The proper efinition of the ae mass for the water entry problem Leonaro Casetta lecasetta@ig.com.br Celso P. Pesce ceppesce@usp.br LIE&MO lui-structure Interaction an Offshore Mechanics Laboratory Mechanical Engineering Department Escola Politécnica University of São Paulo 05508-900, São Paulo, Brazil Introuction The water entry problem is consiere from the point of view of analytic mechanics. It is usual practice to treat potential hyroynamic problems involving motion of soli boies within the frame of system ynamics. This is one whenever a finite number of generalize coorinates can be use as a proper representation for the motion of the whole flui. We refer to this kin of approach as hyro-mechanical. This is mae possible through the use of the well-known concept of ae mass. In the vertical water entry problem the generalize coorinate is the penetration epth of the impacting boy. rom another point of view, a usual way to calculate the impact force is the integration of the pressure-fiel over the wette surface of the impacting boy. The jets are then exclue from the portion of the flui that wets the boy. z or simplicity we take the case of vertical entry only. Uner the hyro-mechanical approach, one may consier only the of the flui to express the impact force as U ), (1) where M is the ae mass associate to the kinetic energy of the of the flui, jets being exclue. However, if the ae mass is interprete as a measure of the kinetic energy of the whole flui, another correct an alternative form for the impact force is given by 1 M U ) U, ()
where M refers to the whole flui omain. Shoul equivocal arguments be use one woul obtain the misleaing expression, 1 M U ) U. (3) The reason for the iscrepancy between expressions (1) an (3) has alreay been aresse an actually emerges from a consistent energy balance; see, e.g., the iscussion in Molin et al. (1996) or in Pesce (003, 006). In fact, via an asymptotic analysis near the contact line, it has been shown, for some particular cases, that a consierable amount 1 of kinetic energy is raine from the of the flui through the jets; Molin et al. (1996), Scolan an Korobkin (003), Casetta an Pesce (005). In other wors, the jets must be consistently consiere if energy arguments are applie in the impact force calculation. In the present work the role of the jets in the impact force calculation is iscusse. We show how one can obtain the correct impact force expression in either way, by consiering the jets, or by not consiering them. In this sense, proper efinitions of the ae mass for the problem are presente an then use in the corresponing proper forms of the Lagrange equation. Analysis consiering the of the flui omain A usual an practical approach in marine hyroynamics is to treat the water entry problem by taking the omain of analysis as the of the flui only. The whole flui omain is therefore ivie into two parts: the an the jets. As the boy penetrates the, jets are expelle from the neighborhoo of the contact line ue to a very rapi expansion of the boy wette surface. The 1 kinetic energy of the of the flui, T = M U, varies explicitly with the penetration epth of the boy, an so oes the ae mass associate to it. In this case, where an out-flux of kinetic energy oes exist from the omain uner analysis (the of flui), there is an effective loss of ae mass through the jets an one must use the extene Lagrange equation vali for systems with mass explicitly epenent on position - see Pesce (003) - that reas M z 1 U = U z. (4) The left-han sie of equation (4) must be interprete as the total force that acts on the of the flui. The first term is the impact force we want to calculate. The secon term, 1 ( ) M / z U, is the reactive force acting on the ue to the out-flux of kinetic energy. In fact, Casetta an Pesce (005) show, for the case of a generic an arbitrary shape of the contact line, that the rate of 1 / z U, being the reactive force kinetic energy raine by the jets is always equal to ( ) 3 term 1 ( / z U obtain the impact force in the form of equation (1), i.e., M 1 M ) also generally vali. Therefore, as T / z ( M /z) U =, one easily U ). (1a) 1 Actually this amount is exactly half the whole kinetic energy of the flui if the impact velocity is enforce to be constant; see Molin et al. (1996), for the case of cyliners; Casetta an Pesce (005), for the general 3D case. Such an extene form of the Lagrange equation shoul also contain a reactive force term ue to a real (physical) mass out-flux. In the water entry problem, the real mass out-flux through the jets is of secon orer, as shown by Cointe an Arman (1978) an Molin et al. (1996), for the particular an important case of a circular cyliner.
