物体から流体への熱伝達（自然対流冷却）¶
シミュレーション名 :: heatTransfer__box_in_water_XY2D
シミュレーション体系¶
基本方程式は、 Navier-Stokes方程式 と 標準k-ε方程式 と 熱輸送方程式
流体は、 非圧縮 , 標準k-εモデル を取扱い、対象は空気とする．
計算対象は、２次元の閉じた立方体の箱の中にバルク鉄塊をおいた際の熱伝達をみる．
2次元 長さ 200 [mm], 幅 200 [mm]、高さ 200 [mm]、厚み 10 [mm] の中空の箱の中央下部に、長さ 50 [mm] 、幅 50 [mm]、高さ 50 [mm] の鉄塊（SS400）を置く．
標準k-εモデル を用いて、壁境界は 垂直方向流速にゼロを課した (Normal-Tangential)．
全領域の初期条件は、流速は0、k=0.00457, epsilon=1e-4 とおいている．
物性条件¶
Target |
Parameters |
Value |
Unit |
Description |
|---|---|---|---|---|
空気 |
Density |
1.2e+0 |
kg/m3 |
密度 |
viscosity |
1.0e-5 |
Pa.s |
粘度 |
|
KE SigmaK |
1.0 |
|||
KE SigmaE |
1.3 |
|||
KE C1 |
1.44 |
係数C1 |
||
KE C2 |
1.92 |
係数C1 |
||
KE Cmu |
0.09 |
係数Cmu |
||
KE Clip |
1.0e-6 |
|||
Viscosity Model |
K-Epsilon |
使用するモデル |
||
Heat Conductivity |
0.0257 |
W/m.K |
熱伝導度 |
|
Heat Capacity |
1.005e+3 |
J/kg.K |
比熱 |
|
Density |
1.166e+0 |
kg/m3 |
密度 |
|
SS400 |
Heat Conductivity |
51.6 |
W/m.K |
熱伝導率 |
Heat Capacity |
473.0 |
J/kg.K |
比熱 |
|
Density |
7.85e+3 |
kg/m3 |
密度 |
メッシュ¶
メッシュ生成スクリプト ( mesh.py )
mesh.py ( heatTransfer__box_in_water_XY2D )¶
import numpy as np
import os, sys
import gmsh
import nkGmshRoutines.define__geometry_2d as dg2
import nkGmshRoutines.load__dimtags as ldt
# ========================================================= #
# === 実行部 === #
# ========================================================= #
if ( __name__=="__main__" ):
# ------------------------------------------------- #
# --- [1] initialization of the gmsh --- #
# ------------------------------------------------- #
gmsh.initialize()
gmsh.option.setNumber( "General.Terminal", 1 )
gmsh.option.setNumber( "Mesh.Algorithm" , 5 )
gmsh.option.setNumber( "Mesh.Algorithm3D", 4 )
gmsh.option.setNumber( "Mesh.SubdivisionAlgorithm", 0 )
gmsh.model.add( "model" )
# ------------------------------------------------- #
# --- [2] Modeling --- #
# ------------------------------------------------- #
dimtags = {}
inpFile = "dat/geometry.conf"
bdrFile = "dat/boundary.json"
dimtags = dg2.define__geometry_2d( inpFile=inpFile, dimtags=dimtags )
dimtags = ldt.load__dimtags( dimtags=dimtags, inpFile=bdrFile )
gmsh.model.occ.synchronize()
gmsh.model.occ.removeAllDuplicates()
gmsh.model.occ.synchronize()
# ------------------------------------------------- #
# --- [3] Mesh settings --- #
# ------------------------------------------------- #
mesh_from_config = True # from nkGMshRoutines/test/mesh.conf, phys.conf
uniform_size = 0.05
if ( mesh_from_config ):
meshFile = "dat/mesh.conf"
physFile = "dat/phys.conf"
import nkGmshRoutines.assign__meshsize as ams
meshes = ams.assign__meshsize( meshFile=meshFile, physFile=physFile, \
dimtags=dimtags, target="surf" )
else:
import nkGmshRoutines.assign__meshsize as ams
meshes = ams.assign__meshsize( uniform=uniform_size, dimtags=dimtags )
# ------------------------------------------------- #
# --- [4] post process --- #
# ------------------------------------------------- #
gmsh.model.occ.synchronize()
gmsh.model.mesh.generate(2)
gmsh.write( "msh/model.msh" )
gmsh.finalize()
geometry.conf
