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C HAPTER 6 External Turbulent Flows 6.1 INTRODUCTION This chapter is concerned with the prediction o f t he heat transfer rate from the surface o f a body placed i n a large fluid stream when the flow over the surface is partly or entirely turbulent. Some examples o f the type o f situation being considered in this chapter are shown in Fig. 6.1. Attention will mainly be given to turbulent boundary layer flows in this chapter. A brief discussiolLOf the analysis o f flows in. which i t is not possible to use these boundary layer assumptions wiP b e given at the e nd o f the chapter. \ <. 6.2 ANALOGY SOLUTIQNS FOR BOUNDARY LAYER FLOWS As discussed in the previous chapter, most early efforts at trying to theoretically predict heat transfer rates in turbulent flow concentrated o n trying to relate the wall heat transfer rate to the wall shear stress [1],[2],[3],[4]. T he reason for this is that a considerable body o f experimental and semi-theoretical knowledge concerning the shear stress in various flow situations is available and that the mechanism o f heat transfer in turbulent flow is obviously similar to t he mechanism o f m omentum transfer. In the present section an attempt will b e m ade to outline some o f t he simpler such analogy solutions for boundary layer flows, attention mainly being restricted to flow over a flat plate. The simplest analogy solution is that due to Reynolds. This analogy has many limitations but does have some practical usefulness and serves as the basis for more refined analogy solutions. Consider turbulent boundary layer flow over a plate whicb has a local temperature o f Tw. T he flow situation is shown i n Fig. 6.2. 254 CHAPTER 6: External Thrbulent Flows 255 F IGURE 6.1 External turbulent flows. a nd T l a re the free-stream velocity ~d temperature respectively. I f T a nd q are defined as the total shear stress and heat transfer rate respectively, at any point in the flow, then, using the definitions given i n Chapter 5, i t follows that: Ul T = Tm + 7'r = au f.L. - a y - pv ,U , ( 6• 1) and: (6.2) These equations c an b e written as: T = p (v au + e)ay (6.3) and: (6.4) . F IGURE 6.2 Flow situation being considered. / 256 Introduction to Convective Heat Transfer Analysis These equations can be rearranged to give: au ay p (v 'T + e) q (6.5) and: - aT ay = - ----':...--- p cp(a + eH) (6.6) Integrating these two equations outward from the wall to some point in the freestream at a distance of.e from the wall then gives: Ul = I e 'T o p (v + e) d y (6.7) and: -(6.8) Because both e and e H depend on the structure of the turbulence it is to be expected that they will have similar values, a conclusion that is confinned by experiment. I n the present simple analysis it will be assumed, therefore, that they ar~ equal, i.e., that: (6.9) This is, o f course, equivalent to assuming that the turbulent Prandtl number, PrT, is equal to 1. I n the Reynolds analogy, attention is further restricted to the flow of fluids for which: . .. a =v (6.10) i.e., to fluids that have a Prandtl number of 1. No such real fluids exist, but as mentioned earlier, most gas~s have Prandtl numbers which are close to unity. Using the assuIllptions given in Eqs. (6.9) and (6.10), Eq. (6.8) can be rewritten as: Tw - Tl = I e q dy o pCp(v + e ) (6.11) Now both the shear stress, 'T, and the heat transfer rate, q, have their highestvalues at the wall and both are zero in the free-stream. I t will therefore be assumed that the distributions of total shearing stress and heat transfer rate are similar, i.e., that at any point in the boundary layer: -=qw 'Tw / q 'T (6.12) where 'Tw and q w are the values o f the shear stress and the heat transfer rate at the wal!. This assumption is supported by available measurements and intuitive reasoning:/ CHAPTER 6: External Turbulent Flows 257 This assumption, i n conjunction with the previous assumptions, i s e quivalent to assuming that the temperature a nd velocity profiles are similar (see Example 6.1 below). Substituting Eq. (6.12) into Eq. (6.11) then gives: Tw - Tl q =~ 'Tw i° f 'T p Cp(v + e) dy (6.13) Dividing this equation b y E q. (6.7) t hen gives: Tw - Tl Ul qw Cp'Tw (6.14) Introducing x, t he distance along the plate from the leading e dge to the point being considered, then allows this equation to b e r ewritten as: Le.: . Pr Nux = (C; )Rex (6.15) where Nux i s the local Nusselt number, Rex i s t he local Reynolds number, a nd Cf is the local shearing stress coefficient, i.e., 'Tw /(ll2p u Now it was assumed i n t he derivation o fEq. (6.15) that the Prandtl number, Pr, was equal to 1. H ence, the final form o f t he Reynolds analogy is: r)· Nux = (C; )Rex (6.16) Therefore the distribution o f Nux c an b e d etermined from a knowledge o f the distribution o f C f . F or example, experiment has indicated that for moderate values of the Reynolds number [1]: C _ 0.058 fReO.2 x (6.17) Substituting this into Eq. (6.16) then gives: . Nux E XAMPLE 6 .1. = 0.029Re~·8 (6.18) Show that the Reynolds analogy indicates that the velocity and temperature profiles in the boundary layer are similar. Solution. I f Eqs. (6.5) and (6.6) are integrated from the wall out to some arbitrary distance, y, from the waIl where the mean velocity is u and the tnean temperature is T, the following are obtained: u -- i 0 y 'T p (v + e) d Y ( i) / 258 Introduction to Convective Heat Transfer Analysis and: Tw - T = F q dy Jo pcp(a + eH) ( ii) I f the assumptions discussed above, Le.: e H=e,a=v q 7' -=- are introduced, Eq. (iz) above becomes: ( iii) Dividing Eq. ( 0 above by Eq. (6.7) and Eq. (iiO above by Eq. (6.13) then gives: -u _ _ JF p(v +e ) d y o 7' Jo and: ( t 7' p (v + e) d y = F 7' d Jo p (v + e ) y Jo Defining: y rt 7' p (v + e ) d y r dy F (y) = Jo p (v + e ) rt 7' Jo Ii 7' p (v + e) d y it will be seen that the above two equations give: =F Ul and: Tw - T = F Tw - Tl Hence, the mean velocity profile and the mean temperature profile are described by , the same function, F , i.e., the two profiles are "similar". A n expression for the loc~ Nusselt number was derived above using the Reynolds analogy. In order to determine the average Nusselt number for· a plate o f length L, i t must be remembered that the flow near the leading edge o f the plate is laminar as shown in Fig. 6.3. As discussed in the previous chapter, transition from laminar to turbulent flow i n the boundary layer does not occur sharply at a point. Instead there is a region o f "mixed" flow over which the transition occurs, the e~tent o f this region being CHAPTER 6: External Turbulent Flows 259 Laminar Transition Region Turbulent F IGURE 6.3 Transition in a turbulent boundary layer on a fiat plate. influenced b y a n umber o f factors. However, for many purposes, such as the present one o f trying to determine the average heat transfer rate for the whole surface, i t is adequate to assume that the flow is laminar up to some distance, X T, from the leading edge and turbulent from there on, X T being chosen roughly as the distance to the center o f t he transition region. In view o f t he above assumption, i f q w€ is the local rate o f h eat transfer a t any point i n the laminar part o f the flow and q wt is the local rate o f h eat transfer a t any point i n the turbulent part o f the flow, t hen the average rate o f h eat transfer for the entire surface is given by:- qw = nr q wtdx+ t*,t - dX ] (6.19) As discussed i n C hapter 3, because a fluid with a Prandtl number o f 1 is being considered, q w€ i s g iven by: q w€ = 0.332 ;~ ( 0.5 ) k (Tw Tl) (6.20) Therefore, u singthe value o f q wt given b y Eq. (6.18), Eq. (6.19) gives, i f t he surface is isothermal: 7iwL k(Tw - Tl) = , [eXT . 10 O.332(~)0.5 d x + fL xv 0.029 XT xO· 2 (Ul )0.8 dX] V i.e.: (6.21) where ReT i s the Reynolds number based o n XT, ReL i s the Reynolds number based on the plate length, L , a nd NUL i s the m ean N usselt number f or t he entire plate. I f ReT is taken as 106 , a n approximate experimental value as discussed i n Chapter 5, then Eq. (6.21) gives: NUL = 0.036Re~8 - 1600 (6.22) I f ReL i s large, the second term is negligible compared to the first and i n this case the above equation reduces effectively to: (6.23) 260 Introduction to Convective Heat Transfer Analysis x Effective Origin o f Turbulent Boundary Layer F IGURE 6.4 Effective origin o f turbulent boundary layer. which is, o f course, the result that would have been obtained had the flow been assumed to b e turbulent from the l~ading edge. , , I n deriving the 'above equations i t was assumed that after transition: qw x = 0.029 ( U IX,) k(Tw ~ T d " ,v 0.8 , where x is the distance from the leading edge o f the plate. However, the effective origin o f t he turbulent boundary layer after transition is not i n general at the leadfig edge o f t he plate, i.e., as indicated i n Fig. 6.4, in the portion o f the flow with a turbulent boundary layer, the heat transfer rate should have been expressed as: qw(x - xo) = 0.029 [ u 1(X - XO)]0.8 k(Tw - Tl) v where Xo is the distance o f the effective origin o f the turbulent boundary layer from the leading edge o f the plate as shown in Fig. 6.4. The effect o f this displaced origin i s however usually small and will n ot b e considered here. E XAMPLE 6 .2. Air flows over a flat plate which has a uniform surface temperature of 50°C, t he temperature o f the air ahead o f t he plate being 30°C. T he air velocity is such that the Reynolds nu:mber based on the length o f the plate is 5 X 106 , the length o f the plate being 3 0 cm. Using the Reynolds analogy, determine the variation o f the local