Localised eigenmodes in a moving frame of reference representing convective instability
LLocalised eigenmodes in a moving frame of reference representing convectiveinstability
Koen J. Groot
1, 3, ∗ and S´ebastien E.M. Niessen
2, 3, † Aerospace Faculty, Texas A&M University, United States Aerospace & Mechanical Engineering, University of Li`ege, Belgium Faculty of Aerospace Engineering, Delft University of Technology, The Netherlands (Dated: January 14, 2020)When representing convective instability mechanisms with the streamwise BiGlobal stability ap-proach, results suffer from a sensitivity to the streamwise domain truncation length and boundaryconditions. The methodology proposed in this paper resolves this sensitivity by considering a movingframe of reference. In that frame, the spectrum features discrete eigenvalues whose correspondingeigenfunctions decay exponentially in both the up- and downstream directions. Therefore, the trun-cation boundaries can be placed far enough that both variations in the domain length and artificialboundary conditions have no impact. The discrete nature of the spectrum enables the use of localand non-local stability methods to perform an independent approximation of the BiGlobal eigenso-lutions via global mode theory. We demonstrate that retrieving set-up-independent solutions in thestationary frame of reference is likely impossible for the considered flow.
I. INTRODUCTION
Linear stability methods are important computational tools for the prediction of the laminar-turbulent transitionof boundary layers. The methods deployed by the industry are LST (Linear Stability Theory, [1]), which accountsfor one-dimensional flow properties only, and PSE (Parabolised Stability Equations, [2]), which improves upon LSTby marching downstream, allowing to account for small streamwise changes in the flow. The streamwise BiGlobalstability approach is superior to both LST and PSE, in the sense that it can represent all linear perturbation dynamicsof practically arbitrary two-dimensional flows [3].Practice has shown, however, that representing convective instability mechanisms, specifically, with the streamwiseBiGlobal stability approach suffers from a notorious sensitivity to the computational set-up. This particularly involvesthe truncation of the domain in the streamwise direction, e.g. the specific choice of the domain length and boundaryconditions [4–6]. The results of Alizard and Robinet [5, § IV.B.2] suggest that the part of the spectrum that is ofinterest tends to a continuum as the domain length tends to infinity. In regard to this expected limit of the spectrum,Theofilis [3] states that: ‘the discretised approximation of the continuous spectrum will always be under-resolved.’The main goal of the proposed methodology in the present paper is to solve the sensitivity issues related to thetruncation of the domain.Next to the convergence problems due to the domain truncation, the physical interpretation of the BiGlobal stabilityresults, again specifically in the case of convective mechanisms, poses an entirely independent issue. The relationshipbetween global and (non-)local stability approaches is known if a global instability exists, e.g. see [7] and [8]. Thisarticle provides a first step in developing the physical interpretation of the BiGlobal results in the case of convectiveinstabilities by establishing the link between the BiGlobal stability results on the one hand and (non-)local methodslike LST and PSE on the other. Alizard and Robinet [5] and Rodr´ıguez [9] have demonstrated the relationshipbetween the solutions corresponding to these different methods by using the complex frequency provided by BiGlobalsimulations as the input for the LST and PSE approaches. Using the currently proposed methodology, the link canbe made without making use of the BiGlobal stability results.The paper is structured as follows: the methodologies used to obtain the BiGlobal and base flow solutions arediscussed in § II, the results are presented in § III and the link with LST and PSE solutions is established in § IV. Thepaper is concluded in § V. ∗ [email protected] † [email protected] a r X i v : . [ phy s i c s . f l u - dyn ] J a n II. METHODOLOGY
