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 α0,that is, small streamwise wave numbers, and Re is only present in the distinguished limit Reα=O(1). This results in a degenerated Orr–Sommerfeld equation and the expanded algebraic eigenvalue problem poses a lower limit of (Reα)min0.84191. 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=6.0×106. Besides the further refined critical values αcr=0.15546,ωcr=0.023297,Recr=54378.62032, discrete spectra and eigenfunctions are examined and ω=ωr+iωi 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.

Ever since Prandtl2 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 Orr3 and Sommerfeld4 who independently modeled perturbations as wave-like solutions of the form g(x,y,z,t)=g̃(y)ei(αx+βzωt) 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 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 Tollmien6 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 V0 superposing a flat-plate boundary layer flow. The result is a laminar base flow with an exponential profile with an asymptotically constant displacement thickness δ1 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 V0, were first analyzed by Chiarulli and Freeman,8 who introduced the idea of transforming the mOSE to a hypergeometric differential equation. Hughes and Reid9 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 Hocking11 who transformed the non-linear disturbed Navier–Stokes equations (NSE) into a non-linear Schrödinger equation, based on techniques employed by Stewartson and Stuart12 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.

FIG. 1.

Schematic visualization of a 3D asymptotic suction boundary layer flow.

FIG. 1.

Schematic visualization of a 3D asymptotic suction boundary layer flow.

Close modal

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 scenarios17 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.

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(1eyV0ν),V0,0)T. It constitutes the base flow to be analyzed below on its stability properties. Non-dimensionalization with the displacement thickness δ1=ν/V0 and the inflow velocity U yields

U(y)=1ey,V(y)=1Re,withRe=Uδ1ν=UV0.
(1)

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

q(x,y,z,t)=q̃(y)ei(αx+βzωt),
(2)

for all perturbations, we obtain the Orr–Sommerfeld equation

[(iω+iαU(y)1ReD)(D2k2)iαd2U(y)dy21Re(D2k2)2]v(y)=0,
(3)

with D expressing the derivative with respect to y and v(y) denoting the wall-normal velocity, α, β denoting the stream- and spanwise wave number, k=α2+β2 and ω=ωR+iωI 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 α=±1, 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 Re1 after non-dimensionalization, namely, Re1D(D2k2)v(y), which does not occur in the classical OSE. By substituting (1) into (3), it was shown by Baldwin10 that the fundamental solution

v(y)=v1(y)+v2(y)+v3(y)+v4(y),
(4)

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

v(y)=C1eky2F3(a1b1;iReαey)+C2eky2F3(a2b2;iReαey)+C3eσ12y2F3(a3b3;iReαey)+C4eσ+12y2F3(a4b4;iReαey).
(5)

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

a1=(k̃kk̃k),a2=(k̃+kk̃+k),a3=(12σ2+k̃12σ2k̃),a4=(12+σ2+k̃12+σ2k̃),
(6)
b1=(12k12+σ2k12σ2k),b2=(1+2k12σ2+k12+σ2+k),b3=(1σ32σ2+k32σ2k),b4=(1+σ32+σ2+k32+σ2k),
(7)

with k̃=k2+1 and

σ=σASBL=4k2+1+4iRe(αω).
(8)

Grosch and Salwen22 labeled analogous solutions for the BBL belonging to e±ky as inviscid solutions, while e±σBBLy were named viscous solutions. Accordingly, the parameter σASBL 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.

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

v(y=0)=0,Dv(y)|y=0=0,
(9a)
v(y)=0,Dv(y)|y=0.
(9b)

Substituting (5) into (9b) gives

v(y)=limy[C1eky+C2eky+C3eσ12y+C4eσ+12y]=0,
(10)
v(y)=limy[C1kekyC2keky+C3(σ12)eσ12yC4(σ+12)eσ+12y]=0.
(11)

As k > 0 the solution v1(y)=C1eky diverges for y, while v2(y)=C2eky converges. Thus, C1=0. The behavior of the two remaining solutions depends exclusively on the real part of σ. In principle, σ=±σR±iσI 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)σ=v(y)σ. Hence, it is sufficient to focus on σ=σR+iσI solely. Regardless of k and Re, the real part σR is always greater than zero, which yields v4(y)=C4eσ+12y convergent.

It can be shown that only the case σR>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

v(y=0)=C22F3(a2b2;iReα)A1(α,β,ω,Re)+C42F3(a4b4;iReα)A2(α,β,ω,Re)=0,
(12)
v(y=0)=C2[a¯2b¯22F3(a2+1b2+1;iReα)iReαk2F3(a2b2;iReα)]A3(α,β,ω,Re)+C4[a¯4b¯42F3(a4+1b4+1;iReα)iReα(σ+12)2F3(a4b4;iReα)]A4(α,β,ω,Re)=0,
(13)

where d2F3(a,b;z)/dz=(a¯/b¯)2F3(a+1,b+1;z) and a¯/b¯=(a1a2)/(b1b2b3) have been used. Resultantly, (12) and (13) constitute a homogeneous linear system of the form

