Introduction to Computational Fluid Dynamics: Governing Equations, Turbulence Modeling Introduction and Finite Volume Discretization Basics.

Transcription

1 Introduction to Computational Fluid Dynamics: Govrning Equations, Turbulnc Modling Introduction and Finit Volum Discrtization Basics. Jol Gurrro Fbruary 13, 2014

2 Contnts 1 Notation and Mathmatical rliminaris 2 2 Govrning Equations of Fluid Dynamics Simplification of th Navir-Stoks Systm of Equations: Incomprssibl Viscous Flow Cas Turbulnc Modling Rynolds Avraging Incomprssibl Rynolds Avragd Navir-Stoks Equations Boussinsq Approximation Two-Equations Modls. Th κ ω Modl Finit Volum Mthod Discrtization Discrtization of th Solution Domain Discrtization of th Transport Equation Approximation of Surfac Intgrals and Volum Intgrals Convctiv Trm Spatial Discrtization Diffusion Trm Spatial Discrtization Evaluation of Gradint Trms Sourc Trms Spatial Discrtization Tmporal Discrtization Systm of Algbraic Equations Boundary Conditions and Initial Conditions Discrtization Errors Taylor Sris Expansions Accuracy of th Upwind Schm and Cntral Diffrncing Schm Man Valu Approximation Gradint Approximation Spatial and Tmporal Linar Variation Msh Inducd Errors Msh Spacing Th Finit Volum Mthod for Diffusion and Convction-Diffusion roblms Stady On-Dimnsional Diffusion Stady On-Dimnsional Convction-Diffusion Stady on-dimnsional convction-diffusion working xampl Unstady On-Dimnsional Diffusion Unstady On-Dimnsional Convction-Diffusion Finit Volum Mthod Algorithms for rssur-vlocity Coupling 56 1

3 Chaptr 1 Notation and Mathmatical rliminaris Whn prsnting th fluid flow quations, as wll as throughout this manuscript, w mak us of th following vctor/tnsor notation and mathmatical oprators. From now on w ar going to rfr to a zro rank tnsor as a scalar. A first rank tnsor will b rfrrd as a vctor. And a scond rank tnsor will associatd to a tnsor. Vctors will b dnotd by minuscul bold lttrs, whras tnsors by majuscul bold lttrs or bold grk symbols. Scalars will b rprsntd by normal lttrs or normal grk symbols. Hraftr, th vctor is almost always a column vctor and a row vctor is xprssd as a transpos of a column vctor indicatd by th suprscript T. Vctors a = a 1 i + a 2 j + a 3 k and b = b 1 i + b 2 j + b 3 k ar xprssd as follows a 1 b 1 a = a 2, b = b 2, a 3 b 3 th transpos of th column vctors a and b ar rprsntd as follows a T = [a 1, a 2, a 3 ], b T = [b 1, b 2, b 3 ], Th magnitud of a vctor a is dfind as a = (a a) 1 2 = (a a a 3 2 ) 1 2. A tnsor is rprsntd as follows A 11 A 12 A 13 A 11 A 21 A 31 A = A 21 A 22 A 23, A T = A 12 A 22 A 32. A 31 A 32 A 33 A 13 A 23 A 33 If A = A T, th tnsor is said to b symmtric, that is, its componnts ar symmtric about th diagonal. Th dot product of two vctors a and b (also known as scalar product of two vctors), yilds to a scalar quantity and is givn by a T b = a b = [ ] a 1, a 2, a 3 b 1 b 2 b 3 = a 1 b 1 + a 2 b 2 + a 3 b 3. 2

4 Th dot product of two vctors a and b is commutativ (a b = b a). Th cross product of two vctors a b (also known as vctor product of two vctors), is th vctor normal to th plan of a and b, and is dfind by th dtrminant i j k a 2 b 3 a 3 b 2 a b = a 1 a 2 a 3 b 1 b 2 b 3 = a 3 b 1 a 1 b 3, a 1 b 2 a 2 b 1 a b and b a rsult in two diffrnt vctors, pointing in opposit dirctions with th sam magnitud (a b = b a). Th tnsor product (also known as dyadic product) of two vctors a b producs a scond rank tnsor and is dfind by a 1 a 1 b 1 a 1 b 2 a 1 b 3 a b = ab T = ab = a 2 [b 1, b 2, b 3 ] = a 2 b 1 a 2 b 2 a 2 b 3, a 3 a 3 b 1 a 3 b 2 a 3 b 3 notic that unlik th dot product, th tnsor product of two vctors is non-commutativ (a b b a). Th doubl dot product (:) of two scond rank tnsors A and B (also known as scalar product of two scond rank tnsors) A 11 A 12 A 13 B 11 B 12 B 13 A = A 21 A 22 A 23, B =, A 31 A 32 A 33 B 21 B 22 B 23 B 31 B 32 B 33 producs a scalar = A:B, which can b valuatd as th sum of th 9 products of th tnsor componnts = A ij B ij = A 11 B 11 + A 12 B 12 + A 13 B 13 + A 21 B 21 + A 22 B 22 + A 23 B 23 + A 31 B 31 + A 32 B 32 + A 33 B 33. Th doubl dot product of two scond rank tnsors is commutativ (A:B = B:A). Th dot product of a tnsor A and a vctor a, producs a vctor b = A a, whos componnts ar A 11 a 1 + A 12 a 2 + A 13 a 3 b = b i = A ij a j = A 21 a 1 + A 22 a 2 + A 23 a 3. A 31 a 1 + A 32 a 2 + A 33 a 3 Th dot product of a non symmtric tnsor A and a vctor a is non-commutativ (A ij a j a i A ij ). If th tnsor A is symmtric thn b = a A = A T a. Th dot product of two tnsors A and B (also known as singl dot product or tnsor product of two tnsors), producs anothr scond rank tnsor C = A B, whos componnts ar valuatd as C = C ij = A ik B kj 3

5 Th dot product of two tnsors is non-commutativ (A B B A). Not that our dfinitions of th tnsor-vctor dot product and tnsor-tnsor dot-product ar consistnt with th ordinary ruls of matrix algbra. Th trac of a tnsor A is a scalar, valuatd by summing its diagonal componnts tr A = A tr = A 11 + A 22 + A 33. Th gradint oprator (rad as nabla) in Cartsian coordinats is dfind by = i + y j + ( z k =, y, ) T. z Th gradint oprator whn applid to a scalar quantity (x, y, z) (whr x,y, z ar th spatial coordinats), yilds to a vctor dfind by = (, y, ) T. z Th notation grad for may b also usd as th gradint oprator, so that, grad. Th gradint of a vctor a producs a scond rank tnsor a 1 a 2 grad a = a = a 3 a 1 y a 2 y a 3 y a 1 z a 2. z a 3 z Th gradint can oprat on any tnsor fild to produc a tnsor fild that is on rank highr. Th dot product of vctor a and th oprator is calld th divrgnc (div) of th vctor fild; th output of this oprator is a scalar and is dfind as div a = a = a 1 + a 2 y + a 3 z. Th divrgnc of a tnsor A, div A or A, yilds to a vctor and is dfind as A 11 + A 12 y + A 13 z A 21 div A = A = + A 22 y + A 23. z A 31 + A 32 y + A 33 z Th divrgnc can oprat on any tnsor fild of rank 1 and abov to produc a tnsor that is on rank lowr. Th curl oprator of a vctor a producs anothr vctor. This oprator is dfind by i j k ( curl a = a = a3 y z = y a 2 z, a 1 z a 3, a 2 a ) T 1. y a 1 a 2 a 3 Th divrgnc of th gradint is calld th Laplacian oprator and is dnotd by. Laplacian of a scalar (x, y, z) yilds to anothr scalar fild and is dfind as Th 4

