A temporal linear stability analysis of the asymptotic suction boundary layer is presented. For this, the Orr–Sommerfeld equation is solved in terms of generalized hypergeometric functions. Together with the corresponding boundary conditions, an algebraic eigenvalue problem is formulated. Thereof we derive the temporal continuous spectrum yielding a rather distinct spectrum if, for example, compared to the one from the Blasius solution. A second key result is that the discrete spectrum in the limits $\alpha \u21920$,that is, small streamwise wave numbers, and $Re\u2192\u221e$ is only present in the distinguished limit $Re\u2009\alpha =O(1)$. This results in a degenerated Orr–Sommerfeld equation and the expanded algebraic eigenvalue problem poses a lower limit of $(Re\u2009\alpha )min\u22480.841\u200991$. We show that this lower bound corresponds to a maximum extension of the viscous eigenfunction in the wall-normal direction. The full algebraic eigenvalue problem is numerically solved for the temporal case up to $Re=6.0\xd7106$. Besides the further refined critical values $\alpha cr=0.155\u200946,\u2009\u2009\omega cr=0.023\u2009297,\u2009\u2009Recr=54\u2009378.620\u200932$, discrete spectra and eigenfunctions are examined and $\omega =\omega r+i\omega i$ is the complex frequency. In particular, eigenvalue spectra are investigated with regard to their behavior due to a variation of the Reynolds number and the wave number, respectively, and only A-modes according to the definition of Mack [“A numerical study of the temporal eigenvalue spectrum of the Blasius boundary layer,” J. Fluid Mech. **73**, 497–520 (1976)] were identified. From these, three different classes of eigenfunctions of the wall-normal disturbance velocity are presented. Finally, we find that the inviscid part of the eigenfunctions is dominant in wall-normal direction and only propagates in streamwise direction, while the viscous part is limited to the vicinity of the wall and propagates toward it in an almost perpendicular direction.

## I. INTRODUCTION

Ever since Prandtl^{2} discovered the reduction of drag by suction of the flow around a cylinder in his groundbreaking works on boundary layer theory, the stabilizing effect of suction has been extensively investigated for boundary layer flows. The theoretical framework of flow stability analysis was established by the works of Orr^{3} and Sommerfeld^{4} who independently modeled perturbations as wave-like solutions of the form $g(x,y,z,t)=g\u0303(y)\u2009ei(\alpha x+\beta z\u2212\omega t)$ of the linearized Navier–Stokes equations (LNSEs), with *g* denoting perturbations of velocity or pressure. This resulted in the Orr–Sommerfeld equation (OSE), which in turn led to a first deep understanding of hydrodynamic stability problems. A major breakthrough in the understanding of flows around bodies was again provided by Prandtl,^{2} who formulated his famous boundary layer equations, in which the stationary wall-parallel flow over a flat plate was physically described via an asymptotic analysis of the scales in the underlying Navier–Stokes equations (NSEs). The ideas were further developed by Blasius,^{5} who examined the case of uniform constant inflow $U(y)=U\u221e$ at the leading edge of a flat plate and thereupon derived a self-similar solution of the corresponding boundary layer equation, later named Blasius boundary layer (BBL) honoring his investigations. First thorough stability analyses of the BBL were published by Tollmien^{6} and Schlichting,^{7} both employing expansions of the OSE for large Reynolds numbers.

The linear stability of suction boundary layers can be studied in an idealized setting by assuming constant wall-normal suction $\u2212V0$ superposing a flat-plate boundary layer flow. The result is a laminar base flow with an exponential profile with an asymptotically constant displacement thickness *δ*_{1} for large enough distances from the leading edge (see Fig. 1 for a schematic visualization). This specific boundary layer flow is usually referred to as the asymptotic suction boundary layer (ASBL). Possible ways of solving the modified Orr–Sommerfeld equation (mOSE), extended by an additional suction term due to the constant suction velocity $\u2212V0$, were first analyzed by Chiarulli and Freeman,^{8} who introduced the idea of transforming the mOSE to a hypergeometric differential equation. Hughes and Reid^{9} derived an analytical solution for the simplified inviscid case constituted by Gaussian hypergeometric functions. A large Reynolds number expansion of the viscous case was presented by Baldwin,^{10} who was the first to derive the complete solution of the viscous mOSE in terms of generalized hypergeometric functions. Improvements on the critical parameters calculated by Hughes and Reid were accomplished by Hocking^{11} who transformed the non-linear disturbed Navier–Stokes equations (NSE) into a non-linear Schrödinger equation, based on techniques employed by Stewartson and Stuart^{12} for plane-parallel flows, which was then solved numerically for its coefficients. A spectral analysis of the stability problem using Chebyshev polynomials in the wall-normal direction was employed by Fransson and Alfredsson,^{13} who investigated the spatial stability problem. A good agreement of the essential numerical results with experimental data was demonstrated.

It is worth noting that modal transition, which is the topic of the present work, is interesting especially in cases with low noise levels. With increasing noise and perturbation levels in the free flow, bypass transition scenarios become increasingly relevant and were previously investigated for the ASBL with regard to transition thresholds,^{14–16} localized edge states in bypass scenarios^{17} as well as in the case of heat transfer over the wall.^{18} In recent years, the effect of porosity of the wall was thoroughly discussed and corresponding studies disclosed a considerable decrease in the critical Reynolds number to roughly *Re* = 800 as well as changes in finite-amplitude transition mechanisms.^{19–21}

In spite of the aforementioned recent extensions of the ASBL stability problem to non-modality, non-linearity, and non-zero porosity, an extensive discussion of the original linear stability problem has not been conducted until today, and even though the solvability of the viscous mOSE for the ASBL has been well known for many decades already, the analytical study of stability has been neglected in favor of the already discussed discrete methods. Evidently, the existence of solutions enables a granular investigation of the stability characteristics of the problem by means of asymptotic techniques as well as parameter studies.

The present work, therefore, presents a thorough stability analysis of the ASBL by studying the solution of the modified Orr–Sommerfeld equation constituted by generalized hypergeometric functions. First, the eigenvalue problem (EVP) and temporal continuous spectra are derived in Secs. II and III, respectively, followed by asymptotic analyses of the eigenvalue problem with respect to the streamwise wave number *α* and the Reynolds number in Sec. IV. Finally, the composition of the temporal stability map, a parameter study of the temporal eigenvalue spectra, wall-normal eigenfunctions as well as the discussion of Tollmien–Schlichting (TS) wave mechanics are presented in Sec. V. Finally, conclusions are drawn in Sec. VI. Various results were aided by Maple and MATLAB.

## II. LINEAR STABILITY OF THE ASYMPTOTIC SUCTION BOUNDARY LAYER

### A. Analytical solution of the modified Orr–Sommerfeld and Squire equations (mSE)

The ASBL is the solution of the NSE for an incompressible flow over a flat-plate subjected to constant suction given by $U(y)=(U\u221e(1\u2212e\u2212yV0\nu ),\u2212V0,0)T$. It constitutes the base flow to be analyzed below on its stability properties. Non-dimensionalization with the displacement thickness $\delta 1=\nu /V0$ and the inflow velocity $U\u221e$ yields

By linearizing the NSE about solution (1) and employing the normal-mode ansatz

for all perturbations, we obtain the Orr–Sommerfeld equation

with $D$ expressing the derivative with respect to *y* and *v*(*y*) denoting the wall-normal velocity, *α*, $\beta \u2208\mathbb{R}$ denoting the stream- and spanwise wave number, $k=\alpha 2+\beta 2$ and $\omega =\omega R+i\omega I\u2208\u2102$ denoting the sought complex wave frequency.

