Conduction Equation. Consider an arbitrary volume V bounded by a surface S. Let any point on the surface be denoted r s

Transcription

1 Conducion Equaion Consider an arbirary volue V bounded by a surface S. Le any poin on he surface be denoed r s and n be he uni ouward noral vecor o he surface a ha poin. Assue no flow ino or ou of V across S (convecive energy ranspor is negligible) and ha eperaures are sufficienly low ha radiaive hea ransfer can be ignored. Under hese condiions, conducion hea ransfer is he only relevan hea ransfer echanis. In he absence of any volueric sinks wihin V, a heral energy balance on V gives he ie rae of change of he heral energy wihin V equals he ne rae a which energy is ransferred across S ino V plus he rae a which hea is generaed wihin V or ) d ρudv = - ( q s n) ds d + q ( r) dv V S V where ρ = Densiy u = Inernal Energy q s = Surface Hea Flux (Vecor Quaniy) q = Volueric Hea Generaion Rae and we have assued a sign convenion where q s ( + ) Ougoing q ( ) Ingoing s Eploying Gauss s heore for vecor fields

2 ) ( B nds ) = ( B) dv S V o conver he surface inegral o a volue inegral, such ha 3) d ρudv + ( q s ) dv d V V - q ( r) dv V = In addiion, Leibniz Rule saes for any paraeer ψ 4) d ψ dv = d V ψ dv + ( ψ vs n) ds V S where v s is he surface velociy. If we assue he boundaries of V do no change wih ie, hen v s = and 5) ( ) ρudv + qdv s V V - q ( r) dv V = Collecing all of he inegrals under one inegral sign 6) ( ρu) + q s q ( r) dv V = which can only be rue for an arbirary volue if he inegrand is zero, i.e. 7) ( ρu) + q s q ( r) = Since we have assued ha conducion is he only significan hea ransfer echanis, he hea flux can be wrien in ers of Fourier s Law such ha 8) q s = - k where k = k ( r, ) is he heral conduciviy and is assued here o be isoropic.

3 9) ( ρu) - k - q ( r, ) = If we furher assue ha densiy is no a funcion of ie (valid for solids in he absence of phase change) ) u ρ - k - q ( r, ) = and expand he ie derivaive using he chain rule u = u Noe: u = C p is he hea capaciy ) ρc p = k + q ( r, ) which is he classic for of he Hea Conducion Equaion. he general for of he Hea Conducion Equaion is a nonlinear, parial differenial equaion in space and ie, for which analyic soluions are available in only special cases. For os applicaions of ineres o nuclear power syses, analyic soluions are eiher ipracical or ipossible, forcing us o rely on nuerical soluions.

4 Nuerical Soluions of he Hea Conducion Equaion Nuerical soluions are based upon convering he coninuous differenial equaions ino a syse of discree algebraic equaions, he soluion of which provides an approxiae soluion o he coninuous equaions a discree poins wihin he proble doain. his is accoplished by subdividing he copuaional doain ino a finie nuber of copuaional volues or nodes. Assuing he boundaries of he volues are fixed, he equaions are discreized by inegraing he conducion equaion over a sub volue Δ V. In doing so, we will eploy any of he sae aheaical ools we used in deriving he original differenial for of he Conducion Equaion. ) ρcp dv = ΔV k dv + q ( r, ) dv ΔV ΔV ) k dv = ΔV k nds S (Gauss s heore) where S is he surface of volue V. We can furher define surface S as he surface connecing volues V and V such ha he surface inegral can be wrien 3) S k nds = k nds S giving 4) ρcp = ΔV k nds S + q ( r, ) dv ΔV Assue ρ and C p are consan wihin he sub volue

5 5) ρcp = ( ρcp) dv ΔV ΔV Since we have assued he boundaries of d 6) ( ρcp) dv = ( ρcp) ( r, ) dv d ΔV ΔV Δ V are fixed, Leibniz s Rule gives We nex inroduce he Mean Value heore (MV), which saes 7) f( ξ ) dξ = f( ξ) Δξ ξ [ Δ ξ ] Δξ where ξ can be posiion, volue, ie, ec. In addiion, since 8) f( ξ ) dξ = f( ξ) Δξ Δξ f ( ξ) dξ = f( ξ) f Δξ Δξ Applying he MV o he inegraed conducion equaion gives d 9) ( ρc ) ( r, ) dv = ( ρc ) ΔV d p p ΔV d d where = ( r ), is he average eperaure of node, and ) q ( r, ) dv = q ( r, ) Δ V = q ( ) Δ V = q ( ) ΔV where q ( ) is he oal hea generaion rae in node. he inegraed conducion equaion is hen d ) Δ V( ρcp) = d k nds S + q ( ) Δ V he inegraed conducion equaion provides he basis for generaing he discreized for of he conducion equaion for any geoery of ineres.

