Large-Eddy Simulation of Turbulent Flow over Wavy Wall with Different Wave Steepness

Large-eddy simulation (LES) with dynamic one-equation subgrid-scale stress model was utilized to study the characteristics of turbulent flow over sea-surface generalized as two-dimensional wavy wall with different wave steepness. The statistical characteristics of turbulent field and pressure distribution were presented in detail. The simulation results showed that the separation zones induce stronger convection as the wave steepness increases. Reynolds shear stress near the wall boundary showed positive correlation with the wave steepness, while in the outer region, time-averaged turbulent fields seemed to be insensitive with the wall boundary. The equivalent velocity profiles obtained from the spatial averaging of the time-averaged velocity indicated that lower velocity occurred as the increase of wave steepness due to the increased pressure-induced form drag. The vortex structures were also visualized and showed vertically-bent characteristics which induced negative Reynolds shear stress at the stoss side as the increase of wave steepness that is equivalent to the effect of wave age.


Introduction
A large variety of dynamic processes in nature such as atmospheric flow over topography, current flow over seabed or airflow over water waves happen naturally (Fernando, 2010). Especially for airflow over sea surface with water waves, the momentum transport become complex due to the impact of complex sea wave. Many studies have been conducted about the flow characteristics over complex geometry boundary by experimental study or numerical simulation. A frequent-used method to simplify the wall geometry is to regard the boundary as two-dimensional wavy wall (Buckles, Hanratty, and Adrian, 1984). Consequently, a large variety of experiments could be conducted to obtain the turbulent fields (Hudson, Dykhno, and Hanratty, 1996, Nakagawa, and Hanratty, 2001, Günther, and von Rohr, 2003. Although these studies gave detailed description of the flow characteristics, however, how the vortex structures impact the turbulent fields with different wave steepness (or the ratio of amplitude to wavelength) was not discussed in detail.
Therefore, Michioka (2018) numerically studied the turbulent flow and gas dispersion over wavy wall with different ratio of amplitude to wavelength, and found the large standard deviations of the fluctuation of streamwise and spanwise velocity appearing obviously at the windward of the wavy wall related to the Görtler vortices. Different from the flat wall boundary layer, Hamed et al. (2015), by experiments, found that negative Reynolds shear stress occurred at stoss side when flowing over wavy wall boundary and it is the reason by using cartesian coordinate system (Hudson, Dykhno, andHanratty, 1996, Cherukat, Hanratty, andMclaughlin, 1998). While another explanation by Yang and Shen (2009) was that quasi-streamwise vortices at stoss side with ejection events induces negative Reynolds shear stress. However, these studies considered just one parameter of wave steepness and lacked of a mechanism about how wave steepness impact the flow characteristics and the vortex structures.
This study presents a large-eddy simulation of turbulent flow over wavy wall with different wave steepness. The statistical characteristics of turbulent field, the time-averaged and spatial-averaged turbulent fields and pressure distribution were discussed in detail. And the vortex structures were extracted to explain the impact on high-order statistics of turbulent fields.

Method
In this study, the large-eddy simulation with dynamic one-equation subgrid-scale stress model was used. The filtered incompressible N-S equations and the continuity equation for describing resolvable turbulent flow field are as follow: x y z directions corresponding to streamwise, spanwise and vertical directions, ( , , ) i u u v w are the filtered velocity components, p is the filtered pressure, ν represents kinematic viscosity and ij τ is subgrid-scale stress tensor modelled by dynamic one-equation model (Kim, and Menon, 1995) that expresses as: where, the 3 x y z ∆ = ∆ ∆ ∆ is the filtered grid scale. K represents subgrid-scale kinetic energy which can be resolved by the transport equation (6). The : represents test filter that °2 ∆ = ∆ , and the u % is the filtered velocity in test filter. The OpenFOAM is used for numerical simulation with PimpleFoam solver that SIMPLE algorithm is carried for pressure-velocity coupling. The numerical verification can be found in detail in our previous work (Zhang et al. 2021).

