The Principle of Turbulent Frame Indifference and new closure relations in LES

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1 Poceedg of he d WSAS I. ofeece o Appled ad Theoecal Mechac Vece Ialy Novebe The Pcple of Tble ae Idffeece ad ew cloe elao S. GARANO G.ANNATA Dpaeo d Idalca Tapo e Sade Uveà d Roa "a Sapeza Va doaa 8 Roa ITAY Abac: - I h pape he pcple of Tble ae Idffeece eved. The pee-day S odel ad he dawbac of he dyac calclao of he cloe coeffce fo he geealzed SGS ble e eo ae aalyzed. A ew cloe elao fo he geealzed SGS ble e eo popoed. The popoed cloe elao fo he geealzed SGS ble e eo: cople wh he pcple of ble fae dffeece; ae o acco boh he aoopy of he blece velocy cale ad blece legh cale; eove ay balace apo bewee he podco ad dpao of SGS ble ec eegy. I he popoed odel he geealzed SGS ble e eo elaed eclvely o he geealzed SGS ble ec eegy (whch calclaed by ea of balace eao) ad he odfed eoad eo; he SGS vco dpao of he geealzed SGS ble ec eegy calclaed by olvg he balace eao. The odelled balace eao of epec he popee of fo-vaace ad fae-depedece of he eac balace eao. Key-Wod: - S Tble fae dffeece cloe elao The pcple of Tble ae Idffeece To dae ay effo have bee ade o eablh ew cloe elao fo he ow ble ae he eolved eao ha gove ble hee-deoal eady flow (S). I he faewo of oday co heodyac hee elao (o blece odel) cold be epeed a cove eao whch ae eceay o cloe he eao of oo ad he eal eegy eao. The cove eao epee a dealzed fo he behavo of he aeal ad coeely hey flfl he pcple of Maeal ae Idffeece []. Th bac wog pcple of co echac ee he cove eao o be he ae fo obeve eal ye ad o-eal oe. The above eoed pcple poe ha he fcoal elao (bewee ow eo ad eac ae) flfl wo dc eee []: a) fo-vaace de he o geeal cla of afoao of he efeece fae; b) fae dffeece pacla depedece of he alaoal ad agla velocy of he fae. The pcple of Tble ae Idffeece he evale blece of he pcple of Maeal ae Idffeece ad ple fo-vaace ad fae dffeece o he ble cloe elao [3]. Sad ad He [] ephazed ha fovaace a) ad fae-depedece b) ae wo dc ae. A ble cloe elao fo-vaa f doe o odfy foal epeo de afoao of he fae ad coced oly wh obecve eo. The obecve eo of a () ae ad o be obecve becae hey afo le geoec obec [] [4]. odeg a eal fae whch a aeal po ha coodae a e ad a o-eal fae whch he ae po ha coodae a e he o geeal law whch gove he afoao of he coodae ad he e epeed he wo fae ha gve by he cldea afoao ( ) b ( ) a () whee () ae he copoe of a e-depede pope ohogoal eo b () he e-depede dace bewee he og of he wo fae ad a ay coa. Teo of a ( ) ae

2 Poceedg of he d WSAS I. ofeece o Appled ad Theoecal Mechac Vece Ialy Novebe ad o be obecve f he copoe afo accodg o S S obecve cala V V obecve veco () A A obecve eo A ble cloe elao fae dffee f epeed e of eo ha ae depede of he agla ad alaoal velocy of he fae. We ad He [4] ephazed he dffeece bewee he fo-vaace ad he fae dffeece by delg he eece of eo ha ae obecve b depede o he agla velocy of he fae. o eaple he ayec pa of he velocy gade o a obecve ay: le W ad W be epecvely he epeeao of h ay a eal ad o-eal fae he law of afoao of hee epeeao gve by W W & (3) I poble o aocae a obecve eo called abole velocy eo W Ω o he ay W. The law of afoao bewee he epeeao of h eo he dffee fae of efeece gve by W Ω W Ω whee W Ω W & (4) The abole vocy eo W Ω a obecve eo (ce epeeao he dffee fae afo accodg o ao ) b fae-depede ce epeeao deped o he fae by ea of he e & aocaed wh he agla velocy of he o-eal fae [3] [4]. oeely alway poble o dedce a obecve b fae-depede eo fo he epeeao a eal fae of a oobecve ay. A ble cloe elao whch epeed e of obecve eo ha ae depede o he agla velocy of he fae doe o flfl he pcple of Tble ae Idffeece becae fo vaa