SECOND-ORDER ADJOINT SENSITIVITY ANALYSIS METHODOLOGY ( nd -ASAM) FOR LARGE-SCALE NONLINEAR SYSTEMS: II. APPLICATION TO A NONLINEAR HEAT CONDUCTION BENCHMARK Dn G. Ccuci Deptment of Mechnicl Engineeing Univesity of South Colin E-mil: ccuci@cec.sc.edu Coesponding utho: Deptment of Mechnicl Engineeing Univesity of South Colin 3 Min Steet Columbi SC 98 USA Emil: ccuci@cec.sc.edu; Phone: (919) 99 964; Oiginlly submitted to J. Comp.Physics on Septembe 7 15 (JCOMP-D-15-1314) but still uneviewed s of Jnuy 1 16. ABSTRACT This wok pesents n illusttive ppliction of the second-ode djoint sensitivity nlysis methodology ( nd -ASAM) to pdigm nonline het conduction benchmk which models conceptul expeimentl test section contining heted ods immesed in liquid led-bismuth eutectic. This benchmk dmits n exct solution theeby mking tnspent the undelying mthemticl deivtions. The genel theoy undelying nd -ASAM indictes tht fo physicl system compising N pmetes the computtion of ll of the fist- nd secondode esponse sensitivities equies (pe esponse) t most N lge-scle computtions using the fist-level nd espectively second-level djoint sensitivity systems (1 st -LASS nd nd -LASS). Fo this illusttive poblem six lge-scle djoint computtions sufficed to compute exctly ll five 1 st -ode nd fifteen distinct nd -ode deivtives of the tempetue esponse to the five model pmetes. The constuction nd solution of the nd -LASS equies vey little dditionl effot beyond the constuction of the djoint sensitivity system needed fo computing the fist-ode sensitivities. Vey significntly only the souces on the ightsides of the het conduction diffeentil opeto needed to be modified; the left-side of the diffeentil equtions (nd hence the solve in lge-scle pcticl pplictions) emins unchnged.
Fo the nonline het conduction benchmk the second-ode sensitivities ply the following oles: () They cuse the expected vlue of the esponse to diffe fom the computed nominl vlue of the esponse; fo the nonline het conduction benchmk howeve these diffeences wee insignificnt ove the nge of tempetues (4-9K) consideed. (b) They contibute to incesing the esponse vinces nd modifying the esponse covinces but fo the nonline het conduction benchmk thei contibution ws smlle thn tht stemming fom the 1 st -ode esponse sensitivities ove the nge of tempetues (4-9K) consideed. (c) They compise the leding contibutions to cusing symmeties in the esponse distibution. Fo the benchmk test section consideed in this wok the het souce the boundy het flux nd the tempetue t the bottom boundy of the test section would cuse the tempetue distibution in the test section to be skewed significntly towds vlues lowe the the men tempetue. On the othe hnd the model pmetes enteing the nonline tempetue-dependent expession of the LBE conductivity would cuse the tempetue distibution in the test section to be skewed significntly towds vlues highe the the men tempetue. These opposite effects ptilly cncel ech othe. Consequently the cumultive effects of model pmete uncetinties on the skewness of the tempetue distibution in the test section is such tht the tempetue distibution in the LBE is skewed slightly towd highe tempetues in the coole pt of the test section but becomes incesingly skewed towds tempetues lowe thn the men tempetue in the hotte pt of the test section. Notbly the influence of the model pmete tht contols the stength of the nonlineity in the het conduction coefficient fo this LBE test section benchmk would be stong if it wee the only uncetin model pmete. Howeve if ll of the othe model pmetes e lso uncetin ll hving equl eltive stndd devitions the uncetinties in the het souce nd boundy het flux diminish the impct of uncetinties in the nonline het conduction coefficient fo the nge of tempetues (4-9K) consideed fo this LBE test section benchmk. Ongoing wok ims t genelizing the nd -ASAM to enble the exct nd efficient computtion of highe-ode esponse sensitivities. The vilbility of such highe-ode sensitivities is expected to ffect significntly the fields of optimiztion nd pedictive modeling including uncetinty quntifiction dt ssimiltion model clibtion nd extpoltion. KEYWORDS: second-ode djoint sensitivity nlysis methodology ( nd -ASAM); nonline het conduction led-bismuth eutectic test section; tempetue esponse covinces; tempetue esponse skewness.
1. INTRODUCTION The ccompnying PART I [1] of this wok hs pesented the mthemticl fomlism of the Second-Ode Adjoint Sensitivity Anlysis Methodology ( nd -ASAM) fo nonline systems. This is new method fo computing exctly nd efficiently second-ode sensitivities (i.e. functionl deivtives) of nonline system esponses (i.e. system pefomnce pmetes in physicl engineeing biologicl systems) to the pmetes chcteizing lge-scle nonline systems. The definition of system pmetes includes ll computtionl input dt coeltions initil nd/o boundy conditions etc. The nd - ASAM builds on the fist-ode djoint sensitivity nlysis methodology (1 st -ASAM) fo nonline systems oiginlly intoduced in [ 3] nd extends the wok pesented in [4]. Fo ech functionl-type esponse of inteest in physicl system compising N pmetes nd N esponses the nd -ASAM equies one lge-scle computtion using the fist-level djoint sensitivity system (1 st -LASS) fo obtining ll of the fist-ode sensitivities i u α R u α i 1... N followed by t most N lge-scle computtions using the second-level djoint sensitivity systems ( nd -LASS) fo obtining exctly ll of the second- ode sensitivities u α j i u α R i j 1... N. In pctice howeve the numbe of lge-scle computtions equied fo computing exctly ll of the second-ode sensitivities j i u α R u α i j 1... N my be considebly smlle thn N s hs been shown in [5-7]. This ppe is stuctued s follows: Section pesents the illusttive pdigm benchmk which models the nonline het conduction in test section of within poposed expeimentl fcility [89] fo investigting theml-hydulics phenomen chcteizing the opetion nd sfety of the conceptully-designed G4M ecto [8] smll modul ecto concept cooled by led-bismuth eutectic (LBE). This pdigm LBE test section benchmk compises the mjo ingedients needed fo highlighting the slient fetues involved in pplying the nd -ASAM fo nonline systems yet is sufficiently simple to dmit n exct solution theeby mking tnspent the mthemticl deivtions pesented in PART I [1]. Section 3 pesents the ppliction of the nd -ASAM fo obtining the exct expessions of both the fist- nd second-ode sensitivities of the tempetue distibution within the test section. 3
