~ SpringerWienNewYork
CISM COURSES AND LECTURES
Series Editors: The Rectors Giulio Maier - Milan Franz G. Rammerstorfer - Wien Jean Salençon - Palaiseau
The Secretary General Bernhard Schrefler - Padua
Executive Editor Paolo Serafini - Udine
The series presents lecture notes, monographs, edited works and proceedings in the field of Mechanics, Engineering, Computer Science and Applied Mathematics. Purpose of the series is to make known in the international scientific and technical community results obtained in some of the activities organized by CISM, the International Centre for Mechanical Sciences.
INTERNATIONAL CENTRE FOR MECHANICAL SCIENCES COURSES AND LECTURES - No. 517
INSTABILITIES OF FLOWS: WITH AND WITHOUT HEAT TRANSFER AND CHEMICAL REACTION
EDITED BY TAPAN K. SENGUPTA HIGH PERFORMANCE COMPUTING LABORATORY, DEPT. OF AEROSPACE ENGG. IIT KANPUR 208 016, INDIA
THIERRY POINSOT CNRS/INST. DE MECANIQUE DES FLUIDES DE TOULOUSE, TOULOUSE, FRANCE
This volume contains 133 illustrations
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. © 2010 by CISM, Udine Printed in Italy SPIN 12822072
All contributions have been typeset by the authors.
ISBN 978-3-7091-0126-1 SpringerWienNewYork
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>R_] ^RO NS]ZO\]SYX \OVK^SYX PY\ /[X ! S] ^RO ]^K^OWOX^ YP QY` O\XSXQ O[_K^SYX SX ^RO ]ZOM^\KV ZVKXO KXN ^OVV] _] ^RK^ ^RO ]MKVO YP ]ZKMO `K\SK^SYX KXN ^RO ]MKVO YP ^SWO `K\SK^SYX K\O XY^ SXNOZOXNOX^ KXN ^ROc K\O \OVK^ON 0Y\ WKXc Y^RO\ Z\YLVOW] ^RO NS]ZO\]SYX \OVK^SYX aSVV LO MYX]O [_OXMO YP LY_XNK\c MYXNS^SYX] K] S] YP^OX NO\S`ON PY\ aK^O\ aK`O] NO`OVYZ SXQ PY\ KX O[_SVSL\S_W ]YV_^SYX QS`OX Lc ^RO 6KZVKMO] O[_K^SYX /[_K^SYX !" SWZVSO] ^RK^ OKMR P\O[_OXMc MYWZYXOX^ aSVV ^\K`OV SX ]ZKMO aS^R ^RO
# "
MY\\O]ZYXNSXQ ZRK]O ]ZOON +Xc ]c]^OW ^RK^ RK] \OVK^SYX]RSZ ]RYaSXQ ^RK^ ^RO ZRK]O MY\\O]ZYXNSXQ ^Y OKMR aK`O X_WLO\ ^\K`OV] Lc ^RO ]KWO ]ZOON S] MKVVON K =>= 38B?4AB8E4 BHBC4< 3X MYX^\K]^ K 38B?4AB8E4 BHBC4< S] YXO PY\ aRSMR NSO\OX^ P\O[_OXMc MYWZYXOX^] ^\K`OV aS^R NSO\OX^ ]ZOON ]_MR ^RK^ KXc MYWZKM^ NS]^_\LKXMO K^ H ) aSVV LO PY_XN NS]ZO\]ON K^ K VK^O\ ^SWO 9XO MKX QOXO\KVSdO ^RO KLY`O PY\ K ]c]^OW ^RK^ NS]ZVKc] L\YKNLKXN NcXKWSM] 9XMO KQKSX PY\ ]_MR K QOXO\KV ]c]^OW ^RO QY`O\XSXQ NSO\OX^SKV O[_K ^SYX] KXN ^RO LY_XNK\c MYXNS^SYX] aSVV NO^O\WSXO ^RO 38B?4AB8>= A4;0C8>= 9XMO aO RK`O ^RS] \OVK^SYX S^ S] ZY]]SLVO ^Y ^KVU KLY_^ ^RO QOXO\KV Z\YZO\ ^SO] YP ^RO \O]ZYX]O OVN 6SQR^RSVV &$% 6O^ _] MYX]SNO\ ^aY XOSQRLY\SXQ aK`O X_WLO\] ? KXN ? YP ^RO \O]ZYX]O OVN 0Y\ ^RO ZRc]SMKV Z\YLVOW \O]ZYX]O OVN NOZOXN] MYX^SX_Y_]Vc YX ^RO PY\MSXQ KXN K_bSVSK\c MYXNS^SYX] 2OXMO ^RO MY\\O]ZYXNSXQ MS\M_VK\ P\O[_OXMSO] KXN K] YL^KSXON P\YW ^RO 38B?4AB8>= A4;0C8>= aSVV KV]Y LO ^aY MVY]OVc ]ZKMON XOSQRLY\SXQ ZYSX^] SX ^RO P\O[_OXMc ZVKXO AO NOXY^O ^RO MY\\O]ZYXNSXQ \O]ZYX]O MYWZYXOX^] Lc : ) 5 MY]? L H
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: ) 5 MY]? L H
!#L
AO MYX]SNO\ RO\O YXVc ^RO \OKV ZK\^ YP ^RO \O]ZYX]O OVN + WY\O QOXO\ KVSdON \O]ZYX]O LK]ON YX ^RO ObZ\O]]SYX ! aSVV RK`O KV]Y KX SWKQSXK\c MYWZYXOX^ ]SQXSPcSXQ ^RO ZRK]O ]RSP^ YP ^RO Y_^Z_^ aS^R \O]ZOM^ ^Y KZZVSON SXZ_^ 3X ^RO KLY`O O[_K^SYX] LY^R ^RO MYWZYXOX^] RK`O LOOX K]]_WON ^Y RK`O ]KWO KWZVS^_NO ^RO NSO\OXMO LOSXQ XOQVSQSLVO K] aO K\O MYX]SNO\ SXQ ^aY MYWZYXOX^] ^RK^ K\O ]OZK\K^ON Lc KX SX XS^O]SWKV KWY_X^ SX ^RO ]ZOM^\KV ZVKXO >RO ^Y^KV MYX^\SL_^SYX MYWSXQ P\YW ^RO]O ^aY MYWZYXOX^] S] ^ROX QS`OX Lc : ) : : ) E 5 MY] ? ? L H G MY] ? ? L H !$ >Y^KV OOM^ YP ^RO]O ^aY MYWZYXOX^] ]RYa] _Z K] K MYWLSXK^SYX YP ^aY PKM^Y\] YXO YX K ]VYa ]MKVO QS`OX Lc ^RO \]^ [_KX^S^c SX]SNO ^RO ][_K\O L\KMUO^ KXN ^RO ]OMYXN [_KX^S^c ^RK^ `K\SO] YX ^RO Y\SQSXKV ]MKVO QS`OX Lc ^RO ]OMYXN MY]SXO P_XM^SYX +] ^RO ^aY MYWZYXOX^] K\O SX XS^O]SWKVVc KZK\^ SX ^RO ]ZOM^\KV ZVKXO ^RO ]VYaVc `K\cSXQ ZK\^ MKX LO `SOaON K] ^RO KWZVS^_NO YP ^RO ^Y^KV OOM^ +] OXO\Qc YP K aK`O ]c]^OW S] Z\YZY\^SYXKV ^Y ^RO ][_K\O YP ^RO KWZVS^_NO YP ^RO aK`O YXO MKX `SOa ^RO OXO\Qc YP ^RO ]c]^OW ^Y `K\c PYVVYaSXQ ^RO ZRK]O `K\SK^SYX YP ^RO KWZVS^_NO >RO\OPY\O ^RO ]ZOON
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YP Z\YZKQK^SYX YP OXO\Qc S] QS`OX Lc ^RO \K^O K^ aRSMR ^RO ZRK]O YP ^RO KWZVS^_NO MRKXQO] KXN S] QS`OX Lc
03 )
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>RS] S] ^RO Q\Y_Z `OVYMS^c YP ^RO ]SWZVO ]c]^OW MYX]SNO\ON RO\O aRO\O ^RO ]ZK^SKV `K\SK^SYX S] KVYXQ K ]SXQVO NS\OM^SYX KXN ^RO Q\Y_Z `OVYMS^c NS\OM^SYX S] QS`OX KVYXQ ^RS] NS\OM^SYX 0Y\ ^RO YXONSWOX]SYXKV ]c]^OW SX ^RO VSWS^ aROX ^RO aK`O X_WLO\] YP ^RO ^aY MYWZYXOX^] K\O YXVc 8? KZK\^ SX ? ]ZKMO ^ROX YXO MKX MYX]SNO\ KX O[_S`KVOX^ \O]ZYX]O OVN MOX^O\ON K\Y_XN ^RS] MV_]^O\ KXN K QOXO\KV ObZ\O]]SYX PY\ ^RO Q\Y_Z `OVYMS^c S] QS`OX Lc 8 !% 8? >RO Q\Y_Z `OVYMS^c PY\ W_V^SNSWOX]SYXKV Z\YLVOW S] K `OM^Y\ NOMSNON Lc ^RO `K\SK^SYX YP MS\M_VK\ P\O[_OXMc aS^R ^RO aK`O X_WLO\ `OM^Y\ 3^ S] YXVc ^RO \OKV ZK\^ YP /[X !% ^RK^ S] ^O\WON ^RO Q\Y_Z `OVYMS^c K] NS]M_]]ON SX ARS^RKW &$% >R_] ^RS] S] ^RO `OVYMS^c K^ aRSMR ^RO OXO\Qc YP K Q\Y_Z YP aK`O] ^\K`OV MOX^O\ON KLY_^ ^RO WSNNVO YP ^RO aK`O X_WLO\ Q\Y_Z 3^ S] XY^ON ^RK^ ^RO YXONSWOX]SYXKV aK`O QS`OX Lc /[X ! S] XYXNS]ZO\]S`O aS^R 03 ) 7
,DB: #CHI67>A>IN &:8=6C>HBH
2O\O ^aY ]SWZVO MK]O] YP SX]^KLSVS^SO] K\O MYX]SNO\ON ^Y OWZRK]SdO ^RO MYXMOZ^] NO]M\SLON KLY`O AO LOQSX Lc NS]^SXQ_S]RSXQ ^RO NSO\OXMO LO^aOOX ]^K^SM KXN NcXKWSM SX]^KLSVS^c Lc MYX]SNO\SXQ ^RO ]^KLSVS^c YP K^WY]ZRO\O K] KX ObKWZVO AROX K ZK\MOV YP KS\ SX ^RO K^WY]ZRO\O S] WY`ON \KZSNVc P\YW KX O[_SVSL \S_W MYXNS^SYX KXN S^] ^OXNOXMc ^Y MYWO LKMU ^Y S^] _XNS]^_\LON ZY]S^SYX S] XY^ON ^ROX aO ^O\W ^RO K^WY]ZRO\O K] ]^K^SMKVVc ]^KLVO >RO WY`OWOX^ YP ^RO ZKMUO^ S] MYX]SNO\ON K] SWZ_V]S`O ^Y Z\OMV_NO KXc ROK^ ^\KX]PO\ P\YW ^RO ZK\MOV ^Y ^RO KWLSOXMO >RS] ^OXNOXMc YP ]^K^SM ]^KLSVS^c aROX ObS]^] S] N_O ^Y ^RO L_YcKXMc PY\MO MK_]ON Lc ^RO NOX]S^c NSO\OX^SKV N_O ^Y ^OWZO\K^_\O `K\SK^SYX aS^R ROSQR^ KXN ]_MR LYNc PY\MO KM^] _ZYX ^RO NS]ZVKMON KS\ZK\MOV
