Rainer Friedrich

Transcription

1 Rainer Frierich et al Rainer Frierich Holger Foysi Joern Sesterhenn FG Stroemngsmechanik Technical University Menchen Boltzmannstr 5 D Garching, Germany Trblent Momentm an Passive Scalar Transport in Spersonic Channel Flow Direct nmerical simlations of compressible trblent channel flow incling passive scalar transport have been performe at five Mach nmbers, M, ranging from 03 to 35 an Reynols nmbers, Re τ, ranging from 8 to 030 The Prantl an Schmit nmbers are 07 an 0, respectively, in all cases The passive scalar is ae to the flow throgh one channel wall an remove throgh the other, leaing to an S-shape mean scalar profile with non-zero graient in the channel centre The paper escribes the set of compressible flow eqations, which is integrate sing high-orer nmerical schemes in space an time Statistical eqations are presente for flly evelope flow, incling bgets for the Reynols stresses, the trblent scalar flxes an the scalar variance Reslts are presente for secon orer moments an the terms in the mentione balance eqations Oter scalings are fon sitable to collapse incompressible an compressible ata The rection in the near-wall pressre-strain an pressre-scalar graient correlations e to compressibility is explaine sing a Green-fnction-base analysis of the flctating pressre fiel Keywors: Direct nmerical simlation, compressible channel flow, passive scalar transport, oter scaling, pressre-strain correlation, pressre-scalar graient correlation Introction Spersonic channel flow is a prototypical example of a highspee flow which allows for a systematic sty of compressibility effects in wall-bone trblence withot other important bt istinct featres sch as streamwise evelopment, shocks, an flow separation Coleman et al (995) were the first to perform irect nmerical simlations (DNS) of channel flow between col isothermal walls with Mach nmbers p to M = 3 They fon that Morkovin s hypothesis, the flow follows an incompressible pattern, hols for the most part, an that the Van-Driest log-law is vali In a companion paper, Hang et al (995) observe that the trblent stresses, Ri, scale with the wall shear stress, τ w, an semi-local scaling is sefl Lechner et al (00), in their sty of M = 5 channel flow, reporte that the anisotropy of the Reynols stresses was change relative to corresponing incompressible vales, bt no explanation was given Morinishi et al (003) simlate spersonic channel flow at M = 5 with one wall isothermal an the other wall aiabatic, fining that the reslting ifferences of the flow between the two halves of the channel are significant The inflence of compressibility on passive scalar transport has not yet been stie so far in wall-bone trblent flows Kim et al (989) performe the first irect simlations of passive scalar transport in incompressible flly evelope trblent channel flow at a Reynols nmber Reτ, base on friction velocity, τ, an channel half with, h, of 80 They se two types of bonary conitions In one case, the scalar (temperatre) was internally generate an isothermal walls of the same temperatre serve as bonary conitions In the other case, a ifferent temperatre was impose at each of the isothermal walls The Prantl nmbers were varie between 0 an Recently, Johansson et al (999) performe a similar DNS of incompressible channel flow, imposing a mean scalar (temperatre) graient at a Reynols nmber of Reτ = 65 an a Prantl nmber of 07 Presente at ETT th Spring School on Transition an Trblence September 7 th - October st, 004, Porto Alegre, RS, Brazil Paper accepte: May, 005 Technical Eitor: Ariste a Silveira Neto In which way compressibility affects passive scalar transport is important to know, before active scalar transport is investigate, the nerstaning of which is prereqisite to the nerstaning of combstion processes Base on the above literatre srvey it appears that, althogh the behavior of the mean velocity profile is well nerstoo in spersonic channel flow, there are open isses regaring the behavior of the trblent stresses an trblent scalar flxes In the case of incompressible channel flow, the flctating velocity scale is = τw, while there are two length scales, the τ / inner length scale, ν / τ, an the oter length scale, h (the channel half with), leaing to well-establishe inner an oter scalings of the trblence Corresponing scalings apply to the trblent scalar flxes The applicability of these scalings to the trblent momentm an scalar flxes an their transport eqations in the context of compressible flow will be evalate At the same time an explanation will be given why the trblence strctre is moifie e to compressibility Nomenclatre c = spee of son, m/s c p, c v = specific heats at constant pressre an volme, J/(kg K) D = Diffsion coefficient, m /s f i = component of boy force in i-irection, kg/(m s ) h = channel half with, m H = total enthalpy, J/kg L x = length of comptational omain in x -irection, m M = Mach nmber, imensionless N x = Nmber of gri points in x -irection p = pressre, Pa Pr = Prantl nmber, imensionless q i = component of heat flx in i-irection, J/(m 3 s) R = specific gas constant, J/(kg K) Re = Reynols nmber, imensionless s = specific entropy, J/(kg K) Sc = Schmit nmber, imensionless s i = eformation tensor, /s t = time, s T = temperatre, K i = component of velocity in i-irection, m/s av = streamwise velocity, average over cross-section, m/s x i = Cartesian coorinate, m 74 / Vol XXVIII, No, April-Jne 006 ABCM

