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Experimental and numerical investigation for a reliable simulation tool for oblique water entry problems
Ocean Engineering 160 (2018) 231–243
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Experimental and numerical investigation for a reliable simulation tool for oblique water entry problems Mohammad Saeed Derakhshanian a, Mobin Haghdel b, *, Mohammad Mahdi Alishahi b, Ali Haghdel c a b c
Aerospace Engineering, Mechanical Engineering Dept., Shiraz University, Shiraz, Iran Mechanic Engineering Dept., Shiraz University, Shiraz, Iran Electrical Engineering, Communication and Electronic Engineering Dept., Shiraz University, Shiraz, Iran
A R T I C L E I N F O
A B S T R A C T
Keywords: Water entry Oblique impact Truncated-nose body Fluid-solid interaction ALE method
Oblique water entry of different marine structures involve several two-phase fluid dynamics phenomena including splash, asymmetrical cavity formation and cavity collapse that have been observed in experimental work. Correct prediction of the applied forces and moments that define the trajectory during water entry is dependent on precise simulation of the mentioned fluid dynamic effects. This study investigates and compares the performance of three different algorithms. i.e.: a) finite volume Eulerian; b) finite difference Eulerian, c) Arbitrary Lagrangian Eulerian (ALE), applied in different software codes. In order to achieve this goal, an experimental task, including normal and oblique water entry cases, was defined to provide the data set for comparing and verifying performance of these different software packages. These experiments have been designed and implemented, concentrating on oblique water entry of blunted nose models with various moment of inertia, center of gravity, and other dynamical properties that affect the water entry problem. Verification of computational results of the three software are carried out through comparison with the experimental results in two stages. Firstly, the fluid dynamics of splash, cavity formation and its collapse for normal water entry of a sphere are compared between experimental and three software numerical results. It is shown that two of the codes, i.e. “a” and “b” as defined above, cannot even predict the much simpler problem of normal water entry problem, therefore, fail to be a candidate for the more involved problem of oblique water entry, i.e., the second stage of the comparison. Through comparison with the experimental results it is shown that ABAQUS software, i.e. the “c” algorithm, is the most capable software among the three compared codes for the correct simulation of the involved fluid dynamic effects. Therefore, the ABAQUS software is applied for simulation of oblique water entry problem in the second stage and the results are compared with experiments. Upon comparison of various experimental results with this software output, it is shown that the numerical algorithm in this computer program (ALE) can be reliably used to simulate this kind of fluid solid interaction problem and determine the impact forces and moments more precisely even in the extreme angles and velocities of water entry problems.
1. Introduction Applications of water entry and water impact problems cover areas such as study of applied hydrodynamic forces and moments in the course of design of the diverse marine vehicles such as ships, vessels and flying boats. Problems of water entry and water impact have been the subject of the research as early as the pioneering work of Von Karman in 1929 (Von Karman, 1929) who were concerned with the hydrodynamics of a
landing seaplane. However, the majority of analytic oblique water-entry solutions considered were either two-dimensional or axisymmetric. Later on, application of potential solution of free surface impact problems for wedges were included in the early work of (Garabedian, 1953) and in a later study of (Xu et al., 2008) as a combined analytical and boundary element method for oblique entry problems. They derived the similarity solution under the assumption that the leading and trailing free surfaces separate from the wedge sides either perpendicularly or tangentially. A
* Corresponding author. E-mail addresses: [email protected] (M.S. Derakhshanian), [email protected] (M. Haghdel), [email protected] (M.M. Alishahi), [email protected] (A. Haghdel). https://doi.org/10.1016/j.oceaneng.2018.04.080 Received 28 June 2016; Received in revised form 26 March 2018; Accepted 20 April 2018 0029-8018/© 2018 Elsevier Ltd. All rights reserved.
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Fig. 1. Stages of normal water entry for a sphere (Rabiee et al., 2010).
were explained in (Laverty, 2005) that will be explained in the next section. A recent application of oblique water entry of a circular cylinder body was presented in (Colicchio et al., 2009). Application of the MSC Dytran software for prediction of water entry and trajectory of a blunt nose body was also shown in (Donea et al., 2004). The basis and formulation of Arbitrary Lagrangian Eulerian (ALE) method as a robust method for dealing with fluid-structure problems is comprehensively discussed in this reference. Analytical derivation of hydrodynamic loading of oblique entry of solid spheres was presented in (Miloh, 1991). Two and three dimensional study of normal and oblique entry of simple objects were numerically studied in (Engle and Lewis, 2003; Kleefsman et al., 2005; Yang and Qiu, 2011, 2012; Yoon and Semenov, 2009; Moore et al., 2012; Gu et al., 2014; Iranmanesh and Passandideh-Fard, 2017; Abraham et al., 2014). However the study of oblique entry of a full nose body in a general situation is not so frequent and (Wei et al., 2012) is one of the rare references in this regard. In the above mentioned numerical investigations various discretization methods i.e. Lagrangian and Eulerian; finite differences, finite volumes and finite elements have been used. Although normal water impact has been explained in more detail in the literature, different aspects of oblique water entry problems remain uncovered thoroughly and precise trajectory prediction of the objects seems out of reach or unexplored at least. Moreover, the forces generated during initial moments of water entry and water impact problems can be significantly large and vary according to the shape of the body nose, surface properties, angle of entry, impact velocity and fluid properties such as density, viscosity and surface tension to a lesser extent. The resulting flow field and the impact forces are also significantly dependent on water free surface deformations and its disintegration, with a full nonlinear inter-relationship. Therefore, prediction of subsequent body motion is quite difficult, due to fluid-solid interactions, especially when phenomenon such as bounce occurs (Rabiee et al., 2011). This is one of the incentives of the researchers to concentrate on this problem during past decades. Presently, more precise simulation of this problem looks feasible and performance of different numerical algorithms and codes if computed and compared would be valuable and useful. Additionally, using the proper algorithm and the code, more understanding of the flow field and body behavior would be provided in water entry conditions that have not studied yet. However, the choice of the proper numerical method for thorough investigation of such problems remains to be answered. This paper is focused on the study of oblique water entry of a blunted nose body at various angles and conditions experimentally and numerically to determine the appropriate software for the correct simulation of such fluid-solid interaction problems. Hence, a set of comprehensive experiments have been taken to
recent review article introduced the work in this area using inviscid incompressible model and BEM for fluid-solid interactions (Sun and Wu, 2014). Although the early works concentrated on the extension of the first analytical approaches, later on, experiments played an important role in illuminating this complex fluid-solid interaction problem for wedges and spheres (Greenhow, 1987). A comprehensive experimental investigation of rigid body slamming was performed by dropping flat-wedge-shaped models with small deadrise angles (Chuang, 1966). A full-scale water-entry test of a pointed nose objects was conducted and the results were compared with scaled model tests equipped with onboard miniature data acquisition system (Cole et al., 1993). Impact loading of prismatic bodies were measured and the effect of tail slamming was studied (New et al., 1993). More complicated situations such as: water entry of spinning spheres and water entry of rigid and deformable bodies were studied by (Truscott and Techet, 2009) and (Nila et al., 2012) respectively, although, more expensive and complex test facility were required in these kind of experimental work. The effects of different nose shapes on the trajectory of the projectiles entering water surface are studied in (Zhang et al., 2017). A series of experiments with PIV technique was performed for an oblique water entry of wedge shape body (Russo et al., 2018). Experimental results with an oblique impact of elliptic paraboloid confirmed the Wagner theory (Yves-Marie, 2014). Based on this experiment, the theory was developed for the body which can move freely in its six degrees of freedom and may also change its shape over time (Scolan and Korobkin, 2015). Oblique water entry problem is so complicated that its investigation cannot be handled using analytical methods alone without simplification of the actual problem. Although various experiments have been reported, however, oblique water entry of blunted nose objects covering a wide area change of parameters had not been observed in the literature that is carried out in this investigation. Full scale experiments are not usually feasible due to being expensive and time consuming and scaled model tests are limited by lack of full similarity and inherent limitation of the test equipment. Therefore, resort to numerical simulation is an attracting remedy for the blunt body oblique entry problems which is further studied in the following. During and after 80's numerical simulations also provided grounds for studying this problem. Ref (Seddon and Moatamedi, 2006). gives a through survey of this subject from 1925 to 2003, emphasizing on water impact of aerospace structures. One of the main suggestions of (Seddon and Moatamedi, 2006) for future investigation in this area was that more experimental and numerical work in oblique water entry is required which is taken in this paper. General features of water entry problems 232
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Fig. 2. Different stages of oblique water entry of the blunted nose model in the present study.
