Similarity transformation methods in the analysis of the two dimensional steady compressible laminar boundary layer

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1 Tm pap:.6 Compssibl Flui Dynamics, Spin 4 Similaity tansomation mtos in t analysis o t to imnsional stay compssibl lamina bounay lay Yunoo Co Anlica Assopos canical Eninin, assacustts Institut o Tcnoloy ABSTRACT T systm o quations in a stay, compssibl, lamina bounay lay is compos o ou unamntal quations. Tos a: t continuity quation, t momntum quation, t ny quation, an t quation o stat. T solutions o ts quations, n solv simultanously o a -imnsional bounay lay, a: t vlocity in t an y iction u an v, t pssu p an t nsity. T systm o quations is a systm o patial intial quations PDE an is usually iicult to solv. To, sopisticat tansomation mtos, call similaity tansomations a intouc to convt t oiinal patial intial quation st to a simplii oinay intial quation ODE st. T solutions o tis oinay intial quation st a usually nonimnsionaliz vlocitis an tmpatu. By pincipl, ts oinay quations a coupl matmatically an usually can b solv by numical mtos. ov, it ut appopiat assumptions lat to t tanspot poptis.. Pantl numb, an lo conitions.. ac numb, omty aoun lo, ts ODEs can b uncoupl matmatically o can av simpl oms, almost simila to t oms obtain om t incompssibl bounay lay analysis... Blasius solution, Falkn-Skan quation. nc, t simplii ODE st maks it possibl to t t solution om t alay istin solutions o t incompssibl analysis an also ucs t computin tim in t numical analysis. In tis pap, t int tansomation mtos ill b scib. A tail ivation o t naliz Lvy-Ilinot tansomation mto an t appopiat assumptions ma uin t ivation ill b plain. T oat tansomation an t Illinot-Statson tansomation ill b scib bily. ITRODCTIO T systm o quations in t incompssibl bounay lay it oc convction, is a PDE systm compos o t continuity, t momntum, an t ny quations. Ts simultanous quations can b uc to to ODEs usin similaity tansomation. In tis cas, continuity quation an momntum quation a uc to a sinl ODE an ny quation is uc to anot ODE. Compa it t incompssibl bounay lay analysis, t ct o compssibility on t nti vlocity an tmpatu il soul b consi. As a sult, t systm o quations in compssibl bounay lay is a mo complicat PDE systm, compos o t continuity quation, t momntum quation, t ny quation an an quation o stat. SYSTE OF EQATIOS OF CORESSIBLE BODARYLAYER T systm o ovnin quations to b solv o a to-imnsional, stay, compssibl, lamina bounay lay itout boy ocs an bulk at tans is as ollos: GOVERIG EQATIOS Continuity quation u v

2 omntum quation P u v µ P Eny quation u v v 5 At t o t bounay lay, t viscous lo insi t bounay lay is qui to smootly tansition into t invisci lo outsi t bounay lay. uy, y 6, t subscipt psnts conition at t o t bounay lay. u P u v An quation o stat P µ ν p RT 4, : Diction alon t suac catin t bounay lay y : Diction nomal to t suac u : Vlocity in t iction v : Vlocity in t y iction : Dnsity p : Pssu µ : Viscosity ν : Kinmatic viscosity p : Pssu : Entalpy R : Gas constant Compain t ny quation 3 to t ny quation A.47 us in incompssibl bounay lay it oc convction son in Appni.4, t ist tm in t ny quation in 3 is tain, ic is u P t compssiv ok tm. T scon tm on t it an si o t ny quation psnts t iusion o at tans to t lui o nat itin t lui. T ti tm psnts t at nat u to viscous stsss itin t lui, i.., viscous issipation. BODARY CODITIOS Ts bounay conitions at t suac, i.., y a ivn by t no-slip vlocity conition it o itout mass tans o at tans. 3 ODIESIOAL FOR OF TE EQATIOS Intoucin t non-imnsional vaiabls: u v y u v y 7 L L µ P µ P µ tn, t oiinal quations ~4 bcom: u v P u v µ R u v γ R, γ v u P L R : Rynols numb µ γ γ C C p v : Spciic at atio : ac numb c P R 8 9 µ

