Aula Teórica 6&7 Princípio de Conservação e Teorema de Reynolds. Derivada total e derivada...
Transcript of Aula Teórica 6&7 Princípio de Conservação e Teorema de Reynolds. Derivada total e derivada...
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
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
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
Fluxo Difusivo
• No caso de o gradiente da propriedade ser uniforme nas faces:
dAndAndifB ..
n
i
esqdiriB lAdif
1
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
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
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
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.
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.
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.
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.
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
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:
l
AQt
VV lllii
ttt
00
xx
xxl
x
xxlttt
xA
xA
xt
100
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:
• (ver capítulo 3 do White)
dSnvdVoldt
ddVol
dt
d
VC SCsistema
.
Sistema e Volume de Control
Volume that flew in Volume that
flew out
Control Volume
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:
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 =>
Fluxo advectivo
dAnvadvB .
Where v velocity relative to the surface. Is the flow velocity if the volume is at rest.
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.
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,
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,
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.
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
• 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
• 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
Friction and pressure loss in a pipe.
D
fV
dx
dpdx
dp
L
p
RLRp
w
w
4
2
1
2
2
2
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.