Notbly this ppliction will show tht the constuction nd solution of the second-level djoint sensitivity system ( nd -LASS) equies vey little dditionl effot beyond the constuction of the djoint sensitivity system needed fo computing the fist-ode sensitivities nd tht the ctul djoint computtions needed fo computing ll of the fist- nd secondode esponse sensitivities e f less thn N pe esponse. In Section 4 the 1 st - nd nd -ode sensitivities e employed to popgte model pmete uncetinties fo computing the uncetinties (i.e. vinces nd skewnesses) in the tempetue distibution esponses in the heted LBE test section benchmk. Pticulizing the genel esults fom [1] Section 4 shows tht the nd -ode sensitivities contibute decisively to cusing symmeties in the tempetue esponse distibution. Finlly Section 5 concludes this wok by highlighting the most significnt esults obtined egding the fetues of the expected tempetue distibution in the LBE test section benchmk nlyzed heein. 4
. A PARADIGM NONLINEAR HEAT CONDUCTION PROBLEM A cylindicl test section fo pefoming het tnsfe expeiments contins electiclly heted ods nd is filled with liquid led-bismuth eutectic (LBE). The length of the cylindicl test section is 1.7m nd its dius is 15 cm. The theml conductivity denoted s the LBE is consideed to depend linely on the tempetue hving the functionl fom k T of k T k 1 ct (1) whee the nominl vlues of the quntities k nd c e: 4.3663 1 1 k Wm K nd.88441 3 1 c K. Thoughout this wok nominl vlues will be denoted by using the supescipt zeo. Fo simplicity the electiclly heted ods e consideed to povide volumeticlly-distibuted het souce of nominl stength 4 Q 1.111 Wm 3. The test section is insulted on its ltel sufce. The tempetue t the bottom of the test section is kept t constnt nominl tempetue T 4 K. At the top of the section t z het is emoved by het exchnge t constnt het flux q Wm hving nominl vlue 3 q 7.441 Wm. To model mthemticlly the het conduction pocess inside the test section descibed in the foegoing it is convenient to tke the cente of the coodinte systems in the cente of the cylinde so tht the test section extends in the xil (veticl) diection fom z. Since the test section is insulted on its dil sufce nd since the length of the cylindicl test section is much gete thn its dius the tempetue vition in the dil diection cn be neglected by compison to the tempetue vitions in the xil diection fo the puposes of this illusttive poblem. Hence the xil tempetue distibution LBE cn be modeled by the following nonline het conduction model: T z in the d dt z k T Q z () 5
dt kt qt z z ; (3) Tz T t z ; (4) Notbly both Eqs. () nd (3) e nonline in T z theeby endeing the bove het conduction benchmk poblem idelly suited fo illustting the ppliction of the nd -ASAM to the genel cse of nonline diffeentil equtions subject to nonline boundy conditions. The solution of the bove system of nonline diffeentil equtions cn be solved by using Kichoff s tnsfomtion to obtin the solution Tz 1 1cT c z c (5) whee the function z is defined s 3 q Q z z z z. k 4 k (6) Note tht T z ttins its mximum vlue denoted s T z mx t the loction z mx Q q Q (7) whee it hs the expession mx with z given by Tzmx 1 1cT mx c z c (8) 6
z mx Q q kq. (9) Fo the nominl pmete vlues povided in the foegoing the xil vition of the nominl tempetue T z is depicted in Figue 1. In pticul T z tkes on the following vlues t the bottom nd top espectively of the test section: T T 7K. 4 K nd Figue 1: Vition of the nominl tempetue T z. It is of inteest to instll themocouples within the test section in ode to mesue the tempetues t vious xil loctions. A typicl themocouple esponse would indicte the tempetue Tz t some esponse loction denoted s z z. Such esponse is mthemticlly epesented in fom R T; T z T z z z (1) whee z z is the customy Dic delt-functionl. A pticully impotnt loction fo instlling mesuing themocouple is the loction mx z whee T z occus. mx 7
Using the nottion fom Pt I [1] the model ( input ) pmetes in this poblem e consideed to be the (five) components of the (column) vecto QqT k c α (11) α Q q T k c with nominl vlues. The dgge will be used thoughout this ppe to denote tnsposition. The model pmetes α e consideed to be fflicted by uncetinties so they cn vy fom thei nominl vlues α by mounts epesented by the component of the vecto of vitions h defined s Q q T k c h (1) In pctice the vitions Q q T k c e usully tken to be the stndd devitions quntifying the uncetinties in the espective model pmetes. 3. APPLICATION OF THE nd -ASAM FOR COMPUTING THE 1 st - AND nd -ORDER SENSITIVITIES Section 3.1 below pesents the computtion of the fist-ode sensitivities long with thei use fo computing the fist-ode contibutions to the stndd devition in the tempetue distibution while Section 3. pesents the computtion of the second-ode sensitivities long with thei use fo computing the second-ode contibutions to the stndd devition nd skewness of the tempetue distibution. 3.1. nd -ASAM Computtion of the Fist-Ode Response Sensitivities As shown in PART I [1] the fist-ode sensitivities of the esponse R T vitions of e e α to the h e genelly obtined by computing the (fist-ode) G-diffeentil Re ; h R e t T e α which is defined s 8
d Re ; h Re h with h h T h. (13) d Applying the bove definition to the esponse defined by Eq. (1) yields the fist-ode diffeentil DR T h of the esponse ( ; ) RT : DR( T α ; h ) T z h z z z h z z z. (14) d T T d Next tking the G-diffeentil of Eqs. () - (4) yields α h z Q1 T ; z d Q d dt k T ht z Qk ck T z k (15) d q dt k T ht z qk ck T z k z z q T α h t z 1 ; ; (16) h z T t z. (17) T z Equtions (15)-(17) coespond to Eqs. (13)-(14) in [1]. Define the Hilbet spce H L to consist of ll sque integble functions f z defined on the domin z nd endowed with the inne poduct [fo two functions f z nd s follows: 1 1 1 f z ] f z f z f z f z. (18) 1 9
Recll fom the genel theoy pesented in [1] tht the my equie two distinct Hilbe spces denoted in [1] s H nd H u x howeve both of these Hilbet spces coincide with L. Q x espectively. Fo this simple iilusttive poblem Foming now the inne poduct of Eq. (15) with yet undefined function z L nd integting the esulting equtions twice by pts to tnsfe the diffeentil opetions fom h z to T z yields: d z k T ht z zq1 T α ; h z d d d T T T z z k T h z h z k T h z k T. (19) The bove Eq. (19) coesponds to Eq. (15) in [1]. Applying the pinciples outlined in Pt I to Eq. (19) yields the following fist-level djoint sensitivity system (1 st -LASS): z d k T z z z () d z t z (1) z t z. () The 1 st -LASS bove cn be edily solved to obtin the 1 st - level djoint function z kt z zz H zz z. (3) whee H z is the customy Heviside unit-step functionl defined s H z if z nd H z 1 if z. Note tht the 1 st - level djoint function to be the Geen`s function z cn lso be intepeted 1
