Transition to turbulence in the boundary layer over a smooth and rough swept plate exposed to free-stream turbulence

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2 Page 1 of 28 Under consideration for publication in J. Fluid Mech. 1 Transition to turbulence in the boundar laer over a smooth and rough swept plate eposed to free-stream turbulence L A R S - U V E S C H R A D E R, S U B I R A M I N AND L U C A B R A N D T Linné Flow Centre, KTH Mechanics, SE-1 44 Stockholm, Sweden (Received 3 November 29) Receptivit, disturbance growth and transition to turbulence of the three-dimensional boundar laer developing on a swept flat plate are studied b means of numerical simulations. The flow is subject to a favorable pressure gradient and represents a model for swept-wing flow downstream of the leading edge and upstream of the pressure minimum of the wing. The boundar laer is perturbed b free-stream turbulence and localized surface roughness with random distribution in spanwise direction. The intensit of the turbulent free-stream fluctuations ranges from conditions tpical for free flight to higher levels usuall encountered in turbo-machiner applications. The free-stream turbulence initiall ecites non-modal streak-like disturbances as in two-dimensional boundar laers, soon evolving into modal instabilities in the form of unstead cross-flow modes. The latter grow faster and dominate the downstream disturbance environment in the laer. The results show that the receptivit mechanism is linear for the disturbance amplitudes under consideration, while the subsequent growth of the primar disturbances rapidl becomes affected b nonlinear saturation in particular for free-stream fluctuations with high intensit. Transition to turbulence occurs in the form of localized turbulent spots randoml appearing in the flow. The main features of the breakdown are presented for the case of travelling cross-flow vortices induced b free-stream turbulence. The flow is also receptive to localized roughness strips, eciting stationar cross-flow modes. The mode with most efficient receptivit dominates the downstream disturbance environment. When both free-stream fluctuations and wall roughness act on the boundar laer at the same time, transition is dominated b stead cross-flow waves unless the incoming turbulence intensit is larger than about.5% for roughness amplitudes of about one tenth of the boundar-laer displacement thickness. The results show that a correct prediction of the disturbance behavior can be obtained considering the receptivit and evolution of individual modes. In addition, we provide an estimate for the amplitudes of the eternal disturbance sources above which a full nonlinear receptivit analsis is necessar. 1. Introduction Receptivit, disturbance growth and breakdown these are the fundamental stages through which laminar flow becomes turbulent. Numerous issues are involved in the transition process, e.g. how eternal perturbations enter the boundar laer and ecite internal disturbances (receptivit), how the latter grow (instabilit), and when and where the flow first becomes turbulent (breakdown). The relevance of these issues is closel related to applications, where for instance drag reduction on aircraft wings b suppression or downstream dela of transition is of interest (flow control).

3 Page 2 of 28 2 Lars-Uve Schrader, Subir Amin and Luca Brandt 1.1. Transition in two-dimensional boundar laers The classical transition scenario in two-dimensional boundar laers originates from eponentiall growing Tollmien-Schlichting (T-S) waves and is observed in flows with low background disturbance levels. When these primar waves grow above a threshold amplitude, the flow becomes susceptible to secondar instabilit which is three-dimensional in nature and characterized b the occurrence of lambda-shaped vortices (cf. the reviews b Herbert 1988; Kachanov 1994). Several eperiments (see e.g. Talor 1939; Klebanoff 1971; Westin et al. 1994) reveal, however, that transition in boundar-laer flows eposed to free-stream turbulence with intensit 1% of the free-stream velocit is initiated b the growth of perturbations elongated in the streamwise direction rather than b T-S waves. These take the form of streaks with high and low streamwise velocit alternating in the spanwise direction. The streamwise velocit perturbations of the streaks are due to the wall-normal displacement of fluid particles in shear flows b weak pairs of counter-rotating streamwise vortices (lift-up mechanism). The lift-up is promoted b low-frequenc oscillations of the turbulent free stream, whereas the high-frequenc components are highl damped inside the shear laer (Jacobs & Durbin 1998; Zaki & Saha 29). As the streaks grow downstream, the become susceptive to high-frequenc secondar instabilit triggered b free-stream turbulence (Zaki & Durbin 25; Hœpffner & Brandt 28) or b streak interactions (Brandt & de Lange 28), and breakdown to turbulent spots is initiated. These spots appear at random locations, grow in size and number and merge with each other, until the flow is full turbulent (bpass transition). Detailed numerical analsis of bpass transition under free-stream turbulence can be found in Jacobs & Durbin (21); Brandt et al. (24) Transition in three-dimensional boundar laers Boundar laers on swept wings, plates and wedges, on cones or on rotating disks are three-dimensional, and the transition mechanisms differ from those in two-dimensional flow owing to the eistence of different instabilit tpes. Saric et al. (23) list four kinds of instabilit, relevant in different regions of the boundar laer: attachment-line, Görtler, streamwise (T-S waves) and cross-flow instabilit. The focus is here on the latter instabilit tpe in accelerated swept-plate flow. Cross-flow instabilit is related to the inflectional velocit profile of the cross component of the mean flow. The base flow is therefore susceptive to strong inviscid instabilit which can be both stead and unstead (see also the review article b Bippes 1999). Cross-flow disturbances are intensified in the forward part of a wing b the favorable pressure gradient, whereas growth of T-S waves is suppressed in this region. These become instead relevant in the decelerating flow in the rear portion of the wing. Whether stead or unstead cross-flow waves lead to transition of swept flow is a relevant issue for the correlation between wind-tunnel eperiments and free-flight tests with significantl lower levels of eternal vortical disturbances. Eperiments b Dehle & Bippes (1996) and White et al. (21) and numerical studies b e.g. Crouch (1993), Choudhari (1994) and Schrader et al. (29) suggest that stead cross-flow modes induced b wall roughness dominate in an environment of low-amplitude free-stream disturbances (at free-flight conditions), whereas travelling modes become dominant at higher intensities of the background disturbance (in turbo-machines or some wind-tunnel tests). As the cross-flow waves grow downstream in amplitude the distort the chordwise mean-velocit profiles in spanwise direction such that the flow becomes susceptive to high-frequenc secondar instabilit. Results on secondar instabilit of swept-wing flow based on the Parabolized Stabilit Equations (PSE) are available in Malik et al. (1999) and Hanes & Reed (2); findings from Direct Numerical Simulation (DNS) are presented in Win-