2 Trblent Momentm an Passive Scalar Transport in X i = wave component in positive i-irection Nmerical Metho an Comptational Details In this section, we escribe the basic eqations for compressible flow of an ieal gas incling passive scalar transport, how these eqations are integrate nmerically an which initial an bonary conitions are se to preict flly evelope trblent channel flow Eqations of Motion Trblent flow of a compressible ieal gas (air) incling passive scalar transport is governe by the following set of transport eqations in Cartesian tensor notation: tp = p γ p ( γ )( Φ q ), () ti = i ( ip τi f δi), () ts= s (R / p)( Φ q ), (3) t= ( D ), where p, i, s,, represent pressre, Cartesian velocity components, entropy, ensity an concentration of a passive scalar, respectively an t, i enote temporal an spatial erivatives This special set of variables has been chosen, in orer to compte the relevant moes of compressible trblence, namely pressre, entropy an vorticity as irectly as possible A vorticity transport eqation has been avoie here, since two of the vorticity bonary conitions cannot be formlate exactly The boy force δ f i in eqation () will be specifie later in sch a way that it replaces the mean streamwise pressre graient In eqations (-3) the components of the heat flx vector q, the viscos stress tensor τi, an the issipation rate Φ, rea: (4) q = λ T, τ = s /3 s δ, (5) i i Φ = τ s, s = / ( ) (6) The thermal eqation of state i i i p = R T, R = c c, (7) an the following laws for ynamic viscosity, heat conctivity λ an iffsivity D close the set of eqations: = (T / T ), λ = c / Pr, D = / Sc, n 07 (8) p n / ref ref p = The Prantl nmber Pr, the Schmit nmber Sc an the ratio of specific heats γ are kept at constant vales in the temperatre an concentration ranges consiere, namely Pr = 07, Sc = 0, γ = c p / c v = 4 Nmerical Metho an Comptational Details The eqations of motion (-4) are cast in a characteristic nonconservative form, following Sesterhenn (00), which allows to formlate wall an free bonary conitions consistently with the i v i kk i fiel eqations A 5 th -orer compact pwin scheme of Aams an Shariff (996) is se to iscretize the hyperbolic (Eler) terms in the eqations of motion The moleclar transport terms are iscretize with a compact 6 th -orer scheme of Lele (99) The soltion is avance in time with a 3 r -orer low-storage Rnge- Ktta scheme, propose by Williamson (980) The boy force term f δ i in the momentm eqation is introce in orer to replace the mean pressre graient in streamwise irection It is niform in 3D space an allows to hanle flly evelope trblent flow sing perioic bonary conitions for all variables in stream- an spanwise irections A mean scalar graient is impose on the flow, sing an initial profile of the form (Johansson, 999): y x (x, x, x, 0) log 0 3 = 0 y0 x y0 = 007 y log 0 0, y0 At the walls the velocity components satisfy no-slip an impermeability bonary conitions The passive scalar is inecte at the lower wall ( x = 0 ) with vanishing momentm an remove throgh the pper wall Its bonary conitions, hence, rea: (9) ( x,0,x3, t ) =, ( x,h, x3, t ) = (0) For a soli isothermal an stationary wall the pressre an entropy bonary conitions have the form: tp = p /(c)(x X ), () ts = R / (c)(x X ) () They follow from Gibbs fnamental relation an the momentm balance in the wall normal irection, which reas in characteristic form: sing waves efine by: X = X ( / ) τ, (3) ± X = ( ± c)(( c) p ± ) (4) / c = ( γ p / ) is the spee of son The bonary conitions () an () show explicitely that pressre an entropy evolve in time at the wall A possible way of initializing compressible trblent channel flow, is as follows: A mean streamwise velocity profile is specifie accoring to a linear law between the wall an x = x τ / νw = 0 an a log-law from there to the centreline The mean temperatre correspons to its wall vale an the mean ensity to its blk-average, see eqation (6) Velocity flctations are given as ranom flctations with zero mean vales The level of the rms-flctations may be of the orer of 5% of the blk-average velocity Temperatre, ensity an entropy flctations are zero Pressre flctations are proportional to velocity flctations times the mean ensity Since flly-evelope shear-trblence is inepenent of initial conitions, the choice of these conitions is not critical They will be swept ot of the flow omain ring an initial transient stage Once a stable trblence fiel has been generate, it can be se as initial conition in the simlation with other flow parameters J of the Braz Soc of Mech Sci & Eng Copyright 006 by ABCM April-Jne 006, Vol XXVIII, No / 75

