1. Numerical Prediction of Steady Flow Around High Speed Vessels with Transom Sterns S.X. Du 1,2, D.A. Hudson 2, W.G. Price 2, P. Temarel 2 and Y.S. Wu 1 1 China Ship Scientific Research Center, Wuxi, PR China. 2 School of Engineering Sciences, Ship Science, University of Southampton, Southampton, UK.

2. Overview 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

3. Motivation Accurate prediction of wave-making resistance Details of pressure and velocity distribution near stern Complex flow phenomena Require efficient method Compromise between theoretical rigour and practical computational method

4. Modelling 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

5. Mathematical Model (1) Assume potential flow –Inviscid, homogenous, irrotational motion flow Outside stern region – satisfy linear free- surface condition Body boundary condition Kelvin wave source potential on surface of hull and appendage

6. Mathematical Model (2) In stern region, free-surface condition is non-linear, giving

7. Mathematical Model (3) Pressure given as, With, Giving wave resistance as,

8. Modelling Requirements Flexible appendage must satisfy 1.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

9. Modelling Flexible Appendage Assume initial form of appendage Calculate velocity and pressure distribution Use pressure to deform appendage shape Re-mesh appendage Until free-surface condition satisfied in cavity

10. Application to Ship Hull NPL mono-hull chosen as demonstration L(m)L/BB/TCBCB CPCP S(m 2 ) 1.69.02.07.420.3970.6930.338 ‘Flat-tailed’ appendage ‘ Canoe-shaped’ appendage

11. Finite 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

12. Shape Variation of Appendage (1) Step 7 Step 15 Step 23 Step 16 Step 24 Flat-tailed appendage, Fn=0.5 Canoe shape appendage, Fn=0.5

13. Variation With Forward Speed Fn=0.5 Fn=0.6 Fn=0.7 Fn=0.9

14. Pressure Distribution (1) Step 7 Step 15 Step 23 Step 1 Pressure distribution adjacent to transom stern

15. Pressure Distribution (2) Pressure distribution at transverse hull sections x’=0.006 x’=0.02 x’=0.04 Initial step Final step

16. Wave Resistance

17. Summary 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

18. 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

19. Future 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