A Numerical Study on Water Waves Generated by A Submerged Moving Body in A Two2Layer Fluid System 3

Transcription

1 China Ocean Engineering, Vol. 23, No. 3, pp Ζ 2009 Chinese Ocean Engineering Society, ISSN A Numerical Study on Water Waves Generated by A Submerged Moving Body in A Two2Layer Fluid System 3 YANGJia2Zhen ( ), NG Chiu2On ( ) 1 and ZHANG Dao2Hua ( ) Department of Mechanical Engineering, The University of Hong Kong, Hong Kong, China (Received 28 July 2008 ; received revised form 23 February 2009 ; accepted 5 May 2009) ABSTRACT This is a numerical study on the time development of surface waves generated by a submerged body moving steadily in a two2layer fluid system, in which a layer of water is underlain by a layer of viscous mud. The fully nonlinear Navier2 Stokes equations are solved on FLUENT with the Volume2of2Fluid (VOF) multiphase scheme in order to simulate the free surface waves as well as the water2mud interface waves as functions of time. The numerical model is validated by mimick2 ing a reported experiment in a one2layer system before it is applied to a two2layer system. It is found that the presence of bottom mud in a water layer can lead to large viscous damping of the surface waves. For the investigation of the problem systematically, the effects of the Froude number and the mud layer thickness, density and viscosity relative to those of water are evaluated and discussed in detail. Key words : wave2mud interaction ; solitary waves ; underwater moving body 1. Introduction Surface waves generated by underwater objects have long been drawing attention owing to the con2 cern over safety of activities in approach channels and harbors. In a shallow coastal environment, the water column is very often underlain by cohesive sediments or marine mud, and the surface waves, which are largely long waves, can be materially affected by the viscous mud. It is important that the bottom mud should be taken into account in the modeling of near2shore water waves. The phenomenon of solitary waves generated by a moving disturbance in a one2layer system was first reported by Thews and Landweber (1935) through a series of towing tank experiments. Some simi2 lar experiments were later carried out by Huang et al. (1982), Sun (1985), Ertekin et al. (1986), and others. Solitary waves were examined numerically by Wu and Wu (1982), who solved the general2 ized Boussinesq (g2b) equation for waves induced by a surface pressure distribution or a bottom distur2 bance. The phenomenon of solitary waves propagating ahead of the disturbance was clearly seen in their study. Akylas (1984) and Cole (1985) used the same model to study various forcing types and discov2 ered similar results as the g2b model. A comprehensive investigation was done by Lee et al. (1989) on nonlinear waves generated by a disturbance in the form of a bump moving at a transcritical velocity 3 The work was supported by the Research Grants Council of the Hong Kong Special Administrative Region, China ( Grant No. HKU 7199/ 03E) 1 Corresponding author. E2mail : hk

2 442 near the bottom in a shallow water system. Both numerical modeling and water flume experiments were conducted to show the phenomena of upstream advancing solitary waves and downstream waves generat2 ed by the underwater disturbance. The numerical methods were based on the generalized Boussinesq (g2b) equations and forced Korteweg2de Vries (f KdV) equation, both with a forcing term. Their ex2 perimental study was focused on cases in which the Froude number was near the critical value and re2 sults were compared with the above2mentioned two inviscid numerical models. Comparison results were found to be good for both g2b and f KdV models. However, many assumptions have been made when the g2b and f KdV models were used by Lee et al. (1989), such as the introduction of an effective bump height to account for the viscous effect. Therefore the actual agreement with experiment remains uncertain. Zhang and Chwang ( 1996) conducted a numerical study based on the fully nonlinear Navier2 Stokes (NS) equations with the exact free2surface boundary conditions for time2dependent water surface waves. Their NS model could accurately simulate the viscous effect on flow around the body and in the gap between the bump base and the channel bottom. The model was validated by comparison of results with the experimental data of Lee et al. (1989), and the numerical results of the f KdV and g2b mod2 els. The agreement with the data was very good, and improvement in prediction was found extensively over the g2b and f KdV models at various speeds. However, a drawback of this method is that the kine2 matic boundary condition, which is imposed by means of an Eulerian method, only allows a single2val2 ued free surface profile. Therefore, it cannot handle cases of strong disturbances, in which wave breaking is likely to occur as reported in the experiment of Lee et al. (1989). Zhang and Chwang (1999) later used their method to study the nonlinear waves generated by a submerged body in a two2 dimensional shallow water channel. The general characters of the solitary waves and the effect of block2 age due to the presence of the body in the channel were discussed in detail in their work. Another mechanism that gives rise to solitary wave generation is bottom topography, i. e., when the bottom is no longer flat but with a certain prescribed profile. Many studies have been focused on positive or negative bottom forcing ( e. g., Grimshaw and Smyth, 1986 ; Wu, 1987 ; Camassa and Wu, 1991 ; Sue et al., 2005 ; Binder et al., 2006 ; Grimshaw et al., 2007). The interaction be2 tween the progressive wave and the bottom topography was extensively discussed in these studies. Zhang and Chwang (2001) carried out a very detailed investigation on the individual effects of a for2 ward2 and backward2step bottom. Their results based on the model of Zhang and Chwang (1996) show that a forward2step is responsible only for the upstream2running solitary waves while a backward2step is responsible only for the generation of a depressed region with the cnoidal waves following it. This im2 plies that the channel bottom geometry can in general have large effects on both the upstream and downstream waves. If the problem is extended to a two2layer system, the influence can be more com2 plicated owing to the additional effects of the displacement of the water2mud interface. Many investigations on water2mud interaction have also been carried out, focusing largely on the bed2induced wave dissipation or the wave2induced mass transport of the mud. Various physical models have been adopted to describe the rheological behaviors of mud, such as viscous Newtonian fluid (Dal2

