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Fluid dynamical equations(Navier-Stokes equations)
€
independent variablesr r = (x,y,z) , t
dynamical variablesr u = (u,v,w) , ρ , p , T (or energy e, or entropy s)
(and for sea water, salinity η )
(and for moist air, water vapor)
Fluid dynamical equations(Navier-Stokes equations)
Dimensional reduction:from QM to Boltzmann eq.
to the fluid equations
Continuum hypothesis:
d
T
Mass conservation(continuity equation)
€
d
dtρ dV
V
∫ =∂ρ
∂tdV
V
∫ = − ρr u ⋅d
r A
A
∫
but
ρr u ⋅d
r A
A
∫ = ∇ ⋅ρr u ( ) dV
V
∫
thus
∂ρ
∂t+∇ ⋅ρ
r u = 0
€
rA
€
V
Mass conservation(continuity equation)
€
from
∂ρ
∂t+∇ ⋅ ρ
r u ( ) = 0
one can also write
Dρ
Dt+ ρ∇ ⋅
r u = 0
D
Dt=
∂
∂t+
r u ⋅∇
Incompressible fluid
€
Dρ
Dt= 0
and from
Dρ
Dt+ ρ∇ ⋅
r u = 0
∇ ⋅r u = 0
Conservation of linear momentum(Navier-Stokes equations)
€
ra =
r F /m (per unit volume)
ρD
r u
Dt≡ ρ
∂r u
∂t+
r u ⋅∇( )
r u
⎡
⎣ ⎢
⎤
⎦ ⎥= force per unit volume
force = body force + surface force
stress tensor for surface forces
commons.wikimedia.org/ wiki/File:Stress_tensor.png
The stress tensor is symmetric!
€
ρ Du
Dt= body force +
+τ xx (x + dx, y,z) − τ xx (x, y,z)
+τ yx (x, y + dy,z) − τ yx (x, y,z)
+τ zx (x, y,z + dz) − τ zx (x,y,z)
ρDu
Dt= body force +
∂ τ xx
∂ x+
∂ τ yx
∂ y+
∂ τ zx
∂ z
ρDu
Dt= −ρ g ˆ z +
∂ τ xx
∂ x+
∂ τ yx
∂ y+
∂ τ zx
∂ z
€
(x, y,z) ≡ (x1,x2, x3) ; (u,v,w) ≡ (u1,u2,u3)
τ ij = − p + 23 μ∇ ⋅
r u ( )δ ij + 2μ eij
eij ≡1
2
∂ ui
∂ x j
+∂ u j
∂ x i
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
constitutive equation(relationship between stress and strain)
for a Newtonian fluid(with the Stokes assumption)
Conservation of linear momentum(Navier-Stokes equations)
€
ρ Dui
Dt= −g ρ δi3 +
∂ τ ji
∂ x j
= −gρ δ i3 +∂ τ ij
∂ x j
ρDui
Dt= −
∂ p
∂ x i
− gρ δi3 +∂
∂ x j
2μ eij − 23 μ ∇ ⋅
r u ( )δ ij[ ]
for incompressible fluid
ρD
r u
Dt≡ ρ
∂r u
∂t+
r u ⋅∇( )
r u
⎡
⎣ ⎢
⎤
⎦ ⎥= −∇p − gρ ˆ z + μ∇ 2 r
u
Thermodynamic equation:first principle of Thermodynamics
€
dE + dW = dQ ⇒ dE + pdV = dQ ⇒ de + pdv = dq
De
Dt+ p
Dv
Dt≡
De
Dt+ p
D
Dt
1
ρ
⎛
⎝ ⎜
⎞
⎠ ⎟=
Dq
Dt= j
where
e specific energy : internal energy per unit mass
v =1