geometry.conf ( heatTransfer__box_in_water_XY2D )¶
# <define> @cntnr.Lx = 200e-3
# <define> @cntnr.Ly = 200e-3
# <define> @cntnr.thick = 10e-3
# <define> @cntnr.x0 = (-0.5) * @cntnr.Lx
# <define> @cntnr.y0 = 0.0
# <define> @cntnr.dx = @cntnr.Lx
# <define> @cntnr.dy = @cntnr.Ly
# <define> @fluid.Lx = @cntnr.Lx - 2.0 * @cntnr.thick
# <define> @fluid.Ly = @cntnr.Ly - 2.0 * @cntnr.thick
# <define> @fluid.x0 = (-0.5) * @fluid.Lx
# <define> @fluid.y0 = @cntnr.thick
# <define> @fluid.dx = @fluid.Lx
# <define> @fluid.dy = @fluid.Ly
# <define> @metal.Lx = 50e-3
# <define> @metal.Ly = 50e-3
# <define> @metal.x0 = (-0.5) * @metal.Lx
# <define> @metal.y0 = @cntnr.thick
# <define> @metal.dx = @metal.Lx
# <define> @metal.dy = @metal.Ly
# <names> key geometry_type x0 y0 dx dy
metal quad @metal.x0 @metal.y0 @metal.dx @metal.dy
fluid.00 quad @fluid.x0 @fluid.y0 @fluid.dx @fluid.dy
cntnr.00 quad @cntnr.x0 @cntnr.y0 @cntnr.dx @cntnr.dy
# <names> key boolean_type targetKeys toolKeys removeObject removeTool
container cut [container.00] [fluid.00] True False
fluid cut [fluid.00] [metal] True False
boundary.json
boundary.json ( heatTransfer__box_in_water_XY2D )¶
{
"metal" : [ [2,1] ],
"fluid": [ [2,2] ],
"container": [ [2,3] ],
"metal_fluid": [ [1,5], [1,6], [1,7] ],
"metal_container": [ [1,13] ],
"fluid_container": [ [1,1], [1,2], [1,3], [1,4], [1,8] ],
"container_outside": [ [1,9], [1,10], [1,11], [1,12] ]
}
phys.conf
phys.conf ( heatTransfer__box_in_water_XY2D )¶
# <names> key type dimtags_keys physNum
metal surf [metal] 201
fluid surf [fluid] 202
container surf [container] 203
metal_fluid line [metal_fluid] 101
metal_container line [metal_container] 102
fluid_container line [fluid_container] 103
container_outside line [container_outside] 104
mesh.conf
mesh.conf ( heatTransfer__box_in_water_XY2D )¶
# <names> key physNum meshType resolution1 resolution2 evaluation
metal 201 constant 0.01 - -
fluid 202 constant 0.01 - -
container 203 constant 0.01 - -
metal_fluid 101 constant 0.0 - -
metal_container 102 constant 0.0 - -
fluid_container 103 constant 0.0 - -
container_outside 104 constant 0.0 - -
生成したメッシュを次に示す．
elmer シミュレーション設定¶
elmer シミュレーション設定ファイルを以下に示す．
steady_ke.sif ( heatTransfer__box_in_water_XY2D )¶
!! ========================================================= !!
!! === time-evolution of natural cooling by Boussinesq === !!
!! ========================================================= !!
include "./msh/model/mesh.names"
Header
CHECK KEYWORDS Warn
Mesh DB "." "msh/model"
Include Path ""
Results Directory "out/"
End
!! ------------------------------------------------- !!
!! --- [1] Simulation --- !!
!! ------------------------------------------------- !!
Simulation
Max Output Level = 3
Coordinate System = "Cartesian"
Coordinate Mapping(3) = 1 2 3
Simulation Type = "Transient"
TimeStepping Method = BDF
BDF Order = 2
Timestep sizes(1) = 1e-1
Timestep Intervals(1) = 1000
Steady State Max Iterations = 200
Post File = heatTransfer.vtu
End
Constants
Stefan Boltzmann = 5.6703e-8 !! -- [ W / m^2 K^4 ] -- !!
gravity(4) = 0 -1 0 9.82 !! -- [ m/s^2 ] -- !!
