Application of Lattice Boltzmann and Cellular Automata in Erosion Modeling for Deep-Water Turbidity Current

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1 Application of Lattice Boltzmann and Cellular Automata in Erosion Modeling for Deep-Water Turbidity Current Siyao Xu 1 Tapan Mukerji 1 Interdisciplinary Program of Earth, Energy and Environment Sciences Energy Resource Engineering Stanford University Abstract The objective of hybrid modeling is to obtain a balance of geological reality, the representation of spatial uncertainty with a fixed set of initial parameters and the capability of being conditioned to the hard data as well. Reasonable algorithms have been constructed in previous works (Pyrcz et al. 003, 005, Michael et al. 008 and Leiva et al. 009). However, realism of erosion and deposition, which were modeled by applying simple geological rules, is still dissatisfying. One reason is that no real physics equation for flow motions area considered, which is an important factor determining erosion. Another reason is that erosion rules should be applied locally, accounting for complex bed surface and flow conditions. In this study, we test a coupled scheme that is expected to incorporate physical equations into hybrid model and is capable to apply erosion rules locally. D Saint Venant Equations are adopted for modeling flow using Lattice Boltzmann Method (LBM). The primary advantage of LBM is the ease of implementing boundary condition with complex geometries. Bed surface processes are modeled in a Generalized Cellular Automata (GCA) model, which can apply specific erosion rules in local neighborhoods. 1D tests have been made to verify feasibility and accuracy of the models. The coupled is scheme is tested to be feasible, however, further studies are required to extended to -D, implement real flow physics of turbidity flow and to improve erosion rules. 1

2 1. Introduction The objective of hybrid modeling is to obtain a balance of geological reality, the representation of spatial uncertainty with a fixed set of initial parameters and the capability of being conditioned to the hard data as well. Previous works include Pyrcz et al. 003, 005, Michael et al. 008, and Leiva and Mukerji 009. These works attempt to model a submarine channel-lobe system. Figure 1.1 demonstrates the modeling scheme (Leiva and Mukerji 009), in which process-based model is used as a database. Distributions of lobe geometry and anchor point are interpreted from the database. The distributions are used to generate training images, by means of objectbased method. Multi Point Statistics (MPS) is applied as conditioning method using training images generated in Object-based method. W L Figure 1.1: Procedure of hybrid modeling (Modified from Michael et al. 009). However, erosion is not well modeled in previous methods. Michael et al. (009) proposed to approximate erosion by rules. The greatest erosion is specified at locations with the maximum depositional thickness and highest underlying topography. This rule generally approximates erosion process within the lobe extent, but it is not realistic within the channel (Figure 1.A). Leiva and Mukerji (009) further developed this idea by using more specific geomorphologic algorithms. In Leiva s scheme, the erosion is assumed to be a function E e( g, c, a), where c represents curvature of topography, and a represents the alignments between assumed flow direction and the gradient of current topography g. For each sediment event, an erosion map is generated with function e. Topography takes more important roles in Leiva s algorithm, and therefore, we can observe more details about erosion in channel area of the erosion map (Figure 1.B). However, since the algorithm does

3 not consider geometries of the geobody, erosion within a lobe does not mimic its shape, which is considered unrealistic as well. Moreover, Leiva and Mukerji applied geomorphologic algorithms to create more realistic geometries for geobodies and erosion zones. But further tests on Leiva and Mukerji s algorithm reveal that the algorithms are not efficient in complex topography. (A) (B) Figure 1.: Comparison of erosion maps generated by Michael s rules (A) and Leiva s (B). In summary, use of geomorphologic rules is a possible method for approximating erosion, but the means of applying rules is difficult. First, if a rule will be applied over the whole geobody area, it should be generalized enough for all topography and flow conditions of that geobody, but this is rarely true in a simulation. Second, geomorphologic algorithms become inefficient and inaccurate in complex topography. Is it possible to apply physical equations for erosion effectively and to apply geomorphologic rules locally? This question leads to the motivation of our study. In this study, we examine a scheme that satisfies three requisites. First, the scheme is simple and is ready to integrate into a geobody of the hybrid model. Second, the scheme is capable of applying geomorphologic rules locally. Finally, the scheme incorporates some physics equations efficiently. Challenges for this study are obvious as well. First is to balance the use of physics equations to the use of local rules. Second, efficiency of the scheme is required to generate multiple realizations in short time. Last, simplicity of the scheme in implementation is needed to be incorporated into the hybrid model. 3

