The oscillatory channel flow with large wall injection

Abstract

In the presence of small-amplitude pressure oscillations, the linearized Navier{Stokes equations are solved to obtain an accurate description of the time-dependent eld in a channel having a rectangular cross-section and two equally permeable walls. The mean solution is based on Taylor's classic pro le, while the temporal solution is synthesized from irrotational and rotational elds. Using standard perturbation tools, the rotational component of the solution is derived from the linearized vorticity transport equation. In the absence of an exact solution to rely on, asymptotic formulations are compared with numerical simulations. In essence, the analytical formulation reveals rich vortical structures and discloses the main link between pressure oscillations and rotational wave formation. In the process, the explicit roles of variable injection, viscosity and oscillation frequency are examined. Using an alternative methodology, both WKB and multiple-scale techniques are applied to the linearized momentum equation. The momentum equation is of the boundary-value type and contains two small perturbation parameters. The primary and secondary parameters are, respectively, the reciprocals of the kinetic Reynolds and Strouhal numbers. The multiple-scale procedure employs two ctitious scales in space: a base and an undetermined scale. The latter is left unspeci ed during the derivation process until ®ow parameters are obtained in general form. Physical arguments are later used to de ne the arbitrary scale, which could not have been conjectured a priori. The emerging multiple-scale solution o¬ers several advantages. Its leading-order term is simpler and more accurate than other formulations. Most of all, it clearly displays the relationship between the physical parameters that control the nal motion. It thus provides the necessary means to quantify important ®ow features. These include the corresponding vortical wave amplitude, rotational depth of penetration, near-wall velocity overshoot and surfaces of constant phase. In particular, it discloses a viscous parameter that has a strong in®uence on the depth of penetration, and furnishes a closed-form expression for the maximum penetration depth in any oscillation mode. These ndings enable us to quantify the location of the shear layer and corresponding penetration depth. By way of theoretical veri cation, comparisons between asymptotic formulations and numeric predictions are reassuring. The most striking result is, perhaps, the satisfactory agreement found between asymptotic predictions and data obtained, totally independently, from numerical simulations of the nonlinear Navier{Stokes equations. In closing, a standard error analysis is used to con rm c

Channel flow induced by wall injection of fluid and particles

“…Currently, the problem is to couple instabilities and acoustic. [6][7][8] Particles are introduced in the propellant because they increase its specific impulse on the one hand, and stabilize possible tangential pressure modes on the other hand. The influence of the particles on the stability of solid-propellant motors has been studied numerically by Dupays,9 but only by considering the vortex shedding resulting from the shear layer initiated by the angular shape of the grain near the nozzle.…”

Some rotational corrections to the acoustic energy equation in injection-driven enclosures

“…14 Analytical, [13][14][15][16][17][18][19][20][21][22] numerical, [23][24][25][26][27][28] and experimental studies [29][30][31][32] have jointly confirmed the essential contributions of pressure-and boundary-driven vortico-acoustic waves. These new findings have triggered a growing trend in the acoustic instability community to incorporate vortical effects in order to refine the irrotational representation of the system energy balance.…”

“…[18][19][20][21][22] For constant ␦ = ͱ / ͑a 0 R͒ = 1/2 M b 3/2 l / ͑m͒ and surface Mach number, S and decrease in more elongated motors. Their physical variation is illustrated in Fig.…”

On a new quasi-linearization method for systems of nonlinear boundary value problems

“…The problem of fluid injection through one side of a long vertical channel was further studied by, among others, Huang [8], who used a generalized Newton-Raphson method to solve the governing equations. An accurate asymptotic and multi-scale solution to the two-dimensional oscillatory flow in a channel with porous was obtained by Majdalani and Roh [9].…”