Stability analysis of three-dimensional stenotic pipe flow

Transcription

1 Arts et Métiers ParisTech DynFluid Laboratory Masters thesis submitted at ÉCOLE POLYTECHNIQUE Specialty: Fluid Mechanics (Fundamentals and application) by Shreyas ACHARYA NEELAVARA under the guidence of M. Jean-Christophe ROBINET, Advisor and M. Jean-Christophe LOISEAU, Co-advisor titled Stability analysis of three-dimensional stenotic pipe flow

2 Contents 1 Introduction 1 2 Methodology Geometry Numerics - Spectral elements code Nek Spatial discretization Temporal discretization Steady states computation Modal decomposition Global linear stability, time-stepping and Arnoldi algorithm Transient growth analysis Results From nearly-axisymmetric solutions to wall-attached ones Subcritical pitchfork bifurcation Lower branch solution s transient growth Secondary bifurcation of the wall-attached solutions Conclusion 22 1

3 List of Figures 1.1 Modification of geometry of arteries. Image courtesy: [1] Growth of stenosis in the artery. Image courtesy: [2] Illustration of the eccentricity in the pipe (a) Elemental mesh for the stenotic pipe flow with eccentricity. (b) Resulting full mesh using an order 8 polynomial reconstruction within each element Block diagram of the time-stepping Arnoldi algorithm implemented around the Nek5000 temporal loop Block diagram of direct-adjoint loop implemented in Nek5000 for the poweriteration method Block diagram of the SVD-Arnoldi algorithm implemented in Nek Bifurcation diagram indicating the hysteresis in the asymmetry values Slices at different y and z locations of the x-velocity of the baseflow for Reynolds number 400 in the forward sequence Slices at different x- locations of the x-velocity of the baseflow for Reynolds number 400 in the forward sequence Slices at different x-locations of the x-velocity of the baseflow for Reynolds number 400 in the backward sequence Slices at different y and z locations of the x-velocity of the baseflow for Reynolds number 400 in the forward sequence The optimal gain curve for flow at Reynolds number The optimal perturbation for flow at Reynolds number The optimal response of the flow at Reynolds number The energy budget of the transient growth at Reynolds number Slices at different y and z-locations of the x-velocity of the baseflow for Reynolds number Slices at different x-locations of the x-velocity of the baseflow for Reynolds number The eigenmode observed at Reynolds number The energy budget of the eigenmode at Reynolds number The recirculation zone at Reynolds number The location of the uv U y 3.16 The location of the uw U z at Reynolds number at Reynolds number

4 Abstract A straight tube with a constriction which is offset from the axis is an idealized representation of a stenosed artery. Baseflows for different Reynolds number are calculated for flows within the eccentric stenotic pipes using the selective frequency damping (SFD) method, and the presence of a hysteresis cycle similar to the axisymmetric stenotic pipe is compared. A subcritical imperfect pitchfork bifurcation is observed with two branches of solution of flow fields the deflected jet and the wall attached flow field. Optimal growth analysis is carried out to determine the optimal perturbation and the optimal response of the flow to this perturbation, and the results are again compared with that of the axisymmetric case. Stability analysis is carried out for flow related to a particular Reynolds number, using the Arnoldi algorithm, and an ustable but non-oscillating eigenmode is observed. The package Nek5000 is used for direct numerical simulations, and also for the implementation of Arnoldi algorithm.

5 Chapter 1 Introduction The association of arterial disease with flow-related mechanisms has motivated the study of flow with arterial model stenoses. Artherosclerosis is a widely spread cardio-vascular disease predominant in the industrialized countries. It is a condition in which an artery wall thickens as a result of the accumulation of fatty materials such as cholesterol, eventually leading to serious health conditions such as break up of the blood vessels walls. Figures 1 and 2 show the modification of the geometry of the arteries in cases of artherosclerosis : Figure 1.1: Modification of geometry of arteries. Image courtesy: [1] The genesis of artherosclerosis is closely related to wall shear rates in the arterial flow. It involves an accumulation of cholesterol in the walls of the large arteries, typically where the local wall shear rate is low, and oscillatory. The process can be considered as a long-period nonlinear instability of the geometry of the arterial wall, wherein a local constriction can grow through promotion of flow separation. The increased pressure losses associated with flow separation can reduce the flow rate and cause problems related to oxygen level in the blood level. Over the past years, a number of studies have been done related to the topic of flow within stenotic pipes including experimental studies in idealized axisymmetric stenotic tubes. Later, computational fluid dynamics began to have an impact, through numerical simulations in idealized stenoses (Varghese, Frankel and Fischer [3]). More recently, studies have been done on three dimensional instabilities and transition of steady axisymmetric stenotic flows (Sherwin and Blackburn [4]) and on transient growth analysis in stenotic flows (Blackburn, Sherwin and Barkley [5]). These studies have used a smooth axisymmetric constriction as an idealized representation of 1

