Towards convergence and consistence for conservative SPH

Towards convergence and consistence for conservative SPH approximation

Towards convergence and consistence for conservative SPH

approximation

Xiangyu Hu

Institute of Aerodynamics and Fluid Mechanics

TU München


Outline

• Problem of covergence and consistence in conservative SPH formulation

– Condition of partition of unity

• Transport-velocity formulation is a simple solution

– The velocity of paticle motion

• Physical background of the transport-velocity formulation

– NS alpha model for simulating turbulent flows

2


Conservative SPH approximation

• Anti-symmetric form

• Force between a pair of particles

• Surface integral around all surfaces between a particle and its neighbors

– Volume intergaral for computing gradient transformed into surface ingtegral for computing force

Inter-particle surface

3


Two types of errors introduced by SPH approximations

Introducing filtering error

Introducing integration error

(Quinlan et al. 2006)

4


Convergence and consistence

Filterring error

• Filtering operation in continuous field

• Consistent if the kernel function has standard properties

• Convergence rate respect to smoothing-length dependent on high-order zero-moments – At least 2nd order

– Able to achieve 2n-th order

Integration error • Discrete particle summation

• Consistence and convergence dependent on particle distribution – Uniform distribution for particles

on grid • Consistent and convergence

respect to number of particle in kernel support

• very high-order dependent on eddy-smoothness of the kernel

– Randomly perturbed particles • NO zero-order consistence, NO

convergence

• Even not able to recover a constant field!

5


„BACKGROUND-PRESSURE DILEMMA“

• Flow around cylinder

0

0

1ii

p p

particle lumping due to negative pressure

0

0

1ii

p p

due to the reference pressure particles fill the region behind the cylinder

6


„BACKGROUND-PRESSURE DILEMMA“ • Taylor-Green flow(2D) with

7

Velocity decay (Re=100)

Background pressure

„Tensile Instability“

Artificial dissipation

“freezing”

• background pressure

strong artificial dissipation


„BACKGROUND-PRESSURE DILEMMA“ • Drop equilibration under surface-tension

8

Pre

ssu

re

Radius

Pre

ssu

re

Radius


What is the problem?

• Non-closure of the inter-particle surfaces

• Error on approximation constant field

– Dependent on the magnitude

• Could this error vanish with more particles within the kernel?

– NO, for randomly perturbed particles (Quinlan et al. 2006)

9


But even an integral with random dots is able to converge?

• Quasi Monte Carlo Method (QMCM)

– Throw dots almost randomly into the integration domain

– Convergence rate

• What is the problem again?

– Implicit constrain in QMCM

– Not satisfied by SPH, because volume is approximation too

10


Achieving partition of unity by particle relaxation

• Equation of motion

– Initially random distribution

– Relaxed after all particle stop to move

Constant pressure

Invariant particle volume

11


Condition for consistence and convergence

• Closure of all surfaces around a particle

• Summation of particle volume

• Leads to partition of unity – Domain is covered by volumes defined by particles without

gap or overlap – Assure zero-order consistence and 1rst-order convergence

and simple rectangle integration rule

Condition for consistence

Condition for convergence

12


Distribution of relaxed particles

Radial distribution function (RDF), which describes how the number density of particles changes as a function of distance from a reference particle.

Typical for liquid molecules in microscopic

13


Convergence property of integration error for relaxed particles

8th-order convergence Same as for uniform particle

14


The transport-velocity formulation to reduce error introduced by

background pressure (Adami et al. 2013)

Transport velocity

Pressure without background pressure

15


Self-relaxation mechanism within SPH simulation?

• Due to the background pressure

– Relaxation interfered by the strain of flow

– But the particle distribution much better than that of randomly perturbed particle without relaxation

• Much better convergence properties in practice

Constant background pressure

16


EXAMPLES

• Taylor-Green flow (2D)

17

Re=100


EXAMPLES • Lid-driven cavity at Re=100

18

Velocity profiles on horizontal and vertical centerline

Velocity field with vectors


EXAMPLES • Lid-driven cavity at Re=1000

19




EXAMPLES

• Lid-driven cavity at Re=10000

20




EXAMPLES

• Cylinder flow (Stokes limit)

21

Particle snapshot colored with velocity Resolution study of drag coefficient


EXAMPLES

• Backward-facing step Re=100

22

Velocity profiles Reattachment point

6.2R

x S 6.3

6.0

FLUENT

Issa

x S

x S

Rx = 6.2 S


Lagrangian Averaged Navier Stokes (LANS) equation

Lagrangian averaged velocity

Eulerian averaged velocity

Eulerian averaged velocity = volume averaged velocity

Lagrangian averaged velocity = regularized Lagrangian velocity

(Holm 2002)

23


Extra stress term (not shown in LANS)

Transport velocity formulation

iviv~

Transport velocity

Momentum velocity

Transport velocity

Momentum velocity

Momentum velocity = volume averaged velocity

Transport velocity = regularized Lagrangian velocity

(Adami, Hu and Adams 2013 JCP)

24


Transport-velocity formulation as a LES model

• LANS can be used as a LES model – NS-a model

– Regularized Lagrangian velocity to prevent small scale flow structures

– However, the performs of the Eulerian formulation not as good as the standard Smagorinsky model

• The transport-velocity formulation is the discretized form of modified LANS equation – Can it also be a turbulence model?

– Possible benefit from the extra stress term?

25


Testing with 3D Taylor-Green Vortex

• A prototype for vortex stretching, instability and production of small-scale eddies to examine the dynamics of transition to turbulence.

• 2D initial condition

(Hu and Adams 2011 JCP)

A direct simulation of TGV http://users.ugent.be/~dfauconn/research.htm 26


Moderate Reynolds numbers

• The transport-velocity formulation avoids the large over prediction of the dissipation, unlike to the classical SPH method.

• For under-resolved case: corrected SPH method comparable to standard Smagorinsky LES model.

time time

Ra

te o

f ki

net

ic e

ner

gy

dec

ay

Ra

te o

f ki

net

ic e

ner

gy

dec

ay

Re = 100 Re = 400 DNS (2563)

classic SPH (643) Corrected SPH (643)

Standard Smagorinsky(643)

DNS (2563) classic SPH (643)

Corrected SPH (643)

27


High Reynolds number, Re = 3000

• Less dissipation rate in the early time and large dissipation rate in the late time compared to the standard Smagorinsky model

• Predicts intermittency which is not presented in the latter

Probability density function of acceleration

PDF

a / arms time

Ra

te o

f ki

net

ic e

ner

gy

dec

ay

DNS (2563) Corrected SPH (643)

Standard Smagorinsky(643)

Corrected SPH (643) Gaussian

(Adami, Hu and Adams 2012 CTR report, Stanford Univ.)

The first time showed that SPH can be better than the standard model for mesh methods on turbulence simulation!

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Conclusion

• Partition of unity is the condition for covergence and consistence in conservative SPH formulation – Relaxation toward the condition – Self-relaxation mechanism in SPH simulation

• Transport-velocity formulation is a simple solution – Keep the self-relaxation machnaism – Decrease the error induced by inconsistancy

• NS alpha model is the physical background of the transport-velocity formulation – Ability to simulate turbulent flows

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Documents

Towards convergence and consistence for conservative SPH