HT TURBULENT NATURAL CONVECTION IN A DIFFERENTIALLY HEATED VERTICAL CHANNEL. Proceedings of 2008 ASME Summer Heat Transfer Conference HT2008

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1 Proceedings of 2008 ASME Summer Heat Transfer Conference HT2008 August 10-14, 2008, Jacksonville, Florida USA Proceedings of HT ASME Summer Heat Transfer Conference August 10-14, 2008, Jacksonville, Florida USA HT TURBULENT NATURAL CONVECTION IN A DIFFERENTIALLY HEATED VERTICAL CHANNEL Abolfazl Siri Turbulence Researc Laboratory Department of Applied Mecanics Calmers University of Tecnology Gotenburg 41296, Sweden abolfazl@calmers.se William K. George Turbulence Researc Laboratory Department of Applied Mecanics Calmers University of Tecnology Gotenburg 41296, Sweden wkgeorge@calmers.se ABSTRACT Te turbulence natural convection boundary layer inside a infinite vertical cannel wit differentially eated walls is analyzed based on a similarity solution metodology. Te differences between mean temperature and velocity profiles in a boundary layer along a vertical flat plate and in a cannel flow, make it necessary to introduce new sets of scaling parameters. In te limit as H, two distinctive parts are considered: an outer region wic dominates te core of te flow and inner constant eat flux region close to te walls. Te proper inner scaling velocity is sowed to be determined by te outer parameters due to momentum integral. Te teory is contrasted wit te one suggested by George & Capp 1), te deficiencies of wic are identified. NOMENCLATURE C p specific eat at constant pressure. F o dimensionless eat transfer rate qw g H H Gr ρc p ). gravitational acceleration. cannel alf-widt and outer lengt scale. H-number based on temperature difference, H-number for constant eat flux walls, Grasof number for constant eat flux walls, ) Nu Nusselt number based on,. Fo T w α ) gβ Tw 3. α ) 2 gβfo 4. α 3 ) gβfo 4 ν 3. Pr Prandtl number α ν. q w wall eat flux. T mean temperature. T It inner temperature scale. T o outer temperature scale. T w wall temperature. T CL center-line temperature. T w walls temperature difference. U mean velocity in x direction. U I inner velocity scale,constant eat flux wall. U o outer velocity scale. V mean velocity in y direction. x distance indirection parallel to walls. y distance indirection perpendicular to walls. ỹ dimensionless outer variable, = y/). y + dimensionless inner variable, = y/η). α termal diffusivity. β termal expansion coefficient. η It inner lengt scale for constant eat flux wall. ρ density. ν viscosity. τ w wall sear stress. ξ lengt scales ratio, = η/). INTRODUCTION Despite its great importance in many industrial and environmental processes and several decades of attention, te basic scal- 1 Copyrigt c 2008 by ASME

2 ing parameters for turbulence natural convection flow near vertical surfaces are still te subject of some debate. In te modeling of suc flows, te near wall beavior of te turbulence quantities is essential in determining te relations among temperature, eat fluxes and velocity. Like oter wall-bounded flows, te problem is usually approaced by seeking scaling laws for te different regions of te flow, ten exploring te consequences of matcing te various regions. For te past tree decades, te primary model under consideration as been tat of George & Capp 1), wo noted te absence of a constant stress layer due to gravity). Tey ten postulated te existence of a buoyant sublayer witin te constant eat flux layer in wic te buoyancy flux from te wall, gβf o, was te crucial parameter, were g is te gravitational acceleration, β is coefficient of termal expansion and F o is te wall eat flux divided by density and specific eat. Using tis assumption tey proposed inner and outer scales for velocity to be gβf o α) 1/4 and gβf o ) 1/3 respectively, were α is te termal diffusivity and is te cannel alf-widt. Te teory as generally been found to be in excellent agreement wit numerous experimental studies for te temperature and eat transfer relations; in particular te temperature varies inversely wit te cube root of distance from wall in te buoyant sublayer, and te Nusselt number is proportional to te cube of te Rayleig number to witin a factor dependent on te Prandtl number. Te