Back to Introduction to Convective Heat Transfer Analysis Home
Table of Contents
C HAPTER 5 a Introduction to Turbulent Flows 5.1 INTRODUCTION As discussed i n C hapter 2, the values o f t he variables such as velocity and temperature fluctuate with time when.the flow i s turbulent. This is illustrated for the case o f temperature i n Fig. 5.1. I t i s not, a t t he present time, possible on a routine basis to solve t he governing equations to obtain t he variation o f t he flow variables with time i n turbulent flow. In most analyses o f turbulent flow i t i s therefore usual to express the variables i n terms o f a time-averaged mean value plus a time-varying deviation from this mean value, i.e., as discussed i n Chapter 2 [1],[2],[3],[4],[5], to express the variables i n the following way: u = I i + u'~ p v . .:. 11 + v', T=T w = w + w', = p+ p', + T' (5.1) where the primes denote the fluctuating components and where the mean values, indicated b y t he over/bar, are defined, for the case where the mean flow is steady, ' . by: Ii = ! (t u dt, t Jo 11 = .!. (t v dt, t Jo w =;= .!. (t w dt, t Jo (5.2) p= ! t Jo (t p dt, T= .!. (t T dt t Jo Here t i s a suitably long integration time. A solution for the mean values o f the flow variables is then sought. In order to illustrate the main features o f t he analysis of turbulent flow, attention will b e restricted to two-dimensional boundary layer flows and to axially symmetric pipe flows. I t w ill also be assumed that t he fluid properties are constant and that the mean flow is steady. / 227 228 Introduction to Convective Heat Transfer Analysis Temperature, T 11ean Temperature, T Time, t FIGURES.1 Variation of temperature with time in turbulent flow. T he equations governing the variation o f the mean flow variables in such flows were discussed i n Chapter 2. T hese equations contain, besides the mean flow variables, additional variables that depend o n t he fluctuating turbulence quantities (see below). There are thus, more variables i n t hese governing sets o f equations than there are equations. This is tenned the "turbulence closure problem". In order to solve turbulent flow problems, it is necessary therefore to relate the extra turbulence terms to the mean flow variables, i.e., to introduce a "turbulence model". T his, at the present, essentially always involves some degree o f empiricism. The degree to which empiricism is introduced and the way in which it is introduced varies considerably between models. Some highly involved equations for these turbulence t enns h ave been developed i n which i t is only necessary to specify the experimentally measured values o f a relatively small number o f fundamental constants. Attention will be restricted to relatively simple turbulence models i n the present book. In discussing these turbulence models it will b e necessary to introduce empiricism at certain points in the discussion. While every effort w in b e made to justify the way i n w hich this empiricism is introduced, this will not always b e possible without a more thorough·discussion of the background experimental results than c an b e given here. In the present chapter and in the following two chapters, which are concerned with turbulent boundary layer flows and with turbulent duct flows, respectively, consideration will b e restricted to forced flows, i.e., the effect o f b uoyancy forces on the mean flow and on the turbulence structure will be assumed to be negligible. Some discussion o f the effect o f buoyancy forces on turbulent flows will be given in Chapter 9. 5.2 G OVEllNING E QUATIONS As mentioned above, attention will be restricted to constant fluid property boundary layer and pipe flows. The equations governing these types o f flow were discussed CHAPfER 5: Introduction to Turbulent Flows 229 briefly in Chapter 2. In the present section these equations will be reviewed and some points concerning their form will be discussed. The equations governing two-dimensional, constant fluid property turbulent boundary layer flow are [2],[3]: a u+av = 0 ax ay _au _au u- + vax ay = --- (5.3) -(v u ) 1 dp p dx + v- - a2u ay2 a - ,-, ay (5.4) _ aT + _aT - (- k ) a2T - -a (v'T') · u- v- _ ax ay pCp ay2 ay (5.5) As discussed above, this set of three equations contains, in addition to the three mean flow variables u ,v, and T, the two turbulence quantities v'u' and v'T', Now since the molecular shearing stress, Tm, and the molecular heat transfer, qm, are given by: Tm and: = f .J,- = au ay au p v- ay (5.6) qm = - k - = - pcp - aT ay aT . ay . (5.7) Eqs. (5.4) and (5.5) can b e written as: _au) p (_au u -+pvax ay and: = d p aTm a - ,-, - -+---(pvu) d xay ay . (5.8) (5.9) A consideration of the right-hand sides of these two equations indicates that the turbulence terms i n these equations have, as discussed i n Chapter 2, the form of additional shearing stress and heat transfer terms although they arise, of course, from the momentum transfer· and enthalpy ·transfer produced by the mixing that arises from the turbulence. Because of their similarity to the molecular terms, the turbulence terms are usually called the turbulent shear stress and turbulent heat transfer rate respectively. Thus, the followitlg are defined: 7'r = - pv'u' (5.10) / and: q r = pcpv'T' (5.11) / 7'r being the turbulent shear stress and q r being the turbulent heat transfer rate. 230 Introduction to Convective Heat Transfer Analysis I n terms o f these quantities, the momentum and energy equations (5.8) and (5.9) can be written as: p (_au + _au) = -dp + -a + TT) u- vax ay dx a(Tm y (5.12) and: (5.13) In most turbulent flows, TT i s much greater than Tm and q T is much greater than q m over most o f the flow field although this will not be the case near the wall. B y analogy with the form o f the molecular shearing stress and molecular heat transfer rate relations, as given i n Eqs. (5.6) and (5.7), i t is often convenient to express the turbulent shearing stress and turbulent heat transfer i n terms o f the velocity and temperature gradients i n the following way: 'TT = - pv u , , = P €au ay - PCp€H (5.14) and: qT = pCpv -,-, T= aT ay (5.15) where € is known as the eddy viscosity and € H is known as the eddy diffusivity (sometimes named eddy conductivity). It must, of course, b e clearly understood that € and € H are not, like v and a , properties o f the fluid involved alone b ut d epend primarily on the turbulence structure at the point under consideration and hence on the mean velocity and temperature at this point and the derivatives o f these quantities as well as on the type o f flow being considered. The use o f € a nd € H does not, i n itself, constitute the use o f a n empirical turbulence model. I t is only when attempts are made to describe the variation of € and € H through the flow field o n t he basis o f certain usually rather limited experimental measurements t hat t he t erm e ddy viscosity turbulence model is applicable. I n fact, even when advanced turbulence models are used, i t is often convenient to express the end results i n terms o f t he eddy viscosity and eddy diffusivity. I f Eqs. (5.14) and (5.15) are used to describe 'TT a nd q T, the momentum and energy equations (5.8) and (5.9) can b e rewritten i n the following way because the fluid properties are being assumed constant: _au +v_au uax ay and: = --- 1 dp a[ au] pdx + - y (v+€)J ay a[ aT] ay (a + €H)ay (5.16) aT aT u - +vax ay =- (5.17) CHAPTER 5: Introduction to Turbulent Flows 231 Now i t w ill b e noted that the Prandtl number is defined such that: P r = /Lcp = /LIp k klpc p € =v a (5.18) B y analogy with this, i t i s often convenient to define the following: PrT =€H (5.19) PrT t hen being termed the turbulent Prandtl number. In order to a void confusion, P r is then sometimes referred to as t he molecular Prandtl number. U sing t hese expressions, the energy equation (5.17) can be written as: u~~ +V~~ = vA(~r +: p~ )~~] : (5.20) Eqs. (5.3), (5.16), and (5.20) are basically the form o f t he governing equations that will b e u sed i n t he analysis o f t urbulent boundary layer flows. As mentioned before, attention will also b e g iven to turbulent pipe flows. I f t he s ame coordinate system that was used i n t he discussion o f l aminar pipe flows is adopted, i.e., i f t he coordinate system shown i n Fig. 5 .2 i s u sed, the equations governing turbulent pipe flow are, i f assumptions similar to those used i n dealing with boundary layer flows are adopted and i f i t i s assumed that the~ i s no swirl, as follows: au + - -(vr) = 0 1a az rar _au +v- = - 1 d- + - - (r- - --(rv,-,) _au p v a au) 1 a - u uaz ar p dz r ar ar r ar ~u -+v-, = ' (k) - - ~r - ---(rvT) ~aT _aT 1 a aT) 1 a -,-, ax ar -pcp r ar ar r ar - (5.21) (5.22) (5.23) The above set o f t hree equations also involves, beside the three m ean flow variables u, v, a nd T, t he two turbulence quantities v'u' and v'T'. T hese quantities also have the form o f additjonal s hear stress and heat transfer, respectively. F or t his reason the momentum a nd energy equations are often written i n t he following way j ust a s FIGURES.2 Coordinate system used i n the analysis o f turbulent pipe flow. / 232 Introduction to Convective Heat Transfer Analysis was done with the boundary layer equations: p(_au + _au) = u- vaz ay and: -- dp + - - [r(Tm + 7'r)] 1a . dz rar (5.24) (5.25) where Tm and 7'r are again the molecular shear stress and turbulent shear stress respectively, and qm and qT are the molecular heat transfer rate and turbulent heat transfer rate respectively. I n turbulent pipe flow it is again also often convenient to write the turbulence quantities i n terms o f the eddy viscosity and diffusivity and when this is done the momentum and energy equations become: _au _au u -+vaz ay and: = -~-.-+-- pdz 1 dp a [, au] r(v+€)rar ar 1 (5.26) aT aT ua- +vz ay 1 = -- a[ aT] r ar rea + € H)ar (5.27) The energy equation, i.e., Eq. (5.25), can alternatively b e written as: aT aT v a [ ( 1 € 1 ) aT] u az +V ay = r ar r Pr + v PrT ar (5.28) where the turbulent Prandtl number as defined i n Eq. (5.19) has been introduced in this equation. . Consider!fuJly developed flow in a plane duct, i.e., essentially fully developed flow between parallel plates. I f the velocity distribution in the flow near the center line can be approximately represented by: E XAMPLE 5 .1. I ~. = (~)ln where u 1 is the mean center line velocity, y is the distance from the wall, and w is the half-width of the duct, find the variation of E and E H in this portion of the flow. Solution. The flow situation being considered is shown in Fig. E5.1a. In fully developed flow, u will not depend on z and hence by virtue of the continuity equation v will be O. Therefore, for the fully developed flow, the momentum equation gives: i.e.: - (7'm d dy + 7'T) - _ dp dz / CHAPTER 5: Introduction to Turbulent Flows 233 Velocity Distribution FIGURE ES.la But dpldz is a constant in fully developed flow, i.e., is independent of z. Therefore, the above equation indicates that the total shear stress, i.e., 'Tm + 'T'r, varies linearly with y. But, on the center line, i.e., at y = w,dU/dy = 0, so the total shear stress is zero on the center line. Therefore, i f the wall shear stress is 'Tw , it follows that: ('Tm+~)=I_l. ' Tw' W Except in the region very close to the wall 'Tm « 'T'r. Hence, in the region here being considered: ' T'r=I_ 'Tw I w I t is next noted that in the case here being considered: du 'TT = P Edy From the above two equations i t follows that: ( 1- ylw) E = 'Tw p(duldy) But the assumed velocity profile gives: From the above equations, it then follows that: E =( =7 7'TWW) 1 - ylw PUl (ylw)-6rl Le.: ~ WUl 'Tw [ 1 - ( I p ui W Y )~ '(w )6rl / The variation of E across the duct as given by this equation is shown in Fig. E5.1b. It should be noted that the equation does not really apply for values of ylw near O,i.e., near the wall. 234 Introduction to Convective Heat Transfer Analysis 2.5 . . . - - - - - - - - - - - . . . . . - - - - - - - - - , 2.0 1.5 1.0 0.5 0.0 I -_ _ _ _ ~ _ _ __J'___ _ _ _ _ __ J M ~ 1~ y /w. F IGURE E5.1b I f the turbulent Prandtl number is assumed constant then since: E EH=-" PrT it follows that: EH _ WUl - PrT 7 . Tw pur 1 [ ( Y )~( Y )6f1 - W w. The variatiOD:9f E H as given by this equation assuming PrT = 0.9 (see later) is also shown in Fig. E5.16. 