The Logarithmic Law of the Wall Explained by a New TBL Conceptual Model

In this Chapter, a new conceptual model for the near-wall viscous region of the turbulent boundary layer is reviewed. The new model offers realistic insights into the origin of the Logarithmic Law of the Wall region. The advantage of this new model is that it directly connects experimental observations to the appearance of the Log Law region making it easy to conceptualize. As a first step, we abandon the unverifiable sub-layer model currently being used to return to a simple two-region viscous and inertial model. The x-momentum equation tells us that the viscous region will be important where the second derivative of the velocity is significant. The new model’s philosophy is to treat the time-averaged second derivative profile as one would treat any statistical function: using the method of moments. In the new model, the thickness and shape of the viscous region is directly measured using velocity profile moments. In the traditional model, one assumes a sub-layer model without actually being able to measure the location and extent of the sublayers. The differing approaches are illustrated by example: testing whether the high Reynolds number pipe and channel flow are equivalent (the results indicate that they are not equivalent).

In the second half of the chapter, we return to our early effort to model the near-wall turbulent boundary layer profiles using an ensemble of perturbed laminar-like profiles. The underpinning of the approach is the assumption that the viscous regions for laminar and turbulent velocity profiles are indistinguishable for similar pressure gradients. The turbulent motion in the outer region induces localized pressure gradients which result in localized wall shear stress fluctuations. The earlier attempt to simulate the turbulent boundary layer inner layer using this spectrum of laminar-like second derivative profiles is further refined to include a spectrum of second derivative profiles attributed to the near-instantaneous breakdowns that result in the streaks or puffs observed in the transitional and turbulent boundary layers. The resulting simulated viscous region, although not complete, is shown to have the right behavior.

8.1 The Traditional Near Wall Turbulent Boundary Layer Model

Before discussing the new model in detail, we first review the traditional model for comparison. The traditional turbulent boundary layer starts by scaling the experimental velocity profiles by the Prandtl Plus parameters. The Prandtl Plus parameters are directly dependent on the one experimental parameter one can easily measure due to the wall-fluid interaction: the wall shear stress. The traditional assumption is that the time-averaged structure of the Prandtl Plus scaled near-wall turbulent boundary layer all appear similar or universal. This "universal" time-averaged scaled boundary layer velocity profile structure is sub-divided into four sublayers: 1) a viscous linear sublayer closest to the wall, 2) a buffer layer, 3) a logarithmic layer, and finally, 4) the wake outer layer (see, for example, Klewicki, et. al. [49]) . The Logarithmic region is characterized by the "Logarithmic Law of the Wall" which describes the behavior of the time-averaged velocity profile for turbulent boundary layer flow along a wall. The Law of the Wall states that the average velocity for turbulent flow at a certain point above the wall is proportional to the logarithm of the distance from that point to the wall. The name "Law of the Wall" comes from the fact that much of the flow community (see, for example, discussion in Marusic, et. al. [50]) believes this logarithmic behavior applies to all turbulent flows along the interior and exterior walls. The original Log Law justification was made independently by Ludwig Prandtl [51] and Theodore von Kármán [52] among others. A review of the historical, as well as recent developments, is provided by Buschmann and Gad-el-Hak [43], Marusic, et. al. [37], and Örlü [53].

Recent revelations about pressure gradient differences (Chauhan, et. al. [54]) and Reynolds number dependence of the Log Law constants (Nagib and Chauhan [55]) have tempered the Log Law outlook as a universal model with fixed sublayer extents. Some believe that those problems go away at a high enough Reynolds number (Marusic, et. al. [50]). Even with these recent revelations, the traditional four sublayer model of the TBL has remained intact. The present near-wall model supposes that while the exact locations may vary, the sublayers occur in all wall-bounded turbulent boundary layers. However, this idea of sublayers is pure conjecture and an untestable conjecture since it is not possible to measure the location and extent of these sublayers. There is no way to manipulate the flow governing equations in some way to allow one to calculate the extents and locations of the sub-layers from experimental data. The controversy over the proper start and end extent of Logarithmic Law region (see Marusic, et. al. [37] and Örlü [53]) is a manifestation of this fact. The different start and end locations are all based on observation of certain experimental profiles and not on some extracted values based on the flow governing equations. The bottom line is that the whole sub-region model has no basis in theory, it is all based on suppositional observations of scaled experimental velocity profile curves.

