A Study for Linear Stability Analysis of Incompressible Flows on Parallel Computers

國立中央大學

數學系碩士論文

A Study for Linear Stability Analysis of

Incompressible Flows on Parallel Computers

研究生：陳信源

指導教授：黃楓南

中華民國九十九年六月

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研究生簽名: 陳信源學號： 972201027

論文名稱: A Study for Linear Stability Analysis of Incompressible Flow on Parallel

Computers

指導教授姓名：黃楓南

系所：數學系所博士班碩士班

日期：民國 99 年 6 月 30 日

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i

摘要此研究主要是想探討不可壓縮流其平衡解的穩定性與分歧現象兩者間的關係，並且偵測流體

發生對稱性破壞的臨界點。首先，使用穩定化有限元素法對二維 Navier-Stokes 方程組施行空

間離散化來描述非穩定、具黏滯性的、不可壓縮之流體。我們使用兩種方法來描述流體粒子的運

動行為。其一，先引入 backward Euler's method 對二維 Navier-Stokes 方程組施行時間離散

化，接著進行時間序列的數值模擬。第二，對流體之平衡解作線性的穩定性分析；在此採用

implicit Arnoldi method 結合 Cayley transformation 來求解一個大型廣義特徵值問題之特

徵根。此外，如何選取 Cayley transformation 的參數使得相對應的線性系統擁有良好的收斂

性亦是非常重要的議題。最後，我們將舉例使用 SuperLU 求解線性系統，並且展示其平行效能。

ii

Abstract

In this study, we focus in investigating the relation between the (linear) stability of stationary

solutions and pitchfork bifurcations of incompressible flows, and detect the critical points of

symmetry-breaking phenomena. First, a stabilized finite element method is used to discretize the 2D

Navier-Stokes equations on the spatial domain for the unsteady, viscous, incompressible flow problem.

There are two approaches used to determine the behavior of the solution. One is via numerical time

integration. Another is to locate the steady-state solutions and then to make the linear stability analysis

by computing eigenvalues of a corresponding generalized eigenvalue problem, for which an implicit

Arnoldi method with the Cayley transformation is used. In addition, it is also an important issue that

how to choose the parameters of the Cayley transformation such that the convergence of the linear

system would be better. Finally, we show a parallel performance of SuperLU, a great parallelable

algorithm which is used to solve the linear system.

iii

Acknowledgements

首先，非常感謝我的指導教授－黃楓南老師不遲辛勞的諄諄教誨；除了數值分析以及平行計

算的專業知識，老師也常常教導我為人處世的方法與態度，兩年來誠實受益良多。其次，非常感

謝王偉仲教授與黃聰明教授百忙之中抽空前來擔任口試委員，並且給予諸多寶貴的建議使得本研

究工作得以完善。再者，我想感謝我們團隊的每個成員，在學習過程中给予我許多的協助，並且

提供豐富的資源讓我可以順利完成此論文。最後，我想與我的家人分享此刻的心情，感謝家人的

支持與鼓勵，讓我得以專心的完成學業。

iv

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 A brief review of bifurcation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.1 Some definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.2 Pitchfork bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3 Application to incompressible flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3.1 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3.2 Governing equation and semi-discrete formulation. . . . . . . . . . . . . . . . . . . . 6

3.3 Numerical tools for detecting bifurcation points . . . . . . . . . . . . . . . . . . . . . . 9

3.3.1 Pseudo-transient Newton-Krylov-Schwarz method . . . . . . . . . . . . . . 10

3.3.2 Linear stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 11

4 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4.1 Setup of Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4.2 Grid resolution testing and parallel code validation . . . . . . . . . . . . . .. . . . . 15

4.3 Predictions of pitchfork bifurcation . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . 18

4.4 The parallel performance of SuperLU factorization . . . . . . . . . . . . . . . . . . 30

5 Conclusions and future works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

Chapter 1

Introduction

The bifurcation phenomena commonly occur in engineering and scientific applications about fluiddynamics. From the intuitive point of view, a bifurcation occurs with the fact that the qualitativestructure of the flow changes. From the mathematical point of view, a bifurcation occur as some physicalparameter varies.

Jet flows are extensively applied in many areas, such as cooling devices, jet engine exhaust, and thrust-vectoring nozzles. The bifurcation phenomena also exist in the jet flows, including single-jet and multi-jetflows. According to the presence and absence of the side-wall in a geometrical arrangement, jet flows canbe classified as confined and free jet flows, respectively. A sudden-expansion flow in a channel can beregarded as a confined single jet flow, and many researches were reported to investigate the bifurcationphenomena in the field [3, 5, 6, 9, 10, 12, 16, 19, 25, 26, 29, 33]. Although there were many studies aboutthe twin-jet flow in the ambient of free air, these focused on a jet flow ejected to a free space withoutconfinement [11, 23, 24]. So far, there has been little paper that describes the bifurcation phenomena orthe mode transition of the flow pattern in a confined twin-jet flow.

Here, we are interested in the two-dimensional, confined, laminar, incompressible twin-jet flows, whosebehavior can be described by time-dependent Navier-Stokes equations. By employing some characteristicparameters, this system can be cast into a dimensionless form. In particular, we use a Reynolds number,simply noted Re, to identify the critical value as a bifurcation occurs. According to the bifurcation theory,only one steady and symmetrical solution is available at a low Reynolds numbers. As the pitchforkbifurcation occurs at a higher Reynolds number, two steady and asymmetry solutions are possible. Atfurther high Reynolds numbers, the flow becomes periodic over time after the Hopf bifurcation occurs.From a previous study [30], we can make a prediction that for the side-wall confinement with S/H = 0.3and S/D = 10, a pitchfork bifurcation occurs at the critical Reynolds number about = 16 and a Hopfbifurcation at the the critical Reynolds number about = 29. Our work makes an attempt to develop aparallel pseudo-transient continuation method to detect the bifurcation phenomena as well as the modetransition of the flow pattern. Moreover, we find the steady-state solutions and then to determine theirlinear stability by computing the left-most eigenvalue of a generalized eigenvalue problem arising fromthe discretized Navier-Stokes equations. On the other hand, it is also the aim of our work that how toaccurately solve a large-scale generalized eigenvalue problem and how to make the linear system efficientlyand rapidly convergent.

In the next chapter, we firstly review some definitions about bifurcations. We apply the ΨNKSalgorithm and the linear stability analysis to solve the problem of confined 2D incompressible twin-jetflow in chapter 3. In chapter 4, we show our numerical results about the 2D single-jet flows and twin-jetflow, particularly in detecting the critical bifurcation points. Finally, we make some conclusions anddescribe our future works in chapter 5.

1

Chapter 2

A brief review of bifurcation theory

2.1 Some definitions

In fluid dynamics, suppose that f is a conservative vector field, x = x(t) is the trajectory of fluidparticles as time t goes on, and

·x= dx

dt . Given a nonlinear system of differential equations·x= f(x, λ)

with a parameter λ, at points where·x= 0, they are stationary flows; such points are called fixed points.

In other words, if x? is a fixed point of f , then f(x?, λ) = 0 for some λ. The collection of x? is calleda stationary orbit.

Fixed points are sometimes called equilibrium solutions or steady solutions. There are two kindsof fixed points : stable or unstable. Here, we take an example as shown in Figure 2.1 from Strogatz’stextbook [31]. It is clear that there are two fixed points at the solid black dot and the open circle,respectively, as the system

·x= f(x) = 0. The solid black dot represents a stable fixed point, which

is defined if all sufficiently small disturbances away from it damp out in time. In other words, a fixedpoint at the solid black dot with small perturbation will come back to the original state after a sufficientlylong time goes. The open circle represents an unstable fixed point, at which all disturbances grow in time.

Figure 2.1: A phase portrait of one dimensional system·x= f(x) [31]

2

2.2 Pitchfork bifurcation

In fluid dynamics, we can image that the structure of flows change as some characteristic parametersare varied. At this moment, fixed points can be created, destroyed, or their stability can change. In par-ticular, these qualitative changes are called bifurcations, and the parameter values, at which bifurcationsoccur, are called bifurcation points.

There are two standard types of pitchfork bifurcations : supercritical pitchfork bifurcations andsubcritical pitchfork bifurcations. In this section, we only consider the so-called supercritical pitchforkbifurcations. We can commonly find such phenomena in symmetry channel flows. The normal form ofa supercritical pitchfork bifurcation is

·x= rx − x3, a 1D example taken from Strogatz’s textbook [31].

