1. 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

2. Monohull
• Single hard chine
hull
• Stepped hull
2
Copyright: 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

3. Planing Hull Regime
• Hydrodynamic forces
dominant
• Froude number, 𝐹𝑟𝐿 > 1.2
• 𝐹𝑟𝐿 =
𝑈
𝑔𝐿
• Applications
• Patrol
• Recreational transport
• Rescue service
• Less wetted area
5
Planing Monohull
Planing Catamaran
Copyright: Allison Ultra Performance Boat and Hypro Marine

4. 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

5. 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
8
• 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)

6. Numerical Model Development
• High velocity flow
• Inertia term dominant
• Steady flow
• Irrotational
Governing equations
9
0 v gPvv
t
v
 








• Viscous force negligible for
hydrodynamic lift and drag
• Working fluid: Water
• Incompressible
• Potential function
and
Continuity eq. Momentum eq.
⇒ 𝛻2 𝜑 = 0
Laplace Eq.
⇒
𝑃
𝜌
+
1
2
𝑣2 + 𝑔𝑧 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
Bernoulli’s Eq.

7. Numerical Model Development (Cont’d)
Linearized Mathematical Model (2D)
10
sources (□) and collocation points (○)
Linearized Bernoulli’s eq.,
𝐶 𝑝
2
+
𝑢′
𝑈0
+
2𝜋𝑦 𝑤
𝜆
= 0
Surface slope eq.,
𝑑𝑦 𝑤
𝑑𝑥
=
𝑣
𝑈0 + 𝑢′
=
𝑄
2𝑈0

A
s
c
dA
r
xQ
xu
)(
2
1
)('

Perturbation velocity,

8. Domain Formation
Incident water flow
0U
X
Z
Top view of wetted plate (3D model)
x x x
z
sources (□) and collocation points (○)
13



j ji
s
j
c
i
ji
jc
i
c
i
r
xx
r
q
zxu
,
2
,4
1
),(

s
i
s
i
s
i
s
i
ii
i
ii
i
xx
yy
U
zx
q
zx
q
1
1
0
11
1
2
2
1















0
),(2),(
),(
2
1
0




 c
i
c
iw
c
i
c
ic
i
c
ip
zxy
U
zxu
zxC

9. Wetted Length
16
• 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
iteration
wL
Point after stopped changingwL
Hull


10. Validation of Monohull Setup
19
• Flow behind a flat bottom hull at finite Froude number
• Good agreement except near transom
• Schmidt’s formula only valid at far stream












 1
8
2cos2
2 2




 x
Fr
y
h
Transom
0U
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

11. 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 (𝐶𝐿𝑤)
Center of pressure (𝐿 𝑃)
Comparison of Numerical Solution with Squire’s Data
20
Validation of Monohull Setup (Cont’d)

12. 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
21
Validation of Monohull Setup (Cont’d)
Comparison of Numerical Solution with Wang & Rispin’s Analytical Sol

13. 24
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
• Modified Savitsky’s correlation from Liu and Wang
(1978)
• For Fr = 2 – 3.5
• AR and Trim angle information not presented
1.1
2
2/5
2/1
]
2
0055.02
012.0[ 


A
FrA
CL 
Correction to the wetted length Interference factor

14. 25
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.”
• Froude number (𝐹𝑟𝑏): 2.73 ~ 2.75 and
3.95 ~ 4.01
• Trim angle: 5.89°~6.14°
• Deadrise angle (𝛽): 18°
• Wake behind a flat plate
• Epstain (1969) & Payne (1984)
• Trim angle: 6°
• Wetted Length (nominal) : 3

15. Conclusions
29
• 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

16. 30
Thank you!
Questions?