EDDY VISCOSITY PROFILES FOR WAVE BOUNDARY LAYERS : VALIDATION AND CALIBRATION BY A kω MODEL

Eddy viscosity in wave boundary layers is a key parameter in coastal engineering. Two analytical eddy viscosity profiles present a particular interest for practical applications: the parabolic-uniform profile (Myrhaug 1982, van Rijn 1993, Liu and Sato 2006) and the exponential-linear profile (Gelfenbaum and Smith 1986, Beach and Sternberg 1988, Hsu and Jan 1998, Absi 2010). The aim of our study is to assess and validate these two profiles by: (1) investigation of eddy viscosity in steady fully developed plane channel flow; (2) comparisons with numerical results of the two equation baseline (BSL) k-ω model (Menter 1994, Suntoyo and Tanaka 2009). Our study shows that these two profiles are able to describe the eddy viscosity distribution in the wave bottom boundary layer but for different wave conditions given by the parameter am/ks, where am is the wave orbital amplitude and ks the equivalent roughness. The exponential-linear profile is adequate for am/ks <500, while the parabolic-uniform profile is more appropriate for am/ks ≥500. We suggest empirical formulations for the different coefficients which appear in these two profiles based on numerical results of the BSL k-ω model.


INTRODUCTION
The resolution of some coastal engineering problems such as coastal erosion, transport of sediments and pollutants needs the knowledge of the boundary layer flow in the vicinity of the sea bed.The turbulent boundary layer generated by a sinusoidal wave has been the subject of many researchers: Kajiura (1968), Brevik (1981), Myrhaug (1982), Sleath (1990), Fredsoe and Deigaard (1992), Nielsen (1992), You et al. (1992), van Rijn (1993), Tanaka and Thu (1994), Madsen and Salles (1998) and others.
The distribution of eddy viscosity within the oscillatory boundary layer generated by waves in shallow waters is a key parameter in coastal engineering.Adequate description of eddy viscosity distribution in boundary layers is needed for accurate predictions of velocity profiles.Eddy viscosity is also related to sediment diffusivity which is involved in the calculation of sediment concentration profiles and therefore for the prediction of sediment transport by waves.For practical applications in coastal engineering, the eddy viscosity is obtained by mathematical models often based on statistical modeling of turbulence as two-equation models (k-ε, k-ω, …), or by simple analytical or empirical models.Different assumptions were made about the variation of eddy viscosity with height in the bottom boundary layer: Kajiura (1968), Lundgren (1972), Smith (1977), Grant and Madsen (1979), Brevik (1981), Myrhaug (1982).We will investigate particularly two analytical eddy viscosity profiles for wave boundary layers namely, the parabolic-uniform profile (Myrhaug 1982, van Rijn 1993, Liu and Sato 2006) and the exponential-linear profile (Gelfenbaum and Smith 1986, Beach and Sternberg 1988, Hsu and Jan 1998, Absi 2010).
In order to assess and validate these profiles, we will first investigate eddy viscosity profiles obtained: (1) for steady flows, as a test case we will consider the fully developed plane channel flow; (2) from a two-equation k-ω model for the case of oscillatory flows.The validation and calibration of analytical profiles need comparisons with the period-averaged eddy viscosity obtained from the BSL kω model for different wave conditions through the parameter s m k a . The aim of is this study is to provide engineers simple analytical tools for practical use.

TIME-INVARIANT EDDY VISCOSITY PROFILES FOR BOTTOM BOUNDARY LAYERS
Eddy viscosity in turbulent oscillatory boundary layers can be assumed to be a time-independent and real-valued parameter (You et al. 1992).Different time-invariant eddy viscosity profiles (figure 1) were proposed for oscillatory flows by Kajiura (1968), Brevik (1981), Myrhaug (1982) and for combined steady and oscillatory flows by Lundgren (1972), Smith (1977), Grant and Madsen (1979).Among the eddy viscosity profiles for wave boundary layers, two seems to present a particular interest.
The first is given by a parabolic variation of eddy viscosity until the half of the boundary layer thickness then a uniform value (Myrhaug 1982, van Rijn 1993, Liu and Sato 2006, van Rijn 2007).We refer to this first profile as the parabolic-uniform profile .It is given by: where κ is the von Karman constant (=0.41), * u the friction velocity, z the distance from the bottom and δ the boundary layer thickness.
The second is given by an exponential law modulated by a linear function (Gelfenbaum and Smith 1986, Beach and Sternberg 1988, Hsu and Jan 1998, Absi 2010).We refer to this second profile as the exponential-linear profile .It is given by: where α and A C are two coefficients.
We write equations ( 1) and ( 2) in dimensionless form respectively as: In equation (3), we introduced an adjustment coefficient L C .The initial profile, given by equation (1), didn't contain this coefficient ( ) In order to assess and validate these profiles, we will investigate eddy viscosity profiles obtained from: (1) Steady flows, we will consider as a test case the fully developed plane channel flow (figure 2) which is considered to be the simplest and most idealized boundary layer flow (2) A two-equation k-ω model applied to oscillatory flows    2) allows accurate description (white dashed lines, figure 4) of DNS in the equilibrium region (Absi et al. 2011, Absi 2012).For 50 < + z , the eddy viscosity profile is based on van Driest mixing length equation (white solid lines, figure 4). Figure (4) shows that, for high friction Reynolds numbers, this layer (z + <50) tends to be very small compared to the main profile given by Eq. 2 (figure 4.d).This could explain why in wave bottom boundary layers with large Reynolds numbers this layer is not observed and the profile is described mainly by Eq. (2).Eddy viscosity profiles of figure (4) are for a flat smooth wall, bottom roughness increases this effect and seems explain clearly why eddy viscosity is described only by Eq.( 2) in wave boundary layers.