Generally, water impact problems can be divided into two major categories: normal water impact and oblique water impact, each of which has its own characteristics. In the moment of impact (Fig. 1-a) sudden change of pressure gradient is formed on the water surface and the water near and under the body is accelerated downwards. Body going down in the water, air bubble is formed behind it and this bubble is drawn into the water (Fig. 1b and c). Spray also would appear on the surface with variable velocity components in a way that the beginning of spray would be radial and outwards (Fig. 1-b and c) and by going upwards it moves radially inward (Fig. 1-c and d) until water level is connected above the bubble with a dome shape; this phenomenon is called surface enclosure (Yang and Qiu, 2011). Enclosure is one of the most important events during bubble growth and it has an important role in the formation of next bubble. This enclosed bubble begins to grow longitudinally and then hydrostatic and
evaluate the performance of the hopeful code. Although, for the sake of space, only some typical comparisons with experimental results have been presented in the article, the overall agreement of the chosen software with widely scattered experiments show the appropriateness of the choice.
1.1. General features of normal and oblique water entry problems Free surface body impact would lead to complex hydrodynamic events such as; spray formation, air bubble formation, enclosure of the water surface and air bubble and finally detachment of the air bubble, shown in Fig. 1 (Rabiee et al., 2010). Different phases of flow establishment as explained in (Laverty, 2005) are; shock wave phase, flow formation phase, phase of open and enclosed bubble and the final phase of complete wetness. 233
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dynamic pressure of the surrounding fluid would cause necking (Fig. 1-c) and finally rupture (Fig. 1-e) of the bubble (Laverty, 2005). Initial velocity of spray for low water entry velocities of sphere has been reported equal to 95 percent of impact velocity (Rabiee et al., 2010). However, during the oblique impact of a blunted nose object with water surface additional complexities arise;
A three piece model, made of polyethylene, prepared, Fig. 4. To control three variables: mass, center of mass location and moment of inertia a plumb bob is provided in the cylindrical part of model. The bob moving on a steel rod, which is threaded in order to set the desired values of the parameters. Properties of different pieces used in construction of the model are mentioned in Table 1. The relevant angles and body coordinates for the following discussion are shown in Fig. 5.
1 In the early impact time, water droplets began to splash outward not necessarily tangent to the nose. Especially, for the truncated nose objects water splashes in forward direction, Fig. 2-a. 2 The flow starts on the nose surface, however, it separates (detaches) in a short distance, Fig. 2-b and 2-c. In this region water bulges outward from the nose and the exerted hydrodynamic force is significantly affected by this part of spray. 3 Subsequently, splash sheath would be formed, Fig. 2-c and 2-d; this part is the main part of the spray which is connected to water surface and the position of formation of meniscus in the edge of air bubble. Spray sheath would not have any effect on the body motion after its formation; it would not affect shape of bubble until it causes the bubble to be closed. 4 The water displaced by the forming air cavity is presented by the spray and some of it is relevant to the raise of water level in a vast surface. 5 Under some circumstances of water entry, bubble enclosure would lead to formation of jet which can be raised to a considerable height above the water (Laverty, 2005).
2.2. Experimental setup These experiments are performed in a water tank with dimensions of 2500*1200*1200 mm containing still water. The shooting mechanism rotates up to 90 , therefore, different entry (pitch) angles from tangential up to normal to water surface may be chosen. An aluminum cylinder connected to the end of the model (with the length of 3 cm, Fig. 4) connects the model to the projecting system that uses compressed air. Different Value of model entry angle velocity magnitude can be chosen by adjusting the projector direction and pressure however exact value of these parameters plus angle of attack can be determined from trajectory data extracted from the images. Using a high speed video camera (SONY FS700) linear and angular position of the model can be extracted from different frames of the movie. This camera is capable of recording video with speed of 800 frames per second with HD quality that proved to be precise enough for the required accuracy of the results, as discussed in following sections. Subsequently, forces and moments acting on the model can be computed from these data, as will be explained later. In order to have a comprehensive set of tests covering various situations in water entry, twenty one test cases have been carried out with different values of mass, center of mass location, moment of inertia, angle of water entry, angle of attack and impact velocities. Some of the conditions of the experiments are briefly presented in Table 2. As can be seen from the last 3 columns of this table, different values of impact velocity, entry angle and angle of attack of the model are produced using different parameters for the flexible model (e.g. plumb bob weight and location, ejection model velocity) as explained above. Thanks to the flexible 3piece model that made various entry conditions mentioned in Table 2 possible. As an example, different frames at different times of test No.5 are shown in Fig. 6. In this Figure all of the stages of water entry including moment of impact, onset of water splash, quality of spray formation, its development and the way air bubble is drawn into the water can be clearly seen.
About oblique water entry of bodies with blunt nose, it has been proved experimentally that some of the water forming the spray is transferred from still water surface (May 1975). After complete submergence of nose, water flow would start to separate in a certain angle from the nose but it would not be separated from the main body; this would connect bubble wall to the spray sheath. Under common circumstances, surface tension would prevent spray sheath rupture and no hole would be seen in the splash sheath. After complete submergence of nose since no extra water would be added to the spray, it would get thinner because it is simultaneously stretched due to acceleration. In the experimental work performed in this article all of the mentioned events have been observed. In Fig. 2-a to 2-e different stages of water entry of constructed model with truncated nose can be seen. 2. Experimental tests 2.1. Model specification To validate numerical results of oblique impact, experiments were designed to throw a variable property model (Fig. 3) and investigate water impact under various conditions. In this way a set of test results is provided for comparison with and evaluation of different software results.
2.3. Image processing Recorded movies have been imported into TRACKER image processing software. This software can import different formats of videos and converts them into a set of constituting frames and subsequently coordinates of special marker points can be measured using the software
Fig. 3. General view of the model.
Fig. 4. The model components, numbers show the part numbers in Table 1. 234
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Table 1 Properties of different model pieces. Part no.
material
Density (kg/m3)
mass(g)
1
polyethylene
1100
34
2
polyethylene
1100
90
3
polyethylene
1100
64
4
steel
7800
41
5
Plumb (lead)
11000
187
6
aluminum
2700
19
Figure
be filtered and the velocities and accelerations be extracted. Three runs carried out for each test case of Table 2 and the average of these three runs (xcg, ycg and θ) are reported as the measured values at each instant of time. Inevitably, some deviations from the average values would occur that would be a measure of experimental errors. Although different effects are present as the error source in
tools. Defining a scale of for the pictures, coordinates of center of mass (xcg, ycg) and angular pitch rotations,θ, of the body with respect to horizontal direction can be measured at different times as shown in the software output window at the left side of Fig. 7. After classifying the experimental results in three matrices of xcg, ycg and θ at different times and various test cases of Table 2, the data should
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Fig. 5. Angles and coordinate directions. Table 2 List of performed experiments and their conditions. TEST NO.
MASS(kg)
CG POS.(m) from nose
MOMMENT OF INERTIA (kg-m2)
IMPACT VELOCITY (m/s)
ENTRY ANGLE (deg)
ANGLE OF ATTACK (deg)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
0.250 0.250 0.250 0.250 0.250 0.250 0.453 0.453 0.453 0.453 0.453 0.453 0.453 0.453 0.453 0.453 0.453 0.453 0.453 0.453 0.453
0.255 0.255 0.255 0.255 0.255 0.255 0.156 0.156 0.156 0.200 0.200 0.200 0.253 0.253 0.253 0.253 0.253 0.253 0.156 0.156 0.156
0.005092 0.005092 0.005092 0.005092 0.005092 0.005092 0.007863 0.007863 0.007863 0.005505 0.005505 0.005505 0.005537 0.005537 0.005537 0.005537 0.005537 0.005537 0.007863 0.007863 0.007863
2.6 3.3 5.7 2.3 4.3 5.4 1.7 3.6 5.2 2.3 4.7 5.0 2.6 2.7 5.5 2.5 3.5 5.6 3.7 4.7 7.1
10.0 10.0 10.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 10.0 10.0 10.0
4.0 3.3 2.3 10.0 6.5 6.4 6.6 0.0 0.0 7.5 0.0 1.0 3.2 0.0 2.5 4.3 2.0 2.5 6.3 1.2 2.5
were almost zero before the impact that is quite expectable, since the aerodynamic forces and moments are negligible while the solid model is free falling in air with small initial angular velocities(see Figs. 11 and 12). It was realized that no noticeable deviations from the vertical plane (xz plane) occurs and the proposition of 3-Degrees of freedom flight would be satisfactory. More results from these experiments are presented and discussed at the final section of results of this paper.