heat transfer rate,irmn the wall, qw, with x lL assuming that (i) the boundary layer flow remains laminar, (ii) the boundary l ayer flow is turbulent from the leading edge of the plate, and (iii) boundary layer transition occurs at ReT o f 106 • x is the distance from the lea~ng e dge o f the plate and L is the length o f the plate.! , Solution. T he mean air temperature ~s ( 30+ 50)/2 = 40°C. A t this temperature, air has a thermal conductivity o f 0.0271 W/m-C. A t any distance, x, from the leading edge, the local Reynolds number will be given by: , Rex x = - ReL = 5x '06 -X1 L L I f the boundary layer flow remains laminar assuming P r = 1: Nux = 0.332Re~·5 CHAPTER 6: External Turbulent Flows 261 i.e.: qwx . (X 6)°·5 k(Tw - T 1) = 0.332 5 I X 10 i.e.: 0.5 L1 qw i.e.: = 0.332 ( 5 ;; X 106) k(Tw - Td xL q . = 0 .332(5 X H l"t' X 0.0271 X ( 50 - 30) ( if' 013 = 1341 ( if' W /m' ( i) I f the boundary layer flow is turbulent from the leading edge: Nux = 0.029Re~·8 Le.: qwx k(Tw - T 1) = ,. (X 6)°·8 0.029 5 I X 10 Le.: Le.: qw = 0.029(5 X 106) 0X 0.0271 X (50 .8, 30) (X)-0.2 01 I . .3 ( ii) = 11980 (;;TO. 2W/m2 18000 16000 14000 W/ql2 12000 10000 8000 6000 4000 2000 0.0 0.5 x /L 1.0 F IGUREE6.2 / 2 62 Introduction to Convective H eat Transfer Analysis I f boundary layer transition occurs a t ReT o f 106 then transition occurs at: T = 5X x XT 106 106 = 0.2 In this case then i t will b e assumed that: L <0.2: qw qw = 1341 ( L >0.2: x = 11980 ( I .W/m2 I W/m2 -05 ) -0.2 ) ( iii) (iv) T he variations o f qw w ith x lL as given b y Eqs. ( iii) a nd (iv) a re shown i n Fig. E6.2 .. Experiments in air suggest that the form o f the relation for N UL given by the Reynolds analogy is approximately correct, i.e., that it does vary approximately as ReL 0.8. T he coefficient, i.e., 0.036, has, however, not been found to be i n good agreement with experiment. For fluids with Prandtl numbers very different from unity, the agreement is even less satisfactory. Thus, the Reynolds analogy gives results that are reasonably satisfactory for fluids with Prandtl numbers near unity but gives relatively poor results for other fluids. For this reason, other analyses have been developed which utilize the same basic ideas as those used in the derivation of the Reynolds analogy but which try to relax some o f the assumptions made i n its derivation. A discussion o f the simplest o f these extended analogy solutions will now be given. I t was previously mentioned that the turbulent shearing stress and heat transfer rate are very much greater than the molecular values over most o f the turbulent boundary layer flow. I t is only in a region close to the surface, i n which the velocities are very low, thatthe molecular terms become more important than the turbulent terms. The extension o f the Reynolds analogy presently being discussed is based on the assumption that~the flow can, therefore, b e split into two distinct regions [1]. I n t he inner region, which lies ~djacent to the wall, it is assumed that the turbu~ent shearing stress and heat transfer rate are negligible compared to the molecular values, while in the outer region, which covers the rest o f the boundary layer, it is assumed that the molecular sh~aring stress and heat transfer rate are negligible compared to the turbulent valy.es. I n reality there are, o f course, not two such clear and distinct regions. The assUmed flow pattern is shown in Fig. 6.5. Because the molecular components are being neglected i n the outer region, the total shearing stress and heat transfer rate i n this region are given by: , 'T = 'TT = p€ ay au (6.24) and: q = qr = aT - PCpEH ay (6.25) These two equations can now be integrated outward from the outer edge o f t he inner layer, i.e., from y = !l.s, to some point i n the free-stream which lies at a I CHAPTER 6: External Turbulent Flows 263 Temperature Profile --., __.I-~-~=:t1 -)-........."._:::j 1-----"'---/)"~,~) )~. ) ·~:i Inner Region F IGURE 6.5 Taylor-Prandtl model for turbulent boundary layer flow. distance, e, from the wall. This integration gives: U l - Us = I i e 'T l1s pE - dy (6.26) - and: Ts - Tl = I q dy (6.27) l1s P CpEH where Us and T s are the mean velocity and temperature at the outer edge of the inner layer and U l and Tt are, as before, the velocity and temperature in the free-stream. Similar assumptions to those used in the Reynolds analogy are now introduced; i.e., it is assumed in the outer layer that: EH = E, - = qs q 'T 'Ts where 'Ts and q s are the total shearing stress and heat transfer rate at the outer edge of the inner layer. Using these assumptions, Eq. (6.27) can be written: Ts - Tl q = -s 'Ts Ii - 'T l1s dy (6.28) P CpE Dividing this equation by Eq. (6.26) then gives: T s-Tl U t - Us =~ Cp'Ts (6.29) Next consider the inner layer. Because it is ·being assumed that turbulent shearing stress and heat transfer rate are negligible in this region, the total shearing stress and heat transfer rate in this layer are given by: 'T = p v- au ay (6.30) / 264 Introduction to Convective Heat Transfer Analysis and: q = - pcpaay aT (6.31) It is next noted that because the velocities are very low in the inner, i.e., nearwall, region, the convective terms in the boundary layer equations can be neglected in this region, i.e., in this region the momentum and energy equations can be assumed to have the form: .0 aT I.e., = a y (6.32) and: .0 aq I.e., = a y (6.33) These equations show that in the inner region the shearing stress and heat transfer rate can be assumed to be constant, i.e., that in the inner region: . ' T = Tw = Ts (6.34) (6.35) and: Using these in Eqs. (6.30) and (6.31) and integrating across the inner layer then gives: - u s = Us Tw A ~ pv (6.36) and: ~!ls p cpa = T w-Ts (6.37) Dividing these two equations then gives: . Tw - Ts qw v - --- (6.38) Now, in view of Eqs. (6.34) and (6.35), the result for the outer region, given in Eq. (6.29), can be written as: U l - Us cpTw (6.39) The temperature, Ts , can now·be eliminated between Eqs. (6.38) and (6.39) to give: Tw - Tl Ul 1 [1 + : : (Pr - 1)1 (6.40) / CHAPTER 6: External Turbulent Flows 265 I t should b e noted that when Pr = 1, this equation reduces to the Reynolds analogy result given i n Eq. (6.14). IT the distance, x, from the leading edge to the point being considered is again introduced, Eq. (6.40) c an b e written as: Nux = (~ )RexPr [1+ : : (Pr - 1) 1 (6.41) where, as before, Nux i s the local Nusselt number, Rex is the local Reynolds number, and Cf i s the local shearing stress coefficient, i.e.; Twl(1I2pur). This is known as the Taylor-Prandtl analogy. I n order to apply it, the. value o f U s/Ul has to b e k nown and the way in which this can be found will now be discussed. As discussed in Chapter 5, the wall region extends out to a distance from the where: wall that is defined b y the value o f y: y s +_ - (TW)O.5 -tJ.s p V From this, i t follows that: tJ. Hence, using Eq. (6.36), i.e.: Us Ys v s - (Twlp)0.5 Tw + (6.42) = - , Us A pv i t follows that: Us =p ( Tw ) 0.5 Ys + i.e.: -- Ul Us _ (Cf)0.5 + 2 Y s (6.43) Substituting this into Eq. (6.40) t hen gives: Nux = (~ )RexPr [1 + (7 Y;(pr-l)] Using Eq. (6.17), i.e., assuming: r (6.44) C f _ 0.029 2"" - Reo.2 x 266 Introduction to Convective Heat Transfer Analysis allows Eq. (6A4) to b e written as: 0.029Re~·8 P r Nux = [ 1+ 0.17 + Re~.l Ys ( Pr - 1)1 (6.45) I f, as discussed i n C hapter 5, i t i s assumed, based on available measurements, that: Y: then Eq. (6.45) gives: = 12 (6.46) 0.029Re~·8 P r Nux = ;::-----::-....:..:....:..---:; [1 + R:~.l (Pr - 1)] = 0.029Re~·8F (6.47) This equation can b e written as: Nux T hus, compared to the Reynolds analogy there is the following additional factor which is dependent on both the Reynolds a nd Prandtl numbers: F= [1 + R:~l Pr ( Pr - (6.49) 1)] (6.50) t' T he variation o f F with P r for SOIne t ypical values o f Rex i s shown i n Fig. 6:6. I t will b e s een from this figure that for Prandtl numbers near one, the factor F is approximately g iven by: 3r-----~----~-r-----~ 2 F 1 o~----~------~------~ 0.1 1 Pr 10 100 F IGURE 6.6 Variation of function F with Prandtl number. I CHAPI'ER 6: External Turbulent Flows 267 and in this case the Taylor~Prandtl analogy for turbulent boundary layer flow over a flat plate gives: Nux = O.029Re~·8Pr°.4 (6.51) Integrating Eq. (6.51) over the entire plate as was done with the Reynolds analogy equation and assuming that the flow is turbulent from the leading edge of the plate then gives the following expression for the mean Nusselt number: NUL = O.036Re~8 Pr°.4 (6.52) The Taylor-Prandtl analogy indicates that the effect of the Prandtl number can be accounted for by splitting the boundary layer into layers in which certain effects predominate. In the Taylor-Prandtl analogy, two such layers were used, However, as indicated in Chapter 5, a better description of the flow is obtained by considering three layers, i.e., the inner or wall region, the buffer region, and the outer or fully turbulent region. An analysis based on the ~se o f such a three-layer mPdel will now be presented. In such an analysis, it is necessary to make assumptions about the nature of the flow in the buffer region. Here, it will be assumed that the mean velocity distribution in this buffer region is described by a universal and ]mown function. It is convenient to write the shear stress and heat transfer equations~ i.e.: T = au p (v + € )ay (6.3) , .. and: q . -; = - pcp(a + € H) aT ay . (6.4) in terms of thefo1lowing variables which were introduced in Chapter 5: u+ U =u*' + yu* Y =- v (6.53) u* being termed the "f9-~tion velocity". I t is also convenient to qefine: T + = T w- T