3. Boundary layer disturbances

Mach number supersonic flow of an ideal gas past an infinitely thin flat plate with

<sup>∞</sup>, temperature T<sup>∗</sup>

<sup>∞</sup> and μ<sup>∗</sup>

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

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>

, T<sup>∗</sup>

Boundary Layer Flows - Theory, Applications and Numerical Methods

<sup>∞</sup>. The velocities, pressure fluctuations, temperature and dynamic viscosity

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

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

but are otherwise arbitrary constants while the acoustic component is governed

, <sup>α</sup>1, <sup>2</sup> <sup>¼</sup> <sup>M</sup><sup>2</sup>

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

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

<sup>α</sup> <sup>¼</sup> <sup>α</sup> <sup>∓</sup> <sup>¼</sup> <sup>M</sup><sup>∞</sup> cos <sup>θ</sup>=ð Þ <sup>M</sup><sup>∞</sup> cos <sup>θ</sup> <sup>∓</sup> <sup>1</sup> , <sup>θ</sup> � tan �<sup>1</sup>

<sup>1</sup> � <sup>α</sup> f g <sup>α</sup>; <sup>γ</sup>; <sup>β</sup>; <sup>1</sup> � <sup>α</sup> <sup>e</sup>

� � where

M<sup>2</sup>

M<sup>2</sup>

v∞=u<sup>∞</sup> ≪ 1 (6)

<sup>∞</sup> �

by the linear wave equation which has a fundamental plane wave solution

^

<sup>∞</sup> to be large, or equivalently requiring the frequency parameter

<sup>∞</sup>, dynamic viscosity μ<sup>∗</sup>

<sup>ε</sup> � <sup>F</sup>1=<sup>6</sup>: (1)

^f g <sup>u</sup>∞; <sup>v</sup>∞; <sup>w</sup><sup>∞</sup> exp ½ � i xð Þ � <sup>t</sup> <sup>þ</sup> <sup>γ</sup><sup>y</sup> <sup>þ</sup> <sup>β</sup><sup>z</sup> , (2)

u<sup>∞</sup> þ γv<sup>∞</sup> þ βw<sup>∞</sup> ¼ 0 (3)

<sup>i</sup>ð Þ <sup>α</sup>xþγyþβz�<sup>t</sup> , (4)

<sup>∞</sup> � 1

<sup>∞</sup> � <sup>1</sup> (5)

ð Þ β=α , (7)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>∞</sup> <sup>þ</sup> <sup>β</sup><sup>2</sup> <sup>M</sup><sup>2</sup>

q � �

<sup>∞</sup>, respectively. The time t is normalized

<sup>∞</sup> and den-

∞=ω<sup>∗</sup>

uniform free-stream velocity U<sup>∗</sup>

<sup>∞</sup>, ρ<sup>∗</sup> <sup>∞</sup> U<sup>∗</sup> ∞ � �<sup>2</sup>

with the coordinate y being normal to the plate.

fluctuations. But only the first two will be considered here.

� � <sup>¼</sup> <sup>δ</sup>

for the velocity and pressure perturbation ua; pa

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>∞</sup> � <sup>1</sup> � �ð Þ <sup>α</sup> � <sup>α</sup><sup>1</sup> ð Þ <sup>α</sup> � <sup>α</sup><sup>2</sup>

for the former disturbances and that

are normalized by U<sup>∗</sup>

∞U<sup>∗</sup>

where δ

γ ¼

requires that

84

q

ua; pa

M<sup>2</sup>

condition

∞L<sup>∗</sup>=μ<sup>∗</sup>

expansion parameter turns out to be

The vortical disturbance u<sup>v</sup> is given

� � <sup>¼</sup> ua; va; wa; pa

u<sup>v</sup> ¼ f g¼ uv; vv; wv δ

sity ρ<sup>∗</sup>

Re <sup>¼</sup> <sup>ρ</sup><sup>∗</sup>

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 <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 therefore require that

$$
\overline{\beta} \equiv \epsilon \beta = \mathcal{O}(\mathbf{1}) \tag{8}
$$

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

$$
\overline{w}\_{\infty} \equiv w\_{\infty}/\varepsilon = O(1), \ \overline{\nu}\_{\infty} \equiv \nu\_{\infty}/\varepsilon = O(1), \overline{\gamma} \equiv \varepsilon \gamma = O(1). \tag{9}
$$

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 be functions of the Dorodnitsyn-Howarth variable

$$\eta \equiv \frac{1}{\varepsilon^3 \sqrt{2\pi}} \int\_0^\gamma \rho(\mathbf{x}, \bar{\mathbf{y}}) d\bar{\mathbf{y}} \tag{10}$$

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 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> ; v<sup>0</sup> ; w<sup>0</sup> ; ϑ<sup>0</sup> f g in this region is given by [14], [17]

$$\mathfrak{u}' = \hat{\delta} \left[ u\_{\infty} \left\{ \overline{u}, \overline{v}, 0, \overline{\mathfrak{d}} \right\} + \overline{\beta} (\overline{w}\_{\infty} + i \overline{v}\_{\infty}) \left\{ \overline{u}^{(0)}, \overline{v}^{(0)}, \overline{w}^{(0)}, \overline{\mathfrak{d}}^{(0)} \right\} \right] e^{i \left( \overline{\beta} \mathfrak{x} / \varepsilon - 1 \right)},\tag{11}$$

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> ð Þ <sup>0</sup> ð Þ <sup>x</sup>; <sup>η</sup> n o satisfies the three dimensional compressible linearized boundary layer equations (with unit spanwise wavenumber) subject to the boundary conditions [14]

$$
\overline{u}^{(0)}, \overline{\theta}^{(0)} \to \mathbf{0}, \ \overline{w}^{(0)} \to \mathfrak{e}^{\rm ir}, \quad \text{as} \ \eta \to \infty,\tag{12}
$$

numerical solutions to the boundary layer problem in Ref. [8]. But we are primarily concerned with the lowest order n ¼ 0 mode because that is the only one that matches onto a spatially growing oblique Tollmien-Schlicting wave further downstream [11]. The receptivity problem can then be solved by combining the numerical computations with appropriate matched asymptotic expansions to relate the instability wave amplitude to that of the free-stream disturbance. But we will analyze the boundary layer disturbances generated by the free-stream acoustic

Leading Edge Receptivity at Subsonic and Moderately Supersonic Mach Numbers

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 α and β satisfy the dispersion relation (7) when the incidence angle γ is equal to zero, which, as noted, above is the case being considered here. Their focus was on hypersonic flows where the most rapidly growing disturbances are usually two dimensional 2nd Mack modes, while, as noted in the introduction, the focus of the present chapter is on the relatively low supersonic Mach number regime (say, less than about 4) where the most rapidly growing instability waves are highly oblique 1st Mack modes. Numerical results [9] show that the obliqueness angle of the most rapidly growing 1st mode lies between 50 and 70

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 <sup>x</sup> <sup>¼</sup> <sup>o</sup> <sup>ε</sup>�<sup>3</sup> ð Þ can be matched onto a 1st Mack mode instability in the downstream region where

takes on Oð Þ1 positive values. The diffraction region has a double layer structure which consists of a region that fills the mean boundary layer and an outer diffraction region of thickness O 1=ε<sup>3</sup>=<sup>2</sup> . (The purely passive Stokes layer near the wall

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 Δθ ¼ Oð Þ1 [14] at the moderately supersonic Mach numbers being considered here. It will however emerge much further upstream when θ is close to the critical angle θc, i.e., when Δθ ≪ 1. But the solution in Ref. [10] does not apply when Δθ ≪ 1 and a new analysis was developed in Ref. [11] to extend their result into the small -Δθ regime.

<sup>α</sup> <sup>¼</sup> <sup>α</sup>~=Δ<sup>θ</sup> <sup>þ</sup> <sup>α</sup>~<sup>1</sup> <sup>þ</sup> …, <sup>β</sup> <sup>¼</sup> <sup>β</sup><sup>1</sup> <sup>¼</sup> <sup>β</sup>~=Δ<sup>θ</sup> (21)

θ<sup>c</sup> (22)

<sup>~</sup> � <sup>1</sup>, <sup>α</sup>~<sup>1</sup> � <sup>1</sup><sup>=</sup> sin <sup>2</sup>

Δθ � θ<sup>c</sup> � θ (19)

cos θ<sup>c</sup> � 1=M<sup>∞</sup> (20)

disturbances before considering these expansions.

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

degrees at Mach numbers between 2 and 6.

of the obliqueness angle θ from the critical angle

does not play a role in the diffraction process and can be ignored).

α~ � 1= tan θc, β

<sup>x</sup> <sup>¼</sup> <sup>O</sup> <sup>ε</sup>�<sup>6</sup> when the deviation

It follows from (7) that

where

87

while u xð Þ ; <sup>η</sup> ; v xð Þ ; <sup>η</sup> ; <sup>0</sup>; <sup>ϑ</sup>ð Þ <sup>x</sup>; <sup>η</sup> � � exp <sup>i</sup> <sup>β</sup>z=<sup>ε</sup> � <sup>t</sup> � � is a quasi-two dimensional solution that satisfies the two dimensional linearized boundary layer equations subject to the boundary conditions

$$
\overline{u} \to e^{i\chi}, \quad \overline{w}, \overline{\theta} \to 0 \text{ as } \eta \to \infty. \tag{13}
$$

The lowest order triple-deck solution will match onto the quasi-two dimensional solution <sup>u</sup>; <sup>v</sup>; <sup>0</sup>; <sup>ϑ</sup> � � exp <sup>i</sup> <sup>β</sup>z=<sup>ε</sup> � <sup>t</sup> � � of the two dimensional boundary layer equations, where the spanwise dependence only enters parametrically through the exponential factor in(11) .

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

$$\overline{u}(\mathbf{x},\boldsymbol{\eta}) = -\frac{B(\boldsymbol{\varkappa})U'(\boldsymbol{\eta})}{T\sqrt{2\boldsymbol{\varkappa}}},\\\overline{\boldsymbol{\vartheta}}(\boldsymbol{\varkappa},\boldsymbol{\jmath}) = -\frac{B(\boldsymbol{\varkappa})T'(\boldsymbol{\eta})}{T(\boldsymbol{\eta})\sqrt{2\boldsymbol{\varkappa}}},\tag{14}$$

$$i\overline{\boldsymbol{\nu}}(\boldsymbol{\varkappa},\boldsymbol{\eta}) = i\boldsymbol{B}(\boldsymbol{\varkappa}) + \frac{d\boldsymbol{B}}{d\boldsymbol{\varkappa}}\boldsymbol{U}(\boldsymbol{\eta}) - \boldsymbol{B}(\boldsymbol{\varkappa})\frac{\boldsymbol{U}'(\boldsymbol{\eta})\boldsymbol{\eta}\_c}{2\boldsymbol{\varkappa}},\tag{15}$$

where

$$\eta\_c \equiv \frac{1}{T(\eta)} \int\_0^{\eta} T(\bar{\eta}) d\bar{\eta} \tag{16}$$

is an exact eigensolution of the two-dimensional linearized unsteady boundary layer equations that satisfies the homogeneous boundary conditions u xð Þ ; η , w xð Þ ; η , ϑð Þ! x; η 0 as η ! ∞ for all B xð Þ, but does not necessarily satisfy the no-slip condition at the wall.