For the inviscid linear NSE, additional viable ansatz functions as alternatives to (2) can be derived via a Lie-symmetry analysis. Such an analysis was conducted for the ASBL, and details may be taken from Appendix B. To summarize the results, due to the traveling wave nature of the respective alternative modes, the boundary conditions (BCs) had to be adjusted toward vanishing perturbations at positive and negative infinity. For the double-exponential ansatz function (see Appendix B 1), this concluded in the very trivial condition $\alpha =\xb11$, yielding no information on the designated eigenvalue *ω*. The alternative exponential ansatz function (see Appendix B 2) allows, under the assumption of vanishing perturbations at infinite distance from the wall, only such modes that lead to linear stability. Hence, the symmetry-induced ansatz functions failed to provide additional instabilities.

In addition to the classical Orr–Sommerfeld equation, (3) is extended by a suction term denoted by a third derivative term, scaling with $Re\u22121$ after non-dimensionalization, namely, $\u2212Re\u22121\u2009D(D2\u2212k2)\u2009v(y)$, which does not occur in the classical OSE. By substituting (1) into (3), it was shown by Baldwin^{10} that the fundamental solution

of the 4th-order ordinary differential equations (ODE) in (3) is given in terms of generalized hypergeometric functions

The parameters $ai=[ai1,ai2]T$ and $bi=[bi1,bi2,bi3]T$ are given as

with $k\u0303=k2+1$ and

Grosch and Salwen^{22} labeled analogous solutions for the BBL belonging to $e\xb1ky$ as inviscid solutions, while $e\xb1\sigma BBL\u2009y$ were named viscous solutions. Accordingly, the parameter $\sigma ASBL$ will hereafter be called *viscous parameter*. This labeling is emphasized due to the influence of *σ* on the stability of the entire flow, as will be seen in Sec. IV.

As the latter analyses in Secs. IV and V are conducted only for the 2D case, the discussion of the Squire equation and its solution may be referred to in Appendix A.

### B. Formulation of the eigenvalue problem for the ASBL

The solution (5) for the wall-normal velocity perturbation *v*(*y*) has to satisfy the BCs both at the wall and at infinity, yielding conditions of the form

As *k* > 0 the solution $v1(y)=C1\u2009eky$ diverges for $y\u2192\u221e$, while $v2(y)=C2\u2009e\u2212ky$ converges. Thus, $C1=0$. The behavior of the two remaining solutions depends exclusively on the real part of $\sigma \u2208\u2102$. In principle, $\sigma =\xb1\sigma R\xb1i\sigma I$ as *σ* is the root of a complex number (see 8). This ambivalence of *σ* can be resolved as the solution *v*(*y*) contains a reflexion symmetry in *σ*, that is, $v(y)\sigma =v(y)\u2212\sigma $. Hence, it is sufficient to focus on $\sigma =\sigma R+i\sigma I$ solely. Regardless of *k* and *Re*, the real part *σ _{R}* is always greater than zero, which yields $v4(y)=C4\u2009e\u2212\sigma +12\u2009y$ convergent.

It can be shown that only the case $\sigma R>1$ is necessary for the evaluation of the BCs. The proof is omitted for the sake of brevity and lack of relevance for the analysis. The remaining BCs (9a) at the wall then reduce to

where $d\u20092F3(a,b;\u2212z)/dz=(a\xaf/b\xaf)\u20092F3(a+1,b+1;\u2212z)$ and $a\xaf/b\xaf=(a1a2)/(b1b2b3)$ have been used. Resultantly, (12) and (13) constitute a homogeneous linear system of the form

for which non-trivial solutions for *C*_{2} and *C*_{4} only exist for

defining an algebraic eigenvalue problem (EVP) of the form

For a given set of parameters $\omega =\omega (\alpha ,Re)$ determined from (16), we may compute the eigenfunction *v*(*y*) from (5). For this, we employ $C1=C3=0$ and express *C*_{2} in terms of *C*_{4} using (12). The final results read

where we have omitted *C*_{4} because of the linearity of the problem and $z(y)=\u2212iRe\u2009\alpha \u2009e\u2212y$. When presented subsequently, eigenfunctions will be normalized to one.

## III. CONTINUOUS SPECTRUM FOR THE ASYMPTOTIC SUCTION BOUNDARY LAYER

For shear flows in an infinite or semi-infinite domain featuring vanishing disturbances at infinity, for example, boundary layer or a jet flows, it is possible to derive an alternative class of physical disturbances, as was demonstrated by Grosch and Salwen.^{22} These solutions appear if the boundary conditions (BCs) (13) at infinity are relaxed, that is,

is assumed instead of strictly vanishing disturbances at infinity.

with

We observe that, due to the reflexion symmetry in *σ*, only $\sigma R>0$ needs to be analyzed and, hence, only the minus sign in (19) remains. This for the viscous modes in (4) and (5) yields

employing

with $\Delta =\alpha \u2212\omega R$, which apparently satisfies (19). Condition (19) corresponds to a continuous set of eigenvalues. Depending on whether the temporal or the spatial stability problem is considered, one may correspondingly obtain a temporal or spatial continuous spectrum. In the following, the continuous spectrum for the temporal problem is derived.

### A. Temporal continuous spectrum

For the temporal stability problem, the real part of viscous parameter *σ* is given in (20). Hence, with $\sigma R=1$ in (19) the condition may be explicitly solved for $\omega I=\omega I(\omega R)$ and hence allows to express $\omega =\omega (\omega R)$ as

This parametrization due to *ω _{R}* denotes a continuous set of eigenvalues $\omega =\omega (\omega R)$ and as such yields $v3(y)$ bounded. For a BBL flow, the continuous spectrum parametrized by $\xi \u2208\mathbb{R}$ is given as

It is common to display spectra utilizing the complex phase speed

Employing (25), the spectrum for the BBL is given as a vertical line in the complex plane with *c _{R}* = 1, and a maximum at $(1,\u2212\alpha /Re)$, while for the ASBL the spectrum is a parabola opened downwards with a maximum also located at $(1,\u2212\alpha /Re)$. A comparison of both spectra can be found in Fig. 2 with

*α*= 1 and

*Re*= 500. Furthermore, for both the BBL and the ASBL the relevant imaginary part

*ω*is always negative, and hence, the eigenfunctions of the temporal continuous spectrum are always stable.

_{I}## IV. ASYMPTOTIC ANALYSIS OF THE EIGENVALUE PROBLEM

It will be demonstrated in Secs. IV A–IV D, how the parameters *Re* and *α* influence one another in both asymptotic limits $\alpha \u21920$ and $\alpha \u2192\u221e$ upon expanding the EVP (16). In accordance with Squire's theorem^{23} which states that the 2D modes are the least stable, the subsequent analyses will only be conducted for the 2D stability problem. To prove that Squire's original analysis applies also for parallel flows superposed by suction, the mOSE (3) is considered. All relevant parameter terms must stay invariant under transformation from 2D to 3D, that is,

which are identical to Squire's transformation rules deduced for parallel flows as suction does not generate any new transformation rule in the mOSE or mSE.

