Equilibrium Convect ion

Similarity Analysis in Forced Turbulent Boundary Layers

With and Without Pressure Gradient Xia Wang*

Luciano Castillot Rensselaer Polytechnic Institute, Troy, NY 12180

A similarity analysis has been developed for a 2-D forced convection turbulent bound- ary layer with and without pressure gradient. Two new inner and outer temperature scalings are derived by means of similarity analysis of the equations of motion. The new scalings will be verified with the experimental data subject to adverse pressure gradient (APG) and zero pressure gradient (ZPG), and will be compared with various scalings. It will be shown that the mean temperature proflles are dependent on the external pressure gradient and the initial conditions. However, using the new scaling in inner variables or in outer variables, the temperature proflles collapse into a single curve. Thus, the true asymptotic solution for the temperature fleld exists even at a flnite Pe'clet number. The asymptotic temperature proflle found in the present analysis are in agreement with the fact that an asymptotic velocity proflle exists if the mean velocity deflcit proflle is normalized by the Zagarola & Smits scaling.'

Nomenclature Introduction

momentum thickness, Jr &(l - &)dy thermal boundary layer thickness thermal displacement thickness, outer temperature scaling inner temperature scaling outer turbulent heat flux scaling inner turbulent heat flux scaling friction velocity, u: = ~ , / p friction temperature, TT = free stream velocity free stream temperature wall temperature local Pkclet number dependence unknown dependence on Zagarola/Smits scaling Stanton Number, St = pcpu,fw-TT,)

Pressure Gradient Favorable Pressure Gradient Adverse Pressure Gradient

Heat transfer in turbulent boundary layers plays a crucial role in industrial applications. For example,

dy hot gases flowing over the turbine's cooler blade, air flowing over computer chips, etc., all involve the in- teraction of heat transfer and momentum transport between the fluid and the solid surface at different temperatures. It is of great interest to understand the physics of this interaction between scales for the tem- perature field and the velocity field, particularly the scales of the nonlinear turbulent quantities. In order to determine these scales, a similarity analysis will be applied to the equations of motion for a 2-D, steady, incompressible flow with constant fluid properties.

Experiments of heat transfer in turbulent boundary layers are rather complex and difficult to control due to the interaction between the temperature field and the velocity field, Kader.2 Based on the analysis by Spald- ing3 the Reynolds analogy has been applied to inves- tigate the heat transfer in turbulent boundary layers. By assuming an analogy between the heat transfer and momentum transport, a convenient approach has been established to study the forced convection turbulent

conditions

boundary layer. However, the Reynolds analogy is a *Graduate Student,Rensselaer Polytechnic Institute, De- very rough method, which breaks down when there

Partment of Mechanical, Aerospace tk Nuclear Engineering, is an external pressure gradient imposed in the outer Troy, NY 12180

nic Institute, Department of Mechanical, Aerospace & Nuclear Moreover, many investigate% such as Blackwell,4 AY- Engineering, Troy, NY 12180 ala et aL5 have shown that the turbulent Prandtl

t Assistant Professor, AIAA memember, RensselMr Polytech- for the adverse pressure gradient flow*

I _. - .

Copyright @ 2003 by the American Institute of Aeronautics and Astronautics, Inc. No copyright is asserted in the United States under Title 17, US. Code. The US. Government has a royalty- free license to exercise all rights under the copyright claimed herein for Governmental Purposes. All other rights are reserved by the

number, prt, is not a constant the turbulent boundary layer, eSpeCidly for those flows subject to external pressure gradient. This is contrary to the im-

copyright owner. plications of the Reynolds analogy. Many advanced

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methods based on the Reynolds analogy have been proposed, and special efforts have been made to im- prove the turbulent Prandtl number P,t. However, there is no satisfying result yet, particularly in the scaling laws.

Similar to the standard law of the wall and the veloc- ity defect law for the inner and outer velocity profiles in turbulent boundary layers, a thermal law of the wall and a thermal defect law were presented for the inner and outer temperature profiles, respectively. However, the thermal law of the wall was born with the faults of the classic “log-law” . For instance, there is only a sin- gle temperature scale for both the inner and the outer flow. Furthermore, the thermal law of the wall cannot describe the thermal boundary layer with the adverse pressure gradient according to Perry et al.6 Also, it does not consider the Pdclet number, Pe, effects.

In 1966, Perry et aL6 presented an inverse half power law for the temperature profile in the adverse pressure gradient (APG) turbulent boundary layer us- ing dimensionless analysis. However, this law cannot describe the heat transfer in the outer region of the turbulent boundary layer. In 1976, Perry & Hoffman7

initially analyzed the scaling for temperature fluctu- ations of the zero pressure gradient (ZPG) boundary layer. In 1991, Kader’s2 investigation made a signif- icant contribution to this field. First, he considered the effect of Prandtl number, Pr, on the temperature profile. Second, two different scalings instead of one as suggested in the classical approach were proposed for both the inner and the outer thermal boundary layer, respectively. However, his analysis did not apply to a sharp variation of the wall temperature.

Most recently, Volino & Simon* proposed a formula- tion for the velocity profile and the temperature profile in the turbulent boundary layer subject to pressure gradient. They showed that their new formulation was deviating from the standard law of the wall, but it was still in agreement with the experimental data stud- ied. However, this agreement was only applicable for the inner thermal boundary layer region. Notice that most of the previous investigations except by Kader2 and Perry et did not answer the question of how to scale the temperature profile. Instead they simply took the friction temperature, T,, as the temperature scale for both the inner and the outer thermal bound- ary layer. Moreover, this single temperature scale failed to collapse the data in APG flows.

