1. Remark: foils with „black background“ could be skipped, they are aimed to the more advanced courses Rudolf Žitný, Ústav procesní a zpracovatelské techniky ČVUT FS 2013 Computer Fluid Dynamics E181107 CFD5 2181106 Transport equations, Navier Stokes equations Fourier Kirchhoff equation

2. Cauchy’s Equations CFD5 mas s acceleration Sum of forces on fluid particle total stress flux MOMENTUM transport Newton’s law (mass times acceleration=force)

3. Pressure forces on fluid element surface CFD5 x xx y z zz S W T E N B xx yy Resulting pressure force acting on sides W and E in the x-direction

4. Viscous forces on fluid element surface CFD5 Resulting viscous force acting on all sides (W,E,N,S,T,B) in the x-direction x xx y z zz xx yy S E T W N B

5. Balance of forces CFD5 [N/m 3 ] Cauchy’s equation of momentum balances (in fact 3 equations)

6. Balance of forces CFD5 Do you remember? This is a generally valid relationship Therefore the following formulations are equivalent (mathematically but not from the point of view of numerical solution - CFD) conservative formulation using momentum as the unknown variable (suitable for compressible flows, shocks…) formulation with primitive variables,u,v,w,p. Suitable for incompressible flows (very large speed of sound)

7. Balance of ENERGY CFD5 TOTAL ENERGY transport Total energy [J/kg] Internal energy all form of energies (chemical, intermolecular, thermal) independent of coordinate system Kinetic energy [J/kg] Heat added to FE by diffusion only (convective transport is included in DE/Dt) Power of mechanical forces is velocity times force. Internal forces are pressure and viscous forces.

8. Balance of ENERGY CFD5 Heat flux by conduction

9. Heat conduction CFD5 x xx y z zz S W T E N B xx yy Heat transfer by conduction is described by Fourier’s law

10. Mechanical work - pressure CFD5 x xx y z zz S W T E N B xx yy

11. Mechanical work - stresses CFD5 x xx y z zz S W E N B xx yy T The situation is more complicated because not only the work of normal but also shear stresses must be included.

12. Total energy transport CFD5 [W/m 3 ] This is scalar equation for total energy, comprising internal energy (temperature) and also kinetic energy.

13. Fourier Kirchhoff equation CFD5 [W/m 3 ] Kinetic energy can be eliminated from the total energy equation using Cauchy’s equation multiplied by velocity vector (scalar product, this is the way how to obtain a scalar equation from a vector equation) (1) (2) Subtracting Eq.(2) from Eq.(1) we obtain transport equation for internal energy

14. Fourier Kirchhoff equation CFD5 Interpretation using the First law of thermodynamics di = dq - p dv Heat transferred by conduction into FE Expansion cools down working fluid This term is zero for incompressible fluid Dissipation of mechanical energy to heat by viscous friction

15. Dissipation term CFD5 Heat dissipated in unit volume [W/m 3 ] by viscous forces

16. Dissipation term CFD5 U=u x (H) y This identity follows from the stress tensor symmetry Rate of deformation tensor Example: Simple shear flow (flow in a gap between two plates, lubrication)

17. Example tutorial CFD5 Gap width H=0.1mm, U=10 m/s, oil M9ADS-II at 0 0 C  =3.4 Pa.s,  =10 5 1/s,  =3.4.10 5 Pa,  = 3.4.10 10 W/m 3 At contact surface S=0.0079 m 2 the dissipated heat is 26.7 kW !!!! U=10 m/s y H=0.1 mm D=5cm L=5cm Rotating shaft at 3820 rpm

18. Fourier Kirchhoff equation CFD5 [W/m 3 ] Internal energy can be expressed in terms of temperature as di=c p dT or di=c v dT. Especially simple form of this equation holds for liquids when c p =c v and divergence of velocity is zero (incompressibility constraint): An alternative form of energy equation substitute internal energy by enthalpy where total enthalpy is defined as Pressur e energy Thermal energy Kinetic energy

19. Example tutorial CFD5 Calculate evolution of temperature in a gap assuming the same parameters as previously (H=0.1 mm, U=10 m/s, oil M9ADS-II). Assume constant value of heat production term 3.4.10 10 W/m 3, uniform inlet temperature T 0 =0 o C and thermally insulated walls, or constant wall temperature, respectively. Parameters: density = 800 kg/m 3, c p =1.9 kJ/(kg.K), k=0.14 W/(m.K). Approximate FK equation in 2D by finite differences. Use upwind differences in convection terms T 0 =0 x y

20. Summary CFD5 Mass conservation (continuity equation) Momentum balance (3 equations) Energy balance State equation F(p,T,  )=0, e.g.  =  0 +  T Thermodynamic equation di=c p.dT formulation with primitive variables u,v,w,p suitable for incompressible flows

21. Constitutive equations CFD5 Macke

22. Constitutive equations CFD5 Constitutive equations represent description of material properties Kinematics (rate of deformation) – stress (dynamic response to deformation) Viscous stresses affected by fluid flow. Stress is in fact momentum flux due to molecular diffusion rate of deformation is symmetric part of gradient of velocity) Gradient of velocity is tensor with components Dynamic viscosity [Pa.s] Second viscosity [Pa.s]

23. Constitutive equations CFD5 Rheological behaviour is quite generally expressed by viscosity function and by the coefficient of second viscosity, that represents resistance of fluid to volumetric expansion or compression. According to Lamb’s hypothesis the second (volumetric) viscosity can be expressed in terms of dynamic viscosity  This follows from the requirement that the mean normal stresses are zero (this mean value is absorbed in the pressure term) This is second invariant of rate of deformation tensor – scalar value (magnitude of the shear rate squared)

24. Constitutive equations CFD5 The simplest form of rheological model is NEWTONIAN fluid, characterized by viscosity independent of rate of deformation. Example is water, oils and air. Can be solved by FLUENT. More complicated constitutive equations (POLYFLOW) exist for fluids exhibiting  yield stress (fluid flows only if stress exceeds a threshold, e.g. ketchup, tooth paste, many food products),  generalized newtonian fluids (viscosity depends upon the actual state of deformation rate, example are power law fluids )  thixotropic fluids (viscosity depends upon the whole deformation history, examples thixotropic paints, plasters, yoghurt)  viscoelastic fluids (exhibiting recovery of strains and relaxation of stresses). Examples are polymers.

25. Constitutive equations CFD5 Ansys POLYFLOW suggests many constitutive models of generalised newtonian fluids and viscoelastic fluids. Differential models of viscoelastic fluids: Upper convected Maxwell model UCM, Oldroyd-B, White Metzner, Phan-Thien-Tanner model and other models. The UCM is typical and the simplest The UCM model reduces to Newtonian liquid for zero relaxation time =0. Integral models of viscoelastic fluids written in a general form:

26. Unknowns / Equations CFD5 There are 13 unknowns: u,v,w, (3 velocities), p, T, , i,  xx,  xy,… (6 components of symmetric stress tensor) And the same number of equations Continuity equation 3 Cauchy’s equations Energy equation State equation p/  =RT Thermodynamic equation di=c p.dT 6 Constitutive equations

27. Navier Stokes equations CFD5 These terms are ZERO for incompressible fluids These terms are small and will be replaced by a parameter s m Using constitutive equation the divergence of viscous stresses can be expressed This is the same, but written in the index notation (you cannot make mistakes when calculating derivatives)

28. Navier Stokes equations CFD5 General form of Navier Stokes equations valid for compressible/incompressible Non-Newtonian (with the exception of viscoelastic or thixotropic) fluids Special case – Newtonian liquids with constant viscosity Written in the cartesian coordinate system