4 . Methodology.1. Erosion in Hybrid Framework Following the idea of hybrid modeling, erosion process is decomposed into three smaller problems: i) flow model; ii) sediment transport processes; iii) bed surface processes. We start combing physics into hybrid framework from flow equations. Two reasons lead to this decision. On one hand, flow motion is too complex to approximate by rules, on the other hand, flow equations are widely studied in fluid mechanics and there are various techniques for their implementation. Finite element method ( FEM ), and finite volume method (FVM) method are used to numerically simulate flow equations conventionally. However, for hybrid models, the complexity of FEM and FVM make them not the best choice. Current hybrid models, demonstrated in Figure 1.1, assumes that erosion and deposition only occur within a sediment event, in other words, they are modeled within a geometry generated by object-based method. Precisely representing boundaries of the geometry is critical for setting boundary conditions in FEM and FVM, which leads to extra computational load and coding complexity. Hence, a simple method for flow equations other than FEM and FVM is needed, and we use the Lattice Boltzmann method. Bed surface processes are modeled by Cellular Automata (CA). CA technique applies intuitive rules in a local window, which perfectly fits our aim of applying erosion and deposition rules locally. Sediment transport processes evolves more complex physics within the scheme. It is not considered yet in our ongoing study... A Coupled Scheme with Lattice Boltzmann and Generalized Cellular Automata A coupled scheme is proposed in Figure.1. Coupled modeling techniques have been applied in different areas of earth sciences such as meteorology (Bjerknes, 1964) and snow particle transport and deposition (Masselot et al. 1998) etc. Advantage of coupled-modeling is that it breaks the erosion problem into two or more simpler problems. In this scheme, a flow model and a bed surface model are constructed independently. The two models respectively correlate to a process that is discussed in the beginning of Section.3 (except for process ii). Each of the models takes values from another as their parameters. The flow is modeled with Saint Venant Equations, through Lattice Boltzmann Method, and bed surface is modeled with simple dune dynamic rules through CA. The interaction between flow model and bed surface model are through flow velocity and bed topography. Variations in water entrainment and volumetric sediment concentration are ignored so far. Sediment erosion and deposition are 4

5 assumed to be a function of flow energy and bed surface roughness, which are simplifications that will be improved with better physics as the study progresses. Analytical Flow Model Discretize Lattice Boltzmann Flow Model Local Flow Energy Local Bed Surface Generalized Cellular Automata Bed Surface Model Figure.1: Coupled scheme for incorporating flow equations with bed surface model..3. The Flow Model.3.1. Saint Venant Equation - Depth Averaged Flow In flow modeling, sometimes the horizontal scale may be much greater than the vertical scale, such as in coastal areas, estuaries and harbors, as well as turbidity currents. In such conditions, problems are characterized by their horizontal motion, and the hydrostatic pressure is often used instead of the momentum equation in the vertical direction, in other words, vertical convection is ignored and the whole flow is modeled as a vertical-averaged layer. Equations are derived by integrating the general flow equations along the vertical direction. Depth-averaged equations have been used to model deepwater turbidity current (Parker et al. 1986), with additional equations on sediment conservation, and turbulence energy. However, in this study, we aim to verify the feasibility of coupled scheme, only the basic Saint Venant equations are considered, which are also known as Shallow Water Equations (Zhou 004), and are given as: 5