5.3 MIXING LENGTH t uRBULENCE MODELS T he simplest turbulence model is based on the assumption that relations between E a nd E H (or P rT) a nd t he mean flow variables c an b e obtained from measurements in relatively simple flow situations and that these relations!.are then more universally applicable. S uch a n approach has, however, not m et with a great deal o f success except i n t he analysis o f free boundary flows, e.g., i n j ets and wakes. Another method o f trying to describe the turbulence terms i n the above equations is b y m eans o f P randtl's mixing length theory. The mixing length concept will be introduced i n this section and some simple turbulence models based on this concept will b e discussed [1],[2],[3],[6];[7]. I n the mixing length theory i t is assumed basically that " lumps" o f fluid ate carried transversely across the fluid flow by t he turbuI.ent eddies and during this motion they preserve their initial momentum a nd enthalpy. T he motion continues over a transverse distance, after which the " lumps" interact with other fluid "lumps" em, , C HAPTER 5: Introduction to Turbulent Flows 235 giving rise to the observed velocity a nd t emperature fluctuations i n t urbulent flow. I n order to derive a n expression for t he t urbulence terms using this idea, consider a two-dimensional flow a nd c onsider conditions on three planes 0 , 1, a nd 2 s hown in Figure 5.3, p lanes 1 a nd 2 lying a t a d istance ~m f rom plane O. I f t he mixing length, ~m, i s assumed to b e s mall compared to the overall extent o f the flow then the velocity on plane 1, i.e., U J, will b e r elated to that on plane 0, Le., Uo, by: Therefore, when the fluid " lump" originating on plane 1 arrives on plane 0 it has, since its x-momentum is assumed to h ave b een conserved during this motion, a velocity excess that is given by: Similarly a fluid " lump" o riginating on plane 2 arrives on plane 0 w ith a v~locity excess which is given by: Auz = -Cm(~~)o I n t he mixing length model it i s n~xt a ssumed that the magnitude o f t he velocity fluctuation a t 0 is proportional to t he a verage o f IAutl a nd IAu21, Le.: lU'l IX "2 [IAUtl +I AU21] Le.: 1 i.e.: lU'l = Ku~m I~~ I (5.29) I n t he mixing length model i t i s n ext a ssumed that the transverse velocity fluctuation v' arises because o f t he " collision" o f t he fluid " lumps" a rriving on plane o from planes 1 and 2 with different momentums. I n order to satisfy continuity Mean ,...- ..- .-....-.,... Plane 1 Mean Velocity Uo --1·-··-·-·····,-·····----·--··::·-······-·····.····-·-_...- ..r.. Plane 0 Mean 1-•••••••••••.••• - ••._ . Plane 2 F IGURE 5.3 Planes in flow considered in mixing length analysis. 236 Introduction to Convective Heat Transfer Analysis requirements, it follows that: Iv'l ex: lu'l i.e., that: (5.30) Lastly, it is assumed that turbulent stress is proportional to Iv'lIu'l, i.e., that: 'IT = - pv'u' ex: plv'lIu'l (5.31) i.e.: I n writing these equations, account has been taken of the fact that the sign o f 'IT depends on the sign of au/ay. Hence, lau/ayl(au/ay) has been used instead of (uu/()y)2. The above equation for the turbulent shear stress is conveniently written as: ' IT-P1- _ o21aul - au ay ay (5.32) where the constants of proportionality K", Ku, and Kv have been combined with em to give e. eis what is conventionally termed the "mixing length"'. O f course, Eq. (5.32) and the analysis that led to it, gives no idea of the distribution of the mixing length in the flow field. However, it has been found that experimental measurements of the distribution of ein certain simple flows can often be used as the basis for deducing the distribution in more complex flows. This point , will be discussed later. A comparison ofEqs. (5.1\4) and (5.32) shows that the eddy viscosity and mixing length are related by: (5.33) Next consider the temperature fluctuations. In moving from plane 1 t o plane 0 shown in Figure 5.3, the fluid "lump" is assumed to preserye its enthalpy and, hence, because the fluid properties are being assumed constant, its temperature. Therefore, the "lumps" arriving at 0 from 1 have a temperature excess of: .1T\ = tm (~~ )0 (~~ f5.34) Similarly, the "lumps" arriving at 0 from 2 have a temperature deficit of: .1Tz = - tm 1 (5.35) / CHAPTER 5: Introduction to Turbulent Flows 237 As with the velocity fluctuation, it is assumed that the magnitude of the temperature fluctuation at 1 is proportional to the average of IdTtl and IdT21, Le.: IT'I ex: 2[ldTd + IdT21l Le.: (S.36) It is also assumed that the turbulent heat transfer rate is is proportional to Iv'IIT'I, i.e., that: qr 1 = - pcpv'T' ex: pCplv'I!T'1 (S.37) Le.: qr = - pcpKr.f2 l aUr aT --m a y ay i.e.: qr = - pcpC.e2 a y a~ Idrilar (S.38) where K r and C are constants of proportionality. In writing Eq. (S.38), accounthas been taken of the fact that the sign o f q r depends on the sign o f a Tlay. A comparison ofEqs. (S.lS) and (S.38) shows that the eddy diffusivity and mixing length areretated'by: €H = c.e21~1 (S.39) Using Eqs. (S.33) ang (S.39) it follows that according to the mixing length theory the turbulent Prandtl ,nU.tltber is given by: 1 (S.40) €H C i.e., according to the mixing length theory, the turbulent Prandtl number is a constant. I n many flows this constant can be taken as 1. However, in some circumstances Prr can be very different from 1 and.can vary significantly through the flow field which is, of course, in disagreement with the simple mixing length theory. / The mixing length analysis presented above does not directly provide a turbulence model. I t is only when the mixing length distribution and the value o f the turbulent Prandtl number in the flow are specified based on existing experimental studies that it yields a turbulence model. The advantage of the mixing length concept is that it should be far easier to deduce fairly general relations for the mixing length distribution from experimental results obtained in relatively simple flows than Prr = - € =- 238 Introduction to Convective Heat Transfer Analysis i t would be to deduce, say, general relations for the eddy viscosity and diffusivity distributions from such measurements . . The mixing length relation can also be deduced by starting with the assumption that the turbulence at any point in the flow can be characterized by a single length scale, eT, and a single velocity scale, UT. From this it follows that: . E· = function (eT, UT) (S.41) (S.42) and: EH = function (eT, UT) Dimensional analysis then indicates from this that: .eTUT - - = constant, i.e., E = CleTUT E (S.43) and: E • - H = constant,' I.e., E H = c2 e UT T eTUT . . (S.44) I f it is then assumed that because the length· scale, eT, is small compared to the overall size of the flow: U T.