o. hassan, k. morgan and n. p. weatherill school of engineering, university of wales

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A Method for Time Accurate Turbulent Compressible Fluid Flow Simulation with Moving Boundary Components

Employing Local Remeshing

O. Hassan, K. Morgan and N. P. Weatherill

School of Engineering, University of Wales

Swansea, United Kingdom

Workshop on Mesh Refinement of Unsteady Flows7 December 2005 - Oxford University

2

Problem of Interest and Adopted Approach

Solution Algorithm

Unstructured Mesh Generation Techniques

Mesh Adaptation Techniques

Mesh Adaptation for Unsteady Flow Problem with Moving Boundary

Parallel Implementation

Conclusion

Outline

3

DASA - F16

Airbus - A380

Dassault - Falcon

Airbus - A340

Industrial End-User Configurations

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The governing equations are the Navier-Stokes equations

The application requires the ability to model complex geometries

The simulation of turbulence affords a real challenge

For many applications, the Euler equations are appropriate

Computational requirements for realistic geometries can be expected to be large

Adopted Approach

Unstructured grid technology provides the required

flexibility for these (and other) applications

5

Governing Equations

The Favre Averaged Navier Stokes Equations

Where

and

Turbulent is modelled by adding the one equation model of Spalart and Allmaras

)()()( t

jj

t

jjj

t

dndnvddt

dxGxUFxU

pu

puu

puu

puu

u

j

jj

jj

jj

j

j

33

22

11

F

w

v

u

U

jjkk

j

j

j

j

qu

3

2

1

0

G

kkv uuTc2

1 RTp

6

Solution Algorithm

Edge Based Data Structure: Typical interior node I

The ALE term for an interior node I is:

Where: should lead to a numerical ALE flux that satisfies GCL

Resulting Equation

)(2

Jj

Ij

ΓJ

IJ

jj UUS

dUnvII

)(2

)(2

Jj

Ij

ΓJ

IJj

jj

Jj

Ij

ΓJ

IJj

jj

II

GGC

dnG

FFC

dnF

UVdt

dUd

dt

d

II

II

I

III RVdt

d][ U

ijS

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Solution Algorithm

The turbulent viscosity equation is discretised in a similar fashion

Stabilisation achieved by replacing the actual flux function by JST flux function

Discontinuity capturing achieved by the addition of a switched artificial diffusion

For steady state Runge-Kutta relaxation and local timestepping is utilised

Convergence acceleration is achieved by using the Full Approximation Storage (FAS) Multigrid scheme• Coarse grids are achieved by agglomeration• Volume weighted operator is used for restriction• Injection is used for prolongation

Parallel implementation allows agglomeration across partitions

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Steady Turbulent Flow

Pressure distribution

Cl vs CdCl vs

AIAA test case: Drag Prediction Workshop 2001M = 0.75 Re = 3 x 106

1.6 million points 35 viscous layers 5 grids levels

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Time Discretisation

For unsteady problems, the second order approximation

is adopted

An implicit formulation is employed and this removes the stability constraints associated with explicit schemes

At each time step, the equation is solved by explicit relaxation with multi-grid acceleration

This approach can be thought of as converging the set of steady state equations with the addition of the time source for every physical timestep

No significant memory penalties compared to explicit procedures

)2

12

2

3(

1 1111

)(1

nI

nI

nI

nI

nI

nI

ttt

i VVVt

dUdt

d

n

UUU

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The surface is defined as a set of:• Surface Components: bi-cubic patches, NURBS• Curve components: cubic splines, NURBS

Mesh control:• Background Mesh• Point, Line, circular and planar sources• Curvature Controlled

Mesh Generation

),max(

)2(2

21p kk

h

: the gap between the element and the surface

k1, k2 : the two

principle curvatures

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Surface mesh generation: Advancing Front

Volume mesh generation: Delaunay Triangulation with automatic point insertion (Requires 100Mb/106 elements)

