20th Australasian Fluid Mechanics ConferencePerth, Australia5-8 December 2016

Pulsatile Flow Through a Compliant Channel: A Comparison of Laminar and Fully DevelopedTurbulent Flow Effects

K. Tsigklifis1 and A. D. Lucey 1

1Fluid Dynamics Research GroupDepartment of Mechanical Engineering

Curtin University,Perth, Westrn Australia 6845, Australia

AbstractWe investigate the fluid-structure interaction (FSI) of planePoiseuille and fully developed turbulent flows through a com-pliant channel with flexible-plate-spring type walls. Althoughfundamental in nature, the study is partially motivated by itsapplication to the biomechanics of blood flow. Our main objec-tive is to assess the effect of modulation frequency and chan-nel compliance on the wall shear stresses due to organized dis-turbances during the fundamental cycle of the unsteady meanflow. For the pulsatile plane Poiseuille mean flow an analyti-cal solution is implemented, while the fully developed turbulentpulsatile mean flow is modelled via the unsteady RANS equa-tions using the Boussinesq hypothesis for the Reynolds stresses.The eddy viscosity is calculated by solving the standard one-equation Spalart-Allmaras model. In order to compare the ef-fects of the two types of mean flow on the FSI, we use the samebulk Reynolds number, the same dimensional applied pressure-gradient pulsation and the same dimensional properties for thecompliant walls at each of a low (Wo = 5) and high (Wo = 15)modulation frequency (where Wo is the Womersely number).Our results show that even though the wall shear stresses ofthe turbulent pulsatile mean flow are larger than those of thelaminar pulsatile mean flow within the pulsation cycle, the wallshear stresses induced by the organized disturbances are largerin the case of the laminar mean flow. Finally, wall compliance isshown to reduce the wall shear stresses due to the organized dis-turbances in all cases but this effect is more pronounced for thelaminar mean flow at the low modulation frequency (Wo = 5).

IntroductionBiomechanical studies of the blood flow in arteries have shownthat the magnitude of wall shear stress plays a significant rolein both plaque formation and rupture. Laminar flow is thoughtto provide an atheroprotective role in mitigating localized ele-vation of the wall shear stresses [2, 3] that cause rupture. How-ever, less is known about the effect of artery-wall complianceon the wall shear stresses induced by the mean pulsatile flowas well as by organised disturbances about the mean flow. Ourvery recent work [8] has systematically studied the stability ofthe pulsatile Poiseuille flow through a compliant channel re-vealing that the pulsatile flow is always stabilised by the ex-istence of relatively stiff compliant walls for the entire rangeof modulation frequencies and dimensionless pressure-gradientmodulation amplitudes in the range Λ ≤ 5 and that this effectcombines favourably with the stabilization effect of the Stokeslayer. In the present paper, we extend this work to study thetime-asymptotic and transient stability of a fully developed tur-bulent pulsatile channel flow with a focus on the size of the wallshear stresses generated by both the mean pulsatile flow and theorganised disturbances during the modulation cycle. Our ob-jective is to assess the effect of modulation frequency and wallcompliance on the magnitudes of the mean and disturbance wallshear stresses and compare these values between laminar andfully developed turbulent pulsatile channel flows.

Figure 1: Schematic of the system studied

System ModellingThe formulation of the FSI problem that follows is based upon[8] in which laminar pulsatile mean flow supporting distur-bances taking a spatial wave form was modelled; we nowterm these organised disturbances. When the pulsatile flow isturbulent, perturbations to the unsteady mean flow compriseboth organised wave disturbances and turbulent fluctuations.It is the effect of the former that we study and therefore em-ploy Reynolds averaged forms of the Navier-Stokes equations(RANS) wherein the turbulent Reynolds stresses contribute toboth the determination of the mean flow and the evolution of theorganised disturbances. With the laminar-flow case detailed in[8], we focus exclusively on the case of turbulent flow throughthe compliant channel.

Mean Turbulent Flow FieldWe consider plane pulsatile viscous flow between two com-pliant walls separated by 2L∗ distance as shown in figure 1.The characteristic length scale is chosen to be the half distanceL∗ between the (undisplaced) compliant surfaces, the veloc-ity scale is the friction velocity, U∗τ , of the underlying steadyfully-developed turbulent flow about which pulsations occur,while the angular frequency of the applied periodic pulses de-fines the time scale, 1/ω∗f . The flow field is then character-ized by the Reynolds number based on the friction velocity,Reτ =U∗τ L∗/ν∗l , where ν∗l is the kinematic viscosity of the fluid,and the Womersley number, Wo = L∗(ω∗f /ν∗l )

1/2. Here andhereafter, ∗ denotes a dimensional quantity.

