1. Introduction
The understanding of the structure of wall-bounded turbulent flows has been a vibrant topic of classical fluid mechanics for almost a century [
1,
2,
3,
4,
5,
6,
7,
8,
9,
10,
11,
12,
13,
14]. The problem that the Reynolds number (
) usually has a strong influence on the flow structure, and our ability to reliably study turbulent flows at very high
using direct numerical simulation (DNS) or experiments is rather limited [
15]. Of specific interest and relevance is the asymptotic structure of wall-bounded turbulent flows at infinite
, and the
scaling of how a potentially existing asymptotic state is reached. Such knowledge can provide valuable guidelines for DNS and experimental studies, the evaluation of promising new developments as determined through minimal error simulation methods [
16,
17,
18,
19,
20], the development of improved turbulence models [
21], the understanding of scaling regimes [
22], and the better understanding of asymptotic structures of other turbulent flows [
20]. There exist prior studies on a potential asymptotic state of canonical wall-bounded flows, but such studies face questions. For example, Kollmann used pipe flow models that include modeling assumptions in contradiction to the universality of the law of the wall [
23]. Pullin et al. assumed log-law mean velocity variations above a certain distance from the wall and developed wake model assumptions in conjunction with debated log-law type assumptions for streamwise turbulence intensities to derive conclusions about asymptotic turbulence [
24].
The motivation for this paper is to address the question about the potential existence of an asymptotic state of canonical wall-bounded turbulent flows on the basis of recent modeling of the mean flow and Reynolds shear stress for channel flow, pipe flow, and the zero-pressure gradient turbulent boundary layer (TBL) (for simplicity, the zero-pressure gradient TBL will be referred to simply as TBL) [
6,
7]. The latter models were obtained through thorough analyses of the physics of these flows up to the highest available
. The model considered is presented next, followed by analyses of outer and inner scaling consequences. Conclusions are presented in the last section.
2. The Probabilistic Velocity Model
An analytical model for the mean velocity
and Reynolds shear stress
introduced by Heinz [
6,
7] for turbulent channel flow, pipe flow, and the TBL is described in
Table 1. In particular, the Reynolds shear stress
for the three flows considered is determined via the momentum balance
used in conjunction with models for the total stress
M; see
Table 1. The momentum balance also involves the characteristic shear rate
. The superscript + refers to inner scaling; we use
and
for the inner scaling wall distance, where
y is normalized by
(the half-channel height, pipe radius, or 99% boundary-layer thickness with respect to channel flow, pipe flow, and the TBL). The friction Reynolds number is defined by
, where
is the friction velocity and
is the constant kinematic viscosity.
The model for the mean velocity
and Reynolds shear stress
presented in
Table 1 was derived for
. A specific feature of the model is the approach to designing it. First, several observational physics requirements were identified. Via analysis of DNS and experimental data, the model was derived by providing explicit evidence that the model satisfies all observational physics criteria. The latter included evidence that both modeled variables and their relevant derivatives accurately represented the corresponding observations in regard to all the relevant scalings. The model’s excellent performance in comparison to DNS [
26,
27,
28,
29,
30] and experimental data [
31,
32,
33,
34] for channel flow, pipe flow, and the TBL is described elsewhere [
6,
7]. These comparisons include a model validation up to
98,190, corresponding to
M [
7]. The velocity model is referred to as the probabilistic velocity model (PVM) because it determines the distribution function for the distribution of mean velocities along the wall-normal direction and its related probability density function (PDF); see the discussion below.
Essential details of the PVM structure are explained in terms of
Figure 1. This figure reveals the mode structure of the PVM given by the contributions
,
, and
to the characteristic shear rate
. The latter mode contributions are related to corresponding velocity contributions
,
, and
. Here,
and
(which are only functions of
) are inner scaling contributions. For all the three flows considered,
and
are the same. In contrast to that,
(which is only a function of
y) is an outer scaling contribution: it depends on the flow considered. There are also two inner scale correction terms,
and
(see
Table 1). They have an irrelevant effect on the mean velocity; these contributions only matter in regard to the correct calculation of turbulent viscosities for channel and pipe flow. A relevant conclusion of
Figure 1a is that the PVM implies a universal log-law. In particular, the PVM implies
for all the three flows considered in absence of boundary effects. This log-law involves a universal von Kármán constant
for the three flows considered. As explained in detail elsewhere [
7], there are critical Reynolds numbers of
20,000,
63,000, and
80,000 for the observation of a strict log-law for channel flow, pipe flow, and the TBL, respectively.