Analogously, as $\beta 2>0$ in the 3D case, (26a) gives $\alpha 3D<\alpha 2D$. Thereupon (26b) yields $Re3D>Re2D$, which of course also applies for the critical Reynolds number so that for the ASBL and similar parallel flows with suction or transpiration the 2D case is still most critical. Hence, Squire's theorem may also be employed without restriction for the temporal stability analysis of the ASBL.

For all, following asymptotic analyses is advisory to compare the results to the computed stability map given in Fig. 5.

### A. Asymptotic analysis for small streamwise wave numbers and large *Re*

The assumption of simultaneously small streamwise wave numbers and large Reynolds numbers is motivated by the fact that a classical asymptotic analysis for small streamwise wave numbers fails to provide results for *ω*. A detailed analysis can be found in Appendix C. Thus, a combined asymptotic analysis for small *α* and large *Re* is attempted in this section. In order to understand the asymptotic behavior of *ω* for small *α*, a general Laurent series expansion of the form

has been employed in the EVP (16). This, however, did not lead to a solution, indicating that irregular singularities or no solutions exist. In fact, computing complex roots with various non-linear root-finding algorithms for sufficiently small streamwise wave numbers *α* and arbitrary Reynolds numbers reveal that no solutions *ω* exist indeed. At first glance, this seems counter-intuitive as there is no plausible reason why the EVP (16) should not allow for large-scale streamwise perturbation modes.

A deeper analysis revealed that when increasing the Reynolds number as such that the product $Re\u2009\alpha $ is of order *O*(1), valid solutions for $\omega (\alpha )$ exist. Hence, if solutions for small *α* only exist for $Re=O(\alpha \u22121)$, it might imply that large-scale streamwise perturbations are only present at high Reynolds numbers. Such an interdependence of large structures and Reynolds number has been observed in many works prior to this.^{24}

At this point, these findings are merely speculations based on computational observations and are yet to be proven from a theoretical point of view. Therefore, a distinguished limit analysis is suggested, in which *α* and $Re\u22121$ simultaneously become asymptotically small, while the product

is of order *O*(1). As done previously, *ω* will be expanded in powers of *α*. In order to prevent a singular *σ*, the constant order in (27) needs to be rejected, that is, $\omega 0=0$. Thus, the highest order *ω* can adopt is $O(\alpha )$ so that

which in turn gives

where the leading order is a constant.

Notice that the necessity of introducing the distinguished limit in (28) as well as cutting the constant order leading to (29) could have also been argued for by attempting an asymptotic expansion of the mOSE in the same limit. Looking closely at the mOSE given in (3), it becomes obvious that terms induced by the base flow $U(y)=1\u2212e\u2212y$ scale with the streamwise wave number *α*. Hence, an expansion for $\alpha \u21920$ would yield these terms of order $O(\alpha )$. Only simultaneously assuming $Re\u2192\u221e$ and, further, taking the distinguished limit $Re\alpha =O(1)$“saves” the base flow terms in the leading order. Expanding (3) in the distinguished limit thereupon gives

where $v\u0303(y)$ denotes the solution arising for *v*(*y*) after the asymptotic expansion in the distinguished limit. Clearly, a term of unknown order is still present in (31), namely, $iRe\alpha \u2009\omega \alpha $. We must make sure that this term is neither of dominant order nor of order $O(\alpha )$. This only allows one inference for the order of *ω*, that is

which coincides exactly with the results obtained previously in (29). Thus, the demand for (28) as well as (29) also occurs upon examination of the mOSE.

With the introduced distinguished limit as well as the ansatz for $\omega (\alpha )$, the expansion of the parameters of the $2F3$ functions are reevaluated. The detailed derivation of the hereupon discussed expansion of (16) may be taken from Appendix D.

Notice that due to the asymptotic expansion, all $2F3$ are reduced to $1F2$ functions as in each function a parameter in the numerator and denominator cancel each other out in the leading order. Substituting all expansions from Appendix D into the dispersion relation (16) and considering only the leading order *O*(1) yields

with $\sigma 0=1+4i\u2009Re\alpha \u2009(1\u2212\omega 1)$. Resultantly, (33) is a degenerated EVP, which determines *ω*_{1} as a function of $Re\alpha $ and is to be solved numerically.

The solutions for the computation of the Laurent-series coefficient $\omega 1=\omega 1(Re\alpha )$ as solutions of (33) can be taken from Fig. 3(a). The parameter is computed analogously to the computation of the original EVP (16). A rather curious feature of the displayed coefficient *ω*_{1} is a lower threshold appearing at $Re\alpha ,min\u22480.841\u200991$, under which no solutions could be computed for *ω*_{1}. As may be taken from Fig. 3(a), we observe that in the limit $Re\alpha \u2192Re\alpha ,min$ we obtain $\omega 1\u21921\u221223i$ or rather $\omega \u2192\alpha \u221223\u2009i\alpha $. For increasingly large $Re\alpha $, the coefficient $\omega 1,R$ decreases monotonically, while $\omega 1,I$ increases monotonically, respectively.

By employing *ω*_{1} into (32) and $Re\alpha $ into (28), the physical parameters *ω* and *α* are obtained, where *α* is linked to the Reynolds number as pointed out earlier. An exemplary re-transformation is illustrated for $Re=1.0\xd7105$ in Fig. 3(b). In order to compare the results obtained by the asymptotic expansion in the distinguished limit, the original EVP was solved for its roots at very small *α* and $Re=1.0\xd7105$ in Fig. 4(a) to enable comparison to the asymptotically derived eigenvalues in Fig. 3(b). A near-perfect agreement may be assumed, which is quantitatively shown in Fig. 4(b), from where the relative errors of the eigenvalues obtained by (33) based on the original EVP (16) may be extracted. Evidently, the error increases the further *α* grows away from the asymptotic limit $\alpha \u21920$. While for $\alpha min=0.841\u200991\xd710\u22125$ the relative error is of order $O(10\u221210)$, it continuously grows to an error of order $O(10\u22126)$ for $\alpha =1.0\xd710\u22123$. This is in very good agreement with what would be expected from asymptotic methods and thus verifies the correctness of the asymptotic expansions conducted in the distinguished limit.

As may be taken from Fig. 3(b), the modes in the distinguished limit are without exception linearly stable. Yet, the distinguished limit and its lower threshold $Re\alpha ,cr\u22480.841\u200991$ play a decisive role in the spatial *y*-decay of the wall-normal (viscous) wave part of the eigenfunctions, which is analyzed in detail in Sec. V C.

### B. Asymptotic analysis for large wave numbers

In the asymptotic limit $\alpha \u2192\u221e$, the eigenvalue $\omega (\alpha )$ is assumed to be a Laurent series in *α* of the form

As is demonstrated in Appendix E, the leading order of the eigenvalue $\omega (\alpha )$ must be of linear order, as otherwise the leading order of the EVP (16) becomes singular. With this, the series (34) reduces to

which yields the solution

and hence, $\omega i(\alpha )<0$ for $\alpha \u2192\u221e$. In summary, the flow is globally stable for large streamwise wave numbers *α* in the case of arbitrary *Re*.

