2.2.1. Leading balance layers

The empirical step involves determining the leading balance structure associated with (2.1). That is, while (2.1) properly identifies the non-zero terms across the half-channel, not all of the terms are important everywhere. Thus, this situation is, for example, distinct from the mathematical representation of flow physics within the laminar boundary layer, where all of the terms in the asymptotically simplified form of the Navier–Stokes equation (i.e. Prandtl's boundary layer equation) retain leading order within the boundary layer. It does, however, bear some resemblance to the relationship between the laminar boundary layer and the outer inviscid flow where the viscous term is absent. Because there are only three terms, the leading balance structure in the channel can be made apparent by examining the ratio of the $VF$ to $TI$ term across the flow. This ratio is shown in figure 1 for three friction Reynolds numbers, $\delta ^+ = \delta u_{\tau }/\nu$, where $u_{\tau } = \sqrt {\tau _w/\rho }$ is the friction velocity, $\tau _w = \mu (\textrm {d}U/{\textrm {d}y})|_{y = 0}$ is the mean wall shear stress and $\nu = \mu /\rho$ is the kinematic viscosity. Superscript ‘+’ denotes normalization by $\nu$ and $u_{\tau }$, and thus $y^+ = yu_{\tau }/\nu$.

Figure 1.

Ratio of the viscous force term ($VF$) to the turbulent inertia ($TI$) term in the turbulent channel flow equation (2.1); $\blacktriangle$, $\delta ^+ = 1993$; $\blacksquare$, $\delta ^+ = 3974$; $\bullet$, $\delta ^+ = 7996$. Vertical dashed lines denote the approximate boundaries of layer III at $\delta ^+ = 7996$, as defined by the criteria of Wei et al. (Reference Wei, Fife, Klewicki and McMurtry2005a). Data are from Yamamoto & Tsuji (Reference Yamamoto and Tsuji2018).

The simulations of Yamamoto & Tsuji (Reference Yamamoto and Tsuji2018) are used throughout the data presentations. This is because these DNS cover a large Reynolds range and maintain high spatial resolution far from the wall. The accuracy of these simulations is thus important to assessing data comparisons with the present theory. A relevant measure of the DNS accuracy is given by the residual error, $E_R^+$, of the total stress balance (Thompson et al. Reference Thompson, Sampaio, de Bragan Alves, Thais and Mompean2016). This error includes statistical convergence and the satisfaction of statistical homogeneity in $x$ and $z$. Yamamoto & Tsuji (Reference Yamamoto and Tsuji2018) report that their $\delta ^+ = 7996$ simulation has an $E_R^+$ that is less than $0.005$ everywhere, and in the region where most comparisons are made herein $E_R^+ \le 0.0015$. Relative to the comparisons in § 3, these uncertainties are close to the differences between the DNS and the analytically based curve fits.

Figure 1 shows that the dominant balance of terms in the channel changes with wall-normal position and Reynolds number. As now described, there are four leading balance layers: $|PG| \simeq |VF|$ (layer I), $|VF| \simeq |TI|$ (layer II), $|VF|, |TI|$ and $|PG|$ all leading order (layer III) and $|TI| \simeq |PG|$ (layer IV).

Evaluation of (2.1) at the wall reveals an exact balance between the $VF$ and $PG$ terms. Owing to this, and through examination of existing low $\delta ^+$ data, it was at first reasoned that for all $\delta ^+$ there exists a layer I of $O(\nu /u_{\tau })$ width that essentially constitutes the traditional viscous sublayer (i.e. a layer over $0 \le y^+ \le 3$) e.g. Wei et al. (Reference Wei, Fife, Klewicki and McMurtry2005a) and Klewicki et al. (Reference Klewicki, Fife, Wei and McMurtry2007). Data at larger $\delta ^+$, such as in figure 1, however, provide evidence that the width of this layer diminishes under inner normalization with increasing $\delta ^+$. Thus, as $\delta ^+$ becomes large, the inner region of the channel apparently becomes more like planar Couette flow, where the dominant balance between the $TI$ and $VF$ terms extends to the wall at all Reynolds numbers. These higher $\delta ^+$ data lead one to surmise that the layer I width in channel flow is $\le O(\nu /u_{\tau })$. Relative to the present study, this further clarification of the layer I structure is of no consequence, as the exclusive focus herein is on mean flow behaviours significantly farther from wall.

The next layer from the wall is where the ratio $VF/TI \simeq -1$. This is called the stress gradient balance layer (layer II) (Fife et al. Reference Fife, Klewicki, McMurtry and Wei2005a). In this layer $|TI| \simeq |VF|$, with the equality becoming more exact with increasing Reynolds number, e.g. Wei et al. (Reference Wei, Fife, Klewicki and McMurtry2005a). The data of figure 1 show that the inner-normalized width of layer II increases with $\delta ^+$. This fact does not comport with the assumption used in the traditional overlap layer based framework that this region has a width that is $O(\nu /u_{\tau })$ (Mellor Reference Mellor1972; Wosnik, Castillo & George Reference Wosnik, Castillo and George2000; Panton Reference Panton2005).

