Characterizing and predicting the turbulent flow in the vicinity of a wall continues to not only be a problem of scientific interest, but also one of great practical relevance. Significant knowledge and insight has been gained from the development of experimental diagnostics in the 1960s to the emergence of direct numerical simulations (DNS) in the 1980s, sophisticated measurements in the 1990s, an emphasis on coherent structures in the 2000s and operator-based dynamical models over the past decade. Against this backdrop of increasingly sophisticated experimentation, computations and analyses, the attached-eddy hypothesis – originally proposed more than 50 years ago by Townsend (Reference Townsend1976) – continues to serve as a simple, yet effective theory to qualitatively and quantitatively describe structural and statistical aspects of turbulent boundary layers, particularly focusing on the logarithmic region.

The attached-eddy model (AEM) was given a firm mathematical footing by Perry & Chong (Reference Perry and Chong1982), and further developed by Marusic and coworkers over the past 25 years (Perry & Marušić Reference Perry and Marušić1995; Marusic et al. Reference Marusic, Monty, Hultmark and Smits2013; Woodcock & Marusic Reference Woodcock and Marusic2015; de Silva et al. Reference de Silva, Woodcock, Hutchins and Marusic2016b ). Among other useful features, the AEM is able to explain scaling behaviours of velocity moments, provide an explanation for uniform momentum zones (de Silva et al. Reference de Silva, Hutchins and Marusic2016a ) and serve as a predictive model for the von Kármán constant as a function of Reynolds number. A review of the theory and developments of the AEM of wall turbulence can be found in Marusic & Monty (Reference Marusic and Monty2019). Over the past few years, theoretical and numerical analyses (e.g. McKeon Reference McKeon2019; Lozano-Durán & Bae Reference Lozano-Durán and Bae2019) have added further credibility to this theory. Despite the success of the AEM, it is pertinent to remember that it is fundamentally a statistical theory, and other hypotheses (e.g. Davidson et al. Reference Davidson, Nickels and Krogstad2006; Davidson & Krogstad Reference Davidson and Krogstad2009) can also be used to explain the statistical characteristics of the log layer.

The attached-eddy hypothesis is based on the principle that the physics and statistical properties of the logarithmic layer can be explained by considering geometrically self-similar eddies that extend from the wall. A foundational assumption is that the length scale of each individual eddy follows a probability distribution that is a function of the distance from the wall. Hence, the term ‘attached’ alludes to the fact that every eddy can be assumed to be randomly placed on the wall. The AEM is effectively an inviscid theory, yet the range of scales is set by the Reynolds number.

A key advance was made by Woodcock & Marusic (Reference Woodcock and Marusic2015) who established a statistical foundation for AEM. They provide a complete derivation for all the velocity moments and demonstrate logarithmic scaling relationships therein. They were also able to provide expressions for the skewness and flatness of the wall-normal and spanwise fluctuations as a function of the Reynolds number. While variants of the AEM continue to be developed in the literature (e.g. Hwang & Eckhardt Reference Hwang and Eckhardt2020), we consider Woodcock & Marusic (Reference Woodcock and Marusic2015) as the starting point of our exploration and exclusively consider zero pressure gradient boundary layers. The main contributions and guiding questions in this work are as follows.

1. (i) Inverse identification of attached-eddy influence functions. Townsend introduced the notion of an ‘eddy intensity’ (or contribution) function: a wall-parallel average of the single-eddy induced velocity and its products. While Townsend gave remarkably insightful descriptions of the nature of this function (figure 5.7, p. 155 of Townsend (Reference Townsend1976)), his and most other treatments are qualitative. Here we pose a concrete inverse problem that infers the ideal influence functions $I_1(y/h)$ and $I_{ij}(y/h)$ implied by reference one-point moments (DNS/experiment). The inferred kernels expose the wall-normal support, sign structure and plateau/decay behaviour required for log-layer mean and Reynolds stresses, and motivate a simplified kernel model.

2. (ii) A minimal Biot–Savart-consistent attached eddy. Woodcock & Marusic (Reference Woodcock and Marusic2015) utilize an eddy that has a complex shape (figure 1 in Woodcock & Marusic (Reference Woodcock and Marusic2015)), presumably configured using insight from DNS and/or particle image velocimetry fields. Guided by the inferred kernels, we construct an attached-eddy template from a small number of Rankine vortex rods (a ‘rectangular hairpin’) together with an inviscid image system. The objective is not an exact Navier–Stokes solution, but a kinematically consistent building block that (i) respects wall-impermeability, (ii) has well-posed planar influence integrals and (iii) admits a transparent connection between morphology and the influence functions.

3. (iii) What aspects of morphology control log-layer statistics? We quantify the sensitivity of $I_1$ , $I_{ij}$ and the resulting moments to eddy geometry (e.g. triangular versus square hairpins, inclination angle and simple packet/nesting configurations treated as composite marks). This clarifies which structural features are essential (e.g. realizing the near-constant portion of $I_1$ for $y/h\lesssim 1$ ) and which details have weak impact once similarity and attachedness are enforced.

4. (iv) A spectral influence kernel and interpretation of attached-eddy spectra. We introduce an explicit spectral influence kernel $I_\phi (\kappa _x,\eta )$ that maps a self-similar wall-parallel footprint to its one-dimensional (1-D) streamwise energy spectrum as a function of relative height $\eta =y/h$ . Combining $I_\phi$ with the scale-invariant population law $p(h)\propto h^{-3}$ yields a compact explanation for the emergence (and limitations) of the $k_x^{-1}$ range, the location of the premultiplied hump and the origin of low- $k_x$ discrepancies under the independent-mark (Poisson) hypothesis. The ideal influence kernel is extracted by defining a Fredholm-type inverse problem.

5. (v) Why the rectangular hairpin is unusually predictive. Exact closed-form expressions for the mean kernel and Fourier-space velocity of a general three-segment hairpin family reveal a clean mean-variance duality: the horizontal head determines the entire mean kernel $I_1$ , while the inclined legs dominate the spectral energy $I_\phi$ . Within the straight segment family, only the rectangular hairpin produces an exact plateau in $I_1$ and hence a purely logarithmic mean.

Forward and inverse modelling tools used here are open sourced (https://github.com/CaslabUM/AttachedEddy).

1. Statistics of attached eddies

We begin with a short description of the attached-eddy hypothesis and modelling. This presentation generally follows Woodcock & Marusic (Reference Woodcock and Marusic2015) with a slightly different pedagogy. As noted above, we adopt the standard coordinate system: $x$ streamwise, $y$ wall-normal and $z$ spanwise. Planar homogenization/influence integrals are taken over the wall-parallel $(x,z)$ plane, so one-point statistics depend only on $y$ .

Consider a collection of $n$ eddies (figure 1) of length scale $h_{e,i}$ that are placed on the wall at locations $\boldsymbol{x}_{e,i}=(x_{e,i},z_{e,i})$ . Since these are attached eddies, it is implicit that only the streamwise and spanwise components of $\boldsymbol{x}_{e,i}$ are variable. Define $\boldsymbol{h}_e \triangleq \{h_{e,1},h_{e,2},..,h_{e,n}\}$ and $\boldsymbol{X}_e \triangleq \{\boldsymbol{x}_{e,1},\boldsymbol{x}_{e,2},..,\boldsymbol{x}_{e,n}\}$ . Assuming self-similarity and linearity, a quantity $q$ (e.g. a velocity component) evaluated at a location $\boldsymbol{x}$ can be determined using the superposition

(1.1) \begin{align} q(\boldsymbol{x},\boldsymbol{X}_e,\boldsymbol{h}_e,n) \triangleq \sum _{i=1}^n q\!\left (\frac {\boldsymbol{x}-\boldsymbol{x}_{e,i}}{h_{e,i}}\right )\!. \end{align}

Figure 1.

Schematic of discrete representation of attached eddies with $n=21$ and $m=3$ .