3 C p : Spciic at at constant pssu C v : Spciic at at constant volum In t non-imnsional ny quation, t ist tm, i.., t ok u to compssion an t ti tm, i.., t at nat by viscous issipation bcom incasinly impotant as t ac numb o t tnal lo incass. BASIC ASSPTOS I TE COPRESSIBLE BODAY LAYER In t PDE systm compos o quations ~4, t inlunc o compssibility is ist contain ictly in t nsity tms in t continuity quation, an mo inictly as a vaiabl coicint in t momntum quation an ny quation 3. T scon inlunc o compssibility is to pouc tmpatu vaiations tat a too la to pmit t assumption o constant poptis µ an k. It is common to us t ny quation ittn in tms o ntalpy in compssibl poblms insta o k as son in t ny quation 3, in ic t Pantl µ c p numb P is son insta o k. To, P t a complity it compssibl, lamina bounay lay poblms is cnt on vaiabl, µ, an P. Fom an quation o stat, t nsity is a unction o tmpatu an pssu, i.., T, P. ov, t pssu is assum constant acoss t bounay lay. To, t nsity can b assum to b a unction o tmpatu only, i.., T. T viscosity µ also can b assum to a unction o tmpatu only, i.., µ µt. Finally, t Pantl numb P is assum naly constant o most ass ov a i an o tmpatu. DERIVATIO OF GEERALIZED SIILARITY TRASFORATIO ILLIGWORT-LEVY OR LEVY-LEE TRASFORATIO T ivation o a naliz similaity tansomation is om t pocu aopt by Li an aamatsu [] an is ll summaiz in []. EERGY EQATIO I TERS OF ETALPY T ny quation can b ittn in tms o t total ntalpy. u 3, u : t vlocity alon t stamlin sin quation 3, t ny quation 3 bcoms: u v u u v P u µ µ uµ P 4 T pssu aint tm in t ny quation 4 can b liminat by multiplyin t momntum quation by u an ain t sult to t ny quation 4. Tis sults in: u P 5 v µ P µ uµ EQATIOS I TERS OF STREA FCTIO Fo t similaity tansomations an t cosponin simila solutions, t compssibl stam unction can b in by: ψ u ψ v 6 7 Equation 6 an 7 automatically satisy t continuity quation. Tn, t momntum quation an ny quation 5 bcom: ψ ψ ψ ψ P ψ µ 8

4 ψ P 9 P ψ ψ µ µ VARIABLE TRASFORATIO Dpnnt vaiabl tansomation ψ ψ µ Fom t pinc it t incompssibl bounay lay quations, t pnnt vaiabl tansomations a intouc as ollos: ψ, y, u, y,, y,, t subscipt inicats patial intiation. T om o t ntalpy tansomation stats tat t compssibl bounay lay is pct to b simila it spct to a non-imnsional total ntalpy poil at tan t static ntalpy o tmpatu poil, as in t cas o t incompssibl constantpopty bounay lay. Inpnnt vaiabl tansomation Inpnnt vaiabl tansomations a intouc as ollos: 3, y 4 Rlation btn inpnnt an pnnt vaiabl tou tnasomation Fom t initions o t stam unctions in 6 an 7: ψ, u, ic sults in: 5 o intatin: y y 7 ill b tmin om t tansom momntum an ny quations. FIRST FOR OF TRASFORED EQATIOS Intoucin quations 3, 4, an 7 into t momntum quation 8 an ny quation 9 sults in: µ P P µ 3 P µ P, 8 9, t subscipts,, an inicat patial intiations. SIPLIFIED FOR OF TRASFORED EQATIOS Capman - Rubsin viscosity assumption In t quations 8 an 9, t nsity an t viscosity, alays appa in t om µ cpt in t pssu aint tm. Tis las to t assumption o a Capman-Rubsin viscosity la, it in quation A.3 as son in Appni. sin t conitions at t o t bounay lay as nc conition sults in: µ T C µ T, ω 3 P ic, om an t quation o stat: µ C µ 3 Substitutin tis sult 3 into quations 8 an 9 sults in: 6

5 µ C 3 P P µ C 3 P µ C 33 P, Lina viscosity la assumption, simila assumption, an iso-ntic assumption T coicint C in quation 3 can vay tou t bounay lay. ov, t constant C assumption is ma, an is valuat at t suac conitions,.., usin t Sutlan viscosity la in Appni. T lo is assum to b simila, in ot os, an suc tat t it an si o t momntum quation 3 an ny quation 33 bcom zo. Finally, it is assum tat t total ntalpy at t bounay lay is constant, i..,,. Tis iso-ntic assumption o t invisci lo at t o t bounay lay, i.., constant, is not stictiv. Sinc, bot t static ntalpy an t vlocity can vay alon t o t bounay lay. Fom t act tat t stanation ntalpy is constant acoss a sock av, t iso-ntic lo assumption is asonabl n t sock av is not siniicantly cuv Fom ts assumptions, quation 3 bcoms olloin 34 by placin pssu aint tm usin Eul s quation at t o t bounay lay, i.., / / : P C µ C µ an, t quation 33 bcoms: P P C µ 34 35, t pim nots oinay intiation it spct to. SIILARITY CODITIOS Fom quations 34 an 35, t similaity conitions a: Conition C µ Conition C µ const Conition 3 const unction o only o P 38 Simpliication om Conition I t constant in conition is t unity, tn, in t absnc o a pssu aint, t momntum quation 34 ucs to t Blasius quation in Appni.. In aition, compain quation 35 it t ny quation A.9 o oc convction in Appni.5, by coosin t constant in conition as unity, t intial quation 35 o t compssibl bounay lay it unit Pantl numb as t sam om as tat o t incompssibl bounay lay it an isotmal all. To, t constant in conition is cosn as unity as ollos: C µ Raanin an intatin o quation 39 sults in: 39 C µ 4 sin quation 4, quation 7 bcoms: y y 4 C µ Sinc, an:

6 C µ 4 sults in: y y 43 T tansomations ivn in 4 an 43 a call t Illinot-Lvy tansomation. Simpliication om conition Fo t cas o. const, usin t inition o in quation an t inition o t stanation ntalpy: u u u 44 Sinc, u /, quation 44 can b ittn as ollos: 45 Finally, om t constant pssu assumption acoss t bounay lay, 46 tn, t tm in conition bcoms: ˆ C β µ 47 Simpliication om conition 3 Conition 3 can b ittn as ollos: γ γ FIAL FOR OF TRASFORED EQATIOS Fom t abov simpliications, t inal ovnin quations a: ˆ β 48 P P σ 49, γ γ σ ASSPTIO FOR TE EXISTECE OF SIILAR SOLTIO Po la vaiation in t ac numb Simila solutions o t quations 48 an 49 ist i ˆ γ β 5 is constant. βˆ can b ittn in tms o t tnal ac numb by intiatin [ ] / / γ an valuatin / usin t act tat t stanation ntalpy is constant at t o t bounay lay. γ γ 5 an 5 Fom quations 5 an 5, quation 5 bcoms: β ln ln ˆ constant 53 Intatin t quation 53 sults in ˆ β const 54 To, t similaity quimnt o t momntum quation 48 is satisi by a po la vaiation o t

7 ac numb in t tansom plan. In aition, t similaity quimnt is satisi by an ponntial ac numb vaiation, ic is son by Li an aamatsu [] an Con []. v µ const 55a, b Ot assumptions Futmo, in o o similaity conitions in quations 48 an 49 to ist, on o t olloin assumptions must also b satisi: γ 3 P 4 const 5 σ, i.., T assumption, i.., γ is unalistic o most ass. T assumption, i.., nlcts bot t viscous issipation an t compssiv ok tms in t ny quation. I it is ut assum tat t is no at tans at t suac, t assumption stats tat t static tmpatu tou t bounay lay is constant. ov, sinc t static tmpatu in t bounay lay soul vay om t suac tmpatu to t static tmpatu at t bounay lay, t assumption is lss alistic tan t unit Pantl numb assumption. T assumption 3, i.., P stats tat t stanation ntalpy o tmpatu o zo at tans at t suac is constant tou t bounay lay. Tis sult is clos to t tu aiabatic all stanation ntalpy vaiation, ic is slit. T assumption 4, i.., const cospons only to t lat plat ˆ β in quation 54. ov, o small valus o t pssu aint paamt, βˆ, t constant tnal ac numb assumption is suicint. T assumption 5, i.., las to t ypsonic lo assumption, i.., σ. Tis appoimation is lss tan iv pcnt in o at t tnal ac numb o tn. In aition, it allos t invstiation o t cts o constant but non-unit Pantl numb on t at tans at t suac. BODARY CODITIOS Bounay conitions at t suac T bounay conitions qui at t suac o simila solutions to ist a: T quation 55a psnts t mass tans nomal to t suac. Tis quation is obtain by intiatin t stam unction, quation, it spct to, usin t inition o ivn by t quation 4 to valuat an t Capman- Rubsin viscosity la to aan t sult. In aition, o simila solutions to ist, must b constant. Tis implis tat: v o v µ 56a, b, as o t Falkn-Skan quation in Appni 3, nativ valus o cospon to mass tans om t suac to t lui, i.., injction o bloin, an positiv valus o cospon to mass tans om t lui into t suac, i.., suction. T bounay conition lat to t ny quation is: const o Out bounay conitions T out bounay conitions a SOLVIG EQATIOS 57a, b 58a, b Simila solutions o quations 48 an 49 subjct ts bounay conitions 55~58 can b obtain accoin to t olloin cass. T sults a summaiz in Appni 6. In all t bllo cass, t unamntal quations a tansom to quations simila to t unamntal quations ovnin t incompssibl bounay lay. Cas : Lo Sp Compssibl Bounay Lay it Vaiabl Poptis ˆ β const,, P const mans nlctin t viscous issipation an compssiv ok in t ny quation an is accptabl n t it an si o quation 49 is small compa to t lt an si o quation 49. T bounay valu poblm o tis cas can b ittn as ollos:

8 ˆ β 59 P 6 it bounay conitions const const o Sinc, γ Fom quation 64, cas sults in 64 Solution o quation 6 psnts nonimnsional static ntalpy poils tou t bounay lay o non-imnsional tmpatu poil o a constant spciic at at constant pssu c. Fut, om quation 5 ˆ β / / an, om quation 4 C µ To, intation yils ˆ ˆ β β βˆ m const const 65 const, m ˆ β / ˆ β βˆ : sam as β, i.., Falkn-Skan pssu aint paamt in Appni 3. Cas. Coupl-quations cas onzo at tans an ˆ β Du to t vaiabl poptis inclu in t solution, t momntum an ny quations a coupl. Wn t is at tans at t suac, t ivn p quations in tis cas as no knon analytical solution. Tis bounay valu poblm ns numical mtos. Cas. ncoupl-quations ˆ β, : Flat plat itout mass tans at t suac Fo tis cas, momntum quation 59 ucs to Blasius quation in Appni. Fut, ny quation 6 as t sam unctional om as t ny quation ovnin t incompssibl constant popty oc convction tmal bounay lay itout viscous issipation, i.., quation A.6 in Appni 4. In paticula, t solution o quation A.6 o abitay but constant Pantl numb is θ 66, θ : non-imnsional solution ivn by quation A.66 in Appni 4 Cas.3 Anot uncoupl-quations - Aiabatic all Fo an aiabatic all, i.., intatin quation 6 tic an usin t bounay conition, sults in. Fo zo ac numb, t static ntalpy is constant tou t bounay lay. Equation 6 is analoous to t Busmann an Cocco intals, ic a, ov, stict to P. sin Equation 6, t momntum quation bcoms ˆ β 67 Tis quation 67 is t Falkn-Skan quation in Appni. Altou t non-imnsional momntum an ny quations a matmatically uncoupl, pysically ty a still coupl tou t tanspot poptis. Consiin t inpnnt vaiabl tansomation o, om quations 43 an 46, t pysical imnsion y bcoms: y J C µ / J 68