velocity profile scaling, owever, as long been recognized as problematical, especially wit more recent experiments and DNS. Altoug te outer velocity scaling matced te experimental results, te inner scaling and cube root velocity profile in te buoyant sublayer seemed to sow systematic departures from te teory wic could not be explained. Experimental data 2) and numerical simulation 3; 4; 5) of te natural convection inside a differentially eated cannel also sows tat te mean profiles of velocity and temperature do not follow te suggested asymptotic beavior in cannel flow. Also, te Reynolds sear stresses at te center of te cannel do not go to zero, unlike te flat plate wit semi-infinite fluid beside it. Terefore te momentum integral equation across te cannel will be seen to depend on te values of te sear stress and viscous stress at te center of cannel. Tis work reconsiders te George & Capp 1) similarity solution metodology and sows teir result to be inconsistent wit te momentum integral in natural convection cannel flow. Te proper inner scaling velocity is in fact determined by outer parameters and te momentum integral to be u i = gβf o ) 1/3. Since te inner and outer velocity profiles are now scaled by te same parameter, te velocity profile in buoyant sublayer must be logaritmic. Te temperature and eat transfer laws are te same as before. GOVERNING EQUATIONS An infinite vertical cannel wit differentially eated walls is sown in figure 1. Te distance between te two walls is 2 Figure 1. SCHEMATIC OF FULLY DEVELOPED TURBULENT BOUND- ARY LAYER INSIDE AN INFINITE VERTICAL CHANNEL WITH DIFFER- ENTIALLY HEATED WALLS. and te temperature difference is T w. Te kinematic viscosity ν, te termal diffusivity α and te termal expansion coefficient β are te fluid properties and considered to be constant. A uniform and steady eat flux is applied across te walls. Assuming te flow to be omogeneous in te streamwise direction, x, and te cannel to be of infinite extent, bot te wall temperature and wall eat flux must independent of x. Te Boussinesq approximation will be assumed to be valid, so density differences can be neglected except as tey affect te gravitational term. Furter, te temperature difference between te two sides is considered to be small enoug tat we can assume at least for te sake of argument ere) tat density and temperature differences are proportional; i.e., ρ/ρ o β T. We are erein interested only in fully-developed cannel flow, so te streamwise gradient of all properties is assumed to be identically zero, so tat all te mean convection terms become identically zero. Also te velocity and temperature profiles are antisymmetric across te cannel, wic gives te boundary condition of zero values at te centerline of te cannel. By contrast, te derivatives of tese profiles and Reynolds stresses are symmetrical wit respect to te center-line. Also te y-momentum equation of te flow can be used to eliminate te pressure in te x-momentum equation. For te limit of very ig Reynolds number, all tese assumptions reduce te equations of motion and energy for entire cannel to: 0 = y ν U ) y uv + gβt T re f ) 1) 2 Copyrigt c 2008 by ASME

3 and 0 = y α T ) y vt Capital letters refer to mean values wile lower case letters are used for fluctuating quantities. Tese equations are valid bot for near wall and te core region of te flow, but te viscous terms and conduction terms in te core or outer) region of te flow ave less effect and no effect at all in te limit of infinite Reynolds number). Te energy or temperature) equation 2 can be integrated across te cannel widt from wall to center-line) to yield 2) vt + α T y = q w ρc p F 0 3) were q w is te wall eat flux and C p is te specific eat at constant pressure. Equation 3 sows tat te entire cannel boundary layer to be a constant eat flux layer and te total kinematic eat flux, F o across te layer is independent of distance from te wall c.f., George and Capp 1)). Te momentum equation 1 can be similarly integrated across te cannel to give momentum integral equation as: unlike forced flow in cannels or pipes, it would appear tat te wall sear stress cannot be imposed as a constant parameter at all distances from te wall. Note tat in general, all of te values at te left-and