comparison of linear and angular positions between experiments and extracted results, however, the main source is due to displacement and angular position extraction errors taken from the images during the water entry. An estimation of this error can be obtained from different runs of the same experiment keeping all conditions constant. The variations of x (horizontal) and θ (pitch) coordinates for three runs of the test number 5 at different times are shown in Figs. 8 and 9, respectively. The vertical model position, z, varies similarly as of x and is not presented here. The error norm used here is the variance with the following definition for any discrete variable x:
3. Governing equations Different formulation and algorithms for solving the flow and solid governing equations are used in the three software packages. The first algorithm is based on finite volume formulation for the governing flow equations and the second uses the finite difference method to discretize the partial differential form of equations. As the final chosen software, i.e. ABAQUS, applies the ALE method and other codes use finite volume and finite difference Eulerian method, a general formulation of ALE method in integral form would encompass other methods and is presented in the following. This general formulation takes both forms of Lagrangian and Eulerian for different values of the parameter (mesh velocity) as explained in this section. The derivation of these equations is fully explained in (Donea et al., 2004) and the same notation is adopted here. The general integral formulation of the ALE method in terms of conservation of mass, momentum and energy for a material volume Vt bounded by a smooth closed surface St are (Donea et al., 2004):
n 1X VarðxÞ ¼ ðxi μÞ: n 1
Where n is the number of discrete values of xi and μ is the average of x. In the present case n is equal 3. Using this definition for the variables shown in Figs. 8 and 9, variances for horizontal displacement and pitch angle are 1.2, mm and 0.3 , respectively. Note that the error involved with force and moment comparison between experiments and computations would be higher since it requires taking second derivative of position with time, that should be carefully handled (see Fig. 10). Initial impact velocity components and pitch and angle of attack are estimated from the few frames collected before the model touching the water surface. It should be noted that in all of the tests angular velocities
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Fig. 6. Water entry stages for experiment No. 5. In this experiment model without bob, initial velocity of 4.3 m/s, angle of impact of 20 and angle of attack of 6 , impacted with water. Interval between images is 0.0125 s.
Fig. 7. TRACKER software environment and its output window.
Fig. 8. X coordinates for three runs of test No. 5.
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v ¼ vD on ΓD n⋅σ ¼ t on ΓN
where vD and t are proposed boundary velocities and tractions respectively; n is the unit outward normal to ΓN , and ΓD and ΓN are two separate subsets which define the piecewise smooth boundary of the domain. If position of some part of the boundary is unknown, e.g. free surface, then a combination of both conditions is required. The necessary conditions on the fluid-structure interface are: (a) no media particle can pass through it, and (b) stresses must be continuous across the surface. The first condition, i.e. the kinematical conditions can be expressed as v ¼ vS continuity of velocities where subscript S denotes the solid medium. Note that continuity of stresses is guaranteed if using the weak form of momentum equations (Donea et al., 2004).
Fig. 9. Pitch angle for three runs of test No. 5.
∂ ∫ ρdV þ ∫ ρc⋅ndS ¼ 0 ∂t χ Vt St
4. Numerical simulations
4.1. A comparison between three software results
∂ ∫ ρvdV þ ∫ ρvc⋅ndS ¼ ∫ ðr⋅σ þ ρbÞ dV ∂t χ Vt St Vt
In this research, the results of three different CFD software packages are compared, in order to choose the most suitable one. There are several software available to solve water impact problem numerically, discretizing the domain by generating computational grid, cell or elements and discretizing the governing integral or differential equations expressed in Eulerian or Lagrangin descriptions by different numerical approaches including finite difference, finite volume or finite element methods. Specifications of the tested software are presented in Table 3. The strategy for choosing between three software include testing them against normal water entry experiments and investigate if they succeed modeling the main flow behavior, as the first step. At the second step the quantitative check of the results of the candidate software with experiments would be carried out. For first step we choose the normal impact of a two dimensional cylinder as observed by (Von Karman, 1929) and Colicchio et al. (2009) while a 30 cm diameter and 40 cm in length cylinder with a specific gravity equal to 0.62 impacts the water surface. The impact speed is 2.55 m/s and flow pattern, object trajectory and velocity have been reported during water entry. This impact problem has been modeled in the three mentioned software and the results of mesh independent solution in terms of phase contours, that differentiate between water, air and solid are presented as follows. The mesh study performed on three different grids from 1 million to 2 million cells and the measures of convergence were based on displacement and body velocities. The third mesh is chosen as the mesh independent solution. As shown in the above Figures, software 1 cannot model the splash and cavity formation precisely neither in laminar nor in turbulent solutions. In the laminar flow the splash has not been formed and no jet exists. Although addition of turbulence model (k-ε two equation model) has forced a cavity to be formed but the general shape of the free surface and splash does not match the experimental data.
∂ ∫ ρEdV þ ∫ ρEc⋅ndS ¼ ∫ ðr⋅ðσ ⋅vÞ þ v⋅ρbÞ dV ∂t χ Vt St Vt Where ρ; v; σ ;b and E represent density, material velocity vector, Cauchy stress tensor, specific body force vector and total specific energy, respectively. In the above formulation, χ denotes the reference coordinates with jχ meaning “holding the reference coordinates fixed”. Also note that the convective velocity c is c ¼ v vb , in which vb is equal to the velocity of computational grid in convective terms. Different forms of the flow equations can be obtained via choosing the appropriate value for vb ¼ v ðc ¼ 0Þ for Lagrangian and vb ¼ 0 ðc ¼ vÞ for Eulerian description. Also the derivation of partial differential equations from the above integral forms, as implemented in the second algorithm, is a straight forward application of Gauss divergence theorem. The above formulation can be applied for discretization of problems in fluid dynamic and solid mechanics as well. As the medium in this study is incompressible, the third equation that is the energy conservation equation is not relevant and the first equation is simplified accordingly. However, regarding the unsteady and deforming mesh issues this simplified form of the continuity equation, also provides guidelines to the geometric conservation law (Donea et al., 2004).
3.1. Boundary conditions Boundary conditions are description independent and same boundary conditions are used in ALE as those applied in Lagrangian or Eulerian descriptions that are
Fig. 10. Phase contour from software 1 (laminar solution). 238
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Fig. 11. Phase contour from software 1 (turbulent solution).
Fig. 12. Phase contour from software 2 (turbulent solution).
The latter Figure compares experimental and different numerical flow pattern at t ¼ 0.1s. As can be seen in Fig. 14-a the first software cannot predict any separation point and therefore, air cavity in the laminar case. Fig. 14-b shows the same software but in turbulent case predicts wrong separation point and no jet like splash which is not acceptable at all. Same situation prevails for the second software, i.e., wrong separation point but with more damping of the jet, Fig. 14-c. Fig. 14-d shows that only the third software has the capability to realize the phenomena involved with normal water impact problems, i.e., correct separation position and jet like splash around the body. It should be mentioned that this agreement prevails with the experiments carried out by the present author shown in Fig. 1 and reported in (Rabiee et al., 2010). The conclusion would be that some codes cannot even predict the much simpler problem of normal water entry problem, therefore, fail to be a candidate for more involved problem of oblique water entry. This discrepancy of two codes includes the inability to model the basic and important events such as separation point of the free jet, the water spray and the size and collapse of the cavity. Through comparison with the experimental results it is shown that the third software, i.e. ABAQUS software is the most capable software among the three compared codes. Regarding the above conclusion, the third software is used to study the oblique entry problem presented in the former section and comparison of the results with experiments will be presented. Although numerical simulations of several cases of Table 2 are carried out, only one of them is discussed in detail here for the sake of brevity. The comparison
Table 3 General description of three commercial software examined in this study. description Software 1
Software 2
Software 3
Uses boundary fitted adaptive grid, dynamic mesh, six degree of freedom equations via user defined function, volume of fluid (VOF) model to capture free surface between water and air. Utilizes Eulerian description of equations and finite volume approach. Uses Cartesian computational grid, general moving object (GMO) algorithm to follow the solid boundary motion, volume of fluid (VOF) model to capture free surface between water and air. Utilizes Eulerian description of equations and finite difference approach. Uses coupled Eulerian-Lagrangian (CEL) formulation to model the physics. The Lagrangian object can move through Eulerian domain using a contact algorithm. Also Eulerian volume fraction (EVF) model is used to follow the free surface between water and void material (this software doesn't solve two phase flow problems). Utilizes ALE description and finite element approach.
In software 2, however the agreement is getting better. The free surface shape has become more accurate while still the splash shape is not correctly presented. Note that again this is a turbulent solution using a two equation model (see Fig. 13). But software 3 can predict the flow pattern more accurately compared to the other two software results. This program with the help of finite element formulation and coupled Eulerian-Lagrangian approach is able to predict the correct behavior of flow. A closer look at Fig. 14, compares the 3 software differences with experiments in more detail.