Boundary layer generation: Hybrid meshes by the Advancing Layer method

Mesh Generation

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Mesh Minimum Volume

Minimum Dihedral Angle

Generated 3.26 x 10-

9

0.098

20-10 2.32 x 10-

7

7.32

70-10 2.36 x 10-

7

12.27

Improved Volume Mesh Quality

• 3D Edge Swap

• 3D Edge Collapse

• 3D Nodal Smoothing

• 3D Local Re-generation

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Super surfaces eliminate small and distorted patches generated by the CAD systems

• Merge neighboring surfaces based on continuity of the normal

Improved Surface Mesh Quality

Patch Independent Remeshing

• Starting from any triangulation, re-triangulate using edge splitting, edge collapse and edge swapping.

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Mesh Adaptation

Error Analysis

Error indicator based on posterior error analysis are also possible, but not very practical for unsteady flow.

A Error indicator based upon interpolation theory and accounts for directionality is employed.

Assuming exact nodal values, estimate the local error for each elements as:

Equidistribution of the error results in a mesh spacing for the new mesh:

In 2D/3D: Apply the 1D criterion separately to each principal direction of the Hessian Matrix

Constantdx

d

2

22

Constantxx ji

ji

22

2

22

11 dx

dhE e

e

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Mesh Adaptation

Mesh Enrichment

• Advantages: Simple and quick to implement Trivial interpolation

• Disadvantages: Multiple refinement results in large

meshes De-refinement require excessive

storage Incorporating stretching results is

distorted elements Not suitable for unsteady flow with

moving components

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Mesh Adaptation

Mesh Enrichment

• Special care is required in 3D to ensure compatibility of adjacent elements

• Special care is also needed to ensure the validity of the mesh after projecting the added points onto the surface

Geometry

Initial Grid Adapted Grid

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Mesh Movement

• Replace the sides of the mesh by spring• Spring stiffness depends on the flow

properties

• Where

• Move the nodes until nodal equilibrium• Solve by iteration

)( KJJKJK Cf rr

N

ji

iJK

iJK

ijJK nnmhC

1;

j

j

S

k

nK

S

k

nKJK

nJ

C

1

11

r

rr

Mesh Adaptation

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Mesh Adaptation

Mesh Movement

• Advantages: Simple and quick to implement Can handle moving components

• Disadvantages: Expensive interpolation Initial mesh may lack the required resolution to resolve all the flow

features Hard to control the quality of the moved mesh

• Coupling of mesh movement and mesh enrichment can over come most of the restrictions.

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Mesh Adaptation

Adaptive Remeshing

• Use the current mesh as a background mesh

• At each node compute the mesh parameters using the equidistribution criterion

• Use the geometry definition to regenerate the mesh

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Adaptive Remeshing Using a Background Mesh

• Advantages: Simple and quick to implement Can handle moving components High quality meshes

• Disadvantages: Expensive to regenerate the

complete mesh

Mesh Adaptation

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Adaptation

Mesh Enrichemnet

Mesh Movement

Remeshing

•No coarsening beyond the initial mesh •Multiple refinement can generate generate distorted elements•Memory intensive in 3D

•Reliable moving methods are expensive•Not easy to guarantee valid mesh•May not have enough initial points

Time consuming in 3DEssential for unsteady flow with large moving

boundaries

Efficient for unsteady flow with small moving

boundaries

Useful for steady state

For unsteady flow with moving boundary components it is essential to develop a scheme which utilise the

advantages of the various methods

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Adaptation for Unsteady Flow

Generate initial mesh Calculate initial quality

measure, qo and spacing,

Loop over timesteps

Update coordinates of moving nodes

Apply deforming mesh algorithm Compute new quality measure, qn

Form holes from marked elements

Remesh hole

Interpolate solutionCompute qo, o

F

Mark element to be deleted

T

Compute new spacing required based on the equidistribution

criterion, n

0nq

NPIP

NEq

qqIE

0

0n

0

0n

*)(

*)(

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Interaction between a strong shock and an object of complex shape

Density contours

1228 elements650 points




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Confined Blast Wave From Rupturing Cylindrical Pressure Vessel