It is this article’s goal to demonstrate that the sensitivity to the computational set-up can be resolved by formulatingthe problem in a moving reference frame. An extensive theoretical motivation is presented by Groot [10, § A. Streamwise BiGlobal stability problem in a moving reference frame
In the streamwise BiGlobal stability problem, one considers an infinitesimally small perturbation to a base flow,whose variables are denoted as Q . Base flows are considered that depend only on the wall-normal y - and streamwise¯ x -coordinate, where the bar denotes the stationary reference frame, in which the base flow is independent of time ¯ t .We reformulate the perturbation problem in a reference frame that moves downstream with the constant speed c g .To this end, the ¯ x - and ¯ t -derivatives in the governing equations have to be transformed as follows: ∂∂ ¯ x = ∂∂x , ∂∂ ¯ t = ∂∂t − c g ∂∂x , (1)where x = ¯ x − c g ¯ t and t = ¯ t correspond to the moving reference frame; y -derivatives remain unchanged. Whileindependent of ¯ t , the base flow does depend on t . Taylor expanding the base flow variables for the elapsed time∆ t = t − t yields: Q (¯ x ( x, t ) , y ) = Q (¯ x ( x, t ) , y ) + c g ∆ t ∂Q∂ ¯ x (¯ x ( x, t ) , y ) + O (cid:16) ( c g ∆ t ) ∂ Q∂ ¯ x (¯ x ( x, t ) , y ) (cid:17) , (2)where t is a reference time. This reveals that the base flow can be assumed to be constant in t when permitting anerror of O (cid:0) c g ∆ t ∂Q/∂ ¯ x (cid:1) , which is small when c g , ∆ t or ∂Q/∂ ¯ x is small. When ∆ t = 0, the solutions are exact. Fornon-zero ∆ t , the solutions of the present approach can be time-integrated such that all unsteady effects due to themoving reference frame are accounted for. Accordingly, the introduced model error can be removed for non-zero ∆ t ,but this lies out of the current scope. Therefore the time-evolution of the solutions is here discarded.Upon neglecting the time-dependence of Q (¯ x ( x, t ) , y ), the perturbation problem in the moving reference frame hasconstant coefficients. In that case, a two-dimensional perturbation variable q (cid:48) can be represented as the product ofan eigenfunction ˜ q = ˜ q ( x, y ) and an exponential function of time t : q (cid:48) ( x, y, t ) = ˜ q ( x, y ) e − i ωt + c.c., (3)where the eigenvalue ω is a complex angular frequency and c.c. the complex conjugate. The subscripts r and i willdenote real and imaginary parts, respectively.Substituting equations (1) and (3) into the linearised incompressible Navier-Stokes equations yields the streamwiseBiGlobal stability equations for a moving reference frame: − i ω ˜ u + (cid:0) U − c g (cid:1) ∂ ˜ u∂x + V ∂ ˜ u∂y + ˜ u ∂U∂x + ˜ v ∂U∂y = − ∂ ˜ p∂x + 1 Re (cid:18) ∂ ∂x + ∂ ∂y (cid:19) ˜ u ; (4a) − i ω ˜ v + (cid:0) U − c g (cid:1) ∂ ˜ v∂x + V ∂ ˜ v∂y + ˜ u ∂V∂x + ˜ v ∂V∂y = − ∂ ˜ p∂y + 1 Re (cid:18) ∂ ∂x + ∂ ∂y (cid:19) ˜ v ; (4b) ∂ ˜ u∂x + ∂ ˜ v∂y = 0 . (4c)The stationary reference frame corresponds to c g = 0. To illustrate how the results in the moving reference framemanifest themselves in the stationary reference frame, define ¯ ω ˜ u and ¯ ω ˜ v as follows: ω = ¯ ω ˜ u ( x, y ) + i c g ˜ u ∂ ˜ u∂x ( x, y ) , where: ¯ ω ˜ u = ¯ ω ˜ u ( x, y ); (5a) ω = ¯ ω ˜ v ( x, y ) + i c g ˜ v ∂ ˜ v∂x ( x, y ) , where: ¯ ω ˜ v = ¯ ω ˜ v ( x, y ) . (5b)By substituting equation (5a) for ω into equation (4a), the c g -term is eliminated from equation (4a). The same canbe done with equations (4b) and (5b). This demonstrates that system (4) governs solutions in a stationary referenceframe, while permitting a ‘varying eigenvalue’: ¯ ω ˜ u and ¯ ω ˜ v are functions of x , y , ˜ u and ˜ v . The right hand sidesof equations (5) add up to a constant ω , such that no extra x - and y -derivatives are introduced when substitutingansatz (3). The real part of equations (5) corresponds to a Doppler shift, while the imaginary part represents theinstantaneous advection-induced ¯ x -translation of the amplitude distribution.The only physically substantiated boundary condition for external incompressible flow problems is given at solidwalls. There no-slip is imposed for ˜ u and ˜ v and the Laplace equation for ˜ p , consistent with the conditions proposedby Theofilis [13]. B. Base flow