First, we can notice that in the equation the variable x is invariant under the change from x to −x.Next, as the parameter r is varied from negative to positive, we plot the coresponding phase portraitsin Figure 2.5 [31]. We can observe that the origin is the only fixed point and stable as r ≤ 0, and theorigin becomes unstable as r > 0 and a pair of stable fixed points, symmetrically located at x? = ±

√r,

simultaneously appear on both side of the origin .

Figure 2.2: Phase portraits of·x= rx− x3 with varied r. [31]

Then, we can make the stability analysis at the origin with varied r and plot a diagram of a supercrit-ical pitchfork bifurcation in Figure 2.6 [31]. But in our mind we should have a concept that sometimesthere occurs imperfect symmetry in practical applications.

Figure 2.3: The diagram of a supercritical pitchfork bifurcation. [31]

3

Chapter 3

Application to incompressible flows

3.1 Problem statement

In our study work, there are two test cases of simulation for two-dimensional, Newtonian, incompress-ible flows: one is a single-jet flow and another is a twin-jet flow.

First, consider a 2D single-jet flow in a long channel with the property of symmetrically suddenexpansion. A typical model of 2D confined single-jet flows is shown in Figure 3.1. A long channel of height2D, with right angles, suddenly expands to a long channel of height H, where H > 2D. The expansionratio ER is defined as the ratio of the downstream channel of height H to the upstream channel height2D. The velocity is u = (u, v) in Cartesian coordinates (X,Y ) We can define a characteristic parameter,Reynolds number, Re = UD/ν, where U is the inlet velocity, D is the half height of the upstream channeland ν represents the kinematic viscosity of fluid.

Figure 3.1: A typical model of 2D confined single-jet flows. [18]

It is clear that the channel has a symmetrical reflection about the centerline. Given a symmetricboundary condition, someone can imagine that the stationary solution is invariant under the transitionof y → −y and v(x, y)→ −v(x,−y) if the other physical quantities are left unchanged.

Next, Figure 3.2 depicts a typical model of 2D confined twin-jet flows. Two parallel jets exhaust fromtwo nozzles of width D separated at a distance S. A dimensionless parameter, S/H, measures the degreeof side-wall confinement. Clearly, a twin-jet flow can be regarded as single-jet if S/H = 0. In weakconfinement, small value of S/H, the flow pattern likes the results of a single-jet. The inlet velocity, U ,at the nozzles exits, is assumed to be uniform. Also, We can define the Reynolds number, Re = DU/ν,where D is the width of nozzles and ν represents the kinematic viscosity of fluid. With a symmetricboundary condition, the stationary solution of a twin-jet flow is also invariant under the transition ofy → −y and v(x, y)→ −v(x,−y) if the other physical quantities are left unchanged.

4

Figure 3.2: A typical model of 2D confined twin-jet flows [30].

Figure 3.3 displays a typical physical feature of the two-dimensional twin-jet flows configuration. Wecan roughly divide the flow field into three regions: the central region and two near-wall regions. Fromthe flow pattern in Figure 3.3 [30], two vortices are formed in the central region and side recirculationsappear near the corner of each side wall. With the effect of strong side-wall confinement, it is possiblethat corner vortices and separation bubbles also occur.

Figure 3.3: Physical feature of the confined plane twin-jet flow configuration [30].

From the bifurcation theory, experiments, and numerical simulations, we can observe the transitionof flow pattern. Similarly to the single-jet flows, at a lower Reynolds number, only one steady state isavailable and the flow remains symmetric with side recirculations on both side walls. With an increasingReynolds number, the symmetric structure of flow may be broken. Two steady states are possible andthe flow becomes asymmetric as the Reynolds number increases above Rep. At this moment, we saythat a pitchfork bifurcation occurs. Except for two asymmetry side recirculation on each side wall, thereare also separation bubbles at a higher Reynolds number. Different from the single-jet flows, confinedtwin-jet flows has central vortices in the central region and corner vortices at both corners. Furthermore,a twin-jet flow turns to an unsteady state and becomes periodic over time if the Reynolds number is aboveReh, at which the Hopf bifurcation occurs. Finally, the flow becomes turbulent at a higher Reynoldsnumber. Notice that here we don’t discuss the Hopf bifurcation in this study.

5

3.2 Governing equation and semi-discrete formulation

The motion of 2D confined, unsteady, viscous incompressible flows can be described by the time-dependent, incompressible Navier-Stokes equations with boundary conditions. Suppose that Ω ∈ R2 be abounded domain with the boundary Γ = ΓD ∪ΓN . By employing D, U , D/U , and ρU2 as characteristicscales of length, velocity, time, and pressure, the two-dimensional Navier-Stokes equations for the laminar,incompressible flow problem can be translated into the dimensionless form of

ut + u · ∇u = −∇p+ 1

Re∇2u,

∇ · u = 0.(3.1)

We impose Derichlet-type boundary condition on ΓD and Neumman-type boundary condition on ΓN .Assume that the flow is at steady state at the beginning of simulation. Thus we have the initial andboundary conditions defined as following :

u(X, t) = uD(X, t), X ∈ ΓD, t ∈ [0, T ]

n · σ(X, t) = ς(X, t), X ∈ ΓN , t ∈ [0, T ]

u(X, 0) = u0(X), X ∈ Ω, t = 0

where Re = Duj/ν denotes the Reynolds number and ν represents fluid kinematic viscosity. Forsimplicity, ς is assumed to be zero.

To discretize the above system (3.1), We apply a P1 − P1 (continuous linear velocity and pressure )stabilized finite element method on the spatial domain with a given triangle mesh, Th = K. Let Sh

and V h be trial and weighting spaces for velocity, respectively, and let Qh be both of trial and weightingspaces for pressure:

Sh = uh ∈ (C0(Ω) ∩H1(Ω))2|uh = uDonΓD

V h = wh ∈ (C0(Ω) ∩H1(Ω))2|wh = 0onΓD

Qh = qh ∈ C0(Ω) ∩ L2(Ω)

Then, the original Navier-Stokes equations (3.1) can be translated into a semi-discrete stabilized finiteelement form: find uh ∈ Sh × [0, T ] and Ph ∈ Qh × [0, T ], for all (wh, qh) ∈ V h ×Qh, such that

(wh,uht ) + a(wh,uh) + c(uh; wh,uh) + b(wh, ph)+∑

K∈Th(uht + (uh · ∇)uh +∇ph, τ(uh · ∇)wh)K + (∇ · uh, δ∇ ·wh) = 0

b(uh, qh) +∑

K∈Th(uht + (uh · ∇)uh +∇ph, τ∇qh)K = 0

(3.2)

where the stabilization parameters δ and τ are suggested in [13]:

6

δ(X,ReK(X)) = |u(X)|2hKξ(ReK(X)),

τ(X,ReK(X)) =hK

2|u(X)|2ξ(ReK(X)).

Here, ReK is an element Reynolds number defined as following:

ξ(ReK(X)) =

ReK(X), 0 ≤ ReK(X) < 11, ReK(X) ≥ 1

This distinguishes the locally convection-dominated flows as ReK(X) ≥ 1 and the locally diffusion-dominated flows as 0 ≤ ReK(X) < 1.

The stabilized finite element discretization (3.2) yield a large sparse nonlinear system of differentialalgebraic equations of the form [15,32]: for t ∈ [0, T ]

(M +Mω

ε )v + [(K +Kωε ) + (C(v, Re) + Cωε (v, Re))] v + (G+Gωε )p = 0,

Mqε v + (GT +Kq

ε + Cqε (v, Re))v +Gqεp = 0,

where v ∈ Rn is the vector of unknown nodal value of uh, v is the time-dependent derivative of v,and p ∈ Rm is the vector of nodal value of Ph. The matrices M , K, C, and G are derived from thetime-dependent, viscous, convective, and pressure terms, respectively. The subscript ε represents thestabilization terms, and the superscripts ω and q distinguish the terms produced by the velocity andpressure test functions, respectively.

