## Abstract

Linear three-dimensional receptivity of boundary layers to distributed wall vibrations in the large Reynolds number limit (*Re*→∞) is studied in this paper. The fluid motion is analysed by means of the multiscale asymptotic technique combined with the method of matched asymptotic expansions. The body surface is assumed to be perturbed by small-amplitude oscillations being tuned in resonance with the neutral Tollmien–Schlichting wave at a certain point on the wall. The characteristic length of the resonance region is found to be O(*Re*^{−3/16}), which follows from the condition that the boundary-layer non-parallelism and the wave amplitude growth have the same order of magnitude. The amplitude equation is derived as a solvability condition for the inhomogeneous boundary-value problem. Investigating detuning effects, we consider perturbations in the form of a wave packet with a narrow O(*Re*^{−3/16}) discrete or continuous spectrum concentrated near the resonant wavenumber and frequency. The boundary-layer laminarization based on neutralizing the oncoming Tollmien–Schlichting waves (or wave packets) is also discussed.

## 1. Introduction

The concept of receptivity proposed by Morkovin (1969) is concerned with the transformation of external disturbances into internal instability oscillations of the boundary layer. The latter may occur in different forms, depending on the magnitude and physical nature of primary perturbations. In this paper, it is assumed that the dominant response of the boundary layer is Tollmien–Schlichting waves. This situation can be met, for example, in the flow over a thin airfoil at a small angle of attack.

Early theoretical studies of Tollmien–Schlichting waves were based on the linear stability theory with a parallel mean flow approximation. Lin (1946) and Smith (1979) proved that in the large Reynolds number limit, lower branch instabilities have a triple-deck structure. Thus, the non-dimensional Tollmien–Schlichting wavelength has the same order of magnitude as the characteristic length of the interaction region, namely O(*Re*^{−3/8}), while the non-dimensional frequency is O(*Re*^{1/4}). The classical perturbation analysis leads to an eigenvalue problem which, when solved, results in a dispersion relation. The latter enables a determination of the frequency and wavenumber of the neutral Tollmien–Schlichting mode at a given location on the body surface. To excite this mode artificially, external disturbances must be tuned in resonance with respect to both frequency and wavenumber.

The first experiments on the generation of Tollmien–Schlichting waves by means of a vibrating ribbon were carried out by Shubauer & Skramsted (1948). In the theoretical analysis of Terent'ev (1981), the vibrator was modelled by a short flexible section of the body surface. Another possible way to excite Tollmien–Schlichting waves was demonstrated by Ruban (1984), Goldstein (1985) and Duck *et al*. (1996), who studied an interaction of free-stream disturbances with an isolated steady hump ‘inscribed’ into the viscous sublayer of a triple-deck region.

However, the fluid flow in boundary layers is not exactly parallel. As a consequence, a neutral Tollmien–Schlichting wave becomes unstable and grows in amplitude after passing the neutral point. To describe this crucial process, the region of interest should be extended in the streamwise direction beyond the standard triple-deck region. Thus, two different longitudinal length-scales arise in the problem formulation. The first of these coincides in order of magnitude with the Tollmien–Schlichting wavelength and serves to determine the ‘fast’ variable. The second length-scale is much longer; it is associated with relatively slow variations of the wave amplitude. The improved theory enables us to analyse the influence of the mean flow non-parallelism and/or nonlinearity to the amplitude growth, as did Smith (1979), Ruban (1983) and Hall & Smith (1984).

The boundary-layer non-parallelism plays a key role in the distributed receptivity process where the perturbation source acts over a distance exceeding O(*Re*^{−3/8}). In this case, the O(*Re*^{−3/16}) neighbourhood of the neutral point is referred to as the resonance region, since the resonance conditions are approximately satisfied only there. A comprehensive mechanism of distributed receptivity, based on the infinite Reynolds number approach, was proposed by Wu (2001), who investigated interaction of steady distributed wall roughness with either acoustic or vortical free-stream disturbances. In particular, it was found that distributed receptivity is much stronger than the localized one, since the Tollmien–Schlichting wave amplitude is accumulated in a much wider region.