Restricting our attention to a streamwise–spanwise square wall patch of side $2L$ in which the eddies are assumed to be independently and uniformly distributed, and assuming that the range of eddy length scales follows a probability density function $p(h)$ , the expectation of $q$ in this region is given by

(1.2) \begin{equation} q(\boldsymbol{x},n) \triangleq \mathbb{E}_{\boldsymbol{X}_e,\boldsymbol{h}_e}\!\left [q(\boldsymbol{x},\boldsymbol{X}_e,\boldsymbol{h}_e,n)\right ] = \frac {1}{4L^2}\int _{h_{\textit{min}}}^{h_{\textit{max}}}\int _{-L}^L\int _{-L}^L \sum _{i=1}^n q\!\left (\frac {\boldsymbol{x}-\boldsymbol{x}_{e,i}}{h}\right )\,p(h)\,\text{d}x_{e,i}\,\text{d}z_{e,i}\,\text{d}h. \end{equation}

Homogenizing in the $(x,z)$ directions, and assuming $L$ is large enough that each eddy centred at the origin has a negligible induced contribution outside an area $4L^2$ (cf. Campbell’s theorem (Rice Reference Rice1944) and the appendix of Woodcock & Marusic (Reference Woodcock and Marusic2015)), it can be shown that

(1.3) \begin{align} q(y,n)\triangleq \mathbb{E}_{x,z}\!\left [q(\boldsymbol{x},n)\right ] \approx \frac {n}{4L^2}\int _{h_{\textit{min}}}^{h_{\textit{max}}}\int _{-L}^L\int _{-L}^L q\!\left (\frac {\boldsymbol{x}}{h}\right )\,p(h)\,\text{d}x\,\text{d}z\,\text{d}h. \end{align}

Note that in the above equation, the entire field is written as a function of one prototypical eddy, scaled by the probability density function of the eddy sizes $p(h)$ and the eddy density $n/(4L^2)$ . If the mean eddy density is $\beta$ , then using Poisson’s law, the expected value (over all numbers of eddies) is

(1.4) \begin{equation} Q(y) \triangleq \mathbb{E}_{n}\!\left [q(y,n)\right ] = \beta \int _{h_{\textit{min}}}^{h_{\textit{max}}}\int _{-L}^L\int _{-L}^L q\!\left (\frac {\boldsymbol{x}}{h}\right )\,p(h)\,\text{d}x\,\text{d}z\,\text{d}h. \end{equation}

Note that all the $q$ defined above are random variables, yet $Q(y)$ is a deterministic quantity. Now we are in a position to define the mean streamwise velocity $U(y)$ as a superposition of eddies of various sizes $h$ . This can be written in terms of the induced velocity field $u_1(\boldsymbol{\cdot })$ of one prototypical eddy,

(1.5) \begin{align} U(y) &\triangleq \beta \int _{h_{\textit{min}}}^{h_{\textit{max}}}\int _{-L}^L\int _{-L}^L u_1\!\left (\frac {\boldsymbol{x}}{h}\right )\,p(h)\,\text{d}x\,\text{d}z\,\text{d}h +U_{\textit{ref}} \nonumber \\ &= \beta \int _{h_{\textit{min}}}^{h_{\textit{max}}} p(h)\,h^2\,I_1\!\left (\frac {y}{h}\right )\,\text{d}h +U_{\textit{ref}}, \end{align}

where the mean-flow eddy contribution (influence) function is

(1.6) \begin{align} I_1\!\left (\frac {y}{h}\right ) &\triangleq \int _{-L/h}^{L/h}\int _{-L/h}^{L/h} u_1\!\left (\frac {\boldsymbol{x}}{h}\right )\,\textit{d}\!\left (\frac {x}{h}\right )\,\textit{d}\!\left (\frac {z}{h}\right )\!. \end{align}

Similarly, we can define the Reynolds stress tensor as

(1.7) \begin{align} R_{ij}(y) &\triangleq \beta \int _{h_{\textit{min}}}^{h_{\textit{max}}} p(h)\,h^2\,I_{ij}\!\left (\frac {y}{h}\right )\,\text{d}h,\nonumber \\ \textrm {where }\; I_{ij}\!\left (\frac {y}{h}\right ) &\triangleq \int _{-L/h}^{L/h}\int _{-L/h}^{L/h} u_i\!\left (\frac {\boldsymbol{x}}{h}\right )\,u_{\!j}\!\left (\frac {\boldsymbol{x}}{h}\right )\,\textit{d}\!\left (\frac {x}{h}\right )\,\textit{d}\!\left (\frac {z}{h}\right ). \end{align}

Note the presence of an additional free stream velocity in the definition of $U(y)$ . This is required because we are working with induced velocity fluctuations.

The final piece we need is the probability distribution of the eddy sizes. Using insight from Townsend (Reference Townsend1976) and Perry & Chong (Reference Perry and Chong1982), Woodcock & Marusic (Reference Woodcock and Marusic2015) propose that $p(h)\propto 1/h^3$ . In particular, for $h\in [h_{\textit{min}},h_{\textit{max}}]$ ,

(1.8) \begin{align} p(h) \triangleq \frac {C}{h^3}, \qquad C \triangleq \frac {2}{1/h_{\textit{min}}^2-1/h_{\textit{max}}^2}. \end{align}

Here $h_{\textit{max}}\approx \delta ^+$ is the outer length scale (boundary layer thickness in wall units) and $h_{\textit{min}}$ is set by the friction Reynolds number in the form $2.6 \sqrt {\textit{Re}_\tau }$ .

1.1. Statistical assumptions and derivation of the Reynolds-stress formula

The attached-eddy framework above is a shot-noise model: the instantaneous velocity perturbation at a point is written as a superposition of contributions from a random collection of eddies. While the mean follows immediately from linearity of expectation, the Reynolds stresses are quadratic and require an explicit statement of the underlying point-process assumptions.

We idealize the set of attached eddies as a stationary marked Poisson point process $\varPhi$ on the wall-parallel plane $(x,z)$ . Each eddy is characterized by a wall-parallel centre location $\boldsymbol{x}_e=(x_e,z_e)$ and a mark $\theta$ that includes its size $h$ (and possibly additional parameters, e.g. orientation, circulation distribution, etc.). The following are assumed:

1. Ai. homogeneity – the process is stationary in $(x,z)$ with constant areal intensity $\beta$ (so the expected number of eddies in area $A$ is $\beta A$ );

2. Aii. independent marking – conditional on the locations, the marks $\{\theta _k\}$ (collectively, all eddy properties beyond the wall-parallel centre location) are independent and identically distributed, and independent of the point locations; in particular the size distribution is $p(h)$ ;

3. Aiii. moment existence – the required single-eddy planar integrals defining $I_i$ and $I_{ij}$ exist in the $L/h\to \infty$ limit (see Appendix A for a convergence discussion);

4. Aiv. ergodic/homogenization interpretation – planar averaging over a sufficiently large patch is identified with the ensemble mean for the stationary process.

Assumptions Ai and Aii are the essential statistical hypotheses behind the Reynolds-stress closure.

Velocity superposition. Let a single prototypical eddy with mark $\theta$ induce a velocity field $\,u_i^{(e)}(\xi ,\zeta ,\eta ;\theta )\,$ in dimensionless coordinates $ (\xi ,\zeta ,\eta )=(x/h,\;z/h,\;y/h).$ Define the single-eddy planar influence integral

(1.9) \begin{equation} I_i(\eta ;\theta ) \;\triangleq \; \int _{\mathbb{R}^2} u_i^{(e)}(\xi ,\zeta ,\eta ;\theta )\,\text{d}\xi \,\text{d}\zeta , \qquad (\eta =y/h). \end{equation}

Under Ai–Aiii, first-order Campbell/Mecke yields

(1.10) \begin{equation} U_i(y)\;\triangleq \;\mathbb{E}[u_i(y)] \;=\; \beta \int p(h)\,h^2\, \mathbb{E}_{\theta |h}\!\left [I_i\!\left (\frac {y}{h};\theta \right )\right ]\,\text{d}h. \end{equation}

Second moment and the cross-term issue. The central subtlety is that $ u_i u_{\!j} = \big(\sum _k u_i^{(k)}\big)\big(\sum _\ell u_{\!j}^{(\ell )}\big)$ contains cross terms with $k\neq \ell$ , so one cannot in general write the second moment as a sum of ‘individual Reynolds stresses’. To make this explicit, expand

(1.11) \begin{equation} \mathbb{E}[u_i(y)u_{\!j}(y)] = \mathbb{E}\!\left [\sum _k u_i^{(k)}u_{\!j}^{(k)}\right ] + \mathbb{E}\!\left [\sum _{k\neq \ell } u_i^{(k)}u_{\!j}^{(\ell )}\right ]\!. \end{equation}

Define the single-eddy quadratic kernel

(1.12) \begin{equation} I_{ij}(\eta ;\theta ) \;\triangleq \; \int _{\mathbb{R}^2} u_i^{(e)}(\xi ,\zeta ,\eta ;\theta )\, u_{\!j}^{(e)}(\xi ,\zeta ,\eta ;\theta )\,\text{d}\xi \,\text{d}\zeta . \end{equation}

For a Poisson process with independent marks (Ai and Aii), the second-order Campbell theorem gives the closed form

(1.13) \begin{equation} \mathbb{E}[u_i(y)u_{\!j}(y)] = \beta \int p(h)\,h^2\,\mathbb{E}_{\theta |h}\!\left [I_{ij}\!\left (\frac {y}{h};\theta \right )\right ]\,\text{d}h \;+\; U_i(y)\,U_{\!j}(y), \end{equation}

where the $U_iU_{\!j}$ term is precisely the contribution of the $k\neq \ell$ cross terms in (1.11). In other words, under Ai and Aii, the cross terms do not vanish individually; rather, they factorize into the product of means.

