Turbulent Boundary Layer Defect Profile Similarity: A Case of Bad Science

In this Chapter, the research work on similarity scaling parameters for the Turbulent Boundary Layer (TBL) for the past 70 years is reviewed and found to be flawed. This work has led to the incorrect identification of the similarity scaling parameters and the mistaken belief that turbulent boundary layer similarity is common. As a remedy, the similarity scaling parameters that were identified by Weyburne [30] are tested and found to give reasonable results for certain experimental wind tunnel results. The same flawed approach is also found in the Thermal TBL literature which is briefly discussed at the end of the Chapter.

The presence of velocity profile similarity and thermal profile similarity in certain boundary layer flows is widely accepted for laminar flow along a wall. For turbulent boundary layer flows along a wall, it is generally acknowledged (Marusic, et. al. [37]) that whole profile similarity, except for sink flow, has never been observed in experimental profiles. The turbulent boundary layer has a viscous inner region and an inertial outer region, so, instead of looking for similarity over the whole profile, researchers started looking for similarity in just the inner or outer regions. The fact that the Navier-Stokes governing equations for turbulent flow do not have a closed-form solution means that the mathematical approach to similarity used by Blasius [9] and Falkner-Skan [26] for laminar flow will not work for turbulent flows. The flow community had some early theoretical guidance beginning with the work of Rotta [31] and Towsend [32] that indicated that the velocity at the boundary layer edge, u_e(x), must be a similarity velocity scaling parameter. However, when no experimental examples emerged, this insight was ignored and “better” alternatives were pursued instead. It turns out that this pursuit for better similarity parameters has been based on a false premise that negates much of the TBL similarity scaling work of the last 70 years. In the following Chapter, the problem with the turbulent boundary layer similarity literature is outlined and discussed. This is followed by a brief outline of a similar problem identified in the temperature profile similarity literature.

To understand the nature of this TBL similarity problem, it is necessary to go back to the work of Fredrick Clauser [38] in the 1950’s. Clauser set out to explore 2-D TBL similarity of the velocity profile. He demonstrated that if he plotted a series to TBL data sets from various groups as scaled velocity profiles, there was no similarity present. However, when he plotted the same data as scaled defect profiles defined as u_e(x)-u(x,y), where u(x,y) is the velocity in the flow direction (x‑direction) and u_e(x) is the corresponding velocity at the boundary layer edge, then the visual inspection of the plotted data sets showed good overlap suggesting that similarity was present. Following closely after Clauser’s work, Rotta [31] and Towsend [32] developed the defect profile-based theory of TBL similarity. Subsequent searches for similarity scaling parameters for the 2‑D TBL have adopted the use of the defect profile as a means of “discovering” similar behavior. What followed has been 70 years of research culminating in work by Castillo and George [33] and others, that indicate that TBL similarity is fairly common and occurs for most TBL data sets if one uses the velocity scaling parameter u_ZS(x) = u_e(x)δ₁(x)/δ₉₉(x) developed by Zagarola and Smits [39]. Prior to the work of Castillo and George, TBL similarity of the outer region was considered nonexistent or, at best, rare.

All of the extensive literature concerning experimental investigations into wall-bounded TBL similarity have been based on the defect profile. To even suggest using the experimentally measured velocity profile to study similarity, as is done for laminar flows, is considered wrong and, according to some, shows a lack of understanding of turbulent boundary layer theory. However, a review of turbulent boundary layer literature reveals there has never been a theoretical justification for this defect profile preference. Therefore, the preference for the defect profile originates solely from the success of experimental comparisons; there is no theoretical preference. Stepping back and looking at the situation from a theoretical perspective, there should be no preference (Weyburne [40]). As we will show below, theory indicates that velocity profile and defect profile similarity must occur simultaneously. This is evident when one considers that the occurrence of similarity is intimately tied to the physics of the flow. It is not possible to change the physics of the flow by simply re-plotting the data set after subtracting off the endpoint. Simple mathematical manipulation of the data does not change the physics.

And yet, that is exactly what appears to be happening in this case. Weyburne [41a, 41b] recently reexamined some of these TBL data sets that others claim to show defect profile similarity and found that the defect profiles did appear to show similar behavior. However, when the same data is re-plotted as standard velocity profiles, visual inspection of the plotted data sets does not show similarity, just as Clauser observed for the data sets he investigated. To understand why the appearance of similarity in one case and not the other is a problem, one can point to the argument that the simple mathematical manipulation of the data set cannot change the physics. However, the flow community has managed to ignore this simple fact for 70 years and this argument is unlikely to change anyone's thoughts today.