= e Tay laul (S.45) Combining the above results then gives: E = C leT ay . zlaul (5.46) and: (5.47) or setting: (5.48) the above equations give: E = e21~~1 Cl (S.49) (S.50) E H· C2 e21aul ay = C2/Cl. These are the same as the equations derived above i f C E XAMPLE 5 .1. Consider flow in the outer portion of the turbulent boundary layer o n a flat plate. I f the velocity distribution i n this flow is approximately given by: ==(*f / CHAPTER 5: Introduction to Turbulent Flows 239 where U1 is the free-stream velocity, y is the distance from the wall, and 8 i s the local boundary layer thickness, find the variation o f E i n this portion o f the flow b y assuming that the mixing length is a constant and equal to 0.098. Solution. Using: gives: i.e.: 5.4 MORE ADVANCED TURBULENCE M ODELS The mixing length model, with the mixing length distribution specified using empirical relations, has given satisfactory solutions to many important problems but has the disadvantage that the mixing length distribution varies considerably from <me type of flow to another and also can vary quite considerably from one part of a flow . to another. It is, therefore, only possible to extrapolate in a very limited way from situations in which the mixing length has been experimentally measured. This was not of great consequence before the widespread availability of digital computers because i t was only possible th~ to solve the governing equations for certain simple flow situations, which were sihriIar to those in which the mixing length· distribution had been experimentally measured. However, since the equations governing turbulent flow can now be I1:umerically solved for complex flows, an obvious need has arisen for a turbulence ~el that is applicable to all flow situations, the only empirical inputs into this/model being the values of one or more universal constants whose values can be determined from measurements in simple flows. No such truly general model has yet been developed but considerable progress has been made in the development of models that are more general than the' mixing length, see [8] to [21]. Almost all o f these advanced turbulence models use additional differential equations to describe the turbulence terms. The most widely used such equation is that governing the turbulent kinetic energy which will here be denoted by K. This quantity is, as discussed in Chapter 2, given by: K = (u lZ + vIZ + W'2)/2 (5.51) / An equation governing this quantity was derived in Chapter 2. I f the mean flow is two-dimensional, this equation is: 240 Introduction to Convective Heat Transfer Analysis a a u -K + v -K = - [ --au + -u'v')- + -u'v')- + -v,2)_ (u,2)_ ( -au ( -av ( -av] ax ay ax ay ax ay - - u,L + v,L + w,L a' p ax ay az 1[4' 4]' l _(u,3) a a a - - [ a - + - (u t2 v') + _ (u,2 w') + _ (v,2 u') 2 ax ay az . ax . a- a a a a-] + _ (v,3) + _ (v,2 w') + _ (w,2 u') + _ (w'2v') + _ (w'3) ay az ax ay az + (~ ) [u'.1.2 u' + v'.1.2v' + w'.1.2w'] (5.52) l it o rder to utilize this equation it i s n ecessary to use other equations to describe some o f t he terms in this equation and/or to model some o f the terms i n t his equation. To illustrate how this is done, attention will b e given to two-dimensional boundary layer flow. F or two-dimensional boundary layer flows the turbulence kinetic energy equation, Eq. (5.52), has the following form, some further rearrangement having been undertaken: _ aK + _aK = - -,v ,)au - -ap'v' - -a (q'2v' ) 1 u- vu (-ax ay ay p ay ay --a2K au' au' av'.av' aw' aw' + v--v - - + - - + - ay2 ay ay ay ay ay ay (5.53) Here: qt2 = u t2 + v t2 + w t2 (5.54) Eq. (5.53) basically states that at a ny p oint in the flow: Rate o f c onvection Of K = R ate o f p roduction o f K / + R ate o f diffusion o f K + Rate o f dissipation o f K (5.55) T he diffusion a nd dissipation terms contain unknown turbulence quantities and m ust b e modeled. Here, a very basic t ype o f model will b e discussed. T he diffusion term is assumed to have the same form as· t he diffusion terms i n t he other conservation equation, i.e., it i s assumed that: . N et r ate o f diffusion o f K = CI aa . y (e aK) ay (5.56) b eing a constant. T he dissipation term. c an b e m odeled b y considering the work done against the drag force o n a fluid lump as i t m oves with the turbulent motion [22]. Now, i f a b ody h as a characteristic size, R, a nd i s m oving with velocity, V, r elative to a fluid, CI / CHAPTER 5: Introduction to Turbulent Flows 241 i t experiences a drag force, D, whose magnitude is given b y a n equation that has the form: (5.57) where CD i s the drag coefficient. The rate a t which work is done on this body will then be given by: W= . W , 1 D V = - CD Py3 R2 2 (5.58) The rate a t w hich work is done per unit mass is then given by: (1I2)CDPV3 R2 = CvpR3 = e lf y3 (5.59) where the volume o f t he body has been expressed as: Volume = CvR3 Cv being a constant whose value depends on the body shape. Now, consider a fluid lump moving in a turbulent flow. Relative to t he mean. flow its m ean velocity is: .J:li + va + W'l = .fiK Therefore, i f t he characteristic size o f t he fluid " lumps" i s L, the rate at which the mean flow does work against the turbulent motion is from Eq. (5.59) g iven by: = :---=---:=' c2 K312 L (5.60) This result could have be~n obtained b y simply assuming that the rate o f