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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.

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Aula Teórica 6&7

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Aula te rica 6 7

AulaTeórica 6&7

Princípio de Conservação e Teorema de Reynolds.

Derivada total e derivadaconvectiva


Princ pio de conserva o

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

ControlVolumneandaccumulation rate

Taxa de acumulação da propriedade B: (Taxa de variação da propriedade )

Definindo a propriedade específica “Beta” :


Fluxo advectivo

Fluxoadvectivo

  • No caso de a propriedadeseruniformenas faces:

  • Se a velocidade for uniformeemcada face:


Fluxo difusivo

FluxoDifusivo

  • No caso de o gradiente da propriedadeseruniformenas faces:


E a equa o de evolu o fica

E a equação de evoluçãofica:

  • Se as propriedadesforemuniformesnas faces e no volume (volume infinitesimal):

  • Que é a forma algébrica do princípio de conservação


Forma diferencial

Forma diferencial


Dividindo pelo volume 1

Dividindopelo volume (1)


Dividindo pelo volume 2

Dividindopelo volume (2)

Fazendo o volume tender para zero, obtém-se umaequaçãodiferencial.


Fazendo tender o volume para zero

Fazendo tender o volume para zero

Divergência da velocidade. Nula emincompressível. Se positiva o volume do fluidoaumenta.


Questions

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)x

u)x+dx

V)y

Is the rate of increase of distance between faces normal to xx axis. The same for other axis.

dy

x

In case of this figure the volume would increase.


Questions1

Questions

  • The rate change of a property conservative property is the symmetrical of the flux divergence?

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

If the fluid is incompressible, the velocity divergence is null


The diffusivity of the specific mass is zero

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

Trabalhocomputacional

  • Casounidimensional, só com difusão:


Referencial euleriano e lagrangeano

ReferencialEuleriano e Lagrangeano

  • O refencialEulerianoestudaumazona do espaço (volume de controlofixo)

  • O referencialLagrangeanoestudaumaporção de fluido “Sistema” (volume de controlo a mover-se à velocidade do fluido).

  • O Teorema de Reynolds relacionaosdoisreferenciais.


Teorema de reynolds

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)


Sistema e volume de control

Sistema e Volume de Control

Control Volume

Volume that flew in

Volume that flew out


Taxas de varia o

Taxas de Variação

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:


Aula te rica 6 7

Fazendo o Balanço por unidade de tempo e usando a definição de propriedade específica (valor por unidade de volume)

=>


Fluxo advectivo1

Fluxo advectivo

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


Balan o integral

Balanço integral

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

Volume infinitesimal

Dividing by the volume,


Derivada total

Derivada total

Shrinking the volume to zero,

But,


Questions2

Questions

  • The velocity of an incompressible fluid in a contraction must increase and consequently the pressure must decrease

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

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


Aula te rica 6 7

  • Fazendoconvergir o volume para zero:


Aula te rica 6 7

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

When r=R, velocity is zero and thus


Friction and pressure loss in a pipe

Friction and pressure loss in a pipe.


About the flow in a pipe

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

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.


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