Currents induced by waves in the surf zone and the pollutant transport analysis

9th International Conference on Hydrodynamics October 11-15, 2010 Shanghai, China

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2010, 22(5), supplement: 1033-1038 DOI: 10.1016/S1001-6058(10)60071-6

Currents induced by waves in the surf zone and the pollutant transport analysis Jing-xin Zhang 1, Hua Liu 2 School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai, China E-mail: [email protected] ABSTRACT: As a long-term hydrodynamic phenomenon

induced by waves, the currents dominate the pollutant transport in the shoaling and surf zones. The present study was focused on the analysis of the cross-shore currents induced by waves and the pollutant transport. The wave parameters were obtained by solving one phase-averaged wave model including wave shoaling and breaking in the surf zone. The effect of the breaking wave rollers in the surf zone was considered by specifying an additional boundary condition on the mean water level (MWL). We introduced a formula for the vertical varied wave radiation stress into the hydrodynamic model. Furtherly, we simulated and analyzed the pollutant transport in the domain for different pollutant release position. KEY WORDS: undertow simulation; vertical varied radiation stress; surf zone; water quality

1 INTRODUCTION Wave breaking is a natural phenomenon that is widespread in the nearshore region. During the process of wave breaking, energy is transferred from the organized wave motion to turbulence motion. Wave breaking provides additional energy for pollutant transport. For a long term water quality evaluation, the wave-induced currents are the dominating dynamics driving the pollutant. Early experimental researches presented the current distribution in the surf zone[1]. Some researches described the flow field characteristics by means of field measurements[2]. Some extensive reviews of the currents induced by waves can be found in several 1 2

[email protected] [email protected]

scientific books[3-4]. In the surf zone, the so-called undertow is the most outstanding characteristics of the currents induced by broken waves, and several numerical models have been established to calculate the undertow distribution[5-10]. In this study, a vertical 2-D hydrodynamic numerical model was established to simulate the currents induced by waves across the surf zone. The traditional 2D radiation stress is actually a depth integration formula, which has been widely coupled into horizontal 2D hydrodynamic models. Advancing the vertical varied radiation stress calculation is necessary in extending 2DV or fully 3D hydrodynamic numerical models. Xia et al.[11] extended the radiation stress to its vertical profile in a new technical way, and the radiation stress formulae can be conveniently introduced into the numerical models. The technical way for calculating the radiation stress is efficient and can be extended to relevant numerical simulations of wave-induced currents[12]. Xia et al.[11] simulated the currents induced by waves in the surf zone, but the results didn’t coincide with the observations, especially, the undertow directed onshore. The neglecting of the breaking roller parameterization in Xia’s model is the main limitation in the undertow simulation. We coupled the breaking roller parameterization in the numerical model by means of specifying one additional shear stress boundary condition on the mean water level (MWL). In the surf zone, the wave-induced currents dominate the long term pollutant transport. A series of simulations were carried out and the results according to different release positions were further analyzed.

9th International Conference on Hydrodynamics October 11-15, 2010 Shanghai, China This study may be helpful in the evaluation of the environmental impact for the polluted water draining in the surf zone. 2 MODEL DESCRIPTION 2.1 Wave module 2.1.1 Wave transformation model The wave height H ( x ) is computed from the linear wave theory by using the wave energy flux balance: ∂ (1) [Cg ( x ) E ( x )] = − Dw ( x ) ∂x where C g ( x ) is the wave group velocity, E ( x ) the wave energy per surface unit and related to the wave height H ( x ) based on the linear wave theory. The wave energy dissipation rate Dw ( x ) induced by wave breaking is calculated by the following formula[13]: Dw ( x ) = WE ( x ) (2) where W is the fraction of broken wave and can be determined by the formula based on the relation between the local water depth D ( x ) , the wave height H ( x ) and the wave breaking criteria coefficient γ :

⎧ KCg ⎪ W =⎨ D ⎪ ⎩

⎛ γ 2 D2 ⎞ ⎜1 − H 2 ⎟ H ≥ γ D ⎝ ⎠ 0

(3)

H <γD

where K is a parameter and C g the wave group velocity. The coefficient γ is the ratio between the local wave height and the mean water depth. 2.1.2 Breaking roller model The breaking roller energy per surface unit Er ( x ) is resolved from the balance between the broken wave energy dissipation rate Dw ( x ) and the broken wave roller energy dissipation rate Dr ( x ) [14]: ∂ [ 2Cϕ ( x ) Er ( x )] = Dw ( x ) − Dr ( x ) (4) ∂x

where Cϕ ( x ) is the wave phase velocity because of the roller traveling on the top of each wave crest. In the present paper, the roller energy dissipation rate Dr ( x ) is parameterized as[15]: Er ( x ) Dr ( x ) = 2 β g (5) Cϕ ( x ) where β is a coefficient related to the wave steepness (usually β = 0.1 ).