A problem with the sublayer conjecture is that this rigid adherence to the four-sublayer model is obscuring rather than illuminating the physics of the turbulent boundary layer. For example, reading the literature, one is led to believe that the velocity profile in the so-called linear sublayer behaves linearly. However, if the profiles behaved linearly in this region, the viscous momentum forces would be zero since the second derivative of the velocity would be zero. The opposite is true, the viscous momentum forces nearly peak in this sub-region. The physics of the Log Law sublayer is similarly opaque. There is no simple physical explanation for the appearance of this sublayer in the traditional model. The theoretical justifications are plentiful but these theoretical justifications do not provide any insight into the origin of this physical behavior. There does not appear to be any way to connect what is physically happening in the turbulent boundary layer to the traditional sublayers.

8.2 The New Turbulent Boundary Layer Conceptual Model

The four-sublayer model is an ill-conceived attempt to assign physical sub-structures to a statistical quantity, the time-averaged velocity profile. Rather than the fixed location sublayer model presently employed, Weyburne [56,57] proposed a new model of the turbulent boundary layer that treats the time-averaged turbulent velocity profile the same way mathematicians treat probability distributions using mathematical moments. The new model returns to the two-layer viscous inner region and the inertial outer region, both of which can be treated as probability distributions. The thickness and location of the regions are not fixed but are defined by the moment-based thickness and shape parameters. These moment-based thickness and shape parameters are experimentally accessible. Rather than fostering a static time-averaged structural model, the new model actively incorporates the notion that the velocity profiles and the wall shear stress are time-dependent. Experimental observations indicate that the wall shear stress for TBLs undergoes rapid changes (many times a second time scale) due to the chaotic turbulence in the outer region of the boundary layer (see, for example, Obi, et.al. [58] or Diaz-Daniel, et. al, [59]). This outer region turbulence causes rapid variations in the wall shear stress which, in turn, causes velocity profile changes.

The first key insight provided by the new TBL model is that this rapid time-varying change in the wall shear stress induces rapid, quantifiable changes in the thickness and shape of the viscous region in the turbulent boundary layer. The relationship between the wall shear stress and the viscous layer of the wall-bounded turbulent boundary layer is detailed in a paper by Weyburne [8] and summarized in the Boundary Layer Thickness Chapter. To understand how the instantaneous viscous region thickness and shape is related to the instantaneous wall shear stress, one has to look at the momentum balance equations. The x-momentum balance equation tells us the viscous forces will be important where the second derivative of the velocity u(x,y) is significant. Although the instantaneous turbulent boundary layer velocity profiles have not been made available at present, useful insights into the behavior of the second derivative profile can be had by examining the laminar flow second derivative profiles. As an example, for laminar flow, the Blasius [9] second derivative profile is plotted in Fig. 8.1.

The Gaussian-like shape in Fig. 8.1 is what led to the adoption of the standard probability moment method for describing the boundary layer thickness and shape (Weyburne [6]). Treated like a probability density function, the second derivative integral moments provide a way to describe the thickness and shape of the region where viscosity is important. The viscous mean location, μ₁, and the viscous boundary layer width, σ_v, are the two main thickness descriptors for the viscous region. The important insight into this new approach is that it is easily verified that the second derivative instantaneous mean location and instantaneous width are both directly (inversely) proportional to the instantaneous wall shear stress value. Following standard probability practice, the mean, or in this case the second derivative mean location μ₁(x,t), at time t can be shown to be given by

where υ is the viscosity, τ_w is the wall shear stress, and u_e is the velocity at the boundary layer edge. Using integration by parts, the viscous boundary layer width σ_v can be shown to be proportional to the instantaneous wall shear stress and the instantaneous displacement thickness as

In principle, it should be possible to measure the instantaneous turbulent boundary layer velocity profile and extract μ₁(x,t) and σ_v(x,t).