# "
3X ]^K^SM ]^KLSVS^c ]^_NSO] aO NY XY^ VYYU PY\ NO^KSVON ^SWONNOZOXNOX^ WY ^SYX YP ^RO ZK\MOV PYVVYaSXQ ^RO NS]ZVKMOWOX^ K] ^RO K]]YMSK^ON KMMOVO\K^SYX] K\O MYX]SNO\ON XOQVSQSLVO
NC6B>8 ,I67>A>IN D; ,I>AA IBDHE=:G: 0Y\ ^RO NcXKWSM ]^KLSVS^c ]^_Nc aO MYX]SNO\ YXMO KQKSX K ZK\MOV YP KS\ ^Y LO K^ O[_SVSL\S_W K^ K ROSQR^ N KXN S] NS]ZVKMON ^Y K ROSQR^ N ^Y PYVVYa S^] NO^KSVON ^SWO RS]^Y\c YP WY^SYX AO XY^O ^RK^ ^RO WY^SYX S] MK_]ON KQKSX Lc ^RO L_YcKXMc PY\MO MK_]ON Lc ^RO ^OWZO\K^_\O `K\SK^SYX YP ^RO KWLSOX^ KS\ QS`OX Lc . ) . N 2O\O ^RO `O\^SMKV NS]ZVKMOWOX^ YP ^RO KS\ZK\MOV S] KXN ^RO NcXKWSM] PYVVYa] ^RO PY\MO LKVKXMO O[_K^SYX
g ) ; F
"
aRO\O g S] ^RO SX]^KX^KXOY_] KMMOVO\K^SYX ObZO\SOXMON Lc ^RO ZK\MOV YP KS\ .OX]S^c YP ^RO NS]ZVKMON ZK\MOV S] MYX]SNO\ON ^Y LO QS`OX Lc aRSVO ^RO KWLSOX^ _SN RK] ^RO NOX]S^c ]Y ^RK^ ^RO \SQR^ RKXN ]SNO YP ^RO KLY`O O[_K^SYX \OZ\O]OX^] ^RO L_YcKXMc PY\MO /[_SVSL\S_W ^RO\WYNcXKWSM] ^OVV] _] ^RK^ PY\ K ]SWZVO MYWZ\O]]SLVO ]_L]^KXMO aS^R YXVc YXO WYNO YP aY\U KXc ]^K^O Z\YZO\^c MKX LO \OZ\O]OX^ON Lc KXc ^aY Y^RO\ Z\YZO\^SO] KXN VO^ _] MYX]SNO\ ^ROW ^Y LO ^RO Z\O]]_\O D KXN ^RO OX^\YZc G >R_] _]SXQ K >KcVY\ ]O\SO] aO MKX \OVK^O ^RO NOX]S^c YP ^RO KWLSOX^ KS\ K^ ^RO ^aY ROSQR^] K]
N ) N D
EDN DNG G ?
EGN GNG " <
9XMO KQKSX aO aSVV K]]_WO ^RK^ ^RO NS]ZVKMOWOX^ Z\YMO]] YP ^RO KS\ ZK\MOV S] S]OX^\YZSM ^RO\O K\O XY `S]MY_] Y\ ROK^ VY]]O] K]]YMSK^ON aS^R ^RO \KZSN WY`OWOX^ YP ^RO ZK\MOV KXN ^R_] PY\ ^RO KS\ZK\MOV EDN DNG "
N ) N D ? 3X /[X] " KXN " WOMRKXSMKV O[_SVSL\S_W OX]_\O] ]KWO D KXN N ) N >R_] ^RO NOX]S^c NSO\OX^SKV MK_]SXQ ^RO L_YcKXMc S] QS`OX Lc 8G "! EGN GNG ) F ) G < G < 8N AO MKX KV]Y \OVK^O ^RO NOX]S^c YP ^RO KS\ZK\MOV K^ ^RO ^aY ROSQR^] K]
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N ) N
D
8D 8D ) N 7 8N ? 8N
aRO\O 7 S] ^RO ]ZOON YP ]Y_XN /[_K^SYX " MKX KV]Y LO a\S^^OX SX ^O\W] YP ^RO ]ZOMS M `YV_WO J ) _]SXQ /[X "! K] J 8D ; J 8G ; g ) F )
J G < 8N 7 8N 0\YW ^RO WOMRKXSMKV O[_SVSL\S_W ZVS ON ^Y
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8. ; ; 8N < 7
3P aO P_\^RO\ MYX]SNO\ ^RO ]ZOON YP ]Y_XN 7 ^Y LO `O\c VK\QO ^ROX ^RO KLY`O O[_K^SYX MKX LO P_\^RO\ KZZ\YbSWK^ON ^Y
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AO MKX MYX]SNO\ ^RO PYVVYaSXQ ZY]]SLSVS^SO]' aRO\O ( -K]O ' 3P ( ^ROX ^RO NcXKWSM] YP ^RO NS]ZVKMOWOX^ aSVV LO Z_\OVc Y]MSVVK^Y\c SWZVcSXQ XO_^\KV ]^KLSVS^c YP ^RO ]^K^SM K^WY]ZRO\O -K]O ' 3P ( ^ROX ^RO `O\^SMKV NS]ZVKMOWOX^ aSVV `K\c K] H ) 9 ! @ 9 ! @ aRO\O ^RO \]^ MYWZYXOX^ MVOK\Vc SXNSMK^O] SX]^KLSVS^c ( S] MKVVON ^RO AD=C EJ 08BJ 0;J 0 Y\ L_YcKXMc P\O[_OXMc +] QS`OX KLY`O PYVVYaSXQ >RYWZ ]YX &$ aO MKX YL^KSX ^RS] P\O[_OXMc aS^R KS\ ^\OK^ON K] KX SNOKV QK] Lc ; ; 8. ( ) "# . 8N < 0Y\ N\c KS\
3
) 5 WO^O\ KXN ROXMO PY\ ]^KLSVS^c YP N\c K^WY
]ZRO\O ^RO ^OWZO\K^_\O NS]^\SL_^SYX RK] ^Y LO ]Y ^RK^ 0' 0F 5 WO^O\ 0' >R_] 0F ) \OZ\O]OX^] ^RO LY\NO\ VSXO YP SX]^KLSVS^c KXN ^RO X_WO\SMKV `KV_O YX ^RO \SQR^ RKXN ]SNO aS^RSX ^RO ][_K\O L\KMUO^ S] ^O\WON K] ^RO N\c 0? KNSKLK^SM VKZ]O \K^O LOMK_]O ^RS] OX]_\O] 0F )
$:AK>C ":AB=DAIO #CHI67>A>IN >RS] K\S]O] aROX ^aY VKcO\] YP _SN] WKc XY^ LO YP ]KWO ]ZOMSO] Y\ NOX ]S^c K\O SX \OVK^S`O WY^SYX >R_] ^RS] S] KX SX^O\PKMSKV SX]^KLSVS^c KXN ^RO \O]_V^KX^ Ya POK^_\O] N_O ^Y SWZY]ON NS]^_\LKXMO aSVV LO W_MR WY\O MYW ZVSMK^ON N_O ^Y \OVK^S`O WY^SYX :Rc]SMKV \OVO`KXMO YP ^RS] Z\YLVOW aK] ]OSdON _ZYX Lc 2OVWRYV^d %#% aRY YL]O\`ON ^RK^ ^RO SX^O\PKMO K] K ]_\ PKMO YP ]OZK\K^SYX ^OK\] ^RO Ya 0BD=34A =YWO^SWO VK^O\ 5OV`SX %$ ZY]ON ^RS] Z\YLVOW K] YXO YP SX]^KLSVS^c KXN ]YV`ON S^ AO PYVVYa ^RS] VK^^O\ KZZ\YKMR RO\O >RO LK]SM O[_SVSL\S_W Ya S] K]]_WON ^Y LO SX`S]MSN KXN SXMYWZ\O]]SLVO K] ^aY ZK\KVVOV ]^\OKW] RK`SXQ NS]^SXM^ NOX]S^c KXN `OVYMS^c YaSXQ YXO Y`O\ ^RO KXY^RO\ K] NOZSM^ON SX Q_\O LOVYa ,OPY\O KXc ZO\^_\LK^SYX S] KZZVSON ^RO SX^O\PKMO S] VYMK^ON K^ N ) KXN ]_L]O[_OX^ NS]ZVKMOWOX^ YP ^RS] SX^O\PKMO S] ObZ\O]]ON ZK\KWO^\SMKVVc K] N? ) HL M H ) L M H
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#
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x Fluid-1
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5F9 6CIB898 5G N
"
>RO Y^RO\ ]O^ YP LY_XNK\c MYXNS^SYX S] KZZVSON K^ ^RO SX^O\PKMO aRSMR S] ^RO XY _SN ^R\Y_QR ^RO SX^O\PKMO MYXNS^SYX SO H f6 H f6 H f6 ) " H N L L M M 3X KNNS^SYX SX ^RO KL]OXMO YP ]_\PKMO ^OX]SYX Z\O]]_\O W_]^ LO MYX ^SX_Y_] KM\Y]] ^RO SX^O\PKMO ?ZYX VSXOK\SdK^SYX ^RO SX^O\PKMO LY_XNK\c MYXNS^SYX " ]SWZVS O] ^Y 6 /6 ) : CF > ) " H L N aRO\O f6 KXN 6 K\O K] \OVK^ON SX /[X "% .O XSXQ ^RO Z\O]]_\O YX ^RO OS^RO\ YaNYWKSX Lc _X]^OKNc ,O\XY_VVS] O[_K^SYX YXO MKX a\S^O f 6 D6 ) 6 6 f6 ; H " H
# "
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?]SXQ /[X " " SX ^RO SX^O\PKMO LY_XNK\c MYXNS^SYX " YXO QO^] I = /
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>RS] Y\NSXK\c NSO\OX^SKV O[_K^SYX PY\ ^RO ^SWO `K\SK^SYX YP ^RO SX^O\PKMO NS]ZVKMOWOX^ MKX LO _XNO\]^YYN LO^^O\ SX ^O\W] YP S^] 0Y_\SO\ ^\KX]PY\W NO XON Lc H 95@ 8 H ) " 9XO YL^KSX] ^RO PYVVYaSXQ NS]ZO\]SYX \OVK^SYX Lc ]_L]^S^_^SYX YP " SX " K] / / ;? / / )
"
>RS] Z\Y`SNO] ^RO MRK\KM^O\S]^SM ObZYXOX^] SX " K]
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,K]ON YX ^RS] NS]ZO\]SYX \OVK^SYX ^RO PYVVYaSXQ ]_LMK]O] MKX LO MYX]SN O\ON' -+=/ ' AROX ^RO SX^O\PKMO S] NS]^_\LON SX ^RO ]ZKXaS]O NS\OM^SYX YXVc SO ) KXN ^ROX ) ;
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# ' !""'
