Instabilities in pulsating pipe flow

of shear-thinning and shear-

thickening fluids

Sasan Sadrizadeh Division of Applied Thermodynamics and Fluid Mechanics

Degree Project

Department of Management and Engineering

LIU-IEI-TEK-A--12/01360--SE

Supervisor: Professor Luca Brandt Examiner: Professor Matts Karlsson

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1

Abstract

n this study, we have considered the modal and non-modal stability of fluids with shear-

dependent viscosity flowing in a rigid straight pipe. A second order finite-difference code is

used for the simulation of pipe flow in the cylindrical coordinate system. The Carreau-

Yasuda model where the rheological parameters vary in the range of and is represents the viscosity of shear- thinning and shear thickening fluids. Variation of

the periodic pulsatile forcing is obtained via the ratio and set between 0.2 and 20. Zero

and non-zero streamwise wavenumber have been considered separately in this study.

For the axially invariant mode, energy growth maxima occur for unity azimuthal wave number,

whereas for the axially non-invariant mode, maximum energy growth can be observed for

azimuthal wave number of two for both Newtonian and non-Newtonian fluids. Modal and non-

modal analysis for both Newtonian and non-Newtonian fluids show that the flow is

asymptotically stable for any configuration and the pulsatile flow is slightly more stable than

steady flow. Increasing the maximum velocity for shear-thinning fluids caused by reducing

power-low index n is more evident than shear-thickening fluids. Moreover, rheological

parameters of Carreau-Yasuda model have ignored the effect on the peak velocity of the

oscillatory components. Increasing Reynolds number will enhance the maximum energy growth

while a revers behavior is observed by increasing Womersley number.

Keywords: Stability, Pulsatile pipe, Transient Growth, Floquet Multiplier, Modal and non-Modal Stability

I

2

Nomenclature ny = Number of point in radial direction

U0 = Maximum velocity of steady component

ν = Kinematic viscosity (

)

Re = Reynolds numbers (

)

Wo = Womersley number ( √ )

Nt = Number of intervals in which the base flow period is divided (computed)

eiglin = Eigenvalues of direct problem (Floquet analysis)

eigadj = Eigenvalues if adjoint problem

eigopt = Eigenvalues of A*At (singular value)

Gamma ( ) = Amplitude of Pulsation (

)

Alpha (α) = Streamwise wavenumber

m = Azimuthal wavenumber

Lambda (λ) = Material coefficients in Carreau-Yasuda model (Material time constant)

n = Material coefficients in Carreau-Yasuda model (Power-low index)

µ0 = Zero shear rate viscosity

µ∞ = Infinite shear rate viscosity

= Strain rate

3

Contents Upphovsrätt ............................................................................................................................................ 2

Copyright ................................................................................................................................................ 2

Abstract .................................................................................................................................................. 1

Nomenclature ......................................................................................................................................... 2

Table of figures ........................................................................................................................................ 4

1. Introduction .................................................................................................................................... 5

1.1 Hydrodynamic instability and pulsatile pipe flow, Newtonian fluid ........................................... 5

1.2 Hydrodynamic instability and pulsatile pipe flow, non-Newtonian fluid .................................... 7

1.3 Eigenvalues and Floquet multiplayer (Modal analysis) .............................................................. 8

1.4 Transient growth (Non-Modal analysis) .................................................................................... 9

1.5 Objective of this work ............................................................................................................ 10

2. Problem definition ......................................................................................................................... 11

2.1 Governing equation ............................................................................................................... 11

2.2 Viscosity model ...................................................................................................................... 12

3. Linear Stability Analysis.................................................................................................................. 13

3.1 Linearized equations .............................................................................................................. 13

4. Numerical solution ........................................................................................................................ 15

4.1 Numerical method ................................................................................................................. 15

4.2 Geometry, meshing and boundary conditions ........................................................................ 16

4.3 Code Validation ...................................................................................................................... 16

5. Results and discussion ................................................................................................................... 18

5.1 Newtonian fluid ..................................................................................................................... 18

5.2 Non-Newtonian fluid .............................................................................................................. 25

6. Conclusion ..................................................................................................................................... 35

Reference .............................................................................................................................................. 36

4

Table of figures Figure 1: Non-Newtonian fluid with different shear strain rate ................................................................. 7 Figure 2: Transient growth of the resulting of two non-orthogonal vectors. ............................................ 10 Figure 3: Viscosity of Carreau-law model versus shear rate for n = 0.5 (left) and λ = 10......................... 13 Figure 4: Radial velocity profiles at different base flow phases (Re = 1000, = 2, Wo = 1, 5, 10) .......... 15 Figure 5: The simple sketch of mesh along pipe radius which is divided in 3 intervals ........................... 16 Figure 6: Transient growth error plotted against reference data (Schmid & Henningson, 2001) .............. 16 Figure 7: Validation of Transient growth versus time for circular pipe flow with reference data (Schmid &

Henningson, 1944) ................................................................................................................................ 17 Figure 8: Validation of Floquet exponents with reference data (Fedele et al., 2003) and (Schmid &

Henningson, 2001) ................................................................................................................................ 18 Figure 9: Dominant Floquet multiplayers as function of Womersley number (Re = 1000, Wo = 10) ....... 19 Figure 10: Plot of Floquet exponent with axisymmetric perturbation (Re=2000, Wo=40, =2 α=1) ........ 19 Figure 11: The optimal transient growth for axially invariant mode as a function of time for different

values of ............................................................................................................................................ 20 Figure 12: The optimal transient growth for axially non-invariant mode as a function of time ........ 21 Figure 13: dominant perturbation structure in the -plane of the pipe (Re=1000, Wo =10, =2) ...... 22 Figure 14: The optimal transient energy growth as a function of time for several phases ................ 23 Figure 15: as a function of for different values of azimuthal wavenumber (Re = 1000, Wo = 10)

.............................................................................................................................................................. 23 Figure 16: as a function of amplitude of pulsation for different values of Womersley number ( Re

= 1000).................................................................................................................................................. 24 Figure 17: as a function of Reynolds number for different values of Womersley number. ........... 24 Figure 18: Maximum transient growth scaled with quadratic Reynolds number plotted against the

Reynolds number .................................................................................................................................. 25 Figure 19: Velocity profiles along the pipe radius for different power-law index n ................................. 26 Figure 20: Effect of power-low index on steady base flow, (Re = 1000, λ=10) ....................................... 26 Figure 21: Effect of λ on peak velocity of base flow (Re = 1000, n=0.9) ................................................ 27 Figure 22: for different values of λ (left) and n (right) with purely pulsatile base flow ............... 27 Figure 23: as function of power-law index n (left) and material time constant λ (right) ............... 28 Figure 24: Dominant Floquet multiplayer as function of Power-law index (left) and λ (right) ................. 28 Figure 25: Plot of Floquet exponent for axially non-invariant, axisymmetric perturbation ...................... 29 Figure 26: Transient Growth versus power-law index with Steady base flow (Line) and pulsatile base

flow (●-marker) ..................................................................................................................................... 30 Figure 27: Transient growth as a function of time with purely pulsatile flow (Re=1000, Wo =10, =2,

λ=10) .................................................................................................................................................... 30 Figure 28: dominant perturbation structure in the -plane of the pipe .............................................. 31 Figure 29: as a function of Reynolds number with non-Newtonian flow ( =2, λ=10, n=0.8) ....... 31 Figure 30: optimal energy density growth as a function time for different phase-shift ............................ 32 Figure 31: The maximum energy growth as a function of Womersley number for several phase-shift ..... 33 Figure 32: The dominant transient energy growth as a function of amplitude of pulsation for several

phase-shifts (Re=1000, Wo =2, =10, n=0.6, )................................................................. 33 Figure 33: Maximum transient growth scaled with quadratic Reynolds number as a function of Reynolds

number .................................................................................................................................................. 34

5

1. Introduction Pipe flows are found in many engineering applications, mainly in nuclear reactors, power

generation and petrochemical industries. Moreover, blood flow through vessels is usually

studied as fluid flow through a pipe (Hale et al., 1955). The pulsatile, incompressible,

Newtonian pipe flow is one of the most studied problems in fluid mechanics over the past

few decades.