Reynolds stresses as covariances (cancellation of the cross term). The one-point Reynolds stress tensor is the covariance of the total field:

(1.14) \begin{equation} R_{ij}(y) \;\triangleq \; \mathbb{E}\!\left [(u_i(y)-U_i(y))(u_{\!j}(y)-U_{\!j}(y))\right ] = \mathbb{E}[u_i(y)u_{\!j}(y)]-U_i(y)U_{\!j}(y). \end{equation}

Substituting (1.13) into (1.14) yields the single-eddy representation

(1.15) \begin{equation} R_{ij}(y) = \beta \int p(h)\,h^2\,\mathbb{E}_{\theta |h}\!\left [I_{ij}\!\left (\frac {y}{h};\theta \right )\right ]\,\text{d}h. \end{equation}

Equation (1.15) is therefore justified only when the eddy field is modelled as a Poisson (or otherwise ‘factorizable’) superposition with independent marks.

If the eddy process is not Poisson (e.g. clustering, exclusion or explicit packet/nesting correlations), then the cross term in (1.11) no longer factorizes into $U_iU_{\!j}$ . In that case, (1.13) acquires an additional correction involving the pair (second factorial) moment measure of the process (equivalently a pair-correlation function), and the covariance is no longer determined solely by the single-eddy kernel $I_{ij}$ . A statistically consistent way to incorporate a correlated internal structure (e.g. a hairpin packet) while retaining (1.15) is to treat the packet as the Poisson object: internal correlations are absorbed into the mark $\theta$ , while packet centres remain a stationary Poisson field.

1.2. Convergence of planar influence integrals

Equation (1.6) defines the mean-flow eddy contribution (influence) function as a planar integral of the streamwise velocity induced by a single prototypical eddy. Writing the dimensionless coordinates $ (\xi ,\zeta ,\eta ) \triangleq (x/h,\;z/h,\;y/h)$ and $R \triangleq L/h$ , we can make the dependence on the truncation radius explicit,

(1.16) \begin{equation} I_1(\eta ;R) \;\triangleq \; \int _{-R}^{R}\int _{-R}^{R} u_1(\xi ,\zeta ,\eta )\,\text{d}\xi \,\text{d}\zeta , \qquad I_1(\eta )\;\triangleq \;\lim _{R\to \infty } I_1(\eta ;R), \end{equation}

whenever the limit exists. In particular, we do not assume compact support of the induced velocity field. Rather, the required condition for (1.6) to be well-defined is that the planar integral converges as $R\to \infty$ , or equivalently that the tail contribution beyond $|(\xi ,\zeta )|\gtrsim R$ becomes negligible for the chosen truncation. A detailed analysis of convergence is presented in Appendix A.

2. The ideal eddy contribution function

The functions $I_1(y/h)$ and $I_{ij}(y/h)$ introduced in (1.6) are key for the moments produced by the attached-eddy model: once these single-eddy planar integrals are specified (together with $\beta$ and $p(h)$ ), the attached-eddy integrals yield the one-point mean and Reynolds stresses, and (through similar constructions) higher-order moments. Rather than prescribing a particular eddy geometry a priori, we first ask a more direct question:

This is an inverse problem for a Fredholm integral equation of the first kind. In its continuous form,

(2.1) \begin{align} U_{\textit {data}}(y)-U_{\textit{ref},\textit{data}} &= \beta \int _{h_{\textit{min}}}^{h_{\textit{max}}} p(h)\,h^2\,I_1\!\left (\frac {y}{h}\right )\,\text{d}h, \\[-12pt]\nonumber \end{align}

(2.2) \begin{align} R_{ij,\textit{data}}(y) &= \beta \int _{h_{\textit{min}}}^{h_{\textit{max}}} p(h)\,h^2\,I_{ij}\!\left (\frac {y}{h}\right )\,\text{d}h, \end{align}

where $U_{\textit{ref},\textit{data}}$ is the outer velocity (e.g. centreline or free stream velocity). For the classical attached-eddy size distribution $p(h)=C/h^3$ , these become

(2.3) \begin{equation} U_{\textit {data}}(y)-U_{\textit{ref},\textit{data}} = \beta \,C \int _{h_{\textit{min}}}^{h_{\textit{max}}} \frac {I_1(y/h)}{h}\,\text{d}h, \qquad R_{ij,\textit{data}}(y) = \beta \,C \int _{h_{\textit{min}}}^{h_{\textit{max}}} \frac {I_{ij}(y/h)}{h}\,\text{d}h, \end{equation}

and, after the change of variables $\eta =y/h$ (so that $\text{d}h/h=-\text{d}\eta /\eta$ ),

(2.4) \begin{align} U_{\textit {data}}(y)-U_{\textit{ref},\textit{data}}= \beta \,C \int _{y/h_{\textit{max}}}^{y/h_{\textit{min}}} \frac {I_1(\eta )}{\eta }\,\text{d}\eta , \qquad R_{ij,\textit{data}}(y) = \beta \,C \int _{y/h_{\textit{max}}}^{y/h_{\textit{min}}} \frac {I_{ij}(\eta )}{\eta }\,\text{d}\eta . \end{align}

Equation (2.4) makes two important points transparent: (i) the mapping $I(\eta )\mapsto U(y)$ is smoothing (ill-conditioned to invert without regularization), and (ii) a plateau of $I_1(\eta )$ over $\eta \lt 1$ immediately yields logarithmic dependence of $U(y)$ on $y$ .

2.1. Discrete formulation and identifiability

In practice, we discretize the eddy-size range using $\tilde {\boldsymbol{h}}=\{h_1,\ldots ,h_m\}$ (log-spaced between $h_{\textit{min}}$ and $h_{\textit{max}}$ ) and sample the wall-normal coordinate at $\tilde {\boldsymbol{y}}=\{y_1,\ldots ,y_{m_y}\}$ . The forward map for the mean can be written compactly as

(2.5) \begin{equation} \boldsymbol{U}_{\textit {data}}(\tilde {\boldsymbol y})-U_{\textit{ref},\textit{data}}\boldsymbol{1}=\boldsymbol{K}(\boldsymbol{c})\boldsymbol{b}, \end{equation}

where $\boldsymbol{c}$ contains the unknown nodal values of $I_1(\eta )$ (in a chosen basis) and $\boldsymbol{K}$ is the corresponding discretized integral operator (assembled from the weights $p(h)\,h^2$ and the ratios $\eta =y/h$ ). An analogous system holds for each component $I_{ij}$ .

The discretized velocity moments are

(2.6) \begin{align} U_{\textit {data}}(y_r; \boldsymbol{d}) &= \beta \sum _{l=1}^{m} p(h_l) h_l^2 \left [\sum _q \sum _p u_1\left (\frac {{x_p,z_q,y_r}}{h_l}; \boldsymbol{d}\right ) \left (\frac {\Delta x}{h_l} \right ) \left (\frac {\Delta z}{h_l} \right ) \right ] \Delta h + U_{\textit{ref},\textit{data}}\nonumber \\ R_{ij,\textit{data}}(y_r; \boldsymbol{d}) &= \beta \sum _{l=1}^{m} p(h_l) h_l^2 \left [\sum _q \sum _p u_i\left (\frac {{x_p,z_q,y_r}}{h_l}; \boldsymbol{d}\right ) u_{\!j}\left (\frac {{x_p,z_q,y_r}}{h_l}; \boldsymbol{d}\right ) \left (\frac {\Delta x}{h_l} \right ) \left (\frac {\Delta z}{h_l} \right ) \right ] \Delta h. \end{align}

Explicitly $K_{ij} = I_1({y}_i/h_{\!j})$ and $b_i = \beta p(h_i) h_i^2 \Delta h.$ We would now like to extract these coefficients. Towards this end, we define the eddy contribution function at selected locations, and interpolate for the values in other locations in a piecewise linear fashion. In other words

(2.7) \begin{equation} \boldsymbol{K}= \begin{bmatrix} I_1(1) & I_1(h_1/h_2) & I_1(h_1/h_3) & .. & I_1(h_1/h_{m-1}) & I_1(h_1/h_m)\\[5pt] I_1(h_2/h_1) & I_1(1) & I_1(h_2/h_3) & .. & I_1(h_2/h_{m-1}) & I_1(h_2/h_m)\\[5pt] .. & .. & .. & .. & .. & ..\\[5pt] I_1(h_m/h_1) & I_1(h_m/h_2) & I_1(h_m/h_3) & .. & I_1(h_m/h_{m-1}) & I_1(1) \end{bmatrix} \end{equation}

(2.8) \begin{equation} \triangleq \begin{bmatrix} c_1 & c_{m+1} & c_{m+2} & .. & c_{2m-2} & c_{2m-1}\\[5pt] c_2 & c_1& c_{m+1} & .. & c_{2m-3} & c_{2m-2}\\[5pt] .. & .. & .. & .. & .. & ..\\[5pt] c_m & c_{m-1} & c_{m-2} & .. & c_2 & c_1\\ \end{bmatrix}\!. \end{equation}

The unknown $I_1(\boldsymbol{\cdot })$ values are then interpolated (for instance, if $h_{\textit{min}}=186$ and $h_{\textit{max}}=5186$ and $m=11$ , then $I_1(h_2/h_3)=0.42106 c_1 + 0.57894 c_{12}$ ) from the nodal locations $\boldsymbol{c}$ . Note that there are more unknowns ( $2m$ ) than equations ( $m$ ), and so while it is possible to determine the unknowns that lead to perfect match of a reference (DNS or experiment) velocity profiles, a choice has to be made on the problem formulation. We choose to solve the following minimum-norm exact interpolation problem:

