Boundary Layer Thickness and Shape

In this Chapter, the new integral moment method [6-8] is outlined for describing the thickness and shape of the velocity, temperature, inertial, and pressure profiles formed by fluid flow along a wall. The boundary layer moment method was developed from the observation that the plot of the second derivative of the Blasius boundary layer for laminar flow over a plate looks very much like a Gaussian distribution curve. This led to the adoption of the moment method usually used to describe probability density functions to describe the fluid boundary layer's thickness and shape. The most important results of this new integral moment approach are:

• A mathematically well-defined measure of the boundary layer thickness that utilizes the entire profile, not just a few tail region data points as is done with δ₉₉.

• Four new parameters that help describe the thickness and shape of the boundary layer. These four parameters are the mean location μ₁, the boundary layer width σ, the velocity profile skewness γ₁, and the velocity profile excess γ₂. The skewness and excess are true shape parameters as opposed to the made-up parameters like H₁₂.

• Applying the moment method to the first and second derivatives of the velocity profile generates additional parameters that, for example, determine the location, shape, and thickness of the viscous forces in a boundary layer. As we show below, the viscous thickness and shape parameters can be calculated reliably without the need to numerically differentiate the velocity profile and can do so even for truncated experimental profiles.

• It is possible to prove that many of these velocity thickness parameters are also similarity scaling parameters. That is, if similarity is present in a set of velocity profiles, then these thickness parameters must also be similarity length scaling parameters.

To demonstrate the usefulness of the new boundary layer description, we consider the advantages of applying the new method to two contemporary boundary layer application areas: Transition Model Development and Assessment and DNS and LES Boundary Layer Assessment. In the process, we show that the well-known T3A transitional profiles [11] show evidence of the ‘streaks’ or ‘puffs’ seen in the bypass transition model.

3.1 Describing the Velocity Boundary Layer formed by Fluid Flow along a Wall

The Prandtl [1] boundary layer conjecture assumes that as a fluid moves along a wall, the velocity at the wall surface will be zero due to friction but then increase to an asymptotic velocity over a relatively small distance above the wall. This "boundary layer" region has been experimentally verified for many fluid flows along a wall. Characterizing the thickness and shape of the boundary layer velocity profile formed due to this flow is therefore important from both a practical as well as a scientific standpoint. For 2-D boundary layer flows measured in wind tunnels and depicted in Fig. 1.5, the velocity profile thickness is considered to be the point where the velocity just reaches the boundary layer edge velocity given by u_e(x). To mathematically define the thickness and shape of the boundary layer, we outline the velocity profile moment method recently developed by Weyburne [6-8]. What is new herein is that the moment method is specifically adapted to the “bounded” and “unbounded” boundary layer concepts discussed in Chapter 1.

3.1.1 The Traditional Method

The traditional methods for describing the thickness and shape of the velocity profile along a wall are rather crude and problematic. The integral-based displacement thickness and the momentum thickness are certainly straightforward to calculate but neither one describes the outer edge boundary layer thickness. As a consequence, the 99% boundary layer thickness parameter, δ₉₉, has become the de facto standard measure. However, this parameter has problems. Since the actual mathematical form of the transition to the free stream for laminar and turbulent flow is unknown, then it is not possible to fit experimental profiles for δ₉₉. To determine δ₉₉ from experimental data one has to interpolate using a few noisy tail region data points. Then there is the problem for external flows. As we saw in Chapter 1.2.2, δ₉₉ is not able to describe an aerodynamically thick object's boundary layer thickness. However, the most serious problem may be theoretical. There is very little chance that a theoretical derivation will prove δ₉₉ is a similarity scaling thickness parameter. This rules out δ₉₉ as a useful parameter for any dimensional analysis of boundary layer flows.

The traditional method to describe the shape of the boundary layer also has limitations. The usual method is to calculate the shape parameter H₁₂, which is the ratio of the displacement thickness to the momentum thickness. This is not a shape parameter in any normal physical or mathematical sense but it was the only parameter available to early practitioners that was able to discriminate between laminar and turbulent boundary layers.