b fae depede. M all he ble cloe elao flfl he Pcple of Tble ae Idffeece? The ble pheoea ae o aocaed o he popee of he aeal: coeely ble cloe elao do o epee he aeal behavo. Tble cloe elao alway be fo vaa b o ecealy be fae dffee [3] [4]. I ohe wod o all he ble cloe elao flfl he pcple of Tble ae Idffeece. I he ble cloe elao he odelled epeo of a ow obecve eo be folaed e of obecve eo (allowg he cloe elao o flfl he eee of fo vaace) ad ea he ae depedece (o he agla velocy of he fae) of he ow eo. A deoaed by Galleao e al. [5] he geealed SGS ble e eo a obecve eo ad depede of he agla ad alaoal velocy of he fae. oeely he cloe elao fo h eo flfl he pcple of Tble ae Idffeece. I ohe wod he cloe elao fo he geealed SGS ble e eo : - be fo vaa (o ahe be epeed e of obecve eo); - be fae dffee (o ahe be epeed e of eo ha ae depede of he agla ad alaoal velocy of he fae). loe elao baed o he Sagoy odel Aog he o coo S odel pee leae ae he dyac ed odel [6] [7] [8] [9] [] baed o he Sagoy cloe elao whch he geealzed SGS ble e eo elaed o he eolved a-ae eo S by ea of a cala eddy vcoy T : S (5) T whee he oveba (.) dcae he fle opeao ad he odfed eoad eo []. I aed hee odel ha he eddy vcoy a cala popooal o he cbc oo of he geealzed SGS ble ec eegy dpao ad ha ch dpao locally ad aaeoly balaced by he podco of he geealzed SGS ble ec eegy (.e. by he ae of ec eegy pe of a afeed fo he lage cale lage ha he fle ze o he eolved oe). oeely he eddy vcoy ae he fo: T ( S Δ) ( S S ) (6) whee Δ he fle wdh ad S he cloe coeffce. I evde ha he dyac ed odel baed o he Sagoy cloe elao ae fagh wh fo eleva dawbac. The f dawbac epeeed by he cala defo of he eddy vcoy; he ecod oe coce he

3 Poceedg of he d WSAS I. ofeece o Appled ad Theoecal Mechac Vece Ialy Novebe local balace apo of he geealzed SGS ble ec eegy podco ad dpao he hd dawbac elaed o he dyac calclao of he coeffce ed o odel he eddy vcoy whl he foh dawbac elaed o he poble ag fo he ecal chee adoped fo he lao of hee-deoal eady flow (S). The cala defo (f coecy) of he eddy vcoy evale o ag ha he pcpal ae of he geealzed SGS ble e eo o he eolved pa of (epeeed by he co ad Reyold e) ae alged wh he pcpal ae of he eolved a-ae eo. Moeove he eddy vcoy popooal o he podc of wo e of whch he deo ae epecvely hoe of a legh ad a velocy []. Thee e whch epee epecvely he blece legh cale ad blece velocy cale ae oe geeally ecod-ode eo of whch he podc a foh-ode eo whch epee he eddy vcoy [3]. The cala defo of he eddy vcoy ed he above-eoed dyac ed odel baed o he Sagoy cloe elao peppoe he eece of a gle blece velocy cale ad a gle blece legh cale. I h ae he blece aoopy dced by he coo afe of eegy fo he ea flow owad he ble flcao whch geeally eeely aoopc o codeed. The ecod coecy of he dyac ed odel baed o he Sagoy cloe elao elaed o he apo of a local ad aaeo balace bewee podco ad dpao of he geealzed SGS ble ec eegy folaed he above-eoed odel o oba he ble vcoy epeo. Th apo cofed acally ad eve aaeoly ad oly locally a he cale aocaed wh wavebe wh he eal bage ad he lae e oly fo oopc blece ad a hgh Reyold be. The hd coecy of hee dyac odel coce he calclao of he above eoed cloe coeffce S. Whe lag cofed flow a hgh Reyold be he el of he dyac pocede ae of dobfl elably he ego cloe o he wall cldg boh he vco blaye ad he bffe laye [4]. I h ego he fle wdh ed he dyac pocede lage ha o edde ha gove he oe ad eegy afe. oeely he dyac pocede ed de hee codo fo he calclao of he coeffce S o able o flly acco fo he local bgd dpave pocee ha affec he ee doa. The foh coecy of he dyac ed odel baed o he Sagoy cloe elao coeced o he poble ag fo he ecal chee adoped fo he lao of hee-deoal eady flow (S). The hee-deoal eady flow lao ee ecal