Thermal energy evolution equation
∂U∂t=−∇⋅q−∇⋅(Uv)−∇⋅(P⋅v)+v⋅(∇⋅P)
\frac{\partial U}{\partial t}=-\bm{\nabla \cdot q}-\bm{\nabla \cdot} (U\bm{v})-\bm{\nabla \cdot }(\bm{{\rm P}\cdot v}) + \bm{v \cdot}(\bm{\nabla \cdot {\rm P}})
∂t∂U=−∇⋅q−∇⋅(Uv)−∇⋅(P⋅v)+v⋅(∇⋅P)
UUU is the thermal energy density. ttt is the time. q\bm{q}q is the heat flux vector. v\bm{v}v is the bulk velocity. P\bm{\rm P}P is the pressure tensor.
Continuity equation
∂n∂t+∇⋅(nv)=0
\frac{\partial n}{\partial t} + \bm{\nabla \cdot} (n \bm{v}) = 0
∂t∂n+∇⋅(nv)=0
Where nnn is the number density of a species.
Because
U=12(Pxx+Pyy+Pzz)
U=\frac{1}{2}(P_{xx}+P_{yy}+P_{zz})
U=21(Pxx+Pyy+Pzz)
We consider the kinetic temperature TTT
T=13Pxx+Pyy+Pzzn
T=\frac{1}{3}\frac{P_{xx}+P_{yy}+P_{zz}}{n}
T=31nPxx+Pyy+Pzz
Then, the relationship between thermal energy UUU and temperature TTT
U=32nT
U=\frac{3}{2}nT
U=23nT
Then
∂U∂t=∂∂t(32nT)=32T∂n∂t+32n∂T∂t=32n∂T∂t−32T∇⋅(nv)
\frac{\partial U}{\partial t} = \frac{\partial}{\partial t}(\frac{3}{2}nT)=\frac{3}{2}T\frac{\partial n}{\partial t} + \frac{3}{2}n\frac{\partial T}{\partial t}=\frac{3}{2}n\frac{\partial T}{\partial t}-\frac{3}{2}T\bm{\nabla \cdot}(n \bm{v})
∂t∂U=∂t∂(23nT)=23T∂t∂n+23n∂t∂T=23n∂t∂T−23T∇⋅(nv)
And
∇⋅(P⋅v)=v⋅(∇⋅P)+(P⋅∇)⋅v
\bm{\nabla \cdot }(\bm{{\rm P} \cdot v})=\bm{v \cdot } (\bm{\nabla \cdot {\rm P}})+(\bm{{\rm P}\cdot \nabla})\bm{\cdot v}
∇⋅(P⋅v)=v⋅(∇⋅P)+(P⋅∇)⋅v
Then
∂U∂t=−∇⋅q−∇⋅(Uv)−(P⋅∇)⋅v
\frac{\partial U}{\partial t}=-\bm{\nabla \cdot q}-\bm{\nabla \cdot} (U\bm{v})-(\bm{{\rm P}\cdot \nabla})\bm{\cdot v}
∂t∂U=−∇⋅q−∇⋅(Uv)−(P⋅∇)⋅v
And
∇⋅(Uv)=∇⋅(32nTv)=32T∇⋅(nv)+32nv⋅∇T
\bm{\nabla \cdot} (U\bm{v})=\bm{\nabla \cdot} (\frac{3}{2}nT\bm{v})=\frac{3}{2}T\bm{\nabla \cdot}(n\bm{v})+\frac{3}{2}n\bm{v \cdot \nabla}T
∇⋅(Uv)=∇⋅(23nTv)=23T∇⋅(nv)+23nv⋅∇T
Then
32n∂T∂t−32T∇⋅(nv)=−∇⋅q−32T∇⋅(nv)−32nv⋅∇T−(P⋅∇)⋅v
\frac{3}{2}n\frac{\partial T}{\partial t}-\frac{3}{2}T\bm{\nabla \cdot}(n \bm{v}) = -\bm{\nabla \cdot q} - \frac{3}{2}T\bm{\nabla \cdot}(n\bm{v}) - \frac{3}{2}n\bm{v \cdot \nabla}T - (\bm{{\rm P}\cdot \nabla})\bm{\cdot v}
23n∂t∂T−23T∇⋅(nv)=−∇⋅q−23T∇⋅(nv)−23nv⋅∇T−(P⋅∇)⋅v
32n∂T∂t=−∇⋅q−32nv⋅∇T−(P⋅∇)⋅v \frac{3}{2}n\frac{\partial T}{\partial t} = -\bm{\nabla \cdot q} - \frac{3}{2}n\bm{v \cdot \nabla}T - (\bm{{\rm P}\cdot \nabla})\bm{\cdot v} 23n∂t∂T=−∇⋅q−23nv⋅∇T−(P⋅∇)⋅v
Then
32n(∂T∂t+v⋅∇T)=−∇⋅q−(P⋅∇)⋅v
\frac{3}{2}n(\frac{\partial T}{\partial t} + \bm{v \cdot \nabla}T) = -\bm{\nabla \cdot q} - (\bm{{\rm P}\cdot \nabla})\bm{\cdot v}
23n(∂t∂T+v⋅∇T)=−∇⋅q−(P⋅∇)⋅v
Let
P=P−pI+pI=P′+pI
\bm{{\rm P=P}}-p\bm{{\rm I}}+p\bm{{\rm I}} = \bm{{\rm P'}}+p\bm{{\rm I}}
P=P−pI+pI=P′+pI
Where I\bm{\rm I}I is the unit tensor, and ppp is the scalar pressure.
Then
32n(∂T∂t+v⋅∇T)=−∇⋅q−(P′⋅∇)⋅v−p(I⋅∇)⋅v
\frac{3}{2}n(\frac{\partial T}{\partial t} + \bm{v \cdot \nabla}T) = -\bm{\nabla \cdot q} - (\bm{{\rm P'}\cdot \nabla})\bm{\cdot v} - p(\bm{{\rm I}\cdot \nabla})\bm{\cdot v}
23n(∂t∂T+v⋅∇T)=−∇⋅q−(P′⋅∇)⋅v−p(I⋅∇)⋅v
Finally, the energy density conservation equation is
32n(∂T∂t+v⋅∇T)+p∇⋅v=−∇⋅q−(P′⋅∇)⋅v
\frac{3}{2}n (\frac{\partial T}{\partial t}+\bm{v} \bm{\cdot \nabla}T)+p\bm{\nabla \cdot v}=-\bm{\nabla \cdot q}-(\bm{{\rm P'} \cdot \nabla})\bm{\cdot v}
23n(∂t∂T+v⋅∇T)+p∇⋅v=−∇⋅q−(P′⋅∇)⋅v
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