3 Rainer Frierich et al The nmerical algorithm has been previosly valiate by Lechner et al (00) whose reslts for a Mach nmber M = 5 case are in excellent agreement with those of Coleman et al (995) The present paper reports on five irect nmerical simlations for ifferent Reynols an Mach nmbers The global Mach an Reynols nmbers for flly evelope trblent channel flow are efine as M = av / cw, Re = m av h / w (5) The spee of son an the ynamic viscosity are evalate at a wall temperatre which has the constant vale T w = 500 K in all cases The blk-average ensity an velocity rea m = x / h, av = x / h 0 0 (6) The overbar enotes Reynols average qantities to be efine below Table smmarizes the flow parameters, box sizes an nmbers of gri points se in the ifferent irect simlations Eqiistant gris are chosen in the (x, x 3 )-, ie the streamwise an spanwise irections of the channel In the wall-normal x irection, points are clstere sing tanh-fnctions (Lechner et al, 00) The friction Reynols nmber Reτ = w τ h / w, with τ = ( τ w / w) /, is a reslt of the comptation Note, that for spatial an velocity coorinates the following notations are se alternately, (x, x, x 3 ) (x, y, z), (,, 3 ) (, v, w) Table Flow an comptational parameters Case M Re Reτ L x/h L x/h L x3/h π π M M M /3 M /3 M /3 N x N x N x3 x x min x max x Statistical Eqations for Flly Develope Channel Flow In this section we efine statistical an flctating qantities an present the statistically average eqations of motion for flly evelope channel flow as well as transport eqations for the Reynols stresses an the trblent scalar flxes It is common practice to work with Reynols- an Favre-average, ie massweighte average qantities simltaneosly Velocity components an temperatre are ecompose in the following way: i = i i = i i, i = i /, (7) T = T T = T T, T = T / (8) Density, pressre, entropy (an the moleclar transport coefficients) follow Reynols ecomposition: T T (, p, s) = (, p, s ) (, p, s ) (9) The statistical operation, represente by the overbar, is achieve by averaging in the streamwise an spanwise irections, ie over 5*56 mesh points (case M35) an in time A typical nmber of statistically inepenent time samples is 0 3 for basic qantities, like mean velocity, pressre, ensity, an 0 4 for correlations like the Reynols stresses Mean Balance Eqations Consiering the above ecompositions an the fact that flly evelope channel flow leas to the following balance eqations for mean streamwise momentm an scalar = ( x), = 3 = 0, () = () = 0, (0) ( τ ) f, = () x 0 0 = x Sc x () In eqation () the mean boy force, f T 3, replaces the mean pressre graient an is relate to the wall shear stress in the following way: p τw f = = (3) h Integrating eqations (-) from the wall (x = 0) to a position in between the wall an the centreline, taking care of the fact that the soli wall inhibits all trblence flctations an that correlations involving flctations of viscosity an iffsivity are negligibly small, provies the following balances for the shear stresses an scalar flxes: x =, w x τw h Sc = w x w (4) (5) The inicates normalization with wall nits, namely the friction velocity, τ, the viscosity at the wall, ν w, an the scalar flctation, τ, efine via the scalar flx at the wall, w, as w = w τ τ Sc = (6) w From (4) we concle that the sm of the viscos an Reynols shear stresses varies linearly in the channel, ecreasing from its wall vale to zero at the centreline (5) inicates that the sm of moleclar an trblent scalar flxes is constant Unlike the viscos stress, the moleclar scalar flx is non-zero at the centreline, which leas to consierable proction of scalar flctations in the channel core The mean balance of the total enthalpy, H = e p i i /, takes the following form in flly evelope channel flow: 76 / Vol XXVIII, No, April-Jne 006 ABCM

4 Trblent Momentm an Passive Scalar Transport in H H = ( i τi q) (7) The first term on the lhs is non-zero since it contains the mean pressre graient Obviosly, p H = x (8) All the other streamwise graients of mean qantities an correlations vanish Now, integrating eqation (7) from the wall to the centreline an sing symmetry conitions for flow variables an correlations, provies the interesting relation h p qw = x = τw av, (9) 0 x which states that the work one by the mean pressre graient in one half of the channel correspons to the heat that leaves the channel throgh the isothermal wall In other wors, in orer to achieve spersonic velocities in a channel, the walls have to be coole Aiabatic walls o not allow for spersonic flow an lea to choking Bgets of the Reynols