6 Exaple Proble For illusraion purposes, consider he seady sae, -D Caresian proble he inegraed conducion equaion for any volue ) k nds + qδv S = Δ V is For his proble, volue inegrals can be wrien Δ V is conneced o volues V ± Δ such ha he surface ) k nds = k nds + k nds S, S, + S, or 3) k nds + k nds = k nds + k nds S, + S, S+ / S / he uni ouward noral vecor is ± î such ha he surface inegrals ay be siplified o 4) k nds + k nds = ds ds S+ / S / S+ / S / = As A s x + / x / For Δ V = AsΔ x where A s is surface area, he inegraed conducion equaion is

7 5) Ak Ak + q AΔx s s s x + / x / = Noe: if k = k( ) hen he proble is nonlinear. 6) k k + q Δx dx x+ / x / = Subec o he -D assupion, Equaion 6 is exac. he approxiae naure of he soluion coes fro he represenaion of he derivaives and he choice for q. Approxiae he derivaives as he siply slope forulae + dx x x + / + x + = Δx+ / + 7) = Δ x / +Δx / + 8) x / Δ x / +Δ x / + Δ = + 9) k+ / k / q x Δ x / +Δx / Δ x / +Δx+ / Collecing like ers, Equaion 9 can be wrien ) c + a + b + q Δ x = where he consans a, b and c are in general funcions of eperaure. he heral conduciviy a he boundaries can be approxiaed by a) assuing k varies linearly beween he nodes, i.e. k+ k + / = k + ( x+ / x) x+ or b) assuing eperaure varies linearly beween nodes, such ha k x

8 k + / = k ( + /) + + / = + ( x+ / x) x+ x which will be he approach assued fro here on. Applying Equaion o each node wihin he copuaional doain produces a syse of equaions of he for ) A ( ) = S where A ( ) is ri-diagonal. ri-diagonal arices are convenien as hey are easily invered and he nuber of operaions scales linearly wih he nuber of unknowns. For heral conduciviy no a funcion of eperaure, he syse is linear and can be solved direcly. For heral conduciviy a funcion of eperaure, he syse is nonlinear and us be solved ieraively. Soluion sill requires boundary condiions. A coon boundary condiion is ha for a convecive surface, and ay be wrien in general as k n = hc( ( rs) ) (For an insulaed condiion h c = ) Siilarly, for radiaive surfaces, he appropriae boundary condiion is k n =σ 4 ( r s ) Exaple: Convecive Boundary a x = = = x =

9 d d k k + q x/= dx dx / d k = h ( () ) c dx hc ( ) + k/ + q x/ = x/ Exaple: Conducing Inerface k k q''' q''' - + d d ka ka + q = s s dx + / dx / + / / ka s ka s + q = x+ / x x x / AΔx AΔx q = q + q s s Exaple: eperaure Disconinuiy a he Inerface k q''' k q''' - + +

10 Even hough he eperaure is disconinuous a he inerface, we sill require coninuiy of hea flux, such ha he boundary condiion a he inerface is given by k n = k n = H Δ Δ across inerface g i where hen H g is he gap conducance. he inegraed conducion equaion for volue is d d Δx ka ka qa dx dx s s + s = / For d k = H ( ) g + dx x Δx H ( ) k + q = g + Δx Siilarly for volue + d d k A k A q A Δx s s + s = dx + 3/ dx + d k = H ( ) g + dx x+ giving Δx k + H ( ) + q = + + g + Δx