9 W, J J Cas : Compssibl Bounay Lay on a Flat Plat ˆ β, const, P const In t tnal invisci lo,, an T a constant, quation 59 ucs to t Blasius quation in Appni. 69 T ovnin intial quations 69 an 49 a uncoupl. Sinc, t ovnin quations a uncoupl, ty a intat squntially in a mann simila to tat us o t incompssibl constant popty oc convction bounay lay in Appni 4. An, t bounay conitions a aain ivn by quations 6, 6, an 63. Cas. P an Aiabatic all Equation 49 ucs to 7 A spciic intal o tis om o t ny quation 6 o zo at tans at t suac, i.., aiabatic all is. ov, sinc, is t atio o stanation ntalpis insta o t atio o static ntalpis. To, in tis cas, t stanation ntalpy is constant tou t bounay lay. Fut, t aiabatic all conition is a. Tis mans tat o t unit Pantl numb, t stanation ntalpy at t suac is qual to t stanation ntalpy at t o t bounay lay. Sinc, t vlocity is zo at t suac, o constant spciic at, t aiabatic all tmpatu is qual to t stanation tmpatu o t lui at t bounay lay, ic mans, consquntly, t unity covy acto. Tis pysical manin is tat t convsion o kintic ny into tmal ny at t suac tou t viscous issipation is as icint as t convsion o kintic ny into tmal ny tou t action o pssu ocs in t invisci lo at t bounay lay. Tis paticula intal o t ny quation is call t Busmann ny intal. Cas. P an Isotmal all const Fom t Blasius quation, i A is multipli to t Blasius quation, A is som constant, an a tat sult to t ny quation in quation 67, t sult bcoms: A A 7 A A 7 Intatin onc: A const 73 sin t bounay conition in quations 6, 6, an 63 at t suac yils const o t isotmal suac. At intatin quation 73 aain an usin quations 6, 6, an 63: A const 74 T constant A is valuat om t bounay conition at ininity, quation 63. Finally, 75 Equation 75 is call t Cocco intal. a an is a solution o t ny quation 75 o unit Pantl numb. Cas.3 P T ny quation P σ P is a lina non-omonous scon o oinay intial quation it vaiabl coicints. T nonomonous tm is a knon ocin-unction tat is pysically attibut to at aition u to viscous issipation. Sinc, t ovnin quation is lina, a solution is obtain as t sum o a complmntay solution o t omonous quation an a paticula solution o t non-omonous quation. 76 K KG σg, G is t solution o t omonous bounay valu poblm, i.. it oiinal bounay conitions: G P G 77 G G 78

10 , G is t solution o t non-omonous bounay valu poblm it omonous bounay conitions: G P G P 79 G G 8 Fom t compaison o quations A.6 & A.63 in Appni 4 it quations 77 & 78, G θ. To, G is ivn by quation A.66 in Appni 4. Compain quations A.64&A.65 in Appni 4 it quations 79&8 vals tat t non-omonous tms a int. By usin t mto o vaiation o a paamt o an intatin acto, P G P τ θ 8 P τ, θ : non-imnsional solution ivn by quation A.7. T constants K an K in quation 76 a valuat usin t bounay conitions quations 6, 6, an 63. T complt solution is to G σ G G G θ σ θ θ θ σ θ 8 Wn σ, quation 8 ucs to quation 66 obtain o. Fom σ γ /[ γ / ], t cts o viscous issipation on t ntalpy poil a siniicant n t tnal ac numb is siniicant as son in Fiu A.4 in Appni 6. As σ incass t maimum ntalpy atio in t bounay lay incass. Tis is a sult o t convsion o kintic ny itin t bounay lay into tmal ny tou viscous issipation. By intiatin t quation 8 an sttin t sult to b zo, aiabatic all tmpatu can b obtain as ollos. θ [ a σ θ / ] 83 o σ θ / 84 a Cas 3: Gnal Simila Compssibl Bounay Lay it nit Pantl umb ˆ ˆ β β const, const, P ˆ β it bounay conitions const const o Tis is t sam non-imnsional bounay valu poblm ovnin t lo sp compssibl bounay lay, i.., cas. ov, t non-imnsional pnnt vaiabl is t / at tan /. atio o stanation ntalpis Tis mans tat in tis cas 3, t cts o viscous issipation a inclu. Cas 3. Aiabatic all Fo zo at tans at t suac, an plicit intal o quation 6 subjct to quations 6, 6, an 63 is. Tis sos tat, o an aiabatic all, t stanation ntalpy is constant tou t bounay lay. To, o zo at tans at t suac, t intnal at nat u to viscous issipation in t vlocity il an t at tans by iusion an conuction in t tmpatu il intact in a pcis mann to maintain t stanation ntalpy constant touout t bounay lay. Tis sult is a consqunc o t unit Pantl numb assumption. As, t non-imnsional momntum quation ucs to t Falkn-Skan quation in Appni 3. To, t nonimnsional momntum an ny quations a uncoupl. Cas 3. on-zo at tans at t suac cas Wn t is at tans at t suac, t ivn quations in tis cas as no knon analytical solution. Tis bounay valu poblm as stui numically by