Arising from the spatial discretization of the unsteady Navier-Stokes equations, we can obtain atime-singular ODEs system with block structure:

B·s +D(s, Re)s = 0, (3.3)

where

B =[M +Mω

ε 0Mqε 0

], s =

[up

],

D(s, Re) =[K +Kω

ε + C(u(t), Re) + Cωε (u(t), Re) G+GωεGT +Kq

ε + Cqε Gqε

]Moreover, a time-dependent system of nonlinear partial differential equations can be rewritten as

B·s= f(s, Re), (3.4)

where s(t) ∈ Rn+m is the solution vector, Re ∈ R is the characteristic parameter of bifurcation,B ∈ R(n+m)×(n+m) is the matrix of time-dependency, and f = −D(s, Re)s is a nonlinear mapping fromRn+m × R→ Rn+m.

7

Matrix/Vector Corresponding term of equation

K a(w,u) =∫

Ω∇w · 1

Re∇udΩ

C C(u; w,u) =∫

Ωw · (u · ∇)u

G b(w, p) = −∫

Ωp∇ ·wdΩ

GT b(u, q) = −∫

Ωq∇ · udΩ

Table 3.1: The matrices and vectors with each corresponding term of equation.

8

3.3 Numerical tools for detecting bifurcation points

For detecting the critical value of pitchfork bifurcations, there are two common approaches :

• Observe the dynamical behavior of the solution via a numerically time-dependent integration of thediscretized equations.

1. Add a periodic perturbation into the stationary solution.

2. Perform a time-dependent simulation with ΨNKS for a long time.

3. Make an examination if the perturbed solution returns to the originally steady state or be-comes periodic over time. If so does it, the solution is stable; if not, the solution is unstable.

4. Vary the Reynolds number until the critical bifurcation point is determined.

• Determine the linear stability analysis of steady-state solutions by computing eigenvalues of a large-scale generalized eigenvalue problem.

1. Solve a generalized eigenvalue problem for the stationary solution.

2. Observe if the selected eigenvalues cross over the imaginary axis. we say that a pitchforkbifurcation occurs when a ( leading ) pure real eigenvalue crosses the imaginary axis.

3. Vary the Reynolds number until the critical bifurcation point is determined.

In our research work, we focus on the second approach because it has the advantage that a steady-statesolution can be efficiently located with robust solver and can be tracked using continuation techniques [21].Besides, an implicit Arnoldi iterative method with the Cayley transformation is a good eigensolver forsolving a large-scale generalized eigenvalue problem. For the iteration of linear system resulting from theCayley transformation when solving the generalized eigenvalue problem, it is particularly interesting forus to study the effect of solving the linear system with a parallelable LU factorization, SuperLU [8]. It isalso a purpose of our work to reduce the cost of solving the linear system. Next, we introduce these twoapproaches.

9

3.3.1 Pseudo-transient Newton-Krylov-Schwarz method Ψ NKS

In section 3.2, arising from the spatial discretization of the unsteady Navier-Stokes equations, we canobtain a time-singular ODE’s system (3.3) with block structure. Employing a backward Euler’s methodfor (3.3) as a temporal discretization at each time step, we can obtain a large-sparse nonlinear algebraicsystem:

Gn+1(x) ≡ B(x)x− snδtn

+D(x,Re) = 0, (3.5)

where x is a new approximation at next time step tn+1, sn and δtn are a current approximationand the time-step size at each tn, respectively. A Pseudo-transient procedure, Ψ NKS, is defined as theinexact Newton method with backtracking iteration. More detail description can be consulted in [18].Here, the algorithm for Ψ NKS. can be written as following:

1: Set n = 0

2: Initialize s0 and δt0

3: Do4: Set k = 0 and x(0) = sn

5: while (‖Gn+1(x(k))‖2 > ε1‖Gn+1(x(0))‖2) do6: Compute G′n+1(x(k))

7: Inexactly solve G′n+1(x(k))y(k) = −Gn+1(xk)) for y(k) by a Krylovsubspace method, such as GMRES with an additive-Schwarz pre-conditioner, M−1

k .

8: Update x(k+1) = x(k) + λ(k)y(k), λ(k) ∈ (0, 1] is a damping param-eter.

9: Set k = k + 1.

10: end while11: Set sn+1 = x(k)

12: Update δtn+1

13: n = n+ 1

14: while (n < nmax) and (‖sn − sn−1‖2 > ε2)

To make ΨNKS more efficient, the time step is chosen as suggested by Hannani et al. [15]

δtn+1 = φ(δtn‖sn+1 − sn‖−12 ),

which is based on the norm of the solution difference ‖sn+1 − sn‖2. Here, φ satisfies the assumption

φ(ξ) =ξ, ξ < δtmaxδtmax, ξ ≥ δtmax,

where δtmax is an upper bound for the time steps δtn.

10

3.3.2 Linear stability analysis

Given a nonlinear system of differential equations Bs = f(s, λ) in block matrix form such as (3.4)with a parameter λ ∈ R, and let x? ∈ Rn+m be a fixed point of f such that f(x?, λ) = 0 for some λ ∈ R.Consider the reduced nonlinear system of ODEs by fixing the parameter λ ∈ R :

Bx = fλ(x). (3.6)

Let z = x − x? ∈ Rn+m denote the perturbed solution by a small time-dependent disturbance fromthe steady solution and x = z+x?, where x? is a fixed point of f

λ, and substitute them into the equation

(3.6), then we have

Bx = B ˙(z + x?) = fλ(z + x?). (3.7)

For the left-hand side of (3.7),

B ˙(z + x?) = Bz +Bx? = Bz + fλ(x?) = Bz, (3.8)

due to the fact that fλ(x?) = 0. Use the Taylor’s expansion at x? for the right-hand side of (3.7),

then

fλ(z + x?) = f

λ(x?) + J(x?)z +O(z2) ≈ J(x?)z (3.9)

Replacing equation (3.7) by the results of (3.8) and (3.9), hence we have the following linearizedsystem

Bz = J(x?)z. (3.10)

The linearized system (3.10) can be settled down to equilibrium through a small perturbation such as

z(t) = ze−λt (3.11)

Finally, let λ = −λ and substitute the perturbed solution (3.11) into the linearized system (3.10), wecan obtain a generalized eigenvalue problem

J(x?)z = λBz (3.12)

Here, we should have in mind that this generalized eigenvalue problem (3.12) arises in the determi-nation of the stability of steady-state flows. In other words, we can determine the stability of stationarysolution to the discretized equation (3.5) by compute the eigenvalues of (3.12). Moreover, the real partof eigenvalue Re(λ) indicates the increasing rate of disturbance. Suppose that these eigenvalues λi, fori = 1, 2, ..., n+m, of J(x?)z = λBz are reordered by Re(λ1) ≤ Re(λ2) ≤ · · · ≤ Re(λn+m), and from thedefinition of stability, a stationary solution is called stable if Re(λ) > 0,otherwise it is called unstableif Re(λ) < 0, it is sufficient that computing the right-most eigenvalue determines the stability of thesteady-state solution. Therefore, we say that a bifurcation phenomenon occurs as Re(λ1) < 0. In fact, apitchfork bifurcation occurs when some pure eigenvalue crosses the imaginary axis, and a hopf bifurcationoccurs if a pair of conjugate complex eigenvalues cross the imaginary axis.

However, in engineering and scientific applications, it is very difficult to solve the system (3.12) sincethe corresponding generalized eigenvalue problem is large-scale with the fact that matrix B is singular.Under the assumption that λ = 0 is not an eigenvalue of system (3.12), then the system (3.12) has aninfinite eigenvalue of algebraic multiplicity 2m, which are defined to be zero eigenvalues of µJ(x?)z = Bzwith corresponding eigenvectors that are null vectors of B [7]. To solve the system(3.12), direct methodsare not viable because they require at least Ø(N2) operations if the system size is of N and need a largeamount of memory, but not scalable to hundreds of processors. Instead, we consider a parallel Arnoldiiteration with a spectral transformation.

11

A shift-invert spectral transformation [22] is typically used to transform equation J(x?)z = λBz intoa standard eigenvalue problem

Tsz = (J− σB)−1Bz = θz, θ =1

λ− σ(3.13)

Notice that the shift-invert spectral transformation (3.13) has some properties [21]:

• It maps the eigenvalues near the pole σ to those of largest magnitude.

• It also maps the eigenvalues far from the pole to zero.

• The spectral condition number ( the ratio of the largest-to-smallest eigenvalues, in magnitude ) ofTs would be quite large.