6 div grad = = 2 = = y z 2. Th Laplacian of a vctor fild a is dfind as th divrg of th gradint just lik in th scalar cas, but in this cas it yilds to a vctor fild, such that 2 a a 1 y a 1 z 2 div grad a = a = 2 a = a = 2 a a 2 y a 2 z 2. 2 a a 3 y a 3 z 2 Th Laplacian transforms a tnsor fild into anothr tnsor fild of th sam rank. As prviously discussd, a tnsor is said to b symmtric if its componnts ar symmtric about th diagonal, i.., A = A T. A tnsor is said to b skw or anti-symmtric if A = A T which intuitivly implis that A 11 = A 22 = A 33 = 0. Evry scond rank tnsor can b dcomposd into symmtric and skw parts by A = 1 ( A + A T ) + 1 ( A A T ) = symm A + skw A. } 2 {{}} 2 {{} symmtric skw Th jacobian matrix of a vctor fild a is givn by a 1 a 2 a 3 a 1 y a 2 y a 3 y a 1 z a 2. z a 3 z Th idntity matrix or unit matrix, is a matrix whos diagonal ntris ar all 1 and th othr ntris ar 0. Th 3 3 idntity matrix I is givn by I = Hraftr w prsnt som usful vctor/tnsor idntitis: a = 0. α = 0. (αβ) = α β + β α. (αa) = a α + α a. ( a)a = a a 2 a ( a)a = a a a 2. a a. (a b) a = b a a 2. (a b) = b a + a b + a b + b a. 5

7 (αa) = α a + a α. a = ( a) ( a). (a b) = b a a b. (a b) = b a + a b. a (b c) = (a b) c + (a b) b. (αa) = A α + α A. (Ab) = ( A T ) b + A T b. (αa) = α a + α a. (a b) = a b + b a ( a)b a b. a (Ab) = A (a b). a (Ab) = (Aa) b if A is symmtric. ab:a = a (b A) A:ab = (A a) b whr α and β ar scalars; a, b and c ar vctors; and A is a tnsor. 6

8 Chaptr 2 Govrning Equations of Fluid Dynamics Th starting point of any numrical simulation ar th govrning quations of th physics of th problm to b solvd. Hraftr, w prsnt th govrning quations of fluid dynamics and thir simplification for th cas of an incomprssibl viscous flow. Th quations govrning th motion of a fluid can b drivd from th statmnts of th consrvation of mass, momntum, and nrgy [1, 2, 3]. In th most gnral form, th fluid motion is govrnd by th tim-dpndnt thr-dimnsional comprssibl Navir-Stoks systm of quations. For a viscous Nwtonian, isotropic fluid in th absnc of xtrnal forcs, mass diffusion, finit-rat chmical ractions, and xtrnal hat addition; th consrvation form of th Navir- Stoks systm of quations in compact diffrntial form and in primitiv variabl formulation (ρ, u, v, w, t ) can b writtn as ρ + (ρu) = 0, t (ρu) + (ρuu) = p + τ, t (ρ t ) + (ρ t u) = q (pu) + τ : u, t (2.0.1) whr τ is th viscous strss tnsor and is givn by τ xx τ xy τ xz τ = τ yx τ yy τ yz. (2.0.2) τ zx τ zy τ zz For th sak of compltnss, lt us rcall that in th consrvation form (or divrgnc form) [4], th momntum quation can b writtn as (ρu) + (ρuu) = p + τ, (2.0.3) t whr th tnsor product of th vctors uu in q is qual to u uu = v [ u v w ] u 2 uv uw = vu v 2 vw. (2.0.4) w wu wv w 2 Lt us rcall th following idntity 7

9 (uu) = u u + u( u), (2.0.5) and from th divrgnc-fr constraint ( u = 0) it follows that u( u) is zro, thrfor (uu) = u u. Hncforth, th consrvation form of th momntum quation q is quivalnt to ( ) (u) ρ + u (u) = p + τ, t which is th non-consrvation form or advctiv/convctiv form of th momntum quation. Th st of quations can b rwrittn in vctor form as follows q t + i + f i y + g i z = v + f v y + g v z, (2.0.6) whr q is th vctor of th consrvd flow variabls givn by ρ ρu q = ρv ρw, (2.0.7) ρ t and i, f i and g i ar th vctors containing th inviscid fluxs (or convctiv fluxs) in th x, y and z dirctions and ar givn by ρu ρv ρw ρu 2 + p i = ρuv ρuw, f ρvu i = ρv 2 + p ρvw, g ρwu i = ρwv ρw 2 + p, (2.0.8) (ρ t + p) u (ρ t + p) v (ρ t + p) w whr u is th vlocity vctor containing th u, v and w vlocity componnts in th x, y and z dirctions and p, ρ and t ar th prssur, dnsity and total nrgy pr unit mass rspctivly. Th vctors v, f v and g v contain th viscous fluxs (or diffusiv fluxs) in th x, y and z dirctions and ar dfind as follows 0 τ xx v = τ xy τ xz, uτ xx + vτ xy + wτ xz q x 0 τ yx f v = τ yy τ yz, uτ yx + vτ yy + wτ yz q y 0 τ zx g v = τ zy τ zz, uτ zx + vτ zy + wτ zz q z (2.0.9) 8