4 Page 3 of 28 Swept-plate boundar-laer transition under free-stream turbulence 3 tergerste & Kleiser (1997) for the temporal framework and in Högberg & Henningson (1998) and Wassermann & Kloker (22, 23) for spatiall evolving flows. Flow control in swept-wing boundar laers has been addressed eperimentall b Saric et al. (1998). These authors used arras of distributed roughness elements with different spanwise spacing near the leading edge of a swept wing. The show that the most amplifing stead cross-flow mode can be suppressed if the spacing between the roughness elements is smaller than the most unstable spanwise wavelength. Under these conditions, a less unstable, useful cross-flow mode becomes dominant, and transistion to turbulence occurs further downstream than in the uncontrolled scenario. Wassermann & Kloker (22) confirmed this passive control strateg b means of DNS. The purpose of the present work is to stud receptivit, growth and breakdown of crossflow instabilit in swept-plate flow under free-stream turbulence, surface roughness and the combination of both b means of nonlinear LES and DNS. This tpe of simulations is presented here for the first time. In a previous investigation, Schrader et al. (29) analzed linear receptivit mechanisms in swept boundar laers. These authors isolated the effect of different components of free-stream disturbances, modeled b single freestream Orr-Sommerfeld modes, and demonstrated that travelling cross-flow modes can be forced b vortical disturbances in the free stream via a scale-conversion process. In addition, scattering of free-stream modes on chordwise localized surface roughness with spanwise periodicit was eamined and shown to become relevant onl for high levels of free-stream turbulence. However, in three-dimensional boundar laers, the interaction between eponentiall growing modes ma continuousl induce waves with large amplification and quickl create a disturbed boundar-laer flow. To better analze this scenario, full nonlinear simulations are needed and presented here. We emplo a more comple model for free-stream vorticit b considering the superposition of a large number of Orr-Sommerfeld/Squire modes with a turbulent energ spectrum. Localized roughness with spanwise random amplitude is modelled through a sum of Fourier modes. These disturbance sources will bring nonlinear effects into pla and provide a more complete picture of receptivit and transition in three-dimensional boundar laers. Moreover, we investigate herein the breakdown of the three-dimensional boundar laer. The first part of the results is meant to illustrate the disturbance features inside the shear laer during growth, saturation and breakdown of the primar instabilities induced b free-stream turbulence. In the second part, the focus is on the earl receptivit process, where receptivit of unstead and stead cross-flow instabilit is compared with the linear results in Schrader et al. (29). 2. Flow configuration and numerical approach 2.1. Base flow Boundar-laer flow over a swept flat plate is herein considered. The mean flow is obtained b solving the Navier-Stokes equations with Falkner-Skan-Cooke velocit profiles as initial condition. This configuration often serves as a prototpe for swept-wing boundar-laer flow, including man of its characteristics such as chordwise pressure gradient, streamline curvature and cross flow, while leading edge and surface curvature are not taken into account. The stream- and cross-wise velocit profiles of swept-plate flow are depicted in figure 1(a), where the coordinate sstem adopted is also shown. The basic flow is completel described b the Renolds number, the Hartree parameter and the sweep angle. These are chosen in correspondence with the values in Schrader et al. (29). This set of parameters defines conditions similar to those of the airfoil eperiments at

5 Page 4 of 28 4 Lars-Uve Schrader, Subir Amin and Luca Brandt (a) 8 (b) U W Re δ Re.1 leading edge φ φ (c) z Figure 1. (a) Wall-normal profiles of streamwise and cross-flow velocit for Falkner-Skan-Cooke boundar-laer flow. (b) Renolds number based on displacement thickness (black) and downstream location (gre) versus chord coordinate. (c) Angle of eternal streamline along chordwise coordinate. Arizona State Universit reported in Reibert et al. (1996). The chordwise pressure gradient is quantified b the Hartree parameter β H, chosen to be β H =.333. This establishes a favorable pressure-gradient boundar laer with chordwise flow acceleration, U ( )=U, ( + ) β H 2 β H and W = U, tan φ, (2.1) where U and W indicate chord- and spanwise mean velocit and the star denotes dimensional quantities. The sweep angle φ = 45 is defined at a reference station, which corresponds to the inflow plane of the computational domain. Sweep together with chordwise flow acceleration causes curved streamlines and a force imbalance inside the boundar laer, driving a secondar mean-flow component in the cross-stream direction, the cross flow. The sweep angle under investigation is characterized b significant cross flow and thus b strong cross-flow instabilit. Lengths are normalized b the chordwise boundar-laer displacement thickness δ δ ( ) at the reference location and velocities b the chordwise free-stream velocit U, U ( ). Reference length and velocit define the Renolds number at the computational inlet, Re δ = U,δ, (2.2) ν where ν is the kinematic viscosit. The local Renolds number Re δ is obtained b replacing δ and U, in (2.2) b their local values. The inflow Renolds number is fied at Re δ = 22. The Renolds number Re based on the chordwise location is also common in literature, related to Re δ via Re =[ U, + U ()]Re δ. (2.3) Figure 1(b) shows the Renolds numbers Re δ and Re and (c) the local angle φ of the eternal streamline versus the chordwise coordinate for the configuration adopted here Numerical method The present results are obtained using a spectral method to solve the three-dimensional time-dependent incompressible Navier-Stokes equations, see Chevalier et al. (27). The simulation code builds on a Fourier representation along the chord- and spanwise coor-