Stability results, obtained with any stability method, are notoriously sensitive to the base flow, see [14]. In additionto this, the solutions of the streamwise BiGlobal problem are also sensitive to the computational set-up. The lattersensitivity is the focal point of this study. Therefore, we consider the self-similar Blasius boundary layer as our baseflow. It does not satisfy the full Navier-Stokes equations, but it is slowly-developing, universal and obtainable toarbitrary precision, eliminating all uncertainty associated with the base flow. We would like to emphasise that thegoal of this paper is not to analyse the stability of the flow over a flat plate, but rather to demonstrate that theinherent methodological sensitivity is removed by considering the proposed technique.The self-similar boundary-layer solution is computed with DEKAF, for all details see Groot et al. [15], settingthe Mach number M = 0. The largest Newton-Raphson residual corresponds to ∂ U /∂y and equals O (10 − ). N η = 500 nodes were used in the wall-normal direction, yielding at most O (10 − ) differences with the solution on agrid with 2 N η − U and V and their y -derivatives are obtained from the self-similar solutions, which arethen ‘GICM-interpolated’ onto the BiGlobal grid’s collocation nodes. GICM (Groot-Illingworth-Chebyshev-Malik)is an iterative procedure that ensures the spectral accuracy of the Chebyshev discretisation after the inversion ofthe Malik collocation point mapping and the Illingworth self-similarity transformation. The x -derivatives of thebase flow quantities are determined with the spectral differentiation matrix corresponding to the BiGlobal problem’sdiscretisation. C. Numerical set-up
The solutions to system (4) are obtained numerically on a domain that is truncated in the up- and downstreamdirections, at x = x in and x = x out , and far from the flat plate at y = y max ( x = 0 corresponds to the leading edgeand y = 0 to the wall). At the latter boundary, all perturbation variables are zeroed. The truncation boundaries at x = x in and x = x out are respectively referred to as the in- and outflow boundaries. The streamwise domain lengthis denoted by L = x out − x in . The literature presents several attempts in prescribing reasonable boundary conditionsat the in- and outflow boundaries, see [5], Rodr´ıguez [9, § et al. [17]. Ourpresent aim is to ensure that the solutions are independent of all truncation boundary conditions through the use ofthe moving reference frame. Unless stated otherwise, we use Neumann conditions at the in- and outflow boundaries.This allows revealing when the solutions become dominant at the boundaries and, in turn, when they do becomedependent on the boundary conditions. The problem is discretised with Chebyshev collocation [18] in both x and y .A BiQuadratic mapping [19] is used in the x -direction, mapping one-third of the collocation points in-between thepoints x i and x i > x i , each lying within [ x in , x out ]. The values x i = x in + ( x out − x in ) and x i = x in + ( x out − x in )are used for all presented results. The Malik mapping [20] is used for the wall-normal direction y , mapping half thecollocation nodes above and below y i .Velocity and length scales are respectively made non-dimensional with the freestream speed U e and the ‘global’Blasius length (cid:96) = ν/U e , where ν is the kinematic viscosity. According to this choice, Re = 1 (using a different scalinghad no impact on the numerical results). Table I presents the parameters used for the selected reference case. TheArnoldi algorithm is used to solve the discretised problem for the 1000 smallest eigenvalues. c g /U e N x x in /(cid:96) x i /(cid:96) x i /(cid:96) x out /(cid:96) N y y i /(cid:96) y max /(cid:96) . × . × . × . ×
50 4 . × . × TABLE I. Reference case parameters (not rounded, yielding largest ω i for a 3 digit c g -value). III. RESULTS
The spectrum and ˜ u -eigenfunctions of interest for the reference case c g /U e = 0 .