10 whr th hat fluxs q x, q y and q z ar givn by th Fourir s law of hat conduction as follows q x = k T, q y = k T y, q z = k T z, (2.0.10) and th viscous strsss τ xx, τ yy, τ zz, τ xy, τ yx, τ xz, τ zx, τ yz and τ zy, ar givn by th following rlationships τ xx = 2 ( 3 µ 2 u v y w ) z τ yy = 2 ( 3 µ 2 v y u w ) z τ zz = 2 3 µ ( 2 w z u v y ( u τ xy = τ yx = µ y + v ), ( u τ xz = τ zx = µ z + w ), ( v τ yz = τ zy = µ z + w ), y,, ), (2.0.11) In quations , T is th tmpratur, k is th thrmal conductivity and µ is th laminar viscosity. In ordr to driv th viscous strsss in q th Stoks s hypothsis was usd [5]. Examining closly quations and counting th numbr of quations and unknowns, w clarly s that w hav fiv quations in trms of svn unknown flow fild variabls u, v, w, ρ, p, T, and t. It is obvious that two additional quations ar rquird to clos th systm. Ths two additional quations can b obtaind by dtrmining rlationships that xist btwn th thrmodynamic variabls (p, ρ, T, i ) through th assumption of thrmodynamic quilibrium. Rlations of this typ ar known as quations of stat, and thy provid a mathmatical rlationship btwn two or mor stat functions (thrmodynamic variabls). Choosing th spcific intrnal nrgy i and th dnsity ρ as th two indpndnt thrmodynamic variabls, thn quations of stat of th form p = p ( i, ρ), T = T ( i, ρ), (2.0.12) ar rquird. For most problms in arodynamics and gasdynamics, it is gnrally rasonabl to assum that th gas bhavs as a prfct gas (a prfct gas is dfind as a gas whos intrmolcular forcs ar ngligibl), i.., p = ρr g T, (2.0.13) whr R g is th spcific gas constant and is qual to 287 m2 for air. Assuming also that th s 2 K working gas bhavs as a calorically prfct gas (a calorically prfct gas is dfind as a prfct gas with constant spcific hats), thn th following rlations hold i = c v T, h = c p T, γ = c p c v, c v = R g γ 1, c p = γr g γ 1, (2.0.14) 9

11 whr γ is th ratio of spcific hats and is qual to 1.4 for air, c v th spcific hat at constant volum, c p th spcific hat at constant prssur and h is th nthalpy. By using q and q , w obtain th following rlations for prssur p and tmpratur T in th form of q p = (γ 1) ρ i, T = p ρr g = (γ 1) i R g, (2.0.15) whr th spcific intrnal nrgy pr unit mass i = p/(γ 1)ρ is rlatd to th total nrgy pr unit mass t by th following rlationship, t = i + 1 ( u 2 + v 2 + w 2). (2.0.16) 2 In our discussion, it is also ncssary to rlat th transport proprtis (µ, k) to th thrmodynamic variabls. Thn, th laminar viscosity µ is computd using Suthrland s formula µ = C 1T 3 2 (T + C 2 ), (2.0.17) whr for th cas of th air, th constants ar C 1 = ms and C K 2 = 110.4K. Th thrmal conductivity, k, of th fluid is dtrmind from th randtl numbr ( r = 0.72 for air) which in gnral is assumd to b constant and is qual to k = c pµ r, (2.0.18) whr c p and µ ar givn by quations q and q rspctivly. Whn daling with high spd comprssibl flows, it is also usful to introduc th Mach numbr. Th mach numbr is a non dimnsional paramtr that masurs th spd of th gas motion in rlation to th spd of sound a, [( ) ] 1 p 2 a = = γ p ρ S ρ = γr g T. (2.0.19) Thn th Mach numbr M is givn by, M = U a = U γ(p/ρ) = U γrg T kg (2.0.20) Anothr usful non dimnsional quantity is th Rynold s numbr, this quantity rprsnts th ratio of inrtia forcs to viscous forcs and is givn by, R = ρ U L µ, (2.0.21) whr th subscript dnots frstram conditions, L is a rfrnc lngth (such as th chord of an airfoil or th lngth of a vhicl), and µ is computd using th frstram tmpratur T according to q Th first row in q corrsponds to th continuity quation. Likwis, th scond, third and fourth rows ar th momntum quations, whil th fifth row is th nrgy quation in trms of total nrgy pr unit mass. Th Navir-Stoks systm of quations , is a coupld systm of nonlinar partial diffrntial quations (DE), and hnc is vry difficult to solv analytically. In fact, to th dat thr 10

12 is no gnral closd-form solution to this systm of quations; hnc w look for an approximat solution of this systm of quation in a givn domain D with prscribd boundary conditions D and givn initial conditions D q. If in q w st th viscous fluxs v = 0, f v = 0 and g v = 0, w gt th Eulr systm of quations, which govrns inviscid fluid flow. Th Eulr systm of quations is a st of hyprbolic quations whil th Navir-Stoks systm of quations is a mixd st of hyprbolic (in th inviscid rgion) and parabolic (in th viscous rgion) quations. Thrfor, tim marching algorithms ar usd to advanc th solution in tim using discrt tim stps. 2.1 Simplification of th Navir-Stoks Systm of Equations: Incomprssibl Viscous Flow Cas Equations , with an appropriat quation of stat and boundary and initial conditions, govrns th unstady thr-dimnsional motion of a viscous Nwtonian, comprssibl fluid. In many applications th fluid dnsity may b assumd to b constant. This is tru not only for liquids, whos comprssibility may b nglctd, but also for gass if th Mach numbr is blow 0.3 [2, 6]; such flows ar said to b incomprssibl. If th flow is also isothrmal, th viscosity is also constant. In this cas, th govrning quations writtn in compact consrvation diffrntial form and in primitiv variabl formulation (u, v, w, p) rduc to th following st u t (u) = 0, + (uu) = p ρ + ν 2 u, (2.1.1) whr ν is th kinmatic viscosity and is qual ν = µ/ρ. Th prvious st of quations in xpandd thr-dimnsional Cartsian coordinats is writtn as follows u + v y + w z = 0, u t + u2 + uv y + uw z = 1 ρ v t + uv + v2 y + vw z w t + uw + vw y + w2 z ( p 2 + ν u u y u z 2 ( = 1 p 2 ρ + ν v v y v z 2 ( = 1 p 2 ρ + ν w w y w z 2 ), ), ). (2.1.2) Equation govrns th unstady thr-dimnsional motion of a viscous, incomprssibl and isothrmal flow. This simplification is gnrally not of a grat valu, as th quations ar hardly any simplr to solv. Howvr, th computing ffort may b much smallr than for th full quations (du to th rduction of th unknowns and th fact that th nrgy quation is dcoupld from th systm of quation), which is a justification for such a simplification. Th st of quations can b rwrittn in vctor form as follow q t + i + f i y + g i z = v + f v y + g v z, (2.1.3) whr q is th vctor containing th primitiv variabls and is givn by 11

13 0 q = u v, (2.1.4) w and i, f i and g i ar th vctors containing th inviscid fluxs (or convctiv fluxs) in th x, y and z dirctions and ar givn by u i = u 2 + p uv, f i = uw v vu v 2 + p vw, g i = w wu wv w 2 + p. (2.1.5) Th viscous fluxs (or diffusiv fluxs) in th x, y and z dirctions, v, f v and g v rspctivly, ar dfind as follows v = τ xx τ xy, f v = τ yx τ yy, g v = τ zx τ zy. (2.1.6) τ xz τ yz τ zz Sinc w mad th assumptions of an incomprssibl flow, appropriat xprssions for shar strsss must b usd, ths xprssions ar givn as follows τ xx = 2µ u, τ yy = 2µ v y, τ zz = 2µ w z, ( u τ xy = τ yx = µ y + v ), ( w τ xz = τ zx = µ + u ), z ( w τ yz = τ zy = µ y + v ), z (2.1.7) whr w usd Stoks s hypothsis [5] in ordr to driv th viscous strsss in q Equation can b writtn in compact vctor form as τ = 2µD, whr D = 1 [ 2 u + u T ] is th symmtric tnsor of th vlocity gradint tnsor u = [D + S], and whr D rprsnts th strain-rat tnsor and S rprsnts th spin tnsor (vorticity). Th skw or anti-symmtric part of th vlocity gradint tnsor is givn by S = 1 [ 2 u u T ]. Equations , ar th govrning quations of an incomprssibl, isothrmal, viscous flow writtn in consrvation form. Hnc, w look for an approximat solution of this st of quations in a givn domain D with prscribd boundary conditions D and givn initial conditions D q. 12