chee wh a hgh ode of accacy: a low ode of accacy of ceeed fe dffeece chee odce ( he heedeoal eady flow lao) a adpave faco whch edce he ably of he geealed SGS ble e eo o epee he ec eegy afe fo he eolved cale o he eolved oe wh a ceae of he eolved ec eegy. The ecal chee bede beg accae flfl he coevao eee. A ggeed [5] ad [6] coevao popee of he a he oe ad he ec eegy eao fo copeble flow ae egaded a aalycal eee fo a pope e of dcee eao. ode he followg goveg eao fo he cala ay φ: φ ( φ ) ( φ ) 3( φ )... (7) he e ( φ ) coevave (coeve φ) f ca be we dvegece fo [8] ( ) ( φ ) (8) Noe ha a coeved a po ce he coy eao appea dvegece fo. o he ae eao he covecve e of he oe eao coevave a po (coeve oe) f we dvegece fo: ( Dv. ) (9) Th defo of he coevao a po dcae he popey of coevg oe ( peodc feld) depedely of he odale by whch he coy eao afed. The goveg eao fo he ec eegy K / ca be developed by ag he veco do podc of he velocy ad he oe eao p T () whee p he pee dvded by he coa dey ad T he vco e. I he above eao he covecve e ca be ewe he

4 Poceedg of he d WSAS I. ofeece o Appled ad Theoecal Mechac Vece Ialy Novebe followg fo coepodg o ha he oe eao / () Th e copoed by wo pa: he f coevave fo ad he ecod volve he coy eao. The covecve e (epeed dvegece fo he oe eao) coeve a po oe b doe o coeve a po ec eegy: fac ao how how he coy eao volved he ec eegy coevao popey of he covecve e ha ae epeed dvegece fo. I ohe wod ec eegy coeved (by dvegece fo of covecve e) oly whe he coy eao pefecly afed. The paage fo he pevo aalycal codeao o he effec ha hey podce o ecal lao poe a efleco o he followg aee: he coy eao cao be pefecly afed by ecal lao. I he lao of hee-deoal eady flow (S) (ha ae ealzed by a hgh-ode fe dffeece chee wh he dvegece fo of he covecve e) he eolved ec eegy o pefecly coeved becae he coy eao o pefecly afed. oeely he eolved ec eegy deed o e log e lao Z T DNS g.. Pofle of eolved ec eegy Reyold aveaged ove cceve eval of e (T T T3 T4). Slao pefoed by g he blece odel of Zag e al. [4]. hael flow Re395. I he dyac ed odel baed o he Sagoy cloe elao he calclao of he cloe coeffce S (by dyac pocede) o able o copeae he effec podced by he covecve e (epeed dvegece fo he eolved oe eao) ha ae o able o pefecly coeve eolved ec eegy. I ode o vefy he above eoed coecy of he dyac ed odel baed T T3 T4 o he Sagoy cloe elao lao of a ble chael flow a Re395 (Re he fco-velocy-baed Reyold be) have bee pefoed by a foh-ode aggeed fe dffeece chee popoed by Moh e al [5]. The geealed SGS ble e eo ha bee calclaed by ea of he ed dyac odel of Zag e al. [6]. The eolved ec eegy ha bee aveaged ove e eval geae ha he egal ble e cale. I ge he ove e aveaged eolved ec eegy pofle ae how. o he fge poble o dedce ha he eolved ec eegy ceae. I h pape deoaed ha ode ovecoe he coece of he dyac ed Sagoy-ype odel ad ode o coa he ceae of eolved ec eegy he ble cloe elao fo he geealed SGS ble e eo be epeed decly a a fco of he geealed SGS ble ec eegy ad of he SGS vco dpao. The geealed SGS ble ec eegy ad he SGS vco dpao ae ow ae ha ae calclaed by olvg he elave balace eao. I hee eao hee ae ow e ha ae calclaed by dyac pocede. I h pape how ha he dyac pocede fo he podco ad dpao e of he SGS vco dpao balace eao ae able o copeae dyacally he ceae of eolved ec eegy ad he allow he lage eddy lao of hee-deoal eady flow alo fo log e lao. 3 The balace eao of he geealed SGS ble ec eegy ad of he SGS vco dpao The balace eao of he geealzed SGS ble ec eegy epeed e of he geealzed ceal oe ae he fo [5]. ( ) ( p ) D D O whee O he foce dey ad ( ) ( f g ) fg f g ( f g h) f g h f ( g h) g ( f h) h ( f g ) fgh ()

