Optimal streaks in a Falkner-Skan boundary layer
aa r X i v : . [ phy s i c s . f l u - dyn ] S e p Optimal streaks
Optimal streaks in a Falkner-Skan boundary layer
Jos´e J. S´anchez- ´Alvarez, Mar´ıa Higuera, and Jos´e M. Vega a) E. T. S. I. Aeron´auticos, Universidad Polit´ecnica de Madrid,Plaza Cardenal Cisneros 3, 28040 Madrid, SPAIN
This paper deals with the optimal streaky perturbations (which maximize the per-turbed energy growth) in a wedge flow boundary layer. These three dimensionalperturbations are governed by a system of linearized boundary layer equationsaround the Falkner-Skan base flow. Based on an asymptotic analysis of this sys-tem near the free stream and the leading edge singularity, we show that for acutewedge semi-angle, all solutions converge after a streamwise transient to a singlestreamwise-growing solution of the linearized equations, whose initial conditionnear the leading edge is given by an eigenvalue problem first formulated in thiscontext by Tumin (2001). Such a solution may be regarded as a streamwise evolv-ing most unstable streaky mode, in analogy with the usual eigenmodes in strictlyparallel flows, and shows an approximate self-similarity, which was partially knownand is completed in this paper. An important consequence of this result is that theoptimization procedure based on the adjoint equations heretofore used to defineoptimal streaks is not necessary. Instead, a simple low-dimensional optimizationprocess is proposed and used to obtain optimal streaks. Comparison with previousresults by Tumin and Ashpis (2003) shows an excellent agreement. The unstablestreaky mode exhibits transient growth if the wedge semi-angle is smaller thana critical value that is slightly larger than π/
6, and decays otherwise. Thus thecases of right and obtuse wedge semi-angles exhibit less practical interest, but theyshow a qualitatively different behavior, which is briefly described to complete theanalysis. a) Corresponding author
I. INTRODUCTION
Internal streaks (or Klebanoff modes ) of a two-dimensional laminar boundary layerdenote three-dimensional disturbances that evolve slowly streamwise and show a fast os-cillation (with a wavelength comparable to the boundary layer thickness); see Refs. forsome recent experiments. The linear stability analysis demonstrated that three dimen-sional perturbations with streaky structure are supported in inviscid shear flows, whoseenergy grows algebraically in time. This initial growth together with a subsequent decaydue to viscous dissipation is known as transient growth . Streaks can be forced eitherinternally, from obstacles near the leading edge or externally, from perturbations in thefree stream, and generally interact with the transversal Tollmien-Schlichting modes, eitherenhancing or delaying transition to turbulence, depending on the streak amplitude. Inthe former case, the effect is known as bypass transition .Luchini proposed in the scope of the Blasius boundary layer an analytical descriptionof streaky perturbations with a small (compared to 1 /δ , where δ is the boundary layerthickness) spanwise wavenumber, which is equivalent to the limit of small distance to theleading edge. The extension of this analysis to the Falkner-Skan profile was made byTumin , who derived an eigenvalue problem describing the velocity components of thestreaky perturbation. He computed the largest eigenvalue of this problem in terms of thewedge angle and found that the unbounded growth is suppressed as the angle exceeds athreshold value, which is slightly larger than π/ and spatial stability .For the non-parallel case, in the scope of Blasius boundary layer, Luchini and Ander-sson et al. calculated optimal streaks using a method based on the adjoint formulation,which is commonly employed in optimal-control problems for distributed parameter sys-tems. Using this method, Tumin and Ashpis computed the optimal streaks in a Falkner-Skan boundary layer and showed the effects of the spanwise wave number and the wedgeangle on transient growth. In particular, they found that an adverse pressure gradientincreases the amplification whereas a favorable presure gradient has the opposite effect(in accordance with the previous asymptotic result by Tumin ). Levin and Henningson also studied the optimal disturbance in a Falkner-Skan base flow, obtaining results thatwere consistent with those by Tumin and Ashpis .In a recent paper , two of the authors analyzed streaks in a Blasius boundary layerand showed that, after an initial transient, they approach a unique (up to a constantfactor) ‘mode’, which was called the unstable streaky mode. This mode is calculated2ptimal streaksfrom the streamwise evolving linearized equations with well-defined initial conditions nearthe leading edge, which are given by the first eigenmode of the eigenvalue problem firstformulated by Luchini . The unstable streaky mode provides the optimal streaks whenthe initial conditions are taken sufficiently close to the leading edge (say, x ≤ − ),making it unnecessary the optimization process in this limit. In fact, the asymptotic valueof the optimal spanwise wavenumber was found to be 0.484, which was slightly different toits counterpart (0.45) calculated by Luchini and Andersson et al. The latter correspondsto an only moderately small value of the initial streamwise coordinate, x ∼ .
