SHIPS, WAVES AND MATH Mathematics & Water, Deltares, 13 November - PowerPoint PPT Presentation

SHIPS, WAVES AND MATH Mathematics & Water, Deltares, 13 November 2014 MARIN, Ed van Daalen

CONTENTS • MARIN • Application of math to ship hydromechanics • Conclusions

MARITIME RESEARCH INSTITUTE NETHERLANDS Wageningen Ede Houston • hydrodynamic research for maritime industry, nonprofit • founded 1929, 7 model basins, 350 employees, 42 M € turnover • model tests, trials & full scale monitoring, simulations • international market: design companies, shipyards, classification, ship operators

MARIN ORGANISATION • Ships: powering & resistance, seakeeping, manoeuvring for all ship types • Offshore: on/offloading, drilling platforms, windmill installation • Nautical Simulator: harbour design, training • Trials and Monitoring: full scale measurements • Software: simulation • Production: model factory, instrumentation • Research and Development: fundamental developments in experiments and simulations

LEARN MORE ABOUT MARIN • www.marin.nl (nice company video!) • www.youtube.com/marinmultimedia

SCHEEPVAART Life can be beautiful …

… BUT SOMETIMES LIFE IS HORRIBLE … • Herald of Free Enterprise • Estonia • Costa Concordia • … How can we help to avoid this?

Very large ships are challenging • Hydrodynamics • Structural • Logistics How can we help ?

LNG carriers • Sloshing in liquid cargo tanks How can we help ?

heavy cargo • structural • (off)loading How can we help ?

bad weather • high waves • high loads How can we help ?

INTRODUCTION TO SHIP DESIGN (2) bad weather • comfort • Designed for specific seas or routes • operability • safety How can we help ?

THRUST ALLOCATION

THRUST ALLOCATION - OBJECTIVES Dynamic Positioning (DP) System Thrust Allocation Algorithm • match required forces • minimize power • account for hydrodynamic interaction effects • respect physical limitations • maximum rpm change • maximum azimuth change

THRUST ALLOCATION – INTERACTION EFFECTS thruster-hull interaction thruster-current interaction thruster-thruster interaction

THRUST ALLOCATION - EFFICIENCY FUNCTIONS η <1 η <1 thruster efficiency • rpm, azimuth η =1 • other thrusters • current η =1 forbidden zones η =1 η <1 η =1 η =1 η =1 η <1 η <1

THRUST ALLOCATION - OPTIMIZATION PROBLEM power minimization generate required forces account for hydrodynamic interactions physical limitations

THRUST ALLOCATION - ALGORITHM

THRUST ALLOCATION - CROSSING FORBIDDEN ZONES 3 5 1 6 2 4

THRUST ALLOCATION – MATCH REQUIREMENTS

THRUST ALLOCATION – RESPECT PHYSICAL LIMITS

MANOEUVERING EQUATIONS research started at SWI 2011

MANOEUVERING EQUATIONS – MATHEMATICAL MODEL • maneuvering model: set of coupled ordinary differential equations (ODEs) describing ship motions in calm water , including nonlinear hull forces and nonlinear propulsion forces • many hull parameters ( ~ 30 ) and propulsion parameters ( ~ 20 ) involved • many of these parameters are determined by experiments (scale models) and CFD 23

MANOEUVERING EQUATIONS – MATHEMATICAL MODEL • Propeller-Rudder Model: used in MARIN maneuvering simulation program S UR S IM           L m ' m ' u m ' r ' v X ' X ' X ' pp uu H R P             L m ' m ' v m ' r ' m ' r ' u Y ' Y ' pp vv vr H R            L m ' v I ' m ' r ' N ' N ' pp rv zz rr H R • (simplified) Thruster Model:            L m ' m ' u m ' r ' v X ' cos pp uu H               L m ' m ' v m ' r ' m ' r ' u Y ' sin pp vv vr H              L m ' v I ' m ' r ' N ' x ' sin pp rv zz rr H T   F u  u ( , ) 24

MANOEUVERING EQUATIONS - SOLUTIONS  F u  u ( , ) obvious thing to do = direct simulation → time integration with initial conditions • constant propulsion parameters: e.g. straight line, turning circle u  0 • NOTE for these motions • time-dependent propulsion parameters: e.g. zig-zag manoeuver u  • 0 NOTE for these periodic motions 25