End
!! ------------------------------------------------- !!
!! --- [2] Solver & Equation --- !!
!! ------------------------------------------------- !!
Solver 1
Equation = k-epsilon
Procedure = "KESolver" "KESolver"
Stabilize = True
Linear System Solver = Direct
Linear System Direct Method = String "umfpack"
!! Linear System Iterative Method = BiCGStab
Linear System Max Iterations = 10000
Linear System Preconditioning = ILUT
Linear System Convergence Tolerance = 1.0e-5
Nonlinear System Max Iterations = 1
Nonlinear System Convergence Tolerance = 1.0e-5
Nonlinear System Relaxation Factor = 0.5
Nonlinear System Newton After Tolerance = 0.0
Nonlinear System Newton After Iterations = 10000
Steady State Convergence Tolerance = 1.0e-5
End
Solver 2
Equation = Navier-Stokes
Procedure = "FlowSolve" "FlowSolver"
Stabilize = True
Linear System Solver = Direct
Linear System Direct Method = String "umfpack"
!! Linear System Iterative Method = BiCGStab
Linear System Max Iterations = 10000
Linear System Convergence Tolerance = 1.0e-5
Linear System Preconditioning = ILUT
Nonlinear System Max Iterations = 1
Nonlinear System Convergence Tolerance = 1.0e-5
Nonlinear System Relaxation Factor = 0.5
Nonlinear System Newton After Tolerance = 0.0
Nonlinear System Newton After Iterations = 10000
Steady State Convergence Tolerance = 1.0e-5
End
Solver 3
Equation = "HeatEquations"
Procedure = "HeatSolve" "HeatSolver"
Variable = "Temperature"
Exec Solver = "Always"
Stabilize = True
Optimize Bandwidth = True
Linear System Solver = Iterative
Linear System Iterative Method = BiCGStab
Linear System Max Iterations = 500
Linear System Convergence Tolerance = 1.0e-8
Linear System Preconditioning = ILU0
Linear System Precondition Recompute = 1
Steady State Convergence Tolerance = 1.0e-5
Nonlinear System Convergence Tolerance = 1.0e-4
Nonlinear System Max Iterations = 1
Nonlinear System Newton After Iterations = 10000
Nonlinear System Newton After Tolerance = 0.0
Nonlinear System Relaxation Factor = 0.5
Steady State Convergence Tolerance = 1.0e-5
End
Equation 1
Name = "metals"
Active Solvers(1) = 3
Convection = Computed
End
Equation 2
Name = "fluid"
Active Solvers(3) = 2 1 3
Convection = Computed
End
!! ------------------------------------------------- !!
!! --- [3] body & materials --- !!
!! ------------------------------------------------- !!
Body 1
Name = "metal"
Target Bodies(1) = $metal
Equation = 1
Material = 1
Initial Condition = 1
End
Body 2
Name = "fluid"
Target Bodies(1) = $fluid
Equation = 2
Material = 2
Initial Condition = 2
Body Force = 1
End
Body 3
Name = "container"
Target Bodies(1) = $container
Equation = 1
Material = 1
Initial Condition = 3
End
Material 1
Name = "SS400"
Heat Conductivity = 51.6 !! W/m.K
Heat Capacity = 473.0 !! J/kg.K
Reference Temperature = 293.15 !! K
Density = 7.85e+3 !! kg/m3
End
Material 2
Name = "Air"
!! fluid property !!
Viscosity = 1.0e-5 !! Pa.s
Density = 1.166e+0 !! kg/m3
KE SigmaK = 1.00
KE SigmaE = 1.30
KE C1 = 1.44
KE C2 = 1.92
KE Cmu = 0.09
KE Clip = Real 1.0e-6
Viscosity Model = K-Epsilon
!! heat property !!
Heat Conductivity = 0.0257 !! W/m.K
Heat Capacity = 1.005e+3 !! J/kg.K
Reference Temperature = 293.15 !! K
Heat Expansion Coefficient = 3.665e-3 !! 1e-6
End
!! ------------------------------------------------- !!
