Skip to main content
SpringerLink
Log in

A parametric study on wave–floating storage tank interaction using coupled VOF-FDM method

  • Original article
  • Published:
Journal of Marine Science and Technology Aims and scope Submit manuscript

Abstract

In this paper, a two-dimensional numerical model was developed for generating wave in a numerical wave tank, and simulating wave–floating storage tank interaction using coupled volume of fluid (VOF) and fast-fictitious domain method (FDM). In the developed model, the fluid flow is considered as viscous and incompressible, and, therefore, Navier–Stokes and continuity equations were used as governing equations. The FDM was used in the VOF technique for tracking the free surface and storage tank motion. The Navier–Stokes equations were discretized using staggered grids finite difference method, and solved by SMAC method. Airy and solitary waves were generated using numerical wave tank, and the results were validated using the wave-maker theory. In this step, floating bodies with different dimensions were modeled under the Airy waves with different amplitudes and periods. Then, the effect of the sloshing phenomenon was considered, and a novel multi-dimensional equation was presented for maximum heave motion prediction of the half-full floating storage tanks. The results show that the sloshing phenomenon increases the maximum have motion of the floating storage tank between 1 and 5%. In the next step, the optimum storage tank dimensions were suggested, considering the heave motion of the tank. Based on the results, the floating storage tank is optimized, when the aspect ratio of the tank is AR = 2. Furthermore, the effect of fill percentage was evaluated on the heave motion of the storage tank. The results show that heave motion is maximized, when the fill percentage of the tank is 60%. Finally, the floating storage tanks were modeled under the solitary waves. The results show that the maximum heave motion of floating body against solitary wave has a linear relation with the wave amplitude of solitary wave.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14

Similar content being viewed by others

Abbreviations

a :

Airy wave amplitude

a sol :

Solitary wave amplitude

c :

Phase speed

Elv-c:

Elevation of the center gravity of the floating storage tank (point C in Fig. 1)

\({F_{i,j}}\) :

Volume fraction in cell (i, j)

Fr :

Froude number

g :

Gravity acceleration

h 0 :

Mean water depth

\(k\) :

Wave number

\(r\) :

Position vector with respect to the solid center of mass

\({I_{\text{s}}}\) :

Moment of inertia

MHM:

Maximum heave motion of point C (see Fig. 1)

\({M_{\text{s}}}\) :

Solid mass

P :

Dynamic pressure

Re :

Reynolds number

t ter :

Terminated time

T :

Time

\({U_{\text{i}}}\) :

Average velocity

V :

Velocity vector

\({\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {V} _{\text{s}}}\) :

Translational velocity

V smax :

Maximum velocity amplitude in heave of the floating storage tank

β :

Outskirt decay coefficient

\(\nu\) :

Kinematic viscosity

\(\rho\) :

Density

\(\Delta x\) :

Mesh sizes in the x direction

\(\Delta y\) :

Mesh sizes in the y direction

\(\forall\) :

Solid volume

\(\omega\) :

Wave angular frequency

\({\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {\omega } _s}\) :

Angular velocity

\(\eta\) :

Free surface elevation

\({\eta _{\hbox{max} }}\) :

Maximum heave motion of the floating storage tank

References

  1. Fochesato C, Grilli S, Dias F (2007) Numerical modeling of extreme rogue waves generated by directional energy focusing. Wave Motion 44:395–416

    Article  MathSciNet  MATH  Google Scholar 

  2. Ducrozet G, Bonnefoy F, Le Touzé D, Ferrant P (2012) A modified high-order spectral method for wavemaker modeling in a numerical wave tank. Eur J Mech B Fluids 34:19–34

    Article  MATH  Google Scholar 

  3. Saghi H, Ketabdari MJ, Booshi S (2012) Generation of linear and nonlinear waves in numerical wave tank using clustering technique-volume of fluid method. Appl Math Mech (English edition) 33(9):1179–1190

    Article  Google Scholar 

  4. Rudman M (1997) Volume-tracking methods for interfacial flow calculations. Int J Numer Methods Fluids 24:671–691

    Article  MathSciNet  MATH  Google Scholar 

  5. Troch P, De Rouck J (1999) An active wave generating-absorbing boundary condition for VOF type numerical model. Coast Eng 38:223–247

    Article  Google Scholar 

  6. Schaffer HA, Steenberg CM (2003) Second-order wave maker theory for multidirectional waves. Ocean Eng 30:1203–1231

    Article  Google Scholar 

  7. Lin P, Karunarathna AS (2007) Numerical study of Solitary wave interaction with porous breakwaters. J Waterw Port Coast Ocean Eng 133(5):352–363

