A Better Alternative to the Prandtl Plus Scaling Parameters

In this Chapter, we re-examine the inner region scaling for turbulent boundary layers. There has been universal consensus that the “Prandtl Plus” scaling developed by Prandtl [51] applies to the inner region of both interior and exterior turbulent boundary layer flows. The Prandtl Plus parameters are based on the wall shear stress, the one experimental parameter that can be traced to the wall-fluid interaction. Prandtl took the wall shear stresses velocity derivative evaluated at the wall and converted it into a length and a velocity scaling parameter. However, the Prandtl method for converting a velocity derivative into a length and velocity scaling parameter is not the only way to do it. Recently, Weyburne [68] developed a new set of TBL inner region scaling parameters based on the integral moment method (Chapter 3). These new parameters have several advantages compared to the Prandtl Plus scaling:

· The New parameter set DOES NOT suffer the Falkner-Skan theoretical failure pointed out by Weyburne [69,70].

· If similarity exists in a set of boundary layer velocity profiles, then the New scaling parameter set can be proven to be similarity scaling parameters. The Prandtl Plus have never been proven to be similarity scaling parameters.

· The New thickness parameter can be directly linked to the physics of the boundary layer. The Prandtl Plus thickness parameter has never been derived from a theoretical or physics-based argument concerning the boundary layer but instead, the origin of the Plus parameters appears to be as a conjecture by Prandtl [51].

· The new thickness parameter is part of a whole set of thickness and shape parameters that define the physical structure of the viscous region for boundary layers.

· The new parameters work for laminar, transitional, and turbulent boundary layers whereas the Prandtl Plus parameters do not.

Despite these advantages, the Prandtl Plus parameters have a decided advantage when it comes to experimentally extracting the wall shear stress. Hence, it depends on what you want to do as to which parameter set is best to use. If one is interested in describing the thickness and shape of the viscous region, then the integral moment-based parameters should be used. If one is trying to experimentally extract the wall shear stress, then use the Prandtl Plus parameters. In what follows, we outline some of these advantages and disadvantages with the intent of better understanding the two sets of parameters. We start by reviewing the traditional Prandtl Plus research and then the case for the new scaling parameter set based on the moment method to augment the Prandtl Plus scaling parameters is outlined.

9.1 The Traditional Turbulent Boundary Layer Inner Region Scaling

The idea of velocity profile similarity is well known for laminar flow along a wall. For turbulent flow along a wall, it is generally acknowledged (Marusic, et. al. [37]) that whole profile similarity does not exist. That is, "... not even the mean velocity can be described from the wall to the free stream by a function of a single similarity variable." So, instead of looking for similarity over the whole profile, researchers started looking for similarity in the inner region and separately, the outer region. For the inner region, there has been no disagreement, until recently, about the proper similarity scaling for the region of the turbulent boundary layer near the wall where viscosity is important. There has been universal consensus that the parameters proposed by Prandtl [51], the so-called Prandtl Plus parameters, are the proper scales.

Recently, Weyburne [68] proposed an alternative to the Prandtl Plus parameters. Rather than launching directly into the new parameter set, we first review the Prandtl Plus parameters since that will provide the lead-in to the New set. The Prandtl Plus parameters are inextricably tied to the Logarithmic Law of the Wall. The belief that the Prandtl Plus scaling is the proper scaling for all exterior and interior turbulent boundary layers is tied to the Logarithmic Law of the Wall. It is the Logarithmic Law of the Wall that is considered universal, at least at high Reynold numbers (Marusic, et. al. [50]), and the Logarithmic Law of the Wall subsumes the Prandtl Plus scaling. A discussion and review of the Log Law issues is provided in Buschmann and M. Gad-el-Hak [43], Örlü [53], and Marusic, et. al. [37].

The boundary layer literature on different aspects of the Logarithmic Law of the Wall is vast, constituting thousands of research papers. Experimentally, the Log Law and the Prandtl Plus parameters have proven to be, at a minimum, a good approximation for the viscous region of the turbulent boundary layer. We say approximately because, despite the vast literature on this topic, the Logarithmic Law of the Wall has been never been verified, either theoretically and experimentally. In the following two sections, a review of the experimental and theoretical foundations of the Logarithmic Law of the Wall is given with the intent of showing that there might be a reason to consider an alternative scaling. This is followed by the known theoretical problems with the Log Law which leads to the introduction of the New alternative to the Prandtl Plus parameters.