Actual TBL experiments indicate that the wall shear stress at any point on the wall is undergoing rapid changes over time (see, for example, Obi, et. al. [58] or Diaz-Daniel, et. al, [59]). This, in turn, means that the second derivative profile width, σ_v(x,t), is undergoing compression and expansion as the mean location, μ₁(x,t), moves toward the wall or away from the wall in lock step with the wall shear stress changes (Eq. 8.1 and 8.2). This is a key link between experimental observations and what is physically happening in the near-wall viscous region of the boundary layer.

Imagine now that the instantaneous second derivative profile is time-averaged. It is the time-averaged profile that is always measured and discussed in the literature. The second key insight of the new model is the observation that the time-averaged tail of the viscous region is decaying very slowly and extends to the Log Law region. Conventional thinking is that the viscous sublayer only extends from the wall to y⁺≅30 into the fluid whereas the Log Law layer starts much further from the wall. However, plots of the velocity profile side-by-side with plots of the second derivative of the velocity, make it clear that the Log Law region is not a separate sublayer but is an integral part of the tail region of the viscous second derivative profile. It is possible to take high-quality experimental turbulent boundary layer velocity profile data sets, numerically differentiate them twice and compare those values to the second derivative of the Log Law velocity profile (see Weyburne [56,57]). An example is shown in Fig. 8.2a for data from Österlund [44]. In Fig. 8.2b the associated velocity profiles are shown. Note that the logarithmic overlap regions are the same in both cases. The key point is that the logarithmic region of the TBL is not some overlap region or a sublayer as traditionally advocated, but it is instead part of the viscous tail region of the second derivative of the time-averaging the instantaneous velocity profile. Hence, the new model puts us back to the two regions, the inner (viscous) and the outer (inertial) region model. It now becomes clear why the first three layers of the traditional four-sublayer TBL model all have the same scaling parameters (Prandtl Plus) and that is because the three closest sublayers to the wall are not separate sublayers but instead are all part of the inner viscous region.

Examination of Fig. 8.2a reveals that the Log Law region is two or three orders of magnitude lower than the viscous peak. How can this region still be considered part of the viscous region if it is so much smaller than the peak? The reason is related to the time-averaging process. At any given time on the turbulent wall, the wall shear stress fluctuations result in a whole spectrum of instantaneous μ₁(x,t) and σ_v(x,t) values, including values that put the instantaneous second derivative curve deep into the Log Law region. However, the probability of having low wall shear stress values over time is small. The net result is that the time-averaged viscous region falls off as ~1/y in the Log Law region since there are fewer instantaneous velocity profiles whose wall shear stress values push the second derivative curve into this region. Hence, when the time average is done, this region appears to be much diminished when in fact it is an artifact of the time-averaging process.

8.3 Why Assume Viscous Regions Thickness and Shape When You Can Measure It?

At the beginning of this Chapter, we mentioned the biggest problem with the traditional viscous region model is that the basis of the four-sublayer model is pure conjecture and an untestable conjecture because it is not presently possible to measure the location and extent of these sublayers. This has forced the community to base the conjectures for the near-wall region on subjective visual comparisons of velocity profile plots. In the new model, we rely on second derivative integral moments to measure the viscous regions thickness and shape. One extracts μ₁(x,t), σ_v(x,t), the viscous skewness, γ_1v(x), and viscous excess, γ_2v(x), for each velocity profile and then compare the numerical values to other profile values. The time-averaged second derivative based viscous mean location, μ₁, and the time-averaged viscous boundary layer width, σ_v, are both experimentally accessible from velocity profile measurements and wall shear stress measurements without having to resort to numerical differentiation (Weyburne [7,8]). The viscous skewness, γ_1v(x), and viscous excess, γ_2v(x), parameters are dimensionless shape parameters that describe the shape characteristics of the time-averaged second derivative profile. Both of these shape parameters can be calculated from the experimentally measured velocity profile and the experimentally measured wall shear stress without having to resort to differentiation.