1 Gz z zzhzzz ktz fo the 1 st -LASS i.e. Eqs. () - () since the point z is bity. (4) As shown in the genel theoy in PART I [1] the 1 st -LASS depends on the esponse cf. Eq.(14) which povides the souce tem z z s shown in Eq. (). Note tht this souce tem does not belong to the Hilbet spce L but belongs to the Sobolev spce 1 H L i.e. 1 d x b H L s usully encounteed when computing Geen s functions. By using the well known Lx-Milgm Lemm it cn be shown tht the biline fom on the ight side of the lst equlity in Eq. (19) coecive so tht the 1 st -LASS cn be solved uniquely s hs been done to obtin the expession fo the djoint function 1 z H H L shown in Eq. (3). The mthemticl techniclities equiing the use of Sobolev spces stemming fom the considetion of distibutions s encounteed bove fo the fist-level djoint sensitivity system (1 st -LASS) will lso ise in the constuction of the second-level djoint sensitivity systems ( nd -LASS) s will be seen in the eminde of this wok. Thus even though the foegoing mthemticl techniclities will not be epeted in the sequel ll of the solutions to such nd -LASS should be intepeted in the wek sense in the ppopite Sobolev spce. No confusion should ise howeve since the espective solutions fo the nd -LASS will be unique nd will be obtined explicitly just s it ws in Eq. (3) fo the fist-level djoint function z. Using the esults in Eqs. (15) - () in Eq. (14) tnsfoms the ltte into the fom: 1 DR( T α ; h ) z Q T α ; h d T kt z q1 T α h z ;. (5) Q T α h nd Using Eqs. (15) nd (16) espectively to eplce the expessions of 1 ; q T α h in Eq. (35) yields 1 ; T z T z T z T z T z DR( T α ; h ) Q q T k c Q q T k c (6) 11
whee the the 1 st -ode sensitivities of T z to the epective model pmetes hve the following expessions: T z S1 T ; α z (7) Q T z S T ; α (8) q 3 T z d S T α k T (9) ; T z T z 1 S4 T ; α Q z q k k (3) T z d dt z S5T ; α z k T z c dt T T z k T z k. k z T z (31) The bove expessions indicte tht ll of the 1 st -ode sensitivities cn be computed excly nd efficiently using qudtues (integtions) once the djoint function hs been obtined by solving the 1 st -LASS. Thus eplcing the expession of z given in Eq. (3) in Eqs. (7) (31) nd cying out the espective integtions ove z yields the following evluted expessions fo the 1 st -ode sensitivities of the esponse T z to the model pmetes: T z 1 3 z z Q k T z 4 (3) T z z q kt z (33) 1
T z 1 ct T 1c T z T z 1 Q 3 z z q z 4 z k kk T z k T z (34) (35). T T z k T z T z z T c k T z 1 c c T z (36) One of the min uses of sensitivities is fo nking the eltive impotnce of pmete vitions in influencing vitions in esponses. Reltive sensitivities e used fo this pupose since they e dimensionless numbes. The eltive sensitivity of esponse the i th -pmete el i is defined s Si R i i R e e R e to. The eltive sensitivities of T z e depicted in Figues -6 s functions of the bity loction z. Figue : Reltive sensitivity of T z to Q. Figue 3: Reltive sensitivity of T z to q. 13
Figue 4: Reltive sensitivity of T z to T. Figue 5: Reltive sensitivity of T z to k. Figue 6: Reltive sensitivity of T z to c. As Figues though 6 indicte the eltive impotnce of the vious model pmetes depends on the position z. Rnking them by the lgest ttinble mximum bsolute vlues of thei eltive sensitivities the impotnce of the model pmetes is s follows: the het souce Q ; the boundy het flux q ; the mbient tempetue T ; the line het conductivity coefficient k ; nd the nonline het conductivity coefficient c. 3.. nd -ASAM Computtion of the Second-Ode Response Sensitivities As discussed in the genel theoy pesented in PART I [1] the fundmentl philosophicl considetion nd stting point fo computing the second-ode esponse sensitivities 14
S R R ws to conside the fist-ode sensitivities to be ij i j j i esponses of the fom α S T ; i 1345. This fct ws explicitly indicted in the i espective definitions povided by Eqs. (7) though (31). Bsed on this fundmentl considetion the nd -ASAM poceeds by computing the fist-ode G-diffeentil S T ; α of ech of the functionls ; i vlues using the definition of the G-diffeentil nmely: o Si T α t the point ; T α of nominl o d o ; ; S i T α ht h h Si T ht h α h d (37) fo n bity scl F nd vectos illusttive exmple thee will be the symmety popety R R ht h h H H H. Fo ou N N 1 15 distinct second-ode deivtives due to i j j i. The sensitivities ij S will be computed next in the ode of pmete impotnce/nking s discussed in the pevious subsection. 3..1. Computtion of the Second-Ode Response Sensitivities S T z Q 1i i Applying the definition shown in Eq. (37) to Eq. (7) yields the G-diffeentil o DS T h h o α T h of the fist-ode sensitivity 1 ; 1 ; ; S T α in the fom o d DS1 T ; α ; ht h h z h z. (38) d The function h in Eq. (38) is the solution of the system of equtions obtined by G- diffeentiting Eqs. () - () nmely 15
dh z d z kt kc h T z d z k 1 c T z ckt z z (39) dh z t z (4) h z t z. (41) The quntity following fom of Eq. (39) d z in Eq. (39) cn be eplced by using Eq. () to obtin the kt h T z dh z c z z 1 c T z k T z c zz Q T α ; h. 1 z k 1 c T z (4) o As Eq. (38) indictes the entie contibution to DS1 T ; ; ht h α h comes fom the indiect-effect tem; thee is no diect-effect tem contibution to o DS T ht h 1 ; α ; h. Applying next the genel theoeticl considetions leding to Eq. (34) of PART I [1] Eqs. (15) nd (4) e witten in the following block-mtix fom: d kt 1 ; ht z Q T α h c z z d ; 1 h z k T Q T α h 1 ct z z. (43) Following the pocedue outlined in [1] intoduce the vecto Ψ H H nd define the inne poduct between two functions () () () 1 11 1 () h z h z h z nd () () z () z T Ψ 1 11 1 s follows: 16
() () () () 1 T 11 1 h z Ψ h z z h z z. (44) Following the sequence of opetions leding to Eq. (37) of [1] fom the inne poduct of z z Ψ () () () 1 11 1 with Eq. (43) to obtin the following sequence of equlities: d kt ht z c z z d h z k T () () 11 z 1 z 1 ct z () () 1 11 1 z z Q T α ; h 1 1 d c z z kt () 11 c T z z ht z h z () 1 d z k T Q T α; h P T α ; h h h ; () () T 11 1 (45) () () whee the biline concomitnt P T ; ht h ; 11 1 α h hs the fom () () () P T α ; h ; T h 11 1 z 11 T T z () () d11 z () dh z d k T 1 z k T h z z k T h z 1 d k T h z z. (46) As genelly shown in [1] the definition of the nd -level djoint function Ψ () () () 1 11 1 is now completed by equiing the tem on the ight side on the lst equlity in Eq. (45) to epesent the sme functionl s the ight side of Eq. (38) which yields the following second-level djoint sensitivity system ( nd -LASS): 17