01. Theanalysis in Ref. was based on three main ingredients:1. The exact self-similarity of the boundary layer, which allows to eliminate thewavenumber from the formulation.2. The modal structure near the leading edge, which allows to expand any initialcondition into a complete system of eigenmodes.3. An asymptotic analysis of the behavior of the solutions near the free stream. Thiswas essential to both understanding the structure of the solutions and formulatinga quite efficient numerical scheme, which provided streaks with initial conditions ata section extremely close to the leading edge.In addition, the approximate self-similarity already detected experimentally in Ref. andconfirmed in Ref. (namely, the wall-normal profile of the streamwise velocity compo-nent of optimal streaks, rescaled with its maximum, remains constant streamwise) wascompleted noting that the wall-normal profile of a certain combination of the cross flowvelocity components shows the same property as the streamwise velocity component. Apart of the present paper is an extension of this previous work to the Falkner-Skan bound-ary layer. In addition, advantage will be taken of the low dimensional nature of streaksto develop a quite efficient and simple method to calculate optimal streaks. This methodconsists in optimizing the perturbed energy gain in a low-dimensional solution manifoldspanned by a few solutions of the streamwise evolving problem.The remaining of the paper starts with the formulation of the problem, in section II,where the asymptotic behaviors near the free stream and the leading edge are analyzed, insubsections II A and II B, respectively. The streamwise evolution of streaks is considered insection III, where a quite efficient numerical scheme is presented that relies on the previousasymptotic results; the modal nature of streaks is considered in subsection III A. Optimalstreaks are studied in section IV, where the above mentioned optimization method ispresented, in subsection IV A. The body of the paper deals with acute wedge semi-angles; the cases of right and obtuse semi-angles are considered in an appendix, at theend of the paper. The paper ends with some concluding remarks, in section V.3ptimal streaks x b)c)a) d) βπ/2 x βπ/2 x βπ/2 xy yy y FIG. 1. The symmetric flow past a wedge with angle πβ , in the cases (a) β = 0 (Blasius flow),(b) 0 < β <
1, (c) β = 1 (stagnation flow), and (d) 1 < β < II. GENERAL FORMULATION
We consider the high-Reynolds-number flow incidenting in a wedge with semi-angle βπ/ β is named after Hartree . In particular, we study thelinear stability of the resulting boundary layer flow, under streaky perturbations that ex-hibit a spanwise period comparable to the thickness of the boundary layer, δ ∗ = L ∗ / √ Re,where L ∗ is the wall wise length of the portion of the wedge under consideration andRe = u ∗ L ∗ /ν ≫ L ∗ and the free stream velocity u ∗ . Nondimensionalization is made according to the usual boundary layer approxima-tion, using the following units: L ∗ and u ∗ for the streamise spatial coordinate x andvelocity u , respectively, and δ ∗ and u ∗ / √ Re for the wall-normal and spanwise coordinates( y, z ) and velocities ( v, w ), respectively; the pressure is scaled with ρ ∗ ( u ∗ ) / √ Re. Thetwo-dimensional, symmetric base flow, ( u, v, w, p ) = ( u b , v b , , p b ), is the Falkner-Skan similarity solution u b ( x, ζ ) = x β − β F ′ ( ζ ) , v b ( x, ζ ) = − F ( ζ ) + ( β − ζ F ′ ( ζ )(2 − β ) g ( x ) , (1)where ζ is the selfsimilar, wall normal coordinate ζ = y/g ( x ) , with g ( x ) = x − β − β , (2)and the streamfunction F is the solution of the Falkner-Skan equation F ′′′ + [ F F ′′ + β (1 − F ′ )] / (2 − β ) = 0 in 0 < ζ < ∞ , (3) F (0) = F ′ (0) = 0 , F ′ ( ∞ ) = 1 . (4)4ptimal streaksThe Falkner-Skan solution includes the cases (Fig.1) of flat plate boundary layer β = 0,flow past an acute wedge (0 < β < β = 1), andflow in an obtuse wedge (1 < β < β involve unphysical reverseflow and are only of academic interest. Note that the scaled boundary layer thickness isproportional to g ( x ), which invoking (2) means that it grows streamwise if β < β >
1, while at β = 1 the boundary layer flow is parallel. The remaining of thepaper will concentrate in the case of acute wedge angle. The case β ≥ u − u b , v − v b , w, p − p b ) = (cid:18) U ( x, ζ ) , g ( x ) V ( x, ζ ) x , i g ( x ) W ( x, ζ ) x , P ( x, ζ ) x √ Re (cid:19) ei αz , (5)to obtain the following linearized boundary layer (LBL) equations x∂ x U = (1 − β ) ζ − β ∂ ζ U − ∂ ζ V − αg ( x ) W, (6) xF ′ ∂ x U = ∂ ζζ U + F − β ∂ ζ U − βF ′ + ( β − ζ F ′′ + (2 − β ) α g ( x ) − β U − F ′′ V, (7) xF ′ ∂ ˆ x V = ∂ ζζ V + F − β ∂ ζ V + ( β − F + (2 β − ζ F ′ + ( β − ζ F ′′ ](2 − β ) U + ( β + 1) F ′ + ( β − ζ F ′′ − (2 − β ) α g ( x ) − β V − ∂ ζ P, (8) xF ′ ∂ x W = ∂ ζζ W + F − β ∂ ζ W + F ′ − (2 − β ) α g ( x ) − β W − αg ( x ) P, (9) U = V = W = 0 at ζ = 0 and ∞ , P = 0 at ζ = ∞ , (10)where ζ is the selfsimilar wall normal coordinate defined in eq.(2) and ∂ x , ∂ y , · · · denotehereafter partial derivatives. Note that the boundary conditions are no-slip at the plateand vanishing at the infinity in the wall-normal direction.If β = 1, then g ( x ) = 1 and the right hand sides of eqs.(6-9) are independent of x .Otherwise, the spanwise wavenumber can be eliminated from the formulation using thescaling(ˆ x, ˆ y, ˆ u b , ˆ v b ) = ( α − β − β , αy, α β − β u b , v b /α ) , ( ˆ U , ˆ V , ˆ W , ˆ P ) = α − β ( U, V, W, P ) . (11)In addition, as in Ref. , we use the new cross flow variableˆ H = ˆ V + ˆ W , (12)which anticipating results in subsection II A below, converges quite fast to zero as ζ →∞ . Thus, we substitute the spanwise velocity component by this new variable in the5ptimal streaksformulation. Using all these, eqs.