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Page 1
Generalized quasi-Newtonian approach for modeling and simulating complex flows
A. Ouazzi, A. Afaq, N. Begum, H. Damanik, A. Fatima, S. Turek Institute of Applied Mathematics, LS III,
TU Dortmund University, Dortmund, Germany
A. Ouazzi | Generalized quasi-Newtonian approach for complex flows
4th Indo-German Workshop on Advances in Materials, Reactions and Separation Processes
February 23-26, 2020 Harnack-Haus,
Conference Venue of the Max Planck Society, Berlin, Germany . Max Planck Institute for Dynamics of Complex Technical System Magdeburg
Page 2 Page 2
• The powder can transit from the quasi-static to the intermediate regime as the shearing rate is increased
Behavior of dense granular material
Shear and pressure dependent viscosity
Axial flow device
Normal Sterss Sensor
Rotating Cylinder
Stationary outer wall
Axial flow experiment in the Couette device: spherical glass beads, 0.1 mm in diameter. (by Gabriel Tardos Group, CCNY)
A. Ouazzi | Generalized quasi-Newtonian approach for complex flows
Page 3 Page 3
Viscoplastic flow
Viscoplastic Lubricate Flow (Yield stress fluids)
Ø Dependent on the stress field Ø Constitutive model is dependent on different flow regimes Ø Non-smooth change in the constitutive relations
Model preserving the sharp changes of the constitutive equations w.r.t. flow regimes
A. Ouazzi | Generalized quasi-Newtonian approach for complex flows
Page 4 Page 4
Thixotropy
Thixotropy concept Ø Based on viscosity Ø Flow induced by time-dependent decrease of viscosity Ø The phenomena is reversible
• Aging / Build-up
Ø At rest or under slow flow: fluid ages Increases of the viscosity in time
• Rejuvenation / Breakdown Ø “Faster” flow: fluid rejuvenates Decreases of viscosity with acceleration of the flow
Investigation of solid/liquid and liquid/solid transitions with non constant yield stress
A. Ouazzi | Generalized quasi-Newtonian approach for complex flows
Page 5 Page 5
Non-Newtonian phenomena • Effects due to normal stresses • Effects due to elongational viscosity • The drag reduction phenomenon
Differential models
A. Ouazzi | Generalized quasi-Newtonian approach for complex flows
Page 6 Page 6
Realization in FeatFlow
Non-Newtonian flow module: • generalized Newtonian model
(Power-law, Carreau,...) • viscoelastic differential model
(Giesekus, FENE, Oldroyd,...)
Multiphase flow module (resolved interfaces): • l/l – interface capturing (Level Set) • s/l – interface tracking (FBM) • s/l/l – combination of l/l and s/l
Numerical features: • Higher order FEM in space & (semi-) Implicit FD/FEM in time • Semi-(un)structured meshes with dynamic adaptive grid deformation • Fictitious Boundary (FBM) methods • Newton-Multigrid-type solvers
Engineering aspects: • Geometrical design • Modulation strategy • Optimization
Here: FEM-based tools for the accurate simulation of (multiphase) flow problems, particularly with complex rheology
HPC features: • Moderately parallel • GPU computing • Open source Hardware
-oriented Numerics
A. Ouazzi | Generalized quasi-Newtonian approach for complex flows
Page 7 Page 7
Governing equations • Generalized Navier-Stokes equations
Ø Viscous stress
Ø Elastic stress
A. Ouazzi | Generalized quasi-Newtonian approach for complex flows
⇢
✓@
@t+ u ·r
◆u�r · � +rp = ⇢f,
r · u = 0,
� = �s + �p,
�p +We�a�p
�t= 2⌘pD(u).
�s = 2⌘s(DII, p)D(u), DII = tr⇣D(u)2
⌘.