6 Figure 1.2: Growth of stenosis in the artery. Image courtesy: [2] a stenosed artery, and have examined the three-dimensional instabilities and transition to turbulence, using both linear stability analysis and direct numerical simulation. It is seen that steady flow undergoes a weak Coanda-type wall attachment and undergoes transition through a subcritical bifurcation, leading to hysteretic behavior with respect changes in Reynolds number. It is also seen that global optimal disturbances to an initially axisymmetric state are three-dimensional wave packets that produce instability in the extended local shear layers. All these studies have been done on an idealized axisymmetric stenotic pipe whereas in reality, artherosclerosis doesnt necessarily lead to the formation of axisymmetric stenoses. The aim of the present work is to compute baseflows, perform stability analysis and transient growth analysis on a stenotic pipe with an eccentricity, i.e, a pipe with a stenosis that is offset from the axis, using fully three dimensional global stability analysis and direct numerical simulations. The baseflows of flows of different Reynolds numbers are to be computed and the presence of a hysteresis cycle similar to the axisymmetric case (Sherwin and Blackburn, 2005) verified. Baseflows are to be calculated using the selective frequency damping (SFD) method detailed in Akervik et al., [6]. The stability of these baseflows is to be computed using three dimensional global stability analysis. Global stability analysis has been described in detail by Schmid and Henningson [7] and Vassilios Theofilis [8]. Since three dimensional flows are being analysed, the full eigenspectrum computation is not feasible because of the huge memory requirement. Hence, iterative global stability analysis is carried out based on time-stepping (Tuckerman and Barkley [9]) and spectral transformations. For the computation of the global eigenspectrum, the Arnoldi algorithm, a member of the Krylov subspace iteration methods is used as detailed in Edwards et al. [10]. 2

7 Transient growth analysis is to be carried out to determine the optimal perturbation and the optimal response of the flow to the perturbation. 3

8 Chapter 2 Methodology 2.1 Geometry A pipe of internal diameter D with a stentoic region with a radius r(z) that varies over an axial length L is considered with r(z) = 0.5D min + 0.5(D D min ) sin 2 (πz/l) where z is centered on the middle of the stenosis (Sherwin & Blackburn [4]). With this geometry, 2 independent dimensionless parameters can be defined which are the ratio of reduction in cross-sectional area to the original area, S = 1 (D min /D) 2 and the stenosis length lamda = L/D. Here, the values considered are S = 0.75 and lamda = 2. The stenosis is given an eccentricity ED to the top with the value being, e = 0.1% of the original diameter. Figure 2.1: Illustration of the eccentricity in the pipe The streamwise direction is considered as the x direction the spanwise direction as the y direction. All the axes pass through the mid point of the throat of the stenosis. The length of pipe is expressed in terms of the diameter of the pipe D, and the pipe extends from 5D to 40D. Another important dimensionless quantity, the Reynolds number is expressed as Re = u m D/ν where u m represents the mean velocity of the flow and ν, the kinematic viscosity. 4

9 2.2 Numerics - Spectral elements code Nek Spatial discretization The non-linear Navier-Stokes equations as well as their linear counterparts are partial differential equations (PDE) and need to be spatially discretized in order to be solved numerically. The code Nek5000 [11] used for this internship is based on the spectral elements method. This method shares close connections with the finite elements discretization and is part of the family of approximation schemes based on the Galerkin method. The details of this discretisation is given in the books by Deville et al. [12] and Karniadakis & Sherwin [13]. About boundary conditions In the spectral elements method, boundary conditions can be imposed in a several different ways. Though the use of Lagrange multipliers is not the way boundary conditions are implemented within Nek5000, this method will be presented here due to the resemblance of the resulting system with the Navier-Stokes equations. Introducing the Lagrange multiplier p, we get [ ] M 0 d 0 0 dt [ ] u + p [ C + K D T D 0 ] [ ] u = 0 (2.1) p Assuming the constraint Du = 0 is similar to u = 0, then the resemblance to the Navier-Stokes equations is striking. From this point of view, the pressure p in the incompressible Navier-Stokes equations can actually be seen to play the role of a Lagrange multiplier in order to impose the divergence-free constraint. One major difference remains however the way this pressure p is treated in the discretization of the Navier-Stokes equations. Further details about this can be found in [12]. The meshing process As for any discretization of partial differential equations, a mesh is a pre-requisite. Meshing a geometry using spectral elements is not as straightforward as in finite differences for instance. Spectral elements mesh relies on two different grid levels: the spectral elements coarse grid itself and the polynomial reconstruction within each element. Figures 2.2(a) and (b) depict a typical elemental mesh for a stenotic pipe with eccentricity along with the resulting mesh when polynomial reconstruction of order 8 is used, respectively Temporal discretization Let us consider the semi-discretized formulation (2.1) established in the previous subsection. Assuming the linear advection operator C is replaced by its non-linear counterpart, then this formulation is equivalent to the semi-discretized formulation of the Navier- Stokes equations. In order to avoid the algorithmic difficulties resulting from an implicit treatment of the non-symmetric non-linear terms C(u i )u i, the remaining possibility is to treat them explicitly. The temporal discretization scheme used in Nek5000 is the semi-implicit scheme BDF k /EXT k : the viscous terms are discretized implicitly using a backward differentiation scheme of order k whereas the non-linear terms are treated 5