415 are shown in Fig. 1. Theattention is restricted to eigenvalues with negative ω r -values (marked blue in Fig. 1( b )). The reference case andeigenmode selection will be justified in § III B. The modes of interest form a branch with 3 sub-branches: the top-left ‘main’ branch, housing modes labelled 1 to 5, the rightward ‘side’ branch, in which mode 9 resides, and thedownward branch, accommodating mode 7, that appears to continue indefinitely into the stable half-plane. Theselected eigenmodes represent wave packets: all eigenfunctions (˜ u , ˜ v and ˜ p ) decay exponentially toward all truncationboundaries. The instantaneous propagation speed of these wave packets is equal to the speed of the reference frame,therefore this justifies referring to the latter speed as a group speed. The solutions along the main branch areindependent of the numerical set-up, this is demonstrated next. FIG. 1. ( a ) ω -spectrum, ( b ) zoom on box and ( c – g ) isocontours of the real part of ˜ u (coloured lines, from min- (blue) tomaximum (red) with ∆ = 2 /
9, scaling the maximum to unity) and | ˜ u | (dotted, level: 1/9) corresponding to the eigenvalueslabelled in ( b ) for the reference case c g /U e = 0 . δ -isocontour (dashed). A. Independence of the numerical set-up and truncation boundary conditions
Convergence information is given in Table II for several of the labelled modes in Figs. 1( a ) and 2( a ). The relativeerror in the eigenvalue’s magnitude is determined by varying the following numerical aspects independently: thestreamwise domain length (indicated by (cid:15) L , fixing the relative resolution N x /L ), the resolution in the streamwisedirection ( (cid:15) N x ), the boundary conditions ( (cid:15) BC ) and the domain height ( (cid:15) y max , fixing the resolution in the boundarylayer by keeping N y and y i constant). Overall, relative errors of O (10 − ) are attained. When representing convectiveinstability mechanisms with the streamwise BiGlobal approach, these small errors are unprecedented in the sense thatspectra computed in the stationary frame of reference presented in literature experienced O (1) errors while changingthe streamwise domain length. Before elaborating further, it should be noted that (cid:15) y max is the largest contributor tothe overall eigenvalue error. Now that the issues related to the streamwise direction are tackled, (cid:15) y max features theslowest convergence rate. Accordingly, the selection of the reference case and the convergence study were approachedby reducing (cid:15) y max to a reasonably low level and using that level as an upper bound for the other errors.The error introduced by the finite domain length, (cid:15) L , representing a primary source of error in the literaturementioned in the introduction, can be made an order of magnitude smaller than (cid:15) y max . Altering the resolution inthe streamwise direction yields a very small error, (cid:15) N x , due to the use of the spectral scheme with N x = 260 to 300 Modeproperties (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
Mode c g /U e ω r (cid:96)/U e − . × − − . × − − . × − ω i (cid:96)/U e +3 . × − +2 . × − − . × − Relative | ω | -errors (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:15) L + 2 . × − − . × − + 4 . × − (cid:15) N x + 4 . × − + 2 . × − − . × − (cid:15) BC + 1 . × − − . × − − . × − (cid:15) y max − . × − − . × − − . × − TABLE II. Mode properties and relative errors in the eigenvalue magnitude for the reference parameters given in Table I withrespect to the parameter changes: x out /(cid:96) = 7 . × (fixing the density N x /L ); N x = 260; the use of Dirichlet in-/outflowboundary conditions; and y max /(cid:96) = 1 . × (fixing N y = 50 and y i = 4 . × ). The reported digits are truncated (notrounded) and those that are tainted by the largest reported error are underlined. nodes. It should be emphasised that these amounts of nodes are not at all necessary to obtain converging solutionsfor the reference case; using N x = 100 nodes for mode 1 results in an error comparable to (cid:15) y max . The truncationboundary conditions represent the other primary uncertainty throughout the literature. By changing from Neumannto Dirichlet conditions, a remarkably small (cid:15) BC is obtained, that is equivalent to (cid:15) N x .These results conclusively demonstrate that the obtained solutions are independent of the numerical set-up. Thenegligible influence of the streamwise domain length and truncation boundary conditions is observed to be directlyrelated to the small amplitude of the eigenfunctions at the truncation boundaries. The spatial decay of the eigen-functions within the domain allows placing the truncation boundaries at a far enough, but finite distance, so that theeigeninformation is virtually unaffected.Only the modes along the main branch are found to converge; the side and downward branches persistently dependon the domain length. For increasing L , the side branch moves downward and modes that are originally positionedwithin the downward branch either merge with the main branch (and do converge thereafter) or they merge with theside branch. The fact that the side and downward branches do not converge is unexpected, because the correspondingeigenfunctions do decay toward all truncation boundaries, e.g. see modes 7 and 9 in Figs. 1( f , g ). This illustrates thatan eigenfunction’s spatial decay toward the truncation boundaries is not a sufficient condition for the eigensolutionto be independent of the numerical set-up. B. Dependence on c g Next, the movement of the converged part of the spectrum is studied while varying c g , see Fig. 2( a ). The Dopplereffect dictates that the frequency ω r should decrease while c g increases. System (4) has real coefficients, renderingthe spectrum symmetric about the ω i -axis. The branch with ω r < c g /U e = 0 .