This paper studies linear three-dimensional receptivity of boundary layers to distributed wall vibrations in the large Reynolds number limit. The oncoming flow is supposed to be ‘clean’, i.e. free of perturbations. The resonance with the Tollmien–Schlichting wave is achieved by an appropriate choice of the wall vibrations' frequency and streamwise and spanwise wavenumbers. It will be shown that for effective receptivity, a small detuning of order O(*Re*^{−3/16}) is allowed. This creates grounds for consideration of Tollmien–Schlichting packets rather than isolated waves.

## 2. Formulation of the problem

Let us consider the upper half of an incompressible fluid flow past a semi-infinite plate oriented parallel to the uniform free stream, as shown in figure 1. Suppose that the plate surface is perturbed by the small-amplitude vertical oscillations being specified later in this section. The density and kinematic viscosity of the fluid are denoted by *ρ* and *ν*, respectively, and the velocity and pressure of the oncoming flow by *U*_{∞} and *p*_{∞}. Furthermore, a distance from the leading edge of the plate to the resonance region is denoted by *l*. We shall suppose that the Reynolds numberis large, and define a small parameter *ϵ*=*Re*^{−1/8}.

Henceforth, the lengths, time , velocities and the pressure increment with respect to *p*_{∞} are non-dimensionalized by *l*, *l*/*U*_{∞}, *U*_{∞} and , respectively. Moreover, the streamwise and spanwise Tollmien–Schlichting wavenumbers, and , are non-dimensionalized by *l*^{−1}, and the wave frequency by *U*_{∞}/*l*. The flow analysis is carried out in the Cartesian coordinate system (, , ) with the origin at the leading edge, where is measured in the direction of the free stream, and , are measured in the normal and spanwise directions, respectively (see figure 1). The corresponding components of the velocity vector are denoted by (, , ).

The fluid motion is governed by the three-dimensional Navier–Stokes equations subjected to the no-slip condition on the body surface,(2.1)and the far field conditions(2.2)Suppose that an oblique neutral Tollmien–Schlichting wave arises in the vicinity of the point . The lower branch stability analysis provides asymptotic expansions for the streamwise and spanwise wavenumbers and frequency of that wave, namelyCorrespondingly, the fast variables *x*, *z* and *t* are defined byThe characteristic length of the resonance region is determined from the condition that the boundary-layer non-parallel effects and the Tollmien–Schlichting amplitude growth have the same order of magnitude (cf. Ruban 1983; Hall & Smith 1984). Based on this length-scale, we introduce the ‘slow’ variables *X*, *Z* and *T* by(2.3)Wu (2001) showed that in the longitudinal direction, the flow field is subdivided into three regions, namely the pre-resonance, resonance and post-resonance regions (see figures 1 and 2). Each of them has a triple-deck structure in the direction normal to the wall. The normal variable is scaled as in the main part of the boundary layer, in the viscous sublayer, and in the upper inviscid deck. Our analysis is based on the multiscale asymptotic technique applied with respect to the streamwise and spanwise coordinates and time. It is combined with the method of matched asymptotic expansions applied with respect to the normal coordinate .

To describe the receptivity process near the neutral point , we shall assume that wall vibrations have the form of a wave packet, which may be represented as a single wave with the carrier Tollmien–Schlichting oscillatory factor:having a slowly modulated order-one amplitudewhere ‘c.c.’ denotes complex conjugate values.

Fourier parameters *k*_{α}, *k*_{β} and *Ω* represent detuning of a particular mode in the wave packet with respect to the corresponding neutral Tollmien–Schlichting wavenumbers *α*, *β* and frequency *ω*. As to the height parameter *h*, it is assumed to be sufficiently small, namely *h*≪*ϵ*^{3}, in order to avoid any nonlinearities, including weak ones.