(A1(α,β,ω,Re)A2(α,β,ω,Re)A3(α,β,ω,Re)A4(α,β,ω,Re))(C2C4)=(00),
(14)

for which non-trivial solutions for C2 and C4 only exist for

D(α,β,Re,ω)=det(A(α,β,ω,Re))=0,
(15)

defining an algebraic eigenvalue problem (EVP) of the form

D(α,β,Re,ω)=2F3(a2b2;iReα)[iReαa¯4b¯42F3(a4+1b4+1;iReα)(σ+12)2F3(a4b4;iReα)]2F3(a4b4;iReα)[iReαa¯2b¯22F3(a2+1b2+1;iReα)α2F3(a2b2;iReα)]=0.
(16)

For a given set of parameters ω=ω(α,Re) determined from (16), we may compute the eigenfunction v(y) from (5). For this, we employ C1=C3=0 and express C2 in terms of C4 using (12). The final results read

v(y)=e12(σ+1)y2F3(a4b4;z(y))eky2F3(a4b4;z(0))2F3(a2b2;z(0))2F3(a2b2;z(y)),
(17)

where we have omitted C4 because of the linearity of the problem and z(y)=iReαey. When presented subsequently, eigenfunctions will be normalized to one.

Further analysis of the EVP (16) is presented in Sec. IV (asymptotic analysis) and Sec. V (stability characteristics). Before proceeding with these analyses, the possible existence of temporal continuous spectra for the ASBL is examined in Sec. III.

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,

v(y),Dv(y)boundedasy
(18)

is assumed instead of strictly vanishing disturbances at infinity.

Employing (10) and (11), we may deduce that (18) is obeyed, if

σR±1=0,
(19)

with

σR=12(4α2+4ReωI+1)2+16Re2(αωR)2+(4α2+4ReωI+1).
(20)

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

v3(y)+v4(y)=C3eiσI2y+C4eyiσI2y,
(21)

employing

σI=sgn(Δ)12(4α2+4ReωI+1)2+(4Reα4ReωR)2(4α2+4ReωI+1),
(22)

with Δ=αωR, 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.

For the temporal stability problem, the real part of viscous parameter σ is given in (20). Hence, with σR=1 in (19) the condition may be explicitly solved for ωI=ωI(ωR) and hence allows to express ω=ω(ωR) as

ω=ωRi(α2Re+Re(αωR)2).
(23)

This parametrization due to ωR denotes a continuous set of eigenvalues ω=ω(ωR) and as such yields v3(y) bounded. For a BBL flow, the continuous spectrum parametrized by ξ is given as

ω=αi(1+ξ2)α2Re.
(24)

It is common to display spectra utilizing the complex phase speed

c=ωα.
(25)

Employing (25), the spectrum for the BBL is given as a vertical line in the complex plane with cR = 1, and a maximum at (1,α/Re), while for the ASBL the spectrum is a parabola opened downwards with a maximum also located at (1,α/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 ωI is always negative, and hence, the eigenfunctions of the temporal continuous spectrum are always stable.

FIG. 2.

Temporal continuous spectrum for the ASBL (line) and the BBL (dashed line) with α = 1 and Re = 500. For both spectra a maximum is located at cR = 1 and cI=α/Re.

FIG. 2.

Temporal continuous spectrum for the ASBL (line) and the BBL (dashed line) with α = 1 and Re = 500. For both spectra a maximum is located at cR = 1 and cI=α/Re.

Close modal

It will be demonstrated in Secs. IV A–IV D, how the parameters Re and α influence one another in both asymptotic limits α0 and α upon expanding the EVP (16). In accordance with Squire's theorem23 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,

α2D2=α3D2+β2,
(26a)
α2DRe2D=α3DRe3D,
(26b)
ω2DRe2D=ω3DRe3D,
(26c)

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 β2>0 in the 3D case, (26a) gives α3D<α2D. 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.

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

ω=n=n=ωnαn
(27)

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 Reα is of order O(1), valid solutions for ω(α) exist. Hence, if solutions for small α only exist for Re=O(α1), 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 Re1 simultaneously become asymptotically small, while the product

Reα:=Reα
(28)

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, ω0=0. Thus, the highest order ω can adopt is O(α) so that

ω(α)=n=1ωnαn,
(29)

which in turn gives

σ(ω(α))=1+4iReα(1ω1)2iReαω21+4iReα(1ω1)α+O(α2),
(30)

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)=1ey scale with the streamwise wave number α. Hence, an expansion for α0 would yield these terms of order O(α). Only simultaneously assuming Re and, further, taking the distinguished limit Reα=O(1)“saves” the base flow terms in the leading order. Expanding (3) in the distinguished limit thereupon gives

d4ṽ(y)dy4+d3ṽ(y)dy3(iReαiReαωαiReαey)d2ṽ(y)dy2iReαeyṽ(y)+O(α)=0,
(31)

where ṽ(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, iReαωα. We must make sure that this term is neither of dominant order nor of order O(α). This only allows one inference for the order of ω, that is

ω=ω1α+O(α2),
(32)

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 ω(α), 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

D(ω1,Reα)O(1)=ω1ω11[iReα(σ01)(1+σ0)(3+σ0)1F2(12(1+σ0)(2+σ0),12(5+σ0);iReα)+12(1+σ0)1F2(12(1+σ0)(1+σ0),12(3+σ0);iReα)]11ω11F2(12(1+σ0)(1+σ0),12(3+σ0);iReα)=0,
(33)

with σ0=1+4iReα(1ω1). Resultantly, (33) is a degenerated EVP, which determines ω1 as a function of Reα and is to be solved numerically.