The outer edge of layer II is characterized by a rapid variation where $VF/TI \to -\infty$. This ratio then reappears from $+\infty$, and with increasing $y^+$ becomes subdominant, approaching zero with increasing $\delta ^+$. The region (layer III) between where $VF/TI \simeq -1$ and $VF/TI \simeq 0$ is characterized by a leading balance that retains all three terms. Here, we note that some have identified the domain in the vicinity of $TI = 0$ as where there is a re-emergence of viscous effects, e.g. Sreenivasan & Sahay (Reference Sreenivasan and Sahay1997). In the mean, however, it is only the $TI$ term that becomes subdominant within this region, and this is only in a diminishingly narrow zone in the immediate vicinity of its zero crossing. Instantaneously, this region is characterized by a complicated mixture of viscous, inertial and pressure forces. On average, across layer III the dynamics transitions from a situation where viscous forces are of leading-order importance to one where they lose this leading-order significance. Regardless of interpretation, one can conclude that there is an exchange of mean balance across layer III: upon entering from layer II the leading balance is $|TI| \simeq |VF|$, while upon exiting into layer IV the leading balance is $|TI| \simeq |PG|$. Thus, beyond layer III the mean dynamical balance is inertially dominated. This feature rationally identifies the onset of the physical space inertial sublayer that is classically associated with a logarithmic mean velocity profile. More specifically, results from the framework employed herein lead to the expectation that logarithmic dependence emerges most assertively with Reynolds number within an interior domain (the inertial sublayer) beginning near the onset of layer IV (Wei et al. Reference Wei, Fife, Klewicki and McMurtry2005a; Klewicki, Fife & Wei Reference Klewicki, Fife and Wei2009). As later shown in § 3.2, this layer IV subdomain ranges over $2.6\sqrt {\delta ^+} \lesssim y^+ \lesssim 0.125\delta ^+$. Analysis of data at large Reynolds number support this prediction (Marusic et al. Reference Marusic, Monty, Hultmark and Smits2013; Vincenti et al. Reference Vincenti, Klewicki, Morrill-Winter, White and Wosnik2013; Morrill-Winter, Philip & Klewicki Reference Morrill-Winter, Philip and Klewicki2017), as do the results presented herein.

Considerably more can be determined empirically from the profiles in figure 1, and this has been done previously, e.g. Wei et al. (Reference Wei, Fife, Klewicki and McMurtry2005a) and Klewicki et al. (Reference Klewicki, Fife and Wei2009). This includes the Reynolds number dependencies of the layer widths and the velocity increments across them. Advancing the analytical treatment of (2.1), however, only requires a knowledge of the leading balance structure. Namely, that there exists a region near the wall where $|VF| \simeq |TI|$, an outer region where $|PG| \simeq |TI|$ and an intermediate region where $|VF| \simeq |TI| \simeq |PG|$ and across which the exchange of balance transition just described occurs. As shown below, the scaling behaviours associated with the layer widths are, in order of magnitude, determinable from analysis of (2.1).

2.2.2. Reconciling the formal and actual orders of magnitude

We now describe how to appropriately normalize (2.1) in light of the leading balance information contained in figure 1. In this regard, the issue of the formal vs actual order of magnitude arises in association with (2.1) being unclosed. The actual order of magnitude of a term is evidenced by the structure indicated in figure 1, while the formal leading balance is that reflected by the specific normalized form of the equation. In this section $u_{\tau }$ is taken to be the only velocity scale, but later we discuss the ramifications associated with other velocity scales. Operationally, the formal leading balance is found by neglecting the term associated with the small parameter, $\epsilon ^2 = 1/\delta ^+$, as $\delta ^+ \to \infty$. The aim of the following is to ensure that within each leading balance layer the normalization employed yields a formal order of magnitude commensurate with the actual order of magnitude. The terms in (2.2) are plotted vs $y^+$ in figure 2. These data are plotted at low $\delta ^+$ in order to see the $TI = PG$ dominant balance in layer IV.

Figure 2.

Terms in (2.2) at $\delta ^+ = 546$. Vertical dashed lines denote the approximate boundaries of layer III. Solid (red) line portion of the $TI$ term profile denotes the region between its inner and outer maximum values, respectively, $y^+_{pi}$ and $y^+_{po}$, where the profile is a monotonically decreasing function. Data are from Hoyas & Jimenez (Reference Hoyas and Jimenez2006).

Since layer II extends very close to the wall as $\delta ^+ \to \infty$, it is natural to expect that, in this region, the leading balance formally indicated by the inner-normalized form of (2.1),

(2.2)\begin{equation} 0 = \frac{\textrm{d}^2{{U^+}}}{{\textrm{d}y}^{{+}2}} + \frac{\textrm{d}T^+}{{\textrm{d}y}^+} + \epsilon^2, \end{equation}

faithfully reflects the actual leading-order balance. Given that $\epsilon \to 0$ as $\delta ^+ \to \infty$, this is indeed the case. Here, the formal leading balance is $|TI| = |VF|$. This formal leading balance agrees with the actual layer II balance indicated in figure 1.

We now consider layer III. Over most of layer II the $TI$ and $VF$ terms are much larger than $PG = \epsilon ^2 = 1/\delta ^+$, see figure 2. With increasing $y^+$, however, $TI$ and $VF$ diminish in magnitude from above and below, respectively. They eventually attain magnitudes comparable to $\epsilon ^2$, and the onset of layer III is marked by this condition. Across layer III, the $VF$ term remains negative and continues to approach zero, while the $TI$ term crosses zero and approaches a magnitude equal to the $PG$ term but of opposite sign. This condition nominally marks the end of layer III (onset of layer IV). Within layer III, all three terms have comparable magnitude, except in the region of diminishing width about the singular point were $TI = 0$ described above. This region centred approximately $TI = 0$ is not singled out for separate mathematical treatment herein.

