Ocean Engng, Vol. 10, No 5, pp. 375382, 1983.
00298018/83 $3.00+.00 Pergamon Press Ltd.
Printed in Great Britain.
ENERGY DISSIPATION IN SLOSHING ROLLING RECTANGULAR TANK III. APPLICATIONS
WAVES IN A RESULTS AND
ZEKI DEMIRBILEK Research and Development Department, Conoco Inc., Ponca City, OK 74603, U.S.A. A l ~ r a e t   T h e numerical results of the theoretical developments introduced in the first two companion papers are presented here. Since the primary focus of the work is on the dissipated energy, an extensive parametric study of the dissipation is carried out. The versatility of the analytical and computational method is demonstrated by investigating the effects of three important parameters of sloshing in a rectangular tank. These are the Reynolds number, the Froude number and the aspect ratio. For the range of parameters studied, the results exhibit an increase in the value of the dissipated energy for the case of shallow liquid depth. This is in conformity with the laboratory experiments showing the formation of a hydraulic jump which could easily turn into a solitary wave with a change in the frequency. The viscous effects play a major role in forming these jumps, and thus, an increase in viscous dissipation must be associated with such cases. However, for moderate and deep fill depths, the present results show that the dissipated viscous energy is less compared to the shallow fill depth case. This decrease is due to the fact that the fluid near the bottom of the tank is essentially motionless for large depths.
1. INTRODUCTION THE DYNAMICS of the sloshing phenomenon are controlled by the geometry of the container, the properties of the liquid and the nature of the external excitation. We shall poin_Lout that the free surface of a liquid under forced oscillations has an infinite number of natural frequencies. However, it is the lowest mode that has the most likelihood of being excited. Consequently, the bulk of the studies in the field of sloshing is concentrated on investigating the forced harmonic oscillations in the neighborhood of the lowest natural frequency. The natural frequencies of liquid in a prismatic, rectangular tank are given by ~On2a = n'tr tanh n~rhh
g
2
(1.1)
2a
where 2a is the tank length, h the equilibrium liquid depth, n the mode number and g the gravitational acceleration. Here, the lowest value of n (n = 1) corresponds to the fundamental frequency at which the resonant phenomenon is predicted by the linear, inviscid solution. No such phenomenon can be expected in the presence of the viscosity. Viscosity dissipates the energy and therefore limits the response amplitude to a finite value. Laboratory experiments show that as the tank oscillates, it can create different types of sloshing waves. The form of these waves depends on the dimensions of the tank as well as the frequency of oscillation according to Equation (1.1). For the case of shallow liquid depth, a standing wave will be formed when the tank is oscillating at a frequency far
375
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ZEKI DEMIRBILEK
below its resonant frequency. An increase in the frequency transforms this wave into a train of progressive waves. With further increase in the frequency, a hydraulic jump might be formed due to any small disturbance, and this jump could easily turn into a solitary wave with a change in the frequency. Naturally, one would anticipate the viscous effects to play a major role in the formation of jumps, and thus, an increase in the value of viscous energy dissipation must be associated with such cases. On the other hand, for moderate and deep fill depths, the sloshing phenomenon is generally characterized by the formation of largeamplitude standing waves near resonance. These waves are usually nonsymmetric and may be combined with traveling waves at largeamplitude tank oscillations. The amplitudes of such waves are generally quite small, except at resonance. The fluid near the bottom of the tank is essentially motionless, and therefore, the dissipated viscous energy is less compared to the shallow fill depth case. As we shall see later, the results of this study are in full agreement with these experimental predictions. 2.
RANGES OF PARAMETERS IN APPLICATIONS
To obtain an idea about the design implications of the present results, an interpetation in terms of fullscale values is necessary. Therefore, the values of Reynolds and Froude numbers for different fields of application of sloshing are needed. These values are given by Demirbilek (1982) with a detailed explanation for numerous areas. No attempt will be made to repeat this information, as it is beyond the objective of this work. It suffices to state that the range for the Reynolds and Froude numbers in practical applications may vary from 102 to 107 for the first and from 103 to 10a for the second. Only the lower bound of the above Reynolds number range has been covered in the present study, as we shall see later from the plotted results. For very viscous liquids, the Reynolds number will be within the range of the present work. But, in general, for the most common liquids and liquid gases used in some applications, the Reynolds number may be quite large and therefore could be beyond the range of this study. Consequently, future studies must concentrate on extending the Reynolds number limit of this work. On the other hand, the complete practical range of the Froude numbers has been fully covered here. The results of the numerical analysis of viscous dissipation are presented next. Throughout the computations, it is assumed that p = 1.0. Furthermore, in all of the plots presented, the values of the dissipated energy are always less than zero, i.e. 6" < 0. 3.
