Steady inflow through a model aneurysm: global and transient stability

The stability of flow through a model aneurysm is numerically computed using a global linear stability analysis and a dire ct transient growth analysis. The geometry features a sinusoi dal bulge in an otherwise uniform circular pipe, with dimension s representative of a human abdominal aortic aneurysm. With a time-independent inflow, the flow is weakly unstable to quasi periodic global eigenmodes of azimuthal wavenumbers 4 and 5 at a Reynolds number (based on area-averaged velocity and pipe diameter)Re≈ 3900. These eigenmodes are concentrated in the outer part of the bulge, in the vicinity of its downstre am end. A transient growth analysis reveals that the flow is sens itive to transient disturbances beyond Re= 33, well below the time-averaged Reynolds numbers of blood flow in the human abdominal aorta. Introduction Pipe flows arise in myriad engineering and biological system , and hence have been a constant focus of fluid mechanics research for more than a century. The stability of flow in pipes is of particular interest, as transition from laminar to tur b lent flow leads to a significant change in the flow dynamics. Moreover, particularly with application to biological flows in t he cardiovascular system, changes in the symmetry of flows through the development of instabilities can markedly alter near-w ll velocity profiles, and hence wall shear stress. This can exac erbate cardiovascular diseases such as atherosclerosis [1 0] and aneurysm [15]. Cardiovascular disease is a major health concern. For insta nce, 18% of Australians suffer a long-term cardiovascular condi tion, and 11% of the National health expenditure is incurred due to this disease [1]. In recent years, studies have elucidated t he stability of flow through constricted pipe geometries designed to model stenosis (e.g. see [5, 7] and references therein). How ever, research into aneurysm disease, which manifests thro ug a localized dilation of an artery due to degradation of wall t issue integrity, has primarily focused on patient-specific si mulation [9]. Exceptions include recent laboratory experiment s [11] and numerical simulations [13] in axisymmetric aneurysm mo dels at physiologically relevant Reynolds numbers. These de t rmined the wall shear stress distribution across, and flow str ucture within, an aneurysm bulge. The laboratory experiments employed particle image velocimetry in a plane bisecting th e bulge, and inspection of their visualizations reveals slig ht asymmetry across the bulge centreline. This hints that these flow s may be susceptible to non-axisymmetric instabilities. The aim of this study is to investigate both the asymptotic st ability and potential for transient growth within the flow thr ough an axisymmetric model aneurysm, shown in figure 1. Fully developed Poiseuille flow with an area-averaged velocity U , flows from left to right. The fluid is Newtonian with kinematic visc osity ν, and a Reynolds number is defined based on the un-dilated pipe diameterD as Re= UD ν . Aneurysm flows in humans can experience time-averaged and Figure 1: The geometry under investigation in this paper: th e length (L) and width (W) of the bulge and the un-dilated pipe diameterD are shown. The bulge profile is sinusoidal. peak Reynolds numbers of Re≈ 300 andRe≈ 2000 to 3000, respectively [11, 9]. These values guide the Reynolds numbe r range ofRe≤ 8000 investigated in the present study. Reproducing the aneurysm geometry considered in [13], the pipe features a smooth bulge with a sinusoidal profile. The bulge length and maximum diameter are 2 . 9D and 1.9D, consistent withModel 3in [11]). Despite linear stability of Hagen– Poiseuille flow, one expects turbulence to appear in the stra ight downstream pipe at higher Reynolds number. Methodology The flows described in this study are computed by solving the incompressible Navier–Stokes equations formulated in cyl i drical coordinates. A nodal spectral-element method is used fo r spatial discretization, and time integration is performed using backwards differencing [8]. The code follows a formulation n cylindrical coordinates [4], and solves the momentum and ma ss conservation equations, ∂tu =−(u ·∇)u−∇p+ν∇2u, ∇ ·u = 0, where∂t denotes a partial