Thermal energy evolution equation
∂t∂U=−∇⋅q−∇⋅(Uv)−∇⋅(P⋅v)+v⋅(∇⋅P)
U is the thermal energy density.
t is the time.
q is the heat flux vector.
v is the bulk velocity.
P is the pressure tensor.
Continuity equation
∂t∂n+∇⋅(nv)=0
Where
n is the number density of a species.
Because
U=21(Pxx+Pyy+Pzz)
We consider the kinetic temperature
T
T=31nPxx+Pyy+Pzz
Then, the relationship between thermal energy
U and temperature
T
U=23nT
Then
∂t∂U=∂t∂(23nT)=23T∂t∂n+23n∂t∂T=23n∂t∂T−23T∇⋅(nv)
And
∇⋅(P⋅v)=v⋅(∇⋅P)+(P⋅∇)⋅v
Then
∂t∂U=−∇⋅q−∇⋅(Uv)−(P⋅∇)⋅v
And
∇⋅(Uv)=∇⋅(23nTv)=23T∇⋅(nv)+23nv⋅∇T
Then
23n∂t∂T−23T∇⋅(nv)=−∇⋅q−23T∇⋅(nv)−23nv⋅∇T−(P⋅∇)⋅v
23n∂t∂T=−∇⋅q−23nv⋅∇T−(P⋅∇)⋅v
Then
23n(∂t∂T+v⋅∇T)=−∇⋅q−(P⋅∇)⋅v
Let
P=P−pI+pI=P′+pI
Where
I is the unit tensor, and
p is the scalar pressure.
Then
23n(∂t∂T+v⋅∇T)=−∇⋅q−(P′⋅∇)⋅v−p(I⋅∇)⋅v
Finally, the energy density conservation equation is
23n(∂t∂T+v⋅∇T)+p∇⋅v=−∇⋅q−(P′⋅∇)⋅v