10 (a) (b) Figure 2.2: (a) Elemental mesh for the stenotic pipe flow with eccentricity. (b) Resulting full mesh using an order 8 polynomial reconstruction within each element. implicitly by an extrapolation of order k as well, with k = 2 or 3. For k = 3, the fully discretized Navier-Stokes problem then reads: ( ) 11 6 t M + K u n+1 i D T i p n+1 = M ( 3u n i 3 t 2 un 1 i + 1 ) 3 un 2 i Re ( (2.2) ) 3Cu n i 3Cu n 1 i + Cu n 2 i Adding D T i p n to both sides and rewritting the right-hand side as a simple forcing vector MF n i, equation (2.2) along with the divergence-free condition can be recast into the following matrix form of the unsteady forced Stokes problem: ( ) ( ) ( ) H D T u n+1 MF n + D D 0 δp n+1 = T p n (2.3) 0 where H is known to be the Helmholtz operator. This matrix problem can be solved using a LU decomposition. Introducing a matrix Q for the sake of projection onto a divergence-free space, the solution to such LU decomposition is a two-step procedure: and ( H 0 D DQD T ) ( u δp n+1 ) = ( ) ( ) I QD T u n+1 0 I δp n+1 = ( MF n i + D T p n 0 ( u δp n+1 ) ) (2.4) (2.5) The choice of the matrix Q then determines which projection method is used. In the present code, the matrix Q is set to be: Q = H 1 (2.6) resulting in the Uzawa algorithm. Extensive details on the resolution of this Helmholtz problem and the choice of both the temporal discretization and the projection method can be found in [12]. 6

11 2.3 Steady states computation Computing steady equilibriums to the non-linear Navier-Stokes equations is a pre-requisite to any linear stability calculation. Several procedures exist to compute such solutions, also called fixed points of the equations or base flows, among which all the techniques derived from the Newton algorithm. Though extremely efficient, the major drawback of these techniques is that they rely on the computation of a Jacobian matrix. Unfortunately, when dealing with fully three-dimensional Navier-Stokes equations, the number of degrees of freedom involved is far too large to enable the computation and storage of such matrices. This problem has been overcome in 2006 by Akervik et al. [6] introducing a method known as selective frequency damping. This technique enables a damping of the oscillations of the unsteady part of the solution using a temporal low-pass filter. This is achieved by adding a forcing term to the right-hand side of the Navier-Stokes equations and extending the system with an extra equation for the filtered state Ū. The extended system is then governed by the following set of equations: U + (U )U = P + Re 1 U χ(u Ū) U = 0 (2.7) Ū = ω c (U Ū) with χ being the strength of the filter and ω c the cutting circular frequency. The choice of these two parameters is crucial for the computation: χ has to be positive and larger than the growth rate of the instability one would like to kill, whereas ω c has to be lower than the eigenfrequency ω I of the instability (usually ω c = ω I /2). It can be seen that, provided the filtered state Ū equals the Navier-Stokes state U, system (2.7) reduces to the steady Navier-Stokes equations showing that this extended system indeed enables the computation of fixed points of the non-linear Navier-Stokes equations. One major limitation of this technique remains however the computation of steady solutions when an eigenvalue of the associated linearised system turns out to be a real eigenvalue. Indeed, in such cases the low-pass filter applied to the non-linear Navier-Stokes equations is totally unable to kill such non-oscillating unstable mode and consequently to stabilize the fully non-linear system. One possibility to overcome this problem is to use existing symmetries of the solution whenever possible though this is not always sufficient. Numerical implementation : The filtered state s evolution equation has been implemented within Nek5000 using a simple explicit Euler temporal scheme. The discretized filtered state s equation thus reads: Ū n+1 = Ūn + t ( ( )) ω c Un Ūn whereas the stabilizing forcing term in the right-hand side of the Navier-Stokes equations is treated explicitly as well: χ(u n Ūn) 7