11 -D Caresian Proble We again sar wih he inegraed for of he conducion equaion ) k nds + q Δ V = S and explicily represen he inegrals over each of he four surfaces ) i+ / + / i+ / + / d d d d k dy+ k dx k dy k dx+ q i Ai = dx dy dx dy i / x / / / / yi / / / yi + i x + Applying he MV o each inegral and approxiaing each of he derivaives wih a sandard difference forula yields 3) i+ i i + i ki+ / ( yi+ / yi / ) + ki+ / ( x+ / x / ) x+ x yi+ yi i i i i ki / ( yi+ / yi /) ki / ( x+ / x / ) + q i Ai = x x yi yi A = ( x x )( y y ) where i + / / i+ / i / Collecing like ers, Equaion 3 ay be wrien as b + a + c + d + e = S 4) i i i i i i+ i i+ i i i For a linear apping where he nodes are nubered consecuively, Equaion 4 is a syse of equaions of he for A ( ) = S where A ( ) is five banded wih srucure illusraed below.

12 Assue for now ha he coefficien arix is independen of eperaure, i.e. A ( )= A. In general A is large and sparse. While direc soluions are possible, arix equaions of his ype are generally solved ieraively. a) Poin Jacobi Marix Soluion echniques Rewrie he linear syse of equaions A = S as A = {L + D + U} = S where L is lower riangular, U is upper riangular and D is a diagonal arix. Rewrie as D = S - L - U which suggess he ieraive schee D = S - L - U p+ p p he ehod is convergen if he Specral Radius is less <. his is guaraneed if he coefficien arix is diagonally doinan. While easy o ipleen, convergence of he poin Jacobi schee can be very slow, paricularly is he specral radius is close o. Noe: Ieraive echniques are inefficien for ridiagonal arices, as he ieraive echniques scale as Order N per ierae, while he direc soluion of a ridiagonal arix scales as Order N b) Poin Gaus-Seidel We again rewrie he linear syse of equaions A = S as A = {L + D + U} = S and

13 [ D+ L] =S-U which suggess he ieraive schee [ + ] p+ p D L =S-U he arix D+L is also lower riangular and easily invered, for exaple p a + = L( ) p p where L( ) is a linear cobinaion of he pas ierae eperaures and a is fro he firs row and colun of A. Siilarly a + a = L( ) p + p + p p Since + p is known fro he previous sep, + ay be solved for direcly. his process is repeaed for he reaining unknowns, such ha in general for any unknown p + k k p+ p p+ kk k L( k ) aknn n= a = For sparse arices, os of he eleens of A will be zero. Relaxaion echniques he rae of convergence can be odified by eploying relaxaion echniques. If p+ is he resul of eiher he noral poin Jacobi or poin Gauss-Seidel ieraion, hen he final value of he ieraion is obained fro = λ + ( λ) p + p + p he relaxaion paraeer λ can be shown o be conained in he inerval [,], wih λ > Over relaxaion (Exrapolaion) λ < Under relaxaion (Inerpolaion) If he Gauss-Seidel ieraion is convergen, hen over relaxaion can resul in significanly iproved ieraion raes. If he ieraion is divergen, hen under relaxaion can soeies be used o produce a convergen ehod.

14 c) Line Seidel Rewrie Equaion 4 b + a + c + d + e = S i i i i i i+ i i+ i i i as b + a + c = S d e i i i i i i+ i i i+ i i which sugges he ieraive schee b + a + c = S d e p p p p p i i i i i i+ i i i+ i i For i =, he i er is no presen. he ieraion equaion is hen a ridiagonal syse for each row which ay be solved direcly. he ieraion sars a he lower boundary and sweeps upward. his ehod eliinaes all non zeros and avoids idenifying he non zero paerns.

15 he previous developen assued he heral conduciviy was consan. For heral conduciviy a funcion of eperaure, he arix coefficiens are funcions of eperaure and he syse of equaions is nonlinear. An addiional level of ieraion is hen required for soluion. Exaine he case of k = k( ), a siple ieraive algorih ay be wrien as b + a + c = S d e p, p, p, p, p, i i i i i i+ i i i+ i i Soluion consiss of an inner and ouer ieraion. A guess is ade for he heral conduciviy (ouer ieraion). he resuling syse of equaions is hen ieraed o convergence for (inner ieraion). he converged eperaures are hen used o produce an updaed esiae for he heral conduciviies and he process repeaed unil convergence. his ieraive schee is easy o ipleen, bu convergence raes can be slow. Alernae Approach: Generalized Newon-Raphson Schee he Newon-Raphson Ieraion is he sandard approach for soluion of syses of non linear equaions. Consider he syse of non linear equaions F( x ) = ( F ) N x = where x = x x x3 x N (,,,..., ) Expand each funcion in a aylor Series abou a known reference poin x, e.g. F F F( x) = F( x ) + ( x x ) + ( x x ) +... x x and runcae he series afer he firs derivaive ers. We seek hose values of x such ha F ( x ) = for k =,, N k F F F( x ) + ( x x ) + ( x x ) +.. x x F F F ( x ) + ( x x ) + ( x x ) +.. x x or in general