11 Con [], Lvy [3], Li & aamatsu [], Con & Rsotko [4], an Ros [5]. Cas 4: Simila ypsonic Compssibl Bounay Lay ˆ β const,, P ˆ β 9 P P 9 it bounay conitions const const o Fo t cass to 3, un ctain conitions, t bounay valu poblm o t compssibl bounay lay coul b uc to an quivalnt incompssibl bounay lay poblm. ov, tis is not possibl o t psnt cas 4. Tis is bcaus t stanation ntalpy is not constant tou t bounay lay vn o an aiabatic all is not an intal o t ny quation. Tus, sinc, ˆ β, t momntum quation 9 cannot b uc to t Falkn-Skan quation in Appni 3. To, t unctions, an qui in t ny intals, quations A.66 an 8, pn on, an bcaus o t couplin btn t momntum an ny quations, 59 an 9. Bcaus o tis couplin, t ny intals cannot b valuat cpt by succssiv appoimations usin t incompssibl Falkn-Skan solutions to bin t appoimation. To, numical mto soul b us to t t act solution. RELIABILITY OF TE SIILARITY TRASFORATIO ETODS Epimntal ata psnt in Fiu A.6 in Appni 6 sust tat t popos tansomations pict t vlocity an ntalpy o t systm it i accuacy.. vlocity poil o t compssibl bounay lay on an aiabatic lat plat It soul b not, ov, tat al li applications a most likly to viat om on o ts ou catois psnt abov. T n o numical simulation is tn bcomin ssntial o mo accuacy. ov, t analytical appoac is citical as it povis t ssntial amok on ic t numical appoimations a built. OTER SIILARITY TRASFORATIOS OWART TRASFORATIO oat tansomation is a stict om o t tansomation simila to t on u to oat [7] an t olloin ivation is aopt om [][8]. Intoucin t compssibl stam unction in by ψ u an ψ v 95, t subscipt inicats som nc conition. Inpnnt vaiabl tansomations a:, y 96, ic a subjct to t conition u ψ. T paticula unctional oms cosn o t inpnnt vaiabl tansomations a bas on t quivalnt oms o incompssibl lo it stiction u ψ ic is also bas on t incompssibl sults. ψ ψ ψ y y 97 ic, yils t qui inpnnt vaiabl tansomation o, i.. y y 98 T omal tansomation quations a y Tansom o t momntum quation 99 sin quations 99 an, t tansom momntum quation bcoms: P µψ ψ ψ ψ ψ

12 sin t Capman-Rubsin viscosity la it in Appni, i.. µ T C µ T an om P /, an usin t quation o stat, µ C µ 3 tn, quation bcoms P ψ ψ ψ ψ ν Cψ 4 I C constant, tn it ψ ψ ψ ψ C, i.., C, P ν ψ C 5 Ecpt o t acto / C / in t pssu aint tm, quation 5 as t sam om as t momntum quation ovnin incompssibl constant popty bounay lay lo. Wn t pssu aint is zo, i.., o a lat plat at zo-incinc, quation 5 as actly t sam om as t incompssibl constant popty momntum quation. In t absnc o pssu aint, t similaity tansomations vlop o t incompssibl bounay lay lo yil t Blasius quation in Appni, i.. ν 6 / / ν ψ, 7 T tansom bounay conitions o an impmabl suac a 8 9, t pim nots intiation it spct to. To, t solution o t momntum quation o t compssibl vaiabl popty bounay lay in t absnc o pssu aint is uc to t solution o an quivalnt incompssibl constant popty quation, i.., t Blasius quation in Appni. In t absnc o a pssu aint, t momntum quation o compssibl bounay lay lo is uncoupl om t ny quation. Fomally tis is tu. ov, tminin t pysical cooinat, y, om t invs o quation 98, quis a knol o t nsity istibution in t bounay lay an to t solution o t ny quation. Tus, t momntum an ny quations o compssibl bounay lay lo, vn in t absnc o a pssu aint, a still tcnically coupl. Tansom o t ny quation Tansomation o t ny quation into, cooinats yils ψ µ C ψ ψ ψ C P C P µ Intoucin t Capman-Rubsin viscosity la in Appni an assumin tat C µ is constant, P ν ψ ψ ψ ν ψ C P In t absnc o a pssu aint, quation 5 is quivalnt to t ny quation ovnin oc convction lo ov a lat plat at zo-incinc. STEWARTSO-ILLIGWORT TRASFORATIO Assumin tat unity Pnatl numb, constant c p, an t viscosity linaly lat to t tmpatu, Statson an Illinot av inpnntly son tat t ists a tansomation om a compssibl lo bounay lay, to a lat incompssibl lo bounay lay [4][8]. A stam unction tat satisis t continuity quation is: ψ u ψ v 3 T ny an momntum quations a tansom to n cooinats X an Y suc tat:

