Critical Roughness Regime

• Critical roughness regime is the transitional zone where surface features and system-specific lengths interact to cause qualitative shifts in transport, mechanical, or scaling properties.
• It is characterized by non-monotonic responses, such as peak drag or heat flux at critical values like Λcr ≈ 0.15–0.18 and k+ ≈ 5–10 in turbulent and convective systems.
• Understanding this regime involves direct numerical simulations, asymptotic homogenization, and fractal analysis to derive effective scaling laws and predict system behavior.

A critical roughness regime delineates the transitional zone in which physical systems—ranging from turbulent flows and thermal transport to fracture mechanics and soft-matter contacts—exhibit a qualitative shift in response due to the geometric scale or density of surface inhomogeneities. This regime is characterized by the crossover between regimes that are governed by distinct mechanisms, often revealed through a non-monotonic response in transport, mechanical, or scaling properties as a function of a roughness parameter, such as height, wavelength, density, or spectral content. The critical regime is distinguished by the emergence of enhanced or suppressed dissipation, anomalous scaling (often saturating theoretical upper bounds), or sharp transitions in structural organization, which are not evident in either the smooth (subcritical) or fully rough (supercritical) limits.

1. Defining Parameters and Phenomenology

Critical roughness arises from the competition between a roughness length scale (height $k$, amplitude $h$, wavelength $\lambda$, or density $\Lambda$) and a system-specific characteristic length: viscous sublayer thickness in turbulence, boundary-layer thickness in convection, elastic screening length in mechanics, or microstructural process zones in fracture. In turbulent wall flows, the roughness solidity $\Lambda$ or roughness Reynolds number $k^+$ controls the transition from hydraulically smooth ($\Delta U^+ \approx 0$) to fully rough (logarithmic velocity deficit) regimes. In thermal convection, the competition between roughness height and thermal/viscous boundary layer thickness generates enhanced transport regimes. In stochastic/random or fractal systems, critical roughness corresponds to the threshold at which self-affinity coincides with marginal differentiability or nontrivial measure-theoretic regularity.

Table: Key roughness parameters and respective critical values (examples):

System/Setting	Roughness Quantifier	Critical Value/Condition
Turbulent channel flow	Solidity $\Lambda$	$\Lambda_{cr} \approx 0.15$–0.18
Pipe/Channel turbulence	$k^+ = k u_\tau/\nu$	$k^+_{cr} \sim 5$–$10$
Rayleigh–Bénard convection	$h/\delta_{BL}$	$h \sim \delta_{th,v}$
Fracture mechanics	Cutoff $\xi$ in $C(\Delta x)$	$\xi \propto$ process zone size
Soft-matter contact (adhesion)	RMS roughness $\sigma$	$\sigma_c \sim (\tfrac{w}{E^*})^{1/2}$
Stochastic/fractal roughness	Series decay $\alpha$	$|\alpha| = 1/b$ (critical)

2. Wall-bounded Turbulence and the Hama Function

Direct numerical simulations of turbulent channel flow over transitionally rough surfaces with varying frontal solidity $\Lambda$ reveal a non-monotonic dependence of the Hama roughness function, $\Delta U^+$, on solidity. In the sparse regime ($\Lambda \lesssim 0.15$), $\Delta U^+$ rises monotonically with $\Lambda$, signifying increasing drag contribution from individual roughness elements. At the critical solidity $\Lambda_{cr} \approx 0.15$–0.18, $\Delta U^+$ attains its maximum (e.g., $\Delta U^+_{max} \approx 4.1$) (MacDonald et al., 2020). Beyond this, in the dense regime, enhanced sheltering between roughness elements suppresses Reynolds shear stress, causing $\Delta U^+$ to decrease. Turbulence statistics concurrently show maximal suppression and subsequent re-energization of the near-wall streaks across $\Lambda_{cr}$, with the wall-normal location of peak streamwise intensity shifting outward. For $\Lambda \to \infty$, the flow approaches that over a smooth wall located at the roughness crests, and $\Delta U^+$ saturates at an asymptotic bound.

