Shireesha, M. ,Harinarayana, T. ,Romashkova, L. ,Peresan, A. ,Arosio, D. ,Longoni, L. ,Papini, M. ,Zanzi, L. ,Kostecki, A. ,Półchłopek, A. ,Abbaszadeh, M. ,Sharifi, M. ,Nikkhoo, M. ,Di Cristo, C. ,Iervolino, M. ,Vacca, A. , Boundary conditions effect on linearized mud-flow shallow model. Acta Geophysica Vol. 61, no. 3/2013
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