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1 Editorial Manager(tm) for Experiments in Fluids Manuscript Draft Manuscript Number: EXIF-D-0-00R Title: Correlating Data; Examples from Turbulent Wall Layers Article Type: Dedication Issue Keywords: Correlation; Data Analysis; Turbulence Corresponding Author: Prof. Ronald L. Panton, Ph.D. Corresponding Author's Institution: University of Texas First Author: Ronald L. Panton, Ph.D. Order of Authors: Ronald L. Panton, Ph.D. Abstract: Flow fields in fluid mechanics often contain regions where different physical events occur. The relative size of the regions changes as a parameter is varied. Correlating field data, either physical or DNS calculation experiments, in these situations can be aided by using ideas from matched asymptotic expansions from applied mathematics. This article discusses how composite expansions and the common part matching behavior is useful in correlating data. Response to Reviewers: See attachment

2 Manuscript Click here to download Manuscript: RockwellExpsFluidsTextR.doc Correlating Data; Examples from Turbulent Wall Layers Abstract Ronald L. Panton J. H. Herring Centennial Professor Emeritus Mechanical Engineering Department University of Texas Austin, Texas rpanton@mail.utexas.edu Flow fields in fluid mechanics often contain regions where different physical events occur. The relative size of the regions changes as a parameter is varied. Correlating field data, either physical or DNS calculation experiments, in these situations can be aided by using ideas from matched asymptotic expansions from applied mathematics. A second situation is when two slightly different processes occur in the same spatial region. For this case a two-term asymptotic expansion is needed. This article discusses how composite expansions and the common part matching behavior is useful in correlating data. Introduction It is a pleasure to write an article for this special issue honoring Don Rockwell. Experiments in Fluids has been very fortunate to have him as the editor for these many years. The present high esteem that the journal has attained is due to the rigorous standards that Don has demanded. It is an accomplishment that benefits the fluid mechanical community, and of which Don can be justly proud. In the title of the article Data means the results of physical experiments and/or computational DNS experiments. In any event, the correlation of data should, if possible, be in terms of nondimensional variables. This implies that one knows all the physical variables that occur in the experiment. This article deals specifically with field problems that have two regions with different dominant physical events, singular perturbation problems. Additionally, sometimes two different processes occur in the same region. Such problems occur very frequently in fluid mechanics, heat transfer, and combustion. The major emphasis is how

3 the ideas of matched asymptotic expansions (inner and outer expansions, common parts, and composite expansions) are useful in correlating data from experiments. Thus, the dominant theme is how to use these tools from applied mathematics to analyze experimental data. The article is intended for experimentalists who deal with singular perturbation events, turbulence being only a vehicle to provide examples. Turbulent wall layers offer good examples because they have two regions with different physics and no complete analytic theory. Experiments are the source of most of our information. More importantly, wall turbulence exhibits many different types of correlation issues. For researchers who do not deal with turbulent flows. This introduction will review many elementary aspects of turbulence, well known to workers in that field. Consider the fully developed turbulent flow in a pipe or channel. The theory is developed for the limit of high Reynolds numbers (Tennekes & Lumley(), Schlichting & Gersten (000), Pope (000)). From our experience with the physics of fluid mechanics, we expect that any dependent variable q, such as mean velocity U or a Reynolds stress uu i j, is a function of the distance from the wall y. Parameters in the problem are the distance to the pipe or channel centerline h, the kinematic viscosity the density, and the pressure gradient dp 0 /dx. Functionally this becomes U dp dx 0 U ( y, h,, ) () For several reasons the pressure gradient is not used in correlating data. Consider that if one evaluates the velocity expression at y = h, the centerline velocity U 0 is obtained. U U y h h dp dx 0 0 (,,, ) () In principle this expression can be used to replace the pressure-gradient variable in Eq. by the free-stream velocity. However, this is not always appropriate. There is another velocity scale, the friction velocity u. It is also related to the pressure gradient by the momentum equation. u h dp dx () 0 0 Thus, u is another candidate to replace the pressure gradient in the list of variables.