Stresses, the Scalar Flxes an the Scalar Variance In orer to analyze effects of compressibility on secon-orer moments of trblent flctations, one has to sty their transport eqations These eqations contain higher-orer correlations an are, hence, nclose It is a real challenge for trblence moellers to evelop proper moels which relate nknown higher-orer correlations to known correlations an variables DNS helps to test these moels an to improve them The transport eqations for the streamwise an spanwise components, R /, R 33 /, an the Reynols shear stress, R rea: 0 = / τ x x τ p p τ, x x τ 0 = / 3 3 3τ 3 3 x x 3 3 p τ 3, 3 0 = p τ τ x x τ p p τ x x x x p τ τ x x (30) (3) (3) They state that there is no mean convective transport of the Reynols stresses in flly evelope flow The rhs of eq (30) contains proction of the streamwise Reynols stress by the action of the shear stress (first term), trblent an viscos transport (secon term), intrinsic compressibility by streamwise trblent mass-flx, = / = /, (33) energy loss by reistribtion (forth term) an issipation (last term) The spanwise Reynols stress (eq (3)) is moifie by trblent an viscos transport as well (first term on the rhs), by streamwise trblent mass flx which is, however, weak It is not irectly proce, bt receives energy from the streamwise component by reistribtion (thir term) an looses energy by issipation (last term) The Reynols shear stress balance (eq (3)) contains corresponing terms The most important ones are proction, reistribtion an trblent transport Viscos iffsion an issipation nearly balance in the wall layer The erivation of transport eqations for the scalar flxes an the scalar variance procees similarly to that of the Reynols stresses Again there is no mean convective transport in flly evelope channel flow The eqations for the streamwise an wallnormal trblent scalar flxes are: 0 = x x x p τ τ, x Sc x Sc 0 = x x p τ Sc τ x Sc x (34) (35) Both flxes are primarily proce by mean scalar graients, the streamwise scalar flx in aition by the mean velocity graient Both balances contain trblent an viscos transport terms, scalar pressre-graient correlations (the analoges of velocity pressregraient terms in the Reynols stress transport eqations) an, finally, estrction terms e to viscos an iffsion effects The transport eqation for the scalar variance, /, reas: 0= / x x (36) / x Sc Sc x Sc x Sc Scalar flctations are proce by the mean scalar graient alone If the mean flow were accelerate or retare, there wol be a proction term by mean ilatation as well We frther note that trblent transport of scalar flctations an viscos iffsion as well as issipation contribte to the ynamics of scalar flctations Diffsion an issipation consist of two terms, respectively, where the terms involving correlations between viscosity an scalar flctations are generally small DNS Reslts In this section we investigate effects of compressibility on the Reynols stresses, scalar flxes an terms in the corresponing balance eqations We also provie an explanation for the strctral changes of correlations involving pressre flctations, like the J of the Braz Soc of Mech Sci & Eng Copyright 006 by ABCM April-Jne 006, Vol XXVIII, No / 77

5 Rainer Frierich et al pressre-strain an pressre-scalar-graient correlations Wherever possible, we se comparisons with DNS ata of incompressible channel flow for a better nerstaning of the high-spee physical mechanisms Mean Flow Properties Compressibility effects in flly evelope channel flow originate mainly from the large changes in fli properties, an, case by viscos heating Figre shows profiles of these qantities for for Mach nmber cases liste in Table The increase in mean temperatre in the channel core as a reslt of mean an trblent issipation, leas to a steep rise in viscosity an entails a simlar ecrease in mean ensity Given the fact that the mean wall-normal pressre graient is negligibly small, corresponing ensity an temperatre graients are eqal in magnite, bt opposite in sign From the average ieal gas law, we obtain, neglecting the contribtion from the ensity-temperatre correlation: T (37) T x x Eqation (37) also contains the assmption that the wall-normal mean pressre graient reslting from the wall-normal momentm balance 0 = p τ (38) is small In (38), τ 4 / 3 / x an after integration w w = Sc, = x x (40) w = Sc (4) Eqation (40) sggests the following transforme velocity an scalar in the viscos sblayer:,v = = x = 0 w Sc w = v w (4) Figre 3 shows both, the Van Driest- an the viscositytransforme mean velocities Figres 4 an 5 present profiles of the local Mach nmber, M = / c w, an the mean scalar, normalize by w From eqation (4) we concle that varies linearly in the viscos sblayer, when plotte against the variable x x w x = (43) 0 In the core region, where w an w both reach a platea, ie nearly constant vales, we note that ( ) h ( x h) w Sc w (44) varies linearly