16 F ( x ) + k N = F δ k x k =,..., N where δ = x x, which is a linear syse of equaions in he individual δ. he linear syse ay be wrien in arix for as Jδ =-F δ = [ δ, δ,..., δ N ] F = [ F( x ), F ( x ),..., F ( x )] N F F F... N x x x F F F... x N x x J = F k x FN FN F N... x N x x he ieraion proceeds by a) Guess x b) Copue δ =-J F c) Copue x = x +δ d) es for convergence Exaple: if converged Sop if no converged le x = x, reurn o b) and repea Consider he wo linear equaions in he unknowns x, x s = ax+ ax s = a x + a x 3 4

17 Cas as he Newon Raphson ieraion F( x, x ) = a x + a x s = F ( x, x ) = a x + a x s = 3 4 x = [ x, x] x = [ x, x] δ = x x F x = a F x = a F x = a 3 F x = a 4 ax ax 3 + ax + ax 4 = ax + ax s = ax+ ax s 3 4 s + ( x x ) a + ( x x ) a s + ( x x ) a + ( x x ) a 3 4 he Newon-Raphson schee when applied o linear syses, is equivalen o solving he original linear syse and converges in ieraion. Exaple: Consider nex he non linear syse s = ax+ ax s = a x + a x + a x x Cas as he Newon Raphson ieraion F( x, x ) = a x + a x s = F ( x, x ) = a x + a x + a x x s = x = [ x, x] x = [ x, x] δ = x x

18 F x = a F x = a F x = a + a x 3 5 F x = a + a x 4 5 = ax + ax s = ax + ax + axx s + ( x x)( a + xa) + ( x x)( a + xa) = ax + ax + ax x s+ ax+ xxa ax x xa+ ax+ xxa ax xxa = s + ax+ xxa + ax + xxa xxa = s + ax+ xxa+ ax+ xxa xxa+ xxa xxa = x a + x a s + x ( x x ) a + x ( x x ) a + x x a Exaine us he nonlinear par of F N ( x, x ) = a x x 5 Expand N ( x ) abou x N N N ( x) = N ( x ) + ( x x ) + ( x x ) x x = x xa 5 + ( x x) xa 5 + ( x x) xa 5 hus if he funcion consiss of linear and nonlinear ers, linearizing he nonlinear ers leads o he sae syse of equaions as applying he Newon-Raphson schee o he enire equaion, i.e. for F ( x) = = L ( x) + N ( x) k k k L ( x) + k N = N δ k x

19 Applicaions o he Hea Conducion Equaion Consider again he -D Caresian for of he discreized hea conducion equaion, where he heral conduciviy is a funcion of eperaure k k q x F Δx Δx + + / / + Δ = = ( ) k = k( ) = k(, ) + / + / i + k = k( ) = k(, ) / / i F ( ) = F ( ) + N δk k = F k where δ = k k = k k + q Δ x+ ( ) + / / + / / Δx Δx Δx Δx ( k + k ) k k + ( ) + Δx Δx Δx + / / + + / / k k + ( + + ) + Δx Δx + / + + / + which is a ridiagonal syse in he new ierae eperaures. If he heral conduciviy is only a weak funcion of eperaure, i.e. k

20 + k / k+ / k / + q Δ x+ ( ) Δx Δx Δx ( k + / + k / ) k + / + ( ) + ( + + ) Δx Δx Collec like ers k + / + Δx Δx Δx k / + + q Δx Δx Δx Δx + = k+ / k / + q Δx Δx Δx which iplies he ieraive schee p+ p+ p + p + / / = k k + q Δx Δx Δx which is equivalen o he siple ieraive schee inroduced earlier. For nonlinear syses, where he nonlineariy is weak, he siple direc ieraion is effecively equivalen o he general Newon-Raphson approach.