ρ specific volume : volume per unit mass
Till now:
€
six dynamical variables :
u,v,w, e, p, ρ
up to now, five equations
Dρ
Dt+ ρ∇ ⋅
r u = 0
ρD
r u
Dt= −∇p − g ρ ˆ z + Du
De
Dt+ p
D
Dt
1
ρ
⎛
⎝ ⎜
⎞
⎠ ⎟= j
Equations of state:
€
p = p ρ,T( )
e = e p,T( )
Perfect gas (e.g., dry air)
€
p = ρ RT R = 287 J kg−1 K−1
e = e(T) ⇒ d e = cV d T , cV =∂ e
∂ T
⎛
⎝ ⎜
⎞
⎠ ⎟V
Perfect gas (e.g., dry air)
€
p = ρ RT R = 287 J kg−1 K−1
d e = cV d T , cV =∂ e
∂ T
⎛
⎝ ⎜
⎞
⎠ ⎟V
cV
DT
Dt+ p
D
Dt
1
ρ
⎛
⎝ ⎜
⎞
⎠ ⎟= j ⇒ cV
DT
Dt−
p
ρ 2
Dρ
Dt= j
cV
DT
Dt+ R
DT
Dt−
1
ρ
D p
Dt= j ⇒ c p
DT
Dt−
1
ρ
D p
Dt= j
c p = cV + R
Fluid dynamical eqns. for a perfect gas
€
ρ= p
RT
Dρ
Dt+ ρ∇ ⋅
r u = 0
ρD
r u
Dt= −∇p − g ρ ˆ z + Du
c p
DT
Dt−
1
ρ
D p
Dt= j
The static solution
€
D
Dt= 0 ,
r u =
r 0
ρ =p
RT
0 = −∂ p
∂ z− g ρ
j = 0
hydrostatic solution
∂ p
∂ z= −g ρ
Adiabatic processes
€
j = 0
c p
DT
Dt−
1
ρ
D p
Dt= 0
c p
DT
Dt−
RT
p
D p
Dt= 0 ⇒
1
T
DT
Dt−
R
c p
1
p
D p
Dt= 0
D
Dtlog T p−R / c p
( ) = 0
θ = Tps
p
⎛
⎝ ⎜
⎞
⎠ ⎟
R / c p
potential temperature
Dθ
Dt= 0
Static stability of a perfect gas(adiabatic processes)
€
ρp
d2z
d t 2= −g ρ p − ρ h (z)( ) ⇒
d2z
d t 2= −g
ρ p − ρ h (z)
ρ p
ρ =p
RT
d2z
d t 2= −g
pp
Tp
−ph
Th (z)pp
Tp
= −g
1
Tp
−1
Th (z)1
Tp
= −gTh (z) − Tp
Th (z), pp = ph
θ = Tps
p
⎛
⎝ ⎜
⎞
⎠ ⎟
R / c p
d2z
d t 2= −g
θh (z) −θ p
θh (z)
Neutral stability of a perfect gas(adiabatic processes)
€
dθN
d z= 0
θ = Tps
p
⎛
⎝ ⎜
⎞
⎠ ⎟
R / c p
0 =d logθN
d z=
d logTN
d z−
R
c p
d log p
d z=
1
TN
d TN
d z−
R
c p
1
p
d p
d z
0 =d logθN
d z=
1
TN
d TN
d z+
R
c p
1
pN
ρ N g =1
TN
d TN
d z+
R
c p
1
pN
ρ N g
0 =d logθN
d z=
1
TN
dTN
d z+
g
c pTN
dTN
d z= −
g
c p
= −Γ
For a general, non adiabatic process
€
c p
DT
Dt−
1
ρ
D p
Dt= j =
Dq
Dt
c p
DT
Dt−
RT
p
D p
Dt=
Dq
Dt⇒ c pT
1
T
DT
Dt−
R
c p
1
p
D p
Dt
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟=
Dq
Dt
c pTD
Dtlogθ =
Dq
Dt, θ = T
ps
p
⎛
⎝ ⎜
⎞
⎠ ⎟
R / c p
c p
D
Dtlogθ =
1
T
Dq
Dt=
Ds
Dt
c pd logθ = ds
Relationship between potential temperature and entropy
Fluid dynamical eqns. for a perfect gas
€
ρ= p
RT
Dρ
Dt+ ρ∇ ⋅
r u = 0
ρD
r u
Dt= −∇p − g ρ ˆ z + Du
c p
DT
Dt−
1
ρ
D p
Dt= c pT
Dlogθ
Dt= j
θ = Tps
p
⎛
⎝ ⎜
⎞
⎠ ⎟
