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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
A. Mehmood Nonlinear Potential Flow Solver Development in OpenFOAM
Nonlinear Potential Flow Solver Development in OpenFOAMMotivation
Wave Structure Interaction Simulation Environment
A. Mehmood Nonlinear Potential Flow Solver Development in OpenFOAM
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
A. Mehmood Nonlinear Potential Flow Solver Development in OpenFOAM
Nonlinear Potential Flow Solver Development in OpenFOAMSolution Methodology
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
Nonlinear Potential Flow Solver Development in OpenFOAMSolution Methodology
Sequence of the Solution Procedure
Generate the grid
Generate the grid
A. Mehmood Nonlinear Potential Flow Solver Development in OpenFOAM
Nonlinear Potential Flow Solver Development in OpenFOAMSolution Methodology
Sequence of the Solution Procedure
Apply the boundary conditions
Apply the boundary conditions at the face center
A. Mehmood Nonlinear Potential Flow Solver Development in OpenFOAM
Nonlinear Potential Flow Solver Development in OpenFOAMSolution Methodology
Sequence of the Solution Procedure
Solve Laplace’s equation
Solve Laplace’s equation for the velocity potential.Compute the required variables (i.e., velocities u = ∇φ,fluxes).
A. Mehmood Nonlinear Potential Flow Solver Development in OpenFOAM
Nonlinear Potential Flow Solver Development in OpenFOAMSolution Methodology
Sequence of the Solution Procedure
Solve Laplace’s equation
Solve Laplace’s equation for the velocity potential.Compute the required variables (i.e., velocities u = ∇φ,fluxes).
A. Mehmood Nonlinear Potential Flow Solver Development in OpenFOAM
Nonlinear Potential Flow Solver Development in OpenFOAMSolution Methodology
Sequence of the Solution Procedure
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
A. Mehmood Nonlinear Potential Flow Solver Development in OpenFOAM
Nonlinear Potential Flow Solver Development in OpenFOAMSolution Methodology
Sequence of the Solution Procedure
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
A. Mehmood Nonlinear Potential Flow Solver Development in OpenFOAM
Nonlinear Potential Flow Solver Development in OpenFOAMSolution Methodology
Sequence of the Solution Procedure
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
A. Mehmood Nonlinear Potential Flow Solver Development in OpenFOAM
Nonlinear Potential Flow Solver Development in OpenFOAMSolution Methodology
Sequence of the Solution Procedure
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
A. Mehmood Nonlinear Potential Flow Solver Development in OpenFOAM
Nonlinear Potential Flow Solver Development in OpenFOAMSolution Methodology
Sequence of the Solution Procedure
Automatic Mesh Motion in OpenFOAM
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:
Laplace equation with constant and variable diffusivitydiffusivity − > quadratic inverseDistance
A. Mehmood Nonlinear Potential Flow Solver Development in OpenFOAM
Nonlinear Potential Flow Solver Development in OpenFOAMSolution Methodology
Sequence of the Solution Procedure
Automatic Mesh Motion in OpenFOAM
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:
Laplace equation with constant and variable diffusivitydiffusivity − > quadratic inverseDistance
A. Mehmood Nonlinear Potential Flow Solver Development in OpenFOAM
Nonlinear Potential Flow Solver Development in OpenFOAMSolution Methodology
Sequence of the Solution Procedure
Automatic Mesh Motion in OpenFOAM
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
A. Mehmood Nonlinear Potential Flow Solver Development in OpenFOAM
Nonlinear Potential Flow Solver Development in OpenFOAMSolution Methodology
Sequence of the Solution Procedure
Automatic Mesh Motion in OpenFOAM
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:
Laplace equation with constant and variable diffusivitydiffusivity − > quadratic inverseDistance
A. Mehmood Nonlinear Potential Flow Solver Development in OpenFOAM
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
A. Mehmood Nonlinear Potential Flow Solver Development in OpenFOAM
Nonlinear Potential Flow Solver Development in OpenFOAMResults and Discussion
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
A. Mehmood Nonlinear Potential Flow Solver Development in OpenFOAM
Nonlinear Potential Flow Solver Development in OpenFOAMResults and Discussion
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
A. Mehmood Nonlinear Potential Flow Solver Development in OpenFOAM
Nonlinear Potential Flow Solver Development in OpenFOAMResults and Discussion
Standing Waves
Standing Wave Animation
H
Initial wave profilewave amplitude
H
Initial wave profilewave amplitude
A. Mehmood Nonlinear Potential Flow Solver Development in OpenFOAM
Nonlinear Potential Flow Solver Development in OpenFOAMResults and Discussion
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
A. Mehmood Nonlinear Potential Flow Solver Development in OpenFOAM
Nonlinear Potential Flow Solver Development in OpenFOAMResults and Discussion
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)
A. Mehmood Nonlinear Potential Flow Solver Development in OpenFOAM
Nonlinear Potential Flow Solver Development in OpenFOAMResults and Discussion
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
A. Mehmood Nonlinear Potential Flow Solver Development in OpenFOAM
Nonlinear Potential Flow Solver Development in OpenFOAMResults and Discussion
Progressive Waves
Comparison with Experiment (F.Gao-2003)
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
Nonlinear Potential Flow Solver Development in OpenFOAMResults and Discussion
Progressive Waves
Comparison with Experiment (F.Gao-2003)
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
Figure: Time history of wave elevation at location x = 3.55 m.A. Mehmood Nonlinear Potential Flow Solver Development in OpenFOAM
Nonlinear Potential Flow Solver Development in OpenFOAMResults and Discussion
Progressive Waves
Comparison with Experiment (F.Gao-2003)
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
Figure: Time history of wave elevation at location x = 5.45 m.A. Mehmood Nonlinear Potential Flow Solver Development in OpenFOAM
Nonlinear Potential Flow Solver Development in OpenFOAMResults and Discussion
Progressive Waves
Comparison with experiment
A. Mehmood Nonlinear Potential Flow Solver Development in OpenFOAM
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
A. Mehmood Nonlinear Potential Flow Solver Development in OpenFOAM
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 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
A. Mehmood Nonlinear Potential Flow Solver Development in OpenFOAM
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 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
A. Mehmood Nonlinear Potential Flow Solver Development in OpenFOAM
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 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
A. Mehmood Nonlinear Potential Flow Solver Development in OpenFOAM
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 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
A. Mehmood Nonlinear Potential Flow Solver Development in OpenFOAM
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