Analysis consiering the whole flui omain The water entry problem can also be treate by taking the whole flui omain. The ae mass 1 M, is now a measure of the kinetic energy of the whole flui omain, i.e., T = M U. In this case, the jets are inclue in the analysis omain an, obviously, there is no out-flux of kinetic energy neither an out-flux of ae mass. In other wors, there is no loss of energy from the system an this is the key point. The extene Lagrange equation, for systems with mass explicitly epenent on position is no longer applicable. One must then apply the usual form 3 of the Lagrange equation, as in Lamb (193), art. 136 an 137, i.e., = t U z. (5) The impact force expression is then obtaine from equation (5) in the form of equation (), 1 M U ) U. (a) The equivalence between impact force expressions By equating the alternative equations (1) an (), we promptly obtain M. (6) 1 ( M U ) U = ( M U ) Shoul a misleaing assumption be taken, by enforcing result woul be foun, i.e., M = M, an obviously erroneous 1 M U = 0, (7) such that 1 ( M / t) U woul be also null. This woul imply the amount of the kinetic energy in the jets to be null, Casetta an Pesce (005), an obviously false assertive. Energy balance rom equations (1) an () one can easily verify that an that T 1 U U = ( M ) 0, (8) T U = 0. (9) 3 Recall that the usual form of Lagrange equation is invariant with respect to systems with mass varying as a function of time - as is the case if the whole flui is taken as the omain; see, e.g., Pesce (003), for a etaile iscussion on this subject.
Both equivalent equations (8) an (9) represent the correct balance of energy between the flui an the impacting boy. The thir term in equation (8) is the time rate of the kinetic energy that fills in the jet. It is also interesting to mention that the general equation (8) can be alternatively obtaine via the classical velocity potential approach, by consiering the water entry problem as a nonlinear bounary-value problem; see Casetta an Pesce (005). Conclusion It was shown how to treat the water entry problem in two equivalent an consistent ways. If one consiers only the of the flui as the omain of analysis, such that there is an out-flux of kinetic energy to the jets, an if an analytic mechanics point of view is followe, the extene form of the Lagrange equation, for systems with mass explicitly epenent on position, must be applie to calculate the impact force. Alternatively, if the whole flui omain is taken, so that no longer exists an out-flux of kinetic energy, the usual form of the Lagrange equation must be applie. The consistency between both approaches lies on a proper omain efinition of the ae mass. Acknowlegments We acknowlege APESP, the State of São Paulo Research ounation, PhD scholarship, process n o 04/04611-5 an CNPq, The National Research Council, Research Grant n o 30450/00-5. The authors are especially grateful to Professor Alexaner Korobkin, for very interesting iscussions uring his visit to Brazil. References Casetta, L., Pesce, C.P., 005, A Noticeable Question of the Water Entry Problem: the Split of Kinetic Energy uring the Initial Stage, COBEM005, 18 th International Congress of Mechanical Engineering, Brazilian Society of Engineering an Mechanical Sciences, Ouro Preto, Brazil. Cointe, R. an Arman, 1987, Hyroynamic Impact Analysis of a Cyliner, Journal of Offshore Mechanics an Arctic Engineering, Vol. 109, pp. 37-43. Cointe, R., ontaine, E., Molin, B. an Scolan, Y.M., 004, On Energy Arguments Applie to the Hyroynamic Impact orce, Journal of Engineering Mathematics, Vol.48, pp. 305-319. Lamb, H., 193, Hyroynamics, Dover Publications, 6 th e., 738 pp. Molin, B., Cointe, R. an ontaine, E., 1996, On Energy Arguments Applie to the Slamming orce, 11 st.international Workshop on Water Waves an loating Boies, Hamburg. Pesce, C.P., 003, The Application of Lagrange Equations to Mechanical Systems with Mass Explicitly Depenent on Position, Journal of Applie Mechanics, Vol. 70, pp. 751-756. Pesce, C.P., 006, A Note on the Classical ree Surface Hyroynamic Impact Problem, Worl Scientific, Hyroynamics Symposium in honor to T.-Y. Wu, to appear. Scolan, Y.M. an Korobkin, A.A., 003, Energy Distribution from Vertical Impact of a Three- Dimensional Soli Boy onto the lat ree Surface of an Ieal Liqui, Journal of luis an Structures, Vol.17, pp. 75-86. Wu, G. X., 1998, Hyroynamic orce on a Rigi Boy uring Impact with Liqui, Journal of luis an Structures, Vol. 1, pp. 549-559.
Casetta, L., Pesce, C.P. The proper efinition of the ae mass for the water entry problem Discusser - R.C.T. Rainey: I isagree with your equation (1). If the flui has infinite epth, the flui momentum is not efine, see beginning of chapter 6 in Lamb (193). If the flui has finite epth, there is a reaction force on the seabe, which oes not ten to zero as the epth increases. Reply: irst, we agree that in the case of finite epth there woul be a reaction force on the seabe. Moreover the ae mass woul be also a function of the proximity to the seabe. Even in that finite epth case, however, the flui omain might be unboune. We note that we are treating the infinite epth case only. The momentum (an the kinetic energy) of the liqui may be efine whenever the velocity fiel is integrable (square integrable, i.e, integrable in the energy norm) over the whole liqui omain. This is the case in the present paper.