    Article  Google Scholar 

  8. Shin S, Bae SY, Kim IC, Kim YJ, Yoo HK (2012) Simulation of free surface flows using the flux-difference splitting scheme on the hybrid Cartesian/immersed boundary method. Int J Numer Meth Fluids 68(3):360–376

    Article  MathSciNet  MATH  Google Scholar 

  9. Mirzaii I, Passandideh-Fard M (2012) Modeling free surface flows in presence of an arbitrary moving object. Int J Multiph Flow 39:216–226

    Article  Google Scholar 

  10. Hu HH, Joseph DD (1992) Direct simulation of fluid particle motions. Theor Comput Fluid Dyn 3:285–306

    Article  MATH  Google Scholar 

  11. Hu HH (1996) Direct simulation of flows of solid–liquid mixtures. Int J Multiph Flow 22:335–352

    Article  MATH  Google Scholar 

  12. Johnson AA, Tezduyar TE (1996) Simulation of multiple spheres falling in a liquid filled tube. Comput Methods Appl Mech Eng 134:351373

    Article  MathSciNet  MATH  Google Scholar 

  13. Jeong KL, Lee YG (2016) A numerical simulation method for the flow around floating bodies in regular waves using a three-dimensional rectilinear grid system. Int J Nav Archit Ocean Eng 8:277–300

    Article  Google Scholar 

  14. Chen Q, Zang J, Dimakopoulos AS, Kelly DM, Williams CJK (2016) A Cartesian cut cell based two-way strong fluid–solid coupling algorithm for 2D floating bodies. J Fluids Struct 62:252–271

    Article  Google Scholar 

  15. Wang WH, Wang LL, Du YZ, Yao YX, Huang Y (2016) Numerical and experimental analysis on motion performance of new sandglass-type floating body in waves. Mar Struct 46:56–77

    Article  Google Scholar 

  16. Lu X, Wang KH (2015) Modeling a solitary wave interaction with a fixed floating body using an integrated analytical–numerical approach. Ocean Eng 109:691–704

    Article  Google Scholar 

  17. Abramson HN (1996) The dynamic behavior of liquids in moving containers. NASA, SP-106. National Aeronautics and Space Administration, Washington D.C.

    Google Scholar 

  18. Frandsen JB (2004) Sloshing motions in excited tank. J Comput Phys 106:53–87

    Article  MATH  Google Scholar 

  19. Ketabdari MJ, Saghi H (2012) Numerical analysis of trapezoidal storage tank due to liquid sloshing phenomenon. Iran J Mar Sci Technol 18(61):33–39

    MATH  Google Scholar 

  20. Ketabdari MJ, Saghi H (2012) Numerical modeling of wave–floating body interaction using fictitious domain method. In: 10th International conference on coasts, ports and marine structures

  21. Ketabdari MJ, Saghi H (2013) Parametric study for optimization of storage tanks considering sloshing phenomenon using coupled BEM–FEM. Appl Math Comput 224:123–139

    MathSciNet  MATH  Google Scholar 

  22. Ketabdari MJ, Saghi H (2013) Numerical study on behavior of the trapezoidal storage tank due to liquid sloshing impact. Int J Comput Methods 10(6):1–22

    Article  MathSciNet  MATH  Google Scholar 

  23. Ketabdari MJ, Saghi H (2013) A new arrangement with nonlinear sidewalls for tanker ship storage panels. J Ocean Univ China 12(1):23–31

    Article  Google Scholar 

  24. Ketabdari MJ, Saghi H, Rezaei H, Rezanejad K (2015) Optimization of linear and nonlinear sidewall storage units coupled boundary element-finite element methods. KSCE J Civ Eng 19(4):805–813

    Article  Google Scholar 

  25. Saghi H, Ketabdari MJ (2012) Numerical simulation of sloshing in rectangular storage tank using coupled FEM-BEM. J Mar Sci Appl 11(4):417–426

    Article  Google Scholar 

  26. Saghi H (2016) The pressure distribution on the rectangular and trapezoidal storage tanks’ perimeters due to liquid sloshing phenomenon. Int J Nav Archit Ocean Eng 8:153–168

    Article  Google Scholar 

  27. Chen BF, Nokes R (2005) Time-independent finite difference analysis of fully nonlinear and viscous fluid sloshing in a rectangular tank. J Comput Phys 209(1):47–81

