Similarity of the emperature Profile formed by Fluid Flow along a Wall David Weyburne 1 AFRL/RYDH 2241 Avionic Circle WPAFB OH 45433 ABSRAC A new approach to the tudy of imilarity of temperature profile i preented. It i applicable for any 2-D fluid flow along an iothermal heated (cooled) wall. he approach i baed on a imple concept; the area under a et of caled temperature profile curve that how imilar behavior mut be equal. hi lead to a new integralbaed definition of temperature profile imilarity. By taking imple area integral of the caled temperature profile and it firt derivative we alo obtain a number of new reult pertaining to imilarity of the temperature profile. For example it i hown that if imilarity exit then: 1) the imilarity temperature and length caling parameter are interdependent 2) the thermal diplacement thickne mut be a imilar length caling parameter and 3) the temperature caling parameter mut be proportional to the free tream minu wall temperature value. 1 David.Weyburne@gmail.com 1
1. INRODUCION Prandtl introduced the concept a boundary layer for fluid flow pat a olid over a hundred year ago [1]. he boundary layer concept for flow over a heated flat plate i depicted in Fig. 1. A the flowing fluid impinge on the flat plate a boundary layer i formed along the plate uch that the velocity and temperature at the wall gradually tranition to the bulk value. Reynold [2] work on dimenional analyi in the 18 led to the concept of profile imilarity (alo called elf-imilarity). hi i the ituation in which for certain fluid flow the downtream velocity or temperature profile along the wall i a imple tretched verion of the uptream profile. For 2-D wall-bounded flow like that depicted in Fig. 1 temperature imilarity i defined a the cae where two temperature profile curve from different tation along the wall in the flow direction differ only by a caling parameter in y and a caling parameter of the temperature profile ( x y ). he temperature profile in thi cae i defined a the temperature taken at all y value tarting from the wall moving outward at a fixed x value. o put thi in formal mathematical term we firt define the length caling parameter a and the temperature caling parameter a. he temperature profile ( x 1 y ) at poition x 1 along the wall will be imilar to the temperature profile ( x2 y ) at poition x 2 if x1 y / x1 x2 y / x2 for all y. (1) An important obervation about the imilarity definition i that in the limit of y the ratio become x y / x w lim. (2) y Since w i a contant (iothermal plate) then imilarity require that mut alo be a contant. hi mean that only the length caling parameter may vary with the flow direction (x-direction). raditionally imilarity of the temperature profile involved finding olution to the flow governing equation. Schlighting [3] for example review ome of the laminar flow temperature profile imilarity olution obtained by thi approach. More recently Wang Catillo and Araya [4] and Saha et al. [5] among other have dicued thi approach for the forced turbulent boundary layer flow cae. Recently Weyburne [6] introduced a new approach for tudying imilarity of the velocity profile for 2-D flow along a wall. In the work herein we apply thi ame equal area integral approach to tudy the temperature profile imilarity formed by 2-D flow over a heated (cooled) iothermal wall. he approach i baed directly on Eq. 1. We tart by manipulating the imilarity definition equation and then integrating the reult. For example in the Section below we differentiate both ide of Eq. 1 with repect to the caled y-coordinate and then integrate the reult from the wall to a poition deep into the freetream. hi area integral i eaily manipulated and can be olved explicitly a a ratio of parameter. Similarity necearily require that the area under the curve mut be equal which then impoe an equivalence condition on the parameter ratio. Since the reult are derived
