The problem of the laminar fluid flow simulations with the lattice Boltzmann method at moderate and at large Reynolds numbers up to 20000 is considered. To avoid the fluctuations that appear in the numerical solutions with the LBM the regularization method that based on the median filtration is proposed. The regularization method was tested on the classical problem of the flow over the circular cylinder. The fluid flow modeling over the Nasa 0012 profile at different angles of attack is carried out at the Reynolds number 1000. The flow patterns, the drag and lift coefficients of the NASA profile studied in detail.
Methods of the computational fluid dynamics (CFD) are classified according to the
different approaches in continuous medium describing. These approaches are the
Euler and the Lagrange ones [
In 1965 as a development of the PiC method was proposed the method of large
particles, developed by Davidov U. M. and Bilatserkovskiy O. M [
In 1973 was appeared another approach in the fluid flow modeling based on the
cellular automata - Lattice Gas Cellular Automata (LGA) [
The LB method becomes widespread. A lot of scientists in Europe and America are
engaged in its development [
• Simplicity in description and programming, since all stages of the method are the
linear equations that solves by the explicit schemes [
• the simplicity in the boundary conditions settings, due to the fact that the method
is based on the kinetic approach in the continuous medium describing [
However, despite the growing popularity, there are still the following problems:
1.Significant calculation time, which increases with Reynolds number increasing
[
These problems make it difficult to obtain numerical solutions for the flows at
moderate Reynolds numbers Re~100 and prevent simulations at large numbers
Re>100. For a partial solution of these problems at small Reynolds numbers Re<100
schemes with several relaxation parameters [
According to the lattice Boltzmann method, the computational domain divides into a
grid which cells are treated as large particles [
The mathematical formulation of the problem consists of the Boltzmann equation
(1), in which the particles collision integral is replaced by the Bhatnagar-Gross-Kruck
approximation (2) [
1 2 Icoll 1 2 f1' f2' f1 f2 dd 2 f eq f
1 fk r,t fkeq r, t fk rVk ts,ttreamt fkr,t collision Numerical model is a system of the lattice equations (3), obtained by sampling the mathematical model (1) - (2) with the absence of the external forces F 0 . where fk r,t – discrete particle distribution function; V k – discrete series of particles possible velocities; t – time step; τ – dimensionless relaxation parameter; fkeq r, t – discrete approximation of the Maxwell-Boltzmann local equilibrium distribution function.
The justification of using the lattice Boltzmann method to model viscous flows is
given in [
u u u p vu ot ; t u 0 ot oM p .
v cs2t 0.5 cs c c 3 d t where d – cell`s size (1) (2) (3) (4) (5) (6) (7) Lattice Mach number
M p Umax cs (8) where Umax – maximum velocity value in the domain
The stability of the LBM numerical scheme was investigated analytically in [
lattice Mach number increasing. The Boltzmann kinetic equation approximates
the Navier-Stokes equation only at small Mach numbers M p 0.3 [
relaxation parameter decreasing. The "safe" value (according to [
speed increasing. The numerical model assumes modeling flows only at low speeds Umax cs .
Reynolds number increasing. As the Reynolds number increases, the flow
becomes turbulent (according to [
Macroscopic fluid parameters: density, velocity and pressure are defined as moments of the distribution function according to the formulas: r,t fk r, t ;ur, t 8
1 8Vk fk r,t ; pr, t cs2 r,t .
r, t k 0
(9)
ux i, j med ux i, j 2, ux i, j 1,ux i, j,ux i, j 1, ux i, j 2
ux i, j ux i, j 1 ux i, j ux i, j 1
(11)
The advantage of such filter is the nonlinearity with the following properties [
Smoothing using (10-11) equations schematically depicted in fig. 2. To verify the proposed method a series of numerical experiments of numerical solutions regularizing was conducted. Considered the flow pattern at Re=500. Fig. 3a illustrates velocity diagram got with the LBM algorithm at the moment t=2.5. The fig. 3a shows pulsations, which arise mainly on the boundary of the region. Further solution is unstable. Figure 3b illustrates velocity diagram of smoothed solution. a b
The regularization method has allowed to smooth the fluctuations and obtain a stable solution over large time intervals, such as the velocity diagram (fig. 4 a) and the streamlines (fig. 4 b).
Fig. 4. Simulation with the LBM with smoothing at Re = 500 at time t = 50 a) velocity diagram b) streamlines
The resulting solution was compared with the solution of a similar problem that was obtained in the Comsol Multiphysics package with the finite element method (fig. 5). Figures 4 and 5 shows the similarity of the obtained patterns and the streamlines.
