A Discrete Flux Scheme for Aerodynamic and Hydrodynamic Flows

2011 ◽  
Vol 9 (5) ◽  
pp. 1257-1283 ◽  
Author(s):  
S. C. Fu ◽  
R. M. C. So ◽  
W. W. F. Leung
Keyword(s):  
Thermal Energy ◽  
Lattice Boltzmann ◽  
Stokes Equations ◽  
Moment Equation ◽  
Zero Order ◽  
Distribution Functions ◽  
Navier Stokes ◽  
Order Moment ◽  
Navier Stokes Equations ◽  
First Order

AbstractThe objective of this paper is to seek an alternative to the numerical simulation of the Navier-Stokes equations by a method similar to solving the BGK-type modeled lattice Boltzmann equation. The proposed method is valid for both gas and liquid flows. A discrete flux scheme (DFS) is used to derive the governing equations for two distribution functions; one for mass and another for thermal energy. These equations are derived by considering an infinitesimally small control volume with a velocity lattice representation for the distribution functions. The zero-order moment equation of the mass distribution function is used to recover the continuity equation, while the first-order moment equation recovers the linear momentum equation. The recovered equations are correct to the first order of the Knudsen number(Kn);thus, satisfying the continuum assumption. Similarly, the zero-order moment equation of the thermal energy distribution function is used to recover the thermal energy equation. For aerodynamic flows, it is shown that the finite difference solution of the DFS is equivalent to solving the lattice Boltzmann equation (LBE) with a BGK-type model and a specified equation of state. Thus formulated, the DFS can be used to simulate a variety of aerodynamic and hydrodynamic flows. Examples of classical aeroacoustics, compressible flow with shocks, incompressible isothermal and non-isothermal Couette flows, stratified flow in a cavity, and double diffusive flow inside a rectangle are used to demonstrate the validity and extent of the DFS. Very good to excellent agreement with known analytical and/or numerical solutions is obtained; thus lending evidence to the DFS approach as an alternative to solving the Navier-Stokes equations for fluid flow simulations.

scholarly journals Involving the Navier–Stokes equations in the derivation of boundary conditions for the lattice Boltzmann method

2011 ◽  
Vol 369 (1944) ◽  
pp. 2184-2192 ◽  
Author(s):  
Joris C. G. Verschaeve
Keyword(s):  
Boundary Condition ◽  
Lattice Boltzmann Method ◽  
Lattice Boltzmann ◽  
Stokes Equations ◽  
Slip Boundary Condition ◽  
Navier Stokes ◽  
Grid Spacing ◽  
Navier Stokes Equations ◽  
Slip Boundary ◽  
Boltzmann Method

By means of the continuity equation of the incompressible Navier–Stokes equations, additional physical arguments for the derivation of a formulation of the no-slip boundary condition for the lattice Boltzmann method for straight walls at rest are obtained. This leads to a boundary condition that is second-order accurate with respect to the grid spacing and conserves mass. In addition, the boundary condition is stable for relaxation frequencies close to two.

scholarly journals A multilayer shallow model for dry granular flows with the -rheology: application to granular collapse on erodible beds

2016 ◽  
Vol 798 ◽  
pp. 643-681 ◽  
Author(s):  
E. D. Fernández-Nieto ◽  
J. Garres-Díaz ◽  
A. Mangeney ◽  
G. Narbona-Reina
Keyword(s):  
Transient Flow ◽  
Stokes Equations ◽  
Low Cost ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Runout Distance ◽  
Multilayer Model ◽  
First Order ◽  
Constant Friction ◽  
Erodible Beds

In this work we present a multilayer shallow model to approximate the Navier–Stokes equations with the ${\it\mu}(I)$-rheology through an asymptotic analysis. The main advantages of this approximation are (i) the low cost associated with the numerical treatment of the free surface of the modelled flows, (ii) the exact conservation of mass and (iii) the ability to compute two-dimensional profiles of the velocities in the directions along and normal to the slope. The derivation of the model follows Fernández-Nieto et al. (J. Comput. Phys., vol. 60, 2014, pp. 408–437) and introduces a dimensional analysis based on the shallow flow hypothesis. The proposed first-order multilayer model fully satisfies a dissipative energy equation. A comparison with steady uniform Bagnold flow – with and without the sidewall friction effect – and laboratory experiments with a non-constant normal profile of the downslope velocity demonstrates the accuracy of the numerical model. Finally, by comparing the numerical results with experimental data on granular collapses, we show that the proposed multilayer model with the ${\it\mu}(I)$-rheology qualitatively reproduces the effect of the erodible bed on granular flow dynamics and deposits, such as the increase of runout distance with increasing thickness of the erodible bed. We show that the use of a constant friction coefficient in the multilayer model leads to the opposite behaviour. This multilayer model captures the strong change in shape of the velocity profile (from S-shaped to Bagnold-like) observed during the different phases of the highly transient flow, including the presence of static and flowing zones within the granular column.

