Roughness-controlled layer in oscillatory turbulent boundary layers over densely packed uniform roughness

Abstract: In coastal wave boundary layers over gravel-scale roughness, with near-bed orbital excursions ten to a hundred times the roughness height, the boundary layer is only a few roughness heights thick. A roughness-controlled layer (RCL) of the steady-flow extent two to five element heights would then leave no room for a logarithmic layer, yet experiments over densely packed marbles recover logarithmic profiles within millimetres of the crests. We resolve this contradiction by re-analysing previous Particle Image Velocimetry (PIV) records with a triple decomposition that separates the marble-locked dispersive motion from the stochastic turbulence, across eleven wave, current and wave-current conditions. The boundary layer organises into an RCL, a transition region, and a logarithmic profile layer, with the RCL only one to two tenths of a marble diameter deep. This thinness is kinematic: above a periodic bed, the dispersive field decays over a length fixed by the element spacing, so close packing caps the layer at a fraction of a diameter. The layer is destroyed and rebuilt every half-cycle, tracking the near-bed velocity quasi-steadily, while the eddies within it stay locked to the inter-crest gap. Thinness does not imply weakness: within the layer, the dispersive kinetic energy rivals the turbulent kinetic energy, and the dispersive stress matches, near the crests exceeds, the Reynolds stress, showing that separated wakes carry organised momentum. The logarithmic layer survives because the decay length imposed by the packing is far smaller than the boundary-layer thickness, a margin set by the bed geometry rather than by the forcing.

• The paper finds that the roughness-controlled layer is only 0.1–0.2 element diameters thick across 11 wave, current, and combined-flow cases, matching a kinematic decay-length prediction.
• Triple decomposition of high-resolution PIV data separates dispersive motion from turbulence, showing that separated wakes can produce dispersive stresses comparable to Reynolds stresses and 3–5 times larger near crests.
• The thin layer is rebuilt twice per cycle while an overlying logarithmic profile extends up to 1.8–2.6 element diameters, resolving why Grant–Madsen-style models remain applicable over densely packed roughness.

The contradiction the paper addresses

In coastal wave boundary layers over gravel-scale roughness, the governing length scales are strongly ordered: the orbital excursion $A_{bm}$ greatly exceeds the wave boundary-layer thickness $\delta_w$, which in turn exceeds the Nikuradse roughness $k_N$. The lower link of this hierarchy is what permits a logarithmic profile layer (LPL) near the bed, and it underpins the Grant–Madsen family of wave–current boundary-layer models. Over a densely packed ceramic-marble bed ($D = 12.5$ mm, $k_N = 20$ mm, $A_{bm}/k_N \sim O(10\text{--}100)$), Yuan and Madsen's oscillating water tunnel experiments recovered accurate logarithmic profiles within millimetres of the marble crests. Yet steady-flow theory places the top of the roughness-controlled layer (RCL) at two to five element heights above the bed — for this geometry, roughly 2.5–6.3 cm, comparable to the entire wave boundary layer. If that estimate held, no logarithmic layer could exist. The paper resolves this contradiction by re-analysing the raw PIV records of those experiments with a triple decomposition across eleven wave, current and wave–current conditions (2608.17543).

Kinematic argument and data reduction

The theoretical core is a kinematic result. Any deviation of the velocity from its bed average over a periodic bed of streamwise spacing $s$ must decay exponentially with height, with the slowest mode decaying over $\ell = s/2\pi$. For close-packed marbles, $\ell \approx 0.16D$, so a velocity-based RCL detected by any threshold criterion is capped at $\ell \ln(R_c/R_{\mathrm{thr}})$ — a fraction of a diameter — regardless of wake dynamics. Crucially, potential flow over fore–aft symmetric elements carries identically zero dispersive stress, so a substantial dispersive stress requires separation; the paper distinguishes an apparent RCL (velocity deviation) from a dynamical RCL (dispersive stress) and measures both.

The re-analysis recomputes velocity fields from raw image pairs on a finer grid (0.20 mm vertical spacing) than the original study, phase-averages over 32 cycles before spatial averaging, and decomposes each velocity into bed-and-phase-averaged, dispersive ($\delta_w$0) and turbulent ($\delta_w$1) parts. The original double-averaged reduction had absorbed the form-induced contribution into an equivalent Reynolds stress; the triple decomposition separates them.