Fig. 5. Simulation in the Comsol Multiphysics package at Re = 500 at time t = 50 a) velocity diagram b) streamlines
A more detailed comparison was made in vertical sections near the streamlined body x=0.8, 1.0 (fig. 6). The graphs of the velocity distribution, obtained by the LBM method with smoothing and the finite elements method (FEM), were constructed.
Consequently, the solutions of the test problem obtained by the lattice Boltzmann method with smoothing showed a good correspondence with the results of other numerical experiments at the Reynolds number Re = 500.
Increase the Reynolds number up to Re = 10,000. The structuring of such flow
disappears and the turbulent is appearing [
Fig. 8. Simulation with the LBM with smoothing at Re = 10000 at time t = 100 a) velocity diagram b) streamlines
As the Reynolds number increases, the periodicity of vortex formation increases: vortices descend from the edges of the cylinder at about t = 2. Such a rapid formation of vortices led to a divergence of numerical solutions by the FEM and LBM methods at about t = 2 (figures 8 and 9).
The results of the fluid flow modeling at Reynolds number Re = 20000, are depicted in fig. 10 with LBM method and fig.11 with the FEM method.
a a b b
The results of numerical experiments for the viscous fluid flow modeling with the lattice Boltzmann method and the finite element method in the Comsol package at Reynolds numbers 500<Re<20000 have been studied in detail. The comparison shows the good correspondence of the results, namely the process of the boundary layer separation and the formation of vortices over the cylinder. The structure of the streamlines also coincides. 4
The fluid flow modeling over the Nasa 0012 profile at different attack angles was carried out at the Reynolds number Re=1000. Figure 12 illustrates the flow scheme: the velocity vector Vin, angle of attack α, drag and lift coefficients F drag , F lift . The influence of the angle of attack on the flow pattern and on the drag and lift coefficients was studied. The flow pattern with the zero of the angle of attack is depicted in fig. 13. b Fig. 13. Flow over the Nasa 0012 profile with the absent angle a) velocity diagram b) streamlines As can be seen from the fig. 12, the flow around the profile is symmetric. Was calculated the average values of the drag Cd and lift Cl coefficients at different times up to t=100. The appropriate graphs are depicted in fig. 14. The change of the flow pattern by the angle of attack increasing α=10°, 20°, 30° was considered. The average values of the hydrodynamic coefficients shown in fig. 15. The angle of attack increasing cause the drag and lift coefficients increasing too as can be seen from the fig. 15. The numerical simulations with the angles of the Nasa 0012 profile more than 30° are unstable with the regularized lattice Boltzmann model. Fig. 16 illustrates the velocity diagrams and streamlines of the flow over the profile with the different angles of attack α=10°, 20°, 30°.
а b
The conducted numerical experiments indicate the growth of the hydrodynamic coefficients and the formation of vortices over the profile with the angle of attack increasing. 5
The problem of the laminar fluid flow simulations with the lattice Boltzmann method at moderate and at large Reynolds numbers up to 20000 was considered. Noticed, that LBM is a new approach in CFD that used the Boltzmann equation to simulate fluid behavior. Although this method becomes more popular there are some problems that make fluid flow modeling at moderate and large Reynolds numbers impossible. They are the significant calculation time, which increases with Reynolds number increasing and conditional stability. The last one may causes fluctuations which are the irregularity of the movement of particles. To avoid the fluctuations the regularization method that based on the median filtration was proposed.
The regularization method was tested on the classical problem of the flow over the circular cylinder. The comparison of the numerical experiments with the lattice Boltzmann method and the finite element method in Comsol Multiphysics package showed the good correspondence of the results, namely the process of the boundary layer separation and the formation of vortices over the cylinder. The structure of the streamlines also coincides. So, the regularization method has allowed to smooth the fluctuations and to obtain a stable solution. This allows to use the LBM for studying the laminar patterns at large Reynolds numbers.
The fluid flow modeling over the Nasa 0012 profile at different angles of attack was carried out at the Reynolds number Re=1000. Declared the vortex formation at the angles more than 7°. Size of the vortex is increasing with the angle increasing as well as the hydrodynamics coefficients. The numerical simulations with the angles of the Nasa 0012 profile more than 30° are unstable even with the regularized lattice Boltzmann model so demands more detail study.