directly from the definition of imilar curve given by Eq. 1 we therefore do not need to know how the curve were generated. Hence the flow governing equation do not have to be invoked. Furthermore the new reult apply to imilar profile curve whether they are laminar tranitional or turbulent fluid flow a long a the temperature i taken a the Reynold averaged temperature. Experimental laminar and turbulent dataet are examined in order to verify the new reult. For the laminar flow cae experimental wind tunnel reult are not available o we turn to the theoretical olution offered by Pohlhauen [7]. For the turbulent boundary layer flow we ue the tabulated data from Blackwell Kay and Moffat [8] and R. du Puit C. Reagk and A. he [9]. 2. Similarity of the emperature Profile 2.1 he Firt Derivative Profile Now we turn to the tak of conidering what can be learned from thi equal area approach to imilarity for 2-D flow along a heated (cooled) iothermal wall. In the analyi below the only aumption that are neceary a to the functional form of the temperature profile i that the form i conitent with a phyically realizable form i.e. no dicontinuitie or ingularitie. he development preented herein i baed on a imple concept; for imilarity the area under the caled temperature profile curve plotted veru the caled y-coordinate mut be equal at each tation along the wall. Rather than tarting directly with the conideration of the temperature profile we will firt conider the implication of imilarity on the firt derivative profile curve ince thi reult i a imple illutration of the new approach and it reult i required later on. We tart with the formal definition of imilarity given by Eq. 1. If imilarity i preent in a et of temperature profile then it i elf-evident that the properly caled firt derivative profile curve (derivative with repect to the caled y-coordinate) mut alo be imilar. It i alo elf-evident that the area under the caled firt derivative profile plotted againt the caled y-coordinate mut be equal for imilarity. In mathematical term the area under the caled firt derivative profile curve i given by h/ y d x y / a( x) d (3) y d where ax ( ) i in general a non-zero numerical contant. For clarity hx ( ) and ( ) have been hortened to h and. For equal area at each meaurement tation it i required that the integral limit condition h( x1) ( x1) h( x2) ( x2) mut hold. However thi requirement i eaily atified ince we can freely chooe hx ( 1) and hx ( 2) a long a they are both located deep into the free tream above the boundary layer edge. Uing the boundary condition ( x) w and ( x h) (where i the free-tream temperature above the boundary layer edge and w i the wall temperature) and a imple variable witch hown to reduce to / 1/ d y dy x Eq. 3 can be 3
h/ y d x y / a( x) d y d a( x) a( x) 1 1 a( x) ( x y) h dy h d ( x y) dy dy d ( x y) dy yh w ax ( ). Similarity require that a( x1) a( x2) contant. he implication of thi imilarity requirement are obviou; the imilarity caling parameter mut be proportional to the temperature difference w. 2.2 Similarity of the Defect Profile Before we turn our attention to the caled temperature profile we firt need to point out that in the limit of y going deep into the free tream Eq. 1 require x1 y / x1 x2 y / x2 lim lim contant. (5) yh( x1) yh( x2) With thi identity we are now in a poition to turn our attention to the caled temperature profile. Starting with the formal definition of imilarity given by Eq. 1 then it i elf-evident that for the profile to be imilar the area under thee caled temperature profile plotted veru the caled y-coordinate mut be equal. he area under the caled temperature profile i not eaily manipulated a i done above. However we can ue the reult from the lat Section to advantage. If we add or ubtract a contant to both ide of Eq. 1 and then integrate the equivalence condition till hold. he Eq. 5 implication i that / mut be a contant if imilarity i preent. Subtracting thi contant value from both ide of Eq. 1 and integrating the area in mathematical term i given by h/ y x y / c( x) d (6) where c ( x ) i in general a nonzero numerical contant. Uing a imple variable witch and imple algebra Eq. 6 can be hown to reduce to (4) 4