6 h ( hui ) 0 t x i ( hu ) ( huiu j) i h ( hui ) g F i t x j xi x j Eq. (.1) zb i bi where Fi gh Ei. In the above equations, h refers to current x i depth, g refers to gravitational acceleration, refers to flow density, t refers to time, xi refers to coordinates in direction i, u i refers to velocity in direction i, refers to kinematic viscosity, and finally the force term F, i i gh z x bi is flow free surface shear stress term, b i F i refers to the force term working on the current. In is the topography term, where zb represents bed surface, is bed shear stress term, i E is the Coriolis term. For the reason of simplification, the shear stress terms and Coriolis terms are not considered in this study. They will be considered in the future. Studies demonstrate that neither 3D nor D flow models can predict the vertical flow velocity profile accurately (Stansby et al. 1998), and for the efficiency requirement of a hybrid model, we only applied D flow model in this study, which effectively reduces the computational costs compared to using 3D models..3.. Lattice Boltzmann Method for Fluid Dynamics Lattice Boltzmann Method (LBM) is an alternative numerical method for simulating a wide range of fluid dynamical and sediment transport phenomena. Opposed to conventional numerical methods, which are based on discretization of macroscopic equations of the conservation laws, LBM describes fluid mechanics at the microscopic or mesoscopic scale in an extremely simplified way. The essential of LBM is to describe fluid particle motions using kinematic theory at microscopic or mesoscopic scale, from which the macroscopic equations are recovered. Particles in LBM for hydrodynamics are not real fluid molecules, but fictitious computational fluid packages. A brief description of LBM is documented in Appendix. For details of LBM in fluid dynamics, readers can refer to Chen et al. 1994, Chorpard et al. 005, and Zhou 004 specifically on LBM for Shallow Water equations. In the LB method, it is easy to implement boundary conditions for complex geometries. Boundary conditions are applied on the mesoscale level, and adaptive meshing is avoided within a LBM lattice at the resolution of a hybrid model lattice. This is one of the advantages of LBM over conventional FEM and FVM methods, which reduces complexity and computational costs. Another potential advantage of 6

7 LBM is that the algorithm is inherently parallel, therefore it is ready to be parallelized..4. Bed Surface Model The bed surface in this study is modeled by generalized cellular automata, of which the rules are intuitive and stochastic. The idea is to find proper rules that approximate erosion and deposition processes. Moreover, the generalized cellular automata framework permits stochastic rules, (which is not permitted in classical cellular automata), hence probability maps, which are proved to be an efficient and simple method (Michael et al. 008, Leiva and Mukerji 009), can be incorporated. Sguanci s (006) ripple dynamics rules are adopted in this study. The model is extremely simplified, in which only saltation and deposition are considered. Again, the variation of sediment concentration is ignored. The model assumes that erosion or deposition is a function of flow energy E i and local topography within an asymmetrical window z ( x R S, x R ), where b W W R W is local window radius; S is the upstream extension on the left window boundary; z b is bed surface (Figure.). Figure.: 1D bed surface scheme. Follows Sguanci et al. With in a local window of grid i, the mean bathymetry zb is calculated. A reasonable assumption is that the probability of grid i to be eroded or deposited is correlated to the relative difference between bathymetry of grid i and the local mean. The difference is defined as z bi z b. If zbi zb 0, erosion occurs, and grid i is eroded by a preset erosion rate. If z z 0, deposition occurs and grid i is deposited by deposition rate d. e b Local bathymetry difference is used to determine potential energy changes in bi 7

8 erosion with equations listed in Table.1. Table.1: Bed Surface Erosion/Deposition Mechanisms Erosion Deposition Eer, i ( zbi zbi 0.5) Ede, i ( zbi zbi 0.5) z ( t 1) z ( t) z ( t 1) z ( t) b b e b b d Then probabilities of erosion and deposition are computed from potential energy change E er, i or E de, i with equations given by Eq. (. ). P 1 Eer, i 0 e Eer, i 0 1 er, i ( Eer, i ) ui P 1 Ede, i 0 e Ede, i 0 1 de, i ( Ede, i ) ui Eq. (.) where i u represents the flow velocity. The final occurrence of erosion and deposition on a grid is determined stochastically with respect to the probability maps generated in Eq. (.). 8

