## Abstract

This chapter is a review of the receptivity and resulting global instability of boundary layers due to free-stream vortical and acoustic disturbances at subsonic and moderately supersonic Mach numbers. The vortical disturbances produce an unsteady boundary layer flow that develops into oblique instability waves with a viscous triple-deck structure in the downstream region. The acoustic disturbances (which have phase speeds that are small compared to the free stream velocity) produce boundary layer fluctuations that evolve into oblique normal modes downstream of the viscous triple-deck region. Asymptotic methods are used to show that both the vortically and acoustically-generated disturbances ultimately develop into modified Rayleigh modes that can exhibit spatial growth or decay depending on the nature of the receptivity process.

### Keywords

- boundary layer
- boundary layer receptivity
- compressible boundary layers
- global instability

## 1. Introduction

This chapter is concerned with the effect of unsteady free-stream disturbances on laminar to turbulent transition in boundary layer flows. The exact mechanism depends on the nature and intensity of the disturbances. Transition at high disturbance levels (say >1%) usually begins with the excitation of low frequency streaks in the boundary layer flow that eventually break down into turbulent spots. This phenomena was initially studied by Dryden [1] and much later for compressible flows by Marensi et al. [2]. But the focus of this chapter is on low free steam disturbances levels (say less than 1%) where the transition usually results from a series of events beginning with the generation of spatially growing instability waves by acoustic and/or vortical disturbances in the free-stream. This so-called receptivity phenomenon results in a boundary value problem and therefore differs from classical instability theory which results in an eigenvalue problem for the Rayleigh or Orr-Sommerfeld equations that only apply when the mean flow can be treated as being nearly parallel (see, for example, Reshotko, [3]). The relevant boundary conditions cannot be imposed on the Orr-Sommerfeld or Rayleigh equations in the infinite Reynolds number limit being considered here but the free-stream disturbances can produce unsteady boundary layer perturbations in regions of rapidly changing mean flow that eventually produce unstable Rayleigh or Orr-Sommerfeld equation eigensolutions further downstream. These regions of nonparallel flow can result from surface roughness elements [4, 5], blowing or suction effects [6] or from the nonparallel mean flow that occurs near the boundary layer leading edge [7, 8].

The mechanism is similar in all cases but the simplest and arguably the most fundamental of these is the one resulting from the nonparallel leading edge flow and the focus here is, therefore, on that case. The initial studies were carried out for two dimensional incompressible flows. Ref. [7] used a low frequency parameter matched asymptotic expansion to show that there is an overlap domain where appropriate asymptotic solutions to the forced boundary layer equations (which apply near the edge) match onto the so-called Tollmien-Schlichting waves that satisfy the Orr-Sommerfeld equation in a region that lies somewhat further downstream. The coupling to the free-stream disturbances turns out to be fairly weak for the two dimensional incompressible flow considered in [7] due to the relatively large decay of boundary layer disturbances upstream of the Tollmien-Schlichting wave region where the Orr-Sommerfeld equation applies.

But there can be a much stronger coupling in supersonic flows which can support a number of different instabilities [9]. The coupling mechanism can be either viscous or inviscid and the instability can either be of the viscous Tollmien-Schlichting type or can be purely inviscid when the mean boundary layer flow has a generalized inflection point. The inviscid coupling, which was first analyzed in [10], tends to be dominant when the obliqueness angle

Fedorov and Khokhlov [10] analyzed the generation of inviscid instabilities in a supersonic flat plate boundary layer by fast and slow acoustic disturbances in the free stream. They showed that the slow acoustic mode propagates downstream/upstream when the obliqueness angle

Smith [13] showed that viscous instabilities, which exhibit the same triple-deck structure as the subsonic Tollmien-Schlichting waves, can also occur at supersonic speeds when the obliqueness angles

The analysis of Ref. [7] was extended to compressible subsonic and supersonic flat plate boundary layer flows by Ricco and Wu [14] who showed that highly oblique vortical disturbances can generate a limiting form of the Smith instability [13]. They found that the instability wave lower branch lies further upstream at supersonic speeds than the subsonic lower branch and much further upstream than the incompressible lower branch considered in [7], which means that the instability wave/free-stream disturbance coupling is much greater at supersonic speeds than it is in the incompressible flow considered in [7]. Goldstein and Ricco [11] show that the instability does not possess an upper branch in this case and matches onto a low frequency (short streamwise wavenumber) Rayleigh instability (that can be identified with the 1st Mack mode) when the downstream distance is slightly smaller than the downstream distance where acoustically generated instability corresponding to the smallest possible