h/ y x y / c( x) d 1 ( x y) c( x) dy w ( x y) c( x) dy c ( x) h h w ( w ) where the i the thermal diplacement thickne given by h ( x y) ( x) dy and where h i located deep in the free tream. Eq. 9 i the direct analog of the velocity profile diplacement thickne and i in fact the area under the caled defect profile. Eq. 7 i an exact reult whether the profile are imilar or not. Similarity require that c ( x ) c ( x ) contant. he importance of Eq. 7 in regard to imilar profile i that it mean 1 2 that the thickne caling factor and temperature caling factor are not independent for 2-D wall-bounded imilarity flow. Uing the reult given by Eq. 7 combined with the reult given by Eq. 4 then it i evident that c( x ) reduce to () x (9) c( x). () x Similarity require that c( x1) c( x2) contant. hi equivalence condition on Eq. 9 mean that if imilarity i preent in a et of temperature profile then the thermal diplacement thickne mut be a length cale that reult in imilarity. 2.3 Integral Definition of Similarity Having equal c( x ) value (Eq. 6) at different tation along the flow i a neceary but not a ufficient condition for imilarity of a et of profile. If the caled temperature profile are imilar then it i elf-evident that the caled temperature profile multiplied by the caled y - coordinate raied to the nth power mut alo be imilar. In mathematical term the area under the caled temperature profile multiplied by the caled y -coordinate raied to the nth power i given by h/ y y x y / (1) cn( x) d where c n are in general non-zero numerical contant. Mathematically it i elf-evident that a ufficient condition for imilarity of a et of profile i that w n (7) (8) 5
cn( x1) cn( x2) for n=123... o long a h i choen appropriately and i located deep in the free tream. 3. Experimental Verification 3.1 Laminar Flow he experimental verification ection i not intended to be an exhautive comparion to variou experimental reult. he above theoretical reult are exact and unimpeachable. So intead thi ection i imply intended to demontrate that there i experimental upport for the theory. For the laminar boundary layer we were not able to find experimental wind tunnel data to tet the new imilarity caling parameter o intead we turned to the imilarity olution to the flow governing equation. For the laminar flow cae the Pohlhauen [7] theoretical olution method for laminar flow on a heated iothermal flat wall i an appropriate firt tep. In Fig. 2a the reult for the Pohlhauen-baed approach to calculating thermal profile for a range of Prandtl number i preented. hi figure i a re-creation of Fig. 12.9 from Schlichting [3] uing a imple FORRAN program to generate the olution. In thi figure the reduced temperature defined a ( ) (13) ( ) w i plotted againt the reduced Blaiu height given by u (14) y x where i the kinematic vicoity. Now conider Fig. 2b. Although not readily apparent all nine curve have collaped on top of one another. It i the ame data plotted in Fig. 2a but uing the imilarity length cale parameter ; the thermal diplacement thickne (Eq. 8). Note that the Phohlhauen olution method uing the reduced temperature ha already fixed the temperature imilarity caling parameter a ( ). 3.2 urbulent Flow w For the turbulent boundary layer the number of available thermal turbulent boundary layer dataet i limited. One of the mot extenive available et i from Blackwell Kay and Moffat [8]. In that tudy an experimental invetigation of the heat tranfer behavior of the near-equilibrium tranpired turbulent boundary layer with advere preure gradient wa carried out. In particular advere preure gradient of the form u ( ) ~ m e x x m = -.15 and -.2 were conidered along with a variety of tranpired condition ( u ( x ) i the velocity at the boundary layer edge). he data which cloely matche the condition depicted in Fig. 1 i the non-tranpired near-equilibrium 91871 12471 and 11871 dataet. he data preented in Fig. 3a repreent the lat ix out nine curve from the m = dataet the lat even out of nine curve from the m = -.15 dataet and the lat three of nine curve from the m = -.2 dataet. In Fig. 3b the ame data i preented a defect profile. It i evident that both the caled temperature profile and the caled defect profile diplay imilarity-like behavior. hee two plot indicate that neither the caled temperature nor the caled defect e (11) 6