MANOEUVERING EQS - NUMERICAL CONTINUATION Alternative: Numerical Continuation Method NCM = a robust and fast method to • find parameter-dependent set of ‘ equilibria ’ of ODEs (equilibrium = steady / stationary state solution) • determine stability properties of equilibria • find bifurcations and e.g. trace periodic solutions (bifurcation = transition from stable to unstable) 26

MANOEUVERING EQS - NUMERICAL CONTINUATION NCM is based on Implicit Function Theorem, stating that « relations can be transformed into functions » u : n-vector (state variables, n=3, 4)      u F u ( , ) 0 F u ( , ) λ : continuation parameters (select 1) 27

MANOEUVERING EQUATIONS - NCM WITH AUTO pseudo arc-length natural parameter continuation (AUTO) continuation Newton iteration       u   u s Newton s iteration         s 28

MANOEUVERING EQUATIONS - TURNING CIRCLE TDS initial condition u=20kn TDS initial condition u=40kn influence of the thruster angle 29

MANOEUVERING EQUATIONS - TURNING CIRCLE time domain simulation  D 3 pp L α =15deg 30 circular motion with noise (stable)

MANOEUVERING EQUATIONS - STRAIGHT LINE straight line motion with noise (unstable) 31

MANOEUVERING EQUATIONS - YAW CONTROL max add yaw as state variable Hopf bifurcation    u v r  u , , ', max add extra ODE:   r max add yaw restoring control:            0 0   0 0 max ( α is a parameter, not a state variable!) 32

MANOEUVERING EQUATIONS - YAW CONTROL direction of increasing course stability ship velocities for several periodic solutions 33

MANOEUVERING EQUATIONS - YAW CONTROL STABLE UNSTABLE with AUTO it is easy to trace out the stability boundary … 34

MANOEUVERING EQUATIONS - ZIGZAG consider half zigzag only and use anti-symmetric boundary conditions Rudder angle Period Time to reach second execute Reach Heading check angle Heading angle Initial course Overshoot angle Overshoot time r max Rate of change of heading Start of test End of test Time 35

PARAMETRIC ROLLING

PARAMETRIC ROLLING What happens if a container ship experiences large roll angles ?

PARAMETRIC ROLLING - SIMPLE ODE MODEL simulation over large time intervals                   ( I A ) B ( ) C ( t ) 0 C ( t ) gV GM GM ( t ) waves wave force n         GM ( t ) A 2 d ( a , b ) cos( t ) j j j j  j 0 transfer coefficients for amplitude change and phase shift of waves acting upon metacentric height: A j = A ( ω j ) β j = β ( ω j ) 38 38

PARAMETRIC ROLLING - EXIT TIME STRATEGY exit time strategy: • very long time domain simulation                 ( I A ) B ( ) C ( t ) 0 C ( t ) C gV GM ( t ) • observe energy 1  C  2 • define critical amplitude and critical energy E crit crit 2

PARAMETRIC ROLLING - EXIT TIME STRATEGY PROBABILITY OF ACHING E CRIT WITHIN TIME T 𝑈 Fraction of runs arriving within time T at E crit : 𝑟 𝑈 = 𝑛𝑗𝑜 𝑈 𝑗𝑜𝑢 , 1 𝑈 Weighted average over all safe zones: 𝑟

PARAMETRIC ROLLING - PROBABILITY stationary solution with resonant forcing and amplitude φ max 41

SHORT CRESTED WAVES long crested waves are ‘easy’: • to analyse • to simulate however: real waves are short crested

SHORT CRESTED WAVES find wave spreading functions that match theoretical and measured wave height distributions

SHORT CRESTED WAVES find wave spreading functions that match theoretical and measured wave height distributions

SHORT CRESTED WAVES wave calibration using Maximum Likelihood Method → find wave spreading functions that match measured cross spectra with theoretical wave height transfer function

NUMERICAL DAMPING AND DISPERSION

NUMERICAL DAMPING AND DISPERSION Consider plane surface waves on 2D uniform flow. Linearise in these waves. They are represented by Fourier components:     u u          kx sz q w w e dkds               Substitute this in the linearised and discretised RANS equations:  ˆ       ˆ kx sz L q L k s ( , ) qe dkds 0 h h = RANS-operator, ˆ L L = discrete Fourier symbol of this operator h h ˆ Non-trivial solution only exists if determinant of vanishes. L h