In 1997, George et aL9 initially applied a similar- ity analysis to study the forced convection turbulent boundary layer. Two different scalings were proposed for both the inner and the outer temperature profiles, respectively. However, their results were not verified for the PG boundary layer. More importantly, the ef- fects of the pressure gradient and upstream conditions were not considered in their analysis. Therefore, the goal of this investigation is to apply the “equilibrium

.

similarity analysis” proposed by George & Castillolo to study forced convection turbulent boundary layers. Attention will be given to forced convection turbulent boundary layers with and without pressure gradient. Using the existing experimental data, comparisons of different scalings will be performed.

The similarity analysis According to the similarity analysis for the turbu-

lent boundary layer with ZPG proposed by George & Castillo” and with pressure gradient (PG) by Castillo & George,’l the scales for both the inner and the outer flows are dictated by equations of motion and their boundary layer conditions alone. Since in the limit as Re + 00, the boundary layer equations become independent of Re, therefore any scale or function rep- resenting the boundary layer solutions must also be independent of the local Re (Le. the Asymptotic In- variance Principle: AIP). Hence, in this limit the inner and outer scales of the turbulent boundary layer will be determined.

Castillo & George” has shown that the outer mean deficit velocity profiles and the outer Reynolds shear stresses are scaled by U, and Ukdhldx , respectively. Moreover, the Reynolds normal stresses in outer vari- ables are scaled only with U&. Using a similar ap- proach to the inner momentum equation, they showed that the velocity profiles and Reynolds stresses are scaled with the friction velocity, uT and u:, respec- tively, which are the same as those in the classical theory. Therefore, the overlap region is characterized by two different velocity scales instead of one as sug- gested in the classical view.

A similar approach will be used here for the forced convection turbulent boundary layer in order to de- termine the scales for the temperature field and the corresponding constraints. The turbulent boundary layer under consideration is 2 - D, steady state on the mean, incompressible flow with heat transfer between the wall and free stream. The physical properties, such as density, p, dynamic viscosity, v and thermal diffu- sivity, a, are assumed to be constant within the small range of temperature difference studied here. The in- tegral form of the inner energy equation is given as

and the outer energy equation is a~ LJT a ax dy a y

u- + v- = -[- < tu >I ,

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The related boundary layer conditions are

y+m,T+T,;<tv>+O. (4) The above equations describe the energy transport in the turbulent boundary layer in the limit as Pe + 00.

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Similarity analysis for the inner flow

Solutions for the inner thermal profiles and the inner turbulent heat flux are sought of the following forms:

where T,i(x) and Fsi(x) are the unknown inner tem- perature scale and inner turbulent flux scale respec- tively, which will be determined from the inner energy equation and its boundary layer conditions. The vari- ables inside the similarity functions gsi and hsi rep- resent the inner similarity length scale, &, the local PCclet number dependence, S; , the Prandtl number, Pr, and any possible dependence on the upstream con- ditions, *.

The Stanton number is defined as St = and the skin friction coefficient is de- eCp CJ- (T, -T,) ’

fined as Cf/2 = &. Notice that the heat flux at the wall, qw, in the thermal boundary layer corresponds to the wall shear stress, 7w, in the momentum bound- ary layer. Therefore, the Stanton number, St, is very similar to the skin friction coefficient, Cf/2. However, the Stanton number and the skin friction coefficient behave very differently if there is a pressure gradient imposed in the outer flow. For instance, for the strong APG flow, the skin friction decreases with increasing the downstream position, 2. Nonetheless, the Stanton number will increase slightly provided that the heat flux on the wall is a constant. In the similarity analy- sis of the momentum equation, the inner length scale y+ can be written in terms of the skin friction coeffi- cient as,

4w

(7)

Hence when choosing the length scale y; for the inner energy equation, the Stanton number St will be used to take the place of the skin friction coefficient Cf/2 as,

y; = ?A,

which is a reasonable assumption since the Stanton number includes the heat transfer information at the wall. In addition, St and C f / 2 behave quite different with changes in the external pressure gradient.15

In the limit as Pe + 00, the similarity solutions of equation (5) and (6) are independent of the local Pdclet number, Pe , as required by the Asymptotic Invariance Principle (AIP). Thus, equations (5) and (6) are reduced to

where gsim and hsim are the asymptotic profiles for the mean temperature and the turbulent heat flux, re- spectively. Notice that the upstream conditions, *, has been retained because they may influence the shape of the profiles even in this limit. Substituting these asymptotic similarity solutions into the inner energy equation (l), the inner energy equation is transformed into

Full similarity exists only if all the terms in the square brackets have the same x dependence, therefore, they must evolve together. In other words, they must be proportional to each other, such as

Consequently, the scales for the inner mean tempera- ture and the inner turbulent heat flux are given as,

Clearly, the inner temperature scale is different from the classical approach. In the classical view, analogous to the friction velocity, u,, the inner mean tempera- ture scale is defined using the friction temperature, T,, defined as

QW T, = - PCP%

(15)

and the inner length scale is given as

The friction velocity in the above equations is deter- mined from the wall shear stress as u: = 7 w / p . This is indeed the Reynolds analogy of the law of the wall, but it breaks down when an external pressure gradient is imposed on the flow. George et al.’ used a similar analysis as the one mentioned above. However, they adopted a different inner length scale and therefore ob- tained a different temperature scale. Here we will use GWC to refer to the scalings proposed by George et

Table 1 shows the comparison of various results for the inner temperature scaling from various investi- gations. The first row represents the scaling by the classical Reynolds analogy, the second row shows the scaling by GWC using similarity analysis, and the last row shows the scaling from the present analysis. No- tice that both the inner length scale and the inner temperature scale are different for all three theories. In addition, it is obvious that the inner temperature scaling of the new theory contains the effects of the Prandtl number and the Stanton number. A compar- ison between these scales will be shown in the subse- quent sections using the experimental data for ZPG and APG flows.

