Numerical Prediction of Steady Flow Around High Speed Vessels

with Transom Sterns

S.X. Du1,2, D.A. Hudson2, W.G. Price2, P. Temarel2 and Y.S. Wu1

1China Ship Scientific Research Center, Wuxi, PR China.

2School of Engineering Sciences, Ship Science, University of Southampton, Southampton, UK.

OverviewOverview

• Introduction• Mathematical Model• Numerical Model of Transom Stern

– Mesh Generation– Finite Element Analysis– Variation of Appendage Shape– Pressure Distribution and Wave Resistance

• Conclusions• Future Work

MotivationMotivation

• Accurate prediction of wave-making resistance

• Details of pressure and velocity distribution near stern

• Complex flow phenomena

• Require efficient methodCompromise between theoretical rigour and

practical computational method

Modelling PhilosophyModelling Philosophy

• Transom ‘runs dry’ at high speed• Extend idea of ‘virtual appendage’

• Use a flexible appendage– Structural deformation with fluid pressure– Iterate towards zero pressure on

appendage– Shape represents steady-state flow

• Three-dimensional Kelvin source for wave resistance of body+appendage

Mathematical Model (1)Mathematical Model (1)

• Assume potential flow– Inviscid, homogenous, irrotational motion flow

• Outside stern region – satisfy linear free-surface condition

0on 0 02

22

zzx

Fn

0 nWBody boundary condition

Kelvin wave source potential on surface of hull and appendage

Mathematical Model (2)Mathematical Model (2)

• In stern region, free-surface condition is non-linear, giving

0 where021

),(21 2 apFnyxaWW

),(on 0 0 yxaz nW

)(

),(

xFn

gLUFn

yxa

W

giving ,

stern transomdry the behind surface free the is

Mathematical Model (3)Mathematical Model (3)

• Pressure given as,

pFnz 2

21

21

WW

• With,

SU

RC ww 2

21

• Giving wave resistance as,

.21

21 23 dsnFnzgLR x

Sw

b

WW

Modelling RequirementsModelling Requirements

• Flexible appendage must satisfy1. Continuous transition from transom stern

to hollow cavity

2. A local non-linear free-surface condition with atmospheric pressure in cavity

3. A linear free-surface condition outside the hollow cavity region

• Appendage of Molland et. al. satisfies 1,3

Modelling Flexible AppendageModelling Flexible Appendage

Assume initial formof appendage

Calculate velocity and pressure distribution

Use pressure to deformappendage shape

Re-mesh appendage Until free-surfacecondition satisfied

in cavity

Application to Ship HullApplication to Ship Hull

• NPL mono-hull chosen as demonstration

L(m) L/B B/T CB CP S(m2)

1.6 9.0 2.0 7.42 0.397 0.693 0.338

31L

‘Flat-tailed’ appendage

‘Canoe-shaped’ appendage

Finite Element AnalysisFinite Element Analysis

• Three-dimensional beam framework

• Nodes coincide with hydrodynamic panel vertices

• Careful choice of Young’s modulus needed

• Maximum displacement limited

• Boundary condition at transom important

Shape Variation of Appendage (1)Shape Variation of Appendage (1)

Step 7

Step 7

Step 15 Step 23

Step 16 Step 24

Flat-tailed appendage, Fn=0.5

Canoe shape appendage, Fn=0.5

Variation With Forward SpeedVariation With Forward Speed

Fn=0.5 Fn=0.6

Fn=0.7 Fn=0.9

Pressure Distribution (1)Pressure Distribution (1)

Step 7

Step 15 Step 23

Step 1

Pressure distribution adjacent to transom stern

Pressure Distribution (2)Pressure Distribution (2)

Pressure distribution at transverse hull sections

x’=0.006 x’=0.02 x’=0.04

Initial step

Final step

Wave ResistanceWave Resistance

SummarySummary

• Method developed to predict flow around transom sterns ‘running dry’

• Source distribution over hull and appendage

• Combined with finite element analysis for deformation of appendage

• Application to NPL hull form

Conclusions Conclusions

• Beam FE model better than shell model

• Good agreement for wave resistance

• Improved evaluation of velocity and pressure around hull

• Important when accounting for influence of steady flow in unsteady hydrodynamic problem

Future WorkFuture Work

• Extend method for sinkage and trim calculation

• Include more robust finite element model• Application to fast catamaran hulls• Validation with experimental data for

– Free-surface elevation– Pressure distribution around hull

• Combine with seakeeping analysis