Having established the actual magnitude ordering of terms, we now seek a rescaling of (2.2) such that all terms are formally leading order in layer III for all $\delta ^+$, i.e. are formally $O(1)$. The simplest set of differential transformations that accomplish this task are used here for clarity,

(2.3a–c)\begin{equation} {\textrm{d}y}^+ = \alpha \,\textrm{d}\tilde{y}, \quad \textrm{d}T^+ = \gamma\, \textrm{d}\tilde{T} \quad \textrm{and} \quad \textrm{d}U^+ = \textrm{d}\tilde{U}, \end{equation}

where $\alpha$ and $\gamma$ are stretching parameters that must be determined. Inserting (2.3a–c) into (2.2) and demanding that all terms are $O(1)$ (i.e. constraining the normalized $PG$ term to equal 1) one obtains

(2.4)\begin{equation} 0 = \frac{\textrm{d}^2{\tilde{U}}}{\textrm{d}\tilde{y}^{2}} + \frac{\textrm{d}\tilde{T}}{\textrm{d}\tilde{y}} + 1, \end{equation}

with $\alpha = \epsilon ^{-1}$ ($= \sqrt {\delta ^+}$) and $\gamma = \epsilon$. From this, one can readily show that the position $y^+ = y_m^+$ of the maximum Reynolds shear stress $T^+ = T_m^+$, and the inner-normalized width of layer II, are $O(\sqrt {\delta ^+})$. More significantly, the scalings reflected in (2.4) also hold over a finite width domain (layer III) that in viscous units is also $O(\sqrt {\delta ^+})$, e.g. Fife et al. (Reference Fife, Klewicki, McMurtry and Wei2005a). All of these findings have been empirically verified, i.e. $y_m^+$ is estimated to be approximately $1.6\sqrt {\delta ^+}$, while the width of layer III is estimated to be approximately $1.0\sqrt {\delta ^+}$, e.g. Wei et al. (Reference Wei, Fife, Klewicki and McMurtry2005a). More will be said approximately these values in § 3. In the presentation below, the rescaling step just described is replicated to derive an invariant form of (2.1) that holds to leading order over a domain of extent $O(\nu /u_{\tau }) < y < O(\delta )$, and within which a similarity solution is analytically shown to exist.

In layer IV the actual leading balance is between the $TI$ and $PG$ terms. Commensurate with this, the traditional outer normalization (using $\delta$ and $u_{\tau }$) reflects the proper formal ordering since it positions the $\epsilon ^2$ term as the leading coefficient on the $VF$ term, i.e.

(2.5)\begin{equation} 0 = \epsilon^2\frac{\textrm{d}^2{{U^+}}}{\textrm{d}\eta^{2}} + \frac{\textrm{d}T^+}{\textrm{d}\eta} + 1, \end{equation}

where, $\eta = y/\delta$. An important consideration pertains to how the different leading-order forms of (2.1) transition from one layer to the next. Relative to the transition from layer III to layer IV, it is a straightforward exercise to show that $\textrm {d}\tilde {T}/\textrm {d}\tilde {y} \equiv \textrm {d}T^+/\textrm {d}\eta$. This is relevant to some of the findings presented in § 3, since, for example, it explains why the layer III scaling (or more generally layer hierarchy scaling) for $T^+$ holds over a domain that starts at a $y^+$ position that is $O(\sqrt {\delta ^+})$ less than $y^+_m$ and extends to the channel centre at $y^+ = \delta ^+$ (Fife et al. Reference Fife, Wei, Klewicki and McMurtry2005b).

2.2.3. Hierarchy of scaling layers

Equation (2.1) is shown in this section to admit an invariant parameter-free form over a mathematically well-defined domain, e.g. Fife et al. (Reference Fife, Klewicki and Wei2009). The structure associated with this invariant form involves a hierarchy of scaling layers. Within well-defined bounds, each member of this layer hierarchy has a finite $\Delta y$ extent. With increasing $y$ the $\Delta y$ width of the layers increases. Thus, in contrast to the classical two length scale construction that associates the physical space inertial sublayer with an overlap layer, the present analytical findings indicate a continuous variation in characteristic length scale. This structure naturally embraces the distance-from-the-wall scaling concept used, for example, in the simple heuristic reasoning (i.e. $\textrm {d}U/{\textrm {d}y} \propto u_{\tau }/y$) to arrive at a logarithmic mean profile, inherently aligns with the key assumption underlying the attached-eddy phenomenology of Townsend (Reference Townsend1976), and explains why logarithmic expansions provide accurate approximations over the hierarchy domain (Sreenivasan & Bershadskii Reference Sreenivasan and Bershadskii2006). Indeed, as demonstrated below, the present framework provides a mean equation basis for distance-from-the-wall scaling, while the results of § 3.4 provide an additional refinement to this scaling.

The layer hierarchy is formulated by recasting the $TI$ term. This reveals the wall-normal variation in the average size of the motions contributing to turbulent momentum transport. The mathematical framework that guides this process, including the steps used to rescale (2.2) above, is codified in the method of scaling patches described by Fife et al. (Reference Fife, Klewicki, McMurtry and Wei2005a, Reference Fife, Klewicki and Wei2009). This method uses admissibility and compatibility criteria that serve to guide multiscale analysis of indeterminate equations by requiring that the normalizations over the relevant domain convey the actual leading balance structure while satisfying empirically testable boundedness criteria.

The mathematical construction begins by writing the Reynolds stress, $T^+$, and pressure gradient, $\epsilon ^2$, in terms of a Reynolds stress-like quantity, $T^+_{\beta }$, plus a small positive parameter $\beta$

(2.6)\begin{equation} T^+_{\beta}(y^+) + {\beta}y^+{-} \epsilon^2y^+ = T^+(y^+). \end{equation}

From (2.6) it is apparent that the positive constant $\beta$ is akin to $\epsilon ^2$ in (2.2). Indeed, when $\beta = \epsilon ^2$ (2.8) below identically recovers (2.2). Within a specific range of wall-normal locations, each value of $\beta$ uniquely corresponds to a $y^+$ value. To show this, we begin by differentiating (2.6) to obtain

(2.7)\begin{equation} \frac{\textrm{d}T^+_{\beta}}{{\textrm{d}y}^+} + \beta = \frac{\textrm{d}T^+}{{\textrm{d}y}^+} + \epsilon^2. \end{equation}

Subsequent insertion into (2.2) gives

(2.8)\begin{equation} 0 = \frac{\textrm{d}^2{{U^+}}}{{\textrm{d}y}^{{+}2}} + \frac{\textrm{d}T^+_{\beta}}{{\textrm{d}y}^+} + \beta. \end{equation}

Here, $\beta$ is a constant for each $T^+_{\beta }$, i.e. like $\epsilon ^2$ is for $T^+$.

