Vassilis Theofilis

Steady laminar flow over a rounded-tip 2 : 1 elliptic cone of 0.86 m length at zero angle of attack and yaw has been computed at Mach number 7.45 and unit Reynolds number Re = 1.015 × 10 7 m −1. The flow conditions are selected to match the planned flight of the Hypersonic Flight Research Experimentation HIFiRE-5 test geometry at an altitude of 21.8 km. Spatial linear BiGlobal modal instability analysis of this flow has been performed at selected streamwise locations on planes normal to the cone symmetry axis, resolving the entire flow domain in a coupled manner while exploiting flow symmetries. Four amplified classes of linear eigenmodes have been unravelled. The shear layer formed near the cone minor-axis centreline gives rise to amplified symmetric and antisymmetric centreline instability modes, classified as shear-layer instabilities. At the attachment line formed along the major axis of the cone, both symmetric and antisymmetric instabilities are also discovered and identified as boundary-layer second Mack modes. In both cases of centreline and attachment-line modes, symmetric instabilities are found to be more unstable than their antisymmetric counterparts. Furthermore, spatial BiGlobal analysis is used for the first time to resolve oblique second modes and cross-flow instabilities in the boundary layer between the major-and minor-axis meridians. Contrary to predictions for the incompressible regime for swept infinite wing flow, the cross-flow instabilities are not found to be linked to the attachment-line instabilities. In fact, cross-flow modes peak along most of the surface of the cone, but vanish towards the attachment line. On the other hand, the leading oblique second modes peak near the leading edge and their associated frequencies are in the range of the attachment-line instability frequencies. Consequently, the attachment-line instabilities are observed to be related to oblique second modes at the major-axis meridian. The linear amplification of centreline and attachment-line instability modes is found to be strong enough to lead to laminar–turbulent flow transition within the length of the test object. The predictions of global linear theory are compared with those of local instability analysis, also performed here under the assumption of locally parallel flow, where use of this assumption is permissible. Fair agreement is obtained for symmetric centreline and symmetric attachment-line modes, while for all other classes of linear disturbances use of the proposed global analysis methodology is warranted for accurate linear instability predictions.

TriGlobal linear instability analysis and direct numerical simulations have been performed to unravel the mechanisms ultimately responsible for transition of steady laminar flow over a long rectangular finite-span open cavity with dimensions L : D : W = 6 : 1 : 2 to unsteadiness. The steady laminar three-dimensional flow loses stability at Re D,cr ≈ 1080 as a consequence of linear amplification of a travelling eigenmode that is qualitatively analogous to the shear-layer mode known from analyses of flow in spanwise-periodic cavities, but has a three-dimensional structure which is strongly influenced by the cavity lateral walls. Differences in the eigenspectrum of the present and the spanwise homogeneous flow configuration are documented. Topological changes exerted on the steady laminar flow by linear amplification of the unstable shear-layer mode are reminiscent of observations in experiments at an order of magnitude higher Reynolds number.

Global and Local Hydrodynamic Stability Analysis as a Tool for Combustor Dynamics Modeling Coherent flow structures in shear flows are generated by instabilities intrinsic to the hydrodynamic field. In a combustion environment, these structures may interact with the flame and cause unsteady heat release rate fluctuations. Prediction and modeling of these structures are thereby highly wanted for thermo-acoustic prediction models. In this work, we apply hydrodynamic linear stability analysis to the time-averaged flow field of swirl-stabilized combustors obtained from experiments. Recent fundamental investigations have shown that the linear eigenmodes of the mean flow accurately represent the growth and saturation of the coherent structures. In this work, biglobal and local stability analyses are applied to the reacting flow in an industry-relevant combustion system. Both the local and the biglobal analyses accurately predict the onset and structure of a self-excited global instability that is known in the combustion community as a precessing vor-tex core (PVC). However, only the global analysis accurately predicts a globally stable flow field for the case without the oscillation, while the local analysis wrongly predicts an unstable global growth rate. The predicted spatial distribution of the amplitude functions using both analyses agrees very well to the experimentally identified global mode. The presented tools are considered as very promising for the understanding of the PVC and physics based flow control.

