Görtler vortices in compressible mixing layers

2001 ◽  
Vol 427 ◽  
pp. 359-388 ◽  
Author(s):  
J. M. SARKIES ◽  
S. R. OTTO
Keyword(s):  
Free Stream ◽  
Flow Direction ◽  
Mixing Layers ◽  
Neutral Stability ◽  
Curvature Condition ◽  
Görtler Vortices ◽  
Analytical Work ◽  
Gortler Vortices ◽  
The Stability ◽  
Parameter Ranges

In experiments, Plesniak, Mehta & Johnson (1994) have noted that curved two-stream mixing layers are susceptible to centrifugal instabilities under the condition that the slower of the streams curves towards the faster one; this condition is analogous to the concave curvature condition for the stability of the flow over a plate. The modes which arise manifest themselves as vortices aligned with the dominant flow direction. Previous numerical and analytical work has elucidated the structure of these vortices within incompressible mixing layers; Otto, Jackson & Hu (1996). In this paper we go on to investigate the rôles of compressibility and heating in determining the streamwise fate of Görtler vortices within these situations.The development of the disturbances is monitored downstream and curves of neutral stability are plotted. The effect of changing the Mach number and free-stream temperatures is studied in detail. It is found that for certain parameter régimes modes can occur within convexly curved, or ‘stable’ mixing layers; these ‘thermal modes’ have no counterpart within incompressible mixing layers. By making use of a large Görtler number analysis we are able to verify our numerical results, and derive a very simple condition which yields information about the parameter ranges for which certain modes are likely to occur. As an aside this method can be used to show that no degree of wall cooling will allow sustained growth of Görtler vortices within boundary layers over convex plates.

scholarly journals Neutral stability curves of compressible Görtler flow generated by low-frequency free-stream vortical disturbances

2019 ◽  
Vol 876 ◽  
pp. 1146-1157
Author(s):  
Samuele Viaro ◽  
Pierre Ricco
Keyword(s):  
Free Stream ◽  
Low Frequency ◽  
Leading Edge ◽  
Neutral Curve ◽  
Neutral Stability ◽  
Numerical Result ◽  
Görtler Vortices ◽  
Compressible Boundary Layers ◽  
Gortler Vortices ◽  
Tollmien Schlichting

Pre-transitional compressible boundary layers perturbed by low-frequency free-stream vortical disturbances and flowing over plates with streamwise-concave curvature are studied via matched asymptotic expansions and numerically. The Mach number, the Görtler number and the frequency of the free-stream disturbance are varied to obtain the neutral stability curves, i.e. curves in the space of the parameters that distinguish spatially growing from spatially decaying perturbations. The receptivity approach is used to calculate the evolution of Klebanoff modes, highly oblique Tollmien–Schlichting waves influenced by the concave curvature of the wall, and Görtler vortices. The Klebanoff modes always evolve from the leading edge, the Görtler vortices dominate when the influence of the curvature becomes significant and the Tollmien–Schlichting waves may precede the Görtler vortices for moderate Görtler numbers. For relatively high frequencies the triple-deck formalism allows us to confirm the numerical result of the negligible influence of the curvature on the Tollmien–Schlichting waves when the Görtler number is an order-one quantity. Experimental data for compressible Görtler flows are mapped onto our neutral-curve graphs and earlier theoretical results are compared with our predictions.

Measurements in a Transitional Boundary Layer With Görtler Vortices

1996 ◽  
Author(s):  
Ralph J. Volino ◽  
Terrence W. Simon
Keyword(s):  
Boundary Layer ◽  
Turbulence Intensity ◽  
Free Stream ◽  
Streamwise Velocity ◽  
Mean Velocity ◽  
Free Stream Turbulence ◽  
Görtler Vortices ◽  
Concave Wall ◽  
The Mean ◽  
Gortler Vortices