Fig. 13. Phase contour from software 3 (laminar solution). 239
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To provide another view of three-dimensional water surface the last frame of Fig. 16, i.e., at time 0.055 s is shown differently. Fig. 16 shows that in all instances of time, the agreement of splash sheath and separation line on model are satisfactory. Actually numerical solution agreement with other cases of Table 2 remains the same which increases confidence on numerical simulation using this software. It should be noticed that other commercial software examined in this study are incapable of producing such agreements with experiments and produced unacceptable results not of any value to be shown here. The coordinates of center of mass at different times obtained from experiments and numerical solution for this test case are shown in Figs. 17 and 18. The agreement is quite good. This agreement is also true for the pitch angle vs. time shown in Fig. 19. This figure shows oscillation of experimental pitch data about a time varying mean that is due to errors of extracting pitch data from video frames (0.3 deg. RMS). However, the time varying mean predictions using the software are satisfactory. 5.1. Various test cases of water entry The results of two tests are presented here to provide insight on the effect of change of various parameters involved. Although, only the empirical results are shown, the agreement of third software, i.e. the ABAQUS software remained satisfactory. The water entry of body was studied at two Froude Numbers, 8.41 and 16.3. Water entry angles of these cases are 30 and the angle of attacks are 0, 7 and 9 which contain two extreme cases. The results are shown in the following Figures. As the reference length is constant in the above tests, Froude number only represents the entry velocity. Although the variation of the axial force coefficient, in Fig. 20 and Fig. 23, is almost negligible with of angle of attack but the normal force and pitching moment coefficients shown in above Figures vary to some noticeable extent with angle of attack. This is true for both Froude Numbers. One interesting point is the remarkable different behavior for positive angle of attack, i.e., 7 Deg. Figs. 21 and 22 show normal force and pitching moment coefficients, respectively, at lower Froude Number and Figs. 24 and 25 show the same at higher Froude number. This distinct behavior at positive angle of attack 7 can be attributed to the vertical force at the nose of the body during the initial stages of water entry that tends to return the body toward the free surface. Regarding the coordinate system shown in Fig. 5, these arguments are substantiated with the negative direction of normal force (almost upward) in Figs. 21 and 23 and positive pitching moment (nose up), in Figs. 22 and 25. Although more discussion can be put forward to connect this force change with the flow field we refrain from doing this for the sake of space. Reminding that other interesting behavior seen among the results also deserve more investigation perhaps in other studies.
Fig. 14. Comparison of numerical phase contours (right) with experiment (left) for three software, at t ¼ 0.1s.
Fig. 15. Grid used for model and water in ABAQUS software.
and more experimental test case are presented afterwards. 5. Comparison of numerical and experimental results for oblique water entry Numerical simulation has been performed for some of the conducted test cases shown in Table 2. In this section, results of the numerical simulation and experiment of test No. 3, are compared. The test conditions include the entry velocity of 5.7 m/s, angle of entry of 10 and angle of attack of 2.3 . The mesh independent solution obtained after increasing the mesh numbers from three million in three steps and reaching 6,000,000 Eulerian cells for water domain that includes 15000 tetrahedral cells for meshing of the model surface, Fig. 15. The measures of convergence of mesh study are based on displacement and body velocities and the third mesh results is chosen as the mesh independent solution. This simulation has been performed on a computer with an 8core processor Intel 3 GHz, accommodating 24 gigabytes of memory for approximately 40 h to simulate about 0.08 s of initial motion of the model in the water. Fig. 16 shows numerical solution of free surface and video frames obtained during test No. 3 at different times. Note that a section of volume fraction of different phases are shown as numerical results and are not exactly the free surface shape seen in video frames.
6. Conclusions The main purpose of this study was to determine the suitable numerical software for simulation of oblique water entry of objects. To provide data for evaluation of three software codes, a set of experiments were designed and implemented, concentrating on oblique water entry of a truncated nose model. These experiments covered a range of varying parameters such as velocity, angle of attack, etc. It is shown that some codes cannot even predict the much simpler problem of normal water entry of a cylinder, therefore, fail to be a candidate for more involved problem of oblique water entry. This discrepancy of two codes includes the inability to model the basic and important events such as separation point of the free jet, the water spray and the size and collapse of the cavity. Through comparison with the experimental results it is shown that ABAQUS software is the most capable software among the three compared codes. Additionally, using the ABAQUS software for the simulation of the oblique water entry problems, it was noticed that main flow phenomena 240
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Fig. 16. Comparison of spray and air bubble shapes in experiment and simulation. 241
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Fig. 17. Comparison of horizontal displacement of center of mass in experiment and simulation. Fig. 21. Normal force coefficient versus depression depth for Froude number 8.41.
Fig. 18. Comparison of vertical displacement of center of the mass in experiment and simulation.
Fig. 22. Pitching moment coefficient versus depression depth for Froude number 8.41.
Fig. 19. Comparison of angular motion of the model in experiment and simulation.
Fig. 23. Axial force coefficient versus depression depth for Froude number 16.3.
effects such as splash and cavity formation and consequently dynamic behavior of the solid body (trajectory and force) are realistically modeled. In addition to qualitative and visual affirmation of phenomena such as splash sheaths of the flow as well as separation of flow on different walls of the body, also quantitative affirmations were shown for Fig. 20. Axial force coefficient versus depression depth for Froude number 8.41. 242
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Ocean Engineering 160 (2018) 231–243 Cole, J.K., Hailey, C.E., Gutierrez, W.T., Ferrario, M.T., 1993. An Experimental Investigation of High-speed Water-entry for Full-size and Scale-model Pointed Nose Vehicles. NASA STI/Recon Technical Report N, p. 94. Colicchio, G., Greco, M., Miozzi, M., Lugni, C. (Eds.), 2009. Experimental and Numerical Investigation of the Water-entry and Water-exit of a Circular cylinder. Proceedings of the 24th Int Workshop on Water Waves and Floating Bodies, Zelenogorsk, Russia, Apr. Donea, J., Huerta, A., Ponthot, J.-P., Rodríguez-Ferran, A., 2004. Arbitrary Lagrangian–eulerian Methods Encyclopedia of Computational Mechanics. Wiley, Hoboken, NJ. Engle, A., Lewis, R., 2003. A comparison of hydrodynamic impacts prediction methods with two dimensional drop test data. Mar. Struct. 16 (2), 175–182. Garabedian, P., 1953. Oblique water entry of a wedge. Commun. Pure Appl. Math. 6 (2), 157–165. Greenhow, M., 1987. Wedge entry into initially calm water. Appl. Ocean Res. 9 (4), 214–223. Gu, H., Qian, L., Causon, D., Mingham, C., Lin, P., 2014. Numerical simulation of water impact of solid bodies with vertical and oblique entries. Ocean. Eng. 75, 128–137. Iranmanesh, A., Passandideh-Fard, M., 2017. A three-dimensional numerical approach on water entry of a horizontal circular cylinder using the volume of fluid technique. Ocean. Eng. 130, 557–566. Kleefsman, K., Fekken, G., Veldman, A., Iwanowski, B., Buchner, B., 2005. A volume-offluid based simulation method for wave impact problems. J. Comput. Phys. 206 (1), 363–393. Laverty, S.M., 2005. Experimental Hydrodynamics of Spherical Projectiles Impacting on a Free Surface Using High Speed Imaging Techniques. Massachusetts Institute of Technology. May, A., 1975. Water Entry and the Cavity-running Behavior of Objects. Navesea Hydroballistics Advisory Committee Silver Spring MD. Miloh, T., 1991. On the oblique water-entry problem of a rigid sphere. J. Eng. Math. 25 (1), 77–92. Moore, M., Howison, S., Ockendon, J., Oliver, J., 2012. Three-dimensional oblique waterentry problems at small deadrise angles. J. Fluid Mech. 711, 259–280. New, A., Lee, T., Low, H. (Eds.), 1993. Impact Loading and Water Entrance Characteristics of Prismatic Bodies. The Third International Offshore and Polar Engineering Conference. International Society of Offshore and Polar Engineers. Nila, A., Vanlanduit, S., Vepa, S., Van Nuffel, D., Van Paepegem, W., Degroote, J., et al. (Eds.), 2012. High Speed Particle Image Velocimetry Measurements during Water Entry of Rigid and Deformable Bodies. Proceedings of the 16th International Symposium on Applications of Laser Techniques to Fluid Mechanics, Lisbon, Portugal. Rabiee, A., Alishahi, M.M., Saranjam, B., Emdad, H., 2010. Experimental and numerical investigation of sphere impact problem. In: Proceedings of ISME. Rabiee, A., Alishahi, M., Emdad, H., Saranjam, B., 2011. Experimental investigation of bounce phenomenon. Sci. Iran. 18 (3), 416–422. Russo, S., Jalalisendi, M., Falcucci, G., Porfiri, M., 2018. Experimental characterization of oblique and asymmetric water entry. Exp. Therm. Fluid Sci. 92, 141–161. Scolan, Y.M., Korobkin, A.A., 2015. Water entry of a body which moves in more than six degrees of freedom. Proc. R. Soc. A-Math Phys. Eng. Sci. 471 (2177), 13. Seddon, C., Moatamedi, M., 2006. Review of water entry with applications to aerospace structures. Int. J. Impact Eng. 32 (7), 1045–1067. Sun, S., Wu, G., 2014. Fully nonlinear simulation for fluid/structure impact: a review. J. Mar. Sci. Appl. 13 (3), 237–244. Truscott, T.T., Techet, A.H., 2009. Water entry of spinning spheres. J. Fluid Mech. 625, 135–165. Von Karman, T., 1929. The Impact on Seaplane Floats during landing. Wei, Z., Shi, X., Wang, Y., 2012. The oblique water entry impact of a body and its ballistic trajectory simulation. Int. J. Numer. Anal. Model. 9 (2). Xu, G., Duan, W., Wu, G., 2008. Numerical simulation of oblique water entry of an asymmetrical wedge. Ocean. Eng. 35 (16), 1597–1603. Yang, Q., Qiu, W., 2011. Numerical solutions of 2D and 3D slamming problems. Int. J. Marit. Eng. 153, A89–A97. Yang, Q., Qiu, W., 2012. Numerical simulation of water impact for 2D and 3D bodies. Ocean. Eng. 43, 82–89. Yoon, B., Semenov, Y. (Eds.), Apr; 2009. Flow Separation at the Initial Stage of the Oblique Water Entry of a Wedge. Proceedings of the 24th Int Workshop on Water Waves and Floating Bodies, Zelenogorsk, Russia. Yves-Marie, S., 2014. Oblique water entry of a three dimensional body. Int. J. Nav. Archit. Ocean Eng. 6 (4), 1197–1208. Zhang, W., Qi, Y., Huang, W., Gao, Y. (Eds.), 2017. Experimental Investigation on Underwater Trajectory Deviation of High-speed Projectile with Different Nose Shape. AIP Conference Proceedings. AIP Publishing.
