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.

Mean flow generation by an intermittently unstable boundary layer over a sloping wall

2018 ◽  
Vol 853 ◽  
pp. 111-149 ◽  
Author(s):  
Abouzar Ghasemi ◽  
Marten Klein ◽  
Andreas Will ◽  
Uwe Harlander
Keyword(s):  
Boundary Layer ◽  
Coriolis Force ◽  
Rotation Rate ◽  
Tangential Component ◽  
Mean Flow ◽  
Unstable Flow ◽  
Görtler Vortices ◽  
Unstable Boundary Layer ◽  
The Mean ◽  
Gortler Vortices

Direct numerical simulations (DNS) of the flow in various rotating annular confinements have been conducted to investigate the effects of wall inclination on secondary fluid motions due to an unstable boundary layer. The inner wall resembles a truncated cone (frustum) whose apex half-angle is varied from $18^{\circ }$ to $0^{\circ }$ (straight cylinder). The large inner radius $r_{1}$, the mean rotation rate $\unicode[STIX]{x1D6FA}_{0}$ and the kinematic viscosity $\unicode[STIX]{x1D708}$ were kept constant resulting in the constant Ekman number $E=\unicode[STIX]{x1D708}/(\unicode[STIX]{x1D6FA}_{0}r_{1}^{2})=4\times 10^{-5}$. Flows were excited by time-harmonic modulation of the inner wall’s rotation rate (so-called longitudinal libration) by prescribing the amplitude $\unicode[STIX]{x1D700}\unicode[STIX]{x1D6FA}_{0}$ and the forcing frequency $\unicode[STIX]{x1D714}=\unicode[STIX]{x1D6FA}_{0}$. By steepening the inner wall and hence reducing the effect of the local Coriolis force in the boundary layer three different flow regimes can be realized: a rotation-dominated, a libration-dominated and an intermediate regime. In the present study we focus on the libration-dominated regime. For small libration amplitudes (here $\unicode[STIX]{x1D700}=0.2$), a laminar Ekman–Stokes boundary layer (ESBL) is realized at the librating wall. With the aid of laminar boundary layer theory and DNS we show that the ESBL exhibits an oscillatory mass flux along the librating wall (Ekman property) and an oscillatory azimuthal velocity, which resembles a radially damped wave (Stokes property). For large libration amplitudes (here $\unicode[STIX]{x1D700}=0.8$), the DNS results exhibit an intermittently unstable ESBL, which turns centrifugally unstable during the prograde (faster) part of a libration period. This instability is due to the Stokes property and gives rise to Görtler vortices, which are found to be tilted with respect to the azimuth when the librating wall is at a finite angle relative to the axis of rotation. We show that this tilt is related to the Ekman property of the ESBL. This suggests that linear and nonlinear dynamics are equally important for this intermittent instability. Our DNS results indicate further that the Görtler vortices propagate into the fluid bulk where they generate an azimuthal mean flow. This mean flow is notably different from the mean flow driven in the case of the stable ESBL. A diagnostic analysis of the Reynolds-averaged Navier–Stokes (RANS) equations in the unstable flow regime hints at a competition between the radial and axial turbulent transport terms which act as generating and destructing agents for the azimuthal mean flow, respectively. We show that the balance of both terms depends on the wall inclination, that is, on the wall-tangential component of the Coriolis force.

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.

Fundamental and subharmonic secondary instabilities of Görtler vortices

1995 ◽  
Vol 297 ◽  
pp. 77-100 ◽  
Author(s):  
Fei Li ◽  
Mujeeb R. Malik
Keyword(s):  
Flow Field ◽  
Vortex Structure ◽  
Streamwise Velocity ◽  
Secondary Instability ◽  
Mean Flow ◽  
Partial Differential ◽  
Relative Significance ◽  
Nonlinear Development ◽  
Görtler Vortices ◽  
Gortler Vortices