The solutions for the computation of the Laurent-series coefficient ω1=ω1(Reα) 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 ω1 is a lower threshold appearing at Reα,min0.84191, under which no solutions could be computed for ω1. As may be taken from Fig. 3(a), we observe that in the limit ReαReα,min we obtain ω1123i or rather ωα23iα. For increasingly large Reα, the coefficient ω1,R decreases monotonically, while ω1,I increases monotonically, respectively.

FIG. 3.

(a) The Laurent-series coefficient ω1 in (29) is plotted vs Reα=Reα. A lower threshold exists at Reα,min0.84191, below which solutions do not exist. (b) The physical parameters ωR, ωI, and α are obtained by re-substituting ω1 and Reα into (32) and (28), respectively, and displayed for Re=1.0×105.

FIG. 3.

(a) The Laurent-series coefficient ω1 in (29) is plotted vs Reα=Reα. A lower threshold exists at Reα,min0.84191, below which solutions do not exist. (b) The physical parameters ωR, ωI, and α are obtained by re-substituting ω1 and Reα into (32) and (28), respectively, and displayed for Re=1.0×105.

Close modal

By employing ω1 into (32) and Reα 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=1.0×105 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=1.0×105 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 α0. While for αmin=0.84191×105 the relative error is of order O(1010), it continuously grows to an error of order O(106) for α=1.0×103. 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.

FIG. 4.

(a) Numerically computed eigenvalues in the vicinity of the distinguished limit Reα are shown for Re=1.0×105. (b) The semilogarithmic error of the asymptotic eigenvalues obtained from the degenerated EVP (33) based on eigenvalues of the original EVP (16) is displayed.

FIG. 4.

(a) Numerically computed eigenvalues in the vicinity of the distinguished limit Reα are shown for Re=1.0×105. (b) The semilogarithmic error of the asymptotic eigenvalues obtained from the degenerated EVP (33) based on eigenvalues of the original EVP (16) is displayed.

Close modal

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 Reα,cr0.84191 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.

In the asymptotic limit α, the eigenvalue ω(α) is assumed to be a Laurent series in α of the form

ω(α)=n=ωnαn.
(34)

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

ω(α)=ω1α+ω0+O(α1).
(35)

Employing (35) into (16) gives the expanded EVP

D(ω(α))=1+iRe(1ω1)+O(α1)=0,
(36)

which yields the solution

ω(α)=ω1α+O(1)=(1iRe)α+O(1),
(37)

and hence, ωi(α)<0 for α. In summary, the flow is globally stable for large streamwise wave numbers α in the case of arbitrary Re.

Analogous to the previous analyses, for the Stokes limit Re0 a Laurent series of the form,

ω(Re)=n=ωnRen,
(38)

is assumed for ω(Re). Employing the series into the viscous parameter σ yields

σ(ω(Re))=4α2+1+4iRe(αn=ωnRen).
(39)

For any arbitrary n < 0 (with n), the leading order of σ is, therefore, always O(Re12(1+n)). A lengthy analysis to derive the necessary leading order of ω(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 ω(Re) must be ω(Re)=ω1Re+O(1), which yields

σ(ω(Re))=4α2+14iω(1)+O(Re12).
(40)

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

D(ω(Re))O(1)=α12(1+4α2+14iω(1))+O(Re12)=0,
(41)

which for the coefficient ω(1) yields

ω(1)=iα,
(42)

giving the leading order of the sought eigenvalue as

ω(Re)=iαRe+O(1).
(43)

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

A thorough analysis of the large Reynolds number case has been covered extensively both for the analytic solution of the mOSE in Baldwin10 as well as in a more recent investigation published by Dempsey and Walton25 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 α=O(Re1/4).

  • For β = 1, the constant of proportionality derived from the ensuing dispersion relation is given as αN=0.617, that is, α0.617Re1/4. These results coincide nicely with the temporal stability map computed in Fig. 5.

FIG. 5.

2D temporal stability map with temporal amplification rate ωi plotted semilogarithmically with 1.0×104Re6.0×106. Blank areas describe linearly stable modes. The asymptotic derived in Dempsey and Walton25 is displayed by the point-dashed line. The critical triplet is given in (44).

FIG. 5.

