Arbitrary Order Boundary Reconstruction Algorithm for Robin Boundary Conditions in Particle-Resolved Direct Numerical Simulations

Abstract

Particle-Resolved Direct Numerical Simulation is a widely used methodology to investigate transport phenomena occurring at the particle scale in a large variety of fluid-particle systems. Such simulations are generally performed using grid-based methods where the governing equations are discretized over grid nodes or finite volumina. The representation of the particulate phase is therefore achieved by (i) creating a mesh conforming to the complex topology of the fluid domain, or by (ii) introducing new terms in the governing equations that account for the presence of the particulate phase. Unfortunately, the generation of high quality conformal grids can be difficult, and hence this approach is often not practical for fully-automated simulation workflows. The latter methodology is more promising, and often referred to as the “Immersed Boundary” (IB) method [1]. While the IB method is more effective for both static and moving particles, it is still not clear how general immersed boundary conditions should be enforced on grid nodes that are not aligned with the immersed surfaces.

In the present contribution, we extend our Hybrid Fictitious Domain Immersed Boundary Method HFDIBM [2] to be able to impose general boundary conditions with – theoretically - arbitrary order. While the HFDIBM was using second order interpolation to impose a Dirichlet boundary condition, the new method performs a Taylor expansion of Eulerian fields at the particle surface. First, a reconstruction of field properties (e.g., velocity, temperature) near the immersed boundary is performed by probing at multiple discrete positions. Subsequently, the terms in the Taylors are evaluated up to the desired order by solving a system of equations built from the probed field properties and the boundary condition to be enforced. These terms are then used to evaluate the field value at the node closest to the immersed surface.

We detail on the implementation of our method in CFDEM® [3], and show that the method is convergent and accurate for a wide range of fluid-particle systems with Dirichlet, Neumann and Robin boundary conditions. Also, we demonstrate how the method can be applied. Specifically, we will show selected results for flow, as well as heat and mass transfer in dense poly-disperse and mono-disperse suspensions in infinitely large and wall bounded systems.

REFERENCES
1. Mittal, G. and G. Iccarino, Immersed boundary methods. Annual Review of Fluid Mechanics, 2005. 37: p. 239–61.
2. Municchi, F. and S. Radl, Consistent closures for Euler-Lagrange models of bi-disperse gas-particle suspensions derived from particle-resolved direct numerical simulations. International Journal of Heat and Mass Transfer (under revision). 2017.
3. Kloss, C., et al., Models, algorithms and validation for opensource DEM and CFD–DEM. Progress in Computational Fluid Dynamics, an International Journal, 2012. 12(2-3): p. 140-152.

Full text

1
F. Municchi, S. Radl1
1Graz University of Technology
Arbitrary Order Boundary Reconstruction
Algorithm for Robin Boundary Conditions in
Particle-Resolved
Direct Numerical
Simulations

2
I - MICRO II - MESO
III -
MACRO
•Meaningful reaction kinetics must be fed
into “micro-scale” models
•Continue with meso and macro scale if necessary.
…but therefore we need closures!
[1] W. Holloway, PhD Thesis, 2012.
Example Application of CxD
closures
closures

3
Micro & Meso Level: Typical Results
Micro-Meso Bridging: data
filtering and statistical analysis
(closure development) using the
tool CPPPO
«Filtered
particle»
Filtering domain
(i.e., coarse cell)
Micro Level: Particle-Resolved Direct
Numerical Simulation (PR-DNS), e.g.,
using the tool CFDEM®
Meso Level:
Closures are
used in Particle-
Unresolved
Euler-Lagrange
models, e.g.,
using the tool
CFDEM®

4
Closures at the Meso Level
Flow
Scalar Transport
•Contact+cohesive
forces and torques
per contact
•Fluid-Particle
interaction (drag)
forces and torques
per particle
•Heat and mass transfer rates
(Nusselt/Sherwood numbers)
per particle
•Dispersion rates (fluid phase)
•Filtration rates per particle
•Liquid transfer rates per
contact
[2] M. Askarhishahi et al., AIChE J (2017) 63:2569-2587