Linear global instability analysis has been performed in the wake of a low aspect ratio threedimensional
wing of elliptic cross section, constructed with appropriately scaled Eppler E387 airfoils. The
flow field over the airfoil and in its wake has been computed by full three-dimensional direct numerical simulation
at a chord Reynolds number of Rec = 1750 and two angles of attack, AoA = 0◦ and 5◦. Point-vortex
methods have been employed to predict the inviscid counterpart of this flow. The spatial BiGlobal eigenvalue
problem governing linear small-amplitude perturbations superposed upon the viscous three-dimensional wake
has been solved at several axial locations, and results were used to initialize linear PSE-3D analyses without
any simplifying assumptions regarding the form of the trailing vortex system, other than weak dependence of
all flow quantities on the axial spatial direction. Two classes of linearly unstable perturbations were identified,
namely stronger-amplified symmetric modes and weaker-amplified antisymmetric disturbances, both peaking
at the vortex sheet which connects the trailing vortices. The amplitude functions of both classes of modes were
documented, and their characteristics were compared with those delivered by local linear stability analysis in
the wake near the symmetry plane and in the vicinity of the vortex core. While all linear instability analysis
approaches employed have delivered qualitatively consistent predictions, only PSE-3D is free from assumptions
regarding the underlying base flow and should thus be employed to obtain quantitative information on
amplification rates and amplitude functions in this class of configurations.

This paper presents an investigation of the origin and evolution of the complex flow pattern on a hemisphere cylinder at separated flow conditions. Three-dimensional numerical simulations have been performed for a range of Reynolds numbers and angles of attack. A critical point theory has been used to analyze the flowfields. This has yielded, for the first time for this geometry, a bifurcation diagram that classifies the different flow topology regimes as a function of the Reynolds number and angle of attack. A complete characterization of the origin and evolution of the complex structural patterns of this geometry is documented. For the higher Reynolds number and angle of attack, a structurally stable topology is found that is associated with the pattern of the horn vortices, usually found on this geometry in a range from low to high Reynolds numbers and from incompressible to compressible regimes. Surface critical points and surface and volume streamlines describe the main flow structures and their strong dependence with the flow conditions.

Three-dimensional Direct Numerical Simulations combined with Particle Image Velocimetry experiments have been performed on a hemisphere-cylinder at Reynolds number 1000 and angle of attack 20 •. At these flow conditions, a pair of vortices, so-called " horn " vortices, are found to be associated with flow separation. In order to understand the highly complex phenomena associated with this fully three-dimensional massively separated flow, different structural analysis techniques have been employed: Proper Orthogonal and Dynamic Mode Decompositions, POD and DMD, respectively, as well as critical-point theory. A single dominant frequency associated with the von Karman vortex shedding has been identified in both the experimental and the numerical results. POD and DMD modes associated with this frequency were recovered in the analysis. Flow separation was also found to be intrinsically linked to the observed modes. On the other hand, critical-point theory has been applied in order to highlight possible links of the topology patterns over the surface of the body with the computed modes. Critical points and separation lines on the body surface show in detail the presence of different flow patterns in the base flow: a three-dimensional separation bubble and two pairs of unsteady vortices systems, the horn vortices, mentioned before, and the so-called " leeward " vortices. The horn vortices emerge perpendicularly from the body surface at the separation region. On the other hand, the leeward vortices are originated downstream of the separation bubble, as a result of the boundary layer separation. The frequencies associated with these vortical structures have been quantified.

Three-dimensional direct numerical simulations (DNS) have been performed on a finite-size hemisphere-cylinder model at angle of attack AoA = 20 • and Reynolds numbers Re = 350 and 1000. Under these conditions, massive separation exists on the nose and lee-side of the cylinder, and at both Reynolds numbers the flow is found to be unsteady. Proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD) are employed in order to study the primary instability that triggers unsteadiness at Re = 350. The dominant coherent flow structures identified at the lower Reynolds number are also found to exist at Re = 1000; the question is then posed whether the flow oscillations and structures found at the two Reynolds numbers are related. POD and DMD computations are performed using different subdomains of the DNS computational domain. Besides reducing the computational cost of the analyses, this also permits to isolate spatially localized oscillatory structures from other, more energetic structures present in the flow. It is found that POD and DMD are in general sensitive to domain truncation and non-educated choices of the subdomain may lead to inconsistent results. Analyses at Re = 350 show that the primary instability is related to the counter-rotating vortex pair conforming the three-dimensional afterbody wake, and characterized by the frequency St ≈ 0.11, in line with results in the literature. At Re = 1000, vortex-shedding is present in the wake with an associated broadband spectrum centered around the same frequency. The horn/leeward vortices at the cylinder lee-side, upstream of the cylinder base, also present finite amplitude oscillations at the higher Reynolds number. The spatial structure of these oscillations, described by the POD modes, is easily differentiated from that of the wake oscillations. Additionally, the frequency spectra associated with the lee-side vortices presents well-defined peaks, corresponding to St ≈ 0.11 and its few harmonics, as opposed to the broadband spectrum found at the wake.