0.4
2P /We
0.3 0.2
Ae
Cl
0.1 0 -0.1 -0.2 -0.3 -0.4
0
100
t
200
300
200
300
Cd
-1.1
-1.2
-1.3
te -1.4
0
100
t
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# "
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550 500 450
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400 350 300 250 200 150 50
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><JG: 'DC 9>B:CH>DC6A I>B: 6I 6BEA>IJ9: H6IJG6I>DC KH +:NCDA9H CJB7:G
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0.6
Present Results (By DNS) Present Results (By Eq. 5.4.2) Norberg (2003)
0.5
Ae
0.4
0.3
0.2
Recr1 = 51.934 Recr2 = 63.868
0.1
0
Recr3 = 84.615
50
100
150 Re
200
250
><JG: /6G>6I>DC D; ,6IJG6I>DC 6BEA>IJ9: D; A>;I 8D:8>:CI L>I= +:NCDA9H CJB7:G
&& SXNSMK^O K ]WYY^R `K\SK^SYX YP > KM\Y]] ,9/> K] aOVV K] ,9/> S^ S] ^RO\OPY\O MY\\OM^ ^Y ]^K^O ^RK^ ^RO NS]MYX^SX_Y_] MRKXQO] YL]O\`ON SX ^RO KWZVS^_NO M_\`O K\O N_O ^Y \KZSN MRKXQO] K]]YMSK^ON aS^R NOM\OK]SXQ `KV_O YP ^RO 6KXNK_ MYO MSOX^ >R_] ^RO [_KVS^K^S`O MRKXQO KM\Y]] M\S^SMKV
_SN NcXKWSMKV ]c]^OW 3^ S] KV]Y XY^ON ^RK^ ? ? KXN ? K\O Y\NO\] YP WKQXS^_NO NSO\OX^ P\YW ? KXN K\O _XSWZY\^KX^ XOK\ \O]ZOM^S`O ,9/> KXN YXVc ROVZ] SX ObZVKSXSXQ ]WYY^R ^\KX]S^SYX P\YW YXO SX^O\`KV YP ) " & ! .SO\OXMO LO^aOOX ^RS] `KV_O KXN ^RO Y^RO\ `KV_O] QS`OX Lc 4KMU]YX &%$ DOLSL &%$ KXN 7Y\dcX]US >RSOVO && S] O]]OX^SKVVc N_O ^Y NSO\OX^ O\\Y\ ]Y_\MO] YP NSO\OX^ WO^RYN] KM M_W_VK^SXQ ^Y ^\SQQO\ `Y\^Ob ]RONNSXQ SX ^RO aKUO +X KMM_\K^O WO^RYN
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3X /[X "!! ^RO 6KXNK_ MYO MSOX^ @> S] ^\_Vc K MYX]^KX^ SXNOZOXNOX^ YP KLVO " 3X 0SQ " MYWZ_^ON 1 S] KV]Y MYWZK\ON aS^R ^RO MY\\OVK^SYX Z\YZY]ON Lc 8Y\LO\Q LK]ON YX RS] ObZO\SWOX^KV \O]_V^] 0Y\ RS] ObZO\SWOX^KV PKMSVS^c ,9/> ) !$ KXN SX ^RO \KXQO !$ ,9 & K]cWZ^Y^SM KWZVS^_NO `K\SK^SYX \W] aS^R RO WKSX ^\OXN] KZZOK\ ^Y LO ]SWSVK\ aS^R NS]MYX^SX_S^SO] SXNSMK^ON SX ^RO \KXQO #" ,9 & SX 8Y\LO\Q] MY\\OVK^SYX N_O ^Y MRKXQO P\YW ZK\KVVOV ^Y YLVS[_O ]RONNSXQ SX ^RO aKUO ]OO ASVVSKW]YX &%& =YWO SX`O]^SQK^Y\] RK`O ]^_NSON ^RO OOM^] YP XYS]O YX ZS^MRPY\U KXN 2YZP LSP_\MK^SYX KXN \OZY\^ON XNSXQ MRKXQON M\S^SMKV ZK\KWO^O\] 0Y\ ObKWZVO 4_OV .K\Lc]RS\O 7_VVSX &&$ \OZY\^ON YXVc MRKXQO SX ^RO ZK\KW O^O\ `KV_O ^YaK\N] ]_LM\S^SMKVS^c aROX WOK]_\ON aS^R \O]ZOM^ ^Y 6KXNK_ WYNOV >RS] S] SXNOON K `O\c ]WKVV MRKXQO K] MYWZK\ON ^Y ObZO\SWOX^KV `KV_O YP M\S^SMKV ZK\KWO^O\ ^RK^ S] WY\O ^RKX " PY\ ^RO Ya ZK]^ K McVSX NO\ >RO O]^SWK^O] SX 4_OV .K\Lc]RS\O 7_VVSX &&$ K\O aS^R \O]ZOM^ ^Y aRS^O XYS]O ^RK^ S] RK\NVc ^cZSMKV YP ObZO\SWOX^KV PKMSVS^SO] 3X^O\O]^ SXQVc ASO]OXPOVN 7M8KWK\K &%# RK`O \OZY\^ON ^RK^ XOK\ ^RO YX]O^ YP NcXKWSM SX]^KLSVS^c KXc ^SWOZO\SYNSM ]c]^OW MKX KM^ ^Y KWZVSPc ]WKVV ZO \SYNSM ZO\^_\LK^SYX] aRY]O NO^KSV] NOZOXN ]YVOVc YX ^RO ^cZO YP LSP_\MK^SYX SX`YV`ON 3X K ^_XXOV Z\O]OXMO YP ]_MR ZO\SYNSM ZO\^_\LK^SYX] MKX XY^ LO
&
# ' !""'
0.21 Present Results (By DNS) Norberg (2003) Henderson (1997)
0.2 0.19 0.18
St
0.17 0.16 0.15 0.14 0.13 0.12 0.11
50
100
150
200
250
Re
><JG: /6G>6I>DC D; ,IGDJ=6A CJB7:G L>I= +:NCDA9H CJB7:G
\_VON Y_^ KXN ^RK^ MKX ObZVKSX K aRYVO \KXQO YP ,9/> \OZY\^ON ObZO\SWOX ^KVVc ,K]ON YX ^RS] NS]M_]]SYX YXO MKX MYXTOM^_\O ^RK^ ^RO KZZOK\KXMO YP ,9/> KXN ,9/> K\O N_O ^Y ^RO Z\O]OXMO YP ]_MR ]WKVV ZO\SYNSM ZO\^_\LK^SYX] 3X 2YWKXX] ObZO\SWOX^ ^RO `S]MY_] NS]]SZK^SYX YP ^RO aY\USXQ _SN MY_VN NOVKc `Y\^Ob ]RONNSXQ aRSVO SX =^\cUYa]US =\OOXS`K]KX && Z\O]OXMO YP ]WKVV MYX^\YV McVSXNO\ KM^ON K] K ]Y_\MO YP NS]]SZK^SYX ^RK^ ROVZON SX ]_Z Z\O]]SYX YP ]RONNSXQ ^Y RSQRO\ R_] ^RO ObS]^OXMO YP KXc Y^RO\ M\S^SMKV S] YP VO]]O\ SX^O\O]^ 3X 0SQ "! ^RO MYWZ_^ON =^\Y_RKV X_WLO\ S] ]RYaX ZVY^^ON KQKSX]^ RO OWZS\SMKV MY\\OVK^SYX] K\O QS`OX
Lc -H ) ## PY\ !$ ,9 & KXN -H ) %& %1 %1 $% ,9 PY\ #" ,9 # >R_] S^ S] `O\c KZZK\OX^ ^RK^ SX ^RO ObZO\SWOX^KV PKMSVS^c YP 8Y\LO\Q ^RO =^\Y_RKV X_WLO\ `K\SK^SYX S] NS]MYX^SX_Y_] Y\ XYX_XS[_O SX ^RO \KXQO #" ,9 & >RS] S] N_O ^Y ^RO NSO\OXMO LO^aOOX ZK\KVVOV KXN YLVS[_O ]RONNSXQ SX ^RO aKUO ^RK^ RK] KV]Y
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1
0.96
Present Results (We obtained by DNS) Morzynski & Thiele (1993) (W by linear stability theory)
W
0.92
0.88
0.84
0.8 50
60
70
80
Re
><JG: >G8JA6G ;G:FJ:C8N D; A>C:6G >CHI67>A>IN 6C9 >IH CDCA>C:6G HJE:G8G>I>86A H6IJG6I>DC EADII:9 6H 6 ;JC8I>DC D; +:NCDA9H CJB7:G
LOOX KNN\O]]ON SX 2KWKMRO 1RK\SL && KXN ASVVSKW]YX &%& 9_\ MKVM_VK^ON =^\Y_RKV X_WLO\ NS]ZVKc] K NS]^SXM^ ]VYZO NS]MYX^SX_S^c K\Y_XN ,9 # KXN S] MYX]S]^OX^ aS^R /[ ! ^RK^ ]RYa] ^RO XYXVSXOK\ OOM^ YP ^RO =^\Y_RKV X_WLO\ ^R\Y_QR S^] NOZOXNOXMO YX KWZVS^_NO 3^ S] SX^O\O]^SXQ ^Y XY^O ^RK^ >Y\NOVVK -KXMOVVS && RK`O \OZY\^ON K =^\Y_RKV X_WLO\ NS]MYX^SX_S^c K^ K\Y_XN ,9 & ^RK^ aK] KV]Y \OZY\^ON Lc >\S^^YX &$ >RO XYXVSXOK\ OOM^] YX `Y\^Ob ]RONNSXQ S] KV]Y NOWYX]^\K^ON SX 0SQ "" aRO\O ^RO Z\O]OX^ MYWZ_^ON =^\Y_RKV X_WLO\ S] MYWZK\ON aS^R VSXOK\ QVYLKV ]^KLSVS^c \O]_V^] YP 7Y\dcX]US >RSOVO && 2K`SXQ YL^KSXON 1 Lc NS\OM^ ]SW_VK^SYX YXO MKX O]^SWK^O ^RO SWKQSXK\c ZK\^ YP ^RO 6KXNK_ MYO MSOX^ Lc _]SXQ /[ " OWZVYcSXQ ^RO VSXOK\ ]^KLSVS^c \O]_V^] YP 7Y\dcX]US >RSOVO && >RO]O \O]_V^] PY\ ^RO `K\SK^SYX YP @5 aS^R