9 3. 1D Simulation Results We implemented a 1D LBM for Saint Venant Equations following Zhou (004). Several tests are performed on the flow model and the bed surface model to check the results. Finally, two models are coupled and results are documented LB Flow Model Tests Steady Flow Over a Bump This test simulates a 1D steady subcritical flow in a 5 m long channel with a bump given by Eq. (3.1) ( x 10) 8 x 1 zb( x) 0 else Eq. (3.1) Zhou (004) tested this problem on his Lattice Boltzmann solver for Saint Venant 3 Equations. The constant discharge 4.4 m / s is set at the upstream end of the model, and the constant current depth h m is set at the downstream end. As simulation reaching the steady state, a surface drop is expected above the bump. Coefficients and parameters are listed in Table 3.1. Table 3.1: Coefficients for Test h( downstream ) m q (1) m / s e 15 m / 1.5 x 0.01m s Current surface for our simulation and Zhou s results are compared in Figure

10 . 1.8 (A) h+z (m) b 0 x(m) (B) Figure 3.1: A) Simulation surface profile; B) Surface profile of Zhou s solution from Zhou 004. The results show reasonable agreement Steady Flow Over Regular Topography 10

11 The second test refers to Bermudez et al It is a 1D problem of coastal wave with channel length L = 14, 000 m, and current depth defined by Eq. (3.). 40x 4x 1 H ( x) sin[ ( )] Eq. (3.) L L The bed surface topography is defined as z H (0) H ( x) b The initial conditions are h( x,0) H ( x) u( x,0) 0 The boundary conditions are 4t 1 h(0, t) H (0) 4 4sin[ ( )] u( L, t) 0 Bermudez also gives out the analytical solution as 4t 1 h( x, t) H ( x) 4 4sin[ ( )] ( x 14000) 4t 1 u( x, t) cos[ ( )] h( x, t) Other coefficients are Table 3.: Coefficients for Test 3.1. e 00 m / s 0.6 x 17.5m Figure 3. compares the results between numerical simulation and the analytical solution. 11

12 V e lo c ity (A) Numeric Results Analytical Results X F low D e p th (B) X Topography Numeric Results Analytical Results Figure 3.: A) Comparison of velocity profile; B) Comparison of flow depth; The relative error between the numerical solution and the analytical solution of velocity is 8.9% and that of current depth is 9.8%. The results are not perfect, but sufficient for generating erosion probability maps. 3.. CA Bed Surface Model With Constant Flow Field For the independent simulation of the bed surface model, we start by overlying a constant velocity field with u( x, t) 0.5 m / s, the initial topography is set the same as in Section Other coefficients are listed in Table 3.3. Table 3.3: Coefficients for Test 3. Lx 5m Grid Information dx 1m dt 1 Simulation Time Simulation Period

13 Erosion/Deposition Parameters Erosion Rate 0.03 Deposition Rate 0.03 Local Window Radius 0.6m Window Upstream shift Local Window Radius/ The initial topography is the same as that in Section Figure 3.3 (A) (D) plots the bed surface evolution at time t=1, 100, 600, 00. Some interesting features are observed from the results. First, observing Figure 3.3 (A), which is at the very beginning of simulation, the topography is characterized by a bump and flat surface. Sharp transitions are at the toes of the bump. We observed that it is more probable for erosion to occur on the upstream side of the bump, but much less at the sharp toes. Opposite phenomena are observed on the depositional probability map. Deposition is more probable to occur at the toes, but much less on the bump. Further examining time series plots at t=100, 600, 00, we find constant erosion on the upstream side of the bump and deposition on the downstream side. The bump evolves from symmetrical to an asymmetrical shape with steady slope on the upstream side and steep slope on the downstream side, while the bump itself migrates downstream. To verify these observations we referred to the scheme of ripple dynamics from Bogs (1995) (Figure 3.4). First of all, comparing the topography generated by our simulation, the upstream side of the bump has a steadier slope than the down stream side, which is consistent with Figure 3.4. Secondly, we plotted the stratigraphy of simulation. Onlap is observed in results, which is caused by migration of the bump. This is consistent with Figure 3.4 as well. Hence we demonstrate that the bed surface model is capable of mimicking some bed surface evolutions under flow. 13

