# 1DV bottom boundary layer modeling under combined wave and current: turbulent separation and phase lag effects

Abstract: On the basis of the Wilcox [1992] transitional k-ω turbulence model, we propose a new k-ω turbulence model for one-dimension vertical (1DV) oscillating bottom boundary layer in which a separation condition under a strong, adverse pressure gradient has been introduced and the diffusion and transition constants have been modified. This new turbulence model agrees better than the Wilcox original model with both a direct numerical simulation (DNS) of a pure oscillatory flow over a smooth bottom in the intermittently turbulent regime and with experimental data from Jensen et al. [1989] , who attained the fully turbulent regime for pure oscillatory flows. The new turbulence model is also found to agree better than the original one with experimental data of an oscillatory flow with current over a rough bottom by Dohmen-Janssen [1999] . In particular, the proposed model reproduces the secondary humps in the Reynolds stresses during the decelerating part of the wave cycle and the vertical phase lagging of the Reynolds stresses, two shortcomings of all previous modeling attempts. In addition, the model predicts suspension ejection events in the decelerating part of the wave cycle when it is coupled with a sediment concentration equation. Concentration measurements in the sheet flow layer give indication that these suspension ejection events do occur in practice.

## Summary (2 min read)

### 1. Introduction

- In coastal zones, the suspension associated to waves and currents in the bottom boundary layer can have an impact on both human activities and ecological equilibrium.
- Moreover suspension can also affect directly the life cycle of some species and hence play a role in their population dynamics.
- Similarly, when using the Wilcox [1992] transitional k-w turbulence model, that includes low-Reynolds-number effect, the eddy viscosity time series for oscillating boundary layers do not present any peak. [9].
- Even though such a sophisticated model is beyond the scope of this paper, it is clear that the strong turbulence activity which takes place during the decelerating phases of the cycle should be taken into account since it contributes to put more sediment in suspension.

### 2.1. Basic Formulation

- The basis of the Reynolds Averaged Navier-Stokes (R.A.N.S.) model the authors use to compute the turbulent bottom boundary layer under an oscillatory flow (with or without current) is the transitional k-w model devised by Wilcox [1992] in its 1DV formulation.
- In addition, turbulence damping by stratification is introduced into the original Wilcox formulation through coupling terms between turbulence and the density field r(z, t) = r0 + C(z, t)(rs r0) resulting from the sediment suspension (r0 is the fluid density, rs is the sediment density and C(z, t) is the sediment concentration per volume).
- The coupling terms are similar to those introduced by Lewellen [1977] in a k-L model.
- Hence, Wilcox [1992] proposed values for RK = 6, Rb = 8 and s = s* = 0.5 that give the best agreement both with experiments and direct numerical simulations of steady boundary layers with and without adverse or favorable pressure gradient.

### 2.2. Modeling of Turbulent Separation Under the Effect of an Adverse Pressure Gradient

- The authors now discuss the modeling of turbulence separation near flow reversal.
- Hence, the authors suggest to model this wall friction enhancement before flow separation under the effect of the adverse pressure gradient for fully developed turbulence and rough walls only, as follows. [14].
- Hence, to define the adverse pressure gradient in oscillatory flow, the authors should compare the pressure gradient action to the near-wall velocity.
- In contrast, a 0.1 phase resolution is required to obtain converged computations with the separation condition. [17].
- On Figure 2b, the authors show the computations with bsep = 20 for wvortex ranging from 30 to 3000 (usual values for wwall for this flow condition is 104).

### 3. Pure Oscillatory Flow Over a Smooth Bottom

- Velocity , Reynolds stress and turbulent kinetic energy vertical profiles through the boundary layer at different phases during half oscillation are also plotted.
- On Figure 6, the nondimensional bottom shear stress time series (bottom shear stress time series divided by the maximum bottom shear stress) computed using the original Wilcox model and the new one are plotted.
- To compare the theoretical predictions with the experimental data, this figure should be compared to Figure 9 of Jensen et al.
- It is then clear that in the original Wilcox model, the laminar-turbulent transition develops much quicker for Re larger than 3.3 104, whereas the new model with modified value for RK and Rb gives results closer to the measurements.

