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Unsteadiness and convective instabilities in two-dimensional flow over a backward-facing step
- Lambros Kaiktsis, George Em Karniadakis, Steven A. Orszag
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- Journal:
- Journal of Fluid Mechanics / Volume 321 / 25 August 1996
- Published online by Cambridge University Press:
- 26 April 2006, pp. 157-187
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A systematic study of the stability of the two-dimensional flow over a backward-facing step with a nominal expansion ratio of 2 is presented up to Reynolds number Re = 2500 using direct numerical simulation as well as local and global stability analysis. Three different spectral element computer codes are used for the simulations. The stability analysis is performed both locally (at a number of streamwise locations) and globally (on the entire field) by computing the leading eigenvalues of a base flow state. The distinction is made between convectively and absolutely unstable mean flow. In two dimensions, it is shown that all the asymptotic flow states up to Re = 2500 are time-independent in the absence of any external excitation, whereas the flow is convectively unstable, in a large portion of the flow domain, for Reynolds numbers in the range 700 [les ] Re [les ] 2500. Consequently, upstream generated small disturbances propagate downstream at exponentially amplified amplitude with a space-dependent speed. For small excitation disturbances, the amplitude of the resulting waveform is proportional to the disturbance amplitude. However, selective sustained external excitation (even at small amplitudes) can alter the behaviour of the system and lead to time-dependent flow. Two different types of excitation are imposed at the inflow: (i) monochromatic waves with frequency chosen to be either close to or very far from the shear layer frequency; and (ii) random noise. It is found that for small-amplitude monochromatic excitation the flow acquires a time-periodic behaviour if perturbed close to the shear layer frequency, whereas the flow remains unaffected for high values of the excitation frequency. On the other hand, for the random noise as input, an unsteady behaviour is obtained with a fundamental frequency close to the shear layer frequency.
Onset of three-dimensionality, equilibria, and early transition in flow over a backward-facing step
- Lambros Kaiktsis, George Em Karniadakis, Steven A. Orszag
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- Journal:
- Journal of Fluid Mechanics / Volume 231 / October 1991
- Published online by Cambridge University Press:
- 26 April 2006, pp. 501-528
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A numerical study of three-dimensional equilibria and transition to turbulence in flow over a backward-facing step is performed using direct numerical solution of the incompressible Navier-Stokes equations. The numerical method is a high-order-accurate mixed spectral/spectral-element method with efficient viscous outflow boundary conditions. The appearance of three-dimensionality in nominally two-dimensional geometries is investigated at representative Reynolds numbers ranging from the onset of three-dimensional bifurcation to later transitional stages. Strongly three-dimensional regions are identified through standard correlation coefficients and new three-dimensionality indices, as well as through instantaneous and time-average streamline patterns and vorticity contours. Our results indicate that onset of three-dimensionality occurs at the boundaries between the primary and secondary recirculating zones with the main channel flow, the latter being the most stable flow component. There is. therefore, strong secondary instability in the shear layers, mainly due to the one emanating from the step corner.
The flow further downstream is excited through the action of the upstream shear layers acquiring a wavy form closely resembling Tollmien–Schlichting waves both spatially and temporally with a characteristic frequency f1; upstream, at the shear layer another incommensurate frequency, f2, is present. The two-frequency flow locks-in to a single frequency if external excitations are imposed at the inflow at a frequency close to f1 or f2; the smaller amplitude excitations, however, may cause a strong quasi-periodic response. Such excitations may significantly increase or decrease (by more than 20%) the length of the primary separation zone XR at lock-in or quasi-periodic states. The equilibrium states resulting from the secondary instability at supercritical Reynolds numbers produce a flow modulated in the spanwise direction, with corresponding variations in the reattachment location XR. While three-dimensionality explains partially the discrepancy between numerical predictions and experimental results on XR at higher Reynolds number Re, the main source of discrepancy is attributed to the inflow conditions, and in particular to external disturbances superimposed on the mean flow, the latter being the main reason also for the somewhat earlier transition found in laboratory experiments.