12 2.4 Modal decomposition Global linear stability, time-stepping and Arnoldi algorithm As shown previously, linear stability analysis introduces the Jacobian matrix J of the Navier-Stokes operator linearised around a given base flow Q 0 = (U 0, P ) T. Unfortunately, eigenvalue problems resulting from fully three-dimensional linear stability analysis involve Jacobian matrices of extremely large dimensions. As a consequence, not only do the classical algorithms for direct computation of the eigenpairs of the Jacobian matrix (e.g. QR or QZ algorithms for instance) become totally ineffective, but the matrix J involved might actually be so large that it cannot be explicitly stored. Because of this inability to store the matrix, even the original Arnoldi algorithm is not very useful. This problem had been overcome in the mid 90 s by Edwards et al. [10] introducing the so-called time-stepper approach and has been greatly popularised since 2009 by Bagheri et al.[14]. This technique relies on the fact that, as shown previously, the linearised Navier-Stokes equations can be recast into the following linear dynamical system form: B q t = J q (2.8) with q = (u, p) T and where B is a singular mass matrix. Projecting the whole dynamics onto a divergence-free vector space, equation (2.8) can be rewritten as: u = Au t (2.9) where A is now the projected Jacobian matrix. Equation (2.9) now accepts the following expression: u( t) = e A t u 0 (2.10) as a formal solution, where e A t, denoted also as M( t), is called the time or exponential propagator. The most interesting feature of the time-stepper approach is based on the fact that, though the action of the Jacobian matrix A onto a vector u 0 cannot be computed, the action of the exponential propagator e A t can be approximated by timemarching the linearised Navier-Stokes equations with u 0 as initial condition. Moreover, it is noteworthy that the eigenpairs (V, Λ) of the Jacobian matrix A and those (V e, Σ) of the associated exponential propagator e A t are related as follows: Λ = log(σ), V = V e (2.11) t Due to this fairly simple relationship linking the eigenpairs of the Jacobian matrix to those of the exponential propagator, and because the action of the latter operator onto a given vector can be approximated relatively easily, the technique introduced in [10, 14] aims at computing an approximation to the exponential propagator s eigenpairs by an Arnoldi algorithm instead of approximating directly those of the Jacobian matrix. The basic Arnoldi iteration thus reads: MU k U k H k (2.12) 8

13 Figure 2.3: Block diagram of the time-stepping Arnoldi algorithm implemented around the Nek5000 temporal loop. with U k being an orthonormal set of vectors spanning a Krylov subspace K of dimension k onto which M is projected, and H k the associated projection. The Hessenberg matrix H k resulting from this Arnoldi iteration is a small k k upper triangular matrix whose eigenpairs (Σ H, X), also called Ritz pairs, can be directly computed and are a good approximation to those of M. As previously, those Ritz pairs are linked to the eigenpairs (Λ, V) of the Jacobian matrix A by the following relationship: Λ = log(σ H), V = UX (2.13) t The actual Arnoldi algorithm used during this internship is presented on figure 2.3 in its block-diagram representation. Here, the Arnoldi iteration (2.12) has been built directly around the temporal loop of the linearised version of the code. As a consequence, the Krylov sequence computed is by construction an orthonormal set of vectors and as such turned out to yield easier convergence of a larger part of the eigenspectrum of the linearised Navier-Stokes operator. 2.5 Transient growth analysis Though eigenspectrum analysis of the linearised Navier-Stokes operator has proven to yield good predictions of the dynamics in numerous cases, it only captures the asymptotic (t ) evolution of an infinitesimal perturbation. Moreover, perturbations governed by the linearised Navier-Stokes operator s eigenvalues turn out to have a relatively small growth rate. As a consequence, it would take quite a large amount of time for such perturbations to grow up to only a few orders of magnitude higher than their initial amplitude. However, numerous experimental investigations have reported instabilities and transition scenarii occurring on a substantially smaller timescale revealing how poor of a proxy the eigenspectrum of the linearised operator is when inferring the short-time evolution of infinitesimal perturbations. As to partially solve this problem, non-modal stability analysis as defined by Schmid & Henningson [7] has to be used. In such 9

14 framework, the short-time evolution of infinitesimal perturbations is no more inferred from the eigenvalues of the Jacobian matrix A but instead from the singular values of the exponential propagator M(t) = e At. The optimal perturbation yielding the largest energy growth at a given time horizon t is given the first right singular vector u solution to the following singular value problem: σv = M(t)u (2.14) where the left singular vector v gives the associated optimal response. As for the linear global stability analysis, due to the large number of degrees of freedom involved in the computation, the exponential propagator e At cannot be explicitly formed and it thus forbids the use of classical singular values algorithms. However, as previously, the action of this operator M(t) onto a given vector can easily be approximated by a linearised Navier-Stokes solver. Moreover, introducing the adjoint operator M (t) = e A t, the singular value problem (2.14) can be recast into the following eigenproblem: M (t)m(t)u = σ 2 u (2.15) where the adjoint operator M just introduced is the time propagator associated to the following adjoint linearised Navier-Stokes equations: u + (U 0 )u u ( U 0 ) T = p + Re 1 u u = 0 (2.16) As for the direct linearised Navier-Stokes equations, this set of equations can be recast into a linear dynamical system form: B q t = J q (2.17) where q = (u, p ) is the adjoint state vector and J the adjoint Jacobian matrix. As previously, projecting onto a divergence-free vector space yields the following linear dynamical system: u t = A u (2.18) with A being the projected adjoint Jacobian and the operator M (t) = e A t its associated time propagator. It is noteworthy that, as for its direct counterpart, the action of this adjoint time propagator onto a given vector can be approximated by using an adjoint linearised Navier-Stokes solver. In the following sections, the algorithm based on such approximation of the action of these operators in order to compute the singular values and associated singular vectors will be presented. For a complete on non-modal stability theory, the reader is refered to Schmid [15]. Because the first singular triplet gives the worst case possible for transient energy amplification at a given time horizon of an otherwise infinitesimal perturbation, computing only this particular triplet is often sufficient to assess the short-time non-modal stability of the flow. As a consequence, the simplest eigenvalue algorithm, i.e. the power iteration method, can be used in order to compute this singular value and associated singular vectors. This simple algorithm consists in repeated applications of the operator 10

