# 후정해변 고파랑 조건하에서 파랑유속 방향전환점에서 발생하는 난류성분의 측정

# Measurement of Turbulence Properties at the Time of Flow Reversal Under High Wave Conditions in Hujeong Beach

## Article information

## Abstract

교란운동에너지(TKE)와 레이놀즈 응력의 수직성분(

## Trans Abstract

The temporal distribution of the turbulence kinetic energy (TKE) and the vertical component of Reynolds stresses (

**Keywords:**레이놀즈 응력; RANS 모델; CADMAS-SURF; 유속방향전환점

**Keywords:**Reynolds stress; RANS model; CADMAS-SURF; flow reversal

## 1. Introduction

The Reynolds stresses (레이놀즈 응력) is the stress tensor that arises when the Navier-Stokes equations are Reynolds averaged by decomposing the velocity field into mean and fluctuation parts as

where *u _{i}* is the fluid velocity in

*i*direction and

where r is the fluid density (밀도), *p* is pressure (압력) and *ν* is the kinematic viscosity (동점성계수).

In Eqn. (2), the product of velocity fluctuation is the Reynolds stresses as

where the overbar denotes the average of the products of velocity fluctuation in time. As shown in Eqn. (2), the Reynolds stresses are the additional forces to the mean flow generated by the turbulent fluctuations. These stresses play important role as they can produce momentum fluxes in the direction perpendicular to the mean flows. In the boundary layer, for example, the vertical component of Reynolds stresses,

Such vertical fluxes are specifically important in the study of sediment transport in coastal regions. In the unsteady boundary layers under oscillating wave conditions, the sediments are suspended from the seabed by the turbulent eddies and the upward sediment fluxes can be described by the Reynolds stress,

where *ν _{t}* is the kinetic eddy viscosity,

*k*is the turbulent kinetic energy (TKE, 교란운동에너지) (

*δ*is the Kronecker delta, and

_{ij}*S*is the mean strain rate tensor (

_{ij}*ν*, which generates the RANS turbulence modeling system (Wilcox, 1998). As one example, the

_{t}*k − ε*RANS model closes the system by solving two additional equations of TKE (

*k*) and dissipation (

*ε*) using

*νt~k*(Lauder and Spalding, 1974).

^{2}/εThe RANS models have successfully applied to simulate various turbulent flows (Rodi, 2017). However, the performance of several RANS models that applied to pulsating turbulent boundary layers showed discrepancies in turbulence quantities such as Reynolds stresses during some part of oscillating cycles while it gave satisfactory description of mean streamwise velocities (Scotti and Piomelli, 2001). In addition, Chang and Scotti (2004) observed that a RANS model underestimated the Reynolds stress,

In the present study, we propose to examine the temporal distribution of

## 2. Field Experiment and Data Analysis

Hujeong Beach (후정 해변) is located in the east coast of Korea between Jukbyeon Port (죽변항) and Bugu River (부구천) (Fig. 1). It is about 3 km long sand beach facing the northeast. In the northern end of the beach, the breakwater to protect the intake of Hanul Nuclear Power Plant (NPP, 한울원자력발전소) is located blocking the sediment discharge from the Bugu River. While it has microtidal environment with tidal range of a few decimeters only, the east coast of Korea suffers from storm waves during winter due to the movement of extratropical cyclone (온대 저기압) from China to the East Sea as the swell-like high waves usually attack the coasts during the period from October to February (Jeong et al., 2007, 2016). For this reason, the East Sea Research Institute (ESRI, 동해연구소) of Korea Institute of Ocean Science and Technology (KIOST, 한국해양과학기술원) that is located in the Hujeong Beach measured waves, currents and turbulence during January, 2017 at the location, ‘L1’ where water depth varies 6.5~7.0 m (Fig. 2(c)). The flow velocities are measured using a Vector, 3-dimensional acoustic velocimeter that is designed for turbulence and wave measurements. The velocimeter was moored at about 0.25 m above the bed while the pressure sensor was moored at about 0.65 m above the bed. The sampling frequency of the flow velocities is 2 Hz while the sampling frequency of pressure sensor is 1 Hz. During the period, the data were sampled for 20 minutes at every hour, which gives 24 data bursts per day as each burst contains the 20-minute data.

