5.3. Direct numerical simulation
To date, there is a significant amount of work in which researchers have studied various problems of physics of two-phase flows using DNS and describing interphase interactions and interphase boundary at various levels.
One of the first papers in which the behavior of point-particle (PP DNS) particles in damped homogeneous isotropic turbulence de-caying homogeneous isotropic turbulence (DHIT) was studied was [
74]. In this study, the motion of 432 particles was studied at a very small Reynolds number (
${\mathrm{Re}}_{\lambda}<35$). Only linear aerodynamic drag was taken into account in the equations of particle motion.
In later studies [
75,
76] devoted to the study of particle motion, both in forced homogeneous isotropic turbulence forced homogeneous isotropic turbulence (FHIT) and in damped homogeneous isotropic turbulence (DHIT), emphasis was placed on the study of the possibilities of various methods of interpolation (linear interpolation, high-order Lagrangian interpolation, high-order Her-mitian interpolation) of the gas velocity at the location of the particle.
A more complex case of turbulent two–phase flow - turbulent flow in the channel is considered in [
77,
78]. In [
77], in addition to the aerodynamic drag force, the Safman force was also taken into account, and in [
78], a more advanced Fourier-Chebyshev pseudo-spectral method was used to interpolate the gas velocity at the particle location. To date, there are numerous studies of two-phase flows by the PP OWC DNS method in the channel [
79], pipes [
80,
81,
82], FHIT [
19,
20] and DHIT [
83].
With an increase in the concentration of particles, they begin to have the opposite effect on the characteristics of the carrier gas flow (see
section 3), so TWC DNS is necessary. This introduces additional difficulties in mathematical modeling. Firstly, in the equation of motion of a particle, not the initial (inherent in a single-phase flow) velocity of the gas should be present, but the "new" velocity of the flow caused by the presence of particles. In [
84], it was suggested that the difference between these velocities is small if the diameter of the particles is smaller than the size of the numerical grid,
${d}_{p}<L$. This condition is almost always satisfied in the case of PP DNS. Secondly, it is necessary to introduce a source term in the equations of gas motion [
85]. If the particle is smaller than the Kolmogorov scale (
${d}_{p}<\eta $), then there are no special problems. Otherwise (
${d}_{p}>\eta $) raises the question of the relevance of the assumption of point particles. In [
86,
87], calculations of a two-phase flow containing a lot of very small particles at
$\mathsf{\Phi}=O(1{0}^{-4})$ were carried out, and the number of particles was commensurate with the number of cells of the computational grid.
Examples of studies in which PP TWC DNS modeling was performed are the works [
88,
89,
90,
91]. In [
90], the turbulent flow in the channel was studied. The volume concentration of particles was equal to
$\mathsf{\Phi}\approx 1{0}^{-4}$. It was assumed that the particles are Stokes (linear law of resistance). It was found that in the case of small particles (
${d}_{p}<\eta $), their presence suppressed turbulence, and the presence of relatively large particles (
${d}_{p}>\eta $), on the contrary, caused turbulence intensification. In [
89,
90], a two-phase flow in the channel at
${\mathrm{Re}}_{\tau}=180$, constructed from the half-height of the channel, was simulated. It is revealed that the presence of particles reduces the resistance and leads to an increase in longitudinal pulsations of the gas velocity. At the same time, the presence of particles caused a decrease in gas velocity pulsations in the other two directions and significantly reduced Reynolds stresses. In [
91], a two-phase turbulent flow in a channel was modeled at the same Reynolds number (
${\mathrm{Re}}_{\tau}=180$), taking into account the nonlinearity in the particle drag law (non-Stokes particles). It is found that particles having small values of the Stokes number (!!!!какoе числo, как oпределялoсь, пoсмoтреть!!!), increased the intensity of turbulence, Reynolds stresses and viscous dissipation. At the same time, particles having large values of the Stokes number (!!!!какoе числo, как oпределялoсь, пoсмoтреть!!!), led to a decrease in the intensity of turbulence.
In [
92], the influence of monodisperse sub-Kolmogorov (
${d}_{p}<{\eta}_{\u041a}$) Stokes (
${\mathrm{Re}}_{p}<<1$) particles on the decay of homogeneous isotropic turbulence (HIT) was studied using PP TWC DNC The article focused on the impact of the numerical concentration of particles. Calculation variables were varied independently, including the Stokes number (
${\mathrm{Stk}}_{K}=0.3-4.8$), the mass concentration of particles (
$M=0.001-0.3$) and the number of particles in the Kolmogorov vortex(
${N}_{\eta}=0.07-17$). The numerical concentration of particles
${N}_{0}$ and the number of particles in the Kolmogorov vortex
${N}_{\eta}$ are related as
${N}_{\eta}={N}_{0}\text{\hspace{0.17em}}{\eta}^{3}$. The calculations carried out allowed for the clear identification of two regimes. In the
${\mathrm{Stk}}_{K}<1$ the presence of particles results in a decrease in the decay of turbulent energy (first mode). In the
${\mathrm{Stk}}_{K}>1$ particles accelerate the decay of turbulence. (second mode).