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Table 1: The comparisons of different inner tempera- ture scalings.

Theory Length scale Temperature scale Classic Theory 5 T, = 9r-

P C l ? U ,

Similarity analysis for the outer flow

The outer scales for the temperature defect and the turbulent heat flux can be determined using the simi- larity analysis of the outer boundary layer equations. The outer solutions of the energy equation can be sought in the forms of

- TCG = T.90(x)g80(gT, 6:, pr, *) - < ut >= F,,(x)hao(gT,6T+,Pr,*)

(17) (18)

where gT = y/S, is the outer similarity length scale and b~ is the outer length scale, which could be defined in terms of the thermal boundary layer thickness or the enthalpy thickness. The similarity functions gg0 and h,, are the outer temperature profile and the turbulent heat flux profile, respectively. The arguments inside the functions g8, and h,, represent the local PCclet number dependence given by S:, the local Prandtl number, P,, effects and any possible dependence on the upstream conditions *. The unknown outer tem- perature scale and the unknown outer turbulent heat flux scale are given as T,, and Fa,, respectively. These unknown scales depend on x only and must be deter- mined from the outer thermal equation.

In the limit as Re + 00 or Pe + 00, the outer momentum equation and thermal boundary layer are independent of Re or Pe as required by the AIP. Hence, in this limit, the outer solution forms of the energy equation reduce to,

- TW = T80(x)g.90M(gT,Pr, *) (19) - < >= F80(x)h.90Xl(gT, pr, *), (20)

where the 0;) is used to represent the asymptotic pro- files of gso, and h,,,, which are independent of Re or Pe. Notice that the upstream dependence has been retained because the flow may depend on it even in this limit.

Also, similar forms exist for the velocity deficit pro- files given as,

- < uv >= R s o ( ~ ) ~ o p m ( j j , A ; *) (22)

where U,, = U, and R,, = U&dG/dx have been determined by the similarity analysis of Castillo & Ge0rge.l' Substituting these asymptotic similarity so- lutions of equations (19), (20), (21) and (22) into the outer energy equation (2), and using the continuity

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equation to get the V component, a transformed outer energy equation in similarity form is obtained as,

(23)

As in the inner similarity analysis, the terms in the square brackets depend on x , and full similarity will exist only if they have the same x dependence. Con- sequently, they must be proportional to each other. Thus,

6T dT80 6 ~ U W d S , db . (24) - - N - - N - N - N -

T,, dx U, dx dx dx UmT80

Taking the similarity of the last two terms, it yields to

d6 F.9, U,T,,--. dx

Matching the heat flux and Reynolds shear stress in inner and outer variables in the limit as Pe + 00 or Re -+ 00, the outer temperature scale can be obtained as,

which is consistent with the GWC scaling. In this limit, the flow loses its dependence on the local Re or Pe. However, at finite Re or Pe, the temperature profiles should not be expected to collapse to one single curve. In addition, the flow may still be affected by the upstream conditions even in this limit. Now it will be of great interest to look for the real asymptotic solution so that the effects of the upstream condition and the local Reynolds number dependence can be removed even at finite Reynolds number or Pdclet number.

The new outer temperature scaling Zagarola & Smits' have shown that the outer ve-

locity scaling for the pipe flow is U,6*/6. Later on, Castillo14 derived the exact same outer velocity scaling for the PG turbulent boundary layer using similarity analysis. He showed that this new velocity scaling could remove the effects of the different upstream con- ditions, Reynolds number changes, and the pressure gradient effects on the outer flow. Thus, the true asymptotic velocity profile for ZPG, APG and FPG flows can be found. Now the attention will be focused on seeking the proper outer temperature scaling such that the true asymptotic temperature profile is found.

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Let us assume that the outer temperature deficit profile of equation (17) can be rewritten as

- The new unknown temperature scale, T,,, needs to be determined using similarity analysis, and gsOw repre- sents the true asymptotic temperature function. No- tice that the new unknown temperature scale Fs, in- cludes the effects of upstream conditions and Pe‘clet number, and ssOw depends on gT only.

The so-called “thermal displacement thickness” S; is defined as

F, rn rn ~~

(28) 1- 1, “ = A T w - T , dY.

Substituting equation(27) into the definition of the “thermal boundary layer thickness”, it follows that,

For the similarity solutions to exist, the bracketed terms should have the same x dependence, thus,

Hence the new outer temperature scaling is given as

TSO(X, 6+, *) = (Tw - Tw)-. G 6T

Interestingly, this outer temperature scaling is very similar to the Zagoralo/Smits velocity scaling, UCC6,/6, and should contain the effects of the unknown upstream conditions and the Pdclet number effect as well. Thus, the true asymptotic temperature profile should exist if this new temperature profile scale is correct.

