Second-Order Symmetric Smoothed Particle Hydrodynamics Method for Transient Heat Conduction Problems with Initial Discontinuity
Abstract
:1. Introduction
2. Transient Heat Conduction Problem
3. SO-SSPH Method
3.1. SPH Method
3.2. SO-SSPH Method
4. Numerical Analysis
4.1. Truncation Error
4.2. Numerical Accuracy
4.3. Convergence Rate
4.4. Stability Analysis
5. Numerical Experiments
5.1. One-Dimensional Case
5.2. Two-Dimensional Case
6. Discussion and Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Nomenclature
Abbreviations | |
CSPM | corrective smoothed particle method |
DNS | direct numerical simulation |
EFG | element-free Galerkin |
FDM | finite difference methods |
FEM | finite element methods |
FO-SSPH | first-order symmetric smoothed particle hydrodynamics |
FTCS | forward time central space |
FVM | finite volume methods |
LBM | lattice Boltzmann method |
MLSPH | moving least squares particle hydrodynamics |
MSPH | modified smoothed particle hydrodynamics |
NHT | Numerical Heat Transfer |
SO-SSPH | second-order symmetric smoothed particle hydrodynamics |
SPH | smoothed particle hydrodynamics |
Symbols | |
bT | heat source on the boundary |
cp | material specific heat capacity |
E | numerical error |
erf | error function |
G | amplification factor |
G0 | source term |
h | smoothing length |
h | vector of particle distance |
i | imaginary unit |
kx | thermal conductivity coefficients in x direction |
ky | thermal conductivity coefficients in y direction |
n | unit normal vector on the boundary |
q | heat flux |
T | temperature |
T0 | initial temperature distribution |
T1 | boundary temperature |
TE | truncation error |
Te | exact temperature |
Tn | numerical temperature |
v | column vector of the wave number in the Fourier analysis method |
W | kernel function |
Wk | compactly supported function |
αx | coefficients of ∂f/∂x |
αy | coefficients of ∂f/∂y |
βx | coefficients of ∂2f/∂x2 |
βy | coefficients of ∂2f/∂y2 |
Dirichlet boundary of the domain | |
Neumann boundary of the domain | |
Δt | time step |
ΔVj | volume of j th particle |
Δx | particle distance in x direction |
Δy | particle distance in y direction |
κx | kx/ρcp, m2/s |
κy | ky/ρcp, m2/s |
ρ | material density |
Ω | computational domain |
approximation operator | |
kernel gradient |
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Method | Δx (m) | |||||
---|---|---|---|---|---|---|
0.01 | 0.02 | 0.05 | 0.10 | 0.25 | 0.5 | |
FO-SSPH | 0.0089 | 0.0178 | 0.0445 | 0.0898 | 0.2297 | 0.3381 |
SO-SSPH | 0.0013 | 0.0051 | 0.0379 | 0.1303 |
Method | Δx (m) | |||
---|---|---|---|---|
0.1 | 0.25 | 0.5 | 1.0 | |
FO-SSPH | 0.0024 | |||
SO-SSPH |
Method | Δx = Δy (m) | |||
---|---|---|---|---|
0.10 | 0.125 | 0.25 | 0.5 | |
FO-SSPH | 0.1548 | 0.1974 | 0.4263 | 0.6867 |
SO-SSPH | 0.0098 | 0.0159 | 0.0773 | 0.2719 |
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Song, Z.; Xing, Y.; Hou, Q.; Lu, W. Second-Order Symmetric Smoothed Particle Hydrodynamics Method for Transient Heat Conduction Problems with Initial Discontinuity. Processes 2018, 6, 215. https://doi.org/10.3390/pr6110215
Song Z, Xing Y, Hou Q, Lu W. Second-Order Symmetric Smoothed Particle Hydrodynamics Method for Transient Heat Conduction Problems with Initial Discontinuity. Processes. 2018; 6(11):215. https://doi.org/10.3390/pr6110215
Chicago/Turabian StyleSong, Zhanjie, Yaxuan Xing, Qingzhi Hou, and Wenhuan Lu. 2018. "Second-Order Symmetric Smoothed Particle Hydrodynamics Method for Transient Heat Conduction Problems with Initial Discontinuity" Processes 6, no. 11: 215. https://doi.org/10.3390/pr6110215
APA StyleSong, Z., Xing, Y., Hou, Q., & Lu, W. (2018). Second-Order Symmetric Smoothed Particle Hydrodynamics Method for Transient Heat Conduction Problems with Initial Discontinuity. Processes, 6(11), 215. https://doi.org/10.3390/pr6110215