Page 8 Page 8
Quasi-Newtonian models • Viscous stress
Ø Power law model Ø Powder flow in the quasi-static and intermediate regimes
A. Ouazzi | Generalized quasi-Newtonian approach for complex flows
⌘s(z) = ⌘0zr� 1
2 (⌘0 > 0, r > 1)
�s = 2⌘s(DII, p)D(u), DII = tr⇣D(u)2
⌘
8<
:⌘s(z, p) =
p2p
⇣sin�z�
12+ cos�zr�
12
⌘if z 6= 0, r > 1
||�s|| p2p sin� else
(� : the angle of internal friction)
Page 9 Page 9
Quasi-Newtonian models Ø Yield stress flow (Bingham Model)
Ø Thixotropic model
Ø Structure parameter equation
A. Ouazzi | Generalized quasi-Newtonian approach for complex flows
@�
@t+ u ·r� = a(1� �)� b�z
12
(a, b are structure parameters)
(⌘s(z,�) = ⌘0 + ⌧0z
� 12 if z 6= 0
||�s|| ⌧0 else
(⌧0 : yield stress)
(⌘s(z,�) = ⌘(�) + ⌧(�)z�
12 if z 6= 0
||�s|| ⌧(�) else
(� : structure parameter)
Page 10 Page 10
Constitutive models • Elastic stress
Ø Upper/Lower convective derivative
ga(�,⇥u) =1� a
2��⇥u + (⇥u)T�
⇥
� 1 + a
2�⇥u � + � (⇥u)T
⇥(a = ±1)
�a⇥
�t=
�⇤
⇤t+ u ·�
⇥⇥ + ga(⇥,⇥u)
A. Ouazzi | Generalized quasi-Newtonian approach for complex flows
�p +We�a�p
�t= 2⌘pD(u).
Page 11 Page 11
Constitutive models • Generalized differential constitutive model
Ø Oldroyd
Ø Giesekus
Ø Phan-Thien and Tanner
Ø White and Metzner
⇤ + We�a⇤
�t+ G(⇤, D) + H(⇤) = 2⇥pD(u)
G = 0, H = 0
G = 0, H = � tr(⇥2)
H = [exp (� tr(⇥))� 1]⇥
G = � (2 D : D)1/2 , H = 0
A. Ouazzi | Generalized quasi-Newtonian approach for complex flows
Page 12 Page 12
Stokes problem
• Two-field formulation
• Three-field formulation
(u, p)
(�, u, p)
8>>>>><
>>>>>:
� � 2⌘D(u) = 0 in ⌦
�r ·✓2⌘(1� ↵)D(u) + ↵�
◆+rp = 0 in ⌦
r · u = 0 in ⌦
u = gD on �D
8>>><
>>>:
�r ·✓2⌘D(u)
◆+rp = 0 in ⌦
r · u = 0 in ⌦
u = gD on �D
A. Ouazzi | Generalized quasi-Newtonian approach for complex flows
Page 13 Page 13
Stokes problem • Two-field formulation
Ø Set
Ø Find s.t.
Ø Compatibily constraints
(u, p)
(u, p) 2 V⇥QDK(u, p), (v, q)
E=
DL, (v, q)
E, 8(v, q) 2 V⇥Q
supv2V
⌦Bv, q
↵
||v||V� � ||q||Q/KerBT , 8q 2 Q
V :=⇥H1
0 (⌦)⇤2
,Q := L20(⌦)
A. Ouazzi | Generalized quasi-Newtonian approach for complex flows
K =
Au BT
B 0
!
Page 14 Page 14
Stokes problem • Three-field formulation
Ø Set
Ø Find s.t.