21 ie Dependen Probles he ie dependen, inegraed hea conducion equaion is d ) Δ V( ρcp) = d k nds S + q ( ) Δ V For sipliciy, apply o node of a -D Caresian geoery wih unifor spacing ) d + Δ x( ρcp) = k+ / k / + q Δx d Δx Δx Furher assue consan aerial properies, giving 3) d + + q = + α d Δx k k where α = is defined o be he heral diffusiviy. ρc p ie Differencing Opions Approxiae he ie derivaive assuing a Forward Difference in ie (Euler s Mehod) d +Δ 4) = + ϑ( Δ) d Δ 5) +Δ + = + q + ( ) α Δ Δx k αδ q () Δx αδ = Δx k Δx +Δ 6) ( + ) Define F o αδ = Fourier Modulus Δx q () = + F F Δ x k +Δ 7) ( )

22 Noe: he only unknown a any spaial node is +Δ. his iplies he soluion for he eperaure disribuion a each ie sep ay be obained by siply arching hrough he spaial grid evaluaing Equaion 7 a each poin. his soluion gives he eperaure profile a each ie sep explicily in ers of known inforaion fro he previous ie sep. hese ypes of soluion echniques are herefore called Explici Mehods. Since he ehod is only firs order correc, Δ us be relaively sall o iniize error in he ie derivaive approxiaion. Nuerical sabiliy Any perurbaion inroduced ino a ie dependen nuerical schee will eiher a) be daped ou as he soluion proceeds fro one ie sep o he nex, b) say a he sae agniude or c) grow fro one ie sep o he nex. Any ie dependen soluion can be driven unsable if driven a is naural frequency. In he absence of any non derivaive source ers (hoogeneous proble) he driving funcion is he nuerical schee iself, and if he perurbaions grow he syse is said o be nuerically unsable. he growh of hese perurbaions is non physical and nuerical insabiliies resul in failure of he soluion. he condiions under which a nuerical schee is unsable can be deerined hrough a sabiliy analysis. Consider again he Explici Schee, +Δ q () = + F + + F Δ x k ( + ) For he schee o be nuerically sable, i us be sable in he absence of any non derivaive source (or sink) er. We herefore consider he hoogeneous proble +Δ ) = + F( + + ) Express each er as a perurbaion abou soe known, consan reference poin ) = + δ +Δ 3) δ = δ + ( δ+ δ + δ ) Assue each perurbaion can be expanded in a Fourier Series 4) δ = δe = ikx We furher assue ha for he syse o be sable, all odes of he expansion us be sable, i.e. if any ode is unsable, hen ha ode will evenually doinae and he syse will be unsable.

23 For any ikx 5) δ δ e ikx ikx ikx + ikx ikx 6) δ e δ e δ F { e e e } +Δ = + + 7) δ +Δ = δ + δ F { e ikδx + e ikδx } ik x e Δ = cos( kδ x) + isin( kδ x) ik x e Δ = cos( kδx) isin( kδ x) 8) δ +Δ = δ + δ F[cos( kδx) ] [ ( cos( ))] = δ Δ F k x 9) δ δ [ F β] β [,] ) λ [ β F ] ) δ λδ +Δ = δ λ = δ +Δ For sabiliy λ In general, λ can be coplex, i.e. λ = a+ ib such ha he agniude of λ is ) a+ ib = ( a + b ) / For his discreizaion λ is real such ha λ iplies 3) 4) 5) 6) / ( Fo β ) ( o ) F β 4βF + 4β F 4β F 4β F

24 β F or F β he os liiing case occurs when β =. he sabiliy crieria is hen αδ Δx and he ehod is said o be Condiionally Sable. I can be shown ha for he hree diensional Caresian case, he sabiliy crieria is F + F + F x y z Noe: As he nuber of diensions increases, he sabiliy crieria becoes ore resricive. Consider nex a Backwards Difference in ie +Δ d ) = + ϑ( Δ) d Δ +Δ he hoogeneous proble can hen be wrien as ) +Δ = + F( ) +Δ + + 3) δ = δ + F ( ) e + e δ +Δ ikδx ikδ x +Δ 4) δ = δ + F (cos( kδx) ) δ +Δ +Δ 5) δ = δ βfδ +Δ +Δ 6) δ δ +Δ = = λ + βf 7) λ + β F his condiion is always rue and he ehod is uncondiionally sable. he Backward Difference schee resuls in a linear syse of equaions ha us be solved a every ie sep. In general he ehod requires ore copuaional effor per

