In[]:=
In[]:=
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(*Triangulatelement*)Needs["NDSolve`FEM`"]Ω=ImplicitRegion[True,{{x,0,1},{y,0,1}}]RegionPlot[Ω,AspectRatio->Automatic]A=Rotate[Ω,15Degree,{0,0}]mesh=ToElementMesh[A]mesh["Wireframe"]
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ImplicitRegion[0≤x≤1&&0≤y≤1,{x,y}]
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ParametricRegion-,+,0≤x≤1&&0≤y≤1,{x,y}
(1+
3
)x2
2
(-1+
3
)y2
2
(-1+
3
)x2
2
(1+
3
)y2
2
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ElementMesh[{{-0.258819,0.965926},{0.,1.22474}},{TriangleElement[<412>]}]
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eq1={u[x,y],v[x,y]}.Inactive[Grad][u[x,y],{x,y}]+Inactive[Div][{p[x,y],0},{x,y}]-Pr*Inactive[Div][{Derivative[1,0][u][x,y],Derivative[0,1][u][x,y]},{x,y}]eq2={u[x,y],v[x,y]}.Inactive[Grad][v[x,y],{x,y}]+Inactive[Div][{0,p[x,y]},{x,y}]-Pr*Inactive[Div][{Derivative[1,0][v][x,y],Derivative[0,1][v][x,y]},{x,y}]-Pr*Ra*θ[x,y]eq3=Derivative[1,0][u][x,y]+Derivative[0,1][v][x,y]eq4={u[x,y],v[x,y]}.Inactive[Grad][θ[x,y],{x,y}]-Inactive[Div][{Derivative[1,0][θ][x,y],Derivative[0,1][θ][x,y]},{x,y}]Pr=0.71;Ra=10^3;OP={eq1==0,eq2==0,eq3==0,eq4==0}bc={DirichletCondition[θ[x,y]==1,-3.73x],DirichletCondition[θ[x,y]==0,3.86-3.73x],DirichletCondition[{u[x,y]==0,v[x,y]==0},True],DirichletCondition[p[x,y]==0,-3.73x||0.267x]}{u,v,p,θ}=NDSolveValue[{OP,bc},{u[x,y],v[x,y],p[x,y],θ[x,y]},{x,y}∈mesh,Method->{"FiniteElement","InterpolationOrder"->{u->2,v->2,p->1,θ->2}}]ContourPlot[θ,{x,-1,1},{y,0,1.5}]
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{u[x,y],v[x,y]}.u[x,y]+·{p[x,y],0}-Pr·[x,y],[x,y]
∇
{x,y}
∇
{x,y}
∇
{x,y}
(1,0)
u
(0,1)
u
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{u[x,y],v[x,y]}.v[x,y]-PrRaθ[x,y]+·{0,p[x,y]}-Pr·[x,y],[x,y]
∇
{x,y}
∇
{x,y}
∇
{x,y}
(1,0)
v
(0,1)
v
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(0,1)
v
(1,0)
u
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{u[x,y],v[x,y]}.θ[x,y]-·[x,y],[x,y]
∇
{x,y}
∇
{x,y}
(1,0)
θ
(0,1)
θ
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{u[x,y],v[x,y]}.u[x,y]+·{p[x,y],0}-0.71·[x,y],[x,y]0,{u[x,y],v[x,y]}.v[x,y]-710.θ[x,y]+·{0,p[x,y]}-0.71·[x,y],[x,y]0,[x,y]+[x,y]0,{u[x,y],v[x,y]}.θ[x,y]-·[x,y],[x,y]0
∇
{x,y}
∇
{x,y}
∇
{x,y}
(1,0)
u
(0,1)
u
∇
{x,y}
∇
{x,y}
∇
{x,y}
(1,0)
v
(0,1)
v
(0,1)
v
(1,0)
u
∇
{x,y}
∇
{x,y}
(1,0)
θ
(0,1)
θ
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{DirichletCondition[θ[x,y]1,-3.73x],DirichletCondition[θ[x,y]0,3.86-3.73x],DirichletCondition[{u[x,y]0,v[x,y]0},True],DirichletCondition[p[x,y]0,-3.73x||0.267x]}
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InterpolatingFunction[x,y],InterpolatingFunction[x,y],InterpolatingFunction[x,y],InterpolatingFunction[x,y]
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Show[RegionPlot[A,PlotStyle->LightBlue],Plot[{-3.73x,3.86-3.73x,0.267x},{x,-1,1},PlotStyle->{Red,Green,Blue}]]
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Show[RegionPlot[A,PlotStyle->LightBlue],Plot[-3.73x,{x,-1,1},PlotStyle->{Red}]]
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