Ø Compatibily constraints
(�, u, p)
(�, u, p) 2 T⇥ V⇥QDK(�, u, p), (⌧, v, q)
E=
DL, (⌧, v, q)
E, 8(⌧, v, q) 2 T⇥ V⇥Q
T :=�L2(⌦)
�4sym
,V :=⇥H1
0 (⌦)⇤2
,Q := L20(⌦)
A. Ouazzi | Generalized quasi-Newtonian approach for complex flows
supv2V
⌦Bv, q
↵
||v||V� � ||q||Q/KerBT , 8q 2 Q
supv2V
⌦Cv, ⌧
↵
||v||V� � ||⌧ ||T/KerCT , 8⌧ 2 T
K =
0
B@
A� C 0
CT Au BT
0 B 0
1
CA
Page 15 Page 15
Approximated Stokes problem
• Conforming approximations
• Non-conforming approximation
• Discrete inf-sup condition
A. Ouazzi | Generalized quasi-Newtonian approach for complex flows
supvh2Vh
⌦Bhvh, qh
↵
||vh||V� �h ||qh||Q/KerBT
h
, 8qh 2 Qh
supvh2Vh
⌦Chvh, ⌧h
↵
||vh||V� �h ||⌧h||T/KerCT
h
, 8⌧h 2 Th
Th ⇢ T, Vh 6⇢ V&Vh ⇢ V, Qh ⇢ QA�h = A�, Auh 6= Au, Bh 6= B, Ch 6= C
Th ⇢ T, Vh ⇢ V = V, Qh ⇢ QA�h = A�, Auh = Au, Bh = B, Ch = C
Page 16 Page 16
Robust non-/conforming FEM Qr/P
discr�1 , r � 2
Qr/Pdiscr�1 , r � 2 (u, p)
(�, u, p)Qr/Qr/Pdiscr�1 , r � 2
Ju(uh, vh) = �uX
e2Eh
2⌘↵h
Z
e[ruh] : [rvh] d⌦
The family of non-/conforming FEM and the family of nonconforming FEM for
Ø Inf-sup stable Ø Arbitrary order with optimal convergence order Ø Discontinuous pressure
Ø Good for the solver Ø Element-wise mass conservation
The family of conforming FEM for with stabilization
Ø Both Inf-sup conditions are satisfied Ø Highly consistent and symmetric stabilization, penelazing any
spurious current, enhancing the preconditioner, which improve accuracy and efficiency
Ø None tensorial FEM approximation for the tensorial field Ø Robust solver w.r.t. the monolitic approach Ø Efficient solver w.r.t. multigird solver
A. Ouazzi | Generalized quasi-Newtonian approach for complex flows
Page 17 Page 17
Monolitic-multigrid linear solver
• Standard geometric multigrid solver
• Full and restriction and prolongation
• Local Multilevel Pressure Schur Complement via Vanka-like smoother
Coupled Monolithic Multigrid Solver !
0
B@
�l+1
ul+1
pl+1
1
CA =
0
B@
�l
ul
pl
1
CA+ !lX
T2Th
0
BB@⇣Kh + Ju
⌘
|T
1
CCA
�1 0
B@R�l
Rul
Rpl
1
CA
|T
Qr P discr�1
A. Ouazzi | Generalized quasi-Newtonian approach for complex flows
Page 18 Page 18
Flow around cylinder Benchmark tests (by Aaqib Afaq)
Ø Two field formulation versus three field formulation
Ø Consistency of the stabilization for three field formulation
Monolithic Multigrid Solver
Three fields Stokes vs Stokes solver in primitive variables
Level Lift Drag NL/LL Lift1 Drag NL/LL
1 0.009498 5.5550 7/2 0.009498 5.5550 9/22 0.010601 5.5722 7/2 0.010601 5.5722 9/23 0.010616 5.5776 7/2 0.010616 5.5776 9/14 0.010618 5.5791 7/1 0.010618 5.5791 8/15 0.010619 5.5794 6/2
Three field solver performance e�cient as primitive Stokes solver
Check robustness and consistency!
1Damanik. H ”FEM Simulation of Non-isothermal Viscoelastic fluids”, PhD ThesisMotivation Governing Equations Variational Formulation Finite Element Approximation Numerical Results Summary
EO-FEM : Consistency
Consistency for case – = 0
No stabilization With stabilizationLevel – Lift Drag NL/LL Lift Drag NL/LL
2 0 0.010601 5.5722 7/2 0.010702 5.5674 7/23 0 0.010616 5.5776 7/2 0.010619 5.5757 7/24 0 0.010618 5.5791 7/1 0.010617 5.5782 7/25 0 0.010619 5.5794 6/2 0.010618 5.5790 6/3
Edge Oriented FEM is consistent
Side e�ect neither on solution nor on the solver
Motivation Summary
A. Ouazzi | Generalized quasi-Newtonian approach for complex flows
Monolithic Multigrid Solver
Three fields Stokes vs Stokes solver in primitive variables
Three-field Two-field1
Level Lift Drag NL/LL Lift Drag NL/LL
1 0.009498 5.5550 7/2 0.009498 5.5550 9/22 0.010601 5.5722 7/2 0.010601 5.5722 9/23 0.010616 5.5776 7/2 0.010616 5.5776 9/14 0.010618 5.5791 7/1 0.010618 5.5791 8/1
Three field solver performance e�cient as primitive Stokes solver
Check robustness and consistency!