The nonlinear development of stationary Görtler vortices leads to a highly distorted mean flow field where the streamwise velocity depends strongly not only on the wall-normal but also on the spanwise coordinates. In this paper, the inviscid instability of this flow field is analysed by solving the two-dimensional eigenvalue problem associated with the governing partial differential equation. It is found that the flow field is subject to the fundamental odd and even (with respect to the Görtler vortex) unstable modes. The odd mode, which was also found by Hall & Horseman (1991), is initially more unstable. However, there exists an even mode which has higher growth rate further downstream. It is shown that the relative significance of these two modes depends upon the Görtler vortex wavelength such that the even mode is stronger for large wavelengths while the odd mode is stronger for short wavelengths. Our analysis also shows the existence of new subharmonic (both odd and even) modes of secondary instability. The nonlinear development of the fundamental secondary instability modes is studied by solving the (viscous) partial differential equations under a parabolizing approximation. The odd mode leads to the well-known sinuous mode of break down while the even mode leads to the horseshoe-type vortex structure. This helps explain experimental observations that Görtler vortices break down sometimes by sinuous motion and sometimes by developing a horseshoe vortex structure. The details of these break down mechanisms are presented.

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.

On the secondary instability of Taylor-Görtler vortices to Tollmien-Schlichting waves in fully developed flows

1988 ◽  
Vol 186 ◽  
pp. 445-469 ◽  
Author(s):  
James Bennett ◽  
Philip Hall
Keyword(s):  
Boundary Layer ◽  
Practical Importance ◽  
Transition To Turbulence ◽  
External Boundary ◽  
Fully Nonlinear ◽  
Boundary Layer Problem ◽  
Görtler Vortices ◽  
Gortler Vortices ◽  
Tollmien Schlichting ◽  
Layer Problem

There are many flows of practical importance where both Tollmien-Schlichting waves and Taylor-Görtler vortices are possible causes of transition to turbulence. In this paper, the effect of fully nonlinear Taylor-Görtler vortices on the growth of small-amplitude Tollmien-Schlichting waves is investigated. The basic state considered is the fully developed flow between concentric cylinders driven by an azimuthal pressure gradient. It is hoped that an investigation of this problem will shed light on the more complicated external-boundary-layer problem where again both modes of instability exist in the presence of concave curvature. The type of Tollmien-Schlichting waves considered have the asymptotic structure of lower-branch modes of plane Poiseuille flow. Whilst instabilities at lower Reynolds number are possible, the former modes are simpler to analyse and more relevant to the boundary-layer problem. The effect of fully nonlinear Taylor-Görtler vortices on both two-dimensional and three-dimensional waves is determined. It is shown that, whilst the maximum growth as a function of frequency is not greatly affected, there is a large destabilizing effect over a large range of frequencies.

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.

Investigation of weakly-nonlinear development of unsteady Görtler vortices

2010 ◽  
Vol 17 (4) ◽  
pp. 455-481 ◽  
Author(s):  
A. V. Boiko ◽  
A. V. Ivanov ◽  
Yu. S. Kachanov ◽  
D. A. Mischenko
Keyword(s):  
Weakly Nonlinear ◽  
Nonlinear Development ◽  
Görtler Vortices ◽  
Gortler Vortices

Data Analysis Methods for Stereo PIV of a High Aspect Ratio Flexible Cylinder

2007 ◽  
Author(s):  
D. Furey ◽  
P. Atsavapranee ◽  
K. Cipolla
Keyword(s):  
Boundary Layer ◽  
Data Analysis ◽  
Aspect Ratio ◽  
High Aspect Ratio ◽  
Particle Image Velocimetry Data ◽  
Flow Fields ◽  
Mean Flow ◽  
Boundary Flow ◽  
The Mean ◽  
Cylinder Flow

