The Law of the Wall and von Kármán Constant: An Ongoing Controversial Debate

: The discovery of the law of the wall, the log-law including the von Kármán constant, is seen to be one of the biggest accomplishments of fluid mechanics. However, after more than ninety years, there is still a controversial debate about the validity and universality of the law of the wall. In particular, evidence in favor of a universal log-law was recently questioned by data analyses of the majority of existing direct numerical simulation (DNS) and experimental results, arguing in favor of nonuniversality of the law of the wall. Future progress requires it to resolve this discrepancy: in absence of alternatives, a reliable and universal theory involving the law of the wall is needed to provide essential guideline for the validation of theory, computational methods, and experimental studies of very high Reynolds number flows. This paper presents an analysis of concepts used to derive controversial conclusions. Similar to the analysis of observed variations of the Kolmogorov constant, it is shown that nonuniversality is a consequence of simplified modeling concepts, leading to unrealizable models. Realizability implies universality: there is no need to adjust simplified models to different flows.


Introduction
One of the biggest challenges of computational fluid dynamics (CFD) is the reliable and efficient prediction of turbulent flows at high Reynolds number (Re), in particular wall-bounded turbulent flows usually seen in reality.Direct numerical simulation (DNS), large eddy simulation (LES), and experimental studies are hardly applicable to extreme Re regimes [1], and relatively cost-efficient hybrid RANS-LES, which combine LES with Reynolds-averaged Navier-Stokes (RANS) equations, suffer from reliability issues [2].There are very promising new developments as given by minimal error hybrid-RANS-LES (which can act as resolving LES) [2][3][4][5][6][7], but these methods also need evidence for their validity at high Re.A closely related challenge is the understanding of the nature of turbulent flow in the limit of infinite Re.
The discovery of the law of the wall, the log-law including the von Kármán constant [8], is of essential relevance in this regard.In particular, a proven universal law of the wall has unique practical benefits which cannot be provided in any other way: B1.It can be used to overcome a very essential problem: it can provide strict guideline for the validation of experiments and computational simulation methods such as DNS, LES, and hybrid RANS-LES for at least several canonical high Re flows [9][10][11].B2.More specifically, minimal error hybrid RANS-LES were developed recently [2,[4][5][6][7].
These methods overcome resolution limitations of existing methods, which offers huge computational cost advantages.A universal law of the wall can provide evidence for the validity of such predictions at very high Re.B3.Usually applied turbulence models are developed on the basis of empirical notions.
A universal law of the wall can be applied for the design of exact turbulence models.This was demonstrated recently by the derivation of an exact transport equation for the turbulent viscosity [12].Such equations can support (not support) empirical turbulence models.B4.A theory involving a universal law of the wall can essentially contribute to our understanding of the structure of turbulent flows at high Re [13].It can explain Re requirements to observe the log-law, the structure of self-similar turbulence characteristics, and convergence toward these structures.Such understanding provides a valuable reference for other turbulent flow studies.
However, after more than ninety years, there is still debate about the validity and universality of the law of the wall.The specific question is whether the mean streamwise velocity U + of at least several canonical wall-bounded flows is characterized by log-law variations in absence of boundary effects, this means whether we have U + = κ −1 ln y + + B including the same von Kármán constant κ and constant B. The superscript + refers to inner scaling; we use U + = U/u τ and y + = Re τ y for the inner scaling wall distance, where y is normalized by δ which is the half-channel height, pipe radius, or 99% boundary layer thickness with respect to channel flow, pipe flow, and the zero-pressure gradient turbulent boundary layer (TBL), respectively (the zero-pressure gradient TBL will be referred to simply as TBL).The friction Reynolds number is defined by Re τ = u τ δ/ν, where u τ is the friction velocity and ν is the constant kinematic viscosity.
Figure 1 illustrates the steadily growing interest in these questions with about 1000 journal publications per year right now.The development of views over 80 years  were reviewed, for example, by Örlü et al. [42], Marusic et al. [43], Smits et al. [44], and Jiménez [45].The Princeton superpipe (PSP) measurements (involving data up to Re τ = 530,023) were available at this time [46], but concerns about the data accuracy still exist [47].Supported by increasing access to high Re data, the controversial debate of the validity and universality of the law of the wall vibrantly continuous over the last 15 years: traditional views in favor of universality are challenged by opposite views [48][49][50][51][52][53][54][55][56][57][58][59][60][61][62][63][64][65].Recent analyses in favor of universality were presented, e.g., in Refs.[56,[66][67][68][69][70].In particular, a comprehensive analysis of available numerical and experimental data up to very high Re was presented recently based on observational physics criteria [9,10].The latter results were recently questioned by data analyses of the majority of existing DNS and experimental results, arguing in favor of nonuniversality of the law of the wall [47,57,[71][72][73].The motivation for this paper is to contribute to the clarification of these questions by an identification of reasons leading to opposite conclusions regarding the universality of the law of the wall and von Kármán constant.The latter clearly matters: this is simply about whether or not there is a reliable universal theory involving the law of the wall which can be used (in absence of alternatives) to validate methods restricted by resolution requirements which are hard to satisfy for very high Re turbulent flows.The approach to address these questions is to compare recent analysis results in favor of nonuniversality of the law of the wall [47,57,[71][72][73] with consequences of observational physics criteria [9,10] in the following.The frame of this discussion is illustrated in Table 1: the physical completeness of models is discussed in conjunction with implications for the model realizability and conclusions about universality.In regard to the model realizability, emphasis will be placed on whether or not the methods considered satisfy Reynolds stress-realizability constraints [1,2,74] and realizability constraints arising from the entropy concept.The latter implies the need that the entropy of physically equivalent flows needs to be the same [9,10].
Table 1.Conceptual features of models considered (MN refers to the model of Monkewitz and Nagib [73]).