u* qw/pcp (6.54) In terms o f the variables defined in Eqs. (6.53) and (6.54), Eqs. (6.3) and (6.4) become: ~ Tw = (1 + ~)au+ ay+ V (6.55) and: q . qw = ( P r + PrT 1 €/V) aT+ ay+ (6.56) '. First, consider the inner layer which, as discussed in Chapter 5, is assumed to COver the region 0 s y+ < 5. I n this region it is again assumed that the effects of 268 Introduction to Convective Heat Transfer Analysis the turbulence on the heat transfer rate are negligible and that the heat transfer rate is constant in this region and equal, therefore, to the heat transfer rate at the wall, qw; i.e., it is assumed that in the inner layer: ! L=l,e=O qw Therefore, in the inner region, Eq. (6.56) gives: (6.57) 1 = --Pray+ 1 a T+ (6.58) This can be integrated to give the temperature distribution in the inner layer as: T + = P ry+ (6.59) Applying this at the outer edge o f t he inner layer where y+ = 5 gives: Ts+ = 5 Pr (6.60) where Ts+ is the value o f T + a t the outer edge o f the inner layer. Next, consider the buffer region. As discussed in Chapter 5, this is assumed to cover the region 5 < y+ < 30. I n this region it is again assumed that the shear stress and the heat transfer rate are constant and, in view o f the assumptions made about the inner layer, they are therefore equal to the shear stress a t t he wall, Tw , and to the heat transfer rate at the wall, qw, respectively; i.e., it is assumed that i n the buffer layer: (6.61) qw I t will also be assumed that the mean velocity distribution in the buffer region is given by: Tw .!. = 1, !L = 1 u+ = 5 + 5 In (Y5 ) + au+ 5 =a y+ y+ (6.62) Differentiating Eq. (6.62) gives: (6.63) Substituting this result into Eq. (6.55) and using Eq. (6.61) then gives: 1 (1 + !)~ v y+ = i.e.: - = --1 v 5 e y+ (6.64) Substituting Eq. (6.64) and Eq. (6.61) into Eq. (6.56) then gives for the buffer regIOn: 1 = [~ + ( y+/5 - 1)] a T+ Pr P rT ay+ (6.65) CHAPTER 6: External Turbulent Flows 269 This equation can be integrated across the buffer region to give: (y+ T + - Ts+ = J5 [2- + (;+/5 - 1)] Pr Prr d+ i.e.: T+ - T,+ - 5prrIn[I + :~ (y;)- 1] = 5Prr I n[I + 5 :~] (6.66) Applying Eq. (6.66) across the entire buffer region, i.e., out to y+ = 30, then gives: T : - T,+ = 5Prr In[ 1 + :~ en€ 1] (6.67) where T t is the value of T + at the outer edge of the buffer region. Lastly, consider the outer, fully turbulent region which is assumed to exist when y+ > 30. In this region it is assumed that € » v and € H » v , i.e., that in this region: ---'Tw 'T au+ v ay+ (6.68) and: = -qw P rr ay+ q € lvaT+ (6.69) It is further assumed that the distributions of shear stress and heat transfer rate are similar in the outer region, i.e., that: -='T q (6.70) where 7'b and q b are the values of the shear stress and heat transfer rate at the outer edge of the buffer layer./Because the shear stress and heat transfer rate were assumed to be constant across the inner two layers, it follows that it is being assumed that: 7'b = 'Tw, qb = qw (6.71) Eq. (6.70) can therefore be written as: -='T q Eqs. (6.68) and (6.69) together then give: - - =Prr-y+ - avay+ I.e.: (6.72) / € au+ € lvaT+ 270 Introduction to Convective Heat Transfer Analysis Integrating this equation across the outer layer then gives: ITt+ T 1 d T+ = PrT iU+ du+ 1 ut (6.73) I.e.: T t - Tt = P rT(ut - ut) where T t and u t are the values of T+ and u+ in the free-stream. But = 30 so Eq. (6.62) gives: yt ut; = 5 + 5 1ne50) = 5(1 + 1n6) Substituting this into Eq. (6.73) then gives: T t - Tt = PrT r ut - 5(1 + ln6)] (6.74) Adding Eqs. (6.60), (6.67), and (6.74) then gives: Tt = 5 Pr + 5PrT In [1 + 5 :~ ] + PrT [ ut = 5 Pr + 5PrT In [ 1 + p rlPrT ] - 5(1 + In 6)] (6.75) - 5) 6 + P rT(ut Now: T+ - Tw - T * _ T w - T - qw1pcp u - qw1pcp VP - f !i _ Pr R Nux e xV rc; 2: (6.76) Eq. (6.75) can therefore be written as: Nux f = - ----::---...:...-..::.-::------- Rex j C l2 5+ i.e., since: 5~: I n[l + 5~rlprT] + ~: ( ut - (6.77) 5) Eq. (6.77) gives: Nux R e x j Cf 12 = - ----....,."..-----=-.....:---....,--------. prIPrT ] + PrT ( + 5 PrT 5 Pr In[1+5 6 Pr j Twlpur 1 .. - 5) . (6.78) Le.: CHAPTER 6: External Turbulent Flows 271 I t will b e noted that i f P r = 1 a nd P rr = 1 Eq. (6.78) gives: Nux = C f Rex 2 which is the Reynolds analogy result. I f the turbulent Prandtl number, P rr, is taken as 1 a nd i f flow over a flat plate is considered, C f then being assumed to b e given b y Eq. (6.17), i.e., by: C _ 0.058 f - ReO.2 x Eq. (6.78) gives: (6.79) This equation c an b e written as: Nux = 0.029Re~·8G (6.80) Thus, compared to the Reynolds analogy there is the following additional factor which is dependent on both the Reynolds and P randtl numbers: G =--;===---------~~~ [spr + S in +6spr )- + I Pr e s] (6.81) The variation o f G w ith P r f9r some typical values o f Rex is shown i n Fig. 6.7. 6.------r----~~----_. 4 G 2 O '--____- --l-_ _ _ _ _ _. ..I.-_ _ _ _- - - J 0.1 1 Pr 10 100 F IGURE 6.7 Variation of function G with Prandtl number. / 272 Introduction to Convective Heat Transfer Analysis 6.3 I NTEGRAL EQUATION SOLUTIONS The analogy solutions discussed in the previous section use the value of the wall shear stress to predict the wall heat transfer rate. In the case of flow over a flat plate, this wall shear stress is given by a relatively simple expression. However, in general, the wall shear stress will depend on the pressure gradient and its variation has to be computed for each individual case. One approximate way of determining the shear stress distribution is based on the use of the momentum integral equation that was discussed in Chapter 2 [1],[2],[3],[5]. As shown in Chapter 2 (see Eq. 2.172), this. equation has the form: :X u: (u, - U)UdY] + ~~ u: dx (u, - u)dY ] = ~ (6.82) where 8 is the local boundary layer thick:p.ess, Ul is the local freestream velocity, and 'Tw is the local wall shear stress. Eq. (6.82) is, for the present purposes, conveniently written as: ~ [UI fc5 (1 _ dx Jo Ul u) u Ul d Y] + Ul d Ul [rc5 Jo (1 _Ul)~~ = u 'Tw P (6.83) The following are then defined: 8, = J: (1 - :.) d Y and: 82 (6.84) = r c5 (1 _ Jo Ul Ul u) u d Ul dx dy (6.85) 81 is termed the boundary layer displacement thickness while 82 is termed the boundary layer momentum thickness. I n terms of these quantities, Eq. (6.83) can be written as: - d U182 dx [2 1 + U1 8 1 - = - 'Tw p Le.: d82 ~ 1d - + (2~ 2+ Ul )- -UI U dx Ul d x _ 'Tw -- pUI Le.: d 82 dx + 82 (2 + H)~ dUI Ul d x = Cf 2 (6.86) where: / CHAPrBR 6: External Turbulent Flows 273 Empirical equations for the variations o f t he wall shear stress a nd o f t he " form factor", H , are now introduced. T he following very approximate relations [1] will be used here i n order to illustrate the method: Cf _ Tand: 0.123 X 10-o.678H pur (u 152/v)O.268 Tw _ (6.88) 12 12 ( U 5 )116 52 d H = e5(H-l.4) [ _ (U 5 )116 V' dx v 52~ dUl Ul d x - 0.0135(H - 1.4)] (6.89) T hese two equations effectively constitute the turbulence model. T hey a re solved simultaneously with Eq. (6.86) to give the variation o f T w/ p ur w ith x. T he selected analogy solution equation is then used to give the local heat transfer rate variation. For flow over a flat plate: dUl = 0 dx and in this case Eq. (6.89) shows that H i s a constant and given by: H = 1.4 Therefore, for flow over a flat plate, Eq. (6.88) gives: (6.90) 0.0138 pur = (u152/ V)O.268 Tw (6.91) and Eq. (6.86) beco1l!es: d52 Tw d x = p ur COJllbining t he above two equations then gives: (6.92) d52 0.0138 d x - (u 152/v)O.268· This equation c an b e integrated using 5 2 _ 2_ 51.268 (6.93) = 0 w hen x = 0 to give: 1.268 = 0 .0138 x (Ul/V)O.268 i.e.: - 82 = 0 .0412 (6.94) / (6.95) . 274 Introduction to Convective Heat Transfer Analysis This equation gives values very close to those given by Eq. (6.17), which was used in the discussion of analogy solutions for flow over a flat plate presented in the previous section. The present section is not really concerned with flow over a flat plate. It is instead concerned with flows in which the free-stream velocity is varying with x. The solution in such a case usually has to be obtained by numerically solving the governing equations. For this purpose, it is convenient to introduce the following dimensionless variables: U =~, u.x, X =L' x (6.96) where, u.x, is some representative free-stream velocity, such as that i n the flow upstream of the surface, and L is some characteristic size of the surface. In terms of these variables, Eqs. (6.86), (6.88), and (6.89) become: -+'-l2 d a2 dx A (2 +'c H ) -d U - 1 - _ 'Tw U dX pu~ _ (6.97) pu~ 'Tw 0.123 X 10-o.678H = Re~268(Ua2)O.268 , (6:98) 1l6 Rell~(Ua2) 1I6a 2 d H = e5(H-1.4) [ -Re L (Ua2).1I6 a2 dd X0' 0 135(H UL dX _ U 14)] . (6.99) where: (6.100) These three equations must be simultaneously solved using a suitable numerical method to give the variations of H a nda2 with X. A simple program, TURBINRK, for obtaining such a solution based on the use of the Runge-Kutta method to numeri- ' cally integrate the simultaneous differential equation is available as discussed in the Preface. This program is written in FORTRAN. This program assumes that the surface is isothermal and that the dimensionless free-stream velocity variation can be represented by a third-order polynomial, i.e., by: U = A + B X + CX2 + D X3 (6.101) where A, B, C, and D are known constants. The inputs to the program are then the values of ReL, A , B, C, D , and Pro , The program assumes the flow is turbulent from the leading edge and that 82 == owhen x = O. The program can easily be modified to use a laminar boundary layer equation solution procedure to provide initial conditions for the turbulent boundary layer solution which would then be started