15 Figure 2.4: Block diagram of direct-adjoint loop implemented in Nek5000 for the poweriteration method. Figure 2.5: Block diagram of the SVD-Arnoldi algorithm implemented in Nek

16 M (t)m(t) onto an initial unit-norm vector x 0 until convergence is achieved. Its associated block-diagram representation is depicted on figure 2.4. Despite the significance of the first singular triplet in the transient dynamics of the flow, one might also be interested in the sub-optimal perturbations. Unfortunately, such sub-optimal perturbations cannot be computed using the simple power-iteration algorithm. Moreover, from a purely computational point of view, the power-iteration method is known to be slowly converging toward the leading singular triplet and can thus be relatively time-consuming. An alternative to overcome these two problems is to use a modified version of the Arnoldi algorithm presented previously. As one can see from the block-diagram depicted on figure 2.5, this SVD-Arnoldi algorithm is very similar to the one presented earlier differing only by the fact that the input vector u k is not only multiplied by the time propagator M(t) but also by its adjoint counterpart M (t). It is noteworthy that the version of the algorithm presented here only enables the computation of the right singular vectors, i.e. the optimal perturbations. A modified version where the operator M (t)m(t) is replaced by M(t)M (t) can be used in order to obtain the left singular vectors, i.e. the optimal responses. However, these responses can also be computed from a simple linearised DNS once the optimal perturbations obtained by the SVD-Arnoldi algorithm. Last but not least, both the Arnoldi and SVD-Arnoldi algorithms presented in this manuscript can be seen as simpler versions of the algorithm introduced by Barkley et al. [5]. 12

17 Chapter 3 Results 3.1 From nearly-axisymmetric solutions to wall-attached ones Subcritical pitchfork bifurcation Baseflows have been computed using selective frequency damping (SFD) as introduced by Akervik et al [6]. The first sequence of baseflows have been calculated by increasing the Reynolds number from 300 up to 540, and a second sequence by decreasing it from 540 down to 300. As to characterize these different solutions, a measurement of their asymmetry is performed using the following formula from Lanzterfor et al. [16]: Asymmetry = sign(x u x l ) x=40 x=1 ( z u z= z u z=0.5 ) 2 dx (3.1) 20 forward sequence backward sequence 15 Asymmetry Reynolds number Figure 3.1: Bifurcation diagram indicating the hysteresis in the asymmetry values Figure 3.1 depicts the evolution of this asymmetry measurement with increasing (blue) and decreasing Reynolds number (red). It can be seen from this figure that a hysteresis cycle sets in, indicating the flow undergoes at some point (Re = 460) a subcritical pitchfork bifurcation. Such bifurcation has already been put in the limelight by Blackburn and Sherwin [4] as well as Griffith et al. [17] in similar geometries. Because of the stenosis offset, it is moreover expected to be imperfect, one side of the stenotic pipe being slightly favored due to the asymmetry of the geometry. 13

18 (a) Slice of x-velocity at y = 0 (b) Slice of x-velocity at z = 0 Figure 3.2: Slices at different y and z locations of the x-velocity of the baseflow for Reynolds number 400 in the forward sequence Figure 3.3 depicts the lower branch solution at a Reynolds number Re = 400. As one can see from these figures, the flow mainly consists in a jet emerging from the stenotic constriction and wrapped by a toric reversed flow region extending down to x = 40. Looking at the flow in the y = 0 plane, only a relatively small asymmetry of the flow can be seen. This small asymmetry can be better observed by plotting the same velocity component in the x = 5 plane (see figure 3.3). (a) Slice of x-velocity at x = 5 (b) Slice of x-velocity at x = 5 (c) Slice of x-velocity at x = 20 Figure 3.3: Slices at different x- locations of the x-velocity of the baseflow for Reynolds number 400 in the forward sequence As one can see, offsetting the stenosis throat upward causes the upper part of the recirculation bubble to shrink a little compared to its lower part. Despite this small asymmetry of the flow, the lower branch solution closely resembles the axisymmetric ones and thus will be qualified as nearly-axisymmetric in the following. Figure 3.4 now depicts the main features of the upper branch solution at Re=400. As one can see, as previously, the flow mainly consists in a jet emerging from the stenotic constriction and wrapped by a toric reversed flow region once again. The asymmetry of the solution is however larger than its nearly-axisymmetric counterpart. Indeed, 14