• Given a vector x, it involves solving a linear system (J − σB)v = Bx so that v = Tsx.

• The resulting linear systems will be difficult to solve because the rate of convergence of a Krylov-based iterative method depends strongly upon the spectral condition number [14,27].

A much better alternation is using a generalized Cayley transformation [22]:

Tcz = (J− σB)−1(J + τB)z = θz, θ =λ+ τ

λ− σ. (3.14)

Then, each of the computed eigenvalues can be back-transformed by the operation

λ =θσ + τ

θ − 1. (3.15)

It is proved that the Cayley transformation is mathematically equivalent to shift-invert. Given avector x, the Cayley system involves solving a linear system (J−σB)v = (J+τB)x so that v = Tcx. Onthe other hand, the process to compute the right-left eigenvalue λ requires that the Cayley transformationmaps these values to θ which are the largest in magnitude. Such situation allows an Arnoldi methodto provide the best approximation to the eigenvalues θi as long as Im(λi) is not large compared withRe(λi)− σ such that ‖θ(λi)‖ ≥ 2. To achive this purpose, Lehoucq et al. [21] supply a good suggestionfor choosing these parameters σ and τ to have the real part of the right-left eigenvalue λ satisfy

−τ < σ < Re(λ) < 2σ + τ. (3.16)

According to the above technique, the spectral condition number of the linear system arising fromthe Cayley transformation can be bounded with a good choice of σ and τ . This implies that the Cayleysystem makes a better condition of linear system. Thus, it is more effective for finding eigenvalues nearσ as well. Unfortunately, there is no positive theory to prove whether the computed eigenvalue is theright-most or not since a rational transformation is used, and the Cayley transformation also performsbadly if the spectral condition number is very large.

12

Chapter 4

Numerical results

The performance and the parallelization of the ΨNKS code is done by PETSc [4], which is a Portable,Extensible, Toolkit for Scientific Computation. SLEPc [17], the Scalable Library for Eigenvalue ProblemComputations, is based on PETSc to compute the eigensolutions of large-scale, sparse, generalized eigen-value problems on parallel computers. For a fixed-size problem, a mesh generation toolkit, CUBIT [1], isused to generate unstructured grids, and a mesh partitioner, ParMetis [20], can make these unstructuredgrids partitioned in equal-size subdomains. After performing computations of the parallel ΨNKS code, avisualization software, Paraview [2], is used to plot the pressure contours, streamline patterns, velocityprofiles in the post-processing stage.

For the linear stability analysis, an Arnoldi method with the Cayley transformation is used to solve ageneralized eigenvalue problem (3.12). The resulting linear system is solved by a parallelable LU factor-ization, SuperLU, which is a library based on PETSc for the direct solution of large, sparse, nonsymmetricsystems of linear equations on high performance machines [8]. Notice that the main purpose of our workis to detect the critical Reynolds number of a pitchfork bifurcation and here we focus on the approach oflinear stability analysis.

4.1 Setup of Numerical experiments

In our simulation, the computational domain only includes the downstream channel. The origin(0, 0) in Cartesian coordinates (X,Y ) is located at the lower-left vertex and the corresponding velocityis u = (u, v) Along the lower and upper side walls, it is imposed the no-slip boundary conditions,u = (u, v) = 0 for all time t. Some parameters in Figure 4.1 are fixed as following:

• H = 6, L = 40., D = 1 and the expansion ratio ER = 3.

• The inlet flow u = (u(0, y), v(0, y)) = ((y − 2)(4− y), 0) for 2 ≤ y ≤ 4.

• The Reynolds number is Re = DUν , where ν is the kinematic viscosity of flows.

13

Figure 4.1: The boundary conditions for confined single-jet flows in computational domain.

Figure 4.2 shows the boundary conditions for confined twin-jet flows in computational domain. Someparameters are fixed as following:

• H = 5, L = 20., S = 1.5 and D = 0.15.

• The side-wall confinement are S/H = 0.3.

• The inlet flow u = (u(0, y), v(0, y)) = (1, 0) for 1.675 ≤ y ≤ 1.825 or 3.175 ≤ y ≤ 3.325.

• The Reynolds number is Re = DUν , where ν is the kinematic viscosity of flows.

Figure 4.2: The boundary conditions for confined twin-jet flows in computational domain.

14

4.2 Grid resolution testing and parallel code validation

Table 4.1 and Table 4.2 show the number of elements and the corresponding ODE’s system for eachmesh. Meshes in Table 4.1 and Meshes in Table 4.2 are used for problems of single-jet flows and problemsof twin-jet flows, respectively. Mesh S1 is the coarsest mesh for problems of single-jet flows, and meshesS2, S3, S4 are generated by uniformly refining Mesh S1. Mesh RS3 is created by locally refining Mesh S3near the inflow region. For the grid resolution test of single-jet flows, we make a comparisons of horizontalvelocity obtained from each mesh at cross-section x/D = 5 and x/D = 10. It can be observed that thepressure and velocity curves obtained from Mesh RS3 and Mesh S4 are indistinguishable in Figure 4.3 toFigure 4.5, respectively.

Mesh label # of elements Size of ODE’s systemS1 1415 2364 × 2364S2 5660 8970 × 8970S3 22640 34917 × 34917S4 90560 137751 × 137751

RS3 56976 68109 × 68109

Table 4.1: Mesh information for single-jet flows.

On the other hand, Mesh T1 is the coarsest mesh for problems of twin-jet flows, and meshes T2, T3,T4 are generated by uniformly refining Mesh T1. For the grid resolution test of single-jet flows, we makea comparisons of horizontal velocity obtained from each mesh at cross-section x/H = 0.1, x/H = 0.5, andx/H = 2. It can be observed that the pressure and velocity curves obtained from Mesh T3 and MeshT4 are indistinguishable in Figure 4.6 to Figure 4.8, respectively. Therefore, two meshes Mesh RS3 andMesh T3 are used for the numerical experiments.

Mesh label ] of elements Size of ODE’s systemT1 7704 12072 × 12072T2 30816 47253 × 47253T3 123264 186951 × 186951T4 493056 743691 × 743691

Table 4.2: Mesh information for twin-jet flows

15

Figure 4.3: Pressure(left) and velocity(right) profiles of stationary single-jet flows for Re = 26 and ER = 3at X/D = 5.



16

Figure 4.6: Pressure(left) and velocity(right) profiles of stationary twin-jet flows for Re = 10, S/H = 0.3and S/D = 10 at X/H = 0.1.

Figure 4.7: Pressure(left) and velocity(right) profiles of stationary twin-jet flows for Re = 10, S/H = 0.3and S/D = 10 at X/H = 0.5.

Figure 4.8: Pressure(left) and velocity(right) profiles of stationary twin-jet flows for Re = 10, S/H = 0.3and S/D = 10 at X/H = 2.

17

4.3 Predictions of pitchfork bifurcation

In this section, two numerical tools introduced in section 3.3 are used for detecting the bifurcationphenomena. First, we directly solve the steady-state discretized Navier-Stokes equations with a symmetricboundary condition, and then plot the streamline patterns in Figure 4.9 and Figure 4.10. It is clear thatthese streamline patterns of steady-state flows are symmetric about the central line of each computationaldomain.

(a)

(b)

Figure 4.9: Streamline patterns of single-jet flows at steady state for (a) Re = 26 and (b) Re = 60. ER= 3 and Mesh RS3 is used.

(a)

(b)

Figure 4.10: Streamline patterns of twin-jet flows at steady state for (a) Re = 10 and (b) Re = 20. S/H= 0.3, S/D = 10 and Mesh T3 is used.

Next, we add a periodic perturbation into the inlet velocity of stationary flows. A periodic perturba-tion can keep the ODE’s system to always satisfy the conservative law. After performing a time-dependentsimulation by using Ψ NKS to solve the perturbed ODE’s system, we can obtain a series of correspondingsolution at each pseudo-time step. The evolution of flow motion can be observed from Figure 4.11 toFigure 4.14. The streamline patterns at time step n = 0 in Figures is the behavior of stationary flows. Atthe first five time steps n = 1, 2, · · · , 5, a constant step size δtn is used during the process of perturbation.As shown in Figure 4.11 and Figure 4.13, the perturbed flows, return the original state at lower Reynoldsnumbers. However, in Figure 4.12 and Figure 4.14 the flows with a small perturbation turn into anotherstate, from symmetric to asymmetric, at higher Reynolds numbers, at which we say that a bifurcationhas occurred. Furthermore, we vary the Reynolds number and solve a perturbed ODE’s system. Wecan estimate the critical Reynolds number by observing whether the streamline patterns of flows becomeasymmetric or not.

