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.

focus of that reference was on hypersonic flows while the interest here is in the moderately supersonic regime (Mach number less than 4), where the so called 1st Mack mode is the dominant instability, but (as shown in Section 6) emerges much too far downstream to be of practical interest when generated by the inviscid mechanism analyzed in [7]. The instability produced by the small Δθ analysis of Ref. [11] can, however, occur much further upstream when Δθ is sufficiently small. But there is a smallest value of Δθ for which the instability wave coupling can occur. 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 θ is greater than the critical angle θc. Their phase speeds are very small and they must therefore be produced by a viscous wall

Low-sweep Aerion AS2 supersonic Bizjet. M∞≤1:5. Posted by Tim Brown on the Manufacturer Newsletter.

Leading Edge Receptivity at Subsonic and Moderately Supersonic Mach Numbers

DOI: http://dx.doi.org/10.5772/intechopen.83672

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 Δθ emerges. It therefore makes sense to consider both of these

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 Δθ acoustic disturbances in a moderately supersonic Mach number free stream. The results are expected to be relevant to transition in the straight wing boundary layers on supersonic aircraft such as the low-sweep Aerion AS2 Bizjet, shown in Figure 2.

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 ω<sup>∗</sup> superimposed on a subsonic or moderately low

layer mechanism similar to the one identified in [7].

Figure 2.

receptivity mechanisms simultaneously.

2. Imposed free-stream disturbances

83

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 θ of the disturbance differs from the critical angle, <sup>θ</sup><sup>c</sup> � cos �<sup>1</sup>ð Þ <sup>1</sup>=M<sup>∞</sup> , where the <sup>M</sup><sup>∞</sup> is the free-stream Mach number, by an Oð Þ1 amount. Figure 1 shows that the theoretical results of Ref. [10] are in good agreement with experimental data when Δθ � θ<sup>c</sup> � θ ¼ Oð Þ1 but the agreement breaks down when θ ! θ<sup>c</sup> [12] and a new rescaled analysis was carried out in Ref. [11] to deal with this case.

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 θ of the acoustic disturbances is smaller/larger than the critical angle θ<sup>c</sup> and that downstream propagating slow acoustic modes with Δθ>0 generate unsteady boundary layer disturbances that match onto the inviscid 1st Mack mode instability without undergoing any significant decay. The

Figure 1. Comparison of the Fedorov/Khokhlov solution with experiment [12].

Leading Edge Receptivity at Subsonic and Moderately Supersonic Mach Numbers DOI: http://dx.doi.org/10.5772/intechopen.83672

Figure 2. Low-sweep Aerion AS2 supersonic Bizjet. M∞≤1:5. Posted by Tim Brown on the Manufacturer Newsletter.

focus of that reference was on hypersonic flows while the interest here is in the moderately supersonic regime (Mach number less than 4), where the so called 1st Mack mode is the dominant instability, but (as shown in Section 6) emerges much too far downstream to be of practical interest when generated by the inviscid mechanism analyzed in [7]. The instability produced by the small Δθ analysis of Ref. [11] can, however, occur much further upstream when Δθ is sufficiently small. But there is a smallest value of Δθ for which the instability wave coupling can occur.

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 θ is greater than the critical angle θc. Their phase speeds are very small and they must therefore be produced by a viscous wall layer mechanism similar to the one identified in [7].

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 Δθ emerges. It therefore makes sense to consider both of these receptivity mechanisms simultaneously.

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 Δθ acoustic disturbances in a moderately supersonic Mach number free stream. The results are expected to be relevant to transition in the straight wing boundary layers on supersonic aircraft such as the low-sweep Aerion AS2 Bizjet, shown in Figure 2.