Universal Velocity Models
A probabilistic velocity model (PVM) was introduced in Refs.[9,10] for Re τ ≥ 500 for turbulent channel flow, pipe flow, and the TBL.The model is provided in Table 2, and a discussion of its mathematical structure can be found in the Appendix A. The model was carefully validated against several observational physics requirements, including evidence that both modeled variables and their relevant derivatives accurately represent corresponding observations in regard to all relevant scalings.The model is supported by a probabilistic interpretation: the probability density function (PDF) related to the distribution function for the distribution of mean velocities along the wall-normal direction represents a statistically most-likely PDF that maximizes the related entropy [9,10].The von Kármán constant κ involved represents an entropy measure.
Figure 2 explains the PVM structure.The model involves the contributions S + 1 , S + 2 , and S + 3 to the characteristic shear rate S + = ∂U + /∂y + .These shear rate contributions imply corresponding velocity contributions U + 1 , U + 2 , and U + 3 .The inner scaling contributions S + 1 and S + 2 (which are only functions of y + ) are the same for all three flows considered.The outer scaling contribution κy + S + 3 (which is only a function of y) depends on the flow geometry.There are two inner scale correction terms S CP 1 and S CP 2 which ensure the correct shear rate limit at the centerline for channel and pipe flow.These contributions provide insignificant corresponding contributions to the mean velocity.As may be seen in Figure 2a, the PVM clearly supports the validity and universality of the log-law.In absence of boundary effects, the PVM implies U + = κ −1 ln y + + 5.03 for all the three flows considered, where κ = 0.40.A relevant conclusion of the PVM is that critical Reynolds numbers for the observation of a strict log-law for channel flow, pipe flow, and the TBL are given by about Re τ = 20,000, Re τ = 63,000, and Re τ = 80,000, respectively.The excellent PVM performance in comparison to DNS and high-Re experimental data is illustrated in Figures 3 and 4.There are other models that support the validity and universality of the log-law.One such model is the model of Luchini [56,66].The characteristic shear rate is described by S + = 1/(κy + ) + A 1 g/Re τ , where κ = 0.392, A 1 = 1, and g = (1, 2, 0) for channel flow, pipe flow, and the TBL, respectively.Luchini's model may be seen as basis of the model presented recently by Monkewitz and Nagib (MN) [73]; see the discussion below.The difference is that Luchini does not make an attempt to introduce different von Kármán constants and other constants for different flows considered.Another model that supports the validity and universality of the log-law is the model of Laadhari [68].The model reads U + = κ −1 ln(y * /a), where y * = y +2 S + and κ = 0.40.This model actually represents an ordinary differential equation (ODE) for U + .It may be seen that this ODE is solved by U + = κ −1 ln(y + /[aκ]) = κ −1 ln(y + ) − κ −1 ln(aκ).The setting a = 0.334 (which is close to a = 0.36 applied by Laadhari [68]) recovers U + = κ −1 ln y + + 5.03 implied by the PVM, i.e., Laadhari's model recovers the implications of the PVM.
A relevant implication of the PVM can be seen by introducing a length scale ℓ = (1 . Figure 5 illustrates the suitability of representing ℓ as ℓ = f d κy + : Figure 5a shows the proportionality to κy + , Figure 5b shows that 2 )/(κy + ) represents a damping function.The definition of ℓ can be used to represent S + 1 + S + 2 as function of ℓ, Hence, the length scale ℓ fully determines the flow-independent inner scaling structure of the velocity field.The exact Equation ( 1) corresponds to an interpolation of limit cases of S + 1 + S + 2 = (1, 0) for ℓ = (0, ∞), respectively.For sufficiently large y + , Equation ( 1) implies the log-law (S + , it is worth noting that the von Kármán constant κ is the essential ingredient of inner scaling velocity variations. An additional conclusion on κ is the following.In outer scaling, the PVM provides y + S + = [1 + κy + S + 3 ]/κ (see Figure 2), so κ can be determined by y + S + → 1/κ for y → 0. The PVM shows that the contribution of outer scaling variations given by y + S + 3 becomes negligible compared to 1/κ for y → 0 (see Figure 2), i.e., the value of κ is independent of y + S + 3 contributions.Hence, κ cannot be determined by the analysis of outer scaling y + S + variations: κ characterizes inner scaling variations (see Equation ( 1)) which are independent of flow-dependent outer scaling y + S + variations.Used in conjunction with models for the total stress M (see Table 2), we note that the PVM also implies analytical models for the Reynolds shear stress, turbulence production, turbulent viscosity, bulk velocity, skin-friction coefficient, and bulk Reynolds number [9,10].Asymptotic limits of these variables are reported elsewhere [10].A closer look at corresponding implications of the PVM for the Reynolds shear stress is beneficial regarding the discussion of nonuniversal velocity models below.By involving ℓ, an asymptotic Reynolds shear stress implied by the PVM for sufficiently high Re τ is given by Hence, ⟨u ′ v ′ ⟩ + is characterized by self-similar separate variations with y (via M) and with y + (via ℓ(S + 1 + S + 2 )).As pointed out in Ref.