As noted above, the present chapter is concerned with the unsteady flow in a flat plate boundary layer generated by mildly oblique vortical disturbance and small

## 2. Imposed free-stream disturbances

Since the boundary layer is believed to be convectively unstable, the receptivity phenomena are best illustrated by considering a small amplitude harmonic distortion with angular frequency

As noted above the phenomenon is analyzed by requiring the Reynolds number

The free-steam disturbances will be inviscid at the lowest order of approximation and, as is well known [15], can be decomposed into an acoustic component that carries no vorticity, and vortical and entropic components that produce no pressure fluctuations. But only the first two will be considered here.

The vortical disturbance

where

but are otherwise arbitrary constants while the acoustic component is governed by the linear wave equation which has a fundamental plane wave solution

for the velocity and pressure perturbation

and, as noted in Section 1,

The leading edge interaction will produce large scattered fields for

for the former disturbances and that

for the latter, where the subscripts −/+ refer to the slow/fast acoustic modes. Eq. (7) shows that the slow mode wavenumber becomes infinite when the obliqueness angle is equal to the critical angle referred to in the introduction.

## 3. Boundary layer disturbances

As indicated above our interest here is in explaining how the incident harmonic distortions generate oblique instabilities at large downstream distances in the viscous boundary layer that forms on the surface of the plate. We begin by considering the fluctuations imposed on this flow by the free-stream vortical disturbance (2).

### 3.1 Boundary layer disturbances generated by the free-stream vorticity

As noted in the introduction, these disturbances will generate oblique Tollmien-Schlichting instability waves which are known to exhibit a triple-deck structure in the vicinity of their lower branch which lies at an

since the spanwise wavenumber must remain constant as the disturbances propagate downstream.

The continuity condition (3) and the obliqueness restriction (6) will be satisfied if we put

The vortical velocity (2) will then interact with the plate to produce an inviscid velocity field [12] that generates a slip velocity at the surface of the plate which must be brought to zero in a thin viscous boundary layer whose temperature, density and streamwise velocity, say

and are determined from the similarity equations given in Stewartson [16] and Ref. [14].

We begin by considering the flow in the vicinity of the leading edge where the streamwise length scale is

where

while

The lowest order triple-deck solution will match onto the quasi-two dimensional solution

Prandtl [18], Glauert [19] and Lam and Rott [20] showed that

where

is an exact eigensolution of the two-dimensional linearized unsteady boundary layer equations that satisfies the homogeneous boundary conditions

Lam and Rott [20], [21] analyzed the two dimensional flat plate boundary layer and showed that the linearized equations possess asymptotic eigensolutions that satisfy a no-slip condition at the wall when

Ref. [14] showed that the Lam and Rott [20, 21] analysis also applies to compressible flows when the full compressible solution (14) and (15) is used in the outer region and the viscous wall layer solution is slightly modified to account for the temperature and viscosity variations. The function

where

The only difference from the Lam-Rott result is the

### 3.2 Boundary layer disturbances generated by the Fedorov/Khokhlov mechanism for obliqueness angles close to critical angle

Fedorov and Khokhlov [10] used matched asymptotic expansions to analyze the generation of Mack mode instabilities by oblique acoustic waves of the form (4) where the wavenumbers

Ref. [10] shows that the boundary layer disturbance produced by diffraction of the slow acoustic wave by the nonparallel mean flow in the region where

of the obliqueness angle

takes on

The instability emerges from the downstream limit of the solution in this region. But as noted in the introduction this occurs too far downstream to be of practical interest when scaled up to actual flight conditions if

It follows from (7) that

where

when

This shows that

The spanwise wavenumber will equal the vortical spanwise wavenumber (8) when

Ref. [11] shows that the smallest value of

The pressure in region 2 is of the form

where

and the surface pressure

where

and the integral and the derivative of the Airy function

which behaves like

as

Inserting (28) and (27) into (25) shows that

where

which is formally the same as the equation considered in [10] who showed that the solution behaves like

The acoustically and vortically generated boundary layer disturbances considered in this section will eventually evolve into propagating eigensolutions in regions that lie further downstream. The resulting flow will have a triple-deck structure of the type considered in [13], [23] and [14] in the former (i.e., vortically generated) case. But the acoustically generated disturbance will only develop an eigensolution structure much further downstream. The minimum distance occurs when

## 4. The viscous triple-deck region

Refs. [13, 14, 23] show that the linearized Navier-Stokes equations possess an eigensolution of the form

in the triple-deck region where

in the main boundary layer where

and

is a scaled transverse coordinate. The complex wavenumber

where the lowest order term in this expansion satisfies the following dispersion relation ([13, 14, 23])

and

whose solution must satisfy the inequality

in order to insure that the eigensolution does not exhibit unphysical wall normal growth.