Therefore, to fully understand why the appearance of similarity in one case and not the other is a problem, it is useful to first review the definition of velocity profile similarity. Recall that similarity of the velocity profile for 2-D wall-bounded flows is defined as the case where two velocity profiles taken from different stations along the flow differ only by simple scaling parameters in y and u(x,y). Assume a set of velocity profiles is discovered that show similarity in the outer region when scaled with the length scaling parameter δ_s(x) and the velocity scaling parameter u_s(x). For the outer region of the TBL, the scaled velocity profile at a station x₁ along the wall will be similar to the scaled profile at x₂ when

The Eq. 7.1 definition is for interior bounded boundary layers and is slightly modified from Schlichting's [2] usual definition of similarity by changing "for all y" to "for all y₁ and y₂ in the outer region" (the exact definition of the outer region extent is not important in the arguments below).

Defect profile similarity is defined in a similar fashion. Using the above notations, defect profile similarity would occur when

for y₁ and y₂ in the outer region. By inspection of Eqs. 7.1 and 7.2, it is apparent that defect profile similarity and velocity profile similarity must occur simultaneously if

This criterion for simultaneous velocity and defect profile similarity also shows up as a criterion for defect profile similarity originally developed by Rotta [31] and Towsend [32]. More recently, Castillo and George [33], and Kitsios, et al. [34] also derived the same criterion for defect profile similarity. Castillo and George’s derivation was specifically developed to consider outer region TBL similarity. All of these theoretical formulations ended up with u_e(x) as a similarity scaling parameter requirement. The theoretical implication is obvious, defect profile similarity must be accompanied by velocity profile similarity.

Consider the implications if Eq. 7.3 is not satisfied but defect similarity Eq. 7.2 is present. If Eq. 7.3 is not satisfied then Eq. 7.1 cannot be satisfied which means the velocity profiles are not similar. This is the situation the literature has been operating in for the last 70 years since Clauser’s work; velocity profile similarity is absent but defect profile similarity is present. However, if the Eq. 7.3 equality condition is not satisfied, then defect profile similarity, Eq. 7.2, would require that u(x₁,y) ≠ u(x₂,y) in the limit x₂ → x₁, a physical impossibility. Therefore, unless one changes the definition of similarity, defect profile similarity and velocity profile similarity must occur simultaneously.

Despite this unassailable theoretical requirement for u_e(x) as a similarity scaling parameter, the literature to date has ignored this theoretical result, without explanation or discussion, and instead looked for other similarity scaling parameters. Using various experimental data sets, several groups have explored different scaling velocities that “appear” to show better similarity behavior than u_e(x) as a similarity scaling parameter. For example, based on experimental defect profile comparisons, Castillo and George seemingly reject their own theoretical derivations for u_e(x) to advocate for Zagarola and Smits [39] scaling u_ZS(x) instead. Others, including Panton [42] and Buschmann and Gad-el-Hak [43], also attempted to use experimental profile comparisons to show that u_ZS(x) is superior to u_e(x) as a similarity scaling parameter.

Hence, the theory says one thing and experimental evidence seemingly says something else. Which is correct? In this case, it is the theory that is correct. To understand why this is the case, one must understand how experimental similarity is usually evaluated. What is normally done is to simply plot all of the scaled profiles onto one graph. If the scaled profiles all overlap using the “chi-by-eye” test, then the profiles are assumed to be similar. Consider, for example, Fig. 7.1a where some of Österlund’s [44] scaled wind tunnel experimental data is plotted at various wall positions. Both Panton and Buschmann and Gad-el-Hak also used some of the Österlund’s data to assert that u_ZS(x) is the better similarity parameter. Examining Fig. 7.1a, one would have to agree that u_ZS(x) is very effective at producing similar-like behavior for the defect profile case. All five profiles plot on top of one another. Now consider Fig. 7.1b in which the same data and scaling parameters are used but the data is plotted as velocity profiles. By any measure, the five velocity profiles in Fig. 7.1b do NOT display similar behavior.

So, what is going on in Figs. 7.1a and 7.1b? How can one plot show similarity but not the other? Consider Fig. 7.1b. Notice that all five tail regions DO NOT overlap. Let us play the devil's advocate. How can the tail regions in Fig. 7.1b be made to appear to be identical without changing the u_ZS(x) scaling parameter? One way to hide the tail region difference is by subtracting the data's end-point from each of the experimental profiles. If this is done, then the defect datasets tail region will automatically be zero for every profile thereby ensuring the tail regions appear to be similar for all the defect profiles. Therefore, if you want to inadvertently dupe people into believing that similarity exists, then plot the data as defect profiles. But of course, the tail regions of the scaled velocity profiles, Fig. 7.1b are not similar. The similarity theory requires the tail regions scaled velocities must be equal which Fig. 7.1b indicates they are not. Similarity must occur in both defect profiles and velocity profiles simultaneously. You cannot subtract off the endpoint value from each profile and change the physics of the flow. This data set does NOT display velocity profile similarity when scaled with u_ZS(x) even though Fig. 7.1a appears to display defect profile similarity. Therefore, this data set is not similar when scaled with the Zagarola and Smits scaling parameters.

