ANALYSIS OF A TRANSITIONAL OSCILLATORY BOUNDARY LAYER BY k-e MODEL

Ahmad Sana and Shuy Eng Ban

School of Civil and Structural Engineering

Nanyang Technological University

Singapore 639798

Tel. (65) 790-5326

Fax. (65) 791-0676

E-mail: cshuyeb@ntu.edu.sg

Abstract: Three versions of low Reynolds number k-e model have been tested against the DNS data for a transitional oscillatory boundary layer. The original model is observed to predict the transitional characteristics better than the other versions. However, the later versions perform better by virtue of the magnitude of velocity, turbulent kinetic energy and bottom shear stress. 

Keywords: oscillatory boundary layer, k-e model, bottom shear stress, turbulence

1    Introduction

The coastal bottom boundary layers undergo transition from laminar to turbulent or vice versa in response to changing field conditions. Due to the importance of transitional characteristics of bottom boundary layer in relation to the sediment movement, a lot of research has been done in the past. Although many experimental and analytical studies have been carried out, the idea of using turbulence models to tackle this phenomenon is relatively new. With the availability of excellent computing facilities at affordable costs, this option is gaining more popularity among the hydraulic and coastal engineers. The benefit of using a good turbulence model is that it produces detailed boundary layer properties to facilitate precise estimation of sediment movement under a variety of hydraulic conditions. For a turbulence model to be good, an essential requirement is computational economy with reasonable accuracy. With the large number of turbulence models available, it is very difficult for practicing engineers to choose the most suitable one. The present study deals with the application of a low Reynolds number k-e model to a transitional oscillatory boundary layer. The term low Reynolds number implies that this model is applicable over the whole cross-stream dimension including the low Reynolds number region (viscous sublayer).

2    k-e model and oscillatory boundary layers

The k-e model was originally developed by Jones and Launder(1972) for steady flow phenomena. Later it was applied to a number of turbulent boundary layers including the oscillatory ones and a number of modifications were proposed to improve its efficiency. Patel et al. (1985) reviewed some of the versions of two-equation models in relation to steady flow phenomena. Tanaka and Sana(1994) reviewed some of the older versions with reference to oscillatory boundary layer properties by using the available experimental data. As a result of this study it was found that the original model by Jones and Launder(1972)(JL Model) performed better than the rest of models tested, especially considering the transitional properties of oscillatory boundary layers. A similar conclusion has been obtained by Sana and Tanaka(2000) after reviewing the original and four newer versions with reference to the available Direct Numerical Simulation (DNS) data for oscillatory boundary layers. Another interesting result of this study concerns the nature of damping functions used in Low Reynolds number k-e models. The damping function used in many of the newer versions involves the wall coordinates y⁺ (=yu_f /n, y = cross-stream distance, u_f = shear velocity and n = kinematic viscosity). In case of oscillatory flow the bottom shear stress goes to zero twice in a wave cycle and at that time the damping function becomes zero in the whole of the cross-stream dimension, which is physically incorrect.

In the present study, two of the available versions of k-e model that meet the requirements of damping function outlined by Sana and Tanaka(2000) have been tested against the DNS data by Spalart and Baldwin(1989) for sinusoidal oscillatory boundary layer on a smooth boundary. Here the k-e models proposed by Yang and Shih(1993)(YS Model) and Abe et al.(1994)(AKN Model) have been tested.

2.1    Governing equations

Using the eddy viscosity concept, the equation of motion, for one dimensional oscillatory boundary layer, may be written as;

                       (1)

where, u; the velocity in x-direction, U; free-stream velocity, t; time, y; cross-stream dimension and n_t; eddy viscosity. According to the general form of k-e model the eddy viscosity is expressed as;

                              (2)

the turbulent kinetic energy (k) transport equation is;

                    (3)

and the transport equation of turbulent kinetic energy dissipation rate  is;

               (4)

Here, C_m , C₁, C₂,  and are model constants and f_m , f₁, f₂, D and E are model functions. The value of C_m is 0.09 and those of other model parameters, for the three versions considered here, are presented in Table 1. In this table, ,  and  (Kolmogorov’s length scale).