!! --- [4] initial boundary condition --- !!
!! ------------------------------------------------- !!
$ T_metal = 473.15
$ T_fluid = 293.15
$ T_container = 293.15
Initial Condition 1
Name = "metal.initial"
Temperature = $T_metal
End
Initial Condition 2
Name = "fluid.initial"
Velocity 1 = 0
Velocity 2 = 0
Temperature = $T_fluid
Kinetic Energy = 0.00457
Kinetic Dissipation = 1.0e-4
End
Initial Condition 3
Name = "container.initial"
Temperature = $T_container
End
Boundary Condition 1
Name = "fluid.boundary"
Target Boundaries(2) = 102 103
Normal-Tangential Velocity = True
velocity 1 = 0.0
!! Noslip Wall BC = True
End
!! ------------------------------------------------- !!
!! --- [6] others --- !!
!! ------------------------------------------------- !!
Body Force 1
Name = "BoussinesqApprox"
Boussinesq = Logical True
End
自然対流冷却のシミュレーション結果¶
温度の時間発展¶
結果について¶
鉄塊内部の温度差が顕著．
容器側へ、熱伝導で大部分の熱が逃げている可能性が高い．
容器なしでシミュレーションして比較検討してみる．
容器なしの場合との比較¶
メッシュ¶
メッシュ生成スクリプト ( mesh.py )
mesh.py ( heatTransfer__box_in_water_XY2D )¶
import numpy as np
import os, sys
import gmsh
import nkGmshRoutines.define__geometry_2d as dg2
import nkGmshRoutines.load__dimtags as ldt
# ========================================================= #
# === 実行部 === #
# ========================================================= #
if ( __name__=="__main__" ):
# ------------------------------------------------- #
# --- [1] initialization of the gmsh --- #
# ------------------------------------------------- #
gmsh.initialize()
gmsh.option.setNumber( "General.Terminal", 1 )
gmsh.option.setNumber( "Mesh.Algorithm" , 5 )
gmsh.option.setNumber( "Mesh.Algorithm3D", 4 )
gmsh.option.setNumber( "Mesh.SubdivisionAlgorithm", 0 )
gmsh.model.add( "model" )
# ------------------------------------------------- #
# --- [2] Modeling --- #
# ------------------------------------------------- #
dimtags = {}
inpFile = "dat/woContainer/geometry.conf"
bdrFile = "dat/woContainer/boundary.json"
dimtags = dg2.define__geometry_2d( inpFile=inpFile, dimtags=dimtags )
dimtags = ldt.load__dimtags( dimtags=dimtags, inpFile=bdrFile )
gmsh.model.occ.synchronize()
gmsh.model.occ.removeAllDuplicates()
gmsh.model.occ.synchronize()
# ------------------------------------------------- #
# --- [3] Mesh settings --- #
# ------------------------------------------------- #
mesh_from_config = True # from nkGMshRoutines/test/mesh.conf, phys.conf
uniform_size = 0.05
if ( mesh_from_config ):
meshFile = "dat/woContainer/mesh.conf"
physFile = "dat/woContainer/phys.conf"
import nkGmshRoutines.assign__meshsize as ams
meshes = ams.assign__meshsize( meshFile=meshFile, physFile=physFile, \
dimtags=dimtags, target="surf" )
else:
import nkGmshRoutines.assign__meshsize as ams
meshes = ams.assign__meshsize( uniform=uniform_size, dimtags=dimtags )
# ------------------------------------------------- #
# --- [4] post process --- #
# ------------------------------------------------- #
gmsh.model.occ.synchronize()