4 The theory for boundary layers is more tedious to derive, but in the end the width h is replaced by a boundary layer thickness. Compared to pipes and channels, the outer layer has different correlations. However, the inner layer for pipes, channels, and boundary layers are the same, or, as some believe, only slightly different. In general, a turbulence quantity is expressed as q f ( y, h,, U or u ) () 0 In a nondimensional form the profile has a distance variable and the Reynolds number as a parameter, Re uh /, q q O _ Scale y F F( Y, Re ) with Y () h Here q O_SCALE is the proper combination of h,, U0 and/or u with the same dimensions as q. A second, and very important, requirement is that q O_SCALE makes F of order one in the limit Re. Although mathematicians assume that the q O_SCALE is known, the correct determination of q O_SCALE is a central issue for engineers and physicists. The parameters U 0 and u are not equivalent as scaling parameters, because they separate as the Reynolds number becomes large according to U u 0 ln(re ) C () Using U 0 or u as a dependent variable scale will lead to different results as Re. Proper scaling requires the correct choice. number. Mathematically, turbulence theory is an asymptotic expansion in the Reynolds F ~ F ( Y ) (Re ) F ( Y)... () 0 Here F 0 (Y) and F (Y) are nondimensional coefficient functions (which we would call correlations ), and (Re ) is a nondimensional gauge function that approaches zero as Re. The correct q SCALE contains proper combinations of U 0 and u to make F 0 of order one. Finding the correct q O_SCALE and gauge function is a trial and error process. In principle F 0 and F are found mathematically by limiting processes.

5 And FY ( ) lim F0 ( Y ) F( Y,Re ) Re () lim F( Y,Re ) F ( Y) 0 (Re) Re Conceptually, q O_SCALE is the scale that makes FY ( ) finite and nonzero for high Reynolds numbers. No trends with Reynolds number still exist. If F ( ) 0 Y has Reynolds number trends, either approaching infinity or zero, then q O_SCALE is not correct. The first order term F is dimensionally like q/ ( q0_ Scale (Re)) so the F scale is q O_SCALE. Hence, it is different from the scale for F 0. This means that the physical processes that determine F 0 and F are slightly different. When two different processes occur in the same region a two-term expansion is required. In principle to find the correlations F 0 and F, we need experiments at a number of Reynolds numbers, and as high as possible. When profiles in Y of FY (, Re ) become independent of Re we will have determined F 0. It is well known that very near the wall the dominant turbulent physical processes change, viscosity becomes important, and h is not the proper distance scale. Here the yu proper distance variable is y Y Re. In this region the expansion for F(Y, Re ) gives the wrong results and a new expansion, with possibly a different scale and gauge function, is required. A inner region expansion is 0 f ~ f ( y ) (Re ) f ( y )... (0) For this inner region it is possible that the dependent variable scaling should also be different. q q I _ Scale f f ( y, Re ) with y () Hence, one must determine both inner and outer functions to have a complete description of the profile. yu ()

6 In addition to the inner and outer regions, there is an overlap region where both the inner expansion and the outer expansion are valid. In some asymptotic sense, the inner and outer representations match in the overlap region. This might be stated as, q F( Y 0, Re ) ~ f ( y, Re ) () I _ Scale qo _ Scale The matching functions are called the common parts. q F Y y f y () I _ Scale cp ( Re ) cp( ) qo _ Scale Often, as in Prandtl s boundary layer theory, the common part is a constant, however, sometimes the common part is a function of y. Turbulent wall layers offers examples of several non-constant common parts. The outer expansion is only good away from the wall, 0Y, and the inner expansion is only good near the wall, 0 y. A composite expansion is a combination that is valid for all values of y. Since the size of the inner region with respect to the size of the outer region changes with Reynolds number, a composite expansion contains a Reynolds number effect. A simple composite expansion is the additive composite expansion. It is formed by adding the inner and outer expansions and subtracting the common part. If one uses only the zero-order terms, a composite is q fcomp ( y,re ) f ( y ) F ( Y) Fcp ( Y) () O _ Scale 0 0 qi _ Scale Typically the result is expressed in the inner variable using Y y / Re. This explicitly introduces the Reynolds number. In the inner region the last two terms cancel each other. In the outer region the first and last terms cancel because of Eq.. In each of the following sections a different type of expansion and common part occurs. These examples show how different techniques are needed to correlate data. Mean Velocity Profile; Logarithmic Common Part The mean velocity has an unusual profile structure with a logarithmic common part. It is discussed for the sake of completeness. Consider the mean velocity profile U(Y,Re ). Obviously in the outer region the centerline velocity U 0 is a scale that

