Nonlinear Potential Flow Solver Development in...

Nonlinear Potential Flow Solver Development in OpenFOAM

Nonlinear Potential Flow Solver Development inOpenFOAM

A. Mehmood

Plymouth University, UK

April 19,2016

A. Mehmood Nonlinear Potential Flow Solver Development in OpenFOAM

Nonlinear Potential Flow Solver Development in OpenFOAM

Table of Contents

1 Motivation

2 Solution MethodologyMathematical FormulationSequence of the Solution Procedure

3 Results and DiscussionStanding WavesProgressive Waves

4 Conclusions and Future Directions


Nonlinear Potential Flow Solver Development in OpenFOAMMotivation

Wave Structure Interaction Simulation Environment


Nonlinear Potential Flow Solver Development in OpenFOAMSolution Methodology

Wave Tank

y axis

x axis0 1 2 3 4 5 6 7 8

0

−1

−2

−3

Free surface



Mathematical Formulation

Mathematical Formulation

∇2φ = 0

∂φ∂t = −gη − 1

2∇φ.∇φ

∂η∂t = ∂φ

∂y −∂φ∂x

∂η∂x

∂η∂t = u.n

ny

∂φ

∂x = F (y , t) ∂φ

∂t + c ∂φ∂n = 0

∂φ

∂y = 0

y axis

x axis0 1 2 3 4 5 6 7 8

0

−1

−2

−3

1 Mayer, S, Garapon, A and Sorensen, LS (1998). “A fractional step method forunsteady free surface flow with applications to non-linear wave dynamics,” Intl JNumerical Methods in Fluids, 28(2), 293–315

1A. Mehmood Nonlinear Potential Flow Solver Development in OpenFOAM


Sequence of the Solution Procedure

Generate the grid

Generate the grid




Apply the boundary conditions

Apply the boundary conditions at the face center




Solve Laplace’s equation

Solve Laplace’s equation for the velocity potential.Compute the required variables (i.e., velocities u = ∇φ,fluxes).




Complications of implementation of the Solver inOpenFOAM

Simplest idea for automatic mesh motion in the FVframework would be to solve an equation to provide pointmotionHowever, as the FVM provides the solution in cell centresand motion is required on the points(vertices), this necessarilyleads to interpolation

∂η∂t = u.n

ny




Automatic Mesh Motion in OpenFOAM

Motion will be obtained by solving a mesh motion equation,where free surface motion acts as a boundary condition

Automatic mesh motion determines the position of internalpoints based on the free surface motionThe role of internal point motion is to accommodatechanges in the domain shape (boundary motion) and preservethe validity and quality of the meshInternal point motion can be specified in a number of ways,without user interactionChoices for a simplified mesh motion equation:

Laplace equation with constant and variable diffusivitydiffusivity − > quadratic inverseDistance





Motion will be obtained by solving a mesh motion equation,where free surface motion acts as a boundary conditionAutomatic mesh motion determines the position of internalpoints based on the free surface motion

The role of internal point motion is to accommodatechanges in the domain shape (boundary motion) and preservethe validity and quality of the meshInternal point motion can be specified in a number of ways,without user interactionChoices for a simplified mesh motion equation:






Motion will be obtained by solving a mesh motion equation,where free surface motion acts as a boundary conditionAutomatic mesh motion determines the position of internalpoints based on the free surface motionThe role of internal point motion is to accommodatechanges in the domain shape (boundary motion) and preservethe validity and quality of the mesh

Internal point motion can be specified in a number of ways,without user interactionChoices for a simplified mesh motion equation:






Motion will be obtained by solving a mesh motion equation,where free surface motion acts as a boundary conditionAutomatic mesh motion determines the position of internalpoints based on the free surface motionThe role of internal point motion is to accommodatechanges in the domain shape (boundary motion) and preservethe validity and quality of the meshInternal point motion can be specified in a number of ways,without user interaction

Choices for a simplified mesh motion equation:Laplace equation with constant and variable diffusivity

diffusivity − > quadratic inverseDistance





Motion will be obtained by solving a mesh motion equation,where free surface motion acts as a boundary conditionAutomatic mesh motion determines the position of internalpoints based on the free surface motionThe role of internal point motion is to accommodatechanges in the domain shape (boundary motion) and preservethe validity and quality of the meshInternal point motion can be specified in a number of ways,without user interactionChoices for a simplified mesh motion equation:



Nonlinear Potential Flow Solver Development in OpenFOAMResults and Discussion

Standing Waves

Standing Waves set up

η(x , t) = a cos(kx) cos(ωt)+πa2

λ

[cos2(ωt)− 1

4 cosh2(kH)+ 3 cos(2ωt)

4 sinh2(kH)

]cos(2kx) (1)

∇2φ = 0

∂φ∂t = −gη − 1

2∇φ.∇φ∂η∂t = u.n

ny

∂φ

∂x = 0 ∂φ

∂x = 0

∂φ

∂y = 0

y axisx axis



Standing Waves

Time Histories of Wave Elevation

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0.02

Time

Surf

ace e

levation

Analtic (linear)−0.005 m

Numerical−0.005 m

Analtic (linear)−0.01 m

Numerical−0.01 m

-1

-0.5

0

0.5

1

1.5

0 1 2 3 4 5

η/a

t/T

Time trace plots

numericaltheoretical 1st-order

theoretical 2nd-order

Figure: Time history of free surface elevation at the centre of the domain



Standing Waves

Variation of wave period

0 0.2 0.4 0.6 0.8

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

H

Tim

e [sec]