18

n = 0

n = 1

n = 3

n = 5

n = 10

n = 20

n = 22

Figure 4.11: Time-simulation of a single-jet flow with small perturbation is obtained by ΨNKS at someselected pseudo-time steps for Re = 26 and ER = 3. Mesh RS3 is used.

19

n = 0

n = 1

n = 3

n = 5

n = 10

n = 20

n = 30

n = 40

Figure 4.12: Time-simulation of a single-jet flow with small perturbation is obtained by ΨNKS at someselected pseudo-time steps for Re = 60 and ER = 3. Mesh RS3 is used.

20

n = 0

n = 1

n = 3

n = 5

n = 10

n = 15

n = 17

Figure 4.13: Time-simulation of a twin-jet flow with small perturbation is obtained by ΨNKS at someselected pseudo-time steps for Re = 20, S/H = 0.3 and S/D = 10. Mesh T3 is used.

21

n = 0

n = 3

n = 5

n = 10

n = 15

n = 20

n = 30

n = 40

Figure 4.14: Time-simulation of a twin-jet flow with small perturbation is obtained by ΨNKS at someselected pseudo-time steps for Re = 20, S/H = 0.3 and S/D = 10. Mesh T3 is used.

22

single-jet flows twin-jet flowsRe 26 60 10 20

# of pseudo timesteps 24 42 18 44avg. # of SNES iterations 2.6 2.8 1.7 2.8avg. # of KSP iterations 50.5 56.4 103.5 113.0

SNESSolve time(s) 8.3827e+01 1.7134e+02 2.9981e+02 1.0596e+03KSPSolve time(s) 6.8610e+01 1.4624e+02 2.3532e+02 8.9170e+02

Table 4.3: Time-simulation of stationary flows.

Table 4.3 shows the results from time-simulation of stationary flows with small perturbation andthe number of used processors is 16. The case of ER = 3 is considered for single-jet flows and thecase of S/H = 0.3 and S/D = 10 is considered for twin-jet flows. The initial pseudo timestep size isδt0 = 0.001 and the maximum pseudo timestep size is δtmax = 500. The relative linear and nonlineartolerances required are 1.0e− 4 and 1.0e− 6, respectively. The periodic perturbation models are definedas following:

1. For single-jet flows and for 0 < t < 45:

– Shift-up models:

u =

(y−2)(4.2−y)

1.21 (1 + | sin πt15 | × 0.2), for y ∈ [2, 3.1],

(y−2.2)(4−y)0.81 (1 + | sin πt

15 | × 0.2), for y ∈ [3.1, 4].

– Shift-down models:

u =

(y−2)(3.8−y)

0.81 (1 + | sin πt15 | × 0.2), for y ∈ [2, 2.9],

(y−1.8)(4−y)1.21 (1 + | sin πt

15 | × 0.2), for y ∈ [2.9, 4].

2. For twin-jet flows and for 0 < t < 40:

– Shift-up models:

u =

1.825−y

3 (1 + sin(πt20 −π2 )) + 1, for y ∈ [1.675, 1.825],

3.325−y3 (1 + sin(πt20 −

π2 )) + 1, for y ∈ [3.175, 3.325].

– Shift-down models:

u =

y−1.675

3 (1 + sin(πt20 −π2 )) + 1, for y ∈ [1.675, 1.825],

y−3.1753 (1 + sin(πt20 −

π2 )) + 1, for y ∈ [3.175, 3.325].

A strategy for characterizing the bifurcation configuration of flows is chosen in [18]. we can detectthe critical bifurcation points of single-jet and twin-jet flows by measuring the difference between tworeattachment points of the side recirculation near the lower and upper side walls on x-axis respectively.It can be determined by estimating the horizontal velocity along the side walls. Because of the no-slipboundary condition on the side walls, the reattachment points can be determined as the correspondinghorizontal velocity approaches to zero. Besides, the difference Dx = xupper − xlower indicates the asym-metry of flows.

Figure 4.15 and Figure 4.16 show the pitchfork diagrams of a single-jet flows and a twin-jet flow,respectively. We can predict that a bifurcation of single-jet flow for ER = 3 occurs at the critical valueof Reynolds number about Re = 42.5, and a bifurcation of twin-jet flows for S/H = 0.3 and S/D = 10occurs at the critical value of Reynolds number about Re = 15.

23

Figure 4.15: Bifurcation diagram of single-jet flows for ER = 3. Mesh RS3 is used.

Figure 4.16: Bifurcation diagram of twin-jet flow for S/H = 0.3 and S/D = 10. Mesh T3 is used.

24

Although it is intuitive for someone to determine the critical value of Reynolds number from thebifurcation diagram, it must take a long time to perform a time-dependent simulation for solving a large-scale ODE’s system. An alternative method is introduced in section 3.3.2. We do the linear stabilityanalysis of symmetry flows by using an implicit Arnoldi method with the Cayley transformation tocompute the eigenvalues of a generalized eigenvalue problem arising from the discretized Navier-Stokesequations. Because the system size is large-scale, a direct method is not viable. It possibly requiresa lot of memory and is not scalable to multi processors. As mentioned in section 3.3.2, the resultinglinear system would be difficult to solve because the rate of convergence of a Krylov-based iterativemethod depends strongly upon the spectral condition number. Therefore, for the linear system, we usethe SuperLU decomposition. Another difficulty is that Lehoucq et al. [14] supply a good suggestion forchoosing these parameters σ and τ of the Cayley system to make a better condition of linear system,but on one knows the location of eigenvalues. A technique of parameter tuning is used here. First, acoarse mesh can be used to solve a small-scale system and compute all of the eigenvalues of the resultinggeneralized eigenvalue problem. Secondly, we choose a value of σ smaller than the real part of the left-most eigenvalue obtained from the coarse-mesh system. Then the parameter τ can be quickly determinedby using the inequality (3.16). Finally, a better condition of linear system can be made by varying theparameters.

Figure 4.17 and Figure 4.18 show the spectrum by using a QZ-algorithm to solve a generalized eigen-value problem with a coarsest Mesh. The leading eigenvalues are showed in the following Table 4.4 andTable 4.5. According to the above mentioned technique, a pair of approximate parameters σ and τ can bechosen for a large-scale system, and the selected eigenvalues can be checked by comparing the eigenvaluesfrom a coarse-mesh system. Notice that a pair of conjugate complex eigenvalues would affect the Cayleytransformation to have a largest (mapped) eigenvalue in magnitude not necessarily be the left-most onein the original system, for which a simple can be found in [21].

Reynolds numbers λ1 λ2 λ3

41 0.0032502880+0i 0.0451539932+0i 0.0580623730+0i42 0.0022257360+0i 0.0435868668+0i 0.0569356850+0i43 0.0012708602+0i 0.0420450479+0i 0.0558492820+0i44 0.0004595486+0i 0.0405040196+0i 0.0547897641+0i45 -0.0003464839+0i 0.0392251198+0i 0.0538349351+0i46 -0.0012155775+0i 0.0377230482+0i 0.0528368163+0i47 -0.0019357095+0i 0.0363196910+0i 0.0518847051+0i48 -0.0025947186+0i 0.0349658596+0i 0.0509654105+0i49 -0.0032105790+0i 0.0336472639+0i 0.0500693443+0i50 -0.0037834877+0i 0.0323615764+0i 0.0492035013+0i

Table 4.4: The leading eigenvalues of single-jet flows problem for ER = 3. The coarsest Mesh S1 is used.

Reynolds numbers λ1 λ2 λ3

10 0.0271300646+0i 0.0387635997+0i 0.0573961558+0i11 0.0183419067+0i 0.0357368811+0i 0.0529748723+0i12 0.0106650512+0i 0.0331637305+0i 0.0492933224+0i13 0.0045380256+0i 0.0307551726+0i 0.0460686319+0i14 0.0000000030+0i 0.0283906425+0i 0.0434075774+0i15 -0.0043023667+0i 0.0259302116+0i 0.0410773702+0i16 -0.0073151673+0i 0.0233118817+0i 0.0391789161+0i17 -0.0094922596+0i 0.0206013458+0i 0.0377205015+0i18 -0.0110457485+0i 0.0178862893+0i 0.0366991392+0i19 -0.0121079431+0i 0.0153204883+0i 0.0362594436+0i20 -0.0128078869+0i 0.0131896086+0i 0.0371180804+0i

Table 4.5: The leading eigenvalues of twin-jet flows for S/H = 0.3 and S/D = 10 . The coarsest Mesh T1is used.