derivative with respect to time t, u is the velocity vector, andp is the kinematic static pressure. Linear stability analysis Asymptotic stability of the axisymmetric base flows ( U, P) to three-dimensional perturbations ( u′, p′) is determined using a linear stability analysis. Linearizing about the base flow y ields the linearized Navier–Stokes equations describing the evo lution of the perturbation field ∂tu =−(U ·∇)u′− (u′ ·∇)U−∇p′+ν∇2u′, ∇ ·u′ = 0. (1) A Fourier decomposition discretizes the three-dimensiona l perturbation field in the azimuthal direction, and linearity de couples individual Fourier modes. This permits the stability o f individual azimuthal wavenumbers ( m) to be computed separately on the same axisymmetric domain as used for the base flow computations. An operatorA (T) is defined describing the action of integrating equation (1) over a time interval T from an initial perturbationu′(t = 0) such thatu′(T) = A (T)u′(0). Solutions of equation (1) can be decomposed into solutions of the form Figure 2: Contour plots showing the azimuthal component of vorticity in the vicinity of the bulge from axisymmetric sim ulations at Reynolds numbers Re= 1 andRe= 5000. 12 equispaced vorticity contour levels are plotted between ωθ ±8U/D, with dark and light shading showing negative and positive vo rticity, respectively. ũ(x)exp(σt), whereσ is the complex growth rate of individual eigenfunctions̃u(x). Stability is dictated by leading eigenmodes of the eigenproblem A (T)ũ j = μj ũ j , whereũ j are eigenvectors corresponding to eigenvalues μj . The eigenvalue relates to the growth rate of an eigenmode throug μ = exp(σT), and instability is predicted when |μ| > 1. The linear stability analysis capabilities of the code have bee n successfully employed in recent studies [14, 3]. Transient growth analysis In contrast to the asymptotic behaviour of linear perturbat ions predicted by linear stability analysis, evolution over sho rt times may result in significant amplification due to the non-normal ity of eigenmodes of A [12]. Transient growth analysis seeks to determine the maximum possible amplification in energy ( G) for a perturbation evolved over a time interval τ, and the corresponding initial field leading to this optimal growth ( v). These have been shown to be equivalent to the square of the leading singular value ( λ) and the leading right singular vector of A , respectively. The square of the principal singular value and right singula r vector ofA are equivalent to the leading eigenvalue and eigenvector ofA ∗A , whereA ∗ is the conjugate transpose (adjoint) of A . This relationship has been used [2] to develop a method for predicting transient growth of a flow without explicitly constructingA or A ∗. They derived the adjoint linearized Navier– Stokes equations −∂tu = (U ·∇)u∗− (∇U) ·u∗−∇p∗+ν∇2u∗, ∇ ·u∗ = 0, (2) such that the evolution of a perturbation backwards in time d escribes the action of A ∗ on a perturbation. Therefore, the action ofA ∗A may be achieved by first integrating a perturbation forward in time overτ using equation (1), and subsequently integrating the result backwards in time using equation (2). T ransient growth is then governed by the leading eigenmodes of A ∗ A (τ)v j = λ jv j . Discretization and grid independence The mesh used in this study discretized the meridional halfplane of the geometry, and comprised 1603 elements. In order to accommodate the perturbation fields arising during tr ansient growth analysis, the domain extended upstream and dow nstream of the bulge for the relatively large distances of 19 D and 44D, respectively. To resolve flow features, elements were concentrated within and downstream of the bulge. To determine grid independence, the convergence of transie nt growth eigenvalues with increasing element polynomial deg re (Np) was computed. Reynolds numbers Re= 300 and 5000 were considered, corresponding respectively to approxima tely the time-averaged Reynolds number for flow through a human abdominal aorta, and a value exceeding the peak Reynolds num ber in these vessels. At Re= 5000, simulations withNp ≤ 5 failed to achieve a time-invariant state. At Np = 6, G converged to within 6 and 3 significant figures for Re= 300 and 5000, respectively. A resolution of Np = 6 is hence used hereafter.