1Damanik. H ”FEM Simulation of Non-isothermal Viscoelastic fluids”, PhD ThesisMotivation Summary
Monolitic-multigrid linear solver
Page 19 Page 19
Flow around cylinder Benchmark tests (by Aaqib Afaq)
Ø Robustness and efficiency of the stabilization for three field formulation
without any viscous contribution
Accurate, robust and efficient monolitic-multigrid Stokes solver in two-field and three-field formulations !
EO-FEM :Robustness
Two extreme cases–=0 æ viscous contribution
–=1 æ no viscous contribution
No stabilization With stabilizationLevel – Lift Drag NL/LL Lift Drag NL/LL
2 0 0.010601 5.5722 7/2 0.010702 5.5674 7/23 0 0.010616 5.5776 7/2 0.010619 5.5757 7/24 0 0.010618 5.5791 7/1 0.010617 5.5782 7/25 0 0.010619 5.5794 6/2 0.010618 5.5790 6/32 1 ———– —— — 0.010588 5.5520 7/23 1 ———– —— — 0.010600 5.5698 7/24 1 ———– —— — 0.010612 5.5756 7/25 1 ———– —— — 0.010617 5.5778 7/3
Robustness w.r.t. problem!
Consistent and grid independent solverMotivation Summary
A. Ouazzi | Generalized quasi-Newtonian approach for complex flows
Monolitic-multigrid linear solver
Page 20 Page 20
Multiphase flow problem
Incompressible N-S Equation)
Ø Viscous stress
Ø Interface boundary conditions
A. Ouazzi | Generalized quasi-Newtonian approach for complex flows
⇢(�)
✓@
@u+ u ·r
◆u�r · �s +rp = 0
�s = 2⌘s(�)D(u)
[u]|� = 0
�[pI + �s]|� · n = �n
Page 21 Page 21
Multiphase flow problem • New extra stress for multiphase flow
• Full set of equations for multiphase flow
A. Ouazzi | Generalized quasi-Newtonian approach for complex flows
8>>>>>>>>>>>>>><
>>>>>>>>>>>>>>:
⇢( )
✓@u
@t+ u ·ru
◆�r · ⌧ +rp = 0
r · u = 0
⌧ � 2⌘s( )D(u) + �
✓r ⌦r
|r |
◆= 0
@
@t+ u ·r +r ·
✓�nc (1� )
r |r |
◆
�r ·✓�nd
✓r · r
|r |
◆r |r |
◆= 0
�m = ��✓r ⌦r
|r |
◆
Page 22 Page 22
Monolitic approach
Oscillating bubble (by Aaqib Afaq)
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0.22
0.24
0.26
100 101 102 103 104
Time
rxrySurface Area
A. Ouazzi | Generalized quasi-Newtonian approach for complex flows
Page 23 Page 23
Generalized Newton’s method Robust nonlinear solver based on Newton’s method following a specefic path of convergence using the residual’s convergence
Ø Robust w.r.t. starting guesses
Ø Dealing with Jacobian’s singularities using generalized deriviatives or approximated one
Ø Full benefit from the quadratic convergence’s region of classical Newton’s method
A. Ouazzi | Generalized quasi-Newtonian approach for complex flows
Page 24 Page 24
Newton’s method Let , , or and be the continuous or the discrete corresponding system’s residum.
Ø Update of the nonlinear iteration with the correction i.e.
Ø The linearization of the residual provides
Ø The Newton’s method assuming invertible Jacobian
(�, u, p,')U = (u, p) (�, u, p) RU (U)
�UUN = U + �U
UN = U � J�1⇣U⌘· RU
⇣U⌘
A. Ouazzi | Generalized quasi-Newtonian approach for complex flows
RU
⇣UN
⌘=RU
⇣U + �U
⌘
=RU
⇣U⌘+ J
⇣U⌘· �U
Page 25 Page 25
Generalized Newton’s method Jacobian calculations
Ø Exact G-Newton based on a priori study of Jacobian’s properties and decompositions
Ø Inexact G-Newton based on the residum‘s convergence
J⇣U⌘=
@RU
�U�
@U
!
