Download Compressible Navier–Stokes Equations with Salinity

Compressible Navier–Stokes Equations with
Salinity Conservation and Free-Surface Boundary
Conditions for Mediterranean Eddies
Omid Khanmohamadi
October 2014
1
Compressible Ocean Model
The governing equations for the compressible ocean model are the compressible
(Euler–)Navier–Stokes1 equations (expressing conservation of mass, momentum,
and energy) together with the conservation of salinity equations
∂ρ ∂ρuj
+
∂t
∂xj
∂ρui uj
∂ρui
+
∂t
∂xj
∂ρe ∂ρeuj
+
∂t
∂xj
∂ρYs
∂ρYs uj
+
∂t
∂xj
= 0,
(1)
∂p
∂τij
− ρgi +
,
∂xi
∂xj
∂msj
∂puj
∂τij uj
∂qj
=−
− ρgj uj +
−
+
,
∂xj
∂xj
∂xj
∂xj
∂msj
=
,
∂xj
=−
(2)
(3)
(4)
1 Euler sits and measures the flow quantities (e.g., the momentum, ρu) at each point as the
fluid goes by. He sees only how fast the quantity is changing at that point as time passes,
∂ρu
. Lagrange follows the fluid particles and measures the flow quantities for them. For
∂t
each particle, he sees how fast the property is changing in total, Dρu
. They get different
Dt
results. Newton (and his second law, of which Navier–Stokes equation is an important special
case) accepts Lagrange’s measurements directly, because both of them work with particles,
and not points in space as Euler does. To be acceptable to Newton and equal to Lagrange,
Euler has to go from the quantity at a point to the quantity for a particle, by correcting for
how much of the quantity he has measured at a point is due just to the differences of the
quantity in the fluid (particles) flowing past, and not due to the quantity changing for any
particular particle that Largrange tracks. Euler does this correction by adding the nonlinear
convective term u · ∇(·), a spatial derivative. Consicely, Lagrange ≡ Euler + u · ∇(·), which
comes from “the total derivative is the sum of the partials” plus the chain rule and is called
Reynolds’ transport theorem. In our example, Dρu
≡ ∂ρu
+ u · ∇ρu. If Lagrange writes the
Dt
∂t
acceleration term more naturally, then why do we usually let Euler write the Navier–Stokes
equations? Compromise: Euler writes the viscous stress term more naturally whereas they
way Lagrange writes it is extremely unwieldy. Incidently, this is why fluid dynamics is hard:
Whichever coordinate system we choose there is a term that is not naturally representable in
that system.
1
where i, j = 1 : 3, s = 1 : 2, ρ is the density, p is the pressure, e is the total specific
energy, Ys and msj are the salinity and xj component of the mass diffusion flux
of the species s, τij is the ij component of the viscous stress tensor, and uj ,
gj , and qj are the xj components of the velocity, gravitational acceleration, and
heat flux, respectively. The total specific energy2 e is defined as
e :=
p
1
uj uj + cv T − ,
2
ρ
(5)
where cv is the heat capacity of water at constant volume. The viscous stress
τij is defined as
∂ui
∂uj
2 ∂uk
τij := µ
+
−
δij ,
(6)
∂xj
∂xi
3 ∂xk
where µ is the kinetic viscosity. The mass diffusion flux msj is given by Fick’s
first law as the scaled gradient of the salinity density,
msj := D
∂ρYs
,
∂xj
(7)
where D is the mass diffusivity.3 The heat flux qj is given by Fourier’s law as
the scaled gradient of the temperature,
qj := −κ
∂T
,
∂xj
(8)
where κ is the heat conduction constant.4 Summation over repeated (dummy)
indices (j and k) is implied.5
The system of equations (1–4) is not yet closed. It consists of seven equations (the conservation conditions for momentum (2) and salinity density (4)
contribute three and two equations, respectively) but involves eight unknowns,
namely the density ρ, pressure p, temparature T , components of momentum
ρui , and species salinity density ρYs . It addition to initial and boundary conditions, to close the system, we need an extra equation that relates p, ρ, and T .
This extra equation is usually called the equation of state.6
2 Specific energy is energy per unit mass; sometimes it is refered to as “energy density,” but
it is more appropriate to reserve the latter for referring to energy per unit volume (ρe in our
notation) just as the (mass) density ρ is used for referring to mass per unit volume.
