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J. Non-Newtonian Fluid Mech. 150 (2008) 32–42
Using Newton-GMRES for viscoelastic flow time-steppers
Zubair Anwar, Robert C. Armstrong ∗77 Massachusetts Avenue, Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Received 20 March 2007; received in revised form 26 September 2007; accepted 7 October 2007
bstract
Kinetic theory models exhibit dynamics that depend on a few low-order moments of the underlying conformational distribution function. Thisependence is exhibited in a compact spectrum of eigenvalues for the Jacobian matrix associated with the dynamical system. We take advantage
f this spectrum of eigenvalues through Newton-GMRES iterations to enable dynamic viscoelastic simulators (time-steppers) to obtain stationarytates and perform stability/bifurcation analysis. Results are presented for three example problems: (1) the equilibrium behavior of the Doi modelith the Onsager excluded volume potential, (2) pressure-driven flow of non-interacting rigid dumbbells in a planar channel, and (3) pressure-drivenow of non-interacting rigid dumbbells through a planar channel with a linear array of equally spaced cylinders.2007 Elsevier B.V. All rights reserved.; New
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eywords: Micro–macro simulations; Computational rheology; Time-steppers
. Introduction
The majority of publications in computational rheology haveeen based on a macroscopic approach that utilizes constitutivequations inspired by kinetic theory. This approach invariablyequires the use of closure approximations in the derivation ofhe constitutive models, which can have a significant qualitativempact on predictions of such simulations. However, the recent,omplementary approach of hybrid simulations circumvents theeed for a closure approximation by directly coupling the macro-copic equations of change with a microscopic kinetic theoryodel. In doing so, the polymer contribution to the stress tensor
s evaluated at each material point by solving the associatedokker–Planck equation or equivalent stochastic differentialquation and evaluating appropriate averages of the distributionunction. To date, available stochastic and Fokker–Planck hybridechniques have been implemented for kinetic theory models thatave relatively few configurational degrees of freedom [1,2]. Inddition, while previous studies [3,4] of simulating complexows with hybrid methods have employed dynamic simula-
ors to converge to steady states and perform linear stability
nalysis, there is no simple method for performing bifurcationnalysis without reverting to long-time simulations. This is pri-arily due to the unavailability of closed equations to which∗ Corresponding author.E-mail address: [email protected] (R.C. Armstrong).
oiflasdi
377-0257/$ – see front matter © 2007 Elsevier B.V. All rights reserved.oi:10.1016/j.jnnfm.2007.10.007
ton-GMRES iteration; Fokker–Planck equation; Bifurcation analysis
xisting numerical techniques for bifurcation analysis may bepplied.
In this paper, motivated by the work of Kevrekidis et al. [5]nd the Recursive Projection Method of Shroff and Keller [6], weropose a computational approach that can converge a dynamicimulation to steady states and perform stability analysis with-ut the need for closure approximations. We will first present anverview of the approach and argue why coarse-grained modelsf kinetic theory are appropriate for its application. This wille followed by three examples: (1) the equilibrium behavior ofhe Doi model with the Onsager excluded volume potential, (2)ressure-driven flow of a dilute solution of non-interacting rigidumbbells in a planar channel, and (3) pressure-driven flow ofon-interacting rigid dumbbells through a planar channel withlinear array of equally spaced cylinders. In the first examplee demonstrate the existence of a compact spectrum of eigen-alues for the Jacobian matrix associated with a well studiedodel from kinetic theory of polymeric liquids. More impor-
antly, we illustrate the ability to obtain stationary states anderform stability/bifurcation analysis of the Doi model with theore realistic Onsager excluded volume potential. With the sec-
nd set of examples we show that the method may also be usedn the context of a hybrid simulation of a non-homogeneousow without any significant modification to the simulation
lgorithm. Although in this paper we focus on converging totable stationary states in order to facilitate comparison withynamic simulations, the results are very encouraging both forncorporation of higher numbers of configurational degrees of
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Z. Anwar, R.C. Armstrong / J. Non-N
reedom for the kinetic theory models and for performing sta-ility/bifurcation analyses with hybrid simulations.