This requirement will be satisfied for all

### 4.1 Matching with the Lam-Rott solution

The dispersion relation (38) and (39) will be satisfied at small values of

The cross flow velocity

### 4.2 Numerical results

The dispersion relation (38) is expected to have at least one root corresponding to each of the infinitely many roots of (18). But only the lowest order

The real and negative imaginary parts of

The triple-deck eigensolution (33) (which contains the Lam-Rott solution as an upstream limit) can undergo a significant amount of damping before it turns into a spatially growing instability wave at the lower branch of the neutral stability curve.

The exponential damping in Eq. (33) is proportional to

## 5. The inviscid triple-deck region

As noted above the acoustically driven solution will only match onto an eigensolution in the downstream region when

into (38), using (29), and taking the limit as

when the square root

### 5.1 Matching with the small Δ θ Fedorov/Khokhlov solution

It can then be shown by direct substitution that the solution

where

The pressure component of the resulting solution will then match onto the downstream limit (32) and (30) of the acoustically generated diffraction region solution when

### 5.2 Numerical results

Figure 6 is a plot of the scaled lowest order wavenumber

## 6. The next stage of evolution

### 6.1 Downstream behavior of the triple-deck solution

Eqs. (29), (38) and (39) show that

when

when

The dashed curves in the main plot of Figure 4 represent the re-scaled large-

As noted in [11], the solution to the reduced dispersion relation (44) satisfies the rescaled version

of (48), which can be considered to be a special case of this result if we put

and allow *r* to be zero or 1/3.

The expansion (37) then generalizes to [11]

where

and

### 6.2 Derivation of the governing equations

Eq. (49) shows, among other things, that the lowest order wave number and streamwise growth rate approach zero but do not become negative as the disturbance propagates downstream. The boundary layer thickness which is

where it becomes of the order of the spanwise length scale, which remains constant at

Eqs. (37), (43), (51) and (52) show that the Tollmien-Schlichting wave becomes more oblique and

as

which means that the solution should be proportional to

in this stage of evolution. The solution should remain inviscid in the main boundary layer and the viscous wall layer (i.e., a Stokes layer) is expected to be completely passive.

The scaled variable

will be

where

whose solution must satisfy the following boundary conditions

Matching with the upstream solution (33) and (37) requires that

Inserting (10) and (57) into (59), using (60) and assuming the ideal gas law

which means that

where

### 6.3 Matching with the triple-deck solution

Eq. (64) clearly approaches zero when

where

Ref. [11] proved that (60)–(64) possess an asymptotic solution of the form

### 6.4 Numerical results

The Rayleigh eigenvalues

Parts (a) and (b) of Figure 7 are plots of the real and imaginary parts respectively of these eigenvalues as a function of *C* are given in the caption of Figure 7. They also show that

## 7. Conclusions

This chapter uses high Reynolds number asymptotics to study the nonlocal behavior of boundary layer instabilities generated by small amplitude free-stream disturbances at subsonic and moderate supersonic Mach numbers. The appropriate small expansion parameter turns out to be

The free-stream vortical disturbances generate unsteady flows in the leading edge region that produce short spanwise wavelength instabilities in a viscous triple-deck region which lies at an

Fedorov and Khokhlov [10] used asymptotic methods to study the generation of inviscid instabilities in supersonic boundary layers by fast and slow acoustic disturbances in the free stream whose obliqueness angle

But, the inviscid instability, which first appears at an

## Acknowledgments

This research was sponsored by NASA’s by the Transformative Aeronautics Concepts Program of the Aeronautics Research Mission Directorate under the Transformational Tools and Technologies (TTT) Project. PR was supported by the Air Force Office of Scientific Research award number AFOSR Grant FA9550-15-1-0248. We would also like to thank Dr. Meelan Choudhari for bringing the photograph in Figure 2 to our attention.