13 X Y P C P a a y C y 4 5 m β : Pssu aint m u : Vlocity atio u, a mans sonic sp an subscipt psnts som nc stat. T ntalpy unction S is in as: S 6 T stam unction is plac by t tansom vlocitis an V tou olloin lations. Y 7 ψ V X 8 Equations -8 a appli to t momntum an ny quations an a n st o quations is obtain. It assum tat t pssu is constant alon t bounay lay an tat all tmpatu is constant. In o to uc tis systm into a systm o oinay intial quation, t olloin lations a assum: a p ψ AX 9 q Y BX b S S, A, B, a, b, p, an, q a untmin vaiabls. Possibl simila solutions a possibl i: m CX o C p C X Tn, t systm o ODEs cosponin to t pola vlocity istibution o quations my b ittn: β S 3 γ S P S P γ 4 SARY Wit t incas complity o t quations o motion o compssibl vaiabl-nsity, vaiabl-popty los, it as natual to sk ays o ioously tnin t matial at an o constant-nsity, constantpopty los to tos cass. Ways sout to tansom a compssibl bounay lay poblm into an quivalnt incompssibl poblm. T istin solutions coul tn b tansom back to a solution o t oiinal compssibl poblm. Tis pocu n in succss it som assumptions. W iscuss t ampls,.. t Illinot-Lvy tansomation, t oat tansomation an t Statson-Illinot tansomation. REFERECES [] Li, T.Y., an aamatsu,.t., Simila solutions o compssibl bounay lay ov a lat plat it suction o injction, JAS, Vol., pp , 955 [] Davi F. Ros, Lamina lo analysis, Cambi nivsity Pss, 99 [3] Con, C., Simila solutions o compssibl lamina bounay lay quations, JAS, Vol., pp.8-8, 949 [4] Lvy, S., Ect o la tmpatu cans incluin viscous atin upon lamina bounay lays it vaiabl -stam vlocity, JAS, Vol., pp , 95 [5] Con, C.B., an Rsotko, E., Simila solutions o t compssibl lamina bounay lay it at tans an pssu aint, ACA TR 93, 956 [6] Ros, D.F, Rvs lo solutions o compssibl lamina bounay lay quations, Pys. o Fluis, Vol., pp , 969 [7] oat, L., Concnin t ct compssibility on lamina bounay lays an ti spaation, Poc. Roy. Soc. Lonon S. A, Vol. 94, pp. 6-4, 948 [8] Stan Sci, Compssibl lo, Wily, 98 [9] A.D. Youn, Bounay lays, Oo, 989,

14 OTER SORCES [] O.A. Olinik, V.. Samokin, atmatical mols in bounay lay toy, Capman&all/CRC, 999 [] Josp A. Sctz, Bounay lay analysis, Pntic- all, 993 [3] mann Sclictin, Klaus Gstan, Bounaylay toy, Spin, APPEDICES Ima mov u to copyit consiations. APPEDIX. PROPERTIES VARIATIO OF TRASPORT T tanspot poptis o impotanc in a viscous compssibl lo a t viscosity, t tmal conuctivity, t spciic at at constant pssu, an t Pantl numb ic is t combination o t ist t poptis. VISCOSITY Fom monatomic as toy, t viscosity o ass pns only on t tmpatu an is inpnnt o t pssu. Epimntal masumnts conim tat tis sult is ssntially coct o all ass. Fo ass, t viscosity incass it incasin tmpatu. In contast, t viscosity o liquis pns on bot tmpatu an pssu an cass it incasin tmpatu. Fiu A.. Absolut viscosity o ctain ass an liquis Ima mov u to copyit consiations. Sutlan viscosity la Epimntal masumnts o t viscosity o ai a lat it tmpatu by t Sutlan quation: Fiu A.. Po la viscosity lationsip µ µ T T 3 / T S T S Fo ai btn 8 R an 34 R, S 98.6 R T 49.6 R µ lbsc/ t A. Capman-Rubsin viscosity la Bcaus o t complity o Sutlan quation, appoimation omula bas on t mpiical quation call Capman-Rbsin viscosity la is us insta. µ µ ω T C A.3 T To, o ai, 3 / T 8 µ.7 lbsc/ t A. T 98.6 Fius A. an A. so t absolut viscosity o ctain ass an liquis an t po la viscosity lationsip spctivly []. A simpl an usul cas o t Capman-Rubsin viscosity la occus n C an in quation A.3. Wit ts valus, an usin t suac as t nc conition, T µ µ T Fo an isotmal all, tis ucs to