### C. Asymptotic analysis for the Stokes limit ($Re\u21920$)

Analogous to the previous analyses, for the Stokes limit $Re\u21920$ a Laurent series of the form,

is assumed for $\omega (Re)$. Employing the series into the viscous parameter *σ* yields

For any arbitrary *n* < 0 (with $n\u2208\mathbb{Z}$), the leading order of *σ* is, therefore, always $O(Re12(1+n))$. A lengthy analysis to derive the necessary leading order of $\omega (Re)$ was conducted by expanding the EVP (16), which can be found in Appendix F. To sum up the key insights, the leading order of $\omega (Re)$ must be $\omega (Re)=\omega 1\u2009Re+O(1)$, which yields

The expansion of the dispersion relation (16) for the Stokes limit is then given as

which for the coefficient $\omega (\u22121)$ yields

giving the leading order of the sought eigenvalue as

As such, in the leading order the imaginary part of the eigenvalue $\omega (Re)$ remains negative for large *Re* and any arbitrary positive valued *α*. Hence, in the Stokes limit the flow remains globally stable.

### D. Asymptotic analysis in the non-viscous limit $Re\u2192\u221e$

A thorough analysis of the large Reynolds number case has been covered extensively both for the analytic solution of the mOSE in Baldwin^{10} as well as in a more recent investigation published by Dempsey and Walton^{25} based on the LNSE for the ASBL.

In the latter work, a triple-deck behavior for the perturbed ASBL has been imposed in the large Reynolds number limit for the lower branch of the neutral stability curve as later displayed in Fig. 5. The theory of triple decks was incidentally developed to better understand boundary layer separation,^{26–28} but the theory may be employed to also derive the asymptotic behavior of the lower neutral branch. The stationary ASBL for this case is separated into three physically differing parts, namely, the lower, main, and upper deck. In the corresponding analysis, asymptotic expansions in each deck in combination with the BCs for the perturbation velocities as well as matching conditions in the transition areas yield that:

The lower branch in the large Reynolds number limit is parametrized by $\alpha =O(Re\u22121/4)$.

For

*β*= 1, the constant of proportionality derived from the ensuing dispersion relation is given as $\alpha N=0.617$, that is, $\alpha \u223c0.617\u2009Re\u22121/4$. These results coincide nicely with the temporal stability map computed in Fig. 5.

To the knowledge of the authors, a similar analysis for the upper branch of the neutral stability curve has not been conducted. It remains interesting to see whether both branches strive to asymptotically small streamwise wave numbers *α* or if this case only applies to the lower branch.

## V. STABILITY CHARACTERISTICS OF THE ASYMPTOTIC SUCTION BOUNDARY LAYER

To the knowledge of the authors, neither the temporal nor the spatial stability map for the ASBL was completely published and thoroughly analyzed previously. For computing solutions of the non-linear complex EVP (16), Muller's algorithm, a 2nd-order secant method, was employed. In the following, only the temporal stability characteristics of the ASBL will be investigated. The critical value triple [$\alpha cr,\omega cr,Recr$] will be compared to the one obtained by a Chebyshev collocation method.^{13}

### A. Complex roots of temporal eigenvalue problem

The temporal stability map in Fig. 5 was computed up to a Reynolds number of $Re=6\xd7106$ and a streamwise wave number of $\alpha =0.2$. Based on the asymptotic analysis conducted for large *α* (see Sec. IV B), the computation was not performed up to larger streamwise wave numbers.

The stabilizing effect of suction on boundary layer stability is best studied by comparing the critical value triple of the ASBL profile to the BBL. Critical values were first calculated by Hocking,^{11} who gave [$\alpha cr,ASBL=0.155\u20095,\u2009\omega cr,ASBL=0.023\u2009325,\u2009Recr,ASBL=54\u2009370$] by solving the linearized Navier–Stokes equations in streamfunction formulation with methods first employed in Stewartson and Stuart,^{12} in which a plane Poiseuille profile was examined. The initial-value problem is investigated with asymptotic and multi-scale methods. For a detailed approach, the reader is referred to the cited papers. In more recent years, a different critical value triple was obtained by Fransson and Alfredsson,^{13} who employed a Chebyshev collocation method in wall-normal direction. The triple computed in this work was given as [$\alpha cr,ASBL=0.155\u20095,\u2009\omega cr,ASBL=0.023\u200931,\u2009Recr,ASBL=54\u2009382$]. Due to the approximating nature of both approaches, the true critical value triple remained unknown. In this work, we use the EVP (16) based on the analytical solution (5) to accurately calculate the critical value triple up to $O(10\u221222)$ and with high decimal accuracy even for the Reynolds number. For this, a 2D bisection method was employed to narrow down the critical point systematically. Such a method is highly dependent on the starting point, the computations were tracked manually to adjust the parameter range when necessary. The critical value triple computed in this work based on (16) is as follows:

of which the *Re _{cr}* is in fact located between the previously given critical Reynolds numbers provided by Hocking

^{11}and Fransson and Alfredsson.

^{13}The residuum of the EVP (16) with the critical values in (44) is of order $O(10\u221222)$, while the positive imaginary part $\omega cr,i$ is of order $O(10\u221214)$. It should be mentioned that due to the analytic nature of the EVP (16), the residuum as well as the decimals in (44) can be refined further, with computation time being the only limiting factor. The point to be made is that numerical schemes, such as Chebyshev collocation schemes, provide results, which are highly dependant on factors such as the used mappings, the number of collocation points, or the cutoff length

*L*.

The form of the stability map in Fig. 5 is reminiscent of a BBL. For increasing Reynolds numbers, the upper and lower branch of the neutrally stable curve relocate to lower streamwise wave numbers. Simultaneously, both branches move closer to each other as the Reynolds number increases. It is worth noticing that global stability is ensured for any streamwise wave number $\alpha \u22650.178$, regardless of the Reynolds number, and this is also validated by the asymptotic analysis for $\alpha \u2192\u221e$.

### B. Temporal eigenvalue spectra and the corresponding wall-normal eigenfunctions *v*(*y*)

For each parameter combination of *Re* and *α*, there exists a spectrum consisting of the TS-mode as well as additional modes representing further solutions of the EVP (33). We aim to answer two questions about these temporal spectra:

How do the spectra behave for varying

*Re*and*α*?How do the TS-waves behave for varying

*Re*and*α*?

With the multiple eigenvalues admitted by (16), computation strongly depends on an initial guess and also delivers only one value at a time. Hence, a two-step scheme is employed. First, all spectra were estimated using a Chebyshev collocation scheme. For the computation, the semi-infinite domain normal to the wall was truncated at *L* = 500 and mapped to $\eta \u2208[\u22121,1]$ using an algebraic linear mapping of the form

where *η* denotes the resulting mapped wall-normal coordinate. The number of collocation points were set to *n* = 600. We note that the emerging spectra were strongly influenced by the truncation length *L* as well as the number of collocation points *n*.

It is important to notice that the sole employment of Chebyshev schemes is not sufficient to acquire physically correct spectra due to the existence of spurious modes. Hence, in order to filter out spurious modes as well as drastically increase the accuracy of higher modes in the spectra, all resulting modes were reiterated on account of the analytical EVP (16). For this purpose, all modes obtained by the Chebyshev method were employed as starting points into the non-linear root finder described in Sec. V A. The residuals for the EVP (16) were set to a very low threshold, until the iteration was halted. It proved necessary to set this threshold to $tol=10\u221280$, except for the case *Re* = 500 000, where a tolerated residual of $tol=10\u2212100$ had to be taken. The reason for these very low residuals was the persistence of some spurious modes, which did not disappear for lower residuals. For this reason, the number of digits had to be increased to 100 and 120, respectively.