dissipation depends on K a nd L alone. Eq. (5.60) could then have been derived by dimensional analysis. ,Substituting Eqs. (5.57) a nd (5.60) into Eq. (5.53) gives the following modeled form o f the turbulent kinetic energy equation for two-dimensional boundary layer flow: aK oK - ,-, au a (OK) c2K312 Uax + v oy = - (u v ) ay + Cl oy \€ oy - L (5.61) I f the turbulent stress in the production term is written in terms o f the eddy viscosity this equation becomes: _oK + _aK u- voxay =€- (a11)2 + C l - (€ - -c- 0 OK) 2K312 ay oy oy L ($.62) In order to utilize this equation to determine the turbulent shear stress, i t is necessary to obtain a n additional equation relating, for example, the eddy viscosity to the quantities involved in the turbulent kinetic energy equation. I f i t is assumed that: € / = function (K, L ) ( 5.63} 242 Introduction to Convective Heat Transfer Analysis then dimensional analysis gives: € = cTKll2L (5.64) where C T is a constant. Eqs. (5.62) and (5.64) together then allow the variation o f € to b e obtained by solving these equations simultaneously with the momentum and continuity equations. The turbulence model discussed above contains three constants C l, C2, and C r together with the unknown length scale, L. I t is assumed that the values o f these constants and the distribution o f L, a re universal, i.e., once their values have been determined from measurements in simple flows i t is assumed that these values will b e applicable in all flows. This has not however, proved to b e the case. The constants have been found to vary slightly with the type o f flow and also to b e Reynolds number-dependent in some circumstances, viscous effects having been neglected in the above derivation. The length scale distribution has also been found to vary quite considerably with the type o f flow. Despite these failings the turbulent kinetic .energy model has proved to be· superior to simpler models in some circumstances. I t is o f interest to note·that i f the convection and diffusion terms are negligible i n t he turbulence kinetic energy equation, i.e., i f t he rate o f production o f kinetic energy is j ust equal to t he rate o f dissipation o f turbulence kinetic energy, Eq. (5.62) reduces to: . (5.65) I f Eq. (5.64) is used to give K Eq. (5.65) gives: = (c~)2 TL C2 € € (a-)2 = L (c)3. ay TL U i.e.: € = (Cf )112 L 21 d u 1 C2 dy . (5.66) This is the same form as the result given by the basic mixing length theory, i.e., the turbulence kinetic energy equation gives the same result as the mixing length model when the convection and diffusion terms are neglected i n t he turbulence kinetic energy equation. Except near a wall, however, all the terms in t he turbulent kinetic equation are significant i n most cases. In applying the turbulence kinetic energy model i t is common to assume that the turbulent Prandtl number, P rr, i s constant. In the case o f axially symmetrical pipe flow, the turbulent kinetic energy equation has the following form when t he terms are modeled i n some way as was done CHAPTER 5: Introduction to Turbulent Flows 243 with the equation for boundary layer flow: u8K + v8K 8x 8r = E (8U)2 8r + CI ~ (rE 8 K) _ cZK3/2 r 8y 8r L (5.67) Eq. (5.64) being applicable without modification in this coordinate system. In an effort to overcome the deficiencies in the turbulent kinetic energy equationbased turbulence model discussed above, particularly with regard to the need to specify the distribution of the length scale, other differential equations have been developed to supplement the kinetic energy equation. The most commonly used such· additional equation is obtained by making the dissipation of turbulent kinetic energy a variable; i.e., i f attention is again given to boundary layer flow, to write the turbulent kinetic energy equation as: u8K + v8K 8x 8y = E (8U)2 8y + CI ~ (E 8K )_ E 8y \ 8y (5.68) where E is the rate of dissipation of turbulent kinetic energy. An additional differential equation for E is then developed, this equation having a similar form to the turbulent kinetic energy equation. For example, for boundary layer flow, the dissipation equation has the form: U 8x 8 E' + v8y 8E = C3 E 8 y (8U)2 E K + C4 8y \ E 8y - • 8 ( 8 E) E2 Cs K (5.69) where C3, C4, 'and Cs are additional empirical constants. Now Eqs. (5.60) and (5.64) give: and: E = cTK 1I2 L Eliminating the length scale, L, between these two equations then gives: E = ce"E K2 (5.70) where Ce = C 2CT. Eq. (5.70) could have been derived by simply assuming that: , E = function (K, E) Dimensional analysis then directly givesEq. (5.70). The K-E turbulence model discussed above, which is often termed the k-E model because of the symbols originally used for K and E, contains a number o f empirical constants. 1)rpically assumed values for these constants are: C I= 1, C3 = 1.44, C4 = 0.77 / Cs = 1.92, Ce = 0.09 244 Introduction to Convective Heat Transfer Analysis I n o rder to u se this model i n t he prediction o f h eat transfer rates i t i s usual to also assume that: PrT = 0.9 T he K-E turbulence model discussed above only applies when E » v. This will not b e t rue n ear t he wall. T he m ost common way o f dealing with this problem is to a ssume that there is a " universal" velocity distribution adjacent to the wall :and the K-E turbulence model is then only applied outside o f t he region i n w hich this wall region velocity distribution applies. Alternatively, more refined versions o f the K-E t urbulence model have been developed that apply under all conditions, i.e.~ across the entire boundary layer. 