Perhaps the best way to illustrate the difference in approaches between the traditional model and the new model is to consider an example. There is an ongoing controversy as to whether turbulent boundary layers Log Law regions in pipe and channel flow are the same. Although it is generally conceded that they are different a low Reynolds number, there is some evidence that they might be similar at high Reynolds number. Recently, Marusic, et. al. [50] used visual comparisons of velocity profile plots from a few different sources to conclude that the results support the existence of a universal logarithmic region for pipe and channel flows at high Reynolds number. Presumably, these few sample sets are representative of all high Reynolds number datasets.

Now consider the new TBL approach. To ascertain whether the pipe channel flows viscous regions are similar, we collected many of the same pipe and channel datasets, plus a few additional sets, and calculated the time-averaged second derivative moment parameters. A preliminary set of plots showing the second derivatives mean location, μ₁, width, σ_v, skewness, γ_1v, and excess, γ_2v, for various turbulent boundary layer datasets, are shown if Fig. 8.3. The mean location and width values are scaled relative to the displacement thickness, δ₁. The datasets, in general, behave as a(Re_θ )^m, where a and m are constants, except for the transitional profiles.

The μ₁/δ₁ results in Fig. 8.3a indicate that all of the datasets are following the same trend line. The mean location μ₁ numerical value is primarily dependent on the estimated wall shear stress value. Hence, the fact that they are all following the same trend line means the estimated wall shear stress values are probably reasonable and therefore the width σ_v values are also reasonably accurate (Eq. 8.2). This is important since differences between pipe and channel flow start to appear for the boundary layer width ratio σ_v/δ₁, for the high Reynolds numbers pipe and channel datasets. The differences for σ_v/δ₁ are on the order of 30% at the same Reynolds number. The differences are even more pronounced for the viscous skewness and the viscous excess where changes are more like 300% for the viscous skewness (Fig. 8.3c) and more than 400%-500% for the viscous excess (Fig. 8.3d) at the same Reynolds numbers. The wide differences in the dataset’s thickness and shape values indicate that the viscous regions for pipe and channel turbulent boundary layers are not the same at high Reynolds numbers.

The parameters δ₁(x), σ_v(x), γ_1v(x), and γ_1v(x) are all calculated by various combinations of displacement thickness-based alpha moments using integration by parts (Weyburne [7,8]). In Chapter 3.2, we pointed out that the expected errors for these parameters in the wind tunnel or pipe experimental datasets ranged from 6% for σ_v(x) to as much as 30% for γ_2v(x). The observed differences in Fig. 8.3 are as much as an order of magnitude larger than the expected error. Hence, the observed differences in the viscous regions for channel flow and pipe flow cannot be attributed to numerical calculation problems.

While the numerical differences are clear, it would be helpful to visualize what is leading to the differences in Fig. 8.3 between the pipe flows and channel flows at high Reynolds number. Plots of the noisy second derivative profiles tend to obscure the differences. However, it is possible to see convincing differences in the plots of the scaled velocity profiles. In Fig. 8.4, Zimmerman, et. al. [63] CoLaPipe pipe profiles are shown with the Bailey, et. al. [60] ICET channel flow data. The observed differences in the velocity profiles are directly coupled to the differences in the second derivative parameters through the alpha moments (Eq. 3.7). Hence, the combination of Figs. 8.3 and 8.4 give us the confidence to be able to say that the viscous turbulent boundary layer regions in pipe and channel flow are not the same. Furthermore, it is simply implausible that the thickness and shape of the viscous regions could be demonstratively different but the Log Law subset region be the same. This would require that the Log Law region be somehow decoupled from the rest of the profile, a physically impossible scenario. Hence, the evidence suggests that the Log Law region is not universal even at high Reynolds number.