RESULTS AND DISCUSSION
The effects of Reynolds and Froude numbers (R and F) and the tank aspect ratio on the viscous dissipation are illustrated in Figs 1  8. In each figure, the dimensionless dissipation, d~*, is plotted against the Reynolds or Froude number..Figs 1 6 correspond to different Froude numbers, and in each, four aspect ratios are considered. Froude number is varied so that the whole possible range of its practical applications is covered. It appears from Figs 1  6 that the behavior of ~b* vs R varies as the Froude number changes, even for a fixed value of the aspect ratio. However, in general, the trend of d~* vs R is smoother for the aspect ratios of 0.25 and 0.50 as compared to the aspect ratios of 1.0 and 2.0. This wellestablished trend for the small aspect ratios becomes an irregular one for large aspect ratios of 1.0 and 2.0, especially when the Froude number is small.
(h/2a)
Energy dissipation in sloshing waves   III
377
Some scatter is displayed in the values of ~b* for these latter two aspect ratios, primarily when Reynolds number is less than 10 1. The scatter is more pronounced at the Froude numbers of 10  3 and 10 2 and diminishes with an increase in values of both the Froude and Reynolds numbers. For Reynolds numbers greater than 20.0, the scatter totally disappears. It is very likely that the observed scatter could be due to a numerical problem, since it shows up primarily for very small values of R and F at large aspect ratios. There is not such a pronounced variation in the values of &* for the other aspect ratios, even when R and F are quite small. It can be seen from Figs 1  6 that d~* generally increases as R increases. This is true in particular for the small aspect ratios. By comparison of Figs 1  6, we note that there is a
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shift in the point at which 4)* begins to increase. This shift is toward the large Reynolds numbers, as the Froude number increases for almost all aspect ratios studied. Furthermore, a sharper increase in ~b* is observed with an increasing Froude number. It should be pointed out that Fig. 4 corresponds to a Froude number of 1.0. The dissipation remains constant up to a Reynolds number of about 1.0 and then starts to increase. However, the magnitude of dissipation in Fig. 4 is much greater than those in the remaining figures. This observation suggests that the Froude number of 1.0 corresponds to an excitation frequency which might be closer to the natural frequency of the fluidtank system than any other frequencies associated with the other Froude numbers.
Energy dissipation in sloshing waves   III
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For the Froude numbers of 101 and 10 2 in Figs 5 and 6, respectively, there is no discernible scatter in the values of ~b* that has been reported for the low Froude number cases. Clearly, (b+ is almost independent of the Froude number for Reynolds number up to 2.0, as shown in Figs 4  6. At a Reynolds number of about 3.0 in Figs 4  6, there is a sudden increase in the values of ~b*. The magnitude of this increase accentuates as the Froude number gets larger. However, Figs 5 and 6, when superposed, show that the values of ~b* are virtually unaffected by the Froude numbers for almost all aspect ratios. In particular, we note that the difference between the values of ~b* for the Froude
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numbers of 101 and 102 is practically negligible for the range of the Reynolds numbers studied. An investigation of Figs 1  6 gives sufficient informtion to construct some ideas on the viscous dissipationFroude number relation. In order to display this relationship more explicitly, Figs 7 and 8 are reconstructed from Figs 1  6 as examples. The aspect ratios and Reynolds number range are selected in order to demonstrate the (5* and F dependence clearly. We note that in Fig. 7, the dissipation depends only on the Reynolds number and remains almost constant for each Reynolds number. This trend persists for the values of Fup to 10  l . However, a very sharp increase in the values of (1)* is found at
Energy dissipationin sloshingwaves III
381