(2.9) \begin{align} \boldsymbol{c}_{\textit {opt}} = \min \boldsymbol{c}^T \boldsymbol{c} \ \ \textrm {such that} \ \ \boldsymbol{U}_{\textit {data}}(\tilde {\boldsymbol y})-U_{\textit{ref},\textit{data}} = \boldsymbol{K} \boldsymbol{b}. \end{align}

Figure 2 shows the optimal eddy contribution functions inferred from the channel flow data (Lee & Moser Reference Lee and Moser2015) at friction Reynolds number $\textit{Re}_{\tau }\approx 5200$ . Figure 3 confirms that the first and second velocity moments are perfectly reproduced across the entire channel. It is emphasized that the perfect reproduction in figure 3 is expected by construction: the system is underdetermined ( $2m$ unknowns, $m$ equations), and the minimum-norm solution exactly satisfies the constraints. The sharp features near $\eta \approx 1$ visible in figure 2 are sensitive to the discretization and the minimum-norm selection criterion; by contrast, the qualitative structure (a near-plateau for $\eta \lesssim 1$ followed by rapid decay) is robust across different discretizations, regularization strategies and data sources. It is this robust qualitative structure, rather than the fine details, that motivates the simplified model in § 2.2. The inversion uses moments across the full channel half-height ( $y^+$ from $h_{\textit{min}}$ to $h_{\textit{max}}$ ), so the inferred $I_1(\eta )$ and $I_{ij}(\eta )$ are constrained by the entire wall-normal profile, not only the log layer.

Figure 2.

Optimal influence functions for $\textit{Re}_\tau \approx 5200$ for the mean flow (a) and Reynolds stresses (b). Here red , streamwise; green spanwise; blue wall-normal and black , shear).

Figure 3.

Reference (symbols) versus optimal attached-eddy statistics for $\textit{Re}_\tau \approx 5200$ . The moments are reconstructed from the inferred $I_1$ and $I_{ij}$ via (1.5) and (1.15). The lowest $y^+$ shown corresponds approximately to $h_{\textit{min}}$ , below which the AEM has no eddies and the model is not expected to apply.

Figure 3. Long description

Panel A: A line graph shows the relationship between U+ and y+. The x-axis is labeled y+ and ranges from 0 to 5000. The y-axis is labeled U+ and ranges from 18 to 26. The data points are marked with red circles and connected by a blue line, showing a linear increase. Panel B: A line graph displays the relationship between Rij+ and y+. The x-axis is labeled y+ and ranges from 0 to 5000. The y-axis is labeled Rij+ and ranges from -1 to 6. Multiple lines, each marked with red circles, show different trends, with some lines decreasing and others increasing as y+ increases.

2.2. A simplified influence-function model and the log law

We remark that the underlying physics is more complex than attached eddies can represent. The inverse problem, however, yields a function that, while smooth, has a visually interpretable structure: a near-plateau over $\eta =y/h\lesssim 1$ followed by a transition and decay for $\eta \gt 1$ . This motivates the simple piecewise model shown in figure 4.

Figure 4.

A hypothetical model of the eddy influence function corresponding to the mean streamwise velocity (blue dashed lines) compared with the optimal influence function.

Consider a collection of attached eddies that result in a simple function as shown in figure 4. It will be shown below that it is indeed possible to construct an attached eddy that corresponds to such a function. Using this influence function, one can reconstruct the mean streamwise velocity as

(2.10) \begin{align} U(y)&= \frac {2 \beta }{1/h_{\textit{min}}^2-1/h_{\textit{max}}^2} \int _{h_{\textit{min}}}^{h_{\textit{max}}} \frac {I_1\left ({y}/{h} \right )}{h} \text{d}h + U_{\textit{ref}}\nonumber \\ &\approx \frac {2 \beta }{1/h_{\textit{min}}^2-1/h_{\textit{max}}^2} \left [\int _{y/h_{\textit{max}}}^{1} \frac {-a_0}{y/h} d(y/h) + \int _1^{a_1/a_2} \frac {a_1-a_2 y/h}{y/h} d(y/h) \right ] + U_{\textit{ref}}\nonumber \\ &= \frac {2 \beta }{1/h_{\textit{min}}^2-1/h_{\textit{max}}^2} \left [a_0 \log [y/h_{\textit{max}}]+a_1 \log [a_1/a_2]-a_1+a_2\right ] + U_{\textit{ref}}. \end{align}

It is thus clear that the von Kármán constant can be constructed as

(2.11) \begin{equation} \kappa = \frac {1/h_{\textit{min}}^2-1/h_{\textit{max}}^2}{2 a_0 \beta }. \end{equation}

Note that Woodcock & Marusic (Reference Woodcock and Marusic2015) use a Taylor series approximation on a generic eddy to derive an alternate expression for $\kappa$ . While that is insightful, it is perhaps a valid approximation only for certain types of eddy influence functions.

Equation (2.11) provides a direct route from measurable quantities ( $\kappa$ , $\textit{Re}_\tau$ ) and the eddy population density $\beta$ to the plateau value $a_0$ of the influence function, or conversely, from the inferred $a_0$ and a prescribed $\beta$ to a prediction of $\kappa$ . As will be shown in the following sections, employing a rectangular hairpin yields a precisely constant $I_1$ for $y/h \leq 1$ . In §§ 3 and 4, $\beta$ is chosen to fit the log law slope. In § 6, a variable $\beta$ model is prescribed.

3. Predictive model and insights

Now, we answer the question whether it is possible to recreate the above hypothetical influence function using an attached eddy. Consider a hairpin (figure 5) with unit circulation as the attached eddy, along with its image across the $y=0$ plane to enforce no-penetration. We consistently achieved good results using Rankine vortex rods instead of vortex filaments. All the results henceforth correspond to vortex segments represented as Rankine vortex rods with core radius $\approx 0.025$ units relative to figure 5.

It is remarked that this eddy corresponds to a closed loop in the sense that the leg of the eddy aligned with the wall is cancelled by an equal and opposite image vortex pair, and is thus consistent with the vorticity kinematics. Using the Biot–Savart law to compute the induced velocity $u_1$ in (1.6), figure 6 shows that such a simple attached eddy can reproduce the mean streamwise velocity accurately. Further, the second moments are shown in figure 6, and the Reynolds stresses trends are well-predicted, though the spanwise stresses are over-estimated in the region $y^+ \lt 2000$ .

Figure 5.

A prototypical hairpin-type eddy.

Figure 6.

Panels (a) and (b) show velocity moments: symbols, data; lines, rectangular hairpin at $\theta = 60^\circ$ . Panels (c) and (d) show eddy influence functions: solid lines, optimal eddy influence function; dashed lines, rectangular hairpin at $\theta = 60^\circ$ . For (a) and (c), colours follow figure 2. Note that the $y^+ \lt 500$ region is not shown here because in the second moments, the log scaling takes hold in the outer part of the log layer. Refer figure 19 for a full view of this phenomenon.

It is noted that an extensive optimization was performed on different eddy shapes (e.g. triangular hairpin) and packet configurations. Yet, these did not yield improved predictions, particularly encountering difficulties in matching the log law slope while providing reasonable second moment and spectral behaviour. The reasons behind this observation will be examined in detail in § 5.

The rectangular hairpin at an angle of $\theta = 60^\circ$ is close to the optimal solution across different metrics and shows comparisons with the experimental measurements from Samie et al. (Reference Samie, Marusic, Hutchins, Fu, Fan, Hultmark and Smits2018). As shown in figures 7, 8 and 9, the model remains remarkably accurate in predicting mean velocity, streamwise Reynolds stress and spectra over a range of Reynolds numbers.

We note that the influence functions were inferred from channel flow DNS at $\textit{Re}_\tau \approx 5200$ (Lee & Moser Reference Lee and Moser2015), whereas the predictions in figures 7, 8 and 9 are compared with zero-pressure-gradient boundary layer measurements at $\textit{Re}_\tau =6000$ – $20\,000$ (Samie et al. Reference Samie, Marusic, Hutchins, Fu, Fan, Hultmark and Smits2018). The fact that the same eddy template transfers successfully across flow geometries supports the universality of the log-layer structure, but this correspondence deserves further scrutiny, particularly for the Reynolds stress components and for the outer region where channel and boundary layer statistics diverge.

We also note that the model predictions align better with the data above the log layer than within it. This is not contradictory despite $p(h) \propto h^{-3}$ being questionable in the outer region: the rapid decay of $I_1(\eta )$ for $\eta \gt 1$ ensures that the outer-region statistics are dominated by the largest eddies ( $h \sim h_{\textit{max}}$ ), whose population amplitude is anchored by $\beta$ through the log-law calibration. The kernel structure naturally suppresses contributions from eddies smaller than the observation height, so the model works above the log layer primarily because of the influence-function shape rather than the population law.