2D temporal stability map with temporal amplification rate ωi plotted semilogarithmically with 1.0×104Re6.0×106. Blank areas describe linearly stable modes. The asymptotic derived in Dempsey and Walton25 is displayed by the point-dashed line. The critical triplet is given in (44).

Close modal

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.

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 [αcr,ωcr,Recr] will be compared to the one obtained by a Chebyshev collocation method.13 

The temporal stability map in Fig. 5 was computed up to a Reynolds number of Re=6×106 and a streamwise wave number of α=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 [αcr,ASBL=0.1555,ωcr,ASBL=0.023325,Recr,ASBL=54370] 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 [αcr,ASBL=0.1555,ωcr,ASBL=0.02331,Recr,ASBL=54382]. 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(1022) 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:

αcr=0.15546,ωcr=0.023297,Recr=54378.62032,
(44)

of which the Recr is in fact located between the previously given critical Reynolds numbers provided by Hocking11 and Fransson and Alfredsson.13 The residuum of the EVP (16) with the critical values in (44) is of order O(1022), while the positive imaginary part ωcr,i is of order O(1014). 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 α0.178, regardless of the Reynolds number, and this is also validated by the asymptotic analysis for α.

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 η[1,1] using an algebraic linear mapping of the form

y=L2(η+1),
(45)

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=1080, except for the case Re = 500 000, where a tolerated residual of tol=10100 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 cR0, a P-branch when cR1 or an S-branch when cR2/3, with cR denoting the real part of the phase velocity. Even though inherently different, the eigenvalue spectra for the BBL29 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.

FIG. 6.

Temporal spectra are displayed at fixed Recr = 54 378.620 32 and varying α: (a) α=0.001; (b) α=0.01; (c)αcr=0.15546; (d) α = 1. Here and in Fig. 6, the red parabola denote the temporal continuous spectra given by (24).

FIG. 6.

Temporal spectra are displayed at fixed Recr = 54 378.620 32 and varying α: (a) α=0.001; (b) α=0.01; (c)αcr=0.15546; (d) α = 1. Here and in Fig. 6, the red parabola denote the temporal continuous spectra given by (24).

Close modal
FIG. 7.

Temporal spectra are displayed at fixed αcr=0.15546 and varying Re: (a) Re = 500; (b) Re = 5000; (c) Recr = 54 378.620 32; (d) Re = 500 000. The spectra were obtained analogous to Fig. 6. Eigenfunctions of the eigenvalues 14 in Fig. 7(c) are shown in Fig. 8.

FIG. 7.

Temporal spectra are displayed at fixed αcr=0.15546 and varying Re: (a) Re = 500; (b) Re = 5000; (c) Recr = 54 378.620 32; (d) Re = 500 000. The spectra were obtained analogous to Fig. 6. Eigenfunctions of the eigenvalues 14 in Fig. 7(c) are shown in Fig. 8.

Close modal

In Fig. 6, the eigensprectra in Recr are compared for varying orders of α. For very small α=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 α=αcr, 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 αcr 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.

FIG. 8.

Eigenfunctions of the wall-normal disturbance vr(y) vs y are plotted for the critical parameters Recr and αcr given in (44). The corresponding ω are given in Fig. 7(c) where eigenfunctions for selected ω are shown: (a)ω1=0.023296, (b)ω2=0.0500410.021402i, (c)ω3=0.0735230.036063i, (d)ω4=0.141200.043418i.

FIG. 8.

Eigenfunctions of the wall-normal disturbance vr(y) vs y are plotted for the critical parameters Recr and αcr given in (44). The corresponding ω are given in Fig. 7(c) where eigenfunctions for selected ω are shown: (a)ω1=0.023296, (b)ω2=0.0500410.021402i, (c)ω3=0.0735230.036063i, (d)ω4=0.141200.043418i.

Close modal

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 αcr [see Fig. 9(a)] and for varying α and fixed Recr [see Fig. 9(b)]. It is quite remarkable that varying the orders of 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 eαy in v(y) as given in (17), dominates.

FIG. 9.

TS eigenfunctions vr(y) parametrized by Re at fixed α=αcr (a) and α at fixed Re=Recr (b). The critical parameters can be taken from (44). For comparison, we have displayed the laminar base solution in the form 1U(y).

FIG. 9.

TS eigenfunctions vr(y) parametrized by Re at fixed α=αcr (a) and α at fixed Re=Recr (b). The critical parameters can be taken from (44). For comparison, we have displayed the laminar base solution in the form 1U(y).

Close modal

It is quite enlightening, however, to extend the analysis to the perturbation velocity v(x,y,t)=v(y)ei(αxωt). When employing the eigenfunction v(y), given in (17), into the normal-mode ansatz (2), one gets

v(x,y,t)=v(y)ei(αxωRt)eωIt=ei(αxσI2yωRt)eσR+12y+ωIt2F3(a4b4;iReαey)c̃ei(αxωRt)eαy+ωIt2F3(a2b2;iReαey),
(46)

with c̃=2F3(a4,b4;iReα)/2F3(a2,b2;iReα) and σR and σI 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 xy 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, the 2F3 hypergeometric functions asymptotically converge to 1, which gives

limyv(x,y,t)=ei(αxσI2yωRt)eσR+12y+ωItc̃ei(αxωRt)eαy+ωIt.
(47)

Comparison with Mack1 unveils that in the limit of large y two analogous wave components also exists in the BBL of the form

limyvBBL(x,y,t)=ei(αxσBBL,RyωRt)eσBBL,Iy+ωIt+c̃ei(αxωRt)eαy+ωIt.
(48)

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

  • The viscous wave propagating in the x–y plane decays with eσR+12y.