A thin, periodically rebuilt layer

The central quantitative finding is that the apparent RCL is only 0.1–0.2$\delta_w$2 thick above the crests in all eleven cases, with no trend in $\delta_w$3. The measured crest-level deviation ratios ($\delta_w$4–1.62) inserted into the kinematic prediction yield tops of 0.1–0.19$\delta_w$5, matching the measurements closely and lying an order of magnitude below the canopy-based estimate for the same geometry. Above the layer sits a transition region and then an LPL occupying up to 1.8–2.6$\delta_w$6. This immediately reconciles the contradiction: because the packing-imposed decay length is far smaller than $\delta_w$7, the logarithmic layer survives with room to spare, and the margin is set by bed geometry rather than forcing.

The layer is not static. It is destroyed and rebuilt twice per cycle, tracking the near-bed velocity quasi-steadily (eddy turnover time $\delta_w$8–0.04): thinnest shortly before free-stream reversal, where the near-bed flow reverses with its characteristic lead (~28°), thickest around the near-bed maximum. A superimposed current raises the period-averaged top by about 0.1$\delta_w$9 and keeps the period-averaged deviation ratio elevated well above the layer — wave–current cases are threshold-sensitive in detection precisely because their ratio decays only gradually. Two-point correlations show eddies locked to the inter-crest gap inside the layer ($k_N$0–0.5$k_N$1 under waves, 0.2$k_N$2 under pure current), while above the layer they settle onto a forcing-dependent plateau (0.57–1.32$k_N$3\kappa z$ growth anywhere in the measured range. Notably, the integral length peaks in mid-deceleration, about 11° before the thickness minimum: thickness answers to instantaneous forcing, eddy length to forcing history.</p> <h2 class='paper-heading' id='dynamical-strength-despite-thinness'>Dynamical strength despite thinness</h2> <p>Thinness does not imply weakness. Within the RCL, the <a href="https://www.emergentmind.com/topics/dispersive-kinetic-energy-dke" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">dispersive kinetic energy</a> rivals the turbulent kinetic energy, with peak ratios ordered pure current (0.5–0.65) &lt; wave–current (0.8–1.15) &lt; pure wave (1.2–1.9). The dispersive stress matches the Reynolds stress through most of the layer and exceeds it near the crests by factors of 3–5. Since attached symmetric flow carries no dispersive stress, these magnitudes establish that separated wakes transport organised momentum within the layer. An independent, stress-based detection of the layer top agrees with the velocity-based one to within 0.05$k_N$4 in every case, establishing that the dynamical and apparent RCLs are the same object. A direct consequence is that a substantial share of the near-crest stress reported as Reynolds stress in the original studies was in fact dispersive.

The paper attributes the residual ordering among forcing types to how the inter-crest cavities are driven: the oscillatory pressure gradient acts at every elevation and drives cavities directly (smallest $k_N$5), a current reaches them only through turbulent shear (larger $k_N$6), and a superimposed current drives the wakes hardest and longest (largest $k_N$7). This explains why the steady-current cases return essentially the same thin layer as the waves — thinness is a property of the bed, not the oscillation.

Limitations and open questions

Several caveats bear directly on these results. The PIV interrogation window low-pass filters the field and attenuates the measured Reynolds stress toward the bed (to about 75% of the momentum-integral estimate at $k_N$8 mm), so the reported stress ratios inside the layer are overestimates. The LPL boundaries respond noticeably to the slope-deviation tolerance, spanning 0.80–1.06$k_N$9 and 0.87–1.27$D = 12.5$0 as the tolerance varies over [0.2, 0.4], so the logarithmic-layer extent is approximate even though the RCL–LPL ordering is robust. The threshold $D = 12.5$1 is a convention; wave–current detections are threshold-sensitive because their ratio decays gradually. The timing relation between the $D = 12.5$2 maximum and the thickness minimum is established observationally but not causally. Finally, the kinematic argument bounds only the marble-periodic part of the deviation field, and the predictions that spacing controls thickness linearly while permeability activates the canopy mechanism remain untested; sparser, permeable and mobile beds, where the margin between RCL and $D = 12.5$3 narrows, are identified as the natural next step.

Conclusion

By separating the element-locked dispersive motion from stochastic turbulence in full-scale oscillatory boundary-layer measurements, this work shows that over densely packed uniform roughness the roughness-controlled layer is one to two tenths of an element diameter deep — set kinematically by the packing-imposed decay length rather than by canopy dynamics — yet dynamically active, with separated wakes carrying organised momentum comparable to or exceeding the Reynolds stress. The logarithmic layer survives because the geometric margin between the packing scale and the boundary-layer thickness exceeds an order of magnitude, a margin controlled by bed geometry rather than forcing.