A. Ouazzi | Generalized quasi-Newtonian approach for complex flows
J⇣U⌘=
@RU
�U�
@U
!+ �
@RU
�U�
@U
!
@ ¯RU
�U�
@U
!
ij
⇡
¯RU
�U + ✏+ej
�� ¯RU
�U � ✏�ei
�
( ✏+ + ✏�)
!,
¯R = R, ˆR, or
˜R .
Page 26 Page 26
Three-field viscoplastic application • Viscoplatic constitutive law
Ø Bingham constitutive law
Ø New extra stress for viscoplastic flow s.t. • Three-field viscoplastic set of equations
A. Ouazzi | Generalized quasi-Newtonian approach for complex flows
8>>><
>>>:
||D(u)||�Y0�D(u) = 0 in ⌦
�r ·✓2⌘D(u) + ⌧0�Y0
◆+rp = 0 in ⌦
r · u = 0 in ⌦
||D(u)||�Y0= D(u)
8<
:� = 2⌘D(u) + ⌧0
D(u)
||D(u)|| if ||D(u)|| 6= 0
||�|| ⌧0 if ||D(u)|| = 0
�Y0
Page 27 Page 27
Generalized Newton’s method
Dynamic path versus static one w.r.t. number of iterations, and the corresponding convergence of the residium (by Arooj Fatima)
10-5
10-4
10-3
10-2
10-1
100
101
0 2 4 6 8 10
spdp
10-8
10-6
10-4
10-2
100
102
104
106
108
1 10 100
sp1sp2sp3sp4sp5dp
A. Ouazzi | Generalized quasi-Newtonian approach for complex flows
Page 28 Page 28
Lid driven cavity benchmark
Unyielded zone for two different yield stresses, , and (by Arooj Fatima)
⌧0 = 2 ⌧0 = 5
A. Ouazzi | Generalized quasi-Newtonian approach for complex flows
Page 29 Page 29
Constitutive models • Generalized differential constitutive model
Ø Oldroyd
Ø Giesekus
Ø Phan-Thien and Tanner
Ø White and Metzner
⇤ + We�a⇤
�t+ G(⇤, D) + H(⇤) = 2⇥pD(u)
G = 0, H = 0
G = 0, H = � tr(⇥2)
H = [exp (� tr(⇥))� 1]⇥
G = � (2 D : D)1/2 , H = 0
A. Ouazzi | Generalized quasi-Newtonian approach for complex flows
Page 30 Page 30
Viscoelastic benchmark • Planar flow around cylinder Oldroyd-B (by Hogenrich Damanik)
A. Ouazzi | Generalized quasi-Newtonian approach for complex flows
Page 31 Page 31
The numerical method do not introduce errors !
• Experimental and numerical results for dry, frictional powder flows in the quasi-static and intermediate regimes
Quasi-Newtonian model for powder flow
A. Ouazzi | Generalized quasi-Newtonian approach for complex flows
Page 32 Page 32
Quasi-Newtonian thixotropic model Ø Viscosity model for thixotropic flow i.e. extended viscosity defined
on all domaine s.t.