3. Convective Heat Transport: Crossovers and Ultimate Regime

In turbulent Rayleigh–Bénard convection, critical roughness demarcates geometrically-induced crossovers in the Nusselt scaling exponent $\beta$ of $\mathrm{Nu} \sim \mathrm{Ra}^\beta$. For sinusoidal or pyramidal roughness, when the feature height $h$ exceeds the smooth-plate thermal (or viscous) boundary layer thickness, an “enhanced-exponent” (sometimes locally up to $\beta \approx 1/2$) regime emerges, reflecting bulk-dominated heat transport (Zhu et al., 2017, Toppaladoddi et al., 2017, Xie et al., 2017). This regime is universal with respect to $(h, \lambda)$ once the aspect ratio matches local BL thickness. However, at higher Rayleigh numbers, the system reverts to a boundary-layer controlled regime ($\beta \approx 1/3$). The critical regime does not correspond to the true Kraichnan–Grossmann–Lohse "ultimate regime" of turbulent convection, but rather to a finite-$\Ra$ bulk–BL crossover set by geometry. Scaling transitions are precisely triggered when $h \sim \delta_{th}^{\rm smooth}$ and, at even larger $\Ra$, when $h \sim \delta_v$.

4. Critical Regimes in Mechanics and Flow in Complex Geometries

Fracture mechanics simulations define the critical roughness regime as the self-affine regime (up to a cutoff $\xi$) with a universal exponent $\beta \simeq 0.54$; the cutoff is determined by the process zone or mean microstructural spacing and scales linearly with fracture toughness $J_{1C}$ (Ponson et al., 2013). In heterogeneous ductile fracture, this links macroscopic toughness to fractographic measurements.

In laminar and turbulent flows through wavy pipes or complex geometries, a critical roughness amplitude triggers the onset of steady recirculation zones at much lower Reynolds numbers ($Re_c \sim 25$ for $k/R_{max} = 0.4$) than in smooth systems. The threshold is set by the balance of adverse-pressure gradient and viscous diffusion (Mellas et al., 21 Nov 2025).

5. Transitional Rheology and Flow in Particulate Media

In dense suspensions of frictional, nearly hard spheres, increasing surface roughness (e.g., increasing $R_q/d$ from $7 \times 10^{-4}$ to $4 \times 10^{-3}$) sharply reduces the jamming packing fraction $\phi_c$, but leaves the quasi-static friction coefficient $\mu_c$ unaltered (Tapia et al., 2019). The critical roughness regime is that in which solid-solid contacts dominate the rheology, demarcated by a shift in the master curves for stress–volume fraction relationships, but with rheological collapse maintained after rescaling by the roughness-dependent $\phi_c$.

6. Fractals, Random Interfaces, and Maximum Dissipation

Mathematical analysis of canonical fractal functions (e.g., Weierstraß, Takagi-van der Waerden) identifies a critical series parameter ($|\alpha| = 1/b$) where all $p$-variation norms with $p>1$ vanish, yet functions remain nowhere differentiable. This “critical roughness regime” is revealed only via gauge $\Phi$-variation, yielding a finite, nonzero, and linear Wiener–Young variation, encapsulating the marginal maximality of oscillatory behavior (Han et al., 2020).

In disordered elastic interfaces (e.g., domain walls), temperature-induced crossovers are marked by a "modified Larkin regime" for $T<T_c$, bounded by a thermal crossover at low scale and a random-manifold regime at large scale. For $T>T_c$, the intermediate regime collapses, resulting in a direct crossover (Agoritsas et al., 2010).

7. Critical Roughness in Contact, Adhesion, and Soft-Matter Systems

In adhesive elastic contacts, a critical RMS roughness $\sigma_c$ maximizes hysteretic energy loss per load–unload cycle. This criticality is calculable from surface energy and spectral shape parameters, following a bell-shaped dependence of dissipation on $\sigma$ (Deng et al., 2018). For liquid foams in contact with rough substrates, frictional regimes (slip, stick–slip, full anchoring) are sharply delineated by the roughness factor $\chi = a/r_{pb}$ (asperity size over Plateau border radius), with critical thresholds $\chi_1 \approx 0.2$ and $\chi_2 \approx 1.9$ (Marchand et al., 2020).

8. Modeling, Scaling, and Universality

Across domains, the critical roughness regime is captured by a variety of analytical and numerical tools, including spectral co-spectral models (Li et al., 2021), periodic unfolding and asymptotic homogenization in multiscale PDEs for thin films and porous flows (Anguiano et al., 21 Dec 2025, Suárez-Grau, 2019), and two-scale expansions in non-isothermal and micropolar lubrication. The commonality is the emergence of new effective equations, scaling regimes, and, often, a sharply defined roughness parameter or threshold at which the dominant physics transitions from one macroscopic behavior to another.

The critical regime thus serves as a locus for maximized or highly nontrivial system responses: peak drag or heat flux, maximal energy dissipation, or onset of new scaling laws and structural reorganizations. Designing or operating systems at or near this regime requires careful characterization of both the geometrical and physical microstructure to accurately predict macroscopic functionality.