  • The inviscid wave propagating in x-direction which decays with eαy.

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.

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

vvis(x,y,t)=ei(αxσI2yωRt)eσR+12y+ωIt2F3(a4b4;iReαey).
(49)

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 σI2. The comparison of the signs of α and σI determines whether the viscous waves move toward or away from the wall. The sign of σI is determined by sgn(αωR), as is given in (22). For the domain 1.0×104Re1.0×106 and 1.0×104α1.0, it turns out that αωR>0 and ωR>0. Thus, α and σI2 without exception bear opposing signs, while ωR>0, due to which the viscous waves defined in (49) in all cases move toward the wall at a propagation angle

θ=arctan(σI2α).
(50)

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

δσ=2σR+1,
(51)

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 δσ is a multiple of the boundary layer displacement thickness δ1=ν/V0. Hence, this quantity hereafter is called wave decay-rate length (WDRL).

Now, σR in (20) by definition is always positive. Hence, δσ reaches a maximum when σR becomes minimal (or ideally zero). Close examination of (20) reveals that σR in fact becomes minimal when α=ωR and ωI is negative. While negative ωI are quite common, the question is pending whether or not there is a point at which α is indeed equal to ωR. 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 α is equal to ωR! 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 Reα=Reα=O(1) do solutions for ω exist in the zero limit of α. A computation of the expanded EVP (33) revealed a lower threshold for Reα,min0.84191, below which no solutions exist for ω. In the limit ReαReα,min, however, the eigenvalue ω converges toward ωmin=αi23α. Thus, for the lower threshold of Reα does the extraordinary case ωR=α occur. At this point, σR becomes minimal and gets very close to zero.

This result may be verified in Fig. 10(a). In this figure, the WDRL δσ is calculated and visualized for the given parameters. In almost the entire parameter range is δσ<1, which means that the WDRL is smaller than the displacement thickness δ1. However, at Re=1.0×104 and α=1.0×104 a maximum for δσ can be witnessed, clearly approaching the theoretical maximum δσ,max=2. At this point, the WDRL becomes larger than δ1. 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.

FIG. 10.

(a) WDRL δσ as given in (51) as a function of α and Re. (b) Wave propagation angle Θ as given in (50) as a function of α and Re (in degrees).

FIG. 10.

(a) WDRL δσ as given in (51) as a function of α and Re. (b) Wave propagation angle Θ as given in (50) as a function of α and Re (in degrees).

Close modal

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 [α,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.

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 cRcI spectrum, as is the case for the BBL, but rather a parabola open downwards with its maximum located at cR = 1 and cI=α/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 α and Re0. However, the expansion for α0 did not provide any solutions for ω in the leading order. It could then be shown that only for the distinguished limit Reα=Reα=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(α1) and only gave solutions if Reα>0.84191. 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 ReαReα,min0.84191 the eigenvalue converges to ω=αi23α. This link between large-scale streamwise structures and large Reynolds numbers was observed in various wall-bounded shear flows24,30 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(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 δσ, which is inversely proportional to the real part of the previously defined viscous parameter σR and thus attained a maximum for σR0. As was shown in a subsequent analysis, this condition is satisfied exactly when ωR=α, which was only the case in the limit Reα0.84191. Hence, the longevity of the oblique viscous wave parts of the TS-waves is highest when approaching this lower thresholds for Reα. 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 1.0×104Re6.0×106, 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, αcr=0.15546 and ωcr=0.023297.

The overall ωRωI 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 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.

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.

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.

While in 2D the streamwise perturbation velocity u(x,y,z,t) is handily obtained by the continuity equation, in 3D the spanwise perturbation velocity component w(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)

[(iω+iαU(y)1ReD)1Re(D2k2)]η(y)=iβdU(y)dyv(y),
(A1)

where η(y)=iαu(y)iβw(y) denotes the wall-normal vorticity. Similar to the mOSE, an analytic solution of the mSE exists and is given as

η(y)=2iReβey2[Iσ((i1)2Reαey2)ey2Kσ((i1)2Reαey2)v(y)dy
(A2)
+Kσ((i1)2Reαey2)ey2Iσ((i1)2Reαey2)v(y)dy]
(A3)
+C5ey2Iσ((i1)2Reαey2)+C6ey2Kσ((i1)2Reαey2),
(A4)

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). Kσ and Iσ 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,

η(y=0)=0,η(y)=0.
(A5)

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

iαu(y)+dv(y)dy+iβw(y)=0,
(A6)