Fig. 2(a)~2(c) show the significant wave heights (*H _{s}*), peak wave period (

*T*) and the peak wave directions (

_{p}*θ*) averaged over each data burst from January 2 to 20.

_{p}*H*are lower than 2 m at most of the dates. However, it sharply increases higher than 2 m at January 14. The high wave condition continues until January 18 during which

_{s}*H*varies between 2.0~2.7 m. After January 18,

_{s}*H*sharply decreases below than 2 m. In case of

_{s}*T*, it increases from about 5 sec at January 2 and, during the period from January 14 to 18, it varies between 8 and 10 sec. In this period,

_{p}*θ*slightly varies around 45o indicating that the waves mainly approached the shore from northeast (90o is the shore normal direction). The high waves with the uniform northeast approaching direction at January 14~18 are likely the storm waves (폭풍파) caused by the extratropical cyclone developed in the northeast of the East Sea as shown in the satellite images of Geostationary Ocean Color Imager (GOCI) in Fig. 3. In the present study, we focus to examine the energy distribution of the waves measured during this storm period. The main reason for the analysis of the data selected at this time is because the characteristics of turbulence energy pattern are more clearly observed under high wave conditions in which the flows are strongly disturbed by the unsteady pulsating wave conditions. In addition, under swell-like high wave conditions, individual waves can be more easily distinguished during each wave period while, under lower wave energy conditions, individual waves are not easily separated because the waves are irregular due to the interference between the waves. For this reason, we selected the data from January 14 to 18 for the investigation of this study as this period is marked with the red vertical lines in Fig. 2.

_{p}The analysis of the wave data is performed using the ensemble average technique (Chang and Hanes, 2004). In the first step of this process, the individual waves that have clear wave pattern are selected. For example, Fig. 4 shows time series of cross-shore velocity and pressure data for 100 seconds during which 16 wave crests are observed. During each wave period, however, most of the waves do not show clear wave pattern that has one crest and one trough with similar magnitude. Instead, many waves show multiple peaks during its period or the magnitude of crest and trough shows severe discrepancies, which may contaminate the energy distribution during one the period of those irregular and asymmetric waves. In spite of the irregularity (불규칙성) of the measured data, we could still find one regular wave (규칙파) during this time that has a clear crest and a trough with symmetric magnitude as marked in red color in Fig. 4. During the storm period between January 14 and 18, we could find total 239 waves that has clear regular wave pattern. These regular waves have similar wave pattern as they have similar wave periods with the mean period (*T _{m}*) to be 10.2 sec and the standard deviation to be 0.9 sec.

In order to capture these individual waves from the time series data, we used the cross-shore velocity time series in Fig. 4(a). First, we found the zero-crossing points of the selected regular waves when the cross-shore velocity changed its sign from positive (onshore) to negative (offshore), which gave one wave period when the two adjacent onshore-tooffshore zero-crossing points were measured. Once the zero-crossing points were obtained, we then interpolated all the selected regular waves that have different wave periods by normalizing the time by the period of each individual regular wave, t/T. Since these selected waves have similar periods around 10 sec, the normalization would not produce serious errors in the ensemble average caused by the difference of the wave periods. Through these processes, we could obtain 239 individual waves (개별파) that have same normalized period as t/T varies 0 to 1. The whole individual regular waves were then divided into 3 groups according to its maximum velocity at wave crest. In Group 1, the 89 waves whose maximum onshore velocities are between 0.4~0.6 m/s were chosen. In Group 2, the 88 waves whose maximum onshore velocities are between 0.6~0.8 were chosen, and the rest 62 waves with onshore maximum velocities between 0.8~1.0 belonged to Group 3. In Fig. 5, the cross-shore velocities and pressure of the individual waves of Group 2 are plotted with solid black solid lines. As shown in the figure, the waves in the same group show similar wave pattern, which enables the measurement of turbulence parameters for each group. The red lines in Fig. 5 show the mean wave patterns of cross-shore velocity and pressure representing the group.