with x The Trblent Stress Tensor At sfficiently large istances from the wall, where the viscos stress is small at high Reynols nmbers, the shear stress balance, eq (4), allows to concle that the qantity R τw is Figre Variation of mean ensity (symbols) an mean viscosity (lines), both normalize with wall vales Figre contains profiles of the Van Driest transforme mean velocity,vd w = 0 (39) Figre Profiles of the Van Driest-transforme mean velocity, eq (39) in a semi-logarithmic plot This transformation provies a reasonable collapse of the varios cases in the oter region alone an there seems to be no way of fining a single transformation which oes a goo ob in the inner as well as the oter layer Neglecting the trblent flxes in eqations (4) an (5), we get for the viscos sblayer: 78 / Vol XXVIII, No, April-Jne 006 ABCM

6 Trblent Momentm an Passive Scalar Transport in Figre 3 Profiles of the Van Driest- an viscosity-transforme mean velocity, eq (4) Figre 6 Oter scaling of the Reynols shear stress, R Figre 4 Profiles of the local Mach nmber M = /cw a linear fnction of x h, inepenent of Mach an Reynols nmber In other wors, τ w, is the proper oter scaling parameter which collapses compressible an incompressible cases on to a niversal profile, see Figre 6 Incompressible channel flow ata of Moser et al (999) at Re τ =80, 395, an 590, enote by cases I, I, an I 3, respectively, have been se for comparison Note that the Reτ vales are similar between cases M03, M5 an I, an between cases M30 an I 3 Figre 7 Oter scaling of the trblent streamwise stress, R profiles in the wall layer are e to mean property variations The linear behavior of R ( M30) starts at x h 0 3 only, while that of R ( I3) starts at x h 0 alreay, becase temperatre an hence viscosity increase with increasing x, extening the importance of the viscos stress to larger istances x Another observation which is not emonstrate here, is the inepenence of the correlation coefficient of from Mach an Reynols nmbers for h 0 3 From this aitional fact we are allowe to concle that x R τw an R τw stresses, ten to a niversal epenence on from the wall, see Figre 7, finally all Reynols x h sfficiently far Figre 5 Profile of the mean scalar Johansson et al (999) normalize by w shows ata of In Figre 6, cases M03 an I are practically inistingishable Given the fact that cases M30 an I 3 have similar Reynols nmbers, Re τ, the ifferences between their The Trblent Scalar Flxes an the Scalar Variance The moleclar scalar flx, being non-zero everywhere in the channel, makes w not an ieal, bt a reasonable oter scale of the trblent scalar flx in wall-normal irection Figres 8 an 9 emonstrate that this is tre in the range of h 0 6 for the streamwise scalar flx as well, irrespective of Mach an Reynols nmbers There is a striking similarity in the behavior of the streamwise scalar flx, w, an the streamwise Reynols stress, This is seen by comparing Figres 7 an 9, bt also by comparing the strctre of the transport eqations for an Introcing mean an flctating qantities into the momentm eqation () for an the scalar transport eqation (4), we obtain: x J of the Braz Soc of Mech Sci & Eng Copyright 006 by ABCM April-Jne 006, Vol XXVIII, No / 79

7 Rainer Frierich et al D Dt x p viscos term D Dt ( ) ( ) ( ) ( ) iff term = x = x x (45) (46) where D Dt = t enotes the sbstantial erivative base on mean convective transport The viscos an iffsion terms have not been written ot, becase we inten to iscss the large scale trblent effects Figre 0 Profiles of the rms scalar flctations w =, normalize by In Figre 0 we present profiles of the rms scalar flctations, normalize by w These profiles reveal peaks in the wall layer an maxima at the channel centreline which are e to non-zero graients of the mean scalar, cf Figre 5 Effects similar to those for the streamwise Reynols stress (an the trblent kinetic energy, not shown here) can be observe, namely: an increase in Reynols nmber intensifies the near-wall peak A simltaneos increase in Mach nmber lowers it again, so that eventally these effects compensate each other Figre 8 Profiles of the wall-normal scalar flx w, normalize by In both eqations there are very similar proction an trblent transport terms ( n an 3 r terms on the lhs) an effects by mean trblent transport The only ifference between both eqations is the flctating streamwise pressre graient As will be shown later, the importance of this term is sbstantially rece e to compressibility The same is tre for the proction terms in the wall layer Hence, the higher the Mach nmber, the stronger is the similarity between an Bgets of Secon-Orer Moments Reynols stresses: Terms in the transport eqations for the Reynols stresses, R i, when normalize with 4 w τ ν w an plotte as fnctions of x, which is cstomary for incompressible flow, o not show a tenency to collapse incompressible an compressible ata Hence, an