Analytical treatment of (2.8) focuses on the monotonically decreasing portion of the $TI$ profile. As denoted in figure 2, the relevant region is situated between the inner and outer extrema that are respectively located at $y^+_{pi}$ and $y^+_{po}$. Limiting attention to the region of decreasing $TI$ ensures a one-to-one correspondence between a point on this curve and a $y^+$ location, i.e. it provides an unambiguous basis for stretching the wall-normal coordinate. From (2.7) it is clear that (2.8) conveys the same information as (2.2). As now illustrated, however, (2.8) allows one to expose additional properties associated (in an average sense) with the motions involved in turbulent momentum transport.

The layer III analysis of (2.2) revealed that in the vicinity of the $TI$ zero-crossing all three terms in (2.1) are leading order, i.e. $O(1)$. Accordingly, a rescaling of (2.2) led to (2.4). This reconciled the formal magnitude ordering of terms in (2.2) with their actual orders of magnitude. For each $\beta$ within a quantifiable range of small values (corresponding to $y^+_{pi} < y^+ < y^+_{po}$), the $T^+_{\beta }$ profile of (2.6) has a single positive maximum, $T^+_{\beta m}$, located at a unique $y^+ = y^+_{\beta }$. Selecting a smaller $\beta$ locates $T^+_{\beta m}$ at larger $y^+ = y^+_{\beta }$, and at any given $T^+_{\beta m}$, $\textrm {d}T^+_{\beta }/{\textrm {d}y}^+ = 0$. Under this condition (2.7) becomes

(2.9)\begin{equation} \beta = \frac{\textrm{d}T^+}{{\textrm{d}y}^+} + \epsilon^2. \end{equation}

This establishes the range of $\beta$ values corresponding to the $y^+$ positions between $y^+_{pi}$ and $y^+_{po}$. In the vicinity of any $y^+_{\beta }$ the balance of terms in (2.8) has a mathematical structure identical to that of (2.2) in the vicinity of $\textrm {d}T^+/{\textrm {d}y}^+ = 0$, i.e. across layer III. Thus, we surmise that every $y^+$ position along the decreasing portion of the $TI$ curve uniquely corresponds to a value of $\beta$, and (2.8) holds at each of these $y^+_{\beta }$ locations. Equation (2.8) is now rescaled at each of these locations.

Because all terms in (2.8) are of equal order in the vicinity of $T_{\beta m}$, we seek a rescaling that formally renders each term $O(1)$, i.e. as was done for layer III. This is accomplished by determining the stretching coefficients, $\alpha (\beta )$, $\gamma (\beta )$ and $\lambda (\beta )$, associated with finite transformations

(2.10a–c)\begin{equation} y^+ = y^+_{\beta} + \alpha \hat{y}, \quad T^+ = T^+_{\beta m} + \gamma \hat{T}(\hat{y}), \quad U^+ = U^+_{\beta} + M(y^+{-} y^+_{\beta}) + \lambda \hat{U}, \end{equation}

where superscript ‘+’ variables are functions of $y^+$, $U^+_{\beta }$ is the value of $U^+$ at $y^+ = y^+_{\beta }$ and $M = \textrm {d}U^+_{\beta }/{\textrm {d}y}^+$. These finite transformations, and their associated differential transformations, are a generalization of those employed in rescaling (2.2) on layer III. In particular, (2.10a–c) allow for the possibility of a velocity scale other than $u_{\tau }$, as reflected by the stretching coefficient $\lambda$ in the expression involving the mean velocity.

Converting (2.10a–c) into differential transformations and inserting into (2.8) leads to

(2.11)\begin{equation} 0 = \frac{\lambda}{\alpha^2}\frac{\textrm{d}^2{\hat{U}}}{\textrm{d}\hat{y}^{2}} + \frac{\gamma}{\alpha}\frac{\textrm{d}\hat{T}}{\textrm{d}\hat{y}} + \beta, \end{equation}

where the mean derivative term containing $M$ is not present owing to multiple differentiations. Requiring that the lead coefficients on the derivative terms in (2.11) equal $\beta$ yields

(2.12)\begin{equation} 0 = \frac{\textrm{d}^2{\hat{U}}}{\textrm{d}\hat{y}^{2}} + \frac{\textrm{d}\hat{T}}{\textrm{d}\hat{y}} + 1. \end{equation}

Equation (2.12) is identical in form to (2.4) and holds over a finite domain in the vicinity of each $y^+_{\beta }$. Because transformations (2.10a–c) include the possibility of a varying velocity scale, the requirement of equal formal orders of magnitude does not uniquely determine the stretching coefficients. Namely, the coefficients $\alpha$ and $\gamma$ are now given by

(2.13a,b)\begin{equation} \alpha = \left(\frac{\lambda}{\beta}\right)^{1/2},\quad \gamma = (\lambda \beta)^{1/2}. \end{equation}

The condition $\lambda = 1$ corresponds to the choice of $u_{\tau }$ as the single velocity scale, and leads to a logarithmic mean profile solution. For $\lambda < 1$, a family of power law mean profile solutions are admitted, with the exponent depending on $\lambda$ (Fife et al. Reference Fife, Klewicki and Wei2009).