3 443 rymple and Liu, 1978 ; Sakakiyama and Bijker, 1989 ; Ng, 2004), Bingham plastics (Mei and Liu, 1987), and viscoelastic medium (MacPherson, 1980 ; Piedra2Cueva, 1993 ; Zhang and Ng, 2006 ; Ng and Zhang, 2007). Although these studies are not concerned with waves generated by a moving ob2 ject, they provide very good references for the mud2water interaction that happens as soon as waves are generated by the disturbance in the present study. Thus far, there has not been any published works for the problem of surface waves generated by a submerged body moving in a two2layer system. 2. Numerical Method In this paper, simulations for both one2layer and two2layer systems are presented. The one2layer system is used to validate the present numerical model by comparing the results with the experiment of Lee et al. (1989) and the numerical model of Zhang and Chwang (1996). The numerical simulation is carried out with the aid of the general purpose CFD package FLUENT. A moving coordinate, which is attached to the submerged body, is employed so that the far2field fluids together with the channel bottom move at a constant speed whereas the submerged body appears stationary in the simulation. As the first step towards a better understanding of the wave generation process in a two2layer system, the mud phase is regarded as a homogeneous Newtonian viscous fluid Governing Equations Similar to Zhang and Chwang (1996), the fully nonlinear NS equations are solved in the present study. The Volume2of2Fluid (VOF) multiphase scheme is employed to simulate the displacements of the free surface and the water2mud interface. The momentum equation, as given below, depends on the phase volume fractions through the fluid properties : density and dynamic viscosity. 5 5 t ( v) + g ( v v) = - g p + [ ( g v + g vt ) ] + g, (1) where = q q, and q stands for the q2th fluid s volume fraction in a cell. The VOF scheme greatly simplifies solving the multiphase flow problem. However, this method is subject to some limitations as well. In order to track the free surface wave, the domain above the water phase cannot be empty other2 wise no volume fraction information can be tracked near the free surface. Therefore, an air layer has to be added above the water, which inevitably introduces additional computational work to the simulation. For computational efficiency as well as accuracy, tests have been performed to determine an optimal air layer that is just thick enough to accommodate the displacement of the free surface. The quantities which are related to length (e. g., wave amplitude A) are normalized with respect to the initial undisturbed water depth h w, while velocities u are normalized with respect to the steady velocity of the moving disturbance u 0. Time t is normalized by h w / u 0, and pressure p is normalized by the pressure scale p 0 = w u 2 0. The normalized/ dimensionless quantities are defined as follows : = A h w, U = u u 0, T = t, P = h w / u 0 p w u 2 0 where Fr is Froude number, and Re is Reynolds number., Fr = u 0, Re = u 0 h w, (2) gh w w