In [
93], the results of PR TWC DNS for direct of turbulent two-phase upward flow in a vertical channel are presented. The influence of particle Reynolds number (
${\mathrm{Re}}_{p}<227$), particle size, bulk (bulk) Reynolds number (
$\mathrm{Re}=5746$ and
$\mathrm{Re}=12000$), phase density ratio (
${\rho}_{p}/\rho =2-100$), particle radius to half channel width ratio (
$2{r}_{p}/H=0.05-0.15$), and particle volume concentration (
$\mathsf{\Phi}=3\cdot 1{0}^{-3}-2.36\cdot 1{0}^{-2}$) on the pulsation velocity intensity of the carrying phase is investigated. The calculations showed that at low values of
${\mathrm{Re}}_{p}$, turbulence intensity decreases across the channel. At moderate values of
${\mathrm{Re}}_{p}$, turbulence intensity increases in the central region of the channel and decreases in the wall region. At high values of
${\mathrm{Re}}_{p}$, turbulence intensity increases across the transverse section of the channel. The critical value of
${\mathrm{Re}}_{p}$ increases with an increase in bulk Reynolds number, particle size, and the ratio of phase densities, as well as a decrease in the volume concentration of particles.
Further increase in particle concentration necessitates accounting for interparticle collisions (see
Section 3), which requires conducting FWC DNS. Intense interparticle collisions influence particle motion statistics and, consequently, their backreaction on gas flow. This greatly complicates mathematical modeling. Currently, several stochastic approaches have been developed to move away from simple deterministic calculations of pairwise particle collisions, which require immense computational time.
Examples of studies in which PP FWC DNS modeling was performed include works [
94,
95]. In [
94], mathematical modeling of turbulent two-phase flow in a vertical pipe in the presence of small heavy particles was carried out over a wide range of variations in particle mass concentration (
$\mathrm{M}=0.1-30$). Various modeling techniques for real wall roughness were used to better match the results with experimental data. It was found that the results strongly depend on the model of wall roughness used, rather than on the variation of parameters characterizing the inter-particle collision process. The calculations also revealed a decrease in turbulence intensity with an increase in particle mass concentration. In [
95], modeling of turbulent two-phase downward flow in a channel was performed at
${\mathrm{Re}}_{\tau}=642$ and particle mass concentration
$\mathrm{M}=0.8$. The calculations were carried out for smooth and rough walls, where roughness was modeled by placing fixed tiny particles on the wall. It was discovered that rough walls enhance the suppression of turbulence caused by the presence of particles in the flow.
In [
96], the interaction between a stationary homogeneous isotropic turbulent (HIT) flow and inertial particles with accounting for inter-particle collisions (PP FWC DNS) is studied via direct numerical simulation (DNS). The calculations were performed for a periodic cubic box of size 128
^{3} for two values of the Reynolds Taylor number (
${\mathrm{Re}}_{\lambda}=35.4$ and
${\mathrm{Re}}_{\lambda}=58$) while varying the volume concentration of particles (from
$\mathsf{\Phi}=1.37\cdot 1{0}^{-5}$ to
$\mathsf{\Phi}=8.22\cdot 1{0}^{-5}$) and the Stokes number (
${\mathrm{Stk}}_{K}=0.19-12.7$). Elastic spherical particles with a diameter of
${d}_{p}=67.6$ μm, corresponding to
${d}_{p}/{\eta}_{\u041a}=0.1$ acted as the dispersed phase. The Stokes number was varied by changing the particle density over a wide range:
${\rho}_{p}=150-18000$ kg/m
^{3}. The results [
96] showed that the dissipation decreases up to 32% with an increase in the Stokes number and volume concentration of particles. It was shown that this maximum reduction in dissipation is overestimated by 7% when accounting for inter-particle collisions. The spectral analysis revealed a transfer of energy from large to small scales due to particle flow, which explains the difference in dissipation.
5.4. Large-eddy simulation
LES is similar to DNS, but the grid used is much larger. Small vortices are approximated a subgrid-scale (subgrid-scale) model of turbulence. The most commonly used model is the dynamic Smagorinsky model of vortex viscosity [
97]. Other well-known models are based on scale-similarity assumption [
98], Teylor series expansion [
99] or approximate deconvolution [
100].
One of the early works that used the PP OWC LES method was the study presented in [
101]. In this work, particle dispersion was investigated for the case of homogeneous shear flow. The authors did not use the term LES, but they considered the spatially-averaged Navier-Stokes equation for the gas and used time- and space-varying coefficients for the small-scale vortices. The calculations were carried out for only 48 passive particles, and the influence of subgrid scales on their motion was not considered.