at some assumed transition point. . ' / CHAPTER 6: External Turbulent Flows 275 T he program as available uses the basic Reynolds analogy, Le.: Tw - Tl =-- qw to obtain the heat transfer rate from the calculated wall shear stress. T he local heat transfer rate is expressed i n terms o f a local Nusselt number based on x. A ir flows through a large plane duct with isothennal walls. The Reynolds number based on the length of the duct and the inlet air velocity is 107• Assuming that the boundary l ayeds turbulent and thin compared to the size ·of the duct, determine how the local Nusselt number varies with distance along the duct i f the duct cross-sectional area is (i) constant, (ii) varies in such a way that the velocity increases linearly by SO% over the length o f the duct. and ( iii) decreases linearly by SO% over the length of the duct. E XAMPLE 6 .3. Solution. In all three cases: Also because: U=~, Uinlet X =L x .. where L is the length of the duct, it follows that in the three~ cases, the variation o f U with X is given by: . Case (i): U =1 Case (ii): u= .Case ( iii): 1 +O.SX u= U ·= A 1 -0.SX Therefore, because the assumed fonn of the variation of U with X is: + B X + ex! + Dx2 it follows that: Case (i): A = 1, B = 0, C = 0, D =0 Case ( ii): A = 1, B = o.s,e = 0, D = 0 Case ( iii): A = 1, B = -O.S, e = 0, D = 0 / The available computer program TURBINRK has been run for each of the three sets ofinput values. The program gives the. variation of the local Nusselt number, Nux. with X and the variations for each of the three cases considered is shown i n Fig. E6.3. 276 Introduction to Convective Heat Transfer Analysis 25000 20000 15000 Nux 'v ~x ~. <01-- \ ')q 10000 5000 0 0.0 ", 0.5 X 1.0 F IGUREE6.3 I t will be seen that when the velocity is increasing along the duct the heat transfer rate is higher than when the velocity is constant. Simultaneously, when the velocity is decreasing along the duct the heat transfer rate is lower than when the velocity is constant. In the latter case, i.e., where the velocity is decreasing along the duct, boundary layer separation occurs at approximately X = 0.73 and the calculation is stopped just before separation occurs. The solution procedure discussed above obtained the heat transfer rate by using the analogy solutions discussed in the previous section. These solutions assume that the velocity and thermal boundary layer thicknesses are of similar orders of magnitude. When this is not the case, the energy integral equation [3] has to be used to determine the variation of the thickness of the thermal boundary layer. To illustrate the procedure consider flow over a flat plate with an unheated leading edge section, often referred to as an unheated starting length, as illustrated in Fig. 6.8. The boundary layer is assumed to be turbulent from the leading edge of the plate. No heating of the flow occurs upstream of the heated section of the plate so the thermal boundary layer starts to grow at x = Xo, i.e., at the beginning of the heated section of the plate. It is assumed that the temperature and velocity profiles in the boundary layer are given by: T - Tl - - - - 1 - ( Y )lIn Tw - Tl 8T (6.102) Thennal Boundary Layer FIGURE 6.8 Turbulent boundary layer flow over a flat plate with an unheated leading edge section. CHAPTER 6: External Turbulent Flows 277 and: ~ =(~r (6.103) Here 8 a nd 8T a re the thicknesses o f t he velocity and thermal boundary layers respectively and n i s an integer constant. These assumed fOfIlls o f t he profiles will not apply very near the wall but this is not o f consequence because o f t he w ay i n which these profiles are applied i n this analysis. In order to decide what value o fn to u se i n Eqs. (6.102) and (6.103), i t is recalled that, as discussed above, experiment indicates that for flow over a flat plate: H( = 8 1/8 2 ) = 1.4 (6.104) Using the assumed velocity profile given i n Eq. (6.103) and recalling that: 81 = i t follows that: I } ~ ~ldY (~ ) ' }Y = U ul 81 = 8 Similarly, recalling that: I.I [ = 1- n! 1 (6.105) .' 8 2 i t follows that: Jo (8 [1 _ U ]dY + 1)(n + 2 ) 8 n Ul 82 = 8 Jo S , - S H (1 [(y)lIn (y)21n] d y = = 81 = n (n (6.106) Using Eqs. (6.105) a nd (6.106) t hen gives: +2 n 82 Hence, for H (6.107) =1.4, it follows that: 2 n = H -l = 5 (6.108) This value will be used i n the present analysis. Using the assumed forms o f the velocity a nd temperature profiles gives: T = . p e au ay _ = peU I (I )lIn-l n8 8 (6.109) and: / ( 6.110)·' 278 Introduction to Convective Heat Transfer Analysis Dividing Eq. (6.110) by Eq. (6.109) then gives: ~ = cp €H (Tw 'T Td (~)lIn OT € Ul Applying this equation at y = 0 (the equation is actually applied at the outer edge o f the inner layer but because the shear stress and heat transfer rate can be assumed constant across the inner layer and equal to their values at the wall the same result is obtained) then gives: qw 'Tw = cp EH (Tw E Tl) (~)lIn OT 1. The above equation then Ul I t will again b e assumed that € H gIves: = E , i.e., P rT = (6.111) . Now the analysis o f the velocity boundary layer on a flat plate presented above gave: 'Tw 0.0324 p ut = (ulxlv)O.211 and: -= x But Eq. (6.106) gave: 0.0412 (ulxlv)O.211 o2 i.e., using n = 5 then gives: = (n + l )(n + 2) n 0 02 Using this i n Eq. (6.94) gives: o =~ 42 0.346 o (6.112) Now in the situation being considered here, OT < 0 so the energy integral equation, i.e., Eq. (2.194) gives: -d dx [i° ST qw u(T-Tl)dy ] = - p Cp This equation can be written as: / CHAPI'ER 6: External Turbulent Flows 279 i.e., because, in the flow being considered, Ul a nd Tw are constants, this equation gives: (6.113) Using Eqs. (6.102), (6.103), and (6.108) then gives: Defining: (6.114) this equation can b e written as: Carrying out the integration and rearranging then gives: ~,1/5d' + ~,6/5d~ 7 dx 42 dx Now Eq. (6.111) c an b e written as: = qw pCpul(Tw - Tl) (6.116) qw 'Tw (~ )1/5 pCpu l(Tw - Tl) = pu~ ~T (6.117) Substituting this into Eq. (6.116) a nd rearranging then gives: ~ 7 dx d, + ~,d~ 42 d x = 'Tw pu~ (6.118) Substituting Eqs. (6.95) a nd (6.112) into this equation then gives: 0.0494 d, 0.0325 0.0324 (Ul x1v )0.211 x d x + (Ul x1v )0.211' = (Ul x1v )0.211 i.e.: dx d, + 0.65S f = 0.658 x x (6.1)9) This equation must b e integrated subject to the condition t hat, as discussed earlier. Eq. (6.119) therefore gives: = 0 w hen x = Xo (6.120) , Jo r' 0.658 -d,0.658' I 'dx = Xo x 280 Introduction to Convective Heat Transfer Analysis Carrying out the integration and rearranging then gives: ? ~ 1 - (~ f58 = 'Tw (6.121) Substituting this result into Eq. (6.117) then gives: qw p Cpul(Tw - Tl) [1 _(xo )0.658]-115 x (6.122) p ur which can b e rearranged to give: This equation shows, o f course, that as x increases, with the result that xol x decreases, Nux tends to the value i t would have i f the plate had been heated from the leading edge. E XAMPLE 6 .4. Air at a temperature of lOo e flows at a velocity o f 100 m1s over a 4-m long wide flat plate which is aligned with the flow. The first quarter o f the plate is unheated and the remainder o f the plate is maintained at a uniform wall temperature of 50o e. Plot the variation o f the local heat transfer rate along the heated section of the plate. Evaluate the air properties at a temperature o f 30°C. Solution. T he flow situation being considered is shown in Fig. E6.4a. At 30°C, the mean temperature, air at standard ambient pressure has the following properties: v = 16.01 X 1 0-6 m 2 /s, k = 0.02638 W/m_oe, Pr In the situation being considered: Xo = 0.71 = 1m Using these values, Eq. (6.122) gives: ~~~~~--~ qwx X = ------~----------~~------- 0.0323 X (100xI16.01 X 1 0-6)°.789 X 0.71 0.02638 (50 - 10) [1 _ (~t'T' lOOmis 4m I: 1m .1 Heated to a Uniform Temperature o f 5 0°C 6SSSssssssssssSSSS5 ·1 Unheated FIGURE E6.4a CHAPTER 6: External Turbulent Flows 281 12000 r-.- - - . . . . , . - - - - , . - - - - - - - , 10000 6000 I [. ~. ~eated F1 • • • • rom Lea":2 ...... ~g.I~<!.ge x -m 4000~--~-~-~----~ . --- - - -- 3 1 4 F IGURE E6.4b i.e.: This equation applies for x > 1. T he variation o f qw with x given b y this equation is shown in Fig. E6.4b. This figure also shows the variation o f qw t hat would exist i f the plate were heated from its leading edge, i.e., i f Xo = o. 6.4 N UMtRICAL S OLUTION O F T HE T URBULENT BOlJl,WARY LAYER EQUATIONS Solutians to the boundaryJayer equations are, today, generally obtained numerically [6],[71;[8],[9],[10],[Ul;[12]. I n order to illustrate how this can be done, a discussion of ho~ the simple numerical solution procedure for solving laminar boundary layer .problePls that was outlined in Chapter 5 can be modified to apply to turbulent bound.ary layer flows. For turbulent boundary layer flows, the equations given earlier in the present chapter can, because the fluid properties are assumed constant, be written as: ax ay _au _au = . - 1 d- + - [(V+E)pa au] u -+vax ay pdx ay ay aT aT aT] u - +v-. = -a [(a + EH)- , ax ay ay ay ~+ UV = 0 (6.123) (6.124) (6.125) 282 Introduction to Convective Heat Transfer Analysis Although not necessary, it is convenient to rewrite these equations in dimensionless form and the following variables are therefore introduced for this purpose: U X I v = -t, V = - ,P = Ur Ur U rX p, P - Pr pu;: = Y = V Ur p Y fJ = T , Twr - Tl T1 (6.126) E = E ,EH = E H V where Ur is some reference free-stream velocity, Pr is some reference pressure, Twr is some reference wall temperature, and T 1 is the free-stream temperature.