1 d c z z kt () c T z 11 z () 1 d z k T 1 (47) As genelly discussed in [1] the boundy conditions fo the bove nd -LASS e obtined by using the boundy conditions given in Eqs. (16) (17) (4) nd (41) in Eq. (46) to eliminte ll unknown vlues of P T α ; h h ; () () T 11 1 conditions fo the nd -LASS: h z nd T h z espectively in the biline concomitnt. These considetions led to the following boundy d () 11 z t z (48) () 11 z t z. (49) d k T () 1 z t z (5) () 1 z t z. (51) Inseting the bove boundy conditions togethe with those those given in Eqs. (16) (17) (4) nd (41) into Eq. (46) educes the biline concomitnt α T to the quntity ˆ ; () () 11 ; 1 P T ; h h ; () () 11 1 following fom: P T α h which hs the () ˆ () () () d 11 PT α ; 11 1 ; hq1t α; h 11 z T. z k T (5) z z Finlly using Eqs. (5) (47) nd (38) in Eq. (45) leds to the following expession fo the second-ode diffeentil expession DS1 T ; α ; ht h h 18
() () ; α ; T h α ; h α ; h 1 DS T h h z Q T z Q T 1 11 1 1 z d ;. () () 11 q1t α h 11 z T z k T z (53) Solving Eqs. (47) (5) nd (51) yields the following expession fo the nd -level djoint function () 1 z : () 1 3 1 z. z z k T z 4 (54) Solving Eqs. (47) - (49) yields the following expession fo the nd -level djoint function () 11 z : with () 13 11 zct z; α z z. 4 z z z H z z (55) kc CT z; α. 3 k T z (56) Replcing Q1 T α ; h nd ; Q T α h with the coesponsing expessions fom Eqs. 1 (15) nd (4) espectively yields the following expession fo the diffeentil DS T ht h 1 ; α ; h : whee T z T z DS1 T ; α ; ht h h Q q Q Qq T z Q T k c T z T z T z QT Qk Qc () 11 z (57) (58) 19
T z Qq () 11 z z (59) T z d z () 11 k T QT z (6) T z 1 () () () Q 11 1 11 z z z z q z Qk z k (61) T z () d dt z k 11 z T z Qc T z dt () () 1 z z z k 11 1 z T z ct z z k T z zt z () () () T d11 1 d 11 1 z k T z. (6) Replcing Eqs. (54) nd (55) in the bove expessions nd cying out the integtions ove z yields the following expessions fo the bove nd -ode sensitivities: T z 1 3 C T z; α z z (63) Q 4 4 T z 1 3 C T z; α z z z (64) Qq 4 T z kt ( ) 3 C T z; α z z (65) QT 4 T z 1 1 3 CT z ; 1 α z c T z z z (66) Qk c 4
T z k Qc ; α C T z T z T z c T z c 3 z z 4. (67) The eltive sensitivities The nd -ode eltive sensitivities S 1i el depicted in Figues 7-11 s functions of the bity loction z. T z Q i Q T z i e Figue 7: Reltive sensitivity T z Q Q Tz. Figue 8: Reltive sensitivity T z Qq Qq T z Figue 9: Reltive sensitivity T z QT QT T z. Figue 1: Reltive sensitivity T z Qk Qk T z 1
Figue 11: Reltive sensitivity T z Qc Qc T z As Figues 7 though 11 indicte the the eltive impotnce (mgnitude) of the nd -ode eltive sensitivities S 1i el T z Q i Q T z i depends on the position z. Rnking them by the lgest ttinble mximum bsolute vlues of these eltive sensitivities the impotnce of the model pmetes is s follows: the het souce Q ; the boundy het flux q ; the line het conductivity coefficient k ; the mbient tempetue T ; nd the nonline het conductivity coefficient c. By compison to the impotnce nking of the 1 st -ode eltive sensitivities the model pmetes k nd T hve switched plces in tht the eltive sensitivity T z QT QT T z T z Qk Qk T z hs become bout 3 times s lge s the eltive sensitivity. Futhemoe the eltive sensitivity impotnce nking of the ptil sensitivities S 1i el T z Qc Qc T z T z Q i. Q T z i hs become thid in the
3... Computtion of the Second-Ode Response Sensitivities S T z q i i Applying the definition shown in Eq. (37) to Eq. (8) yields the G-diffeentil o DS T h h o α T h of the fist-ode sensitivity ; ; ; S T α in the fom o DS T ; α ; ht h h hz z (68) As befoe the function h in Eq. (68) is the solution of Eqs. (4) (4). Following the sme pocedue s in the pevious sub-section leds to the following expession fo the ptil- o diffeentil DS T ; ; ht h α h : () () ; α ; T h α ; h α ; h 1 DS T h h z Q T z Q T 1 1 z d ; () () 1 q1t α h 1 z T z k T z (69) whee the nd -level djoint function () () () 1 second-level djoint sensitivity system ( nd -LASS): Ψ is the solution of the following 1 d c z z kt () c T z 1 z () d z z k T (7) d () 1 z t z (71) () 1 z t z. (7) 3
d k T () z t z (73) () z t z. (74) Solving Eqs. (7) - (74) yields the following expessions fo the components of the nd -level Ψ : djoint function () () () 1 nd 1 z CT z ; α z z z z H z z (75) () () 1 z. z z Hz kt z (76) The second-ode sensitivities S T z q will hve fomlly the sme i i expessions s those shown Eqs. (58) (6) except tht the function () () () 1 11 1 be eplced by the function Ψ () () () 1 to obtin: Ψ will T z qq () 1 z (77) T z q z () 1 z (78) T z d z () 1 k T qt z (79) T z 1 () () () Q 1 1 z z z z q z qk z k (8) 4
T z z T z () () () T d1 1 d 1 k T z qc kt z z. (81) Replcing Eqs. (75) nd (76) in the bove expessions nd cying out the integtions ove z yields the following explicit expessions fo the bove nd -ode sensitivities: T z () () 1 11 z qq Qq T z z z 1 3 CT z; α z z z 4 (8) T z C T z z ; α (83) q T z C T z k T z qt ; α (84) T z 1 C T z c T z z z qk c ; α 1 (85) T z k ; C T z α c T z T z z T z. (86) qc c The symmety of the second-ode sensitivity T z q Q implies the equlity between Eqs. (8) nd (59) which in tun povides stingent independent veifiction of the ccucy of computing the second-level djoint functions () () () 1 11 1 () () () 1 Ψ. The nd -ode eltive sensitivities S i el in Figues 1-15 s functions of the bity loction Ψ nd T z q i q T z z. i e depicted 5
Fig. 1: Reltive sensitivity T z q q Tz Fig. 13: Reltive sensitivity T z qt qt T z Fig. 14: Reltive sensitivity T z qk qk T z Fig. 15: Reltive sensitivity T z qc qc T z Figues 1 though 15 indicte the the eltive impotnce (mgnitude) of the nd -ode eltive sensitivities S i el T z q i q T z i depends on the position z. Rnking them by the lgest ttinble mximum bsolute vlues of these eltive sensitivities the impotnce of the model pmetes is s follows: the het souce Q ; the boundy het flux q ; the line het conductivity coefficient k ; the nonline het conductivity coefficient c ; nd the mbient tempetue T. 6
