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F. Municchi, S. Radl1

1Graz University of Technology

Arbitrary Order Boundary Reconstruction

Algorithm for Robin Boundary Conditions in

Particle-Resolved

Direct Numerical

Simulations

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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

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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®

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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

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Overview

Part I The Method

Part II Application to Bi-Disperse Flows

Part III Application to Wall-Bounded Flows

5 + 5 + 5 = 15 mins

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The Method

Part I

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•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

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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

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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

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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?

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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

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Application: Bi-

Disperse Flows

Part II

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•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

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•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

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•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

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Part III

Application:

Wall-Bounded

Flows

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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

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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

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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

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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

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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

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Conclusions

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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)

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F. Municchi, S. Radl

Graz University of Technology

Arbitrary Order Boundary Reconstruction

Algorithm for Robin Boundary Conditions in

Particle-Resolved

Direct Numerical

Simulations

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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.

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[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!