8.4 The Origin of the Log Law Layer

What is exciting about the new model is that it opens up a whole new way to experimentally explore the physical origins of the observed logarithmic behavior of wall-bounded turbulent boundary layers. The major unknown in the new model is the shape of the instantaneous velocity profile, and in turn, the shape of the instantaneous second derivative profile. Even without experimental profiles for comparison, it should be possible to develop an instantaneous velocity profile theoretical model that takes into consideration the time-varying wall shear stress. The time-based probability density function of the wall shear stress values is experimentally accessible, which, when combined with an instantaneous velocity profile model, should make it possible to construct the time-averaged velocity profile for the TBL. At that point, it might be possible to show the time-averaging process results in logarithmic behavior in the tail region of the viscous inner region.

Along these lines, Weyburne [57] attempted to construct an approximate time-averaged turbulent second derivative profile using laminar-like instantaneous profiles with a range of wall shear stress values. The speculation was that the time-averaged tail region of the turbulent second derivative profile might be formed from an ensemble of laminar-like second derivative profiles having the wall shear stress distribution observed experimentally. From a physical perspective, there is a good reason to assume that the instantaneous second derivative profile for turbulent flow should spacially decay in the same Gaussian-like decay behavior of laminar flow into the fluid. The timescales for the turbulent motion are many orders of magnitude longer than the time scales normally associated with the molecular diffusion time scales of the viscous forces. Hence the main factor affecting the shape of the viscous region of the velocity profile (besides fluid properties) will be the instantaneous wall shear stress value and the free stream velocity. In the end, Weyburne found that the ZPG Gaussian-like model did not produce the slow decay behavior seen in the average TBL velocity profile.

The original model was simplistic in that it was assumed the second derivative profiles are an ensemble of laminar-like ZPG profiles with a spread of wall shear stresses. This approach does not explain how a spread of wall shear stresses is generated. To address this issue, we propose an updated model that incorporates the streak-puff model (see, for example, Durbin [17] and Avila, et. al. [18]) for laminar to turbulent bypass transitional boundary layers (Section 3.1.7.1). The streak-puff model holds that early in the transition process, the flow behaves essentially laminar-like except for occasional streaks (or puffs in pipe flow) that appear in the outer boundary layer region. As the frequency of streaks reaches a spatial and temporal threshold, the flow transitions to fully turbulent (Avila, et. al. [18]). The new model assumes the near-instantaneous initiation of the puff-streaks produces propagating velocity and pressure gradients fluctuations that then result in the relatively long-timescale fluctuations characteristic of turbulent flow. The propagating pressure gradient fluctuations of the puffs-streaks induce wall shear stress fluctuations downstream along the wall. Since the viscous molecular motion time scale is many orders of magnitude faster than the turbulent fluctuations, then the second derivative of the velocity profile should look similar to laminar-like second derivative profiles having the same pressure gradient. This provides a physical basis for using an ensemble of laminar-like profiles to approximate the time-averaged turbulent velocity profile.

This slow-time ensemble model accounts for the response of the fluid to the streaks (or puffs) propagation. However, it does not include the near-instantaneous initial response of the streaks (or puffs). What we propose is that the initiation of the streaks-puffs is a spontaneous short-time, relatively high-pressure pulse to the local fluid in the outer region of the boundary layer. What this does is introduce a second distribution of wall shear stresses that is biased toward the high wall shear stress values. Time and spatial averaging that are observed experimentally washes out the appearance of these high-pressure pulses such that the time-averaged probability distribution of wall shear stress values takes on the observed log‑normal-like distribution (see Diaz-Daniel, et. al. [59]) biased to the high wall shear stress values.

