Introduction Method Formulation Implementation Results Conclusion References

A Hybrid Boundary Element / RANS Approachto Steady Flows in Naval Hydrodynamics

Bill Rosemurgy & Dr. Kevin Maki

Dept. of Naval Architecture and Marine EngineeringUniversity of Michigan, U.S.A.

13 June 2011

Introduction Method Formulation Implementation Results Conclusion References

Outline

Motivation and Background

Velocity DecompositionInviscid SolutionViscous CouplingFree-Surface Boundary ConditionImplementation in OpenFOAM

ResultsWigley HullDTMB #5415

Introduction Method Formulation Implementation Results Conclusion References

Resistance Prediction

The accurate prediction of steady forward speed,calm-water resistance is of great importance to a designer

Early design - prime mover and lightship weight

Goal - Avoid solving fully non-linear unsteady RANSinterface capturing problem

Introduction Method Formulation Implementation Results Conclusion References

Resistance Prediction

ExperimentsCostly, scaling issuesMethodology is well tested

Potential Flow MethodsLow computational cost (boundary vs. volume)Neglect viscous effects

CFDHigh computational cost - grid generation & computationMost accurately models the physicsCaptures non-linear effects - wave-breaking

Empirical Data / RegressionRely on geometric similarity to parent dataPowerful for early design

Introduction Method Formulation Implementation Results Conclusion References

Velocity Decomposition

Formal velocity decomposition

Decompose velocity into rotational and irrotationalcomponents

u ≡ ∇Φ + w (1)

Substitute into RANS equations and simplify to get"complementary" RANS equations

DwDt

+∇p − ν∇2w = −∇ · ∇Φw−∇ ·w∇Φ (2)

Evaluate the irrotational component using a BoundaryElement Method

Introduction Method Formulation Implementation Results Conclusion References

Velocity Decomposition

Solve "complementary" equations for w on full domain OR

Solve decoupled equations for u and use ∇Φ as boundaryconditions on a reduced domain

Introduction Method Formulation Implementation Results Conclusion References

Potential Velocity

The potential (irrotational) velocity is computed using aFree Surface Green Function (Noblesse et al. (2011))

Formulated for a triangular discretization of the hull usingflat, constant strength source panels

Slender body theory to determine panel source strengths

Linearized free-surface and body boundary condition

Far-field radiation condition

Introduction Method Formulation Implementation Results Conclusion References

RANS Solver & Boundary Conditions

Solve the traditional RANS equations with additionalfree-surface boundary condition equation

Discretize below the mean free surface - z = 0

Use the FSGF to calculate the velocity and velocitygradient on domain boundaries

Determine pressure (and pressure gradient) fromBernoulli’s Equation

Introduction Method Formulation Implementation Results Conclusion References

Free-Surface Boundary Condition

Use finite-area method (1.6-ext) to satisfy the linearizedfree-surface kinematic boundary condition at each iterationη represents the free-surface elevation

DDt

(z − η) = 0 (3)

Which linearizes to:

w = Vsηx (4)

We then calculate the pressure on the free-surface from ηand use that as the fixedValue boundary condition

Introduction Method Formulation Implementation Results Conclusion References

FSGF Implementation in OpenFOAM R©

1 Specify hull boundary patch (run-time selectable)

2 Cut hull patch at z = 0 (if needed)

3 Triangularize faces on hull patch

4 Evaluate velocity on specified boundary patches (run-timeselectable)

Introduction Method Formulation Implementation Results Conclusion References

RANS in OpenFOAM R©

Free-surface Green Function velocity and pressure areapplied as boundary conditions on farfield domain extents

Solve free-surface kinematic boundary condition forpressure on free-surface

Generally use fixedValue velocity and fixedGradientpressure boundary conditions

SIMPLE Algorithm to solve RANS equations

k − ω SST turbulence model with wall functions

Convergence is determined by monitoring residuals as wellas the behavior of the caclulated pressure and viscousforce on the hull

Introduction Method Formulation Implementation Results Conclusion References

Test Cases

Wigley Hull DTMB #5415

Compare against SRI 4m experiment

Domain Sizes (in ship lengths)

Grid Inlet Outlet Lateral BottomSmall 0.375 1.25 0.375 0.5

Medium 0.75 1.25 0.375 1.0Large 1.5 2.5 0.75 2.0

Cell Count

Grid Coarse Medium FineSmall 62,400 131,670 441,294

Medium 66,584 165,984 552,102Large 100,254 271,656 887,096

Compare against INSEAN 5.72m experiment

Domain Sizes (in ship lengths)

Grid Inlet Outlet Lateral BottomSmall 0.75 1.15 0.4 0.4

Medium 0.9 1.75 0.7 0.75Large 1.25 3.0 1.4 1.5

Cell Count

Grid Coarse FineSmall 108,210 569,305

Medium 141,025 755,965Large 185,769 959,869

Introduction Method Formulation Implementation Results Conclusion References

Wigley Results - Total Resistance

Introduction Method Formulation Implementation Results Conclusion References

DTMB #5415 Results - Total Resistance

0

0.002

0.004

0.006

0.008

0.01

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

CT

Froude number

INSEAN - exp.DTMB - exp.interFoamvelocity decomposition - coarse small

Introduction Method Formulation Implementation Results Conclusion References

DTMB #5415 Results - Frictional Resistance

Introduction Method Formulation Implementation Results Conclusion References

DTMB #5415 Results - Free-surface Elevation

Gothenburg 2010, Case 3.1b, Re = 5.13E6,Fn = 0.28

Introduction Method Formulation Implementation Results Conclusion References

DTMB #5415 Results - TKE

Gothenburg 2010, Case 3.1b, Re = 5.13E6,Fn = 0.28

Introduction Method Formulation Implementation Results Conclusion References

DTMB #5415 Results - Streamwise Velocity

Gothenburg 2010, Case 3.1b, Re = 5.13E6,Fn = 0.28

Introduction Method Formulation Implementation Results Conclusion References

DTMB #5415 Results - Timing Comparison[c]

Large Mesh - 185,769 cells Small Mesh - 108,210 cells

Introduction Method Formulation Implementation Results Conclusion References

Conclusion

Accomplishments

Able to reduce domain size due to improved boundaryconditionsDrastically decrease computational time

Decrease domain sizeDo not resolve free-surface

Future WorkUse snappyHexMesh & scripting to automatically createresistance curves from .iges filesCalculate sinkage & trimImprove accuracy of ∇Φ term

Introduction Method Formulation Implementation Results Conclusion References

References

Cooperative experiments on Wigley parabolic models in japan.Technical report, Ishikawajima-Harima Heavy Industries Co.,Ltd., Ship Research Institute, University of Tokyo, YokohamaNational University, December 1983.

Kunho Kim, Ana Sirviente, and Robert F. Beck. Thecomplementary rans equations for the simulation of viscousflows. International Journal for Numerical Methods in Fluids,48:199–229, January 2005.

Francis Noblesse, Gerard Delhommeau, Fuxin Huang, and ChiYang. Practical mathematical representation of the flow dueto a distribution of sources on a steadily-advancing ship hull.submitted to the Journal of Engineering Mathematics, 2011.

A Olivieri, F Pistani, A Avanzini, F Stern, and R Penna. Towingtank experiments of resistance, sinkage, and trim, boundarylayer, wake, and free surface flow around a naval combatantinsean 2340 model. Technical Report 421, IIHR, 2001.