The flow behavior is governed mostly by the ratio of viscous forces to the inertial forces

in the flow. According to this, the fluid flow can be either laminar or turbulent. This ratio

is characterized by the Reynolds number, which is high for inertia-dominated turbulent

flows.

Pipe flows are laminar at the Reynolds number less than the critical value (

(Avila et al., 2011)), while beyond the critical value turbulent flow can be sustained.

Instability analysis can reveal the mechanisms which trigger the transition from the

laminar to the turbulent flow.

In this work, we will consider pulsatile pipe flow of non-Newtonian fluids, in particular

pseudo-plastic (shear-thinning) and dilatant fluids (shear thickening) which are in the

class of inelastic fluids. Viscosity of shear- thinning fluids is modeled by the Carreau-law

which represents a decrease/increase of the ratio between shear stress and deformation

rate in the flow.

1.1 Hydrodynamic instability and pulsatile pipe flow, Newtonian fluid The main feature of hydrodynamic instability analysis concerns the stability and

instability of fluid movement. The important problems of hydrodynamic stability were

mathematically characterized in the nineteenth century, particularly by Osborne Reynolds

(Drazin, 2002). Transition from laminar to turbulence in a pipe flow is still considered

among the most reliable studies. This type of flow is interest since there is no critical

Reynolds number above which solutions grow exponentially (Schmid & Henningson,

1944). A large number of numerical and analytical studies have been shown that pipe

Poiseuille flow is linearly stable for all Reynolds numbers (Garg & Rouleau, 1972;

Lessen et al., 1968; Schmid & Henningson, 1944). But experimental investigations show

that, for Reynolds numbers larger than 2000, pipe Poiseuille flow accommodates growing

perturbations (Salwen et al., 1980). Even though, due to lacks a reasonable explanation of

stability problem for pipe flow, several investigators have tried to resolve this

inconsistency between experiments and the linear theory.

(Tatsumi, 1952) investigated the linear stability of the inlet-flow of Poiseuille pipe flow

and found the critical Reynolds number of approximately 9700 for this type of flow.

A reason of inconsistency between experiment and linear theory has been proposed by

nonlinear effects. (Davey & Salwen, 1994) considered the stability of pipe flow to

infinitesimal axisymmetric perturbations and showed that, nonlinear instabilities is causes

by center mode, while wall mode has negligible effect on the stability of this type of

flow.

6

Transient growth of perturbations is associated to instabilities which may initially grow

before the final decay. In stable systems, transient growth can be explained by the non-

normality of the linearized Navier-Stokes operator.

(Schmid & Henningson, 1944) investigated the linear stability of circular pipe flow and

determined that for zero streamwise wavenumber the ratio of energy growth to the square

of the Reynolds number is solely dependent on the azimuthal wavenumber. Also they

found that largest energy density growth is monotonically increasing with Reynolds

number.

(Boberg & Brosa, 1988) studied the nonlinear initial value problem for the pipe flow and

suggested a combination of linear and nonlinear mechanisms in the transition to

turbulence in a pipe. (Bergström, 1993) considers the energy density growth which is

associated with the zero and non-zero streamwise wavenumbers. He shows that, for

axially invariant mode, the large transient growth has been found for unity azimuthal

wavenumber. He reported that if streamwise wavenumber increases the amplification

decreases, although the azimuthal and the radial components are similarly amplified. In a

similar attempt, (Reddy & Henningson, 1993) demonstrated that, the largest energy

density growth has been detected for the perturbations with zero streamwise wavenumber

in accordance with plane shear flows.

Recently, numerous numeric and experimental studies has been carried out to investigate

the stability mechanism of time-dependent flows in the pipes (Trip et al., 2012; Nebauer

& Blackburn, 2009; Yang & Yih, 1977; Zhao et al., 2004). From mathematical point of

view, pulsatile pipe flows are asymptotically stable and therefore, the most dangerous

instability mechanisms are related to its transient dynamics. Usually the pulsatile flow

considered as steady flow plus an oscillatory component, which can be described as the

superposition of steady flow and oscillations, at different temporal harmonics (Smith &

Blackburn, 2010). In linear theory, it suffices to study separately the linear stability of

each component.

Experimental investigation of the stability of pulsatile Pipe flow indicated that, pipe flow

stability depends on the Reynolds number based on steady velocity component of base

flow (Trip et al., 2012).

(Smith & Blackburn, 2010) have shown that, in pulsatile pipe flow, transient growth for

higher azimuthal wavenumbers is produce a larger value over a short time and decay

more quickly. Also, they found the same dependence scaling with quadratic Reynolds

number as (Schmid & Henningson, 1944). In addition (Fedele et al., 2005) illustrated that

pulsatile pipe flows are linearly stable for infinitesimal, axisymmetric perturbations.

Similar studies show that stability of pulsatile pipe flow will be trigger by increasing

Womersley number (Govindarajan, 2002). In a similar attempt, (Nebauer & Blackburn,

2009) illustrated that pulsatile pipe flows are linearly stable to both axisymmetric and

non-axisymmetric disturbances for all finite values of Reynolds and Womersley number.

Some previous studies demonstrate that the purely pulsatile pipe flow is also stable to

both axisymmetric and non-axisymmetric perturbation (Smith & Blackburn, 2010;

Nebauer & Blackburn, 2010). (Fedele et al., 2005) calculated the maximum energy

density growth over a range of Reynolds and Womersley numbers and found an upper

bound of for where beyond that the influence of pulsation forcing on the

stability of the flow can be safely neglected. Moreover, (Sarpkaya, 1966) reported that,

pulsating flow is more stable than the corresponding steady and fully developed

7

Poiseuille flow for the same pressure gradient. It means that, flow was more stable for

higher Womersley numbers. Additionally, they observed that starting transition in higher

mean flow occur with lower oscillating flow component.

1.2 Hydrodynamic instability and pulsatile pipe flow, non-Newtonian fluid There are few studies in the linear stability of pipe flows, which considered shear

dependent viscosity fluids (Pinho & Whitelaw, 1990; Esmael et al., 2010)

The time-independent behavior of fluids categorized into tree general classes. Newtonian

fluids, which the viscosity is independent of shear rate and a plot of shear rate as a

function of shear stress is linear, and passes through the origin (Abramowitz & Stegun,

1964). Shear thinning (also called pseudoplastic) material is one in which viscosity, the

measure of the resistance of a fluid which is being deformed by either shear stress or

tensile stress, decreases with an increasing with applied shear stress and shear thickening

fluids, (also termed dilatant) where the shear viscosity increases with the rate of shear

stress. (Fig. 1).