The proposed new model to explain the physics behind the Log law region, in particular, and the turbulent boundary layer viscous region in general, is:

1) The basis of the new model assumes the localized turbulent second derivative profile consists of two contributions having different time scales. The short-time scale contribution comes from the spatially localized and near-instantaneous fluid breakdown associated with the fluid transitional turbulence effects called puffs or streaks. Initially, the boundary layer is laminar and becomes thicker as it moves along the wall. Eventually, the thickened local velocity profile becomes so stressed due to the high-speed outer flow that the profile structure spontaneously breaks down. These high energy breakdowns in the outer region of the boundary layer take the form of a localized high-pressure spike. The initial pressure spike then propagates downstream as a puff or streak. These types of breakdown events shift the time-averaged second derivative peak closer to the wall. The long-timescale contribution is the fluid's temporal and spatial dissipation effect induced by these high-pressure spikes. The dissipation takes the form a propagating pressure fluctuation that results in wall shear stress fluctuations. It is this type of contribution that causes the broadening of the width of the time-averaged second derivative profile.

2) The localized instantaneous second derivative turbulent boundary layer viscous region is assumed to have the same thickness and shape as the second derivative laminar boundary layer viscous region exposed to the same pressure gradient. That is, we assume an inspection of a turbulent viscous region would be indistinguishable from an inspection of a laminar viscous region for the same pressure gradient. This means that the localized instantaneous zero-pressure gradient (ZPG) second derivative turbulent profile can be assumed to be Gaussian-like of the form

and where μ₁ is the mean location, and σ_v is the profile width (this is the equation that was used to generate Fig. 8.1, see Weyburne [6]). The μ₁ and σ_v parameters are related to the wall shear stress and the displacement thickness through Eq. 8.1 and 8.2. The wall shear stress at any point on the wall for the transitional and turbulent regions is undergoing fluctuations (in the many times a second time frame). The fluctuations follow a roughly log-normal probability function for the wall shear stress values. Hence, the time-averaged second derivative profile consists of an ensemble of the Gaussian-like peaks.

3) The local velocity fluctuations are accompanied by local pressure fluctuations which in turn result in local pressure gradient fluctuations. The instantaneous local non-zero pressure gradient second derivative profile is assumed to be a composite of Eq. 8.3 together with a simple ad hoc analytical correction to account for the non-zero value of the second derivative value at the wall. The relationship between the second derivative at the wall and the pressure gradient is given by the Prandtl x-momentum equation evaluated at the wall as

For the ZPG case, the second derivative value is approximately zero. For FPG case, the wall value can be similar to the second derivative peak value whereas the APG value takes on negative values. Figure 3.2a, for example, shows a series of second derivative FPG, ZPG, and APG profiles on a wing. The fluctuating pressure gradient, therefore, has two effects on the second derivative profile, it changes the second derivative wall value and the wall shear stress value from the nominal ZPG case.

4) To model the relationship between the pressure gradient fluctuations and the wall shear stress, we assume the turbulent wall shear stress and pressure gradient relationship is the same as that that is obtained by laminar flows with an impressed pressure gradient.

8.5 A Laminar to Turbulent Transition Boundary Layer Model

The new approach is intended to model the near-wall Log Law behavior of turbulent boundary layers in terms of an ensemble of laminar-like second derivative profiles having a distribution of wall shear stresses. As an example, we want to be able to model the transitional and turbulent velocity profiles from Wu and Moin’s [58] laminar to turbulent DNS computer simulation. The simulation was able to show the transition from laminar to turbulent boundary layers due to airflow over a flat plate. Our objective is to model the turbulent-like second derivative Re_θ = 800 profile, shown in Fig. 8.5a as the solid green line, by averaging a series of the laminar-like Re_θ = 200 profiles (solid red line). The Re_θ = 800 turbulent-like profile begins to resemble the desired Log Law profile (dashed black line). However, to be able to model the fully turbulent case using the proposed model requires that we know the puff (pipe flow) or streak (channel flow) spatial and temporal initiation probabilities and strength, which are unknown at this point. Therefore, we have chosen a slightly less ambitious first step: can we model the transitional second derivative profile shown as the Re_θ = 300 profile (solid blue line) in Fig. 8.5a using the proposed model?

