Pedro Paredes

Four Decades of Studying Global Linear Instability: Progress and Challenges

Global linear instability theory is concerned with the temporal or spatial development of small-amplitude perturbations superposed upon laminar steady or time-periodic three-dimensional flows, which are inhomogeneous in two (and periodic in one) or all three spatial directions. After a brief exposition of the theory, some recent advances are reported. First, results are presented on the implementation of a Jacobian-free Newton-Krylov time-stepping method into a standard finite-volume aerodynamic code to obtain global linear instability results in flows of industrial interest. Second, connections are sought between established and more-modern approaches for structure identification in flows, such as proper orthogonal decomposition and Koopman modes analysis (dynamic mode decomposition), and the possibility to connect solutions of the eigenvalue problem obtained by matrix formation or time-stepping with those delivered by dynamic mode decomposition, residual algorithm, and proper orthogonal decomposition analysis is highlighted in the laminar regime; turbulent and three-dimensional flows are identified as open areas for future research. Finally, a new stable very-high-order finite-difference method is implemented for the spatial discretization of the operators describing the spatial biglobal eigenvalue problem, parabolized stability equation three-dimensional analysis, and the triglobal eigenvalue problem; it is shown that, combined with sparse matrix treatment, all these problems may now be solved on standard desktop computers.

The PSE-3D instability analysis methodology for flows depending strongly on two and weakly on the third spatial dimension

The present contribution discusses the development of a PSE-3D instability analysis algorithm, in which a matrix forming and storing approach is followed. Alternatively to the typically used in stability calculations spectral methods, new stable high-order finite- dierence-based numerical schemes for spatial discretization 1 are employed. Attention is paid to the issue of eciency, which is critical for the success of the overall algorithm. To this end, use is made of a parallelizable sparse matrix linear algebra package which takes advantage of the sparsity oered by the finite-dierence scheme and, as expected, is shown to perform substantially more eciently than when spectral collocation methods are used. The building blocks of the algorithm have been implemented and extensively validated, focusing on classic PSE analysis of instability on the flow-plate boundary layer, temporal and spatial BiGlobal EVP solutions (the latter necessary for the initialization of the PSE-3D), as well as standard PSE in a cylindrical coordinates using the nonparallel Batchelor vortex basic flow model, such that comparisons between PSE and PSE-3D be possible; excellent agreement is shown in all aforementioned comparisons. Finally, the linear PSE-3D instability analysis is applied to a fully three-dimensional flow composed of a counter-rotating pair of nonparallel Batchelor vortices.

Optimal Growth in Hypersonic Boundary Layers

The linear form of the parabolized linear stability equations is used in a variational approach to extend the previous body of results for the optimal, nonmodal disturbance growth in boundary-layer flows. This paper investigates the optimal growth characteristics in the hypersonic Mach number regime without any high-enthalpy effects. The influence of wall cooling is studied, with particular emphasis on the role of the initial disturbance location and the value of the spanwise wave number that leads to the maximum energy growth up to a specified location. Unlike previous predictions that used a basic state obtained from a self-similar solution to the boundary-layer equations, mean flow solutions based on the full Navier–Stokes equations are used in select cases to help account for the viscous–inviscid interaction near the leading edge of the plate and for the weak shock wave emanating from that region. Using the full Navier–Stokes mean flow is shown to result in further reduction with Mach number in the ma...

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 is 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 analysis is applied to the reacting flow in an industry-relevant combustion system. Both the local and the biglobal analysis accurately predicts the onset and structure of a self-excited global instability that is known in the combustion community as a precessing vortex 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 analysis agree 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.Copyright

High-frequency instabilities along the windward face of a hypersonic yawed circular cone

Instability analysis of the three-dimensional boundary layer over a 7◦ half-angle circular cone at 6◦ angle of attack, flying at Mach 6, is considered. The yawed straight cone has to-date been analyzed using modal local and non-local theories, respectively based on the solution of the one-dimensional eigenvalue problem and the Parabolized Stability Equations (PSE). Both approaches assume the flow to be homogeneous either in one or two spatial directions. The aim of the present work is to analyze this flow using multidimensional stability analysis techniques, namely the spatial BiGlobal analysis, relaxing the azimuthal homogeneity assumption. Multiple instabilities are known to be simultaneously present in a three-dimensional hypersonic boundary layer from earlier stability predictions, as for example the low-frequency disturbances related to traveling crossflow modes or the highfrequency second modes. The multidimensional instability analysis is used to unraveled the high-frequency instability modes occurring on the windward face of the cone, where the main heating happens and therefore it is the most critical for the performance of the modeled high-speed vehicle. Different instability modes are found in the present analysis, namely a two-dimensional second mode peaking on the windward plane and oblique second modes peaking at a certain distance from the windward plane. The instability properties of these instabilities are studied at different axial positions and in a wide range of frequencies. The results are in line with previous numerical and experimental predictions.