Comparisons of the various outer scalings As mentioned previously, the outer scaling in the

classical view is determined using the Reynolds anal- ogy. Since in the classical theory a single velocity scaling is assumed, the mean velocity deficit profiles are normalized by the friction velocity given as,

(32)

Therefore, in a similar manner, the outer mean tem- perature profiles are normalized by the inner friction temperature, T,, given by,

- m

(33)

where T, is given as qw/pCpu,. Table 2 summarizes the outer temperature scaling from various theories.

Table 2: The comparisons of different outer tempera- ture scales.

Theorv Length scale TemDerature scale TT =

P C p U r Classic Theory f

GWC L 6T w w - Tm’+

Present analysis 8 (Tw - Tm)

The first row shows the scaling obtained from the Reynolds analogy, the second row shows the results from the similarity analysis of GWC, and the third row shows the new scaling derived in equation 31. The outer length scale is the same for all three theories, whereas the temperature scale is quite different from each other.

Additional similarity constraints In addition to the previous scaling, there are some

other constraints that follow from the similarity analy- sis of the outer equation. Firstly, using the relationship of equation 23, it follows that

d6T dS dx dx - N - * (34)

Integrating the above equation and dropping the con- stant of integration, it follows that 6~ - 6, which means that the thermal boundary layer thickness and the velocity boundary layer thickness are proportional to each other. Secondly, from

(35)

we can see that for equilibrium boundary layers, the pressure parameter, A, should be a constant given by,

6 dU, A = -- = constant, Uw$ dx

which is exactly the same as the pressure parameter obtained by the Castillo & George’s similarity analysis for the pressure gradient boundary layer. If the mo- mentum thickness, 0, or the displacement thickness, S,, are used instead of the boundary layer thickness, 6, then two other pressure parameters will be obtained in terms of these length scales. Thus, the pressure gra- dient parameter in terms of the momentum thickness, e , is given as,

and the displacement thickness, 6, , as

(37)

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This is exactly the same as the results of Castillo & George” and Castillo & Wang.12 Furthermore, they

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showed that an equilibrium flow exists only if A = constant. They further showed that only three values of the pressure parameters are needed to characterize all equilibrium flows: one for the FPG flow with A = -1.92, one for the ZPG flow with A = 0.0, and one for the APG flow with A = 0.22. The present analysis for the forced convection boundary layer is in agreement with those results, and will be confirmed in the later sections. In addition, a similar constraint exists for the thermal boundary layer and is given as,

(39)

Therefore, a thermal parameter, AT , is then defined as,

6~ d b Tso* dx

AT = -- = constant,

This thermal parameter should be a constant if a ther- mal equilibrium boundary layer exists at all. Also no- tice that this parameter is very similar to the pressure parameter, A, and exists as long as the wall tempera- ture or the free stream temperature are changing with the downstream distance, z. In addition, it is easy to see that a power relation exists between the thermal boundary thickness and the imposed temperature gra- dient. The power represents the thermal parameter, AT. Unfortunately, due to the limited data with such conditions it is impossible to verify this constraint in the present investigation.

Results The scaling from the Reynolds analogy, the GWC

scaling and from the present analysis will be compared for ZPG and APG forced convection boundary layer. The effects of the pressure gradient and upstream con- ditions will be analyzed using the existing data.

The experimental data The ZPG experimental data of Bla~kwell,~ BlomlG

with two different free stream speeds of 6 m/s and 10 m/s (fixed upstream conditions), and the APG experimental data of Orlando17 and Blackwel14 will be used in this investigation. The APG experiments were performed such that a power relationship between the free stream velocity and the streamwise distance exists, thus U , N x m . The power coefficient, m, represents the strength of the pressure gradient. The experimental data from Orlando17 has a power coeffi- cient of m = -0.275, and therefore has the strongest APG, then followed by Blackwel14 with m = -0.2, and m = -0.15. In addition, each of these experimental data has almost the same fixed wind tunnel speed, U,, (upstream conditions) of 9.74 m/s, 10.1 m/s, 11.1 m/s for Bla~kwell.~ The strong adverse pressure gradient data of Orlando17 has a fixed wind tunnel speed, U,, of 11.6 m/s. The maximum Reynolds number based on

the momentum thickness, Re, achieved from the data considered in this investigation, is about 3,000 while the lowest is around 550.

The inner temperature profiles Figure 1 shows the inner temperature profiles in

semi-log scale for all the ZPG data discussed above. Note that the data have different upstream conditions (i.e. in this case wind tunnel speed). Figure l(a) shows the profiles normalized by the classic scaling using the Reynolds analogy. Figure l(b) shows the same ZPG experimental data now normalized by the GWC scal- ing. Figure l(c) shows the profiles normalized by the new scaling. Notice that using the new inner scaling, the profiles collapse into one single curve, regardless of the difference in the upstream speeds. Moreover, the classical scaling collapses the experimental data better than the GWC scaling.

