Introduction to Turbulence Modelling

Transcription

1 Introdcton to Trblence Modellng 1

2 Nmercal methods 0 1 t Mathematcal descrpton p F Reslts For eample speed, pressre, temperatre Geometry Models for trblence, combston etc.

3 Mathematcal descrpton of physcal realty FV, FD, FE? Governng eqatons and bondary condtons Dscretsaton, choce of grd System of algebrac eqatons Eqaton system solver All these steps ntrodces errors! How can we garantee that the appromate solton s close to the eact one and close to realty Appromate solton

4 Dscretsaton and grd Qestons: How comple s the geometry? What accracy s reqred? Grd qalty? What abot stablty? Grd refnement?

5 Modellng of Trblent Flows Isses: Inflow/otflow bondary condtons Handlng walls and near wall effects Accracy

6 Flow vsalsaton of a trblent rond et

7 Scales n a trblent premed flames 3.1m/s 30.5 m/s

8 Scales n Mass- and Momentm Trblent Transport Molecles => contnm m Kn l Kndsen nmber: Bolzmann Eqatons Molecles Eler Naver-Stokes Contnm Kn small large Fld Mech., LTH

9 Combston as an eample Scales n trblent combston: 1. Chemstry 2. Mng 3. Trblence

10 log(e(k)) Trblent knetc energy spectrm (Kolmogorov theory for sotropc & homogenos trblence) Large scales -5/3 Unversal range Inertal sbrange Dsspaton sbrange Hgh Schmdt nmber Sc=/D log(k)

11 Length- and Tme-Scales Length: 1. Integral O(L) 2. Taylor O(Re L -1/2 L) 3. Kolmogorov O(Re L -3/4 L) 4. Batchelor O(Re L -3/4 Sc -1/2 L) Tme: 1. Integral T=O(L/U) 2. Taylor k =O(Re L -1/2 T) 3. Kolmogorov k =O(Re L -1/2 T) 4. Chemstry Ka= c / k Da=T/

12 Tme scales n a GT combston chamber Slow chemstry (NO-formaton) Intermedate scales Fast chemstry (qas-steady-state/ eqlbrm) 1 s 10-1 s 10-2 s 10-4 s 10-6 s 10-8 s Strctre Trblence, Transport Acostc waves/ thermo-acostcs

13 Characterzaton of Trblence Qestons: 1. How to nderstand/nterpret random data? 2. Can we compte a random solton? 3. Is there any meanng n comptng sch soltons?

14 Governng Eqatons Momentm: 0 t t p t R L S h h t h Pr L w Y Sc Y t Y W Y T R p 0 Mass: Energy: Speces: Eqaton of state:

15 Governng Eqatons T Y q q ; ; ; Non-lneartes: 1. Convectve 2. Constttve 3. Eponental O F n O O n F F e W Y W Y T T AT ) ( ) )( / ep( 4 T S or T T C Y D R Y

16 Trblence modellng Drect smlaton of sotropc trblence Doman: Cbc bo of sze 8L11 2 8L 4 0L11 L Reqred resolton: ma 1. 5 In physcal space

17 Trblence modellng

18 Trblence modellng

19 Trblence modellng Drect smlaton of sotropc trblence In 3D: Reqred nmber of grd nodes n each drecton: L 3 4 N ReL N 4.4ReL 0.06 ma L L L ma 11 L 12 L11 N L 9 2 R Trblence Reynolds nmber: Taylor scale Reynolds nmber: Re Lengthscale of large eddes: L R kl g 2 k L 3 2 k L

20 Trblence modellng The Corant nmber: Reqred temporal resolton Drect smlaton of sotropc trblence C kt 1 C 20 Trblence tme scale: Assme that samplng over at least 4 trblence tme scales s needed, then the nmber of tme steps s: M 4 k k L t 120 L 9.2R 3 2 k Comptatonal work: N 3 M 3 L 160Re 0.66R 6

21 Trblence modellng Drect smlaton of sotropc trblence Reqred tme n days at a comptng rate of 1 Gflop T G ReL R 70 6

22 Trblence modellng Drect smlaton of sotropc trblence Reqred tme at a comptng rate of 82 Gflop (standard desktop compter) Re N N 3 M N 3 M CPU tme Memory E06 1.2E03 1.3E09 14s 18 Mb E07 3.3E03 3.2E mn 150 Mb E08 9.2E03 1.1E h 2 Gb E09 2.6E04 5.2E days 30 Gb E10 7.4E04 2.8E years 565 Gb E11 2.1E05 1.6E17 61 years 11 Tb N 3 = nmber of grd ponts M= nmber of tme steps N 3 M= total work reqred Note that sng the fastest spercpmter n Sweden the Re=9600 case wold take abot 22 mn.

