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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
Towards convergence and consistence for conservative SPH approximation
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
Towards convergence and consistence for conservative SPH approximation
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
Towards convergence and consistence for conservative SPH approximation
Two types of errors introduced by SPH approximations
Introducing filtering error
Introducing integration error
(Quinlan et al. 2006)
4
Towards convergence and consistence for conservative SPH approximation
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
Towards convergence and consistence for conservative SPH approximation
„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
Towards convergence and consistence for conservative SPH approximation
„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
Towards convergence and consistence for conservative SPH approximation
„BACKGROUND-PRESSURE DILEMMA“ • Drop equilibration under surface-tension
8
Pre
ssu
re
Radius
Pre
ssu
re
Radius
Towards convergence and consistence for conservative SPH approximation
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
Towards convergence and consistence for conservative SPH approximation
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
Towards convergence and consistence for conservative SPH approximation
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
Towards convergence and consistence for conservative SPH approximation
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
Towards convergence and consistence for conservative SPH approximation
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
Towards convergence and consistence for conservative SPH approximation
Convergence property of integration error for relaxed particles
8th-order convergence Same as for uniform particle
14
Towards convergence and consistence for conservative SPH approximation
The transport-velocity formulation to reduce error introduced by
background pressure (Adami et al. 2013)
Transport velocity
Pressure without background pressure
15
Towards convergence and consistence for conservative SPH approximation
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
Towards convergence and consistence for conservative SPH approximation
EXAMPLES
• Taylor-Green flow (2D)
17
Re=100
Towards convergence and consistence for conservative SPH approximation
EXAMPLES • Lid-driven cavity at Re=100
18
Velocity profiles on horizontal and vertical centerline
Velocity field with vectors
Towards convergence and consistence for conservative SPH approximation
EXAMPLES • Lid-driven cavity at Re=1000
19
Velocity profiles on horizontal and vertical centerline
Velocity field with vectors
Towards convergence and consistence for conservative SPH approximation
EXAMPLES
• Lid-driven cavity at Re=10000
20
Velocity profiles on horizontal and vertical centerline
Velocity field with vectors
Towards convergence and consistence for conservative SPH approximation
EXAMPLES
• Cylinder flow (Stokes limit)
21
Particle snapshot colored with velocity Resolution study of drag coefficient
Towards convergence and consistence for conservative SPH approximation
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
Towards convergence and consistence for conservative SPH approximation
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
Towards convergence and consistence for conservative SPH approximation
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
Towards convergence and consistence for conservative SPH approximation
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
Towards convergence and consistence for conservative SPH approximation
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
Towards convergence and consistence for conservative SPH approximation
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
Towards convergence and consistence for conservative SPH approximation
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
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!
28
Towards convergence and consistence for conservative SPH approximation
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
29
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