5 ae epecvely he geealzed ecod ad hd ode ceal oe elaed o he geec ae f g ad h []. A deoaed [5] he geealzed SGS ble ec eegy balace eao fovaa ad fae-dffee o ch ha each of he e ha appea ae epeeao eal ad o-eal fae of obecve eo ha ae depede of he agla ad alaoal velocy of he fae. The la e o he gh-had de of ao defed a he vco dpao of SGS ble ec eegy. (3) I he popoed S odel a fhe balace eao odced fo he vco dpao. The balace eao of epeed e of he geealzed ceal oe ae he fo p O (4) Th eao obaed fo he Nave-Soe eao ad he fleed Nave-Soe eao. Above all deoaed ha ao 4 fo-vaa ad fae-depede. I a oeal fae ao 4 ae he fo: & p - & & Ol l (5) whch he eo & ha appea he epeeao of he 5h 7h ad h e (wh he epecve boe) aocaed wh he agla velocy of he o-eal fae wh epec o he eal oe. The 5h 7h ad h e of eao 4 ad 5 ae he epeeao ( he eal ad o-eal fae) of zeo-ode obecve eo ha ae fae-depede: he obecvy of hee eo gaaeed by he fac ha he epecve epeeao he dffee fae ae elaed o each ohe accodg o he law of afoao epeed by ao ; he fae-depedece gve by he peece of he above eoed e & aocaed wh he agla velocy of he fae. All he ohe e ha appea ao 4 ad 5 ae epeeao he dffee fae of obecve eo ha ae depede of he agla velocy of he fae. o h codeao ad fo he apo ha a eao fo-vaa f epeed oly e of obecve eo el ha he balace eao of fovaa. Th eao alo fae-depede hogh he appeaace of & he afoao of he epeeao of he 5h 7h ad h e he dffee fae. 4 New cloe elao Accodg o Bada' cale lay apo a cloe elao popoed fo he geealzed SGS ble e eo whch hee appea o coeffce o be calbaed o o be calclaed dyacally: (6) Poceedg of he d WSAS I. ofeece o Appled ad Theoecal Mechac Vece Ialy Novebe

6 The cloe elao 6 obaed who ay apo of local balace bewee he podco ad dpao of geealzed SGS ble ec eegy ad ay h be codeed applcable o S wh he fle wdh fallg o he age of wave be geae ha he wave be coepodg o he a ble ec eegy. The cloe elao 6 fo he geealzed SGS ble e eo: a) epec he pcple of ble fae dffeece; b) ae o acco boh he aoopy of he blece velocy cale ad blece legh cale; c) ae cale lay; d) gaaee a adeae eegy da fo he gd cale o he bgd cale ad gaaee baccae; e) ovecoe he coece led o he dyac calclao of he cloe coeffce ed he odellg of he geealzed SGS ble e eo; f) able o elae he effec podced by he ocoevao a po of he eolved ec eegy. ve hogh he cloe elao blece o ecealy epec he eee of faedffeece ce he geealzed SGS ble e eo a obecve eo ad faedffee [5] he cloe elao fo h eo alo be fo-vaa ad fae-dffee. I ohe wod he pcple of ble fae dffeece o ecealy be appled o all he cloe elao b be appled o all he fcoal elao ha epe he geealzed SGS ble e eo (obecve ad faedffee) e of he eolved eac ae. The cloe elao 6 fo he geealzed SGS ble e eo epec he pcple of ble fae dffeece. I h pape he odelled epeo ed fo he ow e of he eac geealzed SGS ble ec eegy balace eao ae ch a o gaaee he epec of he pcple ha he odelled balace eao of be fovaa ad fae-dffee le he eac balace eao of. The odelled fo of ao Δ D D D (7) whee he cala coeffce D dyacally calclaed by ea of a Geao dey appled o he ad 3d e o he gh-had de of ao [5]. A deoaed eco 3 he balace eao of he SGS vco dpao fo-vaa becae epeed oly e of obecve eo. Th balace eao alo faedepede hogh he appeaace of & he afoao of he epeeao of he 5h 7h ad h e of he balace eao he dffee fae. oeely he cloe elao fo he 5h 7h ad h e of ao 4 be folaed fll epec of he eee ha he odelled eao be fo-vaa ad aa he ae fae-depedece a he eac eao. I h pape he followg odelled balace eao of he vco dpao popoed. l l ( ) P S D (8) whee () ad whch he cloe coeffce ae calclaed dyacally by ea of he Geao dee. Th eao fovaa ad ha he ae depedece o he fae a he eac eao. The 5h 7h ad h e of ao 8 ae he odelled epeo of he coepodg ow e of he ao 4. Thee odelled epeo ae folaed by g he hypohe of cale-lay ad epecg he ae fae-depedece a he epecve ow e 5. o eaple he 5 h e of he balace eao of odelled oly he ow pa. The hypohe of cale lay gve Poceedg of he d WSAS I. ofeece o Appled ad Theoecal Mechac Vece Ialy Novebe