Ø Structure equation
Ø Full set of equations
A. Ouazzi | Generalized quasi-Newtonian approach for complex flows
@�
@t+ u ·r� = a(1� �)� b�z
12
(a, b are structure parameters)
8>>>><
>>>>:
✓@
@t+ u ·r
◆u�r ·
⇣2⌘s(||D(u)||,�)D(u)
⌘+rp = 0 in ⌦
r · u = 0 in ⌦
@�
@t+ u ·r�� a(1� �) + b�||D(u)|| = 0 in ⌦
(⌘s(||D(u)||,�) = ⌘(�) + ⌧(�)||D(u)||�
12 if ||D(u)|| 6= 0
||�s|| ⌧(�) else
(� : structure parameter)
Page 33 Page 33
Thixotropic flow Shear rate in a couette w.r.t. breakdown parameter (by Naheed Begum)
A. Ouazzi | Generalized quasi-Newtonian approach for complex flows
Page 34 Page 34
Thixotropic flow
A. Ouazzi | Generalized quasi-Newtonian approach for complex flows
Structure parameter in a couette w.r.t. breakdown (by Naheed Begum)
Page 35 Page 35
ü shear history effect
ü time history effect
ü Hysteresis
ü stress overhoots
Thixotropy flow
A quasi-Newtonian model for thixotropic phenomena via a time and shear dependent viscosity
A. Ouazzi | Generalized quasi-Newtonian approach for complex flows
Page 36 Page 36
Generalized quasi-Newtonian approach
Ø Include the non-Newtonian stress or any extra stress in diffusion operator
Ø Get rid of a tensorial field
Ø Less constraints for the choices of FE approximation
Ø Robust and efficient numerical algorithms
Ø Simple evolution equations !
Ø Proof of the concept and validation
A. Ouazzi | Generalized quasi-Newtonian approach for complex flows
Page 37 Page 37
Laplacian operator • Divergence form
• Weak form
Benefit of the weak form representation !
Lu =nX
i,j=1
@
@xi
✓aij u
@
@xi
◆
LW u =NX
i,j=1
Aij :⇣r · ej ⌦r · ei
⌘u
A. Ouazzi | Generalized quasi-Newtonian approach for complex flows
Page 38 Page 38
Stokes problem Weak form representation 2D
• Gradient formulation
• Deformation formulation
Different deriviatives combinations accessibilities !
A11 =
"1 0
0 0
#, A12 =
"0 0
0 0
#
A21 =
"0 0
0 0
#, A22 =
"0 0
0 1
#
A11 =
2
41 0
01
2
3
5 , A12 =
2
4 01
20 0
3
5
A21 =
2
40 0
1
20
3
5 , A22 =
2
41
20
0 1
3
5
LW u =2X
k,l=1
NX
i,j=1
[Akl]ij :⇣r · ej ⌦r · ei
⌘ulj , k = 1, 2
A. Ouazzi | Generalized quasi-Newtonian approach for complex flows
Page 39 Page 39
Generalized quasi-Newtonian approach Weak form representation 2D
• Generalized formulation I
More deriviatives combinations accessibilities are allowed !
A11 =
2
64a11
1
2a21
1
2a12
1
4(a11 + a22)
3
75 , A12 =
2
64
1
2a12
1
4(a11 + a22)
01
2a12
3
75
A21 =
2
64
1
2a21 0
1
4(a11 + a22)
1
2a21
3
75 , A22 =
2
64
1
4(a11 + a22)
1
2a21
1
2a12 a22
3
75
LW u =2X
k,l=1
NX
i,j=1
[Akl]ij :⇣r · ej ⌦r · ei
⌘ulj , k = 1, 2
A. Ouazzi | Generalized quasi-Newtonian approach for complex flows
Page 40 Page 40
Viscoelastic benchmark • Planar flow around cylinder Oldroyd-B (by Hogenrich Damanik)
115
120
125
130
135
140
0 0.5 1 1.5 2
Dra
g co
effic
ient
We number
Drag coefficient planar flow around cylinder
H. Damanik, PhD thesis TU Dortmund Generalized quasi-Newtonian approach
Genalized quasi-Newtonian approach for non-Newtonian problem i.e. Oldroyd-B !
A. Ouazzi | Generalized quasi-Newtonian approach for complex flows
Page 41 Page 41
Summary
New generalized quasi-Newtonian approach for modeling and simulating complex flows is introduced and validated. Based on new numerical and algorithmic tools using
ü Monolithic FEM two-field and three-field Stokes solver
ü Generalized Newton’s method w.r.t. singularities with global convergent property
ü Edge Oriented stabilization (EO-FEM)
ü Fast Multigrid Solver with local MPSC smoother Extensively tested from numerical and physical perspectives via the simulations of different flow problems in different formulations to motivate the newly introduced generalized quasi- Newtonian approach.
A. Ouazzi | Generalized quasi-Newtonian approach for complex flows