While the present framework embraces the possibility of power law solutions, from here onward we choose to only consider the $\lambda = 1$ solutions. A primary reason for this is that the $\lambda = 1$ case is the only for which the inner-normalized velocity increment $\Delta U^+$ across layer III remains $O(1)$ as $\delta ^+ \to \infty$, and according to the primary assumption (see (2.17) below) this velocity increment asymptotically becomes constant, $\Delta U^+ \simeq 1.26$, see (2.34). In this regard, we note that the pipe flow measurements of McKeon (Reference McKeon2003) exhibit a $\Delta U^+$ increment across layer III that scatters (with no apparent $\delta ^+$ trend) between approximately $1.0$ to $1.4$ over the Reynolds number range $1800 \lesssim \delta ^+ \lesssim 530\,000$ (Klewicki Reference Klewicki2013b). A second reason is that $\lambda = 1$ is apparently the only case for which the solution to (2.1) becomes universal and logarithmic, while the exponent in a power law fit to data is known to vary continuously with $\delta ^+$, e.g. Wosnik et al. (Reference Wosnik, Castillo and George2000). The results herein pertaining to the geometric properties of the $T^+$ profile also provide compelling evidence that the $\lambda = 1$ solutions to (2.12) coincide with those that are observed. Lastly, it is perhaps relevant to note that the $\lambda = 1$ case corresponds to the simplest set of transformations that yields an invariant form of (2.1) on the indicated domain.

With the specification of $\lambda = 1$, the scaling coefficients, $\alpha$ and $\gamma$, solely depend on $\beta$. They are also of a form identical to those used to arrive at (2.4), except $\epsilon ^2$ is replaced by $\beta$. As in the layer III analysis, the rescaling of (2.8) reveals that at each $y^+_{\beta }$ (i.e. at each $y^+$ between $y^+_{pi}$ and $y^+_{po}$) there exist a finite width domain over which that local scaling holds to leading order. In order of magnitude, the width (characteristic length) of this scaling patch is given by

(2.14)\begin{equation} W^+ = O(\beta^{{-}1/2}), \end{equation}

where at each $y^+_{\beta }$ the value of $\beta$ is given by (2.9), and by virtue of (2.2) $\beta$ is also equal to $-\textrm {d}^2U^+/{\textrm {d}y}^{2+}$. Previous results demonstrate that the equality $W^+ = \beta ^{-1/2}$ can be used without loss of generality (Klewicki et al. Reference Klewicki, Fife and Wei2009, Reference Klewicki, Philip, Marusic, Chauhan and Morrill-Winter2014).

For large $y^+$, the $W^+(y^+)$ profile becomes proportional to $y^+$. This feature is central to the construction of a similarity solution to (2.12), and also indicates that the present analysis framework provides a mean equation based foundation for distance-from-the-wall scaling, including why and where in the flow this scaling is expected to emerge.

2.2.4. Distance-from-the-wall scaling

Showing that $W^+$ becomes proportional to $y^+$ involves determining an expression that quantifies the dependence of $y^+_{\beta }$ on $\beta$, integrating this equation and evaluating its behaviour as $y^+ \to \infty$. Following Fife et al. (Reference Fife, Klewicki and Wei2009), we begin with (2.7) and consider its dependence on $\beta$ at each $y^+_{\beta }$. Noting that $\textrm {d}T^+_{\beta }/{\textrm {d}y}^+ = 0$ at these locations, differentiation of (2.7) with respect to $\beta$ yields

(2.15)\begin{equation} \frac{\textrm{d}^2T^+}{{\textrm{d}y}^{{+}2}}(y^+_{\beta})\frac{{\textrm{d}y}^+_{\beta}}{\textrm{d}\beta} - 1 = 0, \end{equation}

and converting the second derivative to the scaled variables appropriate to each $y^+_{\beta }$ gives

(2.16)\begin{equation} \beta^{3/2}\frac{\textrm{d}^2\hat{T}}{\textrm{d}\hat{y}^{2}}\Bigg|_{(\hat{y} = 0)}\frac{{\textrm{d}y}^+_{\beta}}{\textrm{d}\beta} - 1 = 0, \end{equation}

where $\hat {y} = 0$ is located at $y^+_{\beta }$, see (2.10a–c).

By definition, the parameter-free functions $\hat {T}$ and $\textrm {d}\hat {T}/\textrm {d}\hat {y}$ equal zero at $\hat {y} = 0$, and from (2.16) it is clear that the order of magnitude of $\textrm {d}^2\hat {T}/\textrm {d}\hat {y}^{2}$ with respect to $\beta$ is always $O(1)$. Furthermore, the transformations leading to the invariant form (2.12) satisfy the admissibility and compatibility criteria associated with the scaling patch method, as one can explicitly demonstrate that the values of the scaled functions in (2.12) are invariant at the beginning, middle and end of each $W^+$ layer (Fife et al. Reference Fife, Wei, Klewicki and McMurtry2005b). Morrill-Winter et al. (Reference Morrill-Winter, Philip and Klewicki2017) provide detailed empirical evidence of the invariant structure of the $W^+$ layers in boundary layers at large $\delta ^+$. Collectively, the analysis not only indicates that

(2.17)\begin{equation} -A(\beta) = \frac{\textrm{d}^2\hat{T}}{\textrm{d}\hat{y}^{2}}\Bigg|_{(\hat{y} = 0)} \end{equation}

is $O(1)$, but naturally leads to the expectation that, over a range of $y^+(\beta )$ remote from $y^+_{pi}$ and $y^+_{po}$, $A(\beta )$ should lose any dependence on $\beta$ as $\epsilon \to 0$, i.e. $A(\beta )$ should approach constancy. Namely, the present construction involves an invariant equation (2.12) with invariant boundary conditions on each $W^+$ layer. Thus, the only apparent way to disrupt the non-constancy of $A$ on the hierarchy domain is through contamination at the boundaries of the hierarchy. As $\epsilon \to 0$, however, the number of hierarchy layers becomes large. Thus, any potential edge effects are expected to progressively diminish on an interior portion that is remote from the beginning and end points of the hierarchy domain.