3.1.2 The Moment Method for Bounded Interior Flows

There is a better way to describe the thickness and shape of the fluid boundary layer. A relatively new moment method [6-8] for describing the thickness and shape of the 2-D boundary layer utilizes the integral moment method commonly used to describe a random variable's probability distribution. The moment method for boundary layer flows was developed from the observation that the plot of the second derivative of the Blasius [9] boundary layer for 2-D laminar flow over a flat plate looks very much like a Gaussian distribution curve (Weyburne [6]). In Fig. 3.1a, the second derivative profile is plotted and compared to a Gaussian curve. There are no adjustable parameters; the Gaussian curve uses the mean location and boundary layer width values (defined below) obtained from the exact Blasius curve. In Fig. 3.1b, the velocity profile obtained by twice integrating the Gaussian function is shown along with the Blasius velocity profile. The curves are almost indistinguishable.

It was this Gaussian-like appearance that prompted the idea of adapting the mathematics to describe probability density function technology to boundary layers. Weyburne [6-8] started with the second derivative description but then also applied it to the velocity profile and first derivative profile. It is straightforward to cast the properly scaled velocity profile and its first two derivatives into probability-density-like integral kernels with the zeroth central moment normalized to one. The same methodology will also work for the inertial profiles, the temperature profiles, and the pressure profiles.

To demonstrate the approach, consider interior flow for a 2-D channel with height H that has velocity u(x,y) in the flow direction along the wall (x-direction), and the y-direction is normal to the channel wall. The channel height H is assumed to be much thicker than the maximum viscous boundary layer thickness. Assume the boundary layer asymptotes to a plateau with boundary layer edge velocity, u_e(x). {Strictly speaking, the velocity will not asymptote to a constant velocity value but will peak instead. However, as we discussed in Chapter 1.2.4, wind tunnel data is limited in the y-extent so appears to plateau. Therefore, without recourse, we adopt this assumption.} Casting the velocity profile into a probability-density-like central moment for 2‑D interior flows is done in terms of moments of 1‑u(x,y)/u_e(x). Thus, the velocity boundary layer nth moment ζ̇_n(x) is defined as

where

and where the displacement thickness, δ₁(x), is defined as

The same methodology can also be applied to also calculate the integral moments of the derivatives of the velocity profile. The first derivative nth central moment, 𝞳_n(x), is defined as

so that 𝞳₀(x) normalizes to one. The mean location of the first derivative of the stream-wise velocity profile is formally defined as the first y-moment about zero and, in this case, is just the displacement thickness, δ₁(x).

The viscous second derivative velocity boundary layers nth central moment, λ_n(x), is defined as

where

where ν is the kinematic viscosity and τ_w is the wall shear stress.

With the moments and the mean locations defined, the boundary layer thickness and shape can be described in terms of the boundary layer mean location, width (variance), skewness, and excess (excess kurtosis). Experimentally, it is found [8] that the turbulent boundary layer thickness defined as δ_s = δ₁ + 3σ_s (where σ_s = (𝞳₂)^1/2) tracks the 99% thickness very well.

Taking a cue from the boundary layer momentum balance equations, the second derivative boundary layer moments, λ_n, track the thickness and shape of that portion of the boundary layer where the viscous forces are significant. Weyburne [8] showed that the viscous thickness, given by δ_v(x) = μ₁(x) + 2σ_v(x) (where σ_v = (λ₂)^1/2) approximately tracks the 99% thickness for laminar flow. For turbulent flows, the moment method makes it possible to track and quantify the inner viscous region using λ_n moments whereas the outer region of the turbulent boundary layer can be tracked using ζ̇_n and 𝞳_n moments.

Calculation of the derivative moments without the need to differentiate the u(x,y) profile is simplified by using integration by parts to reduce the moments to simply integrals based on the displacement thickness kernel. Thus, if

then it is possible to calculate the velocity profile moments and the first and second derivative moments of interest using just the first four alpha moments without having to differentiate the velocity profile. The first derivative skewness, for example, can be calculated as

This parameter tracks the boundary layer shape changes that accompany the laminar to turbulent boundary layer transition (Weyburne [8]).

3.1.3 The Moment Method for Unbounded Flows

For exterior flows, there are several ways to modify the above equations to make new moment expressions correctly describe the thickness and shape of the exterior boundary layer. One version (see Weyburne [3-5]) is to divide the boundary layer into two regions around the maximum velocity u_max. For the region between the wall and the u_max location, the above equations can be used but with u_max replacing u_e(x) and δ_max replacing H/2. For the part of the boundary layer between the u_max location and deep into the free stream, the inertial moments described in the Air Force Tech Report [3] are preferred.