Following [6], we decompose the instantaneous velocity andpressure fields (becoming dimensionless through the dynamicpressure based on the friction velocity) as

ui = ui + ui +u′ip = p+ p+ p′, (1)

with ui and p the mean velocity and pressure, u′i and p′ theturbulent velocity and pressure fluctuations, and ui and p thevelocity and pressure of the organised disturbances with prop-

erties p = p′ = ui = u′i = 0, ∂ui/∂xi = ∂ui/∂xi = ∂u′i/∂xi = 0.Then, the time average of the Navier–Stokes equations in di-mensionless form are

Wo2

Reτ

∂ui

∂t+u j

∂ui

∂x j=− ∂p

∂xi+

1Reτ

∂2ui

∂x j∂x j−

∂(u′iu′j)

∂x j, (2)

where it has been assumed that |uiu j| |u′iu′j|, thus uiu j ≈ 0.

The closure problem for the Reynolds stresses in equation (2) ismodeled via the isotropic eddy-viscosity model:

−u′iu′j =

GReτ

(∂ui

∂x j+

∂u j

∂xi

), (3)

with G = ν∗τ/ν∗l being the dimensionless kinematic eddy vis-cosity.

In the two-dimensional system as notated in figure 1, x1 = x,x2 = z, u1 = ux and u2 = uz, the mean flow is entirely in thestreamwise direction x due to a mean pressure gradient in thisdirection and this is changed periodically with a frequency ω∗fand amplitude ΛT following

∂p∂x

=dpdx

∣∣∣∣st.

[1+ΛT

exp(it)+ exp(−it)2

],

dpdx

∣∣∣∣st.

=−1. (4)

Upon application of equation (3) in equation (2) and assum-ing that the streamwise derivatives of the Reynolds stresses dueto turbulent fluctuations are small compared with their cross-stream counterparts, we obtain the equation that determines themean turbulent pulsatile channel flow,

Wo2

Reτ

∂ux

∂t=−∂p

∂x+

1Reτ

∂2ux

∂z2 +G

Reτ

∂2ux

∂z2 +1

Reτ

∂G∂z

∂ux

∂z, (5)

with the no-slip condition, ux = 0, on the walls. The dimension-less kinematic eddy viscosity G = X fu1, fu1 = X3/(X3 + c3

u1),cu1 = 7.1, is modelled via the “standard” one equation ofSpalart-Allmaras [7], which is written for the fully developedpulsatile channel flow in dimensionless form as

Wo2

Reτ

∂X∂t

+ux∂X∂x

=

Wo2

Reτ

Cb1(1− ft2)SX− 1Reτ

[Cw1 fw−

Cb1

κ2 ft2

](Xd

)2

+1

σReτ

∂

[(1+X)

∂X∂x

]∂x

+

∂

[(1+X)

∂X∂z

]∂z

+

Cb2

σReτ

[(∂X∂x

)2+

(∂X∂z

)2], (6)

where S = Ω+( fu2/(Wo2κ2d2))X , Ω = (Reτ/Wo2)√

2ωi jωi j,ωi j =(1/2)(∂ui/∂x j−∂u j/∂xi), d is the dimensionless distancefrom the field point to the nearest wall while fu2, κ, fw, Cb1, Cb2,ft2, Cw1 and σ are functions of X or constants given in [7].

Field of Organised DisturbancesWe proceed by using the decomposition of (1) in the Navier-Stokes equations and applying ensemble averaging, taking intoaccount that 〈ui〉= ui + ui and 〈p〉= p+ p, to obtain

Wo2

Reτ

∂〈ui〉∂t

+ 〈u j〉∂〈ui〉∂x j

=− ∂〈p〉∂xi

+1

Reτ

∂2〈ui〉∂x j∂x j

−∂(〈u′iu′j〉)

∂x j. (7)

Subtracting (2) from (7) we obtain the equation for the organ-ised disturbances,

Wo2

Reτ

∂ui

∂t+u j

∂ui

∂x j+ u j

∂ui

∂x j=− ∂ p

∂xi+

1Reτ

∂2ui

∂x j∂x j+

∂ri j

∂x j, (8)

where ri j = −(〈u′iu′j〉 − u′iu′j) is the effect of the turbulent

Reynolds stresses on the evolution of the organised disturbancesand is of order ri j ∼O(ui). The closure problem for ri j in equa-tion (8) is modeled via the isotropic eddy-viscosity model thatis similar to equation (3):

ri j =G

Reτ

(∂ui

∂x j+

∂u j

∂xi

), (9)

with G, calculated via equation (6).

