Lattice boltzmann method ammar

APPLICATION OF LATTICE APPLICATION OF LATTICE BOLTZMANN METHODBOLTZMANN METHOD

Done by: Ammar Al-khalidiDone by: Ammar Al-khalidiM.Sc. Student M.Sc. Student

University of JordanUniversity of Jordan8/Feb/20068/Feb/2006

Jordanian-Germany winter academy 2006

04/15/2304/15/23 Jordanian-Germany winter academy Jordanian-Germany winter academy 20062006

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Table of contentTable of content• Introduction to Lattice Boltzmann methodIntroduction to Lattice Boltzmann method

– What is the Lattice Boltzmann Method?What is the Lattice Boltzmann Method?– What is the basic idea of Lattice Boltzmann method?What is the basic idea of Lattice Boltzmann method?– What is the basic idea of Lattice Boltzmann method?What is the basic idea of Lattice Boltzmann method?– Why is the Lattice Boltzmann Method Important?Why is the Lattice Boltzmann Method Important?– Comparison between Lattice Boltzmann and conventional numerical schemesComparison between Lattice Boltzmann and conventional numerical schemes– Lattice Boltzmann Lattice Boltzmann important featuresimportant features

• Lattice Boltzmann methodLattice Boltzmann method– Lattice Boltzmann equations.Lattice Boltzmann equations.– Boundary Conditions in the LBMBoundary Conditions in the LBM– Some boundary treatments to improve the numerical accuracy of the LBMSome boundary treatments to improve the numerical accuracy of the LBM

• Application of Lattice BoltzmannApplication of Lattice Boltzmann– Lattice Boltzmann simulation of fluid flowsLattice Boltzmann simulation of fluid flows– Driven cavity flows resultsDriven cavity flows results– Flow over a backward-facing stepFlow over a backward-facing step– Flow around a circular cylinderFlow around a circular cylinder– Flows in Complex GeometriesFlows in Complex Geometries– Simulation of Fluid TurbulenceSimulation of Fluid Turbulence– Direct numerical simulationDirect numerical simulation– LBM models for turbulent flowsLBM models for turbulent flows– LBM simulations of multiphase and multicomponent flowsLBM simulations of multiphase and multicomponent flows


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– What is the Lattice Boltzmann Method?What is the Lattice Boltzmann Method?– What is the basic idea of Lattice Boltzmann method?What is the basic idea of Lattice Boltzmann method?– What is the basic idea of Lattice Boltzmann method?What is the basic idea of Lattice Boltzmann method?– Why Lattice Boltzmann Method Important?Why Lattice Boltzmann Method Important?– Comparison between Lattice Boltzmann and conventional numerical schemesComparison between Lattice Boltzmann and conventional numerical schemes– Lattice Boltzmann Lattice Boltzmann important featuresimportant features

• Lattice Boltzmann methodLattice Boltzmann method– Lattice Boltzmann equations.Lattice Boltzmann equations.– Boundary Conditions in the LBMBoundary Conditions in the LBM– some boundary treatments to improve the numerical accuracy of the LBMsome boundary treatments to improve the numerical accuracy of the LBM

• Application of Lattice BoltzmannApplication of Lattice Boltzmann– Lattice Boltzmann simulation of fluid flowsLattice Boltzmann simulation of fluid flows– driven cavity flows resultsdriven cavity flows results– flow over a backward-facing stepflow over a backward-facing step– flow around a circular cylinderflow around a circular cylinder– Flows in Complex GeometriesFlows in Complex Geometries– Simulation of Fluid TurbulenceSimulation of Fluid Turbulence– Direct numerical simulationDirect numerical simulation– Lbm models for turbulent flowsLbm models for turbulent flows– lbm simulations of multiphase and multicomponent flowslbm simulations of multiphase and multicomponent flows


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What is the Lattice What is the Lattice Boltzmann Method?Boltzmann Method?• The lattice Boltzmann method is a The lattice Boltzmann method is a

powerful technique for the powerful technique for the computational modeling of a wide computational modeling of a wide variety of complex fluid flow problems variety of complex fluid flow problems including single and multiphase flow in including single and multiphase flow in complex geometries. It is a discrete complex geometries. It is a discrete computational method based upon the computational method based upon the Boltzmann equationBoltzmann equation