#### 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 ω<sup>∗</sup> superimposed on a subsonic or moderately low

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

Boundary Layer Flows - Theory, Applications and Numerical Methods

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

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 θ of the disturbance differs from the critical angle, <sup>θ</sup><sup>c</sup> � cos �<sup>1</sup>ð Þ <sup>1</sup>=M<sup>∞</sup> , where the <sup>M</sup><sup>∞</sup> is the free-stream Mach number, by an Oð Þ1 amount. Figure 1 shows that the theoretical results of Ref. [10] are in good agreement with experimental data when Δθ � θ<sup>c</sup> � θ ¼ Oð Þ1 but the agreement breaks down when θ ! θ<sup>c</sup> [12] and a new rescaled analysis was carried

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 θ of the acoustic disturbances is smaller/larger than the critical angle θ<sup>c</sup> and that downstream propagating slow acoustic modes with Δθ>0 generate unsteady boundary layer disturbances that match onto the inviscid 1st Mack mode instability without undergoing any significant decay. The

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.

out in Ref. [11] to deal with this case.

Comparison of the Fedorov/Khokhlov solution with experiment [12].

edge [7, 8].

Figure 1.

82

Mach number supersonic flow of an ideal gas past an infinitely thin flat plate with uniform free-stream velocity U<sup>∗</sup> <sup>∞</sup>, temperature T<sup>∗</sup> <sup>∞</sup>, dynamic viscosity μ<sup>∗</sup> <sup>∞</sup> and density ρ<sup>∗</sup> <sup>∞</sup>. The velocities, pressure fluctuations, temperature and dynamic viscosity are normalized by U<sup>∗</sup> <sup>∞</sup>, ρ<sup>∗</sup> <sup>∞</sup> U<sup>∗</sup> ∞ � �<sup>2</sup> , T<sup>∗</sup> <sup>∞</sup> and μ<sup>∗</sup> <sup>∞</sup>, respectively. The time t is normalized by <sup>ω</sup><sup>∗</sup> and the Cartesian coordinates, say f g <sup>x</sup>; <sup>y</sup>; <sup>z</sup> , are normalized by <sup>L</sup><sup>∗</sup> � <sup>U</sup><sup>∗</sup> ∞=ω<sup>∗</sup> with the coordinate y being normal to the plate.

As noted above the phenomenon is analyzed by requiring the Reynolds number Re <sup>¼</sup> <sup>ρ</sup><sup>∗</sup> ∞U<sup>∗</sup> ∞L<sup>∗</sup>=μ<sup>∗</sup> <sup>∞</sup> to be large, or equivalently requiring the frequency parameter F � 1=Re to be small, and using asymptotic theory to explain how the imposed harmonic distortion generates oblique instabilities at large downstream distances in the viscous boundary layer that forms on the surface of the plate. The natural expansion parameter turns out to be

$$
\kappa \equiv \mathcal{F}^{1/6}.\tag{1}
$$

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 oblique-

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).

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 <sup>O</sup> <sup>ε</sup>�<sup>2</sup> ð Þ distance downstream [13] of the leading edge in the high Reynolds number flow being considered here. The Tollmien-Schlichting waves will have <sup>O</sup> <sup>ε</sup>�<sup>1</sup> ð Þ spanwise wavenumbers and we

since the spanwise wavenumber must remain constant as the disturbances

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

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 Tð Þη , ρ ηð Þ, Uð Þη , respectively, are assumed to

0

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

We begin by considering the flow in the vicinity of the leading edge where the

ð Þ <sup>0</sup> ð Þ <sup>x</sup>; <sup>η</sup> n o satisfies the three dimensional

streamwise length scale is x ¼ Oð Þ1 . Since the inviscid velocity field can only depend on the streamwise coordinate through this relatively long streamwise length

scale the solution for the velocity and temperature perturbation u<sup>0</sup> � u<sup>0</sup>

^ <sup>u</sup><sup>∞</sup> <sup>u</sup>; <sup>v</sup>; <sup>0</sup>; <sup>ϑ</sup> � � <sup>þ</sup> <sup>β</sup>ð Þ <sup>w</sup><sup>∞</sup> <sup>þ</sup> iv<sup>∞</sup> <sup>u</sup>ð Þ <sup>0</sup> ; <sup>v</sup>ð Þ <sup>0</sup> ; <sup>w</sup>ð Þ <sup>0</sup> ; <sup>ϑ</sup> ð Þ <sup>0</sup> h i n o <sup>e</sup>

compressible linearized boundary layer equations (with unit spanwise

w<sup>∞</sup> � w∞=ε ¼ Oð Þ1 , v<sup>∞</sup> � v∞=ε ¼ Oð Þ1 , γ � εγ ¼ Oð Þ1 : (9)

β � εβ ¼ Oð Þ1 (8)

ρð Þ x; ~y d~y (10)

; v<sup>0</sup> ; w<sup>0</sup> ; ϑ<sup>0</sup> f g

<sup>i</sup>ð Þ <sup>β</sup>z=ε�<sup>t</sup> , (11)

ness angle is equal to the critical angle referred to in the introduction.