7 makes U(Y,Re ) of order one. Although the discussion above is general, in this section f and F refer specifically to the velocity profiles. It turns out that F 0 is one, and the gauge function (found through a history of trial and error) is u / U 0. Thus, the outer velocity profile is U Y (,Re) u ~ (Re ) F ( Y )... U0 U0 () The function F (Y), known as the defect velocity profile, is found by solving Eq. (see also Eq. ). F( Y) U ( Y ) U u 0 () Data correlated in this way (see any textbook on turbulence) shows that F becomes a function of Y for high Reynolds numbers. In the inner region near the wall, the velocity scale changes to u. The velocity profile is U y (,Re) ~ 0 ( f y )... u () This correlation is the law of the wall. The common part is the log law for the inner region. Millikan () found this common part by considering how a function of the form of Eq. could match a function of the form of Eq. in an overlap region. U u CP 0_ ln( f CP y ) Ci ; as y () Expressed in the outer variable Y the common part for the defect law is U U u 0 CP F_ CP ln( Y ) Ci C ; as Y 0 () Substituting Eq. into Eq. and recognizing that Y y / Re produces Eq. ; the relation between the scales U 0, u and the Reynolds number. The additive composite expansion is U( y,re) + ~ f ( y ) F ( Y) lny C with Y=y / Re u k 0 0 (0)

8 The last two terms are usually combined into the law of the wake. U y (,Re) ~ + 0 ( f y ) W ( Y ) with Y=y / Re u () This well-known example is plotted in most textbooks on fluid mechanics. The important thing to note in this example is that when a two-term expansion, Eq., the first of which is a constant, is matched to a one-term expansion, Eq., the common part is a logarithmic function. Furthermore, the relationship between the inner and outer scales, u and U 0, and the Reynolds number is firmly established. Reynolds Shear Stress; Composite Expansion Correlation of the Reynolds shear stress in channel flow offers a good example of how the composite expansion can be used to aid in the correlation of data. It is known from theory (for example see Tennekes and Lumley ()) that the outer Reynolds shear stress function G(Y) is uv G( Y ) Y () u From this we see that the common part, G GY 0 CP, is the constant one. G () CP This also indicates the need for an inner layer, because the correct answer would be GY ( 0) 0. The inner Reynolds shear stress function g(y + ) is determined from experiments. There is no change in scaling <uv> between the outer and inner regions because g(y + ) is known to vary between zero and one. Data, taken at several Reynolds numbers, for uv g( y ) () u uv / u is plotted in Fig.. As Re this will become g(y + ). For large y + the data does not collapse together, because the outer behavior becomes dominant. However, since we know the outer function G(Y) and the common part, there is a better way to make the correlation. This is the main idea of this section. In the present

9 case theory provides G(Y) and G CP. In general these items could have been determined experimentally from plots of for all y, is uv u Comp uv / u ( Y, Re ). A composite expansion, which is valid g( y ) G( Y) G g( y ) Y g( y ) y / Re Solving for g(y + ) and identifying the data as the composite gives CP () uv g( y ) y / Re () u Data Now the data is essentially corrected for (the first) Reynolds number effects. Figure shows experimental data processed in this way. The correlation is now much better, and the entire g(y + ) is displayed. This illustrates the use of the composite expansion in correlating data. An added benefit is that once G(Y) and g(y ) are determined, they may be used to predict profiles for any Reynolds number. This is done by substituting the correlations into the composite expansion, Eq.. More details are in Panton (00). Normal Vorticity Fluctuations; Dependent Scale Change In the next example there is a change of scale in the dependent variable between the inner and outer regions. Consider the fluctuating vorticity component normal to the wall, y. The intensity of this is measured by the statistical time average y y, which is a function of the distance from the wall. In the inner region the proper scaling uses the friction velocity. y y () u y y / Experimental measurement of vorticity fluctuations is an extremely difficult task, and as a result, DNS offers the best source of data. Figure gives the vorticity results for the inner region computed at four Reynolds numbers, Re = 0, 0, 0, and 000 (see figure captions for references). While these are not very high, as practical Reynolds numbers go, the results do reveal reasonable correlations.