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What is the basic idea of What is the basic idea of Lattice Boltzmann method?Lattice Boltzmann method?• Lattice Boltzmann method considers a typical Lattice Boltzmann method considers a typical

volume element of fluid to be composed of a volume element of fluid to be composed of a collection of particles that are represented by collection of particles that are represented by a particle velocity distribution function for a particle velocity distribution function for each fluid component at each grid point. The each fluid component at each grid point. The time is counted in discrete time steps and the time is counted in discrete time steps and the fluid particles can collide with each other as fluid particles can collide with each other as they move, possibly under applied forces. The they move, possibly under applied forces. The rules governing the collisions are designed rules governing the collisions are designed such that the time-average motion of the such that the time-average motion of the particles is consistent with the Navier-Stokes particles is consistent with the Navier-Stokes equation. equation.


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Why Lattice Boltzmann Why Lattice Boltzmann Method Important?Method Important?

• This method naturally accommodates a variety This method naturally accommodates a variety of boundary conditions such as the pressure of boundary conditions such as the pressure drop across the interface between two fluids drop across the interface between two fluids and wetting effects at a fluid-solid interface. It and wetting effects at a fluid-solid interface. It is an approach that bridges microscopic is an approach that bridges microscopic phenomena with the continuum macroscopic phenomena with the continuum macroscopic equations. equations.

• Further, it can model the time evolution of Further, it can model the time evolution of systems.systems.


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Comparison between lattice Comparison between lattice Boltzmann and conventional Boltzmann and conventional numerical schemesnumerical schemes• The lattice Boltzmann method is based on The lattice Boltzmann method is based on

microscopic models and macroscopic kinetic microscopic models and macroscopic kinetic equations. The fundamental idea of the LBM equations. The fundamental idea of the LBM is to construct simplified kinetic models that is to construct simplified kinetic models that incorporate the essential physics of incorporate the essential physics of processes so that the macroscopic averaged processes so that the macroscopic averaged properties obey the desired macroscopic properties obey the desired macroscopic equations.equations.

• Unlike conventional numerical schemes Unlike conventional numerical schemes based on discriminations of macroscopic based on discriminations of macroscopic continuum equations. continuum equations.


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lattice Boltzmann lattice Boltzmann important featuresimportant features• The kinetic nature of the LBM introduces The kinetic nature of the LBM introduces

three important features that distinguish three important features that distinguish it from other numerical methods.it from other numerical methods.

1.1.First, the convection operator (or streaming First, the convection operator (or streaming

process) of the LBM in phase space (or process) of the LBM in phase space (or velocity space) is linear.velocity space) is linear.

2.2.Second, the incompressible Navier-Stokes Second, the incompressible Navier-Stokes (NS) equations can be obtained in the (NS) equations can be obtained in the nearly incompressible limit of the LBM.nearly incompressible limit of the LBM.

3.3.Third, the LBM utilizes a minimal set of Third, the LBM utilizes a minimal set of velocities in phase space.velocities in phase space.


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The LBM originated from lattice gas (LG) The LBM originated from lattice gas (LG) automata, a discrete particle kinetics automata, a discrete particle kinetics utilizing a discrete lattice and discrete utilizing a discrete lattice and discrete time. The LBM can also be viewed as a time. The LBM can also be viewed as a special finite difference scheme for the special finite difference scheme for the kinetic equation of the discrete-velocity kinetic equation of the discrete-velocity distribution function. distribution function.

The idea of using the simplified kinetic The idea of using the simplified kinetic equation with a single-particle speed to equation with a single-particle speed to simulate fluid flows was employed by simulate fluid flows was employed by Broadwell (Broadwell 1964) for studying Broadwell (Broadwell 1964) for studying shock structures.shock structures.


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• Application of Lattice BoltzmannApplication of Lattice Boltzmann– Lattice Boltzmann simulation of fluid flowsLattice Boltzmann simulation of fluid flows– driven cavity flows resultsdriven cavity flows results– flow over a backward-facing stepflow over a backward-facing step– flow around a circular cylinderflow around a circular cylinder– Flows in Complex GeometriesFlows in Complex Geometries– Simulation of Fluid TurbulenceSimulation of Fluid Turbulence– Direct numerical simulationDirect numerical simulation– Lbm models for turbulent flowsLbm models for turbulent flows– lbm simulations of multiphase and multicomponent flowslbm simulations of multiphase and multicomponent flows


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LATTICE BOLTZMANN LATTICE BOLTZMANN EQUATIONSEQUATIONS• There are several ways to obtain the lattice There are several ways to obtain the lattice

Boltzmann equation (LBE) from either Boltzmann equation (LBE) from either discrete velocity models or the Boltzmann discrete velocity models or the Boltzmann kinetic equation. kinetic equation.

