The issue of the relevance of modal theory to rectangular duct instability is discussed

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1 1 Supplemental Appendix 4: On global vs. local eigenmodes Rectangular duct vs. plane Poiseuille flow The issue of the relevance of modal theory to rectangular duct instability is discussed in the main part of the article. Within modal linear theory, in the limit of large aspect ratio, A = l z /l y, l z and l y being the width and height of the duct, linear instability of pressure-gradient-driven flow (Tatsumi & Yoshimura, 1990) in a A duct may be related with that in the classic plane Poiseuille flow (PPF) (Orszag, 1971); some discussion of this point may be found in (2000d). When the instability analysis of the duct is performed at A 4 a clear tendency may be observed, whereby members of the discrete eigenspectrum cluster into groups of eigenvalues emanating from each of the discrete PPF eigenvalues ( et al., 2004); in the limit A each group of eigenvalues collapses onto the corresponding PPF eigenmode. This may be explained by the fact that the third term in the streamwise velocity component in the rectangular duct, w(y, z) = 1 y 2 4 ( ) 2 3 π n=0 ( 1) n cos (2n + 1) 3 [ (2n + 1) πy 2 ] [ ] cosh (2n + 1) πz 2 cosh [ (2n + 1) πa ], (1) 2 becomes negligible as A and the PPF parabolic velocity component is recovered in this limit. Likewise, the z derivatives in the linearized operator also tend to zero in this limit. At all finite A values the presence of the duct corners is reflected in the eigenvectors; figure 1 shows the real part of the streamwise component of the amplitude function, ŵ r (y, z), of the leading eigenmode of rectangular duct flow, at a given set of parameters, quoted as neutral by Tatsumi & Yoshimura (1990) and further explored in terms of the frequency of the leading eigenmode by et al. (2004), A = 8, Re = 6800, β = 0.98.

2 2 Interestingly, when the y dependence of both the underlying basic state and this amplitude function at the midplane of the duct, z = 0, are extracted from this data, one finds close coincidence, respectively, with the parabolic profile of plane Poiseuille flow and the corresponding leading eigenfunction of PPF, as also can be seen in the same figure. Qualitatively the same structure of ŵ(z, y) may be seen to persist for quite some distance either side of the z = 0 midplane of the duct (note the different axes scales). This might lead to the conclusion that application of local theory through solution of the Orr-Sommerfeld equation suffices to analyze instability in duct flow in all but the smallest aspect ratio values. Taking a closer look though, a more refined picture emerges. The eigenvalue of the leading rectangular duct mode, obtained using 64 collocation points to discretize each spatial direction, is ω RD,A=8 = i. An Orr-Sommerfeld analysis of the corresponding PPF yields the result ω PPF = i. This implies that, at best, the local approximation may be used in order to obtain an estimate of the frequency of the leading eigenmode, here with relative error of O(1%). However, the amplification rate delivered by the local theory, in this near-neutral case, has an O(1) error when compared with the result obtained from solution of the appropriate two-dimensional eigenvalue problem. The conclusion drawn from this, and a multitude of analogous results not presented here, is that, even when the underlying basic state is (locally) consistent with the parallel-flow approximation, the appropriate partial-derivative-based eigenvalue (or initial-value) problem must be solved.