25

Figure 4.17: The spectrum of a stationary single-jet flow solved by a QZ-algorithm for Re = 45 and ER= 3. The coarsest Mesh S1 is used.

Figure 4.18: The spectrum of a stationary twin-jet flow solved by a QZ-algorithm for Re = 15, S/H =0.3, and S/D = 10. The coarsest Mesh T1 is used.

26

Now, we display how to do parameters tuning of the Cayley transformation. Consider an exampleof a single-jet flow for Re = 45 and ER = 3. Mesh RS3 is used. An implicit Arnoldi method with theCayley transformation is used as a eigensolver. A Krylov-based method GMRES [28] with an Additive-Schwarz preconditioner is used to solve the linear system and the subdomain problems are solved exactLU factorization. The tolerance of eigensolutions and linear system are required to EPS tol = 1e − 5and ST KSP rtol = 1e− 10, respectively. The required number of convergent eigenvalue is nev = 1 andthe number of used processors is np = 4.

Table 4.6 shows that some information from parameters tuning of the Cayley transformation. FromTable 4.3 we can find that for a single-jet flow problem with a coarse Mesh S1, the leading eigenvaluehas crossed the imaginary axis after the Reynolds number is increased higher than Re = 45. It impliesthat there is a pitchfork bifurcation has occurred at Reynolds number Re = 45. To detect the pitchforkbifurcation point for a refined-mesh system, we first choose a smaller parameter σ = −0.004 as theinitial guess, and τ is computed by the suggested inequality (3.16). Fortunately, the selected eigenvaluek = −0.00153410 implies that a pitchfork bifurcation point has occurred. Next, we vary the parameterσ and compute an appropriate parameter τ by the inequality (3.16). Therefore, we can observe someimportant information from Table 4.6:

• As the parameter σ is closer to or very far away from the computed eigenvalue, the condition num-ber of (J − σB) becomes larger and the resulting linear system has bad convergence.

• The cost of solving a large-scale generalized eigenvalue problem is approximate to solve the resultinglinear system.

• One can check the parameters σ and τ for each problems to satisfy the inequality (3.16). It is clearthat all the selected eigenvalues can be bounded by σ.

• The computed leading eigenvalue may not be the left-most if the parameters are selected such asσ = −0.001 and τ = 0.002.

• A good choice of the parameters σ and τ which satisfy the suggested inequality (3.16) would makea better condition of the resulting liner system.

σ τ k linear iters. EPSSolve STApply Cond(J − σB)-0.004 0.008 -0.00153410 1573 8.01e+01 8.01e+01 3.87E+04-0.002 0.004 -0.00153408 2054 1.03e+02 1.03e+02 8.49E+04-0.0016 0.0032 -0.00153407 2668 1.34e+02 1.34e+02 1.12E+05-0.001 0.002 0.04248657 1911 9.69e+01 9.68e+01 2.14E+05-0.01 0.02 -0.00153415 1227 6.29e+01 6.29e+01 1.50E+04-0.1 0.2 -0.00153484 701 3.68e+01 3.67e+01 8.71E+03-0.11 0.22 -0.00153490 678 4.03e+01 4.02e+01 9.20E+03-0.12 0.24 -0.00153495 687 3.56e+01 3.55e+01 9.70E+03-0.13 0.26 -0.00153495 665 3.47e+01 3.46e+01 1.02E+04-0.14 0.28 -0.00153503 648 3.91e+01 3.90e+01 1.07E+04-0.15 0.3 -0.00153514 814 4.23e+01 4.22e+01 1.12E+04-0.2 0.4 -0.00153538 771 4.00e+01 3.99e+01 1.37E+04-0.3 0.6 -0.00153584 1073 5.51e+01 5.49e+01 1.88E+04-0.4 0.8 -0.00153620 1153 6.71e+01 6.69e+01 2.38E+04-0.5 1.0 -0.00153649 1236 6.35e+01 6.33e+01 2.87E+04-1.0 2.0 -0.00153771 2205 1.12e+02 1.12e+02 5.24E+04

Table 4.6: Parameters tuning of the Cayley transformation and resulting information.

27

Apply the strategy for parameters tuning to each problem, and list the computed leading eigenvalues ofsingle-jet flows and twin-jet flows in Table 4.7 and Table 4.8, respectively. By the linear stability analysiswe find that a pitchfork bifurcation of single-jet flows has occurred at Reynolds numbers Re = 43.5 for thecase of refined Mesh RS3 and a pitchfork bifurcation of twin-jet flows has occurred at Reynolds numbersRe = 15 for the case of refined Mesh T3. With the information of eigenvalues from the coarse-meshsystem, we can determine whether the computed leading eigenvalue from a refined-mesh system is theleft-most one. Moreover, for single-jet flows, a system with a coarse mesh seems to be more stable thanthe one with a refined mesh. However, it shows the opposite result for twin-jet flows. It can be kept inmind that the leading eigenvalues may include a pair of conjugate complex eigenvalues, but we show thecomputed eigenvalue in tables just including the left-most pure real eigenvalues. A reason is that thesecomplex eigenvalues would affect the Cayley transformation to have a large spectrum condition number.In other words, the imaginary part of these complex eigenvalues make the linear system badly convergent.

Using an experimental method, Fearn et al. [12] showed that the critical point of pitchfork bifurcationis about Re = 44. By the bifurcation calculation, Hawa et al. [16] and Shapria et al. [29] and Haung [18]showed that it is about Re = 40.35, Re − 41.3 and Re = 44, respectively. Using the time-dependentsimulation, Battaglia et al. [5], Allenborn et al. [3], Mizuahima et al. [26], Drikakis [9], and Mishra et al [25]showed that it is about Re = 42.75−43.5, Re = 40, Re = 40.23, Re = 40, and Re = 40.5, respectively.Onthe other hand, Soong el al. [30] used numerically time-dependent simulations to investigate twin-jetflows for the case of side-wall confinement S/H = 0.3 and S/D = 10, and they showed that a bifurcationoccurs at Reynolds number Re = 16. Compared with others researches, we would confirm our results.

Re λ1 λ2 λ3

40 0.0028164607+0i 0.0491913940+0i 0.0631954533+0i41 0.0018156814+0i 0.0477961406+0i 0.0620788613+0i42 0.0008860862+0i 0.0464209296+0i 0.0610107288+0i43 0.0002435992+0i 0.0450591542+0i 0.0599845436+0i

43.5 -0.0003817988+0i 0.0443959470+0i 0.0594922526+0i44 -0.0007877987+0i 0.0437320271+0i 0.0590062313+0i45 -0.0015420376+0i 0.0424195836+0i 0.0580617006+0i

Table 4.7: The leading eigenvalues of single-jet flows problem for ER = 3. The refined Mesh RS3 is used.

Re λ1 λ2 λ3

11 0.0197867472+0i 0.0346250398+0i 0.0510940422+0i12 0.0120567480+0i 0.0321611358+0i 0.0474803006+0i13 0.0055795432+0i 0.0299306604+0i 0.0443459539+0i14 0.0008213877+0i 0.0277337948+0i 0.0415901796+0i15 -0.0036642536+0i 0.0254819072+0i 0.0392338726+0i16 -0.0067714760+0i 0.0230610396+0i 0.0373090137+0i17 -0.0090903725+0i 0.0204711625+0i 0.0358118911+0i

Table 4.8: The leading eigenvalues of twin-jet flows for S/H = 0.3 and S/D = 10. The refined Mesh T3is used.

From the results, we can find that the eigenvalues seem to move continuously as the Reynolds numbervaries since a flow goes continuously in time if there is no disturbance. Moreover, we compute the leadingeigenvalues for different meshes and list the results in Table 4.9 to Table 4.13. The critical Reynoldsnumber of pitchfork bifurcation for the case of Mesh T2 is different from others. A reason is that theunstructured meshes are used and the coarsest Mesh T1 is locally refined at the inflow region.

