Aula Teórica 6&7
Princípio de Conservação e Teorema
de Reynolds.
Derivada total e derivada convectiva
Princípio de conservação
• A Taxa de acumulação no interior de um volume
de controlo é igual ao que entra menos o que sai
mais o que se produz menos o que se
destrói/consome.
– A propriedade pode entrar por advecção ou por
difusão.
– Os processos de produção/consumo são específicos da
propriedade (e.g. Fitoplâncton cresce por fotossíntese,
o zoo consome outros organismos a quantidade de
movimento é produzida por forças).
Control Volumne and accumulation
rate
Taxa de acumulação da propriedade B:
(Taxa de variação da propriedade )
Definindo a propriedade específica
“Beta” :
  dV 
t0  t

t
B vc 
t0  t
 B vc
t
B 
  dV 
t0
  dV

t0
Fluxo advectivo
 adv
B
  
 
 v .n dA
• No caso de a propriedade ser uniforme nas faces:
n
 
Q i    v .n dA
 adv B    i Q i
i 1
Ai
• Se a velocidade for uniforme em cada face:
Q i  U iAi
Fluxo Difusivo
 dif
B




    .n dA 
 



    .n dA
 
• No caso de o gradiente da propriedade ser uniforme
nas faces:
n
 dif
B


i 1
  dir   esq
 Ai 
l





E a equação de evolução fica:
  dV 
t0  t

  dV 
t0
t


 
 v .n dA 



    .n dA


• Se as propriedades forem uniformes nas faces e
no volume (volume infinitesimal):
 V 
t0  t
  V
t

t0

Q
i
i

  A
 l  l   l
• Que é a forma algébrica do princípio de
conservação
l
Forma diferencial
 V 
t0  t
  V
t

t0

Q
i
i

  A
 l  l   l
l
V   x y z
z
Az   x y
z
Ax   y z
A y   x z
Qi  ui A
  t
0
x
 t
x
y
 
t
t
0
 x y z   u  y z x   u  y z x  x 
  v x z  y   v x z  y  y   w  x y z   w  x y z  z 
 x  x   x 
 x  x   x 




y

z




y

z






x
x

x 
 x  x
 y  y   y 
 y  y   y 


   x z
     y z







y

y

y 
 y  y
 z  z   z 
 z  z   z 


   x y
     z y


z

z

z 
 z  z
y
Dividindo pelo volume (1)
  t
0
 t
 
t
t
0

 u  y z x   u  y z x  x
 x y z
  v x z  y   v x z  y  y
 x y z


 w  x  y z   w  x y z  z
 x y z
 x  x   x 
 x  x   x 


   y z
     y z


x

x

x 
 x  x
 x y z
 y  y   y 
 y  y   y 


   x  z 
     x  z 

y
y

y 
 y  y
 x y z
 z  z   z 
 z  z   z 


   x y
     z y

z
z

z 
 z  z
 x y z



Dividindo pelo volume (2)
  t
0
 t
 
t
t
0

  v  y   v  y  y
y
 u x   u x  x
x


 w z   w z  z
y

 x  x   x 
 x  x   x 



  

x
x

x 
 x  x
x
 y  y   y 
 y  y   y 


  
    

y
y

y 
 y  y
y


 z  z   z 
 z  z   z 











z
z

z 
 z  z
z
Fazendo o volume tender para zero, obtém-se uma equação diferencial.
Fazendo tender o volume para zero


t

t

t

dt
x j
 u j
x j
uj
d
 u j


x j


 
 


 x j   x j 
 
 


 x j   x j 

 
 
 
u j


 x j   x j 
x j
 
u j


 x j   x j 
x j
Divergência da velocidade. Nula
em incompressível. Se positiva
o volume do fluido aumenta.
Questions
• The divergence of the velocity is the rate of
expansion of a volume?
• Let’s consider a volume of fluid in a flow with positive velocity divergence
y
V)y+dy
u j
u)x
u)x+dx
dy
V)y
 u1
x
In case of this figure the
volume would increase.
x j
 x1

 u1
 x1

u 2
x2

u3
x3
Is the rate of increase of
distance between faces
normal to xx axis. The
same for other axis.
Questions
• The rate change of a property conservative
property is the symmetrical of the flux
divergence?

t

 u j
x j

 
 


 x j   x j 
The functions being derivate are the advective flux and the diffusive flux
per unit of area. The operators are divergences of the fluxes.
If the fluid is incompressible, the
velocity divergence is null

t

 u j
x j
u j
x j
 u j
x j
 
0
u j
x j

 
 


 x j   x j 
uj

x j
The diffusivity of the specific mass is
zero!
• That is a consequence of the definition of
velocity.
• Velocity was defined as the net budget of
molecules displacement.
• When molecules move they carry their own
mass and consequently the advective flux
accounts for the whole mass transport.
Trabalho computacional
• Caso unidimensional, só com difusão:
  V t
0
 t
  V
t
  t
0
 t
 
t
t
0
t
0

Q
i
i

  A
 l  l   l
l

 l  x   x 
 l  x   x 
1 


    A

  A 

 x 
x

x
x 
 x  x 
Referencial Euleriano e Lagrangeano
• O refencial Euleriano estuda uma zona do
espaço (volume de controlo fixo)
• O referencial Lagrangeano estuda uma porção
de fluido “Sistema” (volume de controlo a
mover-se à velocidade do fluido).
• O Teorema de Reynolds relaciona os dois
referenciais.
Teorema de Reynolds
• A taxa de variação de uma propriedade num
“sistema de fluido” é igual à taxa de variação
da propriedade no volume de controlo
ocupado pelo fluido mais o fluxo que entra,
menos o que sai:
d
dt
  dVol
sistema

d
dt
  dVol
VC
• (ver capítulo 3 do White)