Figure 7.

Mean velocity prediction using rectangular hairpin at $\theta = 60^\circ$ for $\textit{Re}_\tau = [6000, 10\,000, 14\,500, 20\,000]$ . Symbols: Samie et al. (Reference Samie, Marusic, Hutchins, Fu, Fan, Hultmark and Smits2018).

Figure 7. Long description

The image contains four scatter plots arranged in a 2x2 grid. Each plot shows the relationship between U+ on the vertical axis and y+ on the horizontal axis. The horizontal axis ranges from 200 to 7000, and the vertical axis ranges from 15 to 30. Each plot contains orange circular data points with a central cross, and a blue fitted line. The data points and fitted lines in all four plots show a similar upward trend, indicating a positive correlation between U+ and y+.

Figure 8.

Streamwise velocity fluctuation predictions using rectangular hairpin at $\theta = 60^\circ$ for $\textit{Re}_\tau = [6000, 10\,000, 14\,500, 20\,000]$ . Symbols: Samie et al. (Reference Samie, Marusic, Hutchins, Fu, Fan, Hultmark and Smits2018).

Figure 9.

Premultiplied streamwise energy spectra predictions at $y^+ = 1184$ using rectangular hairpin at $\theta = 60^\circ$ for $\textit{Re}_\tau = [6000, 10\,000, 14\,500, 20\,000]$ . Symbols: Samie et al. (Reference Samie, Marusic, Hutchins, Fu, Fan, Hultmark and Smits2018).

Figure 9. Long description

Panel A: A line graph shows the relationship between lambda_x^+ and k_x^+ phi_xx^+. The x-axis is labeled lambda_x^+ ranging from 10^0 to 10^6, and the y-axis is labeled k_x^+ phi_xx^+ ranging from 0 to 1.4. The graph includes a blue line and orange circles, showing a peak around 10^4. Panel B: Another line graph with the same axes labels and ranges as Panel A. The graph includes a blue line and orange circles, showing a similar peak around 10^4. Panel C: A line graph with the same axes labels and ranges as Panel A. The graph includes a blue line and orange circles, showing a peak around 10^4. Panel D: A line graph with the same axes labels and ranges as Panel A. The graph includes a blue line and orange circles, showing a peak around 10^4.

The inverse problem strongly constrains what any viable attached eddy must do in a wall-parallel plane. In particular, the inferred mean influence function $I_1(y/h)$ exhibits (a) a near-constant region for $y/h\lesssim 1$ and (b) rapid decay for $y/h\gtrsim 1$ , implying that eddies larger than the observation height dominate the log-region population integral. The rectangular hairpin + image realizes these features because the two legs and head generate partially cancelling far-field contributions, while the inviscid image enforces wall compatibility and suppresses the leading-order tail that would otherwise make the planar influence integrals marginal (Appendix A). By contrast, a triangular hairpin produces a more monotone ‘ramp’ in $I_1$ and does not reproduce the inferred plateau as cleanly (figure 6). Experimentation with a pack of triangular hairpins with variable circulation strengths did not improve results beyond the rectangular hairpin which appears optimal.

The inclined hairpin footprint produces elongated streamwise $u$ signatures, which is precisely the kind of wall-parallel signature that feeds the low- $k_x$ premultiplied hump. The log-layer statistics depend weakly on microscopic details once a few structural constraints are met. This eddy structure appears to have some key features that yield accurate streamwise energy spectra in the log layer. We examine this below by providing an explicit similarity map from eddy morphology to the single-eddy spectrum as a function of relative height.

4. The influence kernel

The planar influence functions $I_1(y/h)$ and $I_{ij}(y/h)$ govern one-point moments via (1.5) and (1.15). For spectra, we introduce an analogous spectral influence kernel $I_\phi (\kappa _x,\eta )$ that compactly summarizes how a single self-similar eddy distributes streamwise energy across wavenumbers as a function of relative height $\eta =y/h$ . To the author’s knowledge, making this kernel explicit is not common in the attached-eddy literature; it is useful because it cleanly separates single-eddy geometry (captured by $I_\phi$ ) from population effects (captured by $p(h)$ and $\beta$ ).

Under our geometric and circulation scaling assumptions, the hairpin footprint is self-similar,

(4.1) \begin{align} \mathcal{U}\!\left (\frac {x}{h},\frac {z}{h},\frac {y}{h}\right ) = \mathcal{U}(\xi ,\zeta ,\eta ), \end{align}

where $\mathcal{U}$ is the unit-eddy footprint expressed in dimensionless coordinates. Taking the Fourier transform and changing variables yields the exact scaling $ \widehat {u}_h(k_x,k_z;y) \triangleq \,h^2\,\widehat {\mathcal{U}}(\kappa _x,\kappa _z;\eta ),$ where $ \widehat {\mathcal{U}}(\kappa _x,\kappa _z;\eta ) \triangleq \int _{\mathbb{R}^2}\mathcal{U}(\xi ,\zeta ,\eta )\,e^{-i(\kappa _x\xi +\kappa _z\zeta )}\,\text{d}\xi \,\text{d}\zeta .$ Therefore

(4.2) \begin{align} \varPhi _{xx}^{(h)}(k_x,k_z;y) = \frac {\,h^4}{(2\pi )^2}\, \left |\widehat {\mathcal{U}}(\kappa _x,\kappa _z;\eta )\right |^2. \end{align}

Integrating over $k_z$ and using $k_z=\kappa _z/h$ gives the scalewise spectrum

(4.3) \begin{align} \phi _{xx}^{(h)}(k_x;y) \triangleq \int _{-\infty }^{\infty }\varPhi _{xx}^{(h)}(k_x,k_z;y)\,\text{d}k_z = \,h^3\,I_\phi (\kappa _x,\eta ), \end{align}

where we define the influence kernel as

(4.4) \begin{align} I_\phi (\kappa _x,\eta ) \triangleq \frac {1}{(2\pi )^2}\int _{-\infty }^{\infty } \left |\widehat {\mathcal{U}}(\kappa _x,\kappa _z;\eta )\right |^2\,\text{d}\kappa _z. \end{align}

This identity is a concrete mathematical object that will help decode the energy spectrum in greater detail. In the following subsection, we will formulate an inverse problem to extract the shape of the ideal kernel.

It is noted that the spectral influence kernel is related to the one-point Reynolds stress kernel by $ I_{11}(\eta ) = \int _0^\infty I_\phi (\kappa _x,\eta )\,\text{d}\kappa _x,$ providing a consistency check between the spectral and one-point inverse problems.

4.1. Inverse problem to extract ideal influence kernel

The inversion uses the attached-eddy spectral superposition in the form

(4.5) \begin{equation} \phi _{xx}(k_x;y) = \int _{h_{\textit{min}}}^{h_{\textit{max}}} \psi (k_x h, y/h)\,\text{d}h, \end{equation}

where

(4.6) \begin{align} \psi (\kappa ,\eta ) = \beta C I_{\phi }(\kappa _x,\eta ), \qquad \kappa _x = k_x h, \qquad \eta = y/h. \end{align}

The numerical inversion solves for a gridded approximation to $\psi$ . The kernel is represented on a tensor-product grid in $(\kappa ,\eta )$ , and the unknown vector is the row-major flattening of the $n_\eta \times n_\kappa$ kernel array,

(4.7) \begin{align} \boldsymbol{x} = \operatorname {vec}(\varPsi ) \in \mathbb{R}^{n_\eta n_\kappa }, \qquad \varPsi _{b,a} \approx \psi (\kappa _a,\eta _b), \end{align}

with the index map $ \ell (b,a) = (b-1)n_\kappa + a, \qquad 1 \le a \le n_\kappa , \quad 1 \le b \le n_\eta .$ Hence $ \boldsymbol{x} = [\varPsi _{1,1}, \varPsi _{1,2}, \ldots ,\varPsi _{1,n_\kappa },\varPsi _{2,1}, \ldots ,\varPsi _{n_\eta ,n_\kappa }]^T.$ Equation (4.5) is integrated over $h$ using $n_h$ geometrically spaced nodes $ h_1,\ldots ,h_{n_h} \in [h_{\textit{min}},h_{\textit{max}}].$ The quadrature weights for a finite difference approximation are

(4.8) \begin{align} w_1 &= h_2-h_1, \nonumber \\ w_q &= \frac 12(h_{q+1}-h_{q-1}), \qquad 2\le q\le n_h-1, \nonumber \\ w_{n_h} &= h_{n_h}-h_{n_h-1}. \end{align}

We then solve the non-negative regularized least-squares problem

(4.9) \begin{equation} \min _{\boldsymbol{x}\ge 0} \frac 12\|\boldsymbol{A}\boldsymbol{x}-\boldsymbol{b}\|_2^2 + \frac {\lambda _\kappa ^2}{2}\|\boldsymbol{L}_\kappa \boldsymbol{x}\|_2^2 + \frac {\lambda _\eta ^2}{2}\|\boldsymbol{L}_\eta \boldsymbol{x}\|_2^2 + \frac {\lambda _0^2}{2}\|\boldsymbol{L}_0 \boldsymbol{x}\|_2^2. \end{equation}

