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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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