On the Nonlinear Development of Görtler Vortices in a Compressible Boundary Layer

Wadey, P. D.

doi:10.1002/sapm1991854317

Cited by 5 publications

(13 citation statements)

References 15 publications

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“…In fact an increase in Mach number increases the maximum growth rate of the thermal modes, but only very slightly. In the main however, the variation of the Mach number is seen to have a similar effect to that in most other compressible flows, that is an increase in Mach number tends to produce a stabilizing influence, Wadey (1992). Another result we note concerning the variation of the free-stream temperature ratio, β t , is that for the concave case (χ = 1) the growth rate is of order β −1 t as the upper stream decreases in temperature (which is significant).…”

Section: -1mentioning

confidence: 69%

“…The whole procedure is then repeated, stepping downstream. This means that the scheme is fully implicit unlike that used by Wadey (1992), which relieves any limitation on the streamwise step lengths imposed due to numerical stability requirements.…”

Section: Methodsmentioning

confidence: 99%

“…as the modes progress downstream, Otto et al (1996). Another energy measure we could have used was that employed by Hall (1983) and Wadey (1992). The integrand would then beũ 2 +ṽ 2 +w 2 rather thanũ 2 .…”

Section: Methodsmentioning

confidence: 99%

“…The first two conditions, namely (11) and (12), can be used to determine the Görtler number G 0 and the mode locationη. These conditions have been used in Wadey (1991) and Owen et al (1998) to predict the onset of nonlinear vortex structures. In the asymptote (13) we take m = 0, which corresponds to the first mode.…”

Section: Viscous Görtler Modesmentioning

confidence: 99%

“…The asymptotic results of Hall (1982) were plotted against the right-hand branch of the neutral curves, and good agreement was obtained. The corresponding compressible investigation was carried out by Wadey (1992), wherein the effect of wall cooling was investigated. It was seen that the right-hand branch of the neutral curve moved to the left as the wall was cooled; see Elliott & Bassom (2000) for a full account of the rôle of wall cooling in determining the structure of Görtler vortices.…”

Section: Introductionmentioning

confidence: 99%

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Görtler vortices in compressible mixing layers

Sarkies

Otto

2001

J. Fluid Mech.

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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.

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Section: -1mentioning

confidence: 69%

Section: Methodsmentioning

confidence: 99%

Section: Methodsmentioning

confidence: 99%

Section: Viscous Görtler Modesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Görtler vortices in compressible mixing layers

Sarkies

Otto

2001

J. Fluid Mech.

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Investigation of Görtler vortices in high-speed boundary layers via an efficient numerical solution to the non-linear boundary region equations

Es-Sahli

Sescu

Afsar

et al. 2021

Theor. Comput. Fluid Dyn.

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Streamwise vortices and the associated streaks evolve in boundary layers over flat or concave surfaces due to disturbances initiated upstream or triggered by the wall surface. Following the transient growth phase, the fully-developed vortex structures become susceptible to inviscid secondary instabilities resulting in early transition to turbulence via 'bursting' processes. In high-speed boundary layers, more complications arise due to compressibility and thermal effects, which become more significant for higher Mach numbers. In this paper, we study Görtler vortices developing in high-speed boundary layers using the boundary region equations (BRE) formalism, which we solve using an efficient numerical algorithm. Streaks are excited using a small transpiration velocity at the wall. Our BRE-based algorithm is found to be superior to direct numerical simulation (DNS) and ad-hoc nonlinear parabolized stability equation (PSE) models. BRE solutions are less computationally costly than a full DNS and have a more rigorous theoretical foundation than PSE-based models. For example, the full development of a Görtler vortex system in high-speed boundary layers can be predicted in a matter of minutes using a single processor via the BRE approach. This substantial reduction in calculation time is one of the major achievements of this work. We show, among other things, that it allows investigation

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The mysterious grooves of Volcán Bárcena: a review of the role of streamwise counter-rotating vortices during erosion by dilute pyroclastic density currents

et al. 2021

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Although erosion during high-energy passage of a pyroclastic density current (PDC) causes great damage, analyses of the effects of such erosion are sparse in scientific literature compared to observations and interpretations of depositional processes. In this paper, we review observations of surfaces where PDCs have eroded sets of grooves that provide information on the erosion process. We postulate that in some cases, the grooves were carved by streamwise vortices in the boundary layer of the PDC and review possible fluid dynamic instabilities that can give rise to such vortices. For the prominent grooves at Volcán Bárcena, Mexico, we propose that a fluid dynamic instability, which we dub the "groovy instability," occurred and caused formation of erosive counter-rotating vortices. This instability occurs when the particle concentration boundary layer thickness, δ c , is larger than the velocity (shear) boundary layer thickness, δ u , i.e., L=δ c /δ u >1. In subaqueous turbidity currents, these vortices have a typical wavelength of~25*δ c . If this relation is applied to the grooves formed on Volcán Bárcena, the inferred particle concentration boundary layer is estimated to have been <1 m thick. We postulate that a transition between erosion of grooves and deposition of dunes at Volcán Bárcena occurred when hydraulically supercritical flow on the upper flanks changed to subcritical flow about halfway down the mountain. We call attention to boundary layer dynamics in erosive pyroclastic density currents at a dimension that is difficult to scale quantitatively in laboratory experiments and is usually not resolved computationally and to the need for incorporating such dynamics into models of PDC dynamics.

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On the Nonlinear Development of Görtler Vortices in a Compressible Boundary Layer

Cited by 5 publications

References 15 publications

Görtler vortices in compressible mixing layers

Görtler vortices in compressible mixing layers

Investigation of Görtler vortices in high-speed boundary layers via an efficient numerical solution to the non-linear boundary region equations

The mysterious grooves of Volcán Bárcena: a review of the role of streamwise counter-rotating vortices during erosion by dilute pyroclastic density currents

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