Considering only two-dimensional organised disturbances andthat the mean flow only has a velocity component in the stream-wise direction, the final system of equations for the organiseddisturbances of the turbulent pulsatile channel flow is written as

∂ux

∂x+

∂uz

∂z= 0

Wo2

Reτ

∂ux

∂t+ux

∂ux

∂x+ uz

∂ux

∂z=− ∂ p

∂x+

1Reτ

(∂2ux

∂x2 +∂2ux

∂z2

)+

∂rxz

∂z

Wo2

Reτ

∂uz

∂t+ux

∂uz

∂x=− ∂ p

∂z+

1Reτ

(∂2uz

∂x2 +∂2uz

∂z2

)+

∂rzx

∂x, (10)

having only retained the dominant term rxz [9]. We now assumethat the field of the organised disturbances takes the followingspatial waveforms,

ui(x,z, t) = ui(z, t)exp(iαx)+ c.c.

p(x,z, t) = p(z, t)exp(iαx)+ c.c., (11)

and substitute these in equations (10). We then eliminate thepressure disturbance amplitude, p(z, t), to give the evolutionequation for organised disturbance as

Wo2 ∂

∂tL uz = (L− iαReτux)L uz + iαReτ

∂2ux

∂z2 uz +

GQ 2uz +∂2G∂z2 Q uz +2

∂G∂z

Q∂uz

∂z, (12)

where L = ∂2/∂z2−α2, Q = ∂2/∂z2 +α2.

Using equations (10), (11) and the continuity equation, we canexpress the pressure disturbance amplitude p(z, t) as the follow-ing function of uz(z, t),

p = − Wo2

α2Reτ

∂2uz

∂t∂z− iux

α

∂uz

∂z+

iα

∂ux

∂zuz +

1α2Reτ

L∂uz

∂z+

Gα2Reτ

Q∂uz

∂z+

1α2Reτ

∂G∂z

Q uz. (13)

For the compliant-wall dynamics, we use the one-dimensionalisotropic Kirchhoff plate equation with additional terms to ac-count for a dashpot-type damping and a uniformly distributedspring foundation. Combined with the normal and tangentialforce balance on the compliant walls, we obtain two equa-tions which relate the vertical and axial displacements of thetwo compliant surfaces, ηz,s(x, t) and ηx,s(x, t) (for s = 1,2lower/upper surfaces), with the pressure disturbance p(x,z =∓1, t) and the velocity gradients of the mean and organised dis-turbances on the two interfaces (see [8]). The system of equa-tions is completed with the enforcement of continuity of boththe tangential and vertical disturbance velocities between fluidand solid [8], while the effect of turbulent velocity and pres-sure fluctuations at the interfaces are assumed negligible be-cause their time-average is zero.

For spatial discretization in the direction normal to the meanflow, z, we use the Chebyshev pseudospectral method. The do-main is discretized into M+1 collocation points and the spatialderivatives of a function at the these points are related to the val-ues of the function through the Chebyshev differentiation ma-trices [1].

Solution MethodologyEquations (5) and (6) which describe the mean pulsatile flow aresolved in space with the use of a second-order finite-differencescheme for the streamwise direction and the Chebyshev pseu-dospectral method for the direction perpendicular to the meanflow with periodic inlet-outlet boundary conditions. Both equa-tions are solved using a second-order backward-difference time-stepping scheme using the following iterative algorithm: the lin-ear equation (5) is solved for the mean velocity profile for givendimensionless kinematic eddy viscosity G from the previoustime-step or iteration. The non-linear equation (6) is then solvedfor the eddy viscosity G using the quasi-Newton method byapproximating the Jacobian through a first-order Euler schemeand the whole procedure is repeated until convergence of boththe mean velocity profile and the eddy viscosity G are achievedafter which the whole solution is advanced in time.