Figure 2 shows the APG experimental data of Or- lando17 and Blackwel14 in semi-log scale. Clearly, the new scaling shown in figure 2(c) collapses the data into a single curve. Furthermore, the new scaling suc- cessfully removes the effects of the external pressure gradients and the upstream conditions. The success of this scaling is more evident for the APG data than for the ZPG data. In addition, notice that the Reynolds analogy fails to collapse the APG data as shown in figure 2(a). Furthermore, using the classical scaling, it is clear that the temperature profiles collapse for a given set of upstream conditions (such as wind tunnel speed) and for a given strength of pressure gradient, but these profiles collapse to a different curve. In addi- tion, the collapse of the profiles using the GWC scaling is far better than using the classical scaling, but it is less satisfying than using the present scaling. Conse- quently, it means that at least for APG flows without separation, the inner flow is independent of the pres- sure gradient. In fact, Castillo'* showed that as long as the boundary layer did not approach to the sepa- ration point, the inner flow was not affected by the external pressure gradient.

Figure 3 combined the ZPG experimental data of figure 1 and the APG data of figure 2. As before, the classical scaling profiles are shown in figure 3(a) while the GWC scaling is shown in figure 3(b). The data show an obvious dependence on the pressure gradient and the upstream conditions using the classical scaling or the GWC scaling. However, using the new scaling as shown in figure 3(c), all the data nearly collapse into one single curve, regardless of the strength of the PG and the upstream conditions. Therefore, it is possible to find the asymptotic profile sought in the previous sections at finite Reynolds number but only when the profiles are normalized using the new scaling.

Outer temperature profiles

The ZPG experimental data of Blackwel14 and BlomlG shown in figure 1 are now normalized in outer

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variables using the classical scaling, the GWC scal- ing, and the new scaling as shown in figure 4(a), 4(b), and 4(c), respectively. Notice that using the classical scaling and the GWC scaling, the data of Blom" can not collapse into a single curve. However, using the new scaling, the profiles collapse, but to a different curve from the profile of Bla~kwell.~ This could be explained by the fact that for the experimental data of Blom," the thermal boundary layer develops later than the velocity boundary layer. Thus, the momen- tum boundary layer begins earlier than the thermal boundary layer, which therefore affects the outer flow. Figure 5 includes the APG experimental data of Black- well4 and 0rland0.l~ Clearly, the profiles using the classical scaling or the GWC scaling collapse the ex- perimental data but to different curves depending on the strength of the external PG. However, using the new scaling shown in figure 5(c), the profiles collapse into a single curve regardless of the strength of the APG and the upstream conditions. Thus, the outer asymptotic profile for APG flows is found even at the finite Reynolds number.

Figure 6 shows the above ZPG and APG experimen- tal data plotted together in outer variables. Evidently, the APG profiles collapse to the same curve, and the ZPG collapse to another curve. However, the experi- mental data from Blom shows a different curve from the ZPG data of Blackwell due to the unheated sec- tion. This unheated region has a direct influence on the outer flow but not on the inner flow as shown in figure l(c), this is in agreement with the prediction of Kays.15 Notice that the new scaling removes the effect of the upstream conditions and the strength of pres- sure gradient in the boundary layer, contrary to the classical scaling.

The asymptotic velocity profiles Previous investigations carried out by Castillo &

Walkerlg and Castillo et aI.13 have shown that the outer mean deficit profiles and Reynolds stresses tend to collapse in outer variables as long as the upstream conditions are kept fixed. Upstream conditions, such as wind tunnel speed, tripping wire size, turbulence in- tensity etc, play an important role in the development of the downstream flow, particularly in the apparent Reynolds number dependence observed in turbulent boundary layers. Recently, Zagarola & Smits' found an empirical velocity scaling for pipe flows given by U , %. Using similarity analysis, Castillo14 derived the same scaling for boundary layers with and without pressure gradient given as

Notice that fo is indeed the true asymptotic profile because it is independent of the local Reynolds num- ber and the upstream conditions. This is indeed the

self-preserving solution sought by Townsend.*l More- over, notice that this asymptotic profile is influenced by the pressure parameter, A, only. The remarkable feature of this Zagarola & Smits scaling, U, %, is that it removes the effect of the upstream conditions, the local Reynolds number dependence, and the strength of the pressure gradient. Using the Zagarola & Smits scaling, Castillo14 showed that there are only three profiles in turbulent boundary layers regardless of the strength of the pressure gradient, the local Reynolds number and upstream changes. These results agree with the similarity analysis for equilibrium and non- equilibrium boundary layers carried out by Castillo & George,ll and Castillo & Wang.12

Figure 7 and Figure 8 show the outer velocity deficit profiles for the ZPG experimental data and APG ex- perimental data, respectively. Figure 7(a) and fig- ure 8(a) show the profiles normalized by the clas- sic scaling using the friction velocity as suggested by Clauser." Figure 7(b) and figure 8(b) show the same profiles, but now normalized by the free stream ve- locity, U,, as suggested by the similarity analysis of Castillo & George." Then, figure 7(c) and figure 8(c) show the same ZPG profiles and APG profiles, but now normalized by the Zagarola & Smits scaling, U, $. In the classical theory the mean deficit profiles should col- lapse to one unique profile if normalized by the friction velocity, contrary to the results shown in figure 7(a) and figure 8(a). However, these profiles collapse to different curves as those using the free stream velocity of figure 7(b) and figure 8(b). It is important to re- alize that the reason why these profiles collapse, but to different curves is due to differences in upstream conditions such as trip wire shape, upstream wind tun- nel speed etc. and different strengths of PG. In fact, Castillo & Walkerlg showed that the way the flow is generated affects the way the downstream flow grows.

