Aula Teórica 7
Volume de Controlo e fluxos.
Formulação Euleriana e Lagrangeana
Resultant force applied over a volume
of fluid
 
yx
 
xy
y
 u 

  
 y  y
y  dy
 u 

  
  y  y  dy
Peso   g  dxdydz

General movement equation

du i
dt

p
xi
 ui
2

x
2
j
 g i
2
 u


u

ui

p
i
i


 
uj

 g i
2
 t

dt

x
xi
x j
j 

du i
This equation holds for a material system with a unit of mass. It is written in a Lagrangian
Formulation, i.e. one has to follow that portion of fluid in order to describe its velocity.
That is not easy for us since we use to be in a fix place observing the flow, i.e. we are in
an Eulerian reference .
Lagrangian vs Eulerian Descriptions
• Both describe time derivatives.
• Lagrangian approach describes the rate of change
of a property in a material system, i.e. follows
material as it moves.
• Eulerian describe the rate of change in one point
of space.
• Lagrangian derivative registers changes
independently of velocity. Eulerian registers
changes also if fluid moves.
• In stationary systems local production balances
transport.
Case of velocity
•
•
•
•
•
•
•
Is there acceleration (rate of change of the velocity of a material system)?
How do momentum flux change between entrance and exit?
If the flow is stationary what is the local velocity change rate?
How does momentum inside the control volume change in time?
How does pressure vary along the flow?
What is the relation between momentum production and the divergence of
momentum fluxes?
What is the interest of using a lagrangian and an eulerian description?
Concentration
Fecal Bacteria dies in the environment according to a first order decay, i.e. the
number of bacteria that dies per unit of time is proportional to existing
bacteria. This process is describe by the equation:
dc
  kc
dt
C
c  c0e
 kt
C0
t
This is a lagrangian
formulation. This
solution describes what
is happening inside a
water mass whether is
moving or not.
What happens in an
Eulerian description?
Eulerian description
Let’s consider a river where the contaminated water
would be moving as a patch (without diffusion)
t4
t1
X1
X2
t3
t2
C
X1
X2
t
Concentration decays as the patch
moves. Time series in points x1 and x2
would be:
Maximum concentration difference
depends on decay rate while difference in
time to show up and showing time reduce
as flow velocity increases.
c
t
 u
c
x
 kc
Lagrangian vs Eulerian
• Examples of videos illustrating the difference
between eulerian and lagrangian descriptions
(not always very clear)
http://www.youtube.com/watch?v=mdN8OOkx2ko&feature=related
http://www.youtube.com/watch?v=zk_hPDAEdII&feature=related
Reynolds Theorem
• The rate of change of a property inside a
material system is equal to the rate of change
inside the control volume occupied by the
fluid plus what is flowing in, minus what is
flowing out.
d
dt
  dVol
sistema

d
dt
  dVol
VC


SC
 
 v .n dS
Demonstration of Reynolds Theorem
Let’s consider a conduct and 3
portions fluid (systems), SYS 1,
SYS3 and SYS 3 that are moving.
Time = t
SYS 1
CV
SYS 2
SYS 3
Let’s consider a space control
volume (not moving) that at time
“t” is completed filled by the
fluid SYS 2
Time = t+∆t
SYS 1
CV SYS 2
SYS 3
Between time= t and time =(t+∆t)
inside the control volume
properties changed because
some fluid flew in (SYS1) and
other flew out (SYS2) and also if
properties of those systems have
changed.
Rates of change
 B SYS 
In a material system:
  B SYS
t  t

t
t
Inside the control volume:
 B vc 
t  t
  B vc 
t
t
SYS 2 was coincident with CV at time t:
 B vc 
t
  B SYS 2 
t
At time t+∆t:
 B vc 
t  t
  B SYS 2 
t  t
 flowin  outflow
Computing the budget per unit of time and
using the specific property (per unit of volume)
 B vc t   t   B vc t
t
 
B 
dB
dV
  dV

 B SYS 2 t   t   B SYS 2 t
t
 flowin  outflow
How much is flowing in and out?
 adv B 

 
 v .n dA
The volumetric discharge is the integral
of the volume flowing per unit of area
integrated over the area.
The Mass discharge is the integral of
the mass flowing per unit of area
integrated over the area.
The Mass flowing per unit of area is the
volume per unit of area times the mass
per unit of volume.
 adv B 

 
 v .n dA
Using flux and specific property definitions
integral equations can be written
 
dB
  dV
B 
dV
 B vc t   t   B vc t
t



 dV
 

 B SYS 2 t   t   B SYS 2 t
t





t  t
d
dt

    dV


t




t


 dV

t
  dV
If material is flowing in the internal product is negative and if is flowing
out is positive. As a consequence:
flowin  outflow
  
 
  v .n dA
And finally

t

 dV 
vc
d
dt

 dV 
sistema

 
  v .n dA
surface
Or:
d
dt
  dV
system



t
vc
 
 dV     v .n dA
surface
If the Volume is infinitesimal

 dV

t
d

vc
dt

 
 dV     v .n dA
system
surface
Becomes:

t
  V  
d
dt
 
   V     v .n  Aentrance
 
   v .n  A exit
But:
d ( V )
dt
 V
d(  )
dt

d ( V )
dt
And thus:
 x1  x 2  x 3

t
  x1  x 2  x 3
 V
d 
 x 2  x 3   v1  x 1   x 2  x 3   v1  x

dt
1   x1
 x1  x 3   v 2  x 2   x1  x 3   v 2  x
2
  21
 x1  x 2   v 3  x 3   x1  x 2   v 3  x   x
3
3
Dividing by the volume:
d(  )
dt
 V
   x1  x 2  x 3


u k
x k
u k
xk

Derivada total

t
d
dt
d
dt

d
dt


t

 

v k
xk
 v j

t

x j

x j

v
 v 
j
v k
x k

j
x j
The Total derivative is the rate of change in a material system (Lagrangian description) ;
The Partial derivative is the rate of change in a control volume (eulerian description) ;
The advective derivative account for the transport by the velocity.