3 The mass diffusivity D is proportional to the squared velocity of the diffusing particles of
saline water. It depends on the temperature and viscosity of the saline water and the size of
the diffusing particles, according to Stokes–Einstein relation.
4 We are using Fourier’s law to approximate both the heat flux and mass flux and then take
these approximations as our definitions of the corresponding quantity.
5 This is Einstein’s tensor notation.
6 In the case of an ideal gas, the equation of state has the simple form p = RρT , where R
is the gas constant. In our case, the fluid is the sea water and the equation of state is more
involved.
2
1.1
Temperature from Energy
If we assume for now that energy has no dependence on pressure and multiply
the specific energy (5) through by ρ we get the energy density E (i.e., energy
per unit volume)
E=
ρu2j
+ ρcv T,
2
(9)
which gives the temperature
T =
E−
(ρuj )2
2ρ
ρcv
E
(ρuj )2
=
−
.
ρcv
2ρ2 cv
(10)
From (10) we get
!
(ρu2j ) c0v
E
E0
(ρuj )2 ρ0
(ρuj )(ρuj )0
E ρ0
T =−
−
+
+
−
−
ρ
2ρ2
c2v
ρcv
cv ρ2
cv ρ3
ρ 2 cv
(ρuj )2 ρ0
(ρuj )(ρuj )0
c0
ρ0
−
/ρcv .
= −T v + E 0 − E +
cv
ρ
ρ ρ
ρ
0
2
(11)
Boundary Conditions
Assume that initially the ocean occupies the region between the fixed, impermeable ocean floor at y = −h and the equilibrium sea level at y = 0. After
leaving equilibrium, at any time t and position (x, z), the height of the water
(compared to y = 0) is given by η(x, z, t), where η is an unknown free-surface
(height) function. So, at any time t the ocean occupies the region
Ωt := { (x, y, z) | −h ≤ y ≤ η(x, z, t) }.
(12)
In other words, Ωt is our domain of interest and we refer to its boundaries using
the standard boundary operator ∂.
2.1
Stress-Free Boundary Conditions for Water–Air Interface
If modeled as stress free, the top boundary, henceforth denoted by ∂Ωt |top , im∂u
1
:= ∂u
poses the homogenous Neumann condition ∂n
∂n = 0. Since the momentum
(per unit volume) ρu and not the speed u is the field variable available to us,
∂u
we need to represent ∂n
in terms of ∂ρu
∂n . Staring from the product rule,
∂ρu
∂ρ
∂u
≡
u+ρ ,
∂n
∂n
∂n
3
(13)
we get
∂u
∂ρu
∂ρ
≡
−u
= 0 on ∂Ωt |top .
(14)
∂n
∂n
∂n
Since we are using 2nd-order finite differences as our discretization scheme
(centered in the interior of the domain and one-sided on the boundary), and
the order of accuracy of a finite-difference approximation to a derivative of a
function is independent of the regularity of that function, we can gain efficiency
without losing accuracy if we use a leading-order approximation to (14) by
perturbing the unknown functions involved (ρ and ρu) around their values at
the previous time step (denoted by the superscript (·)p ) and then keep only the
terms which are linear in the unknowns. In short, we linearize (14) around the
previous time step by starting from
ρ
∂ρu (ρu)p + (ρu)0
−
∂n
ρp + ρ0
∂ρu (ρu)p + (ρu)0
−
∂n
ρp
∂ρ0
∂ρp
+
∂n
∂n
ρ0
1− p +O
ρ
≡
ρ0
ρp
2 !! ∂ρp
∂ρ0
+
∂n
∂n
=0
on ∂Ωt |top ,
(15)
where (·)0 := (·) − (·)p denotes the perturbation of (·) around (·)p , and drop the
higher order terms in (15), compactly represented by the “big O ()” notation7 ,
to get
∂ρu (ρu)p ∂ρp
(ρu)0 ∂ρp
(ρu)p ∂ρp 0 (ρu)p ∂ρ0
−
− p
+ p 2
ρ −
≡
p
∂n
ρ
∂n
ρ
∂n
(ρ ) ∂n
ρp ∂n
(ρu)p ∂ρ
∂ρp
∂ρu (ρu)p ∂ρp ρu − (ρu)p ∂ρp (ρu)p ∂ρp
p
− p
−
+
(ρ−ρ
)−
−
≡
∂n
ρ
∂n
ρp
∂n (ρp )2 ∂n
ρp
∂n
∂n
(ρu)p ∂ρp ρ
∂ρ
∂ρu ρu ∂ρp
− p
+
−
= 0 on ∂Ωt |top . (16)
∂n
ρ ∂n
ρp
∂n ρp
∂n
The homogenous Neumann conditions on the temperature T and salinities
Ys (s = 1 : 2) imposed on the top boundary are derived in a similar fashion.