6 Page 5 of 28 N E (a) Swept-plate boundar-laer transition under free-stream turbulence 5 (b) Spec. PSE SEM σ Figure 2. (a) N-factor for energ and (b) growth rate of the stead cross-flow mode with spanwise wavenumber β =.19, initialized at Re δ = 22. Comparison between results from the present spectral code (black), a spectral element method (SEM; black diamonds) and a PSE calculation (gre circles). dinates and z and on Chebshev polnomials in wall-normal direction, along with a pseudo-spectral treatment of the nonlinear terms. A zero-slip condition is imposed along the wall for the base flow and the perturbed flow (base flow plus disturbances). For the simulations with free-stream turbulence von-neumann conditions are applied in Fourier space at the far-field boundar above the plate. To reduce the computational effort the top boundar for the simulations with surface roughness has been placed nearer the plate, achieved b using the asmptotic conditions first proposed b Malik et al. (1985). Periodic boundar conditions are enforced in the spanwise and chordwise directions z and. The swept-plate flow develops along : the boundar laer grows and the streamlines change continuousl direction. To obtain nonetheless the chordwise periodicit required b a Fourier representation, a fringe region is used at the downstream end of the computational domain, as described b Nordström et al. (1999). In this region the velocit field is forced to the desired inflow velocit profiles; the consist of the Falkner-Skan-Cooke similarit profiles in the present case with/without incoming free-stream disturbances. To validate the present implementation we show in figure 2 the chordwise evolution of a stead cross-flow mode as obtained with two different spectral methods. The mode is computed for Re δ = 22, inserted in the fringe region of the present spectral code and prescribed as an inflow condition of a simulation using a spectral element method. The spectral element code (see Tufo & Fischer 1999) builds on an inflow-outflow formulation rather than a fringe technique. Good agreement of the modal N-factor and the growth rate σ is obtained for the two spectral methods. As we will later show results from PSE calculations we also validate the PSE code in figure 2: the PSE results compare well with the numerical solutions of the Navier Stokes equations (see also Tempelmann et al. 29) Sub-grid scale modelling The present stud requires a computational domain of rather large spanwise and wallnormal size to accommodate a free-stream turbulence field with a wide enough range of length scales. The resolution of all scales would, however, result in prohibitivel large computational costs such that the simulations are onl affordable b emploing a Large- Edd Simulation (LES) model. The ADM-RT subgrid-scale model b Schlatter et al. (24) is used for the present simulations, building on the Approimate Deconvolution Model (ADM). This model has been successfull applied in incompressible transitional and turbulent flow (see e.g. the recent work b Monokrousos et al. 28). The main

7 Page 6 of 28 6 Lars-Uve Schrader, Subir Amin and Luca Brandt 75 (a) u ---- v w 5 h.5 (b) u i h z z Figure 3. (a) Turbulent free-stream fluctuations at the inflow plane of the computational domain for a turbulence intensit of 2.53% of U,. (b) Chord- and spanwise contour of the wall roughness. The rms-amplitude is.15 (gre ----). ingredient of the ADM-RT model is the so-called relaation term (RT) used as a closure for the subgrid-scale stresses τ ij, τ ij j = χh N ū i with H N =(I G) N+1. (2.4) ū i indicates the velocit field implicitl filtered b the reduced resolution of the LES grid. H N denotes a three-dimensional high-pass filter of high order derived from the low-order low-pass filter G in Stolz et al. (21), and the star stands for convolution in phsical space. Here, H N is characterized b the eponent N = 5 and the cut-off wavenumber κ c =2π/3 of the filter G. Numerical stabilit is ensured b a model coefficient χ in the range χ 1/ t. Here we chose χ =.2, as in Schlatter et al. (26, 27); Monokrousos et al. (28). The role of the RT term in (2.4) is to drain kinetic energ from the resolved fluctuations at the smallest represented length scale and thereb to model the impact of the unresolved motion on the resolved structures. Note that all simulations were performed adding the sub-grid scale stresses. However, the computations focusing on the boundar-laer receptivit consider onl the initial phase of the disturbance growth. In these cases, the etra relaation term is practicall zero and the computations can be considered as DNS Disturbance generation Two different disturbance sources are considered, a vortical perturbation in the free stream and a roughness element on the wall near the inflow plane of the computational domain. These two tpes of disturbances are listed in the review article b Saric et al. (23) as particularl relevant in swept boundar laers. The free-stream turbulence at the inflow plane and the roughness element are shown in figure 3(a) and (b), and their numerical generation is described below Free-stream turbulence The turbulent inflow disturbances are numericall generated as in Jacobs & Durbin (21) and Brandt et al. (24), i.e. b the superposition of eigenmodes from the continuous spectrum of the Orr-Sommerfeld and Squire operator however, modified here for the Falkner-Skan-Cooke base flow. A wave vector is associated to each free-stream mode, where the comple chordwise wavenumber α is the eigenvalue of the corresponding Orr-Sommerfeld and Squire problem and the wall-normal wave-number γ determines

