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.

On the Görtler instability in hypersonic flows: Sutherland law fluids and real gas effects

1993 ◽  
Vol 342 (1665) ◽  
pp. 325-377 ◽  
Keyword(s):  
Boundary Layer ◽  
Mach Number ◽  
Free Stream ◽  
Layer Growth ◽  
Real Gas ◽  
Görtler Vortices ◽  
Real Gas Effects ◽  
Gortler Vortices ◽  
Local Wavenumber ◽  
Boundary Layer Growth

The Görtler vortex instability mechanism in a hypersonic boundary layer on a curved wall is investigated in this paper. Our aim is to clarify the precise roles of the effects of boundary layer growth, wall cooling and gas dissociation in the determination of stability properties. We first assume that the fluid is an ideal gas with viscosity given by Sutherland’s law. It is shown that when the free-stream Mach number M is large, the boundary layer divides into two sublayers: a wall layer of O ( M 3/2 ) thickness (in terms of the boundary layer variable) over which the basic state temperature is O ( M 2 ), and a temperature adjustment layer of O (1) thickness over which the basic state temperature decreases monotonically to its free-stream value. Görtler vortices which have wavelength comparable with the boundary layer thickness (i.e. have local wavenumber of order M -3/2 ) are referred to as wall modes. We show that their downstream evolution is governed by a set of parabolic partial differential equations and that they have the usual features of Görtler vortices in incompressible boundary layers. As the local wavenumber increases, the neutral Görtler number decreases and the centre of vortex activity moves towards the temperature adjustment layer. Görtler vortices with wavenumber of order one or larger must necessarily be trapped in the temperature adjustment layer, and it is this mode which is the most dangerous. For this mode, we find that the leading-order term in the Görtler number expansion is independent of the wavenumber and is due to the curvature of the basic state. This term is also the asymptotic limit of the neutral Görtler numbers of the wall mode. To determine the higher-order correction terms in the Görtler number expansion, we have to distinguish between two wall curvature cases. When the wall curvature is proportional to (2 x ) -3/2 , where x is the streamwise variable, the Mach number M can be scaled out of the problem and the boundary layer growth takes place over an O (1) lengthscale. The evolution properties of Görtler vortices are then similar to those in incompressible flows. In the more general case when the wall curvature is not proportional to (2 x ) -3/2 , the effect of the curvature of the basic state persists in the downstream development of Görtler vortices; non-parallel effects are important over a larger range of wavenumbers, and they become of second order only when the local wavenumber is of order higher than O ( M 1/4 ). In the latter case the Görtler number expansion has the first two terms independent of non-parallel effects; the first term being due to the curvature of the basic state and the second term due to viscous effects. The second term becomes comparable with the first term when the wavenumber reaches the order M 3/8 , in which case another correction term can also be found independently of non-parallel effects. Next we investigate real gas effects by assuming that the fluid is an ideal dissociating gas. We find that both gas dissociation and wall cooling are destabilizing for the mode trapped in the temperature adjustment layer, but for the wall mode trapped near the wall the effect of gas dissociation can be either destabilizing or stabilizing.

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.

Crossflow effects on the growth rate of inviscid Görtler vortices in a hypersonic boundary layer

1994 ◽  
Vol 276 ◽  
pp. 343-367 ◽  
Author(s):  
Yibin Fu ◽  
Philip Hall
Keyword(s):  
Boundary Layer ◽  
Growth Rate ◽  
Exact Solution ◽  
Eigenvalue Problem ◽  
Hypersonic Boundary Layer ◽  
Minimum Order ◽  
Görtler Vortices ◽  
Neutral Mode ◽  
Gortler Vortices ◽  
The Relationship

The effects of crossflow on the growth rate of inviscid Görtler vortices in a hypersonic boundary layer with pressure gradient are studied in this paper. Attention is focused on the inviscid mode trapped in the temperature adjustment layer; this mode has greater growth rate than any other mode at the minimum order of the Görtler number at which Görtler vortices may exist. The eigenvalue problem which governs the relationship between the growth rate, the crossflow amplitude and the wavenumber is solved numerically, and the results are then used to clarify the effects of crossflow on the growth rate of inviscid Görtler vortices. It is shown that crossflow effects stabilize Görtler vortices in different manners for incompressible and hypersonic flows. The neutral mode eigenvalue problem is found to have an exact solution, and as a byproduct, we have also found the exact solution to a neutral mode eigenvalue problem which was formulated, but unsolved before, by Bassom & Hall (1991).

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.

The fully nonlinear development of Görtler vortices in growing boundary layers

1988 ◽  
Vol 415 (1849) ◽  
pp. 421-444 ◽  
Keyword(s):  
Boundary Layer ◽  
Evolution Equations ◽  
Mean Flow ◽  
Initial Development ◽  
Fully Nonlinear ◽  
Nonlinear Development ◽  
Görtler Vortices ◽  
Starting Point ◽  
The Mean ◽  
Gortler Vortices