15 µ constt TERAL CODCTIVITY T tmal conuctivity o ass k also pns only on t tmpatu an is inpnnt o pssu. T vaiation o t tmal conuctivity o ai it tmpatu is t sam as tat o t ynamic viscosity. SPECIFIC EAT AT COSTAT PRESSRE T spciic at at constant pssu c p o ai is almost constant o a i an o tmpatus. TE PRADTL BER T bavios o tanspot poptis it tmpatu mntion abov mak t Pantl numb P µc p / k ssntially invaiant it tmpatu. To, it is assum tat t Pantl numb o ass is constant. Tis assumption liminats t n to omally spciy t unctional vaiation o t c p an k it tmpatu. Fut, consiabl matmatical simpliication occus i coos a unit Pantl numb an a Capman-Rubsin viscosity la it C. Fiu A.3 sos t vaiation o k, cp an P it tmpatu []. APPEDIX. BLASIS EQATIO- TE FLOW PAST A FLAT PLATE WITOT PRESSRE GRADIET GOVERIG EQATIOS AD BODARY CODTIO Continuity quation v omntum quation A.4 u u v ν A.5 Bounay conition y : u v A.6 y : u A.7 TRASORATIO SIG STREA FCTIO u ψ y v ψ Fom A.5 an A.8, A.8 ψ ψ ψ ψ νψ y y A.9 yy yyy Wit bounay conitions Ima mov u to copyit consiations. y ψ ψ A. y y ψ A. y At similaity tansomation Fiu A.3. Vaiation o tmal conuctivity, spciic at at constant pssu, an t Pantl numb it tmpatu y ψ, y ν A. ν A.3 it A.4 an A.5

16 APPEDIX 3. FALKER-SKA EQATIO - TE FLOW PAST A FLAT PLATE WIT PRESSRE GRADIET GOVERIG EQATIOS AD BODARY CODTIO Continuity quation Fo all >, t bounay conition at ininity y, bcoms: ψ AB, A.6 y v A.6 o, A.7 AB omntum quation P u u v ν Bounay conition y : v A.7 u A.8 y : u A.9 TRASORATIO SIG STREA FCTIO ψ yψ y ψ ψ yy νψ yyy A. Wit bounay conitions y ψ ψ y A. y ψ y A. By intoucin t tansomations Ay A.3 ψ, y B, A.4 Tn t ovnin quations yil: AB [ AB 3 3 νa B νa B AB 3 νa B, ] A.5 In o to non-imnsionaliz quation A.5 an to obtain a simpl numical sult o t bounay conition at ininity, AB an ν A 3 B a cosn. W, : t potntial vlocity upstam o. Solvin o A an B yils A /ν an B ν / A.8 Tn, quation A.5 bcoms: α β, α A.9 β A.3 In o o simila solutions to ist, t tansom stam unction must b a unction o only, i..,. To, t it an si o quation A.9 must b zo. Futmo, α, β must b inpnnt o. Sinc an assum to b unctions o only, α, β a constants. β α A.3 it bounay conitions A.3 A.33 Sinc α, β a assum to b constants, quations A.3 psnt to quations in t to unknon unctions, an. an can tus b tmin. Fom α β A.34

17 Poviin α β, intation o quation A.34 yils α β A.35 A scon albaic quation o an is obtain by consiin α β A.36 ultiplyin bot sis o quation A.36 by an itin sults in: α β β A.37 Intation o quation A.37 sults in: sin t sults o A an in t oiinal tansomations, yils t appopiat inpnnt similaity vaiabl m y A.44 ν In t abov analysis, t cass α an α β clu. APPEDIX 4. FORCED COVECTIO BODARY LAYER WITOOT PRESSRE GRADIET PARALLE FLOW PAST A FLAT PLATE GOVERIG EQATIOS AD BODARY CODITIO Ima mov u to copyit consiations. α β β α β K β A.38 Fiu A.4. Foc convction bounay lay lo past a lat plat [] Simultanous solution o quations A.34 an A.38 yils β α β α β m K [α β ] const A.39 an / α β A.4 Simila solution o t stay to-imnsional incompssibl bounay lay ist i t potntial vlocity vais as a po o t istanc alon t suac. Poviin α, itout loss o nality, α is cosn. In aition, by intoucin β β m o m m an tn bcom m K an, K K m β A.4 m / m A.4 m A.43 Continuity quation v A.45 omntum quation u u v ν A.46 Eny quation T T T c p u v k µ A.47 Bounay conition y u v A.48 T T T o A.49 y u T T A.5

18 , t subscipt inicats conitions in t invisci lo at t o t bounay lay. SIILARITY TRASFORATIO sin t similaity tansomations y ψ, y ν A.5 ν A.5 it A.53 an A.54 T P P c T p Intoucin t non-imnsional tmpatu T T A.55 θ A.56 T T Tn, quation A.55 bcoms θ θ P P E A.57 θ o it θ A.58 an θ A.59, t appopiat Eckt numb is in as E c p T T A.6 Equation A.57 is a scon-o lina nonomonous intial quation subjct to to-point asymptotic bounay conitions. Tis quation can b ivi by to quations by supposition pincipl. θ K θ Eθ A.6 omonous quation θ P θ A.6 θ an it θ A.63 on-omonous quation θ P θ P A.64 it θ an θ A.65 SOLTIO AALYTICAL SOLTIO Bot t omonous an t non-omonous bounay valu poblms a amnabl to analytical solution. omonous solution T solution o t omonous poblm is P θ α P P A.66, α P P P Fo t spcial cas o unit Pantl numb α, quation A.66 ucs to θ A.67 To, n t is at tans at t suac t non-imnsional tmpatu istibution as t sam om as t non-imnsional vlocity istibution. on-omonous solution T non-omonous bounay valu poblm o t aiabatic all is solv usin t mto o vaiation o a paamt. P θ P τ A.68 P τ Fo t spcial cas o unit Pantl numb, tis sult ucs to θ A.69 Total solution sin t conition θ at K Eθ A.7 an quation A.6 bcoms