They are identical to (16), except that in the boundary condition for T we
replace ρu and (ρu)p with ρT and (ρT )p ; in that of Ys we replace them with
∂Ys
and with ρYs and (ρYs )p . That is, ∂T
∂n = 0 and ∂n = 0 are imposed with
∂ρT
ρT ∂ρp
(ρT )p ∂ρp ρ
∂ρ
− p
+
−
= 0 on ∂Ωt |top
(17)
∂n
ρ ∂n
ρp
∂n ρp
∂n
and
ρYs ∂ρp
(ρYs )p
∂ρYs
− p
+
∂n
ρ ∂n
ρp
∂ρp ρ
∂ρ
−
∂n ρp
∂n
= 0 ∂Ωt |top ,
(18)
7 f (x) = O (g(x)) (as x → a) is a slight abuse of notation to indicate that near a, f (x) is
within a multiple of g(x); more precisely, there is some constant C such that limx→a f (x) ≤
C limx→a g(x).
4
respectively.
The boundary condition for ρ on ∂Ωt |top is derived by considering the relation
p = a2 ρ − αs ρYs
(19)
between the pressure, density, and salinity, which leads to the relations
dp = a2 dρ − αs d(ρYs ),
αs
1
dρ = 2 dp + 2 d(ρYs ),
a
a
Dh ρ
1 Dh p αs Dh (ρYs )
= 2
+ 2
,
Dh t
a Dh t
a
Dh t
(20)
(21)
(22)
which, using the hydrostatic pressure relation
p = Patm + ρ0 gη,
(23)
becomes
1
αs Dh (ρYs )
Dh ρ
= 2 ρ0 gv + 2
Dh t
a
a
Dh t
on ∂Ωt |top ,
(24)
(25)
Dh
∂
where D
denotes the (Euler) temporal component ∂t
plus horizontal spatial
ht
∂
∂
hη
components u ∂x + w ∂z of the material derivative and v = D
Dh t on ∂Ωt |top . On
the other hand, from (4) and (7) we have
∂ρYs
∂ρYs
∂
∂Ys
∂uj
+ uj
+ ρYs
=
ρD
,
(26)
∂t
∂xj
∂xj
∂xj
∂xj
∂ρYs
s
+ u ∂ρY
∂x + w ∂z becomes
Dh ρYs
∂ρYs
∂uj
∂
∂Ys
+v
+ ρYs
=
ρD
,
Dh t
∂y
∂xj
∂xj
∂xj
which by putting
Dh ρYs
Dh t
:=
∂ρYs
∂t
(27)
which, after rearranging, finally yields
Dh ρYs
∂u ∂v
∂w
∂ρYs
∂
∂Ys
= −ρYs
+
+
−v
+
ρD
Dh t
∂x ∂y
∂z
∂y
∂xj
∂xj
∂ρYs v
∂
∂Ys
∂u ∂w
+
−
+
ρD
.
= −ρYs
∂x
∂z
∂y
∂xj
∂xj
(28)
Substituting (28) in (24) we get
∂ρ
∂ρ
∂ρ
+u
+w
=
∂t
∂x
∂z
1
αs ∂ρYs v
∂
∂Ys
αs ρYs ∂u ∂w
ρ
gy
−
+
−
+
ρD
0
a2
a2
∂x
∂z
a2 ∂y
∂xj
∂xj
on ∂Ωt |top .
