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.

scholarly journals Analysis of the unstable Tollmien–Schlichting mode on bodies with a rounded leading edge using the parabolized stability equation

2009 ◽  
Vol 623 ◽  
pp. 167-185
Author(s):  
M. R. TURNER ◽  
P. W. HAMMERTON
Keyword(s):  
Asymptotic Theory ◽  
Free Stream ◽  
Leading Edge ◽  
Adverse Pressure Gradient ◽  
Reynolds Numbers ◽  
Neutral Stability ◽  
S Wave ◽  
Stability Equation ◽  
Stability Point ◽  
Tollmien Schlichting

The interaction between free-stream disturbances and the boundary layer on a body with a rounded leading edge is considered in this paper. A method which incorporates calculations using the parabolized stability equation in the Orr–Sommerfeld region, along with an upstream boundary condition derived from asymptotic theory in the vicinity of the leading edge, is generalized to bodies with an inviscid slip velocity which tends to a constant far downstream. We present results for the position of the lower branch neutral stability point and the magnitude of the unstable Tollmien–Schlichting (T-S) mode at this point for both a parabolic body and the Rankine body. For the Rankine body, which has an adverse pressure gradient along its surface far from the nose, we find a double maximum in the T-S wave amplitude for sufficiently large Reynolds numbers.

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.

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.

scholarly journals Compressible unsteady Görtler vortices subject to free-stream vortical disturbances

2019 ◽  
Vol 867 ◽  
pp. 250-299 ◽  
Author(s):  
Samuele Viaro ◽  
Pierre Ricco
Keyword(s):  
Boundary Layer ◽  
Growth Rate ◽  
Mach Number ◽  
Free Stream ◽  
Asymptotic Methods ◽  
Leading Edge ◽  
Vortical Flow ◽  
Initial Development ◽  
Görtler Vortices ◽  
Gortler Vortices

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 Görtler number, the scaled parameter defining the centrifugal effects; we use an eigenvalue formulation where the free-stream forcing is neglected; and we solve the receptivity problem by integrating the compressible boundary-region equations complemented by appropriate initial and boundary conditions that synthesize the influence of the free-stream vortical flow. Near the leading edge, the boundary-layer perturbations develop as thermal Klebanoff modes and, when centrifugal effects become influential, these modes turn into thermal Görtler vortices, i.e. streamwise rolls characterized by intense velocity and temperature perturbations. The high-Görtler-number asymptotic analysis reveals the condition for which the Görtler vortices start to grow. 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. When the Görtler number is large, the theoretical analysis also shows that the vortices move towards the wall as the Mach number increases. These results are confirmed by the receptivity analysis, which additionally clarifies that the temperature perturbations respond to this reversed behaviour 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 Görtler number. The Görtler 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 smaller than the free-stream streamwise wavelength. During the initial development of the vortices, only the receptivity calculations are accurate. At streamwise locations where the free-stream disturbances have fully decayed, the growth rate and wavelength are computed with sufficient accuracy by the eigenvalue analysis, although the correct amplitude and evolution of the Görtler 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. We conclude by qualitatively comparing our results with the direct numerical simulations available in the literature.

Effects of Weak Free Stream Nonuniformity on Boundary Layer Transition (Keynote Paper)

2003 ◽  
Author(s):  
Jonathan H. Watmuff
Keyword(s):  
Boundary Layer ◽  
Free Stream ◽  
Leading Edge ◽  
Boundary Layer Transition ◽  
Mean Flow ◽  
Layer Transition ◽  
Free Stream Turbulence ◽  
Tollmien Schlichting ◽  
Distorted Waves ◽  
Thickness Variations

Experiments are described in which well-defined FSN (Free Stream Nonuniformity) distributions are introduced by placing fine wires upstream of the leading edge of a flat plate. Large amplitude spanwise thickness variations are present in the downstream boundary layer resulting from the interaction of the laminar wakes with the leading edge. Regions of elevated background unsteadiness appear on either side of the peak layer thickness, which share many of the characteristics of Klebanoff modes, observed at elevated Free Stream Turbulence (FST) levels. However, for the low background disturbance level of the free stream, the layer remains laminar to the end of the test section (Rx ≈ l.4×106) and there is no evidence of bursting or other phenomena associated with breakdown to turbulence. A vibrating ribbon apparatus is used to demonstrate that the deformation of the mean flow is responsible for substantial phase and amplitude distortion of Tollmien-Schlichting (TS) waves. Pseudo-flow visualization of hot-wire data shows that the breakdown of the distorted waves is more complex and occurs at a lower Reynolds number than the breakdown of the K-type secondary instability observed when the FSN is not present.