19 θ [ E θ ] θ Eθ A.7 Fo t spcial cas o unit Pantl numb, θ θ θ θ E θ A.7 APPEDIX 5. FORCED COVECTIO BODARY LAYER WIT PRESSRE GRADIET AD OISOTERAL SRFACE CODITIO GOVERIG EQATIOS AD BODARY CODITIO Continuity quation v omntum quation P u u v ν Eny quation ν u v ν P Bounay conition A.73 A.74 A.75 y u v A.76 T T A.77 y u T T A.78 SIILARITY CODITIO FOR TE VELOCITY FIELD FALKER-SKA EQATIO m β β SIILARITY TRASFORATIO m A.79 m y A.8 ν / ψ, y ν m A.8 TRASFORED EQATIO β A.8 T P P c A.84 o p T m T P m P m T A.83 T T T T P P β β P P c A.85 p By intoucin a non-imnsional tmpatu: T T θ A.86 T T Fut assumption is ma suc tat t suac at tans is suicintly small tat it os not can t tmpatu in t invisci lo at t o t bounay lay. Tus, T T θ P θ P β θ T m P c T T p T T A.87 CODITIO FOR TE EXISTECE OF SIILARITY SOLTIO T T T const an m T T const A.88 n T T T A.89, T : a constant associat it t initial tmpatu istibution n : Isotmal conition sin tis assumption, quation A.87 bcoms

20 θ m n P θ n P β θ P E, t Eckt numb is E / c pt T associat bounay conitions a θ θ A.9 SOLTIO Fom quation A.89, t a to classs o simila solutions o t ny quation o oc convction: tos it viscous issipation, an tos itout viscous issipation Lo sp incompssibl lo lct o viscous issipation In tis cas, t Eckt numb is small, sinc is small. n ts conitions, t viscous issipation on t it an si can b nlct. θ P θ n P β θ A.9 n isotmal all Tis quation ucs to t sam om as t omonous solution o t lat plat isotmal all cas, i.., quation A.6. Altou quation A.9 is o t sam om as quation A.6, its solution θ θ is not t sam., t non-imnsional stam unction, ivn by t solution o t Falkn-Skan quation in Appni 3, pns on t valu o β, an, in tun, t solution o quation A.9 also pns on t valu o β. β Simila solutions o t ny quation ist o abitay all tmpatu vaiations. 3 n an β T simila solutions o t ny quation pn on bot t pssu aint β an t suac tmpatu paamt n. Wn t viscous issipation is not nlct Simila solutions o t ny quation its only i m-n. n β / β an β In ot os, simila solutions o t ny quation ist o only on all tmpatu vaiation. < β < T suac tmpatu incass in t iction o t lo β < T suac tmpatu cass in t iction o t lo 3 β an n Constant suac tmpatu APPEDIX 6. SARY OF GOVERIG EQATIOS FOR SIILAR COPRESSIBLE BODARY LAYER Cas : T lo sp compssibl bounay lay ˆ β an Equations ˆ β P. Aiabatic Wall analytical Solutions It can b solv usin t sult o Falkn-Skan quation ˆ β. Isotmal Wall analytical Solutions It can not b solv Commnts: T viscous issipation tms a nlct Cas : T compssibl bounay lay on a lat plat ˆ β an const Equations: σ P. Aiabatic Wall analytical Solutions It can b solv o P: Bussman Intal

21 . Isotmal Wall analytical Solutions It can b solv.. P Cocco intal Ima mov u to copyit consiations... P θ σ θ θ θ σ θ Commnts: T viscous issipation tms a inclu 3 Fo P, Fo a P, a σ θ Fiu A.5 psnts t ct o ac numb on t ntalpy poils o a lat plat []: Fiu A.6. Compaison o pimntal an totical vlocity poils o t compssibl bounay lay on an aiabatic lat plat Cas 3:simila compssibl bounay lay it unit pantl numb ˆ β ˆ β, const, P Equations: β P ˆ 3. Aiabatic Wall analytical Solutions It can b solv: ˆ β Ima mov u to copyit consiations. 3. Isotmal Wall analytical Solutions It cannot b solv Commnts: o analytical solutions o non-unit Pantl numb T viscous issipation tms a inclu Fiu A.5. T ct o ac numb on t ntalpy o ˆ β,. 6, an P.73 Fiu A.6 sos t compaison o pimntal an totical vlocity poils o t compssibl bounay lay on an aiabatic lat plat []: Cas 4: T simila ypsonic compssibl bounay lay it nonunit pantl numb ˆ β Equations: ˆ β P 4. Aiabatic Wall analytical Solutions Cannot b solv:

22 4. Isotmal Wall analytical Solutions Cannot b solv: Commnts: yils σ o t aiabatic all