Concomitant with the decomposition of (1) we write the ver-tical and horizonal components of the complaint-wall defor-mations as ηz,s(x, t) = ηz,s(t)exp(iαx) + c.c. and ηx,s(x, t) =ηx,s(t)exp(iαx) + c.c. in the normal and tangential force bal-ances of the compliant walls (see [8] for details). These are thenused with equation (13), the continuity equation and the systemof equations (12) with the two kinematic interfacial boundaryconditions at each compliant surface, to cast in the followinginitial-value problem for the FSI system,

d~vdt

= Q(t)~v, ~v = cz,s, ηz,s, cx,s, ηx,s,~uTz T , ~v(t = 0) = ~v0, (14)

where cz,s = dηz,s/dt, cx,s = dηx,s/dt and Q is periodic in timewith fundamental period T , due to the imposed periodic meanvelocity ux.

The time-asymptotic stability of the system expressed by equa-tion (14) can be studied through the eigenvalues of the mon-odromy matrix M. This is calculated by solving the followinginitial-value matrix problem [4]:

M = S(t = T ) :dS

dt= Q(t)S , S(t = 0) = I , (15)

where I is the identity matrix. Using a second-order backwardintegration scheme for time, the matrix M is evaluated at the endof one period T from the product of the matrices calculated inthe previous time-steps [8]. Its eigenvalues, the Floquet multi-pliers, k j, j = 1, · · ·M+1 determine the stability of the pulsatilechannel flow over the compliant walls in the following man-ner: Pulsatile flow is stable if |k j|< 1 for j = 1,2, · · ·M+1 andunstable if |k j| > 1 for some j [4]. Alternatively, the systemof equations (14) can be integrated in time with an initial vectorbased on the second-order backward-difference scheme to mon-itor the transient and asymptotic response of the FSI system.

Results and Discussion

Mean Flow FieldThe mean pulsatile velocity profiles were validated throughcomparison with the results of [5]. The main purpose of thepresent work is to compare the wall shear stresses during thefundamental oscillation cycle produced by both the mean pul-satile flow and organized disturbances as generated by laminar

and turbulent pulsatile mean velocity profiles for low (Wo = 5)and high (Wo= 15) modulation frequencies for identical dimen-sional flow conditions. The Reynolds number Reb, based on thebulk velocity, and the dimensional pressure gradient modula-tion amplitude were selected to be those marginally above ofthe critical conditions (instability onset) for laminar pulsatileflow through a compliant channel with properties given in [8].Specifically, for the given dimensional properties of the com-pliant walls, modulation frequency Wo = 5 and dimensionlessamplitude of the pressure-gradient modulation Λ = 3 (made di-mensionless through the centreline velocity of the correspond-ing steady flow; i.e. Λ = 0), the critical Reynolds number basedon the centreline velocity of the steady flow is Re0 = 3775; thisproduces a critical Reb = 2/3Re0 = 2517 or Reτ = 161. Usingthe same dimensional amplitude of the pressure-gradient mod-ulation, its non-dimensional value for steady fully-developedturbulent flow is ΛT = 3ΛReb/Re2

τ = 0.872 (made dimension-less through the friction velocity). Similar calculations forlaminar-flow critical conditions when Wo = 15 and Λ = 5 giveReb = 4333 or Reτ = 260 and the corresponding turbulent-flowvalue ΛT = 0.965.

Figure 2 shows the velocity profiles of the pulsatile mean flowthrough the (rigid-walled) channel for laminar and fully devel-oped turbulent flow during one cycle of the applied pulsationat each of low (Wo = 5) and high (Wo = 15) modulation fre-quency. For Wo = 5, the laminar profile features an inflexionpoint during the cycle that can give rise to an amplified organ-ised disturbance. In contrast, the turbulent profiles and laminarprofile at Wo = 15 do not feature an inflexion point and canbe expected to be stable for the present choice of dimensionalbulk-flow speed and amplitude of the pressure-gradient modu-lation. Figure 3 shows the corresponding time evolution of thewall shear stresses due the mean flows presented in figure 2,τw(z = −1) = (1/Reτ)(∂ux/∂z|z=−1 + ∂uz/∂x|z=−1) over fourperiods of the modulation. It is seen that both the average shearstresses and their amplitudes of oscillation are greater when themean flow is turbulent; this is clearly to be expected given thehigher velocity gradient at the wall as compared with the lami-nar flow. Finally, in both laminar and turbulent cases, the ampli-tudes of the oscillations are larger for the low-frequency modu-lation because the magnitudes of mean-velocity fluctuations aremuch lower at Wo = 15 as can be seen in figure 2.