19 (a) Slice of x-velocity at x = 5 (b) Slice of x-velocity at x = 5 (c) Slice of x-velocity at x = 10 Figure 3.4: Slices at different x-locations of the x-velocity of the baseflow for Reynolds number 400 in the backward sequence as one can see in figure 3.4, depicting the streamwise velocity component of the upper branch solution in various planes, the upper part of the toric recirculation bubble reattaches at x = 15, whereas its lower part reattaches later, at x = 20. The asymmetry of the flow clearly shows that the stenotic jet underwent a strong deflection from the pipe s centerline and has reattached to the upper wall by a weak Coanda effect. Such deflection toward the walls is a common feature of wall-confined jets and has already been observed in two-dimensional geometries such as sudden expansion channels as well as sudden expansion pipes (see [18]) or axisymmetric stenosic pipes (see [4]). In all cases, this is the direct consequence of an unstable real eigenvalue associated to an eigenmode breaking the up-and-down symmetry and thus triggering an infinitesimal, but exponentially growing, deflection of the jet from the channel/pipe s centerline. Once it has achieved a sufficient threshold, this deflection yields the flow to transition from the symmetric/axisymmetric/nearly-axisymmetric lower branch solution to the wall-attached upper branch one. (a) Slice of x-velocity at y = 0 (b) Slice of x-velocity at z = 0 Figure 3.5: Slices at different y and z locations of the x-velocity of the baseflow for Reynolds number 400 in the forward sequence 15

20 It is noteworthy that an interesting, but still not explained, difference between the axisymmetric/nearly-axisymmetric cases and their fully two-dimensional counterparts is the nature of the bifurcation. Indeed, as seen previously, for axisymmetric and nearlyaxisymmetric cases, the first pitchfork bifurcation to occur is a subcritical one, whereas it is a supercritical one in the strictly 2D cases Lower branch solution s transient growth The most interesting feature of subcritical bifurcations as the one just presented is the possibility for the flow to undergo transition though all of the eigenvalues of the associated linear Navier-Stokes operator lies within the lower half-complex plane, i.e. despite the the flow being linearly stable. Despite all infinitesimal perturbations being eventually damped, a few particular perturbations can experience large transient growth. Such perturbations have been computed at several time horizons for the lower branch solution at Re = 400 using the algorithm presented in the last section of the numerics chapter. 11 x G Time Figure 3.6: The optimal gain curve for flow at Reynolds number 400 Figure 3.6 depicts the resulting optimal gain curve. The optimal transient energy growth is achieved for a time τ = 4.5 with an optimal gain G max = As expected from the close resemblance between the two flows, these optimal time and gain closely matches the axisymmetric case one (τ = 4.5, G max = ). Figure 3.7 depicts the optimal perturbation and figure 3.8 represents the associated optimal response streamwise component. Figure 3.7: The optimal perturbation for flow at Reynolds number

21 Figure 3.8: The optimal response of the flow at Reynolds number 400 It is seen that the optimal perturbation is located very close to the throat of the stenosis, and is divergent towards the right, which is very similar to the results obtained by Barkley et al [19]. On the other hand, one can see that the optimal response is spatially located further downstream, between the streamwise stations x = 8 and x = 15. This is very close to the results obtained in the analysis done by Barkley et al. [19], where it is seen that the optimal response is spatially located between an x-distance of 7 to 13. The slight streamwise offset of the optimal response in the present case, as well as the slightly larger optimal time and gain, can be explained as a result of the slightly longer reversed flow region streamwise extent as compared to the one obtained by Barkley et al. [19] in the axisymmetric case. It is observed moreover that whereas the optimal perturbation is divergent to the right, the optimal response is a wave packet that is convergent to the right. This reorientation of the structure, due to the perturbation encountering the baseflow shear, is known as the Orr mechanism. The contribution of different terms to the energy budget of the optimal perturbation is evaluated using the expression for the total energy as : E t = Ω uv V x }{{} I 4 [ u 2 U x uw W }{{ x} I 7 }{{} I 1 2 v V y }{{} I 5 vw U uv y W y }{{} I 8 }{{} I 2 vw V z }{{} I 6 uw U z }{{} I 3 w 2 W z }{{} I ] (u u + v v + w w) dω } Re {{} D Figure 3.9 shows the time evolution of the energy budget of this optimal perturbation. As one can see, the two dominant terms are uv U U and uw, that is the work of the y z Reynolds stresses against the wall-normal gradients of the baseflow streamwise component. It is noteworthy that these shears are maximum in the vicinity of the reversed flow region. Consequently, as long as the perturbation travels along the shear layers bordering the recirculation bubble, it experiences a strong re-orientation and thus transient growth of its kinetic energy. Once the perturbation has been advected further downstream, beyond the recirculation bubble extent, the baseflow shear it encounters weakens and the perturbation cannot extract as much energy as previously. In the meantime, the viscous damping increases, balancing the energy extraction process and eventually causing the perturbation to fade away. The slightly larger amplitude of the uw U term is the di- z 17