(6)-(10) are rewritten asˆ x∂ ˆ x ˆ U = 1 − β − β ζ ∂ ζ ˆ U − ∂ ζ ˆ V − g (ˆ x ) ˆ V + g (ˆ x ) ˆ H, (13) F ′ ˆ x∂ ˆ x ˆ U = ∂ ζζ ˆ U + F − β ∂ ζ ˆ U − βF ′ + ( β − ζ F ′′ + (2 − β ) g (ˆ x ) − β ˆ U − F ′′ ˆ V , (14) F ′ ˆ x∂ ˆ x ˆ V = ∂ ζζ ˆ V + F − β ∂ ζ ˆ V + ( β − F + (2 β − ζ F ′ + ( β − ζ F ′′ ](2 − β ) ˆ U + ( β + 1) F ′ + ( β − ζ F ′′ − (2 − β ) g (ˆ x ) − β ˆ V − ∂ ζ ˆ P , (15)ˆ xF ′ ∂ ˆ x ˆ H = ∂ ζζ ˆ H + F − β ∂ ζ ˆ H + ( β − F + (2 β − ζ F ′ + ( β − ζ F ′′ ](2 − β ) ˆ U + βF ′ + ( β − ζ F ′′ − β ˆ V + F ′ − (2 − β ) g (ˆ x ) − β ˆ H − ∂ ζ ˆ P − g (ˆ x ) ˆ P , (16)ˆ U = ˆ V = ˆ H = 0 at ζ = 0 and ∞ , ˆ P = 0 at ζ = ∞ . (17)These equations will referred to below as the modified linear boundary layer (MLBL)equations. A. Asymptotic behavior near the free stream ( ζ >> ) for β < ζ → ∞ , the streamfunction of the Falkner-Skan base flow behaves as F ( ζ ) ∼ ζ − a β + O ( e − ( ζ − a β ) / ) , (18)where the constant a β depends on Hartree parameter β . Thus both F ′ − F ′′ decayto zero exponentially fast and the LBL equations (6)-(9) can be greatly simplified. Inparticular, the streamwise momentum equation (7) becomes x∂ x U = ∂ ζζ U + ζ − a β − β ∂ ζ U − β + (2 − β ) α g ( x ) − β U. (19)This equation is unforced, which means that U = 0. Using this, the remaining LBLequations (6), (8)-(9) simplify to ∂ ζ V = αg ( x ) W, (20) x∂ x V = ∂ ζζ V + ζ − a β − β ∂ ζ V + β + 1 − (2 − β ) α g ( x ) − β V − ∂ ζ P, (21) x∂ x W = ∂ ζζ W + ζ − a β − β ∂ ζ W + 1 − (2 − β ) α g ( x ) − β W − αg ( x ) P. (22)6ptimal streaksThese equations can be solved in closed form as follows. We first eliminate the pressuremanipulating (21) and (22) as usually, to obtain the following equation for the streamwisevorticity Ω = ∂ ζ W − αg ( x ) Vx∂ x Ω = ∂ ζζ Ω + ζ − a β − β ∂ ζ Ω + 2 − (2 − β ) α g ( x ) − β Ω . (23)This equation is also unforced, which means that Ω = 0, namely ∂ ζ W = αg ( x ) V. (24)Excluding divergent behaviors as ζ → ∞ , eqs.(20) and (24) yield V = − W = V ∞ e − αg ( x )( ζ − a β ) , (25)where V ∞ is a function of x that remains undetermined but will not be necessary below.The pressure P is readily obtained (and seen to behave as ζ e − αg ( x )( ζ − a β ) ) substituting (25)into (22).Since β < g ( x ) = x − β − β is small at small ˆ x and the convergence of V , W , and P tothe final free stream state U = V = W = P = 0 is quite slow (see Fig.3 below), whichexplains the difficulties encountered in former numerical treatments of (6)-(10) thatdid not take into account this behavior. The streamwise velocity component U instead,converges quite fast to zero (see Fig.3 below), as e − ( ζ − a β ) / . Equation (25) shows thatthe same happens with the quantity (cf (12)) H = V + W, (26)as anticipated above. Note using eqs.(20), (24), and (25) that H coincides with thestreamwise vorticity at large ζ . B. Behavior near leading edge ( x ≪ ) in the case β < x at small ˆ x , the relevant behavior isgiven by an eigenvalue problem first formulated and solved by Tumin , whose formulationis re-interpreted here. This will be done taking into account the asymptotic behavior as ζ → ∞ encountered in last sub-section, which in conjunction with the Tumin scalingsuggests the ansatz ( ˆ U , ˆ V , ˆ H, ˆ P ) ∼ ˆ x − λ ( ˜ U , ˜ V , ˜ H/g (ˆ x ) , ˜ P ) e − g (ˆ x )( ζ − a β ) . (27)Substituting this into the MLBL-equations and neglecting O ( g (ˆ x )) = O (ˆ x − β − β )-termsyields the following eigenvalue problem(1 − λ ) ˜ U + ( β − ζ − β ˜ U ′ + ˜ V ′ − ˜ H = 0 , (28)7ptimal streaks β λ T λ T λ CL β c λ ζ ˜ V / ˜ U ˜ H FIG. 2. Left: The two smallest Tumin eigenvalues (subscript T ) and the smallest Chen-Libbyeigenvalue (subscript CL ) in terms of β , in the range 0 ≤ β ≤
1; the first Tumin eigenvalue issmaller than one if 0 ≤ β < β c = 0 . β = 1 come from calculations in theAppendix. Right: ˜ U , ˜ V , ˜ H (solid lines), and ˜ U = −| ˜ H | max ˜ U / | ˜ U | max (dashed lines) for the firstTumin mode and β = 0 . , . , .
353 (that value considered in Ref. ), and 0 .
5; arrows indicateincreasing β . ˜ U ′′ + F − β ˜ U ′ + (1 − β ) ζ F ′′ + [(2 − β ) λ − F ′ − β ˜ U − F ′′ ˜ V = 0 , (29)˜ V ′′ + F − β ˜ V ′ + ( β − F + (2 β − ζ F ′ + ( β − ζ F ′′ ](2 − β ) ˜ U + [(2 − β ) λ + 2 β − F ′ + ( β − ζ F ′′ − β ˜ V = ˜ P ′ , (30)˜ H ′′ + F − β ˜ H ′ + λF ′ ˜ H = 0 , (31)˜ U = ˜ V = ˜ H = 0 , at ζ = 0 , ˜ U , ˜ V , ˜ H → ζ → ∞ , (32)which coincides with that in Ref. except for the fact that we are using the variable˜ H = ˜ U + ˜ W instead of ˜ W . This makes sense since the relevant eigenfuncions are suchthat ˜ H = O (cid:16) e − ( ζ − a β ) / (cid:17) as ζ → ∞ (Fig.2, right), which is consistent with the asymptoticbehavior of the variable H (see Fig.3 below) as explained above, but not with that of ˆ W ,whose decay is much slower, as e − g (ˆ x )( ζ − a β ) (see eq.(25)).Now, eq.