3 3 Lid-driven vs. open cavity flow Continuing with the theme of qualitative analogies in the spatial structure of amplitude functions in certain regions in physical space, attention is now turned to comparison of results of global instability analysis in another closed system, a lid-driven cavity periodically extended in the spanwise spatial direction, against those in the related open cavity flow. The leading eigenmodes of flow in the first geometry, which may be found in the literature under the heuristic denomination of Taylor-Görtler-Like vortices, are now well-documented (, 2000a; Albensoeder et al., 2001). In the open cavity, the leading eigendisturbances are also quasi-stationary modes reaching their maximum amplitude inside the cavity ( & Colonius, 2004; Bres & Colonius, 2008), while a second branch also exists, 1 the amplitude functions of which comprise structure inside the cavity as well as on the downstream cavity wall (De Vicente, 2010). The analogies in the spatial structure of the amplitude functions pertaining to the leading perturbations in lid-driven and open cavity flows, both being related with centrifugal instability of the main vortex, may be seen in the original references. Of interest in the context of comparison with local theory is the second branch, since the basic flow on the downstream cavity wall may be seen as a new steady boundary layer, which starts at the downstream cavity lip. Indeed, figure 2, taken from De Vicente (2010), shows the steady streamwise component of the basic flow velocity, ū(x, y), and the amplitude function, û(x, y), of the streamwise perturbation velocity of one member of the second branch at cavity- 1 The TS-like structure of û(x, y) in the downstream open cavity wall was first seen by Colonius et al. (2001), via application of the residuals algorithm, discussed at the closing section of this appendix.

4 4 depth- and (upstream cavity lip) momentum-thikcness-based Reynolds numbers Re D = 1400 and Re θ = 53.4, respectively, and a spanwise wavenumber β = 0.2; from these results, the profiles ū(x = 4, 0 y 2) and û(x = 4, 0 y 2) are extracted and presented. The profile ū(x = 4, y) is a typical boundary layer streamwise velocity component, while the (normalized) function û(x = 4, y) features the characteristic double-peak structure of a Tollmien-Schlichting wave; however, the existence of a small, but non-negligible wall-normal basic flow velocity component prohibits direct comparisons with solutions of the Orr-Sommerfeld equation at the particular local Reynolds number. What is more interesting is the fact that the Tollmien-Schlichting-like instability developing on the downstream wall is part of a global structure comprising instability of the main vortex inside the cavity. As such, the wave-like perturbation on the downstream wall is intimately linked with the instability of the main cavity vortex, and is expected to grow/decay at the same rate as the latter perturbation. Again, accurate description of this mode requires solution of the appropriate partial-derivative eigenvalue problem, which takes the entire cavity geometry into account. Laminar separation bubble A basic state in which a laminar separation bubble is embedded in a flat-plate boundary layer meets the quasi-parallel assumption of local theory. Consequently, it is reasonable to expect that local theory, through solution of the pertinent Orr-Sommerfeld or Rayleigh equation, may deliver results that are qualitatively analogous with those that may be obtained by application of BiGlobal instability analysis. This section presents some results from a recent global analysis of

5 5 this flow (Rodríguez &, 2010b), which demonstrate that this reasonable expectation is only partly true and needs to be refined in a two-fold manner. First, the stationary global mode of laminar separation, discovered by et al. (2000) by solution of the pertinent eigenvalue problem, has a spatially localized structure which makes its recovery by local analyses impossible 2. Second, a branch of BiGlobal eigenmodes also exists, having amplitude functions of wave-like nature; figure 3, taken from (Rodríguez, 2010), shows the (dominant) streamwise perturbation component of one member of this branch, obtained at inflow displacement-thickness-based Re δ = 450 and spanwise wavenumber parameter β = The wall-normal disturbance of this mode is extracted at three locations, one approximately at the peak of the recirculation region inside the primary laminar separation bubble, and two more at equidistant locations upstream and downstream of this position. The extracted amplitude functions û(x = 175, 0 y 10) and û(x = 295, 0 y 10) outside the bubble have a two-peak structure which may be identified with that pertinent to TS-wave instability of the underlying attached flow, while the three-peak structure of the û(x = 235, 0 y 10), inside the recirculation region is strongly reminiscent of solutions of the Rayleigh equation for the corresponding inflectional profile. Presentation of results of local analyses at the respective x stations and comparisons with the BiGlobal eigenfunction at hand are beyond the scope of the present note. What is of interest here is that, much like the region around the midplane of the rectangular duct and the downstream wall of the open cavity, qualitative local approximations of the two-dimensional amplitude functions may 2 Detailed description of this instability may be found in the original reference, while its consequences from a topological point of view are discussed by Rodríguez & (2010b) and Rodríguez & (2010a)