5
Overview
Part I The Method
Part II Application to Bi-Disperse Flows
Part III Application to Wall-Bounded Flows
5 + 5 + 5 = 15 mins

6
The Method
Part I

7
•Combine Fictitious Domain and Immersed
Boundary algorithm (HFD-IB, [3]).
•Impose the rigidity inside immersed bodies (in
the spirit of the fictitious domain method)
•Enforce a (Dirichlet) boundary condition at the
immersed surfaces: Specifically, we rely on a
reconstruction of the flow and concentration
field in the vicinity of the immersed surface using
an interpolation technique
•Advantage: high order boundary treatment
•Drawback: interpolation points must be in
domain to ensure highest order
The Idea
[3] F. Municchi and S. Radl, Int J Heat Mass Transfer (2017) 111:171–190
Hybrid Fictitious Domain / Immersed Boundary Method

8

 
  


 

 

 
 
 
  Immersed Boundary
The forcing term forces the
solution to a (Dirichlet)
boundary condition at the
immersed body surface [5].
It would be desirable to have a
single flexible formulation that
allows, e.g., Dirichlet and
Neumann BCs.
Fictitious Domain
The forcing term imposes a
rigidity condition inside the
immersed body [4].
Hybrid Fictitious Domain / Immersed Boundary Method
Take into account the presence of
rigid bodies inside the fluid
domain
[4] Smagulov S., Preprint CS SO USSR, N 68 (1979)
[5] Peskin C., Journal of computational physics (1971)
The Idea

9
How to calculate the forcing terms near the surface?
        Rewrite the scalar transport equation: general
partial differential equation of order 

     

    General boundary condition at the particle
surface expressed via a «boundary operator»
The forcing term is calculated from:
          
 

     

 
The boundary operator needs to be inverted!
Remaining challenge: determine the values at the cell
centers z from that at the interface point i.
Algorithmic Details

10
Forced
  
Imposed
  
Not forced
 is calculated from
 and its derivatives











   
  


    
 


  
[3] F. Municchi and S. Radl, Int J Heat Mass Transfer (2017) 111:171–190
use a Taylor expansion to
construct a system of
equations to calculate
 and its derivatives
Algorithmic Details
How to calculate the forcing terms near the surface?

11
Verification - Forced convection around a sphere
Excellent agreement with
existing correlations
Weak mesh dependence
Accurate even on coarse grids
Algorithmic Details
[3] F. Municchi and S. Radl, Int J Heat Mass Transfer (2017) 111:171–190
  
Boundary operator

12
Application: Bi-
Disperse Flows
Part II

13
•One cannot simply re-scale the fluid-
particle interaction force (with 1-
f
p) to
extract the drag force in bi- (and poly)
disperse suspensions
•Fortunately, this can be “repaired” [3]
Bi-Disperse Systems: Mean Drag Coefficient
[3] F. Municchi and S. Radl, Int J Heat Mass Transfer (2017), 111:171–190.
Municchi and Radl (simple re-scaling)
versus Beetstra et al. (simple re-
scaling)
Municchi and Radl (correct pressure
gradient handling) versus Beetstra et al.
(simple re-scaling)
   

 : Total force acting on particle 

 : Drag force acting on particle 

 : Force due to mean pressure gradient

14
•Previous work [6] on per-particle drag
variation attempted to model the total
fluid-particle force (with moderate
success)
•However, when using a correctly-
defined drag coefficient: the scaled
variance for the drag coefficient is
approximately constant: simple closure
possible!
•Particle-individual deviations can be
approximated using a Log-Normal
distribution
[6] S. Kriebitzsch et al., AIChE J 59:316–324, 2012.
Bi-Disperse Systems: Mean versus Per-Particle
Drag Coefficient

15
•Same as for the drag coefficient: scaled
variance for the Nusselt number is
approximately constant: simple closure
possible!
Bi-Disperse Systems: Mean versus Per-Particle
Nusselt Number
[3] F. Municchi and S. Radl, Int J Heat Mass Transfer (2017), 111:171–190.
•Particle-individual deviations again
follow a Log-Normal distribution,
which is a bit more peaked

16
Part III
Application:
Wall-Bounded
Flows

17
Walls
Boundary conditions: velocity field
  
flow is imposed by means of a constant
pressure gradient in x-direction.
  