. Time-steppers and Newton-GMRES
Hybrid methods for simulating complex viscoelastic flowsirectly employ models from kinetic theory for capturing theolymeric contribution to the stress tensor without obtaining alosed form expression for the stress tensor. This approach leadso a dynamical system or “time-stepper” of the form
dx
dt= f (x; μ), (1)
here x represents the state of the system and μ is a parameter ofhe problem. For our work, where closed form constitutive mod-ls cannot be written, the state of the system is a set of momentsf the underlying conformational distribution function. Theseould be obtained either from the conformational distributionunction or from ensemble averaging of polymer conformationsomputed by using stochastic models. Given the state of the sys-em at a moment in time, the time-stepper allows determinationf the state at a later moment. For the hybrid simulations of inter-st here, a closed form expression for f (x, μ) is not available.he action of the time-stepper can only be determined with ablack-box” simulation.
.1. Steady state solutions
We propose recasting the time-stepper as a fixed point solveror the nonlinear system
− �T (x; μ) ≡ F (x; μ) = 0, F : RN → RN, (2)
here F is assumed to be continuously differentiable every-here in RN and
T (x; μ) = x +∫ T
0f (x(t′); μ) dt′, (3)
s the result of integration of Eq. (1) for time T with initial condi-ion x. The key idea is to be able to evaluate F (x; μ) through callso the time-stepper rather than a closed expression for f (x; μ).nce we have formulated the system in Eq. (2), we can then
pply Newton’s method to converge to steady states of the sys-em. Doing so requires, at the kth step, the solution of the linearewton equation for step sk
x(xk)sk = −F (xk), (4)
here xk is the current approximate solution and Fx is the Jaco-ian of the system. Since the system in Eq. (4) is inevitably largeue to discretization of a PDE or a model with large number ofegrees of freedom, we revert to methods of large scale computa-ional linear algebra. To this end we employ GMRES, which is anterative linear solver for determining an approximate solution ofq. (4). GMRES belongs to the general class of Krylov subspace
ethods that approximately solve linear systems of the formy = b by minimizing the norm of the residual r = b − Ay.ne main advantage of such methods is that they are alwaysmplemented as matrix-free methods, since only matrix-vector
mop
t
ian Fluid Mech. 150 (2008) 32–42 33
roducts, rather than details of the matrix itself (in our case theacobian Fx) are needed to implement the method. This is inontrast to direct methods, which require that the computation,torage, and cost of factorizing the Jacobian not be excessive.
More importantly, such methods perform best if the eigen-alues of Fx are in a few tight clusters [7,8]. This is observed inhe context of hybrid simulations of complex viscoelastic flows,n which time-steppers evolve a microscopic description of theystem, whereas the interest of the computational rheologist liesith prediction of macroscopic properties (such as stress) thatepend on certain low order moments of the microscale model.n fact, closed constitutive models are always written in terms offew moments of the underlying microscale model by assuming
hat the remaining higher-order moments quickly become func-ionals of a few, lower-order, slow “master” moments [5]. Thisccurs over timescales that are short compared to the macro-copic observation timescales. It is this separation of timescaleshat leads to the few tight clusters of the eigenvalues of Fx.
.2. Stability
The stability of a steady state x∗ of Eq. (1) can be determinedrom the n eigenvalues (σi, i = 1, . . . , N) of the unavailableacobian fx(x∗; μ). Since we compute the steady state fromq. (2), we note that the Jacobian Fx(x∗; μ) also has N eigen-alues that can be expressed as νi = 1 − eσiT . For a stable steadytate, the σi must all lie in the left half of the complex plane. Theerm eσiT in the expression for νi transforms these stable eigen-alues to within a unit disc centered at the origin. Hence, thetability criterion can be expressed as |1 − νi| < 1. It should beoted that the neutrally stable eigenvalue of 0 for the dynamicalystem corresponds to an eigenvalue of 1 for the system in Eq.4).
.3. Continuation
The task of performing continuation also fits into this frame-ork in the form of an augmented system that contains the
ppropriate continuation algorithm. For example, for pseudo-rclength continuation we add an additional arclength parameterand write the augmented system as
G(x(s), μ(s), s)
=⎛⎝ F (x(s), μ(s))
dx
ds· (x − xj) + dμ
ds(μ − μj) − (s − sj)
⎞⎠ , (5)
here (xj, μj) represents a solution previously calculated dur-ng continuation at arclength sj . This formulation does notequire any modification to the Newton-GMRES method. Moremportantly, assuming that the long-term dynamics of the time-tepper are dictated by p slow moments, where p � N, it can behown that the GMRES iteration for Eq. (4) will converge in at
ost p + 1 iterations, whereas for continuation the dimensionf the slow subspace of moments increases from p to at most+ 2. For a detailed convergence analysis the reader is referred
o the work of Kelly et al. [9].