Linear stability analysis of vortical systems is discussed in a systematic manner using the classic inviscid vortex filament method. Well-known results are recovered as special cases of a unified linearization procedure. The symmetries of the vortex systems analyzed are exploited to obtain an analytical picture of different classes of growing and decaying eigenmodes. Finally, comparisons with viscous global linear theory reveal the limits of applicability of the vortex filament method for the instability analysis of realistic vortex systems.

This paper studies the effect of compressibility on the linear stability of a two-dimensional lid-driven cavity flow in the subsonic regime. The base flow is generated by high fidelity direct numerical simulation and a biglobal mode instability analysis is carried out by a matrix forming approach. The eigenvalue problem is discretized by high-order finite differences and Arnoldi algorithm is used to reduce the size of the problem. The solution procedure uses sparse matrix techniques. Influence of Mach number on the modes known from incompressible calculations is presented, showing that compressibility has a stabilizing effect. New modes that appear only for compressible flows are presented and their relationship with duct acoustics is investigated.

A novel time-stepping shift-invert algorithm for linear stability analysis of laminar flows in complex geometries is presented. This method, based on a Krylov subspace iteration, enables the solution of complex non-symmetric eigenvalue problems in a matrix-free framework. Validations and comparisons to the classical exponential method have been performed in three different cases: (i) stenotic flow, (ii) backward-facing step and (iii) lid-driven swirling flow. Results show that this new approach speeds up the required Krylov subspace iterations and has the capability of converging to specific parts of the global spectrum. It is shown that, although the exponential method remains the method of choice if leading eigenvalues are sought, the performance of the present method could be dramatically improved with the use of a preconditioner. In addition, as opposed to other methods, this strategy can be directly applied to any time-stepper, regardless of the temporal or spatial discretization of the latter.

Linear instability of the three-dimensional boundary-layer over the Hypersonic International Flight Research Experimentation 5 (HIFiRE-5) flight test geometry, i.e. a rounded-tip 2:1 elliptic cone, is analyzed through spatial BiGlobal linear instability analysis, in an effort to understand transition and accurately predict local heat loads on this model of next-generation flight vehicles. The base flow conditions are selected for a Mach 7 flow at altitude of 33.0 km and unit Reynolds number Re 0 ¼ 1:89 Â 10 6 m. The base flow is computed using the US3D solver. The three-dimensionality of the boundary-layer on the elliptic cone produces spanwise pressure gradients, inducing crossflow from the leading edge (major-axis meridian) to the centerline (minor-axis meridian) and producing lift-up of low momentum boundary-layer fluid at the centerline. The present analysis focuses on the resulting mushroom-like structure formed near the minor-axis meridian. An unstable fluid structure, which is composed of a low-velocity streak surrounded by a three-dimensional high-shear layer, is thus generated. Results show that this complex fluid structure can sustain linear growth of a number of instability modes, thus confirming earlier analyses which identified such modes without taking surface curvature of the geometry in question into consideration. These so-called centerline modes are associated with varicose or sinuous deformations of the low-velocity streak and are found to have similar growth rates in the frequency range studied, between 50 and 300 kHz. Furthermore , unstable boundary-layer modes, identified as oblique second Mack modes, are unraveled in the vicinity of the centerline. At the conditions studied, Mack modes are found to be less amplified that the centerline modes.

A theoretical study of linear global instability of incompressible flow over a rectangular spanwise-periodic open cavity in an unconfined domain is presented. Comparisons with the limited number of results available in the literature are shown. Subsequently, the parameter space is scanned in a systematic manner, varying Reynolds number, incoming boundary-layer thickness and length-to-depth aspect ratio. This permits documenting the neutral curves and leading eigenmode characteristics of this flow. Correlations constructed using the results obtained collapse all available theoretical data on the three-dimensional instabilities.

The development of a general Jacobian-free approach for the solution of large-scale global linear instability analysis eigenvalue problems by coupling a time-stepping algorithm with industry-standard second-order accurate aerodynamic codes is presented. The three-dimensional lid-driven cavity, a challenging flow in the context of required computational resources and physical complexity, has been chosen for validation. Results in excellent agreement with the literature have been obtained by using the proposed theoretical methodology coupled with the incompressible solver of the open-source toolbox OpenFOAM. The moderate computational resources required for the solution of the TriGlobal eigenvalue problem using this method opens up a new avenue for the performance of instability analysis of flows of engineering relevance.