8 Page 7 of 28 Swept-plate boundar-laer transition under free-stream turbulence 7 the position along the continuous spectrum (see e.g. Schmid & Henningson 21). After choosing spanwise and wall-normal wavenumber β and γ and angular frequenc ω, the eigenvalue α of the continuous-spectrum modes can be obtained from analtical epressions (see Schrader et al. 29, for swept boundar laers). The velocit profiles pertaining to each eigenfunction are computed numericall, whereas the free-stream behavior can also be obtained analticall. An isotropic perturbation field is obtained in Fourier space b considering 2 concentric spherical shells of radius κ spanning the range of wave vectors of length κ l κ κ u. The limits κ l and κ u depend on the size and resolution of the computational domain and are chosen to be κ l =.5 and κ u =1.1. The frequenc range covered b the turbulence model is ω 1.1. Fort points are distributed with constant spacing on each shell: these define the wave vectors and continuous-spectrum eigenmodes to be included in the epansion for the free-stream disturbance, u F ST = 2 4 a k k=1 ˆ u kl (; γ)e i(α kl+β kl z ω kl t). (2.5) l=1 ˆ u kl denotes the wall-normal disturbance-velocit profiles of the free-stream mode and contains the wall-normal oscillations. The randomness inherent in turbulent fields is obtained through a random rotation of the shells, provided b random phase angles of the comple coefficients a k in (2.5) as well as b the random phase in the comple function ˆ u kl. The energ densit pertaining to the wavenumber κ k of the k-th shell is equall distributed among all modes on the shell, and the energ distribution across the wavenumbers included is approimated b the von-kármán spectrum. Details on the spectrum can be found in Brandt et al. (24). The relevant measure for the disturbance amplitude is the turbulence intensit Tu, defined as Tu = 1 3 (u2 + v 2 + w 2 ). The wallnormal distribution of the free-stream fluctuations at the inlet is depicted in figure 3(a). We conclude b noting that the quantitative results on the boundar-laer response will depend on the turbulence model spectrum chosen; the relevant phsics, on the other hand, will be independent of the spectrum if the receptivit process is linear and if there is no significant nonlinear interaction in the free stream. In that case, the amplitude of the boundar-laer disturbances will be proportional to the amplitude of the free-stream modes, with the turbulence intensit Tu being the factor of proportionalit. We will show below that this holds for most of the Tu values studied. The random phases used for the combination of the continuous modes does not correctl reproduce the energ transfer in a natural turbulent flow. However, these effects appear to be relevant onl for the largest Tu eamined, as shown below; moreover, the snthetic inflow will adjust to the mean-flow conditions further downstream Surface roughness The surface roughness is modelled through non-homogeneous boundar conditions for the disturbance velocities u and w at the wall, u v w wall = h(, z) U h(, z) W wall, h start h end, elsewhere. (2.6)

9 Page 8 of 28 8 Lars-Uve Schrader, Subir Amin and Luca Brandt Table 1. Parameters of the surface-roughness shape. ε h h start h end h rise h fall.375,.75, h, u z Figure 4. Validation of the roughness model for the highest roughness (ε h =.15, dotted line). Comparison between the specified shape h( r,z) at the roughness station r ( -) with the no-slip contour u = (gre ----). u( r, = ) is plotted as well for comparison ( thin). In the epression above, h(, z) is the shape of the roughness bump, h(, z) =ε h h ()h z (z), (2.7) with the amplitude ε h and the functions h () and h z (z) along the chord- and spanwise coordinate respectivel. h () builds on a smooth, localized step function (cf. Schrader et al. 29), rising from = h start along the flank h rise, falling till = h end along h fall and centered at the location r =(h start + h end )/2, the nominal roughness station (see figure 2(b) and table 1). This particular shape is characterized b a broad spectrum of chordwise wavenumbers, giving rise to a broadband disturbance including unstable modes. h z (z) is obtained through an epansion in sinusoidal functions with 16 different spanwise wavenumbers and random phases, h z (z) = 16 n=1 sin (nβ z + φ rand n ), (2.8) where the fundamental spanwise wavenumber β =2π/L z is defined b the spanwise width L z of the computational domain and φ rand n are random angles. The characteristic roughness height ε h in equation (2.7) is ε h = h 2 z, (2.9) r i.e. the rms of the random spanwise hump contour at the roughness station r. Figure 4 gives a validation of the roughness model for the highest roughness (ε h =.15): good agreement is found when comparing the prescribed shape h( r,z) at the roughness station r with the no-slip contour u =. The present roughness model is linear in ε h ; this assumption is valid onl for small enough roughness elements. Figure 8 in Schrader et al. (29) gives the range of ε h for which roughness receptivit is linear in ε h. The present roughness elements similar to shape I in that reference are in the linear range of that figure. Moreover, the individual amplitudes of the 16 modes included in (2.8) are smaller than ε h ; the present bumps can

10 Page 9 of 28 Swept-plate boundar-laer transition under free-stream turbulence 9 Table 2. Size and resolution of the computational domains L and S, length of the fringe region and inflow Renolds number. Bo Re δ, L L L z N N N z L fringe L S c f 1 3 (a) p ma u 2.1 (b) Figure 5. Boundar-laer response to free-stream turbulence with Tu =3.73% and L = 1. Stud of the influence of chordwise resolution of the computational domain on (a) the skin-friction coefficient (dots: Falkner-Skan-Cooke solution) and (b) the evolution of the wall-normal maimum of u rms. =5.86 (N = 256) ( -); 3.91 (N = 384) (gre ----); 2.93 (N = 512) (- - -) and 1.95 (N = 768) (gre ). therefore be considered as low-amplitude roughness. The use of a linear roughness model is hence justified for the values of ε h considered Computational domain Two computational domains with different chordwise length are used: a long one, L, to capture the breakdown of the boundar laer and a short one labelled S for the receptivit stud. The size L L L z, resolution N N N z and length L fringe of the fringe region of the boes L and S are listed in table 2 together with the Renolds number Re δ, defining the location of the inflow plane Domain size and resolution stud The resolution in wall-normal direction is finer than in the wall-parallel planes, see table 2, in particular inside the boundar laer due to the clustering of the Chebshev points near the wall. The normal resolution is in fact comparable to that in Brandt et al. (24), where no sub-grid scale model was used. Since the flow is swept at 45, the resolution requirements in and z direction are epected to be similar. The laminar structures preceding the breakdown the cross-flow modes can thus be full resolved, while the SGS model will compensate mainl for unresolved small wall-parallel scales occurring at and after the turbulent breakdown. Owing to a wall-normal resolution tpical for DNS, the resolution stud is restricted to the chord- and spanwise direction. Figure 5 shows the influence of the chordwise resolution in (a) on the skin-friction coefficient and in (b) on the chordwise fluctuation amplitude in terms of the maimum of the rms. The transition location, identified b the rapid increase of the skin-friction, moves drasticall upstream as the chordwise resolution is refined from =5.86 to This trend becomes slower at a further increase in resolution, and the transition location is observed