3..3. Computtion of the Second-Ode Response Sensitivities S T z T 3i i Applying the definition shown in Eq. (37) to Eq. (9) yields the G-diffeentil o DS T h h o α T h of the fist-ode sensitivity 3 ; 3 ; ; S T α in the fom 3 DS T h h o ; α ; T h d d h 1 k k c c T ht z d o o α T h α T h ; ; ; ; DS T h h DS T h h 3 3 diect indiect (87) whee the diect-effect tem is defined s DS3 T h h o ; α ; T h diect z d k 1 c T z ckt zc kht z z (88) while the indiect-effect tem is defined s o dh 3 ; ; DS T α ht h h k T z z. (89) indiect z The diect-effect tem defined in Eq. (88) cn be evluted immeditely by noting fom Eq. (3) tht z d 1 k T z z (9) nd by using the bove esult in Eq. (88) to obtin o 1 DS3 T ; α ; h 1. T h h k c T c k T T c k diect k T z (91) 7
On the othe hnd the indiect-effect tem defined in Eq. (89) needs to be evluted by constucting the coesponding nd -LASS fo nd -level djoint function Ψ () () () 3 31 3 by following the genel genel pinciples of the nd -ASAM pesented in PART I [1]. Applying these pinciples leds to the following expession fo the indiect-effect tem defined in Eq. (89): DS3 T h h o ; α ; T h indiect α ; h α ; h () () 31 z Q 1 T 3 z Q T 1 z d ; () () 31 q1t α h 31 z T z k T z (9) whee the nd -level djoint function () () () 3 31 3 Ψ is the solution of the following nd - LASS: 1 d c z z kt () c T z 31 z () 3 d z k T (93) d () 31 z t z (94) () 31 z t z. (95) d k T () 3 z t z (96) () 3 z 1 t z. (97) 8
Solving Eqs. (93) - (97) yields the following expessions fo the components of the nd -level djoint function () () () 3 31 3 Ψ : nd 31 z CT z ; α kt z z z Hz z (98) () z k T z k T () 3. (99) Adding Eqs. (88) nd (9) nd identifying the coefficients multiplying the espective pmete vitions yields the following expessions fo the nd -ode sensitivities S T z T : 3i i T z T z d z () () 11 31 z kt T Q Q T z kt ( ) 3 CT z; α z z 4 T z T z d z () () 1 31 z kt T z q qt z CT z; α kt z () T z d z d31 z ck kt T α C T z ; k T z k T T z 1 ct Q T k k T z k () 31 α z () q () 3 z z 31 z z k 1 k C T z ; k T z z (1) (11) (1) (13) 9
T z kt kkt CT z ;. α z T c T z T z (14) T c kt z c The symmety of the second-ode sensitivity T z T Q implies the equlity between the eltions expessed in Eq. (1) nd (65). This equlity povides n independent veifiction of the coectness of the espective expessions s well s veifiction of the solution ccucy of computing the second-level djoint functions () () () 1 11 1 () () () 3 31 3 Ψ nd Ψ. Futhemoe the symmety of the second-ode sensitivity T z T q implies the equlity between the eltions expessed in Eq. (11) nd (79). This equlity povides n independent veifiction of the coectness of the espective expessions s well s veifiction of the solution ccucy of computing the second-level djoint functions Ψ () () () 1 nd Ψ () () () 3 31 3. The nd -ode eltive sensitivities S 3i el T z T i T T z i bity loction z. e depicted in Figues 16-18 s functions of the Fig. 16: Reltive sensitivity T z T T Tz Fig. 17: Reltive sensitivity T z Tk T k T z 3
Fig. 18: Reltive sensitivity T z Tc T c T z As Figues 16 though 18 indicte the the eltive impotnce (mgnitude) of the nd -ode eltive sensitivities S 3i el T z T i T T z i depends on the position z. Rnking them by the lgest ttinble mximum bsolute vlues of these eltive sensitivities the impotnce of the model pmetes is s follows: the boundy het flux q s peviously depicted in Figue 13; nd the het souce Q s peviously depicted in Figue 9. On the othe hnd Figues 16-18 indicte tht the emining pmetes (nmely the line het conductivity coefficient k the mbient tempetue T nd the nonline het conductivity coefficient c ) e much less impotnt. 3..4. Computtion of the Second-Ode Response Sensitivities S T z k 4i i Applying the definition shown in Eq. (37) to Eq. (3) yields the G-diffeentil o DS T h h o α T h of the fist-ode sensitivity 4 ; 4 ; ; S T α in the fom o o ; α ; T h ; α ; T h o DS4 T α ht h h DS T h h DS T h h 4 4 diect ; ; o whee the diect effect tem is defined s DS4 T ; α ; ht h h indiect diect (15) 31
o Q 4 ; ; Q DS T α ht h h k diect z k k q k q k k (16) o while the indiect effect DS4 T ; ; ht h α h tem is defined s indiect o Q q ; ; DS T α ht h h hz h. (17) 4 indiect k k The diect effect tem defined in Eq. (16) cn be evluted by using Eq. (3) to obtin: o 1 3 4 ; ; Q Q DS T α ht h h k z z diect 4 k k k T z q k k k k T z q z. (18) The indiect-effect tem defined in Eq. (17) needs to be evluted by constucting the coesponding nd -LASS fo nd -level djoint function () () 4 41 Ψ by following the genel genel pinciples of the nd -ASAM pesented in PART I [1]. Applying these pinciples leds to the following expession fo the indiect-effect tem defined in Eq. (17): o () () ; α ; T h α ; h α ; h 1 DS T h h z Q T z Q T 4 41 1 4 indiect z d ; () () 41 q1t α h 41 z T z k T z whee the nd -level djoint function () () () 4 41 4 LASS: (19) Ψ is the solution of the following nd - 3
1 d c z z kt () c T z 41 z () 4 Q q d z z k T k k (11) d () 41 z t z (111) () 41 z t z. (11) d k T () 4 z t z (113) () 4 z t z. (114) Solving Eqs (11) (114) yields the following expessions fo the nd -level djoint function Ψ : () () () 4 41 4 nd () 3 41 zct z; α z z z 4 z z z H z z (115) () 1 Q 3 q 4 z z z 4 z z H z kt z k k (116) Replcing Eq. (115) nd (116) in Eq. (19) cying out the integtions ove z nd the ensuing lgeb yields the following expession fo the indiect-effect tem 5 ; α ; h : o DS T ht h indiect 33
o Q ck z 3 4 ; ; DS T α ht h h Q k 3 z z indiect k k T z 4 ck z q ck z T k T k 3 q z k T z k k T z c k z k T z z c T 3 T z c k T z k 1 c T z k T z 3. (117) Summing the expession fo the indiect effect tem given in Eq. (117) with the expession of the diect effect tem given in Eq. (18) nd identifying the coefficients of the espective pmete vitions yields the following expession fo the nd -ode deivtives S T z k : 4i i T z 1 1 3 CT z ; 1 α z c T z z z (118) k Q c 4 T z 1 C T z c T z z z kq c ; α 1 (119) T z k T C T z ; α k T z (1) T z z CT z z k kk T z ; α (11) T z k z ; α (1) CT z c T z T z zt kc c The symmety of the second-ode sensitivity 34 T z k Q implies the equlity between the eltions expessed in Eq. (118) nd (66). This equlity povides n independent veifiction of the coectness of the espective expessions s well s veifiction of the solution ccucy of computing the second-level djoint functions () () () 1 11 1 () () () 4 41 4 Ψ nd Ψ. Futhemoe the symmety of the second-ode sensitivity