Equivalently, this is written as an augmented non-negative least-squares problem

(4.10) \begin{equation} \min _{\boldsymbol{x}\ge 0} \frac 12\|\widetilde {\boldsymbol A}\boldsymbol{x}-\widetilde {\boldsymbol{b}}\|_2^2, \end{equation}

with

(4.11) \begin{align} \widetilde {\boldsymbol A} = \begin{bmatrix} \boldsymbol{A} \\[3pt] \lambda _\kappa \boldsymbol{L}_\kappa \\[3pt] \lambda _\eta \boldsymbol{L}_\eta \\[3pt] \lambda _0 \boldsymbol{L}_0 \end{bmatrix}\!, \qquad \widetilde {\boldsymbol{b}} = \begin{bmatrix} \boldsymbol{b} \\[3pt] \boldsymbol{0} \\[3pt] \boldsymbol{0} \\[3pt] \boldsymbol{0} \end{bmatrix}\!. \end{align}

For the reported results, $ \widetilde {\boldsymbol A} \in \mathbb{R}^{28159\times 5712}, \quad \operatorname {nnz}(\widetilde {\boldsymbol A}) = 1029731$ and $ \lambda _\kappa = 10^{-1}, \quad \lambda _\eta = 4\times 10^{-1}, \quad \lambda _0 = 10^{-5}.$ While these numbers appear arbitrary, the goal is to obtain insight into the shape of the kernel, and thus the primary metric is to obtain a smooth solution that does not overfit to measurement noise.

Figure 10.

Data (symbols) and reconstructed (lines) streamwise energy spectra $\phi _{xx}^+(\kappa _x)$ for a few wall normal locations.

Appendix B provides more detail on the construction of these matrices. This is a more complicated inverse problem than the ones introduced earlier. Equation (4.10) is solved by projected FISTA (Buccini et al. Reference Buccini, Chen, Pasha and Reichel2025). Figure 10 shows the reconstructed streamwise energy spectra (from the inverse solution) and how it compares with the data. The agreement is good, and regularization is shown to smooth the solution to a small degree, but not enough to impact any conclusions.

The spectral data used for this inversion are from the experimental measurements of Samie et al. (Reference Samie, Marusic, Hutchins, Fu, Fan, Hultmark and Smits2018) at $\textit{Re}_\tau =20\,000$ . A brief sensitivity study on the regularization weights $(\lambda _\kappa ,\lambda _\eta ,\lambda _0)$ confirmed that the qualitative shape of $I_\phi$ , in particular the broad plateau at small $\eta$ and the localized hump at $\eta \sim O(1)$ is robust though finer features of the kernel near $\kappa _x\to 0$ are sensitive to the choice of $\lambda _\kappa$ .

Figure 11.

Inferred streamwise energy influence kernel $I_\phi (\kappa _x,\eta )$ (normalized peak value). This figure highlights that the kernel has a broad, elevated region at small $\eta$ and small-to-moderate $\kappa$ , and a more localized hump as $\eta$ increases. Also shown are contributions to $\phi _{xx}^+(k_x^+)$ for $k_x^+ = 1 \times 10^{-5}$ (cyan) and $k_x^+ = 1 \times 10^{-4}$ (magenta) and $k_x^+ = 1 \times 10^{-3}$ (red) at $y^+ = 1184$ for $\textit{Re}_\tau =20\,000$ .

Figure 11. Long description

A 3D surface plot depicts the inferred streamwise energy influence kernel. The horizontal axis is labeled kxh, ranging from 10^-2 to 10^2. The vertical axis is labeled Iphi, ranging from 0 to 1. The depth axis is labeled y/h, ranging from 1 to 3. The plot features three distinct data lines: a cyan line, a magenta line, and a red line. The cyan line shows a broad, elevated region at small and small-to-moderate kxh values. The magenta line indicates a more localized hump as kxh increases. The red line represents contributions to Iphi for specific conditions at y/h equals 1.

The extracted kernel $I_\phi (\kappa _x,\eta )$ , which is a function of the dimensionless wavenumber $\kappa _x=k_x h$ at fixed relative height $\eta =y/h$ and is shown in figure 11. It is immediately clear that eddies of scale $h \lt y$ have negligible influence on the statistics.

For any fixed $\eta$ , typical scaling arguments yield a concentration of the influence $I_\phi (\kappa _x,\eta )$ around $\kappa _x=O(1)$ because $\kappa _x$ is the only dimensionless streamwise wavenumber available to the unit eddy. Therefore, for a given size $h$ and fixed $y$ , the dominant contribution of that eddy appears around

(4.12) \begin{align} k_x h = O(1)\quad \Longleftrightarrow \quad k_x = O(1/h), \end{align}

equivalently $\lambda _x=2\pi /k_x = O(h)$ . This explains the peak of the premultiplied spectra in figure 9.

The rectangular hairpin AEM is able to recover a near-perfect linear behaviour of the streamwise energy spectrum in the intermediate scale range as shown in figure 12. Below, we will give more concrete and quantitative explanations.

Figure 12.

Streamwise energy spectra at $y^+ = 1184$ for $\textit{Re}_\tau =20\,000$ .

4.2. From single-eddy spectra to the attached-eddy spectrum

The attached-eddy prediction for the 1-D spectrum is the population superposition

(4.13) \begin{align} \phi _{xx}(k_x;y) = \beta \int _{h_{\textit{min}}}^{h_{\textit{max}}} p(h)\,\phi _{xx}^{(h)}(k_x;y)\,\text{d}h, \qquad p(h)=\frac {C}{h^3}. \end{align}

Insert the exact single-eddy factorization:

(4.14) \begin{align} \phi _{xx}(k_x;y) = \beta \int _{h_{\textit{min}}}^{h_{\textit{max}}} \frac {C}{h^3}\, \big (\,h^3\,I_\phi (k_x h,y/h)\big )\,\text{d}h = \beta \,C\,\int _{h_{\textit{min}}}^{h_{\textit{max}}} I_\phi (k_x h,y/h)\,\text{d}h. \end{align}

Change variables $\kappa = k_x h$ so $\text{d}h = \text{d}\kappa /k_x$ , and note that $y/h = yk_x/\kappa$ :

(4.15) \begin{align} \phi _{xx}(k_x;y) = \frac {\beta \,C\,}{k_x} \int _{k_x h_{\textit{min}}}^{k_x h_{\textit{max}}} I_\phi \!\left (\kappa ,\frac {y k_x}{\kappa }\right )\,\text{d}\kappa . \end{align}

Define the integral factor

(4.16) \begin{align} F(k_x;y) = \int _{k_x h_{\textit{min}}}^{k_x h_{\textit{max}}} I_\phi \!\left (\kappa ,\frac {y k_x}{\kappa }\right )\,\text{d}\kappa , \end{align}

so that

(4.17) \begin{align} \phi _{xx}(k_x;y)=\frac {\beta \,C\,}{k_x}\,F(k_x;y). \end{align}

The $1/k_x$ scaling in the logarithmic region arises when $F(k_x;y)$ varies slowly with $k_x$ over an intermediate band. This occurs when the $\kappa$ -window $[k_x h_{\textit{min}},k_x h_{\textit{max}}]$ covers the portion of the kernel $I_\phi (\kappa ,\eta )$ that carries most of the contribution, and when the effective $\eta$ values encountered (through $\eta = y/h = yk_x/\kappa$ ) lie in the log-layer-relevant range. In that band,

(4.18) \begin{align} F(k_x;y)\approx \text{constant}\quad \Longrightarrow \quad \phi _{xx}(k_x;y)\propto \frac {1}{k_x} \propto \lambda _x. \end{align}

This is substantiated by the results in figure 13.

Figure 13.

Contribution to $F$ integral from different scales at $y^+ = 1184$ for $\textit{Re}_\tau =20\,000$ . The smallest- $k_x$ trajectory samples the left/low- $\kappa$ edge of the kernel and remains at small $\eta =y/h$ .

The above ‘ $1/k_x$ ’ argument corresponds to the case that $F(k_x;y)$ is approximately constant over an intermediate $k_x$ range. However, the low- $k_x$ behaviour is entirely controlled by how $F(k_x;y)$ varies as $k_x\to 0$ . There is no universality guarantee there: when $k_x$ is small, the $\kappa$ -window and the $\eta$ -trajectory can force the integral to sample the least robust part of the kernel.

This is visible in the kernel visualization. Figure 11 shows the kernel surface, and figure 13 shows three trajectories corresponding to fixed $k_x^+$ values. The low- $\kappa$ region of the kernel is where $I_\phi (\kappa ,\eta )$ typically changes most rapidly. In such a regime, the integral $F(k_x;y)$ cannot be expected to be constant: small shifts in $k_x$ move both the integration bounds and the trajectory through $(\kappa ,\eta )$ , producing non-negligible changes in $F$ .

