We present a simulation-based study of the effect of a passing wave packet on underlying fully developed turbulence. We propose a novel wave-phase-resolved simulation method inspired by Helmholtz decomposition to directly couple the turbulence simulation with instantaneous wave orbital motions without wave-phase averaging. We also introduce a boundary condition treatment for the turbulence at the wave surface, which allows the turbulence simulation to be conducted in a rectangular domain while retaining the wave-phase effect. The results obtained from the proposed method reveal considerable variations in turbulence statistics, including the enstrophy and Reynolds normal stresses, during wave packet passage. Most changes occur rapidly when the narrow bandwidth around the wave packet core passes. Further analyses of the energy spectra indicate that the enhancement of turbulence occurs across a wide range of scales, with the near-surface small-scale motions experiencing the most significant intensification. Meanwhile, large-scale motions with scales comparable to the boundary layer depth are also enhanced. The mechanisms underlying the Reynolds normal stress variation at different length scales are related to the energy transfer from the wave orbital straining to turbulence through production, the pressure–strain effect, the pressure diffusion and the wave advection. By assessing the turbulence statistics and dynamics impacted by a wave packet in detail, this study provides an improved understanding of the response of a developed turbulent flow to a transient wave field. The proposed simulation method also proves to be a promising phase-resolved approach for efficiently modelling the wave effect on turbulence.

In turbulence near a free surface, strong vortices attach to the surface, creating surface imprints visible as nearly circular ‘dimples’. By studying these imprints in direct numerical simulation (DNS) data we make two observations. First, the imprints of surface-attached vortices can be very effectively distinguished from other turbulent surface features using two physical features: they are nearly circular in shape, and persist for a long time compared with other pertinent time scales. Secondly, the instantaneous number of surface dimples from surface-attached vortices in an area, $N(t)$, is intimately related to its mean-square surface divergence, $\beta ^2(t)$. We develop a simple and physically transparent computer vision procedure which, using the properties of low eccentricity and longevity, detects and tracks vortices from their surface features only, with sensitivity and accuracy of $90\,\%$ or better. We compare $N(t)$ and $\beta ^2(t)$, finding a normalised cross-correlation of $0.90$, with changes in $N$ lagging around $0.8T_\infty$ behind those in $\beta ^2$ ($T_\infty$ is an integral time scale), confirming the common observation that vortices are spawned by strong upwelling events where $\beta ^2$ is large. These findings suggest that the rate of mass flux across the surface, being closely related to surface divergence, can be estimated remotely in some natural flows using visible free-surface dimples as proxy.

A model based on a convolutional neural network (CNN) is designed to reconstruct the three-dimensional turbulent flows beneath a free surface using surface measurements, including the surface elevation and surface velocity. Trained on datasets obtained from the direct numerical simulation of turbulent open-channel flows with a deformable free surface, the proposed model can accurately reconstruct the near-surface flow field and capture the characteristic large-scale flow structures away from the surface. The reconstruction performance of the model, measured by metrics such as the normalised mean squared reconstruction errors and scale-specific errors, is considerably better than that of the traditional linear stochastic estimation (LSE) method. We further analyse the saliency maps of the CNN model and the kernels of the LSE model and obtain insights into how the two models utilise surface features to reconstruct subsurface flows. The importance of different surface variables is analysed based on the saliency map of the CNN, which reveals knowledge about the surface–subsurface relations. The CNN is also shown to have a good generalisation capability with respect to the Froude number if a model trained for a flow with a high Froude number is applied to predict flows with lower Froude numbers. The results presented in this work indicate that the CNN is effective regarding the detection of subsurface flow structures and by interpreting the surface–subsurface relations underlying the reconstruction model, the CNN can be a promising tool for assisting with the physical understanding of free-surface turbulence.