2.2    Boundary conditions and numerical method

At the solid boundary, no slip boundary condition and at the free-stream, gradients of velocity, turbulent kinetic energy and its dissipation rate were equated to zero. The wall boundary condition for turbulent kinetic energy dissipation rate is given in Table 1.

   Table 1    Model parameters for the low Reynolds number k-e model

A Crank-Nicolson type implicit finite difference scheme was used here. The grid spacing near the wall was varied exponentially in order to achieve better accuracy. In space 100 and in time 6000 steps per wave cycle were used. The convergence was based primarily on velocity, k and e and then on maximum bottom shear stress. The convergence limit was set to  in the present study.

3    Results and discussion

The velocity profile in oscillatory boundary layers shows a distinct overshooting which is efficiently predicted by the models tested here as shown in Figure 1. The cross-stream co-ordinate is normalized by the Stokes length d_l. The YS and AKN models show better performance in predicting the velocity profile than that of the JL model from wt=0^˚ (t = time) to wt=120^˚, i.e. during acceleration phase and early part of deceleration. At wt=150^˚ all the three models show deviation from the DNS data, which shows the poor performance of these models under deceleration.

Fig. 1    Comparison of velocity profile between JL, YS, AKN model and DNS data

The comparison of turbulent kinetic energy k (normalized by maximum free-stream velocity U_o) profiles is presented in Figure 2. At wt=30^˚ and wt=180^˚, the YS model clearly demonstrates its supremacy over the other two models by reproducing the k profile especially near the wall. During deceleration (wt=150^˚) none of the models can predict the cross-stream peak k value, the AKN model performs better far from the wall, though. An interesting feature of the transitional oscillatory boundary layers, i.e. a sudden increase in wall shear stress (t_o) in deceleration (wt=150^˚) has been very well predicted by the JL model and to some extent by the YS and AKN models as may be observed in Figure 3. The wall shear stress is normalized by mass density r and maximum free-stream velocity U_o.

Fig. 2    Turbulent kinetic energy profiles

Fig. 3    Wall shear stress, free-stream velocity and acceleration

The magnitude of the maximum wall shear stress predicted by the YS and AKN models is closer to the DNS value than that by the JL model. It can be observed that this sudden increase in shear stress occurs at the inflection point in the temporal variation of acceleration (d(U/U_o)/d(wt)).

4    Conclusions

The newer versions of k-e model perform better than the JL model in predicting near wall velocities, turbulent kinetic energy and bottom shear stress are concerned. During deceleration (wt=150^o) none of the tested models could predict the velocity as well as turbulent kinetic energy precisely. Further investigation on the predictive ability of these models has to be carried out in order to propose the possible improvement for better prediction during deceleration.

References

Abe, K., Kondoh, T. and Nagano, Y., 1994, A new turbulence model for predicting fluid flow and heat transfer in separating and reattaching flows-I. Flow field calculations, Int. J. Heat Mass Trans., Vol. 37, No.1, pp.139-151.

Jones, W. P. and Launder, B. E. 1972, The prediction of laminarization with a two-equation model of turbulence. Int. J. Heat Mass Transfer Vol.15, pp.301-314.

Patel, V. C., Rodi, W. and Scheuerer, G. 1985, Turbulence models for near-wall and low Reynolds number flows: A review. AIAA J. Vol. 23, No.9, pp.1308-1319.

Sana, A. and Tanaka, H. 2000, Review of k-e model to analyze oscillatory boundary layers, Journal of Hyd. Engg., Vol. 126, No.9, pp.701-710.

Spalart, P.R. and Baldwin, B. S. 1989, Direct simulation of a turbulent oscillating boundary layer, In Turbulent Shear Flows 6, pp.417-440.

Tanaka, H. and Sana, A. 1994, Numerical study on transition to turbulence in a wave boundary layer. In Sediment Transport Mechanism in Coastal Environments and Rivers, pp.14-25.

Yang, Z. and Shih, T. H., 1993, New time scale based k-e model for near-wall turbulence, AIAA J., Vol. 31, No.7, pp.1191-1198.