Using the Zagarola & Smits scaling shown in fig- ure 7(c) for the ZPG flow and figure 8(c) for the APG flow, all the velocity profiles collapse into one curve re- gardless of the strengths of the pressure gradient and the upstream conditions. Also, all the data with APG collapse to one curve, and the data with ZPG collapse to another curve, which confirms the results by Castillo & George,ll and Castillo & Wan$' that there are only three different velocity profiles in turbulent boundary layers (note that the FPG data is not included here). Obviously the Zagarola & Smits scaling produced the asymptotic profile sought by Castillo & George."

The boundary layer parameters

In this section it will be shown how the upstream conditions and the strength of the pressure gradient influence the growth of the boundary layer parame- ters. Figure 9 shows the relationship of % and Reo or e and Reo for the ZPG and for the APG data. Notice that each individual experiment has fixed the

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upstream conditions. Figure 10 shows the boundary layer growth, Rea versus Reo, or res^ versus Reo for the ZPG flow and for the APG flow, respectively. Ob- viously, for the APG flow shown in figure 10(b), the boundary layer grows differently for each individual experiment considered. This is a consequence of differ- ent strength of PG and different upstream conditions. In addition, the thermal boundary layer grows almost simultaneously with the momentum boundary layer. However for the ZPG data shown in figure lO(a), the growth of the thermal boundary layer is behind the growth of the momentum boundary layer. This is a clear manifestation of the unheated section.

The pressure parameter

An equilibrium boundary layer was defined by Castillo & Georgell as one where the pressure param- eter, A, or AS is a constant. Furthermore, Castillo & George,” Castillo et al.,13 and recently Castillo & Wang12 showed that the pressure parameter for most APG data, including separated flows and non- equilibrium flows, have a constant value of A = 0.22. Therefore, it is of great interest here to explore whether this value is observed in the forced convection turbulent boundary layer subject to an external APG. Figure ll(a) shows the Zog(U,) versus log(6) of the APG experimental data of Blackwel14 and 0rland0.l~ The same data using the momentum thickness as a length scale is shown in figure l l (b) . Firstly, notice that both plots show a linear relationship between the free stream velocity and the length scales, which means that the forced convection turbulent boundary layer is indeed in equilibrium as defined by Castillo & George.” Secondly, the slope represents the pressure parameter A and As, respectively. Finally, the value of the pressure parameter is 0.22, which is the same value as found for equilibrium, and non-equilibrium flows, Castillo & Wang.lg

Conclusion The new inner and outer temperature scales are de-

rived for a 2 - D forced convection turbulent boundary layer subject to external pressure gradient using sim- ilarity analysis of the equations of motion. The new scales are compared with the classical scaling using the Reynolds analogy and the GWC scaling. It was shown that both the inner and outer flows are affected by the external pressure gradient and the upstream con- ditions such as the wind tunnel speed. However, when the experimental data is normalized by the new scal- ings in inner variables, T,i = Pr(T, - T , ) a , or in outer variables, T,, = (T, - T,) the effects of pres- sure gradient and upstream conditions are completely removed from the profiles. The fact that the inner profiles collapse to one single curve for all the ZPG and APG flows means that the inner flow is nearly independent of the pressure gradient; contrary to the

outer flow where the shape of those profiles are differ- ent. Consequently, the external PG influence more the outer flow than the inner flow.

Moreover, using the new scalings the profiles col- lapse into a single curve, thus the asymptotic profiles for thermal boundary layer are found even at finite Reynolds number and finite PCclet number. In addi- tion, using the Zagarola & Smits scaling, the asymp- totic velocity profiles for ZPG and for APG flows are found respectively, regardless of the upstream con- ditions and strength of the pressure gradient. h r - thermore, the pressure parameter, A, for the forced convection turbulent boundary layer was found to be the same as the equilibrium boundary layer with he = 0.22. The strength of the PG and the upstream conditions are shown to affect the shape factor and the boundary layer growth. In summary, the simi- larity analysis applied to forced convection boundary layers enables us to find the asymptotic solutions in thermal boundary layers and it provides new insight into the effects of the upstream conditions and PG on the downstream flow.

ACKNOWLEDGMENTS The authors are very thankful to Dr. William St.

Cyr and Dr. Ramona Travis from NASA Stennis Space Center for their continuous support of many of our projects.

References ‘Zagarola, M.V. and Smits, A.J., “Mean-flow Scaling of

Turbulent Pipe Flow”, J. Fluid Mechanics, Vol. 373, pp. 33-79, 1998.

2Kader, B. A., “Heat and Mass Transfer in Pressure- Gradient Boundary Layers”, Int. J. of Heat and Mass Transfer, Vol. 34, pp. 2834-2857, 1991.

3Spalding D B 1961 ASME Inst. Mech. Eng. 2 439 4Blackwell, B.F., Kays, W.M. and Moffat, R.J., “The

Turbulent Boundary Layer on a Porous Plate: An Experimental Study of the Heat Transfer Behavior with Adverse Pressure Gra- dients”, Report No. HMT-16, Thermosciences Division, Dept. Mech. Eng., Stanford University, Ph.D. thesis, 1972.

5Ayala, A., White, B.R., and Bagheri, N., “Turbulent Prandtl Number Measurements in Adverse Pressure Gradient Equilibrium Boundary Layers”, 2nd Int. Symposium on Turbu- lence, Heat and Mass Transfer, 1997.