side of te equation are constant i.e., x-independent and non-zero). Tus we can define: Φ 2 CL = ν U y uv 8) y= y= Altoug we migt expect te viscous term to be negligible in te limit of infinite Reynolds or Peclet) number, te Reynolds stress terms is most probably not due to te possible presence of igly correlated motions or coerent structures) tat span te cannel. Tis undoubtedly accounts for te observations of Betts & Bokari 2) and also numerical simulations of te natural convection cannel flow 3) were te centerline Reynolds stress is quite significant. It also represents an important difference from te natural convection boundary layer next to a vertical surface in wic te Reynolds stress vanises at large distances from te wall. Recognition of tis difference is part of te purpose of tis paper. And it raises interesting questions about wy some recent LES simulations of low aspect ratio cannels seem to suggest tat te Reynolds sear stress vanises even in cannels e.g., 6)). uv + ν U y + y 0 gβt T re f )dy = ν U y uv 4) y=0 y=0 We can use te wall sear stress, τ w, to define te friction velocity, u by: u 2 = τ w /ρ = ν U y 5) y=0 Also, te kinematic boundary condition at te wall implies: uv = 0. 6) y=0 Using tese conditions and coosing te reference point for te temperature to be te centerline, say T CL, equation 4 can be rewritten as: u 2 ν U ) uv = y y= y= 0 gβt T CL )dy 7) Clearly as first noted by George and Capp 1), unlike forced flow, natural convection flow does not ave a constant stress layer because of te buoyancy term in te momentum equation. Tus INNER AND OUTER SCALING Te Temperature Profile George and Capp 1) argued tat natural convection cannel flow at ig turbulence Reynolds numbers sould in fact be viewed as a classical inner-outer problem, wit viscosity and termal conductivity dominating te inner or near wall) layer but aving no effect on te core region. A direct consequence of tis is tat te viscous and conduction terms in te equations above are negligible in te core region. By contrast, te combination gβ, and te kinematic eat flux, F o, were important everywere. Tus te very near wall region could be completely caracterized by gβ, F o, α and ν. Note tat equation of te near wall region do not really know about te cannel widt,, nor do tey know te net temperature difference T w = T w T cl since at least a small part of it occurs across te outer layer) 1 Te core region in te limit of infinite Reynolds number, by contrast, knows only about te cannel widt,, gβ, and 1 Te coice of inner scales is actually quite problematical. It can be argued tat T w sould be included, since it occurs explicitly in te momentum equation at te wall. Or it migt be argued tat since te temperature and eat flux determine a lengt scale, tese sould be included and gβ ignored c.f. Wosnik and George 7), Wosnik et al. 8). Unfortunately tis does not seem to lead to te expected eat transfer law, peraps because T w and F o are not independent parameters, one being determined if te oter is specified. Hence we ave opted erein for te original George & Capp 1) formulation. 3 Copyrigt c 2008 by ASME

4 F o, te latter because of te constancy of te kinematic eat flux across te flow. It followed immediately on dimensional grounds tat te only coice for an outer temperature scale was T o = gβf o ) 1/3. And on similar grounds, te inner temperature scale was cosen to be T i = Fo 3/4 /gβα) 1/4 altoug ν could ave been used instead of α, implying a residual Prandtl number dependence of te inner layer). Te obvious coice of lengt scale for te core region was te cannel alf-widt,, wile te inner lengt scale was cosen on dimensional grounds as η It = [α 3 /gβf o )] 1/4. Note tat te latter could ave equally been cosen as η Iv = [ν 3 /gβf o )] 1/4, but α was preferred on experimental grounds since it varies wit ν in gases, and as significantly less dependence on temperature in liquids. A major contribution of George & Capp 1) was te recognition tat tere must exist a layer in between te inner and outer layers, te so-called buoyant sublayer, in wic bot te termal diffusivity and kinematic viscosity could be ignored, as well as te outer lengt scale,. It follows on dimensional grounds alone tat dt /dy [Fo 2/3 /gβ) 1/3 ]y 4/3. Tis layer was