Fig. 24. Normal force coefficient versus depression depth for Froude number 16.3.
Fig. 25. Pitching moment coefficient versus depression depth for Froude number 16.3.
displacement and angular position of the body. Therefore it may be concluded that the chosen software is a satisfactory tool for simulations of oblique water entry problems and determination of impact forces, moments that resulted in almost identical model trajectory. One important area for further investigation is to find the connection between capable physical models and numerical algorithms and real world flow behavior in this kind of problems. Second important extension that might be useful if this kind of performance evaluation be applied for other problems of fluid-body impact such as wave body impact problems. References Abraham, J., Gorman, J., Reseghetti, F., Sparrow, E., Stark, J., Shepard, T., 2014. Modeling and numerical simulation of the forces acting on a sphere during earlywater entry. Ocean. Eng. 76, 1–9. Chuang, S.-L., 1966. Slamming of Rigid Wedge-shaped Bodies with Various Deadrise Angles. David Taylor Model Basin Washington DC Structural Mechanics Lab.
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Contents lists available at ScienceDirect
Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
Experimental and numerical investigation for a reliable simulation tool for oblique water entry problems Mohammad Saeed Derakhshanian a, Mobin Haghdel b, *, Mohammad Mahdi Alishahi b, Ali Haghdel c a b c
Aerospace Engineering, Mechanical Engineering Dept., Shiraz University, Shiraz, Iran Mechanic Engineering Dept., Shiraz University, Shiraz, Iran Electrical Engineering, Communication and Electronic Engineering Dept., Shiraz University, Shiraz, Iran
A R T I C L E I N F O
A B S T R A C T
Keywords: Water entry Oblique impact Truncated-nose body Fluid-solid interaction ALE method
Oblique water entry of different marine structures involve several two-phase fluid dynamics phenomena including splash, asymmetrical cavity formation and cavity collapse that have been observed in experimental work. Correct prediction of the applied forces and moments that define the trajectory during water entry is dependent on precise simulation of the mentioned fluid dynamic effects. This study investigates and compares the performance of three different algorithms. i.e.: a) finite volume Eulerian; b) finite difference Eulerian, c) Arbitrary Lagrangian Eulerian (ALE), applied in different software codes. In order to achieve this goal, an experimental task, including normal and oblique water entry cases, was defined to provide the data set for comparing and verifying performance of these different software packages. These experiments have been designed and implemented, concentrating on oblique water entry of blunted nose models with various moment of inertia, center of gravity, and other dynamical properties that affect the water entry problem. Verification of computational results of the three software are carried out through comparison with the experimental results in two stages. Firstly, the fluid dynamics of splash, cavity formation and its collapse for normal water entry of a sphere are compared between experimental and three software numerical results. It is shown that two of the codes, i.e. “a” and “b” as defined above, cannot even predict the much simpler problem of normal water entry problem, therefore, fail to be a candidate for the more involved problem of oblique water entry, i.e., the second stage of the comparison. Through comparison with the experimental results it is shown that ABAQUS software, i.e. the “c” algorithm, is the most capable software among the three compared codes for the correct simulation of the involved fluid dynamic effects. Therefore, the ABAQUS software is applied for simulation of oblique water entry problem in the second stage and the results are compared with experiments. Upon comparison of various experimental results with this software output, it is shown that the numerical algorithm in this computer program (ALE) can be reliably used to simulate this kind of fluid solid interaction problem and determine the impact forces and moments more precisely even in the extreme angles and velocities of water entry problems.
1. Introduction Applications of water entry and water impact problems cover areas such as study of applied hydrodynamic forces and moments in the course of design of the diverse marine vehicles such as ships, vessels and flying boats. Problems of water entry and water impact have been the subject of the research as early as the pioneering work of Von Karman in 1929 (Von Karman, 1929) who were concerned with the hydrodynamics of a
landing seaplane. However, the majority of analytic oblique water-entry solutions considered were either two-dimensional or axisymmetric. Later on, application of potential solution of free surface impact problems for wedges were included in the early work of (Garabedian, 1953) and in a later study of (Xu et al., 2008) as a combined analytical and boundary element method for oblique entry problems. They derived the similarity solution under the assumption that the leading and trailing free surfaces separate from the wedge sides either perpendicularly or tangentially. A
* Corresponding author. E-mail addresses: [email protected] (M.S. Derakhshanian), [email protected] (M. Haghdel), [email protected] (M.M. Alishahi), [email protected] (A. Haghdel). https://doi.org/10.1016/j.oceaneng.2018.04.080 Received 28 June 2016; Received in revised form 26 March 2018; Accepted 20 April 2018 0029-8018/© 2018 Elsevier Ltd. All rights reserved.
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Fig. 1. Stages of normal water entry for a sphere (Rabiee et al., 2010).