6Perry, A.E., Bell, J.B. and Joubert, P.N., “Velocity and Temperature Profiles in Adverse Pressure Gradient Turbu- lent Boundary Layers”, J. Fluid Mech., Vol. 25, pp. 299-320, 1966.

‘Perry, A.E. and Hoffmann P.H., “An Experimental Study of Turbulent Convective Heat Transfer from a Flat Plate”, J. Fluid Mech., Vo1.77, pp. 355-368, 1976.

sVolino, R.J. and Simon, T.W., “Velocity and Temper- ature Profiles in Turbulent Boundary Layer Flows Experiencing Streamwise Pressure Gradients”, Journal of Heat Transfer, Vol.

gGeorge, W.K., Wosnik, M., and Castillo, L., “Similar- ity Analysis for Forced Convection Thermal Boundary Layer”, Transport. Phenomena in Thermal Science and Process Engi- neering. Kyoto, Japan. Vol. 1, pp. 239-244, 1997.

loGeorge, W.K. and Castillo, L., “Zero Pressure Gradi- ent Turbulent Boundary Layer”, Applied Mechanics Reviews,

119, pp. 433-439, 1997.

VOI. 50, pp. 689-729, 1997.

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“Castillo, L. and George, W.K., “Similarity Analysis for Turbulent Boundary Layer with Pressure Gradient: Outer Flow”, AIAA Journal, Vol. 39, No. 1, pp. 41-47, 2001.

‘*Castillo, L. and Wang, X., “Similarity Analysis for Non-equilibrium Turbulent Boundary layer”, 32th AIAA Fluid Dynamics Conference and Exhibit, June 24-26, paper 2002- 3084, St. Louis, Missouri, 2002.

13Castillo, L., George, W.K. and Wang, X., “Charac- terizing Turbulent Boundary Layers Subject to Strong Adverse Pressure Gradient with Eventual Separation”, ASME Fluids Engineering Division Summer Meeting, S-315, FEDSM 2001- 18112, 2001.

14Castillo, L., “Application of Zagarola/Smits Scaling in Turbulent Boundary Layers with Pressure Gradient”, Advances in Fluids Mechanics 3 (Montreal) Canada, May 24-26 2000, ed. by M. Rahman and C.A. Brebbia, pp.275-288.

15Kays W M and Crawford M E 1993 Convective Heat and Mass h n s f e r (McGraw-Hill)

‘“lorn, J . “An Experimental Determination of the Tur- bulent Prandtl Number in A Developing Temperature Boundary Layer”, Ph.D.thesis, Technische Hogeschool, Eindhoven, The Netherlands, 1970.

170rlando, A.F., Kays, W.M. and Moffat, R.J., “Tur- bulent Transport of Heat and Momentum in a Boundary Layer Subject to Deceleration, Suction, and Variable Wall Temper- ature”, Report No. HMT-17, Thermosciences Division, Dept. Mech. Eng., Stanford University, Ph.D. thesis, 1974.

‘sCastillo L., “Similarity Analysis of Turbulent Bound- ary Layers”, Ph.D. Dissertation, Department of Mechanial and Aerospace Engineering, SUNY/Buffalo, Buffalo, NY, 1997.

‘QCastillo, L. and Walker, D., “The Effect of the Up- stream Conditions on the Outer Flow of Turbulent Boundary Layers”, AIAA, Vol. 40, No. 7, pp. 1292-1299, 2002.

2oWalker, D. and Castillo, L., “Turbulent Boundary Layers at Very High Reynolds Number and Its Relation to the Inial Conditions”, 31st AIAA Fluid Dynamics Conference, pa- per AIAA-2001-2913, Anaheim, California, June 11-14, 2001.

21Townsend, A.A., The Structure of Turbulent Shear Flow, Cambridge press, 1976.

22Clauser, F.H., “The Turbulent Boundary Layer in Adverse Pressure Gradient”, Advances in Applied Mechanics, Vo1.4, 1954.

20-

n t* 1 5 -

? : t 1 0 -

I

5 -

1 00 10’ 10‘ 10’

Y’

a) Scaling using Reynolds Analogy

7

0 Bbm19702PG U,r6 WsRo~=1184-1613 Bhc.slcmU 1972 ZPG Uc9.74 mls Rea=516-Z105

B h 1970 ZPG U y l O Ws Re,=1724-2332

I 10’ 10‘ 10’ 1 0‘ 1 0‘

(Y u Jv)S t

b) Scaling using the GWC analysis

7

0

Bla IS7OZPG Up6INsRe,-llW-l613

B k c ( m U 1972ZPG U.19.74 INS Re.1515-2805 Blm, 1970 ZPG Up10 mlS Re,-ll?4-2332

c ) Scaling from the present analysis

Fig. 1: Comparisons of different inner scalings for the ZPG flow.