9.1.1 Experimental Verification of the Logarithmic Law of the Wall

Consider the case for experimental verification. Experimental verification comes down to comparing the wall shear stress value obtained from applying the Log Law to experimental velocity profiles to wall shear stress values obtained by some independent method. The wall shear stress has proven to be very difficult to measure experimentally. For that reason, most experimental determinations of the wall shear stress come in the form of the friction velocity determined by the Clauser [38] Chart method or its equivalent. The Clauser Chart method assumes the Logarithmic Law of the Wall holds so plots of the inner region of turbulent boundary layer velocity profiles using this method will always show similar-like behavior. This has led many in the flow community to falsely believe the Logarithmic Law of the Wall has been verified (see Örlü [53] discussion). The opposite is true, Marusic, et. al. [37] have indicated that at present it is NOT possible to experimentally verify the Logarithmic Law of the Wall due to the limitations of the experimental accuracy of the wall shear stress measurements (the error bars are too large).

The experimental verification problem goes deeper than the error bar problem in the extracted wall shear stress values. The Log Law extracted wall shear values are critically dependent on the start and end locations of the Log Law region experimental profile data (the velocity profile looks like a tipped over S-shaped curve for log scaled profiles so that trying to fit it to a linear line becomes critically dependent on the chosen start and end locations). However, as we discussed in the Log Law Chapter, there is no way to perform some type of manipulation of the flow governing equations that would allow one to pre-determine the extents of the Log Law region. Instead, the start and end locations have been based on speculative observations of experimental profiles and this is reflected in the wide range of start and end location values reported in the literature (Örlü [53]). Therefore, the experimental verification problem is compounded by the lack of a theoretical means to determine the Log Law start and end locations.

9.1.2 Theoretical Verification of the Logarithmic Law of the Wall

Without experimental verification, then one would expect that the Logarithmic Law of the Wall and the Prandtl Plus scaling’s would have been verified theoretically. Although there have been many theoretical justifications of the Logarithmic Law of the Wall [43], the justifications are not the same as theoretical proof that the Logarithmic Law of the Wall is universal for exterior and interior flows or that the Prandtl Plus parameters must be the proper scaling parameters for the viscous region of boundary layer flows. There is no theoretical proof that the Log Law is universal or that the Prandtl Plus scaling’s must be the similarity scaling parameters for the viscous region of the TBL (the proof offered by Jones, et. al. [71] is based on circular logic and is therefore flawed, see Appendix in Weyburne [14]).

Not only is there no theoretical proof but there are several known theoretical problems with the Logarithmic Law of the Wall. The theoretical problem with the start and end location has already been discussed above. More recently, Weyburne [69,70] made a potent theoretical argument against the universality of the Prandtl Plus scalings. Using an α and β based Falkner-Skan momentum equation approach applied to the inner region of the turbulent boundary layer, Weyburne [69] showed that Prandtl Plus scaling’s ONLY works if the wall shear stress behaves as one over the distance along the wall, i.e. like sink flow. According to this theoretical argument, the Prandtl Plus parameters do NOT work for the general Falkner-Skan flow case. These types of flow are fairly common in the boundary layer experimental literature so this theoretical failure is very relevant.

9.1.2.1 The Prandtl Plus Falkner-Skan Problem

To understand the theoretical problem, we briefly review the Falkner-Skan theoretical treatment since it leads directly to the New Prandlt Plus Alternative scalings. Weyburne's [69] Falkner-Skan argument is based on the turbulent boundary layer version of the Falkner-Skan [26] momentum equation. The Falkner-Skan similarity argument starts by constructing a stream function by taking the product of the x‑dependent length and velocity scaling parameters times a scaled y‑dependent functional (see Falkner-Skan development in the y-Momentum Chapter). For the turbulent boundary layer version, the velocities are cast into the average velocities and the deviation from the average value using the Reynolds decomposition approach. The resulting mass and momentum conservation equations for the average velocities reduce to an equation that looks like the laminar flow expressions but has additional terms involving the deviation from the average stress terms. For similarity to be present at various stations along the wall, all of the x‑dependent terms of the nondimensionalized momentum equations must change proportionally as one moves along the wall or, equivalently, the ratios of the x-dependent terms must be constant. The Falkner-Skan flow α and β terms are examples of the constant ratio terms. The turbulent boundary layer version of the Falkner-Skan β term is