were explained in (Laverty, 2005) that will be explained in the next section. A recent application of oblique water entry of a circular cylinder body was presented in (Colicchio et al., 2009). Application of the MSC Dytran software for prediction of water entry and trajectory of a blunt nose body was also shown in (Donea et al., 2004). The basis and formulation of Arbitrary Lagrangian Eulerian (ALE) method as a robust method for dealing with fluid-structure problems is comprehensively discussed in this reference. Analytical derivation of hydrodynamic loading of oblique entry of solid spheres was presented in (Miloh, 1991). Two and three dimensional study of normal and oblique entry of simple objects were numerically studied in (Engle and Lewis, 2003; Kleefsman et al., 2005; Yang and Qiu, 2011, 2012; Yoon and Semenov, 2009; Moore et al., 2012; Gu et al., 2014; Iranmanesh and Passandideh-Fard, 2017; Abraham et al., 2014). However the study of oblique entry of a full nose body in a general situation is not so frequent and (Wei et al., 2012) is one of the rare references in this regard. In the above mentioned numerical investigations various discretization methods i.e. Lagrangian and Eulerian; finite differences, finite volumes and finite elements have been used. Although normal water impact has been explained in more detail in the literature, different aspects of oblique water entry problems remain uncovered thoroughly and precise trajectory prediction of the objects seems out of reach or unexplored at least. Moreover, the forces generated during initial moments of water entry and water impact problems can be significantly large and vary according to the shape of the body nose, surface properties, angle of entry, impact velocity and fluid properties such as density, viscosity and surface tension to a lesser extent. The resulting flow field and the impact forces are also significantly dependent on water free surface deformations and its disintegration, with a full nonlinear inter-relationship. Therefore, prediction of subsequent body motion is quite difficult, due to fluid-solid interactions, especially when phenomenon such as bounce occurs (Rabiee et al., 2011). This is one of the incentives of the researchers to concentrate on this problem during past decades. Presently, more precise simulation of this problem looks feasible and performance of different numerical algorithms and codes if computed and compared would be valuable and useful. Additionally, using the proper algorithm and the code, more understanding of the flow field and body behavior would be provided in water entry conditions that have not studied yet. However, the choice of the proper numerical method for thorough investigation of such problems remains to be answered. This paper is focused on the study of oblique water entry of a blunted nose body at various angles and conditions experimentally and numerically to determine the appropriate software for the correct simulation of such fluid-solid interaction problems. Hence, a set of comprehensive experiments have been taken to
recent review article introduced the work in this area using inviscid incompressible model and BEM for fluid-solid interactions (Sun and Wu, 2014). Although the early works concentrated on the extension of the first analytical approaches, later on, experiments played an important role in illuminating this complex fluid-solid interaction problem for wedges and spheres (Greenhow, 1987). A comprehensive experimental investigation of rigid body slamming was performed by dropping flat-wedge-shaped models with small deadrise angles (Chuang, 1966). A full-scale water-entry test of a pointed nose objects was conducted and the results were compared with scaled model tests equipped with onboard miniature data acquisition system (Cole et al., 1993). Impact loading of prismatic bodies were measured and the effect of tail slamming was studied (New et al., 1993). More complicated situations such as: water entry of spinning spheres and water entry of rigid and deformable bodies were studied by (Truscott and Techet, 2009) and (Nila et al., 2012) respectively, although, more expensive and complex test facility were required in these kind of experimental work. The effects of different nose shapes on the trajectory of the projectiles entering water surface are studied in (Zhang et al., 2017). A series of experiments with PIV technique was performed for an oblique water entry of wedge shape body (Russo et al., 2018). Experimental results with an oblique impact of elliptic paraboloid confirmed the Wagner theory (Yves-Marie, 2014). Based on this experiment, the theory was developed for the body which can move freely in its six degrees of freedom and may also change its shape over time (Scolan and Korobkin, 2015). Oblique water entry problem is so complicated that its investigation cannot be handled using analytical methods alone without simplification of the actual problem. Although various experiments have been reported, however, oblique water entry of blunted nose objects covering a wide area change of parameters had not been observed in the literature that is carried out in this investigation. Full scale experiments are not usually feasible due to being expensive and time consuming and scaled model tests are limited by lack of full similarity and inherent limitation of the test equipment. Therefore, resort to numerical simulation is an attracting remedy for the blunt body oblique entry problems which is further studied in the following. During and after 80's numerical simulations also provided grounds for studying this problem. Ref (Seddon and Moatamedi, 2006). gives a through survey of this subject from 1925 to 2003, emphasizing on water impact of aerospace structures. One of the main suggestions of (Seddon and Moatamedi, 2006) for future investigation in this area was that more experimental and numerical work in oblique water entry is required which is taken in this paper. General features of water entry problems 232
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Fig. 2. Different stages of oblique water entry of the blunted nose model in the present study.
Generally, water impact problems can be divided into two major categories: normal water impact and oblique water impact, each of which has its own characteristics. In the moment of impact (Fig. 1-a) sudden change of pressure gradient is formed on the water surface and the water near and under the body is accelerated downwards. Body going down in the water, air bubble is formed behind it and this bubble is drawn into the water (Fig. 1b and c). Spray also would appear on the surface with variable velocity components in a way that the beginning of spray would be radial and outwards (Fig. 1-b and c) and by going upwards it moves radially inward (Fig. 1-c and d) until water level is connected above the bubble with a dome shape; this phenomenon is called surface enclosure (Yang and Qiu, 2011). Enclosure is one of the most important events during bubble growth and it has an important role in the formation of next bubble. This enclosed bubble begins to grow longitudinally and then hydrostatic and
evaluate the performance of the hopeful code. Although, for the sake of space, only some typical comparisons with experimental results have been presented in the article, the overall agreement of the chosen software with widely scattered experiments show the appropriateness of the choice.
1.1. General features of normal and oblique water entry problems Free surface body impact would lead to complex hydrodynamic events such as; spray formation, air bubble formation, enclosure of the water surface and air bubble and finally detachment of the air bubble, shown in Fig. 1 (Rabiee et al., 2010). Different phases of flow establishment as explained in (Laverty, 2005) are; shock wave phase, flow formation phase, phase of open and enclosed bubble and the final phase of complete wetness. 233
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dynamic pressure of the surrounding fluid would cause necking (Fig. 1-c) and finally rupture (Fig. 1-e) of the bubble (Laverty, 2005). Initial velocity of spray for low water entry velocities of sphere has been reported equal to 95 percent of impact velocity (Rabiee et al., 2010). However, during the oblique impact of a blunted nose object with water surface additional complexities arise;
A three piece model, made of polyethylene, prepared, Fig. 4. To control three variables: mass, center of mass location and moment of inertia a plumb bob is provided in the cylindrical part of model. The bob moving on a steel rod, which is threaded in order to set the desired values of the parameters. Properties of different pieces used in construction of the model are mentioned in Table 1. The relevant angles and body coordinates for the following discussion are shown in Fig. 5.
1 In the early impact time, water droplets began to splash outward not necessarily tangent to the nose. Especially, for the truncated nose objects water splashes in forward direction, Fig. 2-a. 2 The flow starts on the nose surface, however, it separates (detaches) in a short distance, Fig. 2-b and 2-c. In this region water bulges outward from the nose and the exerted hydrodynamic force is significantly affected by this part of spray. 3 Subsequently, splash sheath would be formed, Fig. 2-c and 2-d; this part is the main part of the spray which is connected to water surface and the position of formation of meniscus in the edge of air bubble. Spray sheath would not have any effect on the body motion after its formation; it would not affect shape of bubble until it causes the bubble to be closed. 4 The water displaced by the forming air cavity is presented by the spray and some of it is relevant to the raise of water level in a vast surface. 5 Under some circumstances of water entry, bubble enclosure would lead to formation of jet which can be raised to a considerable height above the water (Laverty, 2005).
2.2. Experimental setup These experiments are performed in a water tank with dimensions of 2500*1200*1200 mm containing still water. The shooting mechanism rotates up to 90 , therefore, different entry (pitch) angles from tangential up to normal to water surface may be chosen. An aluminum cylinder connected to the end of the model (with the length of 3 cm, Fig. 4) connects the model to the projecting system that uses compressed air. Different Value of model entry angle velocity magnitude can be chosen by adjusting the projector direction and pressure however exact value of these parameters plus angle of attack can be determined from trajectory data extracted from the images. Using a high speed video camera (SONY FS700) linear and angular position of the model can be extracted from different frames of the movie. This camera is capable of recording video with speed of 800 frames per second with HD quality that proved to be precise enough for the required accuracy of the results, as discussed in following sections. Subsequently, forces and moments acting on the model can be computed from these data, as will be explained later. In order to have a comprehensive set of tests covering various situations in water entry, twenty one test cases have been carried out with different values of mass, center of mass location, moment of inertia, angle of water entry, angle of attack and impact velocities. Some of the conditions of the experiments are briefly presented in Table 2. As can be seen from the last 3 columns of this table, different values of impact velocity, entry angle and angle of attack of the model are produced using different parameters for the flexible model (e.g. plumb bob weight and location, ejection model velocity) as explained above. Thanks to the flexible 3piece model that made various entry conditions mentioned in Table 2 possible. As an example, different frames at different times of test No.5 are shown in Fig. 6. In this Figure all of the stages of water entry including moment of impact, onset of water splash, quality of spray formation, its development and the way air bubble is drawn into the water can be clearly seen.
About oblique water entry of bodies with blunt nose, it has been proved experimentally that some of the water forming the spray is transferred from still water surface (May 1975). After complete submergence of nose, water flow would start to separate in a certain angle from the nose but it would not be separated from the main body; this would connect bubble wall to the spray sheath. Under common circumstances, surface tension would prevent spray sheath rupture and no hole would be seen in the splash sheath. After complete submergence of nose since no extra water would be added to the spray, it would get thinner because it is simultaneously stretched due to acceleration. In the experimental work performed in this article all of the mentioned events have been observed. In Fig. 2-a to 2-e different stages of water entry of constructed model with truncated nose can be seen. 2. Experimental tests 2.1. Model specification To validate numerical results of oblique impact, experiments were designed to throw a variable property model (Fig. 3) and investigate water impact under various conditions. In this way a set of test results is provided for comparison with and evaluation of different software results.
2.3. Image processing Recorded movies have been imported into TRACKER image processing software. This software can import different formats of videos and converts them into a set of constituting frames and subsequently coordinates of special marker points can be measured using the software
Fig. 3. General view of the model.
