Reynolds Roughness Regime

• Reynolds roughness regime is defined by the critical scaling where surface roughness and film thickness are comparable, requiring explicit multiscale analysis.
• It significantly impacts drag, heat transfer, and momentum exchange in both lubrication and turbulent wall flows through modified effective equations.
• The regime employs two-scale expansion and homogenization techniques to capture the coupling between microscale geometry and macroscale flow dynamics.

The Reynolds roughness regime refers broadly to flow behaviors and effective macroscopic equations that arise when microscale or mesoscale roughness features on a boundary interact strongly with the mean flow, such that the typical roughness wavelength is commensurate with characteristic flow or domain scales (e.g., film thickness in lubrication, or the viscous sublayer in shear flows). The concept has distinct but closely related technical meanings in hydrodynamic lubrication theory, thin-film modeling, and turbulent wall-bounded flows. In each context, the Reynolds roughness regime is centered around a specific scaling limit where the impact of surface roughness is neither negligible (as in hydraulically smooth or classical lubrication limits) nor averages out completely, but must be resolved explicitly using multiscale or homogenization approaches. This regime is associated with intricate coupling between microscopic geometry and macroscopic transport, resulting in modified effective equations and transport coefficients.

1. Scaling, Structural Definition, and Regime Classification

In thin-film flows, the Reynolds roughness regime arises when the roughness period $\varepsilon^\alpha$ and the small parameter $\varepsilon$ (film thickness or gap width) scale together such that $\alpha = \alpha_{cr} = 1$, i.e., both the roughness wavelength and domain height are of the same order as $\varepsilon$. The physical domain can be written as $$\Omega_\varepsilon = \{ (x', x_3) : x' \in \omega \subset \mathbb{R}^2,\, 0 < x_3 < h_\varepsilon(x') \}$$, with $$h_\varepsilon(x') = \varepsilon h(x'/\varepsilon^\alpha)$$. The critical scaling $\alpha = 1$ defines the Reynolds roughness regime (Anguiano et al., 21 Dec 2025).

In this regime, traditional asymptotic reductions (lubrication or Stokes roughness) no longer suffice; rather, the flow and transport must be analyzed using two-scale expansions, introducing both slow (macroscale) and fast (microscale) spatial variables. The local cell flow and the global flow are coupled such that both the momentum and energy equations retain dependence on the roughness geometry through explicit cell problems.

By contrast, when the film is much thinner than the roughness period ($\frac{\eta_\varepsilon}{\varepsilon} \rightarrow 0$, very-thin regime), the problem reduces to a classical Reynolds equation with modified permeability but no strong microscale-macroscale coupling. When the roughness period is much smaller than the film thickness ($$\frac{\eta_\varepsilon}{\varepsilon} \rightarrow \infty$$), roughness effects average out, yielding standard lubrication behavior with minimal effective gap (Suárez-Grau, 2019, Anguiano et al., 21 Dec 2025).

Regime	Scaling	Domain Structure
Very-thin-film (Stokes rough)	$\eta_\varepsilon \ll \varepsilon$	2D cell (in-plane)
Reynolds roughness	$\eta_\varepsilon \sim \varepsilon$	3D cell (full)
High-frequency/lubrication	$\eta_\varepsilon \gg \varepsilon$	1D subfilm

In turbulent wall-bounded flows, the Reynolds roughness regime is commonly identified using the roughness Reynolds number, $k^+ = k U_\tau/\nu$ (where $k$ is the roughness height, $U_\tau$ is the friction velocity, $\nu$ is the kinematic viscosity). Three regimes are delineated:

• Hydraulically smooth: $k^+ \lesssim 5$ (or $k_s^+ \lesssim 5$, for equivalent sand grain roughness), negligible roughness effect.
• Transitionally rough: $5 \lesssim k^+ \lesssim 70$–$100$; roughness impact is nonlinear and geometry-dependent.
• Fully rough: $k^+ \gtrsim 70$–$100$; drag and transport saturate, dominated by form drag (Forooghi et al., 2018, MacDonald et al., 2018, Nilsson-Takeuchi et al., 25 May 2025, MacDonald et al., 2020).