T z k q implies the equlity between the eltions expessed in Eq. (119) nd (85). This equlity povides n independent veifiction of the coectness of the espective expessions s well s veifiction of the solution ccucy of computing the second-level djoint functions Ψ () () () 1 nd Ψ () () () 4 41 4 second-ode sensitivity. Finlly the symmety of the T z k T implies the equlity between the eltions expessed in Eq. (1) nd (13). This equlity povides n independent veifiction of the coectness of the espective expessions s well s veifiction of the solution ccucy of computing the second-level djoint functions Ψ () () () 3 31 3 nd Ψ () () () 4 41 4 eltive sensitivities T z k k Tz T z kc nd k c T z functions of the bity loction z.. The nd -ode e depicted in Figues 19 nd s Fig. 19: Reltive sensitivity T z k k Tz Fig. : Reltive sensitivity T z kc k c T z 3..5. Computtion of the Second-Ode Response Sensitivities S T z c 5i i Applying the definition shown in Eq. (37) to Eq. (31) yields the G-diffeentil o DS T h h o α T h of the fist-ode sensitivity 5 ; 5 ; ; S T α in the fom 35
T T T ht 1 T o 1 d DS5 T ; α ; ht h h z z d c c T h o o α T h α T h DS T ; ; h h DS T ; ; h h 5 5 diect indiect (13) whee o T T z T 5 T diect 1 ct z 1 ct z DS T ; α ; h h h T c T z (14) nd c T T z c 5 ; ; T z o T T indiect DS T α h h h h z z z. 1 ct z (15) The indiect-effect tem defined in Eq. (15) needs to be evluted by constucting the coesponding nd -LASS fo nd -level djoint function () () 5 51 Ψ by following the genel genel pinciples of the nd -ASAM pesented in PART I [1]. Applying these pinciples leds to the following expession fo the indiect-effect tem defined in Eq. (15): o () ; α ; T h α ; h DS T h h z Q T 5 51 1 indiect z d ; () () 51 q1t α h 51 z T z k T z (16) whee the nd -level djoint function () () 5 51 Ψ is the solution of the following nd -LASS: d () 51 z c T T z c T z 3 k zz z k T z (17) d () 51 z t z (18) 36
() 51 z t z. (19) Note tht the bove nd -LASS hs the sme fom s the 1 st -LASS [cf. Eqs. ()-()] except fo diffeent souce tem in Eq. (17). Theefoe the solution of Eqs. (17) (9) cn be witten down by simply modifying ppopitely the solution of the 1 st -level djoint function z in Eq. (3) to obtin () k 51 zct z ; α c T T z c. T z z z H z z z c (13) Replcing Eq. (13) in Eq. (16) cying out the integtions ove z nd the ensuing lgeb yields the following expession fo the indiect-effect tem 5 ; α ; h : o DS T ht h indiect DS5 T α h h o ; ; T h indiect 13 k CT z ; α c T T z c T z Q z z q z c 4 Q 3 q T T z T k T k. z z z c k 4 k (131) Adding Eqs. (131) nd (14) nd identifying the coefficients multiplying the espective pmete vitions yields the following expessions fo the nd -ode sensitivities S T z c : 5i i T z 3 ; k C T z α T z T z c T z z z (13) cq c 4 T z k ; C T z α c T z T z z T z (133) cq c 37
T z kt kkt CT z ; α z T c T z T z (134) ct kt z c T z k z ; α (135) CT z c T z T z zt ck c T z k CT z; α T z T z c c c T z c T T z z T (136) The symmety of the second-ode sensitivity T z c Q implies the equlity between the eltions expessed in Eq. (13) nd (67). This equlity povides n independent veifiction of the coectness of the espective expessions s well s veifiction of the solution ccucy of computing the second-level djoint functions () () () 1 11 1 () () () 5 51 5 Ψ nd Ψ. The symmety of the second-ode sensitivity T z c q implies the equlity between the eltions expessed in Eq. (133) nd (86). This equlity povides n independent veifiction of the coectness of the espective expessions s well s veifiction of the solution ccucy of computing the second-level djoint functions Ψ () () () 1 nd Ψ () () () 5 41 4 sensitivity. Futhemoe the symmety of the second-ode T z ct implies the equlity between the eltions expessed in Eq. (134) nd (14). This equlity povides n independent veifiction of the coectness of the espective expessions s well s veifiction of the solution ccucy of computing the second-level djoint functions Ψ () () () 3 31 3 nd Ψ () () () 5 41 4 symmety of the second-ode sensitivity T z. Finlly the c k implies the equlity between the eltions expessed in Eq. (135) nd (1). This equlity povides n independent veifiction of the coectness of the espective expessions s well s veifiction of the solution ccucy of computing the second-level djoint functions () () () 4 41 4 Ψ () () () 5 41 4. The nd -ode eltive sensitivity s function of the bity loction z. T z c c Tz Ψ nd is depicted in Figue 1 38
Fig. 1: Reltive sensitivity T z c c Tz The computtion of the 1 st - nd nd -ode sensitivities pesented in this Section hve undescoed tht: (i) One djoint computtion ws needed to detemine the 1 st -level djoint function z (ii) (iii) which sufficed to compute using just qudtues ll of the fist-ode sensitivities S R i 1345; i i The mgnitudes of the nd -ode sensitivities e genelly of the sme ode s the mgnitudes of the 1 st -ode sensitivities; Five djoint computtions one ech fo computing the espective nd -level djoint () () () functions Ψ 1 1 3 45 nd subsequently detemining ll of the i i i i nd -ode sensitivities S R ; i j 1 3 45. These computtions lso ij i j povide independent solution veifictions of the nd -level djoint functions Ψ () () () i i1 i i 1 3 45 due to the symmety of the nd -ode sensitivities. Notbly simil diffeentil opetos ppe on the left-sides of the 1 st - nd nd -LASS which theefoe would use the sme solves of diffeentil equtions; only the souces on the ight sides of these suystems diffe fom ech othe. The 1 st - nd nd - ode sensitivities will be used in the following Section to illustte thei essentil ole fo quntifying stndd devitions nd non-gussin fetues (e.g. symmeties) of the vious esponse distibutions. To quntify ssymeties in distibution t (the vey) lest the thid ode ( skewness ) esponse coeltions need to be computed 39
which equie the exct computtion of (t lest) the fist- nd second-ode esponse sensitivities to model pmetes. 4. APPLICATION OF THE nd -ASAM FOR QUANTIFYING NON-GAUSSIAN FEATURES OF THE RESPONSE UNCERTAINTY DISTRIBUTION In genel the model pmetes e expeimentlly deived quntities nd e theefoe subject to uncetinties. Specificlly conside tht the model compises N uncetin pmetes i which constitute the components of the (column) vecto α of model pmetes defined s α 1... N. The usul infomtion vilble in pctice compises the men vlues of the model pmetes togethe with uncetinties (stndd devitions nd occsionlly coeltions) computed bout the espective men vlues. The components of vecto s i nd defined s α... 1 N of men vlues of the model pmetes e denoted i i f f α p α dα. (137) whee the ngul bckets denotes integtion of geneic function f α ove the unknown joint pobbility distibution p α of the pmetes α. The pmete distibution s secondode centl moments ij α e defined s ij α i i j j ij i j; i j 1 N. (138) ii The centl moments ij moments α v i e clled the vince of i while the centl α cov i j ; i j e clled the covinces of i nd j. The stndd 4