(31) decouples from the remaining three equations and yields the eigenvalue λ . These eigenvalues will be called Tumin eigenvalues hereafter, and are all positive. Thetwo smallest eigenvalues are plotted with solid line and indicated with the subscript Tin Fig.2, left. Note that the first eigenvalue is larger than 1 if 0 ≤ β < β c = 0 . β > β c , all eigenvalues arelarger than one and all flow variables decay streamwise. The eigenfunction components˜ U , ˜ V , and ˜ H associated with the first Tumin eigenvalue are plotted in Fig.2, right forvarious representative values of β . Note that ˜ U = −| ˜ H | max ˜ U / | ˜ U | max is always quite closeto ˜ H , which means that ˜ U and ˜ H are always almost linearly dependent.In order to obtain a complete system of eigenfunctions (to, e.g., set all possible initialconditions), a second eigenvalue problem must be considered, which results from thescaling (cf (27)) ( ˆ U , ˆ V , ˆ H, ˆ P ) ∼ ˆ x − λ ( ˜ U , ˜ V , ˜ H, ˜ P ) e − g (ˆ x )( ζ − a β ) . (33)Proceeding as above, a second eigenvalue problem results that is a three-dimensionalextension of its counterpart first considered by Chen and Libby in a two-dimensionalsetting. The resulting continuity and spanwise momentum equations must be replaced by(1 − λ ) ˜ U + ( β − ζ − β ˜ U ′ + ˜ V ′ = 0 , (34)˜ H ′′ + F − β ˜ H ′ + ( β − F + (2 β − ζ F ′ + ( β − ζ F ′′ ](2 − β ) ˜ U + βF ′ + ( β − ζ F ′′ − β ˜ V + (2 − β ) λ + ( β − − β F ′ ˜ H = ˜ P ′ , (35)but the remaining equations are still eqs.(29) and (30), as are the boundary conditions(32). Thus, eqs.(29) and (34) are decoupled and provide the eigenvalues, called hereafterChen-Libby eigenvalues. These eigenvalues are all larger than one (the smallest one isplotted in Fig. 2, left) and thus they promote streamwise decay; they are also larger thanthe first Tumin eigenvalue, which provides the most dangerous behavior.An important property of the two eigenvalue problems considered above is that anyarbitrary initial condition of the MLBL equations, ( ˆ U , ˆ V , ˆ H , ˆ P ), can be expanded interms of the associated modes. In particular, the component of any initial condition onthe first Tumin eigenmode will be relevant in section III. It is given by a = Z ∞ e − β R ζ F dη F ′ ˆ H ˜ H T dζ Z ∞ e − β R ζ F dη F ′ ˜ H T dζ , (36)as obtained multiplying (with the L inner product) the series expansion by the adjoint ofthe first Tumin mode, ( ˜ U ∗ T , ˜ V ∗ T , ˜ H ∗ T , ˜ P ∗ T ) = (0 , , e − β R ζ F dη ˜ H T , L ˜ U T + L ˜ H T = 0, L ˜ H T = 0. The adjoint of this problem is L ∗ ˜ U ∗ T = 0 , L ∗ ˜ U ∗ + L ∗ ˜ H ∗ = 0 , (37)9ptimal streakswhere L ∗ j stands for the adjoint of the operator L j . Since λ is not an eigenvalue associatedwith the Chen-Libby problem, the first equation of (37) implies that ˜ U ∗ T = 0, whichmeans that ˜ V ∗ T = ˜ P ∗ T = 0 and ˜ H ∗ T is given by L ∗ ˜ H ∗ T = 0, where L ∗ is the adjoint ofthe operator defined by the left hand side of eq.(31). Then, ˜ H T is readily found to be˜ H ∗ T = e − β R ζ F dη ˜ H T if the usual L inner product is used. III. STREAMWISE EVOLUTION OF STREAKS FOR β < . Since the basic steady state converges quite fast to itsasymptotic value as ζ → ∞ , the approximation in subsection II A applies for moderatelylarge values of ζ , say ζ > L = 12. In particular, eqs.(20) and (24) can be used, whichexcluding divergent behaviors as ζ → ∞ , lead to ∂ ζ ˆ V + g (ˆ x ) ˆ V = 0 . (38)This equation is independent of all MLBL equations. If we substitute the wall normalmomentum equation by this equation, the resulting system is readily seen to provide thecorrect asymptotic behavior as ζ → ∞ , analyzed in subsection II A. Thus, we selectthe domain of integration L = 20 > L = 12 and consider eqs.(13), (14), and (16) in0 < ζ < L , eq.(15) in 0 < ζ ≤ L , and eq.(38) in L < ζ < L . In addition, second orderspatial derivatives in the resulting system are discretized using second order centereddifferences; first order derivatives of ˆ V and ˆ P are discretized using second order forwarddifferences in eqs.(13) and (38) and second order backward differences in (15); the latterare also used to discretize first order derivatives of ˆ U and ˆ H in all equations. This means(noting that both ˆ V and ˆ P are discretized in L < ζ < L with forward differences) thatno boundary conditions for ˆ V and ˆ P are needed at ζ = L ; instead, ˆ V ( L ) and ˆ P ( L ) areselected by the numerical code itself. Since both ˆ U and ˆ H decay extremely fast as ζ → ∞ ,the boundary conditions for these at ζ = L are ˆ U = ˆ H = 0. The resulting equations canbe written as ˆ x M (ˆ x ) d q /dt = L (ˆ x ) q , where q = ( ˆ U , ˆ V , ˆ H, ˆ P ) ⊤ is the joint flow vectorand the matrices M and L result from the left and right hand sides of the equations.The new logarithmic variable s = log (ˆ x/ ˆ x ) is used to integrate the system from ˆ x = ˆ x to ˆ x = 1, discretizing s -derivatives with second order forward differences, and using animplicit scheme to march in s .The resulting numerical scheme is very efficient because the numerical instabilities areexcluded and the behavior as ζ → ∞ is well captured by (38). The scheme is also robustbecause results are insensitive to both the choice of L and L (provided that L > L >
10, and L is somewhat large compared with L ) and the spatial and s -discretizations(provided that the latter be not too fine compared with the former since the mass matrix M is singular). Note that having solved the problem for ζ → ∞ has allowed to describe10ptimal streaksaccurately the flow very close of the leading edge (ˆ x → ζ , which would have made the method impractical at small ˆ x . Atypical run is carried out in 10 − < ˆ x < L = 12 and L = 20, considering 200equispaced ζ -mesh points and performing 400 s -steps, which requires 20 CPU secondsusing a Fortran90 code in a standard desktop computer. A. The most unstable streaky mode for β < x = ˆ x ≪ λ = λ T . Note that the MUSM is defined up to a constant factor, common to all flowvariables. | ˆ U || ˆ H | | ˆ W || ˆ V | ζζ ζζ FIG. 3. Streamwise, cross flow velocity profiles, and ˆ H -profile for β = 0 .