&
# ' !""'
20
15
li
10
5
0
-5 Recr1 = 51.934
-10 50
60
70
80
Re
><JG: /6G>6I>DC D; #B6<>C6GN E6GI D; %6C96J 8D:8>:CI L>I= +:NCDA9H CJB7:G
VSXOK\ ]^KLSVS^c \O]_V^] K\O K`KSVKLVO 3X PYVVYaSXQ ^RS] aO K\O K]]_WSXQ ^RK^ ^RO QVYLKV VSXOK\ ]^KLSVS^c KXKVc]S] aY_VN Z\YN_MO SNOX^SMKV X_WLO\] PY\ > KXN PY\ NSO\OX^ WO^RYN] NYWKSX] KXN Q\SN] 2YaO`O\ ^RO [_KVS^K^S`O ^\OXN] PY\ ^RO `K\SK^SYX YP @5 aS^R S] WK\UON PY\ \OPO\OXMO K^ aRSMR @5 ]RY_VN LO SX XS^OVc VK\QO 3^ S] KV]Y XY^ON ^RK^ PY\ WY]^ YP ^RO RO]O O[_K^SYX] K\O \OSX^O\Z\O^ON RO\O KXN WYNS MK^SYX] Z\YZY]ON SX ^RO VSQR^ YP W_V^SZVO LSP_\MK^SYX] +V]Y ^RO KWZVS^_NO KXN ^RO ZRK]O `K\SK^SYX aS^R
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YX RS] ObZO\SWOX^KV NK^K RSOVO && K\O _]ON SX YL^KSXSXQ ^RO `K\SK^SYX YP SWKQSXK\c ZK\^ YP 6KXNK_ MYO MSOX^ + NS]MYX^SX_S^c PY_XN ObZO\SWOX^KVVc K^ VYa
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Ya] ^RO SXN_MON LYNc PY\MO] K\O OS^RO\ ZK\KVVOV Y\ KX^SZK\KVVOV ^Y ^RO WOKX MYX`OM^SYX NS\OM^SYX K] SX ^RO MK]O YP WSbON MYX`OM^SYX Ya ZK]^ `O\^SMKV ZVK^O] 2YaO`O\ ^RO Ya KXN ROK^ ^\KX]PO\ Z\YZO\^SO] K\O WY\O MYWZVSMK^ON aROX WSbON MYX`OM^SYX Ya] K\O MYX]SNO\ON ZK]^ SXMVSXON Y\ RY\SdYX^KV ZVK^O] >RS] S] N_O ^Y ^RO PKM^ ^RK^ ^RO L_YcKXMc PY\MO] SXN_MO K VYXQS^_ NSXKV Z\O]]_\O Q\KNSOX^ ^RK^ aSVV NS\OM^Vc KV^O\ Ya KXN ROK^ ^\KX]PO\ \K^O] >RS] S] \OKNSVc O`SNOX^ Lc VYYUSXQ K^ /[X] # KXN # NO]M\SLSXQ ^RO WOKX Ya +NNS^SYXKVVc ]_MR SWZY]ON Z\O]]_\O Q\KNSOX^ KV]Y KV^O\] ^RO SX]^KLSVS^c ZK^^O\X 0Y\ ObKWZVO /MUO\^ =YORXQOX &" K] \OZY\^ON SX ,\Oa]^O\ 1OLRK\^ && RK`O ]RYaX _]SXQ `S]_KVSdK^SYX ^RK^ ^\KX]S^SYX ^Y ^_\L_VOXMO PY\ XK^_\KV MYX`OM^SYX Z\YLVOW PYVVYa] ^RO SX]^KLSVS^c YP ]WKVV NS]^_\LKXMO] 0Y\ XK^_\KV MYX`OM^SYX ZK]^ SXMVSXON K^ ZVK^O] =ZK\\Ya 2_]K\ &#& RK`O \OZY\^ON QOXO\K^SYX YP KX K\\Kc YP VYXQS^_NSXKV `Y\^SMO] 6VYcN =ZK\\Ya &$ D_O\MRO\ 4C 0; &&% RK`O ObZO\SWOX^KVVc SX`O] ^SQK^ON SX]^KLSVS^c YP Ya ZK]^ SXMVSXON ZVK^O KXN \OZY\^ON Z\O]OXMO YP ^aY WYNO] NOZOXNSXQ YX ^RO SXMVSXK^SYX KXQVO YP ^RO ZVK^O AROX ^RO SXMVSXK^SYX KXQVO aS^R \O]ZOM^ ^Y ^RO `O\^SMKV S] VO]] ^RKX !; ^ROX ^RO K_^RY\] \OZY\^ON aK`OVSUO SX]^KLSVS^c =_MR aK`OVSUO SX]^KLSVS^c RK] KV]Y LOOX ]^_NSON SX -ROX 7Y_^]YQV_ &$& 0Y\ SXMVSXK^SYX KXQVO] Q\OK^O\ ^RKX $; PY\ ^RO XK^_\KV MYX`OM^SYX Z\YLVOW S^ S] XY^ON ObZO\SWOX^KVVc ^RK^ ^RO NS]^_\LKXMO OVN S] NYWSXK^ON Lc `Y\^SMO] KXN S] YP^OX ^O\WON K] `Y\^Ob SX]^KLSVS^c >RS] `Y\^Ob WYNO YP SX]^KLSVS^c S] Z\O]OX^ PY\ LY^R ^RO RY\SdYX^KV KXN SXMVSXON ZVK^O] >RS] RK] LOOX `K\SY_]Vc ]^_NSON SX 2KKVKXN =ZK\\Ya &$ A_ -ROXQ &$# =RK_UK^_VVKR 1OLRK\^ &$% 1SVZSX 4C 0; &$% 7Y_^]YQV_ 4C 0; &% KXN AKXQ &% KWYXQ WKXc Y^RO\ ]_MR ]^_NSO] 3X KX KV^O\ XK^O `SOaZYSX^ 2KVV 7Y\\S] && MVK]]SPc SX]^KLSVS^c YP PY\MONMYX`OM^SYX LY_XNK\c VKcO\] Y`O\ RY\SdYX^KV ROK^ON ZVK^O] SX ^O\W] YP ^RO ^aY Z\Y^Y^cZ SMKV SX]^KLSVS^SO]' YVVWSOX =MRVSMR^SXQ ^cZO ^RK^ S] ^cZSMKV YP S]Y^RO\WKV YZOX Ya] K] SX aKVV LY_XNON ]ROK\ VKcO\] ^\SQQO\ON Lc `S]MY_] SX]^KLSVS^c 2KVV 7Y\\S] && SX ^ROS\ ^ROY\O^SMKV KXKVc]S] \OZY\^ ^RO ]SW_V^KXOY_] Z\O]OXMO YP LY^R ^RO WYNO] ^RK^ MYKVO]MO SX ^RO
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FROM SWIRLER
X= 1,5 mm X= 5 mm X= 15 mm X= 25 mm X= 35 mm
TO OUTLET
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#!" !
40
40
20
20
20
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20
0
-20
0
-20
-40
-20
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-30
30
0
Ux [m/s]
30
30
0
-40
-30
Ux [m/s]
x=5 mm
LES LDV
-20
-40
-30
Ux [m/s]
x=1.5 mm
0
-20
-40
-30
Distance from axis [mm]
40
Distance from axis [mm]
40
Distance from axis [mm]
40
Distance from axis [mm]
Distance from axis [mm]
"!
