BiGlobal Stability of Shear Flows: Spanwise & Streamwise Analyses

Laminar-turbulent transition dictates an increase in skin friction. The resulting turbulent skin friction contributes to approximately 40% of the total drag of commercial aircraft. Reducing the turbulent flow region by postponing transition can therefore significantly reduce the carbon footprint and costs of flying. Transition prediction is required in order to do so, which depends on a detailed understanding of the transition process.Aerodynamic

Derivation of and Simulations with BiGlobal Stability Equations

Laminar to turbulent transition has an important role in the aerospace domain in view of its impact on aerodynamic drag and, regarding the high velocity regime, heat transfer. State of the art computational methods, like DNS, LES and RANS are found to be too expensive or rely on case dependent turbulence models to be used for obtaining information regarding the transition phenomenon. Transition is typically initiated by the onset of instability of the laminar flow. Linear stability theory describes the eigenmode growth mechanism. Although this yields a restriction, because additional mechanisms play a role too, the eigenmode growth phase establishes an important base in many practical situations. However, the linearization provides a considerable step in the simplification of the analysis, while the stability theory can be adapted according to the structure of the given mean flow. At the Von Karman Institute (VKI), the VKI Extensible Stability and Transition Analysis (VESTA) toolkit has been developed, which mainly involves methods based on the linear stability theory. In the current project, the main goal was to extend the already present tools to incorporate the BiGlobal stability equations, which, together with appropriate boundary conditions, form an eigenvalue problem. This particular problem is solved for perturbations inhomogeneous in two spatial directions and their complex growth rate and frequency. This extension involved a new version of the tool for the derivation of the BiGlobal stability equations, a tool for their automatic implementation in Matlab via the spectral collocation method and a simulation tool to apply boundary conditions and execute the analysis corresponding to a prescribed mean flow. The derivation of the BiGlobal equations and their verification formed the first part of the project. Both incompressible and compressible versions are derived for different kinds of coordinate systems (e.g. Cartesian and cylindrical) and formulations in the compressible case (e.g. involving temperature and pressure and the energy equation based on static enthalpy). This allowed the verification of the tool with a large number of previously published references. All references, to the knowledge of the current author, that have thus far reported the compressible equations were found to contain errors and had to be cross-verified to yield the ultimate positive outcome. It is hence deemed that the present treatment is the first to report the full compressible BiGlobal stability equations in primitive variable formulation correctly. The second part of the project involved the verification of the performance of the combination of the derivation, implementation and simulation tools. This was done by considering three test cases (mean flows). In all cases, the eigenvalue problem was solved using the QZ algorithm. In cases that required high resolution, the Arnoldi algorithm was used in addition, because of its lean performance with respect to required memory. The first test case was the parallel Blasius boundary layer. Because of its one-dimensional nature, this flow has been intensively analysed in the past by means of the classic local stability analysis type (LST). This allowed the BiGlobal analysis of this mean flow to be thoroughly verified in both the incompressible and supersonic regime. The second case involved the developing incompressible Blasius boundary layer. This flow was chosen because of its better affinity with the actual Blasius boundary layer flow, which has an intrinsic developing nature. The BiGlobal approach involved artificial in- and outflow boundary conditions. Analyses were performed on a domain with a small and large streamwise extent to focus on a flow that is weakly and strongly developing, respectively. The former analyses were again compared to LST simulations to yield an internal verification and consistency check. The results of the analyses on the larger domain could be compared to the literature and were found to agree well in a qualitative sense. The Tollmien-Schlichting branch obtained in this study was found to lie too high with respect to the one reported in the literature. Although the exact reason for this could not yet be established, the most likely cause is a (small) difference in the prescribed mean flow. It is expected that the test case will yield identical results when exactly the same mean flow will be used, as some key differences can be identified in the literature in this regard. It was found that the artificial boundary conditions caused an odd/even effect with respect to the continuous eigenmode branches in the spectrum when the number of points in the streamwise direction was taken to be either odd or even. A similar behaviour was observed when consulting the literature, although the effect was never elaborated on explicitly. Lastly, the incompressible complex lamellar bidirectional vortex was considered. This mean flow is defined on a cylindrical coordinate system and is highly inhomogeneous in at least two spatial directions. Therefore, this case requires the BiGlobal approach and all power of the newly developed tools could be tested. A test case handled in the literature was very precisely reconstructed. Although it was found that no part of the spectrum was converged, the results were nearly identically retrieved. The solutions to all three test cases have been obtained successfully and compare reasonably well with the literature. It is therefore concluded that all capabilities of the newly developed tools have been tested successfully and the tools can be considered to be verified.AerodynamicsAerospace Engineerin