14 Chaptr 3 Turbulnc Modling All flows ncountrd in nginring applications, from simpl ons to complx thr-dimnsional ons, bcom unstabl abov a crtain Rynolds numbr (R = UL/ν whr U and L ar charactristic vlocity and lngth scals of th man flow and ν is th kinmatic viscosity). At low Rynolds numbrs flows ar laminar, but as w incras th Rynolds numbr, flows ar obsrvd to bcom turbulnt. Turbulnt flows ar charactriz by a chaotic and random stat of motion in which th vlocity and prssur chang continuously on a broad rang of tim and lngth scals (from th smallst turbulnt ddis charactrizd by Kolmogorov micro-scals, to th flow faturs comparabl with th siz of th gomtry). Thr ar svral possibl approachs for th numrical simulation of turbulnt flows. Th first and most intuitiv on, is by dirctly numrically solving th govrning quations ovr th whol rang of turbulnt scals (tmporal and spatial). This dtrministic approach is rfrrd as Dirct Numrical Simulation (DNS) [15, 16, 17, 10, 22, 23]. In DNS, a fin nough msh and small nough tim-stp siz must b usd so that all of th turbulnt scals ar rsolvd. Although som simpl problms hav bn solvd using DNS, it is not possibl to tackl industrial problms du to th prohibitiv computr cost imposd by th msh and tim-stp rquirmnts. Hnc, this approach is mainly usd for bnchmarking, rsarch and acadmic applications. Anothr approach usd to modl turbulnt flows is Larg Eddy Simulation (LES) [18, 19, 15, 22, 23]. Hr, larg scal turbulnt structurs ar dirctly simulatd whras th small turbulnt scals ar filtrd out and modld by turbulnc modls calld subgrid scal modls. According to turbulnc thory, small scal ddis ar mor uniform and hav mor or lss common charactristics; thrfor, modling small scal turbulnc appars mor appropriat, rathr than rsolving it. Th computational cost of LES is lss than that of DNS, sinc th small scal turbulnc is now modld, hnc th grid spacing is much largr than th Kolmogorov lngth scal. In LES, as th msh gts finr, th numbr of scals that rquir modling bcoms smallr, thus approaching DNS. Thanks to th advancs in computing hardwar and paralll algorithms, th us of LES for industrial problms is bcoming practical. Today s workhors for industrial and rsarch turbulnc modling applications is th Rynolds Avragd Navir-Stoks (RANS) approach [20, 21, 22, 10, 12, 23]. In this approach, th RANS quations ar drivd by dcomposing th flow variabls of th govrning quations into timman (obtaind ovr an appropriat tim intrval) and fluctuating part, and thn tim avraging th ntir quations. Tim avraging th govrning quations givs ris to nw trms, ths nw quantitis must b rlatd to th man flow variabls through turbulnc modls. This procss introducs furthr assumptions and approximations. Th turbulnc modls ar primarily dvlopd basd on xprimntal data obtaind from rlativly simpl flows undr controlld conditions. This in turn limits th rang of applicability of th turbulnc modls. That is, no 13

15 singl RANS turbulnc modl is capabl of providing accurat solutions ovr a wid rang of flow conditions and gomtris. Two typs of avraging ar prsntly usd, th classical Rynolds avraging which givs ris to th RANS quations and th mass-wightd avraging or Favr avraging which is usd to driv th Favr-Avragd Navir-Stoks quations (FANS) for comprssibl flows applications. In both statistical approachs, all th turbulnt scals ar modld, hnc msh and tim-stp rquirmnts ar not as rstrictiv as in LES or DNS. Hraftr, w limit our discussion to Rynolds avraging. 3.1 Rynolds Avraging Th starting point for driving th RANS quations is th Rynolds dcomposition [20, 3, 22, 10, 12, 23] of th flow variabls of th govrning quations. This dcomposition is accomplishd by rprsnting th instantanous flow quantity by th sum of a man valu part (dnotd by a bar ovr th variabl, as in ) and a tim-dpndnt fluctuating part (dnotd by a prim, as in ). This concpt is illustratd in figur 3.1 and is mathmatically xprssd as follows, (x, t) = }{{} (x) + (x, t) }{{}. (3.1.1) man valu fluctuating part Figur 3.1: Tim avraging for a statistically stady turbulnt flow (lft) and tim avraging for an unstady turbulnt flow (right). Hraftr, x is th vctor containing th Cartsian coordinats x, y, and z in N = 3 (whr N is qual to th numbr of spatial dimnsions). A ky obsrvation in q is that is indpndnt of tim, implying that any quation driving for computing this quantity must b stady stat. In q , th man valu is obtaind by an avraging procdur. Thr ar thr diffrnt forms of th Rynolds avraging: 1. Tim avraging: appropriat for stationary turbulnc, i.., statically stady turbulnc or a turbulnt flow that, on avrag, dos not vary with tim. (x) = 1 t+t lim (x, t) dt, (3.1.2) T + T t 14

16 hr t is th tim and T is th avraging intrval. This intrval must b larg compard to th typical tim scals of th fluctuations; thus, w ar intrstd in th limit T. As a consqunc, dos not vary in tim, but only in spac. 2. Spatial avraging: appropriat for homognous turbulnc. (t) = lim CV 1 (x, t) dcv, (3.1.3) CV CV with CV bing a control volum. In this cas, is uniform in spac, but it is allowd to vary in tim. 3. Ensmbl avraging: appropriat for unstady turbulnc. 1 (x, t) = lim N N N (x, t), (3.1.4) whr N, is th numbr of xprimnts of th nsmbl and must b larg nough to liminat th ffcts of fluctuations. This typ of avraging can b applid to any flow (stady or unstady). Hr, th man valu is a function of both tim and spac (as illustratd in figur 3.1). W us th trm Rynolds avraging to rfr to any of ths avraging procsss, applying any of thm to th govrning quations yilds to th Rynolds-Avragd Navir-Stoks (RANS) quations. In cass whr th turbulnt flow is both stationary and homognous, all thr avraging ar quivalnt. This is calld th rgodic hypothsis. i=1 If th man flow varis slowly in tim, w should us an unstady approach (URANS); thn, quations q and q can b modifid as and (x, t) = 1 T (x, t) = (x, t) + (x, t), (3.1.5) t+t t (x, t)dt, T 1 << T << T 2, (3.1.6) whr T 1 and T 2 ar th charactristics tim scals of th fluctuations and th slow variations in th flow, rspctivly (as illustratd in figur 3.1). In q th tim scals should diffr by svral ordr of magnitud, but in nginring applications vry fw unstady flows satisfy this condition. In gnral, th man and fluctuating componnts ar corrlatd, i.., th tim avrag of thir product is non-vanishing. For such problms, nsmbl avraging is ncssary. An altrnativ approach to URANS is LES, which is out of th scop of this discussion but th intrstd radr should rfr to rfrncs [18, 19, 15, 22, 23]. Bfor driving th RANS quations, w rcall th following avraging ruls, 15