Fig. 4. The model components, numbers show the part numbers in Table 1. 234
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Table 1 Properties of different model pieces. Part no.
material
Density (kg/m3)
mass(g)
1
polyethylene
1100
34
2
polyethylene
1100
90
3
polyethylene
1100
64
4
steel
7800
41
5
Plumb (lead)
11000
187
6
aluminum
2700
19
Figure
be filtered and the velocities and accelerations be extracted. Three runs carried out for each test case of Table 2 and the average of these three runs (xcg, ycg and θ) are reported as the measured values at each instant of time. Inevitably, some deviations from the average values would occur that would be a measure of experimental errors. Although different effects are present as the error source in
tools. Defining a scale of for the pictures, coordinates of center of mass (xcg, ycg) and angular pitch rotations,θ, of the body with respect to horizontal direction can be measured at different times as shown in the software output window at the left side of Fig. 7. After classifying the experimental results in three matrices of xcg, ycg and θ at different times and various test cases of Table 2, the data should
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Fig. 5. Angles and coordinate directions. Table 2 List of performed experiments and their conditions. TEST NO.
MASS(kg)
CG POS.(m) from nose
MOMMENT OF INERTIA (kg-m2)
IMPACT VELOCITY (m/s)
ENTRY ANGLE (deg)
ANGLE OF ATTACK (deg)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
0.250 0.250 0.250 0.250 0.250 0.250 0.453 0.453 0.453 0.453 0.453 0.453 0.453 0.453 0.453 0.453 0.453 0.453 0.453 0.453 0.453
0.255 0.255 0.255 0.255 0.255 0.255 0.156 0.156 0.156 0.200 0.200 0.200 0.253 0.253 0.253 0.253 0.253 0.253 0.156 0.156 0.156
0.005092 0.005092 0.005092 0.005092 0.005092 0.005092 0.007863 0.007863 0.007863 0.005505 0.005505 0.005505 0.005537 0.005537 0.005537 0.005537 0.005537 0.005537 0.007863 0.007863 0.007863
2.6 3.3 5.7 2.3 4.3 5.4 1.7 3.6 5.2 2.3 4.7 5.0 2.6 2.7 5.5 2.5 3.5 5.6 3.7 4.7 7.1
10.0 10.0 10.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 10.0 10.0 10.0
4.0 3.3 2.3 10.0 6.5 6.4 6.6 0.0 0.0 7.5 0.0 1.0 3.2 0.0 2.5 4.3 2.0 2.5 6.3 1.2 2.5
were almost zero before the impact that is quite expectable, since the aerodynamic forces and moments are negligible while the solid model is free falling in air with small initial angular velocities(see Figs. 11 and 12). It was realized that no noticeable deviations from the vertical plane (xz plane) occurs and the proposition of 3-Degrees of freedom flight would be satisfactory. More results from these experiments are presented and discussed at the final section of results of this paper.
comparison of linear and angular positions between experiments and extracted results, however, the main source is due to displacement and angular position extraction errors taken from the images during the water entry. An estimation of this error can be obtained from different runs of the same experiment keeping all conditions constant. The variations of x (horizontal) and θ (pitch) coordinates for three runs of the test number 5 at different times are shown in Figs. 8 and 9, respectively. The vertical model position, z, varies similarly as of x and is not presented here. The error norm used here is the variance with the following definition for any discrete variable x:
3. Governing equations Different formulation and algorithms for solving the flow and solid governing equations are used in the three software packages. The first algorithm is based on finite volume formulation for the governing flow equations and the second uses the finite difference method to discretize the partial differential form of equations. As the final chosen software, i.e. ABAQUS, applies the ALE method and other codes use finite volume and finite difference Eulerian method, a general formulation of ALE method in integral form would encompass other methods and is presented in the following. This general formulation takes both forms of Lagrangian and Eulerian for different values of the parameter (mesh velocity) as explained in this section. The derivation of these equations is fully explained in (Donea et al., 2004) and the same notation is adopted here. The general integral formulation of the ALE method in terms of conservation of mass, momentum and energy for a material volume Vt bounded by a smooth closed surface St are (Donea et al., 2004):
n 1X VarðxÞ ¼ ðxi μÞ: n 1
Where n is the number of discrete values of xi and μ is the average of x. In the present case n is equal 3. Using this definition for the variables shown in Figs. 8 and 9, variances for horizontal displacement and pitch angle are 1.2, mm and 0.3 , respectively. Note that the error involved with force and moment comparison between experiments and computations would be higher since it requires taking second derivative of position with time, that should be carefully handled (see Fig. 10). Initial impact velocity components and pitch and angle of attack are estimated from the few frames collected before the model touching the water surface. It should be noted that in all of the tests angular velocities
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Fig. 6. Water entry stages for experiment No. 5. In this experiment model without bob, initial velocity of 4.3 m/s, angle of impact of 20 and angle of attack of 6 , impacted with water. Interval between images is 0.0125 s.
Fig. 7. TRACKER software environment and its output window.
Fig. 8. X coordinates for three runs of test No. 5.
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v ¼ vD on ΓD n⋅σ ¼ t on ΓN
where vD and t are proposed boundary velocities and tractions respectively; n is the unit outward normal to ΓN , and ΓD and ΓN are two separate subsets which define the piecewise smooth boundary of the domain. If position of some part of the boundary is unknown, e.g. free surface, then a combination of both conditions is required. The necessary conditions on the fluid-structure interface are: (a) no media particle can pass through it, and (b) stresses must be continuous across the surface. The first condition, i.e. the kinematical conditions can be expressed as v ¼ vS continuity of velocities where subscript S denotes the solid medium. Note that continuity of stresses is guaranteed if using the weak form of momentum equations (Donea et al., 2004).
Fig. 9. Pitch angle for three runs of test No. 5.
∂ ∫ ρdV þ ∫ ρc⋅ndS ¼ 0 ∂t χ Vt St
4. Numerical simulations
4.1. A comparison between three software results
∂ ∫ ρvdV þ ∫ ρvc⋅ndS ¼ ∫ ðr⋅σ þ ρbÞ dV ∂t χ Vt St Vt
In this research, the results of three different CFD software packages are compared, in order to choose the most suitable one. There are several software available to solve water impact problem numerically, discretizing the domain by generating computational grid, cell or elements and discretizing the governing integral or differential equations expressed in Eulerian or Lagrangin descriptions by different numerical approaches including finite difference, finite volume or finite element methods. Specifications of the tested software are presented in Table 3. The strategy for choosing between three software include testing them against normal water entry experiments and investigate if they succeed modeling the main flow behavior, as the first step. At the second step the quantitative check of the results of the candidate software with experiments would be carried out. For first step we choose the normal impact of a two dimensional cylinder as observed by (Von Karman, 1929) and Colicchio et al. (2009) while a 30 cm diameter and 40 cm in length cylinder with a specific gravity equal to 0.62 impacts the water surface. The impact speed is 2.55 m/s and flow pattern, object trajectory and velocity have been reported during water entry. This impact problem has been modeled in the three mentioned software and the results of mesh independent solution in terms of phase contours, that differentiate between water, air and solid are presented as follows. The mesh study performed on three different grids from 1 million to 2 million cells and the measures of convergence were based on displacement and body velocities. The third mesh is chosen as the mesh independent solution. As shown in the above Figures, software 1 cannot model the splash and cavity formation precisely neither in laminar nor in turbulent solutions. In the laminar flow the splash has not been formed and no jet exists. Although addition of turbulence model (k-ε two equation model) has forced a cavity to be formed but the general shape of the free surface and splash does not match the experimental data.
∂ ∫ ρEdV þ ∫ ρEc⋅ndS ¼ ∫ ðr⋅ðσ ⋅vÞ þ v⋅ρbÞ dV ∂t χ Vt St Vt Where ρ; v; σ ;b and E represent density, material velocity vector, Cauchy stress tensor, specific body force vector and total specific energy, respectively. In the above formulation, χ denotes the reference coordinates with jχ meaning “holding the reference coordinates fixed”. Also note that the convective velocity c is c ¼ v vb , in which vb is equal to the velocity of computational grid in convective terms. Different forms of the flow equations can be obtained via choosing the appropriate value for vb ¼ v ðc ¼ 0Þ for Lagrangian and vb ¼ 0 ðc ¼ vÞ for Eulerian description. Also the derivation of partial differential equations from the above integral forms, as implemented in the second algorithm, is a straight forward application of Gauss divergence theorem. The above formulation can be applied for discretization of problems in fluid dynamic and solid mechanics as well. As the medium in this study is incompressible, the third equation that is the energy conservation equation is not relevant and the first equation is simplified accordingly. However, regarding the unsteady and deforming mesh issues this simplified form of the continuity equation, also provides guidelines to the geometric conservation law (Donea et al., 2004).
3.1. Boundary conditions Boundary conditions are description independent and same boundary conditions are used in ALE as those applied in Lagrangian or Eulerian descriptions that are
Fig. 10. Phase contour from software 1 (laminar solution). 238
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Fig. 11. Phase contour from software 1 (turbulent solution).
Fig. 12. Phase contour from software 2 (turbulent solution).
