Aula Teórica 6&7

1. AulaTeórica 6&7 Princípio de Conservação e Teorema de Reynolds. Derivada total e derivadaconvectiva

2. 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).

3. ControlVolumneandaccumulation rate Taxa de acumulação da propriedade B: (Taxa de variação da propriedade ) Definindo a propriedade específica “Beta” :

4. Fluxoadvectivo • No caso de a propriedadeseruniformenas faces: • Se a velocidade for uniformeemcada face:

5. FluxoDifusivo • No caso de o gradiente da propriedadeseruniformenas faces:

6. E a equação de evoluçãofica: • Se as propriedadesforemuniformesnas faces e no volume (volume infinitesimal): • Que é a forma algébrica do princípio de conservação

7. Forma diferencial

8. Dividindopelo volume (1)

9. Dividindopelo volume (2) Fazendo o volume tender para zero, obtém-se umaequaçãodiferencial.

10. Fazendo tender o volume para zero Divergência da velocidade. Nula emincompressível. Se positiva o volume do fluidoaumenta.

11. 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.

12. 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.

13. If the fluid is incompressible, the velocity divergence is null

14. 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.

15. Trabalhocomputacional • Casounidimensional, só com difusão:

16. 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.

17. 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)

18. Sistema e Volume de Control Control Volume Volume that flew in Volume that flew out

19. 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:

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

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

22. 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.

23. Volume infinitesimal Dividing by the volume,

24. Derivada total Shrinking the volume to zero, But,

25. 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.

26. 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

27. Fazendoconvergir o volume para zero:

28. 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

29. Friction and pressure loss in a pipe.

30. 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.

31. 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.