Leading Edge Receptivity at Subsonic and Moderately Supersonic Mach Numbers

3.1 Boundary layer disturbances generated by the free-stream vorticity

3. Boundary layer disturbances

DOI: http://dx.doi.org/10.5772/intechopen.83672

therefore require that

propagate downstream.

be functions of the Dorodnitsyn-Howarth variable

in this region is given by [14], [17]

where <sup>u</sup>ð Þ <sup>0</sup> ð Þ <sup>x</sup>; <sup>η</sup> ; <sup>v</sup>ð Þ <sup>0</sup> ð Þ <sup>x</sup>; <sup>η</sup> ; <sup>w</sup>ð Þ <sup>0</sup> ð Þ <sup>x</sup>; <sup>η</sup> ; <sup>ϑ</sup>

wavenumber) subject to the boundary conditions [14]

<sup>η</sup> � <sup>1</sup> ε<sup>3</sup> ffiffiffiffiffi <sup>2</sup><sup>x</sup> <sup>p</sup> ð y

if we put

Ref. [14].

u<sup>0</sup> ¼ δ

85

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 u<sup>v</sup> is given

$$\mathfrak{u}\_v = \{u\_v, v\_v, w\_v\} = \hat{\delta}\{u\_\infty, v\_\infty, w\_\infty\} \exp\left[i(\varkappa - t + \gamma y + \beta \mathbf{z})\right],\tag{2}$$

where δ ^ ≪ 1 is a common scale factor and u∞, v∞, w<sup>∞</sup> satisfy the continuity condition

$$
\mu\_{\infty} + \gamma v\_{\infty} + \beta w\_{\infty} = 0 \tag{3}
$$

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

$$\{\mathbf{u}\_d, p\_a\} = \{u\_a, v\_a, w\_a, p\_a\} = \frac{\hat{\delta}}{\mathbf{1} - a} \{a, \gamma, \beta, \mathbf{1} - a\} e^{i(a\mathbf{x} + \gamma\boldsymbol{\eta} + \beta\mathbf{z} - t)},\tag{4}$$

for the velocity and pressure perturbation ua; pa � � where

$$\gamma = \sqrt{(M\_{\infty}^2 - 1)(a - a\_1)(a - a\_2)}, \quad a\_{1,2} = \frac{M\_{\infty}^2 \pm \sqrt{M\_{\infty}^2 + \rho^2(M\_{\infty}^2 - 1)}}{M\_{\infty}^2 - 1} \tag{5}$$

and, as noted in Section 1, M<sup>∞</sup> denotes the free-stream Mach number.

The leading edge interaction will produce large scattered fields for Oð Þ1 values of the incidence angles tan �<sup>1</sup>ð Þ¼ va=ua tan �<sup>1</sup>ð Þ <sup>γ</sup>=<sup>α</sup> and tan �<sup>1</sup>ð Þ vv=uv of the acoustic and vortical disturbances, respectively. And, in order to focus on the fundamental mechanisms, we assume that the incidence angles of the vortical disturbances are small and that the incidence angles of the acoustic disturbances are zero, which requires that

$$
v\_{\infty}/u\_{\infty} \ll 1\tag{6}$$

for the former disturbances and that

$$a = a\_{\mp} = M\_{\infty} \cos \theta / (M\_{\infty} \cos \theta \mp 1), \ \theta \equiv \tan^{-1}(\beta/a), \tag{7}$$

Leading Edge Receptivity at Subsonic and Moderately Supersonic Mach Numbers DOI: http://dx.doi.org/10.5772/intechopen.83672

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.