2.2 Hydrodynamic module

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2.2.1 Vertical 2D hydrodynamic model The wave-induced currents are the residual flow of wave motion. The continuity and horizontal momentum equations are written as follows: ∂η ∂ ( Du ) ∂ω + + =0 (6) ∂t ∂x ∂σ ∂u ∂t −g

+u ∂η ∂x

∂u ∂x +

+

ω ∂u D ∂σ

=

⎛ υ ∂u ⎞ + 1 ∂ ⎛ υ ∂u ⎜ h ⎟ ⎜ v ∂x ⎝ ∂x ⎠ D ∂σ ⎝ D∂σ ∂

⎞ − ∂S xx (σ ) ⎟ ρ∂x ⎠

(7)

where η ( x, t ) is the surface elevation; u the current speed in x direction, D = h + η the total water depth; σ = ( z − η ) D the relation between σ coordinate and Cartesian coordinate z ; ω the vertical speed in σ direction; υh and υv the horizontal and vertical eddy viscosity, respectively. S xx (σ ) is for the parameterized radiation stress. The vertical velocity w in the physical domain can be calculated by formula:

⎛ ∂D + ∂η ⎞ + ⎛ σ ∂D + ∂η ⎞ ⎟ ⎜ ⎟ ⎝ ∂x ∂t ⎠ ⎝ ∂t ∂t ⎠

w = ω + u ⎜σ

(8)

Vertical integration of Eq. (6) gives the following continuity equation: ∂η ∂ 0 (9) + ∫ Dudσ = 0 ∂t ∂x −1 In the numerical solving process, the equation (9) is firstly integrated to calculate the water elevation η , and then the flow velocities can be resolved from equations (7) and (8)[16]. 2.2.2 Vertical variation of radiation stress The radiation stress formula (10) derived by Xia et al.[11]can be directly coupled into the present hydrodynamic model. 2k Eσ S xx (σ ) = E − + sinh 2 kD D (10) k (1 + σ ) sinh k (1 + σ ) D E ⎡ cosh k (1 + σ ) D ⎤ E

cosh kD

−

D

⎢⎣1 −

cosh kD

⎥⎦

2.2.3 Surface shear stress calculation The contribution of the breaking roller can be taken into account by specifying a shear stress boundary condition on the MWL. In the present hydrodynamic model, the following parameterized formula is adopted[17]: ∂u ( x, 0 ) Dr ( x ) (11) τ s ( x ) = ρυ v ( x, 0 ) = ∂z Cϕ ( x )

9th International Conference on Hydrodynamics October 11-15, 2010 Shanghai, China

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2.3 Pollutant transport module The governing equation is described in the σ coordinate system as the foregoing current model. ∂c ∂ ( uc ) ∂ ( ω c ) + + = ∂t ∂x D∂σ (12) ∂ ⎛ ∂c ⎞ ∂ ⎛ ∂c ⎞ ⎜ ε tv ⎟ + ⎜ ε th ⎟ D∂σ ⎝ D∂σ ⎠ ∂x ⎝ ∂x ⎠

phenomenon of the wave increases with closing to the breaking point. Veeramony et al.[7] simulated the flow in the surf zone by an extended set of Boussinesq equations based on the lowest-order weakly non-linear theory, and presented a better accurate simulation than the present model.

where ε tv and ε th the vertical and horizontal diffusion coefficient assumed to be the same as the flow eddy viscosity in this study. 3 HYDRODYNAMIC MODEL VALIDATION The hydrodynamic model was validated by a series of experiments with monochromatic waves. Wave heights and water set-up measurements were available from the experiments by Hansen and Svendsen[1]. The velocity profiles below the wave trough were available from the measurements by Cox et al.[18]. The schematic computational domain is shown in Fig.1.

Fig. 2 Comparison between simulation and measurement of wave heights (a) and set-up (b) for case 1

Fig. 1 Schematic showing for the computational domain (after from Veeramony[7])

3.1 Wave height and set-up comparison Table 1 presents the wave parameters for each of the cases.

Fig. 3 Comparison between simulation and measurement of wave heights (a) and set-up (b) for case2

Table 1 Wave parameters from Hansen and Svendsen[1] Case

T (s)

H ( m)

T/ gh

1

3.333

4.3

17.4

2

2.5

3.9

13.0

3

2.0

3.6

10.44

Fig.2, Fig.3 and Fig.4 show the comparisons for three cases. The simulated wave heights in the start shoaling zone were well coincided with the measurement. With closing to the wave breaking point, the discrepancy between the simulation and measurement was more obvious, and the most discrepant value appeared at the wave breaking point. After a short distance far away from the wave breaking point, the agreement was again good. The reason for the discrepancy around the breaking point may be that the present wave model is based on the linear wave theory, but the nonlinear

Fig. 4 Comparison between simulation and measurement of wave heights (a) and set-up (b) for case 3

