An Open-Source Code for High-Speed Non-Equilibrium Gas–Solid Flows in OpenFOAM
Abstract
:1. Introduction
2. Methodology
2.1. Algorithmic Overview
- Step 1: Calculation of the gas phase thermophysical properties necessary to calculate the particle drag force and reset the source terms of the particle phase.
- Step 2: Injection of particles from inlet boundaries.
- Step 3: Evolution of the disperse phase, including performing particle movements, updating temperatures, velocities and positions, and conducting Multi-Phase Particle-In-Cell (MPPIC) procedures (optional).
- Step 4: Sampling and calculation of discrete phase properties and returning the desired macroscopic fields, such as number density and surface heat flux.
- Step 5: Updating the particle volume fraction using the particle cell method.
- Step 6: Smoothing the particle volume fraction and source terms using the diffusion-based method (optional).
- Step 7: Solving the continuity, momentum, and energy equations of the gas phase and entering the gas properties in the hy2Foam algorithm.
- Step 8: Return to Step 1 until the time reaches the final calculation time.
2.2. Governing Equations of Gas Phase
2.3. Governing Equations of Discrete Phase
- The solid particles are assumed to be perfectly spherical; for simplicity, the particle geometry effect is neglected during calculation of the gas–particle coupling.
- Because the dimension of particles in this work is small, there is no particle temperature gradient within the solid particles; thus, the particle temperature is spatially uniform.
- There is no mass exchange between the gas phase and solid phase.
- Particle forces such as the Magnus force, virtual mass force, and pressure gradient force are neglected, as particles in high-speed flows are dominated by the drag force.
- Gravitational and buoyancy forceThe particle gravitational and buoyancy force in the gas flow is calculated as follows:
- Drag forceThe drag force of a sphere is the most important force experienced by a particle, as it dominates the momentum exchange between the gas and the solid phase. Inaccurate calculation of the drag force leads to an uncertain particle distribution in space [26]. It is usually formulated asExcept for the built-in Gidaspow drag model [27], which is named ErgunWenYuDrag in OpenFOAM, the drag models added in the solver are listed here:
- (1)
- Singh et al. drag correlation [28]:The particle drag coefficient model proposed by Singh and Kroells et al. [28] has recently achieved some acknowledgement [29,30,31,32] and yields good simulation results. A generalized physics-based spherical particle drag force model with satisfactory accuracy, it takes into account rarefaction effects and covers a wide range of supersonic and hypersonic flow regimes. The coefficients in Equation (18) are long expressions, and can be found in [28]. Note that and are the Mach number and Reynolds number, respectively, and are based on the gas velocity rather than the gas–particle relative velocity.
- (2)
- Henderson drag correlation [33]:Henderson [33] put forward a drag coefficient model for continuum and rarefied flows that has been extensively used in the simulation of high speed gas–particle flows. The expression of the drag coefficient is related to a Mach number based on the relative velocity between the particle and the gas. For < 1:Linear interpolation is used when is in the range of 1 and 1.75:
- (3)
- Clift and Gauvin drag correlation [34]:
- (4)
- Boiko et al. drag correlation [35]:
- Thermophoretic forceThe thermophoretic force is generated in regions characterized by significant temperature gradients, such as those present at the shock and within the thermal boundary layer. It is a result of collisions of molecules on a particle with greater energy on the high-temperature side compared to its low-temperature side [36]. Loth [36] proposed a thermophoretic force model, which is duplicated in the solver. When 0.01, the particle’s thermophoretic force is calculated as follows:
- The MPPIC methodThe MPPIC method, first developed by Andrews and O’Rourke [42], is a highly efficient approach for handling the interactions of particles with moderate and high volume fractions during simulations of multiphase flows. The particle distribution function is obtained by solving the Liouville equation [42]:The particle’s total acceleration is equal to in Equation (13). The acceleration due to the enduring particle–particle contact is
2.4. Interphase Coupling
3. Results and Discussions
3.1. Shock–Particle Curtain Interaction
3.1.1. Low Particle Volume Fraction
3.1.2. Moderate Particle Volume Fraction = 21%
3.2. Two-Phase Flow in JPL Thruster
3.3. MSRO Body: Hypersonic Non-Equilibrium Flow during Mars Entry
4. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Nomenclature
Symbols | |
viscous stress tensor | |
net mass production of gas species i | |
sum of the vibrational source terms | |
specific heat ratio | |
heat transfer coefficient | |
gas dynamic viscosity | |
density | |
volume fraction | |
particle acceleration | |
particle force | |
gravitational and bouyancy force | |
velocity | |
particle position | |
A | particle projection area |
drag coefficient | |
d | particle diameter |
e | total energy per unit mass |
J | diffusion flux |
Knudsen number | |
m | mass |
Mach number | |
Nusselt number | |
p | pressure |
Prandtl number | |
Q | heat transferred from particle to gas |
q | heat transfer |
R | gas constant |
Reynolds number | |
energy exchange source | |
momentum exchange source | |
T | temperature |
Y | gas species mass fraction |
Subscripts | |
∞ | free stream properties |
m | gas molecule |
MPPIC | |
p | particle properties |
thermophoretic fortce | |
trans-rotational energy mode | |
vibro-electronic energy mode | |
Acronyms | |
DSMC | direct simulation Monte Carlo |
MPPIC | multiphase particle-in-cell |
OpenFOAM | Open Field Operation and Manipulation |
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gas total pressure (Pa) | 1.034 × |
gas total temperature (K) | 555 |
particle specific heat capacity (J/kg/K) | 1380 |
particle density (kg/m3) | 4004.62 |
particle diameter (m) | 2 × |
particle inflow mass fraction | 0.3 |
Wall Condition | Non-Catalytic |
---|---|
mass fraction | 1.0 |
Wall temperature | 1500 |
Gas velocity (m/s) | 5223 |
Pressure (Pa) | 7.87 |
Temperature (K) | 140 |
particle density (kg/m3) | 2940 |
particle diameter (m) | |
particle specific heat capacity (J/kg/K) | 1000 |
Reaction | ||||||
---|---|---|---|---|---|---|
−1.5 | 63,275 | −0.75 | 535 | |||
−1.5 | 63,275 | −0.75 | 535 | |||
−1.0 | 129,000 | −1.0 | 0 | |||
−1.0 | 129,000 | −1.0 | 0 | |||
−1.5 | 59,500 | −1.0 | 0 | |||
−1.5 | 59,500 | −1.0 | 0 | |||
0.5 | 65,710 | −0.25 | 0 | |||
−0.18 | 69,200 | −0.43 | 0 | |||
0 | 27,800 | 0.5 | 23,800 |
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Cao, Z.; Zhang, X.; Zhang, Y. An Open-Source Code for High-Speed Non-Equilibrium Gas–Solid Flows in OpenFOAM. Aerospace 2024, 11, 742. https://doi.org/10.3390/aerospace11090742
Cao Z, Zhang X, Zhang Y. An Open-Source Code for High-Speed Non-Equilibrium Gas–Solid Flows in OpenFOAM. Aerospace. 2024; 11(9):742. https://doi.org/10.3390/aerospace11090742
Chicago/Turabian StyleCao, Ziqu, Xiaofeng Zhang, and Yonghe Zhang. 2024. "An Open-Source Code for High-Speed Non-Equilibrium Gas–Solid Flows in OpenFOAM" Aerospace 11, no. 9: 742. https://doi.org/10.3390/aerospace11090742