415 yielded the largest ω i -value.By increasing c g /U e ≥ .
41, the main and side branches increase in extent and the eigenfunctions move upstream,see Figs. 2( e – h ). While moving upstream, the streamwise extent of the eigenfunctions decreases and so does thestreamwise wavelength. This is consistent with the boundary layer becoming thinner.By decreasing c g /U e < .
41, the side branch coalesces with the main branch and the downward branch splits in two.The eigenfunctions move downstream and reach the outflow boundary. When close enough, the functions suddenly‘latch’ onto the outflow boundary and, simultaneously, an artificial structure emerges from the in flow boundary. Thepoint where the downwards branch splits reaches the top of the branch as c g /U e approaches 0.35, which causes thespectrum to have an arc-branch shape, as so described by Lesshafft [21]. As the spectrum attains the arc-branchshape, the latching tail from the inlet reaches the downstream structure, overwhelming the solution throughout theentire domain; all dynamics are then dominated by the artificial truncation boundary conditions. Tests show thatsolutions displaying this feature are strongly dependent on the artificial boundary conditions, domain size and x -resolution. Although the process changes by which the downward branch splits and how the eigenfunctions undergolatching, deploying fourth-order finite differences in the streamwise direction also results in artificial structures thatreach from the in- to the outflow boundary at c g /U e = 0 . § IV.B.2, for c g = 0] suggest that arc-shaped spectra obtained for too small c g approach a continuum as the streamwise domain length tends to infinity. Numerous analyses presented in the FIG. 2. ( a ) Relevant ω -spectrum part and ( b – h ) isocontours of the real part of ˜ u (coloured lines: from min- (blue) to maximum(red) with ∆ = 2 /
9, scaling the maximum to unity) and | ˜ u | (black dotted, level: 1 /
9) corresponding to the maximum ω i -eigenvalues along the labelled branches in ( a ) for indicative values of c g (∆ c g /U e = 0 . δ -isocontour (black dashed). literature are performed in the stationary reference frame and result in arc-shaped spectra. The present analysissuggests that the domain truncation has had a non-negligible artificial impact on these results.It is concluded that, keeping the reference domain length fixed, a large enough c g -value is required to prevent theeigenfunctions from reaching the outlet truncation boundary and eliminate the unwanted dependency on the numericalset-up. C. The limit c g → By increasing the domain length, the eigenfunctions can propagate further downstream and c g could be furtherdecreased, attempting to recover the stationary reference frame.‘Latching’ is from now on identified with the emergence of an artificial inlet structure. It is observed when the˜ u -eigenfunction attains an O (10 − ) relative magnitude at the outflow boundary for the reference domain length.Hence, the most up- and downstream position of the wavepacket, x front and x aft , are respectively defined to be thefirst and last positions where this level is measured. Furthermore, the minimum wavelength λ min (represented bythe real and imaginary part of ˜ u ) for x ∈ [ x front , x aft ] is measured. The domain length is increased (fixing N x /L ) toproperly capture the most unstable solution along the main branch. The measured variation of x aft − x front , x aft and λ min with c g is shown in Fig. 3( b ); all increase with decreasing c g . Their growth rates are quantified by fitting a powerand exponential law; the resulting parameters are reported in Table III.If the power-law trend holds in the limit, the quantities approach infinity when c g →
0, as illustrated in Fig. 3( a ),rendering resolving localised wavepackets in the stationary reference frame impossible. The conjecture that this isimpossible is supported by the theory presented by Groot [10, equation 8.10]. Based on the results that could beobtained with the available computational resources, significantly larger correlations were found for the power versusexponential law, see Table III. Furthermore, independent of the fitting law used, λ min increases at a much lower ratethan x aft − x front . Therefore, both the x -resolution and the domain length have to be increased as c g →