R / c p
Fluid dynamical eqns. for a perfect gas
€
θ =Tps
p
⎛
⎝ ⎜
⎞
⎠ ⎟
R / c p
= Tps
p
⎛
⎝ ⎜
⎞
⎠ ⎟
γ −1( ) /γ
⇒ T = θp
ps
⎛
⎝ ⎜
⎞
⎠ ⎟
γ −1( ) /γ
γ =c p
cv
ρ =p
RT→ ρ =
ps
Rθ
p
ps
⎛
⎝ ⎜
⎞
⎠ ⎟
1/γ
, ρθ =ps
Rθ
Dρ
Dt+ ρ∇ ⋅
r u = 0
ρD
r u
Dt= −∇p − g ρ ˆ z + Du
c pTDlogθ
Dt= j
The Boussinesq approximation(adiabatic process, ideal fluid)
€
ρ =ρ p,θ( ) for a perfect gas : ρ =ps
Rθ
p
ps
⎛
⎝ ⎜
⎞
⎠ ⎟
1/γ ⎧ ⎨ ⎪
⎩ ⎪
⎫ ⎬ ⎪
⎭ ⎪
Dρ
Dt=
∂ρ
∂p
Dp
Dt+
∂ρ
∂θ
Dθ
Dt=
1
c 2
Dp
Dt
c 2 =∂p
∂ρ
⎛
⎝ ⎜
⎞
⎠ ⎟
θ
The Boussinesq approximation(adiabatic process, ideal fluid)
€
ru ≡ (u,v,w) , ρ, p,θ
ρ = ρ 0 + ρ '(z) , p = p0 + p'(z)
ρ '= ρ (z) + ˜ ρ (x, y,z, t) , p'= p (z) + ˜ p (x,y,z, t)
∂p
∂z= −gρ
ρD
r u
Dt≡ ρ 0 + ρ '( )
∂
∂t+
r u ⋅∇
⎛
⎝ ⎜
⎞
⎠ ⎟r u = −∇p'−ρ 'g ˆ z
Dρ '
Dt+ ρ 0 + ρ '( )∇ ⋅
r u = 0
Dρ '
Dt=
1
c 2
Dp'
Dt
The Boussinesq approximation
€
ρ (z) , ˜ ρ (x,y,z, t) << ρ 0
ρ ≈p
c 2≈
ρ 0gH
c 2⇒ ρ << ρ 0 if H <<
c 2
g
˜ p ≈ ˜ ρ gH
1
c 2
D ˜ p
Dt≈
gH
c 2
D ˜ ρ
Dt<<
D ˜ ρ
Dtif H <<
c 2
g
sea water : c 2
g≈ 200 km
air : c 2
g≈10 km
€
ρ (z) , ˜ ρ (x, y,z, t) << ρ 0 if H <c 2
g
Dρ
Dt≈ 0
Dρ
Dt+ ρ∇ ⋅
r u = 0 → ∇ ⋅
r u = 0
ρ 0 + ρ '( )∂
∂t+
r u ⋅∇
⎛
⎝ ⎜
⎞
⎠ ⎟r u = −∇p − ρ g ˆ z
The Boussinesq approximation(adiabatic process, ideal fluid)
The Boussinesq approximation(real fluid)
€
ρ (z) , ˜ ρ (x, y,z, t) << ρ 0 if H <c 2
g
Dρ
Dt≈ 0
Dρ
Dt+ ρ∇ ⋅
r u = 0 → ∇ ⋅
r u = 0
ρ 0 + ρ '( )∂
∂t+
r u ⋅∇
⎛
⎝ ⎜
⎞
⎠ ⎟r u = −∇p − ρ g ˆ z + Du,0 + D'u
↓
ρ 0
∂
∂t+
r u ⋅∇
⎛
⎝ ⎜
⎞
⎠ ⎟r u = −∇p − ρ g ˆ z + μ0∇
2 r u
€
Dρ
Dt≈ 0 , ∇ ⋅
r u = 0
c p,0 +c'p( )DT
Dt−
1
ρ
D p
Dt= j = j0 + j '= −∇ ⋅
r Q 0 +
r Q '( )
↓
j = −∇ ⋅r Q ,
r Q = −k∇T Fourier law
r Q 0 = −k0∇T
DT
Dt= κ 0∇
2 T , κ 0 =k0
c p,0
The Boussinesq approximation(real fluid)
€
∇⋅ r
u = 0
∂
∂t+
r u ⋅∇
⎛
⎝ ⎜
⎞
⎠ ⎟r u = −
1
ρ 0
∇p −ρ
ρ 0
g ˆ z + ν 0∇2 r u
DT
Dt= κ 0∇
2 T
ρ = ρ(T)
The Boussinesq approximation(real fluid)
€
∇⋅ r
u = 0 ,DT
Dt= κ ∇ 2 T
∂
∂t+
r u ⋅∇
⎛
⎝ ⎜
⎞
⎠ ⎟r u = −
1
ρ 0
∇p −ρ
ρ 0
g ˆ z + ν 0∇2 r u
↓
∂
∂t+
r u ⋅∇
⎛
⎝ ⎜
⎞
⎠ ⎟r u = −
1
ρ 0
∇p'−ρ '
ρ 0
g ˆ z + ν 0∇2 r u
↓
ρ '= ρ − ρ 0 = −α ρ 0 T − T0( )
∂
∂t+
r u ⋅∇
⎛
⎝ ⎜
⎞
⎠ ⎟r u = −
1
ρ 0
∇p +α gT ˆ z + ν 0∇2 r u
The Boussinesq approximation(real fluid)