## 3. Model Description and Numerical Experiment

The RANS model used in this study is the two-dimensional CADMAS-SURF developed by the Port and Harbor Research Institute (PHRI), Japan (2001). The model can simulate the wave and structure interaction in a wave channel using *k − ε* turbulence closure model. The model is characterized by the Volume of Fluid (VOF) method that tracks and locates the free surface by solving additional advection equation for a fluid volume function (Hirt and Nichols, 1981; Hieu and Tanimoto, 2006). The CADMASSURF model has been successfully on wave run-up (처오름) problems on porous seabeds (투과성 해저면) (Nam et al., 2002; Yoon et al., 2005).

The continuity as well as momentum equations of the model are the Navier-Stokes equation that is expanded for porous body as

where *u* and *w* are the horizontal and vertical flow velocities, *ρ* is fluid density, *p* is pressure, *g* is gravitational force, *ν _{e}* is eddy viscosity (와동 점성계수),

*D*and

_{x}*D*are the energy attenuation coefficients (에너지 경감 계수),

_{z}*γ*is porosity (다공성),

_{ν}*γ*and

_{x}*γ*are the area transmittance (투과율) in the

_{z}*x*and

*z*direction respectively.

*λ*,

_{ν}*λ*and

_{x}*λ*are functions of

_{z}*γ*,

_{ν}*γ*respectively as they are expressed as

_{x}and γ_{z}where *C _{M}* is the inertia coefficient (관성계수).

*R*and

_{x}*R*are the resistance force (저항력) applying quadratic law as

_{z}where Δx and Δ_{z} are the grid size in the *x* and *z* direction respectively and *C _{D}* is the drag coefficient (항력계수).

*S*,

_{p}*S*and

_{u}*S*are the source terms to generate the wave conditions.

_{z}The VOF method is applied to simulate free surface generation by solving the advection equation of a fluid volume function, *F*, that varies between 0 to 1. The 1 value of *F* at a grid cell indicates that the cell is filled with fluid while 0 value indicates that fluid is empty in the cell. When *F* is greater than 0 but smaller than 1, the grid cell is partly occupied by fluid indicating that the cell is located at the boundary between the fluid and air. The advection equation of *F* is given as

where *S _{F}* is source term to generate wave conditions. In order to close the equations from (5)~(7),

*k − ε*model is employed to calculate the eddy viscosity,

*ν*, as

_{e}where *C _{1}* and

*C*are empirically determined coefficients (경험계수), and the relations between

_{2}*ν*,

_{k}*ν*,

_{ε}*ν*and

_{t}*ν*are given as

_{e}where *C _{μ}*,

*σ*,

_{k}*σ*are empirically determined coefficients.

_{ε}The above RANS equations are solved to simulate the experimental conditions described in section 2. The computational domain is a 2-dimensional rectangular channel with length of 984 m and height of 20 m (Fig. 6). The horizontal grid size varies from 1.0 m to 0.1 m because fine resolution is required near the surf zone. In the vertical direction, the grid size is set to be constant value with 0.2 m throughout the water column, which is fine enough to resolve the turbulence energy near the bed when imposing the no-slip bottom boundary condition. An ideal case of regular wave condition is generated by imposing the wave height to be 2.5 m at the source with 10 sec wave period at 7.3 m water depth in order to generate similar condition to the observed waves shown in Fig. 2. The waves are generated using Stokes wave approximate solutions at the source and the source is located at x = 328 m as it is determined considering the wave length. At the open boundary, the Sommerfeld radiation condition and a damping zone are applied to reduce the reflection effect. At the bottom boundary, the porous seabed condition is not applied as it may affect the near bed flows when applying the inertia and drag coefficients as they are defined in Eqn. (8) and (9).