SC
 
 v .n dS
Sistema e Volume de Control
Control Volume
Volume that
flew in
Volume that
flew out
Taxas de Variação
B sistemaI 
No sistema material de fluido
t0  t
 B sistemaI

t0
t
B vc 
No volume de controlo
t0  t
 B vc

t0
t
No instante inicial o sistema era coincidente com o volume de controlo
B vc 
t0
 B sistema
2

t0
A figura permite relacionar o VC em t+dt:
B vc  t
0
 t
 B sistema
t

2
0
 t
 massa _ que _ entra  massa _ que _ sai
Fazendo o Balanço por unidade de tempo e usando a
definição de propriedade específica (valor por
unidade de volume)
 B vc 
t0  t
  B vc

t0

t
B
sistema
t

2
0
 t
 B sistema
t

2
0
t
 
dB
dV
=>
B 
  quantidade
  dV
_ que _ entra  quantidade
t
_ que _ sai
Fluxo advectivo
 adv B 

 
 v .n dA
Where v velocity relative to the surface. Is the flow
velocity if the volume is at rest.
Balanço integral

t

vc
 dV 
d
dt

sistema
 dV 

 
  v .n dA
surface
The rate of change in the Control Volume is equal to the rate of
change in the fluid (total derivative) plus what flows in minus what
flows out.
Volume infinitesimal

t

t

 dV 
vc
  V  
 x1 x 2  x 3
d
dt
d
dt

 dV 
sistema

 
  v .n dA
surface
 
   V     v .n  Aentrada
 
   v .n  A saida

 d V 

   x 2  x 3   v 1 x 1   x 2  x 3   v 1 x 1   x 1 
t
 t 
 x 1  x 3   v 2 x 2   x 1  x 3   v 2 x
Dividing by the volume,
2
  21
  x 1  x 2   v 3 x 3   x 1  x 2   v 3 x
3
 x 3
Derivada total
 d   V    v 1  x 1    v 1  x1   x1   v 2  x 2    v 2  x 2   2 1   v 3  x 3    v 3  x 3   x 3




t
 x1
x2
 x3
 t 


Shrinking the volume to zero,
But,

t
1 d (V )
V
d
dt

dt
d
dt

t

 vj

t

V d ( )
V

x j
dt

v

v j
x j
V
x j
dt
 d (V )
V


j
1 d (V )
vk
xk
dt



x j
d ( )
dt
 v 
j

u k
xk
Questions
• The velocity of an incompressible fluid in a
contraction must increase and consequently
the pressure must decrease

t
  dV   
  V t
0
 t
 V
t
0
t
uA  x
 
 v .n dA 



    .n dA


   uA  x    uA  x   x
 uA  x   x
u 2  u1
A1
A2
If the velocity increases the acceleration is
positive and so is the applied force.
In a pipe pressure forces plus gravity
forces balance friction forces
• If we consider a control volume (e.g.
with faces perpendicular and parallel to
the velocity it is easy to verify that
acceleration is zero and that forces have
to balance.
• Is the velocity profile a parabola?
•
Let’s consider a “annular control volume” and
perform a force balance
u 
u 


  p ( 2  r  r )  ( 2  r  r )  x  g sin     
2

r

x



2   r   r  x



r  r
 r  r  dr



p
x
  gsen  
 1
1 
u 
u 
u 
















 r 
r  r 
r  r  r  r 
r  r  r
 1
1 
u 
u 
u 


  gsen  















x
 r 
 r  r 
r  r  r  r 
r  r  r
p
• Fazendo convergir o volume para zero:

dp
  gsen  
dx
  u  1  u 

 

r  r  r  r 
1    u     u  1  u 
 r  
  

 

r r   r   r  r  r  r 
dp
1   u 

  gsen  
r 

dx
r r  r 
  u   dp
r
r 




gsen

 C1
 

  r   dx
 2
2
 u 
  

 r 
C1
 dp
r



gsen




r
 dx
 2
• When r is zero the velocity gradient is zero, friction is zero and thus C1
must be zero:
C
  u   dp
r
  
  gsen    1
  
r
  r   dx
 2
u
r

1  dp
r
  gsen  

  dx
2
2
1  dp
r
u  
  gsen  
 C2
  dx
 4
When r=R, velocity is zero and thus
2
1  dp
R
0  
  gsen  
 C2
  dx
 4
2
1  dp
R
C2    
  gsen  
  dx
 4
2

1  dp
r
 2    
u
  gsen   R 1   

4   dx
   R  
Friction and pressure loss in a pipe.
 p  R   w 2  RL
2
p
L

dp
dx
 2 w
4f 
 V 

dx
2
 D 
dp
1
2
About the flow in a pipe
• The velocity profile is a parabola.
• The shear stress is linear.
• The velocity decreases with viscosity and
increases with the radius square and linearly
with the pressure gradient and the gravity.
• Gravity action is equivalent to pressure
gradient action.
Summary
• The conservation principle drives to the advection-diffusion
equation.
• The total derivative represents the rate of change of a portion of
fluid while it is moving. The local temporal derivative represents the
rate of change of a property in a fixed point of the space.
• The laws of physics apply to a portion of fluid. They are responsible
for source and sink terms to be added to the advection diffusion
equation that then becomes a conservation equation.
• The relation between what happens inside a volume of fluid and
what happens inside a fixed volume are the fluxes across its
boundaries.
• The convective derivative represents the contribution of the
transport for what happens in a fixed point.