1DV structure of turbulent wave boundary layers

Abstract

The 1DV structure of turbulent oscillatory boundary layers throughout the full relative roughness range 0<r/A<1 and Reynolds numbers (A2ω/v) up to 6×106 is investigated through experimental data, A is the free stream semi excursion, r the bed roughness, ω the angular frequency and v the kinematic viscosity. It is shown that smooth and rough flows alike can be scaled using the universal vertical scale z1=2v/ω+0.0081rA, which becomes the Stokes' length 2v/ω for smooth beds (r=0) and 0.09rA for roughness dominated flows. Extraction of z1 and hence r from experimental data provides a method for determining the effective hydraulic roughness of live sand-beds under waves in analogy with the log-fit method for determining the r from the zero intercept, z0=r/30 in steady flows. In fairly rough flows, e.g. over vortex ripples, the velocity structure is analogous to laminar flow and hence the eddy viscosity is constant with the value vt=0.004ωrA, corresponding to z1=0.09rA=2vt/ω. In the opposite limit of vanishing roughness and high Reynolds numbers, z1/A<10−3, the velocity structure, which corresponds to von Karman's vt=kufz, agrees reasonably with data. However, this asymptotic velocity structure deviates considerably from measurements through the naturally occurring r/A-range for sand beds. In the intermediate range 10−3<r/A<0.06 the velocity structure does not seem to correspond to any real-valued, time invariant eddy viscosity. Comparison of the Kelvin function solution, which corresponds to vt=κufz, with smooth and almost-smooth, high Reynolds number measurements, shows that, the optimal zero intercepts are not the steady flow values 0.11v/uf respectively r/30, but generally significantly larger. For low-roughness, high-Reynolds number oscillatory flows (z1/A<0.002), the magnitude of the velocity defect function universally follows |D|~exp [−(z/z1)1/3]. This 1/3-power is interesting in that it has previously been shown to correspond maximum energy dissipation. In turbulent ‘2nd order Stokes flows’ the vertical boundary layer scale of the second harmonic is smaller than that of the primary, but by a factor smaller than 21/2, which applies in laminar flow. Density stratification in heavily sediment-laden sheet-flow makes the boundary layer thinner via the power p in |D|~exp [−zp] taking greater values than in fixed bed experiments with similar z1/A.