Lattice Boltzmann Method

Lattice Boltzmann MethodComputational Fluid Dynamics permits the outline of the behaviour of fluid flows in virtual environments and comp ared to traditional experimental tests involves write down time and costs as well as a punter understanding of the studied phenomenon. Indeed, once the solution is obtai- ned all the magnitudes of the fluids lavatory be accurately computed and visualised. Moreover, with CFD there is no interference between the flow reach and the measuring equipment.The of import draw near put ond to study the behaviour of a fluid is the continuum one, which uses the Navier-Stokes equations and discerns the fluid through macroscopic properties as pressure temperature and density. Since it is severe and time-consuming to solve directly the non-li snug partial differential NS equations these are born-again into a system of algebraic equations th- rough finite difference, finite lot or finite element methods. The fluid domain is discretized and each thicke ning or volume contains a huge subprogram of particles, the average pass judgment of pressure, pep pill density etc is computed through an iterative process until convergence is reached.An secondary set about to study the fluid characteristics is the analysis of the microscale beha- viour of the fluid particles. The forces between particles (molecules) describe and determine the flow dynamic and at each time pure tone the position and velocity of each particle is computed using the Newtons second constabulary of momentum conservation. From the kinetic theory macro-scale pro- perties of the fluid can be obtained. This method is clearly impractical because of the extreme tour of particles that constitute even a miniscule volume of fluid.The LBM method is in the middle of these two methodologies and focuses the wariness on di- screte collections of particles whose properties are described through a probability dispersal exercise (PDF). The PDF describes the behaviour of a l arge number of particles using a sta- tistical scattering. This approach hence doesnt request the knowledge of the single particles positions and velocities. The Maxwell-Boltzmann dispersal function can be defined as the probability of finding particles within a certain govern of velocities at a certain range of locations at a certain time f(r,c,t), where r,c,tare the position, velocity and time respectively.The spacial discretization used is named lattice and it is based on a Cartesian distribution of discrete battery-acids with discrete sets of velocity directions. The lattice is determined by the number of dimensions n and discrete velocity directions m (DnQm), a large number of velocities leads to a to a greater extent precise description but also a spicyer computational cost. For each discrete velocity direction a PDF is defined.The Boltzmanns Transport Equation is used to describe the evolution of the PDFs and theirinteractionsft+ cf= (f)The equation states that the tot al derivative of the PDF equals to the collision floozy . This is a Lagrangian approach to the fluid dynamics while the traditional CFD methods use a Eulerian one. The collision operator depends on the distribution functions and it is very composite to compute, a solution was founded by Bhatnagar, Gross and Krook (BGK) who replaced it with a single serenity time (SRT) simplified model = 1 (feqf)where feqis the local anaesthetic symmetricalness distribution function (the distribution which represents the system equilibrium) and is the relaxation time (particle collision frequency). The preceding(prenominal) equation can be written along each velocity direction and can be discretized asteqfi(r+ cit,t+ t) = fi(r,t) +f (r,t)f(r,t) iThis equation can be used to describe many phenomena specifying the proper equilibrium di- stribution function. The two steps represented are the propagation step which models the travail of the distribution functions along discrete directions and the collision step which describes the physical phenomena.The main drawbacks using an SRT scheme are the low Mach number which can be used, the limitations imposed by the value of and the Prandtl number must be near one.For these reasons a multiple relaxation time (MRT) collision operator is used, this approach involves the deliberation of the collision step in the momentum space rather of the velocity space.The CFD software Xflow uses an MRT scheme which improves the stability and enhance the Mach number limitations up to Mach 0.6 for the overall fluid domain and up to Mach 1 in local regions, this means for example that a shock wave in the point of the minimum Cp value in a transonic airfoil can be captured. The lattice structure used in the software is a D3Q27 arran- ged in an octree diagram structure. This method divides the 3D space in a tree data structure where each portion is recursively subdivided in eight equals little parts. In this manner, different spatial scales with di fferent refinements can be obtained in the fluid domain. Each level has spatial and temporal scales twice as smaller than the previous one so the ratio dx/dt and the CFL condition hang in constant allowing a proper time step for each node. This is an emolument towards traditional CFD method where the time step is constant and hence the calculation is inefficient for the coarse part of the mesh. Moreover, adaptive refinement criteria based on the local vorticity level can be used to refine aftermath regions, free rise and interfaces.The model used to simulate subgrid turbulence is the Wall-Adapting local anesthetic Eddy (WALE) vi- scosity model. This approach is the same used for Large Eddy show and introduce an artificial eddy viscosity t. This model appears to be more efficiently applied in the LBM me- thod because the strain rate tensor is available in the local node while it needs information from the neighbours nodes to be evaluated in traditional CFD models. Besides, the proportional aspect ratio of the lattice required for LES turbulence model is another advantage in behalf of LBM technique.The impossibility to represent the leaping layer near walls because of the isotropy of the latti- ce structure and the consequent high number of nodes requested to capture the phenomenon is overcome through the use of the Wall-Modeled LES approach (WMLES). Furthermore, as the turbolence length scale is proportional to the distance from the free surface and the boundary layer thickness is proportional to the Reynolds number, the resolution scale becomes unaccep- tably small near walls. This is why the WMLES approach uses RANS in the proximity of the walls.The discrete velocities projections are also used to calculate the distance between the lattice and the geometry in order to obtain a detailed description of the body breaking ball which is used in WMLES to evaluate the boundary layer behaviour.