## 3. Pre-resonance region

In the pre-resonance region, wall vibrations provoke a weak forced response of the boundary layer. The asymptotic expansions of hydrodynamic functions in regions 4–6 (see figure 2) follow from the linear triple-deck theory. For example, in the viscous sublayer 6 we havewhere is the Tollmien–Schlichting oscillatory factor, and is the Blasius skin friction, with *λ*_{0}≈0.332. It should be noted that the velocity perturbations *U*, *V*, *W* and pressure *P* depend also parametrically on , *X*, *Z*, *T*.

The solution to the pre-resonance problem may be expressed in terms of the Airy function. In particular, it is found thatwhere(3.1)The coefficient stands for the amplitude of forced oscillations and is given by(3.2)where(3.3)The dispersion expression *D*(*λ*) is non-zero everywhere along the plate, except at the neutral point , where the resonance occurs and *λ*=*λ*_{0}. The (real) constants *α*_{1}, *β*_{1} and *ω*_{1} can be found solving the dispersion relation *D*(*λ*_{0})=0. They arewhere *α*_{0}, *β*_{0} and *ω*_{0} are given bywith *d*_{1}≈1.009 and *d*_{2}≈2.29. This provides an infinite set of neutral Tollmien–Schlichting eigenmodes at , propagating in different directions. However, due to the linearity of the problem, these modes do not interact with each other; thereby, only one of them is considered without any loss of generality.

A Taylor expansion of the dispersion expression (3.3) about *λ*_{0},being substituted into equation (3.2), determines the asymptotic behaviour of the forced-oscillation amplitude(3.4)where(3.5)Here and thereafter, *η* and *η*_{0} are evaluated at the neutral point, so we put *λ*=*λ*_{0} into the formulae (3.1).

It follows from the asymptotic formula (3.4) that the response amplitude increases indefinitely on approaching the point . Additionally, when the resonance region is entered, the boundary-layer non-parallelism strongly affects the amplitude growth. It therefore turns out that the preceding analysis is no longer valid in the O(*Re*^{−3/16}) vicinity of the neutral point, where new physical arguments must be included into the model.

## 4. Fluid flow in the resonance region

In the resonance region, fluid oscillations take the form of a growing Tollmien–Schlichting wave. Formulae (2.3) and (3.4) suggest that the wave amplitude can be represented aswhere(4.1)The asymptotic expansions of hydrodynamic functions in regions 1–3 (figure 2) have a similar structure. They are based on the standard triple-deck scalings and the condition of matching to the solution in the pre-resonance region. For example, in the main deck 2, the velocity components and pressure expand as(4.2)where *U*_{B} is the Blasius velocity profile.

The leading order perturbation terms in equation (4.2) represent the Tollmien–Schlichting wave, with the amplitude being a function of slow variables *X*, *Z* and *T*. Note that the physical amplitude of the Tollmien–Schlichting wave is a factor O(*Re*^{3/16}) larger than the wall oscillations' height. The next order terms arise as an order-*ϵ* correction associated with the pressure variations across the boundary layer. However, these terms do not affect derivation of the amplitude equation, and therefore are not considered in the following analysis. Finally, the terms with subscript ‘3’ are driven by the direct forcing from wall vibrations.