5.5 ANALOGY SOLUTIONS F OR HEAT TRANSFER IN TURBULENT F LOW . M any e arly efforts at trying to theoretically p redict h eat t ransfer rates i n turbulent flow concentrated on trying to relate the ·wall h eat transfer rate to the wall shearing stress. T he r eason for this was that a considerable body o f e xperimental a nd semitheoretical knowledge concerning this shearing stress i n various flow situations was available a nd t hat the mechanism o f h eat t ransfer i n t urbulent flow is obviously similar to the mechanism o f m omentum transfer. S uch solutions, which g ive t he heat transfer rate i n t urbulent flow i n terms o f t he wall shearing stress, are term~ analogy solutions [23],[24],[25]. I n outlining t he m ain steps i n o btaining a n analogy solution, attention will here b e g iven to two-dimensional flow. T he " total" s hear stress a nd " total" h eat transfer rate are mad~~up o f t he molecular a nd t urbulent contributions, i.e.: :T=~+~=~~-~u , au, y ( 571) . and: aT - ,-, q = qm + qT = - k ay - pCpv T which c an b e w ritten as: T = p (v (5.72) au + E )ay (5.73) and: q = - pcp(a + EH) a y aT (5.74) / I n t he usual analogy solution approach, Eqs. (5.73) a nd (5.74) a re rearranged to give: CHAPI'BR 5: Introduction to Turbulent Flows 245 au ay and: T p {v + E) (5.75) (5.76) and these equations are then integrated outward from the wall, i.e., y = 0 , to some point in the flow at distance y from the wall. This gives: u -- 1 y T 0 p {v + E) d y (5.77) (5.78) Tw - T = - 1Y q dy o p cp(a + E H) The relationship between E and E H a nd between T, and q is then assumed. The assumed relation between T and q will us~ally involve the values o f these quantities at the wall. The integrals i n Eqs. (5.77) and (5.78) can then be related and eliminated between the two equations leaving a relationship between the wall heat transfer rate· and the wall shear stress and certain mean flow field quantities. The application o f the analogy approach to turbulent boundary flow and to turbulent duct flow will be discussed i n Chrtpters 6 a nd 7, respectively. 5.6 NEAR-WALL R EGION The presence o f the solid wall has a considerable influence on the turbulence structure near the w alt Because there c an b e no flow normal to the wall near the wall, Vi decreases as the wall is a1\>proached a nd as a resUlt t he turbulent stress and turbulent heat transfer rate are negligible i n t he region very near the wall. This region in which the effects o f the turbulent stress and turbulent heat transfer rate can b e neglected is termed the "sublayer" or, sometimes, the "laminar sublayer" [1],[2], [26],[27],[28],[29]. In)his sublayer: T = . pv~ au ay (5.79) and: (5.80) y being measured from the wall into the flow. Further, because the sublayer is normally very thin, the variations o f the shear stress, T and q, through this layer are usually negligible; i.e., i n t he sublayer i t can be assumed that: / (5.81) 246 Introduction to Convective Heat Transfer Analysis the subscript w denoting conditions a t the wall. Substituting these values into Eqs. (5.19) and (5.80) and integrating the resultant equations outward from the wall gives: _ U =-y pv Tw (5.82) and: (5.83) The velocity and temperature distributions i n the sublayer are thus linear. I f the following are defined: . u* = ff, u+ = - U u*' yu* y+ = V (5.84) u* being termed the "friction velocity", then the velocity distribution in the sublayer as given in Eq. (5.82) can be written as: {5.85) , Eqs. (5.82) and (5.83) have been found to adequately describe the mean velocity and temperature distributions from the wall out to y+ = 5; i.e., the sublayer extends . from y+ = 0 to y + = 5. For y* > 5 t he turbulence stress and heat transfer rate become important. However, near the wall the total shear stress and total heat transfer rate will remain effectively constant and equal to the wall shear stress and wall heat transfer rate, re- .. spectively. The size o f the turbulent "eddies" near the wall is determined by the distance from the wall; i.e., near the wall i t i s to b e expected that their size will increase linearly with distance from the wall. Now, the mixing length is related to the scale o f the turbulence,. i.e., to the size o f t he "eddies," and i t i s to b e expected therefore that near the wall: . e = KvY Kv being a constant tenned the von Karman constant. Using the ass~iions discussed above then gives: Tw = (5.86) P( V 2 0u . + Kv y2 - y )Ou a ay . (5.87) I f i t is further assumed that v « E, the above equation becomes: i.e., au K vY- = ay ffw * - (= u ) p / CHAPTER 5: Introduction to Turbulent Flows 247 2 5.--------.---------,---------, Inner Region Buffer Region O uter Region 20 15 • . ,I :" I I I : 10 Eq. (5.90) I I 5 I I o~----~~----~--~--------~ I I I I I I 1 10 100 1000 F IGURE 5.4 M ean velocity distribution near wall. i.e., (5.88) Integrating this equation gives: u+ = - -lny+ 1 Kv +C (5.89) where C is a constant whose value cannot directly b e determined because this expression, which is b asegon the assumption that v « € , does not apply very near the wall. With Kv set equal to 0.4 and C set equal to 5.5, Eq. (5.89) provides a good description o f the mean velocity distribution for y+ > 30 as shown i n Figure 5.4. Thus, Eq. (5.85) applies from the wall out to' y+ = 5 while Eq. (5.89) applies for y+ > 30. Between y+ = 5 and y+ = 30, where both the molecular and t he turbulent stresses are important" experiments indicate that the velocity distribution is given by: u+ = 5 lny+ - 3.05 (5.90) This region between y+ = 5 and y+ = 3 0 is termed the "bUffer" region. The velocity variations given by Eqs. (5.85) and (5.90) are also shown i n Figure '5.4. S~7 -:TRANSITION FROM LAMINAR TO TURBULENT FLOW ~"" An predicting heat transfer rates it is important to know i f the flow remains lami,iar or whether transition to turbulence occurs. I f transition occurs, i t is usually also .. 