11 Page 1 of 28 1 Lars-Uve Schrader, Subir Amin and Luca Brandt.15 (a).15 (b).1 q ma u 2 i.5.1 q ma u 2 i Figure 6. Stud of the influence of resolution and size of the computational domain. Boundar-laer response to free-stream turbulence with L = 1. (a) Tu =2.53%. Variation of chordand spanwise resolution of bo S, see table 2. N N N z = ( ); (----); (- - -). (b) Tu =3.73%. Variation of spanwise size of bo S. L z = 2 ( ); L z = 3 (----). at nearl the same downstream location when the numerical resolution is further refined from =2.93 to Figure 5(b) shows that, while the earl transient and linear growth phase is captured with similar accurac in the different cases, the subsequent rapid amplification observed around 6 is under-predicted on the coarser domains. Small structures responsible for the ecitation of the secondar instabilit need to be represented on the grid for the correct prediction of the transition location. This requirement is met on the fine meshes, and a further refinement below =1.95 will onl weakl affect the results while significantl increasing the computational costs. The differences in the maimum levels of c f and u rms observed in the regime after transition are due to the different range of resolved scales, and uncertainties in the turbulent fluctuations associated with resolution ma therefore be relevant for 7 when Tu =3.73%. However, we do not investigate the turbulent regime here; the main focus of the paper is on the receptivit phase and the structures at the breakdown stage. The SGS model is not designed to effect the phsics of transition but to prevent unphsical parasitic upstream influences from the highl fluctuating turbulent region further downstream for simulations where transition occurs (see also Schlatter et al. 26). On the short domain S, on the other hand, a finer resolution is computationall feasible. Two modifications of bo S in table 2 are considered, first a refinement in the streamwise direction, from =1.95 to 1.46, and second a higher resolution (1.5 times) in the z direction. The vortical perturbation prescribed at the inlet of the original and the modified domains is composed of modes with wavenumbers between κ l =.5 and κ u = 1.1 with a turbulence intensit of Tu =2.53% and a characteristic length of L = 1. It becomes clear from figure 6(a) that the receptivit and the subsequent growth of the instabilit obtained on the coarsest grid do not differ from the results on the two refined meshes. Net, the influence of the width L z of the numerical domain on the boundar-laer response is eamined. This is investigated b considering the shorter bo S with L z = 3 and N z = 192, i.e. the spanwise resolution is kept fied. For a fair comparison, the parameters defining the free-stream turbulence are left unchanged: the turbulence intensit considered is Tu =3.73% and the integral length L = 1. Figure 6(b) shows the downstream evolution of the ecited boundar-laer disturbance in the wider domain as well as in bo S. The initial transient disturbance growth is slightl enhanced in the wider domain whereas the disturbance growth rate further downstream is hardl affected.

12 Page 11 of 28 Swept-plate boundar-laer transition under free-stream turbulence 11.4 (a) Tu = C( r).6.4 (b).3 Tu q u 2 i Figure 7. Downstream deca of free-stream turbulence with inflow turbulence intensit of Tu = 3.73% and integral length L = 1. (a) Downstream evolution of Tu measured at three levels above the plate, = 3 ( ); 35 (----) and 45 (- - -). Comparison with energ deca characteristic for grid turbulence (- - -), see Fransson et al. (25). Here, C = 1.73 and r = 6. (b) Deca of the three fluctuation components measured at = 4 for free-stream turbulence with Tu =3.73%. p u 2 ( ); p v 2 (----) and p w 2 (- - -). This difference can be eplained b the nonlinear interactions occurring between wider structures at the initial receptivit phase. The spanwise scale of the disturbance is not changed b the increased domain size as shown b the identical growth rates downstream. Note, however, that the results are obtained for the second highest free-stream turbulence intensit studied, Tu =3.73%, and the agreement between the curves is epected to improve at lower Tu. In light of the rather small influence of L z on the boundar-laer response and for the sake of the computational costs the subsequent simulations are performed on the default domain in table 2 with L z = Characterization of the free-stream turbulence Due to the distribution of the modal wavenumbers defining the free-stream turbulence on concentric spherical shells the resulting disturbance field is homogeneous and isotropic. In swept-plate flow with favorable pressure gradient, however, the mean flow is subject to chordwise acceleration ( U > ). This gives rise to non-zero production terms in the Renolds-stress transport equation (see for instance Pope 21). Figure 7 displas the behavior of the artificial turbulence field in the free stream. In (a), the downstream deca of the total turbulence intensit at three different levels above the plate is plotted. The turbulent inflow conditions are Tu =3.73% with L = 1. It is apparent that the turbulence intensit decas at a rate similar to that in homogeneous isotropic turbulence, where the energ deca obes a power law. In Fransson et al. (25) the eponent of this law has been eperimentall determined as.6 for grid-generated turbulence, while the parameters C and r depend on the turbulence grid. Here, the curve fit has been done with C =1.73 and r = 6. The plot also shows that the deca rate depends onl weakl on the wall-normal level and hence that homogeneit is maintained across the free stream. However, figure 7(b) reveals that the turbulent kinetic energ is not equall distributed among the fluctuation components, and their individual deca differs. The adjustment to the free-stream conditions occurs alread in the fringe region and in the region near the inflow plane where U and the streamline curvature are maimum. The wall-normal fluctuations are larger than their chordwise counterpart. This can be eplained b considering the production terms in the Renolds-stress transport equation