353 (that value of β con-sidered in Ref. ) rescaled with their maxima in 0 < ζ < ∞ , at ˆ x = 10 − , − , − , − , − and 1; arrows indicate increasing values of ˆ x . In addition, the asymptotic profiles, reconstructedusing (27) and (28)-(32), with λ = λ T , for ˆ x = 10 − , − , − , and 10 − are plotted withdot-dashed lines using the same rescaling. Integration from ˆ x = ˆ x = 10 − to ˆ x = 1 provides the streamwise, cross flow velocities,and the ˆ H -profile (normalized with their maxima in 0 < ζ < ∞ ) illustrated in Fig.3 for β = 0 .
353 (that value considered in Ref. ); the results for other values of β are completelysimilar. Note that our analysis above captures quite well the right behavior at small ˆ x ,plotted with dot-dashed lines in Fig.3. This is possible because of our re-interpretation11ptimal streaksof Tumin asymptotic results in terms of the new variable ˆ H and the use of (27). Alsonote that, as occurred in the case of Blasius boundary layer , the solution exhibits anapproximate selfsimilarity in the streamwise velocity component and the variable ˆ H , which(after rescaling with their maxima) are approximately independent of ˆ x and approximatelyequal to each other. This illustrates the low-dimensional nature of the MLBL equationsin the Falkner-Skan boundary layer. −8 −6 −4 −2 −2 −8 −6 −4 −2 −2 −8 −6 −4 −2 −1 −8 −6 −4 −2 −2 x ^ U| | ^ max V ^ || max U| | ^ max x ^ x ^ x ^ U| | ^ max V ^ || max H| | ^ max W| | max ^ U| | ^ max V ^ || max H| | ^ max W| | max ^ V ^ || max H| | ^ max W| | max ^ W| | max ^ H| | ^ max β=0.1 β=0.2β=0.353 β=0.5 FIG. 4. Maxima (in 0 < ζ < ∞ ) of ˆ U , ˆ V , ˆ W , and ˆ H versus ˆ x (solid lines, as indicated by thelabels), for β = 0 . , . , . .
5. The asymptotic behaviors as ˆ x → Figure 4 shows with solid lines the maxima (in 0 < ζ < ∞ ) of ˆ U , ˆ V , ˆ W , and ˆ H vs. ˆ x for four representative values β , namely β = 0 . , . , . U vs. ˆ x curve in ˆ x < ˆ x < W ∼ ˆ H ∼ ˆ x − β − λ T and ˆ U ∼ ˆ V ∼ ˆ x − λ T (see (27)) are also plotted with dot-dashed linesfor comparison. Note that transient growth is present at β = 0 . , .
2, and 0 . β = 0 .
5, as predicted in section II B.The MUSM is somewhat similar to the standard eigenmodes in strictly parallel flows,except of course for the final viscous dissipation decay at large ˆ x , which is intrinsic to12ptimal streakstransient growth. In particular, the MUSM is the first of a sequence q j = ( ˆ U j , ˆ V j , ˆ H j , ˆ P j ) for j = 1 , . . . , (39)which is obtained taking as initial condition at some small ˆ x the Tumin and Chen-Libbbyeigenmodes associated with the eigenvalues λ , λ , . . . (sorted together in increasing order).Invoking the scalings (27) and (33), such initial conditions are of the form (0 , , ˜ H T ,
0) and( ˜ U CL , ˜ V CL , , ˜ P CL ) for the Tumin and Chen-Libby modes, respectively; in fact, a better(but asymptotically equivalent) definition of the initial conditions are obtained using (27)and (33) with the remaining components of the Tumin and Chen-Libby modes that havebeen set to zero above. The streamwise evolving modes (39) satisfy k q k ≫ k q k ≫ . . . in ˆ x ≤ ˆ x ≤ , (40)provided that ˆ x ≪
1. In addition, any streak can be expanded in series of these modesbecause any initial condition can be expanded in a series of the Tumin and Chen-Libbymodes.All these imply that after a streamwise transient any streak converges to the MUSM,namely q . In addition, the projection of any streak into q is obtained projecting itsinitial condition into the eigenfunction of the first Tumin mode, as given by eq.(36). Thisis illustrated in Fig.4, where various streaks are considered that result from imposing atvarious ˆ x = ˆ x random initial conditions of the type ( ˆ U , ˆ V , ˆ H, ˆ P ) = ( ˆ U , , ˆ H , U = ˆ x − λ T ζ F ′′ X k =0 γ k sin kζ , ˆ H = ˆ x − β − λ T ζ F ′′ X k =1 γ k cos kζ . (41)Here, the coefficients γ k and γ k are chosen randomly between 1 and -1. Note also thatfixing two of the four variables makes sense since the initial condition should satisfy twocompatibility conditions (one obtained multiplying (13) by F ′ and subtracting (14), andthe second one, substituting (15)-(16) and the ˆ x derivative of (14) into (13)), which areselected by the equations after a few integration steps if not satisfy initially. These initialconditions are applied several times for different values of ˆ x . Results (after rescaling)are shown with dashed lines in Fig. 4. Note that transients survive until ˆ x/ ˆ x ∼ x / ˆ x ) µ , where µ = λ T − λ T = 0 . , . , .
13, and 1.25 for β = 0 . , . , . .
5, respectively.
