A dual scaling of the turbulent longitudinal velocity structure function $\overline {{(\delta u)}^n}$, i.e. a scaling based on the Kolmogorov scales ($u_K$, $\eta$) and another based on ($u'$, $L$) representative of the large scale motion, is examined in the context of both the Kármán–Howarth equation and experimental grid turbulence data over a significant range of the Taylor microscale Reynolds number $Re_\lambda$. As $Re_\lambda$ increases, the scaling based on ($u'$, $L$) extends to increasingly smaller values of $r/L$ while the scaling based on ($u_K$, $\eta$) extends to increasingly larger values of $r/\eta$. The implication is that both scalings should eventually overlap in the so-called inertial range as $Re_\lambda$ continues to increase, thus leading to a power-law relation $\overline {{(\delta u)}^n} \sim r^{n/3}$ when the inertial range is rigorously established. The latter is likely to occur only when $Re_\lambda \to \infty$. The use of an empirical model for $\overline {{(\delta u)}^n}$, which complies with $\overline {{(\delta u)}^n} \sim r^{n/3}$ as $Re_\lambda \to \infty$, shows that the finite Reynolds number effect may differ between even- and odd-orders of $\overline {{(\delta u)}^n}$. This suggests that different values of $Re_\lambda$ may be required between even and odd values of $n$ for compliance with $\overline {{(\delta u)}^n} \sim r^{n/3}$. The model describes adequately the dependence on $Re_\lambda$ of the available experimental data for $\overline {{(\delta u)}^n}$ and supports indirectly the extrapolation of these data to infinitely large $Re_\lambda$.

The present study focuses on two-dimensional direct numerical simulations of shallow-water breaking waves, specifically those generated by a wave plate at constant water depths. The primary objective is to quantitatively analyse the dynamics, kinematics and energy dissipation associated with wave breaking. The numerical results exhibit good agreement with experimental data in terms of free-surface profiles during wave breaking. A parametric study was conducted to examine the influence of various wave properties and initial conditions on breaking characteristics. According to research on the Bond number ($Bo$, the ratio of gravitational to surface tension forces), an increased surface tension leads to the formation of more prominent parasitic capillaries at the forwards face of the wave profile and a thicker plunging jet, which causes a delayed breaking time and is tightly correlated with the main cavity size. A close relationship between wave statistics and the initial conditions of the wave plate is discovered, allowing for the classification of breaker types based on the ratio of wave height to water depth, $H/d$. Moreover, an analysis based on inertial scaling arguments reveals that the energy dissipation rate due to breaking can be linked to the local geometry of the breaking crest $H_b/d$, and exhibits a threshold behaviour, where the energy dissipation approaches zero at a critical value of $H_b/d$. An empirical scaling of the breaking parameter is proposed as $b = a(H_b/d - \chi _0)^n$, where $\chi _0 = 0.65$ represents the breaking threshold and $n = 1.5$ is a power law determined through the best fit to the numerical results.