The linear growth of Görtler vortices

doi:10.1016/S0142-727X(97)00022-2

International Journal of Heat and Fluid Flow

Volume 18, Issue 4, August 1997, Pages 389-399

https://doi.org/10.1016/S0142-727X(97)00022-2 Get rights and content

Abstract

Measurements are presented of the streamwise velocity variation within a laminar boundary layer on a concave surface of 4-m radius of curvature for free-stream velocities of 7.5 and 10 m/s. The measured variation was consistent with the presence of counter-rotating vortices resulting from the Görtler instability. Contour plots of velocity and turbulence intensity show that the vortices occur in discrete pairs centred about upwash locations, rather than in a continuous row of counter-rotating vortices modelled by the stability analysis. Comparison with the normal-mode linear stability analysis indicates that the experimental data lie in a region of the stability chart for which the two observed conditions of streamwise development at constant wavelength and constant growth rate coincide. Disturbance velocity profiles compare favourably at streamwise positions for which the linear stability analysis is appropriate. Detailed comparison with the linear stability analysis indicates that the measured growth rates are considerably lower than those obtained from theory. (This is attributed to the limitations of the normal-mode analysis as well as to the fact that the observed vortices occur in discrete pairs often separated by regions of relatively undisturbed flow.) Although measured growth rates obtained by considering individual vortices were found to be greater than those obtained using spanwise-averaged velocity profiles.

References (35)

J.M. Floryan
On the Görtler instability of boundary layers
Prog. Aerospace Sci.
(1991)
S.H. Winoto et al.
Vortex structure in laminar boundary layers on a concave wall
Int. J. Heat Fluid Flow
(1980)
D.H. Zhang et al.
Measurement in laminar and transitional boundary-layer flows on concave surfaces
Int. J. Heat Fluid Flow
(1995)
Y. Aihara
Görtler vortices in the nonlinear region
Y. Aihara et al.
Secondary instability of Görtler vortices: Formation of periodic three-dimensional coherent structure
Trans. Japan. Soc. Aero. Space Sci.
(1981)
H. Bippes
Experimentell untersüchung des laminar turbulenten umschlags an einer parallel angeströmten konkaven wand. Heidelberger Akademie der Wissenschaften, Matematischnaturwissenschaftliche klass, Sitzungsberichte, No. 3
H. Bippes et al.
Dreidimensionale strörungen in der grenzschicht an einer konkaven wand
Acta Mechanica
(1972)
H.P. Day et al.
Comparing local and marching analyses of Görtler instability
AIAA J.
(1990)
M.V. Finnis et al.
Stability of a laminar boundary layer flowing along a concave surface
J. Turbomech.
(1989)
J.M. Floryan et al.
Stability of Görtler vortices in boundary layers
AIAA J.
(1982)

A.E. Forest

Engineering predictions of transitional boundary layers

AGARD CP-224

(1977)

H. Görtler

Uber eine dreidimensionale instabilität laminarer grenzschichten an konkaven wänden

Ges. d. Wiss. Göttingen, Nachr. a. d. Math.

(1940)

M.A. Gregory et al.

The effect on transition of isolated surface excrescences in the boundary layer

ARC TR, Reports and Memoranda No. 2779

(1956)

P. Hall

Taylor-Görtler vortices in fully developed or boundary-layer flows: Linear theory

J. Fluid Mech.

(1982)

P. Hall

The linear development of Görtler vortices in growing boundary layers

J. Fluid Mech.

(1983)

P. Hall

Görtler vortices in growing boundary layers: The leading edge receptivity problem, linear growth and the nonlinear breakdown stage

Mathematika

(1990)

T. Herbert

On the stability of the boundary layer along a curved wall

Archives Mech.

(1976)

Cited by (16)