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

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 Oberlack33 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

[t+(1ey)x1Rey]ΔΨ+eyΨx=0.
(B1)

The underlying Lie algebra is then given by

X1=tX2=xX3=ΨΨX4=e1Re[t+x1Rey],Xθ=θ(x,y,t)Ψ.
(B2)

The symmetries X1 and X2 stand for translations in time and space, while X3 stands for a scaling symmetry in the stream function. These symmetries would also arise in the viscous LNSE. However, the fourth symmetry X4—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 X2, X3, and X4 and following the procedure given in detail in Nold, Oberlack, and Cheviakov,34 we arrive at the ansatz function

Ψ(x,y,t)=Φ(Rey+t)eReωetRe+αx+Reαy.
(B3)

Evidently, the streamfunction would then be comprised of a traveling wave solution Φ(Rey+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

x̃=x,ỹ=Rey+t,t̃=t,
(B4)

then gives

Ψ(x̃,ỹ,t̃)=Φ(ỹ)eReωet̃Re+αx̃+α(ỹt̃).
(B5)

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

(ωeỹReα)Φ(ỹ)+(2αωeỹRe2α2)Φ(ỹ)+(α2(1+1Re2)(1+eỹRe)1)Φ(ỹ)=0.
(B6)

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

Φ(ỹ)=C1[(αe(α+iReα21)ỹωe(1Reα+iReα21)ỹ)2F1(1+i(α+α21),1+i(α+α21)1+2iα21;ωαeỹRe)]+C2[(αe(αiReα21)ỹωe(1ReαiReα21)ỹ)2F1(1+i(αα21),1+i(αα21)12iα21;ωαeỹRe)].
(B7)

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

Ψ(x,y,t)x|y=[αΦ(Rey+t)eReωetRe+αx+Reαy]y=0,
(B8)

requires an expansion of (B5) at large ỹ=Rey+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

Ψ(x,y,t)x|y=C1(ω1Γ1e(iReα)ỹω2Γ2e(iReα)ỹ)+C2(ω3Γ3e(iReα)ỹω4Γ4e(iReα)ỹ)=0,
(B9)

where

ω1=ω(ωα)(1+i(α21+α)),
(B10a)
ω2=ω(ωα)(1+i(α21α)),
(B10b)
ω3=ω(ωα)(1+i(α21+α)),
(B10c)
ω4=ω(ωα)(1+i(α21α)),
(B10d)

and

Γ1=Γ(1+i(α21α))Γ(i(α21α)),
(B11a)
Γ2=Γ(1+i(α21+α))Γ(i(α21+α)),
(B11b)
Γ3=Γ(1+i(α21α))Γ(i(α21α)),
(B11c)
Γ4=Γ(1+i(α21+α))Γ(i(α21+α)),
(B11d)

with Γ(z) denoting the gamma function.

The expanded BC (B9) yields non-trivial solutions for C1 and C2 only if

ω1Γ1ω4Γ4ω2Γ2ω3Γ3=0.
(B12)

Closer examination of (B10) reveals that

ω1ω4=ω2ω3,
(B13)

so that (B12) reduces to

Γ1Γ4=Γ2Γ3.
(B14)

Finally, two identities of gamma functions are utilized:

Γ(z+1)=zΓ(z),
(B15a)
Γ(iz)2=πysinh(πy),
(B15b)

which employed into (B14) yields

sinh(π(α21α))sinh(π(α21α))=sinh(π(α21+α))sinh(π(α21+α)),
(B16)

which is satisfied only if

α=±1.
(B17)

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 X1X4, a second new ansatz function occurs for the 2D non-viscous ASBL given by

Ψ(x,y,t)=φ(y(1λ)Re+tλx)eα(Rey+x),
(B18)

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

x̃=xỹ=y(1λ)Re+tλxt̃=t,
(B19)

transforms (B18) to

Ψ(x̃,ỹ,t̃)=φ(ỹ)eαỹt̃+(1+λ)x̃(1λ)Re.
(B20)

Employing (B20) into (B1) and solving for φ(ỹ) in the arising ODE gives

φ(ỹ)=C1e(κ1+κ2)ỹ+C2e(κ1κ2)ỹ+C3eαλỹ,
(B21)

where

κ1=α(λ1)Re2+λ(λ1)2Re2+λ2,κ2=(Re2+1)λ22Re2λ+(1α2)Re2(λ1)2Re2+λ2.
(B22)

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

Ψ(x,y,t)=C1eRe([1λ](κ1+κ2)+α)ye(αλ(κ1+κ2))xe(κ1+κ2)t+C2eRe([1λ](κ1κ2)+α)ye(αλ(κ1κ2))xe(κ1κ2)t+C3eRe([1λ]αλ+α)yeαλt.
(B23)

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

αλ(κ1+κ2)=iα1,κ1+κ2=β1αλ(κ1κ2)=iα2,κ1κ2=β2,
(B24)

where α1 and α2 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

Ψ(x,y,t)=C1eRe(κ1+iα1)yeiα1xeκ1t+C2eRe(κ2+iα2)yeiα2xeκ2t.
(B25)