Stereo Particle Image velocimetry data was collected over high aspect ratio flexible cylinders (L/a = 1.5 to 3 × 105) to evaluate the axial development of the turbulent boundary layer where the boundary layer thickness becomes significantly larger than the cylinder diameter (δ/a>>1). The flexible cylinders are approximately neutrally buoyant and have an initial length of 152 m and radii of 0.45 mm and 1.25 mm. The cylinders were towed at speeds ranging from 3.8 to 15.4 m/sec in the David Taylor Model Basin. The analysis of the SPIV data required a several step procedure to evaluate the cylinder boundary flow. First, the characterization of the flow field from the towing strut is required. This evaluation provides the residual mean velocities and turbulence levels caused by the towing hardware at each speed and axial location. These values, called tare values, are necessary for comparing to the cylinder flow results. Second, the cylinder flow fields are averaged together and the averaged tare fields are subtracted out to remove strut-induced ambient flow effects. Prior to averaging, the cylinder flow fields are shifted to collocate the cylinder within the field. Since the boundary layer develops slowly, all planes of data occurring within each 10 meter increment of the cylinder length are averaged together to produce the mean boundary layer flow. Corresponding fields from multiple runs executed using the same experimental parameters are also averaged. This flow is analyzed to evaluate the level of axisymmetry in the data and determine if small changes in cylinder angle affect the mean flow development. With axisymmetry verified, the boundary flow is further averaged azimuthally around the cylinder to produce mean boundary layer profiles. Finally, the fluctuating velocity levels are evaluated for the flow with the cylinder and compared to the fluctuating velocity levels in the tare data. This paper will first discuss the data analysis techniques for the tare data and the averaging methods implemented. Second, the data analysis considerations will be presented for the cylinder data and the averaging and cylinder tracking techniques. These results are used to extract relevant boundary layer parameters including δ, δ* and θ. Combining these results with wall shear and momentum thickness values extracted from averaged cylinder drag data, the boundary layer can be well characterized.

The structure of a highly decelerated axisymmetric turbulent boundary layer

2021 ◽  
Vol 929 ◽  
Author(s):  
N. Agastya Balantrapu ◽  
Christopher Hickling ◽  
W. Nathan Alexander ◽  
William Devenport
Keyword(s):  
Boundary Layer ◽  
Pressure Gradient ◽  
Shear Layer ◽  
Layer Thickness ◽  
Boundary Layer Thickness ◽  
Large Scale ◽  
Convection Velocity ◽  
Mean Flow ◽  
Mean Velocity ◽  
The Mean

Experiments were performed over a body of revolution at a length-based Reynolds number of 1.9 million. While the lateral curvature parameters are moderate ( $\delta /r_s < 2, r_s^+>500$ , where $\delta$ is the boundary layer thickness and r s is the radius of curvature), the pressure gradient is increasingly adverse ( $\beta _{C} \in [5 \text {--} 18]$ where $\beta_{C}$ is Clauser’s pressure gradient parameter), representative of vehicle-relevant conditions. The mean flow in the outer regions of this fully attached boundary layer displays some properties of a free-shear layer, with the mean-velocity and turbulence intensity profiles attaining self-similarity with the ‘embedded shear layer’ scaling (Schatzman & Thomas, J. Fluid Mech., vol. 815, 2017, pp. 592–642). Spectral analysis of the streamwise turbulence revealed that, as the mean flow decelerates, the large-scale motions energize across the boundary layer, growing proportionally with the boundary layer thickness. When scaled with the shear layer parameters, the distribution of the energy in the low-frequency region is approximately self-similar, emphasizing the role of the embedded shear layer in the large-scale motions. The correlation structure of the boundary layer is discussed at length to supply information towards the development of turbulence and aeroacoustic models. One major finding is that the estimation of integral turbulence length scales from single-point measurements, via Taylor's hypothesis, requires significant corrections to the convection velocity in the inner 50 % of the boundary layer. The apparent convection velocity (estimated from the ratio of integral length scale to the time scale), is approximately 40 % greater than the local mean velocity, suggesting the turbulence is convected much faster than previously thought. Closer to the wall even higher corrections are required.

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