14 (A) Bed Surface at t = 1 Z b ( m ) E r o s io n P r o b a b ilit y D e p o s iti o n P r o b a b ility Erosional Probability Depositional Probability

15 (B) Bed Surface at t = 100 Z b ( m ) E r o s io n P r o b a b ilit y D e p o s iti o n P r o b a b ility Erosional Probability Depositional Probability

16 (C) Bed Surface at t = 600 Z b ( m ) E r o s io n P r o b a b ility D e p o s i tio n P r o b a b il ity Erosional Probability Depositional Probability

17 (D) Bed Surface at t = 3000 Z b ( m ) E r o s io n P r o b a b ilit y D e p o s iti o n P r o b a b ility Erosional Probability Depositional Probability 0.5 Figure 3.3: (A) Bed surface map, erosion probability and depositional probability map at t=1; (B) Bed surface map, erosion probability and depositional probability map at t=100; (C) Bed surface map, erosion probability and depositional probability map at t=600; (D) Bed surface map, erosion probability and depositional probability map at t=3000; 17

18 Figure 3.4: Scheme of ripple migration. Follows Boggs, Sam, Jr., Stratigraphy at t = Zb(m) Figure 3.5: Stratigraphy of every 500 timesteps of our simulation Coupled Simulation For Steady Flow Over Bed Surface With a Bump as Initial Topography In this section, we couple the LBM flow model with the CA bed surface model. The coefficients used for the flow model are the same as those in Table 3.1, and coefficients for bed surface model are the same as those in Table 3.3. We simulate the flow model first, and the CA bed surface model is coupled after the flow reaches its steady state. In this test, what we are looking to verify is the feasibility of the dynamic 18

19 interaction between the two models. Thus we would like to observe the correlation between the bed evolution and flow evolution. We also would like to verify that the ripple migration could be mimicked under this condition as well. The simulation results at t=38.86s (beginning of the steady state), 40.s, 65.66s, and 79.06s are plotted in Figure 3.6 (A) (B). Observation reveals that as the bed surface evolves under the effect of flow, it also affects flow depth. As the shallow water flow model will keep the discharge conservative, flow velocity is affected by the flow depth. The updated velocity will in turn reshape the bed surface again. The bump migrates downstream similarly in the previous test, and as the bump migrates the fluid velocity and depth changes with the bump. Further examine the interval stratigraphy in Figure 3.7. Because the flow conditions are dynamically changing in this test, the stratigraphy is more complex than the previous case. However, the lamination constructed by migrating ripples is also observed. We should notice that since the variation of sediment concentration is ignored in the model, the only interaction between flow and bed surface are flow depth and velocity. Thus, the topography evolution is essentially dominated by the bed surface model and the simulation results are similar under different flow conditions. 19

20 0. (A) Bed Surface at time = s u at time = s.5.5 h at time = s 1.5 0

21 (B) Bed Surface at time = s 0 3 u at time = s.5.5 h at time = s 1.5 1

22 (C) Bed Surface at time = s 0 3 u at time = s.5.5 h at time = s 1.5

23 (D) Bed Surface at time = s 0 3 u at time = s.5.5 h at time = s 1.5 Figure 3.6: (A) Bed surface map, current velocity and depth map at t=38.86s; (B) Bed surface map, current velocity and depth map at t=40.s; (C) Bed surface map, current velocity and depth map at t=65.66s; (D) Bed surface map, current velocity and depth map at t=79.06s; 3

24 0.4 Stratigraphy Z b (m ) Figure 3.7: Stratigraphy of every 500 timesteps bed surfaces under steady LB flow model; 4

