1 / 36

RiverFLO-2D Model Governing Equations and Numerical Method

elaina
Download Presentation

RiverFLO-2D Model Governing Equations and Numerical Method

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

    1. RiverFLO-2D Model Governing Equations and Numerical Method Reinaldo Garcia FLO-2D Software, Inc. September, 2009

    2. Outline Governing equations Numerical method Sediment transport River FLO-2D sediment transport model Treatment of dry-bed conditions

    3. 2D Shallow water approximation

    4. 2D Shallow water equations

    5. SWE. Friction terms

    6. Sediment transport model for alluvial rivers Sediment continuity equation Sediment load

    7. Sediment transport model Meyer-Peter & Muller:

    8. Sediment transport Bed load, suspended load, wash load. Substrate vs surface based transport. Sediment transport formulas for substrate sediment. “The main source of uncertainty in calculated transport rates arises from uncertainty in the input values of grain size, boundary stress, and hydraulic roughness.” “Too often, the transport formula is blamed for poor results when the real culprit is poor input.” Wilcock et al.

    9. Sediment transport “When calculating transport rates, it is very easy to be very wrong”. (P. Wilcock et al.)

    10. Sediment transport model Required data: Substrate sediment D50 (Also D90 for Van Rijn formula) Sediment specific gravity Bed porosity Inflow boundary conditions for sediment transport assume equilibrium, i.e. Inflow Qs is determined from transport formula.

    11. If this was not enough… “…by making the computations easier, BAGS and similar software makes it possible to produce inaccurate estimates (even wildly inaccurate estimates) very quickly and in great abundance.”, Wilcock et al. Pitlick, John; Cui, Yantao; Wilcock, Peter. 2009. Manual for computing bed load transport using BAGS (Bedload Assessment for Gravel-bed Streams) Software. Gen. Tech. Rep. RMRS-GTR-223. Fort Collins, CO: U.S. Department of Agriculture, Forest Service, Rocky Mountain Research Station. 45 p. http://www.stream.fs.fed.us/publications/software.html.

    12. Sediment transport model Several sediment transport formulas: 1: Meyer-Peter & Muller (1948), 2: Karim-Kennedy (1998), 3: Ackers-White (1975). 4: Yang (Sand), 5: Yang (Gravel), 6: Parker-Klingeman-Mclean (1982), 7: Van Rijn (1984a-c), 8: Engelund Hansen (1967).

    13. Sediment Transport Equations Meyer-Peter & Muller: For steep rivers. Wong and Parker reanalyzed MPM data and adjusted formula. Wong, M.; Parker, G. 2006. Reanalysis and correction of bed-load relation of Meyer-Peter and Müller using their own database. J. Hydr. Engrg. 132(11): 1159-1168. Used for sediment sizes greater than 0.4 mm. Will generate sediment transport rates that approach those of Engelund-Hansen on steep slopes.

    14. Sediment Transport Equations Karim-Kennedy: Smplified Karim-Kennedy equation (F. Karim, 1998). Nonlinear multiple regression relationship based on velocity, bed form, sediment size, and friction factor for a large data set. Use for large rivers with non-uniform sand/gravel conditions. Karim, F. Bed material discharge prediction for non-uniform bed sediments. J. Hyd. Eng., 6, 1998. Sediment sizes 0.08 mm to 0.4 mm (river) and 0.18 mm to 29 mm (flume) and up to 50,000 ppm concentration. Slope range 0.0008 to 0.0243. Will yield similar results to Laursen’s and Toffaleti’s equations.

    15. Sediment Transport Equations Ackers-White Method: Expressed sediment transport based on Bagnold’s stream power concept. Ackers, P., and W.R. White, Sediment transport, new approach and analysis, J. Hyd. Div. ASCE, 99, HY 11, 1975 Assumes that only a portion of the bed shear stress is effective in moving coarse sediment. The total bed shear stress contributes to the suspended fine sediment transport. Dimensionless parameters include a mobility number, representative sediment number and sediment transport function. The various coefficients were determined from laboratory data for Di > 0.04 mm and Froude numbers < 0.8. The condition for coarse sediment incipient motion agrees well with Sheild’s criteria. The Ackers-White approach tends to overestimate the fine sand transport.