The dependence of Reynolds shear stress on wave phase is investigated for initially isotropic turbulence distorted by a progressive surface wave through direct numerical simulations. A wave-following streamline coordinate frame is used to analyse the turbulence dynamics such that the information of the varying direction of wave orbital motions is embedded into the coordinate system, which helps capture the effect of flow curvature on the turbulence and quantify momentum exchange between the near-surface and deep regions in the wavy domain accurately. It is found that the Reynolds shear stress is enhanced under the backward slope of the wave and can be scaled by the wave steepness and the streamwise velocity fluctuations. Analyses of the budget of Reynolds shear stress indicate that such a variation with the wave phase is caused by the variation in the production of the Reynolds shear stress and the effect of pressure fluctuations. Further investigation shows that the production of the Reynolds shear stress is closely associated with the wave surface curvature. A model that includes a correction term for the curvature effect for the pressure–strain correlation term is examined and is found to agree reasonably well with the simulation result. The correction term is found to make an appreciable contribution to the model, further supporting our finding that the wave curvature plays an important role in the turbulence dynamics near the surface.

The dynamics of Reynolds stresses and turbulent kinetic energy (TKE) in Langmuir turbulence are analysed using data of large-eddy simulations with the wave phase resolved. It is found that the streamwise and spanwise Reynolds normal stresses and the Reynolds shear stress vary appreciably with the wave phase, while the vertical normal stress is only weakly dependent on the wave phase. Budget analyses indicate that the production due to wave straining and the effects associated with turbulence pressure are the dominant mechanisms for the wave-phase variation of Reynolds stresses. The accumulative effect of wave–turbulence interactions on TKE is then investigated using the Lagrangian average. It is discovered that the energy transfer from wave to turbulence is contributed by two mechanisms. The first mechanism is the turbulence production by the Lagrangian mean wave shearing and the mean shear stress, which is consistent with the traditional wave-phase-averaged model. The second mechanism, which is not accounted for in previous studies, is the correlation between the wave-phase variation of the Reynolds shear stress and the wave orbital shearing. A model is proposed for the second mechanism. Comparison of the frequency spectrum with Craik–Leibovich simulation results shows that the correlation effect can affect the turbulence fluctuations at time scales around the wave period, indicating the importance of this effect on Reynolds stresses and TKE.

Wall-resolved and wall-modelled large-eddy simulations are performed to study the localizing effect of Langmuir cells (LCs) on small-scale background turbulence in shallow water. The total velocity fluctuations are decomposed into an LC content extracted by streamwise averaging and a background turbulence part. Based on the large-scale motions of LCs, the spanwise domain is divided into three regions dominated by the upwelling, spanwise and downwelling flows of LCs, respectively. The localized Reynolds stresses $\langle u_{i}^{T}u_{j}^{T}\rangle _{xt}$ in different spanwise regions are compared to show the localizing effects of the LCs on the background turbulence quantitatively, where $u_{1}^{T}$ (or $u^{T}$), $u_{2}^{T}$ (or $v^{T}$) and $u_{3}^{T}$ (or $w^{T}$) represent the streamwise, vertical and spanwise components of the background turbulence velocity, respectively, and $\langle \cdot \rangle _{xt}$ denotes time and streamwise averaging. It is shown that the magnitudes of the localized Reynolds stresses in different spanwise regions vary significantly. The transport equations of the localized Reynolds stresses are then analysed to investigate the mechanisms underlying the localizing effects. It is discovered that the difference in the energy production correlated to the shear of the LC content among different regions is the key factor that leads to the localization of background turbulence. In addition, the energy production correlated to the shear of the mean flow, the energy redistribution due to the pressure–strain correlation, and the interaction between the localized Reynolds stresses and the shear of the Stokes drift also play important roles. Based on the results obtained from the analysis of the transport equations, predictive models are proposed for the localizing effects, which assess the spatial dependence of the Boussinesq model for background turbulence in coastal Langmuir turbulence. These models show good scaling performance of $\langle u^{T}u^{T}\rangle _{xt}$ near the water bottom and of $\langle -u^{T}v^{T}\rangle _{xt}$, $\langle -u^{T}w^{T}\rangle _{xt}$ and $\langle -v^{T}w^{T}\rangle _{xt}$ in the central region of the water column under various flow conditions with different values of the Reynolds number, turbulent Langmuir number and wavenumber.