The overprediction of streamwise energy spectra by the AEM in the high wavelength limit is clearly explained by figures 14 and 15. For small $\kappa _x$ , this trajectory spends substantial ‘arclength’ in the region of small $\kappa$ and small $\eta$ and $I_\phi$ does not reduce to zero for small $\kappa _x$ as in figure 11. Nevertheless, the overall shape of the kernel and the scale-by-scale contributions from the hairpin attached eddy appear impressive in relation to the inferred quantities.

Figure 14.

Streamwise energy influence kernel $I_\phi (\kappa _x,\eta )$ (normalized peak value) for hairpin attached eddy at $\textit{Re}_\tau =20\,000$ .

Figure 15.

Contribution to $F$ integral from different scales at $y^+ = 1184$ for $\textit{Re}_\tau =20\,000$ for hairpin attached eddy.

Physically, low $k_x$ corresponds to very long streamwise wavelengths. In the present attached-eddy framework with independent marks, the low- $k_x$ variance is built by incoherently summing contributions of the largest eddies. In a Poisson superposition, long-wavelength modes are fed by many large eddies with essentially random phases, so variances add without cancellation. Real wall turbulence exhibits organization, intermittency and correlations at the largest scales; those effects can reduce energy in the very lowest modes relative to an independent-eddy superposition. Baars & Marusic (Reference Baars and Marusic2020a , Reference Baars and Marusicb ) offer more quantitative insights in the form of three different eddy types, and that attached eddies represent one of these types and contribute a large fraction of the energy only at an intermediate range of scales in the log layer. The latter point has been well argued by Perry & Marušić (Reference Perry and Marušić1995). The independent-mark hypothesis therefore tends to overpopulate the lowest wavenumbers unless correlations are explicitly modelled or packaged into larger marks. It is possible that the influence kernel $I_\phi$ can be used in an inverse problem setting to ‘design’ compositions of prototypical eddies to model desired spectral properties.

5. Why the rectangular hairpin is unusually predictive, and replacing it is difficult

The rectangular hairpin presented above is not just another convenient template that happens to fit well. Rather, it concentrates the entire spanwise displacement at one height, which gives the exact step-shaped mean kernel required by the attached-eddy mean law, while its vertical legs remain spectrally broad and dominate the 1-D streamwise spectrum. In other words, the rectangular hairpin is unusually predictive because it realizes an unusually clean separation of duties. The top segment concentrates the entire spanwise displacement at one height and therefore gives the exact $I_1$ plateau. The legs carry essentially none of $I_1$ , but because they are strongly localized they dominate most of $I_\phi$ . The angle $\theta$ then tunes the spectral peak largely through phase, without spoiling the mean. Replacing the rectangular hairpin is hard because most alternatives appear to spoil at least one of these properties, and many spoil two of them at once. This was a conclusion reached after a number of optimization attempts on configurations such as those shown in figure 16. The optimization set-up and code can be found at https://github.com/CaslabUM/AttachedEddy.

Figure 16.

Set-up of a sample optimization study conducted to find an optimal eddy shape. Design variables $d_i$ are coordinate locations which represent the eddy shape. No combination of design variables was found to result in a better prediction of the mean velocity and streamwise variance than the simple rectangular hairpin.

The author is careful to note that the above statement is not a proof, but a qualitative result. A more rigorous statement can be made in the case of a symmetric three-segment hairpin family whose edge nodes are given by

(5.1) \begin{align} (0,0,-b_0/2) \rightarrow (\lambda ,1,-b_1/2) \rightarrow (\lambda ,1,b_1/2) \rightarrow (0,0,b_0/2). \end{align}

Figure 17 shows the geometry in the $(y,z)$ projection.

Figure 17.

Straight-segment hairpin family and the corresponding exact $I_1(\eta )$ kernels.

Appendix C gives individual segment contributions.

Mean contribution. Summing the three segment contributions from (C7) gives the exact single-hairpin mean kernel

(5.2) \begin{equation} I_1^{\mathit{hp}}(\eta ) = \varGamma \,[b_0+(b_1-b_0)\eta ]H(1-\eta ). \end{equation}

This reduces to a plateau when $b_1=b_0$ and to a ramp when $b_1\ne b_0$ . For the attached-eddy population, the mean profile becomes

(5.3) \begin{equation} U^+(y^+) = U_{\mathit{ref}}^+ - \beta C\kern-1pt\varGamma\kern-1pt F_1(y^+), \end{equation}

with the exact population integral

(5.4) \begin{equation} F_1(y^+) =b_0\log \!\left (\dfrac {h_{\textit{max}}^+}{y^+}\right ) +(b_1-b_0)\left (1-\dfrac {y^+}{h_{\textit{max}}^+}\right ), \ \ h_{\textit{min}}^+\le y^+\le h_{\textit{max}}^+. \end{equation}

Thus, the rectangular hairpin $b_1=b_0=s$ collapses this to the exact logarithmic form

(5.5) \begin{align} F_1(y^+) = s\log (h_{\textit{max}}^+/y^+) \qquad (h_{\textit{min}}^+\le y^+\le h_{\textit{max}}^+). \end{align}

This is the main reason the rectangular head remains so predictive: it gives the best possible morphology for the logarithmic mean.

Streamwise kinetic energy. For the rectangular hairpin with $0\lt \eta \lt 1$ , Appendix C provides exact expressions for the Fourier transform of the streamwise induced velocities. It is instructive to split the contributions from the head of the hairpin and the legs as $\hat {u}_1 \triangleq \hat {u}_h + \hat {u}_\ell$ . Then the spectral kernel can be reduced to an exact quadrature in $\kappa _z$ :

(5.6) \begin{align} I_\phi ^{hh}(\kappa _x,\eta ) &= \frac {1}{2\pi ^2} \int _0^\infty \left | \widehat {u}_h(\kappa _x,\kappa _z;\eta ) \right |^2 \, \text{d}\kappa _z, \\[-12pt]\nonumber \end{align}

(5.7) \begin{align} I_\phi ^{\ell \ell }(\kappa _x,\eta ) &= \frac {1}{2\pi ^2} \int _0^\infty \left | \widehat {u}_{\ell }(\kappa _x,\kappa _z;\eta ) \right |^2 \, \text{d}\kappa _z, \\[-12pt]\nonumber \end{align}

(5.8) \begin{align} I_\phi ^{h\ell }(\kappa _x,\eta ) &= \frac {1}{\pi ^2} \textit{Re} \int _0^\infty \widehat {u}_h(\kappa _x,\kappa _z;\eta )\, \widehat {u}_{\ell }(\kappa _x,\kappa _z;\eta )^* \, \text{d}\kappa _z. \end{align}

For the representative rectangular hairpin at $\theta =60^\circ$ , and $\eta =0.5$ , these exact formulae give at $\kappa _x = 0$

(5.9) \begin{align} I_\phi ^{hh}(0,0.5)=2.74\times 10^{-2}, \quad I_\phi ^{\ell \ell }(0,0.5)=8.36\times 10^{-2}, \quad I_\phi ^{h\ell }(0,0.5)=4.82\times 10^{-2}. \end{align}

This reveals two things: (a) the fact that the kernels (and thus $\phi _{xx}$ ) do not decay to zero at zero streamwise wavenumber; and (b) even at $\kappa _x=0$ , the head contributes only approximately $17\,\%$ of $I_\phi (0)$ , the legs contribute approximately $53\,\%$ , and the cross term contributes approximately $30\,\%$ , even though the head contributes all of $I_1$ . At the premultiplied peak ( $\kappa _x \approx 1.55$ ) the split is approximately $8.5\,\%$ , $71.4\,\%$ and $20.1\,\%$ , and by $\kappa _x=5$ the legs carry approximately $98.1\,\%$ of the spectrum.

Figure 18.

(a) Exact square-hairpin decomposition of the single-eddy spectrum into head, leg, and cross contributions at $\eta =0.5$ . (b) Premultiplied spectrum for the same square geometry at several inclination angles. The mean kernel is fixed by the top segment, while the angle mainly tunes the spectral peak through the leg phase $e^{-i\kappa _x s \cot \theta }$ .

This is the sharpest quantitative statement of the mean-variance duality: the head contributes significantly to the mean kernel, but the legs determine most of the spectral energy. As mentioned earlier, the difficulty of replacing the rectangular head is precise rather than qualitative. Any replacement that spreads the top segment over a vertical band immediately destroys the exact plateau in $I_1$ . Triangles, rounded tops, tilted tops and arch-like heads all behave the same way.

The same replacement also changes the head contribution to $I_\phi$ . Experiments show that downward spreading of the head not only spoils the mean plateau; it also increases the moderate- $\kappa _x$ head content because part of the head is brought closer to the observation plane. This is exactly the wrong direction if one wants to retain the clean head/leg role separation of the rectangular hairpin.