4 Boundary Conditions The boundary condition for a two2layer system is shown in Fig. 1. As mentioned above, for con2 venience of analysis, a moving coordinate is used such that the submerged body appears stationary but the incoming fluids on the inlet boundary as well as the channel bottom itself are specified to move at a constant velocity. Therefore, the results in this study are presented with respect to the body frame. Fig. 1. Boundary conditions for a two2layer system. The boundaries in the present study are specified as follows : velocity inlet, wall/ moving wall, outflow and symmetry. The domain has to be long enough both upstream and downstream to make sure that the surface waves generated will not hit the inlet and outlet boundaries within the simulation peri2 od Model Validation The present model is validated by comparison with the experiment of Lee et al. (1989) and the numerical model of Zhang and Chwang (1996) for a one2layer system. The problem is to consider a two2dimensional bump with an arched cross section moving at a transcritical speed along the channel bottom. The object is placed at a small height above the bottom to prevent the bump base from touching the channel bottom. In order to model the experiment as much as possible, the computational domain is built in a way very similar to that of Zhang and Chwang (1996). The size of the bump, the gap be2 tween the bump and the channel bottom, and the undisturbed water depth h w are exactly of the same values as in the simulations by Zhang and Chwang (1996). Nevertheless, as mentioned before, a layer of air above water has to be brought into the domain. After some numerical tests, an air layer of thick2 ness = 2 has been adopted in this study. A schematic diagram showing the problem domain and dimen2 sions is shown in Fig. 2. The surface wave generated is monitored by a numerical wave gauge which is placed 70 units upstream of the disturbance, and due to the moving coordinate, the numerical wave gauge is to move at the same speed as the flow and channel bottom. The Reynolds number in Lee s ex2 periment was of the order of At this high Reynolds number, the transition to turbulent flow is very likely to occur and therefore it is necessary to find out the significance of turbulence in the present modeling. A standard k - turbulence model is chosen as the solver to produce results to be compared with those by the laminar solver. Fig. 2. Dimensions of computational domain in current numerical model for the one2layer system.

5 445 The current model is capable of revealing the features that have been reported before. For trans2 critical cases, solitary waves are generated periodically ahead of the moving bump. For the trailing waves, a self2elongating depressed region is clearly seen and this region is followed by a train of cnoidal waves diminishing in amplitude with distance from the disturbance and disappearing in the far field. For three values of Froude number, Fr = 0. 89, and 1. 12, the free surface elevation as a function of time (measured by the wave gauge), as obtained by the present model, is compared in Fig. 3 with the experiment of Lee et al. (1989) and the NS model of Zhang and Chwang (1996). It can be seen from Fig. 3 that the overall agreement of the present modeling with the experiment is good especially for the amplitude and phase of the upstream advancing solitary waves. The crest height of the present model is closer to the experimental result compared with the NS model of Zhang and Chwang (1996), but it over predicts the wave trough position and under predicts the trailing wave in the critical and supercritical cases. By and large, the present model and the NS model give very similar results. In addition, the predicted phase of the solitary waves from the present model agrees very well with that of the experiment. All in all, the present model yields quite satisfactory results for transcritical values of the Froude number. It is also shown in Fig. 3 that the laminar and turbulent solvers yield almost the same results for all the cases. The Reynolds numbers of these cases are of order of 10 4, but the results of the laminar and turbulent models are practically very close to each other. In view of the fact that for turbulent flow modeling a much finer grid is needed and an additional equation for k and is solved, the turbulent model is far less time efficient than the laminar model. Therefore, for the two2layer cases to be discussed below, only laminar flow is considered. Fig. 3. Comparison between previous exper2 imental and numerical data and cur2 rent numerical results for both lami2 nar and turbulent solver. 3. Discussion of Numerical Results A two2layer system is where a layer of mud, which is taken as Newtonian fluid with a larger den2 sity and viscosity, is placed under the water column. In order to have a systematic understanding of the problem, a series of numerical simulations are carried out so as to look into the influence of the follow2 ing parameters on the surface waves : (i) the Froude number, (ii) the mud2water thickness ratio, (iii)

