Ressance s obaned from he sm of he frcon and pressre ressance arables o deermne: - eloc ecor, (3) = (,, ) = (,, ) - Pressre, p () ( - Dens, ρ, s defned b he eqaon of sae Ressance and Proplson Lecre 0 4 eqaons o deermne 4 nknons: - Mass conseraon - Momenm balance (Neon s La) - Neonan Fld, shear sresses proporonal o he gradens of he eloc componens Ressance and Proplson Lecre 0
Mass conseraon - Dfferenal form 0 = Ressance and Proplson Lecre 0 3 - Inegral form = 0 ( ) 0 = o ds n = p ρ Momenm balance (Naer-Sokes eqaons) Ressance and Proplson Lecre 0 4 = = p g p ρ ρ m a / Conecon Pressre Dffson egh force scos forces
= p p ρ Momenm balance (=consan) (Naer-Sokes eqaons) Ressance and Proplson Lecre 0 5 m a / = = p g p ρ ρ Conecon Pressre Dffson egh force scos forces Renolds nmber of he flo arond shps s larger han 0 6 and ma reach 0 9 - Mos flos are rblen. Transon from lamnar o rblen flo depends on he Ressance and Proplson Lecre 0 6 lamnar o rblen flo depends on he Renolds nmber and all roghness. The mos sal approach s o assme ha he flo s fll-rblen (he rblence model s spposed o ake care of ranson)
Trblen Flo hp://.obe.com/ach?=xollkedog&feare=relaed hp://br.obe.com/ach?=7kkfgany hp://br.obe.com/ach?=qhxihpc Ressance and Proplson Lecre 0 7 Trblen Flo Ressance and Proplson Lecre 0
Trblen Flo Ressance and Proplson Lecre 0 Trblen Flo Ressance and Proplson Lecre 0
Trblen Flo Ressance and Proplson Lecre 0 Trblen Flo Ressance and Proplson Lecre 0
Trblen Flo Ressance and Proplson Lecre 0 3 Trblen Flo Ressance and Proplson Lecre 0 4
Trblen Flo Ressance and Proplson Lecre 0 5 Trblen Flo. Aleaor. Three-dmensonal 3. Hgh dffson 4. Dsspae 5. Proper of he flo 6. Connos medm 7. Hgh Renolds nmber Ressance and Proplson Lecre 0 6
Trblen Flo Drec Nmercal Smlaon, DNS Naer-Sokes eqaons soled nmercall h a grd spacng and a me sep sffcenl small o resole he smalles eddes of he flo. On he oher hand, smlaon me ms be large enogh o capre he effecs of he larges eddes Nmercal accrac of he solon s er mporan (o aod msleadng resls de o nmercal dffson) Dependen arables change n space and me,.e. he are he nsaaneos ales a a gen flo locaon Ressance and Proplson Lecre 0 7 Trblen Flo Large-Edd Smlaon, LES Naer-Sokes eqaons flered n space. Era mahemacal model reqred o nclde he effec of he flered scales. Tme dependen nmercal solon. Nmercal accrac s also mporan ( on-gong debae abo ho mch ). Correc applcaon o near-all flos s dffcl. Dependen arables change h me, b hae a dfferen meanng of DNS, de o flerng n space Ressance and Proplson Lecre 0 8
Trblen Flo Ml-Scale Smlaon Naer-Sokes eqaons spl n o small and large scales. Analcal mehods sed o oban an appromae solon of he small scales. Nmercal solon of he large scales n me h effec of small scales nclded from s appromae solon. Nmercal accrac s sll mporan. Applcaon near-alls roblesome ( small scales become oo small...) Ressance and Proplson Lecre 0 9 Trblen Flo Renolds-aeraged eqaons Sascs appled o mass conseraon and momenm balance (conn and Naer-Sokes). Tpe of sasc handlng depends on flo properes:. Spaal aeragng. Tme aeragng 3. Ensemble aeragng Insaaneos eloc componens, ~, (dependen arables) spl n o a mean ale,, and a flcaon, ~ = Ressance and Proplson Lecre 0 0