alternate inner scaling is reqire This is obtaine by consiering the kinetic energy proction term, P of the R -balance Away from the viscos layer, the Reynols shear stress scales as R = τw ( x h), while the mean shear is x τ ( κx ) (where τ = τ w ), so that τ w P =, x = x κ x h τ ν (47) Eqation (47) implies that the Reynols stress bgets shol be normalize with τ w an plotte against the semi-local coorinate, x Figre shows profiles of the terms in the R -balance, normalize accoringly The ominant terms in the near-wall region, namely, proction, issipation an viscos iffsion o not change significantly between cases M30 an I 3 However, as shown in Figre (bottom), the pressre-strain correlation, Figre 9 Profiles of the streamwise scalar flx w, normalize by Π i = p s i = p ( i i ), (48) changes significantly between cases Compressible flow, obviosly, reces the pressre-strain correlations in a remarkable way A 80 / Vol XXVIII, No, April-Jne 006 ABCM

8 Trblent Momentm an Passive Scalar Transport in qantitative explanation of this behavior is given below, base on an analysis of the Poisson eqation for the pressre flctations Figre (Contine) Scalar flxes an scalar variance: A sitable normalization of the streamwise an spanwise scalar flx balances mst be fon The streamwise scalar flx balance eg has two proction terms, see eq (34) They scale otsie the viscos layer as: = x x w τ x τ, h τ w (49) x τ w τ τ = (50) κx Sc τ κx w Figre Bget of the R τ -stress, normalize by w an plotte against the semi-local coorinate x Symbols represent the incompressible case I3 an lines the compressible case M30 Top: Proction, issipation an iffsion Bottom: Pressre-strain, trblent iffsion an mass flx variation In Figre we emonstrate for Π an Π33 that τ w av h are oter scaling parameters, which collapse compressible onto incompressible pressre-strain correlation ata As long as the Schmit nmber is or O(), the streamwise scalar flx bget has to be normalize with τw ( τ τ ) This scaling again nerlines the similarity between - an - flctations The proction term for the wall-normal scalar flx in eq (35) scales as shown in eq (49) Finally, we obtain for the scalar variance proction: τ w x Sc w w τ (5) τ Obviosly, the scaling parameters iffer from those for the wallnormal scalar flx only by the factor τ τ Upper bons of proction rates: It is interesting to compte the high Reynols nmber limits of the peak proction rates of the streamwise Reynols stress (respectively the trblent kinetic energy) an the scalar flx an to compare them with their incompressible conterparts In the limit of high Reynols nmber, h = Re τ, the shear stress balance (4) takes the form: = (5) τ w w x The spatial extremm of P is obtaine from Figre Pressre-strain correlation: Streamwise -component (top), Spanwise 33-component (bottom) = 0, x w x x w x τ τ (53) J of the Braz Soc of Mech Sci & Eng Copyright 006 by ABCM April-Jne 006, Vol XXVIII, No / 8

9 Rainer Frierich et al which becomes, sing (5): w = w x x x x For channel flow with wall-cooling, the term in the bracket again as to a qantitiy larger than, since the erivative of, is positive an the secon erivative of w x is negative Hence, The corresponing relations for incompressible channel flow are: = = x τ (54) (55) (56) We therefore concle that in compressible (isothermal) channel flow between coole walls, the Reynols shear stress, R, an the peak proction of, R, are rece with respect to incompressible isothermal flow as a reslt of viscos heating The pper bon ( Re τ ) for the peak proction rate of R is: w x τ Re τ 4 (57) w x Re τ Sc 4 (6) Again, the eqal sign hols for incompressible isothermal flow Viscos heating in compressible flow conseqently reces the proction rate of the scalar variance in a similar way as it reces the streamwise Reynols stress proction Figres 3-5 show bgets of, an as fnctions of x h for case M30 It is obvios that proction an issipation play an important role in the bgets of an Dissipation is less important in the wall-normal scalar flx bget Instea, the pressre-scalar graient correlation balances the proction term there It also plays a remarkable role in the streamwise scalar flx As we will see, it is strongly ampe in compressible flow, very mch like the pressre-strain correlation or velocity-pressre-graient correlation Figres 6-8 emonstrate the effect of Mach nmber on the proction rates of, an There is a clear ecrease of the peak vales as M increases an an increase with Reτ Finally, we emonstrate the oter scaling of the scalar-pressre graient correlations p, p in Figres 9, 0, sing av w h for normalization The eqal sign is vali for incompressible flow In a similar way, we obtain the pper bon for the proction rate of the scalar variance from: = 0, x w x (58) sing the approximation, Sbstitting eq (5) into conition (58), we get: Sc = w x w Sc x x x (59) Figre 3 -bget, normalize by av w h Case M30 Obviosly, the moleclar scalar flx is increase e to viscos heating in compressible flow, as compare to incompressible flow Since the term in the brackets is greater than one, we have Sc w x (60) while in incompressible isothermal flow Sc x = (6) The pper bon ( Re τ ) for the peak proction rate of the scalar variance therefore reas: Figre 4 -bget, normalize by av w h Case M30 8 / Vol XXVIII, No, April-Jne 006 ABCM

