Mathematical Modeling of Gas-Solid Two-Phase Flows: Problems, Achievements and Perspectives (A Review)
Abstract
:1. Introduction
2. Main Problems and Specific Features of Two-Phase Flow Modeling
2.1. Main Problems of Two-Phase Flow Modeling
2.2. Specific Features of Two-Phase Flow Modeling
2.2.1. Multiscale Physics of Two-Phase Flow
2.2.2. Multiplicity of Forces Acting on Particles
2.2.3. Multiplicity of Modeling Parameters
2.2.4. Multiplicity of Collision Processes
2.2.5. Multiplicity of Phase and Chemical Transformations
2.2.6. Multiplicity of Dimensionless Parameters
3. Main Characteristics of Two-Phase Flows
3.1. Particle Concentrations
3.1.1. One-Way Coupling
3.1.2. Two-Way Coupling
3.1.3. Four-Way Coupling
3.2. Particles’ Dynamic Relaxation Time
3.3. Stokes Numbers
3.3.1. Stokes Number in Time-Averaged Motion
3.3.2. Stokes Number in Large-Scale Fluctuation Motions
3.3.3. Stokes Number in Small-Scale Fluctuation Motions
3.4. Classification of Turbulent Two-Phase Flows According to Particle Inertia
4. Lagrangian and Eulerian Modeling of Two-Phase Flows
4.1. Reasons for Considering of the Two-Phase Nature of Tornados
4.2. Lagrangian Modeling
4.3. Eulerian Modeling
4.4. Advantages and Limitations of Lagrangian and Eulerian Modeling
4.5. Description of the Gas Flow Carrying the Particles
4.5.1. Actual Equations
4.5.2. Time-Averaged Equations
4.5.3. Equations for the Reynolds Stresses
5. Methods of Numerically Modeling Two-Phase Flows
5.1. Particle-Resolved DNS
5.2. Particle Point Methods
5.3. Direct Numerical Simulation
5.4. Large Eddy Simulation
6. Conclusions
- (1)
- The development of mathematical modeling methods for two-phase flows with relatively large particles (non-equilibrium flows) that only interact with large energy-carrying vortices and are characterized by dynamic slippage (velocity difference) in relation to average motion.
- (2)
- The development of mathematical modeling methods for two-phase flows with large particles, which form turbulent wakes behind them. With the increase in particle concentration, these turbulent wakes will interfere with each other, and the particles will undergo collisions.
- (3)
- The development of mathematical modeling methods for two-phase flows containing particles of different sizes (polydisperse particles). Such flows are of interest to practicing engineers. Particles of different sizes will have different velocities and different effects on gas flow and tend to collide with each other at lower concentrations.
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Nomenclature
particle diameter, m | |
Kolmogorov length scale, m | |
particle radius vector, m | |
vector of actual velocity of gas, m/s | |
vector of actual velocity of particle, m/s | |
, , | actual velocity components of gas, m/s |
, , | actual velocity components of particle, m/s |
, , | time-averaged velocity components of gas, m/s |
, , | time-averaged velocity components of particle, m/s |
, , | fluctuation velocity components of gas, m/s |
, , | fluctuation velocity components of particle, m/s |
kinematic viscosity of gas, m2/s | |
coefficient of thermal diffusivity, m2/s | |
time, s | |
dynamic relaxation time of particle, s | |
heat relaxation time of particle, s | |
Kolmogorov time scale, s | |
characteristic time of gas in time-averaged motion, s | |
Lagrangian integral time scale, s | |
particle interaction time with energy-containing velocity fluctuations, s | |
particle interaction time with energy-containing temperature fluctuations, s | |
actual temperature of gas, K | |
actual temperature of particle, K | |
time-averaged temperature of gas, K | |
time-averaged temperature of particle, K | |
fluctuation temperature of gas, K | |
fluctuation temperature of particle, K | |
dynamic viscosity of gas, kg/(ms) | |
gas density, kg/m3 | |
particle density, kg/m3 | |
actual pressure of the gas, Pa | |
time-averaged pressure of the gas, Pa | |
fluctuation pressure of the gas, Pa | |
isobaric heat capacity of gas, J/(kg K) | |
heat capacity of material of particle, J/(kg K) | |
actual volumetric concentration of the particles | |
time-averaged volumetric concentration of the particles | |
fluctuation volumetric concentration of the particles | |
time-averaged mass concentration of the particles | |
particles Reynolds number | |
Reynolds number determined via Taylor turbulence scale | |
frictional Reynolds number | |
Stokes number in time-averaged motion | |
Stokes number in large-scale fluctuation motion | |
Stokes number in small-scale fluctuation motion | |
Superscripts | |
fluctuation value | |
time-averaged value | |
Subscripts | |
particle | |
f | fluid (gas) |
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Varaksin, A.Y.; Ryzhkov, S.V. Mathematical Modeling of Gas-Solid Two-Phase Flows: Problems, Achievements and Perspectives (A Review). Mathematics 2023, 11, 3290. https://doi.org/10.3390/math11153290
Varaksin AY, Ryzhkov SV. Mathematical Modeling of Gas-Solid Two-Phase Flows: Problems, Achievements and Perspectives (A Review). Mathematics. 2023; 11(15):3290. https://doi.org/10.3390/math11153290
Chicago/Turabian StyleVaraksin, Aleksey Yu., and Sergei V. Ryzhkov. 2023. "Mathematical Modeling of Gas-Solid Two-Phase Flows: Problems, Achievements and Perspectives (A Review)" Mathematics 11, no. 15: 3290. https://doi.org/10.3390/math11153290
APA StyleVaraksin, A. Y., & Ryzhkov, S. V. (2023). Mathematical Modeling of Gas-Solid Two-Phase Flows: Problems, Achievements and Perspectives (A Review). Mathematics, 11(15), 3290. https://doi.org/10.3390/math11153290