IV. OPTIMAL INTERNAL STREAKS FOR β < G ≡ E in /E out x = x in and x = 1, which invoking (5) is written as G max = max (cid:20)Z ∞ U dy (cid:21) x =1 (cid:20) Re Z ∞ U dy + g ( x in ) x Z ∞ ( V + W ) dy (cid:21) x = x in . (42)A factor Re is omitted in the numerator, where a O (1 / Re) term depending on the crossflow velocities is also neglected. The latter cannot be done in the denominator because,in fact, the O ( Re )-term becomes negligible in the maximizers. Thus, following Luchini ,Tumin and Ashphis set to zero that term imposing the additional condition that ˆ U = 0at x in . Here instead, we shall retain the O ( Re )-term in the denominator, checking (as inRef. ) that results are independent of Re provided that Re be large.Note that maximizers of (42) depend on x in , with the asymptotic result as x in → G max ( α ) = max x β +12 − β in (cid:20)Z ∞ ˆ U dζ (cid:21) ˆ x = α − β − β (cid:20) x − β in Re Z ∞ ˆ U dζ + Z ∞ [ ˆ V + ( ˆ H − ˆ V ) ] dζ (cid:21) ˆ x = α − β − β x in . (43)This quotient could be maximized with the method used in Refs. , who wrote thequotient as ( q ⊤ out · Q out · q out ) / ( q ⊤ in · Q in · q in ), where q is the flow state vector q = ( U, V, H, P ).Using the action U associated with the streamwise evolution dynamical system associatedwith the MLBL equations, defined as q out = U · q in , and its upstreamwise-evolving adjoint,defined such that ( U · q in ) ⊤ = q ⊤ in · U ∗ , the quotient (42) is rewritten as G = ( q ⊤ in · U ∗ · Q out · U · q in ) / ( q ⊤ in · Q in · q in ). Maximizing this leads to the generalized eigenvalue problem U ∗ · Q out · U · q in = G Q in · q in , whose maximum eigenvalue provides the maximum gainand the associated eigenfunctions, the maximizers. This problem is iteratively solved as q in ,n +1 = Q − · U ∗ · Q out · U · q in ,n , which converges quite fast provided that the two firsteigenvalues of the above mentioned generalized eigenvalue problem are not too close eachother. A. A simple optimization method to calculate optimal streaks
The main difficulty in the method outlined above is the need of using the adjoint ofthe MLBL equations, which was made in Refs. for the original linearized boundarylayer equations, written in terms of the unscaled wall normal coordinate y . A similaradjoint equations could be derived for the MLBL equations in this paper, but doing thatin an efficient way would require to analyze the asymptotic behaviors of these equationsas ζ → ∞ and as ˆ x →
0. Instead, we propose here a simpler method that is new inthis context to our knowledge. This method is based on the observation above that the14ptimal streaksinfinitely many solutions of the MLBL equations can be classified as a series of streamwiseevolving ‘modes’ indicated in (39) and that they satisfy (40). Thus, if a particular solutionis written as linear combination of the first n such solutions, the resulting error scales with(ˆ x in / ˆ x ) λ n /λ , which becomes exponentially small as ˆ x/ ˆ x in is moderately large if λ n /λ ismoderately large. This means that higher order modes can be safely neglected whenoptimizing (43) because these will contribute to increase the denominator of (43) but willhave a negligible effect in the numerator. Now, the sequence λ , λ , . . . increases rapidly,meaning that retaining a few modes would be enough to define well generic solutions ofthe MLBL equations. Thus, the method we propose to maximize (43) is as follows:Step 1. Take the first n Tumin and Chen-Libby modes, ordered such that the associatedeigenvalues are sorted in increasing order, as explained right after eq.(39).Step 2. Calculate the n solutions of the MLBL equations,( ˆ U , ˆ V , ˆ H , ˆ P ) , . . . , ( ˆ U n , ˆ V n , ˆ H n , ˆ P n ) . (44)obtained taking as initial conditions at ˆ x = ˆ x in = α − β − β x in the n modes calculatedin step 1.Step 3. Replace the expansion ( ˆ U , ˆ V , ˆ H, ˆ P ) = n X j =1 a j ( ˆ U j , ˆ V j , ˆ H j , ˆ P j ) , (45)into (43), to rewrite the quotient appearing in this equation as G max ( α ) = P nj,k =1 x β +12 − β in E out jk a j a k P nj,k =1 (cid:18) x − β in Re E in1 jk + E in2 jk (cid:19) a j a k , (46)where E out jk = R ∞ ˆ U j ˆ U k dζ at ˆ x = α − β − β , E in1 jk = R ∞ ˆ U j ˆ U k dζ at ˆ x = α − β − β x in , and E in2 jk = R ∞ [ ˆ V j ˆ V k + ( ˆ H j − ˆ V j )( ˆ H k − ˆ V k )] dζ at ˆ x = α − β − β x in .Step 4. Equation (46) is the ratio of two n -th order quadratic forms, which is maximized asusually, solving the generalized eigenvalue problem x β +12 − β in n X k =1 E out jk a k = G n X k =1 (cid:18) x − β in Re E in1 jk + E in2 jk (cid:19) a k , (47)whose maximum eigenvalue G max provides the maximum gain in (43); the eigenvec-tor yields the maximizer of (43) using (45).15ptimal streaks α G a i n α G a i n α G a i n α G a i n FIG. 5. Rescaled (with the maximum in α ) maximun perturbed energy gain (43) for β = 0 . .
353 (bottom, left), and 0.5 (bottom, right), taking the initial stageat x in = 10 − (dot-dashed lines), 10 − (dashed lines), and 0 .
25 (solid lines). Results from Ref. for β = 0 .
353 and x in = 0 .
25 are plotted with plain circles.
The method is tested considering the cases β = 0 . , . , .
353 (the case considered inRef. ), and 0 .
5, with three values of the initial streamwise stage, namely x in = 10 − , 10 − ,and 0.25 (as in Ref. ). The associated rescaled maximum perturbed energy gain curvesobtained applying the method described above are given in Fig.5; in order to facilitatecomparison for the various values of x in , the energy gain is rescaled with its optimal value(plotted vs. β in Fig.6 below). One, three, and five modes are enough for x in = 10 − , − ,and 0 .
25, respectively, at the four considered values of β ; retaining more modes providesresults that are plot indistinguishable. As a reference, the counterpart obtained in Ref. at β = 0 .
353 and x in = 0 .
25 are plotted with plain circles; in fact, Tumin and Ashpis took x in = 0 .
111 and x out = 0 . x in = 0 .
