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Stability And Transition In Shear Flows

The Growth of Gortler vortices in boundary layers on concave walls is investigated. It is shown that for vortices of wavelength comparable to the boundary-layer thickness the appropriate linear stability equations cannot be reduced to ordinary differential equations. The partial differential equations governing the linear stability of the flow are solved numerically, and neutral stability is defined by the condition that a dimensionless energy function associated with the flow should have a maximum or minimum when plotted as a function of the downstream variable X. The position of neutral stability is found to depend on how and where the boundary layer is perturbed, so that the concept of a unique neutral curve so familiar in hydrodynamic-stability theory is not tenable in the Gortler problem, except for asymptotically small wavelengths. The results obtained are compared with previous parallel-flow theories and the small-wavelength asymptotic results of Hall (1982a, b), which are found to be reasonably accurate even for moderate values of the wavelength. The parallel-flow theories of the growth of Gortler vortices are found to be irrelevant except for the small-wavelength limit. The main deficiency of the parallel-flow theories is shown to arise from the inability of any ordinary differential approximation to the full partial differential stability equations to describe adequately the decay of the vortex at the edge of the boundary layer. This deficiency becomes intensified as the wavelength of the vortices increases and is the cause of the wide spread of the neutral curves predicted by parallel-flow theories. It is found that for a wall of constant radius of curvature a given vortex imposed on the flow can grow for at most a finite range of values of X. This result is entirely consistent with, and is explicable by the asymptotic results of, Hall (1982a).

The nonlinear development of Gortler vortices in growing boundary layers

Flow structures in transitional and turbulent boundary layers

To understand transition to turbulence via 3D disturbance growth, we report here results obtained from the solution of Navier-Stokes equation (NSE) to reproduce experimental results obtained by minimizing background disturbances and imposing deterministic excitation inside the shear layer. A similar approach was adopted in Sengupta and Bhaumik [Phys. Rev. Lett. 107, 154501 (2011)], where a route of transition from receptivity to fully developed turbulent stage was explained for 2D flow in terms of the spatio-temporal wave-front (STWF). The STWF was identified as the unit process of 2D turbulence creation for low amplitude wall excitation. Theoretical prediction of STWF for boundary layer was established earlier in Sengupta, Rao, and Venkatasubbaiah [Phys. Rev. Lett. 96, 224504 (2006)] from the Orr-Sommerfeld equation as due to spatiotemporal instability. Here, the same unit process of the STWF during transition is shown to be present for 3D disturbance field from the solution of governing NSE.

http://spectral.iitk.ac.in/hpcl/LH13761E_25April2014.pdf

Precursor of transition to turbulence: spatiotemporal wave front.

This study investigates the stability and transition of Gortler vortices in a hypersonic boundary layer using linear stability theory and direct numerical simulations. In the simulations, Gortler vortices are separately excited by wall blowing and suction with spanwise wavelengths of 3, 6 and 9 mm. In addition to primary streaks with the same wavelength as the blowing and suction, secondary streaks with half the wavelength also emerge in the 6 and 9 mm cases. The streaks develop into mushroom structures before breaking down. The breakdown processes of the three cases are dominated by a sinuous-mode instability, a varicose-mode instability and a combination of the two, respectively. Both fundamental and subharmonic instabilities are relevant in all cases. Multiple modes are identified in the secondary-instability stage, some of which originate from the primary instabilities (first and second Mack modes). We demonstrate that the first Mack mode can be destabilized to either a varicose-mode or sinuous-mode streak instability depending on its frequency and wavelength, whereas the second Mack mode undergoes a stabilizing stage before turning into a varicose mode in the 6 and 9 mm cases. An energy analysis reveals the stabilizing and destabilizing mechanisms of the primary instabilities under the influence of Gortler vortices, highlighting the role played by the spanwise production based on the spanwise gradient of the streamwise velocity in both varicose and sinuous modes. The effects introduced by the secondary streaks are examined by filtering the secondary streaks in two new simulations with nominally identical conditions to those of the 6 and 9 mm cases. Remarkably, the secondary streaks can destabilize the Gortler vortices, therefore advancing the transition. The stability theory results are in good agreement with those from direct numerical simulations.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/5B987AABB27B0C91851BE0642103D88D/S0022112019000247a.pdf/div-class-title-hypersonic-boundary-layer-transition-on-a-concave-wall-stationary-gortler-vortices-div.pdf