The latter Figure compares experimental and different numerical flow pattern at t ¼ 0.1s. As can be seen in Fig. 14-a the first software cannot predict any separation point and therefore, air cavity in the laminar case. Fig. 14-b shows the same software but in turbulent case predicts wrong separation point and no jet like splash which is not acceptable at all. Same situation prevails for the second software, i.e., wrong separation point but with more damping of the jet, Fig. 14-c. Fig. 14-d shows that only the third software has the capability to realize the phenomena involved with normal water impact problems, i.e., correct separation position and jet like splash around the body. It should be mentioned that this agreement prevails with the experiments carried out by the present author shown in Fig. 1 and reported in (Rabiee et al., 2010). The conclusion would be that some codes cannot even predict the much simpler problem of normal water entry problem, therefore, fail to be a candidate for more involved problem of oblique water entry. This discrepancy of two codes includes the inability to model the basic and important events such as separation point of the free jet, the water spray and the size and collapse of the cavity. Through comparison with the experimental results it is shown that the third software, i.e. ABAQUS software is the most capable software among the three compared codes. Regarding the above conclusion, the third software is used to study the oblique entry problem presented in the former section and comparison of the results with experiments will be presented. Although numerical simulations of several cases of Table 2 are carried out, only one of them is discussed in detail here for the sake of brevity. The comparison
Table 3 General description of three commercial software examined in this study. description Software 1
Software 2
Software 3
Uses boundary fitted adaptive grid, dynamic mesh, six degree of freedom equations via user defined function, volume of fluid (VOF) model to capture free surface between water and air. Utilizes Eulerian description of equations and finite volume approach. Uses Cartesian computational grid, general moving object (GMO) algorithm to follow the solid boundary motion, volume of fluid (VOF) model to capture free surface between water and air. Utilizes Eulerian description of equations and finite difference approach. Uses coupled Eulerian-Lagrangian (CEL) formulation to model the physics. The Lagrangian object can move through Eulerian domain using a contact algorithm. Also Eulerian volume fraction (EVF) model is used to follow the free surface between water and void material (this software doesn't solve two phase flow problems). Utilizes ALE description and finite element approach.
In software 2, however the agreement is getting better. The free surface shape has become more accurate while still the splash shape is not correctly presented. Note that again this is a turbulent solution using a two equation model (see Fig. 13). But software 3 can predict the flow pattern more accurately compared to the other two software results. This program with the help of finite element formulation and coupled Eulerian-Lagrangian approach is able to predict the correct behavior of flow. A closer look at Fig. 14, compares the 3 software differences with experiments in more detail.
Fig. 13. Phase contour from software 3 (laminar solution). 239
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To provide another view of three-dimensional water surface the last frame of Fig. 16, i.e., at time 0.055 s is shown differently. Fig. 16 shows that in all instances of time, the agreement of splash sheath and separation line on model are satisfactory. Actually numerical solution agreement with other cases of Table 2 remains the same which increases confidence on numerical simulation using this software. It should be noticed that other commercial software examined in this study are incapable of producing such agreements with experiments and produced unacceptable results not of any value to be shown here. The coordinates of center of mass at different times obtained from experiments and numerical solution for this test case are shown in Figs. 17 and 18. The agreement is quite good. This agreement is also true for the pitch angle vs. time shown in Fig. 19. This figure shows oscillation of experimental pitch data about a time varying mean that is due to errors of extracting pitch data from video frames (0.3 deg. RMS). However, the time varying mean predictions using the software are satisfactory. 5.1. Various test cases of water entry The results of two tests are presented here to provide insight on the effect of change of various parameters involved. Although, only the empirical results are shown, the agreement of third software, i.e. the ABAQUS software remained satisfactory. The water entry of body was studied at two Froude Numbers, 8.41 and 16.3. Water entry angles of these cases are 30 and the angle of attacks are 0, 7 and 9 which contain two extreme cases. The results are shown in the following Figures. As the reference length is constant in the above tests, Froude number only represents the entry velocity. Although the variation of the axial force coefficient, in Fig. 20 and Fig. 23, is almost negligible with of angle of attack but the normal force and pitching moment coefficients shown in above Figures vary to some noticeable extent with angle of attack. This is true for both Froude Numbers. One interesting point is the remarkable different behavior for positive angle of attack, i.e., 7 Deg. Figs. 21 and 22 show normal force and pitching moment coefficients, respectively, at lower Froude Number and Figs. 24 and 25 show the same at higher Froude number. This distinct behavior at positive angle of attack 7 can be attributed to the vertical force at the nose of the body during the initial stages of water entry that tends to return the body toward the free surface. Regarding the coordinate system shown in Fig. 5, these arguments are substantiated with the negative direction of normal force (almost upward) in Figs. 21 and 23 and positive pitching moment (nose up), in Figs. 22 and 25. Although more discussion can be put forward to connect this force change with the flow field we refrain from doing this for the sake of space. Reminding that other interesting behavior seen among the results also deserve more investigation perhaps in other studies.
Fig. 14. Comparison of numerical phase contours (right) with experiment (left) for three software, at t ¼ 0.1s.
Fig. 15. Grid used for model and water in ABAQUS software.
and more experimental test case are presented afterwards. 5. Comparison of numerical and experimental results for oblique water entry Numerical simulation has been performed for some of the conducted test cases shown in Table 2. In this section, results of the numerical simulation and experiment of test No. 3, are compared. The test conditions include the entry velocity of 5.7 m/s, angle of entry of 10 and angle of attack of 2.3 . The mesh independent solution obtained after increasing the mesh numbers from three million in three steps and reaching 6,000,000 Eulerian cells for water domain that includes 15000 tetrahedral cells for meshing of the model surface, Fig. 15. The measures of convergence of mesh study are based on displacement and body velocities and the third mesh results is chosen as the mesh independent solution. This simulation has been performed on a computer with an 8core processor Intel 3 GHz, accommodating 24 gigabytes of memory for approximately 40 h to simulate about 0.08 s of initial motion of the model in the water. Fig. 16 shows numerical solution of free surface and video frames obtained during test No. 3 at different times. Note that a section of volume fraction of different phases are shown as numerical results and are not exactly the free surface shape seen in video frames.
6. Conclusions The main purpose of this study was to determine the suitable numerical software for simulation of oblique water entry of objects. To provide data for evaluation of three software codes, a set of experiments were designed and implemented, concentrating on oblique water entry of a truncated nose model. These experiments covered a range of varying parameters such as velocity, angle of attack, etc. It is shown that some codes cannot even predict the much simpler problem of normal water entry of a cylinder, therefore, fail to be a candidate for more involved problem of oblique water entry. This discrepancy of two codes includes the inability to model the basic and important events such as separation point of the free jet, the water spray and the size and collapse of the cavity. Through comparison with the experimental results it is shown that ABAQUS software is the most capable software among the three compared codes. Additionally, using the ABAQUS software for the simulation of the oblique water entry problems, it was noticed that main flow phenomena 240
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Fig. 16. Comparison of spray and air bubble shapes in experiment and simulation. 241
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Fig. 17. Comparison of horizontal displacement of center of mass in experiment and simulation. Fig. 21. Normal force coefficient versus depression depth for Froude number 8.41.
Fig. 18. Comparison of vertical displacement of center of the mass in experiment and simulation.
Fig. 22. Pitching moment coefficient versus depression depth for Froude number 8.41.
Fig. 19. Comparison of angular motion of the model in experiment and simulation.
Fig. 23. Axial force coefficient versus depression depth for Froude number 16.3.
effects such as splash and cavity formation and consequently dynamic behavior of the solid body (trajectory and force) are realistically modeled. In addition to qualitative and visual affirmation of phenomena such as splash sheaths of the flow as well as separation of flow on different walls of the body, also quantitative affirmations were shown for Fig. 20. Axial force coefficient versus depression depth for Froude number 8.41. 242
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Fig. 24. Normal force coefficient versus depression depth for Froude number 16.3.
Fig. 25. Pitching moment coefficient versus depression depth for Froude number 16.3.
displacement and angular position of the body. Therefore it may be concluded that the chosen software is a satisfactory tool for simulations of oblique water entry problems and determination of impact forces, moments that resulted in almost identical model trajectory. One important area for further investigation is to find the connection between capable physical models and numerical algorithms and real world flow behavior in this kind of problems. Second important extension that might be useful if this kind of performance evaluation be applied for other problems of fluid-body impact such as wave body impact problems. References Abraham, J., Gorman, J., Reseghetti, F., Sparrow, E., Stark, J., Shepard, T., 2014. Modeling and numerical simulation of the forces acting on a sphere during earlywater entry. Ocean. Eng. 76, 1–9. Chuang, S.-L., 1966. Slamming of Rigid Wedge-shaped Bodies with Various Deadrise Angles. David Taylor Model Basin Washington DC Structural Mechanics Lab.
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