3.1.4 Calculating the Moments

Numerical errors encountered in calculating the moments, especially the higher-order moments, are a serious concern. Small experimental variations caused by noise in the tail region or small gradual changes in the plateau velocity u_e(x) can have a large effect on the calculated moments. To avoid this problem, it is advisable to round all the normalized velocities that are nominally in the free stream to one. For the bounded boundary layer, the free stream starting point was taken as the first point at which the velocity u(x,y)/u_e(x) ratio was greater than or equal to one.

The second area of concern has to do with the calculation of the various moments using truncated profiles. Due to experimental difficulties, most wind tunnel experimental velocity profile's first measurement data point starts at about 30% to 60% of the boundary layer thickness, meaning 30% to 60% of the profile closet to the wall is missing. This brings up the obvious question as to how the missing near-wall profile affects the calculated thickness and shape parameters. This would seem to be especially problematic for the second derivative moments since much of the second derivative profile could be missing. To answer the question on the effect of the missing data one has to look at how the moment-based parameters are actually calculated. These parameters are calculated using the wall shear stress and the first couple of alpha moments (Eq. 3.7). It is easily verified that the viscous parameters viscous width σ_v(x), the displacement thickness δ₁(x), the viscous skewness γ_1v(x), and the viscous excess γ_2v(x) can all be calculated by various combinations of α₀, α₁, and α₂ using integration by parts (Weyburne [8]). The key insight is that the alpha moment calculations turn out to be insensitive to truncation of the velocity profile.

To demonstrate this insensitivity, a simple experiment was performed using a Direct Numerical Simulation (DNS) turbulent channel flow dataset from Orlu and Schlatter [13]. The advantage of using the DNS data is that the velocity profile is available all the way to the wall. This means it

is possible to calculate the integral moments using truncated and untruncated profiles (note that the u(x,0)=0 data point is always included whether truncated or not). The results of truncating the second derivative moments so that the first data point starts at 30% of the boundary layer thickness and a second set where the first data point starts at 60% of the boundary layer thickness are shown in Table 3.1. To mimic wind tunnel data, 0.5% full-scale random noise was added to the truncated DNS velocity profile data points. The table results are the percentage difference between the full profile and the truncated profiles and are the mean and standard deviation for ten velocity profiles spanning Reθ = 677 to Reθ = 4061.

The results are clear, even though much of the near-wall second derivative profiles are essentially missing, the calculated second derivative profile thickness and shape parameters can still be calculated with high confidence even for truncated velocity profiles. The viscous excess errors may seem large but are found to be small relative to the large differences observed in turbulent velocity profiles (see Fig. 8.3 in New TBL Chapter). Equally important, these parameters are calculated without having to resort to numerical differentiation of the velocity profile, a process known to be problematic in the presence of noise.

3.1.5 The Adverse Pressure Gradient Second Derivative Problem

The moment method is based on the premise that the various profiles can all be scaled to behave as probability density functions. However, one particular type of profile type cannot be forced into the right form: the second derivative velocity profile for the adverse pressure gradient case. The mean location, μ₁(x), still works but the adverse pressure gradient moments become problematic. To understand why, consider Fig. 3.2 in which a series of second derivative profiles are shown for airflow along a NACA 0012 wing section. The profiles are calculated normal to the flow direction for the 0.5 M, alpha=0, NACA0012 laminar flow simulation by Swanson and Langer [10]. The negative of the second derivative profiles starts to take on negative values in the near-wall region. Negative values are not allowed in the standard probability framework so the application of the moment methodology will result in biased measures for this case. Weyburne [3] pointed out a fix is to simply exclude the negative values and define a new set of moments for a truncated second derivative profile starting at the second derivative peak (Fig. 3.2b). Using the modified moment values, the boundary layer location where the second derivative profile becomes negligible above the wall can still be calculated.

3.1.6 The Moment Thickness Parameters as Similarity Scaling Parameters

One of the advantages of the new thickness parameters is that certain of these parameters are also similarity length scaling parameters if velocity profile similarity is present along the wall. For 2‑D wall-bounded flows, velocity profile similarity is defined as the case where two velocity profiles taken at different stations along the flow differ only by simple scaling parameters in y and u(x,y). Weyburne [14] has presented theoretical proof that some of the above moment parameters, including δ₁, μ₁, and σ_s are similarity length scaling parameters for bounded boundary layer flows. That is, if similarity is discovered in a set of 2-D velocity profiles in bounded boundary layer flows, then δ₁, μ₁, and σ_s must be similarity length (height) scaling parameters. In the Similarity chapter, it is shown that the momentum thickness, δ₂, also is a similarity parameter for bounded boundary layer flows.