Figure 1: Non-Newtonian fluid with different shear strain rate

All the materials that are shear-thinning are thixotropic, in that they will always take a

limited time to bring about the redisposition needed. Shear thinning can take place for

many reasons, such as the arrangement of rod-like particles in the flow direction, failure

of junctions in concentrated polymer solutions, reordering of microstructure in

suspension (Barnes, 1997). Modern paints are examples of shear-thinning materials. By

applying force, the shear caused by the brush will allow them to thin and wet out the

surface consistently.

(Nouar et al., 2007) focus on the linear stability of pseudoplastic fluids modeled by the

Carreau-yasuda law in channel flow, and they found that the most stabilization occurs

with independency of power low index n for the material time constant .

(Pinarbasi & Liakopoulos, 1955) considered two-layer non-Newtonian fluid flow in

channel driven, and the presented that with two shear-thinning fluids, increasing in shear

thinning has a stabilized effect on the flow. In the same attempt, (Ranganathan &

Govindarajan, 2001) studied the stability of the channel flow with two different viscosity

Shear

rate

(

)

Shear Stress ( )

Newtonian

Shear thinning

Shear thickening

8

fluids, and shown that when the mixed layer between two fluids distinct, the flow is

slightly destabilized.

Non-Newtonian fluid flow in circular pipes was considered by (Yurusoy et al., 2006), and

they reported that increasing shear thinning effect will increase the maximum velocity in

the pipes. In the same way, (Pakdemirli & Yilbas, 2006) indicated that increasing non-

Newtonian parameters will reduce the temperature in the pipe as well as velocity. (Wong

& Jeng, 1986) studied the stability of two concentric non-Newtonian fluids in pipe flow

and reported that, the steady flow can become unstable, based on certain combinations of

non-Newtonian parameters, to minuscule axisymmetric perturbations of large

wavelengths, for any Reynolds number however small.

1.3 Eigenvalues and Floquet multiplayer (Modal analysis) Floquet theory is a mathematical framework suited to study the linear stability of a linear

periodic system. It is a part of ordinary differential equation (ODE) theory relating to the

class of solutions to linear differential equations and will designate the stability of the

system. Floquet exponents (multipliers) are related to the eigenvalues of the Jacobian

matrices of equilibrium points (Klausmeier, 2008).

In the stable case, if a system is initially disturbed around its steady position, it will

eventually return to its original location and remain there while unstable systems are

driven away from their steady configurations and cannot return to equilibrium. For

instance, pendulum is a stable system. If disturbed, it will swing around until gravity

brings it to its original position.

The eigenvalues of a system can determine the stability behavior of a system

corresponding to the real and imaginary components of the eigenvalues.

Consider a set of time-periodic linear differential equations

(1)

If is periodic with period T, then X doesn’t need to be periodic (Cantwell, 2009).

The general solution of eq. 1 must be of the form

∑ (2)

where also has period T. Here µ is a complex number called Floquet multiplier.

From Eq. 2, the solution to Eq. 1 is the sum of n periodic functions multiplied by

exponentially decaying or growing terms. The long-term behavior of the system depends

on this Floquet exponent. If all Floquet exponents are real and negative, then the system

will decay exponentially. If all Floquet multipliers real and anyone is positive, then the

system is unstable and ‖ ‖ . Otherwise, if Floquet multipliers are

imaginary, then system will oscillate.

Table 1 shows all possible values for the eigenvalues and the corresponding behavior of

the system.

9

At least one eigenvalues is real and positive. The system disturbance energy

increase exponentially

All eigenvalues are real and negative.

The system decay exponentially

All eigenvalues are complex with zero

real part. The system oscillating around

steady state.

All eigenvalues are complex with

negative real part. The system behaves

as a damped oscillator.

One eigenvalue is complex with positive real part. The system oscillates

with increasing amplitude.

Table 1: System behavior with different eigenvalues

1.4 Transient growth (Non-Modal analysis) Transient growth is a phenomena of instability in which perturbations may initially show

growth, although the flow is linearly stable. This is because of the non-normality of the

linearized Navier-Stokes operator. A normal operator, that is one which commutes has

pairwise orthogonal eigenvectors with its adjoint. Therefore, if all eigenvectors have

negative eigenvalues, and as a result, each eigenvector decays by the action of the

operator, then any vector spanned by the eigenvectors which will essentially decay. Note

that the non-normal operators do not have pairwise orthogonal eigenvectors. This will

increase the possibility of a disturbance vector to grow initially as a consequence of

different decay rates of the constituent eigenvectors. Figure 2 depicts the transient growth

caused by two non-orthogonal vectors.

time

time

time

time

time

10

Figure 2: Transient growth of the resulting of two non-orthogonal vectors.

Decay with different ratio induces an initial growth of a vector due to their non-

orthogonal interaction. This is because the damping rate of two adjacent sides is not the

same (Deshpande et al., 2010).

Transient growth is quantified by the energy perturbation at time t normalized by its

initial energy. As the perturbation equations are linear, we consider normalized initial

perturbations (Barkley et al., 2008).

‖ ‖ (3)

Using evolution operator , ( we will have:

( ) ( ) (4)

where is the adjoint operator to .

We find the dominant eigenvalues of that corresponds to the largest possible

growth at time t. If and characterize eigenvalues and normalized eigenvalues, we

have . The eigenfunction provides an initial perturbation ,

which produces a growth over time t and the maximum growth attainable at time t is

given by:

‖ ‖

(5)

Typically the dimension of the evolution operator, arising from the discretization of the

linearized Navier-Stokes equations, is large and has many thousands of data points.

1.5 Objective of this work The main goal of this work is to determine the neutral conditions and the instability

mechanism for the first unstable mode of shear-thinning and shear- thickening fluids of

pulsatile pipe flow. Pipe flow is asymptotically stable and therefore the most dangerous

instability mechanisms for the pulsating pipe are related to its transient dynamics. The

behavior of the arterial flow in response to the vascular fluctuations can have significant

effects on the vascular wall shear stress and vascular impedance. We want to examine the

pipe flow stability for any configuration considering modal and non-modal analysis. We

11

choose the rheological Carreau law, which is more compatible to bio fluids, to model the

viscous variation due to local shear rate. Additionally the perturbation kinetic energy

budget is considered showing how an additional production term related to the viscosity

variations amplifies the level of energy.

2. Problem definition In this section, we outline the governing equations and develop the mathematical

framework to analyze the linear initial value problem for the evolution of infinitesimal

disturbances for the incompressible pulsatile pipe flow.

2.1 Governing equation We consider a pulsatile pipe flow driven by an axial pressure gradient defined by

[ ] (6)

where is the phase shift and K is chosen in such a way to impose for the steady

component a unity maximum velocity. Thus, K varies depending on the parameter chosen

for the viscosity law. In particular with the Newtonian flow, we will have . is

the amplitude of the pressure oscillation with respect to the steady component ( )

and is the dimensionless angular frequency. The time scale is made dimensionless

using and the radius of the pipe. Here, the radius of the pipe is always taken

as .