The resulting spectra are shown in Figs. 6 and 7. In Fig. 6, the spectra were computed for a fixed $Re=Recr$ and varying *α*. Branches of unfiltered eigenspectra for plane Poiseuille flow were classified in Ref. 1 as an A-branch when $cR\u21920$, a P-branch when $cR\u21921$ or an S-branch when $cR\u21922/3$, with *c _{R}* denoting the real part of the phase velocity. Even though inherently different, the eigenvalue spectra for the BBL

^{29}were in fact classified analogously, in which the continuous spectrum was described as a hybrid P–S-mode family, whereas the scattered modes made up the A-mode family. Due to the conceptual similarity to the eigenspectrum of the BBL, the seemingly scattered modes computed and visualized in Figs. 6 and 7 are identified as modes in the A-mode family.

In Fig. 6, the eigensprectra in *Re _{cr}* are compared for varying orders of

*α*. For very small $\alpha =0.001$ [see Fig. 6(a)], the spectrum is sparse, while already showing a behavior that is maintained for other

*α*; that is, the spectrum in most cases maintains a mode very close to the continuous spectrum with the remaining modes sparsely connecting TS-mode and the mode in the vicinity of the continuous spectrum. In Fig. 6(b), the amount of modes increases, seemingly forming an arch-like structure. For the critical $\alpha =\alpha cr$, however, an additional second quasi-branch seems to emerge [see Fig. 6(c)], featuring two “off-track” modes, while the far right mode is separated from the rest of the spectrum. Finally, the gap between the far right mode and the rest of the modes appears to widen toward the largest

*α*= 1, given in Fig. 6(d).

The comparison of various orders of *Re* at the critical wave number *α _{cr}* curiously reveals a similar behavior of the spectrum. The smallest Reynolds number

*Re*= 500 [see Fig. 7(a)] features a rather sparse spectrum, again containing a TS-mode, a far right mode and modes in between. The number of modes increases at first with increasing Reynolds numbers [

*Re*= 5000, see Fig. 7(b)]. Increasing the orders then again unveils the aforementioned second branch, which for the largest Reynolds number

*Re*= 500 000 gains additional modes [see Fig. 7(d)]. Interesting enough, for this Reynolds number the far right mode seems to disappear, which could be verified by reducing the tolerated residual of the EVP (16) as well as increasing the number of digits in Maple.

For the critical point (44), selected eigenfunctions $vr(y)$ are plotted in Figs. 8(a)–8(d) by substituting the parameters *α* and *Re* as well as the eigenvalue *ω* of interest, taken from Fig. 7(c), into (17). In general, a trend can be observed in the evolution of the eigenfunctions with increasing proximity to the continuous spectrum. While the TS-eigenfunction shows typical behavior in Fig. 8(a), the modes very close to the continuous spectrum induce oscillatory wall-normal eigenfunctions [see Figs. 8(c) and 8(d)]. Interesting enough, these oscillatory eigenfunctions seemingly form a symmetric band with respect to some mid-point in *y*, in which they are non-zero with some distance to the wall.

Due to their relevance for the stability of the flow, it remains to be examined which factors influence the reach of the possibly unstable TS-modes into the far-field. For this, the magnitude of the eigenfunctions $|v(y)|$ is plotted vs *y* for varying *Re* and fixed *α _{cr}* [see Fig. 9(a)] and for varying

*α*and fixed

*Re*[see Fig. 9(b)]. It is quite remarkable that varying the orders of

_{cr}*Re*hardly affects the decay of $|v(y)|$ in wall-normal direction, while varying

*α*influences the reach of $|v(y)|$ into the far-field decisively. This finding describes quantitatively how the magnitude of the eigenfunction

*v*(

*y*) behaves for varying

*Re*and

*α*. In comparison with the laminar base flow, especially such TS-waves comprised by small

*α*severely outreach the boundary layer thickness

*δ*by many factors. The investigation of $|v(y)|$ therefore indicates that the inviscid part of the eigenfunction, that is, the term decaying with $e\u2212\alpha y$ in

*v*(

*y*) as given in (17), dominates.

It is quite enlightening, however, to extend the analysis to the perturbation velocity $v\u2032(x,y,t)=v(y)\u2009ei(\alpha x\u2212\omega t)$. When employing the eigenfunction *v*(*y*), given in (17), into the normal-mode ansatz (2), one gets

with $c\u0303=\u20092F3(a4,b4;\u2212iRe\u2009\alpha )/\u20092F3(a2,b2;\u2212iRe\u2009\alpha )$ and *σ _{R}* and

*σ*as defined in (20) and (22). It becomes apparent that each wall-normal velocity perturbation is comprised by two waves, one wave due to the viscous part of the solution moving in the

_{I}*x*–

*y*plane, that is, the first addend in (46), and the inviscid part moving purely in streamwise direction, which is comprised by the second addend in (46). In the limit $y\u2192\u221e$, the $2F3$ hypergeometric functions asymptotically converge to 1, which gives

Comparison with Mack^{1} unveils that in the limit of large *y* two analogous wave components also exists in the BBL of the form

Now, clearly for the ASBL two specific decay mechanisms occur in *y*-direction:

The viscous wave propagating in the

*x–y*plane decays with $e\u2212\sigma R+12y$.The inviscid wave propagating in

*x*-direction which decays with $e\u2212\alpha y$.

In order to capture the full dynamics, we will conduct a separate analysis of each respective wave as both wave parts have different propagation directions. An analysis of the inviscid waves was already conducted in Figs. 9(a) and 9(b), as the inviscid waves seemingly dominate the magnitude of *v*(*y*), which as we will see in Sec. V C is true for any combination of *α* and *Re*. However, an analysis of the viscous waves needs to be conducted still. For this purpose, the propagation angle and the decay rate of the viscous waves are examined in Sec. V C.

### C. Directional wave propagation of viscous waves and their spatial *y*-decay

From (46), the viscous wave part of the wall-normal perturbation velocity $v\u2032(x,y,t)$ is given by

There are two essential pieces of information that can be extracted from (49), the wave propagation angle *θ* as well as a measure of length characterizing the spatial decaying behavior for large *y*.

On top of the already known streamwise wave number *α*, we may define a wall-normal wave number $\sigma I2$. The comparison of the signs of *α* and *σ _{I}* determines whether the viscous waves move toward or away from the wall. The sign of

*σ*is determined by $sgn(\alpha \u2212\omega R)$, as is given in (22). For the domain $1.0\xd7104\u2264Re\u22641.0\xd7106$ and $1.0\xd710\u22124\u2264\alpha \u22641.0$, it turns out that $\alpha \u2212\omega R>0$ and $\omega R>0$. Thus,

_{I}*α*and $\u2212\sigma I2$ without exception bear opposing signs, while $\omega R>0$, due to which the viscous waves defined in (49) in all cases move toward the wall at a propagation angle

The magnitude of $\sigma R+12$ on the other hand determines at what rate the viscous waves (49) decay in wall-normal direction. In this sense, the inverse,

defines a characteristic length scale, which describes how far into the far-field the viscous waves persist. Further, if transformed back to dimensional quantities, we observe that $\delta \sigma $ is a multiple of the boundary layer displacement thickness $\delta 1=\nu /V0$. Hence, this quantity hereafter is called wave decay-rate length (WDRL).