• There are also several ways to derive the There are also several ways to derive the macroscopic Navier-Stokes equations from macroscopic Navier-Stokes equations from the LBE. Because the LBM is a derivative of the LBE. Because the LBM is a derivative of the LG method.the LG method.– LBE: An Extension of LG AutomataLBE: An Extension of LG Automata


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LATTICE BOLTZMANN LATTICE BOLTZMANN EQUATIONSEQUATIONS

),...,1,0( )),,((),(),( Mitxftxfttxexf iiii

)),(( txfii

where fi is the particle velocity distribution function along the ith direction;

Where Ω is the collision operator which represents the rate of change of fi resulting from collision. Δt and Δx are time and space increments, respectively.


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Boundary Conditions in the LBM• Wall boundary conditions in the LBM were originally Wall boundary conditions in the LBM were originally

taken from the LG method. For example, a particle taken from the LG method. For example, a particle distribution function bounce-back scheme (Wolfram distribution function bounce-back scheme (Wolfram 1986, Lavallée et al 1991) was used at walls to 1986, Lavallée et al 1991) was used at walls to obtain no-slip velocity conditions.obtain no-slip velocity conditions.

• By the so-called bounce-back scheme, we mean By the so-called bounce-back scheme, we mean that when a particle distribution streams to a wall that when a particle distribution streams to a wall node, the particle distribution scatters back to the node, the particle distribution scatters back to the node it came from. The easy implementation of this node it came from. The easy implementation of this no-slip velocity condition by the bounce-back no-slip velocity condition by the bounce-back boundary scheme supports the idea that the LBM is boundary scheme supports the idea that the LBM is ideal for simulating fluid flows in complicated ideal for simulating fluid flows in complicated geometries, such as flow through porous media. geometries, such as flow through porous media.


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Boundary Conditions in the LBM• For a node near a boundary, some of its For a node near a boundary, some of its

neighboring nodes lie outside the flow neighboring nodes lie outside the flow domain. Therefore the distribution functions domain. Therefore the distribution functions at these no-slip nodes are not uniquely at these no-slip nodes are not uniquely defined. The bounce-back scheme is a simple defined. The bounce-back scheme is a simple way to fix these unknown distributions on the way to fix these unknown distributions on the wall node. On the other hand, it was found wall node. On the other hand, it was found that the bounce-back condition is only first-that the bounce-back condition is only first-order in numerical accuracy at the order in numerical accuracy at the boundaries (Cornubert et al 1991, Ziegler boundaries (Cornubert et al 1991, Ziegler 1993, Ginzbourg & Adler 1994). 1993, Ginzbourg & Adler 1994).


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some boundary treatments to some boundary treatments to improve the numerical improve the numerical accuracy of the LBMaccuracy of the LBM• To improve the numerical accuracy of the LBM, To improve the numerical accuracy of the LBM,

other boundary treatments have been proposed. other boundary treatments have been proposed. Skordos (1993) suggested including velocity Skordos (1993) suggested including velocity gradients in the equilibrium distribution function gradients in the equilibrium distribution function at the wall nodes.at the wall nodes.

• Noble et al (1995) proposed using hydrodynamic Noble et al (1995) proposed using hydrodynamic boundary conditions on no-slip walls by enforcing boundary conditions on no-slip walls by enforcing a pressure constraint. a pressure constraint.

• Inamuro et al (1995) recognized that a slip Inamuro et al (1995) recognized that a slip velocity near wall nodes could be induced by the velocity near wall nodes could be induced by the bounce-back scheme and proposed to use a bounce-back scheme and proposed to use a counter slip velocity to cancel that effect.counter slip velocity to cancel that effect.