gmsh.model.mesh.generate(2)
gmsh.write( "msh/model.msh" )
gmsh.finalize()
geometry.conf
geometry.conf ( heatTransfer__box_in_water_XY2D )¶
# <define> @cntnr.Lx = 200e-3
# <define> @cntnr.Ly = 200e-3
# <define> @cntnr.thick = 10e-3
# <define> @cntnr.x0 = (-0.5) * @cntnr.Lx
# <define> @cntnr.y0 = 0.0
# <define> @cntnr.dx = @cntnr.Lx
# <define> @cntnr.dy = @cntnr.Ly
# <define> @fluid.Lx = @cntnr.Lx - 2.0 * @cntnr.thick
# <define> @fluid.Ly = @cntnr.Ly - 2.0 * @cntnr.thick
# <define> @fluid.x0 = (-0.5) * @fluid.Lx
# <define> @fluid.y0 = @cntnr.thick
# <define> @fluid.dx = @fluid.Lx
# <define> @fluid.dy = @fluid.Ly
# <define> @metal.Lx = 50e-3
# <define> @metal.Ly = 50e-3
# <define> @metal.x0 = (-0.5) * @metal.Lx
# <define> @metal.y0 = @cntnr.thick
# <define> @metal.dx = @metal.Lx
# <define> @metal.dy = @metal.Ly
# <names> key geometry_type x0 y0 dx dy
metal quad @metal.x0 @metal.y0 @metal.dx @metal.dy
fluid.00 quad @fluid.x0 @fluid.y0 @fluid.dx @fluid.dy
# <names> key boolean_type targetKeys toolKeys removeObject removeTool
fluid cut [fluid.00] [metal] True False
boundary.json
boundary.json ( heatTransfer__box_in_water_XY2D )¶
{
"metal" : [ [2,1] ],
"fluid": [ [2,2] ],
"metal_fluid": [ [1,5], [1,6], [1,7] ],
"fluid_outside": [ [1,1], [1,2], [1,3], [1,4], [1,8] ],
"metal_outside": [ [1,9] ]
}
phys.conf
phys.conf ( heatTransfer__box_in_water_XY2D )¶
# <names> key type dimtags_keys physNum
metal surf [metal] 201
fluid surf [fluid] 202
metal_fluid line [metal_fluid] 101
fluid_outside line [fluid_outside] 102
metal_outside line [metal_outside] 103
mesh.conf
mesh.conf ( heatTransfer__box_in_water_XY2D )¶
# <names> key physNum meshType resolution1 resolution2 evaluation
metal 201 constant 0.01 - -
fluid 202 constant 0.01 - -
metal_fluid 101 constant 0.0 - -
fluid_outside 102 constant 0.0 - -
metal_outside 103 constant 0.0 - -
容器なしの場合のシミュレーション設定¶
elmer シミュレーション設定ファイルを以下に示す．
steady_ke.sif ( heatTransfer__box_in_water_XY2D )¶
!! ========================================================= !!
!! === time-evolution of natural cooling by Boussinesq === !!
!! ========================================================= !!
include "./msh/model/mesh.names"
Header
CHECK KEYWORDS Warn
Mesh DB "." "msh/model"
Include Path ""
Results Directory "out/"
End
!! ------------------------------------------------- !!
!! --- [1] Simulation --- !!
!! ------------------------------------------------- !!
Simulation
Max Output Level = 3
Coordinate System = "Cartesian"
Coordinate Mapping(3) = 1 2 3
Simulation Type = "Transient"
TimeStepping Method = BDF
BDF Order = 2
Timestep sizes(1) = 5e-2
Timestep Intervals(1) = 500
Steady State Max Iterations = 200
Post File = steady_ke.vtu
End
Constants
Stefan Boltzmann = 5.6703e-8 !! -- [ W / m^2 K^4 ] -- !!
gravity(4) = 0 -1 0 9.82 !! -- [ m/s^2 ] -- !!
End
!! ------------------------------------------------- !!
!! --- [2] Solver & Equation --- !!
!! ------------------------------------------------- !!