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

u(x,y,t)=Ψ(x,y,t)y|y=C1Re(κ1+iα1)eRe(κ1+iα1)yeiα1xeκ1t+C2Re(κ2+iα2)eRe(κ2+iα2)yeiα2xeκ2t=0,
(B26)

and

v(x,y,t)=Ψ(x,y,t)x|y=C1iα1eRe(κ1+iα1)yeiα1xeκ1t+C2iα2eRe(κ2+iα2)yeiα2xeκ2t=0.
(B27)

The only way to satisfy both BCs is to demand κ1,r<0 and κ2,r<0, after which immediately temporal decay can be concluded. Hence, temporal stability may be concluded for the alternative exponential ansatz function.

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

ω=k=0ωkαk,
(C1)

to be employed into the EVP (16).

With this, the viscous parameter σ(ω)α,Re may be expanded to

σ(ω(α))=14iReω0+2iRe(1ω1)14iReω0α+O(α2).
(C2)

Expanding the parameters of the generalized hypergeometric function 2F3 constituting the EVP (16) yields

a2=(α2+1+αα2+1+α)=(1+α+O(α2)1+α+O(α2)),b2=(1+2ασ2+12+ασ2+12+α)=(1+2α1214iReω0+12+(iRe(ω11)14iReω0+1)α+O(α2)1214iReω0+12+(iRe(ω11)14iReω0+1)α+O(α2)),
a4=(α2+1+12+σ2α2+1+12+σ2)=(12(3+14iReω0)+iRe(1ω1)14iReω0α+O(α2)12(1+14iReω0)+iRe(1ω1)14iReω0α+O(α2)),b4=(1+σ32+σ2+α32+σ2α)=(1+14iReω0+2iRe(1ω1)14iReω0α+O(α2)12(3+14iReω0)+(1+iRee(1ω1)14iReω0)α+O(α2)12(3+14iReω0)+(1+iRe(1ω1)14iReω0)α+O(α2)).
(C3)

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

2F3(a2b2;iReα)=n=0(1)n(1+α)n(1)n(12(1σ0))n(12(1+σ0))n(iReα)nn!+O(α2)=1+1ω0α+O(α2),
(C4)

with σ0=14iReω0.

Clearly, 2F3(a2,b2;iReα) asymptotically strives toward unity in the leading order.

Substituting the parameters a4i and b4i into 2F3(a4,b4;iReα) yields

2F3(a4b4;iReα)=n=0(12(3+σ0))n(12(1+σ0))n(1+σ0)n(12(3+σ0))n(12(3+σ0))n(iReα)nn!+O(α)=1+O(α).
(C5)

Analogously, the differentiated hypergeometric functions expand into

2F3(a2+1b2+1;iReα)=n=0(2)n(α)n(2)n(12(3σ0))n(12(3+σ0))n(iReα)nn!+O(α2)=1+O(α2),
(C6)
2F3(a4+1b4+1;iReα)=n=0(12(5+σ0))n(12(1+σ0))n(2+σ0)n(12(5+σ0))n(12(5+σ0))n(iReα)nn!+O(α2)=1+O(α2).
(C7)

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

D(ω(α))=12(1+14iReω0)+O(α)=0,
(C8)

which apparently yields no solution for ω0. As a consequence, the leading order D(ω(α))O(1) is not equal to zero. An ansatz as suggested in (C1) therefore is not viable for α0. In fact, it can be shown analogously that any power series ω=pPωpαp with p0 fails.

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):

a2=(α2+1+αα2+1+α)=(1+O(α)1+α+O(α2)),b2=(1+2ασ2+12+ασ2+12+α)=(1+2α12σ0+12+O(α)12σ0+12+O(α)),a4=(α2+1+12+σ2α2+1+12+σ2)=(12(3+σ0)+O(α)12(1+σ0)+O(α)),b4=(1+σ32+σ2+α32+σ2α)=(1+σ0+O(α)12(3+σ0)+O(α)12(3+σ0)+O(α)),
(D1)

with σ0=1+4iReα(1ω1). Furthermore, the argument of the 2F3 functions is expanded as

z(y=0)=iReα=iReα,
(D2)

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 a41 and b42 are equal, due to which they cancel each other out. Finally, as a22=1+α, the Pochhammer term (a22)n=(1+α)(α)(1+α), that is, the second factor, is of order O(α). Hence, for n2 all series terms of 2F3(a2,b2,z(y=0)) are of order (α). The same applies for 2F3(a2+1,b2+1,z(y=0)), where for n1 the series terms are of order O(α). With this, we can now formally expand the 2F3 function whereupon we obtain

2F3(a2b2;iReα)=n=0(1)n(1+α)n(1)n(12(1σ0))n(12(1+σ0))n(iReα)nn!+O(α)=ω1ω11+O(α),
(D3)

where the series terms for n2 are of order O(α). The same applies for the generalized hypergeometric function 2F3(a2+1,b2+1;iReα), with the only difference being that due to the increment of 1 on all parameters, all series terms for n1 are of order O(α), which gives

