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ABSTRACT
Thermal buoyancy-driven natural convection in partial enclosures with multiple free openings is investigated regarding the coupling of the pressure correction algorithm with inner-loop iterations of the pressure correction equation. Unexpected steady flow phenomena, i.e., nonunique steady flow states, are observed when different inner-loop iterations m and relaxation factors α of the pressure correction equation are implemented, after the appropriate assumptions on the free opening boundary conditions are made. Numerical experiments and benchmark exercises confirm that definitions on the free openings could be too weak to ensure global similarity to the results of extended domain methodology. Furthermore, inner-loop iteration m and relaxation factor α could have similar effects on the nonunique steady flow results. As thermal Rayleigh number is relatively low, a linear relationship could be established among inner-loop iteration m, relaxation factor α, and mass exchange rate, MER. In addition, harmonic arrangements on the iteration residuals of the momentum equation and the pressure correction equation could accelerate convergence of the whole pressure correction algorithm.
Nomenclature
| A | = | coefficient in difference equations |
| B | = | source term in discretized equations |
| Ce, Cn | = | coefficient for pressure, Correction |
| D | = | diffusion term |
| F | = | convection term |
| g | = | gravitational acceleration (m/s2) |
| H | = | height of the rectangle square (m) |
| L | = | length of the boundary edge |
| m, n | = | inner-loop and outer-loop iterations |
| n | = | unit outward normal vector at the boundary |
| MER | = | dimensionless mass flow rate |
| p | = | pressure (N/m2) |
| P | = | dimensionless pressure |
| Pr | = | Prandtl number |
| Ra | = | thermal Rayleigh number |
| t | = | temperature (K) |
| T | = | dimensionless temperature |
| u, v | = | velocity components in x, y |
| U, V | = | dimensionless velocity |
| = | components in X,Y | |
| U0 | = | reference velocity scale (m/s) |
| W | = | width of the enclosure (m) |
| Greek symbols | = | |
| α | = | relaxation factors |
| Λ | = | Thermal conductivity (W/m k) |
| η | = | thermal diffusivity (m2/s) |
| β | = | volumetric expansion coefficient |
| Δ | = | difference value |
| ν | = | kinematic viscosity (m2/s) |
| ρ | = | density (kg/m3) |
| Φ | = | generic intensive variable |
| Ф | = | stream function |
| Θ | = | heat function |
| Ψ | = | generic function for visualization |
| Ω | = | overall length of the enclosure |
| Subscripts | = | |
| e, w, n, s | = | values at control-volume interfaces |
| E, W, N, S, P | = | values at nodal points |
| high (low) | = | higher (lower) value |
| In (out) | = | into(out of) cavity |
| Nb | = | neighboring nodal points |
| 0 | = | reference value or location |
| Superscripts | = | |
| ′ | = | corrected values |
| * | = | uncorrected values |
Nomenclature
| A | = | coefficient in difference equations |
| B | = | source term in discretized equations |
| Ce, Cn | = | coefficient for pressure, Correction |
| D | = | diffusion term |
| F | = | convection term |
| g | = | gravitational acceleration (m/s2) |
| H | = | height of the rectangle square (m) |
| L | = | length of the boundary edge |
| m, n | = | inner-loop and outer-loop iterations |
| n | = | unit outward normal vector at the boundary |
| MER | = | dimensionless mass flow rate |
| p | = | pressure (N/m2) |
| P | = | dimensionless pressure |
| Pr | = | Prandtl number |
| Ra | = | thermal Rayleigh number |
| t | = | temperature (K) |
| T | = | dimensionless temperature |
| u, v | = | velocity components in x, y |
| U, V | = | dimensionless velocity |
| = | components in X,Y | |
| U0 | = | reference velocity scale (m/s) |
| W | = | width of the enclosure (m) |
| Greek symbols | = | |
| α | = | relaxation factors |
| Λ | = | Thermal conductivity (W/m k) |
| η | = | thermal diffusivity (m2/s) |
| β | = | volumetric expansion coefficient |
| Δ | = | difference value |
| ν | = | kinematic viscosity (m2/s) |
| ρ | = | density (kg/m3) |
| Φ | = | generic intensive variable |
| Ф | = | stream function |
| Θ | = | heat function |
| Ψ | = | generic function for visualization |
| Ω | = | overall length of the enclosure |
| Subscripts | = | |
| e, w, n, s | = | values at control-volume interfaces |
| E, W, N, S, P | = | values at nodal points |
| high (low) | = | higher (lower) value |
| In (out) | = | into(out of) cavity |
| Nb | = | neighboring nodal points |
| 0 | = | reference value or location |
| Superscripts | = | |
| ′ | = | corrected values |
| * | = | uncorrected values |
Acknowledgments
This research was financially supported by the National Thousand Youth Talents Program from the Organization Department of CCP Central Committee (Wuhan University, China, Grant No. 208273259), Hunan Provincial Natural Science Foundation for Distinguished Young Scholars supported by Hunan Provincial Government (Grant No. 14JJ1002, Multiple fluid mechanisms of urban ventilation and its safety through source identification), the Natural Science Foundation of China (NSFC, Grant No. 51208192, Instability theory and inverse convection design of air flow patterns in the large space; NSFC, Grant No. 51304233, Multi-physics diffusion of leaked natural gas and backward time inverse identification of leakage sources), Hubei Provincial Natural Science Foundation (Grant No. 2015CFB261, Multiple macro flow states of urban built environment), Scientific Research Foundation for the Returned Overseas Chinese Scholars from the State Education Ministry (Grant No. 230303, Inverse identification of unperceived hazardous sources coupling with organized safe evacuation in the public buildings), the Fundamental Research Funds for the Central Universities (Grant No. 2042016kf0136, Dense pollutant dispersion inside the street canyons, Wuhan University) and the National Key Basic Research Program from Ministry of Science and Technology of China (973 Program, Grant No. 2014CB239203).
In addition, Prof. Di Liu would like to acknowledge the financial support from the Hong Kong Scholar Program (Grant No. XJ2013042), China Postdoctoral Science Foundation (Grant No. 2014M560593), Qingdao Postdoctoral Science Foundation, and Fundamental Research Funding Programme for National Key Universities in China.
Both Prof. Fu-Yun Zhao and Prof. Han-Qing Wang would also like to acknowledge the financial support from the Collaborative Innovation Center for Building Energy Conservation and Environment Control, Zhuzhou, Hunan Province, China.