3.1.7 Integral Moment Method Applications

One way to demonstrate some of the advantages of the new integral moment method is to offer a few examples. To start, we examine the case for the new moment method as an assessment tool for developing approximate shear stress transport models applied to laminar to turbulent transitions on a wall. In the process of comparing the CFD and experimental velocity profiles, it is demonstrated, for the first time, that the well-known Roach and Brierlay [11] T3A transitional velocity profiles shows evidence of the ‘streaks’ or ‘puffs’ bypass transition model. Following that effort, we examine the case for using the moment method as an assessment tool for comparing turbulent boundary layer DNS and LES calculations to wind tunnel experimental results.

3.1.7.1 Turbulent and Transition Model Development

There is a significant need to be able to simulate laminar to turbulent boundary layer transitions and turbulent boundary layers as part of a complete flow modeling software package. An important part of the software development is the need to have some confidence that simulations are accurately modeling real flow situations. The Navier-Stokes closure problem means that transitional and turbulent flows simulations on a wall are problematic. Various approximate computational methods have been developed based on shear stress transport models (see, for example, Fürst, et. al. [15] and Menter, et. al. [16]). To date, the primary means of assessment have been to observe skin friction plots and simple velocity profile plot comparisons. In the following, we show that the integral moment method for describing the thickness and shape of the velocity profiles can add important assessment metrics to the shear stress transport model development and comparisons. As an unexpected highlight, we show that the well-known T3A transitional profiles from Roach and Brierley [11] show evidence of the ‘streaks’ or ‘puffs’ bypass transition model.

The usual method to access approximate shear stress transport models is to calculate the resulting skin friction coefficient and compare it to experimental wind tunnel data. Consider the skin friction results shown in Fig. 3.3. In this figure, the OpenFOAM's T3A turbulence transition tutorial-verification simulation result is compared to the original Roach and Brierley’s [11] T3A wind tunnel data. The simulation was run with the Langtry-Menter k‑omega shear stress transport model. The results in Fig. 3.3 indicate the simulation can reproduce the skin friction experimental results. This would seem to indicate that the wall shear stress model is correctly modeling the laminar to turbulent transition process.

Can the new integral moment method offer any benefit in assessing the shear stress transport models? Consider Fig. 3.4. In this figure, we have plotted the second derivative profile integral moment thickness and shape parameters for the experimental wind tunnel results and the OpenFOAM simulation results Langtry-Menter k‑omega shear stress transport model. The second derivative moments provide a way to describe the thickness and shape parameters for the viscous region.

Based on the plots of the skin friction data, the simulation data using the Langtry-Menter k‑omega shear stress transport model does a good job of following the experimental wind tunnel data. However, the viscous thickness data and the viscous skewness and excess data in Fig. 3.4 show some notable differences between the wind tunnel and simulation data. The shear stress model is not fully capturing the physics correctly. Differences are also seen in the first derivative moments and the velocity profile moments (not shown). These integral moment-based parameters can provide critical guidance for the development of eddy viscosity models for transitional and turbulent boundary layers.

Can the new integral moment method offer any benefit in assessing the shear stress transport models? Consider Fig. 3.4 in which we have plotted the second derivative profile integral moment thickness and shape parameters for the experimental wind tunnel results and the OpenFOAM simulation results using the Langtry-Menter k omega shear stress transport model. These second derivative moment parameters provide a way to describe the thickness and shape parameters for the viscous region. The simulation thickness results seem to compare well to the experimental results. The viscous thickness data and the viscous skewness and excess data in Fig. 3.4 do show some differences in the transitional point thickness values. The experimental values are noticeably larger than the simulation results. This could be due to the experimental shear stress values being calculated as Clauser chart method which may not be accurate in this region. The differences in the shape parameters are harder to explain. The skewness and the excess profile shown in Fig. 3.4d and Fig. 3.4e indicate significant differences in the shape. More work needs to be done on the implications of these results, but overall, it would appear that these integral moment-based parameters can provide useful guidance for the development of eddy viscosity models for transitional and turbulent boundary layers.