3. Outer Scaling Implications
The outer scaling implications of the PVM are considered first by focusing on variations in
y. Apart from considering
y variations, this requires the use of the appropriate outer velocity scale
, as opposed to the use of
for inner scaling variations. The difference between
and
is significant: we have
. An overview of corresponding scaling variables and their relationships is presented in
Table 2. A difference is made in regard to inner and outer scaling and inner-scale and outer-scale variables: inner and outer scaling refers to looking at variations along
and
y, respectively, whereas inner-scale and outer-scale variables refer to the normalization of variables using corresponding characteristic velocity and length scales.
Figure 2 shows the outer scaling variations in the outer-scale velocity
for the three flows considered. It may be seen that
converges with the constant centerline/freestream maximum velocity
, but the convergence is extremely slow. Upon closer inspection, Equation (
1) shows that the plateau value in these plots is given by
. This means that not only
has to become sufficiently large, but
needs to be sufficiently large, too. This figure supports the view that a mean velocity equal to
cannot be realized in reality because such
values cannot be realized.
A different picture of the convergence of the velocity distribution can be seen by considering the asymptotic variation of
implied by the PVM, which is given by
where the definitions
and
are used (
is the wake contribution to
based on
). As shown in
Figure 3,
converges to this asymptotic scaling at a much lower
: at
, there is hardly any visible difference between
and unity anymore. There is also hardly any difference between the flows considered. We note that the neglect of boundary effects (the neglect of
) implies the universal velocity log-law.
The physical relevance of the asymptotic velocity distribution can be seen by introducing the distribution of mean velocities along the wall-normal direction
y,
For the flows and range
, the effect of
on
is smaller than 0.025%. Thus, the neglect of
is well justified, which explains the last expression in Equation (
2). The corresponding PDF
reads
The entropy,
, related to the PDF,
, is defined by
. Using the definition of
for the entropy, we obtain
The last expression results from the neglect of
, as justified above. Hence, the von Kármán constant is an entropy measure,
. It is of interest to compare Equation (
2) and Equation (
3), which apply to the flow-specific asymptotic velocity distributions with corresponding expressions that neglect the flow dependence. According to Equation (
1), we have
with
in the latter case. By referring to the flow-independent distribution function and PDF as
and
, respectively, we find the expressions
The distribution functions
F and
are shown in
Figure 4 for the three flows considered. It can be seen that the influence of the flow considered only modifies
, obtained by the neglect of boundary effects. The structure of
is the simplest possible interpolation between the limit cases at
and
, respectively.
Characteristic properties of turbulence can be well studied by considering characteristic outer-scale velocity, time, and length scales
,
, and
, respectively, which are defined in
Table 2. Instead of directly considering these variables, it is more appropriate to consider the convergence of the Reynolds shear stress
and turbulence Reynolds number
based on
. We note that
is equivalent to the inner-scale turbulence viscosity. Given converged profiles for
and
, asymptotic
,
,
and
,
,
can easily be calculated.
Figure 5 and
Figure 6 present the convergence properties of
and
for the three flows considered. In similarity to the convergence of
to
, it is found that
and
approach their asymptotic values for
. The implied asymptotic profiles for the turbulence velocity, time, and length scales based on
are given by
Using the relationships presented in
Table 2, the implied asymptotic profiles of
,
, and
are found to be given by the following functions of only
y:
Figure 7 presents the corresponding asymptotic distributions of turbulence velocity scales, time scales, length scales, and turbulence Reynolds numbers for the three flows considered. Independent of specific distributions, the most relevant observation is that the turbulence asymptotically decays, as may be seen from the
trends under consideration of the fact that
, where
. In correspondence to that, we find that the turbulence velocity scale vanishes,
, and the time scale
. The structure of
corresponds to the expectations: for channel and pipe flow, we see damping-function-type distributions along
y that approache a constant Reynolds number at the centerline. For the TBL, the flow becomes laminar under freestream conditions.
The distribution of the length scale
seen in
Figure 7 is of particular interest. In contrast to the other variables (
,
, and
),
is finite over most of the domain. In particular, near the wall,
follows
according to Equation (
7) for all the flows considered. The latter provides strong support for the suitability of Prandtl’s debated mixing length concept [
35,
36,
37,
38,
39,
40,
41]. For channel and pipe flow,
diverges for
, and the size of turbulence structures can become unbounded. For the TBL case we see that
approaches zero under freestream conditions, which is consistent with the
behavior showing flow laminarization. An interesting observation is that
is equivalent to the outer turbulence length scale
for
, which shows a mean flow—turbulence balance.