Previous results (Klewicki et al. Reference Klewicki, Fife and Wei2009, Reference Klewicki, Philip, Marusic, Chauhan and Morrill-Winter2014) and those presented herein examine the veracity of $A \to$ constant on an interior domain as $\epsilon \to 0$. Hereafter, the analysis treats $A$ as a constant, but with the recognition that the non-constancy of $A$ will, for example, lead to a logarithmic-like mean profile. For non-constant $A$, previous analyses (Fife et al. Reference Fife, Klewicki, McMurtry and Wei2005a, Reference Fife, Klewicki and Wei2009) show that $U^+$ is bounded from above and below by exactly logarithmic functions, with the width of the bounded region depending on the variation of $A$ from constancy. Overall, while there is a clear rationale for $A$ becoming constant as $\epsilon \to 0$, to date our analyses have only rigorously discerned that $A$ is $O(1)$, and thus the primary assumption of the present approach is that $A$ does indeed attain constancy as $\delta ^+ \to \infty$.

Regardless, (2.16) can be written as

(2.18)\begin{equation} \frac{{\textrm{d}y}^+_{\beta}}{\textrm{d}\beta} ={-}A^{{-}1}\beta^{{-}3/2}. \end{equation}

This expression quantifies the dependence of $y^+_{\beta }$ on $\beta$. Integration of (2.18) for constant $A$ yields

(2.19)\begin{equation} y^+ = y^+_{\beta} ={-}A^{{-}1}\int{\beta}^{{-}3/2}\,\textrm{d}\beta = \frac{2}{A}\beta^{{-}1/2} + C_1 = \frac{2}{A}W^+\,{+}\,C_1, \end{equation}

and solving for $W^+$

(2.20)\begin{equation} W^+ = \frac{A}{2}(y^+{-} C_1), \end{equation}

yields the result that $W^+$ becomes proportional to $y^+$ at sufficiently large $y^+ = y^+_{\beta }$. According to the present framework, this is the origin of distance from the wall scaling; approximate at finite $\delta ^+$ and approaching exact proportionality as $y^+ \to \infty$ ($\delta ^+ \to \infty$).

Figures 3(a) and 3(b) respectively plot $W^+$ vs $y^+$ and $W/y$ vs $y^+$ over a range of existing channel flow DNS. Together these demonstrate how analytically predicted properties of the hierarchy layer width distribution emerge with increasing $\delta ^+$. As examined in greater detail in § 3.3, linear proportionality between $W^+$ and $y^+$ first emerges on the interior part of layer IV, and with increasing $\delta ^+$ the deviation from exact proportionality diminishes. The $W/y$ plot clearly suggests that, interior to $y^+ \simeq 60$, the profiles are invariant with Reynolds number, while the subsequent rise in $W/y$ (beyond $y^+ \simeq 60$) above the approximately constant value in layer IV diminishes with increasing $\delta ^+$. An asymptotic structure composed of only an inner region ($y^+ \lesssim 60$) and a $W/y =$ constant region has apparent consistency with the two length scale approximation employed in the inner/outer plus overlap layer construction. An approach to an inner plus $W/y =$ constant structure as $\delta ^+ \to \infty$ may in fact explain why the asymptotically reduced similarity solutions below are identical to those found using the two layer approximations. The progression toward the asymptotic flow state is, however, conceived to emerge quite differently by the two approaches. As evidenced herein, the hierarchical layer structure inherent to the similarity solutions provides additional results that can be scrutinized.

Figure 3.

Channel flow profiles of the hierarchy layer width distribution: (a) $W^+$ vs $y^+$, (b) $W/y$ vs $y^+$. Profiles increase in $\delta ^+$ from the lightest grey scale to darkest grey scale. Data are from the DNS of Hoyas & Jimenez (Reference Hoyas and Jimenez2006) $\delta ^+ = 547$, $934$ and $2004$; Pirozzoli (Reference Pirozzoli2014) $\delta ^+ = 4073$; Lee & Moser (Reference Lee and Moser2015) $\delta ^+ = 5181$; Yamamoto & Tsuji (Reference Yamamoto and Tsuji2018) $\delta ^+ = 7996$.

2.2.5. Fife similarity parameter

From (2.19) and (2.20) it is clear that, on the interior zone of interest,

(2.21)\begin{equation} \frac{\textrm{d}W^+}{{\textrm{d}y}^+} = \frac{A}{2} = \phi_c^{{-}1}, \quad \text{or equivalently}, \ \frac{{\textrm{d}y}^+}{\textrm{d}W^+} = \frac{2}{A} = \phi_c. \end{equation}

The second of the two forms of (2.21) is made explicit because it literally identifies the stretching of the $y$ coordinate that underlies both the invariant form of (2.1) as given by (2.12), and through the condition for dynamical similarity (2.17) the similarity solutions to (2.12) (Klewicki & Oberlack Reference Klewicki and Oberlack2015).

Owing to the fundamental nature of this coordinate stretching, and following Klewicki (Reference Klewicki2013a) and Klewicki et al. (Reference Klewicki, Philip, Marusic, Chauhan and Morrill-Winter2014), the quantity $\phi$ is herein called the Fife similarity parameter. This parameter generically refers to $\phi ^{-1} = \textrm {d}W^+/{\textrm {d}y}^+$ on the layer hierarchy domain. At any given sufficiently large Reynolds number, however, there is an interior domain where the Fife similarity parameter takes on a nearly constant value. On this domain $\phi$ is called $\phi _c$. With Reynolds number the constancy approximation of $\phi _c$ improves and under inner normalization the domain over which it (approximately) holds increases. Thus, the $\phi _c$ measured at any finite $\delta ^+$ changes slowly with $\delta ^+$ and under the primary assumption approaches a constant value as $\delta ^+ \to \infty$. The conjecture that this ultimate constant value is given by $\phi _c = \varPhi = (1 + \sqrt {5})/2$ is discussed in § 3.4. Here, we note that relative to these naming conventions it is simply a coincidence that the Greek letter for F (as in Fife) is also the same as that typically used for the golden ratio. The following analyses and results demonstrate that the Fife similarity parameter is central to describing the mathematical, physical and geometrical structure of the channel flow mean dynamics.