- It is being assumed that the surface temperature distribution and not the surface heat flux distribution is being specified. The modifications to the procedure that are required to deal with the latter situation are, however, relatively minor and will be discussed later. . In terms of these variables, the governing equations are: au + av = 0 ax ay au au dP a[ a u] U ax + v ay = - dX + ay (1 + E ) ay U afJ (6.127) (6.128) (6.129) v afJ ax + ay = a [( 1 E H ) afJ ] ay Pr + PrT ay The boundary conditions on this set of equations are: Y = 0: U = 0, V = 0, fJ = fJw(X) Y = large: U --"» U 1(X), fJ --"» 0 , ' (6.130) The finite difference forms of the above set of equations and the method of solving them will first be discussed, it being assumed in this discussion that the variation of E and the value of P rr, which will here be assumed to be constant, are determined by the use of a turbulence model. M ter this discussion, one particular way of etrtpir.;, ically describing E, i.e., one particular turbulence model, will be presented. A series of grid lines running parallel to the X - and Y-axes are introduced as was done in dealing with laminar boundary layer flow. The nodal points used in obtaining the solution'lie at the points ofintersection of these grid lines. Because the gradients near the wall in a turbulent boundary layer are very high, a nonuniform grid spacing in the Y -direction will be used in the present finite difference solution, a smaller grid spacing being used near the wall than near the outer edge of the boundary layer. First consider the finite difference form of the momentum equation, i.e., Eq. (6.128). As with laminar boundary layer flow, the four nodal points shown in Fig. 6.9 are used in deriving the fimte difference form of this equation. The following finite-difference approximations, which can be derived in the same way as that used in dealing with laminar boundary layer flow, to the various CHAPTER 6: External TurbulentFlows 283 ----t-----------~- j + 1 I· i ll~_+l-t-_ ~.----4 ll~! ! j ~~ _ _ +_______ I • i -I I I I I I ~ ! t .J.._ j -l . I F IGURE 6.9 'i Nodal points used in obtaining the finite difference , Jonus of the momentum and energy equations for turbulent boundary layer flow. terms i n the momentum equation are introduced: u aul Ui-l,j]' ax 4.}. = u ., -I,}. [ Ui,j -AX , a UI _ " [ A Yj+I V a y i,j - Vi-I,j A Yi(AYj+l + A Yj) ( Ui,j+l - Ui,j) + :Y AYj+l(A~~:1 + AYj) (U,,) - Vi,j-I)] [(1 + E)~~]li,j = AYj+l1+ A Yj [(2 + Ei-I,}+l + E i-l,j)X . . ( Ui,j+l - Ui,j)_ ( 2 + E- -I,}. + E- -l,}-I ) ( Ui,j A Y. A Y'+I , I } } Ui,j-I)~ It will be seen that in'treating the dimensionless eddy viscosity, it has been assumed that, like the coefficient terms~ the values of the dimensionless eddy viscosity can be evaluated on the ( i - I)-line, i.e., the line from which the solution is advancing. Substituting, the above finite-difference approximations into the momentum equation then leads to ltD. equation of the form: A)·U·' I ,} + B}·U,I ,}'+1 + C}·U'I ,}' -I = D}, (6.131) where the coefficients are given by A . = ( Ui-l,j)_ V '-l . [ } AX I,} AYj+1 ' ' A Yj(AYj+l'+ AYj) + AYj.f-l(AYj+l + A Yj) . A Yj ] + [ (2 + E i-l,j+l + E i-l,j) + (2 + E i-l,j + E i-'l,i- 1)]. AYj+l + A Yj " ' AYj +1 ' AYj' 1 , (6.132) A Yj+1, ], ' Bj = Vi -I' [ + - --.....,-,} A Yj(AYj+I + A Yj) A Yj+I + A.Yj 1 [(2 + E i-l,j+l + E i-l,j)] A Yj+I , / (6.133) 284 Introduction to Convective Heat Transfer Analysis (6.134) ° __ DJ - V l-I,i _ d PI ax dX ° I (6.135) The boundary conditions give V i,l the value of V at the wall, and V i,N the value of V at the outermost grid point which is always chosen to lie outside the boundary layer. Therefore, since d PldXli is a known quantity, the application of Eq. (6.131) to each of the points j = 1, 2, 3, . .. , N - 1, N gives a set of N equations in the N unknown values of V , i.e., VI, V2, V 3, V4, . .. V N-I. V N. This set of equations has the following form because V i,l is zero ' V i,l = 0 = D3 A 2 Vi, 2 + B2 V i, 3 + C2 Vi, 1 = D2 A3 V i ,3 + B3 V iA + C3 V i,2 (6.136) = D N-I A N-IVi,N-I + B N-IVi,N + CN-IVi,N-'J.,. V i,N = VI This set of equations thus has the form: 1 C2 0 0 0 0 .B2 A3 A2 C3 0 0 0 B3 0 0 0 B4 C4c" A 4 0 0 0 0 0 0 0 0 0 0 0 0 VOl I, U02 I, V03 I, U04 I, 0 D2 D3 - D4 - oo oo which has the form: o o .. QUVi,i = R u V i,N-I VON I, :(6.137) where Qu is a tridiagonal matrix. This equation can be solved using the standard tridiagonal matrix solver _ lgorithm discussed when considering laminar boundary a layer flow. Next, consider the energy equation (6.129). I t will be assumed that the tUrbulent Prandtl number, P rr is constant i n the flow. The following finite-difference approximations to the various terms i n the energy equation are introduced: V a() I-I,J xo a = VoI ° ° I ,J [()i'i - ()j-I,i] ax (6.138) , CHAPTER 6: External Turbulent Flows 285 (6.139) Substituting these finite-difference approximations into the energy equation then leads to an equation of the form: F ·(J·,J. J' + G ·(J·',J'+1 + H ·(J·!,J' -1 J '" J = L"J (6.141) where the coefficients are given by Ui-1,j) Fj = ( D.X - v ] [ A . Y j +1 A.Yj i-1,j A.Yj(A.Yj+1 + Il.Yj) + A.Yj +1(A.Yj+;1 + A.Yj) + [(21pr+ Ei-1,j+1IPrT + Ei-1,/PrT)' A.Yj +1 + A.Yj Il.Yj+1 1 (6.142) + 21Pr + Ei-1,/PrT + Ei-l,j-1 IprT] A.Yj (6.143) (6.144) (6.145) The case where the wall tempeniture variation is known will be considered here. The boundary conditions then give (Ji,1 and (J i ,N, the outermost grid point being chosen to lie outside both the velocity and temperature boundary layers. The application of Eq. (6.141) to each of the internal points on the i-line, i.e., j = 1, 2, 3, 4, . .. , -- / 286 Introduction to Convective Heat Transfer Analysis N - 2, N - 1, N again gives a set o f N equations in the N unknown values of (J. Because (J i,N is 0 , this set o f equations has the following form: ( hI = (Jw FZ(Ji,Z F3(Ji,3 + GZ(Ji,3 + HZ(Ji,1 + G3(Ji,4 + H3(Ji,z = Lz (6.146) = L N-I = L3 F N-I(Ji,N-I + G N-I(Ji,N + H N-I(Ji,N-Z (Ji,N =0 This set o f equations has the same form as that derived from the momentum equation, i.e., 1 Hz 0 Fz H3 0 Gz F3 H4 0 0 0 0 G3 F4 0 0 0 0 0 0 G4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8·, 1 I 8·, Z I 8·, 3 I ( J'4 I, (Jw Lz LJ L4 0 0 0 0 0 0 H N-I F N-I G N-l 0 0 1 ( Ji,N-l ( J'N I, L i,N-I 0 which has the form: Q r(Ji,j = R r (6.147) where Q r i s again a tridiagonal matrix. Thus, the same form o f equation is obtained from the energy equation as that obtained from the momentum equation. This equation can also b e solved using the standard tridiagonal matrix-solver algorithm. The continuity equation, (6.127) has exactly the same form as that for laminar flow and i t c an therefore b e treated using the same procedure as used i n laminar flow. The grid points shown i n Fig. 6 .10 a re therefore used in obtaining the finite difference approximatio:n to the continuity equation. The continuity equation is applied to the point (not a nodal point) denoted by (i, j - 1/2) i n Fig. 6.10 which lies on the i-line, halfway between the ( j - 1) and the i ...._ -_. i -I I I 1 ! llj/2 !llj' I i ,j-1I2 ...--.......". Continuity !lY·/2 Equation J -.- j-l Applied Here . ...- - . - - - - F IGURE 6.10 I Nodal points used in finite-difference approximation to continuity equation. / CHAPI'ER 6: External Thrbulent Flows 287 j-points. The following is introduced: aVI ay i ,j-1I2 _ y. . I ,} V·I ,}- I . . AYj (6.148) and it is assumed that the X-derivative at the point ( i, j - 112) is equal to the average of the X-derivatives at the points (i, j ) and (i, j - 1), i.e., it is assumed that: ~~ I. '-112 =. 4[~~ I.. + ~~ I. '-1] I ,} I ,} I,} From this it follows that: au a x.. 112 I ,}i.e.: 1 = ! [ Ui,j 2 Ui-I,j + U i,j-I - Ui-I,h I ] AX AX ~~ I. . = 2 lx(Ui,j - I ,}-1I2 Ui~I,j + Ui,j-I ax . Ui-I,j-I) (6.149) Therefore, since the continuity equation can be written as: av au -= - ay the following finite-difference approximation to this continuity equation is obtained: v· . I ,} V·· I 1 AY. "}- = - 2AX(Ui,j - Ui-I,j + U i,j-I - Ui-I,j-I) } This equation c an b e rearranged to give: V i,j = V i,j-I - .(~,y.) 2M ( Ui,j - Ui-I,j+ U i,j-I - Ui-I,j-I) (6.150) Therefore, i f the distribution o f U across the i.;.line is first detennined, this equation can be used to give the distribution o f V across this line. This is possible because Vi,I is given by the boundary conditions and the equation can, therefore, b e applied progressively outward across the i-line starting at point j = 2 and then going to point j = 3 and so on across the line. In order to apply the above equations to the calculation o f turbulent boundary layer flows, the distribution- of the eddy viscosity must be specified. The numerical procedure does not require this to b e done i n any particular way. For the purposes o f illustration and because i t leads to results that are i n quite good agreement with experiment, this will b e done· here using an empirically derived [1],[2],[7],[1:3],[14],[15],[16],[17] distribution o f mixing length, i t being recalled that the mixing length is related to the E by: e, E = e21~~1 / 288 Introduction to Convective Heat Transfer Analysis In terms of the dimensionless variables introduced above this becomes: au m E = L2 1 a y l where the dimensionless mixing length, L m , is defined by: v In a boundary layer flow it is usually adequate to assume that: e = function( Tw , p, IL, y, 8 ) (6.151) Lm = eU r (6.152) (6.153) where, as before, Tw is the local wall shear stress and 8 is the local boundarY layer thickness. Applying dimensional analysis then gives after some rearrangement: ~ Defining: = function [ ; JTwP, * 1 (6.154) (6.155) and recalling that: y+ = L JTwP = Y ( !; IL v~p allows Eq. (6.154) to be written in terms of the dimensionless variables as: Lm T . = f