25 4. Brief Summary As a summary for the whole study, we test a simple scheme that is expected to incorporate physical equations into hybrid model. In the mean time, it is possible to apply erosion rules locally. Again, the focus of this study is on the scheme but not full physics yet. Precise flow equations and whether erosion rules can approximate real processes require further study. D Saint Venant Equations are adopted for modeling flow, and the flow processes is simulated using LBM (Zhou 004). Differentiated from conventional FEM and FVM techniques, LBM describes fluid motion at mesocale and approximate flow equations at macroscale. The primary advantage of LBM is the ease of implementing boundary condition with complex geometries, which is proper for incorporating flow equations in multiple geobodies generated by object-based method. Generalized Cellular Automata (GCA) is used to model bed surface processes. The reason of this choice is that more specific rules could be applied locally within GCA. Erosion and deposition are considered stochastically determined by flow energy and local bed topography. Flow energy is taken from simulation results of LBM flow model. For quantifying local topography, we compute average bathymetry within a local window. The difference between grid bathymetry and its local average represents local topography. The rules are taken from an extremely simple ripple dynamics model (Sguanci et al. 006). The capability for current bed surface model to approximate erosion is under further examination. 5

26 5. Future Works Based on the scheme tested in this study, improvements are required to make the scheme feasible for practical simulation. First and one of the most challenging tasks is to implement flow equations for real gravity driven turbidity currents through LBM, whose motion is significantly different from an open channel flow in this study. In our current scheme, LBM approximates Eq. (.1). However, the motion of self-accelerating turbidity currents are described by Equation (3) and (5) in Parker s four-equation model (1986), listed in Eq. (5.1) here. h ( hu ) ewu t x Eq. (5.1) ( hu ) ( hu ) Ch Rg RgChS u* t x x where h represents flow depth; x is location coordinates vector; t represents time; g is gravitational acceleration rate; U is velocity vector; ew is water entrainment rate from clear ambient water into the turbidity flow; C is sediment volumetric concentration within the turbidity flow; S is the slope of the topography; u represents the friction effect between turbidity flow and the bed surfaces; a * coefficient is defined as R 1; l is the material density of sediment l. Flow l motions described by these equations consider effect of sediment within the turbidity flow C, which causes the start and acceleration of a turbidity flow. The turbidity flow is accelerated when gravitational force works on its sediment. Flow entrainment from clear water dilutes a turbidity flow. The greater the amount of clear water entrained, the slower is the turbidity flow. A turbidity flow event is considered dead when all of itself is clear water and no sediment is included within it. In our next step, we expect that our LBM could be improved to approximate Eq. (5.1) at macroscale. Second, erosion rules of bed surface model require further examinations. In terms of physics, erosion in Parker s model is described by two equations that correlate clear water entrainment and bed sediment entrainment to averaged turbulence energy within a grid. Our future works will test means of approximating effects of Parker s two equations with simple rules. Finally, improvements should be made on channel geometries of the hybrid model. Currently, channel geometry is modeled with a parabolic function that does not consider any bed surface, or flow motions. We will explore geomorphologic and fluid mechanical techniques that may increase realism for channel geometry. 6

27 References Bermudez, A., and Vazquez, M. E., 1994, Upwind methods for hyperbolic conservation laws with source terms, Computers and Fluids, 3: , Bhatnagar, P. L., Gross, E.P., and Krook, M., 1954, A model for collision processes in gases I:small amplitude processes in charged and neutral one-component system, Phys. Rev., 94: Bjerknes, J., 1964, Atlantic air-sea interaction, Vol. 10 of Advances in Geophysics, Academic Press, New York (USA), pp Boggs, S., Jr., 1995 (3rd edition), Principles of Sedimentology and Stratigraphy: Prentice Hall, Englewood Cliffs, NJ, 774 p. Chen, S., and Doolen., G. D., 1998, Lattice Boltzmann method for fluid flows. Annu. Rev. Fluid Mech : Chorpard, B., and Droz, M., 005, Cellular Automata Modeling of Physical Systems, Cambridge University Press. Leiva, A., and Tapan, M., 009, Construction of hybrid geostatistical models combining surface-based methods with object-based simulation: use of flow direction and drainage area, SCRF Annual Meeting 09. Masselot, A., and Chopard, B., 1998, A lattice Boltzmann model for particle transport and deposition, EuroPhys Lett., 100(6), pp 1-infinite. Michael H., Li H., Li T., Boucher A., Gorelick S., and Caers J., 008, Combining methods for geologically-realistic reservoir simulation, Geostat 08, VIII International Geostatistics Congress, 1-5 December, Santiago, Chile. Parker, G., Fukushima, Y., and Pantin, H.M., 1986, Self-accelerating turbidity currents, J. Fluid Mech 171:145{81}. Pyrcz, M. J., and C. V. Deutsch, 003, Stochastic surface modeling in mud rich, fine grained-turbidite lobes (abs.): AAPG Annual Meeting, May 11-14, Salt Lake City, Utah, Extended Abstract, extend/7796, 6p. Pyrcz, M. J., 003, Integration of geologic information into geostatistical models, Ph.D. Thesis, University of Alberta, Edmonton. 7