2 48 Introduction to Convective Heat Transfer Analysis F IGURE 5.5 Laminar flow, transition, and fully turbulent regions. important to know where the transition occurs. I t is also important to realize that there is a transition region b etween the region o f laminar flow and the region o f fully turbulent flow, as illustrated in Figure 5.5. The conditions under which transition occurs depend on the geometrical situation being considered, on the Reynolds number, and on the level o f unsteadiness i n t he flow well away from the surface over which the flow is occurring [2], [30]. For example, in the case o f flow over a flat plate as shown in Figure 5.6, i f the level of unsteadiness i n the freestream flow ahead o f the plate is very low, transition/from laminar to turbulent boundary layer flow occurs approximately when: R ex ( = UIX/V) = 2.8 X 106 a nd fully turbulent flow is achieved approximately when: (5.91) R ex = 3.9 X 106 (5.92). T he level o f unsteadiness i n t he freestream is usually specified using the following quantity:- . ' (5.93) - -.. Flow with - -.. low level of - -.. unsteadiness Laminar Flow Fully Turbulent Flow F IGURE 5.6 Transition i n boundary layer flow over a flat plate. CHAPTER 5: Introduction to Thrbulent Flows 249 where U l i s mean freestream velocity a nd u', v', a nd w' are the instantaneous deviations o f t he velocity from the mean value. J i s often termed the freestream turbulence level and is commonly expressed as a percentage. For the values o f J normally encountered i n practice, i.e., approximately 1%, the transition Reynolds number is usually assumed to be: R ex = 3.5 X 105 to 5 X 105 and fully turbulent flow is usually assumed to b e achieved when: (5.94) R ex = 9 X 105 to 1.3 X 106 (5.95) For boundary layer flows on bodies o f o ther shape, the transition Reynolds number based on the distance around the surface from the leading e dge o f t he body is usually increased i f t he pressure is decreasing, i.e., i f there i s a " favorable" p ressure gradient, a nd is usually decreased i f t he pressure is increasing, i.e., i f t here is an ''unfavorable'' pressure gradient. E XAMPLE 5 .3. A ir at a temperature o f 30°C flows at a velocity o f 100 m ls over ~ wide flat plate that is aligned with the flow. T he surface o f the plate is maintained at a unifonn wall temperature o f 50°C. Estimate the distance from the leading edge o f the plate at which transition to turbulence begins and when transition is complete.. Solution. The mean air temperature is (30 + 50)/2 = 40°C. A t 40°C, for air at standard ambient pressure: ; v = 16.96 X 1 0-6 m2/s The Reynolds number at distance, x, from the leading edge is then given by: Rex = u lxlv = 100x116.96 X 1 0-6 = 5.896 X 106 x I t will be assumed that transition begins when: Rex = 4 X 105 Therefore, the distance from the leading edge at which transition begins is given by: 5.896 X 106 x =4X Rex = 105, i.e., x = 0.0678 m I t will be assumed that transition is complete when: 1 X 106 Therefore, the distance from the leading edge at which tr~sition is complete is given . by: 5.896 X 106 X = 106, i.e., x = 0.1696 m Therefore, transition begins a t 0.0678 m from the leading edge and is complete at a distance of 0.1696 m from the leading edge o f the plate. Attention is next turned to transition i n p ipe flow. It i s usual to assume that, with the level o f unsteadiness in t he flow that is usually encountered i n p ractice, transition will occur if: R eD(= UmDlv) = 2300 (5.9(j) 250 Introduction to Convective Heat Transfer Analysis Here, Um is the mean velocity across the section of the pipe, i.e.: U _ Volume flow rate through pipe m- 7T'D2/4 (5.97) For ducts with other cross-sectional shapes, it is common to assume that the same criterion as that given in Eq. (5.96) applies provided the "hydraulic diameter" is used in place of the diameter. The hydraulic diameter, as previously discussed, is defined by: D = 4 X Flow area (5.98) h Wetted perimeter This is only a relatively rough guide as very complex flows in the transitional region can arise in some cases involving noncircular ducts. 5.S CONCLUDING REMARKS This chapter has been concerned with certain fundamental aspects of turbulent flows. The time-averaged forms o f the governing equations have been reviewed. The concept of a turbulent stress and a turbulent heat transfer rate have been introduced and the eddy viscosity and diffusivity have been defined together with the turbulent Prandtl number. A turbulence model has been defined and some basic turbulence models, including those based on the mixing-length concept, have been considered. The idea. o f a universal near-wall region has been introduced and the· basic ideas involved in an analogy-type solution have been discussed. Some attention has also been given to the conditions under which transition from laminar to turbulent flow occurs under various circumstances. . P ROBLEMS 5.1. Assuming that the velocity distribution in fully developed pipe flow near the center line can be approximately represented by: ~ . :, = (~)l where is the mean center line velocity, y is the distance from the wall, R is the radius of the pipe, and n is a constant, find the variation of € and € H in this portion of the flow. Assume a turbulent Prandtl number of 0.9. 5.2. The mean velocity distribution in the outer portion of the turbulent boundary .layer on a flat plate is approximately given by: "1 :, =M where U1 is the freestream velocity, y is the distance from the wall, and l ) is the local boundary layer thickness. Find the variation of € , € H , the turbulence shear stress; and / CHAPTER 5: Introductionto Turbulent Flows 251 the turbulent heat transfer rate in this portion o f the flow b y assuming that the mixing length is a constant and equal to 0.09 0 and that the turbulent Prandtl number is 0.9. 5.3. Consider air flow a t a velocity o f 35 mls over a wide flat plate that is aligned with the flow. T he air is a ta temperature o f 20°C a nd the plate is kept at a unifonn wall temperature o f 40°C. Estimate the distance from the leading edge o f the plate at which transition to turbulence begins and when transition is complete. 5.4. Air flows a t a velocity o f 25 mls over a wide flat plate that is aligned with the flow. The mean air temperature i n the boundary layer is 30°C. Plot the mean, velocity distribution near the wall i n the boundary layer assuming that flow is turbulent and that the wall shear stress is given by: Tw Pu2 1 = 0.029 ReO.2 x 5.5. Consider transition in the boundary layer flow over a flat plate. U sing t he expression for the thickness o f a laminar boundary layer on a flat plate given i n C hapter 3, find the value o f the Reynolds number based on the boundary layer thickness a t which transition begins. 