Starting from the Navier-Stokes and the continuity equation, the following equations for

pipe flow in a cylindrical coordinate with non-Newtonian fluid can be derived.

( [

] )

( [

])

( [

])

( [

])

( [

])

( [

])

(7)

( [

] )

( [

])

( [

])

where is the material derivative in cylindrical coordinate and denote

the velocities in radial, azimuthal and axial direction. Navier–Stokes equations have been

non-dimensionalized by the radius of the pipe (R) and centerline velocity ( ) as below:

(8)

12

In terms of these variables, we will have the following dimensionless Navier–Stokes

equations:

(

( *

+ )

( *

+)

( *

+))

(

( *

+)

( *

+)

( *

+))

(9)

(

( [

] )

( [

])

( [

]))

where

are non-dimensional velocities components and Reynolds number define

as

(10)

2.2 Viscosity model Over the last decades, a few non-Newtonian models have been developed, describing the

shear thinning and shear thickening properties of different flows. Many non-Newtonian

constitutive models exist, tuned for specific fluids such as blood, polymers and paint. To

quantify the viscosity (μ) dependence on the flow, the Carreau-Yasuda rheological

model, probably the most developed and widely used, has been employed during this

study. This model was initially developed for polymers and describes the reaction

kinetics between particle chain formation and chain structure rupture due to varying

shear. It has enough flexibility to fit a wide range of experimental data describing the

relation between viscosity and rate of strain (Pinarbasi & Liakopoulos, 1955). The

relation between viscosity and deformation rate is:

[ ][ ]

(11)

Dividing equation (11) by yields the non-dimensional viscosity:

*

+ [ ]

(12)

where and is the viscosity at zero and infinite shear rate respectively, set to 1 and

0.001 in this study and is the second invariant of the strain rate tensor. This is

determined by the dyadic product

where . The infinite

shear-rate viscosity , which is typically related with a breakdown of the fluid, is often

considerably smaller than (Tanner, 2000). An example of non-Newtonian parameters

for polymer solutions is given by (Carreau, 1972). The predictions depend on the material

13

time constant λ, which reproduces the onset of shear thinning (thickening), and the

dimensionless power-law index n, which describes the degree of shear thinning ( )

or shear thickening ( ). Note that, for and/or , the Carreau-Yasuda

rheological model reproduces a Newtonian fluid of viscosity . If λ become very large,

the model reduces to the power-law ( )

( and is dimensional quantities).

"a" is a dimensionless parameter which describes the transition behavior between the zero

and infinite shear rate viscosity. For the Carreau-Yasuda model can be fitted to the

rheological behavior of many polymeric solutions (Pinarbasi & Liakopoulos, 1955).

A logarithmic plot of the viscosity as a function of shear rate is reported in figure 3 for

the Carreau-law model; this provides intuition on how the viscosity of a shear-thinning

fluid decreases when increasing the shear rate. The viscosity is equal to one at zero shear

rates in both cases and it tends to for extremely large shear rates. For fixed power-law

index ( ), the shear-thinning effects become more evident when increasing λ. We

observed reverse viscosity behavior by increasing power-law index n for a fixed material

time constant ( ).

3. Linear Stability Analysis

3.1 Linearized equations To study the stability of a base flow field, a minuscule perturbation is superimposed and

the equations have been linearized around the steady flow.

⏟

⏟

(13)

Figure 3: Viscosity of Carreau-law model versus shear rate for n = 0.5 (left) and λ = 10

14

Where and denotes the base flow and infinitesimal perturbation respectively.

Substitution the eq. (13) in Eq. (9) and neglecting nonlinear terms, leads to the following

equations governing the linear dynamics flow perturbation.

(

* (

)+

( *

+)

( *

+))

(

( *

+)

( *

+

)

( *

+)) (14)

(

( *

+ )

( *

+)

(

))

*

+

The fully-developed streamwise velocity satisfies the following initial boundary

value problem

(

) (15)

Considering no-slip boundary condition and boundedness of the velocity field at the

centerline of the pipe, the solution for the radial velocity profile for Newtonian

fluid is given by:

[

(

) ] (16)

Where Jo is the Bessel function of the first kind of zero order (Abramowitz & Stegun,

1964), μ is the viscosity. The amplitude of pulsation ( ) is defined as the ratio of

and the parameter Wo, known as the Womersley number, is defined by √ .

It may be seen as either the ratio of oscillatory inertia to viscous forces or as a Reynolds

number for the flow using ωR as the velocity scale. Figure 4 shows radial profiles of the

axial velocity, twenty profiles during one period of oscillation, for three different

Womersley numbers and Newtonian fluid.

15

Figure 4: Radial velocity profiles at different base flow phases (Re = 1000, = 2, Wo = 1, 5, 10)

4. Numerical solution

4.1 Numerical method The numerical computations in this thesis have been performed using a second order

finite-difference code. The CPL code was developed by Flavio Giannetti at University of

Salerno. To obtain more accurate results, stretching is implemented in the code to cluster

grid cells near the pipe wall. Eigenvalues and Eigen modes of both the direct and adjoint

linearized stability problem are computed by employing the Arnoldi shift and invert

method and sparse-matrix memory storage. The adjoint modes are computed as left

eigenvector of the system together with the direct modes. All the equations are

discretized in space by using a second order finite difference scheme on a smoothly

varying mesh. Base flow is obtained by marching the equation in time over the period

imposed by the pressure gradient several times until a periodic solution is obtained. The

time integration used is a standard Crank-Nicholson method. The eigenvalues are found

by discretizing the linearized equation with the same scheme used for the base flow.

Integration in time over a period is then linked to the Arpack package to find the

eigenvalues. Arpack requires the action of the Floquet transition matrix on a vector. This

is simply obtained as the output of the time integration over a period of the linearized

equation. The adjoint equations are obtained using the numerical adjoint (transposition of

the matrix). The adjoint code is obtained by making the adjoint of the single subroutines

(seen as input-output) which are used to build the direct linearized code and then calling

them in a reverse order. To find the optimal disturbance, the strategy used is to march

from [0...T] the direct linear equation and then with that solution go back from T to 0

using the adjoint equations. The main difference with the code based on the stream

function vorticity formulation is based on the fact that here an additional constrain is

imposed to the initial condition to make it divergence free. This is obtained by the

0 0.5 1-1

0

1

t/T

Pip

e D

iam

eter

Wo = 1

0 0.5 1-1

0

1

t/T

Pip

e D

iam

eter

Wo = 5

0 0.5 1-1

0

1

t/T

Pip

e D

iam

eter

Wo = 10

16

optimization at the end of the each direct-adjoint iteration. In this way, the new starting

solution for the next iteration is guaranteed to be divergence free.

4.2 Geometry, meshing and boundary conditions The geometry of the problem consists of a circular pipe with radius unity. The cylindrical

coordinate system has its origin in the center of the pipe. We use uniform and non-

uniform grid stretching. The pipe radius is divided in three intervals, to have better

control on the grid stretching. We use the uniform grid for first two intervals starting

from the center of the pipe ( and ) and adding more points in the last interval ( ) in

order to reduce the grid spacing near the wall. In this study, one dimensional grid along

the radius of the pipe has been used, and Fourier modes are assumed in the axial and the

azimuthal direction.