where ν is the kinematic viscosity, δ_s(x) is the similarity length scaling parameter, and u_s(x) is the similarity velocity scaling parameter. This term is identical to the Falkner-Skan laminar flow β term. With the α and β terms in hand, Weyburne [69] observed that if the Prandtl Plus scaling was truly universal, then Prandtl Plus scaling parameters should work for the inner region of the Falkner-Skan turbulent flow case. That is, if we assume Prandtl Plus scaling parameters are correct scaling parameters for the inner region, i.e.

where u_𝞃 is the friction velocity, then Eq. 9.1 becomes

The solution to this differential equation for a constant β is that u_𝞃 must behave as 1/x where x is the distance along the wall in the flow direction. This behavior, together with the calculated α= 0 result (not shown, see Weyburne [69]), is characteristic of sink flow, flow in a converging-diverging channel. The Prandtl Plus scaling’s, therefore, are NOT theoretical a solution for general near-wall turbulent Falkner-Skan flows along a wall.

The result is incontrovertible: either the Falkner-Skan approach to similarity is flawed OR the Prandtl Plus scaling’s are NOT similarity scaling parameters for general Falkner-Skan flows along a wall. Although there has been no existence proof offered for the Falkner-Skan approach to similarity or the equivalent defect profile approach, there has been no literature indicating that these approaches are flawed or have problems. Therefore, the Prandtl Plus scaling has a serious theoretical shortcoming.

9.2 An Alternative to the Prandtl Plus Scaling's

In the same paper (Weyburne [69]) that the Falkner-Skan problem with the Prandtl Plus scaling was discussed, a new inner region similarity scaling parameter set was introduced by essentially reverse-engineering the Prandtl Plus parameter failure. Both the new parameter set and the Prandtl “Plus” scaling parameters are based on the one wall parameter that is experimentally accessible; the wall shear stress. The wall shear stress is directly proportional to the derivative of the velocity evaluated at the wall. Prandtl [51] converted this velocity derivative into a length scale and a velocity scale by combining it with the kinematic viscosity. However, the Prandtl method of creating a length scale from the wall shear stress is not the only way to do it. The development of the new reverse engineered parameters begins with the realization that although the Prandtl Plus parameters have a big theoretical problem, experimentally, the wall shear stress determined by the Clauser chart method is reasonably consistent with other direct experimental measurements. This appears to be true even for general turbulent boundary layer Falkner-Skan type flows. So, how do we resolve the theoretical failure but the experimental success?

Weyburne [69] approached this problem by first reverse-engineering the theoretical failure. The first step is to define a new velocity scaling parameter u₀(x), and a new length scaling parameter δ₀(x), based on a length to velocity parameter ratio expression found Weyburne [14]. Weyburne proved that for similarity to occur in a set of velocity profiles for 2-D boundary layer flow, the ratio of the similarity velocity scaling parameter to the similarity length scaling parameter must be proportional to the friction velocity squared divided by the kinematic viscosity (see Weyburne [14], Eq.10). That is, the ratio must be given by

where c is a proportionality constant. There are two facts to notice here. First, the Prandtl Plus scaling's also satisfies this condition. Secondly, this definition by itself does not fully define the new parameters. To do that, Weyburne revisited the Prandtl Plus failure. The failure came in the form of not satisfying the Falkner-Skan α and β terms. So, in addition to Eq. 9.4, u₀(x) and δ₀(x), are required to make the Falkner-Skan α and β terms be constants. Substituting Eq. 9.4 into Eq. 9.1, we require that u₀(x) be given by

where β, ν , and c are constants. The new scaling parameters given by Eqs. 9.4 and 9.5, therefore, satisfy the part of the flow governing equations approach to similarity which the Prandtl Plus scaling’s do not, i.e., they will work for general Falkner-Skan flows.