unction y , [+ a j Y (6.156) Many experiments have established that, as mentioned before, there is a region near the wall where the local turbulent shear stress depends on the wall shear stress and the distance from the wall alone and is largely independent of the nature of the rest of the flow. In this region, the mixing length increases linearly with distance from the wall except that near the wall there is a "damping" of the turbulence due to . ' viscosity. In the wall'region, i t is assumed therefore that: e = 0.41y[1 Lm exp(-y+/26)] (6.157) The term in the bracket is known as the van Driest damping factor. In terms of the dimensionless variables introduced above this equation is: = 0.41Y[1 - exp(-y+/26)] (6.158) Experimentally it has been found that Eq. (6.157) applies from the wall out to about 10% of the boundary layer thickness. Hence, it will here be assumed that: For Yltt.. < 0.1 : Lm = 0.41Y[1 - exp(-y+/26)] (6.159) I In the outer portion of the boundary layer, experiment indicates that the mixing length has a constant value of about 0.089 o f the boundary layer thickness, this result CHAPTER 6: External Turbulent Flows 289 being applicable over approximately the outer 40% of the boundary layer. Thus, it will be assumed that: For yI8 > 0.6 : e = 0.0898 , (6.160) which can be written in terms of the dimensionless variables introduced above as: For YIll. > 0.6 : Lm = 0.089ll. (6.161) Over the remaining 50% of the boundary layer, i.e., from y = 0.18 to y = 0.68, it will be assumed that evaries smoothly from the value given by Eq. (6.157) to the value given by Eq. (6.160) and that the variation can be described by a fourth-order polynomial in y18, i.e., it will be assumed that: ForO.l < y l8 < 0.6: e= 0.41y[1 - exp(-y+/26)] - [1.54(yI5 - 0.1)2 - 2.76(yI8 - 0.1)3 + 1.88(y15 - 0.1)4]8 (6."162) which can be written in terms of the dimensionless variables introduced above as.: ForO.l :s; YIll. :s; 0.6: Lm = 0 .41Y[1 - exp(-y+126)] - [1.54(Y/~ - 0.1)2 - 2.76(YI6. - 0.1)3 + 1.88(YI6. - 0.1)4]6. (6.163) Eqs. (6.159), (6.161), and (6.163) together describe the variation of the dimensionless mixing length, Lm , across the boundary layer, the variation being as shown in Fig. 6.11. In order to utilize this set of equations to determine the mixing length distribution from the velocity distribution it is necessary, of course, to determine y+ . Now: y+ - - Yffw - y~w -v p pu~ (6.164) r Ill 0.5 0.0 . ......::;;-----'--------' 0.00 0.10 F IGURE 6.11 Assumed variation of dimensionless mixing length across boundary layer. / 290 Introduction to Convective Heat Transfer Analysis I t is thus necessary to determine the value of Twl pu;. To do this, it is noted thatthe turbulent stress is zero at the wall and, therefore, that: 'Tw = f.L - all ay y=o Le.: ;:; ~ ~~ 1M U. 1,2 (6.165) Applying a Taylor expansion to give the velocity at the first nodal point away from the wall in terms of the velocity at the wall, i.e., zero, gives on neglectirigthe higher order terms: = 0 + au (y _ Y ) a y. 2 I I, I I " + a f2 a2u, (Y2 - YI)2 2 (6.166) i,1 But YI = 0 and applying the full momentum equation, i.e., Eq. (6.128), to conditions at the wall gives: a2u a f2 Hence, Eq. (6.166) g~ves: a - UI . i, I _ d PI dX i ay i,l Y2 d PI =U i 2 - - - -' Y2 2 dX i (6:167) Substituting Eq. (6.167) into Eq. (6.165) and the result into Eq. (6.164) then gives the following finite-di(ferenceapproximation: y+ = y .[Ui,2 _ Y2 d PI ]112 J Y2 2 dX i (6.168) The second term in the bracket will usually be much smaller than the first and is often negligible. It is also necessary to determine the local value of the dimensionless boundary layer thickness in order to find the mixing length distribution. It is usually adequate to take this as the value of Y at which U reaches 0.99 of its free-stream value. Once the dimensionless mixing length distribution has been found using the above equations, the dimensionless eddy viscosity distribution, in terms of which the numerical method has been presented, can be found using the finite-difference approximation for the velocity derivative that waS introduced earlier, i.e., using: E = Lm 2[ d Yj +l d Yj(dYj +l + d Yj) ( Ui,j+l - Ui,j) + d Yj +1(dYj +l + d Yj) ( Ui,j.- ' d Yj " , )] Ui,j-l (6.169) CHAPTER ·6: . External Turbulent Flows 291 I n many cases, a region o f l aminar flow will have to b e allowed for over the initial part o f t he body. The calculation o f this portion o f the flow can b e accomplished by using the same equations as for turbulent flow with E set equal to O. This calculation can be started using initial conditions o f t he type discussed i n C hapter 3 for purely laminar flows. From the point where transition is assumed to occur, the calculation can be continued using the equations presented above to describe E. A computer program, T URBOUND, a gain written i n FORTRAN, b ased on the numerical procedure described above is available as discussed i n the Preface. It is, of course, essentially the same program as discussed earlier for the calculation o f laminar boundary layer flows. ' . T he program, as available, will calculate flow over a surface with a varying freestream velocity and varying surface temperature. These variations are both assumed to b e described b y a third-order polynomial, i.e., by: Ul = Au and: + BUXR + c uxi + DUXR3 (~.170) (6.171) where: X XR = - - . . Xmax I t follows from the assumed form o f t he free-stream velocity distribution th~t since: - dp d UI = - PU l dx dx the following appli~s: d~ d = - Ul dd~1 .~ U l(Bu . + 2CUXR + 3DiJXR2) X 1 (6.172) max The program finds the distribution o f t he dimensionless temperature. The wall he.at transfer-rate is then fOllnd b y noting that t he turbulenthe~t transfer rate is 0 at the wall and, therefore, that: qw = - k ay y=o aT i.e.: qw x k(Twr - Tt} _X = aOI ' .: a y y=o i.e.: Nux = : w ;~I y=o . (6.173) / 2 92 Introduction to Convective Heat Transfer Analysis where Nux i s the local Nusselt number, i.e.: Nux qw x = k(Tw - TI) (6.174) Applying a Taylor expansion to give the temperature at the first nodal point away from the wall gives on neglecting the higher order terms: (6.175) But YI = 0 a nd applying the full energy equation, i.e., Eq. (6.128), to conditions a t the wall gives: =0 Hence Eq. (6.175) gives: ao I a y i,l = ( h,2 - Oi,l YI Y2 (6.176) - Substituting Eq. (6.176) into Eq. (6.173) then gIves the following finitedifference approximation: l vUx 71.T = -X [fh2 - Oi,l] Y2 -:- Yl Ow (6.177) As with the program for laminar boundary layer flow discussed i n Chapter 3, the turbulent boundary flow program calculates the velocity and temperature boundary layer thicknesses using, as discussed above, the assumption that A is the value of Y at which U = 0 .99Ul a nd AT is the value o f Y at which 0 = 0.010 w • I f either 11 or I1T is greater than YN -5, the number o f grid points is increased b y ten. The program is easily modified to utilize other turbulence models. E XAMPLE 6 .5. NUmerically determine the dimensionless heat transfer rate variation with two-dimensional turbulent boundary layer air flow over an isothermal flat plate. Assume that the boundary layer is turbulent from the leading edge. Compare the numerical results with those given b y the three-layer analogy solution discussed previously (see Eq. 6.79). plate with an isothermal surface is being considered, the free-stream velocity and surface temperature are constant. Therefore, .because the boundary· is assumed to be turbulent from the leadirig edge, the inputs to the program are: Solution. A solution for values o f X up to 107 will be considered. Because flow over a flat X MAX = 10000000, X TRAN = 0, Pr = 0.7, AT, BT, CT, D T = 1, 0, 0, 0, A V; BV; CV; D V = 1, 0, 0, ° T he program, when run with these inputs, gives the values o f Nux a t various X values along the plate. These are shown in Fig. E6.5. CHAPTER 6: External Turbulent Flows 293 12000 r -----,----,---.-----.----, 8000 4000 OL--~-~--L--~-~ 0 e+0 2 e+6 4 e+6 X 6 e+6 8 e+6 1 e+7 F IGURE E6.5 . T he variation given b y u sing Eq. (6.79) i s also shown i n this figure. I t w ill be seen that there is quite good agreement between t he two results a t Reynolds numbers (Le., X values) up to about 5 X 106 b ut the numerically determined values o f Nux a re higher than those given b y the three-layer equation a t t he higher Reynolds numbers. E XAMPLE 6 .6. A ir flows over a flat plate which has a uniform surface temperature o f 50°C, t he temperature o f the air a head o f t he plate being 10°C. T he a ir velocity is such that the Reynolds number b ased o n the length o f t he plate i s 6 X 106 , the l ength o f the plate being 2 m. Numerically determine the variation o f the local heat transfer rate from the wall, qw, w ith x assuming that (i) t he boundary layer flow remains laminar, (ii) the boundary layer flow is turbulent from the leading edge o f the plate, and (iii) boundary layer transition occurs a t ReT o f 1 06 • x i s the distance from the leading edge o f the plate. Solution. T he definition o f dimensionless X is such that: x = X X Xmax L = 6000000 x 2 = 3 000000 m X ( i) T he local Nusselt number i s defined by: qw x Nux = k (Tw - Tl) Therefore, since: T w - Tl = 50 - 10 = 40°C + 10)/2 = 30°C: and, i f t he properties o f the air a re evaluated a t (50 k Hence: = 0 .02638W/m-oC,Pr = 0.7 Nux = qw x 0 .02638 X 4 0 Le.: qw Nux = 1 .0552x ( ii) / 2 94 Introduction to Convective Heat Transfer Analysis 10000 . ------,----,----,..----., 8000 6000 N .