10 The same nondimensional form is displayed in Fig. for the outer region. Since the curves are tending toward zero as Re, this is not the proper order-one scaling. It is not reasonable that the outer region answer is zero as vorticity fluctuations are an essential characteristic of turbulence. A scaling change between the inner and outer regions is required. The matching and the character of the common part function can help in determining the proper scaling change. Assume that the common part is a power law, that is; ~ ( ) n y y CP C y () The common part of the outer function is a function of Y alone and equal to the inner common part. Re-expressing the inner common part in the outer variable, Y y / Re gives ~ ( Re ) n () y y CP CY n In order to be a function of Y alone, the outer common part would be ~ CY. And the proper outer dependent variable scaling is / Re n y y. That is n n y y CP yy Re ~ CY (0) These matching characteristics will be used to determine n. Figure is a log-log plot of y y verses y. Also shown is a line of slope minus one. The overlap region goes to higher y + as Re becomes higher. The question is what is the correlation as the Reynolds number becomes high? The slope of the curves in the overlap region is nearly minus one as the Reynolds number increases. Assuming that n implies that the proper scaling in the outer region is Or yy yy Re () y y y y u / ( ) () h Data correlated in this variable is displayed in Fig.. The curves do collapse together nicely in this variable. This scaling yields a variable that is of order one in the outer region as the Reynolds number increases. Curves scaled with any other combinations of

11 u and/or U 0 will tend toward zero or infinity as the Reynolds number increases. Readers familiar with turbulence theory may note that scale. 0 ( h) /u is the Kolmogorov time This section is an example of how matching behavior between inner and outer functions can be used to fine the change in scale of the dependent variable. An alternate method is to try different values of n in / Re n y y until one is found that produces finite curves as Re. Spanwise Vorticity Fluctuations; An Expansion Requiring Two Terms It has recently been observed, but not universally accepted, that several fluctuating turbulence quantities do not scale solely with u, but also involve U 0. It is possible that the physical explanation is associated with Townsend s concept of active and inactive turbulent motions. Assuming that these motions scale differently. This is a case where different physics occur in the same spatial region. Consider the vorticity fluctuation v/ x - u/ y. It involves the streamwise z velocity fluctuation u, which according to Townsend has both types of motion. In the inner region a two-term expansion is needed for the statistical average. Let z z 0 be the inactive component and z z be the active portion. A dimensionally consistent expression is u U u U u U zz z z 0 zz u 0 / 0 / / 0 u This is the form of an expansion with U 0 as a gauge function ~ u ( y ) (Re ) ( y ) () z z z z 0 z z U0 Here the symbol indicates a mixed scaling, z z () u U z z 0 /

12 The + symbol indicates inner scaling, z z () u z z / Subscripts 0 and now stand for the zero and first-order terms. This example uses the fact that a two-term expansion has coefficient functions with essentially different scalings. Data from DNS calculation is given in Fig. in the scaling. In the domain of y + from 0 to 0 there is a decrease in the curves as Re increases. Incidentally this is the region where the y vorticity fluctuations reach a maximum; see Fig.. The question is, what is the limiting behavior of these curves as Re? According to Eq. this would be 0 ( z z y ). A dashed curve is drawn on Fig. that is conjectured to be the limiting value 0 ( z z y ) as Re. This is only a guess as some Re trends are still observed in the curves. The first-order term is found by solving Eq., inserting the data for and the conjectured curve for z z 0. ( ) u y ( y ) / (Re ) z z z z z z 0 U0, Figure displays the data processes in this manner. Ideally it is a function of only y +, and Fig. shows this tendency. In fact, the conjectured curve was chosen considering that the difference should correlate. If the data is precise enough, the limit process definitions of finding successive terms in an asymptotic expansion can be used to find a two-term expansion. Streamwise Reynolds Stress; Employing Physical Assumptions. Ultimate goal of experiments is an interpretation of physical events. If there are two terms that substantially contribute to a quantity, it is desirable to realize this situation to aid in the physical understanding. As noted above, this is the situation with regard to the streamwise velocity fluctuations, u. z z ()