Trblen Flo Renolds-aeraged eqaons. Spaal aeragng n = = lm n ~ (,, ) n Homogeneos rblence Ressance and Proplson Lecre 0 Trblen Flo Renolds-aeraged eqaons. Tme aeragng = lm T o T ~ o T d Sascall sead flo Ressance and Proplson Lecre 0
Trblen Flo Renolds-aeraged eqaons 3. Ensemble aeragng = = lm n n ( ~ ( ) ) n Mean ales are me dependen. Sable for perodc flos. Ressance and Proplson Lecre 0 3 Trblen Flo Renolds-aeraged (RANS) eqaons Tme aeragng appled o he dependen arables and o conseraon prncples ~ = lm T o T o ~ d T = Φ ~ sands for an of he dependen arables (ncompressble flo,,,p) Ressance and Proplson Lecre 0 4
Trblen Flo Renolds-aeraged (RANS) eqaons Decomposon of he nsananeos arables ~ = Φ ~ Φ Insananeos arable Mean ale Flcaon arond he mean ale Ressance and Proplson Lecre 0 5 Trblen Flo Renolds-aeraged (RANS) eqaons Conseqences of me aeragng Φ = 0 = 0 = 0 Tme derae of mean ale s ero Mean ale of he me derae of he flcaons s ero Ressance and Proplson Lecre 0 6
Lnear erms ~ ~ Φ _ Trblen Flo Renolds-aeraged (RANS) eqaons Ressance and Proplson Lecre 0 7 = 0 Φ = Φ = Φ = ~ ~ _ Non-lnear erms = ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ Trblen Flo Renolds-aeraged (RANS) eqaons Ressance and Proplson Lecre 0 8 = ~ ~ ~ Φ = ~ ~ _
Conn eqaon Trblen Flo Renolds-aeraged (RANS) eqaons Ressance and Proplson Lecre 0 9 0 0 = = - eloc flcaons also sasf = P ρ Momenm eqaons Trblen Flo Renolds-aeraged (RANS) eqaons Ressance and Proplson Lecre 0 30 = = P P ρ ρ Renolds sresses The nmber of eqaons s smaller han he nmber of nknons ρ
Transpor eqaon of D D Trblen Flo Renolds-aeraged (RANS) eqaons = k k = k ρ _ p ρ k k ( ) Ssem remans h less eqaons han nknons k k p ρ k Ressance and Proplson Lecre 0 3 k p Trblen Flo Renolds-aeraged (RANS) eqaons Renolds sress models - 6 addonal ranspor eqaons - Mos of he erms of he Renolds sresses ranspor eqaons ms be modeled, ncldng pressre flcaons - There are alernae Eplc Algebrac Sress Models (EARSM) aalable - Trblence ansorop s aken no accon Ressance and Proplson Lecre 0 3
Edd-scos models - Bossnesq hpohess: he Renolds sresses are proporonal o mean eloc deraes Trblen Flo Renolds-aeraged (RANS) eqaons Ressance and Proplson Lecre 0 33 are proporonal o mean eloc deraes - The proporonal consan s he edd-scos - I s dffcl o nclde he ansorop of rblence. Mos models are soropc,.e. edd-scos s a scalar qan ef ef ef P ρ = Trblen Flo Renolds-aeraged (RANS) eqaons Momenm eqaons ncldng an effece scos Ressance and Proplson Lecre 0 34 ef ef ef ef ef ef ef P P ρ ρ ρ = = =
Trblen Flo Renolds-aeraged (RANS) eqaons Momenm eqaons ncldng an effece scos s he edd-scos - Edd-scos s obaned from eloc and lengh scales of rblence - Seeral models proposed n he las 50 ears Ressance and Proplson Lecre 0 35 Trblen Flo Renolds-aeraged (RANS) eqaons Edd-scos rblence models - Algebrac models - Trblence lengh scale l = κ Mng lengh - Trblence eloc scale l ω ω s he orc ecor = l ω Ressance and Proplson Lecre 0 36
