Simple aerodynamic configurations under even modest conditions can exhibit complex flows with a wide range of temporal and spatial features. It has become common practice in the analysis of these flows to look for and extract physically important features, or modes, as a first step in the analysis. This step typically starts with a modal decomposition of an experimental or numerical dataset of the flowfield, or of an operator relevant to the system. We describe herein some of the dominant techniques for accomplishing these modal decompositions and analyses that have seen a surge of activity in recent decades [1–8]. For a nonexpert, keeping track of recent developments can be daunting, and the intent of this document is to provide an introduction to modal analysis that is accessible to the larger fluid dynamics community. In particular, we present a brief overview of several of the well-established techniques and clearly lay the framework of these methods using familiar linear algebra. The modal analysis techniques covered in this paper include the proper orthogonal decomposition (POD), balanced proper orthogonal decomposition (balanced POD), dynamic mode decomposition (DMD), Koopman analysis, global linear stability analysis, and resolvent analysis.

https://arc.aiaa.org/doi/pdf/10.2514/1.J056060

Modal Analysis of Fluid Flows: An Overview

Results are reported from a three-dimensional computational stability analysis of flow over a backward-facing step with an expansion ratio (outlet to inlet height) of 2 at Reynolds numbers between 450 and 1050. The analysis shows that the first absolute linear instability of the steady two-dimensional flow is a steady three-dimensional bifurcation at a critical Reynolds number of 748. The critical eigenmode is localized to the primary separation bubble and has a flat roll structure with a spanwise wavelength of 6.9 step heights. The system is further shown to be absolutely stable to two-dimensional perturbations up to a Reynolds number of 1500. Stability spectra and visualizations of the global modes of the system are presented for representative Reynolds numbers.

Three-dimensional instability in flow over a backward-facing step

The linear global instability and resulting transition to turbulence induced by an isolated cylindrical roughness element of height and diameter immersed within an incompressible boundary layer flow along a flat plate is investigated using the joint application of direct numerical simulations and fully three-dimensional global stability analyses. For the range of parameters investigated, base flow computations show that the roughness element induces a wake composed of a central low-speed region surrounded by a three-dimensional shear layer and a pair of low- and high-speed streaks on each of its sides. Results from the global stability analyses highlight the unstable nature of the central low-speed region and its crucial importance in the laminar–turbulent transition process. It is able to sustain two different global instabilities: a sinuous and a varicose one. Each of these globally unstable modes is related to a different physical mechanism. While the varicose mode has its root in the instability of the whole three-dimensional shear layer surrounding the central low-speed region, the sinuous instability turns out to be similar to the von Karman instability in the two-dimensional cylinder wake and has its root in the lateral shear layers of the separated zone. The aspect ratio of the roughness element plays a key role on the selection of the dominant instability: whereas the flow over thin cylindrical roughness elements transitions due to a sinuous instability of the near-wake region, for larger roughness elements the varicose instability of the central low-speed region turns out to be the dominant one. Direct numerical simulations of the flow past an aspect ratio (with ) roughness element sustaining only the sinuous instability have revealed that the bifurcation occurring in this particular case is supercritical. Finally, comparison of the transition thresholds predicted by global linear stability analyses with the von Doenhoff–Braslow transition diagram provides qualitatively good agreement.

/pdf/investigation-of-the-roughness-induced-transition-global-5am5bs42rq.pdf

Investigation of the roughness-induced transition: global stability analyses and direct numerical simulations