the Froude number of 1.0. Since the Froude number of 1.0 corresponds to a frequency in the neighborhood of the resonance frequency, the jump in the values of d~* at this particular Froude number should be expected. Consequently, the effects of Froude number on d~* become more significant as the Froude number of 1.0 is approached, either from the lower or upper limits. As pointed out by Demirbilek (1982), the jump is more pronounced for the smaller aspect ratios, and its magnitude increases with an increasing Reynolds number. Although we have shown only two figures to illustrate the cb* and F relationship, it is possible to construct similar plots for other aspect ratios or for any possible range of Reynolds number. Finally, the variation of the viscous dissipation with respect to the aspect ratio can also be explained from the information given in Figs 1  6. As can be seen from these, each figure includes values of 4" for four aspect ratios at a fixed Froude number. The general trend in these figures indicates that the viscous dissipation increases as the aspect ratio increases. Some large and irregular variations can be seen in the values of 4" for a Reynolds number less than 10t in Figs 1 and 2 when the aspect ratio is 1.0. As suggested earlier, a numerical problem may be the cause of this irregularlity. Repeated computations supported this expectation to be true. Moreover, there are a few missing data points in Fig. 1 for h/2a of 1.0 when R ~< 10 1. These points corresponded to the cases in which the numerical values of d~* were positive. Such cases had to be rejected, since they violate the physical laws because of the fact that the dissipation is an irreversible process and the dissipated energy must be converted into heat. Thus, ~b* cannot have a positive value. It should also be pointed out that the said irregularities of d~* that were peculiar to very small Froude numbers did not exist for Froude numbers greater than 101 . In fact, no such variations or inconsistencies can be observed in Figs 36. The reported increase in the values of ~b* with respect to the aspect ratio could possibly be related to the motion of the fluid contained in the tank. For the small aspect ratio, the bulk of the fluid is set in motion when the tank is excited externally. In such cases, the fluid not only sloshes against the side walls and moves vertically up and down, but it also demonstrates some movement near the tank bottom. An increase in the motion manifests itself by producing more viscous dissipation through an increase in the viscous shear stresses. As the fill depth gets larger, the motion of the fluid at the bottom of the tank subsides considerably. This in turn causes a decrease in the viscous dissipation, since the contribution of the viscous shear stresses at bottom will be reduced. At the same time, more fluid is in vertical motion at the side walls, which results in an increase in the side wall shear stresses. However, the latter has a net increasing effect on the total viscous dissipation as the aspect ratio increases. In other words, the increase in the side wall shear stresses is predominant over the decrease in the bottom shear stresses as the aspect ratio increases. The results presented in Figs 1  6 fully concur with this explanation. 4.
CONCLUSIONS
An attempt has been made to provide an explanation for the motion of a fluid in a rectangular tank subject to a roll excitation. The main objective of the work was to study the viscous dissipation in liquid sloshing. Field equations and the boundary conditions
382
ZEKI DEMIRBILEK
were linearized, and the viscosity was included in the problem formulation. The analytical solution obtained was based on a truncated infinite series. The results of this study have tried to demonstrate how the viscous dissipation varies as a function of the Reynolds and Froude numbers as well as the aspect ratio. Unfortunately, there is no experimental data available for the viscous dissipation of liquid sloshing in rectangular tanks against which to test directly the accuracy of the results of the present investigation. The presented results are very promising and suggest that viscous dissipation may not be negligible. A complete and realistic study of liquid sloshing in containers must take into account the energy dissipation process. It should further be pointed out that there is a spectrum of problems of great practical importance in various engineering disciplines in which the results of this work are either directly applicable or a possible extension is feasible. For more details on this subject, the reader is referred to Demirbilek (1982).
AcknowledgementsThis work was carried out in the Ocean Engineering Program of the Department of Civil Engineering of the Texas A&M University. The author is grateful to all of his doctoral committee members for much helpful criticism and advice. Special thanks are expressed to Professors T.C. Su. G. Venezian and R.O. Reid for their valuable suggestions.
REFERENCES DEMmBILEK, Z. 1982. A linear theory of viscous liquid sloshing. Ph.D. thesis, Texas A&M University, College Station, Texas. DEMIRBILEK, Z. 1983a. Energy dissipation in sloshing waves in a rolling rectangular tank   I. Mathematical theory. Ocean Engng. 10, 347358. DEMIRBILEK, Z. 1983b. Energy dissipation in sloshing waves in a rolling rectangular tank   II. Solution method and analysis of numerical technique. Ocean Engng. 10, 359374.