As a first step, consider the Re_θ = 200 profiles in Fig. 8.5b. The dashed red line is the Gaussian approximate profile using the μ₁ and σ_v values obtained from the experimental (solid red line) profile. The Gaussian profile is a good approximation for the experimental ZPG profile. With that in hand, we attempted to model the long-timescale transitional Re_θ = 300-second derivative velocity profile (solid blue line) as a sum of Gaussian profiles with a spread of wall shear stress values. As a quick preview, we show the results of our new model as the dashed blue line and dashed-dot blue line in Fig. 8.5b (the details are explained below).

The dashed blue line in Fig. 8.5b corresponds to the long-timescale contribution to the fluid’s temporal and spatial response to the puff-streak events. To model this contribution, it is necessary to know the wall shear stress probability distribution. The wall shear stress probability distribution was not reported by Wu and Moin and is unknown at this time. Instead, what is available are similar DNS results from Diaz-Daniel, et. al. [59] that have reported four turbulent wall shear stress probability distribution functions (PDFs) spanning the range Re_θ = 790 to 1820. Their results indicate that, at least for turbulent-like flows, the distribution does not change significantly with Reynolds number. Therefore, without recourse, we assume the transitional Re_θ = 300 wall shear stress PDF has the same distribution as the Diaz-Daniel, et. al. [59] turbulent Re_θ = 790 case. A reproduction of Diaz-Daniel, et. al. [59] turbulent Re_θ = 790 case is depicted as the solid blue line in Fig. 8.6a (ignore the red lines for now).

The new approach is intended to model the near-wall Log Law behavior of turbulent boundary layers in terms of an ensemble of laminar-like second derivative profiles having a distribution of wall shear stresses. As an example, we want to be able to model the transitional and turbulent velocity profiles from Wu and Moin’s [60] laminar to turbulent DNS computer simulation. The simulation was able to show the transition from laminar to turbulent boundary layers for to airflow over a flat plate. Our objective is to model the turbulent-like second derivative Re_θ = 800 turbulent-like second derivative profile as a sum of ZPG laminar second derivative profiles with a range of wall shear stress values. For the updated model, the assumption is that the wall shear stress fluctuations result from pressure gradient fluctuations (Eq. 8.4) in the upper region of the boundary layer. Thus, for the new model, we need: 1) a relationship between the wall shear stress and the pressure gradient at the wall, and 2) approximate models for the non-zero pressure gradient second derivative velocity profiles. To obtain the wall shear stress to dP/dx at the wall correlation, a series of CFD laminar channel flow simulations were run that started with the same Wu and Moin Re_θ = 300 flow conditions. The different pressure gradient conditions were generated by tilting the top channel wall. The wall shear stress to dP/dx at the wall results for the laminar flow simulations are shown as red dots in Fig. 8.6b. The wall shear stress values are scaled by Wu and Moin’s Re_θ = 300 value . The dashed blue line is the quadratic fit to the data.

We have the shear stress PDF and the shear stress to pressure gradient correlation, now we need approximate models for the laminar APG and FPG second derivative of the velocity profile in addition to the ZPG model (Eq. 8.3). Our working assumption is that the second derivative profiles are all laminar-like. Using the laminar flow simulation profiles as a guide, simple ad hoc analytical corrections (linear for FPG profiles and Logistic for APG) to the second derivative Gaussian profile (Eq. 8.3) were generated. The APG series of CFD simulations were found to diverge at high-pressure gradient values. As a remedy, these results were supplemented with the APG profiles from the NACA0012 laminar airfoil simulation by Swanson and Langer [10]. The final approximate profiles were set up so that the pressure gradient at the wall was the only input variable. Sample results for a FPG and APG second derivative approximations are shown in Fig. 8.7. These approximate FPG and APG second derivative profiles do a reasonable job of approximating the laminar simulation results.