17 = 0, =, = + =, =, + ϕ = + ϕ, ϕ = ϕ = ϕ, ϕ = ϕ, ϕ = ϕ = 0, (3.1.7) ϕ = ( + )( ϕ + ϕ ) = ϕ + ϕ + ϕ + ϕ = ϕ + ϕ + ϕ + ϕ = ϕ + ϕ, 2 0, ϕ Incomprssibl Rynolds Avragd Navir-Stoks Equations Lt us rcall th Rynolds dcomposition for th flow variabls of th incomprssibl Navir- Stoks quations q , u(x, t) = ū(x) + u (x, t), p(x, t) = p(x) + p (x, t), (3.2.1) w now substitut q into th incomprssibl Navir-Stoks quations q and w obtain for th continuity quation Thn, tim avraging this quation rsults in (u) = (ū + u ) = (ū) + (u ) = 0. (3.2.2) and using th avraging ruls statd in q , it follows that (ū) + (ū ) = 0, (3.2.3) (ū) = 0. (3.2.4) W nxt considr th momntum quation of th incomprssibl Navir-Stoks quations q W bgin by substituting q into q in ordr to obtain, (ū + u ) t + (( ū + u ) ( ū + u )) = ( p + p ) ρ + ν 2 ( ū + u ), (3.2.5) by tim avraging q , xpanding and applying th ruls st in q , w obtain ū t + (ūū + u u ) = p + ν 2 ū. (3.2.6) ρ 16

18 Grouping quations and 3.2.6, w obtain th following st of quations, ū t (ū) = 0, + (ūū) = p ρ + ν 2 ū 1 ρ τ R. (3.2.7) Th st of quations q ar th incomprssibl Rynolds-Avragd Navir-Stoks (RANS) quations. Notic that in q w hav rtaind th trm ū/ t, dspit th fact that ū is indpndnt of tim for statistically stady turbulnc, hnc this xprssion is qual to zro whn tim avrag. In practic, in all modrn formulations of th RANS quations th tim drivativ trm is includd. In rfrncs [20, 22, 3, 16, 25], a fw argumnts justifying th rtntion of this trm ar discussd. For not statistically stationary turbulnc or unstady turbulnc, a tim-dpndnt RANS or unstady RANS (URANS) approach is rquird, an URANS computation simply rquirs rtaining th tim drivativ trm ū/ t in th computation. For unstady turbulnc, nsmbl avrag is rcommndd and oftn ncssary. Th incomprssibl Rynolds-Avragd Navir-Stoks (RANS) quations q ar idntical to th incomprssibl Navir-Stoks quations q with th xcption of th additional trm τ R = ρ ( u u ), whr τ R is th so-calld Rynolds-strss tnsor (notic that by doing a chck of dimnsions, it will show that τ R it is not actually a strss; it must b multiplid by th dnsity ρ, as it is don consistntly in this manuscript, in ordr to hav dimnsions corrsponding to th strsss. On th othr hand, sinc w ar assuming that th flow is incomprssibl, that is, ρ is constant, w might st th dnsity qual to unity, thus obtaining implicit dimnsional corrctnss. Morovr, bcaus w typically us kinmatic viscosity ν, thr is an implid division by ρ in any cas). Th Rynolds-strss tnsor rprsnts th transfr of momntum du to turbulnt fluctuations. In 3D, th Rynolds-strss tnsor τ R consists of nin componnts τ R = ρ ( u u ) ρu u ρu v ρu w = ρv u ρv v ρv w. (3.2.8) ρw u ρw v ρw w Howvr, sinc u, v and w can b intrchangd, th Rynolds-strss tnsor forms a symmtrical scond ordr tnsor containing only six indpndnt componnts. By inspcting th st of quations q w can count tn unknowns, namly; thr componnts of th vlocity (u, v, w), th prssur (p), and six componnts of th Rynolds strss ( τ R = ρ ( u u )), in trms of four quations, hnc th systm is not closd. Th fundamntal problm of turbulnc modling basd on th Rynolds-avragd Navir-Stoks quations is to find six additional rlations in ordr to clos th systm of quations q Boussinsq Approximation Th Rynolds avragd approach to turbulnc modling rquirs that th Rynolds strsss in q to b appropriatly modld (howvr, it is possibl to driv its own govrning quations, but it is much simplr to modl this trm). A common approach uss th Boussinsq hypothsis to rlat th Rynolds strsss τ R to th man vlocity gradints such that τ R = ρ ( u u ) 2 [ = 2µ T DR 3 ρκi = µ T u + ( u) T] 2 ρκi, (3.3.1) 3 whr D R dnots th Rynolds-avragd strain-rat tnsor ( 1 2 ( u + ut )), I is th idntity matrix, µ T is calld th turbulnt ddy viscosity, and κ = 1 2 u u, (3.3.2) 17

19 is th turbulnt kintic nrgy. Basically, w assum that this fluctuating strss is proportional to th gradint of th avrag quantitis (similarly to Nwtonian flows). Th scond trm in q ( 2 3ρκI), is addd in ordr for th Boussinsq approximation to b valid whn tracd, that is, th trac of th right hand sid in q must b qual to that of th lft hand sid ( ρ(u u ) tr = 2ρκ), hnc it is consistnt with th dfinition of turbulnt kintic nrgy (q ). In ordr to valuat κ, usually a govrning quation for κ is drivd and solvd, typically two-quations modls includ such an option. Th turbulnt ddy viscosity µ T (in contrast to th molcular viscosity µ), is a proprty of th flow fild and not a physical proprty of th fluid. Th ddy viscosity concpt was dvlopd assuming that a rlationship or analogy xists btwn molcular and turbulnt viscositis. In spit of th thortical waknss of th turbulnt ddy viscosity concpt, it dos produc rasonabl rsults for a larg numbr of flows. Th Boussinsq approximation rducs th turbulnc modling procss from finding th six turbulnt strss componnts τ R to dtrmining an appropriat valu for th turbulnt ddy viscosity µ T. On final word of caution, th Boussinsq approximation discussd hr, should not b associatd with th compltly diffrnt concpt of natural convction. 3.4 Two-Equations Modls. Th κ ω Modl In this sction w prsnt th widly usd κ ω modl. As might b infrrd from th trminology (and th tittl of this sction), it is a two-quation modl. In its basic form it consist of a govrning quation for th turbulnt kintic nrgy κ, and a govrning quation for th turbulnt spcific dissipation rat ω. Togthr, ths two quantitis provid vlocity and lngth scals ndd to dirctly find th valu of th turbulnt ddy viscosity µ T at ach point in a computational domain. Th κ ω modl has bn modifid ovr th yars, nw trms (such as production and dissipation trms) hav bn addd to both th κ and ω quations, which hav improvd th accuracy of th modl. Bcaus it has bn tstd mor xtnsivly than any othr κ ω modl, w prsnt th Wilcox modl [24]. Eddy Viscosity µ T = ρκ ω (3.4.1) Turbulnt Kintic Enrgy ρ κ t + ρ (ūκ) = τ R : ū β ρκω + [(µ + σ µ T ) κ] (3.4.2) Spcific Dissipation Rat ρ ω t + ρ (ūω) = αω κ τ R : ū βρω 2 + [(µ + σµ T ) ω] (3.4.3) Closur Cofficints α = 5 9, β = 3 40, β = 9 100, σ = 1 2, σ = 1 2 (3.4.4) 18