An unintended benefit of the assessment process is that for the first time, we can document the step-by-step processes occurring in the laminar to turbulent transition. To understand what we mean, consider Fig. 3.4c. The purely laminar portion of the depicted flow consists of the first two simulation points as determined by comparing the μ₁(x) T3A values to the Blasius laminar value (Fig. 3.4a). The next four data points constitute the start of transition. These transition data points are also showing square root behavior, but this square root behavior does not follow the same square root behavior as the Blasius laminar flow result. The next two transition data points no longer show square root behavior. The remaining data points have a distinctly different slope and constitute the purely turbulent boundary layer data points.

This transition behavior in Fig. 3.4 supports the evolving model for bypass transition (see discussion in Durbin [17]). Bypass transition supposes that transition begins with the appearance of boundary layer ‘streaks’ or ‘puffs’ in the other-wise laminar-like wall flow. According to Avalia, et. al., [18] the transition to fully turbulent flow occurs when the temporal and spatial frequency of the puffs reaches a certain threshold (puffs for pipe flow and, presumably, steaks for wall flow). The transition regime starts as laminar-like with relatively isolated puffs or streaks which dissipate resulting in the flow returning to laminar-like behavior. As the temporal and spatial frequency increases, the flow does not have time to fully return to laminar-like behavior and starts to behave more like turbulent flow. Once a certain threshold is reached, the flow turns completely turbulent. The second derivative profile moment method provides, for the first time, a way to observe this behavior in experimental velocity profiles.

The second conclusion we can make

The data may also indicate that the transition behavior may occur as a two-step process with a possible inflection point seen in the viscous skewness (Fig. 3.4d) and the viscous excess (Fig. 3.4e). The results are difficult to tell conclusively and will require further study to confirm.

3.7.1.2 DNS and LES Boundary Layer Assessment

Direct numerical simulation (DNS) and Large Eddy Simulation (LES) have emerged as powerful tools for investigating the nature of turbulence and turbulent boundary layers. It is important to have some confidence that the simulations are accurately modeling real fluid flow situations. To date, the primary means of assessment have been to observe skin friction plots and simple velocity profile plots. In the following, we show that the integral moment method for describing the thickness and shape of the velocity profiles can add important assessment metrics to the DNS and LES computations.

In Fig. 3.5 the thickness ratios for Orlu and Schlatter [13] the wind tunnel and DNS values are displayed. The displacement thickness δ₁ is used as the normalization thickness. The second derivative integral moments provide a way to describe the thickness and shape of the region where viscosity is important (Weyburne [7,8]). In Fig. 3.5a, the location of the viscous mean location, μ₁(x), ratio and the viscous boundary layer width, σ_v(x), ratio are displayed. The μ₁(x) parameter (Eq. 3.6) is the viscous mean location and σ_v(x) is the viscous width. In Fig. 3.5b, the velocity profile mean location, m, (Eq. 3.2) ratio and the velocity profile width, σ_u, ratio are displayed.

The skewness and excess shape parameters for the second derivative profiles for these two datasets are shown in Fig. 3.6a and 3.6b. The viscous skewness, γ_1v(x), as the name implies, provides guidance as to the shape of the second derivative profile. The nearly Gaussian-like Blasius curve has a low skewness value with higher values indicating a distorted asymmetrical shape. The viscous excess, γ_2v(x), gives information about the wings of the profile. In Fig. 3.6c and 3.6d, the skewness and excess shape parameters are shown for the velocity profile.

Overall, Figs. 3.5 and 3.6 indicate the wind tunnel and DNS show good consistency for the thickness and shape of the velocity profile as well as the second derivative profile. The second derivative results are particularly significant in that they provide evidence that the DNS process is correctly simulating the viscous portion of the boundary layer. The use of thickness ratio and shape plots would appear to provide a powerful way to assess the DNS process for simulating turbulent boundary layers.

All of the above parameters can be calculated with only six measured or calculated parameters. The viscous mean location μ₁(x) and the viscous width σ_v(x) parameters can both be calculated by knowing the wall shear stress and the displacement thickness (Weyburne[7,8]). The various skewness and excess parameters also require various combinations of α₁ , α₂ , α₃ , and α₄ where the alpha moments (Eq. 3.7) are given by the integral of yⁿ times the displacement thickness integrand.