    16. Sediment Transport Equations Yang: Total sediment concentration is a function of the potential energy dissipation per unit weight of water (stream power ~ f(velocity and slope)) Yang, C.T. Sediment Transport Theory and Practice. McGraw Hill, New York, 1996. Sediment concentration is a series of dimensionless regression relationships. Based on field & flume data with sediment particles ranging from 0.137 mm to 1.71 mm and flows depths from 0.037 ft to 49.9 ft. Mostly limited to medium to coarse sands and flow depths less than 3 ft Can be applied to sand and gravel

    17. Sediment Transport Equations Parker-Klingeman-McLean developed and tested using gravel or sandy gravel transport data. Applied to a large number of different gravel-bed rivers.

    18. Sediment Transport Equations Parker-Klingeman-McLean: computes transport rates on the basis of a single grain size: the median grain size of the substrate, D50 Parker, G.; Klingeman, P.C.; McLean, D.L. 1982. Bedload and size distribution in paved gravel bed streams. Journal of Hydraulics Division, ASCE. 108: 544-571. Recognizes role of armor layer in bed load transport rates.

    19. Sediment Transport Equations Van Rijn: Computes suspended load and bed load separately. Van Rijn, L.C., Sediment Transport, Part I: Bed load transport, J. Hyd. Eng., ASCE, no 10. 1984a Van Rijn, L.C., Sediment Transport, Part II: Suspended load transport, J. Hyd. Eng., ASCE, no 11. 1984b Van Rijn, L.C., Sediment pick-up functions, J. Hyd. Eng., ASCE, no 10. 1984c Estimates sediment concentration. Qb = Qs+Qb

    20. Sediment Transport Equations Engelund-Hansen Method: Bagnold’s stream power concept was applied with the similarity principle to derive a sediment transport function. Engelund, F. and Hensen, E. A Monograph on Sediment Transport to Alluvial Streams. Copenhagen: Teknique Vorlag, 1967. Uses energy slope, velocity, bed shear stress, median particle diameter, specific weight of sediment and water, and gravitational acceleration Can be used in both dune bed forms and upper regime (plane bed) D50 > 0.15 mm

    21. Finite element method Why Finite Elements? Solid mathematical theory Meshes can adapt to irregular bottom and boundaries Improved computational efficiency

    22. Finite element method Galerkin weighted residual method Converts the PDEs to a system of ODEs Triangular 3 nodes elements

    23. RiverFLO-2D finite element method What is new? Explicit four-step time stepping algorithm based on matrix lumping Solve on an element-by-element basis Parallelized Fortran 95 using OpenMP

    24. Finite element method What improvements does it bring over traditional FEM? Does not require matrix assembly. No upwind required. Allows larger time steps than previous explicit FE models. Faster execution. Easy addition of h-adaptive refinement. Highly parallelizable code.

    25. Finite element method Linear interpolation functions

    26. Finite element method System of ODEs

    28. Finite element method

    29. Finite element method

    30. Finite element method Lumping: Lumping improves numerical stability but introduces numerical damping Selective lumping e in [0.9,0.98] is the selective lumping parameter Selective lumping reduces damping while preserving stability

    31. FEM. Four-Step time stepping scheme

    32. FEM. Von Newmann stability analysis Linearize equations (1D) Discretize with FEM the linearized equations Assume form for solution Substitute in discretized eqn’s Spectral radius if amplification matrix less than 1 CFL condition

    33. FEM. Von Newmann stability analysis

    34. Model parallelization Subdomain decomposition. Seamless parallel computation using OpenMP OpenMP: An API for multi-platform shared-memory parallel programming in C/C++ and Fortran. www.openmp.org, 2009. Each Processor/Core computes one subdomain Model speedup depends on number of processors, processor cache memory, etc., and is case specific.

    35. Model Speedup

    36. Treatment of dry-wet areas At the beginning of each time step all elements are evaluated to see if they are wet or dry. A completely dry element is defined when all nodal depths are less than a user defined minimum depth Hmin, that can be zero. A partially dry element has at least one node, where depth is less than or equal to Hmin. If one element is completely dry, equations are locally modified and only d?/dt=0, dU/dt=0, dV/dt=0 is solved for the element. If an element is partially dry, the full equations are solved and velocity components are set to zero for all nodes on the element. Water surface elevations are not altered for dry elements.

More Related