6 446 the mud2water density ratio, and (iv) the mud2water viscosity ratio Computational Details The dimensions of the two2layer system are shown in Fig. 4, where the mud layer thickness varies with the cases. The disturbance is placed in the water column at a height that is closer to the water2 mud interface than the free surface. A two2dimensional body with an elliptical cross section is used in2 stead of the bump in the one2layer system. The body has a major chord of 0. 5 and a minor chord of 0. 2, and is placed 0. 6 away from the water surface and 0. 4 away from the water2mud interface. The upstream and downstream domain lengths are each specified to be 100 units long. The grids are built in a way similar to that in the one2layer problem, and two rules are observed in order to achieve the desired convergence and stability : (i) as shown in Fig. 5, grids are more con2 centrated around the body, near the water free surface and water2mud interface, where the velocity/ pressure gradient is expected to be larger ; (ii) the size of cells located on either side of a boundary be2 tween two regions is the same, and there is no abrupt change of grid size. Fig. 4. Dimensions of the two2layer system. Fig. 5. A partial view of the grid for the two2layer system. In the present study, a moving horizontal coordinate is employed as in the one2layer problem. In the first step of simulation, the flow field is set into motion at a certain speed in a manner similar to the experiment of Lee et al. (1989). The discretization scheme for pressure is PRESTO ( PREssure STag2 gering Option), First Order Upwind for momentum discretization and Geo2Reconstruct for Volume Fraction are adopted for the numerical simulation. The PISO ( Pressure2Implicit with Splitting of Oper2 ators) method is used for the pressure2velocity coupling. Altogether twenty eight cases, as listed in Table 1, are simulated. The simulations can be organized into five groups : group 1 (Cases 1 to 7) is to look into the general feature of the wave generated in a two2layer system and the behavior of the waves at different Froude numbers ; group 2 (Cases 8 to 18) is to investigate the effect of the mud2wa2 ter thickness ratio on the generation of solitary waves ; group 3 (Cases 21 to 27 together with Case 13) is to investigate the influence of the density ratio ; group 4 (Cases 13, 19 and 20) is to examine the in2 fluence of the viscosity ratio ; and finally, the limiting case (Case 28) is to consider a one2layer system again, but with an elliptical cylinder instead of a bump as the forcing.

7 447 Table 1 Computational parameters for the two2layer system Case Object height Fr m / w m / w h m / h w / / / N/ A N/ A Effect of Froude Number In this section, the characteristic solitary wave phenomena and wave behaviors at different Froude numbers are discussed. Many studies for one2layer systems have pointed out the importance of the Froude number in the problem (e. g., Kevorkian and Yu, 1989 ; Torsvik et al., 2006). Before look2 ing into the two2layer cases, it is necessary to first review the one2layer counterparts so that the differ2 ence between the two systems can be identified. A 32D (amplitude2distance2time) plot of wave devel2 opment for Fr = 1 is shown in Fig. 6. It can be seen that in this exact resonant case, solitary waves

8 448 are generated periodically ahead of the disturbance. The amplitudes of these solitary waves are large and the amplitudes for the following solitary waves are almost the same as the first one. The self2elon2 gating depressed region right behind the disturbance is clearly seen from the plot and the trailing waves have relatively small amplitudes. Fig. 6. Surface waves generated by an ellip2 tic cylinder versus time in the one2 layer system. When the problem is extended to the two2layer system, as can be seen from Fig. 7a, which is a 32D plot of the surface and interface wave development versus time at the Froude number equal to uni2 ty, the overall wave pattern of surface wave is more like that of a sub2critical case in a one2layer sys2 tem. Advancing solitary waves with amplitude that decreases with time are observed. The trailing waves have relatively large amplitude, and the first trailing wave moves very slowly down the channel. Also, the depressed region right after the disturbance hardly elongates. The difference between the one2and two2layer cases can be identified by comparing the trailing wave pattern in Fig. 7a with that in Fig. 6. A train of gradually diminishing cnoidal2like waves is also observed. Meanwhile, the water2mud inter2 face deforms under the influence of the disturbance near X = 0, and also under the surface waves that move away from the disturbance. The propagation and dispersion of the first generated trailing wave can be also seen in Fig. 7b. The advancing waves on the free surface and those on the interface are in phase and the interface waves have much smaller amplitude due to the larger density and viscosity of the mud layer. Fig. 7. Surface and interface waves versus time, Fr = The leading advancing solitary wave will be attenuated steadily owing to the damping effect of the mud layer. The attenuation of the first solitary wave is shown in Fig. 8, in which 0 is the amplitude when the wave just leaves the disturbance and X is the distance between the wave crest and the dis2 turbance. It can be seen that the amplitude of the solitary wave starts to decrease as soon as the solitary