€ ~ J 4000 2000 Transition Reynolds Number > 6x 106 0.5 o~--~~-~--~--~ 0.0 1.0 x -m 1.5 2.0 FIGUREE6.6 T he program T URBOUND has been run with the following three sets o f input values: XM AX = 6000000, XTRAN = 6000000, P r = 0.7, A~B~C~DT=L~~~A~B~C~DV=L~~O = 6000000, XTRAN = 0, P r = 0.7, A T, B T, C T, D T = 1 ,0, 0 ,0, A~ B~ C~ D V = 1, 0, 0, 0 X MAX = 6 00oooo,XTRAN = 1000000, P r = 0.7, XM AX A T, B T, C T, D T = L 0, 0, 0, A ~ B~ C~ D V = 1, 0, 0, 0 T he calculated variations o f Nux with X have then been used i n conjunction with Eqs. (i) and (ii) to derive the variations o f qw with x for the three cases. The variations so obtained are shown in Fig. E6.6. In the above discussion it was assumed that the surface temperature variation was specified. The procedure is easily extended to deal with other thermal boundary conditions at the surface. For example, i f the heat flux distribution at the surface is specified, it is convenient to define the following dimensionless temperature: 0* = T - Tl (qwRvlku r ) (6.178) where qwR is some convenient reference wall heat flux. The energy equation in terms of this dimensionless temperature has the same form as that obtained in the specified surface temperature case; i.e., the dimensionless energy equation in this case is: ao* . ao* u ax + v ay 1 a2 o* = Pr ay2 (6.179) Using Fourier's law, the boundary condition on temperature at the wall in the specified heat flux case is: (6.180) CHAPI'ER 6: External Turbulent Flows 295 i.e., (6.181) where QR = qwl qwR Now, following the same procedure as used i n deriving Eq. (6.176) gives the following finite-difference approximation: a(}~ = i ,l ( }\2 - (}i,l ay Y2 - YI (6.182) Substituting this result into Eq. (6.181) then gives: _ (}~2 - (}~1 I, I, Y2 - YI = QR Le.: (6.183) This boundary condition i s easily incorporated into the solution procedure t hat was outlined above, the set o f equations governing the dimensionless temperature in this case having the form: F2(}~2 " + G28~3 + H2(}~1 = 14. I, I, (6.184) F N-l(}i,N-l + GN-18'i,N + H N-18'i,N-2 = L N-l (}~N = I, 0 the coefficients having the same values as previously defined. The matrix equation that gives t he dimensionl,ess temperature therefore has the form. 1 H2 1 F2 H3 0 G2 0 0 G3 0 0 0 0 0 0 0 F3 H4 0 0 0 G4 F4 0 0 0 0 0 0 0 0 0 0 0 0 ()~, 1 I \ Q Ray " " (}~4 " (}~2 ()~ 3 14. ~ L4 0 0 0 0 0 0 . H N-l F N-l G N-l 0 0 1 8 'i,N-l 8~ N L N-I 0 Thus, as before, a tridiagonal matrix is obtained. The program discussed above is therefore easily modified to deal with the specified wall h eat flux case. A program with this wall thermal boundary case, / " / 296 Introduction to Convective Heat Transfer Analysis T URBOUNQ, is also available as discussed i n t he Preface.' In this program, the dimensionless wall heat flux variation is assumed to b e o f the form: (6.185) 6.5 E FFECTS O F DISSIPATION ON T URBULENT BOUNDARY LAYER F LOW O VER A F LAT PLATE Consider two-dimensional boundary layer flow over a flat plate as shown in Fig. 6.12. I f the effects o f fluid property variations are neglected, the governing equations are: au+av =0 ax ay au au u] u - + v - = -a [(v + Eay )ax ay ay a aT aT aT] au]2 u - + v - = -a [(a + E H)- + ( v + E) [-ay ax ay ay ay (6.186) (6.187) (6.188) where E a nd E H are the eddy viscosity and eddy diffusivity as previously defined in Chapter 5. T he l ast term on the right-hand side o f t he energy equation is, o f course, the dissipation term, dissipation here arising both as a result o f the presence o f the viscous stress and as a result o f the effective turbulent stress. The above equations can be solved provided the turbulence terms i n the momentum a nd energy equatioQ,s are related to the other flow variables, i.e., provided a turbulence model is introduced. For example, a mixing-length model o f the type introduced in Chapters 5 and 6 could b e used, i t b eing assumed that the viscous dissipation has no effect on the equation that describes the variation o f t he mixing length in the boundary layer: /Using such a turbulence model, the adiabatic wall temperature and hence the recovery factor for turbulent boundary flow can b e determined [18],[19],[20],[21],[22],[23],[24],[25],[26],[27]. This procedure gives: r = Pr1l3' (6.189) t tv u ~~= Plate at Temperature Tw or Plate Adiabatic ~r6.rm.ry~ F IGURE 6.12 Turbulent boundary layer flow with viscous dissipation. / Turbulent .g , , ! CHAPTER 6: External Turbulent Flows 297 T he effects o f fluid property variations on heat transfer i n turbulent boundary layer flow over a flat plate have also been numerically evaluated. This evaluation indicates that i f the properties are as with laminar boundary layers evaluated at: Tprop = Tl + O.5(Tw - Tt} + O.22(Twad - Tl) the effects o f these property variations can b e neglected. E XAMPLE 6 .7. A ir at a temperature of O°C flows at a velocity of 600 m ls over a wide flat plate that has a length of 1 m. The pressure in the flow is 1 atm. The flow situation is therefore as shown in Fig. E6.7. , Find: 1. The wall temperature i f the plate is adiabatic. 2. The heat transfer rate from the surface per unit span of the plate if the plate surface is maintained at a unifonn temperature, T w, o f 60°C. S olution P art 1 I n the free-stream: a = J 'YRT = J 1.4(287)(273) = 332 mls Hence: M= V a= 600 331 = 1.81 Now: TWad Tl = 1 + r ('Y - 2 1 )M2 I n order to find the Reynolds number, the surface temperature must be known. How- ever, in order to find the w~ temperature, the recovery factor must be known and its value is different in laminat and turbulent flow. Therefore, an assumption as to the nature o f the flow, i.e., laminar or turbulent, will be made. The wall temperature and then the gas properties will be found and then the Reynolds number, i.e.: Re = p VL /L will be evaluated and the initial assumption about the nature of the flow can be checked. Here, i t will b e assumed that the flow i n the boundary layer is turbulent. Experience suggests that this is very likely to be a correct assumption. Since the flow is assumed turbulent, it follows that since the Prandtl number of air can b e assumed to be equal to 0.7: r V =600m/s = Prl13 = 0.7 113 = 0.89 --.... T=O°C p =latm. FIGUREE6.7 / 298 Introduction to Convective Heat Transfer Analysis Using this value then gives: TWad _ 273 - 1 + 0.89 Hence: TWad = X 0.2 X 1.81 2 432 K = 159°C Since the adiabatic surface case is being considered, the air properties are found at: Tpr~p = Tl + 0.5(Tw = 0 + 0.5(159 - T 1) + 0.22(Twad - T 1) 0) + 0.22(159 - 0) = 114°C Now at a temperature of 114°C, air has the following properties when ~e pressure is 1 atm: p /.L = 0.9 kglm3 1 0-7 Ns/m2 k = 33 x 10:::'3 W /mK = 225 X I f the pressure had not been 1 atm, the density would have had to be modified using the perfect gas law. Using these values then gives: R e = p VL /.L = 0.9 x 600 x 1 225 X 1 0-7 = 2 4 107 . X At this Reynolds number, the flow in the boundary layer will indeed be turbulent so the assumed recovery factor is, in fact, the correct value. Hence, the adiabatic wall temperature is 159°C. . P artZ When the wall is at 60°C the air properties are found at the following temperature: Tprop = Tl + 0.5(Tw-:- Tl) + 0.22(Twad = 0 + 0.5(60) + 0.22(159) = 65°C Tl) Now at a temperature of 65°C, air has the following properties at a pressure of 1 atm: p = 0.99 kglm3 /.L = 208 X 1O~7 Ns/m2 k = 30 X 1 0-3 W /mK = p VL = 4.3 X 107 /.L I n this case then: Re so the flow is again turbulent. The same equation for the Nusselt number as derived for flow without dissipation can be used here. The Nusselt number is therefore given by: Nu / = 0.037Reo. 8 Pr1l3 = 0.37 X (4.3 X 107)°.8 X (0.7)113 = 41980 CHAPTER 6: External Turbulent Flows 299 From this it follows that: h L = 41980 k so: h = 41980 X (30 x 10-3) = 1259 W/m20C 1 Therefore, considering both sides o f the plate, the heat transfer rate from the plate ' is given by: Q = h A(Tw - T wad ) = 2 X 1259 X 1 X 1 X ( 60- 159) = - 250,000W The negative sign means that heat is transferred to the plate. Hence, the net rate of heat transfer t o the plate is 250 kW. 6.6 SOLUTIONS T O T HE F ULL TURBULENT F LOW EQUATIONS This chapter has mainly been devoted to the solution o f the boundary layer form of the governing equations. While these boundary layer equations do adequately describe a number o f problems o f g reat practical importance, there are many other problems that can only b e adequately modeled b y u sing the full governing equations. I n such cases, i t is necessary to obtain the solution numerically and also almost always necessary t o u se a more advanced type o f turbulence model [6],[12],[28],[29]. Such nu,merical solutions are most frequently obtained using the commercially available software based on the finite volume or the finite element method. 6.7 CONCLUDING REMARKS While i t m ay one day b e possible to numerically solve, on a routine basis, the full unsteady f orm o f the· governing equations for turbulent flow, m ost solutions undertaken a t the present time are based on the use o f t he time-averaged form o f the governing equations together with a turbulence model. Such solutions for the flow over the outer surface o f a body immersed i n a fluid stream have been discussed i n this chapter. So-called analogy solutions for predicting the heat transfer rate from a knowledge o f the wall shear stress distribution were first discussed. The initial discussion was o f the simplest such analogy solution, the Reynolds analogy, which only really applies to fluids with a P randtl number near one. Multi-layer analogy solutions which apply for all Prandtl numbers were then discussed. I n order to use these analogy solutions for flow over bodies o f complex shape i t is necessary to solve for the surface shear stress distribution. The use o f t he integral equation method for this purpose was discussed. . 