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 x becomes large. These solutions exhibit a two-layer structure consisting of an outer region that encompasses the main part of the boundary layer and a thin viscous region near the wall. The outer solution is given by (14) and (15) with the arbitrary function B xð Þ determined by matching with the viscous wall layer flow.

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 B xð Þ is then given by

$$B(\mathbf{x}) = \mathbf{x}^{3/2} \mathbf{B}\_n \exp\left[ -\frac{2^{3/2} e^{i\mathbf{x}/4}}{3\lambda \xi\_n^{3/2}} \left( \frac{T\_w}{\mu\_w} \right)^{1/2} \mathbf{x}^{3/2} \right] + \dots \tag{17}$$

where Tw � Tð Þ 0 , μ<sup>w</sup> � μð Þ Tð Þ 0 , λ � U<sup>0</sup> ð Þ 0 and ζ<sup>n</sup> is a root of

$$Ai'(\xi\_n) = 0, \text{ for } n = 0, 1, 2, 3... \tag{18}$$

The only difference from the Lam-Rott result is the ð Þ Tw=μ<sup>w</sup> <sup>1</sup>=<sup>2</sup> factor in the exponent. The asymptotic solution to the full inhomogeneous boundary value problem can now be expressed as the sum of a Stokes layer solution plus a number of these asymptotic eigensolutions. The first few Bn were determined from

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

numerical solutions to the boundary layer problem in Ref. [8]. But we are primarily concerned with the lowest order n ¼ 0 mode because that is the only one that matches onto a spatially growing oblique Tollmien-Schlicting wave further downstream [11]. The receptivity problem can then be solved by combining the numerical computations with appropriate matched asymptotic expansions to relate the instability wave amplitude to that of the free-stream disturbance. But we will analyze the boundary layer disturbances generated by the free-stream acoustic disturbances before considering these expansions.

## 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 α and β satisfy the dispersion relation (7) when the incidence angle γ is equal to zero, which, as noted, above is the case being considered here. Their focus was on hypersonic flows where the most rapidly growing disturbances are usually two dimensional 2nd Mack modes, while, as noted in the introduction, the focus of the present chapter is on the relatively low supersonic Mach number regime (say, less than about 4) where the most rapidly growing instability waves are highly oblique 1st Mack modes. Numerical results [9] show that the obliqueness angle of the most rapidly growing 1st mode lies between 50 and 70 degrees at Mach numbers between 2 and 6.

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 <sup>x</sup> <sup>¼</sup> <sup>o</sup> <sup>ε</sup>�<sup>3</sup> ð Þ can be matched onto a 1st Mack mode instability in the downstream region where <sup>x</sup> <sup>¼</sup> <sup>O</sup> <sup>ε</sup>�<sup>6</sup> when the deviation

$$
\Delta\theta \equiv \theta\_{\hat{\epsilon}} - \theta \tag{19}
$$

of the obliqueness angle θ from the critical angle

$$\cos \theta\_{\varepsilon} \equiv \mathbf{1}/M\_{\infty} \tag{20}$$

takes on Oð Þ1 positive values. The diffraction region has a double layer structure which consists of a region that fills the mean boundary layer and an outer diffraction region of thickness O 1=ε<sup>3</sup>=<sup>2</sup> . (The purely passive Stokes layer near the wall does not play a role in the diffraction process and can be ignored).

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 Δθ ¼ Oð Þ1 [14] at the moderately supersonic Mach numbers being considered here. It will however emerge much further upstream when θ is close to the critical angle θc, i.e., when Δθ ≪ 1. But the solution in Ref. [10] does not apply when Δθ ≪ 1 and a new analysis was developed in Ref. [11] to extend their result into the small -Δθ regime.

It follows from (7) that

$$a = \tilde{a}/\Delta\theta + \tilde{a}\_1 + \dots, \quad \beta = \beta\_1 = \tilde{\beta}/\Delta\theta \tag{21}$$

where

$$
\bar{a} \equiv \mathbf{1}/\tan\theta\_c,\\
\bar{\beta} \equiv \mathbf{1},\\
\bar{a}\_1 \equiv \mathbf{1}/\sin^2\theta\_c \tag{22}
$$

uð Þ <sup>0</sup> , ϑ

u xð Þ¼� ; η

u ! e

Boundary Layer Flows - Theory, Applications and Numerical Methods

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

B xð ÞU<sup>0</sup>

v xð Þ¼ ; <sup>η</sup> iB xð Þþ dB

ð Þη T ffiffiffiffiffi

<sup>η</sup><sup>c</sup> � <sup>1</sup> Tð Þη

layer equations that satisfies the homogeneous boundary conditions

to the boundary conditions

exponential factor in(11) .

the no-slip condition at the wall.

with the viscous wall layer flow.

B xð Þ¼ <sup>x</sup><sup>3</sup>=<sup>2</sup>

where Tw � Tð Þ 0 , μ<sup>w</sup> � μð Þ Tð Þ 0 , λ � U<sup>0</sup>

where

86

ð Þ <sup>0</sup> ! <sup>0</sup>, <sup>w</sup>ð Þ <sup>0</sup> ! <sup>e</sup>

while u xð Þ ; <sup>η</sup> ; v xð Þ ; <sup>η</sup> ; <sup>0</sup>; <sup>ϑ</sup>ð Þ <sup>x</sup>; <sup>η</sup> � � exp <sup>i</sup> <sup>β</sup>z=<sup>ε</sup> � <sup>t</sup> � � is a quasi-two dimensional solution that satisfies the two dimensional linearized boundary layer equations subject

The lowest order triple-deck solution will match onto the quasi-two dimensional solution <sup>u</sup>; <sup>v</sup>; <sup>0</sup>; <sup>ϑ</sup> � � exp <sup>i</sup> <sup>β</sup>z=<sup>ε</sup> � <sup>t</sup> � � of the two dimensional boundary layer equations, where the spanwise dependence only enters parametrically through the

<sup>2</sup><sup>x</sup> <sup>p</sup> , <sup>ϑ</sup>ð Þ¼� <sup>x</sup>; <sup>y</sup>

ð η

0

is an exact eigensolution of the two-dimensional linearized unsteady boundary

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 x becomes large. These solutions exhibit a two-layer structure consisting of an outer region that encompasses the main part of the boundary layer and a thin viscous region near the wall. The outer solution is given by (14) and (15) with the arbitrary function B xð Þ determined by matching

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 B xð Þ is then given by

> ei<sup>π</sup>=<sup>4</sup> 3λς<sup>3</sup>=<sup>2</sup> <sup>n</sup>

Tw μw � �<sup>1</sup>=<sup>2</sup>

" #

x<sup>3</sup>=<sup>2</sup>

ð Þ 0 and ζ<sup>n</sup> is a root of

Ai<sup>0</sup> ς<sup>n</sup> ð Þ¼ 0, for n ¼ 0, 1, 2, 3:… (18)

þ :… (17)

<sup>1</sup>=<sup>2</sup> factor in the

Bn exp � <sup>23</sup><sup>=</sup><sup>2</sup>

The only difference from the Lam-Rott result is the ð Þ Tw=μ<sup>w</sup>

exponent. The asymptotic solution to the full inhomogeneous boundary value problem can now be expressed as the sum of a Stokes layer solution plus a number

of these asymptotic eigensolutions. The first few Bn were determined from

u xð Þ ; η , w xð Þ ; η , ϑð Þ! x; η 0 as η ! ∞ for all B xð Þ, but does not necessarily satisfy

dx <sup>U</sup>ð Þ� <sup>η</sup> B xð Þ <sup>U</sup><sup>0</sup>

ix, as <sup>η</sup> ! <sup>∞</sup>, (12)

ix, w, <sup>ϑ</sup> ! 0 as <sup>η</sup> ! <sup>∞</sup>: (13)

B xð ÞT<sup>0</sup>

ð Þη η<sup>c</sup>

ð Þη <sup>T</sup>ð Þ<sup>η</sup> ffiffiffiffiffi

Tð Þ ~η d~η (16)

<sup>2</sup><sup>x</sup> <sup>p</sup> , (14)

<sup>2</sup><sup>x</sup> , (15)

when <sup>Δ</sup><sup>θ</sup> <sup>≪</sup> 1 since tan ð Þ¼ <sup>θ</sup><sup>c</sup> � <sup>Δ</sup><sup>θ</sup> tan <sup>θ</sup><sup>c</sup> � <sup>Δ</sup>θ<sup>=</sup> cos <sup>2</sup>θ<sup>c</sup> <sup>þ</sup> <sup>O</sup>ð Þ <sup>Δ</sup><sup>θ</sup> <sup>2</sup> in that case. This shows that α also becomes large when Δθ ≪ 1 and that α will expand in powers of Δθ as indicated in (21) if β is fixed at the indicated value to all orders in Δθ (which we now assume to be the case).

The spanwise wavenumber will equal the vortical spanwise wavenumber (8) when Δθ ¼ Oð Þε and as in that case the diffraction wave solution will eventually develop a triple-deck structure but the resulting solution will (as shown in [11]) not decay at large wall normal distances and is therefore invalid. This means that the diffraction region solution cannot be continued downstream for Δθ ¼ Oð Þε .

Ref. [11] shows that the smallest value of <sup>Δ</sup><sup>θ</sup> is <sup>Δ</sup><sup>θ</sup> <sup>¼</sup> <sup>O</sup> <sup>ε</sup>2=<sup>3</sup> � � and the diffraction region will then occur at an O ε�4=<sup>3</sup> � � distance downstream. The relevant solution will have the triple-deck structure shown in Figure 3: a main boundary layer region that fills the mean boundary layer (region 1), an outer diffraction region of thickness <sup>O</sup> <sup>ε</sup>�1=<sup>3</sup> � �(region 2) and an <sup>O</sup> <sup>ε</sup><sup>3</sup> ð Þ thick viscous wall layer in which the unsteady, convective and viscous terms all balance.