30
-30
Ux [m/s]
x=15 mm
30 Ux [m/s]
x=25 mm
x=35 mm
40
40
20
20
20
20
20
0
-20
0
-20
-40
-20
-40
0
5 10 15 Ux' [m/s]
x=1.5 mm
20
0
5 10 15 Ux' [m/s]
x=5 mm
20
0
-20
-40
0
Distance from axis [mm]
40
Distance from axis [mm]
40
Distance from axis [mm]
40
Distance from axis [mm]
Distance from axis [mm]
><JG: +`O\KQO KbSKV `OVYMS^c Z\Y VO] -S\MVO]' 6.@( ]YVSN VSXO' 6/=
5 10 15 Ux' [m/s]
x=15 mm
20
0
-20
-40
0
LES LDV
-40
0
5 10 15 Ux' [m/s]
x=25 mm
20
0
5 10 15 Ux' [m/s]
20
x=35 mm
><JG: <7= KbSKV `OVYMS^c Z\Y VO] -S\MVO]' 6.@( ]YVSN VSXO' 6/=
""
40
40
20
20
20
20
20
0
-20
0
-20
-40
-20
-40
-30
30
0
Ut [m/s]
30
30
0
-40
-30
Ut [m/s]
x=5 mm
LES LDV
-20
-40
-30
Ut [m/s]
x=1.5 mm
0
-20
-40
-30
Distance from axis [mm]
40
Distance from axis [mm]
40
Distance from axis [mm]
40
Distance from axis [mm]
Distance from axis [mm]
! "
30
-30
Ut [m/s]
x=15 mm
30 Ut [m/s]
x=25 mm
x=35 mm
40
40
20
20
20
20
20
0
-20
0
-20
-40
-20
-40
0
5 10 15 Ut' [m/s]
x=1.5 mm
20
0
5 10 15 Ut' [m/s]
x=5 mm
20
0
-20
-40
0
Distance from axis [mm]
40
Distance from axis [mm]
40
Distance from axis [mm]
40
Distance from axis [mm]
Distance from axis [mm]
><JG: +`O\KQO ^KXQOX^SKV `OVYMS^c Z\Y VO] -S\MVO]' 6.@( ]YVSN VSXO' 6/=
5 10 15 Ut' [m/s]
x=15 mm
20
0
-20
-40
0
LES LDV
-40
0
5 10 15 Ut' [m/s]
x=25 mm
20
0
5 10 15 Ut' [m/s]
20
x=35 mm
><JG: <7= ^KXQOX^SKV `OVYMS^c Z\Y VO] -S\MVO]' 6.@( ]YVSN VSXO' 6/=
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40
40
20
20
20
20
20
0
-20
0
-20
-40
-20
-40
-30
0 30 Ux [m/s]
60
0 30 Ux [m/s]
60
0 30 Ux [m/s]
60
0
-40
-30
x=15 mm
LES LDV
-20
-40
-30
x=5 mm
0
-20
-40
-30
x=1.5 mm
0
Distance from axis [mm]
40
Distance from axis [mm]
40
Distance from axis [mm]
40
Distance from axis [mm]
Distance from axis [mm]
! "
0 30 Ux [m/s]
60
-30
x=25 mm
0 30 Ux [m/s]
60
x=35 mm
40
40
20
20
20
20
20
0
-20
0
-20
-40
-20
-40
0
5 10 15 Ux' [m/s]
x=1.5 mm
20
0
5 10 15 Ux' [m/s]
x=5 mm
20
0
-20
-40
0
Distance from axis [mm]
40
Distance from axis [mm]
40
Distance from axis [mm]
40
Distance from axis [mm]
Distance from axis [mm]
><JG: 7OKX KbSKV `OVYMS^c SX ^RO MOX^\KV ZVKXO -S\MVO]' 6.@( ]YVSN VSXO' 6/=
5 10 15 Ux' [m/s]
x=15 mm
20
0
-20
-40
0
LES LDV
-40
0
5 10 15 Ux' [m/s]
x=25 mm
20
0
5 10 15 Ux' [m/s]
20
x=35 mm
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#!" !
40
40
20
20
20
20
20
0
-20
0
-20
-40
-20
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-30
0 Ut [m/s]
30
0 Ut [m/s]
30
0 Ut [m/s]
30
0
-40
-30
x=15 mm
LES LDV
-20
-40
-30
x=5 mm
0
-20
-40
-30
x=1.5 mm
0
Distance from axis [mm]
40
Distance from axis [mm]
40
Distance from axis [mm]
40
Distance from axis [mm]
Distance from axis [mm]
#
0 Ut [m/s]
30
-30
x=25 mm
0 Ut [m/s]
30
x=35 mm
40
40
20
20
20
20
20
0
-20
0
-20
-40
-20
-40
0
5 10 15 Ut' [m/s]
x=1.5 mm
20
0
5 10 15 Ut' [m/s]
x=5 mm
20
0
-20
-40
0
Distance from axis [mm]
40
Distance from axis [mm]
40
Distance from axis [mm]
40
Distance from axis [mm]
Distance from axis [mm]
><JG: 7OKX ^KXQOX^SKV `OVYMS^c SX ^RO MOX^\KV ZVKXO -S\MVO]' 6.@( ]YVSN VSXO' 6/=
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x=15 mm
20
0
-20
-40
0
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-40
0
5 10 15 Ut' [m/s]
x=25 mm
20
0
5 10 15 Ut' [m/s]
20
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2
1
3
4
5
6
Percent
80 40 0 -40 0.1640
0.1645
0.1650 0.1655 Time (s)
0.1660
0.1665
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800
Spark power Combustion heat release Mean temperature
700 800 600
Temperature (K)
Integral power (kW)
1200
400
0
500 0.0
0.1
0.2
0.3 0.4 Time (ms)
0.5
0.6
0.7
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9P MY_\]O 6/= .8= ]YV_^SYX] K\O UXYaX ^Y RK`O K WOKXSXQ YXVc SX K ]^K^S]^SMKV WKXXO\ E ! G ]Y ^RK^ YL]O\`SXQ ^RK^ ^RO ]YV_^SYX YP K QS`OX 6/= .8= K^ K QS`OX SX]^KX^ MRKXQO] aROX ^RO \Y_XNSXQ O\\Y\] Y\ ^RO SXS ^SKV MYXNS^SYX] MRKXQO S] XY^ \OKVVc ]_\Z\S]SXQ 3^ S] RYaO`O\ K \OKV NS M_V^c SX ^RO Z\KM^SMKV _]O YP 6/= .8= LOMK_]O S^ WOKX] ^RK^ \_XXSXQ ^RO ]KWO
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]OX^ON ]SW_VK^SYX] K\O NS]ZVKcON SX >KLVO KXN >RO SX]^KX^] MY\\O ]ZYXN ^Y SX aKVV _XS^] H ) $#% H ) %! KXN H ) # \O]ZOM^S`OVc aRO\O H ) I H 9L`SY_]Vc ^RO ^aY Ya OVN] YL]O\`ON K^ H ) $#% K\O SNOX^SMKV 2YaO`O\ K^ H ) %! NSO\OXMO] ]^K\^ ^Y LOMYWO `S]SLVO 0SXKVVc K^ H ) # ^RO SX]^KX^KXOY_] Ya OVN] YL^KSXON SX >- KXN >- K\O ^Y^KVVc NSO\OX^ /`OX ^RY_QR ^RO SX]^KX^KXOY_] Ya OVN] K\O NSO\OX^ ]^K^S]^SM] \OWKSX ^RO ]KWO' WOKX KXN \YY^ WOKX ][_K\O KbSKV `O VYMS^c Z\Y VO] K`O\KQON Y`O\ H # K\O SNOX^SMKV PY\ LY^R ]SW_VK^SYX] K] MKX LO ]OOX SX 0SQ] " >RS] `O\c ]SWZVO ObKWZVO SVV_]^\K^O] ^RO WKSX [_O]^SYX YP ^RO Z\O]OX^ aY\U' K\O ^RO \O]_V^] YP 0SQ] e ! \OK]YXKLVO* 3P S^ S] XY^ K ]SWZVO Z\YQ\KWWSXQ O\\Y\ ^RO XOb^ ]OM^SYX aSVV ]RYa ^RK^ S^ S] XY^ ]Y MKX Y^RO\ ZK\KWO^O\] Z\YN_MO ]SWSVK\ OOM^]*
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# !
Velocity (m/s) 50
Velocity (m/s) 5
45
4.5 TC1 TC2
40 35
3.5
30
3
25
2.5
20
0
0.1
0.2
0.3
Normalized wall distance
0.4
TC1 TC2
4
0.5
2 0
0.1
0.2
0.3
Normalized wall distance
0.4
0.5
><JG: -YWZK\S]YX YP ^RO WOKX VOP^ \YY^ WOKX ][_K\O \SQR^ `O VYMS^c Z\Y VO] PY\ >- ! Z\YMO]]Y\] KXN >- % Z\YMO]]Y\] ]SW_VK^SYX] Y`O\ RKVP MRKXXOV ROSQR^
,:CH>I>K>IN D; % , >C %6B>C6G 6C9 -JG7JA:CI
ADLH
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aRO\O KXN 0 \O]ZOM^S`OVc NOXY^O ^RO MYWZ_^K^SYXKV NYWKSX KXN S^] `YV_WO ,Y^R XY\W] SX W ] aSVV LO KZZVSON ^Y ^RO KbSKV `OVYMS^c OVN ]Y ^RK^ (9-D Z\Y`SNO] ^RO WKbSW_W VYMKV `OVYMS^c NSO\OXMO SX ^RO OVN LO^aOOX ^aY ]YV_^SYX] aRSVO (91-: cSOVN] K `YV_WO^\SMKVVc K`O\KQON NSO\ OXMO LO^aOOX ^RO ^aY ]YV_^SYX] >RO Q\Ya^R YP (9-D KXN (91-: `O\]_] ^RO
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3N >- >- >>-! >-" >-# >-$ >-% >-& >- >-