2.2.6. Similarity solutions

The condition for asymptotic self-similarity, $A =$ constant (or $\phi = \textrm {constant}$), allows one to solve for $U^+$ or $T^+$ via direct integration and application of boundary conditions. Because the $A =$ constant condition closes the mean equation, arriving at one of these solutions does not involve first solving for the other. Both are independently determinable solutions to (2.1), and thus one is not subordinate to the other. (Of course, as required, both solutions adhere to all the expressions that connect $U^+$ and $T^+$.) This feature is apparently distinct from other approaches, and it provides clear justification for one to draw upon the properties of both solutions to empirically estimate $\phi _c$ at finite $\delta ^+$, or potentially determine the asymptotic value of $\phi _c$.

For the mean velocity we follow the approach of Klewicki & Oberlack (Reference Klewicki and Oberlack2015). In terms of inner variables, $A$ is given by

(2.22)\begin{equation} A = \frac{2}{\phi_c} ={-}\frac{\textrm{d}^2\hat{T}}{\textrm{d}\hat{y}^{2}} ={-}\frac{\textrm{d}^2T^{+}}{{\textrm{d}y}^{{+}2}}\beta^{{-}3/2}. \end{equation}

Noting that $\beta = - \textrm {d}^2U^+/{\textrm {d}y}^{+2}$, and using the mean vorticity transport equation for channel flow, one arrives at

(2.23)\begin{equation} \frac{2}{\phi_c} = \frac{\textrm{d}^3U^+}{{\textrm{d}y}^{{+}3}}\left(-\frac{\textrm{d}^2U^+}{{\textrm{d}y}^{{+}2}}\right)^{{-}3/2}, \end{equation}

which is an ordinary differential equation with a single dependent variable.

To solve (2.23), we treat $\phi _c$ as a constant, and let $f = -\textrm {d}^2U^+/{\textrm {d}y}^{+2}$. This yields

(2.24)\begin{equation} \frac{2}{\phi_c} f^{3/2} ={-}\frac{\textrm{d}f}{{\textrm{d}y}^+}. \end{equation}

Separating variables and integrating yields,

(2.25)\begin{equation} f^{{-}1/2} = \frac{1}{\phi_c}(y^+{-} C_1). \end{equation}

The process is completed by squaring and inverting (2.25) and integrating twice to obtain

(2.26)\begin{equation} U^+(y^+) = \phi_c^2\ln(y^+{-} C_1) + C_2y^+\,{+}\, C_3, \end{equation}

where at finite $\delta ^+$ $C_1, C_2$ and $C_3$ are $O(1)$ and nearly constant, and, like $\phi _c$, as $\delta ^+ \to \infty$ they approach fixed values. Because $\textrm {d}U^+/{\textrm {d}y}^+ \to 0$ as $y^+ \to \infty$ it is known that $C_2$ also asymptotically approaches zero. Recovery of the traditional version of the logarithmic mean profile equation at finite $\delta ^+$, however, requires $C_2 \to 0$ faster than $1/\delta ^+$. This, and other properties of (2.26), are explored empirically in § 3. Lastly, note that $\phi _c^2$ appears as the leading coefficient on the logarithmic term, and thus (2.26) clarifies that $\phi _c^2 = 1/\kappa$, where $\kappa$ is the von Kármán constant.

To obtain the solution for $T^+$ we follow the approach of Fife et al. (Reference Fife, Klewicki, McMurtry and Wei2005a, Reference Fife, Klewicki and Wei2009). First recall that $W^+ = \beta ^{-1/2}$, $\phi _c = 2/A$, and that on the hierarchy domain each $y^+_{\beta }$ corresponds uniquely to a $y^+$. Using (2.19) to solve for $\beta$ yields

(2.27)\begin{equation} \beta = \left( \frac{2}{A} \right)^2 (y^+{-} C_1)^{{-}2} = \frac{\phi_c^2}{(y^+\,{-}\, C_1)^2}. \end{equation}

Then, using the definition of $\beta$ and the fact that $\textrm {d}T^+/{\textrm {d}y}^+ = 0$ at $y^+ = y_m^+$ (i.e. where $T^+ = T^+_m$) gives

(2.28)\begin{equation} \frac{\textrm{d}T^+}{{\textrm{d}y}^+} = \frac{\phi_c^2}{(y^+{-} C_1)^2} - \frac{1}{\delta^+} = \frac{\phi_c^2}{(y^+{-} C_1)^2} - \frac{\phi_c^2}{(y^+_m - C_1)^2}, \end{equation}

and subsequent integration yields

(2.29)\begin{equation} T^+(y^+) = C_4 - \frac{\phi_c^2}{(y^+{-} C_1)} - \frac{y^+\phi_c^2}{(y^+_m - C_1)^2}. \end{equation}

The analysis of Wei et al. (Reference Wei, McMurtry, Klewicki and Fife2005b) indicates that (2.29) is valid to leading order over a domain that begins $O(\sqrt {\delta ^+})$ interior to $y_m^+$ and extends to $y^+ = \delta ^+$.

From the first equality of (2.28) it is straightforward to show that $y^+_m = \lambda _m\sqrt {\delta ^+}$, and subsequent use of (2.29) allows one to show that $\lambda _m \to \phi _c$ like $1/\sqrt {\delta ^+}$ (Klewicki et al. Reference Klewicki, Philip, Marusic, Chauhan and Morrill-Winter2014). Recalling that $\kappa = \phi _c^{-2}$ one can see that this asymptotic result is identical to that found by Afzal (Reference Afzal1982) (i.e. $\lambda _m = \sqrt {1/\kappa }$) using matched asymptotic methods. It is similarly straightforward to verify that the expressions for $\textrm {d}U^+/{\textrm {d}y}^+$ associated with (2.26) and for $T^+$ in (2.29) satisfy the exact relation

(2.30)\begin{equation} \frac{\textrm{d}U^+}{{\textrm{d}y}^+} = 1 - T^+{-} \frac{y^+}{\delta^+}, \end{equation}

that derives from integrating (2.2) and applying the boundary condition at $y^+ = 0$.

