Introduction of compressible turbulence

Transcription

1 Introduction of compressible turbulence 1

2 Main topics Derive averaged equations for compressible turbulence Introduce a math. technique to perform averaging in presence of density variation Favre average Leads to compactness of notation Difference between incompressible and compressible turbulence New terms due to compressibility. Old terms, with new consideration. Models those terms 2

3 Compressible turbulence: examples 1: Shock/turbulence interaction 3

4 Compressible turbulence: examples Acoustic waves. Combustion, heat release due to exothermic reaction Heat and mass transfer T, density, change. Mode of fluctuation decomposed into: Vorticity mode + acoustic mode + entropy mode Dynamic decoupled (1 st order) in fluctuation amplitude 4

5 Governing equations Equations Incompressible Conservation of: Mass Momentum Compressible Conservation of: Mass Momentum Energy Equation of State Unknown variables Velocity, (u,v,w) pressure (p) Velocity (u,v,w) Pressure(p) Density (ρ), Energy/Enthalpy/temperature (e/h/t)

6 Incompressible Equations Incompressibilty: =. or =0 Mass = 1 + = 1 =0 Momentum + = Unknown :, = 1,2,3 Num. of Eq. : 3+1 = 4 6

7 Compressible Equations Mass Momentum Energy + =0 ( = 0, ) = + = 1 = h+ 2 +,.) = = e is the internal energy and h is the enthalpy h = + Eq. of state = 7

8 momentum + = + energy h+ 2 = Volume dilation Assuming Newtonian fluid: =2 + The second viscosity is often related to the dynamic viscosity by: = 2 3 Valid for monoatomic gases. However, in CFD it is in principle used for all gases 8

9 energy = Assuming perfect gas with constant C v and C p = h= The heatflux term: Fouriers law then reformulated by h. = = h Prandtl number = 9

10 What we plan to do now Use Reynold average directly on all the governing equations For example: Take average Problems Average breaks, enters derivative Solved problem for incompressible eq. + =

11 Introduce Favre averaging ( ) Previously: =. apply Reynold average ( ) =(+ )( + ) = = ( ) Now : =+ = = ( ) = Still want to keep as simple as the Reynold average? = + Too many non-zero terms! Solution: Faver average ( ): A new decompsoition ( ): 11

12 Introduce favre averaging ( ) = = + Rule 1: = 0 ( = Rule 2: = + =/=( + )/ =+ / = = / 0) Rule 3: = (/) = (/) = Now =[ + + ] =( +b +a +a b ) = +b +a +a b Not Expand r.3 =+( )+( )+ r.2 = r.1 = Old Reynold average =(+ )( + ) = Apply to bi-product = 1= = = 12

13 Let s derive averaged equations Equations + Mean mass (Easy, =0) Mean momentum ( to be shown), Mean Kinetic energy equation ( = Turbulent kinetic energy (k, to be shown), Mean energy. one more math: = ( ) () = = () +() = ( ) (), to be shown), 13

14 Derive (compressible) averaged momentum equation Expand + = + ( + ) + ( + )( + ) = (Eq.1) Mean: + + ( ) = (Eq.2) Easy cancel using = + + = = + Reynold stress, also written with favre average as: 14

15 Mean mass Eq. Mean momentum Eq. (Conservative form) 1 non-conservative form Eq(2) + 1 Conservative form Derive mean kinetic energy equation ( = ) = /2 + / = + = + (Eq.1) = + = + = + = + (Eq.2) + + = + 15

16 Derive compressible turbulent kinetic energy (k) eq. Instantaneous continuity Eq. Instantaneous momentum Eq. (Non-Conservative) Remind: here is conservative form 2 : + = + Expand Keep ( + ) ( )+ + + =0 ( + ) + = + Expand Expand. + = + (Eq.1) (Eq.2) + = ( + ) = + = 2 / /2 + + / + + Expand /2 + = + = Expand +0+( + ) /2 +( + ) = ( + ) Take average + 16

17 Compressible turbulent kinetic energy (k) equation +0+( + ) /2 +( + ) = + +. h.. = =. h.. = ( ) = + + Transport Production Disppation Pressure work 0 = = Pressure dilatation 0 17

18 Averaged equations for compressible turbulence Applying Reynold average to whole equations. rewrite using two sets of decompositions: Reynold-Ave. :,,(, ) Favre-Ave.: (, ), (, ), (h, h ), (, ) Mass Eq. : + =0. = Momentum Eq. + = + = k- Eq. + K-Eq. : + Energy Eq h h = = + h + + = + = + = 2 Turbulent transport of heat Molecular diffusion of k Turbulent transport k 18

19 Models Old terms in k- equation + = Production Disppation Transport Pressure work Transport: Production: The Reynolds stresses model: Disspation: = Solenodial Dilatation 2 = + Pressure dilatation = put in epilson equation = 4 3 ~ ( )~2 Sarkar s Model turbulent Mach number = 19

20 Model New terms in k- equation + = Production Disppation Transport Pressure work 0 Pressure work ~ Pressure dilatation 0 Pressure dillation: ~ + turbulent Mach number = 20

21 Model Terms in energy- equation h h + 2 h = + Turbulent heat flux is modelled using the gradient diffusion hypothesis. Pr T is the turbulent Prandtl number, usually with a value of about 0.9 h = = h 21

22 Summary: compressible turbulence models Compressibility caused extra terms: Solenoidal dissipation / dilation dissipation + Model Pressure dilation term 0 Pressure work term 0 22

23 = 2 = 2 23