Eddy viscosity distribution obtained by a two-equation k-ω model
Eq. ( 2) for eddy viscosity is validated for the case of steady plane channel flow.However for use in wave boundary layers, we need to assess this equation for the case of oscillatory flows.Eq. ( 2) is therefore analyzed by the baseline (BSL) k-ω model proposed by Menter (1994).This model allows accurate prediction of velocity profiles in oscillatory boundary layers (Suntoyo and Tanaka 2009).The BSL k-ω model is a two-equation model that gives results similar to the k-ω model of Wilcox (1988) in the inner boundary layer but changes gradually to the k-ε model of Jones and Launder (1972) towards the outer boundary layer and the free stream velocity.The blending between the two regions is done by a blending function F1 changing gradually from one to zero in the desired region.Equations of turbulent kinetic energy k, specific dissipation ω and eddy viscosity are given respectively by: (5)   Even if the eddy viscosity is highly time-dependent (figure 6), the period-averaged dimensionless eddy viscosity (Figure 7) has a shape which is well described by the exponential-linear analytical profile given by Eq. ( 2) for z/z h < 0.6 (figure 7) where z h is the water depth or the distance from the wall to the axis of symmetry (for oscillatory water tunnel experiments) or free surface.At the present stage, the exponential-linear profile (Eq.2) is validated by both eddy viscosity profiles for steady plane channel flow (figure 4) and with period-averaged eddy viscosity obtained by the BSL k-ω model for sinusoidal wave for 115 = s m k a (figure 7).However, it is important to provide a validation through comparisons for different wave conditions.

CALIBRATION OF THE TWO ANALYTICAL EDDY VISCOSITY PROFILES
Our study shows that the coefficients of the two analytical eddy viscosity profiles (parabolicuniform and exponential-linear) are not constants and should depend on wave conditions through the parameter (2) (dashed line).Even for the case of asymmetric wave, the period-averaged dimensionless eddy viscosity has a shape which is well described by Eq. (2) for z/z h < 0.5 (figure 11.b).Figures (7) and (11.b) shows that the period-averaged eddy viscosity profile for sinusoidal wave is different from the profile of asymmetric wave.This indicates that the period-averaged eddy viscosity profile should depend on the wave non-linearity parameter given by N i =(U 1 +U 2 )/(2U 1 ) or N i =U c /û, where U c is the velocity at wave crest and û is the total velocity amplitude.We need therefore, for asymmetric waves, a specific calibration for parameters of Eq. ( 2).

CONCLUSIONS
The aim of is this study was to provide simple analytical tools for practical use in coastal engineering.We validated and calibrated two analytical eddy viscosity profiles by numerical results of a two-equation k-ω model for the case of sinusoidal wave for different wave conditions (figure 12).Our study shows that: -For practical applications the period-averaged eddy viscosity could be described by simple analytical formulations -The parabolic-uniform profile allows a good description of eddy viscosity for 500 ≥

Figure 1 .
Figure 1.Assumptions for the variation of eddy viscosity with height in the bottom boundary layer (adapted from Fredsoe and Deigaard, 1992).

Figure 2 .(
Figure 2. Test case: steady fully developed plane channel flow with smooth walls

Figure 3 .
Figure3shows TKE profiles given byNezu and Nakagawa (1993) andAbsi (2008), while figure (4) presents eddy viscosity profiles.In these figures, variables with the superscript of + are those nondimensionalized by the friction velocity and the kinematic viscosity as ν / * u z z = +

Figure 5 .
Figure 5. Summary of the approach used for the present study.

Figure ( 6
Figure (6) presents temporal and spatial variation of dimensionless eddy viscosity for a sinusoidal wave given by ( ) ( ) t U t U ω sin 0 =

Figure 7 .
Figure 7. Period-averaged dimensionless eddy viscosity for a sinusoidal wave.

Figure ( 8
Figure (8) shows that the parabolic-uniform profile (Eq. 1) allows a good description of eddy viscosity for 500 ≥ s m k a

Figure
Figure 8. Validation of analytical eddy viscosity profile for

Figure
Figure 9. Validation of analytical eddy viscosity profile for

Figure 10 .
Figure 10.Calibration of the two eddy-viscosity profiles

Figure 11 .
Figure 11.Dimensionless eddy viscosity for asymmetric wave Ni=0.67;(a) Temporal and Spatial Variation obtained by BSL k-ω model; (b) Period-averaged dimensionless eddy viscosity parabolic-uniform profile needs an adjustment parameter L C -Coefficients in the two analytical eddy viscosity profiles (parabolic-uniform and exponentiallinear) are not constants and depend on parameter s m k a , we proposed equations for these coefficients -For asymmetric waves, eddy viscosity profile depends on the wave non-linearity parameter and requires therefore a specific calibration.

Figure 12 .
Figure 12.Summary of the present study