presumed to lie between te molecular viscosity/diffusivity dominated region very close to te wall te viscous and conductive sublayers respectively) and te peak in te mean velocity profile. Integration across te inner layer leads immediately to two equivalent forms for te temperature in te buoyant sublayer, one in inner variables and one in outer variables, bot of wic are equally valid if te wole idea of te overlap region is sound. For te inner profile te result is: [ ] T T w y 1/3 = K 2 + APr) 9) T It η It were te Prandtl number dependent additive constant reflects te Prandtl number dependence of te inner layer. By contrast, integration from te centerline yields te same profile in outer variables as: η It = η Iv = [ gβfo 4 α 3 [ gβfo 4 were H and G are defined respectively by: ν 3 ] 1/4 = H 1/4 11) ] 1/4 = Gr 1/4 12) H = gβf o 4 α 3 13) Gr = gβf o 4 ν 3 14) Tus unlike forced boundary layers were te local Reynolds number δ + = u δ/ν is te ratio of lengt scales, ere it is te 1/4-root of H or te corresponding Raleig, Ra or Grasof, Gr, numbers). Many ave concluded tat values of H of order 10 6 are sufficient to acieve an asymptotic state, but in fact it is clear from equations 11 and 12 tat muc iger values of tese are necessary to acieve even a modest separation of scales. E.g., typically /η It or /η Iv greater tan 10 would be considered to be an absolute minimum for any asymptotic teory to even begin to be valid.) Unfortunately tere is very little experimental data and no DNS data satisfying tese conditions; in fact, recent attempts ave typically been /η It < 10 and sometimes even as low as 3-4 e.g., refs. 2; 3)). So in spite of te advances of experimental and computational tecniques over te past few decades, tere as been to te best of our knowledge) virtually noting to contribute to our understanding of a ig H- number buoyant sublayer, or te applicability of te teoretical deductions from it. Te Heat Transfer Law An immediate consequence from matcing equations 9 and 10 is tat te asymptotic eat transfer law given by: T T CL T o = K 2 [ y ] 1/3 + A1 10) were A 1 is expected to be universal, or at most dependent on te nature of te core flow i.e., coerent versus incoerent motions). Suc a buoyant sublayer region can be expected to exist only wen tere exists a region for wic y << and simultaneously y >> η It and y >> η Iv. Tus a necessary condition for te existence of a buoyant sublayer is tat te ratio of inner to outer lengt scales be muc greater tan unity; i.e., H 1/4 >> 1 since: Gr 1/4 >> 1 and Nu 1 H 1/4 = APr) + A 1 H 1/12 15) So te corresponding eat transfer law is given by: Nu = APr)H 1/4 + A 1 H 1/6 16) Clearly APr) < 0 is te only pysically realistic possibility. In te limit as H, te first term dominates so te asymptotic eat transfer law is given by: 4 Copyrigt c 2008 by ASME

5 Nu = APr)H 1/4 17) But tis sould only be expected for very large H ; i.e., wen H 1/6 >> 1 H > 10 8 or greater), because of te 1/6t root dependence of te additive term. Note tat te eat transfer law could equally well be rewritten in te more familiar form as: Nu = [ APr)] 4/3 H 1/3 18) were H is defined using te temperature difference T w as: H = gβ T w 3 α 2 19) Te eat transfer result of George & Capp 1) was particularly significant, since it accorded wit te long-standing observation of Arpaci and oters tat te eat transfer in cannels appeared to be independent of te cannel widt Note te same factor of on bot sides of equations 18 and 19.) Te sligtly varying values of te coefficient can probably be attributed to te absence of data at ig enoug values of H, and fitting te eat transfer law witout te extra H 1/6 term. Surprisingly tere seems to ave been little effort over te past few decades to relate te parameters in tese equations to te actual temperature profiles as te teory suggests. Peraps tis is because of te concentration of effort on relatively low H-number flows. THE MEAN VELOCITY PROFILE Te Problematical George & Capp Inner Scaling Te second major point of tis paper is tat te George & Capp 1) analysis of te mean velocity profile for te inner region is incorrect. Wile te results for te temperature and eat transfer laws above ave been reasonably well-received by te researc community recent low H-number