devition of i is defined s i ii α. When the model unde considetion is used to compute N esponses (o esults) denoted in vecto fom s N 1... ech of these esponses will be implicit functions of the model s pmetes i.e. α. It follows tht α will be vecto-vlued vite which obeys (genelly intctble) multivite distibution in α. Fo lge-scle systems the pobbility distibution pα is not known in pctice nd even if it wee known the induced distibution in α would still be intctble since p α could not be popgted exctly though the lge-scle models used in fo simulting elistic physicl systems. The uncetinties in esponse α ising fom uncetinties in the pmetes α cn be computed by expnding fomlly the esponse α in Tylo seies ound α constucting ppopite poducts of such Tylo seies nd integting fomlly the vious poducts ove the unknown pmete distibution function p α to obtin esponse coeltions. This method fo constucting esponse coeltions stemming fom pmete coeltions is known s the popgtion of eos o popgtion of moments method [see e.g. Ref 7]. Fo illustting the effects of second-ode esponse sensitivities fo the pdigm nonline het conduction benchmk consideed in this wok it suffices to tke fom Ref. [7] esponse coeltions up to thid-ode fo the vey simple cse when: (i) the pmetes e uncoelted nd nomlly distibuted; nd (ii) only the fist- nd second-ode esponse sensitivities e vilble. Fo these pticul conditions the esponse coeltions deived in [7] educe to the following expessions fo the fist thee esponse moments: 41
(i) The expected vlue of esponse k denoted hee s UG E k which ises due to uncetinties in uncoelted nomlly-distibuted model pmetes (the supescipt UG indictes uncoelted Gussin pmetes) is given by the expession E N k k k i i1 i UG 1 α (139) whee k α denotes the computed nominl vlue of the esponse; (ii) The covince cov k between two esponses k nd ising fom nomllydistibuted uncoelted pmetes is given by N N UG 1 k k 4 cov k i. i (14) i1i i i1i i The vince v k of esponse k is obtined by setting k in the bove expession l to obtin N N UG 1 k k 4 k i i i1 i i1 i v. (141) As indicted by the expessions in Eqs. (139) - (141) the second-ode sensitivities hve the following impcts on the esponse moments: () They cuse the expected vlue of the esponse UG computed nominl vlue of the esponse k α ; E k to diffe fom the (b) They contibute to the esponse vinces nd covinces; howeve since the contibutions involving the second-ode sensitivities e multiplied by the fouth powe of the pmetes stndd devitions the totl of these contibutions is 4
expected to be eltively smlle thn the contibutions stemming fom the fist-ode esponse sensitivities. 4.1. Computtion of Response Stndd Devitions To illustte the impct of 1 st -ode vesus nd -ode sensitivities in contibuting to the uncetinty in the tempetue esponse T z t n bity loction z conside tht the model pmetes Q q T k c e uncoelted nd nomlly distibuted ll hving eltive stndd devitions of 1% i.e. Q Q q q T T k k c c whee Q 1% T k nd q c denote the espective bsolute stndd devitions. It follows fom Eq. (141) tht the contibutions stemming fom the 1 st -ode pmete deivtives (in the bsence of nd - nd highe-ode deivtives) yields the following 1 st -ode stndd devition denoted s Std.Dev T z of FO T z : Std.Dev FO T z T z Tz Q q Q q T z T z T z T k c T k c 1/. (14) The mgnitudes of ech of the tems on the ight side of Eq. (14) quntifies the contibution mde by ech of the model pmetes consided to by uncoelted nd nomlly distibuted to 1 st -ode the stdd devition T z function of z in Figues 6. Std.DevFO. These mgnitudes e plotted s 43
T z Q Q K T z q q K z z Fig. : Contibutions to Std. Dev. in T z Fig. 3: Contibutions to Std. Dev. in T z of 1 st -ode sensitivities of Q. of 1 st -ode sensitivities of q. K T z T T K T z k k z z Fig. 4: Contibutions to Std. Dev. in T z Fig. 5: Contibutions to Std. Dev. in T z of 1 st -ode sensitivities of T of 1 st -ode sensitivities of k. 44
K T z c c K Std.DevFO T z z z Fig. 6: Contibutions to Std. Dev. in T z Fig. 7: Contibutions to Std. Dev. in T z of 1 st -ode sensitivities of c. of 1 st -ode sensitivities of in ll pmetes. As shown in Figues though 6 the quntities QT z Q nd e monotoniclly incesing the quntity quntities k T z k nd q T z q T T z T displys minimum while the c T z c disply mxim s functions of z. Ech of these behvios is govened by the behvio of the espective 1-st ode deivtives of couse. Since the deivtives T z Q nd T z contibutions cuse the quntity of q e the lgest the espective Std.DevFO T z to incese monotoniclly s function z. Noticbly the smllest contibution to Std.DevFO T z stems fom c T z c indicting tht the pmete c which ctully contols the stength of the nonlineity in the conduction eqution is not vey impotnt in the tempetue nge unde considetion (4 8 K). It follows fom Eq. (141) tht the contibutions stemming fom the 1 st -ode pmete deivtives (in the bsence of nd - nd highe-ode deivtives) yields the following 1 st -ode stndd devition denoted s Std.Dev T z of FO T z : 45
The quntities contin the contibutions involving the nd -ode deivtives i k i to the totl stndd devition denoted s Std.Dev T z of T z. The mgnitudes of ech of these nd -ode contibutions e plotted s function of z in Figues 8 3. T z Q K Q T z q K q z z z Fig. 8: Contibutions fom nd -ode sensitivities of Q to Std. Dev. in Fig. 9: Contibutions fom nd -ode T z. T z. sensitivities of q to Std. Dev. in K T z T T T z k K k z z Fig. 3: Contibutions fom nd -ode sensitivities of T to Std. Dev. in Fig. 31: Contibutions fom nd -ode T z. T z. sensitivities of k to Std. Dev. in 46
K T z c c K Std.Dev.T z z z Fig. 3: Contibutions of nd -ode sensitivities of c to Std. Dev. in Fig. 33: Sptil vition of the stndd devition T z. T z. of the tempetue distibution The esults plotted in Figues -6 to the coesponding esults plotted in Figues 8-3 indicte tht lthough some of the nd -ode sensitivities hve mgnitudes compble to the 1 st -ode ones the contibutions stemming fom the 1 st -ode sensitivities e much smlle thn those stemming fom the nd -ode sensitivities. This is becuse the contibutions stemming fom the nd -ode sensitivities e multiplied by the fouth-powe of the espective stndd devitions the thn by the second-powe s e the 1 st -ode sensitivities. The contibutions to the totl stndd devition in T z stem pedominntly fom the het souce Q nd the boundy het flux q which cuse this stndd devition to incese monotoniclly s function of the loction z fom the inlet to the outlet eching its mximum vlue of 17K t the outlet. Equivlently the eltive stndd devition of T z inceses fom 1% the inlet to 4% t the outlet; ecll tht ll of the model pmetes wee ssumed to hve eltive stndd devition of 1%. Tble 1 pesents the ctul vlues t the inlet outlet nd t z zmx =8 1 cm which denotes the loction whee the tempetue distibution Tz eches its mximum vlue T mx (z mx ) = 874.7K. Notbly the tems involving the nd -ode sensitivities do not contibute t ll t thebottom of the test section but thei contibutions incese monotoniclly fom the bottom to the test section s top. 47