20 Auxiliary Rlations ɛ = β ωκ and l = κ 1 2 (3.4.5) ω Equations q and q , ar th govrning quations of an incomprssibl, isothrmal, turbulnt flow writtn in consrvation form. Hnc, w look for an approximat solution of this st of quations in a givn domain D, with prscribd boundary conditions D, and givn initial conditions D q. 19

21 Chaptr 4 Finit Volum Mthod Discrtization Th purpos of any discrtization practic is to transform a st of partial diffrntial quations (DEs) into a corrsponding systm of discrt algbraic quations (DAEs). Th solution of this systm producs a st of valus which corrspond to th solution of th original quations at som prdtrmind locations in spac and tim, providd crtain conditions ar satisfid. Th discrtization procss can b dividd into two stps, namly; th discrtization of th solution domain and th discrtization of th govrning quation. Th discrtization of th solution domain producs a numrical dscription of th computational domain (also known as msh gnration). Th spac is dividd into a finit numbr of discrt rgions, calld control volums (CVs) or clls. For transint simulations, th tim intrval is also split into a finit numbr of tim stps. Th govrning quations discrtization stp altogthr with th domain discrtization, producs an appropriat transformation of th trms of th govrning quations into a systm of discrt algbraic quations that can b solv using any dirct or itrativ mthod. In this sction, w brifly prsnts th finit volum mthod (FVM) discrtization, with th following considrations in mind: Th mthod is basd on discrtizing th intgral form of th govrning quations ovr ach control volum of th discrt domain. Th basic quantitis, such as mass and momntum, will thrfor b consrvd at th discrt lvls. Th mthod is applicabl to both stady-stat and transint calculations. Th mthod is applicabl to any numbr of spatial dimnsions (1D, 2D or 3D). Th control volums can b of any shap. All dpndnt variabls shar th sam control volum and ar computd at th control volum cntroid, which is usually calld th collocatd or non-staggrd variabl arrangmnt. Systms of partial diffrntial quations ar tratd in a sgrgatd way, maning that thy ar ar solvd on at a tim in a squntial mannr. Th spcific dtails of th solution domain discrtization, systm of quations discrtization practics and implmntation of th FVM ar far byond th scop of th prsnt discussion. Hraftr, w giv a brif dscription of th FVM mthod. For a dtaild discussion, th intrstd radr should rfr to rfrncs [7, 3, 10, 11, 12, 9, 13]. 20

22 4.1 Discrtization of th Solution Domain Discrtization of th solution domain producs a computational msh on which th govrning quations ar solvd (msh gnration stag). It also dtrmins th positions of points in spac and tim whr th solution will b computd. Th procdur can b split into two parts: tmporal discrtization and spatial discrtization. Th tmporal solution is simpl obtaind by marching in tim from th prscribd initial conditions. For th discrtization of tim, it is thrfor ncssary to prscrib th siz of th tim-stp that will b usd during th calculation. Th spatial discrtization of th solution domain of th FVM mthod prsntd in this manuscript, rquirs a subdivision of th continuous domain into a finit numbr of discrt arbitrary control volums (CVs). In our discussion, th control volums do not ovrlap, hav a positiv finit volum and compltly fill th computational domain. Finally, all variabls ar computd at th cntroid of th control volums (collocatd arrangmnt). Figur 4.1: Arbitrary polyhdral control volum V. Th control volum has a volum V and is constructd around a point (control volum cntroid), thrfor th notation V. Th vctor from th cntroid of th control volum V (point ), to th cntroid of th nighboring control volum V N (point N), is dfind as d. Th fac ara vctor S f points outwards from th surfac bounding V and is normal to th fac. Th control volum facs ar labld as f, which also dnots th fac cntr. A typical control volum is shown in figur 4.1. In this figur, th control volum V is boundd by a st of flat facs and ach fac is shard with only on nighboring control volum. Th shap of th control volum is not important for our discussion, for our purposs it is a gnral polyhdron, as shown in figur 4.1. Th control volum facs in th discrt domain can b dividd into two groups, namly; intrnal facs (btwn two control volums) and boundary facs, which coincid with th boundaris of th domain. Th fac ara vctor S f is constructd for ach fac in such a way that it points outwards from th control volum, is locatd at th fac cntroid, is normal to th fac and has a magnitud qual to th ara of th fac (.g., th shadd fac in figur 4.1). Boundary fac ara vctors point outwards from th computational domain. In figur 4.1, th point rprsnts th cntroid of th control volum V and th point N rprsnts th cntroid of th nighbor control volum V N. Th distanc btwn th 21

23 point and th point N is givn by th vctor d. For simplicity, all facs of th control volum will b markd with f, which also dnots th fac cntroid (s figur 4.1). A control volum V, is constructd around a computational point. Th point, by dfinition, is locatd at th cntroid of th control volum such that its cntroid is givn by V (x x ) dv = 0. (4.1.1) In a similar way, th cntroid of th facs of th control volum V is dfind as S f (x x f ) ds = 0. (4.1.2) Finally, lt us introduc th man valu thorm for th transportd quantity ovr th control volum V, such that = 1 (x)dv. (4.1.3) V V In th FVM mthod discussd in this manuscript, th cntroid valu of th control volum V is rprsntd by a picwis constant profil. That is, w assum that th valu of th transportd quantity is computd and stord in th cntroid of th control volum V and that its valu is qual to th man valu of insid th control volum, = = 1 (x)dv. (4.1.4) V V This approximation is xact if is constant or vary linarly. 4.2 Discrtization of th Transport Equation Th gnral transport quation is usd throughout this discussion to prsnt th FVM discrtization practics. All th quations dscribd in sctions 2 and 3 can b writtn in th form of th gnral transport quation ovr a givn control volum V (as th control volum shown in figur 4.1), as follows ρ V t dv + (ρu) dv (ργ }{{} V }{{} ) dv = S () dv. (4.2.1) V }{{} V }{{} convctiv trm diffusion trm sourc trm tmporal drivativ Hr is th transportd quantity, i.., vlocity, mass or turbulnt nrgy and Γ is th diffusion cofficint of th transportd quantity. This is a scond ordr quation sinc th diffusion trm includs a scond ordr drivativ of in spac. To rprsnt this trm with accptabl accuracy, th ordr of th discrtization must b qual or highr than th ordr of th quation to b discrtizd. In th sam ordr of idas, to conform to this lvl of accuracy, tmporal discrtization must b of scond ordr as wll. As a consqunc of ths rquirmnts, all dpndnt variabls ar assumd to vary linarly around th point in spac and instant t in tim, such that (x) = + (x x ) ( ) whr = (x ). (4.2.2) ( ) t (t + δt) = t + δt whr t = (t). (4.2.3) t 22