• Other boundary treatments Other boundary treatments


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• Application of Lattice BoltzmannApplication of Lattice Boltzmann– Lattice Boltzmann simulation of fluid flowsLattice Boltzmann simulation of fluid flows– driven cavity flows resultsdriven cavity flows results– flow over a backward-facing stepflow over a backward-facing step– flow around a circular cylinderflow around a circular cylinder– Flows in Complex GeometriesFlows in Complex Geometries– Simulation of Fluid TurbulenceSimulation of Fluid Turbulence– Direct numerical simulationDirect numerical simulation– Lbm models for turbulent flowsLbm models for turbulent flows– LBM simulations of multiphase and multicomponent flowsLBM simulations of multiphase and multicomponent flows


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LATTICE BOLTZMANN LATTICE BOLTZMANN SIMULATION OF FLUID FLOWSSIMULATION OF FLUID FLOWS• DRIVEN CAVITY FLOWS:DRIVEN CAVITY FLOWS:

The fundamental characteristics of the 2-D cavity flow are The fundamental characteristics of the 2-D cavity flow are the emergence of a large primary vortex in the center the emergence of a large primary vortex in the center and two secondary vortices in the lower corners. and two secondary vortices in the lower corners.

The lattice Boltzmann simulation of the 2-D driven cavity The lattice Boltzmann simulation of the 2-D driven cavity by Hou et al (1995) covered a wide range of Reynolds by Hou et al (1995) covered a wide range of Reynolds numbers from 10 to 10,000. They carefully compared numbers from 10 to 10,000. They carefully compared simulation results of the stream function and the simulation results of the stream function and the locations of the vortex centers with previous numerical locations of the vortex centers with previous numerical simulations and demonstrated that the differences of simulations and demonstrated that the differences of the values of the stream function and the locations of the values of the stream function and the locations of the vortices between the LBM and other methods were the vortices between the LBM and other methods were less than 1%. This difference is within the numerical less than 1%. This difference is within the numerical uncertainty of the solutions using other numerical uncertainty of the solutions using other numerical methods. methods.


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DRIVEN CAVITY FLOWS RESULTS


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FLOWOVER A BACKWARD-FACING STEP• The two-dimensional symmetric sudden expansion

channel flow was studied by Luo (1997) using the LBM. The main interest in Luo’s research was to study the symmetry-breaking bifurcation of the

flow when Reynolds number increases.• In this simulation, an asymmetric initial

perturbation was introduced and two different expansion boundaries, square and sinusoidal,

were used. This simulation reproduced the symmetric-breaking bifurcation for the flow

observed previously, and obtained the critical Reynolds number of 46.19. This critical Reynolds

number was compared with earlier simulation and experimental results of 47.3.


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FLOW AROUND A CIRCULAR FLOW AROUND A CIRCULAR CYLINDER CYLINDER • The flow around a two-dimensional circular cylinder The flow around a two-dimensional circular cylinder

was simulated using the LBM by several groups of was simulated using the LBM by several groups of people.people.

• The flow around an octagonal cylinder was also The flow around an octagonal cylinder was also studied (Noble et al 1996). Higuera&Succi (1989) studied (Noble et al 1996). Higuera&Succi (1989) studied flow patterns for Reynolds number up to 80. studied flow patterns for Reynolds number up to 80. – At Re =52.8, they found that the flow became At Re =52.8, they found that the flow became

periodic after a long initial transient. periodic after a long initial transient. – For Re = 77.8, a periodic shedding flow emerged.For Re = 77.8, a periodic shedding flow emerged.– They compared Strouhal number, flow-They compared Strouhal number, flow-

separation angle, and lift and drag coefficients separation angle, and lift and drag coefficients with previous experimental and simulation with previous experimental and simulation results, showing reasonable agreement.results, showing reasonable agreement.


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Flows in Complex Geometries• An attractive feature of the LBM is that the An attractive feature of the LBM is that the

no-slip bounce-back LBM boundary no-slip bounce-back LBM boundary condition costs little in computational time. condition costs little in computational time. This makes the LBM very useful for This makes the LBM very useful for simulating flows in complicated geometries, simulating flows in complicated geometries, such as flow through porous media, where such as flow through porous media, where wall boundaries are extremely complicated wall boundaries are extremely complicated and an efficient scheme for handing wall-and an efficient scheme for handing wall-fluid interaction is essential.fluid interaction is essential.


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Simulation of Fluid Turbulence

• A major difference between the LBM A major difference between the LBM and the LG method is that the LBM and the LG method is that the LBM can be used for smaller viscosities. can be used for smaller viscosities.

