Numerical Solutions of Navier-Stokes Equations: Visualization of Turbulence
A coherent structure corresponds to a solitary soliton solution of flow-dynamics equations (Navier-Stokes equations):
where u is velocity vector, t is time, p is pressure, v is kinematic viscosity, p is density, and f are accelerations due to external forces.
The solution of these equations allows us to model coherent structures. This makes it possible further investigation of the properties of turbulence, which is formed by different coherent structure (including processes of turbulence formation and evolution). These processes can be visualized by flow lines of medium motion in solution of Navier-Stokes equations.
The analytical solution of Navier-Stokes equations is a complicated problem because of their nonlinearity. Therefore, numerical methods are usually used for solution of such problems. In connection with the recent advent of high- performance computers by democratic prices, the numerical solution of flow- dynamic equations with a good accuracy within reasonable time became possible.
Now it is possible to solve the problems of flow dynamics numerically with the aid of free specialized software called “the Gerris Flow Solver” [185,186] as a free software for solution of partial differential equations describing problems of flow dynamics. This open-source software was developed by Stephane Popinet [185,186,189,190]. The efficiency and needed accuracy of the software  were tested and confirmed at a rather wide class of 100 typical test problems [186-190], the solution of which gives good results.
Atmospheric Turbulence in the Anisotropic Boundary Layer ? 161
To describe the convection in fluids and gases, Navier—Stokes equations are usually written in Boussinesq approximation. The software  solves system (3.33) in the following form:
where D is deformation tensor D^ = (dujdxt + du Jdxj), a is the surface tension coefficient, к is curvature of the surface, 6S is delta function of the surface, n is normal to the surface, g is acceleration due to gravity (g = —ge3, e3 is the unit vector of the vertical axis ox3), T is deviation of absolute temperature from equilibrium temperature T0, p0 is density at equilibrium temperature T0, x is thermal diffusivity, and в is the thermal expansion coefficient (usually в = 1/T0).
Our contribution to numerical solutions of system of Equation 3.34 consists in the imaginative complementation of the software  with tools for visualization and spectral analysis. As a result, several computer animations have been created from the data of numerical solution of the Navier—Stokes equations.
As an example, we present the experimental data and the corresponding numerical solution of the boundary-value problem from Navier—Stokes equations (3.34).
In 2012 , experimental studies of the daytime astroclimate were carried out in the specialized (dome) room of the Big Alt-azimuthal Telescope (BAT, Special Astrophysical Observatory of the Russian Academy of Sciences, Figure 3.36a). The diameter of the BAT receiving mirror is 6.05 m. The results of measurements have shown that two large contra-rotating vortices with vertical axes and maximal diameters of about 16 m are observed in the BAT dome space (Figure 3.36b). At every measurement point, the spectrum of temperature fluctuations in the inertial range is characterized by the 8/3 power-law decrease (with the following
Figure 3.36 (a) Big alt-azimuthal telescope (BAT). (b) Experimentally recorded
approximate pattern of air motion in the BAT dome space: two air contra-rotating vortices with vertically oriented axes. Side view.
faster decrease), which corresponds to coherent turbulence. The spectra become Kolmogorov (5/3-law decrease) only at the measurements directly in the open slit of the telescope.
For numerical simulation, we have formulated the following boundary-value problem corresponding to conditions of the experiment.
The Dirichlet boundary conditions are the following: zero velocities at the boundaries: U = V = W = 0 (U is the x-component of velocity; V is the y- component of velocity; W is the z-component), for the parameters: temperature
Tmin = 293 K; Tmax = 300 K; Tdome surface = Т(в) is temperature distribution; Tbase is
not specified (T~); в = 60° is sun zenith angle.
Initial conditions are: T0 = 290 K; pressure P0 = 94 kPa. In the dome, there is the air-like medium. The dome model (with diameter of 45.2 м) is empty, that is, without telescope and other equipment.
The dome side facing the sun (Figure 3.37a) is heated up to the maximal temperature Tmax at the point under direct sunlight; with the distance from the point of temperature maximum, the temperature decreases gradually down to Tmip the temperature of other unlighted sides is Tmin (Figure 3.37b).
As a result of solution of the formulated bound ary-value problem, we have obtained the pattern of air motion inside the dome model in parameters of the vector velocity field and scalar temperature and pressure fields (Figure 3.38).
From the comparison of the data in Figures 3.36b and 3.38, it can be seen that the pattern of air motion obtained as a result of numerical simulation of coherent structures (Figure 3.38) practically coincides with the independently experimentally observed pattern of average air motion in the BAT dome space (Figure 3.36b).
Thus, the results of numerical simulation (for conditions of the experiment) confirm the presence of two contra-rotating air vortices with vertically oriented axes in the BAT dome space.
Figure 3.39 shows the calculated and experimental temporal frequency spectra of temperature fluctuations WT in the BAT dome (at point P02, Figure 3.37c).
Figure 3.37 (a) Model (diagram) of the telescope dome with the side heated
by the sun, right view; (b) Dome surface heated by the sun, right view; (c) Arrangement of measurement points, top front view.