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4 Z. Anwar, R.C. Armstrong / J. Non-N
To illustrate the use of the method described above, we firstresent the Doi model from kinetic theory of polymeric liquids inection 3 to demonstrate the existence of a compact spectrum ofigenvalues for the linearized system in Eq. (4). We then exploithis property not only to converge to both stable and unstabletationary states but also to perform continuation to constructhe bifurcation diagram without invoking any closure approxi-
ations. This is followed in Section 4 by a set of examples toemonstrate that we can also employ Newton-GMRES to obtaintationary states of a hybrid simulation in which a linear diffu-ion equation for the conformational distribution function ofigid rods is coupled with the macroscopic conservation of massnd momentum equations in two spatially non-homogeneousows.
. Equilibrium bifurcation diagram of the Doi modelith the Onsager excluded volume potential
The Doi diffusion equation [10] coupled with suitable repre-entations of the interbody excluded volume potential has beensed to study problems involving rigid, rodlike nematic poly-ers (e.g., liquid crystalline polymers). In the absence of flow,
he orientation distribution function f (u, t) for a spatially homo-eneous solution of rigid rods with infinite aspect ratio followshe diffusion equation given by
∂f
∂t= 1
6λ
∂
∂u·[
∂f
∂u+ f
kT
∂V (u)
∂u
], (6)
here u is the radial unit vector in spherical coordinates, kT ishe Boltzmann factor, λ is the rotational time constant of a rigidod, and V (u) is a mean field interaction potential. In this studye use the Onsager mean field potential
(u) = UkT
∫|u × u′|f (u′, t) du′, (7)
here U is the dimensionless potential intensity.For this work we choose to study the scalar structure
arameter S, which represents a scalar measure of the degreef order of the sample, as a function of the dimension-ess potential U. The structure parameter is defined as S =(9/2)(S · S) : S]1/3, in which the structure tensor S = 〈uu〉 −δ/3), 〈uu〉 = ∫
uuf (u) du and δ is the unit tensor.For real-valued f satisfying f (u) = f (−u) we substitute a
pherical harmonic expansion for the distribution function intoq. (6) such that
(u, t) =∞∑
l = 0l=even
m=+l∑m=−l
aml (t)Pm
l cm + bml (t)Pm
l sm, (8)
here Pml are the Legendre polynomials Pm
l (cos θ), cm =os mφ, and sm = sin mφ. Normalization of f,
∫f (u) du = 1,
ields a00 = (4π)−1 ∀t, whereas a−m
l = (−1)maml , and b−m
l =
−1)mbml for all m ≥ 0. Truncating the expansion in Eq. (8) at aertain level, M, and using the orthogonality property of spher-cal harmonics transforms Eq. (6) into a set of first-order ODEsor the spherical harmonic coefficients am
l and bml . This system
aia
ian Fluid Mech. 150 (2008) 32–42
f ODEs resembles Eq. (1) and can be recast as a “black-box”imulator of the form of Eq. (3), such that it takes a set of spher-cal harmonic coefficients and returns the evolved coefficientsfter a specified time interval. Approximate models of such aystem are often written for the evolution of the structure ten-or S or the second moment 〈uu〉 of the distribution function bynvoking various closure approximations for the fourth moment.y choosing the complete set of spherical harmonic coefficientss defining the state of the system, we capture the full distri-ution function without any closure approximation in terms ofhe second moment, which is equivalent to writing the aboveystem in terms of just the am
2 and bm2 coefficients. Moreover, as
e will show next, this system exhibits a compact spectrum ofigenvalues that suggests a closure, albeit an unknown one, thateed not be invoked when using the framework of Section 2.