9 449 wave departs and it decreases quickly. The comparison of the two2layer and one2layer systems is also shown in Fig. 8. In the one2layer system, the maximum amplitude is observed around 18 unit lengths upstream of the disturbance, which is regarded as the forcing zone. However the phenomenon cannot be seen in the two2layer system, which suggests that the damping effect due to the mud layer is domi2 nant after the solitary wave is generated. Fig. 8. Wave attenuation versus distance from the disturbance for one2layer and two2layer systems, Fr = For the subcritical and supercritical cases, the free surface and interface waves are quite different from the critical case ( Fr = 1. 0). For Fr = 0. 7, as shown in Fig. 9a, the advancing solitary waves on the free surface have a very small amplitude whereas the trailing waves are dramatically larger in amplitude and stay closer to the disturbance. As for the water2mud interface given in Fig. 9b, the in2 terfacial waves are in phase with the free surface waves. Fig. 9. Surface and interface waves versus time at Fr = For a supercritical case Fr = 1. 3, the wave pattern is opposite to the low subcritical case. On the free surface as shown in Fig. 10a, the upstream advancing solitary waves have relatively large ampli2 tude whereas the downstream waves have very small amplitude. The generation period of both the up2 stream and downstream waves are dramatically increased. The first trailing wave behind the object moves but very slowly away from the disturbance. It may be due to the fact that when the object moves at a faster speed, the upstream wave is hard to get away from it and stays in the disturbing zone in the

10 450 proximity of the moving object. As shown in Fig. 10b, the water2mud interface waves have the similar features as the surface waves. However, the downstream wave crest is sharper and flattens more quick2 ly than the critical and subcritical cases. Fig. 10. Surface and interface waves versus time at Fr = It has been remarked by Lee et al. (1989) that the curve of drag coefficient versus time has as many maxima as the number of upstream2advancing solitary waves. It was observed that each relative maximum of C d, noted as C dmax here, corresponds to the birth of a new solitary wave. One may therefore identify the generation period by referring to the time change of the drag coefficient. The gen2 eration period of upstream solitary wave generation at different Froude numbers is given in Fig. 11a and the C dmax value at different Froude numbers is shown in Fig. 11b. It can be seen that the generation period T s increases as the Froude number increases and it increases much faster as the Froude number exceeds The first maximum of drag coefficient C dmax grows almost linearly as the Froude number increases. Fig. 11. Wave properties at different Froude numbers Effect of Thickness Ratio It has been pointed out in Section 3. 2 that the mud will deform in the vicinity of the moving dis2 turbance, and the interface away from the disturbance changes only slightly under the solitary waves. However, this is only for cases in which the mud layer thickness is equal to 1, which is considered to