28 Pyrcz, M. J., Cateneanu O., and C. V. Deutsch, 005, Stochastic surface-based modeling of turbidite lobes: AAPG Bulletin, V. 89, no., February 005, pp Rothman, D. H., and Zaleski, S., 1997, Lattice-Gas Cellular Automata, Cambridge University Press, London. Stansby,. P. K., and Zhou, J. G., Shallow Water flow solver with non-hydrostatic pressure: D vertical plane problems, Int. J. Num. Meth. Fluids, 8: , Sguanci, L., Bagnoli, F., and Fanelli, D., 006, A Cellular Automata Model for Ripple Dynamics, Lecture Notes in Computer Science, 006, Volume 4173/006, , DOI: / _48 Zhou, J. G., 004, Lattice Boltzmann Method for Shallow Water Flows, Springer. 8

29 Appendix: Brief review of Lattice Boltzmann Method for Saint Venant Equations LBM includes three components: Lattice Boltzmann equation, lattice pattern, and local equilibrium functions. The first two are common for all fluid problems, as long as the lattice pattern is fixed. Local equilibrium functions are specified to different flow equations. In this study, we carry out the local equilibrium functions for shallow water equations. Lattice Boltzmann equation The lattice Boltzmann method involves two parts: streaming and collision (Zhou 004) (Figure A1). (A) (B) 9

30 (C) Figure A1: A) Initial States; B) After collision; C) After streaming. The streaming process is modeled as Eq. (A1) t f ( x e t, t t) f ( x, t) e F ( x, t) ' i i Ne Eq. (A1) where f is the local distribution function after streaming; ' f is the local distribution function before streaming but after collision; e is lattice velocity which x is defined as e ; x is lattice size; t is time step; F i is the component of t force term in Eq. (.1) in direction i ; e i is the particle velocity to the th neighbor of a grid in direction i ; N is a constant defined by the lattice pattern N 1 e i e. Collision is modeled as Eq. (A) f ' ( x, t) f ( x, t) [ f ( x, t)] Eq. (A) where refers to collision operator. Skipping mathematical derivations, the most widely utilized collision operator is defined as Eq. (A3) from the famous BGK model (Bhatnagar et al., 1954) 1 eq ( f ) ( f f ) Eq. (A3) 30

31 where is time relaxation coefficient; function derived from certain flow equations. eq f is the local equilibrium distribution Substituting Eq. (A), Eq. (A3) into Eq. (A1), we achieve the equation that combines both collision and streaming process as Eq. (A4). This equation has been widely used in modern LBM. 1 t f x e t t t f x t f f e F x t eq (, ) (, ) ( ) (, ) i i N e Lattice Pattern Eq. (A4) Lattice pattern is the organization of lattices. Local lattice pattern is a factor determining the mesoscapic/microscopic kinetics of fictitious fluid packages and the constant N in Eq. (A4). Generally, two types of lattice patterns are utilized in D LB models: square lattice and hexagon lattice. In this study, the 9-direction square lattice is utilized (Figure A) Figure A: 9-direction square lattice pattern Flow packages move along the lattice pattern in direction 1-8. For direction 0, the flow package has zero velocity and stays at the previous grid. The particle velocity vectors for each direction are defined as Eq. (A5). 31