Before proceeding further, we must first address the degen-racy in the problem with regards to the nematic states thatifurcate from the trivial isotropic state. The nematic state ofhe system is characterized by a director vector that correspondso the peak in the orientation distribution function. For thequilibrium problem being studied, this director is rotationallyegenerate. Therefore, we restrict the director based on the workf Gopinath et al. [11,12] by setting all l = odd and bm
l coeffi-ients to zero when initializing the time-stepper. This restrictionoes not alter the prediction of the structure parameter or thetability of the computed steady state. It only prevents the con-inuation algorithm from exploring states with the same value ofthat differ by a rotation. Even if this restriction were removed,
he time-stepper based algorithm would converge to the correctteady state. However, due to the inherent degeneracy in theroblem, the associated director vector would be non-unique.or our simulations we set M = 10, T = 0.1λ, and μ = U andse pseudo-arclength continuation to trace the solution branches.xcluding the odd and bm
l coefficients, we obtain a set of 20 real,rst order ODEs. A fourth-order Runge–Kutta scheme was used
o formulate the time-stepper with a time-step size of 0.005λ.tarting with the prolate (S > 0) steady state solution at U = 13
he curve shown in Fig. 1 was obtained via continuation. For theurpose of comparison, we also show the results obtained byopinath et al. [11] in which isolated integrations to steady stateere used. In Fig. 1, solid lines represent stable stationary solu-
ions whereas dashed lines are unstable stationary solutions. Twoolution branches (one isotropic and the other nematic) cross attranscritical bifurcation point, Uc = 10.19. This is in excellentgreement with the prediction of Uc = 32/π from linear stabilitynalysis [11]. A turning point is obtained on the nematic S > 0ranch at U = 8.87. Thus, for U < 8.87 only a stable isotropichase is predicted, whereas for 8.87 < U < 10.19, one stablesotropic solution, and two nematic solutions (one stable and thether unstable) are predicted. Both these solutions are prolate.or U > 10.19 three solutions coexist: a stable prolate nematicolution (S > 0), an unstable isotropic solution (S = 0), and annstable oblate nematic solution (S < 0).
In Table 1 we present a list of the five eigenvalues |1 − νi| thatre farthest from zero. For a parameter value such as U = 12 thats far from the turning point, we see that most of the eigenvaluesre close to zero except for three at approximately 0.1, 0.2 and
Z. Anwar, R.C. Armstrong / J. Non-Newton
Fig. 1. Equilibrium phase diagram for the Doi model with the Onsager excludedvolume potential. The nematic branches bifurcate from the critical pointUce10.19 for the Doi equation. The turning point on the S > 0 branch occurs atUe8.87. (–) Stable, (- ·-) unstable, and (◦) data from Gopinath et al. [11].
Table 1List of five eigenvalues, |1 − νi|, farthest from 0 on the prolate branch (T = 0.1λ)
U = 12 U = 8.877
0.0495 0.02020.0579 0.11760.0663 0.15990.1048 0.201100
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.5. The fact that most of the eigenvalues are clustered aroundero allows GMRES to converge to the steady state. Increasing
would improve GMRES performance as more eigenvaluespproach zero, but this would occur at the cost of longer simu-ation time for the Doi model. For the second parameter value
= 8.877, which lies near the turning point, we see that eigen-alues start leaving the cluster at zero, though they still existn distinct groups with the largest eigenvalue approaching the
nit circle. The change in stability at the turning point corre-ponds to the crossing over the boundary of the unit circle ofhis largest eigenvalue. The corresponding eigenmode in con-guration space is shown in Fig. 2.cWtv
ig. 2. (a) Distribution function and (b) contour plot for the eigenmode correspondistribution function was computed with M = 10.
ian Fluid Mech. 150 (2008) 32–42 35
. Pressure-driven flow of non-interacting rigidumbbells in a planar channel and through a planarhannel with a linear array of cylinders
To demonstrate that a hybrid simulation can be cast in theramework of Section 2 we study pressure-driven flow of ailute solution of non-interacting rigid dumbbells both in annfinitely wide planar channel and through an infinitely widelanar channel with an infinite linear array of cylinders orientederpendicular to the flow direction and equally spaced alonghe centerline in the flow direction. We solve the momentumEq. (9)) and continuity (Eq. (10)) equations for the velocitynd pressure fields in the Stokes limit, whereas the polymericontribution to the stress tensor τp (Eq. (11)) is computed fromhe moments of the orientational distribution function obtainedy solving the diffusion equation for rigid rods (Eq. (12)). Wehus use
∇ · (∇v + ∇vT ) − ∇p − ∇ · τp = 0, (9)
· v = 0, (10)
p = 1 − β
λV/L
(δ − 3〈uu〉 − 6
(λV
L
)κ : 〈uuuu〉
), (11)
∂f
∂t+v · ∇f = 1
6λ
(∂
∂u· ∂
∂u
)f − ∂
∂u· ([κ · u−κ : uuu])f,
(12)
here f = f (r, u, t), β = ηs/η0 (ratio of solvent to solu-ion zero-shear-rate viscosity), κ = ∇vT , and V and L are theharacteristic velocity and length scales. The momentum andontinuity equations are solved with the Discrete Elastic Vis-ous Split Stress-Gradient (DEVSS-G) formulation of Szady etl. [13] through introduction of a new variable G = ∇v, whereashe discontinuous Galerkin (DG) method is used to solve theiffusion equation. For the planar channel and the planar chan-el with a linear array of cylinders, the strength of the flow is
haracterized by the Deborah number, De = 3λQ/L2, and theeissenberg number, We = λ〈v〉/L, respectively, where Q ishe flow rate per unit width of the channel and 〈v〉 is the averageelocity.