25 ie sep, bu ie sep size is no liied by sabiliy consideraions. he ehod is sill only firs order correc in ie, and so relaively sall ie seps are sill required for accuracy. Consider nex he cenral difference in ie +Δ Δ d = + ϑ Δ d Δ ( ) +Δ Δ + + = α Δ Δx { + } = + F + he cenral difference schee is a second order, explici, uli-sep ehod. o analyze his ehod, assue he perurbaions grow a he sae rae fro Δ +Δ δ δ δ δ +Δ Δ = λ = λ A sabiliy analysis of his ehod produces a quadraic equaion in he growh paraeer λ. Boh roos us saisfy he condiion λ. Analysis of he roos shows his ehod o be uncondiionally unsable. Consider nex he alernae nd order correc cenral difference schee. Le +Δ d = + ϑ Δ d Δ +Δ/ ( ) +Δ + + = α Δ Δx +Δ/ A his poin, his ehod is equivalen o he uncondiionally unsable, Muli-Sep ehod considered previously, wih he ie sep siply reduced by a facor of. We have seen ha iplici ehods can increase sabiliy. his oivaes us o find a linear relaionship beween he ers a + Δ / and he ers a and + Δ. An obvious relaionship is

26 +Δ / +Δ Δx Δx Δx I can be shown ha +Δ / +Δ ϑ( ) = + + Δ Δx Δx Δx In general, for any funcion [ ] Δξ F ξ + = F( ξ +Δ ξ) + F( ξ) + ϑ( Δξ ) where ξ can be posiion, ie, energy, angle, ec. his produces he iplici schee +Δ +Δ = + α Δ Δx Δx which can be shown o be uncondiionally sable. his ie differencing schee is known as he Crank-Nicholson Mehod. In general we ay eploy a hea differencing schee o produce a general ie advanceen ehod = θ Δ + ( θ) Δ α Δ +Δ +Δ where Δ represens any second derivaive difference esiaor. Noe, ha for θ (,] i.e. θ > an iplici ehod θ = Produces he condiionally sable, explici forward difference schee θ = Produces he uncondiionally sable, iplici backward difference schee θ = / Produces he uncondiionally sable, Crank-Nicholson schee he wo iplici ehods we have exained are uncondiionally sable. Once could hen ask: a) are all iplici ehods ( θ > ) uncondiionally sable

27 b) if no, is here a iniu value of θ such ha for all θ > θin he ehod is uncondiionally sable c) and finally, for θ < θin wha is he appropriae sabiliy crieria? he previous exaple assued consan aerial properies. Consider nex he case of heral conduciviy a funcion of eperaure. d k k =Δxρc Δx Δx d + + / / p Express boh he eperaures and he heral conduciviies in ers of a perurbaion abou a known reference value = + δ k = k + δ k ± / ± / +Δ δ+ δ δ δ δ δ k + δk k δk + ρc = p Δx Δx Δ ( + /) ( /) Neglec ers of δ and higher (local linear sabiliy analysis) +Δ δ+ δ δ δ δ δ k k ρc = p Δx Δx Δ his equaion is idenical in for o he linear case, iplying ha he sabiliy crieria is unchanged.

28 Convecive Syses Consider he siple, -D ass and energy equaions Mass Energy ρ ( ρv) + = d dz ρu ( ρuv) + = z where u is he inernal energy. Expand he energy equaion, giving u u v ) u ρ ρ + ρ + ρv + u = z z ) v du u u ρ ρ + + ρ + ρv = z dz u u 3) + v = z 4) u u + v = z 5) + v = z Assuing he copuaional esh illusraed above, inegrae he energy equaion over he volue cenered a and bounded by ± / 6) dz + v dz = z z+ / z + / z / z /