Now, *σ _{R}* in (20) by definition is always positive. Hence, $\delta \sigma $ reaches a maximum when

*σ*becomes minimal (or ideally zero). Close examination of (20) reveals that

_{R}*σ*in fact becomes minimal when $\alpha =\omega R$ and

_{R}*ω*is negative. While negative

_{I}*ω*are quite common, the question is pending whether or not there is a point at which

_{I}*α*is indeed equal to

*ω*. If such a case occurs, it would have great effect on the longevity of the viscous modes into the far-field, and indeed, there is a very specific point at which

_{R}*α*is equal to

*ω*! Such an occurence was witnessed in Sec. IV A when the necessity of a distinguished limit emerged in order to expand the EVP (16) for very small wave numbers. It became apparent that only for $Re\alpha =Re\u2009\alpha =O(1)$ do solutions for

_{R}*ω*exist in the zero limit of

*α*. A computation of the expanded EVP (33) revealed a lower threshold for $Re\alpha ,min\u22480.841\u200991$, below which no solutions exist for

*ω*. In the limit $Re\alpha \u2192Re\alpha ,min$, however, the eigenvalue

*ω*converges toward $\omega min=\alpha \u2212i23\alpha $. Thus, for the lower threshold of $Re\alpha $ does the extraordinary case $\omega R=\alpha $ occur. At this point,

*σ*becomes minimal and gets very close to zero.

_{R}This result may be verified in Fig. 10(a). In this figure, the WDRL $\delta \sigma $ is calculated and visualized for the given parameters. In almost the entire parameter range is $\delta \sigma <1$, which means that the WDRL is smaller than the displacement thickness *δ*_{1}. However, at $Re=1.0\xd7104$ and $\alpha =1.0\xd710\u22124$ a maximum for $\delta \sigma $ can be witnessed, clearly approaching the theoretical maximum $\delta \sigma ,max=2$. At this point, the WDRL becomes larger than *δ*_{1}. In other words, the viscous waves clearly reach well into the far-field. It is quite remarkable that the distinguished limit theorized in Sec. IV A eventually plays a key role for the WDRL. This shows that this limit indeed has an actual physical significance for the perturbation waves.

The wave propagation angle *θ* defined in (50) is additionally plotted for the same parameter range in Fig. 10(b). It is important to notice that the wave propagation angle *θ* remains close to 90° throughout the entirety of the analyzed parameter region [$\alpha ,Re$]. The viscous part of the wave (46) as a result has almost no streamwise variation for the Reynolds numbers investigated in Fig. 10(b), being emitted perpendicular to the wall into the far-field, where the viscous wave eventually decays.

## VI. CONCLUSION

Despite the existence of a closed form analytic solution of the extended Orr–Sommerfeld equation for the ASBL, its analytical study has been neglected ever since Baldwin derived a large Reynolds number expansion of the solution in 1970.

In our work, a variety of new results have been derived for the ASBL among which is the temporal continuous spectrum with methods described by Ref. 22 by expanding the analytic solution for large *y* and demanding boundedness of the solution at infinite distance from the wall. The spectrum itself is not a vertical line in the $cR\u2212cI$ spectrum, as is the case for the BBL, but rather a parabola open downwards with its maximum located at *c _{R}* = 1 and $cI=\u2212\alpha /Re$.

We could further show that an exhaustive investigation of the analytic solution provided granular insights on both the linear stability of the ASBL as well as the dynamics of the perturbation waves. The solution of the mOSE comprised of $2F3$ generalized hypergeometric functions was required to fulfill the boundary conditions at wall and at infinity, which yielded an algebraic EVP for the complex eigenvalue *ω*. For this EVP, an asymptotic analysis disclosed linear stability in the limits $\alpha \u2192\u221e$ and $Re\u21920$. However, the expansion for $\alpha \u21920$ did not provide any solutions for *ω* in the leading order. It could then be shown that only for the distinguished limit $Re\alpha =Re\u2009\alpha =const.$ solutions for *ω* exist. Indeed, the computation of solutions for the EVP (16) failed to converge toward solutions for *ω* if $O(Re)<O(\alpha \u22121)$ and only gave solutions if $Re\u2009\alpha >0.841\u200991$. This unique threshold was reproduced analytically by expanding the EVP in the distinguished limit and computing solutions *ω* for the degenerated EVP (33). Even more interesting, it transpired that for $Re\alpha \u2192Re\alpha ,min\u22480.841\u200991$ the eigenvalue converges to $\omega =\alpha \u2212i23\u2009\alpha $. This link between large-scale streamwise structures and large Reynolds numbers was observed in various wall-bounded shear flows^{24,30} and for the ASBL specifically.^{31,32} We believe in the universality of the analytic interdependence of large *Re* and small *α* we derived for the ASBL, as the mOSE analysis in (31) showed.

Further, these results bear great importance for the dynamics and the wall-normal decay of the TS-waves. A detailed break-down of the wall-normal perturbation $v\u2032(x,y,t)$ for the TS-modes showed that each perturbation wave is made up of an inviscid wave part propagating in *x*-direction and an viscous wave part moving in the *x–y* plane, which was also observed for the BBL for a far-wall expansion.^{1} The oblique viscous waves in fact turned out to propagate at an almost right-angle toward the wall. Now, it is the wall-normal decay rate of the viscous waves, which is connected to the distinguished limit theorized before. We defined a length scale $\delta \sigma $, which is inversely proportional to the real part of the previously defined viscous parameter *σ _{R}* and thus attained a maximum for $\sigma R\u21920$. As was shown in a subsequent analysis, this condition is satisfied exactly when $\omega R=\alpha $, which was only the case in the limit $Re\alpha \u21920.841\u200991$. Hence, the longevity of the oblique viscous wave parts of the TS-waves is highest when approaching this lower thresholds for $Re\alpha $. This finding may describe quantitatively how large-scale structures (i.e., small streamwise wave numbers) decay in wall-normal direction

*y*. In any case, the interdependence of

*Re*and

*α*in the distinguished limit may provide new insights on the relationship between large structures and Reynolds number—one of the driving questions of turbulence research.

^{24}

The computation of the least stable TS-mode solutions was arranged together in a stability map ranging from $1.0\xd7104\u2264Re\u22646.0\xd7106$, that is, covering even very large Reynolds numbers. The critical point at which the flow becomes unstable was previously calculated by Ref. 11 and most recently by Ref. 13, who employed Chebyshev collocation methods. We could demonstrate, however, that the critical values provided even by the very refined Chebyshev collocation methods deviated from the exact solution, yielding deviations for the critical Reynolds number even at unit place. We corrected the critical value triple by utilizing a 2D bisection method in the vicinity of the critical value to *Re* = 54 378.620 32, $\alpha cr=0.155\u200946$ and $\omega cr=0.023\u2009297$.

The overall $\omega R\u2212\omega I$ spectra were examined for different orders of *Re* and *α*. For this, the spectra emerging from a Chebyshev collocation scheme were reiterated with the EVP (16) upon which modes emerging exactly on the continuous spectra disappear, whereas only modes on the $A\u2212$ branch remained in the spectra, which was previously described in Ref. 29 for the BBL. The eventually discrete spectra were sparse but saw accumulations in the region between the TS-mode and the continuous spectrum. For increasing *Re*, the majority of modes shifted toward the TS-mode, an effect also witnessed for increasing *α*. For moderately large *Re* and arbitrary *α*, one single mode persistently emerged very close to the continuous spectrum. For the critical Reynolds number, we plotted eigenfunctions of the TS-mode, the mode very close to the continuous mode and two modes located in between. In all eigenfunctions, a distinctive maximum occurred at 1–3 times the displacement thickness, after which all eigenfunctions decayed. Interesting enough, the eigenfunctions became more oscillatory the closer their respective modes were located to the continuous spectrum. Finally, we could show that the wall-normal decay of the magnitude of the TS-eigenfunctions is controlled mainly by the order of *α*, meaning that it is the inviscid wave parts which persist longest at increasing distance from the wall.