profile collape to ome univeral function of the caled height a i ometime claimed in the literature (ee for example [4]). Finally R. du Puit C. Reagk and A. he [9] have made highly reolved temperature meaurement in turbulent Rayleigh-Bénard convection in air at a Prandtl number Pr =.7. Although not technically forced flow the reulting gravity induced recirculation cloely match the forced flow condition at certain location. In Fig. 4 we plot the reduced temperature veru the caled height for one of their dataet that matche thi condition. We note that the reduced temperature data wa corrected for the fact that the a reported data never aymptote to the bulk temperature. hi correction wa done by imply recaling the reported data o that it pan to 1. We note that R. du Puit C. Reagk and A. he [9] did in fact report that the caled temperature howed imilar behavior when caled by the thermal diplacement thickne. he theoretical reult herein confirm their experimental obervation. 4. Dicuion An important point about the above derivation i that although the reult are not preented formally the reult above are mathematically rigorou and can be eaily ubtantiated in the form of mathematical proof. he above reult apply to any 2-D fluid flowing along a heated (cooled) iothermal wall including all laminar flow tranitional flow or turbulent fluid flow a long a the temperature i taken a the Reynold averaged temperature. he ignificance of the equivalence condition on Eq. 4 i that it indicate that if imilarity exit in a et of temperature profile then the appropriate temperature caling parameter for 2-D wall-bounded flow along an iothermal wall i w. In addition the equivalence condition on Eq. 9 indicate that if imilarity exit in a et of temperature profile then the thermal diplacement thickne mut be a imilarity length cale factor. o our knowledge thi i the firt time the thermal diplacement thickne ha been theoretically identified a a imilarity length caling parameter in the literature. he experimental verification preented above i intended to imply demontrate that the new imilarity caling do work for certain data et. he reult for the laminar flow cae are epecially ignificant in that Pholhauen [7] laminar flow work appear in every major textbook that deal with heat and fluid flow. he collape of the figure to one curve (Fig. 2b) uing a the tretching parameter i dramatic. hi the firt time the thermal diplacement ha been even conidered a a imilarity caling parameter to our knowledge. he reult herein prove that that mut be the cae for imilarity to be preent in any 2-D wall-bounded flow. Although thee reult apply to any flow howing imilar behavior the cae of the turbulent boundary layer i intereting. Prior to the work herein the theoretical approach to imilarity for turbulent boundary flow ha been hampered by the fact that the flow governing equation do not have a cloed form for turbulent flow. Hence cloed form mathematical olution do not exit a i the cae for laminar flow. herefore what i uually done i to make imple educated guee for what might be the proper imilarity caling and then compare thoe 7
guee to experimental reult. hi i the path taken by Wang Catillo and Araya [4] and Saha et al. [5] among other. We have not attempted to compare the reult herein to thee previou guee preciely becaue the reult herein are mathematically derived not gueed. Mathematically Eq. 1 i the traditional definition of temperature profile imilarity but a mathematically equivalent definition i to require the condition given by Eq. 11 exit for all n. For ue on experimental dataet thi latter method would eem to have the untenable requirement of calculating a very large (infinite) et of integral. However for flow imilarity of temperature profile taken at variou tation along the wall in the flow direction the profile curve are not arbitrary and in general are not changing rapidly. Hence it may only be neceary to enure that the firt couple of c( x1) c( x2) value are contant a oppoed to the infinite et. In any cae thi approach to imilarity ha an advantage from an experimental tandpoint ince the equal area tet method would allow for tatitical teting for imilarity by comparing cn( x ) value at variou tation along