22 7 x uu du/dx uv du/dy uw du/dz 5 Energy budget Time Figure 3.9: The energy budget of the transient growth at Reynolds number 400 rect consequence of the stenosis throat offset in the z-direction causing the associated gradients to be slightly larger than their y counterparts Secondary bifurcation of the wall-attached solutions It has been seen in the previous section that the nearly-axisymmetric solution experiences a subcritical pitchfork bifurcation at Re = 460 yielding a wall-attachment of the stenotic jet. As a consequence of this, a strong asymmetry of recirculation bubble can be observed in the steady solutions belonging to the upper branch solution. Further increasing the Reynolds number causes this family of solutions to become unstable as well. (a) Slice of x-velocity at y = 0 (b) Slice of x-velocity at z = 0 Figure 3.10: Slices at different y and z-locations of the x-velocity of the baseflow for Reynolds number

23 (a) Slice of x-velocity at x = 5 (b) Slice of x-velocity at x = 5 (c) Slice of x-velocity at x = 10 Figure 3.11: Slices at different x-locations of the x-velocity of the baseflow for Reynolds number 540 Figure depicts the baseflow for Re = 540. As one can see, this steady state exhibits similar features as the Re = 400 wall-attached solution presented previously: a wall-attached stenotic jet strongly asymmetric reversed flow region in the z-direction the plane y = 0 is a symmetry plane. The major effect of the Reynolds number increase is the streamwise extension of the recirculation bubble as well as its stronger asymmetry. Indeed, whereas the lower recirculation extends down to x = 20 in the Re = 400 wallattached solution, in the present case it extends down to x = 30, that is a 50% increase of the reversed flow region length. Similar behavior holds for the upper recirculation length with an increase of 55%. A linear global stability analysis of the present steady state has been carried out using the Arnoldi algorithm presented in the numerical chapter of this report. A Krylov subspace of dimension 500 and sampling period T = of the snapshots have been considered for this eigenvalue computation. This time sample enables the convergence the leading eigenvalues below the frequency of 2.5. Only one single real eigenvalue has been observed to lie in the upper-half complex plane (λ = ) indicating that a secondary pitchfork bifurcation of the flow actually takes place at a slightly lower Reynolds number. The eigenmode results in an unstable but non-oscillating flow pattern. The eigenmode breaks the last remaining right-left symmetry of the stenotic pipe. It is observed that the eigenmode lies entirely in the recirculation region and doesn t affect the rest of the flow field. This eigenmode is very similar to the one observed by Theofilis [20] in 2-D flows where the eigenmode causes spanwise modulation of the 2-D recirculation zone. Figure 3.12 shows a slice in the y = 0 plane and the x = 2 plane of the spanwise velocity of the eigenmode (where the similarity to the eigenmode observed by Theofilis [20] is observed most clearly). The energy budget of the eigenmode has been evaluated and is shown in the following figure : 19

24 (a) Slice of y-velocity at y = 0 (b) Slice of y-velocity at x = 2 Figure 3.12: The eigenmode observed at Reynolds number I 1 I2 I3 I4 I5 I6 I7 I8 I9 D Figure 3.13: The energy budget of the eigenmode at Reynolds number 540 It is once again noticed that the major contribution to the energy of the eigenmode is from the work done by the Reynolds stress terms against the wall normal gradients to the baseflow streamwise velocity. Figure 3.14 shows the lengths of the upper and the lower recirculation zones for the present case. 20

25 Figure 3.14: The recirculation zone at Reynolds number 540 The two figures 3.15 and 3.16 represent respectively the spatial location of the components which contribute the most to the energy of the eigenmode. Figure 3.15: The location of the uv U y at Reynolds number 540 Figure 3.16: The location of the uw U z at Reynolds number 540 It is probable that, if this eigenmode is allowed to evolve non-linearly, it saturates eventually, with the breaking of the recirculation region. Since all the symmetries in the pipe are broken down, there is no more counter-balancing of the shear strains that are produced on the recirculation bubble, and hence, the bubble might break leading to transition to turbulence. 21

