Evolution of compressible Gortler vortices subject to free-stream vortical disturbances

The perturbations triggered by free-stream vortical disturbances in compressible boundary layers developing over concave walls are studied numerically and through asymptotic methods. We employ an asymptotic framework based on the limit of high Gortler number, the scaled parameter defining the centrifugal effects, we use an eigenvalue formulation where the free-stream forcing is neglected, and solve the receptivity problem by integrating the compressible boundary-region equations complemented by appropriate initial and boundary conditions which synthesize the influence of the free-stream vortical flow. In the limit of high frequencies, the triple-deck equations are also solved and their results compared with the solution of the boundary-region equations. The boundary-layer perturbations, in the proximity of the leading edge, develop as thermal Klebanoff modes and, when centrifugal effects become influential, these modes turn into thermal Gortler vortices, i.e., streamwise rolls characterized by intense velocity and temperature perturbations. The high-Gortler-number asymptotic analysis reveals the condition for which the Gortler vortices start to grow and that the Mach number is destabilizing when the spanwise diffusion is negligible and stabilizing when the boundary-layer thickness is comparable with the spanwise wavelength of the vortices. The theoretical analysis also shows that the vortices move towards the wall as the Mach number increases when the Gortler number is large. These results are confirmed by the receptivity analysis, which additionally clarifies that the temperature perturbations respond to this reversed behavior further downstream than the velocity perturbations. A matched-asymptotic composite profile, found by combining the inviscid core solution and the near-wall viscous solution, agrees well with the receptivity profile sufficiently downstream and at high Gortler number. The Gortler vortices tend to move towards the boundary-layer core when the flow is more stable, i.e., as the frequency or the Mach number increase, or when the curvature decreases. As a consequence, a region of unperturbed flow is generated near the wall. We also find that the streamwise length scale of the boundary-layer perturbations is always lower than the free-stream streamwise wavelength. During the initial development of the vortices, only the receptivity calculations are accurate. Downstream where the free-stream disturbances have fully decayed, the growth rate and wavelength are computed accurately by the eigenvalue analysis, although the correct amplitude of the Gortler vortices can only be determined by the receptivity calculations. It is further proved that the eigenvalue predictions of the growth rate and wavenumber worsen as the Mach number increases, as these quantities show a dependence on the wall-normal direction. The receptivity analysis is also used to compute the neutral curves generated by free-stream disturbances, i.e., curves that identify the region of growth and decay of the boundary-layer perturbations, for different Gortler numbers, Mach numbers, wavelengths, and low frequencies of the free-stream disturbance. The growth rate of the perturbation is used to identify if the boundary-layer instability is in the form of Klebanoff modes or Gortler vortices. A critical Gortler number can be identified below which Klebanoff modes are the only source perturbations, even when curvature is present. From the receptivity and eigenvalue formulation we define a streamwise-dependent receptivity coefficient and discuss the N-factor approach for transition prediction. Finally, the equations the triple-deck analysis reveals that the curvature effects do not play a role in the limit of high frequencies, which is also confirmed by the boundary-region results

Direct numerical simulation of supersonic turbulent flows around a tandem expansion-compression corner

© 2015 AIP Publishing LLC. The M = 2.9 supersonic turbulent flows over a tandem expansion-compression corner configuration with a sharp deflection angle of 25° at three Reynolds numbers Reδ = 20 000, 40 000, and 80 000 were studied by using direct numerical simulation. The flow statistics were validated against available experimental measurements and other numerical predictions. The flow structures and turbulence statistics were detailed visualized and analysed for the Reδ = 40 000 case, especially in the interaction region where flow separation and reattachment occurred. It was found that during the expansion process, the boundary layer exhibited a characteristic two-layer structure also discovered in previous experimental studies, and the turbulence evolved differently within these two layers. In the outer layer, the turbulence was consistently suppressed along the ramp to a large extent, while in the inner layer, it was suppressed only in a small region around the expansion corner, and the near-wall quasi-streamwise vortices were well preserved. Flow patterns near the reattachment line have shown the existence of the Görtler-type vortices, which would largely amplify turbulence fluctuations in the near-wall region, thus promoting the regeneration of wall turbulence that in turn contributed to the redevelopment of a downstream turbulent boundary layer. The Reynolds number effects and the characteristics of coherent structures were also discussed. With the increase of the Reynolds number, the separation bubble size decreased, but the pattern and the characteristic size of wall streamlines near the reattachment line were preserved