There is one additional factor (Weyburne [14]) particular to the u_ZS(x) scaling parameter that tends to ensure the defect profiles at every scaled y-location look similar and not just in the tail region. Plots of y/δ₉₉(x) versus u(x,y)/u_ZS(x) where u_ZS(x) = u_e(x)δ₁(x)/δ₉₉(x) means all of the curves are normalized by the displacement thickness which is the area under the defect profile. Hence, the area under the plots like Fig. 7.1a all have areas equal to one. For profiles that are taken from locations downstream of each other, plotting scaled profile data with equal areas tends to make the profiles appear to overlap whether they are similar or not. Therefore, the combination of eliminating the tail region disparity combined with the equal area factor means that most data sets will “appear” to be similar when the Zagarola and Smits scaling is used.

When Clauser [38] discovered that defect profiles showed similarity but the velocity profiles did not, the theoretical work of Rotta [31], Towsend [32], Castillo and George [33], and Kitsios, et al. [34] was not yet in place. All of the theoretical results indicate that if defect profile similarity is present, then u_e(x) must be a similarity scaling parameter (Eq. 7.3). This means that defect profile similarity and velocity profile similarity must occur simultaneously (Eqs. 7.1 and 7.2). In every data set that Weyburne [41a, 41b] examined that other groups have claimed to show defect profile similarity, velocity profile similarity was absent. The bottom line is that much of the research on turbulent boundary layer similarity for the last 70 years is seriously flawed and requires a thorough review. Perhaps the most important conclusion in regards to the search for TBL similarity is that contrary to Castillo and George, Turbulent Boundary Layer similarity is NOT widespread but rare. Unfortunately, this means that developing a generalized approximate TBL velocity profile based on similarity scaling is unlikely to work.

The TBL similarity problem is extensive. The Zagarola and Smits [39] papers, for example, have been referenced over 900 times according to Google Scholar. The visual proof offered by defect profile plots is so strong that the flow community has never attempted to explain how the theories of Rotta [31], Towsend [32], Castillo and George [33], and Kitsios, et al. [34] are somehow flawed in requiring u_e(x) be a similarity scaling parameter instead of u_ZS(x). Common sense indicates that the flow physics of profile similarity should not be changed by simply replotting the DC shifted data. Yet, the flow community’s acceptance of Clauser’s defect profile preference over velocity profile similarity has been universal.

7.1 Alternative Outer Region TBL Similarity Scaling Parameters

In the same set of papers, Weyburne [41a,41b] showed how the Zagarola and Smits scaling parameters do not work, Weyburne also showed how the whole profile similarity parameters δ₁(x) and u_e(x) appear to give reasonable results for a certain limited set of TBL data sets. Four examples were demonstrated. One example is the five Österlund [44] TBL profiles used in Fig. 7.1. In Fig. 7.2, the data is replotted using the δ₁(x)and u_e(x) scaling parameters. The overlap is not perfect but certainly better than the Zagarola and Smits scaling parameter set result shown in Fig. 7.1b. In Fig. 7.2b, eight ZPG turbulent boundary layers from Smith and Smits [70] are plotted using δ₁(x) and u_e(x) scaling parameters. In this case, the overlap is very good. Additional examples are offered in a second set of papers in Weyburne [35,45] in which a simple comparison test using the δ₁(x) and u_e(x) scaling parameters, the Prandtl Plus scaling patrameters, and the Zagarola and Smits scaling parameters. The former paper [35] also offers a numerical method for determining whether similarity is present in a set of velocity profiles as opposed to the present “chi-by-eye” examination of graphed data sets. The results indicate that strict whole profile similarity is not evident in any of the datasets that were searched. However, ten datasets were found that displayed “similar-like” behavior when scaled with δ₁(x) and u_e(x) scaling parameters (Weyburne [35,45] ) but not the other parameter sets.

7.2 The Thermal Profile TBL Similarity Scaling Fiasco

The same type of similarity fiasco that has been playing out in the velocity profile literature has also been repeated for the thermal TBL profile case (Weyburne [47]). Wang and Castillo [48] have developed empirical parameters based on the thermal displacement thickness for scaling the temperature profile of the turbulent boundary layer flowing over a heated wall. They presented experimental data plots that showed similarity type behavior when scaled with their new scaling parameters. However, what was actually plotted, and what actually showed similarity type behavior, was not the temperature profile but the defect profile formed by subtracting the temperature in the boundary layer from the temperature in the free stream. Recently, Weyburne [47] showed that if the same data and same scaling is replotted as just the scaled temperature profile, similarity is no longer prevalent. This failure to show both defect profile similarity and temperature profile similarity is indicative of the same type of failed similarity discussed above for the velocity profile similarity case. The arguments leading to this conclusion are identical to those discussed above. Rather than repeating the arguments, refer to the Wang and Castillo Rebuttal paper (Weyburne [47]) instead. The bottom line is that the thermal similarity claims of Wang and Castillo [48] are flawed and the paper should be retracted.