This article contains a review of modal stability theory. It covers local stability analysis of parallel flows including temporal stability, spatial stability, phase velocity, group velocity, spatio-temporal stability, the linearized Navier–Stokes equations, the Orr– Sommerfeld equation, the Rayleigh equation, the Briggs–Bers criterion, Poiseuille flow, free shear flows, and secondary modal instability. It also covers the parabolized stability equation (PSE), temporal and spatial biglobal theory, 2D eigenvalue problems, 3D eigen-value problems, spectral collocation methods, and other numerical solution methods. Computer codes are provided for tutorials described in the article. These tutorials cover the main topics of the article and can be adapted to form the basis of research codes.

Three-dimensional instabilities arising in open cavity flows are responsible for complex broad-banded dynamics. Existing studies either focus on theoretical properties of ideal simplified flows or observe the final state of experimental flows. This paper aims to establish a connection between the onset of the centrifugal instabilities and their final expression within the fully saturated flow. To that end, a linear three-dimensional modal instability analysis of steady two-dimensional states developing in an open cavity of aspect ratio L/D = 2 (length over depth) is conducted. This analysis is performed together with an experimental study in the same geometry adding spanwise endwalls. Two different Reynolds numbers are investigated through spectral analyses and modal decomposition. The physics of the flow is thoroughly described exploiting the strengths of each methodology. The main flow structures are identified and salient space and time scales are characterised. Results indicate that the structures obtained from linear analysis are mainly consistent with the fully saturated experimental flow. The analysis also brings to light the selection and alteration of certain wave properties, which could be caused by nonlinearities or the change of spanwise boundary conditions.

The linear instability and breakdown to turbulence induced by an isolated roughness element in a boundary layer at Mach 2.5, over an isothermal flat plate with laminar adiabatic wall temperature, have been analysed by means of direct numerical simulations, aided by spatial BiGlobal and three-dimensional parabolized (PSE-3D) stability analyses. It is important to understand transition in this flow regime since the process can be slower than in incompressible flow and is crucial to prediction of local heat loads on next-generation flight vehicles. The results show that the roughness element, with a height of the order of the boundary layer displacement thickness, generates a highly unstable wake, which is composed of a low-velocity streak surrounded by a three-dimensional high-shear layer and is able to sustain the rapid growth of a number of instability modes. The most unstable of these modes are associated with varicose or sinuous deformations of the low-velocity streak; they are a consequence of the instability developing in the three-dimensional shear layer as a whole (the varicose mode) or in the lateral shear layers (the sinuous mode). The most unstable wake mode is of the varicose type and grows on average ∼17 % faster than the most unstable sinuous mode and ∼30 times faster than the most unstable boundary layer mode occurring in the absence of a roughness element. Due to the high growth-rates registered in the presence of the roughness element, an amplification factor of N = 9 is reached within ∼50 roughness heights from the roughness trailing edge. The independently performed Navier–Stokes, spatial BiGlobal and PSE-3D stability results are in excellent agreement with each other, validating the use of simplified theories for roughness-induced transition involving wake instabilities. Following the linear stages of the laminar–turbulent transition process, the roll-up of the three-dimensional shear layer leads to the formation of a wedge of turbulence, which spreads laterally at a rate similar to that observed in the case of compressible turbulent spots for the same Mach number.

Flows of practical significance exist, like systems of trailing vortices, which are inhomogeneous in two directions and weakly dependent along the third spatial direction. Exploiting these characteristics, an integration of the Navier– Stokes equations using the parabolic Navier–Stokes concept is proposed for the recovery of steady solutions that might be used subsequently in the scope of primary instability analyses. The parabolic Navier–Stokes equations are first formulated in a cylindrical coordinate frame and used to calculate the solution of an isolated, axisymmetric nonparallel (axially developing) vortex. Then, the fully three-dimensional flow corresponding to a counter-rotating pair of nonparallel vortices is obtained by parabolic Navier–Stokes formulated in Cartesian coordinates. Stable high-order finite-difference-based numerical schemes are used for the spatial discretization to exploit the benefits of using a sparse direct solver for the inversion of the large matrices resulting from the discretization of the partial-derivative-based parabolic Navier–Stokes equations. Solutions recovered converge to the analytically obtained evolution of an isolated vortex in the limit of large distance between vortices and depart from this idealized situation when the distance becomes of the same order as the radii of the vortices. Nomenclature Γ = vortex circulation γ = vortex axial velocity defect δ = vortex core radius ζ = axial vorticity κ = vortex swirl strength parameter