### 4. Oscillatory Flow Plus Current Over a

- The k-W Model Versus Tunnel Experiments 4.1. Dohmen-Janssen [1999] ClearWater Experiments [27], also known as Rough Bottom.
- In addition, the authors think the values they suggest for wwall and bsep to model secondary humps at the end of the decelerating phase will give physical and realistic results for usual field conditions since experiments G4 and G5 correspond to drastic field conditions.
- Nevertheless, concentration peaks are also observed in time series measured using optical conductivity probes further from the bottom.
- A significant discrepancy still remains between the predicted and the measured values.
- The model predictions can be improved at all levels by taking into account the intergranular forces in the ‘‘sheet flow’’ layer (highly concentrated bottom layer).

### 5. Conclusions

- A new transitional k-w model has been devised introducing a turbulent separation condition under adverse pressure gradient and modifying the diffusion and transition constants of the Wilcox [1992] original k-w transitional model.
- The authors are thus able to reproduce the wall shear stress sharp increase, which takes place at transition in good agreement with Jensen et al. data.
- The change of the diffusion constants improves also the description of the vertical distribution of both velocity and Reynolds stress compared to the original transitional Wilcox [1992] model.
- This feature has never been reproduced in standard R.A.N.S. models.
- This work was funded by the EC through a MAST-III project SEDMOC (contract MAS3-CT97-0115).

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### Cites background or methods from "1DV bottom boundary layer modeling ..."

...(ii) The effects of high sediment concentration on flow turbulence are not modelled [17, 39, 53] or turbulence suppression is modelled in a simple way by adding a buoyancy term in the kinetic energy equation [36, 54, 55]....

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...[54] using a k–ω model and by Holmedal et al....

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...The 1DV time-varying Reynolds averaged Navier–Stokes (RANS) model is based upon the k–o turbulent model, for oscillatory flow plus current by Guizien et al. (2003); in this, turbulence damping by stratification is taken into account in a linearized form (O 1⁄4 0)....

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### Additional excerpts

...All these features are typical of oscillatory wave-current boundary layers and have been observed in clear fluids both experimentally [e.g., Jensen et al., 1989] and numerically [e.g., Guizien et al., 2003 ]....

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##### References

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### "1DV bottom boundary layer modeling ..." refers methods in this paper

...[16] Second, we model the wall shear stress enhancement, when these conditions are fulfilled, prescribing a much lower value for the energy dissipation rate at the wall, wwall, than the one given by the Wilcox condition cited above [Wilcox, 1998, p. 177]....

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...At the bottom, we prescribe the true value of k and u [Saffman, 1970], meanwhile the value of w is fixed depending on whether a smooth or rough wall should be modeled [Wilcox, 1998, p. 177]....

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...In addition, under stationary conditions with an adverse pressure gradient and for low-Reynolds-numbers, standard k-w models perform better than standard k- models [Wilcox, 1998]....

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...The equations (1)–(3) for u, k, and w are solved using the implicit finite control volume method of Patankar [1980]...

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### "1DV bottom boundary layer modeling ..." refers background or methods in this paper

...In this paper, starting from the Wilcox [1992] transitional k-w model, a new transitional k-w turbulence model is proposed in order to improve the 1DV modeling of oscillating bottom boundary layers....

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...change of the diffusion constants improves also the description of the vertical distribution of both velocity and Reynolds stress compared to the original transitional Wilcox [1992] model....

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...Hence, Wilcox [1992] proposed values for RK = 6, Rb = 8 and s = s* = 0....

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...A much smaller narrow peak, is also present near flow reversal in the eddy viscosity time series computed using a standard k-w turbulence model [Wilcox, 1988], whereas a k-L turbulence model [Huynh Than et al., 1994] does not produce such peaks....

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...boundary layer under an oscillatory flow (with or without current) is the transitional k-w model devised by Wilcox [1992] in its 1DV formulation....

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