Effects of Passive Porous Walls on Boundary-Layer Instability

Abstract

A theoretical study is described of the effects of a passive porous wall on boundary-layer instability. The passive porous wall is conceived of as a thin porous sheet stretched over a plenum chamber. When disturbances in the form of Tollmien-Schlichting waves propagate along the boundary layer, the uctuating pressure forces air in and out of the plenum chamber. The basic approach is based on classic linear stability theory for the at-plate boundary layer with modi ed wall boundary conditions. The wall response is represented by a complex admittance (ratio of uctuating ow rate per unit area to wall pressure) and, therefore, applies to a general class of passive porous, and other interactive, walls. The effects of a wide range of admittance values are studied. A stabilizing effect is obtained when the admittance phase is close to ¼/2, and an optimum value of admittance magnitude is also found. A theoretical model is introduced and used to calculate the admittance of the porous panels used in a parallel experimental study. It is shown that it may be possible to manufacture stabilizing porous panels. The stabilizing mechanism is due to the production of a near-wall region of negative Reynolds stress. Nomenclatureboundary-layer displacement thickness = dimensionless disturbance vorticitý = dimensionless compliant wall displacement ¹ = dynamic viscosity º = kinematic viscosity ½ = density ¾ = wall porosity, that is, proportion of area occupied by pores ' = admittance phase ! = dimensionless disturbance frequency Subscripts c = quantities evaluated in the plenum chamber i = imaginary part r = real part w = quantities evaluated at the wall ¾ D 1 = quantities evaluated with ¾ D 1 Superscripts O = disturbance amplitude N = average over a disturbance cycle

Stability and admittance of a channel flow over a permeable interface

“…A comparison between the data of Ref. 1 and the values calculated here shows a good agreement as far as Y y is concerned whereas Y x is not considered in that paper. However, the way the problem presented in Ref.…”

“…A number of papers on this particular subject appeared in the last decade and we just recall a few of them. [1][2][3] These last considerations take us to the second case of a permeable wall which is made of a porous slab. In a rather old article, 4 Beavers and Joseph first treated the flow in a channel over and through a porous matrix.…”

Numerical modelling and passive flow control using porous media

“…An other form of such boundary conditions is proposed in [15] and a mathematical analysis of a similar conditions is performed in [22]. In some cases it is necessary to compute the flow inside the porous domain and so this approach is not satisfactory.…”

Numerical Study of Mach 6 Boundary-Layer Stabilization by Means of a Porous Surface

“…The equations to be solved are the compressible Navier-Stokes equations for flow of a perfect gas with density , velocity components u i , pressure p, and internal energy e written in conservationlaw form as @ @t @u j @x j 0 (1) @u i @t @u i u j @x j @p @x i @ ij @x j f i (2) @E @t @E pu i @x i @q i @x i @u i ij @x j g (3) where E e u i u i =2. Forcing terms f i and g are included in the right-hand side of the equations such that a specified parallel base flow y, u i y, and Ey is time-independent.…”

“…These methods are complex and there is interest in technologically simpler methods, such as the use of a passive porous wall. The modeling of such surfaces for incompressible flow was discussed by Carpenter and Porter [3]. If the technique can be extended to the hypersonic flow regime, there would be a considerable return in terms of the vehicle weight savings.…”