25 and x out = 1 withthe adimensionalization in the present paper.This method provides the maximum gain in a quite fast and robust way. Note that themethod does not require any calculation of adjoint equations. Instead, only a few Tuminand Chen-Libby eigenmodes are needed. But these are not really necessary, noting thatthe n solutions of the MLBL equations considered in step 2 can be replaced in the methodby any set of n solutions with initial conditions at ˆ x = ˆ x in ≡ α − β − β x in that are linearlyindependent and exhibit a significant projection into the n first Tumin and Chen-Libbyeigenmodes. For instance, we can integrate the MLBL equations taking at some smaller16ptimal streaksvalue of ˆ x (say, ˆ x = ˆ x in /
10) the following n initial conditions (cf (41))( ˆ U k , ˆ V k , ˆ H k , ˆ P k ) = (cid:18) ˆ x − λ T ζ F ′′ sin kζ , , ˆ x − β − λ T ζ F ′′ cos kζ , (cid:19) , (48)for k = 1 , . . . , n , and considering in step 2 these solutions for ˆ x ≥ ˆ x in . The resultingmodified method, retaining the same numbers of modes as above produce the same resultsin Fig.5 to plot accuracy. Since this modification provides a simpler method, it is theresulting modified method that we propose in this paper.Note that the number of required modes in Fig.5 decreases as x in decreases. In par-ticular, just one mode is enough if x in ≤ − , and furthermore, the resulting maximizercoincides with the MUSM defined in section III A, as was to be expected. In fact, takinginto account the behavior of the MUSM as ˆ x → G max ( α ) = x β +12 − β in (cid:20)Z ∞ ˆ U dζ (cid:21) ˆ x = α − β − β (cid:20)Z ∞ ˆ H dζ (cid:21) ˆ x = α − β − β x in ∼ x λ T + β − − β in as x in → . (49)Here, ˆ U and ˆ H are the associated components of the MUSM, at ˆ x = α − β − β and ˆ x = α − β − β x in ,respectively. The indicated asymptotic behavior as x in → λ T . β α op t FIG. 6. Optimal spanwise wavenumber α opt vs. β at x in = 10 − (dot-dashed line), 10 − (dashedline), and 0.25 (solid line). Plain circles at β = 1 result from calculations in the Appendix. To complete the results above, the optimal gain and the associated value of the spanwisewavenumber are plotted vs. β in Fig.6 for x in = 10 − , − , and 0.25. The limiting valuesat β = 1 result from the analysis in the Appendix, at the end of the paper. Note that α opt depends non-monotonously both on β and on x in .Summarizing, a simple optimization method has been proposed to calculate optimalstreaks that does not rely on the adjoint formulation. Instead, minimization is made17ptimal streakson the low dimensional manifold spanned by a few solutions of the direct problem. Infact, five solutions are enough in the streaks of the Falkner-Skan boundary layer for x in ≤ .
25. In addition, only one such solution is enough for small x in ( x in ≤ − ),meaning that no optimization process is necessary in this limit. We believe that themethod proposed above is also useful to treat related transient growth problems in FluidDynamics, since the main ingredient that allowed constructing such method is usual inthese problems. This ingredient is that the behavior in the growth stage of transientgrowth are described by an eigenvalue problem (Tumin problem in the case consideredin this paper), whose eigenvalues are somewhat separated. And furthermore, the methodcould work in nonlinear, time dependent parabolic problems as well, since the dynamicsof parabolic problems is low dimensional at large time . V. CONCLUDING REMARKS
An analysis of optimal streaks in the Falkner-Skan boundary layer has been performedconcentrating in various issues that are now summarized, and are expected to apply torelated transient growth boundary layer problems: • The careful analysis of the behavior of streaks near the free stream was necessaryto construct a quite efficient numerical scheme, which allowed for integrating fromextremely small values of the streamwise coordinate. The analysis of the free streambehavior also allowed for making the correct interpretation of the behavior near theleading edge, in terms of a new variable ˆ H that behaves as the streamwise vorticitynear the leading edge. • The already known approximate selfsimilarity of the solution has been completedin terms of the new variable ˆ H . • Streaks behave as ‘modes’, which can be classified according to their behavior nearthe leading edge. The analogy with standard modes in strictly parallel flows becomesclear comparing the cases β < β = 1(considered in the Appendix). The most dangerous mode was called the MUSMand played an essential role in understanding the streamwise evolution of streaks. • A simple optimization method has been proposed that does not rely on the adjointequations and allows for the fast and precise computation of optimal streaks. • Optimal streaks can be directly defined from the MUSM, without the need of anyoptimization process, if the initial stage is sufficiently close to the leading edge. • All the above is somehow a pre-requisite to derive simple, yet sufficiently precise de-scriptions of the interaction between longitudinal streaks and transversal Tollmien-Schlichting modes, which can be still considered a major open problem in the field.18ptimal streaks • The analysis above applies to internal streaks, resulting from, e.g., obstacles nearthe leading edge. External streaks forced by perturbations in the outer stream showa different behavior near the leading edge and require a different treatment, whichis currently under research.
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Laminar-Turbulent Transition , NASA Conf. Pub. , vol. 2386, pp. 161–204 (1984). P. Luchini, “Reynolds-number-independent instability of boundary layer over a flat sur-face”, J. Fluid Mech. , 101 (1996). A. Tumin, “A model of spatial slgebraic growth in a boundary layer subjected to astreamwise pressure gradient”, Phys. Fluids , 1521 (2001). H. Schlichting,