8L\ Z\YM ! % ! % ! % %
3XS^ MYXN 0SbON 0SbON 0SbON 7YNSP 0SbON 0SbON 0SbON 0SbON 0SbON 0SbON 0SbON
:\OMS]SYX .Y_LVO .Y_LVO .Y_LVO .Y_LVO .Y_LVO .Y_LVO .Y_LVO =SWZVO =SWZVO ;_KN\ ;_KN\
1\KZR Y\NO\SXQ -7 -7 -7 -7 <-7 -7 -7 -7 -7 -7 -7
-06 $ $ $ $ $ " " $ $ $ $
X_WLO\ YP S^O\K^SYX] aSVV LO _]ON K] K NS\OM^ SXNSMK^Y\ PY\ ^RO NS`O\QOXMO YP ^RO ]YV_^SYX]
JAAN :I:GB>C>HI>8 % ,
0S\]^ S^ S] _]OP_V ^Y SXNSMK^O ^RK^ ZO\PY\WSXQ KXc YP ^RO 6/= YP >KLVO ^aSMO YX ^RO ]KWO WKMRSXO aS^R ^RO ]KWO X_WLO\ YP Z\YMO]]Y\] ^RO ]KWO SXS^SKV MYXNS^SYX] KXN ^RO ]KWO ZK\^S^SYX KVQY\S^RW VOKN] ^Y ObKM^Vc ^RO ]KWO ]YV_^SYX (9-D KXN (91-: LOSXQ dO\Y ^Y WKMRSXO KMM_\KMc 3X ^RK^ ]OX]O ^RO 6/= \OWKSX] P_VVc NO^O\WSXS]^SM 2YaO`O\ ^RS] S] ^\_O YXVc SP ^RO Y\NO\ YP YZO\K^SYX] K^ SX^O\PKMO] S] XY^ NO^O\WSXON Lc ^RO Y\NO\ YP WO]]KQO K\\S`KV ]Y ^RK^ ]_WWK^SYX] K\O KVaKc] MK\\SON Y_^ SX ^RO ]KWO Y\NO\ 9^RO\aS]O ^RO \KXNYWXO]] SXN_MON Lc ^RO XYX NO^O\WSXS]^SM Y\NO\ YP WO]]KQO K\\S`KV S] OXY_QR ^Y SXN_MO NS`O\QSXQ ]YV_^SYX]
#CJ:C8: D; -JG7JA:C8: >RO \]^ ^O]^ S] ^Y MYWZK\O K ^_\L_VOX^ MRKXXOV Ya ]^_NSON SX ^RO Z\O`SY_] ]OM^SYX KXN K VKWSXK\ Ya + ^R\OO NSWOX]SYXKV :YS]O_SVVO Ya SX K ZSZO QO YWO^\c aK] _]ON K] ^O]^ MK]O >RO Ya S] VKWSXK\ KXN ^RO RO LY_XN
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# ! -67A: =_WWK\c YP VKWSXK\ \_X] :YS]O_SVVO Ya
3N 6: 6:
8L\ Z\YM ! %
3XS^ MYXN 0SbON 0SbON
:\OMS]SYX .Y_LVO .Y_LVO
1\KZR Y\NO\SXQ -7 -7
-06 $ $
K\c MYXNS^SYX] K\O ]O^ ZO\SYNSM K^ ^RO SXVO^ Y_^VO^ KXN XY ]VSZ K^ ^RO N_M^ aKVV] K MYX]^KX^ KbSKV Z\O]]_\O Q\KNSOX^ S] SWZY]ON SX ^RO OX^S\O NYWKSX Differences (log scale) 1e+00 1e-03
Nmax TC1/TC2 Nmean TC1/TC2 Nmax LP1/LP2 Nmean LP1/LP2
1e-06 1e-09 1e-12 1e-15 0
1e+05
2e+05
Number of iterations
3e+05
4e+05
><JG: /OM^] YP ^_\L_VOXMO .SO\OXMO] LO^aOOX ]YV_^SYX] WOK ]_\ON Lc (9-D YZOX ]cWLYV] KXN (91-: MVY]ON ]cWLYV] `O\]_] S^O\K^SYX =[_K\O]' NSO\OXMO] LO^aOOX >- KXN >- ^_\L_VOX^ MRKXXOV -S\MVO]' NSO\OXMO] LO^aOOX 6: KXN 6: VKWSXK\ :YS]O_SVVO Ya 0SQ_\O # ]RYa] ^RO O`YV_^SYX] YP (9-D KXN (91-: `O\]_] S^O\K^SYX PY\ \_X] >- >- KXN 6: 6: 8Y^O ^RK^ ^RO \]^ ZYSX^ YP ^RO Q\KZR S] ^RO O`KV_K^SYX YP ^RO NSO\OXMO KP^O\ YXO S^O\K^SYX >RO YXVc ZK\KWO ^O\ ^O]^ON RO\O S] K MRKXQO YP ^RO X_WLO\ YP Z\YMO]]Y\] +] ObZOM^ON P\YW ^RO ]XKZ]RY^] YP 0SQ] e ! ^RO ^_\L_VOX^ MRKXXOV ]SW_VK^SYX] K\O `O\c ]OX]S^S`O ^Y K MRKXQO SX ^RO X_WLO\ YP Z\YMO]]Y\] KXN ^RO ]YV_^SYX] YP >- KXN >- NS`O\QO \KZSNVc VOKNSXQ ^Y K WKbSW_W NSO\OXMO YP W ] KXN K WOKX NSO\OXMO YP ! W ] KP^O\ & S^O\K^SYX] 9X ^RO Y^RO\ RKXN ^RO NSO\OXMO LO^aOOX 6: KXN 6: RK\NVc SXM\OK]O] KXN VO`OV] Y aROX \OKMRSXQ `KV_O] YP ^RO Y\NO\ Y\ >RS] S] ObZOM^ON ]SXMO ^RO\O S] YL`S Y_]Vc YXVc YXO ]^KLVO ]YV_^SYX PY\ ^RO :YS]O_SVVO Ya PY\ SX XS^O ^SWO] KXN
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VKWSXK\ Ya] NY XY^ SXN_MO ObZYXOX^SKV NS`O\QOXMO YP ^\KTOM^Y\SO] 2Ya O`O\ ^RS] ]SWZVO ^O]^ MK]O MYX \W] ^RK^ ^RO ^_\L_VOX^ MRK\KM^O\ YP ^RO Ya S] ^RO ]Y_\MO YP ^RO NS`O\QOXMO YP ]YV_^SYX] >RS] ZROXYWOXYX W_]^ XY^ LO MYXP_]ON aS^R ^RO Q\Ya^R YP K RcN\YNcXKWSM WYNO aRSMR S] SXN_MON Lc ^RO LSP_\MK^SYX SX ZRK]O ]ZKMO YP KX O[_SVSL\S_W ]^K^O YP K QS`OX ZRc]SMKV ]c]^OW 9L`SY_]Vc ]_MR KX O[_SVSL\S_W ]^K^O NYO] XY^ ObS]^ PY\ K P_VVc NO `OVYZON ^_\L_VOX^ MRKXXOV Ya 3X ^RS] MK]O ^RO ]OZK\K^SYX YP ^\KTOM^Y\SO] S] MK_]ON Lc `Y\^SMS^c aRSMR VOKN] ^Y KX SXM\OK]O SX ^RO X_WLO\ YP NOQ\OO] YP P\OONYW SX ZRK]O ]ZKMO E "$G KXN ^R_] RSQR ]OX]S^S`S^c ^Y SXS^SKV MYXNS ^SYX] 7Y\OY`O\ ^RO ]^KQXK^SYX YP KL]YV_^O KXN WOKX NSO\OXMO] LO^aOOX >- >- ]SWZVc SWZVSO] ^RK^ KP^O\ & S^O\K^SYX] ]YV_^SYX] RK`O LOMYWO P_VVc _XMY\\OVK^ON KXN ]RY_VN XY^ LO WS]SX^O\Z\O^ON K] ^RO ]K^_\K^SYX YP KX ObZYXOX^SKVVc Q\YaSXQ WYNO >RO LK]SM WOMRKXS]W VOKNSXQ ^Y 0SQ] e ! S] ^RK^ ^RO ^_\L_VOX^ Ya KM^] K] KX KWZVS O\ PY\ \Y_XNSXQ O\\Y\] QOXO\K^ON Lc ^RO PKM^ ^RK^ ^RO WO]R S] NOMYWZY]ON NSO\OX^Vc SX >- KXN >- >RO ]Y_\MO YP ^RS] NSO\OXMO S] ^RO XOa Q\KZR \OY\NO\SXQ YL^KSXON PY\ LY^R NOMYWZY]S^SYX] >RS] SWZVSO] K NSO\OX^ Y\NO\SXQ aROX KNNSXQ ^RO MYX^\SL_^SYX] ^Y K MOVV \O]SN_KV PY\ XYNO] SX]SNO ^RO ]_LNYWKSX] L_^ WKSXVc K^ ZK\^S^SYX SX^O\PKMO] >RS] \KXNYW XYS]O \Y_QRVc ]^K\^] K^ WKMRSXO KMM_\KMc 0SQ # K^ K POa ZYSX^] SX ^RO
Ya KXN Q\Ya] MYX^SX_Y_]Vc SP ^RO Ya S] ^_\L_VOX^ >RO Q\Ya^R \K^O YP ^RO NSO\OXMO] LO^aOOX ]YV_^SYX] SX ]SW_VK^SYX] >- KXN >- MKXXY^ LO O]^SWK^ON SX K ]SWZVO WKXXO\ + ]SWZVS ON NO ]M\SZ^SYX PY\ ^RO NO^O\WSXK^SYX YP Q\Ya^R \K^O] YP ^\KTOM^Y\c ]OZK\K^SYX S] PY_XN SX 4SWOXOd E &&G S^ S] L\SO c ]_WWK\SdON SX ^RO PYVVYaSXQ 0Y\ ^aY NSWOX]SYXKV `Y\^SMKV Ya] K]]_WSXQ K NO]M\SZ^SYX YP `Y\^SMO] K] ZYSX^] aS^R K]]YMSK^ON MS\M_VK^SYX] KXN XOQVOM^SXQ `S]MY]S^c K ]O^ YP VSXOK\SdON Y\NSXK\c NSO\OX^SKV O[_K^SYX] MKX LO NO\S`ON ^Y O`KV_K^O ^RO NSO\OXMO LO^aOOX ^aY XOSQRLY_\SXQ Ya OVN ^\KTOM^Y\SO] NSO\SXQ Lc KX K\LS^\K\c SX XS^O]SWKV ZO\^_\LK^SYX / SX ^RO KbSKV `OVYMS^c >RS] ]c]^OW KNWS^] ObZYXOX^SKV ]Y V_^SYX] ^RO Q\Ya^R \K^O] YP aRSMR K\O NO^O\WSXON Lc ^RO \OKV ZK\^ YP ^RO OSQOX`KV_O] >RO O`YV_^SYX YP SX`S]MSN MYX]O\`K^S`O ]c]^OW] MYX]O\`O] `YV _WO SX ZRK]O ]ZKMO +] ^RO \OKV ZK\^ YP ^RO OSQOX`KV_O] NO]M\SLO] ^RO ]OZK \K^SYX YP ^\KTOM^Y\SO] SX ^SWO S^ \OZ\O]OX^] K WOK]_\O YP ^RO O`YV_^SYX YP ^RO `YV_WO SX ZRK]O ]ZKMO >R_] SP ^RO ]_W YP ^RO \OKV ZK\^] `KXS]RO] ^ROX K^ VOK]^ YXO YP ^ROW RK] ^Y LO ZY]S^S`O +^ ^RS] ]^KQO ^RO X_WLO\ YP NOQ\OO] YP P\OONYW YP ^RO ]c]^OW SWZY]O] ^YZYVYQSMKV MYX]^\KSX^] YX ^RO ^\KTOM^Y\SO] KXN MKX Z\O`OX^ ^ROS\ ]OZK\K^SYX L_^ K POa NOQ\OO] YP P\OONYW ]_ MO PY\ ]_MR ]c]^OW] ^Y ObRSLS^ MRKY^SM LORK`SY\ K] aK] NOWYX]^\K^ON Lc ^RO PK WY_] 6Y\OXd K^^\KM^Y\ E "G >RS] K\Q_WOX^ SVV_]^\K^O] ^RK^ ^RO ]OZK\K^SYX YP ^\KTOM^Y\SO] S] K Z\YZO\^c \OVK^ON ^Y ^RO XK^_\O YP `Y\^SMS^c KXN WKSXVc
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# !