Airy 2nd−order

ampl−005

ampl−01

Figure: Variation of wave period against mean water depth normalized bywavelength



Standing Waves

Standing Wave Animation

H

Initial wave profilewave amplitude

H

Initial wave profilewave amplitude



Progressive Waves

Progressive Waves set up

∇2φ = 0

∂φ∂t = −gη − 1

2∇φ.∇φ

∂η∂t = u.n

ny

∂φ

∂x = ux (y , t) ∂φ

∂t + c ∂φ∂n = 0

∂φ

∂y = 0

y axis

x axis0 1 2 3 4 5 6 7 8

0

−1

−2

−3



Progressive Waves

Progressive Waves

-1.5

-1

-0.5

0

0.5

1

1.5

0 2 4 6 8 10 12 14

η/a

t/T

x=6.0 m

(a) a = 0.01 m, H = 1.5 m, T=1.5 s

-1.5

-1

-0.5

0

0.5

1

1.5

0 2 4 6 8 10 12 14

η/a

t/T

x=6.0 m

(b) a = 0.06 m, H = 1.0 m, T=1.5 s.Figure: Time history of free surface elevation at location x = 6.0 m (frominlet boundary)



Progressive Waves

Comparison with Experiment (F.Gao-2003)

8.85m

0.28m∂φ

∂x = ux (t) zeroGrad

1 Gao, F, (2003). “An efficient finite element technique for free surface flow,”Ph.D.thesis, Brighton University, UK.

1



Progressive Waves


0.55m8.85m

0.28m

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0 2 4 6 8 10 12

Surf

ace e

levation (

m)

Time (s)

x = 0.55 m

numerical simulationsexperiment by Gao-2003

Figure: Time history of wave elevation at location x = 0.55 m.A. Mehmood Nonlinear Potential Flow Solver Development in OpenFOAM


Progressive Waves


3.55m0.28m

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

0 2 4 6 8 10 12

Surf

ace e

levation (

m)

Time (s)

x = 3.55 m

numerical simulations-Gradedexperiment by Gao-2003



Progressive Waves


5.45m8.85m

0.28m

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

0 2 4 6 8 10 12

Surf

ace e

levation (

m)

Time (s)

x = 5.45 m

numerical simulations-Gradedexperiment by Gao-2003



Progressive Waves

Comparison with experiment


Nonlinear Potential Flow Solver Development in OpenFOAMConclusions and Future Directions

Conclusions and Future Directions

Developed a free surface tracking solver for numericalsimulation of unsteady irrotational fully non-linear water waves

The solver has been validated by application to a number oftest cases, ranging from shallow water standing waves todifferent wave amplitudes progressive wavesSolution of Laplace’s equation for the velocity potential, thenon-linear free surface boundary conditions, the wavegeneration and the absorption boundary conditions are all notpart of the standard OpenFOAM R© distributionCoupling to available Navier-Stokes solvers inOpenFOAM R©The developed solver and the associated boundaryconditions will be released as an open-source for themarine and offshore community




Developed a free surface tracking solver for numericalsimulation of unsteady irrotational fully non-linear water wavesThe solver has been validated by application to a number oftest cases, ranging from shallow water standing waves todifferent wave amplitudes progressive waves

Solution of Laplace’s equation for the velocity potential, thenon-linear free surface boundary conditions, the wavegeneration and the absorption boundary conditions are all notpart of the standard OpenFOAM R© distributionCoupling to available Navier-Stokes solvers inOpenFOAM R©The developed solver and the associated boundaryconditions will be released as an open-source for themarine and offshore community




Developed a free surface tracking solver for numericalsimulation of unsteady irrotational fully non-linear water wavesThe solver has been validated by application to a number oftest cases, ranging from shallow water standing waves todifferent wave amplitudes progressive wavesSolution of Laplace’s equation for the velocity potential, thenon-linear free surface boundary conditions, the wavegeneration and the absorption boundary conditions are all notpart of the standard OpenFOAM R© distribution

Coupling to available Navier-Stokes solvers inOpenFOAM R©The developed solver and the associated boundaryconditions will be released as an open-source for themarine and offshore community




Developed a free surface tracking solver for numericalsimulation of unsteady irrotational fully non-linear water wavesThe solver has been validated by application to a number oftest cases, ranging from shallow water standing waves todifferent wave amplitudes progressive wavesSolution of Laplace’s equation for the velocity potential, thenon-linear free surface boundary conditions, the wavegeneration and the absorption boundary conditions are all notpart of the standard OpenFOAM R© distributionCoupling to available Navier-Stokes solvers inOpenFOAM R©

The developed solver and the associated boundaryconditions will be released as an open-source for themarine and offshore community




Developed a free surface tracking solver for numericalsimulation of unsteady irrotational fully non-linear water wavesThe solver has been validated by application to a number oftest cases, ranging from shallow water standing waves todifferent wave amplitudes progressive wavesSolution of Laplace’s equation for the velocity potential, thenon-linear free surface boundary conditions, the wavegeneration and the absorption boundary conditions are all notpart of the standard OpenFOAM R© distributionCoupling to available Navier-Stokes solvers inOpenFOAM R©The developed solver and the associated boundaryconditions will be released as an open-source for themarine and offshore community



Thank you


Nonlinear Potential Flow Solver Development in...

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