In summary, the results from the asymptotic analysis, the computation of solutions for the EVP as well as the investigation of spectra and eigenfunctions synthesized a very thorough understanding of the linear perturbation mechanisms of the ASBL. Several topics remain to be investigated, such as a spatial stability analysis of the ASBL as well as an extension to 3D ASBL flows. Furthermore, the thoroughly analyzed perturbation wave dynamics and their implications on the stability of the flow should be studied in a direct numerical simulation. It remains exciting to see how the viscous and inviscid wave parts influence the stability of the flow and whether the theorized interdependence of Reynolds number and large structures may be visualized. We are optimistic that these future research topics will provide an even deeper understanding of how transition in the ASBL and similar canonical flows is triggered in detail.

## ACKNOWLEDGMENTS

Alparslan Yalcin was financially supported by the German Research Foundation (DFG) through the grant OB96/45-1 within the Priority Programme SPP 1881: Turbulent Superstructures.

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request. Eigenvalue data sets used in this paper available at https://tudatalib.ulb.tu-darmstadt.de/handle/tudatalib/2773.

### APPENDIX A: MODIFIED SQUIRE EQUATION FOR THE ASYMPTOTIC SUCTION BOUNDARY LAYER

While in 2D the streamwise perturbation velocity $u\u2032(x,y,z,t)$ is handily obtained by the continuity equation, in 3D the spanwise perturbation velocity component $w\u2032(x,y,z,t)$ demands an additional equation in order for the entire perturbation field to be acquired. For this purpose, the Squire equation may be obtained via cross-differentiation and summation of the *x*- and *z*-components of the LNSE, which after employment of the normal-mode ansatz (2) gives the modified Squire equation (mSE)

where $\eta (y)=i\alpha u(y)\u2212i\beta w(y)$ denotes the wall-normal vorticity. Similar to the mOSE, an analytic solution of the mSE exists and is given as

with the integral terms representing the particular solutions arising from the inhomogeneous ODE in (A1) due to the Wronskian while *v*(*y*) is to be taken from (5). $K\sigma $ and $I\u2212\sigma $ denote modified Bessel functions of the first and second type.

Analogous to the mOSE, homogeneous BCs arise for the wall-normal vorticity upon demanding vanishing perturbations at the wall and at infinity, that is,

These results now suffice, in combination with (5) and the continuity equation for the perturbations

to calculate the entire perturbation field once valid solutions are computed from (16).

### APPENDIX B: NEW SYMMETRY-INDUCED ANSATZ MODES

Throughout this work, the ansatz of choice for solving the linearized Navier–Stokes equation (LNSE) was the classical normal mode ansatz, leading to the mOSE as given in (3). It was shown previously in Nold and Oberlack^{33} that for linear plane shear flows a specific symmetry induces the Kelvin mode as well as a continuation of Kelvin and normal modes. Also, for shear flows of algebraic, exponential and logarithmic form additional symmetries exist, which induce other alternative ansatz functions for the eigenfunctions.^{33,34} Based on the methods presented in these works, two new invariant ansatz functions could be derived for the ASBL. The analysis was conducted for the 2D inviscid LNSE with a ASBL base profile, which in stream function formulation is given as

The underlying Lie algebra is then given by

The symmetries *X*_{1} and *X*_{2} stand for translations in time and space, while *X*_{3} stands for a scaling symmetry in the stream function. These symmetries would also arise in the viscous LNSE. However, the fourth symmetry *X*_{4}—only present in the inviscid case—is novel and lays the ground for the derivation of new invariant ansatz function. The detailed derivation of the now presented Lie algebra as well as the ansatz functions can be found in Mirzayev.^{35} Notice that that results here are different, as Nold only considered the case $V0=0$.

##### 1. Double-exponential ansatz function

Employing the symmetries *X*_{2}, *X*_{3}, and *X*_{4} and following the procedure given in detail in Nold, Oberlack, and Cheviakov,^{34} we arrive at the ansatz function

Evidently, the streamfunction would then be comprised of a traveling wave solution $\Phi (Re\u2009y+t)$ and an exponential part in which the temporal growth or decay due to *ω* is of double-exponential type. It is immediately apparent that the wave-type behavior is not consistent with the BCs at the wall, as for *y* = 0 the time *t* remains as a variable. An attempt is made to instead introduce a generic set of homogeneous BCs at negative and positive infinity. The solution satisfying these BCs shall than be superposed to achieve an additional satisfaction of the wall BCs.

A variable transformation of the form

then gives

Substituting (B5) into (B1) indeed leads to a variable reduction and the ensuing ODE is then given as

This 2nd-order ODE can be solved in terms of Gaussian hypergeometric functions of the form

Demanding vanishing wall-normal velocity perturbations at positive infinity in the original variable space, that is,

requires an expansion of (B5) at large $y\u0303=Re\u2009y+t$. Asymptotic expansions of Gaussian hypergeometric functions are taken from Olver *et al.*^{36} in the case of large arguments. The rather lengthy calculations are omitted at this point for the sake of clarity whereupon BC (B8) is expanded as

where

and

with $\Gamma (z)$ denoting the gamma function.

The expanded BC (B9) yields non-trivial solutions for *C*_{1} and *C*_{2} only if

Closer examination of (B10) reveals that

so that (B12) reduces to

Finally, two identities of gamma functions are utilized:

which employed into (B14) yields

which is satisfied only if

Obviously, this result is too restricting in the sense of a proper stability analysis, especially as no further information on the actual eigenvalue *ω* is encapsulated. Hence, further analysis of this ansatz is omitted and declared not expedient in the context of linear stability analysis.

##### 2. Alternative exponential ansatz function

Upon utilization of all symmetries *X*_{1}–*X*_{4}, a second new ansatz function occurs for the 2D non-viscous ASBL given by

where *α* and *Re* denote the wave number in streamwise direction and the Reynolds number, respectively, while *λ* consists of the group parameters generated in the Lie symmetry analysis. In analogy with the previous example, the variables are transformed to simplify the argument of the amplitude function so that

transforms (B18) to

where

Employing (B21) into (B20) and transforming the variables back into the original coordinate system gives

In order to analyze the BCs at infinity, it is convenient to introduce parameter transformations of the form

where *α*_{1} and *α*_{2} are assumed to be real. The remaining solution needs to be rejected as the dependence with respect to *x* is lost in the process of retransformation. Employing (B24) into (B23) then gives rise to

Finally, the BCs at infinity can be evaluated. We have

and

The only way to satisfy both BCs is to demand $\kappa 1,r<0$ and $\kappa 2,r<0$, after which immediately temporal decay can be concluded. Hence, temporal stability may be concluded for the alternative exponential ansatz function.

### APPENDIX C: ASYMPTOTIC EXPANSION FOR SMALL STREAMWISE WAVE NUMBERS

The influence of large wavelength perturbations in the streamwise direction is attempted to be understood by an asymptotic analysis for small streamwise wave numbers *α*. For this we introduce a formal Poincaré expansion for the eigenvalue *ω* of the form

to be employed into the EVP (16).