7 (9) By odcg he followg cloe elao fo he ow e of ao 9 () he fal odelled fo of he 5 h e () whee he cloe coeffce calclaed dyacally by ea of a Geao dey. Ude a cldea afoao of he fae he eac 5 h e of he balace eao of afo a. & () Ude he ae cldea afoao of he fae he odelled fo of he 5 h e of he apo eao of afo a & (3) The odelled e el a beg depede o he fae of efeece he ae ae a he eac e. A la el obaed by epeag he ae pocede fo he ohe e of ao 8. The deoao oed fo he ae of bevy. 5 Rel ad Dco Tble chael flow (bewee wo fla paallel plae placed a a dace of ) ae laed wh he popoed age ddy Slao odel a dffee fco-velocy-baed Reyold be (Re) agg fo 395 o 34. I ode o valdae he popoed cloe elao fo he geealzed SGS ble e eo he ecal el obaed wh he popoed odel ae copaed wh DNS el [9] ad wh epeeal daa [] g.. Te-aveaged eawe veloce. opao bewee DNS ad S el obaed wh DMM ad he popoed odel (TM). hael flow Re 395. g. 3. Te-aveaged eawe veloce. opao bewee epeeal eaee ad S el obaed wh he popoed odel (TM). hael flow Re 34. I ge ploed he pofle of he eaveaged eawe velocy copoe obaed wh he popoed odel copaed wh he pofle obaed wh DNS [9] ad he Dyac Med Model DMM [4] fo chael flow a Re 395. Poceedg of he d WSAS I. ofeece o Appled ad Theoecal Mechac Vece Ialy Novebe

8 Poceedg of he d WSAS I. ofeece o Appled ad Theoecal Mechac Vece Ialy Novebe The fge how ha he pofle obaed wh he popoed odel agee oe wh he DNS velocy pofle ha wh he pofle obaed wh he DMM boh he boday laye ad he ego de he chael. ge 3 how he pofle of he e-aveaged eawe velocy copoe fo a chael flow a Re34 obaed wh he popoed odel copaed wh he pofle of he aalogo velocy copoe eaed epeeally []. The ageee bewee he wo velocy pofle vey good. e of ceae l eache a vale (z ) ad he e of ble apo ad vco dffo of ae copaable wh he podco e of. I he ego bewee he bffe laye ad he log laye (<z<4) he covecve ad ble apo e ad he vco dffo e ae eglgble copaed wh he podco ad dpao e. Oly h led ego hee a balace bewee he podco ad he dpao of. Towad he cee of he chael (z>3) he vco dpao ed owad a b o eglgble vale. I h ego he podco e of balaced o oly by he dpao b alo by he ble apo of. g. 4. Reyold e < ' 3 '>. opao bewee epeeal eaee ad S el obaed wh he popoed odel (TM). hael flow Re 34. ge 4 copae he pofle of he copoe < ' 3 '> of he Reyold e eo (whee dee () ad (3) deoe epecvely he eawe ad wall-oal deco) calclaed wh he popoed odel wh he pofle of he la copoe of he Reyold e eo obaed fo epeeal eaee [] fo a chael flow a Re 34. ge 4 how ha a Re 34 he popoed odel povde a pofle of he copoe < ' 3 '> ageee wh ha of he coepodg copoe of he Reyold e eo obaed fo he epeeal eaee. ge 5 how he aaeo pofle of he e of he balace eao of aveaged ove hoogeeo plae fo chael flow a Re34. ge 5 deoae ha he balace bewee podco ad dpao of he geealzed SGS ble ec eegy cofed oly a led ego bewee he bffe laye ad he log laye (<z<4) whl o cofed ohe ego of he doa. The vco dpao of balaced he vco blaye (z<5) by he vco dffo e whl he podco of paccally eglgble. Movg away fo he wall he f pa of he bffe laye he podco g. 5. Iaaeo geealzed SGS ble ec eegy balace e aveaged ove hoogeeo plae. Podco: P ; Tble apo: T ; oveco: ; Vco dffo: D ; Vco dpao: ep. hael flow Re X ge 6: Voe defcao wh λ ehod [] -z plae I ge 6 he ea wall voe ce (de he ble boday laye) ae clealy defed by he λ ehod []: he deo of he paal dceao ep allow he opal lao of he above eoed voe ce ha gove he apo he podco ad he dpao of he ble ec eegy.