Boundary Layer Theory , McGraw-Hill, 1968. L. S. Hultgren and L. H. Gustavsson, “Algebraic growth of disturbances in a laminarboundary layer”, Phys. Fluids , 1000 (1981). L. Boberg and U. Brosa, “Onset of turbulence in a pipe”, Z. Naturforsch., A: Phys. Sci. , 697 (1988). L. H. Gustavsson, “Energy growth of three-dimensional disturbances in plane Poiseuilleflow”, J. Fluid Mech. , 241 (1991). 19ptimal streaks K. Butler and V. Farrell, “Three-dimensional optimal perturbations in viscous shearflow”, Phys. Fluids A , 1637 (1992). D. Henningson, A. Lundbladh, and A. Johansson, “ A mechanism for by-pass transitionfrom localized disturbances in wall-bounded shear flows”, J. Fluid Mech. , 169(1993). S. C. Reddy and D. S. Henningson, “Energy growth in viscous channel flows”, J. FluidMech. , 209 (1993). L. N. Trefethen, A. E. Trefethen, and S. C. R. T. A. Driscoll, “Hydrodynamic stabilitywithout eigenvalues”, Science , 578 (1993). A. Tumin and E. Reshotko, “Spatial theory of optimal disturbances in boundary layers”,Phys. Fluids , 2097 (2001). P. Luchini, “Reynolds-number-independent instability of boundary layer over a flat sur-face: optimal perturbations”, J. Fluid Mech. , 289 (2000). P. Andersson, M. Berggren, and D. Henningson, “Optimal disturbances and bypasstransition in boundary layers”, Phys. Fluids , 134 (1999). A. Tumin and D. E. Ashpis, “Optimal disturbances in boundary layers subject to stream-wise pressure gradient”, AIAA , 2297 (2003). O. Levin and D. S. Henningson, “Algebraic growth and transition prediction in boundarylayer flow”, Flow, Turbulence and Combustion , 183 (2003). M. Higuera and J. M. Vega, “Modal description of internal optimal streaks”, J. FluidMech. , 21 (2009). K. K. Chen and P. A. Libby, “Boundary layers with small departures from the Falkner-Skan profile”, J. Fluid Mech. , 273 (1968). V.M. Falkner and S.W. Skan, “Some approximations of the boundary layer equations”,Phil. Mag. , 865 (1931). D.R. Hartree, “On an equation occurring in Falkner and Skan’s approximate treatmentof the equations of the boundary layer”, Proc. Cambridge Phil. Soc. , 273 (1937). C. Foias, G. R. Sell, and R. Temam, “Inertial manifolds for nonlinear evolutionaryequations”, J. Diff. Eqns. , 309 (1988). APPENDIX: THE CASE β ≥ β = 1 the spanwise wavenumber α cannot be eliminated from the LBLequations (6)- (9), but the right hand sides of these are independent of x and thus can beexpanded in terms of normal modes as (cf eq.(27))( U, V, W, P ) = ∞ X j =1 a j ( ˜ U j , ˜ V j , ˜ W j , ˜ P j )ˆ x − λ j + c . c ., (50)20ptimal streakswhere c.c. stands for the complex conjugate and the modes and exponents are given bythe following eigenvalue problem(1 − λ ) ˜ U + ˜ V ′ − α ˜ W = 0 , (51)˜ U ′′ + F ˜ U ′ + [( λ − F ′ − α ] ˜ U − F ′′ ˜ V = 0 , (52)˜ V ′′ + F ˜ V ′ + [( λ + 1) F ′ − α ] V = ˜ P ′ , (53)˜ W ′′ + F ˜ W ′ + ( λF ′ − α ) ˜ W = α ˜ P , (54)˜ U = ˜ V = ˜ W = 0 at ζ = 0 and ∞ , ˜ P = 0 at ζ = ∞ . (55)Note that as α → W as ˜ W /α and setting α = 0,(51) and (54) lead to the Tumin eigenvalue problem (28)-(31), while setting α = 0 in(51) and (55), the Chen-Libby eigenvalue problem (29)-(30), (34), and (35) results. Thismeans that the limiting values of the eigenvalues of (51) and (55) as α → αλ α I m ( λ ) × FIG. 7. The real (left) and imaginary (right) parts of the first three eigenvalues of (51)-(55) vs. α . Plain circles at α = 0 correspond to the Tumin and Chen-Libby eigenvalues at β = 1, plottedwith plain circles in Fig.2. The eigenvalues of (51) and (55) are generally complex, as shown in Fig.7, where thereal and imaginary parts of first three eigenvalues are plotted vs. α ; note that the secondand third eigenvalues are complex in the intervals 0 . < α < .
96 and 0 . < α < . β → n of the modes (50), which is G max ( α ) = P nj,k =1 x in E out jk ¯ a j a k P nj,k =1 (cid:0) x in Re E in1 jk + E in2 jk (cid:1) ¯ a j a k , (56)where overbar stands for the complex conjugate, E out jk = R ∞ ¯˜ U j ˜ U k dζ at ˆ x = α , E in1 jk = R ∞ ¯˜ U j ˜ U k dζ at ˆ x = αx in , and E in2 jk = R ∞ [ ¯˜ V j ˜ V k + ( ¯˜ H j − ¯˜ V j )( ˜ H k − ˜ V k )] dζ at ˆ x = αx in . The21ptimal streaksresulting optimal gain calculated from the generalized eigenvalue problem (cf (47)) n X k =1 x in E out jk a k = G n X k =1 (cid:0) x in Re E in1 jk + E in2 jk (cid:1) a k , (57)whose maximum eigenvalue G max provides the maximum of the energy gain (57). Notethat although the amplitudes a k are complex, the eigenvalues are real. This is becausethe matrices appearing in (57) are Hermitian. Using these, the maximum perturbedenergy gain is calculated from the initial stages x in = 10 − , 10 − , and 0.25, which peakat α = 0 .
38, 0.45, and 0.60, respectively. These are precisely the limiting values of theircounterparts calculated for β < β > x →
0. This is because g (ˆ x ) = ˆ x − β − β → ∞ as ˆ x → g (ˆ x ) is large, ˆ U , ˆ V , ˆ H , and ˆ P converge to zero exponentially fast as g (ˆ x ) ζ is large, which means that activity is concentrated in a thin wall-normal layer near thewall. The ˆ x -behavior in this stage is given by an eigenvalue problem that is omitted herefor the sake of brevity. As ˆ x → ∞ instead g (ˆ x ) is small and the Tumin and Chen-Libbyeigenvalue problems are recovered. Thus, the solution converges to the first Tumin modeas ˆ x → ∞ . In order to illustrate all these, the MLBL equations are integrated for β = 1 . x = ˆ x = 0 . . The maxima of | ˆ U | , | ˆ V | , | ˆ H | , and | ˆ W | are plotted withsolid lines in Fig.8 left. Note that all these decay quite fast in an initial stage, followed bya transition to the behavior given by the first Tumin eigenmode, plotted with dot-dashedlines. The irregular behavior at small ˆ x is not a numerical artifact, but is due to the factthat we are plotting maximum values, and the maxima alternated between two positions,as illustrated in the right plot, where the rescaled profiles of | ˆ U | are plotted for variousvalues of ˆ x , as indicated. 22ptimal streaks −1 −20 −10 ˆ x = 0 .
138 ˆ x | ˆ V | max | ˆ U | max | ˆ H | max | ˆ W | max ζζ ζζ | ˆ U | × − | ˆ U | × − | ˆ U | × − | ˆ U | × − FIG. 8. The case β = 1 .
5. Left: Maxima (in 0 < ζ < ∞ of | ˆ U | , | ˆ V | , | ˆ H | , and | ˆ W | vs. ˆ x .The upper and lower straight dot-dashed lines provide the asymptotic behaviors of | H | and | U | ,which invoking (11), (12), and (27) are x − λ and x − λ /g ( x ), respectively. Right: A sequence ofprofiles of | ˆ U | at nine equispaced values of x between x = 0 . . x , . . . , x . Dueto the exponential decay, these profiles cannot be plotted in the same plot. Instead, two ofthem are given in each plot, which also contains the last profile in the former subplot, namelythe four subplots provide the profiles at x = x , x , x (top, left), x = x , x , x (top, right), x = x , x , x (bottom, left), and x = x , x , x (top, right). Arrows indicate increasing valuesof ˆ x ..