The pressure in region 2 is of the form

$$p = \mathbf{1} + \hat{\delta} p\_2(\mathbf{x}\_2, y\_2) e^{i\left[\left(\bar{a}/\Delta\theta + \bar{a}\_1\right)\mathbf{x} + \bar{\beta}\mathbf{x}/\Delta\theta - t\right]},\tag{23}$$

<sup>v</sup>1ð Þ <sup>x</sup>2; <sup>∞</sup>ffiffiffiffiffiffiffi 2x<sup>2</sup>

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

as x<sup>2</sup> ! ∞ since ([22], pp. 446–447)

which behaves like

where

the solution behaves like

p ¼ ip1ð Þ x<sup>2</sup>

<sup>v</sup>1ð Þ <sup>x</sup>2; <sup>∞</sup> <sup>=</sup> ffiffiffiffiffiffiffi

Ai0 ð Þξ = ∞ð

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

p1ð Þ¼ x<sup>2</sup> 1 � γ0x<sup>2</sup>

2x<sup>2</sup>

Leading Edge Receptivity at Subsonic and Moderately Supersonic Mach Numbers

ξ

ð

ffiffiffi σ p ffiffiffiffiffiffiffiffiffiffiffi

<sup>α</sup>~<sup>2</sup> <sup>þ</sup> <sup>β</sup>~<sup>2</sup> � �α~<sup>1</sup>=<sup>2</sup>

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

<sup>0</sup>πð Þ x<sup>2</sup>

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

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

i <sup>1</sup> ε3 Ð x1 0

κð Þ x1;ε dx1þβz�t � �

^ ≪ 1 is the common scale factor introduced at

(33)

^Πð Þ <sup>y</sup>; <sup>ε</sup> <sup>e</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2πi M<sup>2</sup>

<sup>∞</sup> � 1

T2 w

01

λ

structure much further downstream. The minimum distance occurs when

γ<sup>0</sup> �

<sup>p</sup>1ð Þ� <sup>x</sup><sup>2</sup> exp <sup>γ</sup><sup>2</sup>

<sup>Δ</sup><sup>θ</sup> <sup>¼</sup> <sup>O</sup> <sup>ε</sup><sup>2</sup>=<sup>3</sup> � �. We begin by considering the triple-deck region.

f g u; v; w; p ¼ δ

4. The viscous triple-deck region

in the triple-deck region where δ

eigensolution of the form

the beginning of Section 2,

89

<sup>α</sup>~<sup>2</sup> <sup>þ</sup> <sup>~</sup><sup>β</sup> <sup>2</sup> � �T<sup>2</sup>

λAi<sup>0</sup> ξ<sup>2</sup> ð Þ

<sup>p</sup> � �i p1ð Þ <sup>x</sup><sup>2</sup> <sup>α</sup>~<sup>2</sup> <sup>þ</sup> <sup>β</sup>~<sup>2</sup> � �T<sup>2</sup>

<sup>w</sup>ξ<sup>2</sup>

∞ð

Aið Þξ dξ, (27)

<sup>w</sup>=λ (28)

ξ2

Ai qð Þdq ! �ξ as ξ ! ∞: (29)

<sup>1</sup> � <sup>σ</sup> <sup>p</sup> <sup>p</sup>1ð Þ <sup>σ</sup>x<sup>2</sup> <sup>d</sup>σ, as <sup>x</sup><sup>2</sup> ! <sup>∞</sup> (30)

<sup>q</sup> � � , (31)

<sup>2</sup> h i as <sup>x</sup><sup>2</sup> ! <sup>∞</sup>: (32)

where

$$\varkappa\_2 \equiv \varkappa \epsilon^{4/3} = O(\mathbf{1}), \quad \jmath\_2 \equiv \jmath \epsilon^{1/3} = O(\mathbf{1}) \tag{24}$$

and the surface pressure p2ð Þ x2; 0 is related to the up-wash velocity v1ð Þ� x2; ∞ lim<sup>η</sup>!<sup>∞</sup> v1ð Þ x2; η at the outer edge of the boundary layer by

$$p\_2(\mathbf{x}\_2, \mathbf{0}) = p\_1(\mathbf{x}\_2) = \mathbf{1} - \frac{\mathbf{x}\_2}{\sqrt{2\pi i \bar{a} \left(\mathcal{M}\_{\infty}^2 - 1\right)}} \prod\_{0}^{1} \frac{\sqrt{\sigma}}{\sqrt{1 - \sigma}} i\tilde{a} \left[\frac{v\_1(\mathbf{x}\_2 \sigma, \infty)}{\sqrt{\mathbf{x}\_2 \sigma}}\right] d\sigma,\tag{25}$$

where p1ð Þ x<sup>2</sup> denotes the pressure in the boundary layer region 1 (which is independent of the wall normal direction) and the wall normal velocity v1ð Þ x2; ∞ is given in terms of

$$\xi\_2 \equiv -i^{1/3} \left(\sqrt{2\kappa\_2}/\tilde{a}\lambda\right)^{2/3} \left(T\_w/\mu\_w\right)^{1/3} \tag{26}$$

and the integral and the derivative of the Airy function Aið Þξ by

Figure 3. Structure of diffraction region for <sup>Δ</sup><sup>θ</sup> <sup>¼</sup> <sup>O</sup> <sup>ε</sup><sup>2</sup>=<sup>3</sup> � �.

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

$$\frac{w\_1(\infty\_2, \infty)}{\sqrt{2\overline{\alpha\_2}}} = \dot{p}\_1(\infty\_2) \frac{\left(\ddot{a}^2 + \ddot{\beta}^2\right) T\_w^2 \xi\_2}{\lambda A \dot{i}'(\xi\_2)} \int\_{\xi\_2}^{\infty} Ai(\xi) d\xi,\tag{27}$$

which behaves like

when <sup>Δ</sup><sup>θ</sup> <sup>≪</sup> 1 since tan ð Þ¼ <sup>θ</sup><sup>c</sup> � <sup>Δ</sup><sup>θ</sup> tan <sup>θ</sup><sup>c</sup> � <sup>Δ</sup>θ<sup>=</sup> cos <sup>2</sup>θ<sup>c</sup> <sup>þ</sup> <sup>O</sup>ð Þ <sup>Δ</sup><sup>θ</sup> <sup>2</sup> in that case. This shows that α also becomes large when Δθ ≪ 1 and that α will expand in powers of Δθ as indicated in (21) if β is fixed at the indicated value to all orders in

The spanwise wavenumber will equal the vortical spanwise wavenumber (8) when Δθ ¼ Oð Þε and as in that case the diffraction wave solution will eventually develop a triple-deck structure but the resulting solution will (as shown in [11]) not decay at large wall normal distances and is therefore invalid. This means that the diffraction region solution cannot be continued downstream for Δθ ¼ Oð Þε .

Ref. [11] shows that the smallest value of <sup>Δ</sup><sup>θ</sup> is <sup>Δ</sup><sup>θ</sup> <sup>¼</sup> <sup>O</sup> <sup>ε</sup>2=<sup>3</sup> � � and the diffraction region will then occur at an O ε�4=<sup>3</sup> � � distance downstream. The relevant solution will have the triple-deck structure shown in Figure 3: a main boundary layer region that fills the mean boundary layer (region 1), an outer diffraction region of thickness <sup>O</sup> <sup>ε</sup>�1=<sup>3</sup> � �(region 2) and an <sup>O</sup> <sup>ε</sup><sup>3</sup> ð Þ thick viscous wall layer in which the unsteady,

<sup>i</sup> <sup>α</sup>~=Δθþα<sup>~</sup> ð Þ<sup>1</sup> ½ � <sup>x</sup>þβ~z=Δθ�<sup>t</sup> , (23)

<sup>x</sup><sup>2</sup> � <sup>x</sup>ε<sup>4</sup>=<sup>3</sup> <sup>¼</sup> <sup>O</sup>ð Þ<sup>1</sup> , y<sup>2</sup> � <sup>y</sup>ε<sup>1</sup>=<sup>3</sup> <sup>¼</sup> <sup>O</sup>ð Þ<sup>1</sup> (24)

ffiffiffi σ p ffiffiffiffiffiffiffiffiffiffiffi

ð Þ Tw=μ<sup>w</sup>

<sup>1</sup> � <sup>σ</sup> <sup>p</sup> <sup>i</sup>α<sup>~</sup> <sup>v</sup>1ð Þ <sup>x</sup>2σ; <sup>∞</sup> ffiffiffiffiffiffiffi

x2σ p � �

<sup>1</sup>=<sup>3</sup> (26)

dσ, (25)

Δθ (which we now assume to be the case).

Boundary Layer Flows - Theory, Applications and Numerical Methods

convective and viscous terms all balance. The pressure in region 2 is of the form

where

given in terms of

Figure 3.

88

Structure of diffraction region for <sup>Δ</sup><sup>θ</sup> <sup>¼</sup> <sup>O</sup> <sup>ε</sup><sup>2</sup>=<sup>3</sup> � �.

p ¼ 1 þ δ

<sup>p</sup>2ð Þ¼ <sup>x</sup>2; <sup>0</sup> <sup>p</sup>1ð Þ¼ <sup>x</sup><sup>2</sup> <sup>1</sup> � <sup>x</sup><sup>2</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ξ<sup>2</sup> � �i

^p<sup>2</sup> <sup>x</sup>2; <sup>y</sup><sup>2</sup> � �e

and the surface pressure p2ð Þ x2; 0 is related to the up-wash velocity v1ð Þ� x2; ∞ lim<sup>η</sup>!<sup>∞</sup> v1ð Þ x2; η at the outer edge of the boundary layer by

2πiα~ M<sup>2</sup>

<sup>1</sup>=<sup>3</sup> ffiffiffiffiffiffiffi 2x<sup>2</sup> p =αλ~ � �<sup>2</sup>=<sup>3</sup>

and the integral and the derivative of the Airy function Aið Þξ by

<sup>∞</sup> � 1 q � �

where p1ð Þ x<sup>2</sup> denotes the pressure in the boundary layer region 1 (which is independent of the wall normal direction) and the wall normal velocity v1ð Þ x2; ∞ is

ð 1

0

$$(\nu\_1(\varkappa\_2, \infty) / \sqrt{2\varkappa\_2} \sim -i p\_1(\varkappa\_2) \left(\bar{a}^2 + \bar{\beta}^2\right) T\_w^2 / \lambda \tag{28}$$

as x<sup>2</sup> ! ∞ since ([22], pp. 446–447)

$$Ai'(\xi) / \int\_{\xi}^{\infty} Ai(q) dq \to -\xi \text{ as } \xi \to \infty. \tag{29}$$

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

$$p\_1(\mathbf{x}\_2) = \mathbf{1} - \chi\_0 \mathbf{x}\_2 \int \frac{\sqrt{\sigma}}{\sqrt{\mathbf{1} - \sigma}} p\_1(\sigma \mathbf{x}\_2) d\sigma,\text{ as } \mathbf{x}\_2 \to \infty \tag{30}$$

$$\mathbb{I}\_1$$

where

$$\chi\_0 \equiv \frac{\left(\bar{a}^2 + \tilde{\boldsymbol{\beta}}^2\right) \bar{a}^{1/2} T\_w^2}{\lambda \sqrt{2\pi i \left(M\_\infty^2 - 1\right)}},\tag{31}$$

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

$$p\_1(\varkappa\_2) \sim \exp\left[\chi\_0^2 \pi (\varkappa\_2)^2\right] \text{ as } \varkappa\_2 \to \infty. \tag{32}$$

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 <sup>Δ</sup><sup>θ</sup> <sup>¼</sup> <sup>O</sup> <sup>ε</sup><sup>2</sup>=<sup>3</sup> � �. We begin by considering the triple-deck region.