Leading-edge receptivity of a hypersonic boundary layer on a flat plate

2001 ◽  
Vol 426 ◽  
pp. 73-94 ◽  
Author(s):  
A. A. MASLOV ◽  
A. N. SHIPLYUK ◽  
A. A. SIDORENKO ◽  
D. ARNAL
Keyword(s):  
Boundary Layer ◽  
Acoustic Waves ◽  
Free Stream ◽  
Three Dimensional ◽  
Leading Edge ◽  
Experimental Investigations ◽  
Two Dimensional ◽  
Inclination Angles ◽  
Tollmien Schlichting ◽  
Radiation Conditions

Experimental investigations of the boundary layer receptivity, on the sharp leading edge of a at plate, to acoustic waves induced by two-dimensional and three- dimensional perturbers, have been performed for a free-stream Mach number M∞ = 5.92. The fields of controlled free-stream disturbances were studied. It was shown that two-dimensional and three-dimensional perturbers radiate acoustic waves and that these perturbers present a set of harmonic motionless sources and moving sources with constant amplitude. The disturbances excited in the boundary layer were measured. It was found that acoustic waves impinging on the leading edge generate Tollmien–Schlichting waves in the boundary layer. The receptivity coefficients were obtained for several radiation conditions and intensities. It was shown that there is a dependence of receptivity coefficients on the wave inclination angles.

Receptivity to free-stream vorticity of flow past a flat plate with elliptic leading edge

2010 ◽  
Vol 653 ◽  
pp. 245-271 ◽  
Author(s):  
L.-U. SCHRADER ◽  
L. BRANDT ◽  
C. MAVRIPLIS ◽  
D. S. HENNINGSON
Keyword(s):  
Boundary Layer ◽  
Flat Plate ◽  
Free Stream ◽  
Three Dimensional ◽  
Low Frequency ◽  
Leading Edge ◽  
Sound Waves ◽  
Streamwise Vorticity ◽  
Low Frequencies ◽  
Free Stream Turbulence

Receptivity of the two-dimensional boundary layer on a flat plate with elliptic leading edge is studied by numerical simulation. Vortical perturbations in the oncoming free stream are considered, impinging on two leading edges with different aspect ratio to identify the effect of bluntness. The relevance of the three vorticity components of natural free-stream turbulence is illuminated by considering axial, vertical and spanwise vorticity separately at different angular frequencies. The boundary layer is most receptive to zero-frequency axial vorticity, triggering a streaky pattern of alternating positive and negative streamwise disturbance velocity. This is in line with earlier numerical studies on non-modal growth of elongated structures in the Blasius boundary layer. We find that the effect of leading-edge bluntness is insignificant for axial free-stream vortices alone. On the other hand, vertical free-stream vorticity is also able to excite non-modal instability in particular at zero and low frequencies. This mechanism relies on the generation of streamwise vorticity through stretching and tilting of the vertical vortex columns at the leading edge and is significantly stronger when the leading edge is blunt. It can thus be concluded that the non-modal boundary-layer response to a free-stream turbulence field with three-dimensional vorticity is enhanced in the presence of a blunt leading edge. At high frequencies of the disturbances the boundary layer becomes receptive to spanwise free-stream vorticity, triggering Tollmien–Schlichting (T-S) modes and receptivity increases with leading-edge bluntness. The receptivity coefficients to free-stream vortices are found to be about 15% of those to sound waves reported in the literature. For the boundary layers and free-stream perturbations considered, the amplitude of the T-S waves remains small compared with the low-frequency streak amplitudes.

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.

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