10 Trblent Momentm an Passive Scalar Transport in Figre 5 Scalar variance bget, normalize by w w h Case M30 Figre 8 Proction of, normalize by w w h ( ) i i i τ i { p = i A A A3 ( ) B B i i i tt C C C3 ( ) ( ) D = { f (63) Figre 6 Proction of, normalize by av w h Case M30 Figre 7 Proction of, normalize by av w h Case M30 Correlations Involving Pressre Flctations A qantitative explanation for the rection of pressre-strain correlations an scalar pressre-graient correlations e to compressibility starts from an eqation for the pressre flctations It is erive from the momentm eqation, by taking its ivergence an incorporating mass conservation Splitting flow variables into mean an flctating variables, we get the following relation (see Foysi et al (004)), vali for flly evelope channel flow: Note that the operator Dtt = tt t i i is Galilean invariant In incompressible isothermal flow, the first term on the rhs of eq (63), labelle A (nonlinear flctation), an the secon, A (mean shear), srvive In compressible flow, there are aitional terms, A3 (viscos stress, thir term), B (ensity graient), B (ensity secon-graient), an the last three terms involving ensity flctations, C, C an C3 From the DNS ata base we concle that all terms involving ensity flctations can be neglecte in the above eqation This allows s to interpret eq (63) as a Poisson eqation for the pressre flctation A Green s-fnction base analysis of the Poisson eqation for p will now be performe In incompressible flow the exact wall bonary conition for p is p = In compressible channel flow the ilatation flctation is small an, hence, the wall bonary conition is given by p w Let s enote the rhs of eq (63) by f an Forier-transform the whole eqation in the homo-geneos x, x 3 irections, eg p ( x, x, x3 ) pˆ ( k, x, k3) Then, the transforme Poisson eqation, after normalizing length scales with the channel half with, becomes, pˆ with ( k k ) 3 pˆ = fˆ pˆ û = w x x = ± x = ± (64) If we replace fˆ in this eqation by the point sorce, δ ( x x ), then the Green fnction, ( k, x ; ) soltion of eq (64) ( k, x ; ) Ĝ x with k = k k 3 is the Ĝ x can be erive for the homogeneos bonary conition, ( pˆ ) w = 0, sing stanar methos an it trns ot that it is as given by eq (7) of Kim (989): J of the Braz Soc of Mech Sci & Eng Copyright 006 by ABCM April-Jne 006, Vol XXVIII, No / 83

11 Π Π Ĝ Ĝ ( k,x ;x ) ( k,x ;x ) cosh = cosh = [ k ( x ) ] cosh [ k ( x ) ] ksinh k [ k ( x ) ] cosh [ k ( x ) ] ksinh k, x x,, x x (65) Rainer Frierich et al vale an, accoring to eqations (69, 70) are the correlations smaller than the corresponing incompressible ones The soltion of eq (64) incling the non-homogeneos bonary conition is: pˆ ( k, x, k ) Ĝ ( k,x, x ) ( x ) fˆ ( k, x, k ) Bˆ ( k, x ), with Bˆ given by 3 = 3 x (66) Bˆ = pˆ cosh x = pˆ x = ksinh k ( k( x )) cosh( k( x )) (67) The inverse Forier transform of (66) provies the pressre flctation in physical space, p ( x, x, x3) = ( x ) G f ( x, x, x3;x ) x (68) B ( x, x, x3), where the convoltion G f is the inverse Forier transform of Ĝ fˆ From eq (68) all correlations involving pressre flctations can be constrcte The pressre-strain correlation, eg, is given by Figre Comparison between DNS ata an eqation (69) for the pressre-strain correlation (top) In the figre below, symbols illstrate the effect of mean ensity on Contribtions of ifferent sorce terms, f, to eqation (69) are shown by lines for case M5 Πi ( x ) = ( x ) G f ( x,x,x3;x ) s i x B s i, (69) an the pressre-scalar-graient correlation reas Πi ( x ) = ( x ) G f ( x,x,x ;x ) B i 3 i x (70) The pper parts of Figres an show a comparison of the analytical soltions, eqs(69) an (70), an the DNS ata for cases M03 an M5 On the lower parts of these figres we se the varios sorce terms in the pressre Poisson eqation an investigate their contribtions to the convoltion terms in (69, 70) The proof that the changes in the correlations are mainly e to the variation in the mean ensity is given by replacing by w an taking the DNS velocity fiel of case M5 to compte the rhs of eqations (69) an (70) The fact that the reslts (sqares in the figres) compare well with the qasi-incompressible case (triangles) in the region x 30, confirms the hypothesis that the variableensity extension of the Poisson eqation is sfficient for obtaining the pressre-strain an pressre-scalar-graient term The main reslt of the Green fnction analysis, hence, is to nerline the non-local effect of on correlations involving pressre flctations an to explain the observe rection of these correlations compare to their incompressible conterparts The fli in the interior of a spersonic channel is hotter than that at the coole isothermal walls so that ( x ) is smaller than the wall Figre Comparison between DNS ata an eqation (70) for the pressre-scalar-graient correlation (top) In the figre below, symbols illstrate the effect of mean ensity on Contribtions of ifferent sorce terms, f, to eqation (70) are shown by lines for case M5 84 / Vol XXVIII, No, April-Jne 006 ABCM