32 e (0,0) 0, ( 1) ( 1) e[cos,sin ] 4 4 1, 3,5, 7, ( 1) ( 1) e[cos,sin ] 4 4, 4,6,8, Eq. (A5) Now we can determine the constant 1 1 N e e 6 x y e e Local Equilibrium Functions This is the key component of LBM, because a local equilibrium function is specific to specific flow equations. According to the theory of Lattice Boltzmann, the local equilibrium function is a Maxwell-Boltzmann equilibrium distribution, which is true to all fluid flows. One assumes that the equilibrium functions could be expanded as a power series of the macroscopic velocity and particle velocity as Eq. (A6) (Rothman et al., 1997) eq f A B e iui Ce ie juiu j D uiui Eq. (A6) Considering the lattice pattern in Figure A, 1,3,5,7 and,4,6,8, hence we have eq f has the same symmetry for A0 D0uiui 0 eq f A Be iui Ce ie juiu j Du iui 1,3,5,7 Eq. (A7) A Be iui Ce ie juiu j Duiui, 4,6,8 For solving the coefficients, we must apply constraints on Eq. (A7). These constraints are from the conservation laws (mass and momentum) represented by the Saint Venant equations Eq. (.1). eq f ( x, t) h( x, t) Eq. (A8) eq e f ( x, t) h( x, t) u ( x, t) i i 1 e e f x t gh x t h x t u x t u x t eq i j (, ) (, ) ij (, ) i(, ) j(, ) Eq. (A9) Eq. (A10) Now we have constructed a system with Eq. (A7), Eq. (A8), Eq. (A9), Eq. (4.10). In addition, we have the particle velocity definition Eq. (A5). The solution starts by substituting Eq. (A7) respectively into Eq. (A8), Eq. (A9), Eq. (A10), then we have 3

33 three equations involves e, which could be simplified using Eq. (A5). Finally, equating coefficients on both sides in the three simplified equations, we solve the coefficients for a Lattice Boltzmann Saint Venant Equations Solver as Eq. (A11). 5gh h h u iui 6e 3e 0 eq gh h h h f e iui e 4 ie juiu j u iui Eq.(A11) 6e 3e e 6e 1,3,5,7 gh h h h e iui e 4 ie juiu j u iui 4e 1e 8e 4e,4,6,8 Readers interested in the details of the derivation may refer to (Zhou, 004). To reconstruct the macroscopic velocity u i and current depth h from microscopic variables, we have h( x, t) f ( x, t) 1 eq ui( x, t) e i f ( x, t) h( x, t) Boundary Conditions Eq. (A1) In this study, two simple boundary conditions are applied on the lattice: elastic collision boundary condition (Figure A3), and influx-outflux boundary conditions (Figure A4). The elastic collision boundary condition ignores frictions at boundaries. This condition is only applied at channel side walls to get a 1D model. It is not correct for simulating D hydrodynamic problems. Further information is in Zhou 004. At model boundaries (Figure A3), components, 3, 4 of local distribution function are unknown. The elastic collision boundary condition is applied as Eq. (A13). f f8 f3 f7 f4 f6 Eq. (A13) Hence the momentum of each direction is conserved, in other words, friction is ignored. 33

34 Solid Walls Figure A3: elastic collision boundary condition at a side wall At the upstream boundary, components 1,, 8 of local distribution function are unknown. If we use f ( i, j) to represent fictitious flow package in the th direction of a grid at grid ( i, j ), the influx boundary condition is set as Eq. (A14) (Refers to Figure A4 with left hand side as upstream end and right hand side as downstream) f (1, j ) f (, j) 1,,8 Eq. (A14) At the downstream boundary, components 4, 5, 6 of local distribution function are unknown, the outflux boundary condition is set as Eq. (A15) f ( end, j) f ( end 1, j ) 4,5,6 Eq. (A15) 34

35 Influx outflux j j 8 j-1 1 i end-1 end Figure A4: influx-outflux boundary conditions 35