Graphics Reference
In-Depth Information
The key to connecting the two viewpoints is the material derivative.
We'll start with a Lagrangian description: there are particles with positions
x
and velocities
u
. Let's look at a generic quantity we'll call
q
: each particle
has a value for
q
.(Quantity
q
might be density, or velocity, or temperature,
or many other things.) In particular, the function
q
(
t, x
) tells us the value
of
q
at time
t
for the particle that happens to be at position
x
:thisisan
Eulerian variable since it's a function of space, not of particles. So how
fast is
q
changing for the particle that happens to currently be at
x
, i.e.,
the Lagrangian question? Just take the total derivative (a.k.a. the Chain
Rule):
dt
q
(
t, x
)=
∂q
dx
dt
d
∂t
+
∇
q
·
∂q
∂t
+
=
∇
q
·
u
Dq
Dt
.
≡
This is the material derivative!
Let's review the two terms that go into the material derivative. The
first is
∂q/∂t
, which is just how fast
q
is changing at that fixed point in
space, an Eulerian measurement. The second term,
u
, is correcting for
how much of that change is due just to differences in the fluid flowing past
(e.g., the temperature changing because hot air is being replaced by cold
air, not because the temperature of any molecule is changing).
Just for completeness, let's write out the material derivative in full,
with all the partial derivatives:
Dq
Dt
∇
q
·
=
∂q
∂t
+
u
∂q
∂x
+
v
∂q
∂y
+
w
∂q
∂z
.
Obviously in 2D, we can just get rid of the
w
-and
z
-term.
Note that I keep talking about how the quantity, or molecules, or parti-
cles, move with the velocity field
u
. Thisiscalled
advection
(or sometimes
convection
or
transport
; they all mean the same thing). An
advection equa-
tion
is just one that uses the material derivative, at its simplest setting it
to zero:
Dq
Dt
=0
,
∂q
∂t
+
u
i.e.,
·∇
q
=0
.
This just means the quantity is moving around but isn't changing in the
Lagrangian viewpoint.