Accurate numerical approximation of the absolute stability of unbounded flows

The initial stage of the laminar–turbulent transition of semi-infinite flows can be characterized as either an absolute or convective instability, naturally associated with localized wave packets. A convective instability is directly linked to an absolute instability in a different reference frame. Therefore, our aim is to determine the absolute stability of a flow in a given but arbitrary reference frame, which can only be directly inferred from the absolute eigenvalue spectrum. If advective processes are present, the associated absolute eigenfunctions grow exponentially in space in the advective direction. The eigenvalue spectrum is usually computed numerically, which requires truncating the domain and prescribing artificial boundary conditions at these truncation boundaries. For separated boundary conditions, the resulting spectrum approaches the absolute spectrum as the domain length tends to infinity. Since advective processes result in spatially exponentially growing eigenfunctions, it becomes increasingly difficult to represent these functions numerically as the domain length increases. Hence, a naive numerical implementation of the eigenvalue problem may result in a computed spectrum that strongly deviates from the (mathematically correct) absolute spectrum due to numerical errors. To overcome these numerical inaccuracies, we employ a weighted method ensuring the convergence to the absolute spectrum. From a physical point of view, this method removes the advection-induced spatial exponential growth from the eigenfunctions. The resulting (absolute) spectrum allows for a direct interpretation of the character of the pertinent perturbations and the eigensolutions can be used to construct and analyse the evolution of localized wave packets in an efficient way.AerodynamicsDelft Institute of Applied MathematicsMathematical Physic

On Closing the Streamwise BiGlobal Stability Problem: The Effect of Boundary Conditions

The modal streamwise BiGlobal stability approach introduces problems regarding the specification of in- and outflow boundary conditions (BCs). Several conditions linked to lower hierarchical stability frameworks are elaborated and are applied to a freestream and boundary layer flow. The former case is used to demonstrate the odd-even decoupling of the spectrum with the streamwise node number. The latter case illustrates that the spatial growth varies widely with different BCs, while specific Robin BCs yield the largest amplification near a target frequency. Combined, the cases show that the spectra might vary widely while corresponding to very similar spatial growth characteristics and vice versa.Aerodynamics, Wind Energy & PropulsionAerospace Engineerin

Secondary crossflow instability through global analysis of measured base flows

A combined experimental and numerical approach to the analysis of the secondary stability of realistic swept-wing boundary layers is presented. Global linear stability theory is applied to experimentally measured base flows. These base flows are three-dimensional laminar boundary layers subject to spanwise distortion due to the presence of primary stationary crossflow vortices. A full three-dimensional description of these flows is accessed through the use of tomographic particle image velocimetry (PIV). The stability analysis solves for the secondary high-frequency modes of type I and type II, ultimately responsible for turbulent breakdown. Several pertinent parameters arising from the application of the proposed methodology are investigated, including the mean flow ensemble size and the measurement domain extent. Extensive use is made of the decomposition of the eigensolutions into the terms of the Reynolds-Orr equation, allowing insight into the production and/or destruction of perturbations from various base flow features. Stability results demonstrate satisfactory convergence with respect to the mean flow ensemble size and are independent of the handling of the exterior of the measurement domain. The Reynolds-Orr analysis reveals a close relationship between the type I and type II instability modes with spanwise and wall-normal gradients of the base flow, respectively. The structural role of the in-plane velocity components in the perturbation growth, topology and sensitivity is identified. Using the developed framework, further insight is gained into the linear growth mechanisms and later stages of transition via the primary and secondary crossflow instabilities. Furthermore, the proposed methodology enables the extension and enhancement of the experimental measurement data to the pertinent instability eigenmodes. The present work is the first demonstration of the use of a measured base flow for stability analysis applied to the swept-wing boundary layer, directly avoiding the modelling of the primary vortices receptivity processes.AerodynamicsEducation A

Effectivity and efficiency of selective frequency damping for the computation of unstable steady-state solutions

Selective Frequency Damping (SFD) is a popular method for the computation of globally unstable steady-state solutions in fluid dynamics. The approach has two model parameters whose selection is generally unclear. In this article, a detailed analysis of the influence of these parameters is presented, answering several open questions with regard to the effectiveness, optimum efficiency and limitations of the method. In particular, we show that SFD is always capable of stabilising a globally unstable systems ruled by one unsteady unstable eigenmode and derive analytical formulas for optimum parameter values. We show that the numerical feasibility of the approach depends on the complex phase angle of the most unstable eigenvalue. A numerical technique for characterising the pertinent eigenmodes is presented. In combination with analytical expressions, this technique allows finding optimal parameters that minimise the spectral radius of a simulation, without having to perform an independent stability analysis. An extension to multiple unstable eigenmodes is derived. As computational example, a two-dimensional cylinder flow case is optimally stabilised using this method. We provide a physical interpretation of the stabilisation mechanism based on, but not limited to, this Navier–Stokes example.AerodynamicsControl & Simulatio