Systematic study of distributed excitation of unsteady Görtler modes by freestream vortices
2018, European Journal of Mechanics, B/Fluids
The paper is devoted to experimental investigation of an efficient mechanism of distributed excitation of nonstationary Görtler vortices in a boundary layer on concave wall due to scattering of three-dimensional, streamwise oriented freestream vortices on natural two-dimensional base-flow nonuniformity associated with the boundary layer growth. The investigations are performed in a broad range of disturbance frequencies and spanwise wavenumbers. The measurements have shown that the distributed receptivity mechanism is able to modify significantly the disturbance growth rates. This mechanism is found to be able to compete successfully with the linear instability mechanism. It is found that at relatively low frequencies and not too large freestream disturbance amplitudes, the Görtler modes’ development is dominated basically by the corresponding linear instability mechanism. Meanwhile, this mechanism becomes weaker with growth of frequency and, simultaneously, the distributed receptivity mechanism gets stronger. This leads to enhancement of role of the latter in amplification of Görtler instability modes, and at high frequencies the role of the distributed receptivity mechanism becomes very important and often even decisive one. Based on deep processing of the experimental data, quantitative values of the distributed linear receptivity coefficients are obtained for the case of excitation of unsteady Görtler modes and estimated for the case of steady Görtler vortices by means of extrapolation to the zero frequency. A complementary experiment devoted to investigation of the problem of linear unsteady Görtler instability is performed for the particular conditions of the distributed receptivity experiments. All main stability characteristics are obtained and compared with calculated ones. In general, the experimental information presented in this paper can be used for verification of various instability and vortex receptivitytheories.
Steady and unsteady Görtler boundary-layer instability on concave wall
2010, European Journal of Mechanics, B/Fluids
Citation Excerpt :
In experiments of Ref. [13,14] a set of thin plates located in free stream was used to generate steady Görtler vortices; some of the results obtained in Ref. [12] were reproduced and also additional points of the neutral stability curve were documented. However, the disturbance growth rates obtained experimentally did not agree with those calculated by the linear-stability theory in the whole range of spanwise wavenumbers and Görtler numbers studied [3,15]. One of the causers of the discrepancy was associated with nonlinear effects because the amplitudes of the streamwise velocity variations caused by the investigated vortices were quite large (often about 10% of free-stream speed, as in Ref. [12]).
The subject of the present combined experimental and theoretical investigation is the steady and unsteady linear Görtler instability. The majority of previous experiments were devoted to the steady Görtler vortices, despite the unsteady ones are also observed in real transitional flows. Moreover, even for the steady Görtler vortices no quantitative agreement between the experimental and theoretical linear-stability characteristics was obtained, especially for disturbance amplification rates. The experimental difficulties were connected, in particular, with a rather poor accuracy of measurements at zero disturbance frequency, a possible influence of nonlinearity, and an admixture of non-modal (transient) growth mechanism. All these difficulties have been overcome in the experimental part of the present study by means of: (i) tuning-out of the exact zero frequency of Görtler vortices and working, instead, with quasi-steady perturbations of very low frequencies, (ii) performing measurements at low disturbance amplitudes, and (iii) minimization and careful estimation of the disturbance-source near-field by means of utilizing a special controlled disturbance source and performing special numerical computations for exact experimental conditions. A detailed study of all linear-stability characteristics for essentially unsteady Görtler vortices was performed in this paper as well. The results are obtained in a range of Görtler numbers 13 ≲ Gö ≲ 17.3, frequency parameters F = 0.56–22.70, and spanwise wavelength parameters Λ = 149–775 (close to the most amplified Görtler modes).
Appropriate calculations based on locally-parallel and non-local non-parallel linear-stability theories were performed and compared quantitatively with experimentally obtained linear-stability characteristics. For the first time all stability characteristics measured for steady Görtler vortices (in quasi-steady regimes) are found to agree very well with those calculated for the most amplified first discrete-spectrum mode of the linear Görtler-instability problem. Similar good agreement is obtained for essentially unsteady Görtler vortices. The roles of effects of the base-flow non-parallelism and the disturbance-source near-field are examined.
Numerical simulation of Görtler vortices in hypersonic compression ramps
2005, Computers and Fluids
A numerical study of the formation and evolution of Görtler type vortices over a compression ramp under an hypersonic flow is presented. Numerical predictions in laminar compression ramps (body/body-flaps configurations) usually require the inclusion of three-dimensional effects to correlate separation lengths with experimental measurements. Moreover, in the presence of large recirculation regions, the heat transfer calculated over the compression ramp severely underestimate experimental values. This discrepancy could be explained at least in part by the presence and effects of centrifugal disturbances. Indeed the regular striation patterns in heat transfer rates seen in experiments suggest this. In the present work numerical simulations were carried out over three-dimensional 15° compression ramps in a laminar hypersonic flow at Mach 6.85 and a Reynolds number per unit length of 2.45 × 10⁶ m⁻¹. The relatively high Görtler numbers obtained suggested that Görtler vortices might be important. In most cases, a spanwise periodic velocity perturbation is imposed on the flat plate ahead of separation. However for one case, which had a grid with a higher spanwise resolution, a regular striation pattern emerged without forcing. In this case, the pattern had a wavelength comparable to that found in experiments and as predicted from linear theory. A spanwise variation of 5–15% in the heat transfer close to the centreline was obtained in this case.
Görtler vortices and streaks in boundary layer subject to pressure gradient: Excitation by free stream vortical disturbances, nonlinear evolution and secondary instability
2020, Journal of Fluid Mechanics
Compressible unsteady Görtler vortices subject to free-stream vortical disturbances
2019, Journal of Fluid Mechanics
Görtler vortices in low-Reynolds-number flow over multi-element airfoil
2018, Journal of Fluid Mechanics

View all citing articles on Scopus

View full text

PaperThe linear growth of Görtler vortices

Abstract

Prog. Aerospace Sci.

Int. J. Heat Fluid Flow

Int. J. Heat Fluid Flow

Görtler vortices in the nonlinear region

Secondary instability of Görtler vortices: Formation of periodic three-dimensional coherent structure

Trans. Japan. Soc. Aero. Space Sci.

Experimentell untersüchung des laminar turbulenten umschlags an einer parallel angeströmten konkaven wand. Heidelberger Akademie der Wissenschaften, Matematischnaturwissenschaftliche klass, Sitzungsberichte, No. 3

Dreidimensionale strörungen in der grenzschicht an einer konkaven wand

Acta Mechanica

Comparing local and marching analyses of Görtler instability

AIAA J.

Stability of a laminar boundary layer flowing along a concave surface

J. Turbomech.

Stability of Görtler vortices in boundary layers

AIAA J.

Engineering predictions of transitional boundary layers

AGARD CP-224

Uber eine dreidimensionale instabilität laminarer grenzschichten an konkaven wänden

Ges. d. Wiss. Göttingen, Nachr. a. d. Math.

The effect on transition of isolated surface excrescences in the boundary layer

ARC TR, Reports and Memoranda No. 2779

Taylor-Görtler vortices in fully developed or boundary-layer flows: Linear theory

J. Fluid Mech.

The linear development of Görtler vortices in growing boundary layers

J. Fluid Mech.

Görtler vortices in growing boundary layers: The leading edge receptivity problem, linear growth and the nonlinear breakdown stage

Mathematika

On the stability of the boundary layer along a curved wall

Archives Mech.

Paper
The linear growth of Görtler vortices