Tble 1: Individul nd totl contibutions of pmetes stndd devitions to the 1 st -ode nd totl stndd devition in Tz t selected loctions. Loction z l z zmx z l Std.Dev Q 1 T z [K] 88 1 Std.Dev Q Std.Dev q 1 Std.Dev q Std.Dev T 1 Std.Dev T Std.Dev k 1 Std.Dev k Std.Dev c 1 Std.Dev c T z [K] 5 1 T z [K] 5 96 T z [K] 1.4 6 T z [K] 4 4 9 T z [K].5.6 T z [K] 38 6 T z [K] 3 5 T z [K] 5 16 T z [K] 1.3 Std.Dev FO T z [K] 4 117 16 Std.Dev T z [K] 4 15 17 4.. Computtion of Response Skewness The thid-ode esponse coeltion the following expession: 3 k l m mong thee esponses ( k nd m ) hs N UG k l m k l m k l m 4 3 k l m. i i1 i i i i i i i i (143) i In pticul the thid-ode centl moment setting k l m in Eq. (143) to obtin 3 k UG 4 3 3 N k k k. i i1 i i of the esponse k is detemined by (144) 48
The skewness 1 k of esponse k cn be computed using the customy definition 3/ 1 k 3 k v k. (145) Recll tht the skewness of distibution quntifies the deptue of the subject distibution fom symmety. Symmetic univite distibutions (e.g. the Gussin) e chcteized by 1 k. A distibution with long ight til would hve positive skewness while distibution with long left til would hve negtive skewness. In othe wods if then the espective distibution is skewed towds the left of the men UG lowe vlues of k eltive to E UG k. On the othe hnd if k distibution is skewed towds the ight of the men UG eltive to UG E k. 1 1 k E k fvoing then the espective E k fvoing highe vlues of k As Eq. (143) indictes neglecting the second-ode sensitivities fo nomlly distibuted model pmetes would nullify the thid-ode esponse coeltions nd hence would nullify the skewness 1 k of esponse k cf. Eq. (144). Consequently ny events occuing in esponse s long nd/o shot tils which e chcteistic of e but impotnt events would likely be missed. It is of inteest to quntify the skewness induced in the tempetue distibution Tz by ech model pmete consideed septely. These individullyinduced skewnesses in T z will be denoted s T z T z T z nd T z espectively. In othe wods the quntity Q q T T z k c Q T z denotes the skewness tht would be induced in the tempetue distibution T z if only the het souce Q wee nomlly distibuted vite with eltive stndd devition of 1%. The 49
quntity q T z denotes the skewness tht would be induced in the tempetue distibution Tz if only the boundy het flux q wee nomlly distibuted vite with eltive stndd devition of 1% nd so on. The espective 3 d -ode moments of the tempetue distibution Tz e plotted long with the coesponding skewnesses in Figues 34-38 s functions of the loction z fom the bottom to the top of the test section. z z Tz T z 4 3 Q Q Q 3 K Fig. 34: 3 d -ode moment of T z Fig. 35: Skewness of T z ising solely fom the vite Q. due soley to the vite Q. Q T z z z Tz T z 4 3 q q q 3 K q T z Fig. 36: 3 d -ode moment of T z Fig. 37: Contibutions to skewness in T z ising solely fom the vite q. due solely to the vite q. 5
3 K T T z Tz T z 4 3 T T T z Fig. 38: 3 d -ode moment of T z Fig. 39: Contibutions to skewness in T z ising solely fom the vite T. due solely to the vite T. 3 K Tz T z 4 3 k k k k T z z z Fig. 4: 3 d -ode moment of Tz Fig. 41: Contibutions to skewness in Tz ising solely fom the vite k. due solely to the vite k. 51
3 K c T z Tz T z 4 3 c c c z z Fig. 4: 3 d -ode moment of Tz Fig. 43: Contibutions to skewness in T z ising solely fom the vite c. due solely to the vite c. 1 T z z Fig. 44: Sptil vition of the skewness in T z Figues 35 nd 37 indicte tht T z nd T z Q q. Hence if only the het souce Q nd/o the boundy het flux q wee nomlly distibuted vites ech hving eltive stndd devition of 1% they would cuse the tempetue distibution Tz to be skewed significntly towds vlues lowe the the men tempetue (i.e. the distibution would be skewed to the left of the men vlue). On the othe hnd Figues 41 nd 43 5
indicte tht T z nd k c T z. Hence if only the model pmetes k nd/o c wee nomlly distibuted vites ech hving eltive stndd devition of 1% they would cuse the tempetue distibution T z to be skewed significntly towds vlues highe the the men tempetue (i.e. the distibution would be skewed to the ight of the men vlue). Finlly Figue 39 indictes tht T T z but hs the smll positive vlues. Hence if the tempetue T t the bottom of the test section wee the sole vite nomlly distibuted nd hving eltive stndd devition of 1% (nd ll othe pmetes wee pefectly known tking on exctly thei nominl vlues) then the tempetue distibution T z would disply slight symmety towds tempetues highe thn the men tempetue. The totl skewness of the tempetue distibution T z eflecting the cumultive effects of the model pmetes (ssumed to be nomlly distibuted nd hving ll eltive stndd devitions of 1%) is depicted in Figue 43. This figue indictes tht the negtive contibutions stemming fom the 3 d -moments of het souce Q nd the boundy flux q e initilly smll so the positive contibutions fom the 3 d -ode moments of the othe model pmetes dominte t the bottom of the test section. Theefoe the tempetue distibution Tz is skewed slightly towd highe tempetues in the egion extending bout cm fom the bottom of the test section. Fo the eminde of the test section howeve the negtive contibutions stemming fom the 3 d -moments of het souce Q nd the boundy flux q become dominnt so the tempetue distibution T z becomes incesingly skewed towds tempetues lowe thn (i.e. to the left of) the men tempetue towds the top of the test section. Notbly the influemce of the model pmete c which contols the stength of the nonlineity in this illusttive benchmk poblem would be stong if it wee the only uncetin model pmete. Howeve if ll of the othe pmetes e lso uncetin ll hving equl eltive stndd devitions the uncetinties in the het souce Q nd boundy het flux q estompte the impct uncetinties in c fo the nge of tempetues (4-9K) consideed fo this benchmk poblem. The numeicl vlues of the impct of the individul pmetes s well s thei cumultive impct on the tempetue distibution T z t the bottom nd top of the test section s well s t the loction z zmx =8 1 cm e pesented in Tble. 53