9.2.1 The Integral Moment Parameters Identified

Eqs. 9.4 and 9.5 address the expected x-behavior of the new velocity scaling parameter, u₀(x), and the new length scaling parameter, δ₀(x), in terms of the x-behavior of the wall shear stress. They do not address the identity of these parameters. In a follow-on paper, Weyburne [68] went about identifying these new inner region scaling parameters. It turns out the new length scaling parameter is one of the integral moment method parameters that define the thickness and shape of any 2-D wall-bounded boundary layer region. Using the second derivative-based moment method, Weyburne [68] showed that the identity of the length scaling parameter is the second derivative mean location μ₁(x) and the velocity at the boundary layer edge u_e(x) is the velocity scaling parameter for the inner region of the turbulent boundary layer, including the Log Law region.

The second derivative mean location parameter μ₁(x) is formally defined as the first y-moment about zero of the second derivative moments (see Eq. 5.5 and Eq. 5.6). It is easily verified (Weyburne [7,8]) that μ₁(x) is given by

where υ is the viscosity. This means that the new length scaling parameter is inversely proportional to the wall shear stress. Comparing Eq. 9.4 to Eq. 9.6, it is evident that the identity of u₀(x) and δ₀(x) are μ₁(x) and u_e(x).

The New parameter set μ₁(x) and u_e(x) is designed to satisfy the Falkner-Skan similarity condition for general Falkner-Skan flows. For the New parameter set, the equivalent Falkner-Skan β term (Eq. 9.1) is

Assuming the resulting β is a constant, then Eq. 9.7 represents a similarity requirement for the x‑behavior of u_e(x) and u_𝞃(x). To solve this equation for the friction velocity, u_𝞃(x), one needs to know the x‑behavior of the boundary layer edge velocity u_e(x). This is in contrast to Eq. 9.3 for the Prandtl Plus Falkner-Skan parameters which have a simple solution for u_𝞃(x). Comparing Eqs. 9.3 and 9.7, it is apparent that the Prandtl Plus Falkner-Skan problem comes about due to Prandtl expressing both the length scale and the velocity scale in terms of the friction velocity, u_𝞃(x). In effect, Prandtl over-specified the problem by trying to generate two independent parameters from one measured parameter, τ_w. In contrast, the new parameter set generates two parameters from two independent measured parameters, u_e and τ_w.

9.2.2 The Physical Origin of the Length Scale

It is not just in the Falkner-Skan application where the new set show advantages compared to the Prandtl Plus set. Consider the length scaling parameters. On the one hand for the Prandtl Plus scaling, we have the length parameter “ν/u_𝞃”. It is not a parameter that is derivable or that falls out of a theoretical treatment or physical manipulation of some boundary layer equation. Instead, the origin of the parameter appears to be a conjecture by Prandtl [51]. There is not a simple physical interpretation that can be assigned to it in terms of its role in the near-wall region. The best that can be said about this parameter from a mechanistic standpoint is that it has the right units and is related to the wall shear stress. On the other hand, for the New set, the length parameter μ₁(x) is intimately tied to the physics of the viscous flow region. It has a well-defined physical meaning and derivation: it is the mean location of the second derivative of the velocity profile. It also represents the area under the scaled second derivative velocity profile (actually one over the area, see Eq. 9.6). The second derivative of the velocity term in the x‑momentum balance equation tells us where the viscous forces are important in the boundary layer. The length parameter μ₁(x) therefore tells us where the mean location of the viscous region of the boundary layer is located. Taking the integral of the second derivative of the velocity and then evaluating at the wall results in the wall shear stress-related mean location μ₁(x) (Eq. 9.6). Thus, the new length scale can be traced directly to the physics of the viscous region. It is part of a whole system of related length and shape parameters that describe the thickness and shape of the viscous region formed by 2-D fluid flow along a wall.

