*6.2. Taylor-Green Vortex*

Similarly to the previous problem, we consider a square domain, and the initial velocity field is given by the formula

$$u\_x(\mathbf{x}, y, t = 0) = -lL\_0 \cos(k\mathbf{x})\sin(ky), \quad u\_y(\mathbf{x}, y, t = 0) = lL\_0 \sin(k\mathbf{x})\cos(ky), \quad 0 < y < \pi$$

where the size of the domain is *L* × *L* (or *N* × *N* in lattice units, where *N* is the number of the lattice nodes) and *k* = <sup>2</sup>*<sup>π</sup> <sup>L</sup>* . The periodic boundary conditions are applied. For the present problem we set *U*<sup>0</sup> = 0.01, *ν* = 0.001, *N* = 51, the time step *δt* = 1. The analytical solution to the problem is as follows

$$u\_x(\mathbf{x}, y, t) = -\mathcal{U}\_0 \cos(k\mathbf{x}) \sin(ky) e^{-2\nu k^2 t}, \quad u\_y(\mathbf{x}, y, t) = \mathcal{U}\_0 \sin(k\mathbf{x}) \cos(ky) e^{-2\nu k^2 t},$$

one can see that the initial structure of the velocity field persists in time, and only uniform decay of the velocity amplitudes is observed. The numerical simulations for the model (29)–(37) (implemented in the form (42)) show that the form of the velocity field does not change. We also present the behavior of the velocity *ux*(*x*, *y* = *L*/2, *t*) over time, obtained analytically and numerically for three different moments of time; obviously, both approaches give very similar profiles (Figure 3).

Finally, we consider the convergence rates of the numerical simulation results to the benchmark solutions. This can be performed by considering the logarithms of the simulation errors (44) for the different values of log(*h*) = log(1/*N*). In the present case, we take *N* = 25, 49, 73, 101. In Figure 4, the logarithms of the errors of the velocities are presented for DV and the conventional LB *D*2*Q*9 models; the results are very similar for

both models. One can see that the estimated slope values are close to 2; this indicates that the proposed scheme is accurate in the second-order.

**Figure 3.** Taylor–Green vortex. The velocity streamlines are presented in the (**first slide**). The velocity profiles *ux*(*x*, *y* = *<sup>L</sup>*/2, *<sup>t</sup>*) for three different moments of time *<sup>t</sup>* <sup>=</sup> <sup>2</sup> <sup>×</sup> 104, *<sup>t</sup>* <sup>=</sup> <sup>4</sup> <sup>×</sup> 104, *<sup>t</sup>* <sup>=</sup> <sup>6</sup> <sup>×</sup> 104 obtained analytically and numerically are presented (**second slide**), and the spatial variables *x*, *y* are normalized on the domain length *L*.

**Figure 4.** Convergence rates for the shear wave decay and Taylor–Green vortex problems are shown. The results are obtained by applying DV and the conventional LB *D*2*Q*9 models. In the (**first slide**) (shear wave decay), the logarithms of the errors (44) for the velocity *ux*(*y*, *t*) computed at the moment of time *t* = 1/(*νk*2) are presented; in the (**second slide**) (Taylor–Green vortex), the logarithms of the errors of the velocity *ux*(*x*, *y* = *L*/2, *t*) computed at the moment of time *t* = 1/(2*νk*2) are presented, where the variable *h* is proportional to the lattice spacing. The slope estimates are obtained by fitting the values of log(*error*) using the linear regression.

### **7. Results and Discussion**

In this paper, we have considered the DV Boltzmann model applicable to the modeling of viscous quasi-incompressible flows at a small Mach number limit. The presented model has the same discrete velocity structure and absolute equilibrium as LB *D*2*Q*9, but the collision rules for the particles are postulated exactly. There are four types of collision and ten possible different collisions; the unique transition probability corresponds to all possible reactions in the group. Moreover, these collisions conserve only mass, momentum and energy (spurious invariants do not exist). In terms of LB theory, this model can be considered as a scheme with multiple relaxation times. Note that the H-theorem is valid for the model by construction (at least for the continuous space–time variables).

We have demonstrated that DV Boltzmann equations can be a viable tool in modeling of hydrodynamic flows. The shear wave decay and Taylor–Green vortex have been considered as benchmark problems. The comparison of the simulation results with the analytical solutions has shown good accuracy.

One of the most intriguing problems is the evaluation of the stability properties of the presented DV Boltzmann system and the optimal choice of transition probabilities. One can expect that the DV Boltzmann model for *D*2*Q*9 lattice has a better stability than the conventional LB *D*2*Q*9 model, since the H-theorem is satisfied. In order to elucidate this issue, one can consider additional problems like Sod shock tube, double shear layer and lid-driven cavity. These problems are left for future study.

**Funding:** This research was supported by the Ministry of Science and Higher Education of the Russian Federation, project No 075-15-2020-799.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The simulation code that supports the findings of this study is available from the author upon reasonable request.

**Conflicts of Interest:** The author declares no conflict of interest.

#### **Abbreviations**

The following abbreviations are used in this manuscript:


## **References**