ing to the most dangerous eigenvalue of 0.9967 (Table 1) at U = 8.877. The
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6 Z. Anwar, R.C. Armstrong / J. Non-N
.1. Weak form of the diffusion equation
The PDE for the distribution function for polymer conforma-ions (Eq. (12)) is first rewritten in the form
∂f
∂t+ v · ∇f = 1
6λ∇2
uf + �(u, κ)f, (13)
here
(u, κ) =[
3(κ : uu) − κ : u∂
∂u
].
e then solve this time dependent PDE by using a sphericalarmonic-Galerkin method to discretize the equation in orienta-ion space (u or θ, φ), and the discontinuous Galerkin method toiscretize the equation in physical space. The DG method usesasis functions that are discontinuous across element bound-ries. Each basis function is associated with a node within anlement, such that it has a value of unity at its own node and aalue of zero at all other nodes. Outside the element under con-ideration, the basis function is identically zero. This decoupleshe problem on every element from that on every other element,o that the problem can be solved element by element. It ismportant to note that the element-by-element solution can onlye carried out if the solution on the inflow element has alreadyeen computed. Two options are available: either the elementsre solved according to their ordering along a streamline or theroblem is solved in a time-dependent context, treating the con-ection term (v · ∇f ) explicitly. We reject the first option, sincet involves computing an element ordering for every new floweld. The second option allows us to obtain maximum benefitf the elemental decoupling that results from the DG method.
We use the following expansion for f
(r, θ, φ, t) =#of nodes∑
i
M∑n=0
n∑m=0
fmn,1i(t)ΨDG,i(r)Pm
n cm
+fmn,2i(t)ΨDG,i(r)Pm
n sm. (14)
he ΨDG,i(r) are discontinuous basis functions for physicalpace, and M denotes some level of truncation for the expansionn orientation space. We designate f k(r, u) = f (r, u, tk) as thealue of f at time t = t0 + k t. Since it is computationally effi-
ient to avoid performing a matrix inversion for each element atach time-step, all operators that vary in space or time are treatedxplicitly. Only the diffusion operator in orientation space (∇2u)s treated implicitly.
gmfs
Fig. 3. Computational domain for processor 1 with boun
ian Fluid Mech. 150 (2008) 32–42
Introducing test/weight functions ΨDG,jPlscl and ΨDG,jP
lssl
long with time discretization leads to the following weak formver an element A∫A
∫u
(1 − t
6λ∇2
u
)f kΨDG,j
{Pl
scl
Plssl
}du dA
= t
∫A
∫u
�(u, κ)f k−1ΨDG,j
{Pl
scl
Plssl
}du dA
+∫
A
∫u
f k−1ΨDG,j
{Pl
scl
Plssl
}du dA
− t
∫A
∫u
(v · ∇f k−1)ΨDG,j
{Pl
scl
Plssl
}du dA
− t
∫δA−
∫u
(n · v)[f e − f i]k−1
ΨDG,j
{Pl
scl
Plssl
}du dl.
(15)
n this formulation the weak form of the convective term,∫A
(v ·f )ΨDG dA, has been expressed as the sum of an area integral
ver the element and a line integral involving the jump in f acrosshe inflow element boundary
A
(v · ∇f )ΨDG dA =∫
A
(v · ∇f i)ΨDG dA
+∫
δA−(n · v)[f e − f i]ΨDG dl, (16)
uch that f e and f i represent the external value of f convectednto the element A from the adjacent upstream element and thenternal elemental value, respectively, and δA− represents thenflow boundary. It is this line integral that conveys informationn a streamwise direction and embodies all the communicationetween the element of interest and the ‘upstream’ elements.his inclusion of a ‘jump’ term at the inflow boundary aloneerves the same purpose as do the various types of upwindingchemes used in solution of hyperbolic PDEs.
.2. Parallelization
The DG formulation yields equations that are local to the
enerating element without dependence on neighboring ele-ents. The only exception is the need to obtain boundary datarom its neighbors. This renders the problem particularly welluited for parallelization. For pressure-driven flow solved on a
daries that communicate with adjacent processors.