Physical model and flow parameter
In present study, the wall boundary was generalized as sinusoidal wave with four different wave steepness defined as ( 0.157, 0.314, 0.471, 0.628) ak = . Here, a is the amplitude and k is the wave number. The flow above the boundary wall is driven by external force with periodic boundary conditions along , x y directions. No-slip wall is applied at top and bottom boundaries. The computational domain is ( , , that is large enough to capture flow structures by two-points spatial correlation. The Reynolds number is defined as , the h′ is the half height of the domain and in U is bulk velocity. The first layer of grids is set as 0.5 1 z + ∆ = < to make sure enough accuracy of boundary layer. After the turbulent flow fully develops, time-averaging is carried out for another additional thirty flow periods. Figure 1 illustrates the contour of time-averaged turbulent fields. The distribution of streamwise velocity near the wall show wavy-spatial distribution as the increase of wave steepness. In the separation zones, the vertical velocity shows different direction at trough with downwelling motion at lee side, while upwelling motion at stoss side (behind the reattachment point). As shown in figure 1(c1)-(c4), the turbulent kinetic energy shows high correlation with the ak , and the region of higher turbulent level corresponds to the separation zones, revealing the relation between turbulent level with strong convection. In addition, the Reynolds shear stress shows similar characteristics with turbulent kinetic energy as illustrated in figure 1(d1)-(d4), but it should be noted that the Reynolds shear stress at stoss side near the wall is negative that is related with the quasi-streamwise vortices (Yang, and Shen, 2009). ( )      In order to compare the velocity profile of wavy wall with flat wall boundary layer, spatial averaging of time-averaged velocity was implemented along one wavelength as figure 5 shows. But attention should be paid that the spatial averaging of velocity is an equivalent variable instead of real quantity. The dashed line represents linear law in viscous 5 E3S Web of Conferences 299, 03012 (2021) https://doi.org/10.1051/e3sconf/202129903012 EDEP 2021 sublayer for flat wall boundary layer at 5 z + 〈 〉 < (Kim, Moin, and Moser, 1987), and the dot dashed line is the logarithmic law in logarithmic region (+ represents dimensionless by τ u for velocity scale and τ u ν for length scale). As the increase of wave steepness, logarithmic region extends into the range of 50 150 z + 〈 〉 = :

Results
, besides, profile of velocity decreases with down shift compared with flat wall boundary layer in both viscous sublayer and logarithmic region.
In addition, the pressure distribution for different wave steepness is given in figure 6. Reduced positive pressure region can be observed at stoss side as the increase of wave steepness, while the enhanced negative pressure region locates at lee side. The dimensionless pressure-induced form drag and viscous drag then can be obtained by ∫ . As figure 7 shows, the pressure-induced form drag increases as the increase of wave steepness, while the magnitude of viscous drag can be ignored. The form drag at wall boundary reduces the flow velocity in viscous sublayer, while the different profile of velocity in logarithmic region is owing to the recirculation bubble in separation zones.  , and the vector in 3D view is ( , , ) u v w . Section locates at 1 x λ = with vector ( , ) v w shown. The vortex pair is depicted by A and B that the vorticity is The turbulent field is known to be largely affected by the large-scale vortex structures. To clearly understand the impact of vortex on the turbulent field, the vortex structures are extracted by Q criterion in the instantaneous velocity field. And the Q is defined as the residual of the vorticity tensor norm squared subtracted from the strain-rate tensor norm squared that can be expressed as: where, the 0.5( ) Τ = ∇ + ∇ S u u is strain-rate tensor and 0.5( ) Τ = ∇ − ∇ Ω u u is vorticity tensor. The vortex occurs at a region with 0 Q > . Figure 8 shows the vortex structures of instantaneous turbulent field with 5 Q = . The quasi-streamwise vortices dominate under the condition of lower wave steepness, while vertically-bent vortices occur especially at trough for higher wave steepness as shown in figure 8(d). Where vertically-bent vortex striding across the crest can be clearly observed at stoss side as the increase of wave steepness. To discuss how the vortex structure impact the turbulent field at stoss side for large wave steepness case, figure 9 shows the vortex of 10 Q = and a section located at 1 x λ : with instantaneous streamwise velocity and vector ( , ) v w to depict the vorticity near the wall boundary at stoss side. The vortex pair A and B show counter-rotating feature with A 0 x ω < and B 0 x ω > (the vorticity can be calculated by inducing the ejection event along z + direction, consequently, stimulating the enhanced negative Reynolds shear stress.

Discussion
The simulation results indicate that wave steepness impacts significantly on the flow characteristics that stronger convection occurs with high wave steepness due to flow separation. Besides, the enhanced turbulent level with higher turbulent kinetic energy and Reynolds shear stress also shows high correlation with the wave steepness. The increased wave steepness results in higher pressure-induced form drag, giving rise to reduced spatialaveraged velocity with profile shifting downward. By visualizing instantaneous vortex structures, quasi-streamwise vortex gradually bents vertically as the increase of wave steepness which is equivalent to the impact of wave age.

Conclusions
The study presents a numerical simulation to analyse the impact of wave steepness on turbulent flow. In conclusion, the wave steepness highly impact turbulence flowing over wavy wall boundary, especially at trough with strong convection due to flow separation and at stoss side with enhanced shear layer due to the vertically-bent vortex structures. Therefore, the impact of wave steepness that is as equivalent as the wave age's effect should be more considered.