11 451 be a moderate value. One can expect that the thickness ratio h m / h w will affect the interfacial displace2 ment in the vicinity of the disturbance, and as a result, will also affect the generation of the surface waves. As in the one2layer water system, it has been remarked by Zhang and Chwang (1996) that much of the viscous damping is due to the bottom. In the two2layer system, the viscous effect is even more pronounced because of the large viscosity of mud. In order to investigate the influence of the thickness ratio on the surface wave generation, twelve cases including the one2layer case (Cases 8 18, 28) are simulated as mentioned in Section The properties of the mud are specified to be moderate relative to those of water. The density of the mud is times that of water, and the viscos2 ity is 100 times that of water. The typical Reynolds number of these cases is in terms of the water viscosity. It is interesting to see from Fig. 12a that the generation period of the upstream surface waves has a maximum at a thickness ratio approximately equal to h m / h w = The vertical intercept is the gen2 eration period for the one2layer ( h m / h w = 0) case. The generation period increases as the thickness ratio increases for h m / h w < 0. 7, but decreases for h m / h w > The amplitude of the first solitary wave upon departure from the disturbance decreases as the thickness ratio increases as shown in Fig. 12b. This indicates that the existence of a mud layer causes energy dissipation and greatly reduces the amplitude of the upstream advancing waves. Even a very thin mud layer will cause a large reduction of the amplitude of the advancing solitary wave. The amplitude then decreases more gradually as the thickness ratio h m / h w further increases. The damping effect due to the mud layer approaches a con2 stant value as the mud layer thickness becomes large. The interfacial displacement offers information about the strength of the forcing relative to the water depth and it is useful to take a closer look at the interface. Fig. 13 shows snapshots of the water2mud interface at T = 10, which is an early stage of the generation ; here by early stage it is meant to be a time before the emergence of the first solitary wave ( usually the first solitary wave emerges at T > 20). The mud layer thickness does not have appreciable effects on the interface upstream of the disturbance. It is interesting to find that the interface near X = 0 is not smooth and distorted under the influence of the object. For a thin mud layer ( h m / h w < 0. 7), there is a small depressed region right at X = 0 as can be seen from Fig. 13b, and the crest next to it is of relatively large amplitude. As the mud layer becomes thicker ( h m / h w > 0. 7), the crest becomes flatter and the small depressed region becomes less conspicuous while the depressed region further downstream gradually becomes dominant. Moreover, as can be seen from Fig. 13b, the h m / h w = 0. 2 curve has a steeper slope upstream of X = 0 than the h m / h w = 0. 5 curve, and the depressed region behind it creates a negative topography forcing to the system. This region increases the resistance for the first solitary wave to detach from the disturbance as can be seen from Fig. 14. The drag coefficient at the moment when the first solitary wave is generated is the largest for h m / h w = In other words, at this particular thickness ratio, the blockage effect is the largest. The steepness of the slope increases until the thickness ratio reaches 0. 7, and then the depressed region becomes less conspicuous and the depressed region further downstream forms a flatter slope. This difference in local interfacial profile leads to variations in the generation period of the upstream solitary waves.

12 452 Fig. 12. Wave period and maximum amplitude of free surface waves T s, versus thickness ratio h m / h w. Fig. 13. Snapshots of water2mud interface at an early stage ( T = 10) of the simulation. Fig. 14. Drag coefficient at different thick2 ness ratio h m / h w Effect of Density Ratio It is reasonable to expect that the density of mud is also an important factor. A mud layer of dif2 ferent density will deform to a different extent under the same forcing. For the sake of analysis we here consider only the thicker mud cases, in which the mud layer thickness is specified to be 1. 0 ( > 0. 7) so that the small region of depression at the interface near X = 0 is minimal as discussed in Section Altogether eight cases (Cases 21 to 27 and Case 13 as in Table 1) with mud2water density ratio

13 453 m / w ranging from to 1. 8 are simulated with Fr = 1. In this critical case, the general surface wave and interface wave patterns are similar to those al2 ready seen in Section At an early stage, the water2mud interface forms a slope in the vicinity of the disturbance, and solitary waves start to emerge and then travel ahead of the disturbance. The dif2 ference in density will, however, cause the solitary waves to have different generation periods and am2 plitudes. Firstly, as can be seen from Fig. 15b, in the vicinity of the disturbance, the mud layer with a smaller density deforms more, and vice versa. The water2mud interface upstream of X = 0 is not af2 fected by the density ratio, but the steepness of the interface near X = 0 varies with the density ratio. The interface slope in the vicinity of the disturbance becomes milder as the density ratio increases. Fig. 16a shows a relationship between the generation period and the density ratio. The period increases as the density increases. The interfacial displacement downstream of the disturbance is responsible for the slope formed under the disturbance and leads to variations in the generation period and amplitude of the upstream solitary wave. This once again confirms the importance of the local displacement of the interface to the generation of the solitary waves on the free surface. It is clear from Fig. 16b that the amplitude grows almost linearly as the density ratio increases. Fig. 15. Snapshots of water2mud interface at an early stage ( T = 10) of the simulation. Fig. 16. Wave period T s and amplitude versus density ratio m / w : (a) period ; (b) amplitude. As for the attenuation of the surface waves due to the damping effect from the mud layer, the re2