13 Traditional turbulence theory regarded all fluctuations as scaling on the friction velocity u. However, Degraaff and Eaton (000) claimed that in their boundary layer experiments, the proper scaling for / uu is the mixed scale u U 0. Another instance is the paper of Johansson and Alfredsson () who employed a mixed scale in another setting. The Degraaff and Eaton s laboratory experiments went to a high Reynolds number, but practical situations are much higher. Metzer and Klewicki (00) confirmed the mixed scaling at extremely high Reynolds numbers by testing in the atmospheric boundary layer. Accepting the possibility that the proper scale for inconsistency. The Reynolds shear stress uv uu is uu 0produces an scales solely on u, both theoretically and experimentally. Another physical consideration is that Townsend proposed that u fluctuations contained active motions that contribute to the shear stress, and inactive motions that do not. He did this to explain differences in intensities, however, he thought that both components scaled on u. Based on these physical reasons it is logical to propose that <uu> is composed of two nondimensional terms. (, Re ) u uu y uu ( y, Re ) uu ( y, Re ) (Re ) () 0 U0 Again the symbol indicates a mixed scaling, And the + symbol indicates u scaling, uu uu () u U 0 uu uu () u Essentially the assumption is that the parts with different scalings are distinct and additive. The second term is a small contribution compared to the first, and this contribution becomes smaller as the Reynolds number increases. Experimental measurements are not accurate enough to use the limiting definitions Eqs. and to determine the two parts. On the other hand, the data is taken at Reynolds numbers where the second term is

14 somewhat important. Also, the physical understanding is that two physical processes are involved. To account for the second term a physical assumption is interjected. The second term has the same scaling as the Reynolds shear stress so they are in some sense related. Assume that the profiles are related by some number P. uu P u (0) v For statistical reasons P is probably negative and P=- is chosen. This may be a crude approximation, but it is undoubtedly better than ignoring the fact that u has two scalings. Consider the outer region of a zero-pressure-gradient boundary layer. The first term in Eq. is computed from u v uu 0 uu u () U0 Data is used for uu and uv profiles were established by previous correlations. Figure from McKee (00) shows experimental data from various sources plotted in the outer variable. For boundary layers the outer variable is the wall distance normalized by the Rotta-Clauser distance, y / where u / U and is the 0 displacement thickness (there are several equivalent choices for the outer length scale). Ideally, if the assumptions are correct and the experiments accurate, these curves would show no Reynolds number effects for in the outer region. No definite Reynolds number trends are apparent. Scatter on the figure indicates how difficult it is to accurately measure uu. A curve has been fitted to the data and is shown as a solid line. The value at =0 is the common part, which in this instance is a constant. The function for the inner region is found by solving the composite expansion for uu (,Re ), as done previously in Eq.. 0 y uu ( y,re ) uu ( y ) uu ( ) uu () 0_ Comp 0_ In 0_ Out 0_ CP uu ( y ) uu ( y,re ) uu ( ) uu () 0_ In 0_ Comp 0_ Out 0_ CP Data is used for the composite, and an equation fitted to Fig. for the outer and its common part. A plot of the results, displayed in the inner variable, is given in Fig. 0.

15 The correlation is as good as can be expected with current day instruments and the crude assumption of Eq. 0. Once again solving a composite expansion for the inner function has corrected the data for a Reynolds number effect. In this instance the outer function was determined from the data after a physical assumption was applied to the data. Summary This article illustrates how ideas from applied mathematics are useful in analyzing and correlating experimental data. Singular perturbation situations, situations with two regions having different physical processes, are considered. Also the inner region displays a situation where two processes with different scalings occur simultaneously. Matched asymptotic expansions, and composite expansions with different types of common parts are illustrated by examples. Knowledge of the matching behavior is shown to be helpful in analyzing and correlating experimental data. It is well known that the turbulent mean velocity profile requires that two terms in the outer expansion, one of which is constant, match to a single term in the inner profile. This situation requires a logarithmic matching function. An additional result is that the ratio of the velocity scales is a logarithmic function of the Reynolds number. The Reynolds shear stress example shows how a composite expansion can be employed to correct the data for Reynolds number effects. In this way a correlation is produced that is not otherwise apparent in the data. The streamwise Reynolds stress is an example of a quantity that requires a two-term expansion in both regions. This means that there are two scalings and two physical processes that should be recognized. Another example that needs two terms in the inner region occurs in the x and z- direction vorticity fluctuation components. Physical arguments indicate that two scalings must be sought in the analysis of the data. Another aspect of the vorticity fluctuation profiles is a change in the scale between the inner and outer regions. The y-component vorticity fluctuation scale in the inner region is u, while in the outer region it is u / ( ) h. Their ratio is the Reynolds / number Re, the perturbation parameter. This means that the matching function between