3.2 The Thermal Boundary Layer Moment Equations

The power of the moment method is not limited to the velocity profile. It can also be applied to any boundary layer quantity that can be put into probability density-like functional form. The thermal profile formed by fluid flowing above a heated or cooled wall is one such example. For laminar flow on a heated flat plate, Weyburne [6] showed that the second derivative of the scaled temperature profile shows Gaussian-like behavior. Hence, the same moment method technology for describing the thickness and shape of the velocity profiles can also be used to describe the thermal boundary layer profile (Weyburne [19]). This "thermal boundary layer" can have a significant impact on the efficiency of heating/cooling equipment, among other things. Characterizing the thickness and shape of the thermal boundary layer is therefore important from both a practical as well as a scientific standpoint.

To demonstrate the thermal profile method, consider a semi-infinite 2-D channel with height H in which the fluid flowing along the inside walls has velocity u(x,y), where x is the flow direction, and y is the normal direction to the plate. The height H is a set value that must be much thicker than the maximum velocity or temperature boundary layer thicknesses. The fluid inside the channel has a temperature T(x,y), the temperature at the channel walls is T_w, and the free stream fluid temperature is T₀. Applying the integral moment method, the thermal boundary layer can be described in terms of the central moments, ξ_n(x), given by

where the thermal displacement thickness β₀(x) is the normalizing constant. The first moment about zero, which is called the mean location, is defined as

such that the thermal displacement thickness, β₀(x), is defined as

There are some advantages to also calculating the integral moments of the derivatives of the thermal profile. Let

then the thermal first derivative central moments, ε_n(x), are defined as

where the thermal displacement thickness β₀(x) (Eq. 3.11) is the mean location.

The thermal second derivative thermal profile central moments, χ_n(x), are defined as

where, μ_T(x), is both the normalizing constant and is also the second derivatives mean location. The mean location, μ_T(x), is formally defined as the first y-moment about zero but its actual value is deduced by normalizing χ₀(x) to one. Thus,

With the moments and the thermal mean location defined, the thermal boundary layer thickness and shape can be described in terms of the thermal boundary layer mean location and the width (variance), σ_T. For the Pohlhausen [20] solution for laminar flow on a heated flat plate, for example, the thermal boundary layer thickness, defined as δ_T = m_T(x)+ 4σ_T (where σ_T=(χ₂)^1/2), tracks the 99% thickness very well (Weyburne [19]). The boundary layer shape is tracked with the thermal skewness, and thermal excess (excess kurtosis).

For laminar flow, the thermal profile and the first two derivative moment cases all give similar values for the thermal boundary layer thickness. For turbulent flow, the thermal boundary layer can be divided into a region near the wall where thermal diffusion is important and an outer region where thermal diffusion effects are mostly absent. Taking a cue from the boundary layer energy balance equation, the second derivative boundary layer moments, χ_n, track the thickness and shape of that portion of the thermal boundary layer where the thermal diffusivity, α, is significant. Hence the moment method makes it possible to track and quantify the region where thermal diffusivity is important using χ_n moments whereas the overall thermal boundary layer is tracked using ε_n and ξ_n moments.

Calculation of the derivative moments without the need to take numerical derivatives is simplified by using integration by parts to reduce the moments to simply integrals based on the thermal displacement thickness kernel auxiliary integrals (Weyburne [19]). If we define the set of auxiliary integrals as

The above thermal boundary layer development applies to bounded interior flows. The equivalent development for unbounded exterior flows has not been formally presented. There does not appear to be a reason the same trick of dividing the boundary layer into two regions should not work. This would involve an inner region where thermal diffusivity, α, is significant and an outer region where thermal diffusivity is not significant.

These thermal-displacement-like integrals let us calculate the higher-order χ_n and ε_n moments without having to differentiate the thermal profile. Using integration by parts, the first derivative thermal boundary layer width, σ_T=(ε₂)^1/2 , can be conveniently calculated as σ_T= (β₀² - 2β₁)^1/2 for example. Other first and second derivative parameters such as the thermal skewnesses and excesses can be calculated without having to numerical differentiate the experimental dataset using this technique.

The above thermal boundary layer development applies to bounded interior flows. The equivalent development for unbounded exterior flows has not been formally presented. There does not appear to be a reason the same trick of dividing the boundary layer into two regions should not work. This would involve an inner region where thermal diffusivity, α, is significant and an outer region where thermal diffusivity is not significant.