Figure 5: The simple sketch of mesh along the pipe radius which is divided in 3 intervals

4.3 Code Validation Maximum error in energy growth with different grid stretching obtained by CPL code

compare with reference data (Schmid & Henningson, 2001) is depicted in figure 6. The

maximum error in all cases is less than 0.2 %.

Figure 6: Transient growth error plotted against reference data (Schmid & Henningson, 2001)

Solid Line (Uniform grid) Dashed Line (Non-uniform grid)

20 40 60 80 100 120 140 160 180 200 2200

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

time

Gro

wth

Rate

Err

or

(%)

ny=100

ny=200

ny=300

17

The linear stability analysis has been validated against the results by (Schmid &

Henningson, 2001; Schmid & Henningson, 1944) and (Fedele et al., 2003) for Newtonian

fluid. It can be seen in Table 2 that the present results based on all stretched grids

represent an error less than 0.1%.

In the figures bellow, transient growth versus time validated against (Schmid &

Henningson, 1944). The CPL data are in a good agreement with the reference data.

In addition, eigenvalues for steady and oscillatory base flow validated against (Schmid &

Henningson, 2001) and (Fedele et al., 2003).

α=1 m=0

α =0.5 m=1

α =0.25 m=2

ny=100 ny=200 ny=300

ny=100 ny=200 ny=300

ny=100 ny=200 ny=300

Eigenvalue 1 0.013041 0.003259 0.001451

0.000103 0.000553 0.000529

0.004249 0.000903 0.000327

Eigenvalue 2 0.005266 0.001334 0.00059

0.013212 0.0025 0.000864

0.006016 0.001392 0.000503

Eigenvalue 3 0.030154 0.007525 0.003377

0.000199 0.000815 0.000397

0.072121 0.018382 0.008138

Eigenvalue 4 0.022432 0.005615 0.002476

0.021932 0.004251 0.001688

0.003735 0.001017 0.000575

Eigenvalue 5 0.059986 0.015075 0.006704

0.038188 0.00799 0.003163

0.050612 0.012823 0.005814

Eigenvalue 6 0.052672 0.013108 0.005856

0.036037 0.009113 0.00416

0.008447 0.002166 0.001213

Eigenvalue 7 0.04612 0.011279 0.005036

0.023037 0.005756 0.002329

0.007293 0.00198 0.001111

Eigenvalue 8 0.039851 0.009962 0.004446

0.004803 0.001046 0.000746

0.004668 0.001134 0.000492

Table 2: Relative error (%) for magnitude of eigenvalue with respect to reference data (Schmid & Henningson, 2001)

Figure 7: Validation of Transient growth versus time for circular pipe flow with reference data (Schmid & Henningson, 1944)

Solid line: Reference data, Dots: CPL Code, Re = 2000, Steady (𝝎 𝟎 𝚪 𝟎)

(a) 𝜶 𝟎 (b) 𝜶 𝟎 𝟏 (c) 𝜶 𝟏

18

5. Results and discussion We consider the linear stability of steady and pulsatile flow. Results obtained with shear

dependent viscosity are compared with those for Newtonian fluids. The comparison is

done by keeping the same pressure gradient and same zero-shear-rate viscosities

in the Carreau-Yasuda model.

5.1 Newtonian fluid In this part, we investigate the effect of different parameters (azimuthal (m) and streamwise

(α) wavenumber, Reynolds (Re) and Womersley number (Wo), Amplitude of pulsation ( )) on

transient growth and Floquet exponents. First we consider modal analysis of steady and

pulsatile pipe flow. A plot of the dominant Floquet multipliers as a function of

Womersley number for different values of the azimuthal wave number is depicted in

figure 9 for both axially invariant and non-invariant modes.

Figure 8: Validation of Floquet exponents with reference data (Fedele et al., 2003) and (Schmid & Henningson, 2001)

Left (Re = 3000, 𝚪 𝟐 𝛚 𝟏 𝛂 𝟏 𝐦 𝟎) Right (Re = 2000, 𝚪 𝟎 𝛚 𝟎 𝛂 𝟏 𝐦 𝟎)

19

First we see that the flow is asymptotically stable for any configuration as also shown in

previous studies (Zhao et al., 2004; Schmid & Henningson, 1944). This figure also

illustrates that dominant axially invariant, axisymmetric mode is significant less stable

than non-axisymmetric Floquet modes, although the axially non-invariant mode, with a

non-axisymmetric perturbation ( ) is less stable than the other axisymmetric and

non-axisymmetric modes. Increasing the azimuthal wavenumber we see a more stable

flow both for and .

To investigate the effect of pulsatile base flow on eigenvalues, as an example, we

consider an axial, axisymmetric perturbation with unity wavenumber and a time-periodic

forcing of the base flow characterized by . The characteristic Floquet exponents are

plotted in Figure 10 in the complex plane for and . For comparison

purposes, a plot of the eigenvalues of the steady Poiseuille flow is also shown in the

figure 10.

Figure 10: Plot of Floquet exponent with axisymmetric perturbation (Re=2000, Wo=40, =2 α=1)

-1 -0.5 0 0.50

0.5

1

1.5

2

2.5

Imaginary ( )

-Real (

)

Steady

pulsatile

Figure 9: Dominant Floquet multiplayers as function of Womersley number (Re = 1000, Wo = 10)

axially invariant mode α=0 (left) and axially non-invariant mode α=1 (right)

20

As depicted here, the real part of the Floquet exponents corresponding to the pulsatile

base flow are slightly more negative than their steady equivalents, indicating that the

oscillatory flow is somewhat more stable than the steady Poiseuille flow (Fedele et al.,

2005).

Next, we study the non-modal behavior of the system, which is observed to be relevant

for subcritical transition in pipe flows. In figure 11, we report the optimal transient

growth as a function of time for different values of for axially invariant modes.

Obviously the largest transient growth is attained at azimuthal wave number equal to

unity ( ). As the azimuthal wave number is increased, the maximum transient

growth is decreased. For , reducing the amplitude of pulsation ( ) causes an

increase in the maximum transient growth ( ), whereas for , has a negligible

influence on the transient growth of the pulsatile flow. Previous studies have shown that

the transient growth of the axially invariant modes is dependent only upon a single

control parameter, the azimuthal wave number, and partly independent of the Womersley

number which is also the only parameter needed to describe the radial velocity profiles of

the base flows (Nebauer & Blackburn, 2009; Nebauer & Blackburn, 2010).

Figure 11: The optimal transient growth 𝑮 𝒕 for axially invariant mode as a function of time for different values of 𝚪

(Re = 1000, Wo = 10) (a) m = 1, (b) m =2, (c) m =3, (d) m =4

21

The same analysis is repeated for the axially non-invariant modes. Figure 12 illustrates

the optimal energy growth as a function of time for different values of .

The figure demonstrates that for an axial perturbation with wavenumber equal to one, the

most dangerous disturbance has an azimuthal wave number . In addition,

increasing the amplitude of the pulsation causes a significant reduction of the peak value

of transient growth.

Zero and non-zero streamwise perturbation structure was depicted for the least stable

perturbations in figure 13.