16 the regions, the common part, decreases as / y. In fact the analysis is used in the reverse direction. The common part behavior was determined from the data, in order to find the change in scaling. The use of concepts from applied mathematics- matched asymptotic expansions, common parts, composite expansions-can be employed in analyzing experimental data to produce a sharper result and better physical understanding. References Antonia, R.A., M. Teitel, J. Kim, & L.W. B. Browne Low Reynolds-number effects in a fully developed turbulent channel flow, J. Fluid Mech., pp-0. Carlier, J. & Stanislas, M. 00 "Experimental study of eddy structures in a turbulent boundary layer using PIV," J. Fluid Mech., pp-. Del Álamo, J. C. & Jimenez, J. 00 Spectra of very large anisotropic scales in turbulent channels. Phy. Fluids pp L-L. Del Álamo, J. C. Jimenez, J. Zandonade, P. & Moser, R. D. 00 Scaling the energy spectra in turbulent channels. J. Fluid Mech. 00, pp-. DeGraff, D. B. & J. K. Eaton 000 Reynolds number scaling of the flat plate turbulent boundary layer, J. Fluid Mech., pp-. Harder, K. J. & Tiederman, W. G., "Drag reduction and turbulent structure in twodimensional channel flow," Phil.Trans. Roy. Soc. Series A, Vol., pp-. Hoyas, S. & Jimenez, J. 00, "Scaling the velocity fluctuations in turbulent Channels up to Re = 00," Annual Research Briefs-00, Center for Turbulence Research, pp -. Johansson, A. V. & Alfredsson, P. H. "Effects of imperfect spatial resolution on measurements of wall-bounded turbulent shear flows," J. Fluid Mech., pp0. McKee, R. J. 00 Composite Expansions for Active and Inactive Motions in the Streamwise Reynolds Stress of Turbulent Boundary Layers PhD dissertation University of Texas McKeon, B. 00 editor "Scaling and structure in high Reynolds wall-bounded flows," Phil. Trans. Roy. Soc.,, pp -.

17 Metzger, M. M. Klewicki, J. C. Bradshaw, K. L. & Sadar, R. 00 "Scaling the nearwall turbulent stress in the zero pressure gradient boundary layer," Phys. Fluids,, pp-. Metzger, M. M. & Klewicki 00 "A comparative study on near-wall turbulence in high and low Reynolds number boundary layers," Phys. Fluids,, pp-0. Millikan, C. B. Proc. th International Conference on Applied Mechanic, Cambridge, MA, p. Moser, R. D. Kim, J. Mansour, N. N. Direct numerical simulation of turbulent channel flow up to Re_tau = 0, Phys. Fluids,, p. Panton, R. L. A Reynolds stress function for wall layers, Journal of Fluid Engineering,, pp. Panton, R. L. 00 "Review of wall turbulence as described by composite expansions," Applied Mechanics Reviews,, pp -. Panton, R. L. 00 "Composite asymptotic expansions and scaling wall-turbulence," Phil. Trans. Roy. Soc.,, pp -. Panton, R. L. 00 "Scaling and correlation of vorticity fluctuations in turbulent channels," Phys. Fluids,, 0 (pp). Pope, S. B. 000 Turbulent Flows Cambridge University Press Schlichting, H. & K. Gersten 000 Boundary layer Theory, Springer, th edition Tennekes, H. & J. L. Lumley A First Course in Turbulence MIT Press. Townsend, A. A. The Structure of Turbulent Shear Flow Cambridge University Press. second edition Wei, T. & Willmarth, W. W. Reynolds Number Effects on the Structure of a Turbulent Channel Flow J. Fluid Mech. 0, pp -. Zanoun, E. Nagib, H. Durst, F. & P. Monkewitz 00 "Higher Reynolds number channel data and their comparison to recent asymptotic theory," AIAA Zanoun, E.-S. 00 "Answers to some open questions in wall-bounded laminar and turbulent shear flows," PhD thesis, Inst. Fluid Mech. Friedrich-Alexander University of Erlangen- Nuremberg.