2F3(a2+1b2+1;iReα)=n=0(2)n(α)n(2)n(12(3σ0))n(12(3+σ0))n(iReα)nn!+O(α)=1+O(α).
(D4)
2F3(a4b4;iReα)=n=0(12(1+σ0))n(12(3+σ0))n(1+σ0)n(12(3+σ0))n(12(3+σ0))n(iReα)nn!+O(α)=n=0(12(1+σ0))n(1+σ0)n(12(3+σ0))n(iReα)nn!+O(α)=1F2(a41,0b41,0,b42,0;iReα)+O(α),
(D5)

where a42,0,b41,0, and b43,0 denote the leading orders of the corresponding parameters expanded in (D1). As is displayed in (D5), the leading orders of a41 and b42 cancel each other out, yielding a 1F2 generalized hypergeometric function to leading order. An analogous expansion is conducted for the last generalized hypergeometric function 2F3(a4+1,b4+1;iReα) resulting in

2F3(a4+1b4+1;iReα)=n=0(12(5+σ0))n(12(1+σ0))n(2+σ0)n(12(5+σ0))n(12(5+σ0))n(iReα)nn!+O(α)=n=0(12(1+σ0))n(2+σ0)n(12(5+σ0))n(iReα)nn!+O(α)=1F2(a41,0+1b41,0+1,b42,0+1;iReα)+O(α).
(D6)

Employing all expansions derived above into the EVP (16) gives the degenerated EVP (33), for which results are computed and discussed in Sec. IV A. It is noteworthy that the results of (33) agree perfectly with numerical computations of (16) conducted in the distinguished limit.

The asymptotic analysis for the case α is rather straightforward. The EVP (16) is transformed by redefining

αT:=1α,
(E1)

which now for the examined limit yields αT0, allowing us to employ classical tools of asymptotic theory. The EVP then is written as

D(α,β,Re,ω)=2F3(a2b2;iRe1αT)[a¯4b¯4i1αTRe2F3(a4+1b4+1;iRe1αT)(σ+12)2F3(a4b4;iRe1αT)]2F3(a4b4;iRe1αT)[a¯2b¯2i1αTRe2F3(a2+1b2+1;iRe1αT)1αT2F3(a2b2;iRe1αT)]=0,
(E2)

with the parameters

a2=(α̃T+1αTα̃T+1αT),a4=(12+σ2+α̃T12+σ2α̃T),
(E3)
b2=(1+21αT12σ2+1αT12+σ2+1αT),b4=(1+σ32+σ2+1αT32+σ21αT),
(E4)

with α̃T=(1/αT)2+1 and σ=4/αT+1+4iRe(1/αTω). Asymptotic theories are now used to derive a leading order solution for ω(αT). For this, we assume a Laurent series of the form

ω(αT)=n=ωnαTn.
(E5)

Without going into detail, the following can be observed:

  • In the above series, for n1 the parameter quotients ai/bi are of order O(αT) 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(αT1). Hence, the leading order of (E5) must be
    ω(αT)=ω(1)1αT+O(1),
    (E6)
    which yields
    σ(αT)=2αT+iRe(1ω(1))+O(αT).
    (E7)
  • 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:
    T1=a4¯b4¯iRe1αT,
    (E8)
    T2=1αT(σ+12)=12(iRe(1ω(1))1)+O(αT),
    (E9)
    T3=a2¯b2¯iRe1αT.
    (E10)

As already pointed out, the leading orders of the parameter quotients, respectively, are O(αT). Hence, T2 embodies the leading order term and has to disappear in order to yield (E2) zero in the leading order. This results in

ω(1)=1iRe,
(E11)

which substituted into (E5) gives

ω(α)=(1iRe)α+O(1).
(E12)

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 α for arbitrary Reynolds numbers.

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

ω(Re)=n=ωnRen.
(F1)

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¯bi¯ must therefore at least be of order O(Re1) in order to cancel out with the argument of each respective series. With this, the Laurent series (F1) reduces to

ω(Re)=n=2ωnRen.
(F2)

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

σ(Re)=2ω(2)1Re+O(1),
(F3)

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

ω(2)=0,
(F4)

which further reduces the Laurent series to

ω(Re)=ω(1)1Re+O(1).
(F5)

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 ω(1). 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(·)=1+O(Re). Again, the leading order of the EVP comes down to evaluating the three pre-factor terms T1, T2, and T3 given in (E10). With (F3) the terms T1 and T3 are of order O(Re). Consequently, the leading order of (16) is given by

D(Re)O(1)=α(σ+12)=α12(1+4α2+14iω(1))=0,
(F6)

which gives

ω(1)=iα.
(F7)

Substituted into (F5), we obtain

ω(Re)=iα1Re+O(1).
(F8)

As was claimed at the beginning of this analysis, the flow is globally stable in the Stokes limit for any α0.

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