29 d Δ + = d 7) z v( + ) / / where we have assued he velociy is independen of posiion. We nex apply a derivaive esiaor o he ie derivaive + = Δ Δz +Δ v + / / 8) ( ) * where * is an as of ye undefined ie level. vδ Δz +Δ 9) ( ) * + + / / = he diensionless group vδ C Δz v is he aerial Couran Nuber. + / / +Δ ) C ( ) * + = v o solve Equaion requires ha we express he boundary valued eperaures ± / in ers of he cell cenered values, ±. An obvious approach is o esiae he boundary valued eperaures as he ariheic average of he cell cenered values, i.e. ) ) + / / = = C +Δ v 3) ( ) * + + = Noe: his is equivalen o approxiaing he spaial derivaive as he cenral difference 4) z + = + ϑ( Δz ) Δz

30 Consider firs he explici schee obained by seing * = C + = +Δ v ) ( + ) o deerine he nuerical sabiliy of his ehod, we again express he eperaure as a perurbaion abou a known reference value = + δ C + = +Δ v ) δ δ ( δ+ δ ) Express each perurbaion as a Fourier coponen +Δ ikz ikz C ikz ikz 3) v + δ e δ e δ { e e } + = C v +Δ ikδz ikδz 4) δ δ + δ { e e } = 5) δ δ δ { i k z } C v +Δ + sin( Δ ) = 6) δ δ { C i k z } Sabiliy requires +Δ = v sin( Δ ) = 7) λ = Cisin( kδz) v λ 8) λ = [+ C sin ( kδ z)] Uncondiional Unsable / v C (, ) v sin ( kδz) (,) Assue nex, he iplici schee obained fro * = +Δ C + = +Δ +Δ v ) ( + ) C v + sin( Δ ) = +Δ +Δ ) δ δ δ { i k z }

31 +Δ 3) δ = δ ( + C isin( kδ z) ) v 4) +Δ δ = δ + Cvisin( k Δ z) λ Sabiliy requires 5) λ = + Cisin( kδz) v Noe: / + id c+ id c c + d = = a+ ib a+ ib a + b hus, λ = < 6) / [+ Cv sin ( kδz)] Uncondiionally sable Fro a nuerical sabiliy poin of view, he iplici backwards difference schee is accepable. Consider he behavior of his schee is he presence of a convecive er ) Axρcp + ρvac x p = hcpw( w ) z which has he seady sae soluion d ) ρ vaxc p = hcpw( w ) dz Discreize he spaial derivaive 3) + ρvaxc p = hcpw( w ) Δz 4) = ( ) + w 5) = ( ) + w

32 For probles where he fluid is being cooled, if > w hen + can becoe negaive. his is clearly non physical. Alernaive Approach Approxiae he spaial derivaive by he backward difference approxiaion ) d dz ( z) = + Δ Δz ϑ (Firs order correcion in space) he seady sae soluion in he presence of he convecive er hen has he for ) = ( w ) 3) + = w + 4) = w For his schee, will always >, and as w which is physically correc.

33 Sabiliy We nex consider he sabiliy of he backward spaial difference schee. d d ) + v = d dz ) 3) d + v = d Δz C +Δ * + v[ ] = Le * = explici schee +Δ 4) Cv( ) + = ikz ikz ikz ikz v 5) δ +Δ e δ e + C δ ( e e ) = 6) δ +Δ δ + C δ ( e ikδz ) = v +Δ ikδz 7) δ = δ [ Cv( e )] 8) λ = C [ cos( kδ z) + isin( kδ z)] v λ 9) λ = Cv[ cos( kδ z)] + Cv( isin( kδz)) β β [,] ) ) / λ = [( βcv) + Cv sin ( kδz)] β( β) ( βcv) + Cvβ( β) ) ( βc ) + C β( β) = [ βc + β C v v v v + βcv β Cv ] = [ βcv + βcv] 3) 4) βcv + βcv βcv = βc v

34 vδ 5) Cv = Δz such ha he ehod is condiionally sable and is said o be Couran Liied. Wha if * = +Δ Iplici Backward Difference Schee Equaion 6 becoes ) δ +Δ δ + C δ +Δ ( e ikδz ) = v +Δ Δ ) δ [ + C ( e )] = δ ik z v 3) δ = δ +Δ ikδz [ + Cv( e )] λ = = 4) λ / / [( + βcv) + Cv sin ( kδ z)] [( + βcv) + Cvβ( β)] he ehod is uncondiionally sable, and resuls in a linear syse of equaions ha us be solved a each ie sep.