Experimental Apparatus

Comparison of pressure history at various transducers

1499 < Number of elements < 21022

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Surface Adaptation

The Geometry definition is utilised for the regeneration of the surface portion of the hole

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Surface Adaptation

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Unsteady Simulation

B60 Configuration M = 0.801 0 = 2.780

m = 10 Zm = 1m

745198 Elements 135760 Points

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Unsteady Inviscid Flow

Store Separation Simulation

init = zero degrees

M= 0.5 DegreeContainer motion computed

2.7 million tetrahedra

15 time steps 50 multigrid cycle per time step

Geometry for a completeF16 Configuration

8h Wall clock timeSolver: 16 R14000 CPUs

Preprocessing and adaption : 1 CPU

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Surface Pressure Distribution

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Adaptation for Unsteady Turbulent Flow

• Generate initial mesh • Calculate minimum dihedral angle (Dio)

• Store the layer number for all nodes in the

boundary layer

Loop over physical time-steps

Update coordinates of moving nodes and the boundary layer

nodes

Apply deforming mesh algorithm

Calculate new dihedral angles (Din)

Din<0

NEdi

didiIE

0

0n *)(

Form holes from marked elements

Interpolate solution & Compute Dio

F

Mark elements to be deleted

T• Check elements

intersection• Check if any boundary

layer node can grow further• Remesh hole

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Unsteady Turbulent Flow

NACA64A010 Aerofoil:

• Prescribed sinusoidal oscillation

Amplitude 1.01 degrees

init = zero degrees

St = 15.567

• 3D stacked hybrid mesh

173720 nodes

• 300 multigrid cycle/time step

32time steps per cycle

16 R14000 processors

One movement cycle 5 h clock time

Estimated speed up 175

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Lift Polar

k = 90

k = 0.

k = 270


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Shuttle Booster Separation Simulation

init = zero degrees

M= 0.85 DegreeRe = 3 * 10 6

Prescribed Shuttle movement

Initial mesh: 2.9 million elements

Final mesh: 3.3 million elements

20 time steps 300 multigrid cycle per time

step36h Wall clock time

Solver: 16 R14000 CPUsPreprocessing and adaption : 1

CPU


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Meshes on the symmetry plane



Cut through the volume mesh

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Shuttle Booster Separation SimulationUnsteady Turbulent Flow

Meshes of the symmetry plane after remeshing

Cut through the volume mesh after remeshing

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Elements are select to be remeshed in each domain separately • Selection Based on Deviation from Prescribed Spacing• Selection Based on Element Quality• Selection Based on Intersection Tests

To determine intersection of elements due to moving geometries, one ovelapping ghost layer of elements is used.

If intersection with the ghost element has occurred, the search will also take place in to the domain which own the ghost cell.


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1

3

12

1

1

2

2

Domain 4 Domain 3

Domain 2Domain 1

Interfaces

Constraint Repartitioning is employed to ensure that each region to be remeshed will be contained completely on one process

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init = 0.46 degrees

M= 0.96Container motion computed

2.1-2.3 million Nodes12.1-13.4 million Elements

40 Physical timesteps with sub-cycling

10.4 h on 24 Processors

Geometry for a completeF18 Configuration

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CFD Solution 40% Motion Application 57%

• Mesh Deformation 10.3%• Volume mesh Analysis 3.7%• Volume remeshing 37%• Re-partitioning 6%

I/O 3%


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Conclusions

A hybrid unstructured finite volume method for aerodynamic flows has been presented

Turbulent flows are treated via the one equation Spalart and Allmaras model

Computational performance is enhanced by the use of multigrid acceleration and parallelisation

Transient moving boundary flows are treated by an ALE approach

Mesh movement and adaptive remeshing have been employed to handle the deformation due to the moving components

Adaptive remeshing was extended to meshes with stretched elements in the boundary layers

Parallel implementation of the adapted remeshing has been completed

A number of challenging problems have been simulated and the agreement with available experimental observations is good

o. hassan, k. morgan and n. p. weatherill school of engineering, university of wales

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