Mécanique

22 févr. 2014 (il y a 7 années et 5 mois)

283 vue(s)

By Santhanu Jana

Talk Overview

Motivation

Techniques in Grid Movement

Physical and Numerical Implications in time

dependent PDE‘s

Outlook and Conclusions

Motivation

Applications in Physics

Fluid Structure

Aerostructures and Aeroacoustics

Moving Elastic Structures

eg. Simulation of Heart

Thermodynamical Considerations

Phase Change Phenomena

Free Surfaces

Material Deformations

Multiphase Flows

Some Examples
-

Fluid Structure Interactions

Source:
http://www.onera.fr/ddss
-
en/aerthetur/aernummai.html

:http://www.erc.msstate.edu/simcenter/04/april04.html

Some Examples
-

Phase Change Phenomena

Source: Work At LSTM

Crystal Growth

Czochralski Crystal Growth

Simulation of Free Surface

What Is Moving Grid ?

PDE’s must be satisfied on each side of the

interface (often different equations on each side)

Solutions coupled through relationships or jump

conditions that must hold at the interface

These conditions may be in the form of differential

equations

Movement of the interface is unknown in advance

and must be determined as part of the solution

Computational Techniques in Moving
Grids

Lagrangian Methods

Eulerian Methods

Mixed Lagrangian and

Eulerian Methods

Lagrangian Methods(1)

Explicit Tracking of the Interface Boundary

No Smearing of Information at the boundary

No Modeling is necessary to define the interface

Un/structured boundary Conforming Grids

No modelling to define the interface

Grid Regeneration

Requires redistribution of field information

Problems in Lagrangian Methods(2)

Grid Distortion

Solution: Grid Sliding

Problems in Lagrangian Methods(3)

Resolving Complex
Structures near the
interface

Solution: Local Grid

Refinement

Increase the

Convergence order

Necessary Modifications in the
Conservation Equations(4)

Eg: Solution of Navier Stokes Equations

1 ) Momentum Equation:

2 ) Energy Equation
:

3 ) Mass Conservation Equation:

Necessary Modifications in the
Conservation Equations(5)

4) Geometric Conservation

NOTE: Grid Velocities should satisfy Geometric

Conservation Equation

References:

1) Thomas, P.D., and Lombard, C.K.,

Geometric Conservation Law and Its Applications to Flow Computations on Moving Grids,"

AIAA Journal, Vol. 17, No. 10, pp. 1030
-
1037.

2) Weiming Caso, Weizhang Huang and Robert D. Russel

A Moving mesh Method based on the Geometric Conservation Law,

SIAM J. SCI. COMPUTING Vol24, No1, pp.118
-
142

Eulerian Methods(1)

Boundary is derived from a Field Variable

eg: VOF, Level Set

Interface is diffused and occupies a few grid cells in

practical calculations

Strategies are necessary to sharpen and physically

reconstruct the interface

Boundary Conditions are incorporated in the

governing PDE.

Grid Generation: Grid is created once

Basic Features of Eulerian Methods(2)

Grid Topology remains simple even though the interface

may undergo large deformations

Two Basic Approaches

Immersed Boundary Method

Without explicit tracking

Interface Cut
-
Cell Method

Interface tracked explicitly

(Reconstruction procedures to calculate coefficients

in the Solution Matrix)

Ref: 1) C.S.Peskin, Numerical Analysis of blood flow in the heart,

Journal of Computational Physics, 25, (1977), 220
-
252

2) H.S.Udaykumar, H.C.Kan, W.Shyy, and R.Tran
-
Son
-
Tay,

Multiphase dynamics in arbitrary geometries on fixed cartesian grids,

Journal of Computational Physics, 137, (1997), 366
-
405

Eulerian Methods: Immersed Boundary
Method (3)

Marker Particles

FLUID 2

ds

n

FLUID I

The Interface between Fluid 1

and Fluid 2 is represented by

curve C is marked by

particles (dots) that do not

coincide with the grid nodes

C

Important Considerations:

Interface Representation

Assignment of Material Properties

(Change of Contants in PDE)

Immersed Boundary Treatment

Immersed Boundary Method: Interface
Representation(4)

Immersed boundary represented by C(t)

Curve in 2D and Surface in 3D.

Markers or interfacial points of coordinates

Markers are regularly distributed along C(t) at a

fraction of grid spacing (ds).

The interface is parameterised as a function of

arclength by fitting a quadratic polynomial.

The normal vector and curvature

(divergence of normal vector) is evaluated.

Immersed Boundary Method: Material
Properties(5)

Should handle the transition zone.

Treatment handles improved Numerical

Stability and solution smothness

Immersed Boundary Method: Boundary
Treatment(6)

Facilitates Communication between the moving markers

(interface) and the fixed grid.

Evaluation of the forces acting on the interface

Estimation of interface velocity

To improve accuracy of the interface tracking, a local

grid refinement aroung the interface can be used
.

Ref:
H. S. Udaykumar, R. Mittal, P. Rampunggoon and A. Khanna,

A Sharp Interface Cartesian Grid Method for Simulating Flows with Complex Moving
Boundaries

Journal of Computational Physics, Volume 174, Issue 1, 20 November 2001, Pages 345
-
380

Interface Cut Cell Method(6)

Improvement over Immersed Boundary

method :

Summary of the Procedure

Location of Interface Marker.

The interfacial marker closest to mesh point.

Material parameters.

Interface Cell Reconstruction :

Geometric details.

Intersection of the immersed boundary with the

Fixed grid mesh.

Suitable stencil and evaluate coefficients

Example: Stencil to evaluate variables

Mixed Eulerian
-
Lagrangian methods

Combines features of Eulerian and Lagrangian
methods.

Solver doesnot see discontinuity (Eulerian
Methods)

Solver experiences distributed forces and
material properties on the vicinity of the
interface

No smearing of interface

Ref: S. Kwak and C. Pozrikidis

-
Front Method

Journal of Computational Physics, Volume 145, Issue 1, 1 September 1998, Pages 61
-
88

Outlook and Conclusion

Lagrangian Methods are physically consistent over
Eulerian Methods but suffers when grid distortion is
severe.

In Eulerian Methods mergers and break ups are tackled
automatically.

Interface Reconstruction in Eulerian Methods may be
very complicated on nonorthogonal un/structured grid.
Extension to 3D might be a problem.

Local Refinement may be used to the capture the
interface more accurately.