Transitional Flow Dynamics Behind a Micro-Ramp

Micro-ramps are popular passive flow control devices which can delay flow separation by re-energising the lower portion of the boundary layer. We compute the laminar base flow, the instantaneous transitional flow, and the mean flow around a micro-ramp immersed in a quasi-incompressible boundary layer at supercritical roughness Reynolds number. Results of our Direct Numerical Simulations (DNS) are compared with results of BiLocal stability analysis on the DNS base flow and independent tomographic Particle Image Velocimetry (tomo-PIV) experiments. We analyse relevant flow structures developing in the micro-ramp wake and assess their role in the micro-ramp functionality, i.e., in increasing the near-wall momentum. The main flow feature of the base flow is a pair of streamwise counter-rotating vortices induced by the micro-ramp, the so-called primary vortex pair. In the instantaneous transitional flow, the primary vortex pair breaks up into large-scale hairpin vortices, which arise due to linear varicose instability of the base flow, and unsteady secondary vortices develop. Instantaneous vortical structures obtained by DNS and experiments are in good agreement. Matching linear disturbance growth rates from DNS and linear stability analysis are obtained until eight micro-ramp heights downstream of the micro-ramp. For the setup considered in this article, we show that the working principle of the micro-ramp is different from that of classical vortex generators; we find that transitional perturbations are more efficient in increasing the near-wall momentum in the mean flow than the laminar primary vortices in the base flow.AerodynamicsWind Energ

Linear and Non-Linear Dynamics of a Micro-ramp Wake

Micro-ramps are deployed to prevent boundary layer separation by creating a momentum excess close to the wall. Through Direct Numerical Simulations (DNS) of the base, instantaneous and mean flow, we identify that the perturbation dynamics in the wake of the microramp play an essential role in creating the near-wall momentum excess. To identify the origin of the perturbations, we deploy BiGlobal stability analysis on the laminar base flow. We demonstrate that the amplification of the most unstable linear mode is closely related to the time-averaged amplitude of the unsteady perturbations. The flow structure corresponding to this mode has a varicose symmetry with respect to the symmetry plane and matches with the early development of the hairpin vortices in the instantaneous flow field. It is concluded that the varicose instability supported by the laminar base flow represents the mechanism that generates the hairpins.Aerodynamic

Convective instabilities in a laminar shock-wave/boundary-layer interaction

Linear stability analyses are performed to study the dynamics of linear convective instability mechanisms in a laminar shock-wave/boundary-layer interaction at Mach 1.7. In order to account for all two-dimensional gradients elliptically, we introduce perturbations into an initial-value problem that are found as solutions to an eigenvalue problem formulated in a moving frame of reference. We demonstrate that this methodology provides results that are independent of the numerical setup, frame speed, and type of eigensolutions used as initial conditions. The obtained time-integrated wave packets are then Fourier-transformed to recover individual-frequency amplification curves. This allows us to determine the dominant spanwise wavenumber and frequency yielding the largest amplification of perturbations in the shock-induced recirculation bubble. By decomposing the temporal wave-packet growth rate into the physical energy-production processes, we provide an in-depth characterization of the convective instability mechanisms in the shock-wave/boundary-layer interaction. For the particular case studied, the largest growth rate is achieved in the near-vicinity of the bubble apex due to the wall-normal (productive) and streamwise (destructive) Reynolds-stress energy-production terms. We also observe that the Reynolds heat-flux effects are similar but contribute to a smaller extent. Funding Information: The authors acknowledge the funding provided to Sébastien E.M. Niessen by the Fonds National de la Recherche Scientifique de Belgique (F.R.S.-FNRS) under Grant No. FC27285 and the computational resources provided by the Consortium des Équipements de Calcul Intensif (CÉCI), funded by the F.R.S.-FNRS under Grant No. 2.5020.11 and by the Walloon Region (Belgium). Publisher Copyright: © 2023 Author(s).Aerodynamic

Secondary instabilities in swept-wing boundary layers: Direct Numerical Simulations and BiGlobal stability analysis

The evolution of secondary instabilities in a three-dimensional stationary-crossflow-domina- ted boundary layer is investigated by means of Direct Numerical Simulations (DNS) and linear spanwise BiGlobal stability analysis. Single-frequency unsteady disturbances and a critical stationary crossflow mode are considered. Unsteady perturbation content at 1 kHz manifests in the form of the type-III secondary instability mechanism in the lower portion of the boundary layer in the both the DNS and the stability approach. Considering disturbances at 6 kHz, the results from the stability analysis reveal the existence of largely amplified type-I and type-II secondary instability mechanisms. Strong growth displayed by the former is measured in the DNS, which potentially overshadows manifestations of the type-II mechanism. Laminar- turbulent transition primarily induced by the growth of type-I disturbances is captured in the 6 kHz case. Overall, we report good agreement between DNS and stability analysis in terms of perturbation organization and growth rate for all cases studied.Aerodynamic