Figure 3.38 Pattern of air motion inside the dome model (simulation): (a) top view, (b) horizontal cross section of the velocity field (top view), (c) bottom view, (d) general side view. Solid lines are flow lines.
Figure 3.39 Temporal frequency spectrum of temperature fluctuations WT. Simulation. Experimental WT spectrum (right top inset).
It can be seen from Figure 3.39 that the theoretical spectrum appears to be practically identical to the experimental spectrum recorded. There is a small 8/3 power-law decrease part (in the inertial range) with the further, as expected, faster decrease at high frequencies. Both the theoretical and the experimental spectra correspond to the coherent turbulence.
The processes of turbulence formation and evolution were visualized in References 144—157 through drawing of the flow lines (motion of the medium, more often, the air) in numerical solutions of the Navier—Stokes equations for different boundary-value problems.
The structure of turbulent air motions in closed volumes over a heated surface was studied numerically in References 147—150, 152—155. It was shown that solitary toroidal vortices (coherent structures or topological solitons) arise over the irregularly heated surface. The number of vortices and their internal structure depend on the shape and size of heated spots. For complex shapes of the heating pattern (thermal mottling), toroidal vortices become markedly deformed. Vortices can both be elongated along the surface and have spiral flow lines.
In the process of evolution, the vortices mix markedly. This yields the Kolmogorov’s (incoherent) turbulence.
The results presented in References 147-150, 152-155 allow to follow the evolution of the turbulence structure formed over homogeneously and irregularly heated surfaces.
Thus, at the initial phase of turbulence formation over the heated surface, a family of small (in comparison with the size of the considered volume) convective toroidal (mushroom-like) vortices, usually referred to as thermics, arise. Thermics are formed over both homogeneously and irregularly heated surfaces (for example, over individual heated spots on the surface). The rate of formation of thermics and their sizes depend on viscosity of the medium, shape of the volume, and the degree of heating of its surfaces. In the worldwide scientific literature, the phase of formation of a family of thermics is often called the near-wall turbulence.
In the absence of spatial limits, thermics rise up, increasing their volume. The presence of limits (closed volume) leads to the situation that small vortices (thermics), distorting their shapes, join in large vortices-cells, which form a certain order in the volume (which is characteristic of Benard cells). At this stage, first, the stationary pattern of motion (topological precursors) arises for the short time, and then it is alternated by the period of chaotization. In the following, the transition from chaos to stationary motions, being stable (or slowly breaking-down) vortices (coherent structures or topological solitons) occur. The cascade breakdown of these vortices shows itself as the coherent turbulence.
As follows from above discussions, the transition from small thermics (through the period of chaotization) to large stable vortices is inverse with respect to the usually observed cascade coherent breakdown of large vortices into smaller ones.
Consequently, the formation of near-wall turbulence (thermics) and its further evolution can serve an example of the inverse cascade process of energy transfer by the spectrum of motion scales.
Numerical calculations confirm our experimental conclusion  that the mixing of coherent structures with different close sizes (and with close frequencies of main vortices) yields the Kolmogorov turbulence. In addition, rather extended inertial ranges of the spectrum with Kolmogorov 5/3-law decrease are observed in media with high viscosity. Experimental data discussed earlier in the rooms of astronomical telescopes and in open air confirm our numerical calculations. The spectra of single coherent structures, observed experimentally in special campaigns of 2005—2007 and 2010—2016, are also confirmed.
Thus, the obtained results of numerical solution of Navier—Stokes equations in closed volumes allow us to state that the processes of turbulence formation and evolution, including formation of coherent turbulence, can be described based on solution of boundary-value problems of fluid dynamics (Navier—Stokes equations).
Numerical solutions of Navier—Stokes equations explain the mechanism of formation and existence of hydrodynamic turbulence, in which the role of stochas- tization decreases considerably.
In general, the accomplished analysis of all the accumulated data of measurement of the turbulence parameters both in open air and in closed volumes (as well as the data of numerical solution of the Navier—Stokes equations, presented above) shows that:
The laminar and turbulent flows can be considered as different phases of the single process of turbulence formation and evolution, which is also the single process of formation and breakdown of hydrodynamic topologicalsolitons.
Within the framework of this main conclusion it can be stated that:
Non-Kolmogorov coherent turbulence and Kolmogorov incoherent turbulence can be considered as varieties (particular cases) of the single process of formation and breakdown of hydrodynamic topological solitons.
These varieties are characterized by different intensities and different tpatiotemporal resolution of energy sources of turbulence (thermodynamic gradients) at boundaries of continuous fluid medium.
As was already noted, we have revealed the properties ofcoherent structures through processing of accumulated experimental data by the methods of spectral analysis of random processes with the following formulation of logical conclusions. However, these methods do not allow one to see coherent structures and their mixtures. The possibility of seeing the coherent structures by visualizing them by flow lines in numerical solutions of the Navier—Stokes equations we demonstrated above in this section.