Investigation of Görtler vortices in high-speed boundary layers via an efficient numerical solution to the non-linear boundary region equations

Streamwise vortices and the associated streaks evolve in boundary layers over flat or concave surfaces due to disturbances initiated upstream or triggered by the wall surface. Following the transient growth phase, the fully developed vortex structures become susceptible to inviscid secondary instabilities resulting in early transition to turbulence via ‘bursting’ processes. In high-speed boundary layers, more complications arise due to compressibility and thermal effects, which become more significant for higher Mach numbers. In this paper, we study Görtler vortices developing in high-speed boundary layers using the boundary region equations (BRE) formalism, which we solve using an efficient numerical algorithm. Streaks are excited using a small transpiration velocity at the wall. Our BRE-based algorithm is found to be superior to direct numerical simulation (DNS) and ad hoc nonlinear parabolized stability equation (PSE) models. BRE solutions are less computationally costly than a full DNS and have a more rigorous theoretical foundation than PSE-based models. For example, the full development of a Görtler vortex system in high-speed boundary layers can be predicted in a matter of minutes using a single processor via the BRE approach. This substantial reduction in calculation time is one of the major achievements of this work. We show, among other things, that it allows investigation into feedback control in reasonable total computational times. We investigate the development of the Görtler vortex system via the BRE solution with feedback control parametrically at various freestream Mach numbers M∞ and spanwise separations λ of the inflow disturbances

Analysis of Instabilities in Non-Axisymmetric Hypersonic Boundary Layers Over Cones

Hypersonic flows over circular cones constitute one of the most important generic configurations for fundamental aerodynamic and aerothermodynamic studies. In this paper, numerical computations are carried out for Mach 6 flows over a 7-degree half-angle cone with two different flow incidence angles and a compression cone with a large concave curvature. Instability wave and transition-related flow physics are investigated using a series of advanced stability methods ranging from conventional linear stability theory (LST) and a higher-fidelity linear and nonlinear parabolized stability equations (PSE), to the 2D eigenvalue analysis based on partial differential equations. Computed N factor distribution pertinent to various instability mechanisms over the cone surface provides initial assessments of possible transition fronts and a guide to corresponding disturbance characteristics such as frequency and azimuthal wave numbers. It is also shown that strong secondary instability that eventually leads to transition to turbulence can be simulated very efficiently using a combination of advanced stability methods described above

Control of Gortler Vortices by Means of Wall Deformations and Blowing/Suction

Görtler vortices evolve in boundary layers over concave surfaces as a result of the imbalance between centrifugal forces and radial pressure gradients. Depending on various geometrical and free-stream flow conditions, these instabilities may lead to secondary instabilities and early transition to turbulence. In this thesis, a control algorithm based on the boundary region equations is applied to reduce the strength of the Görtler instabilities by controlling the energy of the fully developed vortices, using either local wall deformations or blowing/suction at the wall. A proportional-integral control scheme is utilized to deform the wall or to provide transpiration velocity, where the inputs are either the wall-normal or streamwise velocity components in a plane that is parallel to the wall. The results show that the control based on wall deformation using wall-normal velocity components is more effective in tempering the vortex during its streamwise growth by almost one or two orders of magnitude

Nonlinear centrifugal instabilities in curved free shear layers

Curved free shear layers exist in many engineering problems involving complex flow geometries, such as the backward facing step flow, flows with wall injection, the flow inside side-dump combustors, or flows generated by vertical axis wind turbines, among others. Most of the studies involving centrifugal instabilities have been focused on wall flows where Taylor instabilities between two rotating concentric cylinders or Görtler vortices in boundary layers resulting from the imbalance between centrifugal effects and radial pressure gradients, are generated. Curved free shear layers, however, did not receive sufficient attention. An examination of the stability characteristics and the flow structures associated with curved free shear flows should provide a better understanding of these complex flow problems. In this work, we study the development of Görtler vortices inside a curved shear layer in both the incompressible and compressible regimes using a numerical solution to a parabolized form of the Navier-Stokes equations, in the assumption that the streamwise wavenumber associated with the vortex flow is much smaller than the crossstream wavenumbers. Various results consisting of contour plots of centrifugal instabilites in crossflow planes, and energy and streak amplitude distributions along the streamwise direction are reported and discussed. In addition, we conduct a biglobal stability analysis to study the growth rates and the eigenmodes associated with these flows

