Reynolds wave and turbulent stresses

Figure 4.11: Experiments PoW1–5, paddle waves with opposing wind (different reflective conditions, same LDV section). Mean horizontal u (red filled triangles) and vertical w (empty circles) velocities, non dimensional with respect to the velocity scale H_tT⁻¹. Error bars are one standard deviation.

Figure 4.12: Experiments PoW2a–d, paddle waves with opposing wind (same re-flective conditions, different LDV measurement sections). Mean horizontal u (red filled triangles) and vertical w (empty circles) velocity, non dimensional with re-spect to H_tT⁻¹. Error bars are one standard deviation.

The theoretical model of Chapter §1 gives the tools to analyse the Reynolds wave stresses. From (1.13), the vertical profile of the Reynolds wave shear stress can be expressed at the first order as

−euw = gkae ²_isinh [2k (z + h)]

sinh (2kh) Krsin(2kx + ∆ϕ). (4.3) We highlight the importance of the term −uew, since for progressive wavese it is null and appears with partial reflection conditions. We have shown in Chapter §3 that neglecting this term, also for small values of the re-flection coefficient and phase shift, could lead to significant errors in the interpretation of the data in laboratory and field campaigns.

Figure 4.13 shows the comparison between theoretical and experimental values of −uew, for different reflective conditions (experiments (PoW1–5).e Figure 4.14, reports the results ofeuw for experiments 2a–d, which were donee in similar reflective conditions but different LDV measurement sections in order to observe the spatial variation. As found in the previous Chapter (§3) and also in literature (see Olfateh et al., 2017), the Reynolds wave shear stress euw determines momentum transfer with different sign and intensity,e depending on reflective conditions and measurement position in x, and its value is also predictable with an appropriate reflection analysis.

We remind that the reflection analysis (see 2.3.3) is based on the US signals. Since we had two different US acquisitions for each experiment, one with only paddle waves (no wind) and one with paddle waves plus opposing wind, two different vertical profiles of the theoretical euw are calculated.e The reflection parameters used to obtain a good theoretical fit with the experimental velocity correlation are those with paddle waves plus opposing wind; for only-paddle waves the vertical profile changes significantly. It is a further checked that the presence of opposing wind modifies the reflection coefficient K_r and the phase shift ∆ϕ, as reported in §4.2), and it affects also the wave shear stress (thus the net momentum transfer at the interface between air and water).

The termsueeu andwew represent the diagonal components (i.e., the nor-e mal stresses) of the second-order Reynolds wave stress tensor, represented in equation (1.25). Their theoretical relation derives from the solution of

Figure 4.13: Experiments PoW1–5, paddle waves with opposing wind (different reflective conditions). Vertical profile of the wave velocity covariance uew. Solide lines represent theoretical velocity correlation at first order, while symbols are experimental data. Dashed lines are confidence interval with probability of 95%.

In blue, reflection parameters K_r and ∆ϕ are evaluated in the presence of wind, while in green they are evaluated without wind.

Figure 4.14: Experiments PoW2a–d, paddle waves with opposing wind (same re-flection conditions, different LDV section). Vertical profile of the Reynolds wave shear stress uew. Solid lines represent theoretical velocity correlation at first or-e der, symbols are experimental data. Dashed lines are confidence interval with probability of 95%, error bars two standard deviations. In blue solid line, reflec-tion parameters K_rand ∆ϕ are evaluated in the presence of wind, while in green dash-dotted line Kr and ∆ϕ are evaluated without wind.