Trblen Flo Renolds-aeraged (RANS) eqaons Edd-scos rblence models - Algebrac models - Dampng fncon s appled o he rblence lengh scale n he near-all regon. Lengh scale ms be changed for he oer regon of he bondar-laer, akes and es - Smples model aalable, b h seere lmaons. Nmercal mplemenaon n comple flos ma be roblesome Ressance and Proplson Lecre 0 37 Trblen Flo Renolds-aeraged (RANS) eqaons Edd-scos rblence models - -eqaon models ( old sle ) - Trblence lengh scale s dencal o ha sed n he algebrac models - Trblence eloc scale s he sqare roo of he rblence knec energ, hch has s on ranspor eqaon k = Ressance and Proplson Lecre 0 38
Trblen Flo Renolds-aeraged (RANS) eqaons Trblence knec energ, k - Transpor eqaon (k balance) Dk D k = k = p ρ k _ Ressance and Proplson Lecre 0 39 Trblen Flo Renolds-aeraged (RANS) eqaons Trblence knec energ, k p ρ k _ Conecon Prodcon Trblen dffson Ressance and Proplson Lecre 0 40
Trblen Flo Renolds-aeraged (RANS) eqaons Trblence knec energ, k k scos dffson Dsspaon rae, e Mos of he erms nclde nknon qanes and so he ms be modeled Ressance and Proplson Lecre 0 4 Trblen Flo Renolds-aeraged (RANS) eqaons Edd-scos rblence models ~ c b, c b, c - -eqaons models Spalar & Allmaras ~ = c ~~ b S σ Consans s [ ( ~ ) ~ c ( ~ ~ )] f, = ~ f f b Fncons c f ~ d Ressance and Proplson Lecre 0 4
Trblen Flo Renolds-aeraged (RANS) eqaons Edd-scos rblence models - -eqaon model of Spalar & Allmaras - ald don o he all - Edd-scos s proporonal o he dependen arable of he model - I reqres he dsance o he all,d, and n s orgnal erson he locaon of ranson Ressance and Proplson Lecre 0 43 Trblen Flo Renolds-aeraged (RANS) eqaons Edd-scos rblence models - -eqaon models: rblence eloc scale s k k - k-ε model = Cµ ε k k = S k ε σ k ε ε ε = C S k σ C C, C, σ, σ Consans µ, k ε ε C Ressance and Proplson Lecre 0 44 ε ε k
Trblen Flo Renolds-aeraged (RANS) eqaons Edd-scos rblence models - k-ε model - del sed, specall n flos ho alls and n hea ransfer problems - Poor resls for flos h aderse pressre gradens Ressance and Proplson Lecre 0 45 Trblen Flo Renolds-aeraged (RANS) eqaons Edd-scos rblence models - k-ε model - Can no be appled n he near-all regon - To-laer models combne he k-ε model n he oer regon h a -eqaon model n he near-all regon - There are (oo man) Lo-Renolds nmber ersons of he model for s eenson o he near-all regon Ressance and Proplson Lecre 0 46
- k-ω model ω k = Trblen Flo Renolds-aeraged (RANS) eqaons Edd-scos rblence models Ressance and Proplson Lecre 0 47 ( ) = = ω ω ω ω σ σ α β β βω ω ω ω σ α ω ω ω β σ F k F S k k S k k k k,,,, * * Consans Fncon - k-ω model Trblen Flo Renolds-aeraged (RANS) eqaons Edd-scos rblence models Ressance and Proplson Lecre 0 48 - Ma be appled don o he all - ω goes o nfn a he all (smooh alls) - Seeral formlaons aalable. One of he mos poplar s he SST (Shear-Sress Transpor) erson ha ncldes a lmer for he edd-scos