13 Page 12 of Lars-Uve Schrader, Subir Amin and Luca Brandt Table 3. Amplitude of the forcing disturbance. Turbulence intensit of the free-stream turbulence at the inflow plane and rms-height of the surface-roughness strip. Disturbance source Amplitude Free-stream turbulence Tu ;.42;.84; 1.26; 1.69; 2.11; 2.53; 2.95; 3.73; 5.6 Surface roughness ε h ; 7.5; 15 for the chordwise u and wall-normal v fluctuations in accelerating flows, P 11 = 2u 2 U P 22 = 2v 2 V (2.1a) =2v 2 U. (2.1b) A negative production term is seen in the first equation, whereas positive production of wall-normal fluctuations is caused b the flow acceleration, where the continuit equation has been used to relate the mean-flow gradients. Energ is drained from the u-component of the free-stream turbulence into the v-component b the chordwise acceleration of the underling mean flow. The spanwise velocit fluctuations, initiall strong, deca rapidl until the reach amplitudes similar to those pertaining to u. The relative enhancement of the wall-normal fluctuations can also be related to the stretching of chordwise vorticit, as observed e.g. in the convergent section of a wind tunnel, while the drop of u and w is associated with the decrease of wall-normal vorticit induced b the negative V. 3. Results The response of the three-dimensional swept-plate boundar laer to free-stream turbulence and surface roughness is studied. The disturbance amplitude is epressed in terms of the turbulence intensit for the free-stream disturbance and the rms-height for the roughness. The values considered are compiled in table 3. The statistics presented in the following are obtained b averaging in time and in the spanwise direction. Note that the rms-values are indicative of the total disturbance energ, sum of the energ in the modes with different wavenumbers. The phase relation between the individual components, however, determines the local distortion. rms-values can therefore mask large localized distortions, origin of secondar instabilities Part I: Laminar-turbulent transition Results on the transition under a high-amplitude disturbance environment are presented first. The skin-friction coefficient provides a good indication of the transition location. This is shown in figure 8(a) for free-stream turbulence with inflow intensities of Tu =2.53% and 3.73%. The completion of transition is observed further upstream when increasing the level of the eternal disturbance. Bonfigli (26) pointed out that the skin friction does not ehibit an overshoot before approaching the value for full turbulent flow, in contrast to the case of Blasius flow (cf. Monokrousos et al. 28, for instance), and this observation is confirmed here. In figure 8(c) and (d) the boundar laer is characterized in terms of displacement and momentum-loss thickness as well as shape factor. These quantities are computed both from the chord- and the spanwise mean flow. A substantial rise of the thicknesses and drop of the shape factor is observed around 7 and 9 respectivel, indicating the region of laminar-turbulent

14 Page 13 of 28 Swept-plate boundar-laer transition under free-stream turbulence (a).2 (b) c f 4 q ma u 2 i.1 δ θ (c) 5 1 H δθ (d) Figure 8. Visualization of breakdown to turbulence of swept-plate flow eposed to free-stream turbulence with inflow turbulence intensit of Tu =2.53% (gre) and 3.73% (black) through (a) skin-friction coefficient and (b) downstream evolution of the rms of chordwise ( ), vertical (----) and spanwise disturbance velocit ( ). (c) Evolution of displacement ( ) and momentum-loss thickness (----) and (d) shape factor, based on chord- (top) and spanwise mean velocit (bottom). The laminar solution (thin gre) and the theoretical curves for Falkner-Skan Cooke flow (dots) are given as well. transition. In agreement with eperimental and numerical findings for bpass transition in two-dimensional boundar laers (Matsubara & Alfredsson 21; Brandt et al. 24), the displacement thickness is seen to decrease slightl below the laminar values in the transitional region, and then to increase faster in the turbulent flow. The momentum loss is, on the contrar, alwas larger than in the laminar case owing to the increase of skin friction caused b the occurrence of the turbulent spots. Note in figure 8(d) that the shape factor computed from the chordwise mean velocit slightl increases in the laminar region rather than being constant. This shows that the chordwise base-flow profiles, solution to the Navier-Stokes equations, are not eactl self-similar, in contrast to the initial Falkner-Skan profiles. The wall-normal maimum of the chordwise, wall normal and spanwise velocit fluctuations is reported in figure 8(b). Upstream of 2 the v component is weak while the horizontal components rapidl grow. This indicates that the boundar laer is initiall subject to non-modal instabilit, as tpicall observed in two-dimensional boundar laers. In Blasius flow, onl the streamwise velocit component grows at the initial stages of the transition while the cross-stream components deca (Brandt et al. 24). In threedimensional boundar laers, in contrast, the basic flow is characterized b wall-normal shear both in the chordwise and spanwise directions; therefore the lift-up effect induces

