Numerical investigation of incompressible flow in grooved channels. Part 1. Stability and self-sustained oscillations

Abstract

Incompressible moderate-Reynolds-number flow in periodically grooved channels is investigated by direct numerical simulation using the spectral element method. For Reynolds numbers less than a critical value Rc the flow is found to approach a stable steady state, comprising an ‘outer’ channel flow, a shear layer at the groove lip, and a weak re-circulating vortex in the groove proper. The linear stability of this flow is then analysed, and it is found that the least stable modes closely resemble Tollmien–Schlichting channel waves, forced by Kelvin–Helmholtz shear-layer instability at the cavity edge. A theory for frequency prediction based on the Orr–Sommerfeld dispersion relation is presented, and verified by variation of the geometric parameters of the problem. The accuracy of the theory, and the fact that it predicts many qualitative features of low-speed groove experiments, suggests that the frequency-selection process in these flows is largely governed by the outer, more stable flow (here a channel), in contrast to most current theories based solely on shear-layer considerations. The instability of the linear mode for R > Rc is shown to result in self-sustained flow oscillations (at frequencies only slightly shifted from the originating linear modes), which again resemble (finite-amplitude) Tollmien-Schlichting modes driven by an unstable groove vortex sheet. Analysis of the amplitude dependence of the oscillations on degree of criticality reveals the transition to oscillatory flow to be a regular Hopf bifurcation.

Flow instability and wall shear stress variation in intracranial aneurysms

“…The Reynolds numbers in our study based on the flow rate and the kinematic viscosity are 877 and 1271 on average and at the systolic peak, respectively. They are in good agreement with the critical Reynolds number for the onset of instability which is close to Re ¼ 975 (Ghaddar et al 1986). …”

“…The reasons for the discrepancy seem to be the different flow conditions such as the much higher Reynolds number 1000-2000 of the flow in the aorta or the stenosed carotid. In the numerical study of flow over periodic grooves in a channel by Ghaddar et al (1986), the frequency of the selfsustained oscillation corresponding to the least unstable Tollmien -Schlichting mode was predicted from linear stability analysis. Our peak frequencies of patient C (around 24-58 Hz) seem to agree well with a frequency range (16 -24 Hz) found in their study; this frequency range corresponds to frequencies in grooves with non-dimensional separation distance L ¼ 5 and non-dimensional groove length l ¼ 2.5 with different depth (non-dimensionalized with the half-channel width).…”

“…Self-sustained oscillations including the cavity flow were reviewed in Rockwell & Naudascher (1979). Cavity flow and its stability were studied numerically by Ghaddar et al (1986). In aneurysms in cerebral arteries, owing to the effects of complicated geometry, we expect to find distinct regions of inflow and outflow at the mouths of wide-necked aneurysms.…”

An Overlapping Schwarz Method for Spectral Element Simulation of Three-Dimensional Incompressible Flows

“…Such a n a p p r o a c h w as suggested by M a d a y, P atera, and R nquist 24] and analyzed by P erot 4] and Couzy 7] it follows classical splitting approaches (e.g., 18,26]) that lead to a Poisson equation for the pressure except that, in the present case, the splitting is e ected in the discrete form of the equations. The correct boundary conditions are preserved and no steady-state temporal errors are introduced.…”

Stability analysis forn-periodic arrays of fluid systems

“…The ubiquity of fluid systems characterized by an n-periodic arrangement of identical units or by multi-periodic geometric features has spawned a great deal of analyses and simulations: flow in wavy or grooved channels [9,15,16,40] or past arrays of roughness elements and vortex generators [6], acoustics in periodic wave-guides [1], energy extraction from an buoy array [12] and, of course, flow in turbomachines [8,13,14,20,23,24,30] and combustors [5,28,29,31,38,39,45] are but a few examples that fall under this category. Not surprisingly, specific analysis and simulation techniques that efficiently address this periodicity have been developed, in particular for turbomachinery applications, typically describing blade-to-blade dynamics, aeroelastic properties, and rotor-stator interactions.…”