Görtler instability and transition in compressible flows

We present a discussion on theoretical, experimental, and computational research studies on Görtler instability and the related transition to turbulence occurring in compressible boundary layers over concave surfaces. We first examine the theoretical results on primary and secondary instabilities, emphasizing the role of receptivity, the mechanism by which external agents, such as freestream fluctuations or wall roughness, act on a boundary layer to trigger Görtler vortices. We review experimental findings obtained from measurements in supersonic and hypersonic wind tunnels and discuss studies employing numerical methods, focusing on the direct numerical simulation approach. The research in these two last sections is surveyed according to the geometrical configuration, from simple concave walls to more complex surfaces of hypersonic vehicles. The experimental investigations have been successful in the visualizations of Görtler vortices, in the measurement of the wall-heat transfer in the transitional region, and in the computation of the Görtler-vortex growth rates, although detailed boundary-layer velocity measurements are still missing. Direct numerical simulations have confirmed instability results emerging from stability theories and revealed nonlinear interactions between Görtler vortices and other disturbances. The established initial-boundary-value receptivity theory can certainly benefit from more advanced experimental measurements, and receptivity results should be used in combination with direct numerical simulations. A major conclusion of our review is therefore that the understanding of Görtler vortices should be pursued by a combined methodology including theoretical analysis based on the receptivity formalism, direct numerical simulation, and experiments. Highly desirable outcomes of such endeavor are the prediction of the location and extension of the transition region, and a model for the transition process. We finally highlight further prospects and challenges on fundamental and applied research on Görtler instability and transition in compressible flows

Secondary Instability of Second Modes in Hypersonic Boundary Layers

Second mode disturbances dominate the primary instability stage of transition in a number of hypersonic flow configurations. The highest amplification rates of second mode disturbances are usually associated with 2D (or axisymmetric) perturbations and, therefore, a likely scenario for the onset of the three-dimensionality required for laminar-turbulent transition corresponds to the parametric amplification of 3D secondary instabilities in the presence of 2D, finite amplitude second mode disturbances. The secondary instability of second mode disturbances is studied for selected canonical flow configurations. The basic state for the secondary instability analysis is obtained by tracking the linear and nonlinear evolution of 2D, second mode disturbances using nonlinear parabolized stability equations. Unlike in previous studies, the selection of primary disturbances used for the secondary instability analysis was based on their potential relevance to transition in a low disturbance environment and the effects of nonlinearity on the evolution of primary disturbances was accounted for. Strongly nonlinear effects related to the self-interaction of second mode disturbances lead to an upstream shift in the upper branch neutral location. Secondary instability computations confirm the previously known dominance of subharmonic modes at relatively small primary amplitudes. However, for the Purdue Mach 6 compression cone configuration, it was shown that a strong fundamental secondary instability can exist for a range of initial amplitudes of the most amplified second mode disturbance, indicating that the exclusive focus on subharmonic modes in the previous applications of secondary instability theory to second mode primary instability may not have been fully justified

Boundary-Layer Instabilities on a Cooled Flared Cone at Mach 6

There are significant challenges involved in the design of hypersonic vehicles/projectiles, one of them being the fluid dynamics on the surface. The fluid dynamics can affect the thermal load, structural load and handling of the vehicle. Boundary-layer transition alone can account for an order of magnitude higher heating and thus it is important to understand how this transition process occurs. In this experiment the boundary layer on a sharp-tip, part-straight, part-flared actively cooled cone is being studied in a quiet Mach 6 wind tunnel. The Mack mode and its associated secondary instability dominate above Mach 4 as a cause of boundary-layer transition and is being focused on in this study. The role of curvature is also included in order to study the effects of Görtler vortices. Boundary-layer transition was tracked using embedded thermocouples, focusing schlieren and Focused Laser Differential Interferometry (FLDI). The Mack-mode disturbances were observed using a high-speed camera and spectral data were taken at a point in the boundary layer using focusing schlieren and FLDI. Constant Temperature Anemometry (CTA) was used in the form of hot-film probes that were traversed in the boundary layer to do azimuthal sweeps and showed periodic mass-flux variations with a wavenumber of 90, which were attributed to Görtler vortices. Distributed Roughness Elements (DREs) were placed at the neutral point with the same wavenumber seen from the hot-film data to force a 3-D breakdown. FLDI was used to probe six points in the boundary layer simultaneously and captured the Mack-mode instability and its harmonics. Computational work confirmed the experimental findings. This work creates a database for Direct Numerical Simulations (DNS) and Non-linear Parabolized Stability Equations (NPSE) computations and adds to the current knowledge of hypersonic boundary-layer transition