12 Trblent Momentm an Passive Scalar Transport in Conclsions Direct nmerical simlations of compressible trblent channel flow incling passive scalar transport have been performe in orer to investigate in which way compressibility affects the mean flow qantities an the trblence strctre in wall-bone flows The following finings are reporte: - Spersonic, trblent channel flow exists only when the heat generate by issipation within the flow fiel is remove throgh the walls This nees wall cooling an entails strong near-wall mean temperatre, viscosity an ensity graients - There is no similarity transformation which collapses velocity profiles for varios Mach an Reynols nmbers onto one profile in the whole omain It is observe that viscositytransforme mean velocities work well in the near-wall region an ensity-transforme Van Driest velocities are site for the log layer - All the Reynols stress components scale with the wall shear stress, τ w, in the channel core region, inepenent of Mach an Reynols nmber In the wall layer, semi-local coorinate scaling at least provies a proper collapse of the positions where most of the trblence is proce, bt not of the amplites themselves - Other than the mean streamwise velocity, the mean scalar (introce on one sie an remove from the other) has a non-zero graient on the channel centreline, which leas to a non-zero mean moleclar flx an estroys the analogy between the Reynols shear stress an the wall-normal scalar flx The mean scalar graient in trn proces a peak in the scalar flctations on the channel centreline - The streamwise scalar flx on the other han scales properly in the core region of the channel with the moleclar scalar flx at the wall, w - The ratio of the wall shear stress sqare an the local mean viscosity ( τ w ) provies a proper scaling for all terms in the Reynols stress bgets, except the pressre-strain correlation terms Similarly, the ratio, w av h, is nsite to properly collapse all scalar pressre-graient correlations - The rection in pressre flctations observe in the wall layer of compressible trblent channel flow that is responsible for the ramatic rection of pressre-strain an scalar pressre-graient correlations col be relate to the variation of the mean ensity normal to the wall This is emonstrate by a Green s fnction approach of the Poisson eqation for the pressre flctations These finings shol have an impact on the moeling of correlations involving pressre flctations Acknowlegment One of the athors (HF) grateflly acknowleges the financial spport of the German Research Association (DFG) ner grant nmber FR 478/- References Aams, N A, Shariff, K, 996, A high-resoltion hybri compact- ENO scheme for shock-trblence interaction problems J Compt Phys Vol 7, pp 7 Coleman, G, Kim, J an Moser, R, 995, Trblent spersonic isothermal-wall channel flow J Fli Mech, Vol 305, pp Foysi, H, Sarkar, S an Frierich, R, 004, Compressibility effects an trblence scalings in spersonic channel flow J Fli Mech, Vol 509, pp 07-6 Hang, P, Coleman, G an Brashaw, P, 995, Compressible trblent channel flows: DNS reslts an moelling J Fli Mech, Vol 305, pp 85-8 Johansson, A V, Wikström, P M, 999, DNS an moelling of passive scalar transport in trblent channel flow with focs on scalar issipation rate moelling Flow, Trblence an Combstion, Vol 63, pp 3-45 Kim, J, 989, On the strctre of pressre flctations in simlate trblent channel flow J Fli Mech, Vol 05, 4-45 Kim, J an Moin, P, 989, Transport of passive scalars in a trblent channel flow Trblent Shear Flows, Vol 6 Springer, pp Lechner, R, Sesterhenn, J an Frierich, R, 00, Trblent spersonic channel flow J Trblence, Vol, pp -5 Lele, S K, 99, Compact finite ifference schemes with spectral-like resoltion J Compt Phys, Vol 03, pp 6-4 Morinishi, Y, Tamano, S an Nakabayashi, K, 003, A DNS algorithm sing a B-spine collocation metho for compressible trblent channel flow Compters & Flis, Vol 3, pp Moser, R, Kim, J an Mansor, N N, 999, Direct nmerical simlation of trblent channel flow p to Reτ = 590 Phys Flis, Vol 9, pp Sesterhenn, J, 00, A characteristic-type formlation of the Navier- Stokes eqations for high orer pwin schemes Compt Flis, Vol 30, pp 37 Williams, J K, 980, Low-storage Rnge-Ktta schemes J Compt Phys, Vol 35, pp J of the Braz Soc of Mech Sci & Eng Copyright 006 by ABCM April-Jne 006, Vol XXVIII, No / 85