Field of Organized DisturbancesIn the present work, we are mainly interested in investigatingthe effect of the mean velocity profile on the stability of the FSIsystem and the wall shear stresses produced by the organiseddisturbances. For this reason we solve the Orr-Sommerfeld typeequation (12) without the contribution of the turbulent terms onthe stability problem. The time-asymptotic stability of the fullydeveloped turbulent pulsatile channel flow, studied through theextraction of the eigenvalues of the Monodromy matrix M viathe Floquet multipliers, reveals that the flow is stable with re-spect to growth of organised wave-like disturbances at bothlow (Wo = 5) and high (Wo = 15) modulation frequencies forboth rigid and the relatively stiff compliant-channel walls usedherein. Figures 4(a) and (b) show the time evolution of the wallshear stresses due to the organised disturbances over four peri-ods of the modulation for the laminar and the turbulent pulsatilemean flow respectively and for Wo = 5, while figure (c), showsthe same variables for Wo = 15. These results were generatedby integration of equation (14) with the same dimensional ini-tial vector, ~v0∗.

Figures 4(a) and (b) show that the disturbance wall shearstresses are significantly larger, by almost four orders of magni-tude, for the laminar as compared with the turbulent pulsatile

(a)

ux

zt = 0

t = π/2

t = π

t = 3π/2

(b)

ux

zt = 0

t = π/2

t = π

t = 3π/2

(c)

ux

zt = 0

t = π/2

t = π

t = 3π/2

(d)

ux

zt = 0

t = π/2

t = π

t = 3π/2

Figure 2: Velocity profiles during one cycle of the disturbancepressure gradient modulation for Wo = 5, Reτ = 161 for (a) pul-satile Poiseuille flow with Λ = 3 , (b) fully developed turbu-lent pulsatile channel flow with ΛT = 0.872. Velocity profilesfor Wo = 15, Reτ = 260 for (c) pulsatile Poiseuille flow withΛ = 5, (d) fully developed turbulent pulsatile channel flow withΛT = 0.965.

t

τw(z=−

1)

Reτ = 161,Wo = 5, turbulentReτ = 161,Wo = 5, laminarReτ = 260,Wo = 15, turbulentReτ = 260,Wo = 15, laminar

Figure 3: Time evolution of the mean wall shear stresses τw ofthe bottom wall during four periods of the modulation, corre-sponding to the velocity profiles of figure 2.

mean flow. The effect of the wall-compliance is stabilisingmainly for the laminar pulsatile flow as seen through both thetime-asymptotic behaviour (details not shown here) and time-integrated initial-value problem that generated the data in fig-ures 4(a) and (b) as evidenced by the reduction in amplitude ofthe fluctuating wall shear stresses in the laminar pulsatile flow.We also remark that the time-amplification of successive wavepackets for the laminar-flow case reflects the fact that the or-ganised disturbances are (marginally) unstable for the systemparameters selected and discussed in the sub-section above. Forhigh-frequency modulation, Wo = 15, the results of figure 4(c)show that the disturbance wall shear stresses produced by thelaminar pulsatile flow are still larger than those of the turbu-lent pulsatile mean flow, while the compliant wall is much lesseffective in reducing the wall shear stresses in both cases; thismight be expected because a relatively stiff compliant wall hasbeen used in this preliminary investigation.

ConclusionsWe have studied the time-asymptotic stability and compared thewall shear stresses generated by both the mean pulsatile channelflow and the organised disturbances for laminar and turbulentpulsatile flows through both rigid and flexible channels. Our re-sults show that although the shear stresses due to the pulsatile

(a)

t

τw(z=−

1)

rigid, turbulentcompliant, turbulent

(b)

τw(z=−

1)

rigid, laminarcompliant, laminar

t(c)

t

τw(z=−

1)

rigid, turbulentcompliant, turbulentrigid, laminarcompliant, laminar

Figure 4: Time evolution of the wall shear stresses due to theorganized disturbances for rigid and compliant channel flow for(a) and (b) Wo = 5,Reτ = 161, (c) Wo = 15,Reτ = 260

turbulent mean flow are larger than those of the equivalent lam-inar flow, the shear stresses due to organised wave-like distur-bances are larger for the laminar flow. This is because inflec-tional velocity profiles occur during the deceleration phase ofthe modulation cycle in laminar flow; these are absent in thecorresponding turbulent-flow profiles. Finally, wall-compliancehas been shown to reduce disturbance wall shear stresses mainlyfor the laminar flow at low modulation frequency.

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