experiments and DNS notwitstanding), te George & Capp 1) results for te mean velocity profile ave always been problematical. First te mean velocity near te wall does not seem to scale very universally wit teir proposed inner velocity scale, gβf o α) 1/4. Second te predicted cube root region corresponding to te buoyant sublayer does not seem to be te best fit to te data. First let us note tat almost all recent investigations including DNS and LES) conclude tat it is te friction velocity, u, and te corresponding viscous lengt scale, ν/u, wic correctly scales te velocity profile closest to te wall. In fact, tis is not a conclusion but a necessity. Since u is defined from te wall sear stress, wic is turn proportional to te mean velocity gradient at te wall, te only way scaling te mean velocity profile tere wit u and ν would NOT work is if te experiments or computations were incorrect. 2 So if scaling by u and ν is not te rigt question, wat is? Te real problem is ow to relate tese parameters to te remaining parameters of te flow! In oter words, wat is te friction law in terms of te boundary conditions and parameters of te problem? Clearly te friction law proposed by George & Capp 1) is incorrect, at least based on te data, and in fact in principle as well. Tis can be demonstrated using te momentum integral equation 7 as sown below. First split te temperature integral in te buoyant sublayer in inner variables at y/η It = A so it can be written in two parts as: u 2 ν U ) A uv = gβt It η It y y= y= + gβt o 0 1 Aη/ T T w d y T It η It gβ T T CL d y T o 20) In te limit as H, te two integrals are simply numbers, say I and II. Moreover, from te definitions above it follows tat T It η It = gβf o α) 1/2 = u 2 I and T o = gβf o ) 2/3 = u 2 o, wic are in fact just te squares of te inner and outer velocity scales proposed by George & Capp 1). Tus te integral of equation 20 can be rewritten as: u 2 ν U ) uv = I u 2 IGC + II u 2 ogc 21) y y= y= Dividing by u 2 IGC and using te definitions yields te friction law as: u 2 u 2 IGC 1 u 2 ν U ) { } uv = I + II H 1/6 IGC y y= y= 22) But te rigt-and side blows up in te limit as H. Tus regardless of te problems presented by te second term on te rigt-and-side, te George & Capp 1) coice of an inner scaling velocity is clearly not correct. In particular te dominant 2 Te same observation can be made for te mean temperature profile near te wall as well: it must by definition collapse wen normalized using T w if te lengt scale is defined as α T w /F o. So collapse in tese variables only implies tat te experimental measurements or computations are correct in tis region. 5 Copyrigt c 2008 by ASME

6 contribution to momentum integral is not near te wall, but far from te wall. Hence te velocity near te wall is not a consequence primarily of te local buoyancy tere, but is in fact largely a flow driven from te outside by te effects of buoyancy tere. Tis is actually somewat counterintuitive, because almost all of te temperature profile occurs near te wall; but it is te residual outer temperature profile tat dominates te integral because of te muc greater distance over wic it must be integrated. 3 Te Friction Law In fact it appears tat te proper coice for an inner scaling velocity parameter sould ave been te outer scaling velocity, at least in te limit as H and including te extra terms from te centerline. To see tis, divide equation 21 by u 2 o = gβf o ) 2/3 to obtain: u 2 u 2 1 o u 2 ν U ) { uv = o y y= y= I H 1/6 } + II 23) It is easy to see tat tis indeed slowly approaces a constant, II, in te limit as H. Tus, as noted above te friction is primarily determined by te outer velocity scaling, and te inner velocity would be properly cosen proportional to it. Implications for Te Velocity Profile In Te Buoyant Sublayer Tus it appears tat at least in te limit as H, te inner and outer velocity scales are te same; i.e., u gβf o ) 1/3. It is well-known c.f., George and Castillo 9)) tat if te inner and outer velocity scales are te same, an immediate consequence is tat te mean velocity profile in te overlap region must be logaritmic. Tus an immediate and somewat surprising) conclusion of te above is tat te mean velocity profile in te overlap region sould vary logaritmically. Tus