Z. Anwar, R.C. Armstrong / J. Non-Newtonian Fluid Mech. 150 (2008) 32–42 37
ization
tbctsc
(
(
(
(
(
(
F
4
tataafon the distribution function to be imposed at the inflow bound-ary only. The one-dimensional, fully developed (v · ∇f = 0)diffusion equation is, therefore, solved at the inflow boundaryyielding an essential boundary condition for the solution of the
Fig. 4. Flowchart for problem parallel
wo-dimensional rectangular domain, each element only needsoundary information from the upstream element. The physi-al domain is decomposed uniformly among processors, suchhat each processor solves the distribution function on only oneubdomain. With this decomposition, the steps for the parallelomputation are:
1) Processor 0 computes the velocity and pressure fields orsolves the v − p − G problem in the entire domain. Thecomputation of the flow field is cheaper than solving for thedistribution function.
2) Processor 0 then sends the nodal velocity in each subdomainto the appropriate processor.
3) Each processor integrates the diffusion equation by one timestep on each element within its subdomain in order to updatethe distribution function f and polymeric contribution to thestress τp.
4) Each processor then sends and receives the solution of thediffusion equation to and from other processors that share acommon boundary. This is shown in Fig. 3 for processor 1,which solves the diffusion equation on subdomain 1. Onceprocessor 1 has updated the solution for f in its domain,it sends the new solution for f on its upper boundary toprocessor 2 and receives the corresponding solution for f onthe lower boundary of processor 2. A similar communicationis also performed at the lower boundary of processor 1. Thiscommunication is essential in order to deal with flows inwhich streamlines are not parallel to the x-axis or there isa small upwinding term from elements that reside on other
processors.5) Processors 1 to n − 1 then send the τp information to pro-cessor 0, which computes nodal averages from adjacentelements and updates the flow field.
Fi
on nodes of a computational cluster.
6) Steps 1 through 5 are repeated until convergence to steadystate.
A flow chart for the parallel computation is presented inig. 4.
.3. Boundary conditions
The boundary conditions and the computational domain forhe planar channel and the channel with linear array of cylindersre shown in Figs. 5 and 6, respectively. For the planar channelhe computational domain has length and width equal to the char-cteristic length scale of the geometry with a flow rate specifiedt the inlet, x = 0. As a result, the hyperbolic character of the dif-usion equation in physical space requires boundary conditions
ig. 5. Computational domain and boundary conditions for flow through annfinitely wide planar channel.
38 Z. Anwar, R.C. Armstrong / J. Non-Newtonian Fluid Mech. 150 (2008) 32–42
Fig. 6. Computational domain and boundary conditions for flow around a lineararray of cylinders in a planar channel. Since the cylinders are placed periodicallyalong the centerline of the channel and since the flow is assumed symmetric aboutthe midplane of the channel, computation is restricted to the unit cell shown.The cylinders have radius L and the cylinder-to-cylinder spacing is 2.5L.
dobsfladi
Fig. 8. Pressure drop across channel as a function of (a) distribution function tru
Fig. 9. Velocity profile for varying (a) distribution truncation parameter M (200 elemM = 12.
Fig. 7. Newton-GMRES solver for viscoelastic flow time-stepper.
iffusion equation in the bulk flow. Similarly for velocity, thene-dimensional momentum equation is solved at the inflowoundary by using the one-dimensional solution of the diffu-ion equation. The inflow velocities obtained from this solutionurnish essential boundary conditions for the bulk flow prob-em. The x-component of the velocity at the outflow boundary is
ssumed to be fully developed, whereas a no-slip boundary con-ition is imposed at the wall, and a symmetry boundary conditions imposed at the centerline.ncation parameter M (200 element mesh), and (b) mesh size with M = 12.
ent mesh), and (b) mesh size (M3 = 200 elements, M5 = 400 elements) with
Z. Anwar, R.C. Armstrong / J. Non-Newtonian Fluid Mech. 150 (2008) 32–42 39
Fig. 10. Contour plots for the distribution function at y = 0.2 as a function of the truncation parameter M for a 200 element mesh.
Fig. 11. Contour plots for the distribution function at y = 0.2 as a function of mesh size (M1 = 50 elements, M2 = 100 elements, M3 = 200 elements, M4 = 300elements, M5 = 400 elements) with M = 12.