15 Page 14 of Lars-Uve Schrader, Subir Amin and Luca Brandt the growth of both components, as observed in the figure. Upstream transient growth is thus found to be a precursor of downstream cross-flow modes, providing the latter with initial energ. After the initial phase of primar growth, a fast increase in amplification rate is observed in 8(b) first for the case with Tu =3.73%, 6, and later for the lower intensit Tu =2.53%, 8. This is associated to the rapid rise in skin friction coefficient in figure 8(a) and is due to the appearance of high-frequenc secondar instabilities and turbulent spots in the flow. The weak overshoot in the curves for the velocit fluctuations can be related to the maimum values of skin friction. The maimum amplitude prior to breakdown is slightl larger for Tu =2.53% than for Tu =3.73%, i.e. the breakdown is not initiated at the same disturbance level inside the shear laer. This suggests that the transition location can onl approimatel be predicted from a certain threshold amplitude of the primar cross-flow mode alone. At the end of the computational domain the level of velocit fluctuations is almost the same for the two cases: the flow has reached an equilibrium turbulent state. It is worth pointing out the difference between the disturbance evolution depicted in figure 8(b) and, for instance, the data in figure 1(a) of Fransson et al. (25) pertaining to nonmodal growth of streaks in two-dimensional boundar-laer flow: In that reference, a strong overshoot in the curves for the streamwise fluctuation energ, intensifing with increasing Tu, is reported before the energ level sets to a constant value in the full turbulent boundar laer. This behavior suggests a pronounced region of intermittenc in two-dimensional flow, which is not observed here in the swept boundar laer. This discrepanc can be eplained with the different dominating instabilit tpes prior to transition: the cross-flow vortices in swept flow are destabilized much faster b secondar instabilit than the streaks in two-dimensional flow. Therefore, the amplitude of the crossflow modes (upon nonlinear saturating effects) and the average transition location ma be closer correlated in three-dimensional boundar laers than in their two-dimensional counterpart. An overall picture of the transition in swept boundar laers eposed to free-stream turbulence is provided in figure 9. This figure displas a time series of snapshots of the flow in a wall-parallel plane inside the boundar laer, = 2, together with a view of the free-stream fluctuations in a plane located at = 4 well above the boundar laer, figure 9(a). The results are obtained with inflow turbulence of intensit Tu =3.73% and length scale L = 1, and the snapshots are taken after the flow field has reached a statisticall converged state. The flow is swept from left to right and bottom to top in the figure. As seen in (a), the finer length scales of the eternal disturbance disappear further downstream, indicating deca of the free-stream turbulence. Plots (b) through (g) show that instabilities in the laminar region of the boundar laer appear in the form of long structures, tilted about 45 with respect to the chord of the plate. Around = 5 these structures have reached a threshold amplitude such that the boundar laer becomes susceptive to secondar instabilities: high-frequenc modes are triggered at random locations and grow rapidl in amplitude. Instabilities are most likel triggered b high-frequenc components of the free-stream turbulence. An alternative mechanism ma be the nonlinear interaction between low-frequenc modes alread in the boundar laer (see also Brandt & de Lange 28, for two-dimensional flow). Patches of irregular motion are seen to appear further downstream, forming local regions of turbulence called spots. The spot seen in figure 9(b) grows in size with time; a second spot occurs in plot (c) and both become wider and longer as the travel downstream. At time t = 623, figure 9(e), a third spot can be identified in the lower part of the domain, and later, image (g), the three turbulent spots have almost merged to form one large region of turbulent motion. Downstream of = 7 the boundar laer

16 Page 15 of 28 1 (a) Swept-plate boundar-laer transition under free-stream turbulence (b) t = 611 (c) t = 615 (d) t = 619 (e) t = 623 (f) t = 627 (g) t = 631 Figure 9. Free-stream turbulence with intensit Tu = 3.73% and integral length L = 1 and the response of the swept-plate boundar laer to it. The flow is from left to right. (a) Downstream evolution of the chordwise fluctuation u in the free stream at = 4. (b)-(g) Boundar-laer response in terms of the chordwise fluctuation u at = 2 at different instants of time. is full turbulent, and the turbulent region is constantl fed b merging spots incident from the upstream laminar part of the laer. It becomes apparent that the dominant disturbance structures in the full turbulent boundar laer are still elongated and tilted in the free-stream direction (Schlatter & Brandt 28). Note finall that all spots in figure 9 form in a rather limited chordwise region. This provides further indication for a less pronounced intermittenc phase of this transition scenario as compared with bpass transition in two-dimensional flow, as discussed above. Figure 1 characterizes the wall-normal and spanwise length scales of the disturbance

17 Page 16 of Lars-Uve Schrader, Subir Amin and Luca Brandt 1 5 (a) (b) (c) (d) z Figure 1. Boundar-laer response to free-stream turbulence with intensit Tu =3.73% and integral length L = 1. -z planes showing contour lines of instantaneous chordwise velocit in the boundar laer for time t = 627 (compare with figure 5(f)) at four chordwise stations upstream of the breakdown location. (a) = 3, (b) = 4, (c) = 5 and (d) = 6. The contour lines show levels between.5 and 1.25 (from black to gre) with spacing.15. Note that the ais is enlarged b a factor of structures inside the boundar laer b showing contour lines of instantaneous chordwise velocit in lateral planes at four downstream locations. At = 3, plot (a), the deformation of the mean flow is still moderate, and the smallest observed spanwise length scale is about 1, i.e. comparable to the integral length scale of the free-stream turbulence. Further downstream, at = 4 and 5, contour lines tpical for cross-flow modes are identified, the gradients in instantaneous chordwise velocit have become steeper and the structures have grown in size in wall-normal direction with the boundar laer. Their spanwise size has in contrast decreased relative to the laer thickness, see plots (b) and (c). Owing to the nois environment and random superposition of different modes, the mean flow deformation does not resemble that usuall observed under controlled disturbance generation. Cross-flow modes of different amplitude, spanwise and wall-normal scale appear side b side in an unpredictable fashion. However, the formation of strong shear laers supporting rapid secondar instabilities is clearl observed. In figure 1(d) regions of strong flow deformation, steep velocit gradients and fine spanwise scales are identified near the edges of the depicted domain, while the flow field in the neighborhood is smoother. These highl perturbed regions are referred to as turbulent spots (cf. figure 9(f)) and indicate where the boundar laer first approaches the turbulent state. Note that these turbulence patches are located closer to the wall than the original cross-flow modes. As in two-dimensional boundar laers, the late-stage high-frequenc instabilit