(29)
5
2.2
2.2.1
Free-Surface Boundary Conditions for Water–Air Interface
Kinematic, Free-Surface Boundary Condition
Let us now change our model and consider free-surface conditions for the water–
air interface. At a fixed, impermeable boundary (such as the sea bottom) we
would require that the velocity of water normal to the boundary be zero. At a
moving free sufrace such as the water–air boundary the corresponding requirement is that the velocity of water normal to the boundary must be equal to the
velocity of boundary normal to itself.8 This condition is equivalent to requiring
that the material water elements on the free surface must remain on the free
surface. In other words, if at some time t a particular water particle identified
by its position vector (x(t), y(t), z(t)) is on the surface, meaning that x(t), y(t),
z(t), and t are related according to y(t) = η(x(t), z(t), t)), then it must remain
on the surface for all time t ≥ 0.9 In symbols, this implies
Dη
Dy
(t) =
(x(t), z(t), t)
Dt
Dt
for all t ≥ 0
(30)
if
y(t) = η(x(t), z(t), t)
for some t.
(31)
Expanding the material derivatives in (30), and suppressing the arguments, we
get
∂η
∂η
∂η
+u
+w
on ∂Ωt |top .
(32)
v=
∂t
∂x
∂z
Equation (32) is a kinematic boundary condition, since it prescribes motion at
the interface without considering the forces which cause that motion.
2.2.2
Dynamic, Free-Surface Boundary Condition
However, the kinematic boundary condition (32) is not enough by itself. It
expresses one unknown, namely the velocity normal to the free surface, v on
∂Ωt |top , in terms of another, namely the position of free surface, η. A force
balance on the free surface gives us another boundary condition: The water–air
interface has zero mass and hence it cannot support any force, according to
Newton’s Second Law (F = ma, where m is zero and hence so is F ). This
8 Otherwise either water would rise up faster than the boundary and flow through it or
the boundary would rise up faster than water and a vacuum would be left behind, neither of
which makes sense.
9 Notice how the free-surface height function η is being evaluated here for a particle, by
Lagrange! Using the argument (x(t), z(t), t) he is tracking the particle (since he has made x
and z functions of time t) and requires that it remains on the surface if it is there. Looking
back at our definition (12) of the region occupied by the ocean we should see that there η was
evaluated at (x, z, t), a fixed point in space at different times t, by Euler! We are abusing the
notation here (or there) slightly. Lagrange’s x and z are different from Euler’s.
6
implies that there must be no pressure jump at the water–air interface; pressure
is continuous at the interface:
p = Patm
on ∂Ωt |top ,
(33)
where Patm (assumed constant) is the atmospheric pressure at sea surface.10
Equation (33) is a dynamic boundary condition, since it prescribes force at the
interface (or more specifically force per unit area, namely pressure).
From a different point of view, we expect to have this dynamic boundary
condition for the pressure on the surface. Otherwise, since only the pressure
gradient (and not the pressure itself) appears in the Navier–Stokes equations,
we would only be able to determine the pressure up to an additive constant
(i.e., relatively) using our model, whereas in reality we are able to measure
the pressure at each point in the sea with no additive constants involved (i.e.,
absolutely).
2.2.3
Kinematic and Dynamic Free-Surface Boundary Conditions,
Combined
From the dynamic, free-surface boundary condition (33) and a hydrostatic assumption for pressure near the free surface we have
p = Patm + ρ0 gη + ∆ρgη,
(34)
which after neglecting the change in the density ∆ρ over the small height η
becomes
p = Patm + ρ0 gη,
(35)
where ρ0 is the density of sea water on the free surface. Equation (35) gives us
a clue how to eliminate the unknown η in the kinematic, free surface boundary
condition (32): After being multiplied by ρ0 g each partial derivative of η is the
same as the corresponding partial derivative of p. That is,
∂ρ0 gη
∂ρ0 gη
∂ρ0 gη
+u
+w
∂t
∂x
∂z
∂p
∂p
∂p
=
+u
+w
∂t
∂x
∂z
∂p ρu ∂p ρw ∂p
+
+
on ∂Ωt |top .
=
∂t
ρ ∂x
ρ ∂z
ρ0 gv =
(36)
10 In general, in addition to the external (atmospheric) pressure term P
atm there is a surface
tension term 2γH contributing to the stress on the free surface (γ is the surface tension of
water and H is the mean curvature of the free surface). At the length scales we are trying to
resolve the curvature is negligible and so is the surface tension term.
7