#### 4. The viscous triple-deck region

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

$$\{\{u,v,w,p\}\} = \hat{\delta}\Pi(\mathcal{y},\varepsilon)e^{i\left[\frac{1}{\varepsilon^3}\int\_0^{\mathbf{x}\_1} \kappa(\mathbf{x}\_1,\varepsilon)d\mathbf{x}\_1 + \overline{\beta}\overline{\mathbf{x}} - t\right]}\tag{33}$$

in the triple-deck region where δ ^ ≪ 1 is the common scale factor introduced at the beginning of Section 2,

$$\Pi(\boldsymbol{y},\varepsilon) = \left\{ \frac{\overline{A}(\boldsymbol{\varkappa}\_{1})U'(\boldsymbol{\eta})}{T(\boldsymbol{\eta})}, -i\kappa\_{0}\overline{A}(\boldsymbol{\varkappa}\_{1})U(\boldsymbol{\eta})\sqrt{2\boldsymbol{\varkappa}\_{1}} - \frac{\varepsilon^{2}\overline{\beta}\overline{P}}{\kappa\_{0}U(\boldsymbol{\eta})}, \varepsilon^{2}\overline{P} \right\} \tag{34}$$

in the main boundary layer where η ¼ Oð Þ1 ,

$$\mathbf{x}\_1 \equiv \varepsilon^2 \mathbf{x} = \mathcal{O}(\mathbf{1}) \tag{35}$$

4.2 Numerical results

streamwise coordinate x†

transverse wavenumber β

changes at small x†

parts of the small-x†

ε�<sup>2</sup>Im Ð ð Þ x<sup>1</sup> LB 0

Figure 4. Re κ † 0

91

� � as a function of x†

main graph is the rescaled large-x†

is relatively small.

κ †

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

<sup>0</sup> <sup>¼</sup> <sup>κ</sup>0T1=<sup>2</sup>

The real and negative imaginary parts of κ

†

<sup>w</sup> μ1=<sup>6</sup> <sup>w</sup> , x†

<sup>1</sup> asymptotic formula (41).

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

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 n ¼ 0 root can produce the spatially growing modes of (38). The wall temperature Tw and viscosity

> <sup>w</sup>=μ<sup>w</sup> 2=3 , β †

> > †

<sup>¼</sup> <sup>β</sup>T1=<sup>2</sup> <sup>w</sup> μ1=<sup>6</sup>

<sup>1</sup> in Figures 4 and 5 for three values of the frequency scaled

≥2. The insets are included to more clearly show the

<sup>1</sup> . The dashed curves in the insets denote the real and imaginary

<sup>0</sup> calculated from (38) together with

xLB 0

κð Þ x<sup>1</sup> dx ¼

<sup>1</sup> ¼ 0:01

<sup>w</sup> : (42)

μ<sup>w</sup> can be scaled out of this equation by introducing the rescaled variables.

Leading Edge Receptivity at Subsonic and Moderately Supersonic Mach Numbers

<sup>1</sup> <sup>¼</sup> <sup>x</sup>1T<sup>2</sup>

the n ¼ 0 Lam-Rott initial condition (41) are plotted as a function of the scaled

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.

κð Þ x<sup>1</sup> dx1, where ð Þ x<sup>1</sup> LB and xLB denote the scaled and unscaled streamwise location of the lower branch of the neutral stability, which implies that the total damping is proportional to the area under the growth rate curve between

of this upstream region is very short and therefore that the total amount of damping

<sup>1</sup> calculated from (38) together with the initial condition (41) for M<sup>∞</sup> ¼ 2, 3, 4

†

≥2. The dashed curve in the

(double dot dashed, dot dashed, and solid lines, respectively) and three values of β

<sup>1</sup> asymptote (49).

zero and the lower branch in Figure 5. The inset shows that the length Δx†

and

$$\overline{z} \equiv z/\varepsilon = z^\* a^\*/\varepsilon \mathcal{U}\_{\infty}^\* \tag{36}$$

is a scaled transverse coordinate. The complex wavenumber κ has the expansion [11].

$$\kappa(\mathbf{x}\_1, \varepsilon) = \kappa\_0(\mathbf{x}\_1) + \varepsilon \kappa\_1(\mathbf{x}\_1) + \varepsilon^2 \kappa\_2(\mathbf{x}\_1) + \dots,\tag{37}$$

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

$$\kappa\_0^2 + \overline{\rho}^2 = \frac{1}{(i\kappa\_0)^{1/3}} \left(\frac{\lambda}{\sqrt{2\varkappa\_1}}\right)^{5/3} \left(\frac{\mu\_w}{T\_w^{\overline{\gamma}}}\right)^{1/3} \frac{\left[\overline{\rho}^2 - (\mathcal{M}\_\infty^2 - 1)\kappa\_0^2\right]^{1/2} \text{Ai}'(\xi\_0)}{\int\_{\xi\_0} \text{Ai}(q) dq} \tag{38}$$

and

$$\xi\_0 = -i^{1/3} \left(\frac{\sqrt{2\kappa\_1}}{\kappa\_0 \lambda}\right)^{2/3} (T\_w/\mu\_w)^{1/3} \tag{39}$$

whose solution must satisfy the inequality

$$\operatorname{Re}\left[\overline{\beta}^2 - (\mathcal{M}\_{\infty}^2 - \mathbf{1})\kappa\_0^2\right]^{1/2} \ge \mathbf{0} \tag{40}$$

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

This requirement will be satisfied for all M<sup>∞</sup> < 1 but will only be satisfied at supersonic Mach numbers when the obliqueness angle θ is greater than the critical angle <sup>θ</sup><sup>c</sup> � cos �<sup>1</sup>ð Þ <sup>1</sup>=M<sup>∞</sup> [11, 13]. The dispersion relation (38) and (39) reduces to the dispersion relation given by Eqs. (4.52), (5.2) and (5.3) of [7] when β and M<sup>∞</sup> are set to zero.

#### 4.1 Matching with the Lam-Rott solution

The dispersion relation (38) and (39) will be satisfied at small values of x<sup>1</sup> if κ<sup>0</sup> � ffiffiffiffiffi x1 <sup>p</sup> and <sup>ξ</sup><sup>0</sup> ! <sup>ζ</sup>n, for <sup>n</sup> <sup>¼</sup> <sup>0</sup>, <sup>1</sup>, <sup>2</sup>… as <sup>x</sup><sup>1</sup> ! <sup>0</sup>, where <sup>ς</sup><sup>n</sup> is the nth root of the Lam-Rott dispersion relation (18). Inserting this into (38) shows that

$$\kappa\_0 \to \frac{1}{\lambda \mathfrak{s}\_n^{3/2}} \left( \frac{2T\_w \varkappa\_1}{i\mu\_w} \right)^{1/2} \text{as } \varkappa\_1 \to \mathbf{0}. \tag{41}$$

The cross flow velocity w drops out of (33) as x<sup>1</sup> ! 0 and the flow in the main deck is therefore compatible with the quasi-two dimensional Lam-Rott solution (14)–(17).

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

### 4.2 Numerical results

<sup>Π</sup>ð Þ¼ <sup>y</sup>; <sup>ε</sup> A xð Þ<sup>1</sup> <sup>U</sup><sup>0</sup>

and

expansion [11].

κ2 <sup>0</sup> þ β

and

growth.

κ<sup>0</sup> � ffiffiffiffiffi x1

(14)–(17).

90

relation ([13, 14, 23])

<sup>2</sup> <sup>¼</sup> <sup>1</sup> ð Þ iκ<sup>0</sup> 1=3 ð Þη

Boundary Layer Flows - Theory, Applications and Numerical Methods

in the main boundary layer where η ¼ Oð Þ1 ,

λ ffiffiffiffiffiffiffi 2x<sup>1</sup> p � �<sup>5</sup>=<sup>3</sup>

ξ<sup>0</sup> ¼ �i

Re β 2 � <sup>M</sup><sup>2</sup>

whose solution must satisfy the inequality

4.1 Matching with the Lam-Rott solution

1=3

<sup>T</sup>ð Þ<sup>η</sup> ; �iκ0A xð Þ<sup>1</sup> <sup>U</sup>ð Þ<sup>η</sup> ffiffiffiffiffiffiffi

<sup>x</sup><sup>1</sup> � <sup>ε</sup><sup>2</sup>

<sup>z</sup> � <sup>z</sup>=<sup>ε</sup> <sup>¼</sup> <sup>z</sup><sup>∗</sup>

is a scaled transverse coordinate. The complex wavenumber κ has the

where the lowest order term in this expansion satisfies the following dispersion

2 � <sup>M</sup><sup>2</sup>

<sup>∞</sup> � <sup>1</sup> � �κ<sup>2</sup>

h i<sup>1</sup>=<sup>2</sup>

Ð ∞ ξ0

ð Þ Tw=μ<sup>w</sup>

0

Ai qð Þdq

<sup>κ</sup>ð Þ¼ <sup>x</sup>1; <sup>ε</sup> <sup>κ</sup>0ð Þþ <sup>x</sup><sup>1</sup> εκ1ð Þþ <sup>x</sup><sup>1</sup> <sup>ε</sup><sup>2</sup>

μw T7 w

!<sup>1</sup>=<sup>3</sup> β

ffiffiffiffiffiffiffi 2x<sup>1</sup> p κ0λ � �<sup>2</sup>=<sup>3</sup>

<sup>∞</sup> � <sup>1</sup> � �κ<sup>2</sup>

This requirement will be satisfied for all M<sup>∞</sup> < 1 but will only be satisfied at supersonic Mach numbers when the obliqueness angle θ is greater than the critical angle <sup>θ</sup><sup>c</sup> � cos �<sup>1</sup>ð Þ <sup>1</sup>=M<sup>∞</sup> [11, 13]. The dispersion relation (38) and (39) reduces to the dispersion relation given by Eqs. (4.52), (5.2) and (5.3) of [7] when β and M<sup>∞</sup> are set to zero.

The dispersion relation (38) and (39) will be satisfied at small values of x<sup>1</sup> if

2Twx<sup>1</sup> iμ<sup>w</sup> � �<sup>1</sup>=<sup>2</sup>

The cross flow velocity w drops out of (33) as x<sup>1</sup> ! 0 and the flow in the main deck is therefore compatible with the quasi-two dimensional Lam-Rott solution

Lam-Rott dispersion relation (18). Inserting this into (38) shows that

<sup>κ</sup><sup>0</sup> ! <sup>1</sup> λς<sup>3</sup>=<sup>2</sup> <sup>n</sup>

<sup>p</sup> and <sup>ξ</sup><sup>0</sup> ! <sup>ζ</sup>n, for <sup>n</sup> <sup>¼</sup> <sup>0</sup>, <sup>1</sup>, <sup>2</sup>… as <sup>x</sup><sup>1</sup> ! <sup>0</sup>, where <sup>ς</sup><sup>n</sup> is the nth root of the

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

h i<sup>1</sup>=<sup>2</sup>

0

2x<sup>1</sup>

ω<sup>∗</sup>=εU<sup>∗</sup>

� �

<sup>p</sup> , � <sup>ε</sup>2βPTð Þ<sup>η</sup>

<sup>κ</sup>0Uð Þ<sup>η</sup> ; <sup>ε</sup><sup>2</sup>

x ¼ Oð Þ1 (35)

P

<sup>∞</sup> (36)

κ2ð Þþ x<sup>1</sup> :…, (37)

Ai<sup>0</sup> ξ<sup>0</sup> ð Þ

<sup>1</sup>=<sup>3</sup> (39)

≥0 (40)

as x<sup>1</sup> ! 0: (41)

(34)

(38)

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 n ¼ 0 root can produce the spatially growing modes of (38). The wall temperature Tw and viscosity μ<sup>w</sup> can be scaled out of this equation by introducing the rescaled variables.

$$\kappa\_0^\dagger = \kappa\_0 T\_w^{1/2} \mu\_w^{1/6}, \mathfrak{x}\_1^\dagger = \mathfrak{x}\_1 T\_w^2 / \mu\_w^{2/3}, \overline{\beta}^\dagger = \overline{\beta} T\_w^{1/2} \mu\_w^{1/6}.\tag{42}$$

The real and negative imaginary parts of κ † <sup>0</sup> calculated from (38) together with the n ¼ 0 Lam-Rott initial condition (41) are plotted as a function of the scaled streamwise coordinate x† <sup>1</sup> in Figures 4 and 5 for three values of the frequency scaled transverse wavenumber β † ≥2. The insets are included to more clearly show the changes at small x† <sup>1</sup> . The dashed curves in the insets denote the real and imaginary parts of the small-x† <sup>1</sup> asymptotic formula (41).