4. Inner Scaling Implications
In regard to the inner scaling
variations, there are no wake contributions
such that
, and the momentum balance
reduces to
. Using the abbreviation
, the inner-scale characteristic turbulence velocity, time, and length scales and
read
Using the definition of
, the latter relations can be also written as
The corresponding outer-scale variables are then given by
The asymptotic distributions of the velocity and turbulence characteristics are illustrated in
Figure 8.
Figure 8a shows that the convergence of
to
with increasing
is clearly a characteristic feature of outer scaling: there is no such convergence with respect to inner scaling. This figure also shows the difference between
and
contributions: the effect of
is rather little for the
range considered. There is a remarkable agreement between the variations in
and
. Both
and
are driven by the damping of the Reynolds shear stress due to the presence of the wall.
Figure 8b shows the near-wall variations in
,
, and
. In agreement with Equation (
9), we see only minor differences between
,
, and
. For a sufficiently large
, we have
. Because of
being combined with the asymptotic
, the values of
,
, and
asymptotically approach
, as may be seen in
Figure 8b. The latter is consistent with the corresponding transition into outer scaling variations given in
Figure 7. The implications for the outer-scale variables given in Equation (
10) are consistent with the corresponding implications of outer scaling:
and
asymptotically vanish, and
goes to infinity. On the other hand, we find
, i.e., finite
variations, controlled by the distance to the wall.
Figure 8c shows the asymptotic distribution of the production of kinetic energy,
. The analysis of
variations shows that
has a maximum of
at
corresponding to
. Thus, turbulence is still present in inner scaling at infinite
, although the turbulence decays in outer scaling.
5. Summary
The asymptotic structure of wall-bounded turbulent flows is reported here for the first time for three canonical flows independent of a modeling assumption in conflict with the universality of the law of the wall and other modeling assumptions with uncertain support. The results obtained can be summarized as follows.
In regard to outer scaling considered to be function of
y, there is a trend that the mean velocity
approaches the constant
. However, this convergence is so slow that there are clear differences between
and
, even for
. It has to be expected, therefore, that
is still different from
under conditions of practical relevance. On the other hand,
converges to
for about
. It is beneficial to discuss this asymptotic velocity distribution in terms of the implied PDF of the distribution of mean velocities along the wall-normal direction
y. In absence of boundary conditions (in absence of wake contributions), a linear mean velocity PDF was found to be equivalent to the length scale distribution of turbulence. The wake effect adjusts the PDF to the boundary conditions. Considered again in outer scaling, asymptotic outer-scale turbulence characteristic velocity, time, and length scales observed for about
reveal features in consistency with the mean velocity trend toward a spatial smoothing. The turbulence decays:
and
approach zero. Simultaneously, the turbulence time scale
approaches infinity, which indicates frozen turbulence structures. In contrast to the other variables, it is of interest to note that the turbulence length scale
is finite throughout the domain except at the centerline for channel and pipe flows. The latter provides strong support for the suitability of Prandtl’s debated mixing length concept [
35]. In particular, not too far from the wall
is proportional to the distance
y from the wall with the von Kármán constant
as a proportionality constant.
For infinite , inner scaling reveals flow features in an infinitesimally thin layer close to the wall. Inner scaling features (considered as function of ) of the variables considered are the following ones. The mean velocity is finite and characterized by the damping effect of the wall. The behavior of the main component in this region is very similar to the corresponding behavior of the turbulence velocity . The correlation between and can be explained by the wall-damping effect on the Reynolds shear stress. Turbulence survives in this infinitesimally thin layer close to the wall, as can be seen from the distribution of production and the distribution. The characteristic time and length scales and show trends in consistency with their outer scaling trends: approaches infinity and approaches (corresponding to ).
The results reported here are very beneficial in regard to several questions.
DNS and experimental studies are supposed to provide essential contributions to the validation of simpler computational methods. Unfortunately, such studies suffer significantly from the uncertainty of their predictions for high
[
6,
7,
8,
12]. The results reported here are, therefore, essential to understand the requirements for accurate DNS and experimental studies.
One of the basic problems of turbulence modeling is the uncertainty of the scale (
or
) equation: existing equations are considered to have a rather weak theoretical basis. Similar to recent work [
21], the distributions of turbulence variables determined here can be used for the validation or improvements of scale equations.
The existence and structure of asymptotically stable turbulence regimes is debated in regard to many turbulent flows (e.g., for complex hump-type flows involving flow separation [
19,
20]). The identification of asymptotic
regimes as reported here matters to such discussions. The latter provides insight into
values needed to observe asymptotic regimes, and insight of which mean velocity and turbulence structures enable asymptotically stable turbulent flows.