in inner variables: U = K 1 ln yu + BPr) 24) u ν were B 1 sould be universal only if te coerent structures of te core region are te same. Clearly tis sould be a matter for investigation. Note tat it migt be tempting to associate K 1 wit te usual von Karman parameter, 1/κ. As noted by George 10), tere is really no reason to believe tis parameter to be te same for different flows, and most certainly not for tis one for wic te constant stress layer does not exist. SUMMARY AND CONCLUSION A new teory as been proposed for te velocity profiles of a turbulent natural convection boundary layer in a differentially eated vertical cannel. Te mean velocity and temperature scaling parameters for inner and outer region of te flow ave been derived using local similarity solutions at te limit of infinite local H-number. Te inner and outer scaling parameters for te mean temperature profile is te same as suggested by George & Capp 1), as is te outer scale for te mean velocity. Te inner velocity scale and friction law, owever, are quite different. In particular bot te inner and outer velocity scales are te same in te limit; and given in te limit of infinite H by u gβf o ) 1/3. An immediate consequence is tat te mean velocity profile in te buoyant sublayer is logaritmic. Te sear stress is primarily determined by te buoyancy integral and te value of te total stress at te centerline, neiter of wic can be assumed to be zero. Te latter represents a significant difference between cannel flow and te natural convection boundary layer next to a vertical surface, and opens te possibility for te dependence of te flow on te nature of te turbulence in te core region; i.e., coerent structures versus incoerent motions. ACKNOWLEDGMENT Te autors would like to acknowledge te support of Vetenskapsrådet te Swedis Researc Foundation). In addition, WKG would like to express is gratitude to Professor Martin Wosnik of te University of New Hampsire for many elpful discussions over te past 15 years. were BPr) may vary from fluid to fluid. In outer variables, te corresponding mean velocity profile would be: U gβf o ) 1/3 = u { gβf o ) 1/3 K 1 ln y } + B 1 25) 3 Interestingly, te George & Capp 1) mistake is similar to tat often made for forced boundary layers were it is argued tat most of te energy dissipation is near te wall, wen in fact te opposite is true. Even toug indeed te peak in te dissipation is very near te wall, te integral of te dissipation is dominated by te overlap region and outer flow. REFERENCES [1] George, W. K. J., and Capp, S. P., A teory for natural convection turbulent boundary layers next to eated vertical surfaces.. International Journal of Heat and Mass Transfer, 226), June, pp [2] Betts, P., and Bokari, I., Experiments on turbulent natural convection in an enclosed tall cavity. International Journal of Heat and Fluid Flow, 216), pp [3] Versteeg, T., and Nieuwstadt, F., Turbulent budgets of natural convection in an infinite, differentially eated, 6 Copyrigt c 2008 by ASME

7 vertical cannel. International Journal of Heat and Fluid Flow, 192), pp [4] Versteeg, T., and Nieuwstadt, F., A direct numerical simulation of natural convection between two infinite vertical differentially eated walls scaling laws and wall functions. International Journal of Heat and Mass Transfer, 4219), pp [5] Ince, N., and Launder, B., On te computation of buoyancy-driven turbulent flows in rectangular enclosures. International Journal of Heat and Fluid Flow, 102), pp [6] Baragi, D., and Davidson, Natural convention boundary layer in a 5:1 cavity. Pysics of Fluids, 1912), p [7] Wosnik, M., and George, W. K., Anoter look at te turbulent natural convection boundary layer next to eated vertical surfaces. In Int. Symp. of Turb. Heat and Mass Transfer, Vol. 1, pp [8] Wosnik, M., Castillo, L., and George, W., A teory for turbulent pipe and cannel flows. Journal of Fluid Mecanics, 421, pp [9] George, W. K., and Castillo, L., Zero-pressuregradient turbulent boundary layer. Applied Mecanics Reviews, 5012 pt 1), pp [10] George, W. K., Is tere a universal log law for turbulent wall-bounded flows?. Pilosopical Transactions of te Royal Society London, Series A Matematical, Pysical and Engineering Sciences), ), p Copyrigt c 2008 by ASME