9 Poceedg of he d WSAS I. ofeece o Appled ad Theoecal Mechac Vece Ialy Novebe oclo I h pape he pcple of Tble ae Idffeece eved. The pee-day S odel ad he dawbac of he dyac calclao of he cloe coeffce fo he geealzed SGS ble e eo ae aalyzed. A ew cloe elao fo he geealzed SGS ble e eo popoed. The popoed cloe elao fo he geealzed SGS ble e eo: cople wh he pcple of ble fae dffeece; ae o acco boh he aoopy of he blece velocy cale ad blece legh cale; eove ay balace apo bewee he podco ad dpao of SGS ble ec eegy. I he popoed odel he geealzed SGS ble e eo elaed eclvely o he geealzed SGS ble ec eegy (whch calclaed by ea of balace eao) ad he odfed eoad eo; he vco of he geealzed SGS ble ec eegy calclaed by olvg he balace eao. The odelled balace eao of epec he popee of fo-vaace ad fae-depedece of he eac balace eao. The popoed odel ha bee eed fo a ble chael flow a Reyold be (baed o fco velocy ad chael half-wdh) agg fo 395 o 34. The popoed odel pove he ageee bewee he el obaed wh S ad hoe obaed wh DNS. Refeece: []. Tedell A f coe aoal co echac Acadec Pe 993. [] A. Sad K. He O he fae depedece ad fo vaace of he apo eao fo he Reyold e eo ad he ble hea fl veco: coeece o cloe odel blece odellg o Mechac ad Theodyac pp [3] K. He K. Joe o Mehod of Phycal Modelg Spge 4. [4] J. We K. He O cldea vaace of algebac Reyold e odel blece J. ld Mech pp [5]. Galleao. Paeo G. aaa A dyac wo-eao Sb Gd Scale odel o Mechac ad Theodyac Vol. 7 No 5 pp. -3. [6] Y. Zag R.. See J.R. Koeff A dyac ed bgd-cale odel ad applcao o ble ecclag flow Phyc of ld A pp [7] B. Vea B Ge H. Kee O he folao of he dyac ed bgdcale odel Phyc of ld pp [8]. Meevea TS d W.H. abo A lagaga dyac bgd-cale odel of blece. J.ld Mech pp [9] V.M. Salve S. Baeee A po e of a ew dyac bgd-cale odel fo fe dffeece lage-eddy lao Phyc of ld pp [] M. Geao U. Poell P. Mo W.H. abo A dyac bgd cale eddy vcoy odel. Phy. ld A3 99 pp [] M. Geao Tblece: he fleg appoach Joal of ld Mechac pp [] K. Teee J.. ley A f coe blece MIT Pe 97. [3] A.S. Mo A.M.Yaglo Sacal ld Mechac: Mechac of Tblece MIT Pe 97. [4]. Sagh U. Poell. Balaa Scalela odel fo lage-eddy lao Phy. ld 999 pp [5] Y. Moh T.S. d O. Valyev P. Mo lly oevave Hghe Ode e Dffeece Schee fo Icopeble low Joal of opaoal Phyc pp [6] O. Valyev Hgh Ode e Dffeece Schee o No-fo Mehe wh Good oevao Popee Joal of opaoal Phyc 57 pp [7] B. Tao J Kaz. Meevea Geoey ad cale elaohp hgh Reyold be blece deeed fo hee-deoal hologaphc velocey Phyc of ld pp [8]. Meevea J Kaz Scale-vaace ad blece odel fo lage eddy lao. A. Rev. ld Mech pp. -3. [9] N.N. Mao R.D. Moe J. K lly developed ble chael flow lao AGARD Advoy Repo 343. [] G. oe-bello Hgh Reyold be chael flow epee AGARD Advoy Repo 343. [] J. Joeg. Ha O he defcao of a voe J.ld Mech pp

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