    Article  MATH  Google Scholar 

  28. Saghi H, Lakzian E (2017) Optimization of the rectangular storage tanks for the sloshing phenomena based on the entropy generation minimization. Energy 128:564–574

    Article  Google Scholar 

  29. Saghi H (2018) Entropy generation minimization for the sloshing phenomenon in half-full elliptical storage tanks. Phys A Stat 491:972–983

    Article  MathSciNet  Google Scholar 

  30. Yu ZS, Shao X (2010) A three-dimensional fictitious domain method for the simulation of fluid–structure interactions. J Hydrodyn Ser B 22(5):178–183

    Article  Google Scholar 

  31. Kadapa C, Dettmer WG, Peric D (2016) A fictitious domain/distributed Lagrange multiplier based fluid–structure interaction scheme with hierarchical B-Spline grids. Comput Methods Appl Mech Eng 301:1–27

    Article  MathSciNet  Google Scholar 

  32. Padak A, Raessi M (2016) A 3D, fully Eulerian, VOF-based solver to study the interaction between two fluids and moving rigid bodies using the fictitious domain method. J Comput Phys 311:87–113

    Article  MathSciNet  MATH  Google Scholar 

  33. Roshchenko A, Minev P, Finlay WH (2015) A time splitting fictitious domain algorithm for fluid–structure interaction problems (A fictitious domain algorithm for FSI). J Fluids Struct 58:109–126

    Article  Google Scholar 

  34. Shen YM, Ng CO, Zheng YH (2004) Simulation of wave propagation over a submerged bar using the VOF method with a two-equation turbulence modeling. Ocean Eng 31:87–95

    Article  Google Scholar 

  35. Ketabdari MJ, Saghi H (2011) Large eddy simulation of laminar and turbulent flow on collocated and staggered grids. ISRN Mech Eng 2011:1–13

    Article  Google Scholar 

  36. Ketabdari MJ, Saghi H (2013) Development of volume of fluid methods to model free surface flow using new advection algorithm. J Braz Soc Mech Sci Eng 35(4):479–491

    Article  Google Scholar 

  37. Saghi H, Ketabdari MJ, Booshi S (2013) A novel algorithm based on parameterization method for calculation of curvature of the free surface flows. Appl Math Model 37:570–585

    Article  MathSciNet  MATH  Google Scholar 

  38. Saghi H, Ketabdari MJ (2014) A modification to SLIC and PLIC volume of fluid models using new advection method. Arab J Sci Eng 39:669–684

    Article  MathSciNet  MATH  Google Scholar 

  39. Geuyffier D, Li J, Nadim A, Scardovelli R, Zaleski S (1999) Volume-of-fluid interface tracking with smoothed surface stress methods for three-dimensional flows. J Compt Phys 152(2):423–456

    Article  MATH  Google Scholar 

  40. Sharma N, Chen Y, Patankar NA (2005) A distributed Lagrange multiplier based computational method for the simulation of Particulate-Stokes flow. Comput Methods Appl Mech Eng 194:4716–4730

    Article  MATH  Google Scholar 

  41. Boussinesq MJ (1871) Thèorie de l’intumescence liquide, appelèe onde solitaire ou de translation, se propageant dans un canal rectangulaire. Comptes Rendus de l’Acad´emie des Scinces 72:755–759

    MATH  Google Scholar 

  42. Clamond D, Germain JP (1999) Interaction between a Stokes wave packet and a solitary wave. Eur J Mech B Fluids 18:67–91

    Article  MathSciNet  MATH  Google Scholar 

  43. Temperville A, Contribution A, (1985) àl’ètude des Ondes de Gravitè en Eau Peu Profonde. Thèse d’Etat,Universitè Joseph Fourier

  44. Xing-Kaeding Y (2004) Unified approach to ship seakeeping and maneuvering by a RANSE method. Ph.D. thesis. Teschnichen Universitat Hamburg-Harburg

  45. Yeung RW, Ananthakrishan P (1992) Oscillation of a floating body in a viscous fluid. J Eng Math 26:211–230

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Civil Engineering, Hakim Sabzevari University, Sabzevar, Iran

    Hassan Saghi

Authors
  1. Hassan Saghi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hassan Saghi.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Saghi, H. A parametric study on wave–floating storage tank interaction using coupled VOF-FDM method. J Mar Sci Technol 24, 454–465 (2019). https://doi.org/10.1007/s00773-018-0564-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00773-018-0564-0

Keywords

Search

Navigation