24 Equations and ar obtaind by using Taylor Sris Expansion (TSE) around th nodal point and tim t, and truncating th sris in such a way to obtain scond ordr accurat approximations. A ky thorm in th FVM mthod is th Gauss thorm (also know as th divrgnc or Ostrogradsky s thorm), which will b usd throughout th discrtization procss in ordr to rduc th volum intgrals in q to thir surfac quivalnts. Th Gauss thorm stats that th volum intgral of th divrgnc of a vctor fild in a rgion insid a volum, is qual to th surfac intgral of th outward flux normal to th closd surfac that bounds th volum. For a vctor a, th Gauss thorm is givn by, V adv = V nds a, (4.2.4) whr V is th surfac bounding th volum V and ds is an infinitsimal surfac lmnt with th normal n pointing outward of th surfac V. From now on, ds will b usd as a shorthand for nds. By using th Gauss thorm, w can writ q as follows (ρ) dv + ds (ρu) ds (ργ t V V }{{} ) = S () dv. (4.2.5) V }{{} V convctiv flux diffusiv flux Equation is a statmnt of consrvation. It stats that th rat of chang of th transportd quantity insid th control volum V is qual to th rat of chang of th convctiv and diffusiv fluxs across th surfac bounding th control volum V, plus th nt rat of cration of insid th control volum. Notic that so far w hav not introduc any approximation, quation is xact. In th nxt sctions, ach of th trms in q will b tratd sparatly, starting with th spatial discrtization and concluding with th tmporal discrtization. By procding in this way w will b solving q by using th Mthod of Lins (MOL). Th main advantag of th MOL, is that it allows us to slct numrical approximations of diffrnt accuracy for th spatial and tmporal trms. Each trm can b tratd diffrntly to yild to diffrnt accuracis Approximation of Surfac Intgrals and Volum Intgrals In q , a sris of surfac and volum intgrals nd to b valuatd ovr th control volum V. Ths intgrals must b approximatd to at last scond ordr accuracy in ordr to conform to th sam lvl of accuracy of q To calculat th surfac intgrals in q w nd to know th valu of th transportd quantity on th facs of th control volum. This information is not availabl, as th variabls ar calculatd on th control volum cntroid, so an approximation must b introducd at this stag. W now mak a profil assumption about th transportd quantity. W assum that varis linarly ovr ach fac f of th control volum V, so that may b rprsntd by its man valu at th fac cntroid f. W can now approximat th surfac intgral as a product of th transportd quantitis at th fac cntr f (which is itslf an approximation to th man valu 23

25 ovr th surfac) and th fac ara. This approximation to th surfac intgral is known as th midpoint rul and is of scond-ordr accuracy. It is worth mntioning that a wid rang of choics xists with rspct to th way of approximating th surfac intgrals,.g., midpoint rul, trapzoid rul, Simpson s rul, Gauss quadratur. Hr, w hav usd th simplst mthod, namly, th midpoint rul. For illustrating this approximation, lt us considr th trm undr th divrgnc oprator in q and rcalling that all facs ar flat (that is, all vrtxs that mad up th fac ar containd in th sam plan), q can b convrtd into a discrt sum of intgrals ovr all facs of th control volum V as follows, adv = ds a, V V = ( ) ds a, f f (S f a f ) = f f (S f a f ). (4.2.6) Using th sam approximations and assumptions as in q , th surfac intgrals (or fluxs) in q can b approximat as follow V ds (ρu) }{{} convctiv flux = f ds (ργ ) V }{{} diffusiv flux = f f ds (ρu) f f f ds (ργ ) f f S f (ρu ) f = f S f (ργ ) f = f S f (ρu) f. (4.2.7) S f (ργ ) f. (4.2.8) To approximat th volum intgrals in q , w mak similar assumptions as for th surfac intgrals, that is, varis linarly ovr th control volum and =. Intgrating q ovr a control volum V, it follows (x) dv = [ + (x x ) ( ) ] dv, V V [ ] = dv + (x x ) dv ( ), V V = V. (4.2.9) Th scond intgral in th RHS of q is qual to zro bcaus th point is th cntroid of th control volum (rcall q ). This quantity is asily calculatd sinc all variabls at th cntroid of V ar known, no intrpolation is ndd. Th abov approximation bcoms xact if is ithr constant or varis linarly within th control volum; othrwis, it is a scond ordr approximation. Introducing quations into q w obtain, t ρv + f S f (ρu) f f S f (ργ ) f = S V. (4.2.10) 24

26 Lt us rcall that in our formulation of th FVM, all th variabls ar computd and stord at th control volums cntroid. Th fac valus apparing in q ; namly, th convctiv flux F C = S (ρu) through th facs, and th diffusiv flux F D = S (ργ ) through th facs, hav to b calculatd by som form of intrpolation from th cntroid valus of th nighboring control volums locatd at both sids of th facs, this issu is discussd in th following sction Convctiv Trm Spatial Discrtization Th discrtization of th convctiv trm in q is obtaind as in q , i.., (ρu) dv = V f = f = f S f (ρu) f, S f (ρu) f f, F f, (4.2.11) whr F in q rprsnts th mass flux through th fac, F = S f (ρu) f. (4.2.12) Obviously, th flux F dpnds on th fac valu of ρ and u, which can b calculatd in a similar fashion to f (as it will b dscribd in th nxt sction), with th cavat that th vlocity fild from which th fluxs ar drivd must b such that th continuity quation is obyd, i.., udv = ds u = ( ) ds u = S f (ρu) f = F = 0. (4.2.13) V V f f f f Bfor w continu with th formulation of th intrpolation schm or convction diffrncing schm usd to comput th fac valu of th transportd quantity ; it is ncssary to xamin th physical proprtis of th convction trm. Irrspctiv of th distribution of th vlocity in th domain, th convction trm dos not violat th bounds of givn by its initial condition. If for xampl, initially varis btwn 0 and 1, th convction trm will nvr produc valus of that ar lowr than zro or highr that on. Considring th importanc of bounddnss in th transport of scalar proprtis, it is ssntial to prsrv this proprty in th discrtizd form of th trm Convction Intrpolation Schms Th rol of th convction intrpolation schms is to dtrmin th valu of th transportd quantity on th facs f of th control volum V. Thrfor, th valu of f is computd by using th valus from th nighbors control volums. Hraftr, w prsnt two of th most widly usd schms. For a mor dtaild discussion on th subjct and a prsntation of mor convction intrpolation schms, th intrstd radr should rfr to rfrncs [7, 3, 10, 11, 9, 13, 14]. Cntral Diffrncing (CD) schm. In this schm (also known as linar intrpolation), linar variation of th dpndnt variabls is assumd. Th fac cntrd valu is found from a simpl wightd linar intrpolation btwn th valus of th control volums at points and N (s figur 4.2), such that 25