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AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS PAPER 2003-0640

BLKlnnB1072APG m 4 . 2 U y l l . 1 mlaRme=17184533 II B*Clnnl1972 APG m d . 1 5 UylO.1 mh Ray20064734 I' Ohndo1974APG m 4 . 2 7 5 U~=ll.6ml.Re,=33(MlbS17

I 4 0 ' lo' 1 0' 10' l@

Y'

a) Scaling using Reynolds Analow

c 8-1 1972 APG -4.2 U.=ll.l nrhRe,=l7184533 > BWOW11972 APG m 0 . 1 5 U.=lO.l mlmRm,=2008-3734 ' Ohndol974APG -4.275 U.=11.6 nrlsRmp3Wl-5517 'I

oL 10'

(Y U_/v)St

b) Scaling using the GWC analysis

B h S ( a * n U l O n bPGmdJ.15 U.rlO.1 mlsRa,-200+3734 L B h ~ l l 1 9 7 2 ~ G m d J ~ U . - 1 1 . 1 mlsRep1716-4533 j, OhndolO74bPG m-Q.275 U.-ll.S mle Rn..381-5517

25

' T O . ' "10' 1; '"'';A' ' I f f ' Ib 10' ' "- Y UJvSflR

c ) Scaling from the present analysis

Fig. 2: Comparisons of different inner scalings for the APG flow.

a) Scaling using Reynolds Analom

I # , , I 10' 10' 10' 10' 1 0.

YUJvSt

b) Scaling using the GWC analysis

I I

I I 40' lo" 10' lb 1 0' 1 0,

Y u JvSt-ln

c ) Scaling from the present analysis

Fig. 3: Comparisons of different inner scalings for the ZPG and APG flow.

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AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS PAPER 2003-0640

I

o Blom 1970zPG U-lomla Re.11724-2332 Bbclaall1972 ZPG U,19.74 mla ROp816-2805

Blom 1970 ZPC Uo-6 ml8 Re,=1184-1613

Y16,

a) Scaling using Reynolds Analogy

1

i . 8lom197OZPG UO-BmlaRep1184-1613

BIIcIEyeII 1972 ZPG U.19.74 ml8 Rl,d16-2805 Blom 1970ZPG UollO mfa Re.-1724-2332

Y16,

b) Scaling using the GWC analysis

'I!

..

Yl6,

c ) Scaling from the present analysis

Fig. 4: Comparisons of different outer scalings for the ZPG flow.

'et

Yl6,

a) Scaling using Reynolds Analogy

1.5

- a i w z * P ~ ~ 4 . 2 u.=ii.i m l s R e ~ = 1 7 1 8 ~ 5 3 3 I BhckwmY 1972 APG w 4 . 1 6 UplO.l mlS R@,=2008-3734 I.: . Ol(lndo1974APG m=4.276 U,=11.6 mfsRe,JO61-5517

0 05 5

Y16,

b) Scaling using the GWC analysis

Yl6,

c ) Scaling from the present analysis

Fig. 5: Comparisons of different outer scalings for the APG flow.

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AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS PAPER 2003-0640

Y16,

a) Scaling using Reynolds Analog?,

b) Scaling using the GWC analysis

c ) Scaling from the present analysis

Fig. 6: Comparisons of different outer scalings for the ZPG, APG and FPG flow.

1 z Ebm 1970ZPG U 9 mlORe,=1184-1613 , 3 Ebm 1970 ZPG Uk10 mfs Re.=1724-2332

0 0.5

YIF

a) Classical scaling

> Ebm1970ZPG U.=6mhRae=l184-1613 o 7

Ebm 1970 ZPG U.=lO mh R*.=1724-2332 Ebc*urllS72 ZPG Uy9.74 mfs R*,=515-2805

Od

o s

? 0 4

02

0

0 0 5

Y16

b) WC Similarity analysis scaling

> Ebm 1970ZPG U.=6 mlsRee=1184-1813 L?

,r Ebm 1870 ZPG U$O mls Rm,rl724-2332 E*C*ml l972 ZPG U,=9.74 mfs R~,=515-2806

c ) Zagarola/Smits scaling

Fig. 7: Comparisons of different outer velocity scalings for the ZPG flow.

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AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS PAPER 2003-0640

L Bbc*nU1972APG m = 4 2 U.=ll.l m l~R*~=17184533 . Bbclmrll1972 APG m 4 . 1 5 U,=10.1 mlsRe,=2008-3734

Ohndo 1874 APG m 4 . 2 7 5 Uf11.6 mls R*.=3061-6517

Yl6

a) Classical scaling

PG m=-0.2 U y l l . 1 mlsRae=1718-45X PO m=4.16 U.sl0.1 mlsRee=2W8-37: G m 4 . 2 7 6 U y l l . 6 WsRa,=JM116S'

b) WC Similarity analysis scaling

e

.: BbclrmUl972 APG m d . 2 U,=ll.l ml8 Rap17184533 Bbclmll1872 APG m-4.15 U,=lO.l mls Rae=200&3734 O*ndo 1974 APG m-4.275 Up11.6 14s R~,=3081-5617

k

0.35 1 0.30 1

0

O n o O 0 ,

C

O C

0

0

3

. I

a) 9 versus Re@

0.16 -

0.14 -

5 0.12 L a

0.10 -

'i 5

0 ".

0 ,I .' 0

0 -

0

00

3 3

- 5000 6

YE

c ) Zagarola/Smits scaling

Fig. 8: Comparisons of different outer velocity scdings for the APG flow.

b) e versus Reo

Fig. 9: Comparisons of different upstream conditions for the APG and ZPG flow.

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AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS PAPER 2003-0640

4oooo 4oooo

- 35000 35000 ,' /

,' // / -

,' /

I 4 -

2 1 3 - 4

.

a) APG flow

A d 2 2

?.

b) ZPG flow

Fig. 10: The boundary layer growth for the APG and ZPG flow.

a) A

b) Ae

Fig. 11: The boundary layer growth for the APG and ZPG flow.

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AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS PAPER 2003-0640