Nonuniversal Velocity Models
A nonuniversal velocity model was presented recently by Cantwell [47]; see also Ref. [71].The model uses classical mixing length theory and an ad hoc model for the mixing length λ, By using Equation (3) in the momentum equation S + − ⟨u ′ v ′ ⟩ + = M we can find a model for S + , where M = 1 − y for the pipe flow considered.This model involves five adjustable parameters, k, a, m, b, n, where k corresponds to the von Kármán constant.Figure 15d in Ref. [47] reveals that there is no strict log-law region with the influence of wall and outer length scales on the intermediate region of the velocity profile persisting to all Re τ [47].A comparison of model predictions of the normalized turbulent viscosity ν + t = −⟨u ′ v ′ ⟩ + /S + and turbulence production P + = −⟨u ′ v ′ ⟩ + S + with DNS data and PVM predictions is shown in Figure 6: it may be seen that the model does not accurately reflect the flow structure.An interesting model feature is the following.The y contribution in S + is very small for sufficiently high Re τ , and its influence decreases with increasing Re τ .For Re τ > 5000, there is no observable difference anymore between S + calculated by Equation ( 4) and S + calculated by neglecting y.In this case, analysis of dP + /dλ = 0 shows that the P + maximum appears at λ = 2 1/2 (at S + = 1/2), which provides a maximum P + = 1/4.A similar PVM analysis leads to a maximum at ℓ = 2 1/2 corresponding to S + 1 + S + 2 = 1/2 and also P + = 1/4.The nonuniversality of Cantwell's model [47] is reflected by the need to provide k, a, m, b, n (which are determined from an analysis based on the whole velocity profile) as functions of Re τ .The reason for this nonuniversality is the empirical introduction of Re τ effects via y = y + /Re τ in λ leading to an unphysical dependence of ⟨u ′ v ′ ⟩ + on Re τ .The following provides evidence for this claim.
O1.The simplest way to support this claim is the comparison of Equation ( 3) with the physics-based Equation ( 2): although the structures of Equation ( 2) and ( 3) are similar, Equation (3) does not ensure a self-similar structure of the Reynolds shear stress in contrast to Equation (2).O2.The turbulence production peak position is known to be y + = 11.07,unaffected by Re τ for sufficiently high Re τ [9,10]: see Figure 6.For Re τ > 5000, Cantwell's model provides the production peak position at λ = 2 1/2 .According to the definition λ = ky + [1 − e −(y + /a) m ]/([1 + y n /b n ] 1/n ) and the Re τ dependence of model coefficients, this implies y + peak positions which vary (randomly) with Re τ : see the inset in Figure 6b.This behavior is unphysical and in contrast to DNS and experimental results.O3.More specifically, the model's entropy is given by S E = 1 − ln(κ) = 1 − ln(k) [9].We find, therefore, random entropy changes for each Re τ and flow (see Figure 6c).This is unphysical; the entropy needs to be the same under physically equivalent conditions.
Another nonuniversal velocity model was presented recently by Monkewitz and Nagib (MN) [73].Figure 7 demonstrates the model concept by a comparison with PVM results: MN approximate the log-law indicator y + S + in regard to outer scaling.By following MN, the TBL results are shown in dependence on the boundary layer thickness Y, which differs from the 99% boundary layer thickness [73].The structure of MN assumptions is illustrated in the insets of Figure 7: linear functions are used to characterize y + S + .

Figure 1 .
Figure 1.New journal publications per year that mention "law of the wall" and Kármán.
flow, pipe flow, and TBL.Corresponding Reynolds shear stress models are given via the momentum balance S + − ⟨u ′ v ′ ⟩ + = M. Here, M refers to the total stress given by M = (M CP , M CP , M BL ) used in conjunction with M CP = 1 − y and M BL = e −y 6 −1.57y 2 .

Figure 5 .
Figure 5. Characteristic features of the length scale ℓ = f d κy + (a) length scale and (b) damping function involved.

Figure A1 .
Figure A1.The transition functions f , H, and 1 − w are shown in (a-c), respectively.Figure (a) also shows U + 1 /U + 1∞ (blue line).Figure (c) also shows 1 − w for channel and pipe flow (black line) and the TBL (blue line).The dashed line shows 1 − w = 1 as a reference.