23 Trblence Trblence data have meanng ONLY n a statstcal sense: Averages: Ensemble Tme Space (Phase average) (wth/wthot densty weght) Other representatons! (e.g. POD) 23

24 Trblence Trblence data have meanng ONLY n a statstcal sense: Take average ether a-posteror (on the DNS data) a-pror (on the eqatons before solvng them) Note: All epermental data are averaged n space-tme 24

25 Propertes of averagng operators Characterzaton of Trblence ) ( ) ( ) ( dz z f z z G z f 1 ) ( d G ) ( ) ( ) ( ) ( g f g f z f z f

26 Reynolds Averaged Eqatons Ensemble-averagng q (, t) qn (, t) n Tme-averagng q( ) lm T 1 T t 0 T t o q(, t) dt

27 Averagng the governng eqatons 0 p t 2 1 Propertes of tme and ensemble averagngs: v' ' v v v v (,t) (,t) (,t) 0 Mass Momentm

28 0 p t 2 1 Reynolds Averaged Naver-Stokes (RANS) eqatons Reynolds stress tensor, symmetrc 2nd rank tensor We now have: 4 eqatons 4+6 nknowns Leads to the closre problem

29 Trblence Modellng N-eq. Models (e.g. k-) - Short comptatonal tme - Smple, robst - Lmted range of valdty Reynolds Stress Models, RSM - More general, stll not nversal - More comple, seven PDE:s - Etensve modelng - Longer comptatonal tme Reynolds Averaged Naver-Stokes: RANS

30 Spatal Averagng (LES) Bo flter G( z z ) 1/ 0 3 for for / 2 / 2 Gassan flter 6 3/ 2 6 G( z z ) ( ) ep z z 2 2 2

31 Spatal Averagng (LES) Decompose: q(, t) q(, t) q (, t) Propertes: q(, t) q(, t) q (, t) 0 s q s q s q s q sq

32 Fltered governng eqatons Mass: 0 Momentm t 1 p

33 Fltered governng eqatons Momentm p t 1 Sb-grd stresses: 0 Mass:

34 Germano s decomposton Leonard tensor L o Interacton among resolved scales Cross-term tensor C o Interacton between resolved and non-resolved scales Reynolds stresses R o Interacton among non-resolved scales

35 Large-Eddy Smlaton for Trblent Flows Spatal flterng rather than ensemble average G(, ' ) ( ' ) d' Comptatonal Grd

36 log(e(k)) Trblent knetc energy spectrm (Kolmogorov theory for sotropc & homogenos trblence) Large scales -5/3 Unversal range Inertal sbrange Flter ctoff Dsspaton sbrange LES-spectrm log() log(k)

37 Trblence Modellng k- Models - Short comptatonal tme - Smple, robst - Lmted range of valdty Large Eddy Smlaton, LES - Resolvng large scales - Modelng small scale effects - Long comptatonal tme Reynolds Stress Models, RSM - More general, stll not nversal - More comple, seven PDE:s - Etensve modelng - Longer comptatonal tme NOT a pre model!!

38 Trblence Modellng k- Models - Short comptatonal tme - Smple, robst - Lmted range of valdty Large Eddy Smlaton, LES - Resolvng large scales - Modelng small scale effects - Long comptatonal tme Reynolds Stress Models, RSM - More general, stll not nversal - More comple, seven PDE:s - Etensve modelng - Longer comptatonal tme Appromatons Drect Nmercal Smlaton, DNS - No Model, resolves all scales - Very long comptatonal tme - Lmted to low Re

39 Modellng of Trblent Flows Qeston: What s the qalty of the reslts? => VALIDATION

40 Comptatonal Valdaton Assessment of nmercal accracy Assessment of model contrbtons Senstvty to parameters -Comptatonal -Problem related Comparson wth eperments Relevant (?) data - Mean and rms qanttes - Correlatons (trblent fles, mlt-pont) - Flow/flame dynamcs (freqences, modes)