The fully nonlinear development of small-wavelength Görtler vortices in a growing boundary layer is investigated by a combination of asymptotic and numerical methods. The starting point for the analysis is the weakly nonlinear theory of Hall ( J. Inst. Math. Applies 29, 173 (1982)) who discussed the initial development of large-wavenumber small-amplitude vortices in a neighbourhood of the location where they first become linearly unstable. That development is unusual in the context of nonlinear stability theory in that it is not described by the Stuart-Watson approach. In fact, the development is governed by a pair of coupled nonlinear partial differential evolution equations for the vortex flow and the mean flow correction. Here the further development of this interaction is considered for vortices so large that the mean flow correction driven by them is as large as the basic state. Surprisingly it is found that such a nonlinear interaction can still be described by asymptotic means. It is shown that the vortices spread out across the boundary layer and effectively drive the boundary layer. In fact, the system obtained by the equations for the fundamental component of the vortex generates a differential equation for the basic state. Thus the mean flow adjusts so as to make these large amplitude vortices locally neutral. Moreover in the region where the vortices exist the mean flow has a ‘square-root’ profile and the vortex velocity field can be written down in closed form. The upper and lower boundaries of the region of vortex activity are determined by a free-boundary problem involving the boundary-layer equations. In general it is found that this region ultimately includes almost all of the original boundary layer and much of the free stream. In this situation the mean flow has essentially no relation to the flow that exists in the absence of the vortices.

Compressor blade leading edges in subsonic compressible flow

2000 ◽  
Vol 214 (1) ◽  
pp. 221-242 ◽  
Author(s):  
L Tain ◽  
N. A. Cumpsty
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Mach Number ◽  
Free Stream ◽  
Separation Bubble ◽  
Leading Edge ◽  
Inviscid Flow ◽  
Compressor Blade ◽  
Free Stream Turbulence ◽  
Mach Numbers

The flow around the leading edge of a compressor blade is interesting and important because there is such a strong interaction between the viscous boundary layer flow and the inviscid flow around it. As the velocity of the inviscid flow just outside the boundary layer is increased from subsonic to supersonic, the type of viscous-inviscid interaction changes; this has important effects on the boundary layer downstream and thus on the performance of the aerofoil or blade. An investigation has been undertaken of the flow in the immediate vicinity of a simulated compressor blade leading edge for a range of inlet Mach numbers from 0.6 to 0.95. The two-dimensional aerofoil used has a circular leading edge on the front of a flat aerofoil. The incidence, Reynolds number and level of free-stream turbulence have been varied. Measurements include the static pressure around the leading edge and downstream and the boundary layer profile far enough downstream for the leading edge bubble to have reattached. Schlieren pictures were also obtained. The flow around the leading edge becomes supersonic when the inlet Mach number is 0.7 for the zero-incidence case; for an inlet Mach number of 0.95 the peak Mach number was approximately 1.7. The pattern of flow around the leading edge alters as the Mach number is increased, and at the highest Mach number tested here the laminar separation bubble is removed. Positive incidence, raised free-stream turbulence or increased Reynolds number at intermediate inlet Mach numbers tended to promote flow patterns similar to those seen at the highest inlet Mach number. Both increased free-stream turbulence and increased Reynolds number, for the same Mach number and incidence, produced thinner shear layers including a thinner boundary layer well downstream. The measurements were supported by calculations using the MSES code (the single aerofoil version of the MISES code); the calculations were helpful in interpreting the measured results and were demonstrated to be accurate enough to be used for design purposes.

Effects of Görtler vortices, wall cooling and gas dissociation on the Rayleigh instability in a hypersonic boundary layer

1993 ◽  
Vol 247 ◽  
pp. 503-525 ◽  
Author(s):  
Yibin Fu ◽  
Philip Hall
Keyword(s):  
Boundary Layer ◽  
Growth Rate ◽  
Large Amplitude ◽  
Hypersonic Boundary Layer ◽  
Rayleigh Instability ◽  
Variable Curvature ◽  
Görtler Vortices ◽  
Gortler Vortices

In a hypersonic boundary layer over a wall of variable curvature, the region most susceptible to Görtler vortices is the temperature adjustment layer sitting at the edge of the boundary layer. This temperature adjustment layer is also the most dangerous site for Rayleigh instability. In this paper, we investigate how the existence of large-amplitude Görtler vortices affects the growth rate of Rayleigh instability. The effects of wall cooling and gas dissociation on this instability are also studied. We find that all these mechanisms increase the growth rate of Rayleigh instability and are therefore destabilizing.

scholarly journals Boundary-layer receptivity for a parabolic leading edge

1996 ◽  
Vol 310 ◽  
pp. 243-267 ◽  
Author(s):  
P. W. Hammerton ◽  
E. J. Kerschen
Keyword(s):  
Boundary Layer ◽  
Acoustic Waves ◽  
Free Stream ◽  
Asymptotic Methods ◽  
Leading Edge ◽  
Flow Speed ◽  
The Body ◽  
Large Reynolds Number ◽  
Mean Flow ◽  
Nose Radius

The effect of the nose radius of a body on boundary-layer receptivity is analysed for the case of a symmetric mean flow past a body with a parabolic leading edge. Asymptotic methods based on large Reynolds number are used, supplemented by numerical results. The Mach number is assumed small, and acoustic free-stream disturbances are considered. The case of free-stream acoustic waves, propagating obliquely to the symmetric mean flow is considered. The body nose radius, rn, enters the theory through a Strouhal number, S = ωrn/U, where ω is the frequency of the acoustic wave and U is the mean flow speed. The finite nose radius dramatically reduces the receptivity level compared to that for a flat plate, the amplitude of the instability waves in the boundary layer being decreased by an order of magnitude when S = 0.3. Oblique acoustic waves produce much higher receptivity levels than acoustic waves propagating parallel to the body chord.

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 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.

Sign in / Sign up

Export Citation Format

Share Document