A PLATFORM FOR DEVELOPING NEW LATTICE BOLTZMANN MODELS

2005 ◽  
Vol 16 (01) ◽  
pp. 61-84 ◽  
Author(s):  
H. W. ZHENG ◽  
C. SHU ◽  
Y. T. CHEW ◽  
J. QIU
Keyword(s):  
Lattice Boltzmann ◽  
Stokes Equations ◽  
Cavity Flow ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Driven Cavity ◽  
Velocity Models ◽  
Equilibrium Function ◽  
New Models ◽  
Lattice Boltzmann Models

This paper presents a platform to develop new lattice Boltzmann models. It gives a general framework for different applications. It also presents basic velocity models and a set of basic conditions to construct new models which can recover Navier–Stokes equations. Besides, the equilibrium function can be easily obtained through a set of equations. By using the platform, we can easily recover the existing models. Some new models are derived from the platform and validated by their application to simulate the two-dimensional driven cavity flow. The obtained numerical results agree very well with available data in the literature.

scholarly journals Dynamic Domain Decomposition Strategy Coupling Lattice Boltzmann Methods With Finite Differences Approximations of the Navier-Stokes Equations to Study Bodies in Current

2015 ◽  
Author(s):  
Giuseppina Colicchio ◽  
Claudio Lugni ◽  
Marilena Greco ◽  
Odd M. Faltinsen
Keyword(s):  
Domain Decomposition ◽  
Lattice Boltzmann ◽  
Stokes Equations ◽  
Dynamic Change ◽  
Accurate Solution ◽  
The Body ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Coupling Strategy ◽  
Hybrid Solver

A Domain-Decomposition (DD) strategy is proposed for problems involving regions with slow variations of the flow (A) and others where the fluid features undergo rapid changes (B), like in the case of steady current past bodies with pronounced local unsteadiness connected with the vortex shedding from the structures. For an efficient and accurate solution of such problems, the DD couples a Finite Difference solver of the Navier-Stokes equations (FD-NS) with a Multiple Relaxation Time Lattice Boltzmann method (MRT-LBM). Regions A are handled by FD-NS, while zones B are solved by MRT-LBM and the two solvers exchange information within a strong coupling strategy. Present DD strategy is able to deal with a dynamic change of the sub-domains topology. This feature is needed when regions with vorticity shed from the body vary in time for a more flexible and reliable solution strategy. Its performances in terms of accuracy and efficiency have been successfully assessed by comparing the hybrid solver against a full FD-NS solution and experimental data for a 2D circular cylinder in an impulsively started flow.

A New Partial Slip Boundary Condition for the Lattice-Boltzmann Method

2013 ◽  
Author(s):  
Marc-Florian Uth ◽  
Alf Crüger ◽  
Heinz Herwig
Keyword(s):  
Boundary Condition ◽  
Lattice Boltzmann ◽  
Stokes Equations ◽  
Slip Boundary Condition ◽  
Navier Stokes ◽  
Slip Model ◽  
Navier Stokes Equations ◽  
Navier Slip ◽  
Slip Boundary ◽  
Boltzmann Method

In micro or nano flows a slip boundary condition is often needed to account for the special flow situation that occurs at this level of refinement. A common model used in the Finite Volume Method (FVM) is the Navier-Slip model which is based on the velocity gradient at the wall. It can be implemented very easily for a Navier-Stokes (NS) Solver. Instead of directly solving the Navier-Stokes equations, the Lattice-Boltzmann method (LBM) models the fluid on a particle basis. It models the streaming and interaction of particles statistically. The pressure and the velocity can be calculated at every time step from the current particle distribution functions. The resulting fields are solutions of the Navier-Stokes equations. Boundary conditions in LBM always not only have to define values for the macroscopic variables but also for the particle distribution function. Therefore a slip model cannot be implemented in the same way as in a FVM-NS solver. An additional problem is the structure of the grid. Curved boundaries or boundaries that are non-parallel to the grid have to be approximated by a stair-like step profile. While this is no problem for no-slip boundaries, any other velocity boundary condition such as a slip condition is difficult to implement. In this paper we will present two different implementations of slip boundary conditions for the Lattice-Boltzmann approach. One will be an implementation that takes advantage of the microscopic nature of the method as it works on a particle basis. The other one is based on the Navier-Slip model. We will compare their applicability for different amounts of slip and different shapes of walls relative to the numerical grid. We will also show what limits the slip rate and give an outlook of how this can be avoided.

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