Nonlinear instability of a supersonic boundary layer with two-dimensional roughness

2014 ◽  
Vol 752 ◽  
pp. 497-520 ◽  
Author(s):  
Olaf Marxen ◽  
Gianluca Iaccarino ◽  
Eric S. G. Shaqfeh
Keyword(s):  
Boundary Layer ◽  
Large Amplitude ◽  
Supersonic Boundary Layer ◽  
Secondary Instability ◽  
Secondary Wave ◽  
Nonlinear Instability ◽  
Transient Growth ◽  
Primary Wave ◽  
Two Dimensional ◽  
Lower Amplitude

AbstractNonlinear instability in a supersonic boundary layer at Mach 4.8 with two-dimensional roughness is investigated by means of spatial direct numerical simulations (DNS). It was previously found that an important effect of a two-dimensional roughness is to increase significantly the amplitude of two-dimensional waves downstream of the roughness in a certain frequency band through enhanced instability and transient growth, while waves outside this band are damped. Here, we investigate the nonlinear secondary instability induced by a large-amplitude two-dimensional wave, which has received a significant boost in amplitude from this additional roughness-induced amplification. Both subharmonic and fundamental secondary excitation of the oblique secondary waves are considered. We found that even though the growth rate of the secondary perturbations increases compared to their linear amplification, only in some of the cases was a fully resonant state attained by the streamwise end of the domain. A parametric investigation of the amplitude of the primary wave, the phase difference between the primary and the secondary waves, and the spanwise wavenumber has also been performed. The transient growth experienced by the primary wave was found to not influence the secondary instability for most parameter combinations. For unfavourable phase relations between the primary and the secondary waves, the phase speed of the secondary wave decreases significantly, and this hampers its growth. Finally, we also investigated the strongly nonlinear stage, for which both the primary and the subharmonic secondary waves had a comparable, finite amplitude. In this case, the growth of the primary waves was found to vanish downstream of the transient growth region, resulting in a lower amplitude than in the absence of the large-amplitude secondary wave. This feedback also decreases the amplification rate of the secondary wave.

Direct numerical simulations of hypersonic boundary-layer transition with finite-rate chemistry

2014 ◽  
Vol 755 ◽  
pp. 35-49 ◽  
Author(s):  
Olaf Marxen ◽  
Gianluca Iaccarino ◽  
Thierry E. Magin
Keyword(s):  
Large Amplitude ◽  
High Speed ◽  
Nonlinear Effects ◽  
Linear Instability ◽  
Boundary Layer Transition ◽  
Nonlinear Instability ◽  
Primary Wave ◽  
Two Dimensional ◽  
Finite Rate ◽  
Layer Transition

AbstractThe paper describes a numerical investigation of linear and nonlinear instability in high-speed boundary layers. Both a frozen gas and a finite-rate chemically reacting gas are considered. The weakly nonlinear instability in the presence of a large-amplitude two-dimensional wave is investigated for the case of fundamental resonance. Depending on the amplitude of this two-dimensional primary wave, strong growth of oblique secondary perturbations occurs for favourable relative phase differences between the two. For essentially the same primary amplitude, secondary amplification is almost identical for a reacting and a frozen gas. Therefore, chemical reactions do not directly affect the growth of secondary perturbations, but only indirectly through the change of linear instability and hence amplitude of the primary wave. When the secondary disturbances reach a sufficiently large amplitude, strongly nonlinear effects stabilize both primary and secondary perturbations.

Nonlinear Interaction of High-Intensity Disturbances into the Supersonic Boundary Layer

2012 ◽  
Vol 7 (1) ◽  
pp. 38-52
Author(s):  
Natalya Terekhova
Keyword(s):  
Boundary Layer ◽  
Nonlinear Model ◽  
Nonlinear Interaction ◽  
High Intensity ◽  
Plane Waves ◽  
Nonlinear Process ◽  
Supersonic Boundary Layer ◽  
Two Dimensional ◽  
Longitudinal Dynamics ◽  
Second Order Nonlinearity

A nonlinear model of interaction of disturbances in the regime of coupled combinatorial relations is used to explain the dynamics of unstable waves. The model includes effects of self-action and combinatorial interaction of unstable waves. Considered effects in the boundary layer with M = 2 controlled disturbance large enough intensity. In the second case when M = 5,35 examines the interrelationship of two-dimensional perturbations of various nature – vortex and acoustic. Shows the direction of impact of the different components of the nonlinear process. Found that this model of the second order nonlinearity can accurately describe the features of longitudinal dynamics of plane waves

Nonlinear evolution and breakdown in unstable boundary layers

1980 ◽  
Vol 99 (2) ◽  
pp. 247-265 ◽  
Author(s):  
A. D. D. Craik
Keyword(s):  
Boundary Layer ◽  
Laminar Flow ◽  
Flat Plate ◽  
Boundary Layers ◽  
Nonlinear Evolution ◽  
Secondary Instability ◽  
Nonlinear Instability ◽  
Weakly Nonlinear ◽  
Theory And Experiment

The nonlinear evolution and breakdown of laminar flow in the boundary layer on a flat plate is examined with the aim of making a closer comparison of theory and experiment than has been attempted previously. The importance of three-dimensionality is emphasized. It is concluded that many features of the nonlinear instability are consistent with existing linear and weakly nonlinear theories even as breakdown is approached. The development of the secondary instability, or ‘spike’, is also considered and suggestions for an improved theory of its growth are made.