14 454 sults for the density ratio equal to and 1. 3 are shown in Fig. 17. It shows the attenuation of the first solitary wave with respect to its largest amplitude after it emerges. The amplitude decreases very fast at m / w = and decreases slower for a higher density ratio. It also can be seen that the soli2 tary wave amplitude decreases faster as it moves out of the forcing zone as mentioned above. The re2 sults suggest that a layer of mud with a smaller density will lead to a larger damping effect on the gen2 eration of solitary waves. Generally speaking, a softer and less dense mud bed will lead to larger ener2 gy dissipation in the system and the solitary waves generated will have smaller amplitude. Fig. 17. Surface wave attenuation for differ2 ent density ratios Effect of Viscosity Ratio In a one2layer system, the viscous effect mainly arises from the development of a boundary layer and flow separation near the body. In a two2layer system, much additional viscous effect is due to the mud. As a result, the viscous damping and energy dissipation can be much greater than that in a one2 layer system. The effect of the viscosity ratio, defined to be the ratio of kinematic viscosity of mud to water, is our focus here. For the investigation of the influence of viscosity ratio, three cases of simulation are performed. They are Cases 13, 19 and 20 as listed in Table 1. It is found that the generation period of the up2 stream solitary wave T s changes very slightly. For viscosity ratio m / w = 10, 100 and 1000, the gen2 eration period T s are , and , respectively. Although no significant difference is found for the generation period, it is interesting to find that the surface wave patterns are very different for the three cases. As in Figs. 18a and 18b, the waves for m / w = 10 and m / w = 100 look similar to each other. However, Fig. 18c shows that when the viscosity ratio reaches 1000, the advancing wave stays close to the disturbance and mingles with the subsequent solitary wave as the wave flattens out. It is hard for the solitary wave generated to depart from the disturbance. As a result, the subse2 quent solitary wave is barely seen and the wave pattern near the disturbance appears to change little with time. As can be seen from Fig. 19, the drag coefficient curves for m / w = 10 and m / w = 100 are similar except for the magnitude and the drag coefficient being larger for a smaller viscosity ratio. However, the drag coefficient curve of m / w = 1000 is quite different from the above two curves. The coefficient drops faster with time and local maxima are hardly seen after the appearance of the first one.

15 455 This agrees with the observation that no obvious crest is seen after the first solitary wave is generated. In all the three cases, the trailing waves have relatively large amplitude and they remain close to the disturbance. Fig. 18. Surface wave development with dif2 ferent viscosity ratios m / w. Fig. 19. Drag coefficient for different viscosity ratios m / w. Fig. 20. Attenuation of the first solitary wave for differ2 ent viscosity ratios. The attenuation curves are shown in Fig. 20. As the viscosity ratio increases, the amplitude of the first solitary wave decreases more when moving farther away from the disturbance. It is remarkable that the m / w = 1000 curve is much steeper than the other two. This reflects that the increase in vis2 cosity of the mud will greatly increase the energy dissipation, and hence greatly change the surface wave pattern.

16 Conclusions A numerical study on waves generated by a submerged moving disturbance in a water2mud fluid system has been performed by means of the general2purpose computational fluid dynamics package FLUENT. The fully nonlinear Navier2Stokes equations are solved with the Volume2of2Fluid (VOF) multiphase scheme in order to capture the displacements of the water free surface and the water2mud interface as functions of time. The numerical model has been validated by comparison of results with previous experimental and numerical studies for a one2layer fluid system. The generation and time2development of surface and interfacial waves with the presence of a mud layer has been studied in detail. A series of systematic numerical simulations have been carried out to examine the influence of the following parameters on the surface waves : (i) the Froude number, (ii) the mud2water thickness ratio, (iii) the mud2water density ratio, and (iv) the mud2water viscosity ra2 tio. Because of the presence of a mud layer, the overall surface wave pattern at the critical speed in a two2layer system resembles that of a subcritical case in a one2layer system. The surface waves generat2 ed at subcritical and supercritical speeds are quite different from the waves generated at the critical speed as can be seen in Figs. 7, 9 and 10. The generation period of the advancing solitary waves T s increases as the Froude number increases in the near2critical regime. The first maximum of the drag coefficient C dmax increases almost linearly with respect to the Froude number. It is also found that the damping caused by the mud layer is influential in the development of upstream solitary waves after they are generated. Even though the wave crest is still within the forcing zone of the disturbance, the ampli2 tude of the wave keeps decreasing rapidly. It is interesting to find that the generation period has a maximum value at a thickness ratio ap2 proximately equal to Also, the drag coefficient reaches a maximum value at this critical thickness ratio. A closer look reveals that the local water2mud interface acts like a flexible bottom. The mud2wa2 ter thickness ratio therefore affects the blockage, which in turn affects the surface wave generation. This agrees with the statement made by Ertekin et al. (1984) that the blockage coefficient is a domi2 nant parameter governing the generation of the solitary waves. On the other hand, the amplitude of solitary wave generated decreases as the thickness ratio increases. This is due to the increasing damp2 ing effect for a thicker mud layer. The influence of the density ratio m / w has been investigated while keeping the thickness ratio constant. It is found that the displacement of the local water2mud interface is appreciably affected by the density ratio. Also, the generation period T s of the solitary wave increases as the density ratio in2 creases. The solitary wave amplitude increases as the density ratio increases, reflecting that the damping effect is larger for a smaller density contrast between mud and water. It also can be seen from Fig. 17 that for a solitary wave being generated, the amplitude decreases faster for a smaller density ratio. The wave attenuation with distance, however, does not change appreciably with the density ra2 tio.