0, so that theproblem becomes computationally intractable when c g becomes small. An insufficient x -resolution was also observedto cause latching to the in-/outflow boundaries. FIG. 3. For the most unstable ˜ u -eigenfunction: aft-most location ( x aft , black), streamwise extent ( x aft − x front , red) andminimum wavelength (10 × λ min , blue) versus c g . Measured values (symbols), power (solid lines) and exponential (dashed) fitsand outflow boundary x out (dash-dotted). ( a ) Trend-extrapolation as c g → . U e ) and ( b ) data and fits. x aft x aft − x front λ min Power law: − . − . − . − . − . − . c g data points in Fig. 3 for x aft , x aft − x front and λ min of the wavepacket: p in ac p g for the power law, ε in b e εc g for the exponential law, the Pearson correlation coefficients are given in brackets. IV. LINK WITH LOCAL AND NON-LOCAL METHODS
As opposed to the approach used in literature [5, 9], this section demonstrates the link between local (LST), non-local (PSE) and global stability methods for Blasius flow without making use of the BiGlobal eigeninformation. Theconverged modes appear as discrete (i.e. not forced continuum ) modes, which permits their approximation via theglobal mode theory developed by Monkewitz et al. [7]. From a physical perspective, this can be justified as follows.A convective instability appears as an absolute instability in a moving reference frame [22, 23]. In turn, the existenceof an absolute instability is a necessary condition for the existence of a global instability [24]. To demonstrate theseinstability natures, a spatio-temporal stability framework must be used. For conciseness, this framework will herebe recited in recipe form, after describing the used (non-)local stability approaches, see Monkewitz et al. [7] for alldetails.The LST and PSE problems are discretised as consistently as possible with respect to the BiGlobal problem. ForPSE, the stabilised discretisation method proposed by Andersson et al. [25] is used. The streamwise wavenumber σ equals: LST: σ = α ;PSE: σ = α + α aux , where: α aux = − i (cid:90) y max ˜ u ∗ ∂ ˜ u∂x d y (cid:44) (cid:90) y max | ˜ u | d y, (6)and α is the streamwise wavenumber in the standard perturbation ansatzes [1, 2] and the star denotes complexconjugation. For PSE, the growth in the shape function is accounted for with α aux , which is minimised up to O (10 − ) relative errors. Both problems are solved for a frequency ¯ ω and location ¯ x corresponding to the stationaryframe of reference, such that σ = σ (¯ x, ¯ ω ). The frequency in the moving frame of reference is obtained throughthe Doppler shift formula: ω (¯ x, ¯ ω ) = ¯ ω − σ (¯ x, ¯ ω ) c g , equivalent to equation (5). In the moving reference frame, thesolutions have a non-convective nature. Therefore, the PSE problem would diverge if solved in that reference frame;resorting to the stationary reference frame allows circumventing this issue entirely.The global frequency is obtained by manipulating the LST/PSE solutions as follows:1. Find ¯ ω ∈ C for which | d ω/ d σ | = 0, while fixing ¯ x . This is equivalent to | d ω/ d¯ ω | = 0, because d ω/ d σ = c g / ((d¯ ω/ d ω ) − ω = ω represents a saddlepoint when ω is graphed versus σ and σ is a double root of the dispersion relation. Note that ω and σ areboth a function of ¯ x .2. Find ¯ x for which | d ω / d¯ x | = 0. For this ¯ x -value, ω (¯ x ) displays a cusp in the ω -plane and this cusp-point,denoted by ω g , approximates one BiGlobal eigenvalue.The criteria (i) and (ii) were checked numerically by evaluating the solutions for increasingly denser ¯ ω - and ¯ x -sequences, respectively. The criteria were deemed satisfied if the magnitude of the derivatives | d ω/ d¯ ω | and | d ω / d¯ x | was of O (10 − ) and O (10 − U e /(cid:96) ) , respectively. No ω g -cusp could be found for the real-valued c g = 0 . U e .Therefore c g ,r was fixed (to the reference value 0.415 U e ) and non-zero c g ,i -values were permitted. This is equivalent tothe use of a complex spatial ¯ x -coordinate by Monkewitz et al. [7]. This did yield LST and PSE solutions satisfying bothcriteria (10 c g ,i /U e = − . − . c g demonstratesthat complex group speeds can be circumvented completely through the use of the moving reference frame. Aninterpretation of the complex group speeds and complex spatial coordinates is therefore immaterial. Fig. 4( a ) displaysthe cusped ω (¯ x )-branches and the BiGlobal spectra for the corresponding complex c g -values. FIG. 4. ( a ) Comparison of the BiGlobal spectra (open symbols) for the complex c g -values for which cusp-branches could beobtained with PSE (red dots) and LST (black dots); equal colours indicate equal c g ,i -values, ω -values for LST and PSE cuspsand most unstable BiGlobal eigenvalues are presented in the legend (for c g ,i = 0: ω (cid:96)/U e = ( − .