Solver 1
Equation = k-epsilon
Procedure = "KESolver" "KESolver"
Stabilize = True
Linear System Solver = Direct
Linear System Direct Method = String "umfpack"
!! Linear System Iterative Method = BiCGStab
Linear System Max Iterations = 10000
Linear System Preconditioning = ILUT
Linear System Convergence Tolerance = 1.0e-5
Nonlinear System Max Iterations = 1
Nonlinear System Convergence Tolerance = 1.0e-5
Nonlinear System Relaxation Factor = 0.5
Nonlinear System Newton After Tolerance = 0.0
Nonlinear System Newton After Iterations = 10000
Steady State Convergence Tolerance = 1.0e-5
End
Solver 2
Equation = Navier-Stokes
Procedure = "FlowSolve" "FlowSolver"
Stabilize = True
Linear System Solver = Direct
Linear System Direct Method = String "umfpack"
!! Linear System Iterative Method = BiCGStab
Linear System Max Iterations = 10000
Linear System Convergence Tolerance = 1.0e-5
Linear System Preconditioning = ILUT
Nonlinear System Max Iterations = 1
Nonlinear System Convergence Tolerance = 1.0e-5
Nonlinear System Relaxation Factor = 0.5
Nonlinear System Newton After Tolerance = 0.0
Nonlinear System Newton After Iterations = 10000
Steady State Convergence Tolerance = 1.0e-5
End
Solver 3
Equation = "HeatEquations"
Procedure = "HeatSolve" "HeatSolver"
Variable = "Temperature"
Exec Solver = "Always"
Stabilize = True
Optimize Bandwidth = True
Linear System Solver = Iterative
Linear System Iterative Method = BiCGStab
Linear System Max Iterations = 500
Linear System Convergence Tolerance = 1.0e-8
Linear System Preconditioning = ILU0
Linear System Precondition Recompute = 1
Steady State Convergence Tolerance = 1.0e-5
Nonlinear System Convergence Tolerance = 1.0e-4
Nonlinear System Max Iterations = 1
Nonlinear System Newton After Iterations = 10000
Nonlinear System Newton After Tolerance = 0.0
Nonlinear System Relaxation Factor = 0.5
Steady State Convergence Tolerance = 1.0e-5
End
Equation 1
Name = "metals"
Active Solvers(1) = 3
Convection = Computed
End
Equation 2
Name = "fluid"
Active Solvers(3) = 2 1 3
Convection = Computed
End
!! ------------------------------------------------- !!
!! --- [3] body & materials --- !!
!! ------------------------------------------------- !!
Body 1
Name = "metal"
Target Bodies(1) = $metal
Equation = 1
Material = 1
Initial Condition = 1
End
Body 2
Name = "fluid"
Target Bodies(1) = $fluid
Equation = 2
Material = 2
Initial Condition = 2
Body Force = 1
End
Material 1
Name = "SS400"
Heat Conductivity = 51.6 !! W/m.K
Heat Capacity = 473.0 !! J/kg.K
Reference Temperature = 293.15 !! K
Density = 7.85e+3 !! kg/m3
End
Material 2
Name = "Air"
!! fluid property !!
Viscosity = 1.0e-5 !! Pa.s
Density = 1.166e+0 !! kg/m3
KE SigmaK = 1.00
KE SigmaE = 1.30
KE C1 = 1.44
KE C2 = 1.92
KE Cmu = 0.09
KE Clip = Real 1.0e-6
Viscosity Model = K-Epsilon
!! heat property !!
Heat Conductivity = 0.0257 !! W/m.K
Heat Capacity = 1.005e+3 !! J/kg.K
Reference Temperature = 293.15 !! K
Heat Expansion Coefficient = 3.665e-3 !! 1e-6
End
!! ------------------------------------------------- !!
!! --- [4] initial boundary condition --- !!
!! ------------------------------------------------- !!
$ T_metal = 473.15
$ T_fluid = 293.15
Initial Condition 1
Name = "metal.initial"
Temperature = $T_metal
End
Initial Condition 2
Name = "fluid.initial"
Velocity 1 = 0
Velocity 2 = 0
Temperature = $T_fluid
Kinetic Energy = 0.00457
Kinetic Dissipation = 1.0e-4
End
Boundary Condition 1
Name = "fluid.boundary"
Target Boundaries(2) = 101 102
Normal-Tangential Velocity = True
velocity 1 = 0.0
!! Noslip Wall BC = True
End
!! ------------------------------------------------- !!
!! --- [6] others --- !!
!! ------------------------------------------------- !!
Body Force 1
Name = "BoussinesqApprox"
Boussinesq = Logical True
End
容器の有無による温度分布の差異 ( t=25 (s) 経過後 )¶
比較した結果について¶
明らかに容器を伝って熱伝導により、熱が逃げているのがわかる．