28

Re λ1 λ2 λ312 0.0106650512+0i 0.0331637305+0i 0.0492933224+0i13 0.0045380256+0i 0.0307551726+0i 0.0460686319+0i14 0.0000000030+0i 0.0283906425+0i 0.0434075774+0i15 -0.0043023667+0i 0.0259302116+0i 0.0410773702+0i16 -0.0073151673+0i 0.0233118817+0i 0.0391789161+0i17 -0.0094922596+0i 0.0206013458+0i 0.0377205015+0i

Table 4.9: The leading eigenvalues of twin-jet flows for S/H = 0.3 and S/D = 10. The coarsest Mesh T1is used.

Re λ1 λ2 λ312 0.0096956378+0i 0.0326812064+0i 0.0486973765+0i13 0.0034545795+0i 0.0302996506+0i 0.0455207281+0i

13.5 0.0010759216+0i 0.0291071564+0i 0.0441031251+0i14 -0.0013975864+0i 0.0278929268+0i 0.0427635625+0i15 -0.0052605183+0i 0.0253764871+0i 0.0404994381+0i16 -0.0081105780+0i 0.0226880996+0i 0.0387038545+0i17 -0.0101960065+0i 0.0198512325+0i 0.0373651125+0i

Table 4.10: The leading eigenvalues of twin-jet flows for S/H = 0.3 and S/D = 10 . The refined Mesh T2is used.





Mesh The Critical Reynolds numberT1 13 ≤ Rep ≤ 15T2 13.5 ≤ Rep ≤ 14T3 14 ≤ Rep ≤ 15T4 14 ≤ Rep ≤ 15

Table 4.13: Prediction of the critical Reynolds number for pitchfork bifurcation.

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4.4 The parallel performance of SuperLU factorization

The main purpose of this study is using the technique of linear stability analysis to determine thecritical Reynolds number at which a bifurcation appears. Moreover, we are interested in the large-scalegeneralized eigenvalue problems with the matrix size O(106). Thus, a direct method like QZ algorithm isnot viable because it requires a pretty amount of memory and not scalable to multi-processors. An Arnoldimethod with the Cayley transformation is a good choice for such problems, but it necessarily involvessolving linear systems. As mentioned in previous sections, the rate of convergence of a Krylov-basediterative method depends strongly upon the spectral condition number of the resulting linear system.Instead, we apply a parallelable SuperLU factorization.

Here we take two examples to test the parallel performance of SuperLU factorization for the linearsystem. We use an implicit Arnoldi method with the Cayley transformation to solve the problem onparallel computers. For the linear system, A LU factorization is used on parallel computers to force theconvergence of solutions can satisfy tolerance = 1e-5 and a relative tolerance 1e-10 for the linear system.In the test case, we require the number of converged eigenvalues nev = 1 and use the number of columnvectors ncv = 16.

np k ‖Ax− kBx‖/‖kx‖ # of iters # of linear iters EPSSolve STApply1 -0.0015405397 6.17e-14 1 16 2.3858 2.25522 -0.0015405397 6.26e-14 1 16 1.7592 1.68274 -0.0015405397 6.74e-14 1 16 1.5897 1.52648 -0.0015405397 6.15e-14 1 16 1.3766 1.335016 -0.0015405397 6.95e-14 1 16 2.5154 2.455632 -0.0015405397 3.84e-14 1 16 122.97 117.59

Table 4.14: The information of solving a single-jet flow for Re = 45 and ER = 3 on varied number ofprocessors.

np k ‖Ax− kBx‖/‖kx‖ # of iters # of linear iters EPSSolve STApply1 -0.0036642536 6.75e-15 1 16 8.5234 8.13732 -0.0036642536 6.59e-15 1 16 6.0972 5.84724 -0.0036642536 1.18e-14 1 16 5.1836 4.92958 -0.0036642536 7.97e-15 1 16 395.81 395.6416 -0.0036642536 5.98e-15 1 16 269.16 267.7732 -0.0036642536 7.15e-15 1 16 48.145 46.82964 -0.0036642536 5.89e-15 1 16 134.50 126.84

Table 4.15: The information of solving a twin-jet flow for Re = 15, S/H = 3 and S/D = 10 on variednumber of processors.

As the results showed in Table 4.14 and Table 4.15, the selected eigenvalue is left-most with smallrelative error and the cost of solving the linear system is not much. Finally, we compute the speedup andefficiency by the following relation:

• The parallel speedup for n processors is defined as :

Sp(n) = T1

Tn,

where T1 is the running time for a single processor and Tn is the running time for n processors.

• The efficiency is determined by

ε(n) = T1

nTn.

30

Figure 4.19: The parallel speedup for n processors. A single-jet flow for Re = 45, ER = 3 and Mesh RS3is used.

Figure 4.20: The parallel efficiency for n processors. A single-jet flow for Re = 45, ER = 3 and MeshRS3 is used.

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Figure 4.21: The parallel speedup for n processors. A twin-jet flow for Re = 15, S/H = 3, S/D = 10,and Mesh T3 is used.

Figure 4.22: The parallel efficiency for n processors. A twin-jet flow for Re = 15, S/H = 3, S/D = 10,and Mesh T3 is used.

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Chapter 5

Conclusions and future works

From the numerical results, we have some conclusions as following:

• The grid resolution test is necessary. From the pressure and velocity profiles, we can take less costto detect the pitchfork bifurcation points by choosing an appropriate mesh.

• A time integration approach may take a long time to investigate the behavior of solutions and ishighly dependent on initial conditions. Comparing with others researches, we can confirm thatthe linear stability analysis is a good choice to detect the critical bifurcation point. Moreover, asteady-state solution can be efficiently located with robust solver and can be tracked using continu-ation techniques. Hence, the qualitative information of flows as parameter changing can be readilydetermined.

• A large-scale eigenvalue problem can not be solved by a direct method, so a iterative method mustbe used. An implicit Arnoldi method with a Cayley transformation is a good approach for linearstability analysis. But the resulting linear systems will be difficult to solve because the rate of con-vergence of a Krylov-based iterative method depends strongly upon the spectral condition number.Instead, a SuperLU factorization shows a good performance of solving the linear system.

• Information resulting from a small-scale system with a coarse mesh is useful. With the spectrum,it can supply a direction for choosing the parameters σ and τ in the application of the Cayleytransformation such that the linear system has a good spectral condition number.

In the future, we hope to use the technique of linear stability analysis in more complex applications.In particular, we are interested in detecting the Hopf bifurcation of incompressible flows. Finally, wemention the concept of Hopf bifurcation for further research study in the future.

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Hopf bifurcation

First, we consider one complex dynamical system, whose phase portrait is shown in Figure 2.2 [31].The fixed points A, B, and C represent equilibria of the system and are classfied unstable, whereas theclosed orbit D herein is stable.

Figure 5.1: A phase portrait of one more complex dynamical system [31]

A closed orbit, like D in Figure 2.2, corresponds to periodic solutions. i.e., solutions for whichx(t+T ) = x(t) for all t, for some T > 0 . Moreover, a limit cycle is defined to be an isolated closed orbit.The word ” isolated ” means that neighboring trajectories are not closed; they spiral either toward oraway from the limit cycle. The Figure 2.3[1] just illustrate three kinds of limit cycles.

Figure 5.2: Limit cycles [31].

In general, limit cycles are inherently nonlinear phenomena; they can’t occur in linear systems. Ofcourse, a linear system

·x= Ax can have closed orbits, but they won’t be isolated. For example, if x(t) is

a periodic solution, then so is cx(t) for and constant c 6= 0. Hence, x(t) is surrounded by one-parameterfamily of closed orbits, such as in Figure 2.4 [31].

Figure 5.3: Closed orbits are not isolated [31].

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Now, suppose we have a physical system added small disturbation, through an exponentially dampedperturbation, at fixed points for a while . Suppose that the decay rate depends on a characteristic pa-rameter µ. If the decay becomes slower and slower after a sufficiently long time, and finally changes togrowth at a critical value µc, such as respectively shown in Figure 2.7(a) and (b) [31], the fixed pointswill lose stability. Then we say that the system has undergone a supercritical Hopf bifurcation.

Figure 5.4: The effect of small disturbation for a physical system after a long time [31].