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 Im Ð xLB 0 κð Þ x<sup>1</sup> dx ¼

ε�<sup>2</sup>Im Ð ð Þ x<sup>1</sup> LB 0 κð Þ x<sup>1</sup> dx1, where ð Þ x<sup>1</sup> LB and xLB denote the scaled and unscaled streamwise location of the lower branch of the neutral stability, which implies that the total damping is proportional to the area under the growth rate curve between zero and the lower branch in Figure 5. The inset shows that the length Δx† <sup>1</sup> ¼ 0:01 of this upstream region is very short and therefore that the total amount of damping is relatively small.

#### Figure 4.

Re κ † 0 � � as a function of x† <sup>1</sup> calculated from (38) together with the initial condition (41) for M<sup>∞</sup> ¼ 2, 3, 4 (double dot dashed, dot dashed, and solid lines, respectively) and three values of β † ≥2. The dashed curve in the main graph is the rescaled large-x† <sup>1</sup> asymptote (49).

where <sup>α</sup>^<sup>0</sup> � <sup>M</sup><sup>2</sup>

(43) and (45) that

<sup>1</sup>=ε<sup>3</sup> ð Þ xð1

0

5.2 Numerical results

upstream.

Figure 6.

93

Scaled wavenumber κ0=β ¼ κ0=β as a function of βTw

limit.

∞T<sup>2</sup>

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

<sup>w</sup>= M<sup>2</sup>

<sup>κ</sup>0ð Þ <sup>x</sup><sup>1</sup> dx<sup>1</sup> <sup>¼</sup> <sup>1</sup>=ε<sup>4</sup> � �x^ð<sup>1</sup>

function of the scaled streamwise coordinate βTw

<sup>¼</sup> ð Þ <sup>α</sup>~=Δ<sup>θ</sup> <sup>x</sup> � <sup>β</sup><sup>5</sup>

<sup>∞</sup> � <sup>1</sup> � �7=<sup>4</sup>

0

α^0 <sup>2</sup> <sup>ε</sup><sup>3</sup> ð Þ <sup>x</sup> 2

λ h i. The square root <sup>β</sup>

Leading Edge Receptivity at Subsonic and Moderately Supersonic Mach Numbers

satisfies the inequality (40) when x^<sup>1</sup> ! 0 and (44) therefore remains valid in this

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 <sup>β</sup> <sup>¼</sup> <sup>O</sup> <sup>ε</sup>2=3=Δ<sup>θ</sup> � � and <sup>x</sup><sup>2</sup> is given by (24) since it follows from (8),(35),

Figure 6 is a plot of the scaled lowest order wavenumber κ0=β ¼ κ0=β as a

various values of the free-stream Mach number M<sup>∞</sup> calculated from the inviscid triple-deck dispersion relation(44) together with the asymptotic initial condition (45) which is shown by the dashed curves in the figure. The lowest order wave number κ<sup>0</sup> is purely real which means that exponential growth (if it occurs) can only occur at higher order. This suggests that the acoustically generated instabilities will be less significant than the vortically-generated instabilities which appear

� �<sup>4</sup>

solid lines represent the numerical solution. Dashed lines are the asymptotic solution(45) .

<sup>x</sup>^1=λ<sup>2</sup> <sup>¼</sup> <sup>β</sup>Tw � �<sup>4</sup>

x1=λ<sup>2</sup> for various values of M∞. The

κ0ð Þ x^<sup>1</sup> dx^<sup>1</sup> ! ð Þ α~=Δθ x � εβ

=2 ¼ α~=ΔθÞx � α^<sup>0</sup>

� �<sup>4</sup>

2 � <sup>M</sup><sup>2</sup>

> 5 α^0 2 x2=2

2 x2

<sup>x</sup>^1=λ<sup>2</sup> <sup>¼</sup> <sup>β</sup>Tw

� �<sup>4</sup>

x1=λ<sup>2</sup> for

<sup>2</sup>=2: �

<sup>∞</sup> � <sup>1</sup> � �κ<sup>2</sup>

� �1=<sup>2</sup>

0

still

(46)

#### Figure 5.

�Im κ † 0 � � as a function of x† <sup>1</sup> calculated from (38) together with the initial condition (41) for M<sup>∞</sup> ¼ 2, 3, 4 (double dot dashed, dot dashed and solid lines, respectively) and three values of the frequency scaled transverse wavenumber.

#### 5. The inviscid triple-deck region

As noted above the acoustically driven solution will only match onto an eigensolution in the downstream region when <sup>O</sup>ð Þ <sup>Δ</sup><sup>θ</sup> <sup>≥</sup>ε<sup>3</sup><sup>=</sup>2. This region will lie downstream of the viscous triple-deck region considered above and will be closest to that region when <sup>O</sup>ð Þ¼ <sup>Δ</sup><sup>θ</sup> <sup>ε</sup><sup>3</sup><sup>=</sup>2. It will have an inviscid triple- deck structure and the relevant dispersion relation can be obtained by putting <sup>ε</sup>=Δ<sup>θ</sup> <sup>¼</sup> <sup>O</sup> <sup>ε</sup><sup>1</sup>=<sup>3</sup> � � in (21), inserting the rescaled variables

$$
\overline{\beta} = \overline{\beta}/\varepsilon^{1/3}, \overline{\kappa}\_0 = \kappa\_0/\varepsilon^{1/3}, \hat{\mathfrak{x}}\_1 = \mathfrak{x}\_1 e^{4/3} \tag{43}
$$

into (38), using (29), and taking the limit as ε ! 0 with β, κ<sup>0</sup> and x^<sup>1</sup> held fixed, to show that the rescaled wavenumber κ<sup>0</sup> satisfies the inviscid dispersion relation

$$\overline{\kappa}\_0^2 + \overline{\overline{\beta}}^2 = \frac{\lambda \left[ \overline{\overline{\beta}}^2 - (\mathcal{M}\_\infty^2 - 1)\overline{\kappa}\_0^2 \right]^{1/2}}{\overline{\kappa}\_0 \sqrt{2\hat{\kappa}\_1} T\_w^2} \tag{44}$$

when the square root β 2 � <sup>M</sup><sup>2</sup> <sup>∞</sup> � <sup>1</sup> � �κ<sup>2</sup> 0 � �<sup>1</sup>=<sup>2</sup> is required to remain finite as ε ! 0.

#### 5.1 Matching with the small Δθ Fedorov/Khokhlov solution

It can then be shown by direct substitution that the solution κ<sup>0</sup> behaves like

$$
\overline{\kappa}\_0 \to \frac{\overline{\beta}}{\left(M\_\infty^2 - 1\right)^{1/2}} - \overline{\overline{\beta}}^\sharp \widehat{a}\_0^2 \widehat{\kappa}\_1 \quad \text{as } \widehat{\kappa}\_1 \to 0,\tag{45}
$$

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

where <sup>α</sup>^<sup>0</sup> � <sup>M</sup><sup>2</sup> ∞T<sup>2</sup> <sup>w</sup>= M<sup>2</sup> <sup>∞</sup> � <sup>1</sup> � �7=<sup>4</sup> λ h i. The square root <sup>β</sup> 2 � <sup>M</sup><sup>2</sup> <sup>∞</sup> � <sup>1</sup> � �κ<sup>2</sup> 0 � �1=<sup>2</sup> still satisfies the inequality (40) when x^<sup>1</sup> ! 0 and (44) therefore remains valid in this limit.

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 <sup>β</sup> <sup>¼</sup> <sup>O</sup> <sup>ε</sup>2=3=Δ<sup>θ</sup> � � and <sup>x</sup><sup>2</sup> is given by (24) since it follows from (8),(35), (43) and (45) that

$$\begin{split} \left( \mathbf{1}/\varepsilon^{3} \right) \Bigg\limits\_{\mathbf{0}}^{\mathbf{x}\_{1}} \Bigg( \kappa\_{0} (\mathbf{x}\_{1}) d\mathbf{x}\_{1} = \left( \mathbf{1}/\varepsilon^{4} \right) \Bigg\limits\_{\mathbf{0}}^{\hat{\mathbf{x}}\_{1}} \Bigg( \tilde{\kappa}\_{0} (\hat{\mathbf{x}}\_{1}) d\hat{\mathbf{x}}\_{1} \rightarrow \left( \tilde{\alpha}/\Delta \theta \right) \mathbf{x} - \varepsilon \overline{\beta}^{5} \hat{\alpha}\_{0} \mathbf{x}^{2} / 2 \Bigg) \\ = \left( \hat{\alpha}/\Delta \theta \right) \mathbf{x} - \beta^{5} \hat{\alpha}\_{0} \prescript{2}{}{\mathbf{c}}^{\mathbf{x}} \big( \mathbf{c}^{3} \big)^{2} / 2 = \left( \tilde{\alpha}/\Delta \theta \right) \mathbf{x} - \hat{\alpha}\_{0} \prescript{2}{}{\mathbf{x}}\_{2}^{2} / 2. \end{split} \tag{46}$$

#### 5.2 Numerical results

5. The inviscid triple-deck region

Figure 5. �Im κ † 0

wavenumber.