3.2 Velocity comparison In the experiment by Cox et al.[18], the horizontal

9th International Conference on Hydrodynamics October 11-15, 2010 Shanghai, China velocities were gathered. Horizontal velocities in the vertical direction at six locations (Table 2) were measured. Fig. 5 shows the simulated velocity field in the vertical section. In the surf zone, the upper water flow onshore and the bottom water flows offshore, which is coincide with observations. Fig. 6 shows the comparison of the horizontal velocity between simulations and measurements. Locations L1 and L2 is outside the surf zone, Location L3 is in the transition region and Locations L4, L5 and L6 are in the inner surf zone. At locations L1 and L2, the measured velocities increased very quickly in a short distance away from the bottom, and then varied slowly in the vertical direction. Near the MWL, the velocities turned to the onshore direction. The simulation presented the basic characteristics of the vertical varied velocity, but gave a sharper velocity gradient in vertical direction than measurements. In the transition region (L3), the model underestimated the current. At locations L5 and L6, the accuracy of the simulation was the best. Table 2 Locations of measuring lines by Cox et al. (1995) Line b h(cm)

L1

L2

L3

L4

L5

L6

28

21.4

17.71

14.29

10.86

7.43

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as a kind of phase-averaged model, it is convenient to simulate the whole velocity field in the vertical section. The fluid circulation in the surf zone is reasonably presented, and it is helpful to illustrate the pollutant transport in the surf zone.

Fig. 6 Comparison of horizontal velocity profile between simulations and measurements. The ordinate z ′ = z + h is zero at the bottom

4 POLLUTANT TRANSPORT SIMULATIONS The numerical model was used to simulate the long term pollutant transport in the surf zone. The physical setup was as same as the experiments from Cox et al.[18]. Two different horizontal release points were specified across the surf zone, respectively. For each horizontal position, the release point was respectively set at the upper, the middle and the bottom of the water. The simulated cases are listed in Table 3. Table 3 Lists of the simulation cases Case1 Csee2 Case3

Fig. 5 Wave-induced currents simulation

In the present model, the calculated MWL is the wave period-averaged free surface, so no details about the flow between wave crest and trough can be presented. Although the zero flux condition is automatically met by the present model, the undertow is underestimated because of the onshore fluid between crest and trough being compulsively pushed down under MWL. The present model is not advanced in accuracy compared to some phase-averaged models[19], Boussenesq models[7] and NS equations models[8]. But

X=9m, Upper X=9m, Middle X=9m, Bottom

Case4 Case5 Case6

X=13m, Upper X=13m, Middle X=13m, Bottom

The concentration of the pollutant drained was specified as 1.0, and the total simulation lasted about 30 min which was about hundreds of wave periods. For each simulation case, the pollutant concentrations at four different moments were investigated (Fig.7). For each case, the polluted domain expanded gradually, and the concentration in the surf zone was obviously higher than that outside the surf zone. It is well known from the simulations that the efficiency of the diluting of the pollutant in the surf zone is lower. For pollutant draining in the nearshore zone, it is better to arrange the inlet outside the surf zone. According the simulations, the final polluted statuses are obviously different for vertical varying positions. For the upper setting of release point, the pollutant is transported onshore by the onshore current, and is transported offshore by the undertow. As a result, the polluted domain for this case is obviously wider than those for other two cases. The currents induced by waves in the surf zone are the main forces for the

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9th International Conference on Hydrodynamics October 11-15, 2010 Shanghai, China

pollutant transport, which is illustrated by the analysis of the velocity field and the pollutant transport simulations. Among the three vertical source positions, the bottom setting is the best one for pollutant to be transported outside the surf zone because of the offshore currents near the bottom appearing not only inside but also outside the surf zone.

e) Pollutant concentration evolution for case5

a) Pollutant concentration evolution for case1 f) Pollutant concentration evolution for case6 Fig. 7 Pollutant transport by wave-induced currents

5 CONCLUSION

b) Pollutant concentration evolution for case2

c) Pollutant concentration evolution for case3

We coupled the vertical varied radiation stress into the shallow water equation to simulate the wave-induced currents in the surf zone. The breaking wave roller was parameterized, which was introduced into the model by means of specifying one additional stress condition on the MWL. Based on the model analysis, this boundary condition was significant in the simulation of undertow, which was outstanding hydrodynamics in the surf zone. As a phase-averaged model, the present model is more efficient to simulate the velocity field in the vertical section. Because of the absence of the detailed wave breaking process simulated, it is difficult to improve the accuracy of the undertow simulation by the present model. In the surf zone, the currents induced by broken waves dominate the pollutant transport. The vertical circulation makes an outstanding difficulty for the pollutant to be transported outside the surf zone. According to this analysis, it is better to establish the pollutant draining inlet outside the surf zone. ACKNOWLEDGEMENTS This work was jointly sponsored by the National Natural Science Foundation of China (No.10702042) and the Shanghai Leading Academic Discipline Project, Project Number B206. The authors wish to thank the support of these funds.

d) Pollutant concentration evolution for case4

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