The effects of a water surface wave on the vorticity in the turbulence underneath are studied for Langmuir turbulence using wave-phase-resolved large-eddy simulation. The simulations are performed on a dynamically evolving wave-surface-fitted grid such that the phase-resolved wave motions and their effects on the turbulence are explicitly captured. This study focuses on the vorticity structures and dynamics in Langmuir turbulence driven by a steady and co-aligned progressive wave and surface shear stress. For the first time, the detailed vorticity dynamics of the wave–turbulence interaction in Langmuir turbulence in a wave-phase-resolved frame is revealed. The wave-phase-resolved simulation provides detailed descriptions of many characteristic features of Langmuir turbulence, such as elongated quasi-streamwise vortices. The simulation also reveals the variation of the strength and the inclination angles of the vortices with the wave phase. The variation is found to be caused by the periodic stretching and tilting of the wave orbital straining motions. The cumulative effect of the wave on the wave-phase-averaged vorticity is analysed using the Lagrangian average. It is discovered that, in addition to the tilting effect induced by the Lagrangian mean shear gradient of the wave, the phase correlation between the vorticity fluctuations and the wave orbital straining is also important to the cumulative vorticity evolution. Both the fluctuation correlation effect and the mean tilting effect are found to amplify the streamwise vorticity. On the other hand, for the vertical vorticity, the fluctuation correlation effect cancels the mean tilting effect, and the net change of the vertical vorticity by the wave straining is negligible. As a result, the wave straining enhances only the streamwise vorticity and cumulatively tilts vertical vortices towards the streamwise direction. The above processes are further quantified analytically. The role of the fluctuation correlation effect in the wave-phase-averaged vorticity dynamics provides a deeper understanding of the physical processes underlying the wave–turbulence interaction in Langmuir turbulence.

Langmuir circulations (LCs) generated by the interaction between wind-driven currents and surface waves can engulf the whole water column in neutrally stratified shallow water and interact with the turbulence in the bottom boundary layer. In this study, we perform a mechanistic study using wall-resolved large-eddy simulations (LES) based on the Craik–Leibovich equations to investigate the effects of LCs on turbulence statistics in the bottom half of shallow water. The highest Reynolds number considered in this paper, $Re_{\unicode[STIX]{x1D70F}}=1000$, is larger than the values considered in wall-resolved LES studies of shallow-water Langmuir turbulence reported in literature. The logarithmic layer is diagnosed based on a plateau region in the profile of a diagnostic function. It is found that the logarithmic layer disrupted at $Re_{\unicode[STIX]{x1D70F}}=395$ reappears at $Re_{\unicode[STIX]{x1D70F}}=1000$, but the von Kármán constant is slightly different from the traditional value $0.41$. To study the effects of LCs on turbulence statistics, LCs are extracted using streamwise averaging. The velocity fluctuations $u_{i}^{\prime }$ are decomposed into a LC-coherent part $u_{i}^{L}$ and a residual turbulence part $u_{i}^{T}$. It is found that the profiles of LC-coherent Reynolds shear stress $-\langle u^{L}v^{L}\rangle$ obtained at various Reynolds numbers are close to each other in the water-column coordinate $y/h$, with $h$ being the half-water depth. As the Reynolds number (or, by definition, the ratio between the outer and inner length scales) increases, the influence of LCs on the near-bottom momentum transfer is reduced, which is responsible for the reappearance of the logarithmic layer. At all of the Reynolds numbers under investigation, the peaks of $\langle u^{L}u^{L}\rangle$ are collocated in the water-column coordinate $y/h$, while those of $\langle u^{T}u^{T}\rangle$ are collocated in the inner-scale coordinate $y/(\unicode[STIX]{x1D708}/u_{\unicode[STIX]{x1D70F}})$. Due to the increase in the distance between the peaks of $\langle u^{L}u^{L}\rangle$ and $\langle u^{T}u^{T}\rangle$ with the Reynolds number, the profile of $\langle u^{\prime }u^{\prime }\rangle$ forms a bimodal shape at $Re_{\unicode[STIX]{x1D70F}}=700$ and $1000$.