Parallel Computation of Viscous Incompressible Flows Using GodunovProjection Method on Overlapping G

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Parallel Computation of Viscous Incompressible
Flows Using Godunov-ProjectionMethod on
Overlapping Grids

• H. Pan
• Institute of High Performance Computing
• 1Science Park Road 01-01, The Capricorn
• Singapore 117528
• Email panhua_at_ihpc.a-star.edu.sg

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Outline

• Introduction
• Formulation
• Numerical Method
• Overlapping Grids
• Parallel Computation
• Numerical Results
• Conclusions

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Underwater Robotic Vehicle
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Motivation

• To estimate the hydrodynamic force acting on the
  vehicle in order to establish the dynamic model
  of underwater vehicle.
• To develop a numerical method to model the
  interaction between underwater vehicle and
  underwater environment.

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Introduction(1)

• fixed-boundary problem
• vehicle moving far away from underwater structure
• solving the unsteady incompressible NS equation
  where the grid has no change

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Introduction(2)

• moving-boundary problem
• vehicle moving in the vicinity of underwater
  structure
• solving the Navier-Stokes equation in the moving
  grid system

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Coordinate Systems
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Formulation

• Unsteady Incompressible Navier-Stokes equation in
  the non-inertial coordinate system

where
is hydrodynamic pressure
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Godunov-Projection Algorithm

• Originally designed by Bell,Colella and Glaz
• Time-accurate algorithm
• Second-order accuracy both in time and space
• Godunov procedure is incorporated in order to
  remove the cell Reynolds number restriction
• Applications in several complex flows, such as
  various density flow and reactive flow have been
  done

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Numerical Scheme(1)

• Convection term is
  calculated by using Godunov procedure

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Numerical Scheme(2)

• Intermediate velocity field is computed by
  solving the following equation

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Numerical Scheme(3)

• Projection is performed to obtain the
  divergence-free velocity field.

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Overlapping Grid(1)

• Chimera grid
• Flexible to generate structured grids for complex
  geometric configurations
• Reduce computation overhead for solving
  moving-boundary problems
• Overture used as grid generator

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Overlapping Grid(2)

• Computational domain is covered by the union of
  several sub-domains, which overlap each other.
• There is no requirement that the component grids
  match up at their boundaries and only sufficient
  overlap is required.
• Component grid is constructed by transforming the
  each sub-domain into a unit computational space.
• Each component grid can be constructed almost
  independently

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overlapping grid for flow past a sphere
Overlapping Grid(3)
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Parallel Computation(1)
P number of processors

• Box-wise partition for component grid
• Scaleable
• Load-balancing

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Grid partitions for the overlapping grid for flow
past sphere
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Parallel Computation(2)

• Master/Slave model
• Master takes control the execution and perform
  the I/O
• Slaves perform the computations and signal the
  master when its ready
• Slaves exchange data directly under the
  supervision of the master
• MPI
• PETSc

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Master/Slave Model
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Numerical Results (1)

• Lid-driven cubic cavity flow
• classical benchmark problems
• Re100,400,1000,3200

lid
U
0
z
1
y
1
x
1
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Lid-driven cubic cavity flow
Re400
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Lid-driven cubic cavity flow
Re1000
x0.5
y0.5
z0.5
Velocity
vorticity
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Lid-driven cubic cavity flow
Comparison of velocity component distributions
along central lines
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Numerical Results (2)

• Flow around moving body in a tank -
  moving-boundary problem
• Rens Problem
• A cylinder moves in a tank.
• Initially, the fluid is static. The cylinder
  begins to move with constant velocity.
• Reynolds number

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Grids and Streamlines of Flow around moving body
in a tank
t3.45
t5.41
t7.48
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Drag coefficient
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Numerical Results (3)

• Flow past an oscillating cylinder
• Karanths case
• transverse, in-line and combined oscillation

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Flow Past an oscillating cylinder
Transverse oscillation
pressure
force coefficients
vorticity
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Flow Past an oscillating cylinder
In-line oscillation
pressure
force coefficients
vorticity
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Flow Past an oscillating cylinder
Combined oscillation
pressure
force coefficients
vorticity
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Numerical Results (4)

• Flow past sphere
• Re100,400
• performance of parallel computation on
  overlapping grid

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Flow past a sphere Re100
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Flow past a sphere - Re100
Drag coefficient on three different grids
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Flow past a sphere Re400
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Speedup and Efficiency of Flow past a sphere
Efficiency
Speedup
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Numerical Results (5)

• Flow past underwater vehicle
• multi-body configuration
• hydrodynamic coefficients for added-mass effect,
  etc

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Flow past underwater vehicle - hierarchical
structure of overlapping grids
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Flow past underwater vehicle- Generated
Overlapping Grid
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Flow past underwater vehicle - Re200
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Model for Underwater Rigid Body

• Equations of motion of underwater rigid body

• where

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Numerical Method for Coupled System(1)

• rigid body subjected to control forces
• trajectory has to be determined
• multi-discipline problem
• dynamics of rigid body
• solvable provided the hydrodynamics forces
• fluid dynamics
• solvable provided the motion of rigid body

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Numerical Method for Coupled System(2)
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Numerical Method for Coupled System(3)

• Runge-Kutta method to solve equation of motion of
  rigid body
• Godunov-projection method to solve the flow field
• Coupling strategy
• loose coupling method
• explicit algorithm
• implicit algorithm

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Numerical Method for Coupled System(4)
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Numercial Problem (6)

• Free motion of circular cylinder
• coupled problem
• neutrally buoyant cylinder
• cylinder is immersed in a uniform flow
• cylinder is subjected to specified control forces
• initially, vortex shedding is fully developed

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Free motion of circular cylinder
pressure contour plot
vorticity contour plot
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Free motion of circular cylinder
drag and lift coefficient
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Free motion of circular cylinder
position
acceleration
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Conclusion

• Godunov-projection method has been improved and
  implemented on overlapping grids for both
  inertial and non-inertial coordinate systems.
• It has been verified that overlapping grid can be
  used to solve the moving boundary problems.
• The coupling system, which contains both the
  rigid body dynamics and fluid dynamics, is solved
  by using loose-coupling method
• Parallel fluid solver has been developed and
  verified.

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Future Work

• The Godunov-projection method should be improved
  so that the adaptive overlapping grids can be
  applied.
• Three-dimensional moving-boundary problems should
  be solved by extending current numerical
  algorithm.
• More efficient methods are required for
  addressing the coupled system.

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Thanks