Blow-up and control of marginally separated boundary layers

Author(s):  
Stefan Braun ◽  
Alfred Kluwick

Interactive solutions for steady two-dimensional laminar marginally separated boundary layers are known to exist up to a critical value Γ c of the controlling parameter (e.g. the angle of attack of a slender airfoil) Γ only. Here, we investigate three-dimensional unsteady perturbations of such boundary layers, assuming that the basic flow is almost critical, i.e. in the limit Γ c − Γ →0. It is then shown that the interactive equations governing such perturbations simplify significantly, allowing, among others, a systematic study of the blow-up phenomenon observed in earlier investigations and the optimization of devices used in boundary‐layer control.

2013 ◽  
Vol 432 ◽  
pp. 168-172
Author(s):  
Y. Zhou ◽  
Y.H. Fang

In this paper, the coupling method of PSE and FLUENT was experimented for predicting the laminar-turbulent transition. The software FLUENT was used to get the basic flow over a flat plate. A two-dimensional T-S wave and a pair of three-dimensional T-S waves were fed in at the entrance. The transition criterion was verified by DNS results. The availability of the coupling methodology has been evaluated.


1987 ◽  
Vol 109 (3) ◽  
pp. 388-397 ◽  
Author(s):  
A. J. Wennerstrom

Between 1970 and 1974, ten variants of a supersonic axial compressor stage were designed and tested. These included two rotor configurations, three rotor tip clearances, addition of boundary-layer control consisting of vortex generators on both the outer casing and the rotor, and the introduction of slots in the stator vanes. Design performance objectives were a stage total pressure ratio of 3.0 with an isentropic efficiency of 0.82 at a tip speed of 1600 ft/s (488 m/s). The first configuration passed only 70 percent of design flow at design speed, achieving a stage pressure ratio of 2.25 at a peak stage isentropic efficiency of 0.61. The rotor was grossly separated. The tenth variant passed 91.4 percent of design flow at design speed, producing a stage pressure ratio of 3.03 with an isentropic efficiency of 0.75. The rotor achieved a pressure ratio of 3.59 at an efficiency of 0.87 under the same conditions. Major conclusions were that design tools available today would undoubtedly permit the original goals to be met or exceeded. However, the application for such a design is currently questionable because efficiency goals considered acceptable for most current programs have risen considerably from the level considered acceptable at the inception of this effort. Splitter vanes placed in the rotor permitted very high diffusion levels to be achieved without stalling. However, viscous effects causing three-dimensional flows violating the assumption of flow confined to concentric stream tubes were so strong that a geometry optimization does not appear practical without a three-dimensional, viscous analysis. Passive boundary-layer control in the form of vortex generators and slots does appear to offer some benefit under certain circumstances.


Kelvin showed that a two-dimensional vortex under a two-dimensional disturbance in incompressible flow responds at a discrete set of eigenvalues, which were found by Broadbent & Moore ( Phil. Trans. R. Soc. Lond. A 290, 353-371 (1979) to become unstable in a compressible fluid. It is now shown that three-dimensional perturbations are also unstable provided the wavelength is greater than some critical value that depends on the Mach number of the vortex. A critical boundary dividing stable from unstable modes is defined. Most of the results relate to a Rankine vortex, as in the previous work mentioned above, but some results are also given for a vortex with a different velocity profile within the core; qualitatively the same kind of behaviour is found.


Author(s):  
Jack Schaeffer

SynopsisThe equation utt − Δu = |u|p is considered in two and three space dimensions. Smooth Cauchy data of compact support are given at t = 0. For the case of three space dimensions, John has shown that solutions with sufficiently small data exist globally in time if but that small data solutions blow up in finite time if Glassey has shown the two dimensional case is similar. This paper shows that small data solutions blow up in finite time when p is the critical value, in three dimensions and in two.


The present theoretical article considers the nonlinear interaction of oblique three dimensional Tollmien-Schlichting waves and induced or input longitudinal vortex motion, mainly for channel flow at large Reynolds numbers. Both the waves and the vortices are controlled by viscous-inviscid balancing but their respective flow structures are rather different because of the different typical timescales involved. This leads to the vortex-wave interaction being governed by nonlinear evolution equations on the vortex timescale, even though the wave amplitudes are notably small. The analogue in boundary-layer transition, addressed in a previous paper, is also re-considered here. Computational and analytical properties of the interaction equations for both channel flows and boundary layers are investigated, along with certain connections with companion studies of other vortex-wave interactions in channel flow. The nonlinear interactions in channel flow are found to lead to finitetime blow-up in amplitudes or to sustained vortex flow at large scaled times, depending on the input conditions. In particular, increasing the input amplitudes of the vortex or the wave can readily provoke blow-up even in the linearly stable regime; whereas in the case of sustained vortex flow new physical effects come into play on slightly longer timescales. Again, a very interesting feature is that the blowup response is found to be confined to a small range of wave angles near 45° relative to the original flow direction.


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