9.2.3 Dimensionless Numbers in Boundary Layer Flow

The advantages of the New parameter set do not end there. A further advantage of the New parameter set is that it is possible to prove that if similarity is present in a set of velocity profiles, then the length scaling parameter μ₁(x) must be a similarity scaling parameter for 2-D bounded boundary layer flow (Weyburne [14]). The importance of similarity scaling parameters is related to the use of dimensionless numbers in fluid mechanics. Dimensionless numbers in fluid mechanics are a set of dimensionless parameters that are useful in analyzing the behavior of fluids. A common example is the Reynolds number which describes the fluid velocity as a ratio of the relative magnitude of fluid and physical system characteristics, in this case, the density, viscosity, and flow speed. Presumably, the advantage of a dimensionless number is that it applies to any Newtonian fluid. For unbounded boundary layer flows, if similarity is present in a set of velocity profiles, then μ₁(x) and u_max(x) are possible similarity scaling parameters. On the other hand, the Prandtl Plus scaling’s have never been proven to be similarity scaling parameters for any set of velocity profiles (the proof offered by Jones, et. al. [71], is based on circular logic and is flawed, see the Appendix of Weyburne [14]).

It should be emphasized that strict similarity can only be expected for data sets displaying whole profile similarity. Whole profile similarity of the turbulent boundary layer, except for sink flow, has never been observed (Marusic, et. al. [37] However, whole profile similarity for exterior laminar flow over a flat plate is accepted as a normal condition for 2-D flow. For laminar flow, the viscous forces are important through the entire boundary layer region. The New parameter set and the Prandtl Plus scaling parameters are supposed to work for the viscous region. Theoretically, the new set should show whole profile similarity for the laminar flow case but what about the Prandtl Plus case? Weyburne [69] has already shown that the Prandtl Plus parameters do work theoretically for the laminar flow and turbulent flow Falkner-Skan cases. Experimentally, a simple way to answer this laminar flow similarity question is to examine experimental CFD simulations⁵ for exterior-like unbounded laminar flow on a thin flat plate. In Fig. 9.1a and 9.1b, the results for exterior-like laminar flow are displayed for the two-parameter sets. The results are striking. All seven profiles for the New set show complete overlap. Fig. 9.1b indicates that the Prandtl Plus parameters do NOT show similarity for laminar flow.

This puts Plus parameter advocates in a difficult situation. They need to explain why the Prandtl Plus does not work for laminar flows but the New parameters do work. It is now apparent why the Blasius scaling parameters are used for laminar exterior flows and the Prandtl Plus scaling for turbulent flows. The Prandtl Plus parameters simply do not work for laminar flows. As opposed to the Prandtl set, for the New set, it is possible to show that the ratio between the Blasius thickness parameter and μ₁(x) must be a constant (Weyburne [8]). The bottom line is that the New set works for laminar flow and the Plus parameters do not. This means the New set can be used for laminar, transitional, and turbulent boundary layers, as opposed to the Prandtl Plus parameters, which cannot.

9.3 The Prandtl Plus Scaling Advantage

Despite the many theoretical problems with the Prandtl Plus scaling, it must be stated that the Prandtl Plus, in combination with the Logarithmic Law of the Wall, appears to have one advantage over the new scaling in terms of extracting the wall shear stress from experimental turbulent boundary layers velocity profiles. It is very difficult to measure the wall shear stress experimentally. It is also difficult to measure the velocity profile very close to the wall. However, since the logarithmic region extends further out from the wall due to the slow decay of the averaged second derivative velocity profile, then this makes it possible to extract the wall shear stress assuming the Log Law holds. Extensive experimental results comparing the Clauser extracted values to experimental wall shear stress measurements confirm that the Log Law extracted values are usually good approximations to the actual values. This approach is an appropriate way to obtain approximate experimental length scaling parameter μ₁(x) values.

Weyburne (unpublished) attempted to generate a Log Law type approximation using μ₁(x) and u_e(x) to extract approximate wall shear stress values. The results were not very satisfying. It appears that μ₁(x) and u_e(x), which can be shown to be similarity parameters for the whole profile case, do not work as well as the Prandtl parameters when whole profile similarity is NOT present. This makes sense in that μ₁(x) and u_e(x) are a mix of an inner and outer region parameter whereas the Prandtl Plus parameters are both inner region based parameters. However, in spite of the advantage Prandtl Plus parameters seem to have for extracting approximate wall shear stress values, the theoretical problems mean that the extracted wall shear stress values have to be treated carefully. There may be circumstances where the flow situation is too different from the sink flow underpinning of the Plus parameters to yield reliable approximations for the wall shear stress.