• Consequently the LBM can be used Consequently the LBM can be used for direct numerical simulation (DNS) for direct numerical simulation (DNS) of high Reynolds number fluid flows.of high Reynolds number fluid flows.


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DIRECT NUMERICAL DIRECT NUMERICAL SIMULATION SIMULATION • To validate the LBM for simulating turbulent flows, To validate the LBM for simulating turbulent flows,

Martinez et al (1994) studied decaying turbulence of Martinez et al (1994) studied decaying turbulence of a shear layer using both the pseudospectral method a shear layer using both the pseudospectral method and the LBM. and the LBM.

• The initial shear layer consisted of uniform velocity The initial shear layer consisted of uniform velocity reversing sign in a very narrow region. The initial reversing sign in a very narrow region. The initial Reynolds umber was 10,000. they carefully Reynolds umber was 10,000. they carefully compared the spatial distribution, time evolution of compared the spatial distribution, time evolution of the stream functions, and the vorticity fields. Energy the stream functions, and the vorticity fields. Energy spectra as a function of time, small scale quantities, spectra as a function of time, small scale quantities, were also studied. The correlation between vorticity were also studied. The correlation between vorticity and stream function was calculated and compared and stream function was calculated and compared with theoretical predictions. They concluded that the with theoretical predictions. They concluded that the LBE method provided a solution that was accurate .LBE method provided a solution that was accurate .


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DIRECT NUMERICAL DIRECT NUMERICAL SIMULATION ResultsSIMULATION Results


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LBM MODELS FOR TURBULENT FLOWS• As in other numerical methods for solving the Navier-Stokes As in other numerical methods for solving the Navier-Stokes

equations, a subgrid-scale (SGS) model is required in the equations, a subgrid-scale (SGS) model is required in the LBM to simulate flows at very high Reynolds numbers. LBM to simulate flows at very high Reynolds numbers. Direct numerical simulation is impractical due to the time Direct numerical simulation is impractical due to the time and memory constraints required to resolve the smallest and memory constraints required to resolve the smallest scales (Orszag & Yakhot 1986). scales (Orszag & Yakhot 1986).

• Hou et al (1996) directly applied the subgrid idea in the Hou et al (1996) directly applied the subgrid idea in the Smagorinsky model to the LBM by filtering the particle Smagorinsky model to the LBM by filtering the particle distribution function and its equation in distribution function and its equation in particle velocity distribution Equation using a standard box filter. Equation using a standard box filter.

• This simulation demonstrated the potential of the LBM SGS This simulation demonstrated the potential of the LBM SGS model as a useful tool for investigating turbulent flows in model as a useful tool for investigating turbulent flows in industrial applications of practical importance.industrial applications of practical importance.


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LBM SIMULATIONS OF MULTIPHASE AND MULTICOMPONENT FLOWS• The numerical simulation of multiphase and

multicomponent fluid flows is an interesting and challenging problem because of difficulties in modeling interface dynamics and the importance of related engineering applications, including flow through porous media, boiling dynamics, and dendrite formation. Traditional numerical schemes have been successfully used for simple interfacial boundaries .

• The LBM provides an alternative for simulating complicated multiphase and multicomponent fluid flows, in particular for three-dimensional flows.


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• Method of Gunstensen et al:Method of Gunstensen et al:Gunstensen et al (1991) were the first to develop Gunstensen et al (1991) were the first to develop

the multicomponent LBM method.the multicomponent LBM method.• It was based on the two-component LG model It was based on the two-component LG model

proposed by Rothman & Keller (1988). Method of proposed by Rothman & Keller (1988). Method of Shan & ChenShan & Chen

Later, Grunau et al (1993) extended this model Later, Grunau et al (1993) extended this model to allow variations of density and viscosity.to allow variations of density and viscosity.

Shan & Chen (1993) and Shan & Doolen (1995) Shan & Chen (1993) and Shan & Doolen (1995) used microscopic interactions to modify the used microscopic interactions to modify the surface-tension–related collision operator for surface-tension–related collision operator for which the surface interface can be maintained which the surface interface can be maintained automatically.automatically.


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• Free Energy Approach– The above multiphase and multicomponent

lattice Boltzmann models are based on phenomenological models of interface dynamics and are probably most suitable for isothermal multicomponent flows.