N\S`OX Lc ^RO X_WLO\ YP NOQ\OO] YP P\OONYW SX ZRK]O ]ZKMO >RO\OPY\O K ]SWZVO O]^SWK^O YP ^RO Q\Ya^R \K^O P\YW Ya ZK\KWO^O\] NYO] K Z\SY\S XY^ ]OOW ZY]]SLVO +V^RY_QR ^RO ]SWZVS MK^SYX] SX ^RO NO]M\SLON KXKVc]S] K\O ]O`O\O YXO MKX ]_ZZY]O ^RK^ SXNOZOXNOX^ YP ^RO ]ZK^SKV NS]^\SL_^SYX KXN KWZVS^_NO aS^RSX ^RO VSWS^ YP ^RO VSXOK\S^c K]]_WZ^SYX YP ZO\^_\LK^SYX] KZZVSON ^Y K QS`OX ^_\L_VOX^ Ya OVN ^RO ]OZK\K^SYX YP ^\KTOM^Y\SO] PY\ `K\SY_] ]SW_VK^SYX] cSOVN] ]SWSVK\ ObZYXOX^SKV Q\Ya^R \K^O] aRSMR S] MYX \WON SX ^RO PYVVYaSXQ 7Y\OY`O\ S^ S] K Z_\O ZRc]SMKV ZROXYWOXYX KXN ^RY_QR SXN_MON Lc \Y_XNSXQ O\\Y\] ^RO Q\Ya^R \K^O ]RY_VN XY^ NOZOXN YX X_WO\SMKV ZK\KWO^O\] ]_MR K] WKMRSXO Z\OMS]SYX Y\ ^SWO ]^OZ
#CJ:C8: D; #C>I>6A DC9>I>DCH >RO Z\O`SY_] ]OM^SYX RK] ]RYaX ^RK^ ^_\L_VOXMO MYWLSXON aS^R K NSO\OX^ NYWKSX NOMYWZY]S^SYX SO K NSO\OX^ X_WLO\ YP Z\YMO]]Y\] PY\ ^RO PYVVYa SXQ S] ]_ MSOX^ ^Y VOKN ^Y ^Y^KVVc NSO\OX^ SX]^KX^KXOY_] Ya \OKVSdK^SYX] 3^ S] ObZOM^ON ^RK^ K ZO\^_\LK^SYX SX SXS^SKV MYXNS^SYX] aSVV RK`O ^RO ]KWO OOM^ K] NYWKSX NOMYWZY]S^SYX >RS] S] `O\S ON SX \_X] >- KXN >-! aRSMR K\O \_X YX YXO Z\YMO]]Y\ YXVc ^RO\OLc OVSWSXK^SXQ S]]_O] VSXUON ^Y ZK\KVVOV SWZVOWOX^K^SYX >RO YXVc NSO\OXMO LO^aOOX >- KXN >-! S] ^RK^ SX >-! ^RO SXS^SKV ]YV_^SYX S] SNOX^SMKV ^Y >- ObMOZ^ K^ YXO \KXNYW ZYSX^ aRO\O K ZO\^_\LK^SYX S] KZZVSON ^Y ^RO ]^\OKWaS]O `OVYMS^c MYWZYXOX^ =SW_ VK^SYX] aS^R NSO\OX^ VYMK^SYX] YP ^RO ZO\^_\LK^SYX aO\O \_X ^Y OX]_\O ^RK^ ^ROS\ ZY]S^SYX NSN XY^ KOM^ \O]_V^] 0SQ_\O $ ]RYa] ^RK^ ^RO Q\Ya^R \K^O YP ^RO NSO\OXMO LO^aOOX >KXN >-! S] ObKM^Vc ^RO ]KWO K] ^RO YXO YL]O\`ON LO^aOOX >- KXN >- KV]Y NS]ZVKcON SX 0SQ $' ^aY ]YV_^SYX] ]^K\^SXQ P\YW K `O\c ]VSQR^Vc ZO\^_\LON SXS^SKV MYXNS^SYX NS`O\QO K] PK]^ K] ^aY ]YV_^SYX] ]^K\^SXQ P\YW ^RO ]KWO ]YV_^SYX L_^ \_XXSXQ YX NSO\OX^ X_WLO\] YP Z\YMO]]Y\] 8Y^O ^RK^ ^RO NSO\OXMO LO^aOOX \_X] >- KXN >- MYWO] P\YW \KXNYW \Y_XNSXQ O\\Y\] SX^\YN_MON K^ OKMR ^SWO ]^OZ aRSVO >- KXN >-! NSO\ YXVc ^R\Y_QR ^RO SXS^SKV MYXNS^SYX' XY ZO\^_\LK^SYX S] KNNON N_\SXQ ^RO ]SW_VK^SYX =^SVV ^RO NSO\OXMO] LO^aOOX >- KXN >-! SXM\OK]O K] PK]^ K] ^RY]O LO^aOOX >- KXN >- ' ^RS] MYX \W] ^RK^ K ^_\L_VOX^ Ya KWZVS O] KXc NSO\OXMO SX ^RO ]KWO WKXXO\ aRO^RO\ S^ S] N_O ^Y \Y_XNSXQ O\\Y\] Y\ ^Y K ZO\^_\LK^SYX YP ^RO SXS^SKV MYXNS^SYX]
:8IH D; !G6E= (G9:G>C< 3^ RK] KV\OKNc LOOX SXNSMK^ON ^RK^ ZO\PY\WSXQ ^RO ]KWO ]SW_VK^SYX ^aSMO aS^R ^RO ]KWO X_WLO\ YP Z\YMO]]Y\] KXN ]KWO SXS^SKV MYXNS^SYX] VOKN] ^Y
! "
&$
Differences (log scale) 1e+00 1e-03
Nmax TC1/TC2 Nmean TC1/TC2 Nmax TC3/TC4 Nmean TC3/TC4
1e-06 1e-09 1e-12 1e-15 0
1e+05
2e+05
Number of iterations
3e+05
4e+05
><JG: /OM^] YP SXS^SKV MYXNS^SYX] .SO\OXMO] LO^aOOX ]YV_^SYX] WOK]_\ON Lc (9-D YZOX ]cWLYV] KXN (91-: MVY]ON ]cWLYV] `O\]_] S^ O\K^SYX =[_K\O]' NSO\OXMO] LO^aOOX >- KXN >- NSO\OX^ X_WLO\] YP Z\YMO]]Y\] -S\MVO]' NSO\OXMO] LO^aOOX >- KXN >-! NSO\OX^ SXS^SKV MYXNS^SYX]
ObKM^Vc ^RO ]KWO \O]_V^ 2YaO`O\ ^RS] S] YXVc ^\_O K] VYXQ K] ObKM^Vc ^RO ]KWO MYNO S] _]ON 3^ S] XY^ `O\S ON KXc WY\O K] ]YYX K] K WYNS MK^SYX KOM^SXQ \Y_XNSXQ O\\Y\] S] NYXO SX ^RO MYNO +^ ^RS] ZYSX^ ]Y WKXc PKM^Y\] KOM^SXQ \Y_XNSXQ O\\Y\] MKX LO MS^ON ^RK^ K QOXO\KV NS]M_]]SYX S] ZYSX^VO]] >RS] MRKZ^O\ aSVV PYM_] YX P_VVc ObZVSMS^ MYNO] KXN YX YXO ObKWZVO YXVc' ^RO Y\NO\ _]ON ^Y KNN \O]SN_KV] K^ XYNO] SX K MOVV `O\^Ob ]MROWO >RS] Y\NO\ S] MYX^\YVVON Lc ^RO NO`OVYZO\ 0Y\ ]SW_VK^SYX >-" ^RO Y\NO\SXQ YP ^RS] KNNS^SYX aK] MRKXQON \O`O\]O -_^RSVV7M5OO KVQY\S^RW' ^RO \O]SN_KV K^ K QS`OX WO]R XYNO aK] K]]OWLVON Lc KNNSXQ ^RO MYX^\SL_^SYX] ^Y K MOVV \O]SN_KV SX K NSO\OX^ Y\NO\ >RS] MRKXQO NYO] XY^ KOM^ ^RO Ya NK^K' SX >-" ^RO XYNO \O]SN_KV SX K \OQ_VK\ ^O^\KRON\KV WO]R S] YL^KSXON Lc !, , , , aRO\O ^RO ,5 ] K\O ^RO \O]SN_KV] YP ^RO MOVV] ]_\\Y_XNSXQ ^RO XYNO KXN Lc !, , , , SX >- 3^ RK] KX OOM^ RYaO`O\ YX \Y_XNSXQ O\\Y\] KXN ^RO M_W_VK^ON OOM^] YP ^RS] XYXK]]YMSK^S`S^c O\\Y\ K\O aRK^ ^RS] ^O]^ ^\SO] ^Y S]YVK^O >-" KXN >- K\O ZO\PY\WON aS^R ^RO ]KWO SXS^SKV MYXNS^SYX KXN \_X YX YXO Z\YMO]]Y\ YXVc >RO YXVc NSO\OXMO S] ^RO Q\KZR \OY\NO\SXQ ]^\K^OQc +] ]RYaX Lc 0SQ % ^RO NSO\OXMO] LO^aOOX >-" KXN >- K\O KQKSX ]SWSVK\ ^Y ^RY]O YL]O\`ON LO^aOOX >- KXN >- YL^KSXON Lc MRKXQSXQ ^RO X_WLO\ YP Z\YMO]]Y\] >RS] MYX \W] ^RK^ \Y_XNSXQ O\\Y\] KXN XY^ ^RO ZK\KVVOV MRK\KM^O\ YP ^RO MYNO K\O ^RO ]Y_\MO YP ^RO ]YV_^SYX NS`O\QOXMO 3^
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# !