Figure 12: The optimal transient growth 𝑮 𝒕 for axially non-invariant mode as a function of time

(Re = 1000, Wo = 10) (a) m = 1, (b) m =2, (c) m =3, (d) m =4

22

The structure within the -plane perpendicular to the streamwise coordinate axis of

the pipe was observed to be a counter rotating vortex pair (CRVP) close to the pipe center

for both zero and non-zero streamwise wavenumber. The non-modal stability analysis for

pipe Poiseuille flow similarly identified optimal perturbations for unity azimuthal

wavenumber with slight streamwise dependence (Schmid & Henningson, 1944).

Furthermore, the axially invariant modes have been recognized as the least stable modes

in the asymptotic stability analysis of pulsatile flows (Nebauer & Blackburn, 2009).

For axially invariant mode, the flow field is characterized by a pair of stronger counter-

rotating vortices compare to the axially non-invariant mode near the center of the pipe.

When the base flow is time-dependent, an additional parameter, phase , should be

entered into consideration (Blackburn et al., 2008). The phase , at which the

perturbation is initiated relative to that of the base flow. This is accommodated solely by

time-shifting the axial pressure gradient (see eq. 6). In the previous figures we first fixed

attention on the case where . Figure 14 shows optimal energy growth as a function

of time for different values of time-shift.

Figure 13: dominant perturbation structure in the 𝒓 𝜽 -plane of the pipe (Re=1000, Wo =10, 𝜞=2)

𝜶 𝟎 𝒎 𝟏 (left) 𝜶 𝟏 𝒎 𝟐 (right)

23

For both zero and non-zero streamwise wavenumbers, energy density growth maxima

was achieved as the curve emanating from phase-shift .

Figure 15 summarizes the previous results and displays the effect of on the maximum

transient growth for axially invariant and axially non-invariant modes.

Figure 15: as a function of for different values of azimuthal wavenumber (Re = 1000, Wo = 10)

α = 0 (left) and α = 1 (right)

For the axially non-invariant mode, increasing the amplitude of oscillation exhibits a

large variation in , while for the axially invariant mode, the maximum transient

growth is weakly changing with . This seems to indicate that, the long perturbations are

not sensitive to the pulsating components of the flow. Most importantly, the largest

transient growth is observed for .

0 10 2020

30

40

50

60

70

= 0

Gm

ax

m = 1

m = 2

m = 3

m = 4

0 10 20

10

15

20

25

30

35

40

45 = 1

Gm

ax

Figure 14: The optimal transient energy growth 𝑮 𝒕 as a function of time for several phases-shift

(Re=1000, Wo =2, 𝜞=2), 𝜶 𝟎 𝒎 𝟏 (left) 𝜶 𝟏 𝒎 𝟐 (right)

24

Figure 16 shows contour plots of versus Womersley number and amplitude of

pulsation Γ for the two azimuthal wave-numbers yielding the largest possible growth.

Figure 16: as a function of amplitude of pulsation for different values of Womersley number ( Re = 1000).

Left (α=0, m=1) Left (α=1, m=2)

As shown before, axially invariant modes are not affected by the Womersley number and

the amplitude Γ, whereas for streamwise-dependent modes the largest transient growth

can be seen when the base flow is closer to the non-pulsatile case, indicating that the

periodic flow is more stable than the steady flow.

To illustrate the effect of the Reynolds number on the maximum energy growth,

versus Re is plotted in figure 17 for different values of Womersley number and a strongly

oscillatory forcing of the basic flow characterized by .

Figure 17: as a function of Reynolds number for different values of Womersley number .

For comparison purposes, the plot of for the case of steady Poiseuille flow is also illustrated.

The largest energy growth occurs with the steady flow. Increasing the Reynolds number

casus to increases quadratically, same as previous studies (Fedele et al., 2005). Since

the Womersley number is reduced, the flow perturbation results in smaller than its

steady counterpart. We found an upper bound of twenty on Womersley number for the

Wo

= 0, m = 1

0 5 10 15 20

10

15

20

25

30

35

40

67

68

69

70

71

72

73

Wo

= 1, m = 2

0 5 10 15 20

5

10

15

20

25

30

35

40

5

10

15

20

25

30

35

40

0 1000 2000 3000 4000 50000

250

500

750

1000

1250

1500

1750

2000

Reynolds number

Gm

ax

= 0, m = 1

Steady

Wo = 5

Wo = 10

Wo = 20

0 1000 2000 3000 4000 50000

50

100

150

200

250

300

350

Reynolds number

Gm

ax

= 1, m = 2

Steady

Wo = 5

Wo = 10

Wo = 20

Wo = 30

25

axial invariant mode which the effect of oscillatory forcing has negligible influence on

maximum energy growth beyond the upper bound. Furthermore, there is an upper bound

of thirty on the Womersley number with the same behavior for the case of axially non-

invariant mode.

Maximum energy growth was illustrated to scale with quadratic Reynolds number and is

shown in Figure 18.

Figure 18: Maximum transient growth scaled with quadratic Reynolds number plotted against the Reynolds number

α=0 (left) α=1 (right)

Transient growth dependence scaling with Reynolds number squared has been found in

computational and analytical studies for parallel shear flows (Schmid & Henningson,

1944; Smith & Blackburn, 2010; Kreiss et al., 1994).

We found that, for the zero streamwise wavenumber, a negligible dependence of the

Reynolds number ratio is observed when scaling optimal transient growth with Re2. The

energy growth ratio to the square of the Reynolds number is only dependent on azimuthal

wave number, whereas the maximum energy growth is not constant by changing

Reynolds number for the axially non-invariant mode.

5.2 Non-Newtonian fluid The Carreau-Yasuda model introduced above models the shear-thinning behavior of the

flow. First of all, we want investigate the effect of the power-law index n and the material

time constant λ on the base flow. Figure 19 shows the velocity profile for several phase-

shifts for three different values of the power-law index.

2000 40002

3

4

5

6

7

8x 10

-5 = 0

Re

(Gm

ax)/

Re2

2000 4000

0.03

0.04

0.05

0.06

0.07

Re

(Gm

ax)/

Re

= 1

m = 1

m = 2

m = 3

m = 4

0 T/2 T-1

0

1

Pip

e D

iam

eter

Power = 1

26

Figure 19: Velocity profiles along the pipe radius for different power-law index n

Re = 1000, = 2, Wo = 2, λ = 10

Reducing the power-law index n will increases the shear-thinning properties of the fluid

and thus the maximum velocity of the base flow will increase. Reducing the viscosity

increases the rate of shear strain near the pipe wall and enhances the peak velocity inside

the pipe. Consequently, low rate of fluid strain leads to low mean and maximum

velocities in the pipe flow (Yurusoy et al., 2006; Pakdemirli & Yilbas, 2006).

As mentioned before, we consider the flow as created by a steady pressure gradient and

an oscillatory component. Although the two components cannot be decoupled in the case

of non-Newtonian fluids (nonlinear equation for the parallel base flow), so we look at

them separately to try to understand more. Figure 20 focuses solely on the influence of

power-law index n on the steady component of the base flow.

