aula teórica 6&7 princípio de conservação e teorema de reynolds. derivada total e derivada...
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Aula Teórica 6&7
Princípio de Conservação e Teorema de Reynolds.
Derivada total e derivada convectiva
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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).
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Control Volumne and accumulation rate
t
BB 00 tvc
ttvc
dVB
Taxa de acumulação da propriedade B: (Taxa de variação da propriedade )
Definindo a propriedade específica “Beta” :
t
dVdVttt
00
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Fluxo advectivo
• No caso de a propriedade ser uniforme nas faces:
• Se a velocidade for uniforme em cada face:
dAnvadvB .
i
n
iiB Qadv
1
dAnvQiA
i .
iii AUQ
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Fluxo Difusivo
• No caso de o gradiente da propriedade ser uniforme nas faces:
dAndAndifB ..
n
i
esqdiriB lAdif
1
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E a equação de evolução fica:
• Se as propriedades forem uniformes nas faces e no volume (volume infinitesimal):
dAndAnv
t
dVdVttt
..
00
l
AQt
VV lllii
ttt
00
• Que é a forma algébrica do princípio de conservação
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Forma diferencial
lAQ
t
VV lllii
ttt
00
AuQ
zxA
zyA
yxA
zyxV
ii
y
x
z
x
y
z
y
z
x
zz
zzz
z
zzz
yy
yyy
y
yyy
xx
xxx
x
xxx
zzzyyy
xxx
ttt
zyz
zyx
yzy
yzx
xzy
xzy
yxwyxwzxvzxv
zyuzyuzyxt
00
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Dividindo pelo volume (1)
zyx
zyz
zyx
zyx
yzx
yzx
zyx
xzy
xzy
zyx
yxwyxw
zyx
zxvzxv
zyx
zyuzyu
t
zz
zzz
z
zzz
yy
yyy
y
yyy
xx
xxx
x
xxx
zzzyyy
xxxttt
00
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Dividindo pelo volume (2)
z
zz
y
yy
x
xx
y
ww
y
vvx
uu
t
zz
zzz
z
zzz
yy
yyy
y
yyy
xx
xxx
x
xxx
zzzyyy
xxxttt
00
Fazendo o volume tender para zero, obtém-se uma equação diferencial.
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Fazendo tender o volume para zero
jjj
j
xxx
u
t
j
j
jjjj
jjj
j
x
u
xxxu
t
xxx
u
t
j
j
jj x
u
xxdt
d
Divergência da velocidade. Nula em incompressível. Se positiva o volume do fluido aumenta.
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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
x
y
V)y
V)y+dy
dy
u)x+dx
u)x3
3
2
2
1
1
x
u
x
u
x
u
x
u
j
j
1
1
x
u
Is the rate of increase of
distance between faces normal to xx axis. The same for other axis.
In case of this figure the volume would increase.
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Questions
• The rate change of a property conservative property is the symmetrical of the flux divergence?
jjj
j
xxx
u
t
The functions being derivate are the advective flux and the diffusive flux per unit of area. The operators are divergences of the fluxes.
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If the fluid is incompressible, the velocity divergence is null
0
j
j
jj
j
j
j
j
jjj
j
x
u
xu
x
u
x
u
xxx
u
t
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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.
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Trabalho computacional
• Caso unidimensional, só com difusão:
l
AQt
VV lllii
ttt
00
xx
xxl
x
xxlttt
xA
xA
xt
100
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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.
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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:
• (ver capítulo 3 do White)
dSnvdVoldt
ddVol
dt
d
VC SCsistema
.
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Sistema e Volume de Control
Volume that flew in Volume that
flew out
Control Volume
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Taxas de Variação
t
BB 00 tsistemaI
ttsistemaI
t
BB 00 tvc
ttvc
00 t2sistema
tvc BB
sai_que_massaentra_que_massaBB tt2sistema
ttvc
00
No sistema material de fluido
No volume de controlo
No instante inicial o sistema era coincidente com o volume de controlo
A figura permite relacionar o VC em t+dt:
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Fazendo o Balanço por unidade de tempo e usando a definição de propriedade específica (valor por unidade de volume)
t
BB tvc
ttvc
00
t
sai_que_quantidadeentra_que_quantidade
t
BB 00 t2sistema
tt2sistema
dB
dV dVB =>
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Fluxo advectivo
dAnvadvB .
Where v velocity relative to the surface. Is the flow velocity if the volume is at rest.
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Balanço integral
dAn.vdVdt
ddV
t surfacesistemavc
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.
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Volume infinitesimal
saidaentrada AnvAnvVdt
dV
t
..
dAn.vdVdt
ddV
t surfacesistemavc
3312
11
321332122312231
132113221 3
xxxxx
xxx
vxxvxxvxxvxx
vxxvxxtVd
txxx
Dividing by the volume,
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Derivada total
3
333
2
2222
1
111 331211
x
vv
x
vv
x
vv
t
Vd
txxxxxxxx
jj
vxdt
Vd
Vt
)(1
k
k
j
j
jj x
v
x
v
xv
tdt
d
jj xv
tdt
d
Shrinking the volume to zero,
k
k
x
u
dt
d
dt
Vd
Vdt
d
V
V
dt
Vd
V
)()()()(1
But,
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Questions
• The velocity of an incompressible fluid in a contraction must increase and consequently the pressure must decrease
xxx
ttt
uAuAt
VV
00
dAndAnvdVt
..
2
112 A
Auu
uAuA xxx
If the velocity increases the acceleration is positive and so is the applied force.
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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?
xrrr
uxr
r
ugxrrrrp
drrr
22sin)2()2(
• Let’s consider a “annular control volume” and perform a force balance
rrrrr r
u
rr
u
r
u
rgsen
x
p
11
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• Fazendo convergir o volume para zero:
rrrrr r
u
rr
u
r
u
rgsen
x
p
11
r
u
rr
u
rgsen
dx
dp 1
r
u
rr
u
rr
ur
rr 11
r
ur
rrgsen
dx
dp 1
r
Crgsen
dx
dp
r
u
Cr
gsendx
dp
r
ur
1
1
2
2
2
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• When r is zero the velocity gradient is zero, friction is zero and thus C1 must be zero:
r
Crgsen
dx
dp
r
u 1
2
2
2
4
1
2
1
Cr
gsendx
dpu
rgsen
dx
dp
r
u
When r=R, velocity is zero and thus
22
2
2
2
2
14
1
4
1
4
10
R
rRgsen
dx
dpu
Rgsen
dx
dpC
CR
gsendx
dp
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Friction and pressure loss in a pipe.
D
fV
dx
dpdx
dp
L
p
RLRp
w
w
4
2
1
2
2
2
2
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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.
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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.