2. Asymptotic Reduction and Effective Equations in the Reynolds Roughness Regime

In the Reynolds roughness regime for thin films and porous media, the governing transport equations (Darcy-Brinkman, Stokes, or power-law Navier-Stokes) undergo a formal two-scale expansion:

• Introduce fast microstructure variables $y' = x'/\varepsilon$, stretched normal variable $z_3 = x_3/\varepsilon$.
• Expand the flow, pressure, and temperature fields as $$u_\varepsilon(x',z_3) \approx u^0(x',y',z_3) + \varepsilon u^1(x',y',z_3)$$, $$p_\varepsilon(x',z_3) = \varepsilon^{-2} p^0(x') + \varepsilon^{-1} p^1(x',y')$$, $T_\varepsilon(x',z_3) \approx T^0(x',y',z_3)$ (Anguiano et al., 21 Dec 2025).

The cell problem is posed on $Y := Z' \times (0, h(y'))$, $Z' = [-1/2, 1/2]^2$. For a Newtonian fluid:

$-\mu \Delta_y w^i + \nabla_y \pi^i = e_i, \qquad \text{div}_y w^i = 0, \qquad w^i = 0 \text{ at } z_3 = 0, h(y'), \text{ $y'$-periodic}$

Where $w^i$ is the cell-scale velocity in direction $e_i$. The effective macroscopic equation is then:

$$- \nabla_{x'} \cdot \left( \mathbb{A}_{\mathrm{eff}}(h) \, (\nabla_{x'} p^0 - f') \right) = 0$$

with the permeability tensor

$$(\mathbb{A}_{\mathrm{eff}})_{ij}(x') = \int_{Z'} \int_{0}^{h(y')} w^i_j(y',z_3) \, dz_3 \, dy'$$

This tensor encodes the non-local, geometry-dependent permeability imposed by roughness whose scale matches the film thickness (Anguiano et al., 21 Dec 2025).

In the non-Newtonian (power-law) case, the effective equation is a nonlinear Reynolds law, with the local cell problem and the averaged conductivity tensor incorporating power-law exponents ($p$), yielding modified flow indices and fully nonlinear Darcy-type equations (Anguiano et al., 2017).

Micropolar and other complex fluids yield parallel developments, with coupling between microrotation and macroscopic flow parameters in the critical (Reynolds roughness) regime (Suárez-Grau, 2019).

3. Momentum and Heat Transfer Regimes in Turbulent Flows

In turbulent wall-bounded flows, the Reynolds roughness regime is primarily classified in terms of changes to the inner-layer structure, expressed as a log-law shift in the mean streamwise velocity profile:

$$U^+(y^+) = \frac{1}{\kappa} \ln y^+ + B - \Delta U^+(k_s^+)$$

where $\Delta U^+(k_s^+)$ is the Hama or roughness function, $k_s^+$ is the inner-normalized equivalent sand-grain roughness, and $(\kappa,B)$ are the von Kármán constants ($\kappa \approx 0.384$, $B \approx 4.27$).

• Hydraulically smooth: $\Delta U^+ \approx 0$, $C_f$ and $St$ (Stanton number) match smooth-wall fits (Forooghi et al., 2018, MacDonald et al., 2018, Nilsson-Takeuchi et al., 25 May 2025, MacDonald et al., 2020).
• Transitionally rough: $\Delta U^+$ rises smoothly with $k^+$ (surface-specific), $C_f$ and $St$ increase above smooth-wall, the Reynolds analogy factor $RA=2\,St/C_f$ decreases but does not reach plateau (Forooghi et al., 2018, MacDonald et al., 2018, MacDonald et al., 2020).
• Fully rough: $\Delta U^+$ becomes $k^+$-independent, matching log law:

$$\Delta U^+_{\rm FR}(k_s^+) = \frac{1}{\kappa} \ln k_s^+ + B - B_s(\infty)$$

with $B_s(\infty)\approx 8.5$ (Nikuradse), or, numerically, $\Delta U^+_{\rm FR} \simeq 2.60 \ln k_s^+ - 4.23$ (Nilsson-Takeuchi et al., 25 May 2025).

These regimes can have different thresholds depending on roughness morphology, but typically occur at $k^+\sim50$ (smooth/transitional) and $k^+\sim70$–$100$ (transitional/fully rough) (Forooghi et al., 2018, MacDonald et al., 2018, MacDonald et al., 2020, Nilsson-Takeuchi et al., 25 May 2025).