� � as a function of x†

inserting the rescaled variables

when the square root β

92

As noted above the acoustically driven solution will only match onto an eigensolution in the downstream region when <sup>O</sup>ð Þ <sup>Δ</sup><sup>θ</sup> <sup>≥</sup>ε<sup>3</sup><sup>=</sup>2. This region will lie downstream of the viscous triple-deck region considered above and will be closest to that region when <sup>O</sup>ð Þ¼ <sup>Δ</sup><sup>θ</sup> <sup>ε</sup><sup>3</sup><sup>=</sup>2. It will have an inviscid triple- deck structure and the relevant dispersion relation can be obtained by putting <sup>ε</sup>=Δ<sup>θ</sup> <sup>¼</sup> <sup>O</sup> <sup>ε</sup><sup>1</sup>=<sup>3</sup> � � in (21),

(double dot dashed, dot dashed and solid lines, respectively) and three values of the frequency scaled transverse

, <sup>κ</sup><sup>0</sup> <sup>¼</sup> <sup>κ</sup>0=ε<sup>1</sup>=<sup>3</sup>

λ β 2 � <sup>M</sup><sup>2</sup>

<sup>∞</sup> � <sup>1</sup> � �κ<sup>2</sup>

� �<sup>1</sup>=<sup>2</sup>

<sup>∞</sup> � <sup>1</sup> � �<sup>1</sup>=<sup>2</sup> � <sup>β</sup>

into (38), using (29), and taking the limit as ε ! 0 with β, κ<sup>0</sup> and x^<sup>1</sup> held fixed, to show that the rescaled wavenumber κ<sup>0</sup> satisfies the inviscid dispersion relation

κ0

0

It can then be shown by direct substitution that the solution κ<sup>0</sup> behaves like

5 α^2

<sup>∞</sup> � <sup>1</sup> � �κ<sup>2</sup>

<sup>1</sup> calculated from (38) together with the initial condition (41) for M<sup>∞</sup> ¼ 2, 3, 4

� �<sup>1</sup>=<sup>2</sup>

ffiffiffiffiffiffiffi 2x^<sup>1</sup> <sup>p</sup> <sup>T</sup><sup>2</sup> w 0

, <sup>x</sup>^<sup>1</sup> <sup>¼</sup> <sup>x</sup>1ε<sup>4</sup>=<sup>3</sup> (43)

is required to remain finite as ε ! 0.

<sup>0</sup>x^<sup>1</sup> as x^<sup>1</sup> ! 0, (45)

(44)

<sup>β</sup> <sup>¼</sup> <sup>β</sup>=ε<sup>1</sup>=<sup>3</sup>

Boundary Layer Flows - Theory, Applications and Numerical Methods

κ2 <sup>0</sup> þ β 2 ¼

> 2 � <sup>M</sup><sup>2</sup>

<sup>κ</sup><sup>0</sup> ! <sup>β</sup> M<sup>2</sup>

5.1 Matching with the small Δθ Fedorov/Khokhlov solution

Figure 6 is a plot of the scaled lowest order wavenumber κ0=β ¼ κ0=β as a function of the scaled streamwise coordinate βTw � �<sup>4</sup> <sup>x</sup>^1=λ<sup>2</sup> <sup>¼</sup> <sup>β</sup>Tw � �<sup>4</sup> x1=λ<sup>2</sup> for various values of the free-stream Mach number M<sup>∞</sup> calculated from the inviscid triple-deck dispersion relation(44) together with the asymptotic initial condition (45) which is shown by the dashed curves in the figure. The lowest order wave number κ<sup>0</sup> is purely real which means that exponential growth (if it occurs) can only occur at higher order. This suggests that the acoustically generated instabilities will be less significant than the vortically-generated instabilities which appear upstream.

#### Figure 6.

Scaled wavenumber κ0=β ¼ κ0=β as a function of βTw � �<sup>4</sup> <sup>x</sup>^1=λ<sup>2</sup> <sup>¼</sup> <sup>β</sup>Tw � �<sup>4</sup> x1=λ<sup>2</sup> for various values of M∞. The solid lines represent the numerical solution. Dashed lines are the asymptotic solution(45) .

## 6. The next stage of evolution

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

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

$$\overline{\beta} \to \frac{1}{\kappa\_0^{1/3}} \left(\frac{\lambda}{\sqrt{2\varkappa\_1}}\right)^{5/3} \left(\frac{1}{T\_w^2}\right) \left(\frac{\sqrt{2\varkappa\_1}}{\varkappa\_0 \lambda}\right)^{2/3} = \frac{\lambda}{\kappa\_0 T\_w^2 \sqrt{2\varkappa\_1}}\tag{47}$$

when x<sup>1</sup> ! ∞ and, therefore, that

$$\kappa\_0 \to \frac{\lambda}{\overline{\beta}T\_w^2\sqrt{2\kappa\_1}},\tag{48}$$

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

ε3 1ð Þ <sup>þ</sup><sup>r</sup>

3 5 ! e

<sup>z</sup> <sup>¼</sup> <sup>z</sup>=ε<sup>1</sup>�<sup>r</sup>

x^ð1

κ0ð Þ x^1; ε dx^<sup>1</sup> þ

xð1

0

, where αð Þ x<sup>1</sup> is an Oð Þ1 function of x<sup>1</sup> that

þ … as x<sup>1</sup> ! 0 (56)

<sup>y</sup> � <sup>y</sup>=ε<sup>1</sup>�<sup>r</sup> (57)

� �

3 5…

� <sup>β</sup><sup>2</sup> � �<sup>v</sup> <sup>¼</sup> <sup>0</sup> (59)

<sup>w</sup>ð Þ <sup>y</sup>; <sup>x</sup><sup>1</sup> ; <sup>ε</sup>2 1ð Þ �<sup>r</sup> <sup>p</sup>ð Þ <sup>y</sup>; <sup>x</sup><sup>1</sup>

(58)

<sup>v</sup>ð Þ <sup>y</sup>; <sup>x</sup><sup>1</sup> , <sup>ε</sup><sup>1</sup>�<sup>r</sup>

1 ε2þ4<sup>r</sup>

αð Þ x1; ε dx<sup>1</sup> þ β <sup>z</sup>�<sup>t</sup>

, (55)

x^ð1

κ1ð Þ x^1; ε dx^<sup>1</sup>

(54)

3 5

0

0

2 4

i ε�2 2ð Þ <sup>þ</sup><sup>r</sup>

more oblique and

0

x^ð1

0

exp i ε�ð Þ <sup>4</sup>þ2<sup>r</sup> Ð

completely passive. The scaled variable

should expand like

95

behaves like

x1 0

κð Þ x1; ε dx<sup>1</sup> þ βz � t

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

<sup>κ</sup>2ð Þ <sup>x</sup>^1; <sup>ε</sup> dx^<sup>1</sup> <sup>þ</sup> <sup>O</sup> <sup>ε</sup>�4<sup>r</sup> ð Þþ <sup>ε</sup><sup>r</sup>

3

Leading Edge Receptivity at Subsonic and Moderately Supersonic Mach Numbers

<sup>5</sup> <sup>¼</sup> exp <sup>i</sup> <sup>1</sup>

βz � t

as x^<sup>1</sup> ! ∞, where αð Þ x<sup>1</sup> is an Oð Þ1 function of x<sup>1</sup> (given by (53)) and

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

will be Oð Þ1 in the main boundary layer since its thickness is now of the order of the spanwise length scale, <sup>O</sup> <sup>ε</sup><sup>1</sup>�<sup>r</sup> ð Þ. It therefore follows from (53) and (57) that the transverse pressure gradients will come into play and the solution in this region

αð Þ x1; ε dx<sup>1</sup> þ βz � t

dU=dy T � �

<sup>z</sup> � <sup>ε</sup><sup>r</sup>

which means that the solution should be proportional to

<sup>α</sup> ! <sup>λ</sup> βT<sup>2</sup> w ffiffiffiffiffiffiffi 2x<sup>1</sup> p

αð Þ x1; ε dx<sup>1</sup> þ β z � t

f g <sup>u</sup>; <sup>v</sup>; <sup>w</sup>; <sup>p</sup> <sup>¼</sup> f g <sup>U</sup>; <sup>0</sup>; <sup>0</sup>; <sup>0</sup> <sup>þ</sup> ^δAð Þ <sup>x</sup><sup>1</sup> <sup>u</sup>ð Þ <sup>y</sup>; <sup>x</sup><sup>1</sup> ; <sup>ε</sup><sup>1</sup>�<sup>r</sup>

T d dy 1 T dv dy þ

exp <sup>i</sup> <sup>1</sup> ε<sup>4</sup>þ2<sup>r</sup>

2 4 xð1

0

Tα 1 � αU

v is determined by the incompressible reduced Rayleigh equation

where Að Þ x<sup>1</sup> is a function of the slow variable x1. Substituting (58) into the linearized Navier-Stokes equations shows that the wall normal velocity perturbation

> d dy

" #

2 4

exp <sup>i</sup> <sup>1</sup> ε3 xð1

þ 1 ε1þ5<sup>r</sup>

2 4

when κ<sup>0</sup> is allowed to approach zero as x<sup>1</sup> ! ∞.

The dashed curves in the main plot of Figure 4 represent the re-scaled large-x† 1 asymptote (48). It confirms that the numerical results are well approximated by the (appropriately rescaled) large-x<sup>1</sup> asymptote (48).

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

$$\overline{\kappa}\_0 \to \frac{\lambda}{\overline{\beta}T\_w^2\sqrt{2\hat{\mathbf{x}}\_1}} \text{ as } \hat{\mathbf{x}}\_1 \to \infty \tag{49}$$

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

$$
\overline{\beta} = \overline{\beta}/\varepsilon^r, \overline{\kappa}\_0 = \kappa\_0/\varepsilon^r, \hat{\mathbf{x}}\_1 = \mathfrak{x}\_1 \varepsilon^{4r} \tag{50}
$$

and allow r to be zero or 1/3. The expansion (37) then generalizes to [11]

$$
\overline{\kappa}(\mathbf{x}\_1, \varepsilon) = \overline{\kappa}\_0(\hat{\mathbf{x}}\_1) + \varepsilon^{1-r} \overline{\kappa}\_1(\hat{\mathbf{x}}\_1) + \varepsilon^{2(1-r)} \overline{\kappa}\_2(\hat{\mathbf{x}}\_1) + \dots,\tag{51}
$$

where

$$
\overline{\kappa}, \overline{\kappa}\_1, \overline{\kappa}\_2 \dots \equiv \kappa / \varepsilon^r, \kappa\_1, \kappa\_2 \varepsilon^r \dots \tag{52}
$$

and x^<sup>1</sup> is defined in (43).

#### 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 O ε<sup>3</sup> ffiffiffi <sup>x</sup> <sup>p</sup> ð Þ continues to increase and the triple-deck scaling breaks down at the streamwise location

$$
\overline{\mathfrak{X}}\_1 = \mathfrak{x}e^{4+2r} = O(1), \tag{53}
$$

where it becomes of the order of the spanwise length scale, which remains constant at <sup>O</sup> <sup>ε</sup><sup>1</sup>�<sup>r</sup> ð Þ. This region is located well upstream of the region where the unsteady flow is governed by the full Rayleigh equation considered in [9].