17 457 The surface wave patterns also vary significantly depending on the viscosity ratio, especially when the ratio is large. It is found for m / w = 1000 that the advancing solitary wave stays close to the dis2 turbance and mingles with the subsequent solitary waves as it flattens out quickly. As a result, the se2 ries of upstream solitary waves form a single flattened wave ahead of the submerged body. The drag co2 efficient curves for m / w = 10 and 100 have similar shapes, but for m / w = 1000, the drag coeffi2 cient drops appreciably after the second solitary wave is generated, and the drag coefficient finally ap2 proaches a rather small limit. The viscosity ratio also has influence on the attenuation of the solitary wave. It is found that for a large viscosity ratio ( m / w = 1000), the amplitude of the first solitary wave decreases rapidly as the wave moves away from the disturbance. The damping effect and energy dissipation are considered to be significant for a viscosity ratio of this magnitude. It is desirable if the present finding can be confirmed by comparison with that of experiments. This will form the scope of our next study. References Akylas, T. R., On the excitation of long nonlinear water waves by a moving pressure distribution, J. Fluid Mech., 141, Binder, B. J., Dias, F. and vanden2broeck, J.2M., Steady free2surface flow past an uneven channel bottom, Theor. Comput. Fluid Dyn., 20 (3) : Cole, S. L., Transient waves produced by flow past a bump, Wave Motion, 7 (6) : Camassa, R. and Wu, T. Y., Stability of forced steady solitary waves, Phil. Trans. R. Soc. Lond., A337, Dalrymple, R. A. and Liu, P. L.2F., Waves over soft muds : a two2layer fluid model, J. Phys. Oceanogr., 8 (6) : Ertekin, R. C., Webster, W. C. and Wehausen, J. V., Ship2generated solitons, Proceedings of the 15 th Symposium on Naval Hydrodynamics, Hamburg, Lee, S. J., Yates, G. T. and Wu, T. Y., Experiments and analyses of upstream2advancing solitary waves gen2 erated by moving disturbances, J. Fluid Mech., 199, Grimshaw, R. H. and Smyth, N. F., Resonant flow of a stratified fluid over topography, J. Fluid Mech., 169, Grimshaw, R. H., Zhang, D. H. and Chow, K. W., Generation of solitary waves by transcritical flow over a step, J. Fluid Mech., 587, Huang, D. B., Sibul, O. J., Webster, W. C., Wehausen, J. V., Wu, D. M. and Wu, T. Y., Ships moving in the transcritical range, Proceedings of the Conference on Behavior of Ships in Restricted Waters, Varna, II, 26/ 1226/ 10. Kevorkian, J. and Yu, J., Passage through the critical Froude number for shallow2water waves over a variable bottom, J. Fluid Mech., 204, MacPherson, H., The attenuation of water waves over a non2rigid bed, J. Fluid Mech., 97, Mei, C. C. and Liu, K.2F., A Bingham2plastic model of a muddy seabed under long waves, J. Geophys. Res., 92 (C13) : Ng, C. O., Mass transport and set2ups due to partial standing surface waves in a two2layer viscous system, J. Fluid Mech., 520, Ng, C. O. and Zhang, X. Y., Mass transport in water waves over a thin layer of soft viscoelastic mud, J. Flu2 id Mech., 573, Piedra2Cueva, I., On the response of a muddy bottom to surface water waves, J. Hydraul. Res., 31 (5) :

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