Boundary conditions: temperature field
artificial heat sink to sustain fluid-
particle temperature gradient
  
  
[7] Municchi et al., Int J Heat and Mass Transfer (2017), submitted
Particle-Resolved DNS to identify Modeling Needs
•Particle bed generated via bi-axial
compaction in the xy plane using
LIGGGHTS®
•Flow and temperature fields are
solved in a xy periodic domain.
Particles are isothermal.
•CFDEM®Coupling to solve the
governing equations for the
continuum phase
•Particles are represented by forcing
terms in the governing equations,
Hybrid Fictitious Domain-Immersed
Boundary method

18
Walls
Particle-Resolved DNS to identify Modeling Needs
particle center =
center of filter
filter box (native)
Lagrangian filtering: bulk particles
filter box (shrunken)
Lagrangian filtering: wall particles
center of filter
•CPPPO is also employed to draw more
“conventional” statistics (e.g., profiles in
wall-normal direction, “pancake filter”)
•Filter boxes are shrunk in the
vicinity of wall boundaries, same as
done for wall bounded single phase
turbulent flow
 
 Dimensionless
filter size
•We make use of the filtering toolbox
CPPPO to spatially average (“filter”) the
continuum phase properties around each
particle

19
Walls
Particle-Resolved DNS to identify Modeling Needs
Local Voidage and Speed
•General correlation proposed for
f
(z)
•Fluid speed fluctuates strongly, but with small wavelength  we
expect a filter-size independent near-wall correction

20
Walls
Particle-Resolved DNS to identify Modeling Needs
Local Temperature
•Fluid temperature shows a “boundary layer” behavior due to “near-wall convection”
•Significant and systematic decrease of temperature when approaching the
(adiabatic!) wall  we expect a filter-size dependent near-wall correction
Re = 100 Re = 400

21
Walls
Particle-Resolved DNS to identify Modeling Needs
Local Drag Correction and Nusselt Number
•<
f
p> = 0.4: substantial negative drag correction for “2nd layer” particles
•For the Nusselt number, the situation is more complex (due to temperature
profile!), and even higher (mixed) heat flux corrections are necessary
Force Nusselt

22
Conclusions

23
Conclusions
Adopting the boundary operator concept allows one to
implement flexible BCs. Order of boundary treatment
depends on number of reconstruction points
Mean drag coefficient and Nusselt number in bi-disperse
systems: worth to recheck existing closures
Closures for drag and heat/mass transfer are still poor on a
per-particle level. Particle (thermal) inertia “irons out” this
problem.
A first set of near wall corrections ready to use! …but there
are still many improvements necessary near walls (e.g., wall-
fluid heat transfer rates, polydispersity)

24

25
F. Municchi, S. Radl
Graz University of Technology
Arbitrary Order Boundary Reconstruction
Algorithm for Robin Boundary Conditions in
Particle-Resolved
Direct Numerical
Simulations

26
Acknowledgement and Disclaimer
Parts of the “CPPPO” code were developed in the frame of the “NanoSim” project funded by the
European Commission through FP7 Grant agreement no. 604656.
http://www.sintef.no/projectweb/nanosim/
©2017 by TU Graz, DCS Computing GmbH, Princeton University, and NTNU Trondheim. All rights
reserved. No part of the material protected by this copyright notice may be reproduced or utilized in any
form or by any means, electronically or mechanically, including photocopying, recording or by any
information storage and retrieval system without written permission from the author.
LIGGGHTS® is a registered trade mark of DCS Computing GmbH, the producer of the LIGGGHTS®
software. CFDEM® is a registered trade mark of DCS Computing GmbH, the producer of the
CFDEM®coupling software.

27
[5] F. Municchi and S. Radl, Int J Heat Mass Transfer (2017) 111:171–190
Saturation
z
•For small Re and high
f
p  fluid phase is quickly
saturated with the transferred quantity (i.e., small
zsat)
•Fluid field quickly relaxes to equilibrium value
provided at particle surface
•In a meso-scale simulation, Nu would NOT matter!