On substituting equation (4.2) into the Navier–Stokes equations and setting *Re*→∞, it is simple to arrive at the general form of the main deck solution. In particular, we find that(4.3)where *A*_{1} is a constant and *A*_{3} is a function of slow variables, both appearing in the asymptotic expansion of the displacement functionSimilarly, the solution in region 1 satisfying the far field conditions (2.2) may be obtained. Then, matching the pressure and vertical velocity in the main and upper decks, we deduce the interaction law formulae(4.4)

### (a) Viscous sublayer

In the near-wall viscous sublayer 3 the solution expands asSubstitution of these expansions into the Navier–Stokes equations and setting *Re*→∞ result in a system of linear equations. They are subject to the boundary conditions (2.1) and the condition of matching to the solution (4.3) in region 2. The leading order solution in the viscous sublayer is found to bewhereThe Tollmien–Schlichting wave amplitude remains unknown but can be determined through consideration of the terms with subscript ‘3’. The latter are governed by the following inhomogeneous system of equations:whereThe linearized no-slip condition on the wall implies that(4.5)while matching to the main-deck solution requires(4.6)After eliminating *U*_{3}, *V*_{3} and *P*_{3} among the equations of motion, we arrive at the single differential equationthe general solution of which, satisfying the no-slip condition, may be written as(4.7)where is some unknown function of slow variables.

Applying the boundary conditions (4.5), (4.6) to the solution (4.7), and making use of the interaction law formulae (4.4) and the dispersion relation *D*(*λ*_{0})=0, we obtain a solvability condition which appears to be a first order partial differential equation for the amplitude .

### (b) The amplitude equation

We finally arrive at the equation for the Tollmien–Schlichting wave amplitude(4.8)whereand *σ*, *a*, *G* are given by equations (3.5). Obviously, the left-hand side of equation (4.8) is negligible for large negative *X*, and this is in line with the asymptotic formula (4.1).

The amplitude equation (4.8) can be solved by means of the Fourier transform method applied with respect to the variables *Z* and *T*. For this purpose, let us define the Fourier transforms of functions and *G* asThus, in the Fourier space equation (4.8) takes the formThe general solution of the transformed amplitude equation is(4.9)with the downstream asymptotic behaviour beingThe first term in the formula (4.9) represents the Tollmien–Schlichting wave packet arising spontaneously in the resonance region, while the second term stands for the packet generated by wall vibrations.

The physical amplitude can be obtained by taking inverse Fourier transform of equation (4.9) and introducing the functionIn particular, the downstream asymptotic behaviour of the wave amplitude is found to be(4.10)whereDownstream of the resonance region, the direct external forcing becomes unimportant, and the Tollmien–Schlichting wave is described by the free-evolution theory, based on the WKB method.

## 5. Concluding remarks

The generation of Tollmien–Schlichting wave packets by distributed wall vibrations has been studied, and the explicit formula (4.10) for the downstream asymptotic behaviour of the wave amplitude has been obtained. Compared with other cases, this type of receptivity appears to be more efficient for the following reasons. On one hand, distributed receptivity takes place over a much wider region and is, therefore, a factor O(*Re*^{3/16}) stronger than that due to the localized vibrator, considered by Terent'ev (1981). On the other hand, the perturbation source (i.e. wall oscillations) lies entirely inside the boundary layer. Hence, there is no waste of energy for penetrating of disturbances into the boundary layer, which occurs when external perturbations are present in the free stream (see Wu 2001).

From the practical standpoint, wall vibrations are easier to control. Formula (4.10) suggests the method of boundary-layer laminarization, based on the possibility of mutual cancellation of Tollmien–Schlichting wave packets. The condition for cancellation is *K*_{∞}=0; this can be achieved if the function *G*(*X*,*Z*,*T*) satisfies the integral equationWe can see that for effective neutralization of spontaneous instabilities, it is sufficient to apply wall vibrations in the resonance region only. The decay rate of the residual perturbations as *X*→+∞ depends on the shape of the function *G*. In particular, if *G* is constant, the amplitude of the neutralized Tollmien–Schlichting wave decreases as O(*X*^{−1}).

Finally, it should be noted that our receptivity analysis is applicable not only to wall vibrations but also to any other perturbation source on the body surface, e.g. periodic suction and blowing.

## Footnotes

One contribution of 19 to a Theme ‘New developments and applications in rapid fluid flows’.

- © 2005 The Royal Society