Hydrodynamic Interaction between Rigid Surfaces Planing on Water

Ghazi Bari &

Konstantin Matveev

69th Annual Meeting of the APS Division of Fluid Dynamics, Portland, OR 

November 21, 2016

Monohull• Single hard chine

hull• Stepped hull

2Copyright: Offshore only, Maritime Journal, Boat International Home, & Military Sealift Command

Different Types of hull

Multi Hull (Catamarans)• Advantages

• Speed• Safety/Stability • Fuel efficiency• Spacious decks

Planing Hull Regime

• Hydrodynamic forces dominant

• Froude number, > 1.2

• Applications • Patrol • Recreational transport• Rescue service

• Less wetted area

5

Planing Monohull

Planing Catamaran

Copyright: Allison Ultra Performance Boat and Hypro Marine

BEM in Planing Hull Hydrodynamics

7

• Boundary Element Method (BEM) is used in our research

• BEM is useful on very large domain

• Only boundary needs to be discretized

• Disadvantages • Non-linear flow problem• Require the explicit knowledge of a fundamental solution of the

differential equation

General:• BEM involves placing

singularities (solutions of the fundamental equation) with unknown magnitudes on the domain boundaries

• Boundary conditions are satisfied in collocation points to determine those magnitudes

BEM in Potential Flow models• Low-cost computation• Simplicity of implementation• Sufficient accuracy

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• Use special staggered arrangement for sources and collocation points to suppress effects of the finite domain (i.e., wave reflection from the domain downstream end)

Source pointsCollocation points

BEM in Planing Hull Hydrodynamics

Specific to this study:• Use point (discrete) hydrodynamic

sources along the domain boundary • Use linearized theory (assume small

boundary deformations)

Numerical Model Development

• High velocity flow• Inertia term dominant• Steady flow• Irrotational

Governing equations

9

0 v gPvvtv

• Viscous force negligible for hydrodynamic lift and drag

• Working fluid: Water• Incompressible

• Potential function

and

Continuity eq. Momentum eq.

⇒𝛻2𝜑=0

Laplace Eq.

⇒ 𝑃𝜌 +

12 𝑣

2+𝑔𝑧=𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡

Bernoulli’s Eq.

Numerical Model Development (Cont’d)Linearized Mathematical Model (2D)

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sources (□) and collocation points (○)

Linearized Bernoulli’s eq., Surface slope eq.,A

sc dA

rxQxu )(

21)('

Perturbation velocity,

Domain Formation

Incident water flow0U

X

Z

Top view of wetted plate (3D model)

x x x

z

sources (□) and collocation points (○)

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j ji

sj

ci

ji

jci

ci r

xxrq

zxu,

2,4

1),(

si

si

si

si

ii

i

ii

i

xxyy

Uzx

qzx

q

1

10

11

1 221

0),(2),(),(21

0

ci

ciw

ci

cic

icip

zxyUzxuzxC

Wetted Length

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• Water raise is accounted after each iteration• Calculations repeated until wetted length stops changing• CP from the Linearized Bernoulli’s Eq.• Coefficient of Lift and Drag, Center of pressure related

with Lw

A simple picture shows how to find final wetted length

Initial guess of wetted length,

Free surface water

nL

Intersection of hull and water surface

(initial guess)

Final wetted length, wL

wL after 1st iteration

after 2nd iterationwL

Point after stopped changingwL

Hull

Validation of Monohull Setup

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• Flow behind a flat bottom hull at finite Froude number• Good agreement except near transom• Schmidt’s formula only valid at far stream

1

82cos2

2 2

xFr

yh

Transom0U

Schmidt, G., 1981, "Linearized stern flow of a two-dimensional shallow draft ship," J Ship Res(25), pp. 236-242.

Schematic of (a) 2D problem and (b) 3D problem

1. Squire, H. B., 1957, "The Motion of a Simple Wedge along the Water Surface," Proceedings of the Royal Society, 342, pp. 48-64.

Lift coefficient (

Center of pressure ()

Comparison of Numerical Solution with Squire’s Data

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Validation of Monohull Setup (Cont’d)

1. Wang, D. P., and Rispin, P., 1971, "Three-dimensional planing at high Froude number," Journal of Ship Research, 15(3), pp. 221-230.

Pressure coefficient along the center of hull

Pressure coefficient in a longitudinal section at 90% of the plate

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Validation of Monohull Setup (Cont’d) Comparison of Numerical Solution with Wang & Rispin’s Analytical Sol

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Validation of Symmetric Catamaran

1. Bari, G.S., Matveev, K.I., 2016. Hydrodynamic modeling of planing catamarans with symmetric hulls. Ocean Engineering 115, 60-66.2. Liu, C. Y., and Wang, C. T., 1978, "Interference effects of catamaran planing hulls," Journal of Hydronautics, 13(1), pp. 31-32.3. Savitsky, D., and Dingee, D., 1954, "Some interference effects between two flat surfaces planing parallel to each other at high speed," Davidson Laboratory, Technical Note No. 247

• Modified Savitsky’s correlation from Liu and Wang (1978)

• For Fr = 2 – 3.5 • AR and Trim angle information not presented

1.12

2/52/1 ]

20055.02012.0[ AFrA

CL

Correction to the wetted length

Interference factor

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Validation of Asymmetric Catamaran

1. Bari, G.S., Matveev, K.I., 2016. “Hydrodynamics of Single-Deadrise Hulls and Their Catamaran Configurations.” International Journal of Naval Architecture and Ocean Engineering

2. Morabito, M.G., 2011. “Experimental investigation of the lift and interference of asymmetric planing catamaran demi-hulls.” 3. Epstein, L.A., 1969. “Determination of the depression behind a finite-span underwater wing and gliding flat plate.”4. Payne, P.R., 1984. “On the shape of the wake behind a planing boat.”

• Froude number (): 2.73 ~ 2.75 and 3.95 ~ 4.01

• Trim angle: • Deadrise angle ():

• Wake behind a flat plate• Epstain (1969) & Payne (1984)• Trim angle: • Wetted Length (nominal) : 3

Conclusions

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• Developed: 2D and 3D model based on point-source BEM• Good agreement with empirical data is found• Robust design tool for transitional (between hydrostatic

and hydrodynamic support) and early planing regimes of fast boats

• Fast, numerically inexpensive, and sufficiently accurate model

• Applicable to different types of hull setup

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Thank you!

Questions?