Nosetip bluntness effects on transition at hypersonic speeds: experimental and numerical analysis under NATO STO AVT-240

The existing database of transition measurements in hypersonic ground facilities has established that the onset of boundary layer transition over a circular cone at zero angle of attack shifts downstream as the nosetip bluntness is increased with respect to a sharp cone. However, this trend is reversed at sufficiently large values of the nosetip Reynolds number, so that the transition onset location eventually moves upstream with a further increase in nosetip bluntness. This transition reversal phenomenon, which cannot be explained on the basis of linear stability theory, was the focus of a collaborative investigation under the NATO STO group AVT-240 on Hypersonic Boundary-Layer Transition Prediction. The current paper provides an overview of that effort, which included wind tunnel measurements in three different facilities and theoretical analysis related to modal and nonmodal amplification of boundary layer disturbances. Because neither first and secondmode waves nor entropy-layer instabilities are found to be substantially amplified to initiate transition at large bluntness values, transient (i.e., nonmodal) disturbance growth has been investigated as the potential basis for a physics-based model for the transition reversal phenomenon. Results of the transient growth analysis indicate that disturbances that are initiated within the nosetip or in the vicinity of the juncture between the nosetip and the frustum can undergo relatively significant nonmodal amplification and that the maximum energy gain increases nonlinearly with the nose radius of the cone. This finding does not provide a definitive link between transient growth and the onset of transition, but it is qualitatively consistent with the experimental observations that frustum transition during the reversal regime was highly sensitive to wall roughness, and furthermore, was dominated by disturbances that originated near the nosetip.

Traveling global instabilities on the HIFiRE-5 elliptic cone model flow

The linear instability of the three-dimensional boundary-layer over the HIFiRE-5 flight test geometry, i.e. a rounded-tip 2:1 elliptic cone, has been analyzed through spatial BiGlobal analysis, in a effort to understand transition and accurately predict local heat loads on this configuration. The base state flow conditions of Mach 7 flow at altitude of 33.0 km and unit Reynolds number 1.89 × 10 /m have been computed using the US3D non-equilibrium solver by Gosse & Kimmel. The stability analysis results reveal that the leading unstable modes peak on the mushroom-like structure formed near the minor-axis meridian. On account of the three-dimensionality of the elliptic cone boundary layer, this produces spanwise pressure gradients, inducing crossflow from the leading edge (major-axis meridian) to the centerline (minor-axis meridian), resulting in a lift-up of low momentum boundary layer fluid at the centerline. An unstable fluid structure, which is composed of a low velocity streak surrounded by a three-dimensional high-shear layer, is thus generated and is found to sustain the rapid growth of instability modes. Furthermore, unstable crossflow modes are documented and are found to be less amplified that the centerline modes at these conditions. Attachment-lines instabilities are not present in the studied case. The range of unraveled frequencies coincides with that measured in flight testing.

Transient Growth Analysis of Compressible Boundary Layers with Parabolized Stability Equations

The linear form of parabolized linear stability equations (PSE) is used in a variational approach to extend the previous body of results for the optimal, non-modal disturbance growth in boundary layer flows. This methodology includes the non-parallel effects associated with the spatial development of boundary layer flows. As noted in literature, the optimal initial disturbances correspond to steady counter-rotating stream-wise vortices, which subsequently lead to the formation of stream-wise-elongated structures, i.e., streaks, via a lift-up effect. The parameter space for optimal growth is extended to the hypersonic Mach number regime without any high enthalpy effects, and the effect of wall cooling is studied with particular emphasis on the role of the initial disturbance location and the value of the span-wise wavenumber that leads to the maximum energy growth up to a specified location. Unlike previous predictions that used a basic state obtained from a self-similar solution to the boundary layer equations, mean flow solutions based on the full Navier-Stokes (NS) equations are used in select cases to help account for the viscous-inviscid interaction near the leading edge of the plate and also for the weak shock wave emanating from that region. These differences in the base flow lead to an increasing reduction with Mach number in the magnitude of optimal growth relative to the predictions based on self-similar mean-flow approximation. Finally, the maximum optimal energy gain for the favorable pressure gradient boundary layer near a planar stagnation point is found to be substantially weaker than that in a zero pressure gradient Blasius boundary layer.

Instability Study of the Wake Behind a Discrete Roughness Element in a Hypersonic Boundary-Layer

The linear instability induced by an isolated roughness element in a boundary-layer at Mach 6 has been analysed through spatial BiGlobal and three-dimensional parabolised (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 critical 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 an convectively unstable wake where different instability modes develop. Furthermore, at this high Mach number, boundary-layer modes develop at high frequencies and are also covered here. Important discrepancies are observed between BiGlobal and PSE-3D predictions, mainly for the roughness-induced wake modes. Results are in qualitative agreement with a full Navier-Stokes receptivity study of the same flow.

Three-Dimensional Solutions of Trailing-Vortex Flows Using Parabolized Equations

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 discre...