6 6 be obtained by application of local theory. However, quantitative information on both the spatial structure of the amplitude functions and, more importantly, the frequency and amplification/damping rate of the related global eigenmode, may only be obtained by solution of the pertinent partial-derivative based eigenvalue problem. In addition, as the case of the stationary three-dimensional global mode of laminar separation ( et al., 2000) shows, global instability analysis is capable of unraveling results inaccessible to local analysis. Residuals in time-accurate numerical simulations near convergence to a steady state We close this section by complementing the aspects of physical knowledge provided by global (eigen-)modes, as briefly introduced above, with a discussion on numerical algorithms for the recovery of steady-state solutions from transient data, based on exploitation of global mode information. The key idea, first articulated in (2000c), is the identification of residuals in time-accurate numerical simulations close to convergence with least-damped global flow eigenmodes. An extensive discussion may be found in (2000b); here, the main points are highlighted. A numerical solution q = (ρ, u, v, w, p) T is sought by direct time-integration of the equations of motion, in an arbitrarily complex domain in two- or three spatial dimensions. Depending on parameters, a steady-state solution may exist; in its neighborhood the decomposition q(x, y, z, t) = q(x, y, z) + ɛ R(x, y, z, t), ɛ 1, (2) holds, q denoting the steady state sought and R being the unsteady (vector)

7 7 deviation from it - the residual. The small-parameter ɛ indicates closeness of q to q. In view of the separability of spatial from temporal derivatives in the equations of motion, R may be written as R = (ˆq r cos ω r t ˆq i sin ω r t) e σt, (3) where ω r and σ are real constants, while ˆq r and ˆq i are real vector functions. Close to convergence, all unknown vectors, q, ˆq r and ˆq i may be delivered by simple algebraic operations (, 2000c,b). If the signal q(x, y, z, t) contains a harmonic component, ω r may be readily identified, otherwise it is zero. The damping rate σ, may be calculated from 1 2 σ = q ttt q tt qt=0, ω r 0, q tt q t, ω r = 0, (4) subscripts denoting partial differentiation. With ω r and σ known, Equation 2 may be written at a number of consequtive times within the time-integration process equal to the number of unknowns sought. The resulting linear system delivers the real and imaginary parts of ˆq, as well as the steady-state q toward which the time-integration converges; one of the related formulae (see the original references for a complete discussion) is q = { 1 ωr 2 + σ 2 (ωr 2 + σ 2 )q 2σ q } t + 2 q t 2. (5) Application of this so-called residuals algorithm eliminates the arbitrariness typically invoked in order to devise stopping criteria of the time-integration process. More importantly, use of this idea delivers machine-precision accurate solutions q from transient data, the latter extracted at a fraction (typically onequarter to one-half) of the time needed to drive residuals to machine-zero level. Figure 4 shows results of application of the algorithm for the recovery of steady

8 8 state and amplitude functions in square lid-driven cavity two-dimensional direct numerical simulations at two Reynolds numbers at which steady states exist, Re = 100 and Machine precision accuracy in the respective predictions, in addition to savings of 75% of the total simulation cost have been obtained in both these sets of results. The residuals algorithm is yet to be demonstrated in three-dimensional flows, but could be used to obtain accurate and inexpensive TriGlobal modal stability predictions by straightforward use of existing direct numerical simulation codes. References Albensoeder S, Kuhlmann HC, Rath HJ Three-dimensional centrifugalflow instabilities in the lid-driven-cavity problem. Phys. Fluids 13: Bres GA, Colonius T Three-dimensional instabilities in compressible flow over open cavities. J. Fluid Mech. 599: Colonius T, Rowley CW, V Global instabilities and reduced-order models of cavity flow oscillations. Sept , 2001, Crete, Greece: ISBN-13: Fundación General UPM De Vicente J Spectral multidomain methods for global instability analysis of complex cavity flows. Ph.D. thesis, School of Aeronautics, Universidad Politécnica de Madrid Orszag SA Accurate solution of the Orr-Sommerfeld stability equation. J. Fluid Mech. 50: Rodríguez D Global instability of laminar separation bubbles. Ph.D. thesis, School of Aeronautics, Universidad Politécnica de Madrid