A unified solution framework is presented for one-, two-or three-dimensional complex non-symmetric eigenvalue problems, respectively governing linear modal instability of incompressible fluid flows in rectangular domains having two, one or no homogeneous spatial directions. The solution algorithm is based on subspace iteration in which the spatial discretization matrix is formed, stored and inverted serially. Results delivered by spectral collocation based on the Chebyshev-Gauss–Lobatto (CGL) points and a suite of high-order finite-difference methods comprising the previously employed for this type of work Dispersion-Relation-Preserving (DRP) and Padé finite-difference schemes, as well as the Summation-by-parts (SBP) and the new high-order finite-difference scheme of order q (FD-q) have been compared from the point of view of accuracy and efficiency in standard validation cases of temporal local and BiGlobal linear instability. The FD-q method has been found to significantly outperform all other finite difference schemes in solving classic linear local, BiGlobal, and TriGlobal eigenvalue problems, as regards both memory and CPU time requirements. Results shown in the present study disprove the paradigm that spectral methods are superior to finite difference methods in terms of computational cost, at equal accuracy , FD-q spatial discretization delivering a speedup of Oð10 4 Þ. Consequently, accurate solutions of the three-dimensional (TriGlobal) eigenvalue problems may be solved on typical desktop computers with modest computational effort.

ABSTRACT Linear instability of the three-dimensional boundary-layer over the Hypersonic International Flight Research Experimentation 5 (HIFiRE-5) flight test geometry, i.e. a rounded-tip 2:1 elliptic cone, is analyzed through spatial BiGlobal linear instability analysis, in an effort to understand transition and accurately predict local heat loads on this model of next-generation flight vehicles. The base flow conditions are selected for a Mach 7 flow at altitude of 33.0 km and unit Reynolds number . The base flow is computed using the US3D solver. The three-dimensionality of the boundary-layer on the elliptic cone produces spanwise pressure gradients, inducing crossflow from the leading edge (major-axis meridian) to the centerline (minor-axis meridian) and producing lift-up of low momentum boundary-layer fluid at the centerline. The present analysis focuses on the resulting mushroom-like structure formed near the minor-axis meridian. An unstable fluid structure, which is composed of a low-velocity streak surrounded by a three-dimensional high-shear layer, is thus generated. Results show that this complex fluid structure can sustain linear growth of a number of instability modes, thus confirming earlier analyses which identified such modes without taking surface curvature of the geometry in question into consideration. These so-called centerline modes are associated with varicose or sinuous deformations of the low-velocity streak and are found to have similar growth rates in the frequency range studied, between 50 and 300 kHz. Furthermore, unstable boundary-layer modes, identified as oblique second Mack modes, are unraveled in the vicinity of the centerline. At the conditions studied, Mack modes are found to be less amplified that the centerline modes.

ABSTRACT Three-dimensional direct numerical simulations (DNS) have been performed on a finite-size hemisphere-cylinder model at angle of attack AoA = 20° and Reynolds numbers Re = 350 and 1000. Under these conditions, massive separation exists on the nose and lee-side of the cylinder, and at both Reynolds numbers the flow is found to be unsteady. Proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD) are employed in order to study the primary instability that triggers unsteadiness at Re = 350. The dominant coherent flow structures identified at the lower Reynolds number are also found to exist at Re = 1000; the question is then posed whether the flow oscillations and structures found at the two Reynolds numbers are related. POD and DMD computations are performed using different subdomains of the DNS computational domain. Besides reducing the computational cost of the analyses, this also permits to isolate spatially localized oscillatory structures from other, more energetic structures present in the flow. It is found that POD and DMD are in general sensitive to domain truncation and non-educated choices of the subdomain may lead to inconsistent results. Analyses at Re = 350 show that the primary instability is related to the counter-rotating vortex pair conforming the three-dimensional afterbody wake, and characterized by the frequency St ≈ 0.11, in line with results in the literature. At Re = 1000, vortex-shedding is present in the wake with an associated broadband spectrum centered around the same frequency. The horn/leeward vortices at the cylinder lee-side, upstream of the cylinder base, also present finite amplitude oscillations at the higher Reynolds number. The spatial structure of these oscillations, described by the POD modes, is easily differentiated from that of the wake oscillations. Additionally, the frequency spectra associated with the lee-side vortices presents well-defined peaks, corresponding to St ≈ 0.11 and its few harmonics, as opposed to the broadband spectrum found at the wake.

Abstract Highly resolved solutions of the two-dimensional incompressible Navier-Stokes and continuity equations, describing the evolution of vortex systems, have been obtained accurately and efficiently by spectral collocation methods. Such solutions have formed the basic ...