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

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

$$\exp i \left[ \frac{1}{\varepsilon^3} \int\_0^{\hat{\mathbf{x}}\_1} \kappa(\mathbf{x}\_1, \varepsilon) d\mathbf{x}\_1 + \overline{\rho}\overline{\mathbf{z}} - t \right] = \exp i \left[ \frac{1}{\varepsilon^{3(1+r)}} \int\_0^{\hat{\mathbf{x}}\_1} \overline{\kappa}\_0(\dot{\mathbf{x}}\_1, \varepsilon) d\dot{\mathbf{x}}\_1 + \frac{1}{\varepsilon^{2+4r}} \int\_0^{\hat{\mathbf{x}}\_1} \overline{\kappa}\_1(\dot{\mathbf{x}}\_1, \varepsilon) d\dot{\mathbf{x}}\_1 \right]$$

$$+ \frac{1}{\varepsilon^{1+5r}} \int\_{\overline{\kappa}}^{\hat{\mathbf{x}}\_1} \overline{\kappa}\_2(\dot{\mathbf{x}}\_1, \varepsilon) d\hat{\mathbf{x}}\_1 + O(\varepsilon^{-4r}) + \varepsilon^r \overline{\overline{\rho}} \overline{\mathbf{z}} - t \right] \longrightarrow e^{\left[ \varepsilon^{-2(+r)} \left[ \overline{\alpha}(\overline{\mathbf{x}}\_1, \varepsilon) d\overline{\mathbf{x}}\_1 + \overline{\overline{\beta}} \overline{\overline{\overline{\varepsilon}}} - t \right] \right]} \tag{54}$$

as x^<sup>1</sup> ! ∞, where αð Þ x<sup>1</sup> is an Oð Þ1 function of x<sup>1</sup> (given by (53)) and

$$
\overline{\overline{z}} \equiv \epsilon^r \overline{z} = z/\epsilon^{1-r},\tag{55}
$$

which means that the solution should be proportional to exp i ε�ð Þ <sup>4</sup>þ2<sup>r</sup> Ð x1 0 αð Þ x1; ε dx<sup>1</sup> þ β z � t " #, where αð Þ x<sup>1</sup> is an Oð Þ1 function of x<sup>1</sup> that behaves like

$$
\overline{\alpha} \to \frac{\lambda}{\overline{\beta} T\_w^2 \sqrt{2 \overline{\mathbf{x}}\_1}} + \dots \text{ as } \ \overline{\mathbf{x}}\_1 \to 0 \tag{56}
$$

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

6. The next stage of evolution

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

<sup>β</sup> ! <sup>1</sup> κ 1=3 0

when x<sup>1</sup> ! ∞ and, therefore, that

6.1 Downstream behavior of the triple-deck solution

Boundary Layer Flows - Theory, Applications and Numerical Methods

λ ffiffiffiffiffiffiffi 2x<sup>1</sup> p

when κ<sup>0</sup> is allowed to approach zero as x<sup>1</sup> ! ∞.

(appropriately rescaled) large-x<sup>1</sup> asymptote (48).

The expansion (37) then generalizes to [11]

<sup>κ</sup>ð Þ¼ <sup>x</sup>1; <sup>ε</sup> <sup>κ</sup>0ð Þþ <sup>x</sup>^<sup>1</sup> <sup>ε</sup><sup>1</sup>�<sup>r</sup>

and allow r to be zero or 1/3.

and x^<sup>1</sup> is defined in (43).

6.2 Derivation of the governing equations

rescaled version

where

location

94

� �5=<sup>3</sup> 1

<sup>κ</sup><sup>0</sup> ! <sup>λ</sup> βT<sup>2</sup> w ffiffiffiffiffiffiffi 2x^<sup>1</sup> p

<sup>β</sup> <sup>¼</sup> <sup>β</sup>=ε<sup>r</sup>

T2 w

<sup>κ</sup><sup>0</sup> ! <sup>λ</sup> βT<sup>2</sup> w ffiffiffiffiffiffiffi 2x<sup>1</sup>

! ffiffiffiffiffiffiffi

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

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

asymptote (48). It confirms that the numerical results are well approximated by the

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

, <sup>κ</sup><sup>0</sup> <sup>¼</sup> <sup>κ</sup>0=ε<sup>r</sup>

<sup>κ</sup>, <sup>κ</sup>1, <sup>κ</sup>2… � <sup>κ</sup>=ε<sup>r</sup>

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 O ε<sup>3</sup> ffiffiffi

continues to increase and the triple-deck scaling breaks down at the streamwise

where it becomes of the order of the spanwise length scale, which remains constant at <sup>O</sup> <sup>ε</sup><sup>1</sup>�<sup>r</sup> ð Þ. This region is located well upstream of the region where the unsteady flow is governed by the full Rayleigh equation considered in [9].

, κ1, κ2ε<sup>r</sup>

2x<sup>1</sup> p κ0λ � �2=<sup>3</sup>

<sup>¼</sup> <sup>λ</sup> κ0T<sup>2</sup> w ffiffiffiffiffiffiffi 2x<sup>1</sup>

p , (48)

as x^<sup>1</sup> ! ∞ (49)

, <sup>x</sup>^<sup>1</sup> <sup>¼</sup> <sup>x</sup>1ε<sup>4</sup><sup>r</sup> (50)

… (52)

<sup>x</sup> <sup>p</sup> ð Þ

<sup>κ</sup>1ð Þþ <sup>x</sup>^<sup>1</sup> <sup>ε</sup>2 1ð Þ �<sup>r</sup> <sup>κ</sup>2ð Þþ <sup>x</sup>^<sup>1</sup> :…, (51)

<sup>x</sup><sup>1</sup> <sup>¼</sup> <sup>x</sup>ε<sup>4</sup>þ2<sup>r</sup> <sup>¼</sup> <sup>O</sup>ð Þ<sup>1</sup> , (53)

p (47)

1

$$\overline{\mathcal{Y}} \equiv \mathcal{Y}/\mathfrak{e}^{1-r} \tag{57}$$

will be Oð Þ1 in the main boundary layer since its thickness is now of the order of the spanwise length scale, <sup>O</sup> <sup>ε</sup><sup>1</sup>�<sup>r</sup> ð Þ. It therefore follows from (53) and (57) that the transverse pressure gradients will come into play and the solution in this region should expand like

$$\{\underline{u}, \underline{v}, \underline{w}, \underline{p}\} = \{U, \mathbf{0}, \mathbf{0}, \mathbf{0}\} + \hat{\delta}\mathcal{A}(\overline{\mathbf{x}}\_{1}) \{\overline{u}(\overline{y}; \overline{\mathbf{x}}\_{1}), \varepsilon^{1-r}\overline{v}(\overline{y}; \overline{\mathbf{x}}\_{1}), \varepsilon^{1-r}\overline{w}(\overline{y}; \overline{\mathbf{x}}\_{1}), \varepsilon^{2(1-r)}\overline{p}(\overline{y}; \overline{\mathbf{x}}\_{1})\}$$

$$\exp i \left[ \frac{1}{\varepsilon^{4+2r}} \prod\_{0}^{\overline{\varpi}\_1} \overline{\alpha}(\overline{\varpi}\_1, \varepsilon) d\overline{\varpi}\_1 + \overline{\overline{\rho}\overline{\varpi}} - t \right] \dots \tag{58}$$

where Að Þ x<sup>1</sup> is a function of the slow variable x1. Substituting (58) into the linearized Navier-Stokes equations shows that the wall normal velocity perturbation v is determined by the incompressible reduced Rayleigh equation

$$T\frac{d}{d\overline{\eta}}\frac{1}{T}\frac{d\overline{v}}{d\overline{\eta}} + \left[\frac{T\overline{a}}{1-\overline{a}U}\frac{d}{d\overline{\eta}}\left(\frac{dU/d\overline{\eta}}{T}\right) - \overline{\overline{\rho}}^2\right]\overline{v} = 0\tag{59}$$

whose solution must satisfy the following boundary conditions

$$
\overline{v} \sim e^{-\overline{\overline{\beta}}\overline{\overline{\gamma}}} \quad \text{as} \ \overline{y} \to \infty, \quad \overline{v} = 0 \text{ at} \ \overline{y} = 0. \tag{60}
$$

Matching with the upstream solution (33) and (37) requires that αð Þ x<sup>1</sup> satisfy the matching condition (56) as x<sup>1</sup> ! 0.

Inserting (10) and (57) into (59), using (60) and assuming the ideal gas law ρT ¼ 1 shows that

$$\frac{d}{d\eta} \frac{1}{T^2} \frac{d\overline{\boldsymbol{v}}}{d\eta} + \left[ \frac{\overline{\boldsymbol{a}}}{1 - \overline{\boldsymbol{a}} \boldsymbol{U}} \left( \frac{\boldsymbol{U}'}{T^2} \right)' - \left( \overline{\overline{\boldsymbol{\beta}}} \sqrt{2\overline{\boldsymbol{\kappa}}} \right)^2 \right] \overline{\boldsymbol{v}} = \mathcal{O} \left( \varepsilon^{2(1-r)} \right),\tag{61}$$

$$
\overline{v} = \mathbf{0} \quad \text{at } \eta = \mathbf{0}, \tag{62}
$$

which means that

$$
\overline{a} = f\left(\hat{\boldsymbol{\beta}}\right),
\tag{63}
$$

where

$$
\hat{\beta} \equiv \overline{\overline{\beta}} \sqrt{2 \overline{\overline{\mathbf{x}\_1}}}.\tag{64}
$$

some finite value of β^ which is consistent with the fact that U<sup>0</sup>

Leading Edge Receptivity at Subsonic and Moderately Supersonic Mach Numbers

for M<sup>∞</sup> ¼ 2C ¼ 129:4 for M<sup>∞</sup> ¼ 3 and C ¼ 340:1 for M<sup>∞</sup> ¼ 4.

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

spanwise wavelength at an <sup>O</sup> <sup>ε</sup>�<sup>4</sup> ð Þ distance downstream.

the typical supersonic flight conditions at

there.

97

Figure 7.