40 Z. Anwar, R.C. Armstrong / J. Non-Newton
Fig. 12. Steady state of τEp,yx = −3(1 − β)/(λV/L)〈uyux〉 with β = ηs/η0 =
0sa
d2dtbna
4
tsatatcbtG
d
4
4
cawtba
.9, De = 3.0, T = λ, and t = 0.05λ. (–) Solution obtained from dynamicimulation. Solution obtained from Newton-GMRES simulation at (•) x = 0.2,nd (♦) x = 0.8.
For flow through the channel with a linear array of cylin-ers, the problem is solved on a periodic domain of length.5L, which is also the inter-cylinder spacing, by specifying theimensionless pressure drop across the domain. Here the charac-
eristic length L is taken to be the radius of the cylinder. No-slipoundary conditions are imposed on the cylinder and the chan-el wall (y = 2L) along with symmetry boundary conditionst y = 0.pbcs
Fig. 13. Contour plots for the distribution function across th
ian Fluid Mech. 150 (2008) 32–42
.4. Newton-GMRES wrapper
Given that we have a dynamic simulator for the viscoelas-ic flow problem, we can then wrap it in a Newton-GMRESolver to obtain the steady state of the system. This can bechieved by treating the time-stepper as a black box integra-or that takes a given distribution function for the flow domainnd returns an evolved distribution function, f (T ), after integra-ion over time horizon T. In doing so, the integrator computesonsistent flow fields at each intermediate time-step, which isuilt into the integrator. The task of the Newton-GMRES solverhen is to solve for the steady state f (ss) of the nonlinear system
(f ) = f − f (T ) = 0 given an initial guess f (0). A schematiciagram of the method is shown in Fig. 7.
.5. Results
.5.1. Convergence in physical and configuration spaceTo demonstrate convergence in both configuration and physi-
al space, the dynamic simulation was run for the planar channelt De = 3.0 and with β = 0.9. A stable time-step of t = 0.05λ
as chosen, while steady state was defined as the point at whichhe L2 norm of the change in the distribution function waselow 10−8. The steady-state pressure drop and velocity profilere shown as a function of the distribution function truncation
arameter, M, and mesh size in Figs. 8 and 9, respectively. It cane observed that the steady state pressure drop converges in bothonfiguration and physical space whereas there is no change intreamline velocity with either mesh size or truncation parame-e channel, with wall at y = 0 and centerline at y = 1.
ewtonian Fluid Mech. 150 (2008) 32–42 41
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er M. This is most probably due to the large value of β used inhis study. Additionally, Figs. 10 and 11 show that the distribu-ion function near the wall also converges with M and the meshize. These results suggest that it is sufficient to use M = 12 and200 element mesh to obtain converged results, since the error
n the pressure drop for the 200 element mesh relative to the 400lement mesh is only approximately 0.03%.
.5.2. Planar channelFor the planar channel the time-horizon of the Newton-
MRES solver was set to T = λ, with convergence defined byG(f )‖2/‖G(f (0))‖2 < 10−8. An isotropic distribution func-ion was chosen both as an initial condition and initial guessor the dynamic and Newton-GMRES simulations, respectively.he polymer contribution to the stress tensor consists of bothlastic and viscous components. In Fig. 12, the steady state ofhe yx component of the elastic part of the polymer contributiono the stress tensor, τE
p = (1 − β)/(λV/L)(δ − 3〈uu〉), is showns a function of y for two values of x. There is excellent agree-ent between the steady states computed from the dynamic andewton-GMRES simulations. We also plot the steady-state dis-
ribution function across the channel in Fig. 13 to demonstratehe degree of alignment with varying shear rate across the chan-el. As expected, the distribution function is peaked near the wally = 0) and is nearly isotropic, f = 1/4π, near the symmetrylane (y = 1).
.5.3. Linear array of cylinders in a planar channelThe steady-state results of dynamic and Newton-GMRES
imulations were also compared for flow through a planar chan-el with a linear array of cylinders. Previously, Liu et al. [14]ave presented results for the flow of flexible polymer solu-ions through this geometry at β = 0.59 and We = 0.5 and.0. They describe the polymer solutions with the Giesekusodel, the finitely extensible, nonlinear elastic dumbbell modelith Peterlin’s approximation (FENE-P), and the FENE dumb-ell model of Chilcott–Rallison (CR). These three constitutivequations can be derived from kinetic theory models of diluteolymer solutions; the latter two invoke closure approximationsn their derivations. Because Liu et al. [14] show stresses onlyor We = 0.5, we chose this value of We for our simulations.
e run both the dynamic and Newton-GMRES simulations ate = λ〈v〉/L = 0.5201 and β = 0.59 for a qualitative compari-
on of polymeric contribution to the stress tensor with the resultsf Liu et al. [14].