the analytical model of Chapter §2; in particular, the horizontal wave nor-mal stressueeu is the time average of the product of the horizontal velocity of equation (1.10), and at the first order reads

ueeu = gka²_icosh [k (z + h)]

sinh (2kh) 1 + K_r²− 2K_rcos(2kx + ∆ϕ) . (4.4) Analogously, time averaging the product of the vertical velocity of equa-tion (1.11) yields the vertical wave normal stresswew, which at the first ordere results

wew = gkae ²_isinh [k (z + h)]

sinh (2kh) 1 + K_r²+ 2K_rcos(2kx + ∆ϕ) . (4.5) Figures 4.15 and 4.16 show the wave normal stresses of the wave-induced Reynolds stress tensor for experiments PoW1–5 (different partial reflection conditions, same LDV position) and 2a–d (same partial reflection condi-tions, different LDV positions), respectively. Both the Reynolds wave nor-mal stresses ueu ande wew, show good agreement with the theoretical modele within the statistical and experimental uncertainties. The results show also that reflection induces a spatial variation of the normal stresses (they are both modulated along x) and exerts a stronger modulation on the horizon-tal componentueeu than on the vertical normal stresswew.e

Figure 4.17 reports the Reynolds turbulent shear stress gu⁰w⁰ for experi-ments PoW1–5 (different reflective conditions and same LDV measurement section). Results show that opposing wind causes a negative turbulent shear stress near the surface, which suggest a momentum transfer down-wards (from air to water). Furthermore, the turbulent shear stress goes to zero at ≈ 0.5 z/h, i.e. half the water column. For decreasing reflection (from PoW1 to 5), the turbulent shear stress is larger and reach lower water levels.

Since the forcing of the process at the surface is the wind action, we are not surprised that also for paddle waves with opposing wind the Reynolds turbulent shear stress yields a momentum transfer from the wind to the wave.

Figure 4.15: Experiments PoW1–5 (different reflective conditions, same LDV sec-tion): vertical profile of the Reynolds wave normal stresses. Symbols are experi-mental the horizontalueu (blue stars) and verticale ueeu (red triangles) components, solid lines are theoretical values. Error bars are two standard deviations.

Figure 4.16: Experiments PoW2a–d (same reflection, different LDV sections):

vertical profile of the Reynolds wave normal stresses. Symbols are experimental the horizontal euu (blue stars) and verticale euu (red triangles) components, solide lines are theoretical values. Error bars are two standard deviations.

Figure 4.17: Experiments PoW1–5, paddle waves with opposing wind (different reflective conditions, same LDV position): turbulent Reynolds shear stress gu⁰w⁰ along the vertical. Symbols are experimental data, error bars are two standard deviations.

Figure 4.18: Experiments PoW2a–d, paddle waves with opposing wind (same re-flection, different LDV sections): turbulent Reynolds shear stress gu⁰w⁰ along the vertical. Symbols are experimental data, error bars are two standard deviations.

Figure 4.18 shows the Reynolds turbulent shear stress in the vertical for experiments PoW2a–d (same reflective conditions, different LDV section).

The vertical profile of u⁰w⁰ differs for each test and suggests some depen-dence on the LDV measurement section, even though the trend is not clear.

Also in these experiments, observations indicate a net momentum transfer from wind to the water field.

Figures 4.19 and 4.20 show the diagonal terms of the turbulent Reynolds stresses tensor for tests PoW1–5 and PoW2a–d, respectively. Their values increase approaching the air/water boundary layer; it means that much of the kinetic energy is concentrated at the interface between air and water.

In particular, it is observed that the horizontal normal stress gu⁰u⁰ is always larger than the vertical gw⁰w⁰, suggesting a dominant component in the wind direction. At lower levels, the normal stresses decrease and become uniform in the vertical, indicating a rotation of the Reynolds turbulent stress ten-sor. For increasing reflection, normal stresses show a reduction of their values in the measured water column and have less variability (deviations are smaller). Experiments PoW2a–d show similar vertical profiles at differ-ent LDV sections and also a slight spatial variation of the normal stresses appears, more evident near the surface.

In general, observations of the non-dimensional Reynolds turbulent stress tensor components indicate that reflection acts as a constraint on the bulent flow field induced by an opposing wind; for lower reflection, the tur-bulent stresses are larger with higher deviations and diffuse at lower water levels, while for higher reflection the turbulent stresses have less intensity and smaller variability. The vertical profile of the turbulent stresses, and in particular the shear component, shows a transfer of momentum from the air flow to the water column which deserves further investigations through a quadrant analysis.