the wall. From a practical tandpoint thi method i uperior to the ue of Eq. 1 ince to ue Eq. 1 the experimentalit need to inure the meaured temperature at each y/ ( x) value i equal at both meaurement tation. hi i a very difficult tak ince x i changing with x and in general it value i not known a priori. A a reult of the y/ ( x) iue the uual imprecie method the flow community preently ue to judge whether a et of experimental temperature profile are imilar i to plot the profile and ue the ubjective chi-by-eye method to judge the ucce or failure (do they plot on top of one another?). In contrat the equal area tet method would allow for tatitical teting by performing imple numerical integral of the profile data. 5. Concluion A new approach for tudying imilarity of the temperature profile i outlined. It tart from the equation ued to define imilarity of the temperature profile. hi method wa ued to dicover fundamentally new reult for the imilarity of 2-D flow along a heated (cooled) iothermal wall. It wa hown that if imilarity exit then the imilarity temperature and length caling contant cannot be independent. Furthermore it wa hown that if imilarity exit the thermal diplacement thickne mut be a length caling variable and the temperature difference mut be a temperature caling parameter. w ACKNOWLEDMENS hi work wa upported by AFOSR PM Dr. Gernot Pomrenke and the Air Force Reearch Laboratory. he author thank the variou experimentalit for making their dataet available for incluion in thi manucript. Reference [1] L. Prandtl Über Flüigkeitbewegung bei ehr kleiner Reibung Verhandlungen de Dritten Internationalen Mathematiker-Kongree in Heidelberg 194 A. Krazer ed. eubner Leipzig pp. 484 491 195. [2] O. Reynold An experimental invetigation of the circumtance which determine whether the motion of water in parallel channel hall be direct or inuou and of the law of reitance 8
in parallel channel Philoophical ranaction of the Royal Society of London. 174 935-982(1883). [3] H. Schlichting 7th ed. Boundary Layer heory (McGraw-Hill New York ) 1979. [4] X. Wang L. Catillo and G. Araya emperature Scaling and Profile in Forced Convection urbulent Boundary Layer. Journal of Heat ranfer 13 2171 (28). [5] S. Saha J. Klewicki A. Ooi H. Blackburn and. Wei Scaling propertie of the equation for paive calar tranport in wall-bounded turbulent flow International Journal of Heat and Ma ranfer 7 779(214). [6] D. Weyburne Similarity of the Velocity Profile AF echnical Report AFRL-RY-HS-R-21-14 214 (available from www.dic.mil). [7] E. Pohlhauen Der Wärmeautauch zwichen feten Körpern und Flüigkeiten mit kleiner reibung und kleiner Wärmeleitung Z. Angew. Math. Mech. 1 115(1921). [8] B. Blackwell W. Kay and R. Moffat "he urbulent Boundary Layer on a Porou Plate: An Experimental Study of the Heat ranfer Behavior With Advere Preure Gradient" Report No. HM-16 hermocience Diviion Dept. of Mechanical Engineering Stanford Univ. Stanford CA 1972 available a NASA NRS report NASA-CR-13291. [9] R. du Puit C. Reagk and A. he hermal boundary layer in turbulent Rayleigh-Bénard convection at apect ratio between 1 and 9 New J. of Phyic 15 134 (213). Figure Fig. 1: A chematic diagram howing the flat plate 2-D flow geometry and variable. 9
/( w - ) 1. a) 1. b).8.6 Pr =.6.8 1 3 7 15 5 3 1.8.6 Pr =.6.8 1 3 7 15 5 3 1.4.4.2.2...8 1.6 2.4 3.2 4. Fig. 2a: Scaled temperature profile for laminar flow over a plate for a range of Pr number (after Schlichting [3])....8 1.6 2.4 3.2 4. / Fig. 2b: he ame temperature data from Fig. 2a but with the new length cale. All nine curve are collaped onto one another. 4 a) 1 b) ZPG Re = 1534-2971 APG m= -.2 Re = 3659-4533 APG m= -.15 Re = 1545-3734 2 ZPG Re = 1534-2971 APG m= -.2 Re = 3659-4533 APG m= -.15 Re = 1545-3734 5 1 y/ Fig. 3a: Scaled temperature profile for Blackwell Kay and Moffat [5] turbulent boundary layer flow. 5 1 y/ Fig. 3b: he ame caled temperature profile data from Fig. 3a but caled a defect profile. 1
1 Ra = 5e1 Heating Plate =1.13 =1.25 =1.5 =1.75 =2. =2.25 =2.5 =2.75 1 z/ Fig. 4: Eight caled defect temperature profile from R. du Puit C. Reagk and A. he [9] Rayleigh-Bénard turbulent boundary layer flow. 11