## 4. Results

### 4.1 Reynolds stress distribution

The 3 groups of wave data sets obtained from the ensemble average method described in Section 2 are used to calculate the turbulence properties such as turbulence kinetic energy (*u _{max}*) not to be larger than 0.2 m/s. Moreover, we also consider the wave asymmetry (비대칭도) and skewness (왜곡도) as [(

*u*)

_{max}*− (*

_{onshore}*u*)

_{max}*]/(*

_{offshore}*u*)

_{max}*of each selected regular wave are monitored to be less than 10% and the times of offshore-to-onshore zero-crossing points of the waves are also to be concentrated within 10% range from the mean values.*

_{onshore}From the data sets of the waves selected based on the above criteria, we calculated the TKE and
*u _{max}*)

*− (*

_{onshore}*u*)

_{max}*, is also about 0.1 m/s for all three groups. In addition, the times of offshore-to-onshore zerocrossing points occur at t/T~0.55 for all three groups. Another similarity is also found in the oscillating periods as the averaged times for the cross-shore velocity to cycle between the two adjacent zero-crossing points are 10.1, 10.1 and 10.3 sec for Group 1, 2 and 3 respectively. These similarities between the cross-shore velocity patterns between the three groups of the selected waves indicate that the only significant difference is the maximum velocity magnitude, and the similar wave pattern enhances the accuracy of turbulence measurement at this time.*

_{offshore}In Fig. 7(b), the TKE distribution is compared between the three groups. The maximum TKE magnitudes also increase as the maximum velocity magnitude increases. In addition, the TKE of all three groups show two peaks during the normalized period as they roughly correspond to the time of maximum offshore and onshore cross-shore velocities, showing that the turbulence energy varies correspondingly to the mean flow energy during the wave period. This pattern of TKE with two peaks at maximum crossshore flow rates is similar to the model results over flat bottom by Chang and Hanes (2003), which indicates that the morphological effect of seabed does not significantly affect the turbulence measurement of this study.

One interesting founding that is distinguished from the velocity pattern is observed in the vertical component of Reynolds stress distribution (Fig. 7(c)). Unlike the TKE distribution,

In measuring the turbulence quantities, one of the weaknesses of the ensemble average technique used in this study is the possible underestimation/overestimation of the turbulence magnitude by the pseudo-turbulence. As shown in Fig. 5, the turbulence quantities are calculated by measuring the difference between the selected wave velocities (black lines) and their phase-averaged value (red line). This may still involve the mean components in the turbulence measurements as they are caused by the shift in the velocity magnitudes, which needs to be distinguished from the pure turbulent components. Another common approach to measure the turbulence quantities is the high-pass filtering that removes the components that are lower than the cut-off frequency. By doing this, the mean velocity components are filtered and the high frequent turbulent motions are to be measured. The green lines in Fig. 7(b) and (c) are the TKE and

### 4.2 Model comparison

Fig. 8 shows a schematic of the wave propagation calculated by the CADMAS-SURF by contouring the fluid volume function (*F*), water pressure (*P*) and the cross-shore velocity (*u*) as they are calculated from Eqn. (6) and (10). The red vertical line in the figure denotes the location of the source where the wave generation condition is imposed. The figures show that the model nicely simulates the propagated wave trains on the flat bed. Near the bottom boundary layer below 1 m, the cross-shore velocities show time lags from the velocities in the upper layers due to the bottom drag.

In order for a comparison with the observed data, we sampled the CADMAS-SURF data at the location of x = 500 m and z = 0.4 m above the bed as the grid cell of this location is closest to the location of the observation as described in section 2. Fig. 9 shows the cross-shore velocity, TKE and the magnitude of vertical component of Reynolds stress distribution (

## 5. Conclusion

In this study, we compared the temporal distributions of turbulence quantities of TKE and

The result shows an important indication in the study of transport of materials such as sediments in the nearshore region. Nielsen (1992) proposed a concept of turbulence convection process by which sediments are picked from the bed by flow turbulence in more organized way than the diffusion process by which the sediments are dispersed into the water column by the near-bed sediment concentration gradient. While the diffusion process is strong during the time of maximum flow rates, the convection process may be strong when the flow turbulent structures are active at such times of flow reversal. The observation of the peak

Another important result of this study is that a failure of a RANS model to produce the turbulence convection at the time of flow reversal as *S _{ij}*). Therefore, the development of these turbulence quantities strongly depends on the mean flow pattern, which may not correctly describe the turbulence pattern at the time of flow reversal when the mean velocity diminishes. To overcome this problem, additional contributions may be necessary in formulating the turbulence quantities under high wave energy conditions, which is suggested for future studies.

## Acknowledgements

This study was funded by Korea Institute of Ocean Science and Technology (KIOST, PE99523).