Figure 20: Effect of power-low index on steady base flow, (Re = 1000, λ=10)

Shear thickening (left) and Shear thinning (right)

Furthermore, figure 21 illustrates the effect of λ on the maximum velocity. Comparing

with figure 20, we see that the peak velocity is affected more by the power-law index n,

while the material time constant λ has little effect on the maximum centerline velocity of

the base flow. Increase of the peak velocity for shear-thinning fluids is considerably more

0 T/2 T-1

0

1

Pip

e D

iam

eter

Power = 0.8

0 T/2 T-1

0

1

Pip

e D

iam

eter

Power = 0.6

0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

UBase

Flow

Pip

e D

iam

ete

r

= 10

n=1.5, Umax=0.44

n=1.4, Umax=0.49

n=1.3, Umax=0.56

n=1.2, Umax=0.66

n=1.1, Umax=0.80

n=1.0, Umax=1.00

10-1

100

101

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

UBase

Flow

Pip

e D

iam

ete

r

= 10

n=1.0, Umax=1.00

n=0.9, Umax=1.32

n=0.8, Umax=1.88

n=0.7, Umax=2.97

n=0.6, Umax=5.48

n=0.5, Umax=13.0

n=0.4, Umax=44.6

27

evident than shear-thickening fluids. Reducing power-low index less than 0.6 will

enhance the maximum velocity drastically.

Figure 21: Effect of λ on peak velocity of base flow (Re = 1000, n=0.9)

Figure 22 shows the effect of the power-law index and the time constant λ on the velocity

of the purely pulsatile base flow at the instant of maximum positive velocity. It is clear

that the power-law index n and λ in the Carreau-Yasuda model have no effect on the peak

velocity of the oscillatory components.

Figure 22: for different values of λ (left) and n (right) with purely pulsatile base flow

(Re = 1000, Wo = 10, )

Figure 23 illustrates the maximum velocity of the base flow as a function of power-low

index n and material time constant λ.

0 0.5 1 1.5-1

-0.5

0

0.5

1

UBase

Flow

Pip

e D

iam

ete

r

Power = 0.9

=0.1, Umax=1.00

=1.0, Umax=1.06

=10, Umax=1.32

=20, Umax=1.43

=30, Umax=1.49

=40, Umax=1.54

=50, Umax=1.58

=60, Umax=1.61

=70, Umax=1.64

=80, Umax=1.66

=90, Umax=1.69

=100, Umax=1.71

0 0.05 0.1-1

-0.5

0

0.5

1

Velocity

Pip

e D

iam

eter

n = 0.9

=0.1

=1.0

=10

=20

=50

=80

=100

0 0.05 0.1-1

-0.5

0

0.5

1

Velocity

= 10

n=1.5

n=1.2

n=1.0

n=0.8

n=0.4

28

Figure 23: as function of power-law index n (left) and material time constant λ (right)

Re=1000, Steady base flow

More investigation shows that with shear-thinning fluids, increasing λ until twenty causes

a drastic increase of the maximum velocity, with less evident effect when further

increasing λ.

Next, we want to investigate the modal analysis of the non-Newtonian flow. As depicted

in figure 24, the flow is stable for all power-law indexes n and λ, investigated in this

study.

Figure 24: Dominant Floquet multiplayer as function of Power-law index (left) and λ (right)

Re = 1000, line: Steady flow, ●-marker: Pulsatile flow Wo=10, =2

So we confirm that, steady and pulsatile pipe flows within both Newtonian and non-

Newtonian fluids are linearly stable, and pulsatile forcing has a negligible influence on

the modal stability analysis of the pipe flows.

0.5 1 1.5

100

101

102

n

Um

ax

= 0.1

= 1

= 10

= 100

0 50 10010

-1

100

101

102

103

Um

ax

n = 1.5

n = 1.4

n = 1.3

n = 1.2

n = 1.1

n = 1.0

n = 0.9

n = 0.8

n = 0.7

n = 0.6

n = 0.5

n = 0.4

n = 0.3

0.511.50

0.2

0.4

0.6

0.8

1

Power

Flo

quet

multip

lier

(

)

= 10

= 0 m = 1

= 0 m = 2

= 1 m = 1

= 1 m = 2

0 20 40 60 80 10010

-3

10-2

10-1

100

Flo

quet

multip

lier

(

)

power = 0.9

29

We compare the Floquet multiplier for steady and oscillatory base flow in figure 9,where

we consider an axial perturbation with wavenumber zero, amplitude of pulsation

and . The characteristic Floquet exponents are plotted in Figure 25 in the

complex plane for . A plot of the eigenvalues of the steady Poiseuille flow is

also shown in this figure. The fluid is considered as non-Newtonian with

and .

Figure 25: Plot of Floquet exponent for axially non-invariant, axisymmetric perturbation

(Re=2000, Wo=40, =2, λ=10, n=0.7) For comparison purpose the plot of steady base flow is also shown.

Same as before (see fig. 10), the real part of the Floquet exponents are slightly more

negative than their steady counterparts, indicating that the oscillatory flow is somewhat

more stable than the steady Poiseuille flow.

In the next step, we are going to consider non-modal stability for both shear-thinning and

shear-thickening fluid for steady Poiseuille and pulsatile flows.

As aforementioned, with the Newtonian fluid in the axially invariant mode, the most

unstable transient growth is observed for unity azimuthal wave number, while for the

axially dependent mode, the maximum transient growth is observed for . For the

non-Newtonian fluid, we consider the critical cases which are investigated in the

Newtonian section.

Figure 26 depicts the effect of power-law index n, on the maximum energy growth.

Obviously, reducing power-law index n from 1.5 until to 0.4, i.e. the flow properties from

shear-thickening to Newtonian ( ) and then to shear-thinning fluid, increases the

maximum transient growth for all cases. Same as Newtonian fluid, the largest transient

growth for the axially invariant mode is observed for unity azimuthal wavenumber,

whereas it occurs for an azimuthal wave number equal to two for the axially-dependent

mode.

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.40

0.2

0.4

0.6

0.8

1

Imaginary ( )

-Rea

l (

)

Steady

Pulsatile

30

Figure 26: Transient Growth versus power-law index with Steady base flow (Line) and pulsatile base flow (●-marker)

(Re=1000 =2, Wo=10, =10, n =0.8)

The results indicate that, the largest possible amplification of external disturbances,

although transient occurs for the steady flow and not in the oscillatory case.

Figure 27 shows transient growth as a function of time for different values of the power-

law index n for both the axially invariant and non-invariant modes with the purely

pulsatile base flow.

With purely pulsatile base flow, for the high Womersley number which we consider in

this study (Wo=10), there is no amplification of initial disturbances and the maximum

transient growth is always . This is because of very small oscillatory component

of base flow with higher values of the Womersley numbers.

Vector flow fields of the dominant perturbation for the axially invariant and non-invariant

modes were portrayed in figure 28.

0.511.5

102

104

106

n

Gm

ax

= 0, m = 1

= 0, m = 2

= 1, m = 1

= 1, m = 2

Figure 27: Transient growth as a function of time with purely pulsatile flow (Re=1000, Wo =10, 𝚪=2, λ=10)

31

For axially invariant mode, the vector flow field of the least stable disturbance is remarkably

similar to the one in figure 13, although the flow field of the dominant disturbances for axially

non-invariant mode is substantially different from the previous cases and corresponding counter

rotating vortex shows a shrunk in dimension.