The work presented in [
102] investigated particle dispersion in a turbulent pipe flow using PP OWC LES and DNS methods for different Reynolds numbers. The equation of particle motion took into account the drag force, lift force, and buoyancy force. Due to very low values of particle volume concentration, their back-reaction on the gas and interparticle collisions were not considered. Moreover, the influence of subgrid scales of the gas velocity was also not taken into account. The main conclusion of this work was that the dynamic relaxation time of particles plays an important role in their sedimentation.
In [
103] studied particle motion in a vertical channel with very low particle volume concentration using the PP OWC LES method. The dynamic Smagorinsky approach, previously developed in [
104], was used as subgrid-scale model. A comparison of the results obtained with those of DNS-based modeling showed good agreement. It should be noted that this work investigated the influence of subgrid-scale velocities on particle settling. For this purpose, an additional equation for the transport of kinetic energy of subgrid-scale turbulence was used, revealing only a minor effect on the calculation results.
In [
105] performed calculations of a two-phase flow for the case of forced homogeneous isotropic turbulence (FHIT) with the reverse influence of particles on gas taken into account, i.e. using the PP TWC LES method. The authors applied various subgrid-scale models to the equations of motion of the carrying gas. A very important conclusion was drawn that an increase in particle mass concentration leads to a decrease in the weighting coefficients in the dynamic model of vortex viscosity. As a consequence, the calculation error due to the use of subgrid-scale models for the two-phase flow is reduced compared to the single-phase flow.
The PP FWC LES method was used to account for particle collisions in [
106] when studying a two-phase flow in a vertical channel at
${\mathrm{Re}}_{\tau}=644$ and volume concentration up to
$\mathsf{\Phi}=1.4\times 1{0}^{-4}$. The impact of drag force, gravitational force, and lift forces (due to the presence of gas velocity shear and particle rotation) on particle behavior was taken into account in the work. A deterministic model was used to account for particle collisions. The conclusion was drawn about the significant influence of inter-particle collisions on the statistical characteristics of particle motion, including the concentration magnitude.
In [
107], two-phase flow calculations were performed using the PP FWC LES method in a channel with a very high particle volume concentration
$\mathsf{\Phi}=1.3\times 1{0}^{-2}$. Among all the forces, only the drag force and gravitational force were considered. The calculations showed that particles have a colossal effect on turbulence, leading to a thinning of the boundary layer, an increase in gas velocity fluctuations in the longitudinal direction and, conversely, a reduction in gas fluctuations in the two other directions.
In [
108], the parameters of a two-phase flow in a channel were calculated at a particle volume concentration of
$\mathsf{\Phi}=4.8\times 1{0}^{-4}$ and a Reynolds flow rate of
$\mathrm{Re}=\mathrm{42,000}$, which was based on the height of the channel. The authors separately considered the effects of particle back-influence on gas and inter-particle collisions (PP TWC LES and PP FWC LES). They also emphasized the use of various particle collision models (hard-sphere and soft-sphere), different wall conditions (smooth and rough), and different subgrid viscosity models (Smagorinsky model and dynamic model). The calculation results showed that differences when using different collision and subgrid models are insignificant. At the same time, the consideration of particle collisions and wall roughness leads to better agreement with available experimental data.
In [
109], PP FWC LES was performed for a two-phase flow with particles at a volume concentration of
$\mathsf{\Phi}=7.3\times 1{0}^{-5}$ and a Reynolds flow rate of
$\mathrm{Re}=\mathrm{11,900}$, which was based on half the height of the channel. The authors used a subgrid model developed earlier in [
110] for the particle motion equation, as well as a deterministic model to calculate inter-particle collisions. It was shown that with such a small volume concentration of particles, their influence on gas turbulence is negligible. At the same time, it was found that the influence of particle collisions plays a significant role. A good agreement was found between the results and the DNS data described in [
86], as well as with experimental data.
The authors of [
109] later performed PP FWC LES simulations of a two-phase flow [
111] in a horizontal pipe at a Reynolds number of
$\mathrm{Re}=\mathrm{120,000}$, based on the pipe diameter. The peculiarity of this study was the consideration of particle polydispersity and their rotation, as well as the inclusion of not only the drag force but also the lift force of Saffman and the Magnus force. Wall roughness was modeled by introducing coefficients of normal and tangential velocity restitution that differ from unity, as well as by taking into account the so-called shadow effect at small wall collision angles.
In [
112], PP FWC LES of a two-phase flow in a channel was performed with the presence of particle agglomeration effects. The main technique that allows taking into account the appearance of particle agglomerates in the flow after their collision is the introduction of the van der Waals force, which is responsible for the phenomenon of cohesion. As examples of the use of two-phase flows of the future, various aerodynamic and energy systems can serve [
113,
114,
115,
116,
117,
118,
119].