– One important improvement in models using the free-energy approach (Swift et al 1995, 1996) is that the equilibrium distribution can be defined consistently based on thermodynamics.

– Consequently, the conservation of the total energy, including the surface energy, kinetic energy, and internal energy can be properly satisfied (Nadiga & Zaleski 1996).


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Numerical Verification and Numerical Verification and ApplicationsApplications• Two fundamental numerical tests associated with interfacial

phenomena have been carried out using the multiphase and multicomponent lattice Boltzmann models.

• The first test, the lattice Boltzmann models were used to verify Laplace’s formula by measuring the pressure difference and surface tention between the inside and the outside of a droplet . The simulated value of surface tension has been compared with theoretical predictions, and good agreement was reported (Gunstensen et al 1991, Shan and Chen 1993, Swift et al 1995).

• In the second test of LBM interfacial models, the oscillation of a capillary wave was simulated (Gunstensen et al 1991, Shan & Chen 1994, Swift et al 1995). A sine wave displacement of a given wave vector was imposed on an interface that had reached equilibrium. The resulting dispersion relation was measured and compared with the theoretical prediction (Laudau & Lifshitz 1959) Good agreement was observed, validating the LBM surface tension models.


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SIMULATION OF HEAT SIMULATION OF HEAT TRANSFER AND REACTION-TRANSFER AND REACTION-DIFFUSIONDIFFUSION• The lattice Bhatnagar-Gross-Krook (LBGK) The lattice Bhatnagar-Gross-Krook (LBGK)

models for thermal fluids have been models for thermal fluids have been developed by several groups. To include a developed by several groups. To include a thermal variable, such as temperature, thermal variable, such as temperature, Alexander et al (1993) used a two-Alexander et al (1993) used a two-dimensional 13-velocity model on the dimensional 13-velocity model on the hexagonal lattice. In this work, the internal hexagonal lattice. In this work, the internal energy per unit mass was defined through energy per unit mass was defined through the second-order moment of the distribution the second-order moment of the distribution function.function.


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• Two limitations in multispeed LBM thermal models severely Two limitations in multispeed LBM thermal models severely restrict their application. First, because only a small set of restrict their application. First, because only a small set of velocities is used, the variation of temperature is small. Second, velocities is used, the variation of temperature is small. Second, all existing LBM models suffer from numerical instability all existing LBM models suffer from numerical instability (McNamara et al 1995), (McNamara et al 1995),

• On the other hand, it is difficult for the active scalar approach to On the other hand, it is difficult for the active scalar approach to incorporate the correct and full dissipation function. Two incorporate the correct and full dissipation function. Two dimensional Rayleigh-Bénard (RB) convection was simulated dimensional Rayleigh-Bénard (RB) convection was simulated using this active scalar scheme for studying scaling laws using this active scalar scheme for studying scaling laws (Bartoloni et al 1993) and probability density functions (Massaioli (Bartoloni et al 1993) and probability density functions (Massaioli et al 1993) at high Prandtl numbers.et al 1993) at high Prandtl numbers.

• Two dimensional free-convective cavity flow was also simulated Two dimensional free-convective cavity flow was also simulated (Eggels & Somers 1995), and the results compared well with (Eggels & Somers 1995), and the results compared well with benchmark data. Two-D and 3-D Rayleigh-Bénard convections benchmark data. Two-D and 3-D Rayleigh-Bénard convections were carefully studied by Shan (1997) using a passive scalar were carefully studied by Shan (1997) using a passive scalar temperature equation and a Boussinesq approximation. This temperature equation and a Boussinesq approximation. This scalar equation was derived based on the two-component model scalar equation was derived based on the two-component model of Shan & Chen (1993). The calculated critical Rayleigh number of Shan & Chen (1993). The calculated critical Rayleigh number for the RB convection agreed well with theoretical predictions. for the RB convection agreed well with theoretical predictions. The Nusselt number as a function of Rayleigh number for the 2-D The Nusselt number as a function of Rayleigh number for the 2-D simulation was in good agreement with previous numerical simulation was in good agreement with previous numerical simulation using other methodssimulation using other methods


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Applications in commercial programs

•ex direct building energy simulation based on large eddytechniques and lattice boltzmann methods


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Two orthogonal slice planes of the averaged velocity field


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Turbulent flow, orthogonal slice planes of the averaged velocity field


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Thank You

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Lattice boltzmann method ammar