Properties of (2.26) and (2.29), and their simplified forms as $\epsilon \to 0$, are investigated in detail in § 3.

2.2.7. Discrete hierarchy construction

The analysis above shows that each $\beta$ value (within in its allowable range) uniquely locates both the anchor point and width of any given hierarchy layer. The $\Delta y^+$ widths of the layers are quantified by the $W^+ (y^+)$ distribution of figure 3(a). Over the layer hierarchy domain $y^+_{\beta }$ varies continuously, and thus there are an infinite number of layer anchor points on the interval $y^+_{pi} \lesssim y^+ \lesssim y^+_{po}$. Each $W^+(y^+)$, however, defines the finite width over which (2.12) holds to leading order. This attribute can be leveraged to significantly reduce the number of anchor points required to describe the solution properties on the hierarchy domain. Here a number of possibilities exist. An obvious constraint is that gaps between adjacent layers are not permitted. In general, a permissible construction thus requires that the anchor point of one layer resides within the bounds (end points inclusive) of at least one nearby layer, i.e. adjacent layers will generally have a finite degree of overlap. Finite overlap of adjacent layer is, however, not required. In fact, the simplest valid discrete representation of the layer hierarchy is the one in which adjacent layers abut. Attributes of this construction are now described.

A few actual layers of the abutting layer construction are depicted in figure 4 for the $\beta$ profile from the DNS of Yamamoto & Tsuji (Reference Yamamoto and Tsuji2018) at $\delta ^+ = 7996$. With increasing $\delta ^+$, $W^+$ most accurately gains proportionality with $y^+$ on the inertial domain. Thus, near to and within this region the $W^+$ layers appear approximately equal width when plotted on a logarithmic scale. The layers on the hierarchy are self-similar, and layer III is one of these layers – specifically the one for which $\beta = \epsilon ^2$. The asymptotic $y^+ = y_m^+$ position of $T_m^+$ is analytically known to reside at the start of layer III, and thus we accordingly choose the anchor point of each layer to be its starting point (Klewicki et al. Reference Klewicki, Philip, Marusic, Chauhan and Morrill-Winter2014). For this same reason, $y_m^+$ is a natural choice for one of the boundaries between abutting layers, although any $y^+$ on the hierarchy domain is equally valid. This fact is exploited later. As revealed through the analysis above, the coordinate transformations at any $y^+ = y^+_{\beta }$ retain leading-order validity over the associated $W^+$ extent. Under the discrete construction, the number of layers on the hierarchy increases logarithmically with Reynolds number, becoming countably infinite as $\delta ^+ \to \infty$.

Figure 4.

Abutting hierarchy layers in the vicinity of $y_m^+$ in the $\delta ^+ = 7996$ channel DNS of Yamamoto & Tsuji (Reference Yamamoto and Tsuji2018). Note that $\beta = \textrm {d}T^+/{\textrm {d}y}^+ + \epsilon ^2$. Coloured dots/dashed lines denote the anchor points for the layers having the same colour $W^+$.

Figure 5 illustrates the basic layer-to-layer construction. As depicted, under asymptotic conditions at any given location on the hierarchy

(2.31)\begin{equation} y^+_i + W^+_i = y^+_{i+1}. \end{equation}

By restricting attention to the $\phi _c = 2/A$ portion of the hierarchy and neglecting the additive constant $C_1$ in (2.20) for asymptotically large $\delta ^+$, Klewicki et al. (Reference Klewicki, Philip, Marusic, Chauhan and Morrill-Winter2014) show that

(2.32)\begin{equation} y^+_i = \phi_c W^+_i, \end{equation}

and

(2.33)\begin{equation} \frac{y^+_i}{y^+_{i+1}} = \frac{W^+_i}{W^+_{i+1}} = \frac{\phi_c}{\phi_c+1}. \end{equation}

These relations reveal that on the $\phi = \phi _c$ domain the layer widths and their anchor points comprise nested geometric series with each series having the same common ratio, $(\phi _c + 1)/\phi _c$.

Figure 5.

Sketch of the $y^+$ anchor point and associated $W^+$ arrangement used in the present discrete hierarchy construction.

Asymptotically, there are well-defined $\Delta U^+|_{W^+}$ and $\Delta T^+|_{W^+}$ increments on any given $W^+$ layer. Owing to the non-overlapping property of the discrete hierarchy construction these increments respectively sum to give the overall mean velocity and Reynolds stress variations. The asymptotic reduction for the mean velocity gradient is $\textrm {d}U^+/{\textrm {d}y}^+ = \phi _c^2/y^+$, and use of (2.33) yields the constant velocity increment

(2.34)\begin{equation} \Delta U^+|_{W_i^+} = \int_{y_1^+}^{y_i^+\,{+}\, W_i^+} \frac{\textrm{d}U^+}{{\textrm{d}y}^+}\,{\textrm{d}y}^+ = \phi_c^2\ln\left(\frac{\phi_c + 1}{\phi_c}\right ). \end{equation}

This relation holds on the domain where $\phi = \phi _c$ for any given $y^+_i$. Similarly, use of $y_m^+ = \phi _c\sqrt {\delta ^+}$, (2.32) and the asymptotic reduction of the Reynolds stress gradient equation (2.28) gives

(2.35)\begin{equation} \Delta T^+|_{W_i^+} = \int_{y_i^+}^{y_i^+\,{+}\, W_i^+} \frac{\textrm{d}T^+}{{\textrm{d}y}^+}\,{\textrm{d}y}^+ = \frac{\phi_c^2}{y_i^+(\phi_c + 1)} - \frac{y_i^+}{\phi_c \delta^+}. \end{equation}

Properties derived from the discrete layer hierarchy formulation, including those associated with (2.34) and (2.35) are explored and leveraged throughout § 3.