27 f = f x + (1 f x ) N. (4.2.14) In q , th intrpolation factor f x, is dfind as th ratio of th distancs fn and N (rfr to figur 4.2), i.., f x = fn N = x f x N. (4.2.15) d A spcial cas ariss whn th fac is situatd midway btwn th two nighboring control volums V and V N (uniform msh), thn th approximation rducs to an arithmtic avrag f = ( + N ). (4.2.16) 2 This practic is scond ordr accurat, which is consistnt with th rquirmnt of ovrall scond ordr accuracy of th mthod. It has bn notd howvr, that CD causs nonphysical oscillations in th solution for convction dominatd problms, thus violating th bounddnss of th solution ([7, 3, 10, 11, 9, 13, 14]). Figur 4.2: Fac intrpolation. Cntral Diffrncing (CD) schm. Upwind Diffrncing (UD) schm. An altrnativ discrtization schm that guarants bounddnss is th Upwind Diffrncing (UD). In this schm, th fac valu is dtrmind according to th dirction of th flow (rfr to figur 4.3), f = { f = for F 0, f = N for F < 0. (4.2.17) This schm guarants th bounddnss of th solution ([7, 3, 10, 11, 9, 13, 14]). Unfortunatly, UD is at most first ordr accurat, hnc it sacrifics th accuracy of th solution by implicitly introducing numrical diffusion. In ordr to circumvnt th numrical diffusion inhrnt of UD and unbounddnss of CD, linar combinations of UD and CD, scond ordr variations of UD and boundd CD schms has bn dvlopd in ordr to conform to th accuracy of th discrtization and maintain th bounddnss and stability of th solution [7, 3, 10, 11, 9, 13, 14]. 26

28 Figur 4.3: Fac intrpolation. Upwind Diffrncing (UD) schm. A) F 0. B) F < Diffusion Trm Spatial Discrtization Using a similar approach as bfor, th discrtization of th diffusion trm in q is obtaind as in q , i.., V (ργ ) dv = f = f S f (ργ ) f, (ργ ) f S f ( ) f, (4.2.18) Th Intrfac Conductivity In q , Γ is th diffusion cofficint. If Γ is uniform, its valu is th sam for all control volums. Th following discussion is, of cours, not rlvant to situations whr th Γ is uniform. For situations of non-uniform Γ, th intrfac conductivity (Γ ) f can b found by using linar intrpolation btwn th control volums V and V N (s figur 4.4), Figur 4.4: Diffusion cofficint Γ variation in nighboring control volums. (Γ ) f = f x (Γ ) + (1 f x ) (Γ ) N whr f x = fn N = x f x N. (4.2.19) d If th control volums ar uniform (th fac f is midway btwn V and V N ), thn f x is qual to 0.5, and (Γ ) f is qual to th arithmtic man. 27

29 Howvr, th mthod abov dscribd suffrs from th drawback that if (Γ ) N is qual to zro, it is xpctd that thr would b no diffusiv flux across fac f. But in fact, q approximats a valu for (Γ ) f, namly (Γ ) f = f x (Γ ), (4.2.20) whr w normally would hav xpctd zro. Similarly, if (Γ ) N is much lss that (Γ ), thr would b rlativly littl rsistanc to th diffusiv flux btwn V and fac f, compard to that btwn V N and th fac f. In this cas it would b xpctd that (Γ ) f would dpnd on (Γ ) N and invrsly on f x. A bttr modl for th variation of Γ btwn control volums is to us th harmonic man, which is xprssd as follows, (Γ ) N (Γ ) (Γ ) f = whr f x = fn f x (Γ ) + (1 f x )(Γ ) N N = x f x N. (4.2.21) d This formulation givs (Γ ) f qual to zro if ithr (Γ ) N or (Γ ) is zro. For (Γ ) >> (Γ ) N givs as rquird. (Γ ) f = (Γ ) N f x, (4.2.22) Numrical Approximation of th Diffusiv Trm From th spatial discrtization procss of th diffusion trms a fac gradint aris, namly ( ) f (s q ). This gradint trm can b computd as follows. If th msh is orthogonal, i.., th vctors d and S in figur 4.5 ar paralll, it is possibl to us th following xprssion Figur 4.5: A) Vctor d and S on an orthogonal msh. B) Vctor d and S on a non-orthogonal msh. S ( ) f = S N d. (4.2.23) By using q , th fac gradint of can b calculatd from th valus of th control volums straddling fac f (V and V N ), so basically w ar computing th fac gradint by using a cntral diffrnc approximation of th first ordr drivativ in th dirction of th vctor d. This mthod is scond ordr accurat, but can only b usd on orthogonal mshs. 28

30 An altrnativ to th prvious mthod, would b to calculat th gradint of th control volums at both sids of fac f by using Gauss thorm, as follows ( ) = 1 (S f f ). V f (4.2.24) Aftr computing th gradint of th nighboring control volums V and V N, w can find th fac gradint by using wightd linar intrpolation. Although both of th prviously dscribd mthods ar scond ordr accurat; q uss a largr computational stncil, which involvs a largr truncation rror and can lad to unboundd solutions. On th othr hand, spit of th highr accuracy of q , it can not b usd on non-orthogonal mshs. Unfortunatly, msh orthogonality is mor an xcption than a rul. In ordr to mak us of th highr accuracy of q , th product S ( ) f is split in two parts S ( ) f = Th two vctors introducd in q. condition ( ) f }{{} + k ( ) f }{{}. (4.2.25) orthogonal contribution non-orthogonal contribution , namly; and k, nd to satisfy th following S = + k. (4.2.26) If th vctor is chosn to b paralll with d, this allows us to us q on th orthogonal contribution in q , and th non-orthogonal contribution is computd by linarly intrpolating th fac gradint from th cntroid gradints of th control volums at both sids of fac f, obtaind by using q Th purpos of this dcomposition is to limit th rror introducd by th non-orthogonal contribution, whil kping th scond ordr accuracy of q To handl th msh orthogonality dcomposition within th constraints of q , lt us study th following approachs ([14, 8, 10]), with k calculatd from q : Minimum corrction approach (figur 4.6). This approach attmpts to minimiz th nonorthogonal contribution by making and k orthogonal, = d S d. (4.2.27) d d In this approach, as th non-orthogonality incrass, th contribution from dcrass. and N Orthogonal corrction approach (figur 4.7). This approach attmpts to maintain th condition of orthogonality, irrspctiv of whthr non-orthogonality xist, = d S. (4.2.28) d 29