Differences (log scale) 1e+00 1e-03
Nmax TC1/TC2 Nmean TC1/TC2 Nmax TC3/TC5 Nmean TC3/TC5
1e-06 1e-09 1e-12 1e-15 0
1e+05
2e+05
Number of iterations
3e+05
4e+05
><JG: /OM^] YP KNNS^SYX Y\NO\ .SO\OXMO] LO^aOOX ]YV_^SYX] WOK ]_\ON Lc (9-D YZOX ]cWLYV] KXN (91-: MVY]ON ]cWLYV] `O\]_] S^O\K^SYX =[_K\O]' NSO\OXMO] LO^aOOX >- KXN >- -S\MVO]' NSO\OXMO] LO^aOOX >- KXN >-"
KV]Y ]RYa] ^RK^ KXc WYNS MK^SYX YP ^RO MYNO aRSMR MKX SX^\YN_MO K ]WKVV WYNS MK^SYX YP ^RO \O]_V^] aS^RSX WKMRSXO KMM_\KMc MKX ^ROX VOKN ^Y ]_MR K NS`O\QOXMO ]_QQO]^SXQ ^RK^ \OZOK^SXQ KX 6/= ]SW_VK^SYX aS^R ^RO ]KWO MYNO KP^O\ K POa WYNS MK^SYX] aSVV Z\YLKLVc XO`O\ cSOVN ^RO ]KWO SX]^KX^KXOY_] Ya OVN] ZY^OX^SKVVc VOKNSXQ ^Y NS]M_]]SYX] YX ^RO `KVSNS^c YP ^RO WYNS ON MYNO
:8IH D; ->B: ,I:E 3^ S] SX^O\O]^SXQ ^Y `O\SPc ^RK^ X_WO\SMKV K]ZOM^] NY XY^ SX _OXMO ^RO Q\Ya^R \K^O YP ^RO ]YV_^SYX] NSO\OXMO KXN ^RK^ ^RO Q\Ya^R \K^O S] YXVc NO^O\WSXON Lc ^RO ZRc]SMKV KXN QOYWO^\SMKV ZK\KWO^O\] YP ^RO MYX Q_\K^SYX 9X ^RK^ KMMY_X^ ]SW_VK^SYX] >-# KXN >-$ K\O ZO\PY\WON aS^R K ^SWO ]^OZ \ON_MON Lc K PKM^Y\ MYWZK\ON ^Y ]SW_VK^SYX] >- KXN >- >-# KXN >-$ K\O MK\\SON Y_^ YX \O]ZOM^S`OVc ! KXN % Z\YMO]]Y\] >RO XY\W] LO^aOOX >-# KXN >-$ K\O NS]ZVKcON SX 0SQ & KXN MYWZK\ON ^Y ^RO XY\W] LO^aOOX >- KXN >- 0\YW ^RO ObZVKXK^SYX] QS`OX KLY`O ]SWSVK\ Q\Ya^R \K^O] K\O ObZOM^ON aROX MYWZK\SXQ ^RO Q\Ya^R \K^O] Y`O\ ZRc]SMKV ^SWO >RO Q\Ya^R \K^O] YL]O\`ON SX 0SQ & K\O SXNOON `O\c ]SWSVK\ >RO ]VSQR^ NSO\OXMO S] Z\YLKLVc N_O ^Y ^RO `K\SK^SYX YP ^RO X_WO\SMKV NS]ZO\]SYX KXN NS]]SZK^SYX Z\YZO\^SO] YP ^RO ]MROWO aS^R ^RO ^SWO ]^OZ E &#G
! "
&&
Differences (log scale) 1e+00
Nmax TC1/TC2 Nmean TC1/TC2 Nmax TC6/TC7 Nmean TC6/TC7
1e-06
1e-12
1e-18 0
0.1
0.2
0.3
Physical time (s)
0.4
0.5
><JG: /OM^] YP ^SWO ]^OZ .SO\OXMO] LO^aOOX ]YV_^SYX] WOK]_\ON Lc (9-D YZOX ]cWLYV] KXN (91-: MVY]ON ]cWLYV] `O\]_] ZRc]SMKV ^SWO =[_K\O]' NSO\OXMO] LO^aOOX >- KXN >- ^SWO ]^OZ H -S\MVO]' NSO\ OXMO] LO^aOOX >-# KXN >-$ ^SWO ]^OZ H
:8IH D; &68=>C: )G:8>H>DC + VK]^ ^O]^ ^Y `O\SPc ^RK^ ^RO NS`O\QOXMO LO^aOOX ]YV_^SYX] S] XY^ N_O ^Y K Z\YQ\KWWSXQ O\\Y\ L_^ NOZOXN] Z\SWK\SVc YX \Y_XNSXQ O\\Y\] S] ^Y ZO\PY\W ^RO ]KWO MYWZ_^K^SYX aS^R ]SWZVO [_KN\_ZVO Z\OMS]SYX SX]^OKN YP NY_LVO Z\OMS]SYX =SW_VK^SYX] >- KXN >- aO\O \OZOK^ON _]SXQ ]SXQVO Z\OMS]SYX SX \_X] >-% KXN >-& >KLVO KXN [_KN\_ZVO Z\OMS]SYX SX >- KXN >- >Y MYWZOX]K^O PY\ ^RO SXM\OK]O SX MYWZ_^K^SYXKV ^SWO PY\ [_KN\_ ZVO Z\OMS]SYX ]SW_VK^SYX] \Y_QRVc K PKM^Y\ ^OX MYWZK\ON ^Y NY_LVO Z\OMS]SYX >- KXN >- aO\O MK\\SON Y_^ YX \O]ZOM^S`OVc % KXN Z\YMO]]Y\] SX Y\ NO\ ^Y cSOVN K \OK]YXKLVO \O]^S^_^SYX ^SWO - KXN >- 0SQ_\O ]RYa] ^RK^ ^RO ]YV_^SYX NSO\OXMO] PY\ >-% >-& KXN >- >- \Y_QRVc ]^K\^ P\YW ^RO \O]ZOM^S`O WKMRSXO KMM_\KMSO] NSO\OXMO] YP PY\ ]SXQVO Z\OMS]SYX KP^O\ YXO S^O\K^SYX NSO\OXMO] YP PY\ [_KN\_ZVO Z\O MS]SYX KP^O\ YXO S^O\K^SYX KXN SXM\OK]O ObZYXOX^SKVVc aS^R ^RO ]KWO Q\Ya^R \K^O LOPY\O \OKMRSXQ ^RO ]KWO NSO\OXMO VO`OV] PY\ KVV ^R\OO MK]O] >RS] ]RYa] ^RK^ RSQRO\ Z\OMS]SYX MYWZ_^K^SYX] MKXXY^ Z\O`OX^ ^RO ObZYXOX^SKV NS`O\QOXMO YP ^\KTOM^Y\SO] L_^ YXVc NOVKc S^
# !
Differences (log scale) 1e+00
1e-06
Nmax TC1/TC2 Nmean TC1/TC2 Nmax TC8/TC9 Nmean TC8/TC9 Nmax TC10/TC11 Nmean TC10/TC11
1e-12 1e-18 1e-24 1e-30 0
1e+05
2e+05
Number of iterations
3e+05
4e+05
><JG: /OM^] YP WKMRSXO KMM_\KMc .SO\OXMO] LO^aOOX ]YV_^SYX] WOK]_\ON Lc (9-D YZOX ]cWLYV] KXN (91-: MVY]ON ]cWLYV] `O\]_] S^ O\K^SYX =[_K\O]' NSO\OXMO] LO^aOOX >- KXN >- NY_LVO Z\OMS]SYX -S\MVO]' NSO\OXMO] LO^aOOX >-% KXN >-& ]SXQVO Z\OMS]SYX >\SKXQVO]' NSO\OXMO] LO^aOOX >- KXN >- [_KN\_ZVO Z\OMS]SYX
DC8AJH>DCH
>RS] MRKZ^O\ PYM_]ON YX ^RO ]OX]S^S`S^c YP SX]^KX^KXOY_] 6/= OVN] ^Y W_V^SZVO ZK\KWO^O\] ]_MR K] X_WLO\ YP Z\YMO]]Y\] SXS^SKV MYXNS^SYX ^SWO ]^OZ MRKXQO] SX KNNS^SYX Y\NO\SXQ YP MOVV \O]SN_KV] PY\ MOVV `O\^Ob WO^RYN] >RO LK]OVSXO ]SW_VK^SYX _]ON PY\ ^RO ^O]^] aK] K P_VVc NO`OVYZON ^_\L_VOX^ MRKXXOV >RO MYXMV_]SYX] K\O ^RO PYVVYaSXQ' +Xc ^_\L_VOX^ Ya MYWZ_^ON Lc 6/= ObRSLS^] ]SQXS MKX^ ]OX]S^S`S^c ^Y ^RO]O ZK\KWO^O\] VOKNSXQ ^Y SX]^KX^KXOY_] ]YV_^SYX] aRSMR MKX LO ^Y^KVVc NSO\OX^ 6KWSXK\ Ya] K\O KVWY]^ SX]OX]S^S`O ^Y ^RO]O ZK\KW O^O\] >RO NS`O\QOXMO YP ]YV_^SYX] S] N_O ^Y ^aY MYWLSXON PKM^]' ^RO Ob ZYXOX^SKV ]OZK\K^SYX YP ^\KTOM^Y\SO] SX ^_\L_VOX^ Ya] KXN ^RO XYX NO^O\WSXS]^SM \Y_XNSXQ O\\Y\] SXN_MON Lc NSO\OX^ NYWKSX NOMYWZY ]S^SYX] Y\ NSO\OX^ Y\NO\SXQ YP YZO\K^SYX] 7Y\O QOXO\KVVc KXc MRKXQO SX ^RO MYNO VSXO] KOM^SXQ \Y_XNSXQ O\\Y\] aSVV RK`O ^RO ]KWO OOM^] =WKVV MRKXQO] SX SXS^SKV MYXNS^SYX YP ^RO Y\NO\ YP WKMRSXO KMM_\KMc K^ YXO ZYSX^ YP ^RO Ya YXVc Z\YN_MO ]SWSVK\ NS`O\QOXMO YP ]YV_^SYX] AY\USXQ aS^R RSQRO\ Z\OMS]SYX WKMRSXO] NYO] XY^ ]_ZZ\O]] ^RO NS`O\ QOXMO YP ]YV_^SYX] L_^ NOVKc] S^ >RO]O \O]_V^] MYX \W ^RO ObZOM^ON XK^_\O YP 6/= E ! G SX aRSMR ]YV_^SYX] K\O WOKXSXQP_V YXVc SX K ]^K^S]^SMKV ]OX]O KXN SX]^KX^KXOY_] `KV_O] MKX XY^
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