Hypersonic boundary layer transition on a concave wall: stationary Görtler vortices

The present experimental study is devoted to examination of the vortex receptivity mechanism associated with excitation of unsteady cross-flow (CF) waves due to scattering of unsteady free-stream vortices on localized steady surface non-uniformities (roughness). The measurements are carried out in a low-turbulence wind tunnel by means of a hot-wire anemometer in a boundary layer developing over a swept-wing model. The harmonic-in-time free-stream vortices were excited by a thin vibrating wire located upstream of the experimental-model leading edge and represented a kind of small-amplitude von Karman vortex street with spanwise orientation of the generated instantaneous vorticity vectors. The controlled roughness elements (the so-called ‘phased roughness’) were placed on the model surface. This roughness had a special shape, which provided excitation of CF-waves having basically some predetermined (required) spanwise wavenumbers. The linearity of the stability and receptivity mechanisms under study was checked accurately by means of variation of both the free-stream-vortex amplitude and the surface roughness height. These experiments were directed to obtaining the amplitudes and phases of the vortex-roughness receptivity coefficients for a number of vortex disturbance frequencies. The vortex street position with respect to the model surface (the vortex offset parameter) was also varied. The receptivity characteristics obtained experimentally in Fourier space are independent of the particular roughness shape, and can be used for validation of receptivity theories.

Receptivity coefficients at excitation of cross-flow waves by free-stream vortices in the presence of surface roughness

This experimental work is devoted to a parametric study of characteristics of the laminar-turbulent transition occurred in a model of a crossflow-dominated swept-wing boundary layer. The experiments are performed in 39 different regimes with low and elevated freestream turbulence levels both in the absence and in the presence of steady freestream vortices of various spanwise scales. The measurements are performed at several values of freestream velocity for two types of distributed surface roughness. Transition scenarios and criteria of turbulence onset are investigated and the problem of transition prediction is examined. It is shown that in all studied cases, the turbulence onset starts with the appearance of local high-frequency secondary instability of the base flow perturbed by primary instability modes. It is found that the 30% level of the threshold zero-to-peak amplitude of combined boundary-layer disturbances (i.e., the steady ones plus unsteady ones) can be used in all studied cases as a simplif...

Swept-wing boundary-layer transition at various external perturbations: Scenarios, criteria, and problems of prediction

Influence of Surface Roughness and Free-stream Turbulence on Crossflow-instability Transition☆

The paper is devoted to experimental investigation of an efficient mechanism of distributed excitation of nonstationary Gortler 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 Gortler 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 Gortler 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 Gortler modes and estimated for the case of steady Gortler vortices by means of extrapolation to the zero frequency. A complementary experiment devoted to investigation of the problem of linear unsteady Gortler 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.

Systematic study of distributed excitation of unsteady Görtler modes by freestream vortices

Extensive combined experimental and theoretical investigations of the linear evolution of unsteady (in general) Cross-Flow (CF) and three-dimensional (3D) Tollmien-Schlichting (TS) instability modes of 3D boundary layers developing on a swept airfoil section have been carried out. CF-instability characteristics are investigated in detail at an angle of attack of −5° when this kind of instability dominates in the laminar-turbulent transition process, while the 3D TS-instability characteristics are studied at an angle of attack of +1.5° when this kind of instability is predominant in the transition process. All experimental results are deeply processed and compared with results of calculations based on several theoretical approaches. For the first time, very good quantitative agreement of all measured and calculated stability characteristics of swept-wing boundary layers is achieved both for unsteady CF- and 3D TS-instability modes for the case of a boundary layer developing on a real swept airfoil. The first part of the present study (this paper) is devoted to the description of the case of CF-dominated transition, while the TS-dominated case will be described in detail in a subsequent second part of this investigation.

/pdf/experimental-and-theoretical-study-of-swept-wing-boundary-1ipt2hj2om.pdf

V. I. Borodulin

Papers

Receptivity coefficients at excitation of cross-flow waves by free-stream vortices in the presence of surface roughness

Swept-wing boundary-layer transition at various external perturbations: Scenarios, criteria, and problems of prediction

Influence of Surface Roughness and Free-stream Turbulence on Crossflow-instability Transition☆

Systematic study of distributed excitation of unsteady Görtler modes by freestream vortices

Experimental and theoretical study of swept-wing boundary-layer instabilities. Unsteady crossflow instability