One might try to keep the top segment at a single height so that $I_1$ remains exact, and instead replace only the legs. This is a different failure mode. If the legs are broadened, bowed, packetized or rounded in a way that gives them an effective spanwise width $w_\ell$ , then the mean kernel is still unchanged to leading order because the leg contribution is small, but the spectrum is no longer the same. Even when the mean is preserved exactly, broadening the legs narrows the Cauchy spectrum and shifts the premultiplied peak to lower $\kappa _x$ . In other words, keeping the right $I_1$ is not enough: the rectangular hairpin also succeeds because its legs are highly localized and therefore spectrally broad. That is the main reason the rectangular hairpin is unusually predictive: it is one of the very few simple filament templates in which the mean and spectral constraints are nearly decoupled.

Figure 19.

(a,b) Joint optimal solution for eddy shape and density at $\textit{Re}_\tau =20\,000$ . (c) Inferred eddy density functions for $\textit{Re}_\tau = [6000, 10\,000, 14\,500, 20\,000]$ and piecewise linear approximation.

6. Inferring and modelling the eddy density

Once the eddy shape is chosen and $\beta$ is fixed, there are no degrees of freedom left. Yet, the comparisons in the previous section suggest that a minimal hairpin-type eddy can reproduce not only the one-point moments at $\textit{Re}_\tau \approx 5200$ but also the evolution of $U^+$ , $u^{\prime 2}$ and the premultiplied spectrum across Reynolds numbers. This section provides a more substantial interpretation of why this occurs, which ingredients appear essential, and which limitations are inherent to an independent-eddy superposition.

It is useful to separate (i) the population law $(\beta ,p(h),h_{\textit{min}},h_{\textit{max}})$ from (ii) the single-eddy template (geometry, inclination angle $\theta$ , circulation scaling and vortex-core regularization). Under these conventions, the only parameter used to anchor the data is the population amplitude $\beta$ .

It is, however, possible to improve the accuracy of the model by making the eddy density a function of the eddy size. A joint optimization run was performed for the eddy width, angle and $\beta (h^+)$ . The optimization for $\textit{Re}_\tau =20\,000$ converged to $\theta =75^\circ$ and an aspect ratio (ratio of hairpin width and height) of 1.8, and an eddy density as shown in figure 19, which imply that it is possible to get a near-perfect prediction. Keeping the eddy shape and angle fixed, and examining the behaviour of the eddy density function, the following piecewise linear fit was found to be accurate:

(6.1) \begin{equation} C \beta (\bar {h}) = \begin{cases} 1.25+3.5 \bar {h}& 0\le \bar {h} \le 0.5,\\[1.2em] 3.0-2.5 (\bar {h}-0.5)/0.5 &0.5 \le \bar {h} \le 1, \end{cases} \end{equation}

where $\bar {h} = (({h^+ - h_{\textit{min}}^+})/({h_{\textit{max}}^+ - h_{\textit{min}}^+}))$ .

Physically, the inferred $C\beta (\bar {h})$ peaks at midrange eddy sizes ( $\bar {h}\approx 0.5$ ) and decreases towards both the smallest and largest eddies. This is qualitatively consistent with the expectation that the largest eddies are geometrically constrained by the boundary layer thickness and may be less numerous than a pure $p(h)\propto h^{-3}$ law would predict, while the smallest eddies in the log layer interact with the buffer region where the attached-eddy hypothesis breaks down. However, we note that the piecewise linear form was chosen for simplicity and not derived from physical consistency requirements.

We also note that the jointly optimized angle ( $\theta =75^\circ$ ) and aspect ratio (1.8) differ from the values used in § 3 ( $\theta =60^\circ$ , aspect ratio 1); this suggests that when $\beta (h)$ absorbs additional modelling flexibility, the optimal eddy shape shifts accordingly and the two parameterizations should not be directly compared.

Using this analytical eddy density function, figure 20 shows predictions at 4 different Reynolds numbers.

Figure 20.

Improved predictions using analytical eddy density function $C \beta (h^+)$ for $\textit{Re}_\tau = [6000, 10\,000, 14\,500, 20\,000]$ .

7. Related approaches and connections

The present work occupies a complementary position within the broader landscape of attached-eddy theories and coherent-structure studies. An important parallel development is the hierarchical random additive process (HRAP) model introduced by Yang et al. (Reference Yang, Marusic and Meneveau2016) which provides an alternative statistical reinterpretation of Townsend’s attached-eddy hypothesis. This formulation elegantly recovers logarithmic scaling laws for moments, structure functions, and generalized two-point correlations without requiring explicit specification of eddy geometry. We note that the HRAP framework is a higher level abstraction than the present work, as it encodes only the statistical consequence (additive contributions from a hierarchy of scales) without committing to any particular morphology.

Davidson & Krogstad (Reference Davidson and Krogstad2009) propose a deliberately simple log-layer model for streamwise fluctuations which reproduces the $k_x^{-1}$ behaviour of the 1-D longitudinal spectrum in the log region and provides reasonable predictions for the variance $u'u'$ (up to an additive constant). The existence and criteria for a $k_x^{-1}$ spectral dependence have been investigated extensively, more recently by Nickels et al. (Reference Nickels, Marusic, Hafez and Chong2005). Notably, the $k_x^{-1}$ trend is most apparent very close to the surface, whereas the present comparisons are shown at an intermediate wall-normal distance. The Davidson–Krogstad viewpoint supports the observation that once the mean and a small set of scaling assumptions are fixed, the streamwise spectrum in the log region has limited freedom.

The above works reinforce a key message: different formulations can reproduce expected log-layer statistics. The present work attempts to provide a concrete mechanistic underpinning within a specific formulation of the attached-eddy hypothesis, and thus the conclusions of this study are limited purely to the specific formulation at hand.

The inverse-problem analysis (§ 2) demonstrates that the inferred influence functions $I_1(\eta )$ and $I_{ij}(\eta )$ represent the essential kinematic constraints that any viable eddy must satisfy. As shown in § 5, the rectangular hairpin yields a precise plateau and functions as a minimal template that satisfies these influence-function requirements while remaining consistent with Biot–Savart kinematics and wall boundary conditions.

It is important to distinguish the present statistical/kinematic AEM from the growing body of work that seeks to identify wall-attached eddies as instantaneous, material features in DNS or experimental data. Beginning with del Álamo et al. (Reference del Álamo, Jiménez, Zandonade and Moser2006) who identified self-similar vortex clusters in the logarithmic region of turbulent channels, a series of studies have extracted three-dimensional (3-D) coherent structures using various detection criteria. Dennis & Nickels (Reference Dennis and Nickels2011) provided detailed experimental measurements of vortex packets in turbulent boundary layers via tomographic particle image velocimetry. Lozano-Durán et al. (Reference Lozano-Durán, Flores and Jiménez2012) characterized the 3-D structure of momentum-transferring Q2/Q4 events, demonstrating that attached structures become geometrically self-similar with sizes proportional to their distance from the wall. More recently, Hwang & Sung (Reference Hwang and Sung2018) developed a clustering methodology to extract wall-attached structures of velocity fluctuations, and Cheng et al. (Reference Cheng, Li, Lozano-Durán and Liu2019, Reference Cheng, Li, Lozano-Durán and Liu2020) employed bidimensional empirical mode decomposition to isolate attached eddies and quantify their contribution to skin-friction generation. Additionally, several studies have demonstrated that statistical properties can be recovered from stochastic representations of eddies with wall-distance scaling (Cuevas Bautista et al. Reference Bautista, Carlos, Ebadi, White, Chini and Klewicki2019; Ehsani et al. Reference Ehsani, Heisel, Li, Voller, Hong and Guala2024; Nath & Hickey Reference Nath and Hickey2026), and the clustering and larger-scale organization of attached eddies has been investigated by Hutchins et al. (Reference Hutchins, Chauhan, Marusic, Monty and Klewicki2012) and Deshpande et al. (Reference Deshpande, van den Bogaard, Vinuesa, Lindić and Marusic2023), among others.

These data-driven approaches validate the physical premise of wall-attached, self-similar hierarchies, providing essential structures that models such as the AEM seek to represent statistically. However, they also reveal complexity that the minimal AEM intentionally abstracts away: instantaneous eddies exhibit significant variability in shape, orientation and internal structure; they interact nonlinearly; and their dynamics involve processes (generation, merging, cascade) not captured by easily interpretable models. The present work thus occupies a middle ground: it provides more geometric specificity than purely statistical models (HRAP) while remaining deliberately simpler than the full complexity revealed by structure-extraction studies. The spectral influence kernel $I_\phi (\kappa _x, \eta )$ introduced in § 4.1 offers a bridge between these perspectives and encodes the spectral signature of a single eddy template and can, in principle, be compared with conditionally averaged spectra from identified structures. Such comparisons, along with extensions to two-dimensional (2-D) $(k_x, k_z)$ spectra and cross-component correlations, represent promising directions for strengthening the connection between kinematic models and data-driven structure identification.

We also note a connection to the resolvent framework (McKeon Reference McKeon2019): the self-similar hierarchy of response modes identified by resolvent analysis shares the same wall-attached scaling as the AEM, and the resolvent modes can be viewed as a dynamically informed basis for the eddies that the AEM treats kinematically. Integrating the spectral influence kernel $I_\phi$ with resolvent-based predictions of the 2-D spectral tensor represents a promising path towards a dynamically grounded attached-eddy model.