46 + 3 . × − ). ( b , c )Comparison of the reconstruction of the BiGlobal | ˜ u | -eigenfunction (red solid, levels: 1, 3, . . .
9, scaling the maximum to 10)with LST ( b , black dashed) and PSE ( c , black dashed) for the respective c g ,i -values used in ( a ). The value c g ,r /U e = 0 .
415 isused for all presented cases.
Comparing the ω g -values to the BiGlobal eigenvalues (given in the legend), the minimum distance is smaller forPSE (red) than for LST (black). Similarly, a smaller c g ,i -value is required for PSE than LST. Both the smallerdistance of the ω g -value and smaller c g ,i -value are argued to be caused by the smaller model error in the PSE overthe LST approach, i.e. due to non-parallel effects. The reconstruction of the BiGlobal eigenfunction with the (non-)local solutions is shown in Figs. 4( b , c ), obtained by fixing the frequency in the moving reference frame to ω g andselecting the α -branches that represent decaying solutions in the up- and downstream directions [7, equation 3.16].The obtained | ˜ u | -functions closely resemble the BiGlobal equivalents. The structure found with LST lies significantlyupstream of the BiGlobal one, while a striking match is obtained with PSE. These comparisons reflect the founddifferences in the ω g - and c g ,i -values and they confirm that the most unstable BiGlobal modes are recovered.These results establish the link between the BiGlobal and (non-)local stability approaches for convective mecha-nisms, specifically. It moreover demonstrates that, while Tollmien-Schlichting waves are a convective instability inthe stationary reference frame, they represent a global instability mechanism in a moving reference frame.In conclusion, the BiGlobal stability method formulated in the moving reference frame does exactly what it issupposed to do: it localises global instability mechanisms without having to go through the complicated σ -saddle-and ω -cusp-point-finding algorithms and without having to deal with esoteric complex group speeds or complex spatialcoordinates. V. CONCLUSION
By solving the BiGlobal problem in a moving frame of reference, we obtain eigensolutions that converge numeri-cally for a sufficiently large, but finite resolution and domain length. These solutions appear as discrete modes in theeigenvalue spectrum, which enabled us to independently approximate the BiGlobal eigenvalues with (non-)local sta-bility methods, i.e. without using the BiGlobal results as input. Moreover, we demonstrate that retrieving convergedeigensolutions in the stationary reference frame is likely impossible for the examined base flow case.A moving reference frame renders developing base flows unsteady. While currently not accounting for the related ef-fects, the present methodology and results establish a reliable point of departure for their quantification by performingtime-integration in the future. Further investigations should be focused on more closely establishing the link betweenthe absolute and global instability characteristics and reproducing the equivalents of the neutral and amplification( N -factor) curves. ACKNOWLEDGEMENTS
The authors acknowledge the funding provided by the FNRS-FRIA fellowship granted to Sbastien E.M. Niessenand thank Henk Schuttelaars and Stefan Hickel at Delft University of Technology, Fabio Pinna at the Von K´arm´anInstitute for Fluid Dynamics, Vincent Terrapon at Universit´e de Li`ege and Ethan Beyak, Andrew Riha and HelenReed at Texas A&M University for the useful discussions. [1] L. M. Mack, Special course on stability and transition of laminar flow, in
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