The results are encouraging but there is still work that needs to be done to fill out the model. For one, the DNS-based wall shear stress distribution Re_θ = 300 would be welcomed. The speculation concerning the contributions of the spatially localized and near-instantaneous fluid breakdown associated with the fluid transitional turbulence effects called puffs or streaks needs to be explored and documented. Finally, the transitional Re_θ = 300 simulated result needs to be extended to the turbulent Re_θ = 800 result.

With the approximate second derivative curves for the FPG, APG, and ZPG in hand, the last step is to put this all together with the shear stress PDF distribution and the to dP/dx correlation to calculate an approximate “averaged” second derivative profile. The wall shear stress distribution (blue line Fig. 8.6a) is divided into one thousand equally spaced values and then the approximate second derivative profiles for the corresponding dP/dx value are generated. The one thousand profiles are then averaged using the PDF distribution in Fig. 8.6a as the weights. The result is the dashed blue line profile shown in Fig. 8.5b. This profile does a reasonable job of simulating the DNS Re_θ = 300 result, solid blue line, except in the near-wall region. This is an important result in that it shows that observed time-averaged second derivative profile broadening can be explained as an ensemble of laminar-like second derivative velocity profiles with different pressure gradients. This broadening is what ultimately results in the logarithmic behavior seen in fully turbulent boundary layers.

A close look at the near-wall region of the dashed blue line profile in Fig. 8.5b indicates that it fails to address the peaking observed in the near-wall Re_θ = 300 and Re_θ = 800 second derivative velocity profiles. We speculate that this peaking behavior is a result of the short-time scale contribution caused by the spatially localized and near-instantaneous fluid breakdown associated with the fluid transitional turbulence effects called puffs or streaks. Imagine we take a fast snapshot of a patch on the wall surface. If we measured the wall shear stress at multiple locations in the patch, the wall shear stress distribution might look something like the solid red line and the dashed red line in Fig. 8.6a. The Gaussian-shaped solid red line is a result of propagating pressure gradient fluctuations due to the puff and streaks. The dashed red line is a single initiation, fluid break-down event. If one were to continue to take snapshots and average the result, one would presumably obtain the solid blue line distribution in Fig. 8.6a. The breakdown events shift the PDF distribution from a Gaussian-like to a Log-Normal-like distribution.

To try to simulate the puff and streak initiation, fluid break-down event, we assume the pressure gradient amplitude of the breakdown event is larger than the slow time response pressure gradient response to the propagation of the puff-streaks. This was accomplished by artificially changing the quasi-equilibrium laminar-like response (dashed blue line) to the orange dotted line in Fig. 8.6b labeled Streak Peak Adjustment. The adjustment is only made to the high-energy side to be consistent with our breakdown assumption and also consistent with the observation that the laminar channel flow simulations will not converge for the high-energy, high APG flow cases. The process of generating a composite second derivative profile was then repeated using the orange dotted line distribution in Fig. 8.6b. The result shown as the dashed-dot blue line in Fig. 8.5b indicates that the new TBL model does a good job of simulating the DNS Re_θ = 300 result.

The results are encouraging but there is still work that needs to be done to fill out the model. For one, the DNS-based wall shear stress distribution Re_θ = 300 would be welcomed. Comparing the Re_θ = 300 and the Re_θ = 800 second derivative profiles in Fig. 8.5a indicates that the spatially localized, high-energy fluid breakdown events become more and more dominant as the transition progresses. Thus, it would be helpful to try to verify and flesh out the properties of these fluid breakdown events. This would make it possible to extend the transitional Re_θ = 300 simulated results to the turbulent Re_θ = 800 result.