The laminar-turbulent transition process has been documented in a concave-wall boundary layer subject to low (0.6%) free-stream turbulence intensity. Transition began at a Reynolds number, Rex (based on distance from the leading edge of the test wall), of 3.5×105 and was completed by 4.7×105. The transition was strongly influenced by the presence of stationary, streamwise, Görtler vortices. Transition under similar conditions has been documented in previous studies, but because concave-wall transition tends to be rapid, measurements within the transition zone were sparse. In this study, emphasis is on measurements within the zone of intermittent flow. Twenty-five profiles of mean streamwise velocity, fluctuating streamwise velocity, and intermittency have been acquired at five values of Rex, and five spanwise locations relative to a Görtler vortex. The mean velocity profiles acquired near the vortex downwash sites exhibit inflection points and local minima. These minima, located in the outer part of the boundary layer, provide evidence of a “tilting” of the vortices in the spanwise direction. Profiles of fluctuating velocity and intermittency exhibit peaks near the locations of the minima in the mean velocity profiles. These peaks indicate that turbulence is generated in regions of high shear, which are relatively far from the wall. The transition mechanism in this flow is different from that on flat walls, where turbulence is produced in the near-wall region. The peak intermittency values in the profiles increase with Rex, but do not follow the “universal” distribution observed in most flat-wall, transitional boundary layers. The results have applications whenever strong concave curvature may result in the formation of Görtler vortices in otherwise 2-D flows. Because these cases were run with a low value of free-stream turbulence intensity, the flow is not a replication of a gas turbine flow. However, the results do provide a base case for further work on transition on the pressure side of gas turbine airfoils, where concave curvature effects are combined with the effects of high free-stream turbulence and strong streamwise pressure gradients, for they show the effects of embedded streamwise vorticity in a flow that is free of high-turbulence effects.

Hypersonic boundary layer transition on a concave wall: stationary Görtler vortices

2019 ◽  
Vol 865 ◽  
pp. 1-40 ◽  
Author(s):  
X. Chen ◽  
G. L. Huang ◽  
C. B. Lee
Keyword(s):  
Boundary Layer ◽  
Numerical Simulations ◽  
Stability Theory ◽  
Direct Numerical Simulations ◽  
Boundary Layer Transition ◽  
Hypersonic Boundary Layer ◽  
Mode Instability ◽  
Görtler Vortices ◽  
Gortler Vortices ◽  
The Stability

This study investigates the stability and transition of Görtler vortices in a hypersonic boundary layer using linear stability theory and direct numerical simulations. In the simulations, Görtler vortices are separately excited by wall blowing and suction with spanwise wavelengths of 3, 6 and 9 mm. In addition to primary streaks with the same wavelength as the blowing and suction, secondary streaks with half the wavelength also emerge in the 6 and 9 mm cases. The streaks develop into mushroom structures before breaking down. The breakdown processes of the three cases are dominated by a sinuous-mode instability, a varicose-mode instability and a combination of the two, respectively. Both fundamental and subharmonic instabilities are relevant in all cases. Multiple modes are identified in the secondary-instability stage, some of which originate from the primary instabilities (first and second Mack modes). We demonstrate that the first Mack mode can be destabilized to either a varicose-mode or sinuous-mode streak instability depending on its frequency and wavelength, whereas the second Mack mode undergoes a stabilizing stage before turning into a varicose mode in the 6 and 9 mm cases. An energy analysis reveals the stabilizing and destabilizing mechanisms of the primary instabilities under the influence of Görtler vortices, highlighting the role played by the spanwise production based on the spanwise gradient of the streamwise velocity in both varicose and sinuous modes. The effects introduced by the secondary streaks are examined by filtering the secondary streaks in two new simulations with nominally identical conditions to those of the 6 and 9 mm cases. Remarkably, the secondary streaks can destabilize the Görtler vortices, therefore advancing the transition. The stability theory results are in good agreement with those from direct numerical simulations.

Receptivity, instability and breakdown of Görtler flow

2011 ◽  
Vol 682 ◽  
pp. 362-396 ◽  
Author(s):  
LARS-UVE SCHRADER ◽  
LUCA BRANDT ◽  
TAMER A. ZAKI
Keyword(s):  
Boundary Layer ◽  
Free Stream ◽  
Low Frequency ◽  
Boundary Layer Transition ◽  
Transition To Turbulence ◽  
Flat Plates ◽  
Frequency Spectra ◽  
Free Stream Turbulence ◽  
Görtler Vortices ◽  
Gortler Vortices

Receptivity, disturbance growth and breakdown to turbulence in Görtler flow are studied by spatial direct numerical simulation (DNS). The boundary layer is exposed to free-stream vortical modes and localized wall roughness. We propose a normalization of the roughness-induced receptivity coefficient by the square root of the Görtler number. This scaling removes the dependence of the receptivity coefficient on wall curvature. It is found that vortical modes are more efficient at generating Görtler vortices than localized roughness. The boundary layer is most receptive to zero- and low-frequency free-stream vortices, exciting steady and slowly travelling Görtler modes. The associated receptivity mechanism is linear and involves the generation of boundary-layer streaks, which soon evolve into unstable Görtler vortices. This connection between transient and exponential amplification is absent on flat plates and promotes transition to turbulence on curved walls. We demonstrate that the Görtler boundary layer is also receptive to high-frequency free-stream vorticity, which triggers steady Görtler rolls via a nonlinear receptivity mechanism. In addition to the receptivity study, we have carried out DNS of boundary-layer transition due to broadband free-stream turbulence with different intensities and frequency spectra. It is found that nonlinear receptivity dominates over the linear mechanism unless the free-stream fluctuations are concentrated in the low-frequency range. In the latter case, transition is accelerated due to the presence of travelling Görtler modes.

