TY - JOUR

T1 - Two asymptotic forms of the rotational solution for wave propagation inside viscous channels with transpiring walls

AU - Majdalani, J.

AU - Rienstra, S.W.

PY - 2002

Y1 - 2002

N2 - In this article, a long rectangular channel is considered with two transpiring walls that are a small distance apart. The channel's head end is hermetically closed while the aft end is either open (isobaric) or acoustically closed (choked). A mean flow enters uniformly across the permeable walls, turns, and exits from the downstream end. The slightest unsteadiness in flow velocity is inevitable and occurs at random frequencies. Small pressure disturbances are thus produced. Those waves with oscillations matching the natural frequencies of the enclosure are promoted. Inception of small pressure perturbations alters the flow character and leads to a temporal field that we wish to analyse. The mean flow is of the Berman type and can be obtained from the Navier–Stokes equations over different ranges of the cross-flow Reynolds number. The unsteady component can be formulated from the linearized momentum equation. This has been carried out in numerous studies and has routinely given rise to a singular, boundary-value, double-perturbation problem in the cross-flow direction. The current study focuses on the resulting second-order differential equation that prescribes the rotational wave motion in the transverse direction. This equation exhibits unique features that define a general class of ordinary differential equations. Due to the oscillatory behaviour of the problem, two general asymptotic formulations are derived, for an arbitrary mean-flow profile, using WKB and multiple-scales expansions. The fundamental asymptotic solutions reveal the same similarity parameter that controls the rotational wave character. The multiple-scales solution unravels the problem's characteristic length scale following a unique, nonlinear variable transformation. The latter is derived rigorously from the problem's solvability condition. The advantage of using a multiple-scales procedure lies in the ease of construction, accuracy, and added physical insight stemming from its leading-order term. For verification purposes, a specific mean-flow solution is used for which an exact solution can be derived. Comparisons between asymptotic and exact predictions are gratifying, showing an excellent agreement over a wide range of physical parameters.

AB - In this article, a long rectangular channel is considered with two transpiring walls that are a small distance apart. The channel's head end is hermetically closed while the aft end is either open (isobaric) or acoustically closed (choked). A mean flow enters uniformly across the permeable walls, turns, and exits from the downstream end. The slightest unsteadiness in flow velocity is inevitable and occurs at random frequencies. Small pressure disturbances are thus produced. Those waves with oscillations matching the natural frequencies of the enclosure are promoted. Inception of small pressure perturbations alters the flow character and leads to a temporal field that we wish to analyse. The mean flow is of the Berman type and can be obtained from the Navier–Stokes equations over different ranges of the cross-flow Reynolds number. The unsteady component can be formulated from the linearized momentum equation. This has been carried out in numerous studies and has routinely given rise to a singular, boundary-value, double-perturbation problem in the cross-flow direction. The current study focuses on the resulting second-order differential equation that prescribes the rotational wave motion in the transverse direction. This equation exhibits unique features that define a general class of ordinary differential equations. Due to the oscillatory behaviour of the problem, two general asymptotic formulations are derived, for an arbitrary mean-flow profile, using WKB and multiple-scales expansions. The fundamental asymptotic solutions reveal the same similarity parameter that controls the rotational wave character. The multiple-scales solution unravels the problem's characteristic length scale following a unique, nonlinear variable transformation. The latter is derived rigorously from the problem's solvability condition. The advantage of using a multiple-scales procedure lies in the ease of construction, accuracy, and added physical insight stemming from its leading-order term. For verification purposes, a specific mean-flow solution is used for which an exact solution can be derived. Comparisons between asymptotic and exact predictions are gratifying, showing an excellent agreement over a wide range of physical parameters.

U2 - 10.1093/qjmam/55.1.141

DO - 10.1093/qjmam/55.1.141

M3 - Article

VL - 55

SP - 141

EP - 162

JO - The Quarterly Journal of Mechanics and Applied Mathematics

JF - The Quarterly Journal of Mechanics and Applied Mathematics

SN - 0033-5614

IS - 1

ER -