Also, there are two standard types of Hopf bifurcations : supercritical and subcritical Hopf bifurca-tions. Here we just consider the supercritical Hopf bifurcations. To illustrate this concept of a supercriticalHopf bifurcation, we take an example given by the following system from Strogatz’s textbook [31] :

·x= µx− x3

·θ= w + bx2

Figure 2.8 [31] shows the phase portraits for µ. We can observe some interesting Phenomena : first,for µ < 0, at the origin r = 0 there is a stable spiral whose sense of rotation depends on ω. For µ = 0,the origin is still a stable spiral, although a very weak one. Finally, for µ > 0, there is an unstable spiralat the origin and a stable circular limit cycle at r =

√µ.

Figure 5.5: The phase portraits for µ above and below the supercritical Hopf bifurcation [31].

From this example we can obtain an important information : In terms of the flow in phase space, asupercritical Hopf bifurcation occurs when a stable spiral changes into an unstable spiral surrounded bya small limit cycle.

Note that in the example there are two much more important parameters : µ controls the stabilityof the fixed point at the origin, and ω gives the frequency of oscillations, Moreover, we can rewrite thesystem in Cartesian coordinates and then translate it into an eigenvalue problem. Let x = rcosθ andy = rsinθ. Then

35

·x =

·r cosθ − r

·θ sinθ

= (µr − r3)cosθ − r(ω + br2)sinθ

=[µ− (x2 + y2)x

]−[ω − b(x2 + y2)

]y

= µx− ωy + cubic terms.

and similarly,

·y = ωx− µy + cubic terms.

So the Jacobian at the origin is

J =[µ −ωω µ

],

which has eigenvalues λ = µ± iω.

It is clear that the eigenvalues of the Jacobian are dependent upon the parameter µ. For this two-dimensional system, we have known that fixed points can lose stability as parameter µ varies from µ < 0to µ > 0. At the moment, the sign of real part of eigenvalues change from negative to positive. In brief,we can compute the corresponding eigenvalues of the origin system to predict if the structure of flowschange, or to predict if a bifurcation occur.

Because bifurcations may occur in phase spaces of any dimension n ≥ 2, we extend the techniqueof computing the corresponding eigenvalues of the origin system for any dimension n ≥ 2 in the latersection.

36

Bibliography

[1] Online CUBIT user’s manual. http://cubit.sandia.gov/documentation.html.

[2] ParaView homepage. http://www.paraview.org.

[3] N. Alleborn, K. Nandakumar, H. Raszillier, and F. Durst. Further contributions on the two-dimensional flow in a sudden expansion. Journal of Fluid Mechanics, 330:169–188, 1997.

[4] S. Balay, K. Buschelman, W.D. Gropp, D. Kaushik, M.G. Knepley, L.C. McInnes, B.F. Smith, andH. Zhang. Petsc web page. http://www.mcs.anl.gov/petsc.

[5] F. Battaglia, S.J. Tavener, A.K. Kulkarni, and C.L. Merkle. Bifurcation of low Reynolds numberflows in symmetric channels. AIAA Journal, 35:99–105, 1997.

[6] W. Cherdron, F. Durst, and JH Whitelaw. Asymmetric flows and instabilities in symmetric ductswith sudden expansions. Journal of Fluid Mechanics Digital Archive, 84(01):13–31, 1978.

[7] K.A. Cliffe, T.J. Garratt, and A. Spence. Eigenvalues of block matrices arising from problems influid mechanics. SIAM Journal on Matrix Analysis and Applications, 15:1310–1318, 1994.

[8] J. Demmel, J. Gilbert, and X. Li. Superlu homepage. http://www.cs.berkeley.edu/ dem-mel/SuperLU.html.

[9] D. Drikakis. Bifurcation phenomena in incompressible sudden expansion flows. Physics of Fluids,9:76–87, 1997.

[10] F. Durst, J.C.F. Pereira, and C. Tropea. The plane symmetric sudden-expansion flow at low Reynoldsnumbers. Journal of Fluid Mechanics, 248:567–581, 1993.

[11] J. A. Elbanna, H.; Sabbagh. Interaction of two nonequal plane parallel jets. AIAA Journal, 25:12–13,1987.

[12] R. M. Fearn, T. Mullin, and K. A. Cliffe. Nonlinear flow phenomena in a symmetric sudden expan-sion. Journal of Fluid Mechanics, 211:595–608, 1990.

[13] L.P. Franca and S.L. Frey. Stabilized finite element methods. II: The incompressible Navier-Stokesequations. Computer Methods in Applied Mechanics and Engineering, 99:209–233, 1992.

[14] A. Greenbaum. Iterative Methods for Solving Linear Systems. SIAM, Philadelphia, PA, 1997.

[15] S.K. Hannani, M. Stanislas, and P. Dupont. Incompressible Navier-Stokes computations with SUPGand GLS formulations - a comparison study. Computer Methods in Applied Mechanics and Engi-neering, 124:153–170, 1995.

[16] T. Hawa and Z. Rusak. The dynamics of a laminar flow in a symmetric channel with a suddenexpansion. Journal of Fluid Mechanics, 436:283–320, 2001.

[17] V. Hernandez, J.E. Roman, and V. Vidal. SLEPc: A scalable and flexible toolkit for the solution ofeigenvalue problems. ACM Trans. Math. Software, 31:351–362, 2005.

[18] C.Y. Huang and F.N. Hwang. Parallel pseudo-transient newton–krylov–schwarz continuation al-gorithms for bifurcation analysis of incompressible sudden expansion flows. Appl. Numer. Math.,60(7):738–751, 2010.

37

[19] M. Kadja and G. Bergeles. Numerical investigation of bifurcation phenomena occurring in flowsthrough planar sudden expansions. Acta Mechanica, 153:47–61, 2002.

[20] G. Karypis. METIS homepage. http://cubit.sandia.gov/documentation.html.

[21] R.B. Lehoucq and A.G. Salinger. Large-scale eigenvalue calculations for stability analysis of steadyflows on massively parallel computers. International Journal for Numerical Methods in Fluids,36:309–327, 2001.

[22] K. Meerbergen, A. Spence, and D. Roose. Shift-invert and cayley transforms for the detection ofeigenvalues with largest real part of nonsymmetric matrices. BIT, 34:409–423, 1994.

[23] David R. Miller and Edward W. Comings. Force-momentum fields in a dual-jet flow. Journal ofFluid Mechanics, 7:237–256, 1960.

[24] David R. Miller and doi:10.1017/S0022112057000440 Edward W. Comings Static pressure distribu-tion in the free turbulent jet. Journal of Fluid Mechanics. Static pressure distribution in the freeturbulent jet. Journal of Fluid Mechanics, 3:1–16, 1957.

[25] S. Mishra and K. Jayaraman. Asymmetric flows in planar symmetric channels with large expansionratio. International Journal for Numerical Methods in Fluids, 38:945–962, 2002.

[26] J. Mizushima and Y. Shiotani. Structural instability of the bifurcation diagram for two-dimensionalflow in a channel with a sudden expansion. Journal of Fluid Mechanics, 420:131–145, 2000.

[27] Y. Saad. Iterative Methods for Sparse Linear Systems. PWS, Boston, MA, 1996.

[28] Y. Saad and Martin H. Schultz. Gmres: a generalized minimal residual algorithm for solving non-symmetric linear systems. SIAM Journal on Scientific and Statistical Computing, 7(3):856–869,1986.

[29] M. Shapira, D. Degani, and D. Weihs. Stability and existence of multiple solutions for viscous flowin suddenly enlarged channels. Computing Fluids, 18(3):239–258, 1990.

[30] C.Y. Soong, P.Y. Tzeng, and C.D. Hsieh. Numerical investigation of flow structrue and bifurcationphenomena of confined plane twin-jet flow. Physics of Fluids, 10:2909–2921, 1998.

[31] S.H. Strogatz. Nonlinear Dynamics and Chaos. Addison-Wesley Reading, 1994.

[32] T.E. Tezduyar. Stabilized finite element formulations for incompressible flow computations. Adv.Appl. Mech., 28:1–44, 1991.

[33] E.M. Wahba. Iterative solvers and inflow boundary conditions for plane sudden expansion flows.Appl. Math. Model., 31:2553–2563, 2007.

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A Study for Linear Stability Analysis of Incompressible Flows on Parallel Computers