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

tBB 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

tVV lll

ii

ttt

00

AuQ

zxAzyAyxAzyxV

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)

zyxz

yzz

yx

zyx

yzx

yzx

zyxx

zyx

zy

zyxyxwyxw

zyxzxvzxv

zyxzyuzyu

t

zz

zzz

z

zzz

yy

yyy

y

yyy

xx

xxx

x

xxx

zzzyyy

xxxttt

00

Dividindo pelo volume (2)

zzz

y

yy

xxx

yww

yvv

xuu

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

xxxu

t

j

j

jjjj

jjj

j

xu

xxxu

t

xxxu

t

j

j

jj xu

xxdtd

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

xu

xu

xu

xu

j

j

1

1

xu

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

xxxu

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

xu

xu

xu

xu

xxxu

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)

dSnvdVoldtddVol

dtd

VC SCsistema

.

Sistema e Volume de Control

Volume that flew in Volume that

flew out

Control Volume

Taxas de Variação

tBB 00 tsistemaI

ttsistemaI

tBB 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)

tBB tvc

ttvc

00

t

sai_que_quantidadeentra_que_quantidadetBB 00 t

2sistematt

2sistema

dBdV dVB =>

Fluxo advectivo

dAnvadvB .

Where v velocity relative to the surface. Is the flow velocity if the volume is at rest.

Balanço integral

dAn.vdVdtddV

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 AnvAnvVdtdV

t

..

dAn.vdVdtddV

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

xvv

xvv

tVd

txxxxxxxx

jj

vxdt

VdVt

)(1

k

k

j

j

jj x

vxv

xv

tdtd

jj xv

tdtd

Shrinking the volume to zero,

k

k

xu

dtd

dtVd

Vdtd

VV

dtVd

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 AAuu

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?

xrrruxr

rugxrrrrp

drrr

22sin)2()2(

• Let’s consider a “annular control volume” and perform a force balance

rrrrr ru

rru

ru

rgsen

xp

11

• Fazendo convergir o volume para zero:

rrrrr ru

rru

ru

rgsen

xp

11

ru

rru

rgsen

dxdp 1

ru

rru

rrur

rr 11

rur

rrgsen

dxdp 1

rCrgsen

dxdp

ru

Crgsendxdp

rur

1

1

2

2

2

• When r is zero the velocity gradient is zero, friction is zero and thus C1 must be zero:

rCrgsen

dxdp

ru 1

2

2

2

41

21

Crgsendxdpu

rgsendxdp

ru

When r=R, velocity is zero and thus

22

2

2

2

2

141

41

410

RrRgsen

dxdpu

RgsendxdpC

CRgsendxdp

Friction and pressure loss in a pipe.

DfV

dxdp

dxdp

Lp

RLRp

w

w

421

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.