Nonlinear evolution and secondary instability of steady and unsteady Görtler vortices induced by free-stream vortical disturbances

2017 ◽  
Vol 829 ◽  
pp. 681-730 ◽  
Author(s):  
Dongdong Xu ◽  
Yongming Zhang ◽  
Xuesong Wu
Keyword(s):  
Boundary Layer ◽  
Free Stream ◽  
Nonlinear Evolution ◽  
Low Frequency ◽  
Secondary Instability ◽  
Moderate Intensity ◽  
Mean Flow ◽  
Görtler Vortices ◽  
Gortler Vortices ◽  
The Impact

We study the nonlinear development and secondary instability of steady and unsteady Görtler vortices which are excited by free-stream vortical disturbances (FSVD) in a boundary layer over a concave wall. The focus is on low-frequency (long-wavelength) components of FSVD, to which the boundary layer is most receptive. For simplification, FSVD are modelled by a pair of oblique modes with opposite spanwise wavenumbers $\pm k_{3}$, and their intensity is strong enough (but still of low level) that the excitation and evolution of Görtler vortices are nonlinear. For the general case that the Görtler number $G_{\unicode[STIX]{x1D6EC}}$ (based on the spanwise wavelength $\unicode[STIX]{x1D6EC}$ of the disturbances) is $O(1)$, the formation and evolution of Görtler vortices are governed by the nonlinear unsteady boundary-region equations, supplemented by appropriate upstream and far-field boundary conditions, which characterize the impact of FSVD on the boundary layer. This initial-boundary-value problem is solved numerically. FSVD excite steady and unsteady Görtler vortices, which undergo non-modal growth, modal growth and nonlinear saturation for FSVD of moderate intensity. However, for sufficiently strong FSVD the modal stage is bypassed. Nonlinear interactions cause Görtler vortices to saturate, with the saturated amplitude being independent of FSVD intensity when $G_{\unicode[STIX]{x1D6EC}}\neq 0$. The predicted modified mean-flow profiles and structure of Görtler vortices are in excellent agreement with several steady experimental measurements. As the frequency increases, the nonlinearly generated harmonic component $(0,2)$ (which has zero frequency and wavenumber $2k_{3}$) becomes larger, and as a result the Görtler vortices appear almost steady. The secondary instability analysis indicates that Görtler vortices become inviscidly unstable in the presence of FSVD with a high enough intensity. Three types of inviscid unstable modes, referred to as sinuous (odd) modes I, II and varicose (even) modes I, are identified, and their relevance is delineated. The characteristics of dominant unstable modes, including their frequency ranges and eigenfunctions, are in good agreement with experiments. The secondary instability is intermittent when FSVD are unsteady and of low frequency. However, the intermittence diminishes as the frequency increases. The present theoretical framework, which allows for a detailed and integrated description of the key transition processes, from generation, through linear and nonlinear evolution, to the onset of secondary instability, represents a useful step towards predicting the pre-transitional flow and transition itself of the boundary layer over a blade in turbomachinery.

Excitation of steady and unsteady Görtler vortices by free-stream vortical disturbances

2011 ◽  
Vol 682 ◽  
pp. 66-100 ◽  
Author(s):  
XUESONG WU ◽  
DIFEI ZHAO ◽  
JISHENG LUO
Keyword(s):  
Boundary Layer ◽  
Flat Plate ◽  
Numerical Solutions ◽  
Free Stream ◽  
Plate Boundary ◽  
Modal Shape ◽  
Görtler Vortices ◽  
Concave Wall ◽  
Gortler Vortices ◽  
The Impact