The flow response to axially invariant and non-invariant mode is depicted in figure 29. The

maximum transient growth is plotted as a function of Reynolds number for different values of

Womersley number ranging from five to twenty. For comparison purpose, for the case of steady

Poiseuille flow, a plot of is also displayed.

Figure 28: dominant perturbation structure in the 𝒓 𝜽 -plane of the pipe

(Re=1000, Wo =10, 𝜞=2, λ=10, n = 0.6) 𝜶 𝟎 𝒎 𝟏 (left) 𝜶 𝟏 𝒎 𝟐 (right)

Figure 29: 𝐆𝐦𝐚𝐱 as a function of Reynolds number with non-Newtonian flow (𝚪=2, λ=10, n=0.8)

For comparison purposes, the plot of 𝐆𝐦𝐚𝐱 for the case of steady Poiseuille flow is also illustrated.

32

For the axially invariant mode, the maximum transient growth is independent of the

Womersley number while, for the axially non-invariant mode same as previous (see fig.

17), the flow was more stable for lower Womersley numbers, and by decreasing it, the

flow perturbation is illustrated smaller than its steady counterpart. Increasing the

Womersley number will drives the maximum transient growth to its steady values

until . Increasing greater than this value, has an insignificant effect on the

stability of the oscillatory pipe flow. For larger Womersley numbers, stability indicates of

the pulsatile flow is recovered. For both cases, the most unstable energy growth could be

seen with steady base flow, characterized pulsatile flow is more stable.

Here we want to examine the dependence of transient energy growth maxima on the

phase for shear thinning fluids. Figure 30 shows dominant energy growth as a function

of Reynolds number for several phase-shifts.

For both zero and non-zero streamwise wavenumbers, maximum energy growth was achieved as

the curve emanating from phase-shift and respectively with shear-thinning fluid,

whereas we found energy growth maxima for the phase-shift for both mentioned cases

with Newtonian fluid (see fig. 14).

Additionally figure 31 shows the transient energy growth maxima as a function of Womersley

number.

Figure 30: optimal energy density growth as a function time for different phase-shift

(Re=1000, Wo =2, 𝜞=2 𝝀=10, n=0.6) 𝜶 𝟎 𝒎 𝟏 (left) 𝜶 𝟏 𝒎 𝟐 (right)

33

When the Womersley number increases more than 15, the differences between dominant

transient growths emanating different phases is decrease. Increasing the Womersley number

casus to reduce the oscillatory forcing, therefore, different phases have no effect on the transient

growth for higher the Womersley number.

Figure 32 shows the dominant energy density growth as a function of amplitude of pulsation for

different phases.

Figure 32: The dominant transient energy growth as a function of amplitude of pulsation for several phase-shifts

(Re=1000, Wo =2, =10, n=0.6, )

5 10 15 2010

3

104

105

106

107

Re = 1000,Wo = 2, = 0,m = 1

Gm

ax

= 0

= /3

= 2/3

=

= 4/3

= 5/3

Figure 31: The maximum energy growth as a function of Womersley number for several phase-shifts

(Re=1000, 𝚪=2, 𝛌=10, n=0.6) 𝜶 𝟎 𝒎 𝟏 (left) 𝜶 𝟏 𝒎 𝟐 (right)

34

Increasing pulsatile forcing cause to was increase the maximum energy growth. Also the

differences between the dominant transient growths will increases by increasing the amplitude of

pulsation. This is because, when the amplitude of pulsation increases, the pulsatile forcing

increases and the phases affect more the transient growth.

Finally, the maximum transient growth to the quadratic Reynolds number plotted against the

Reynolds number for both zero and non-zero streamwise wave number in figure 33.

Figure 33: Maximum transient growth scaled with quadratic Reynolds number as a function of Reynolds number

Solid line (Steady base flow), dashed line (Pulsatile base flow, Wo=10, =2)

For the axially invariant mode, the Reynolds number has negligible effect on the maximum

transient growth and it is solely depends on azimuthal wave number. For the axially non-

invariant mode, the transient growth curve maxima influenced by Reynolds number.

1000 2000 3000 4000 50000

0.2

0.4

0.6

0.8

1x 10

-3

Re

(Gm

ax)/

Re

2

= 10, n = 0.8

=0, m=1

=0, m=1

=1, m=2

=1, m=2

35

6. Conclusion In this study, we have investigated the linear stability of pulsatile pipe flow of Newtonian and

non-Newtonian fluids. The shear-dependent viscosity is modeled by the Carreau-Yasuda law and

the rheological parameters, the power-index and the material time examined in the range

and . A second order finite difference code is used for the

simulation of pipe flow. The main conclusions can be summarized as follow:

Analysis of the Floquet exponents shows that the flow is asymptotically stable for any

configuration with both shear-thinning and shear-thickening fluids. In addition, the real part of

the Floquet exponents corresponding to oscillatory base flow are slightly more negative compare

to their steady counterpart, indicating the oscillatory flow is somewhat more stable.

For the axially invariant mode ( ): maximum energy growth occurs at azimuthal

wavenumber . Amplitude of pulsation has a negligible effect for azimuthal wavenumbers

less than two ( ). Reduction of the oscillating component causes an increase in maximum

energy growth for . Moreover for non-Newtonian fluids, in the range of , there exists

an upper bound of where beyond that the influence of pulsation forcing on the Stability

of the flow can be neglected. A negligible dependence of Reynolds number can be observed

when optimal transient growth scaling with quadratic Reynolds number. The transient growth is

shown to scale with as for steady Newtonian flow.

For the axially non-invariant mode ( ): the most dangerous disturbance has an azimuthal

wave number equal to two. Increasing the pulsatile forcing can lead to the significant reduction

of the maximum transient growth. Furthermore, for non-Newtonian fluids, in the range of ,

there exists an upper bound for the Womersley number where beyond which the

influence of oscillatory forcing on the stability of the flow can be ignored. Increasing Reynolds

number will increase the energy growth while the opposite behavior is observed by increasing

the Womersley number for both zero and non-zero streamwise wavenumber.

As concerns, with non-Newtonian fluids, reducing the power-law index n will increases the

maximum velocity of the base flow at constant pressure gradient. In fact, reducing the viscosity

increases the rate of shear strain near the pipe wall and enhances the peak velocity inside the

pipe. The increase of the peak velocity for shear-thinning fluids is more evident than the decrease

for shear thickening fluids. Reducing the power-low index n lower than 0.6 enhances the peak

velocity drastically. We found that, the rheological parameters of the Carreau-Yasuda model

have negligible effect on the peak velocity of the oscillatory component.

In the case of phase-shift, maximum energy growth was achieved for axially invariant and non-

invariant modes as the curve emanating from phase-shift and respectively for

shear thinning fluid and for Newtonian fluids we found maximum energy growth for the curve

which is initiating for the phase-shift for both zero and non-zero streamwise

wavenumber.

The present work can be extending the scope of linear theory for pulsatile pipe flow to include

transient energy growth of minuscule two and three dimensional perturbations. Additionally

considering a flexible pipe instead of the rigid pipe, which we consider here, and examine the

fluid-structure interaction (FSI) would be beneficial.

36

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