For this study we used a 1896 element mesh, a stable time-stepf t = 0.01λ, and a dimensionless pressure drop of P = 9o obtain converged results. An isotropic distribution functionas once again chosen as the initial condition for the dynamic
imulation and as the initial guess for the Newton-GMRESolver, whereas the time-horizon was set to T = λ. The steadytate for τE
p,yx obtained from both the dynamic simulation andewton-GMRES solver is shown in Fig. 14. Once again excel-
ent agreement is observed between the solutions of the dynamicnd Newton-GMRES simulations. In addition, there is goodualitative agreement with the results of Liu et al. [14](cf. Fig.1 in that reference). First, flow for an inter-cylinder spacing
hcaw
ig. 14. Steady state of τEp,yx for β = 0.59, P = 9, and We = 0.5201.
f 2.5L is characterized by the development of a recirculationegion between adjacent cylinders with small fluid velocity andelocity gradient. This results in stresses that are near equi-ibrium, in contrast with a larger inter-cylinder spacing whereolymer molecules along the centerline of the geometry are farrom equilibrium. Second, the largest stresses exist at the solidoundaries in the gap between the cylinder and the channel wallhere the flow is shear dominated with the extrema occurringp- and down-stream of x = 0.
Most importantly, however, in contrast with the planar chan-el, which is an inhomogeneous flow in one dimension, thiseometry represents a complex, inhomogeneous flow in two spa-ial dimensions, demonstrating the applicability of the method toybrid simulations of general complex flows. With either geom-try, the Newton-GMRES solver obtains the steady state of theroblem by computing the function G(f ) (cf. Fig. 7), whichs obtained from running the “black-box” dynamic simulation,ithout any modification, over a short time horizon. For the
xample problems presented here, the main goal was to demon-trate that the framework introduced in this paper can enable a
ybrid simulation to converge to steady state. The steady stateomputed here is a stable steady state as it is also accessible viadynamic simulation. However, these results when combinedith those presented in Section 3 indicate that the method pre-
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2 Z. Anwar, R.C. Armstrong / J. Non-N
ented in this paper may also be used to obtain unstable steadytates and to perform bifurcation analysis of a viscoelastic flow.
. Conclusions
In this paper we have presented a method to enable dynamicimulators or time-steppers from kinetic theory to obtain sta-ionary states and perform stability/bifurcation analysis. Theeparation in the time scales of evolution of kinetic theoryoments leads to a linearization with a compact spectrum of
igenvalues. This allows the use of matrix-free iterative meth-ds to locate steady states and perform continuation/bifurcationnalysis of the unavailable closed-form macroscopic system. Weemonstrate this, first by obtaining the equilibrium bifurcationiagram for the structure parameter for the unclosed Doi-modelith the Onsager excluded volume potential. We show that mostf the eigenvalues of the linearized system lie in a tight clusterbout zero, with only a few eigenvalues leaving this cluster nearfold bifurcation.
The second set of examples involve dynamic hybrid imple-entations of pressure-driven flow of non-interacting rigid
umbbells in a planar channel and through a planar channel withlinear array of equally spaced cylinders. We show that short
ursts of a state-of-the-art parallel implementation with the dis-ontinuous Galerkin method can be used to obtain steady statesf the system with our method. Most importantly, we did notave to modify the hybrid simulation algorithm, which indicateshat such an approach can be quickly adapted to other state-of-he-art simulators. In addition, since the method requires onlyhort bursts of a viscoelastic time-stepper, it presents a feasiblepproach of studying macroscopic flows through hybrid simula-ions that incorporate more configurational degrees of freedomn the kinetic theory description. Finally, these results are par-icularly encouraging, as the method presented in this paper,ontrary to previous methods, also allows for convergence tonstable steady states, opening up the possibility of perform-ng stability/bifurcation analysis of viscoelastic flows via hybrid
ime-steppers. We are currently in the process of studying flowsuitable for demonstrating the applicability of this method foricro-mechanical models with more configurational degrees ofreedom than typically employed in such hybrid simulations.
[
ian Fluid Mech. 150 (2008) 32–42
cknowledgements
This work was supported in part by the ERC Program of theational Science Foundation under Award No. EEC-9731680
nd by an ITR Collaborative Research grant of the Nationalcience Foundation under Award No. 0205411.
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