18 Manuscript Click here to download Manuscript: RockwellExpsFluidsfigCap.doc Figure Captions Fig Experimental data for uv / u taken at several Reynolds numbers As Re this becomes g(y + ). Data from Antonia et al. (), Harder & Tiederman (), Wei &Willmarth (), and Zanoun et al. ( 00). Fig Data of Fig. processed according to a composite expansion, Eq. Fig DNS data for y-vorticity fluctuations in the inner region scaled as /( / ) y y y y u. DNS data from Hoyas, & Jimenez (00), Del Álamo, et al. (00, 00), Moser et al. (). Fig DNS data for y-vorticity fluctuations of Fig. in the outer region scaled as yy yy u /( / ) Fig DNS data for y-vorticity fluctuations of Fig. on a log-log plot of verses y. Also shown is a line with slope minus one y y Fig DNS data for y-vorticity fluctuations (sources same as Fig.) with / / ( ) Re outer scaling u h y y y y y y Fig DNS data for z-vorticity fluctuations (sources same as Fig.) in the inner region scaled as zz zz / u U0 / conjectured first term. The dashed line is a z z 0 (y ) Fig DNS data for second term for z-vorticity fluctuations (sources same as Fig.) ( ) z z y processed according to Eq. Fig Experimental data processed to yield uu ( y ) and plotted in the outer variable (from McKee(00)). Data from DeGraaff & Eaton (000) and Castello and Johansson. 0

19 Fig 0 Experimental data for uu 0_ In (same source as Fig.) processed according to Eq.

20 Figure Reynolds Stress <uv>+ = <uv>/u^ Antonia Re= Re= Re= Re= H&T Re=0 Re=0 Re=0 Re=0 W&W Re= Re=0 Re=00 Re= Zanoun Re= Re= Re= Distance ~ y+

21 Figure Reynolds Stress <uv>+ = <uv>/u^ Antonia Re= Re= Re= Re= H&T Re=0 Re=0 Re=0 Re=0 W&W Re= Re=0 Re=00 Re= Zanoun Re= Re= Re= Curve Fit Distance ~ y+

22 Figure y-vorticity Fluctuation ~ < y y> DNS Re= Distance ~ y+

23 Figure y-vorticity Fluctuations ~ < y y> DNS Re = Distance ~ Y = y / h

24 Figure y-vorticity Fluctuation ~ < y y> DNS Re= Slope = Distance ~ y+

25 Figure y-vorticity Fluctuation ~ < y y>+ Re 0 0 DNS Re = Distance ~ Y = y/h

26 Figure z-vorticity Fluctuations ~ < z z> Re= Conjectured High Re Limit , Distance ~ y+

27 Figure z-vorticity Fluctuations ~ < z z>+_ DNS Re= Distance ~ y+

28 Figure

29 Figure 0. Streamwise Inner Function ~ <uu>_0_in Distance ~ y+ D&E Re= Re= Re= Re= Re=000 C&J Re= Re= Re= Re=0 Re=

30 Manuscript Click here to download Manuscript: PantonExpFluidsReply.doc Reply to referees for Experiments in Fluids article: Correlating Data; Examples from Turbulent Wall Layers, by R. Panton My answers to the reviewer comments are in larger type just below the comment. Reviewer Reviewer : Overall, this is an interesting and informative paper. At first, I was concerned whether a paper on composite expansion methods is suitable for Experiments in Fluids. However, the author does well in explaining the relevance of the topic for analysis of experimental data, and how it can be used to gain physical insights. The paper could be improved and I recommend a minor revision taking into account the following comments:. The acknowledgement of Don Rockwell is well justified but would be better placed at the end of the article rather than at the start of the Introduction. I see no problem with the present location. I ll let the editor decide.. The article would be more useful if it discussed openly the limitations or implicit assumptions that need to be made in matched expansion analysis. For example, one needs to know/assume what the two layers scale on, as well as choose the gauge function for the expansion. For example, does one use /Re_tau, or C_f, or sqrt(c_f) etc? This is a difficult question. I have added the comment: Finding the correct q O_SCALE and gauge function is a trial and error process.. The discussion in the Introduction about turbulent wall layers having "two regions with different physics" is very brief and the some references could be given here to highlight the latest studies on how these regions interact. The Phil. Trans. Society of R. Soc Volume 00 in which the author contributed is a good source as are several recent papers in JFM. Good suggestion. I have added the reference McKeon, B. 00 editor "Scaling and structure in high Reynolds wall-bounded flows," Phil. Trans. Roy. Soc.,, pp -.