9 Rodríguez D, V. 2010a. On the birth of stall cells on airfoils. Theor. Comp. Fluid Dyn. doi: /s Rodríguez D, V. 2010b. Structural changes of laminar separation bubbles induced by global linear instability. J. Fluid Mech. 655: Tatsumi T, Yoshimura T Stability of the laminar flow in a rectangular duct. J. Fluid Mech. 212: V. 2000a. Globally-unstable flows in open cavities. AIAA Paper V. 2000b. On numerical residuals and physical instabilities in incompressible steady-state fluid flow calculation. Tech. Rep. F WE090, EOARD V. 2000c. On steady state flow solutions and their nonparallel global linear instability. In 8 th European Turbulence Conference, June 27 30, 2000, ed. C Dopazo. Barcelona, Spain, pp V. 2000d. On the spatial structure of global linear instabilities and their experimental identification. Aerosp. Sci. Technol. 4: V, Colonius T Three-dimensional instabilities of compressible flow over open cavities: Direct solution of the BiGlobal eigenvalue problem. AIAA Paper V, Duck PW, Owen J Viscous linear stability analysis of rectangular duct and cavity flows. J. Fluid Mech. 505: V, Hein S, Dallmann U On the origins of unsteadiness and threedimensionality in a laminar separation bubble. Phil. Trans. Roy. Soc. London (A) 358:

10 Ŵ r (y,z) ŵ(-1 y 1,z=0) W(y,z) w(-1 y 1,z=0) Figure 1: Left: basic state, w(y, z), (lower) and real part of the spanwise component of the leading eigenvector, ŵ r (y, z), (upper) in A = 8

10 10 Ŵ r (y,z) ŵ(-1 y 1,z=0) W(y,z) w(-1 y 1,z=0) Figure 1: Left: basic state, w(y, z), (lower) and real part of the spanwise component of the leading eigenvector, ŵ r (y, z), (upper) in A = 8 constant-pressuregradient-driven rectangular duct flow at Re = 6800, β = Right: The y dependence of both functions at the duct midplane, z = 0, is extracted and superposed upon the result of solution of the Orr-Sommerfeld equation at the same parameters; line-thickness agreement is obtained.

second (TS-like) branch in open cavity flow at Re D = 1400, Re θ = 53.4, β = 0.

11 11 û(x,y) û(x=4,0 y 2) ū(x=4,0 y 2) ū(x,y) Figure 2: Streamwise velocity component of the steady basic state and that of the leading two-dimensional perturbation of the second (TS-like) branch in open cavity flow at Re D = 1400, Re θ = 53.4, β = 0.2 (De Vicente, 2010). The wallnormal dependence of the amplitude function, û(x = 4, 0 y 2) is highlighted.

12 12 û(x,y)! û(x=175,y)! ū(x,y)! ū(x=175,y)! û(x,y)! û(x=235,y)! ū(x,y)! ū(x=235,y)! û(x,y)! û(x=295,y)! ū(x,y)! ū(x=295,y)! Figure 3: Basic state streamwise velocity component, ū(x, y), and corresponding amplitude function û(x, y) of the laminar separated boundary layer flow analyzed by Rodríguez (2010). Wall-normal y dependence of both functions is extracted and shown at three streamwise locations, x/δ = 175, 235 and 295.

13 13 Figure 4: Amplitude functions of the scaled eigenvector components, û (upper left), ˆv (upper right), ŵ, (middle left) and ˆp (middle right) of the vector ˆq = (û, ˆv, ŵ, ˆp) T, obtained by application of the residuals algorithm, in lid-driven square cavity flow at Re = The values of the indicated levels are to be found on the table shown. Lower: Relative error in the estimation of the steady state, q at Re = 100, by application of the residuals algorithm (, 2000b).

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