4. Influence of Roughness Morphology and Geometry

The dependence on roughness geometry is prominent in the transitional and Reynolds roughness regimes. For turbulent channel flows with "ridge-type" roughness, the roughness function and the onset of the fully rough regime differ notably from canonical homogeneous roughness:

• Transitionally rough regime is extended to much higher $k^+$ than for sand-grain or grit roughness.
• The fully rough asymptote is reached at unusually low $\Delta U^+$ and low $k_s^+$ ($\Delta U^+ \approx 3$ at $k^+ \approx 300$ for certain ridge-type surfaces) due to enhanced viscous effects and secondary motions (Nilsson-Takeuchi et al., 25 May 2025).
• Much of the drag arises from viscous shear on side faces and ridge-induced secondary flows, not from form drag.
• Collapse onto the Nikuradse law for $\Delta U^+(k_s^+)$ requires rescaling with a geometry-dependent factor $C$.

In three-dimensional sinusoidal and random roughness, changes in slope or density at fixed roughness height produce a sharp transition from rough to smooth regimes. Sparse packing and steep geometry approach the fully rough plateau ($RA/RA_0 \rightarrow 0.55$), whereas extreme sparsity or density can restore smooth-wall behavior, as mutual sheltering or insufficient interaction limits roughness impact (Forooghi et al., 2018, MacDonald et al., 2020).

5. Macroscopic Manifestations: Drag, Heat Transfer, and Reynolds Analogy

The Reynolds roughness regime affects both drag and scalar transport (heat, mass):

• Skin-friction coefficient $C_f$ becomes geometry-pinned and independent of Reynolds number in the fully rough regime, with $\Delta U^+$ following the log law asymptote (Forooghi et al., 2018, MacDonald et al., 2018, Nilsson-Takeuchi et al., 25 May 2025).
• The Stanton number $St$ also attains an asymptotic limit, after peaking in the transitional regime.
• The Reynolds analogy factor $RA = 2\,St/C_f$ drops from $RA_0\approx0.86$ (Prandtl number 0.71) to $RA_\infty \approx 0.47$, or more generally $RA/RA_0 \rightarrow 0.55$ in the fully rough regime, indicating a reduction of turbulent heat-to-momentum transfer ratio because of form drag and saturation of near-wall mixing (Forooghi et al., 2018, MacDonald et al., 2018).

The empirical fit

$\frac{RA}{RA_0} = 0.55 + 0.45 \exp(-k_s^+/130)$

describes the evolution from smooth to rough-wall turbulent heat transfer over a large range of surface morphologies (Forooghi et al., 2018).

6. Generalization to Non-Newtonian, Micropolar, and Porous Flows

The Reynolds roughness regime extends to a broader class of fluids and transport processes:

• Non-Newtonian (power-law) fluids: the regime persists, with the effective macroscopic equations taking the form of nonlinear Reynolds-type laws, in which the local roughness-induced cell problem determines a generalized, often nonlinear, conductivity tensor that explicitly depends on the shear index $p$. The classical $h^3/(12\mu)$ law is recovered for $p=2$, and shear-thinning/thickening effects modulate the flow in the Reynolds roughness regime (Anguiano et al., 2017).
• Micropolar fluids: The flow equations couple microstructure rotation into the cell and macroscopic equations, with the Reynolds roughness regime (matched film thickness and roughness wavelength) producing full 3D cell problems and effective coefficients sensitive to both geometry and rotational viscosity (Suárez-Grau, 2019).
• Non-isothermal Darcy-Brinkman and porous flows: The critical scaling ($\alpha_{cr}$) produces a geometry-sensitive permeability tensor and induces a source term in the macroscopic energy equation due to roughness-induced viscous dissipation concentrated near the microscale structure (Anguiano et al., 21 Dec 2025).

7. Physical Interpretation and Significance

The Reynolds roughness regime marks a crossover between “flow forgetting” the roughness (high-frequency oscillations averaged out) and flow dominantly shaped by it (film thickness large relative to roughness wavelength). In this intermediate regime, the interplay between microscale geometry and macroscale transport is nontrivial and must be resolved using homogenization and multiscale analysis.

In practical turbulent flows, this regime corresponds to the region where roughness neither leaves the near-wall dynamics undisturbed (hydraulically smooth) nor saturates all turbulent mixing and drag (fully rough), but instead produces strongly geometry- and scale-dependent shifts in drag, heat transfer, and turbulence structure (Nilsson-Takeuchi et al., 25 May 2025, Forooghi et al., 2018, MacDonald et al., 2018, MacDonald et al., 2020, Anguiano et al., 21 Dec 2025, Suárez-Grau, 2019, Anguiano et al., 2017). The correct identification of this regime is critical for accurate modeling and prediction in engineering, geophysics, and applied physics, especially for surfaces with complex multi-scale topography.