Excitation of Görtler vortices in a boundary layer over a concave wall by free-stream vortical disturbances is studied theoretically and numerically. Attention is focused on disturbances with long streamwise wavelengths, to which the boundary layer is most receptive. The appropriate initial-boundary-value problem describing both the receptivity process and the development of the induced perturbation is formulated for the generic case where the Görtler number GΛ (based on the spanwise wavelength Λ of the disturbance) is of order one. The impact of free-stream disturbances on the boundary layer is accounted for by the far-field boundary condition and the initial condition near the leading edge, both of which turn out to be the same as those given by Leib, Wundrow & Goldstein (J. Fluid Mech., vol. 380, 1999, p. 169) for the flat-plate boundary layer. Numerical solutions show that for a sufficiently small GΛ, the induced perturbation exhibits essentially the same characteristics as streaks occurring in the flat-plate case: it undergoes considerable amplification and then decays. However, when GΛ exceeds a critical value, the induced perturbation exhibits (quasi-) exponential growth. The perturbation acquires the modal shape of Görtler vortices rather quickly, and its growth rate approaches that predicted by local instability theories farther downstream, indicating that Görtler vortices are excited. The amplitude of the Görtler vortices excited is found to decrease as the frequency increases, with steady vortices being dominant. Comprehensive quantitative comparisons with experiments show that the eigenvalue approach predicts the modal shape adequately, but only the initial-value approach can accurately predict the entire evolution of the amplitude. An asymptotic analysis is performed for GΛ ≫ 1 to map out distinct regimes through which a perturbation with a fixed spanwise wavelength evolves. The centrifugal force starts to influence the generation of the pressure when x* ~ ΛRΛG−2/3Λ, where RΛ denotes the Reynolds number based on Λ. The induced pressure leads to full coupling of the momentum equations when x* ~ ΛRΛGΛ−2/5. This is the crucial regime linking the pre-modal and modal phases of the perturbation because the governing equations admit growing asymptotic eigensolutions, which develop into fully fledged Görtler vortices of inviscid nature when x* ~ ΛRΛ. From this position onwards, local eigenvalue formulations are mathematically justified. Görtler vortices continue to amplify and enter the so-called most unstable regime when x* ~ ΛRΛGΛ, and ultimately approach the right-branch regime when x* ~ ΛRΛG2Λ.

The excitation of Görtler vortices by free stream coherent structures

2017 ◽  
Vol 826 ◽  
pp. 60-96 ◽  
Author(s):  
L. J. Dempsey ◽  
P. Hall ◽  
K. Deguchi
Keyword(s):  
Boundary Layer ◽  
Coherent Structures ◽  
Free Stream ◽  
Nonlinear Regime ◽  
Weakly Nonlinear ◽  
Stream Structure ◽  
Görtler Vortices ◽  
Gortler Vortices ◽  
Asymptotic Suction ◽  
Imperfect Bifurcation

The effect of free stream coherent structures in the asymptotic suction boundary layer on the initiation of Görtler vortices is considered from both the ‘imperfect’ bifurcation and receptivity viewpoints. Firstly a weakly nonlinear and a full numerical approach are used to describe Görtler vortices in the asymptotic suction boundary layer in the absence of forcing from the free stream. It is found that interactions between different spanwise harmonics occur and lead to multiple secondary bifurcations in the fully nonlinear regime. Furthermore it is shown that centrifugal instabilities of the asymptotic suction boundary layer behave quite differently than their counterparts in either fully developed flows such as Couette flow or growing boundary layers. A significant result is that the most dangerous disturbance is found to bifurcate subcritically from the unperturbed state. Within the weakly nonlinear regime the receptivity of Görtler vortices to the free stream exact coherent structures discovered by Deguchi & Hall (J. Fluid Mech., vol. 752, 2014, pp. 602–625; J. Fluid Mech., vol. 778, 2015, pp. 451–484) is considered. The presence of free stream structures results in a resonant excitation of Görtler vortices in the main boundary layer. This leads to imperfect bifurcations reminiscent of those found by Daniels (Proc. R. Soc. Lond. A, vol. 358, 1977, pp. 173–197) and Hall & Walton (Proc. R. Soc. Lond. A, vol. 358, 1977, pp. 199–221; J. Fluid Mech., vol. 90, 1979, pp. 377–395) in the context of transition to finite amplitude Bénard convection in a bounded region. In order to understand the receptivity problem for the given flow the spatial initial value problem for this interaction is also considered when the free stream structure begins at a fixed position along the wall. Remarkably, it will be shown that free stream structures are incredibly efficient generators of Görtler vortices; indeed the induced vortices are found to be larger than the free stream structure which provokes them! The relationship between the imperfect bifurcation approach and receptivity theory is described.

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