Oblique-mode breakdown and secondary instability in supersonic boundary layers

1994 ◽  
Vol 273 ◽  
pp. 323-360 ◽  
Author(s):  
Chau-Lyan Chang ◽  
Mujeeb R. Malik
Keyword(s):  
Boundary Layer ◽  
Rapid Growth ◽  
Plate Boundary ◽  
Streamwise Vortex ◽  
Supersonic Boundary Layer ◽  
Secondary Instability ◽  
Single Frequency ◽  
Small Scale ◽  
Instability Mechanism ◽  
Oblique Waves

Laminar–turbulent transition mechanisms for a supersonic boundary layer are examined by numerically solving the governing partial differential equations. It is shown that the dominant mechanism for transition at low supersonic Mach numbers is associated with the breakdown of oblique first-mode waves. The first stage in this breakdown process involves nonlinear interaction of a pair of oblique waves with equal but opposite angles resulting in the evolution of a streamwise vortex. This stage can be described by a wave–vortex triad consisting of the oblique waves and a streamwise vortex whereby the oblique waves grow linearly while nonlinear forcing results in the rapid growth of the vortex mode. In the second stage, the mutual and self-interaction of the streamwise vortex and the oblique modes results in the rapid growth of other harmonic waves and transition soon follows. Our calculations are carried all the way into the transition region which is characterized by rapid spectrum broadening, localized (unsteady) flow separation and the emergence of small-scale streamwise structures. The r.m.s. amplitude of the streamwise velocity component is found to be on the order of 4–5 % at the transition onset location marked by the rise in mean wall shear. When the boundary-layer flow is initially forced with multiple (frequency) oblique modes, transition occurs earlier than for a single (frequency) pair of oblique modes. Depending upon the disturbance frequencies, the oblique mode breakdown can occur for very low initial disturbance amplitudes (on the order of 0.001% or even lower) near the lower branch. In contrast, the subharmonic secondary instability mechanism for a two-dimensional primary disturbance requires an initial amplitude on the order of about 0.5% for the primary wave. An in-depth discussion of the oblique-mode breakdown as well as the secondary instability mechanism (both subharmonic and fundamental) is given for a Mach 1.6 flat-plate boundary layer.

Secondary instabilities in the wake of an elongated two-dimensional body with a blunt trailing edge

2018 ◽  
Vol 846 ◽  
pp. 578-604 ◽  
Author(s):  
B. Gibeau ◽  
C. R. Koch ◽  
S. Ghaemi
Keyword(s):  
Boundary Layer ◽  
High Speed ◽  
Energy Content ◽  
Streamwise Vortex ◽  
Secondary Instability ◽  
Two Dimensional ◽  
Cylinder Wake ◽  
Trailing Edge ◽  
Piv Measurements ◽  
Blunt Trailing Edge

The secondary instability in the wake of a two-dimensional blunt body with a chord to thickness ratio of 46.5 was experimentally investigated for Reynolds numbers of 3500, 5200 and 7000 based on the blunt trailing edge height $h$. Planar, stereoscopic and high-speed particle image velocimetry (PIV) measurements were performed to characterise the wake and upstream boundary layer. The same mode B secondary instability that is found in the cylinder wake was found to be present in the wake of the elongated body studied here. The most probable wavelength of the secondary instability, defined as the spanwise distance between adjacent streamwise vortex pairs in the wake, was found to range from $0.7h$ to $0.8h$ by applying a spatial autocorrelation to the spanwise–wall-normal instantaneous fields of the $Q$-criterion. The temporal evolution of the secondary wake vortices was investigated using time-resolved stereoscopic PIV measurements and it was shown that the vortices maintain both their directions of rotation and spanwise positions during the primary vortex shedding cycles. In agreement with previous literature, the secondary instability did not greatly change as the upstream boundary layer transitioned from laminar to turbulent. Moreover, any upstream boundary layer structures were found to rapidly evolve into wake structures just past the blunt trailing edge. The wavelength of the secondary instability was shown to match the spanwise distance between adjacent low-speed zones of streamwise velocity in the wake. These undulating velocity patterns proved to be a viable method for determining the secondary instability wavelength; however, this type of analysis is highly sensitive to the energy content used for data reconstruction when proper orthogonal decomposition is applied beforehand.

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