5.6. Air flows through a duct with a square cross-sectional shape, the side length of the square section being 3 cm. I f the mean air temperature in the duct is 40°C, find the air velocity i n the duct at which the flow becomes turbulent. 5.7. In t he mixing length turbulence model as discussed in this chapter, i t is assumed that the " lumps" o f fluid maintain their temperature while they move through a distance .em. In fact, because, during this motion, the " lumps" are at a different temperature from the surrounding fluid, there is heat transfer from the "lumps" leading to a temperature change during the motion. Assuming that the heat transfer, and therefore the temperature change, is proportional to the mean temperature difference, examine the changes that are required to the mixing length model to account for this effect. 5.8. Consider fully developed flow between parallel plates. The velocity distribution i n the flow near the center line can b e approximately represented by: li l ie = 1 - w (y)l where lie is the mean center line velocity, y is the distance from the center line, and w is half the distance between the plates. Determine the variations o f € a nd € H in this portion o f the flow. Assume a turbulent Prandtl number o f 0.9 5.9. Air at a temperature o f 20°C flows at a velocity o f 100 mls over a wide flat plate that is aligned with the flow. The surface o f the plate is maintained a t a uniform wall temperature. Plot the variation of the distances from the leading edge o f the plate a t which transition to turbulence begins and when transition is complete against surface temperature for surface temperatures between 20°C and 100°C. 5.10. Derive an expression for the time-averaged mass flow in the sublayer o f a turbulent boundary layer on a plane surface. How does this mass flow rate vary with the dis~ce along the surface? 2 52 Introduction to Convective Heat Transfer Analysis REFERENCES 1. Hinze, J.O., Turbulence, 2nd ed., McGraw-Hill, New York, 1975. 2. Schlichting, H., Boundary Layer Theory, 7th ed., McGraw-Hill, New York, 1979. 3. Tennekes, H. and Lumley, J.L., A F irst Course in Turbulence, T he M IT Press, Cambridge, MA, 1972. 4. Townsend, A .A, The Structure o f Turbulent S hear Flow, 2 nd ed., Cambridge University Press, London, 1976. 5. Bradshaw, P., Editor, Turbulence, Springer-Verlag, Berlin, 1978. 6. Adams, J.C., J r and Hodge, B.K., " The Calculation o f Compressible, Transitional, Turbulent, and Relaminarizational Boundary Layers over Smooth a nd Rough Surfaces Using a n Extended Mixing Length Hypothesis", A IAA Paper 77-682, Albuquerque, NM, 1977. 7. Lin, C.C., Editor, "Turbulent Flows and Heat Transfer", i n High S peed Aerodynamics a nd J et Propulsion, Vol. V, Princeton University Press, Princeton, N.J., 1959. 8. Bradshaw, P.T., Cebeci, and Whitelaw, J., Engineering Calculational M ethodsfor Turbulent Flow, Academic Press, New York, 1981. 9. Chyou, Sh.-W., and Sleicher, C .A," Call: One- and Two-Equation Turbulence Models be Modified to Calculate Turbulent Heat Transfer with Variable Properties?" Industrial and Engineering Chemistry Research, Vol. 31, pp. 756-759,1992. 10. Kolmogorov, A N., "Equations o f Turbulent Motion o f a n Incompressible Turbulent Fluid",Izv. Akad, Nauk SSSR, Ser. Fiz., VI, No. 1 -2, pp. 5 6-58, 1942. 11. Launder, B.E. and Spalding, D.B., Mathematical Models o fTurbulence, Academic Press, New York, 1972. 12. Launder, B.E. and Spalding, D.B., ''The Numerical Computation o f Turbulent Flows", Comput. Meth. Appl. Meek. Eng., Vol. 3, pp. 269-289, 1974. 13. Launder, B.W., " On the Computation o f Convective Heat Transfer i n Complex Turbulent Flows", A SME J. H eat Transfer, 110, pp. 1112-1128, 1988. 14. Maciejewski, P.K. and Anderson, A.M., "Elements o f a General Correlation for Turbulent Heat Transfer", Journal o f H eat Transfer, Vol. 118, pp. 287-293, May 1996. 15. Patankar, S.v., Numerical Heat Transfer a nd F luid Flow, Hemisphere Publ., Washington, D.C., 1980. 16. Pletcher, R H., "Progress i n Turbulent Forced Convection", A SME J. H eat Transfer, 110, 1129-1144, 1988. 17. Rodi, W., ''Examples o f Turbulence Models for Incompressible Flows", AlAA J., Vol. 20, pp. 872-879, 1982.. 18. Saffman, P.G. atl,<fWi1cox, D.C., ''Thrbulence Model Prediction for Turbulent Boundary Layers", A IAA J., Vol. 12, pp. 541-546, 1974. 19. Wilcox, D.C. and Traci,R.M., " A Complete Model o f Turbulence", A IAAPaper76-351, San Diego, CA, 1976. 20. Youssef, M.S., Nagano, Y., and Tagawa, M., " A Two-Equation Heat Transfer Model for Predicting Turbulent Thermal Fields U nder Arbitrary Wall Thermal Conditions", Int. 1. H eat Mass Transfer, Vol. 35, pp. 3 095-3104, 1992. 21. Cho, J.R. and Chung~ M.K., " A k-€ Equation Turbulence Model", J. F luid Mech., Vol. 2 37,pp.301-322,1992. 22. Eckert, E.R.G. and Drake, R M., Jr., A nalysis o fH eat a nd Mass Transfer, McGraw-Hill, New York, 1973. 23. Boelter, L.M.K., Martinelli, R.C., and Jonassen, E , "Remarks o n the Analogy between Heat and Momentum Transfer", Trans. ASME, Vol. 63, pp. 447-455, 1941. 24. Colburn, A.P., A Method for Correlating Forced Convection Heat Transfer Data and a Comparison with Fluid Friction, Trans. Am. [nst. Chem. Eng., Vol. 2 9, p p.174-ZlO, 1933; reprinted i n Int. J. H eat Mass Transfer, Vol. 7, pp. 1359-1384, 1964. CHAPTER 5: Introduction to Turbulent Flows 253 25. von Kannan, T., ''The Analogy Between Fluid Friction and Heat Transfer", Trans. ASME, Vol. 61, pp. 705-710, 1939. 26. Spalding, D.B., " A Single Fonnula for the Law o f the Wall", J. Appl. Mech., Vol. 28, pp. 455-457, 1961. 27. Kozlu, H., Milcic, B.B., and Patera, A.T., ''Turbulent Heat Transfer Augmentation Using Microscale Disturbances Inside the Viscous Sublayer", J. H eat Transfer, Vol. 114, pp. 348-353, 1992. 28. van Driest, E.R., " On Thrbulent Flow Near a W all",J. Aero. Sci., Vol. 23, pp. 1007-1011, 1956. 29. Kasagi, N., Kuroda, A., and Hirata, M., "Numerical Investigation o f Near-Wall Thrbulent Heat Transfer Taking Into Account the Unsteady H eat Conduction in the Solid Wall", J. o f H eat Transfer, Vol. 111, pp. 385-392, 1989. 30. Chandrasekhar, S ., Hydrodynamic a nd Hydromagnetic Stability, Clarendon Press, Oxford,1961.