Trblen Flo Renolds-aeraged (RANS) eqaons Edd-scos rblence models - k-ω model - del sed for he calclaon of aderse pressre graden flos - Nmercal mplemenaon s no ral (ω all bondar condon) and some ersons (as for eample SST) reqre he dsance o he all Ressance and Proplson Lecre 0 49 Trblen Flo Renolds-aeraged (RANS) eqaons Bondar condons - Sold alls τ a) No slp condon appled drecl a he all = µ = 0 b) all fncons µ, <, τ =, τ = ln C, > 30 50, <? κ τ τ ρ Ressance and Proplson Lecre 0 50 τ =
Trblen Flo Renolds-aeraged (RANS) eqaons Bondar condons - Eernal bondar a) ndsrbed Flo b) Poenal flo solon Ressance and Proplson Lecre 0 5 Trblen Flo Renolds-aeraged (RANS) eqaons Bondar condons - Free srface a) Srface rackng: doman fed o he free srface (erae procedre) here pressre s fed. A he free srface, normal eloc componens s ero and shear-sress s dencal for aer and ar b) Srface caprng: doman coers ar and aer olme of fld (OF) or Leel Se Ressance and Proplson Lecre 0 5
Nmercal solon of RANS eqaons Flo arond he KLCC anker Ressance and Proplson Lecre 0 53 Nmercal solon of RANS eqaons Flo arond he KLCC anker Ressance and Proplson Lecre 0 54
Nmercal solon of RANS eqaons Flo arond he KLCC anker Ressance and Proplson Lecre 0 55 Nmercal solon of RANS eqaons Flo arond he KLCC anker Ressance and Proplson Lecre 0 56
Nmercal solon of RANS eqaons Flo arond he KLCC anker Model k-ω SST 3.5 Rn=4,6 0 6 C F 0 3 3.6 k-ω ahbh 3.4 3.3 3. 0 0.5.5 h /h Ressance and Proplson Lecre 0 57 Nmercal solon of RANS eqaons Flo arond he KLCC anker 0.8 k-ω Model k-ω SST Rn=4,6 0 6 C P 0 3 0.7 0.6 0 0.5.5 h /h Ressance and Proplson Lecre 0 58
Nmercal solon of RANS eqaons Flo arond he KLCC anker.55 Model k-ω SST Rn=,03 0 9 k-ω SST p=.8 C F 0 3.5.45 0 0.5 0.5 0.75.5.5.75 Ressance and Proplson Lecre 0 59 h /h Nmercal solon of RANS eqaons Flo arond he KLCC anker Model k-ω SST Rn=,03 0 9 C P 0 3 0.3 k-ω p=.6 0. 0. 0 0.5.5 h /h Ressance and Proplson Lecre 0 60
Nmercal solon of RANS eqaons Flo arond he KLCC anker Cp: -0.4-0.3-0. -0. 0 0. 0. 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Model k-ω SST Rn=4,6 06 Cp: -0.4-0.3-0. -0. 0 0. 0. 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Rn=,03 09 Ressance and Proplson Lecre 0 6 Nmercal solon of RANS eqaons Flo arond he KLCC anker Rn=4,6 06 Model k-ω SST Rn=,03 09 Ressance and Proplson Lecre 0 6
Nmercal solon of RANS eqaons Flo arond he KLCC anker : 0.05 0. 0.5 0. 0.5 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 0 Model k-ω SST -0.0-0.0 Z/Lpp -0.03-0.04-0.05-0.06 Rn=4,6 0 6-0.07 0 0.03 0.06 0.09 Y/Lpp Ressance and Proplson Lecre 0 63 Nmercal solon of RANS eqaons Flo arond he KLCC anker : 0.05 0. 0.5 0. 0.5 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 0 Model k-ω SST -0.0-0.0 Z/Lpp -0.03-0.04 Rn=,03 0 9-0.05-0.06-0.07 0 0.03 0.06 0.09 Y/Lpp Ressance and Proplson Lecre 0 64
0 Nmercal solon of RANS eqaons Flo arond he KLCC anker Model k-ω SST -0.0-0.0 /L PP -0.03-0.04-0.05 Rn=4,6 0 6-0.06-0.07 0 0.03 0.06 0.09 Ressance and Proplson /L PP Lecre 0 65 0-0.0 Model k-ω SST -0.0 /L PP -0.03-0.04 Rn=,03 0 9-0.05-0.06-0.07 0 0.03 0.06 0.09 /L PP Ressance and Proplson Lecre 0 66