26 Chapter 4 Conclusion In summary, the effect of the eccentricity of the stenosis on a pipe is studied in detail, and has been compared with the results obtained on a pipe with axisymmetric stenosis. All the studies in this particular project have been done on a stenotic pipe with an eccentricity of 10 %. It is observed while calculating the baseflows that a hysteresis is observed in measuring the asymmetry between the top and the bottom recirculation regions. This hysteresis is typical of a subcritical bifurcation, and it is seen that indeed an imperfect pitchfork bifurcation takes place at Re = 460 which is subcritical in nature. This leads to branches of solutions of the baseflows - the lower branch with nearly-axisymmetric flows with a slight deflection and the upper branch with wall attached flow fields and strong recirculation regions. Further, transient growth analysis is done in order to observe the optimal perturbation and the optimal response of the flow to this perturbation. The values observed here are quite close to those observed in the case of the axisymmetric stenotic pipe. An energy budget analysis is carried out to understand the contributions of various terms to the perturbation energy. Linear stability analysis is performed at Re = 540 which yields an unstable but non-oscillating eigenmode which in turn breaks the last remaining symmetry in the flow. It is also observed that the eigenmode is localised in the recirculation region. The package Nek5000 has been used for the direct numerical simulations and the application of the Arnoldi algorithm. Also, the behaviour of flowfields at higher eccentricities need to be observed so as to understand if similar bifurcations are encountered. At Re = 540, all remaining symmetries in the flowfield are broken, and there is no counterbalance to the high shear strain experienced by the lower recirculation region. Hence, with the availability of more computational power, a non-linear evolution of the flow-field needs to be carried out to understand if the flow indeed transitions to turbulence. Ackowlegements I would like to thank Prof. Jean-Christophe Robinet for giving me the opportunity to work under his guidance, and for his continuous help and support during the course of the internship. I also thank Jean-Christophe Loiseau who was very supportive and patiently answered each and every one of my doubts with a smile. I am indebted to Ganga for his enormous last-minute help (as usual) in using latex. 22

27 Bibliography [1] W. page of webmd, Web page of webmd. heart-disease/atherosclerosis [2] W. page of depositphotos, Web page of depositphotos. com/ /stock-illustration-atherosclerosis.html. [3] S. S. Varghese, S. H. Frankel, P. F. Fischer, et al., Direct numerical simulation of stenotic flows. part 1. steady flow, Journal of Fluid Mechanics, vol. 582, no. 1, pp , [4] S. Sherwin and H. M. Blackburn, Three-dimensional instabilities and transition of steady and pulsatile axisymmetric stenotic flows, Journal of Fluid Mechanics, vol. 533, pp , [5] D. Barkley, H. Blackburn, and S. Sherwin, Direct optimal growth analysis for timesteppers, International journal for numerical methods in fluids, vol. 57, pp , [6] E. Akervik, L. Brandt, D. S. Henningson, J. Hoepffner, O. Marxen, and P. Schlatter, Steady solutions of the navier-stokes equations by selective frequency damping, Physics of fluids, vol. 18, [7] P. J. Schmid and D. S. Henningson, Stability and transition in shear flows. Springer, [8] V. Theofilis, Global linear instability, Annual Review of Fluid Mechanics, vol. 43, pp , [9] L. S. Tuckerman and D. Barkley, Bifurcation analysis for timesteppers. Springer, [10] W. Edwards, L. Tuckerman, R. Friesner, and D. Sorension, Krylov methods for the incompressible navier-stokes equations, Journal of computational physics, vol. 110, pp , [11] P. F. Fischer, J. W. Lottes, and S. G. Kerkemeier, nek5000 Web page. https: //nek5000.mcs.anl.gov/index.php/main_page, [12] M. Deville, P. Fischer, and E. Mund, High-Order methods for incompressible fluid flow. Cambridge University Press, [13] G. E. Karniadakis and S. Sherwin, Spectral/hp element methods for computational fluid dynamics. Oxford science publications,

28 [14] S. Bagheri, E. Akervik, L. Brandt, and D. S. Henningson, Matrix-free methods for the stability and control of boundary layers, AIAA Journal, vol. 47, [15] P. J. Schmid, Nonmodal stability theory, Annual Review of fluid mechanics, [16] D. Lanzerstorfer and H. C. Kuhlmann, Global stability of multiple solutions in plane sudden-expansion flow, Journal of Fluid Mechanics, vol. 702, pp , [17] M. D. Griffith, T. Leweke, M. C. Thompson, and K. Hourigan, Steady inlet flow in stenotic geometries: convective and absolute instabilities, Journal of fluid mechanics, vol. 616, pp , [18] E. Sanmiguel-Rojas and T. Mullin, Finite-amplitude solutions in the flow through a sudden expansion in a circular pipe, Journal of Fluid Mechanics, vol. 691, pp , [19] H. Blackburn, S. J. Sherwin, and D. Barkley, Convective instability and transient growth in steady and pulsatile stenotic flows, Journal of Fluid Mechanics, vol. 607, no. 1, pp , [20] V. Theofilis, S. Hein, and U. Dallmann, On the origins of unsteadiness and threedimensionality in a laminar separation bubble, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, vol. 358, no. 1777, pp ,