3 00 Introduction to Convective Heat Transfer Analysis PROBLEMS 6.1. A ir flows over a wide 2-m long flat plate which has a uniform surface temperature o f 80°C, the temperature o f the air ahead o f t he plate being 20°C. T he air velocity is such that the Reynolds number based on the length o f the plate is 5 X 106. D erive an expression for the local wall heat flux variation along the plate. U se the Reynolds analogy and assume the boundary layer transition occurs at a Reynolds number o f 106 . 6.2. Air at a temperature o f 50°C flows over a wide flat plate at a velocity o f 6 0 mls. The plate is kept at a uniform temperature o f 10°C. I f the plate is 3 m long, plot the variation o f local heat transfer rate per unit area along the surface o f the plate. Assume that transition occurs at a Reynolds number of 3 X 105. 6.3. In the discussion o f the use o f the Reynolds analogy for the prediction o f t he heat transfer rate from a flat plate i t was assumed that when there was transition on the plate, the xcoordinate in the turbulent portion o f the flow could be measured from the leading edge. Develop an alternative expression based o nthe assumption that the momentum thickness before and after transition is the same. This assumption allows an effective origin for the x-coordinate in the turbulent portion o f the flow to be obtained. 6.4. Using the Taylor-Prandtl analogy, determine the relation between the velocity and temperature profiles in the boundary layer. 6.5. Derive a modified version o f the Reynolds analogy assuming the Prandtl number and turbulent Prandtl number are equal but are not equal to one. 6.6. A ir flows over a flat plate which has a uniform surface temperature o f 50°C, the temperature o f the air ahead o f the plate being 30°C. T he air velocity is such that the Reynolds number based on the length o f the plate is 5 X 106, the length o f the plate being 2 m. U sing the Reynolds analogy, plot the variation o f the local heat transfer rate from the wall, qw, with x lL assuming that (i) the boundary layer flow remains laminar, (ii) the boundary layer flow is turbulent from the leading edge o f t he plate, and (iii) boundary layer transition occurs a t ReT o f 106 • x is the distance from the leading edge o f the plate and L is the length o f the plate. 6.7. Air flows through a large plane duct with isothermal walls. The Reynolds number based on the length o f the duct and the inlet air velocity is 107 • Using t he integral equation method and assuming that the boundary layer is turbulent from the inlet and thin compared to the size o f the duct, determine how the local Nusselt varies with distance along t he duct i f the duct cross-sectional area varies in such a way that the velocity increases linearly by 50% over the length o f the duct. 6.8. Modify the integral equation computer program to use the Taylor-Prandtl analogy. Use this modified program to determine the local Nusselt number variation for the situation described in Problem 6.6. 6.9. A ir a t a temperature o f 2 0°C flows a t a velocity o f 100 mls over a 3 -m long wide flat plate which is aligned with the flow. T he first fifth o f the plate is unheated and the remainder o f the plate is maintained at a uniform wall temperature o f 60°C. Plot the variation o f the local heat transfer rate along the heated section o f the plate. Evaluate the air CHAPTER 6: External Turbulent Flows 301 properties at a temperature o f 300C and assume that the boundary layer is turbulent from the leading edge. 6.10. Modify the integral equation analysis o f flow over a plate with an unheated leading edge section that was given in this chapter to apply to the case where the plate has a heated leading edge section followed by an adiabatic section. 6.11. Numerically determine the local Nusselt number variation with two-dimensional turbulent boundary layer air flow over an isothermal flat plate for a maximum Reynolds number o f 107 • Assume that transition occurs at a Reynolds number of 5 X 105 . Compare the numerical results with those given by the Reynolds analogy. 6.12. In the numerical solution for boundary layer flow given in this chapter it was assumed that transition occurred at a point; i.e., the eddy viscosity was set equal to zero up to the transition point and then the full value given by the turbulence model was used. Show how this numerical method and the program based on it can b e modified to allow for a transition zone in which the eddy viscosity increases linearly from zero at the beginning o f the zone to the full value given by the turbulence model at the end o f the zone. 6.13. Air flows over a 3-m long flat plate which has a uniform surface temperature of 60°C, the temperature o f the air ahead of the plate being 20°C. The air velocity is 60 mls . Numerically determine the variation o f the local heat transfer rate from the wall, qw, with x assuming that boundary l ayer transition occurs at R~T.of 106 • . 6.14. Numerically determine the local Nusselt number variation for the situation described in Problem 6.6. 6.15. Air flows over a wide flat plate which is aligned with the flow. T he Reynolds number based on the length o f the plate and the free-stream air velocity is 1 07 • A specified heat flux is applied at the surface o f the plate, the surface heat flux increasing linearly from 0.5 qwm at the leading edge o f the plate to 1.5 qwm at the trailing edge of the plate, qwm being the mean surface heat flux. Assuming that the boundary layer is turbulent from the leading edge o f the plate, numerically determine how the dimensionless wall temperature varies with distance along the plate. 6.16. Consider air flow over a wide flat plate which is aligned with the flow. T he Reynolds number based on the length o f the plate and the free-stream air velocity is 6 X 106. The first third of the plate is adiabatic, the second third of the plate has a uniform heat flux applied at the surface, while the last third o f the plate is again adiabatic. Assuming that the boundary layer is turbulent from the leading edge o f the plate, numerically determine how the dimensionless wall temperature varies with distance along the plate. 6.17. Air at a temperature of O°C and standard atmospheric pressure flows at a velocity of 50 mfs over a wide flat plate with a total length of 2 m. A u nifonn surface heat flux is applied over the first 0.7 m o f the plate and the rest o f the surface o f t he plate is adiabatic. Assuming that the boundary layer is turbulent from the leading edge, use the numerical solution to derive an expression for the plate temperature at the trailing edge o f the plate i n terms of the applied heat flux. What heat flux i s required to ensure that the trailing edge temperature is 5°C? 3 02 Introduction to Convective Heat Transfer Analysis 6.18. Discuss how the computer program for calculating heat transfer from a surface with a specified surface temperature would have to be modified to incorporate the effect of suction a t the surface. 6.19. Air at standard atmospheric pressure and a temperature o f 30°C flows over a flat plate at a velocity o f 20 mls. T he plate is 6 0 c m square and is maintained at uniform temperature o f 90°C. T he flow is normal to a side o f the plate. Calculate the heat transfer from the plate assuming that the flow is two-dimensional. 6.20. T he roof o f a building is flat and is 2 0 m wide and long. I f the wind speed over the roof is 10 mis, determine the convective heat transfer rate to the roof (i) on a clear night when the roof temperature is 2°C a nd the air temperature is 12°C a nd (ii) on a hot, sunny day when the roof temperature is 46°C a nd the air temperature is 28°C. Assume two-dimensional turbulent boundary layer flow. 6.21. A rocket ascends vertically through the atmosphere with a velocity that c an b e assumed to increase linearly with altitude from zero a t sea level to 1800 mls a t an altitude of 30,000 m. I f the surface o f this rocket is assumed to be adiabatic, estimate the variation o f the skin temperature with altitude at a point on the surface o f t he rocket a distance of 3 m from the nose o f the rocket. U se the flat plate equations given in this chapter and assume that at the distance from the nose considered, the Mach number and temperature outside the boundary layer are the same as. those in the free-stream ahead o f the rocket. 6.22. A flat plate with a length o f 0.8 m a nd a width o f 1.2 m is placed i n the working section o f a w ind tunnel in which the Mach n umber is 4, the temperature is - 70°C, and the pressure is 3 kPa. I f the surface temperature o f t he plate is kept a t 30°C b y an internal cooling system, find the rate a t which heat must b e added to or removed from the plate. Consider both the top and the bottom o f the plate. 6.23. A t a n altitude o f 30,000 m the atmospheric pressure is approximately 1200 P a and the temperature is approximately - 4SoC. Assuming a turbulent boundary layer flow over an adiabatic flat plate, plot the variation o f the adiabatic wall temperature with Mach number for Mach numbers between 0 a ndS. 6.24. Air at a pressure o f 29 kPa and a temperature o f - 3SoC flows a t a M ach number of 4 over a flat plate. The plate is maintained a t a uniform temperature o f 90°C. I f the plate is O.S m long, find the mean rate o f h eat transfer per unit surface area assuming a two-dimensional turbulent boundary layer flow. REFERENCES 1. Schlichting, H., Boundary Layer Theory, 7 th ed., McGraw-Hill, New York, 1979. 2. 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