Nmercal solon of RANS eqaons Flo arond he KLCC anker 0.5 0.5 =-0.075L PP =-0.09L PP Model k-ω SST 0.5 =-0.0365L PP 0.5 0.5 0.5 Epermenal G G G3 G4 =-0.0435L PP =-0.05075L PP =-0.058L PP 0 0 0.05 0.05 0.075 0. /L PP Rn=4,6 0 6 Ressance and Proplson Lecre 0 67 Nmercal solon of RANS eqaons Flo arond he KLCC anker 0.6 k-ω ah bh 3 0.58 Model k-ω SST 0.56 f 0.54 0.5 0.5 0 0.5.5 h /h ake fracon Rn=4,6 0 6 Ressance and Proplson Lecre 0 68
Nmercal solon of RANS eqaons Flo arond he KLCC anker 0.8 Model k-ω SST k-ω p=.6 f 0.75 0.7 Rn=,03 09 0.65 0.6 0 0.5 h/h.5 Ressance and Proplson Lecre 0 ake fracon 69 Roghness effecs Ressance and Proplson Lecre 0 70
Roghness effecs C F 0 3 Nmercal nceran h R =0 h.8 R /L=50µm p=.8 p= 0.9 h R /L=30µm h R /L=00µm.6 p=.0 h R /L=50µm X p * = h R /L=50µm h R /L=75µm p * = h R /L=300µm p * =.4 p * = p * = h R /L=00µm p=.9..8.6 X X X X X X.4 0 0.5 0.5 0.75.5.5.75 h /h Tanker C P 0 3 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0. 0. h R =0 p=.5 h R /L=30µm p=.5 h R /L=50µm p=.6 h R /L=75µm p=.6 h R /L=00µm p=.6 X X h /h h R /L=50µm p=.6 h R /L=00µm p=.6 h R /L=50µm p=.7 h R /L=300µm 0 0 0.5 0.5 0.75.5.5.75 X X X X X Frcon ressance, C F Pressre ressance, C P Ressance and Proplson Lecre 0 7 Roghness effecs Roghness effecs on he flo feld Frcon ressance, C F C F /C F (h R =0).5 Tanker, Rn=4.6 0 6 Tanker, Rn=.03 0 9 Conaner shp, Rn=.77 0 7 Conaner shp, Rn=. 0 9.5.4 Car carrer, Rn=. 0.4 9.3.. C F /C F (h R =0).3.. Tanker, Rn=4.6 0 6 Tanker, Rn=.03 0 9 Conaner shp, Rn=.77 0 7 Conaner shp, Rn=. 0 9 Car carrer, Rn=. 0 9 0 00 00 300 h R ( µm) 0 500 000 500 000 500 = h Ressance and Proplson Lecre 0 h R R 7 R
Roghness effecs on he flo feld.5 Tanker, Rn=4.6 0 6 Tanker, Rn=.03 0 9 Conaner shp, Rn=.77 0 7 Conaner shp, Rn=. 0 9.4 Car carrer, Rn=. 0 Roghness effecs Pressre ressance, C P Car carrer, Rn=. 0 9.4.5 Tanker, Rn=4.6 0 6 Tanker, Rn=.03 0 9 Conaner shp, Rn=.77 0 7 Conaner shp, Rn=. 0 9.4 Car carrer, Rn=. 0 9 C P /C P (h R =0).3.. C P /C P (h R =0).3.. 0 00 00 300 h R ( µm) 0 500 000 500 000 500 = h Ressance and Proplson Lecre 0 h R R 73 R f / f (h R =0) Roghness effecs on he flo feld.5 Tanker, Rn=4.6 0 6 Tanker, Rn=.03 0 9 Conaner shp, Rn=.77 0 7 Conaner shp, Rn=. 0 9.4 Car carrer, Rn=. 0.3.. Roghness effecs ake fracon, f.5 Car carrer, Rn=. 0 9.4 0 00 00 300 h R ( µm) f / f (h R =0) Tanker, Rn=4.6 0 6.4 Tanker, Rn=.03 0 9 Conaner shp, Rn=.77 0 7 Conaner shp, Rn=. 0 9 Car carrer, Rn=. 0 9.3.. 0 500 000 500 000 500 = h Ressance and Proplson Lecre 0 h R R 74 R
Roghness effecs Roghness effecs on he flo feld Non-dmensonal roghness hegh, h R Ressance and Proplson Lecre 0 75 0.03 0.0 RANS, Eamples of Nmercal Solons Seres 60 nscd flo `h / L;Y/L;3 scos (model) `h / L;Y/L;3 scos (shp) `h / L;Y/L;3 epermens (model) `h / L;Y/L;3 / L pp = 0.0755 0.0 / L pp 0 0.0 Ressance and Proplson Lecre 0 76
5 RANS, Eamples of Nmercal Solons Dna Tanker nscd ;Y/L;3 flo scos ;Y/L;3 (model) scos ;Y/L;3 (shp) epermens ;Y/L;3 (model) / L pp = 0.0900 5 0 5 Ressance and Proplson Lecre 0 77 RANS, Eamples of Nmercal Solons Ressance and Proplson Lecre 0 78
RANS, Eamples of Nmercal Solons Ressance and Proplson Lecre 0 79