31 Reviewer Correlating Data; Examples from Turbulent Wall Layers, R. Panton The present study describes how ideas of matched asymptotic expansions can be used to correlate data from turbulent wall layer experiments (both physical and computational). The author presents compelling results based on the application of this method toward understanding Reynolds number effects in the wall-normal profiles of the Reynolds shear stress, vorticity fuctuations, and streamwise velocity fuctuations. The paper will be of general interest especially to those in the scientific community studying turbulent wall bounded flows, and, as such, is deemed worthy of publication in this dedication issue of Experiments in Fluids, with some minor revisions as listed below. In my version of the manuscript, Figure numbers and captions were missing. In addition, several of the plots lacked a y-axis label. Finally, I believe that the paper would make a stronger contribution and have a broader impact if the author were to elaborate on some of the scaling choices made in The various examples. Other specific concerns or questions are listed below. _ pg., after eqn.: How does one select the proper gauge function in general? Finding the correct q O_SCALE is a trial and error process. _ pg., eqn.: Along the same line as the question above, how does one know that the correct gauge function in this case is u_=u? It turns out that F 0 is one, and the gauge function (found through a history of trial and error) is u / U 0. _ pg., eqn.: State in the text that F in this form is obtained directly from () once q0 scale =_(Re_) has been chosen. The function F (Y), known as the defect velocity profile, is found by solving Eq. (see also Eq. ). _ pg., eqn.: What is the function f0(y+)? In the inner region near the wall, the velocity scale changes to u. The velocity profile is U y (,Re) ~ 0 ( f y )... u This correlation is the law of the wall. () _ pg., eqn.: Please provide details as to how this common part is obtained. The common part is the log law for the inner region. Millikan () found this common part by considering how a function of the form of Eq. could match a function of the form of Eq. in an overlap region.

32 _ pg., eqn.: Please provide reference for the theory mentioned. (for example see Tennekes and Lumley ()) _ pg., eqn.: How does one know that there is no change in scaling between the outer and inner regions in this example? There is no change in scaling <uv> between the outer and inner regions because g(y + ) is known to vary between zero and one. _ pg., end of section: Can you show how well and to what extent this works by curve fitting low-moderate Re_ data and then testing the resultant composite expansion against higher Re_ data. It seems like this would be a nice test of the methodology proposed. A reference has been given for this. This is done by substituting the correlations into the composite expansion, Eq.. More details are in Panton (00). _ pg., before eqn.: How do you know which range of y+ constitutes the overlap region because the slope so determined appears to be sensitive to this? The overlap region goes to higher y + as Re becomes higher. The question is what is the correlation as the Reynolds number becomes high? The slope of the curves in the overlap region is nearly minus one as the Reynolds number increases. Assuming that n implies that the proper scaling in the outer region is _ pg., eqn.: This results supporting this in Fig. are quite striking. Yes. An alternate method is to try different values of n in / Re n y y until one is found that produces finite curves as Re. _ pg., last sentence of section: change fine" to find". Done _ pg., eqn.: Why is mixed scaling selected for the first term in the expansion? Let z z 0 be the inactive component and z z be the active portion. A dimensionally consistent expression is

33 u U u U u U zz z z 0 zz u 0 / 0 / / 0 u This is the form of an expansion with U 0 as a gauge function _ Fig. : What is the basis for the conjecture of the limiting value of h!z!zi 0 and how was this curve determined? This is only a guess as some Re trends are still observed in the curves. Ideally it is a function of only y +, and Fig. shows this tendency. In fact, the conjectured curve was chosen considering that the difference should correlate. _ pg., after eqn.: Why was chosen as the correct outer variable and not simply? For boundary layers the outer variable is the wall distance normalized by the Rotta- Clauser distance, y / where u / U0 and is the displacement thickness (there are several equivalent choices for the outer length scale). _ Fig.0: Can you please comment on the slight, but apparent, Re_ trend in the peak value? A plot of the results, displayed in the inner variable, is given in Fig. 0. The correlation is as good as can be expected with current day instruments and the crude assumption of Eq. 0.