Gortler vortices developed in laminar boundary layer experience remarkable changes when the flow is subjected to compressibility effects. In the present study, five numbers, covering incompressible to hypersonic flows, at , 1.5, 3.0, 4.5 and 6.0 are specified to illustrate these effects. Gortler vortices in subsonic and moderate supersonic flows ( , 1.5 and 3.0) are governed by the conventional wall-layer mode (mode W). In hypersonic flows ( , 6.0), the trapped-layer mode (mode T) becomes dominant. This difference is maintained and intensifies downstream leading to different scenarios of secondary instabilities. The linear and nonlinear development of Gortler vortices which are governed by dominant modal disturbances are investigated with direct marching of the nonlinear parabolic equations. The secondary instabilities of Gortler vortices set in when the resulting streaks are adequately developed. They are studied with Floquet theory at multiple streamwise locations. The secondary perturbations become unstable downstream following the sequence of sinuous mode type I, varicose mode and sinuous mode type II, indicating an increasing threshold amplitude. Onset conditions are determined for these modes. The above three modes can each have the largest growth rate under the right conditions. In the hypersonic cases, the threshold amplitude is dramatically reduced, showing the significant impact of the thermal streaks. To investigate the parametric effect of the spanwise wavenumber, three global wavenumbers ( , 1.0 and ) are specified. The relationship between the dominant mode (sinuous or varicose) and the spanwise wavenumber of Gortler vortices found in incompressible flows (Li & Malik, J. Fluid Mech., vol. 297, 1995, pp. 77–100) is shown to be not fully applicable in high-speed cases. The sinuous mode becomes the most dangerous, regardless of the spanwise wavelength when . The subharmonic type can be the most dangerous mode while the detuned type can be neglected, although some of the sub-dominant secondary modes reach their peak growth rates under detuned states.

Secondary instabilities of Görtler vortices in high-speed boundary layer flows

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 × 107 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.

Linear modal instabilities of hypersonic flow over an elliptic cone

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 growthrates registered in the presence of the roughness element, an amplification factor of N D 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.

/pdf/laminar-turbulent-transition-induced-by-a-discrete-roughness-4bgyslh1ak.pdf

Laminar-turbulent transition induced by a discrete roughness element in a supersonic 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.

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

Direct numerical simulations are run for a NACA0012 airfoil at Mach 0.4 and Reynolds number 50,000. The flow fields are characterised by transitional separation bubbles that occupy 50–60% of the suction surface and are followed by a region of attached turbulent flow. At \(20^\circ \) sweep the dominant structures are oblique relative to the leading edge, while at \(40^\circ \) they are parallel to the leading edge. The latter case is also characterised by the emergence of a broadband peak in the energy spectrum. The changes in the flow are further studied by a global stability analysis based on the time and span-averaged flowfields. At the largest sweep angle a strongly unstable global mode emerges, peaking at a Strouhal number \(\mathrm {St}=3\) and with a range \(1< \mathrm{St} <5\) that is in good agreement with the DNS spectra. From the global eigenmode it is possible to discover the mechanism responsible for the mode, which involves a combination of shear layer convective instability, leading-edge receptivity and trailing edge sound radiation. The consistency of the global analysis and the DNS is notable, considering that the base flow is not a steady solution of the governing equations.

Global Stability Analysis and DNS of a Swept Airfoil Section in Subsonic Flow

The weight and structure of civil transport aircraft are mainly dictated by the loads they experience at the limit of the design envelope (e.g. turbulence, gusts and manoeuvres). During the design process aircraft loading is often predicted using simplified models. However, under extreme operating conditions aircraft experience erratic unsteady loads that are not well understood, and hence difficult to predict using state of the art reduced models. Achieving a more profound understanding of the flow structures dictating the aerodynamic loads under extreme conditions will be crucial for the design of efficient aircraft. An example of such conditions occurs when a wing starts heaving or encounters a gust near its stall angle (e.g. during landing); unsteady flows and transient effects, including flow instabilities and vortex shedding, take place, making the aerodynamic loads highly unpredictable (see for example Von Ellenrieder et al, J Fluid Mech, 490:129–138, 2003, [1]). These complex physical mechanisms can only be fully captured by experiments and accurate numerical simulations, both of which are currently expensive to be used during the design stages, but can provide useful insight. Direct numerical simulations (DNS), being free from simplified modelling assumptions, have the advantage of including all the relevant flow physics. On the other hand, the large eddy simulation (LES) technique can be used to simulate realistic flow conditions, but its accuracy in predicting complex flow phenomena including flow instability and vortex shedding is not clear. This work focuses on assessing the performance of OpenFoam’s LES solver for the prediction of the aerodynamic loads acting on a heaving wing at incidence through comparisons with DNS and experimental results.

Performance Analysis of a Heaving Wing Using DNS and LES

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Papers

Laminar-turbulent transition induced by a discrete roughness element in a supersonic boundary layer

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

Global Stability Analysis and DNS of a Swept Airfoil Section in Subsonic Flow

Performance Analysis of a Heaving Wing Using DNS and LES