With this, the viscous parameter $\sigma (\omega )\alpha ,Re$ may be expanded to

Expanding the parameters of the generalized hypergeometric function $\u20092F3$ constituting the EVP (16) yields

Now, only the leading order terms are considered. The first $2F3$ function then expands into

with $\sigma 0=1\u22124iRe\u2009\omega 0$.

Clearly, $2F3(a2,b2;\u2212iRe\u2009\alpha )$ asymptotically strives toward unity in the leading order.

Substituting the parameters $a4i$ and $b4i$ into $2F3(a4,b4;\u2212iRe\u2009\alpha )$ yields

Analogously, the differentiated hypergeometric functions expand into

Substituting the leading order of all expansions into the eigenvalue problem in (16) results in

which apparently yields no solution for *ω*_{0}. As a consequence, the leading order $D(\omega (\alpha ))O(1)$ is not equal to zero. An ansatz as suggested in (C1) therefore is not viable for $\alpha \u21920$. In fact, it can be shown analogously that any power series $\omega =\u2211pP\omega p\u2009\alpha p$ with $p\u22650\u2208\mathbb{R}$ fails.

### APPENDIX D: ASYMPTOTIC EXPANSION FOR DISTINGUISHED LIMIT $Re\alpha =Re\u2009\alpha $

In order to expand (16) asymptotically in the distinguished limit (28), it is necessary to expand each parameter of the $2F3$ generalized hypergeometric functions given in (7):

with $\sigma 0=1+4i\u2009Re\alpha \u2009(1\u2212\omega 1)$. Furthermore, the argument of the $2F3$ functions is expanded as

which in the distinguished limit is now constant.

We observe that for all $2F3$ functions, the expansions of their parameters as well as their respective arguments are constant to leading order. Further, the $2F3$ functions constituted by $a4$ and $b4$ will be reduced to $1F2$ functions as the leading orders of *a*_{41} and *b*_{42} are equal, due to which they cancel each other out. Finally, as $a22=\u22121+\alpha $, the Pochhammer term $(a22)n=(\u22121+\alpha )\u2009(\alpha )(1+\alpha )\u2026$, that is, the second factor, is of order $O(\alpha )$. Hence, for $n\u22652$ all series terms of $2F3(a2,b2,z(y=0))$ are of order $(\alpha )$. The same applies for $2F3(a2+1,b2+1,z(y=0))$, where for $n\u22651$ the series terms are of order $O(\alpha )$. With this, we can now formally expand the $2F3$ function whereupon we obtain

where the series terms for $n\u22652$ are of order $O(\alpha )$. The same applies for the generalized hypergeometric function $\u20092F3(a2+1,b2+1;\u2212iRe\u2009\alpha )$, with the only difference being that due to the increment of 1 on all parameters, all series terms for $n\u22651$ are of order $O(\alpha )$, which gives

where $a42,0,\u2009b41,0$, and $b43,0$ denote the leading orders of the corresponding parameters expanded in (D1). As is displayed in (D5), the leading orders of *a*_{41} and *b*_{42} cancel each other out, yielding a $1F2$ generalized hypergeometric function to leading order. An analogous expansion is conducted for the last generalized hypergeometric function $\u20092F3(a4+1,b4+1;\u2212iRe\u2009\alpha )$ resulting in

### APPENDIX E: ASYMPTOTIC EXPANSION FOR LARGE STREAMWISE WAVE NUMBERS ($\alpha \u2192\u221e$)

The asymptotic analysis for the case $\alpha \u2192\u221e$ is rather straightforward. The EVP (16) is transformed by redefining

which now for the examined limit yields $\alpha T\u21920$, allowing us to employ classical tools of asymptotic theory. The EVP then is written as

with the parameters

with $\alpha \u0303T=(1/\alpha T)2+1$ and $\sigma =4/\alpha T+1+4iRe\u2009(1/\alpha T\u2212\omega )$. Asymptotic theories are now used to derive a leading order solution for $\omega (\alpha T)$. For this, we assume a Laurent series of the form

Without going into detail, the following can be observed:

- In the above series, for $n\u2265\u22121$ the parameter quotients $ai/bi$ are of order $O(\alpha T)$ regardless of the series count of the corresponding hypergeometric series. Together with the argument of the series, each term of these series is of
*O*(1). For*n*< 1, however, these series diverge heavily as then each term of the series is at least of order $O(\alpha T\u22121)$. Hence, the leading order of (E5) must be(E6)$\omega (\alpha T)=\omega (\u22121)1\alpha T+O(1),$which yields(E7)$\sigma (\alpha T)=2\alpha T+iRe\u2009(1\u2212\omega (\u22121))+O(\alpha T).$ - The order of the dispersion relation (E2) with (E6) purely depends on the factors of each series product. Concretely, we need to compare the leading orders of:(E8)$T1=a4\xafb4\xaf\u2009iRe\u20091\alpha T,$(E9)$T2=1\alpha T\u2212(\sigma +12)=12(iRe\u2009(1\u2212\omega (\u22121))\u22121)+O(\alpha T),$(E10)$T3=a2\xafb2\xaf\u2009iRe\u20091\alpha T.$

As already pointed out, the leading orders of the parameter quotients, respectively, are $O(\alpha T)$. Hence, *T*_{2} embodies the leading order term and has to disappear in order to yield (E2) zero in the leading order. This results in

which substituted into (E5) gives

The real part of the eigenvalue thus is equal to the wave number, while the imaginary part is always negative, regardless of the Reynolds number. The flow therefore is stable at the infinity limit $\alpha \u2192\u221e$ for arbitrary Reynolds numbers.

### APPENDIX F: ASYMPTOTIC EXPANSION FOR THE STOKES-LIMIT ($Re\u21920$)

The Stokes limit can be treated qualitatively, claiming that flows at very small Reynolds numbers are naturally globally stable. For the ASBL, it is still possible to demonstrate this with the already employed asymptotic methods rather rigorously. First, we assume a Laurent series of the form

In analogy to the large *α* case, we examine the hypergeometric series with regard to their asymptotic behavior. To be specific, it is generally necessary for the series terms to not diverge, as then the series surely diverges. Since all four series in (16) should not be equivalent in the Stokes limit, it is obligatory to prevent any of these series from diverging. The parameter quotients $ai\xafbi\xaf$ must therefore at least be of order $O(Re\u22121)$ in order to cancel out with the argument of each respective series. With this, the Laurent series (F1) reduces to

In fact, with this reduction every term in (16) is of order *O*(1), except

which in conclusion constitutes the leading order of the EVP (16), which must be zero. This demands

which further reduces the Laurent series to

So far the considerations helped to determine the leading order of the ansatz (F1). It is still necessary, however, to determine the coefficient of the leading order $\omega (\u22121)$. This is also achieved by doing a leading order consideration. All series are now of order *O*(1). It can even be shown that all hypergeometric series $2F3(\xb7)=1+O(Re)$. Again, the leading order of the EVP comes down to evaluating the three pre-factor terms *T*_{1}, *T*_{2,} and *T*_{3} given in (E10). With (F3) the terms *T*_{1} and *T*_{3} are of order $O(Re)$. Consequently, the leading order of (16) is given by

which gives

Substituted into (F5), we obtain

As was claimed at the beginning of this analysis, the flow is globally stable in the Stokes limit for any $\alpha \u22650$.