7. Conclusions

accurate numerical predictions.

at some finite value of η and Eq. (61) therefore has a generalized inflection point

(a) Reð Þ <sup>α</sup> and (b) j j Imð Þ <sup>α</sup> vs. <sup>β</sup>^ calculated from the modified Rayleigh solution. The red dashed curves are calculated from the asymptotic formula(56) . The red dashed lines in the inset are j j Im ð Þ <sup>α</sup> <sup>¼</sup> <sup>C</sup>β^, where <sup>C</sup> <sup>¼</sup> <sup>36</sup>

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 <sup>ε</sup> <sup>¼</sup> <sup>F</sup><sup>1</sup>=<sup>6</sup>, where <sup>F</sup> denotes the frequency parameter. The oblique 1st Mack mode instabilities generated by free-stream acoustic disturbances are compared with those generated by elongated vortical disturbances. The focus is on explaining the relevant physics and not on obtaining

The free-stream vortical disturbances generate unsteady flows in the leading edge region that produce short spanwise wavelength instabilities in a viscous tripledeck region which lies at an <sup>O</sup> <sup>ε</sup>�<sup>2</sup> ð Þ distance downstream from the leading edge. The mechanism was first considered for two dimensional incompressible flows in Ref. [7], but the instability onset occurs much further upstream in the supersonic case and is, therefore, much more likely to be important at the higher Mach numbers considered in this chapter. The lowest order triple-deck solution does not possess an upper branch and evolves into an inviscid 1st Mack mode instability with short

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 θ deviated from its critical value by an Oð Þ1 amount and showed that slow acoustic disturbances generate unsteady boundary layer disturbances that produce Oð Þ1 spanwise wavelength inviscid 1st Mack mode instabilities a much larger O ε�<sup>6</sup> distance downstream. But the calcu-

lations in Ref. [11] show that the physical streamwise distance <sup>x</sup><sup>∗</sup> <sup>¼</sup> <sup>U</sup><sup>∗</sup>

corresponding to this scaled downstream location is at least equal to about 7 m for

=T<sup>2</sup> <sup>0</sup>

is equal to zero

∞ <sup>3</sup>

<sup>=</sup> <sup>ω</sup><sup>∗</sup> ð Þ<sup>2</sup> ν∗ ∞

#### 6.3 Matching with the triple-deck solution

Eq. (64) clearly approaches zero when x<sup>1</sup> ! 0, which means that α will be consistent with the matching condition (54) if we require that it behave like

$$
\overline{a} = \lambda / \mathbf{T}\_w^2 \hat{\boldsymbol{\beta}} + a\_1 + a\_2 \hat{\boldsymbol{\beta}} + \dots \text{ as } \overline{\mathbf{x}}\_1 \to \mathbf{0} \tag{65}
$$

where α1, α2… are (in general complex) constants such that

$$a\_1 = \lim\_{\hat{\mathfrak{x}}\_1 \to \infty} \overline{\kappa}\_1(\hat{\mathfrak{x}}\_1), a\_2 = \lim\_{\hat{\mathfrak{x}}\_1 \to \infty} \overline{\kappa}\_2(\hat{\mathfrak{x}}\_1) / \overline{\tilde{\beta}} \sqrt{2\hat{\mathfrak{x}}\_1}.\tag{66}$$

Ref. [11] proved that (60)–(64) possess an asymptotic solution of the form <sup>v</sup> <sup>¼</sup> <sup>U</sup>ð Þþ <sup>η</sup> <sup>β</sup>^v<sup>1</sup> <sup>þ</sup> <sup>β</sup>^<sup>2</sup> <sup>v</sup><sup>2</sup> <sup>þ</sup> ::… as <sup>β</sup>^ ! 0 when <sup>α</sup> satisfies (65) which implies that their solutions are able to match onto the lowest order triple-deck solution upstream and are consistent with the higher order solutions in this region.

#### 6.4 Numerical results

The Rayleigh eigenvalues α are determined by the boundary value problem (60), (61) and (62). We assume in the following that the Prandtl number is equal to unity and that the viscosity μð Þ T satisfies the simple linear relation μð Þ¼ T Tð Þη .

Parts (a) and (b) of Figure 7 are plots of the real and imaginary parts respectively of these eigenvalues as a function of β^. They show that the numerical solution for α will be consistent with the matching conditions (65)and(66) if the higher order terms in the triple-deck expansion(51) satisfy Im limx^1!<sup>∞</sup> κ1ð Þ¼ x^<sup>1</sup> 0 and limx^1!<sup>∞</sup> <sup>κ</sup>2ð Þ <sup>x</sup>^<sup>1</sup> <sup>=</sup><sup>β</sup> ffiffiffiffiffiffiffi 2x^<sup>1</sup> <sup>p</sup> <sup>=</sup>�iC, where the values of <sup>C</sup> are given in the caption of Figure 7. They also show that α is initially real and eventually becomes complex. But these eigenvalues must occur in complex conjugate pairs since the coefficients in (61) are all real. The computations show that Imð Þ α eventually goes to zero at

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

#### Figure 7.

whose solution must satisfy the following boundary conditions

Boundary Layer Flows - Theory, Applications and Numerical Methods

U0 T2 � �0

� � <sup>p</sup> <sup>2</sup> � �

<sup>α</sup> <sup>¼</sup> <sup>f</sup> <sup>β</sup>^

<sup>β</sup>^ � <sup>β</sup> ffiffiffiffiffiffiffi 2x<sup>1</sup>

Eq. (64) clearly approaches zero when x<sup>1</sup> ! 0, which means that α will be consistent with the matching condition (54) if we require that it behave like

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

solutions are able to match onto the lowest order triple-deck solution upstream and

The Rayleigh eigenvalues α are determined by the boundary value problem (60), (61) and (62). We assume in the following that the Prandtl number is equal to unity

Parts (a) and (b) of Figure 7 are plots of the real and imaginary parts respectively of these eigenvalues as a function of β^. They show that the numerical solution for α will be consistent with the matching conditions (65)and(66) if the higher order terms in the triple-deck expansion(51) satisfy Im limx^1!<sup>∞</sup> κ1ð Þ¼ x^<sup>1</sup> 0 and

Figure 7. They also show that α is initially real and eventually becomes complex. But these eigenvalues must occur in complex conjugate pairs since the coefficients in (61) are all real. The computations show that Imð Þ α eventually goes to zero at

<sup>p</sup> <sup>=</sup>�iC, where the values of <sup>C</sup> are given in the caption of

Matching with the upstream solution (33) and (37) requires that αð Þ x<sup>1</sup> satisfy the

� <sup>β</sup> ffiffiffiffiffiffiffi 2x<sup>1</sup>

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

�β<sup>y</sup> as <sup>y</sup> ! <sup>∞</sup>, <sup>v</sup> <sup>¼</sup> 0 at <sup>y</sup> <sup>¼</sup> <sup>0</sup>: (60)

<sup>v</sup> <sup>¼</sup> <sup>O</sup> <sup>ε</sup>2 1ð Þ �<sup>r</sup> � �

� �, (63)

p : (64)

<sup>w</sup>β^ <sup>þ</sup> <sup>α</sup><sup>1</sup> <sup>þ</sup> <sup>α</sup>2β^ <sup>þ</sup> … as <sup>x</sup><sup>1</sup> ! <sup>0</sup> (65)

2x^<sup>1</sup>

p : (66)

<sup>x</sup>^1!<sup>∞</sup> <sup>κ</sup>2ð Þ <sup>x</sup>^<sup>1</sup> <sup>=</sup><sup>β</sup> ffiffiffiffiffiffiffi

<sup>v</sup><sup>2</sup> <sup>þ</sup> ::… as <sup>β</sup>^ ! 0 when <sup>α</sup> satisfies (65) which implies that their

v ¼ 0 at η ¼ 0, (62)

, (61)

v � e

<sup>þ</sup> <sup>α</sup> 1 � αU

6.3 Matching with the triple-deck solution

<sup>α</sup> <sup>¼</sup> <sup>λ</sup>=T<sup>2</sup>

α<sup>1</sup> ¼ lim

where α1, α2… are (in general complex) constants such that

are consistent with the higher order solutions in this region.

<sup>x</sup>^1!<sup>∞</sup> <sup>κ</sup>1ð Þ <sup>x</sup>^<sup>1</sup> , <sup>α</sup><sup>2</sup> <sup>¼</sup> lim

and that the viscosity μð Þ T satisfies the simple linear relation μð Þ¼ T Tð Þη .

matching condition (56) as x<sup>1</sup> ! 0.

d dη 1 T2 dv dη

which means that

where

<sup>v</sup> <sup>¼</sup> <sup>U</sup>ð Þþ <sup>η</sup> <sup>β</sup>^v<sup>1</sup> <sup>þ</sup> <sup>β</sup>^<sup>2</sup>

6.4 Numerical results

limx^1!<sup>∞</sup> <sup>κ</sup>2ð Þ <sup>x</sup>^<sup>1</sup> <sup>=</sup><sup>β</sup> ffiffiffiffiffiffiffi

96

2x^<sup>1</sup>

ρT ¼ 1 shows that

(a) Reð Þ <sup>α</sup> and (b) j j Imð Þ <sup>α</sup> vs. <sup>β</sup>^ calculated from the modified Rayleigh solution. The red dashed curves are calculated from the asymptotic formula(56) . The red dashed lines in the inset are j j Im ð Þ <sup>α</sup> <sup>¼</sup> <sup>C</sup>β^, where <sup>C</sup> <sup>¼</sup> <sup>36</sup> for M<sup>∞</sup> ¼ 2C ¼ 129:4 for M<sup>∞</sup> ¼ 3 and C ¼ 340:1 for M<sup>∞</sup> ¼ 4.

some finite value of β^ which is consistent with the fact that U<sup>0</sup> =T<sup>2</sup> <sup>0</sup> is equal to zero at some finite value of η and Eq. (61) therefore has a generalized inflection point there.

#### 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 <sup>ε</sup> <sup>¼</sup> <sup>F</sup><sup>1</sup>=<sup>6</sup>, where <sup>F</sup> denotes the frequency parameter. The oblique 1st Mack mode instabilities generated by free-stream acoustic disturbances are compared with those generated by elongated vortical disturbances. The focus is on explaining the relevant physics and not on obtaining accurate numerical predictions.

The free-stream vortical disturbances generate unsteady flows in the leading edge region that produce short spanwise wavelength instabilities in a viscous tripledeck region which lies at an <sup>O</sup> <sup>ε</sup>�<sup>2</sup> ð Þ distance downstream from the leading edge. The mechanism was first considered for two dimensional incompressible flows in Ref. [7], but the instability onset occurs much further upstream in the supersonic case and is, therefore, much more likely to be important at the higher Mach numbers considered in this chapter. The lowest order triple-deck solution does not possess an upper branch and evolves into an inviscid 1st Mack mode instability with short spanwise wavelength at an <sup>O</sup> <sup>ε</sup>�<sup>4</sup> ð Þ distance downstream.

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 θ deviated from its critical value by an Oð Þ1 amount and showed that slow acoustic disturbances generate unsteady boundary layer disturbances that produce Oð Þ1 spanwise wavelength inviscid 1st Mack mode instabilities a much larger O ε�<sup>6</sup> distance downstream. But the calculations in Ref. [11] show that the physical streamwise distance <sup>x</sup><sup>∗</sup> <sup>¼</sup> <sup>U</sup><sup>∗</sup> ∞ <sup>3</sup> <sup>=</sup> <sup>ω</sup><sup>∗</sup> ð Þ<sup>2</sup> ν∗ ∞ corresponding to this scaled downstream location is at least equal to about 7 m for the typical supersonic flight conditions at

<sup>M</sup><sup>∞</sup> <sup>¼</sup> <sup>3</sup> <sup>U</sup><sup>∗</sup> <sup>∞</sup> <sup>¼</sup> <sup>888</sup> <sup>m</sup>=s; <sup>ν</sup><sup>∗</sup> <sup>∞</sup> <sup>¼</sup> <sup>0</sup>:<sup>000264</sup> <sup>m</sup>2=<sup>s</sup> end an altitude of 20 km with an upper bound of 100 kHz for the characteristic frequency. This means that this instability occurs too far downstream to be of any practical interest at the moderately low supersonic Mach numbers considered in this chapter.

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But, the inviscid instability, which first appears at an O ε�ð Þ <sup>4</sup>þ2=<sup>3</sup> distance downstream when Δθ is reduced to O ε2=<sup>3</sup> can be significant when scaled to flight conditions. It is therefore appropriate to compare the vortically-generated instabilities with the instabilities generated by oblique acoustic disturbances with obliqueness angles in this range as done in this chapter.