18 Page 17 of 28 Swept-plate boundar-laer transition under free-stream turbulence 17 8 (a) (b) z Mean flow Figure 11. Instantaneous flow field upstream of the transition location (t = 627, Tu =3.73%). (a) -z plane ( = 6) showing contour lines of chordwise velocit with levels between.8 and 1.28 and spacing of.8. The shaded areas identif regions of high-intensit secondar disturbances (chordwise component). (b) Contours of positive and negative chordwise disturbance velocit (gre: u =.12; black: u =.12 and of λ 2 (light gre: λ 2 = ). moves from the upper part of the boundar laer towards the wall (Jacobs & Durbin 21; Brandt et al. 24; Zaki & Durbin 25). Net, the disturbance structures caused b the secondar instabilities shortl upstream of the transition location are eamined. Secondar instabilit of cross-flow modes has been considered both theoreticall (Malik et al. 1999), eperimentall (White & Saric 25, more recentl) and numericall (Bonfigli & Kloker 27, for a comparison with stabilit analsis). These instabilities are of inviscid inflectional tpe and associated with strong shear laers of the mean flow induced b saturated cross-flow modes. Following Malik et al. (1999), three different classes of modes are identified, (i) mode I or z, associated with the minimum of the spanwise gradient of the streamwise velocit and characterized b high-frequenc oscillations; (ii) mode II or, associated with the maimum of the wall-normal gradient, also characterized b high-frequenc oscillations; (iii) mode III, associated with the maimum of the spanwise gradient, characterized b lower frequencies and located closer to the wall. These modes have been clearl identified when considering secondar modal stabilit of saturated stead cross-flow modes, whereas a less clear-cut distinction has been observed for travelling waves, which are more relevant for the transition scenario in the presence of free-stream turbulence. A combination of modes z and seems to appear in the eperiments reported in Bippes (1999) under natural conditions. To identif the relevant flow structures of the spot precursors, numerous velocit fields have been eamined in detail. Note that the identification of the various secondar instabilit modes of travelling cross-flow vortices presented in Wassermann & Kloker (23) required a spanwise Galileian transformation to travel with the primar wave and an adapted temporal-spanwise Fourier decomposition of the disturbance (see reference above for details). Such an analsis is impossible in the case of disturbances randoml induced b free-stream turbulence. The primar cross-flow modes are neither periodic in time nor in space; the appear as elongated structures of finite length. Similarl, the high-frequenc perturbations leading to formation of turbulent spots appear as localized, rapidl amplifing wave-packets. As a consequence, visual analsis is adopted to characterize the late stages of the laminar-turbulent breakdown.

19 Page 18 of Lars-Uve Schrader, Subir Amin and Luca Brandt An eample of tpical pre-transitional disturbance structures is provided in figure 11. Plot (a) is a close-up view from figure 1(d) displaing contour lines of instantaneous chordwise velocit tpical for a saturated cross-flow vorte. In addition, regions of the most intense secondar perturbation riding on top of the cross-flow vorte are shown as shaded areas. These are obtained b following the primar instabilit mode over a short downstream distance and correlating the instantaneous chordwise velocit at two locations 1 and 2, u( 1 )u( 2 )/u 2 ( 1 ). The perturbation determined in this wa represents streamwise fluctuations of the chordwise velocit component along the crossflow vorte. The fluctuations will inevitabl contain also the streamwise variation of the underling primar mode due to nonparallel effects and inhomogeneities; however this contribution turns out negligible when considering onl the regions of largest disturbance. In the figure, two regions are identified where the chordwise component of the secondar disturbance velocit is larger than 5% of the maimum. These two areas can be related (i) to strong shear at the updraught side of the primar vorte in what appears as a combination of modes z and and (ii) to the region of positive spanwise gradient located closer to the wall, as tpicall observed for mode III. The first tpe of highfrequenc perturbation is observed to appear more frequentl and to dominate during the breakdown. This can be eplained b the fact that the secondar instabilit is most likel triggered b high-frequenc disturbances in the free stream: modes located in the upper part of the boundar laer are thus most easil ecited. Note in addition that largest growth rates are obtained for modes z and when eamining the linear instabilit of individual stead cross-flow modes or packets of them (Wassermann & Kloker 22). Finall, vortical structures tpical of spot precursors are displaed in figure 11(b). These are identified b the λ 2 criterion (Jeong & Hussain 1995). Finger vortices located on the updraught side of the primar vorte are clearl visible, in agreement with previous numerical studies with controlled disturbances (Wintergerste & Kleiser 1997; Wassermann & Kloker 22, 23). Tpicall, packets of two to three vortices are observed in our simulations Part II: Receptivit First, snthetic turbulence with different intensities is prescribed at the inflow plane of the computational domain, from where it is convected b the free stream while decaing. This disturbance source will occup the entire free stream and therefore act non-locall on the boundar laer. Secondl, a thin roughness strip with a step-like contour in - and a random shape in z direction is placed parallel to the leading edge near the inflow plane of the domain. This disturbance is confined to a small downstream region and hence interacts locall with the boundar laer. Finall, we combine both disturbance sources to determine whether stead or unstead modes dominate the boundar-laer response. The following results are obtained on the short domain, bo S, sufficientl long to include receptivit, primar disturbance growth and nonlinear interaction even for the lowest amplitudes of the forcing disturbance Response to free-stream turbulence Figure 12 depicts the downstream evolution of the boundar-laer disturbance forced b free-stream turbulence with different intensities in the range Tu [.17, 5.6]%. Tu is defined b its value at the inflow plane. In figure 12(a) and (b) the wall-normal maimum of the rms of the chordwise fluctuation u is shown versus for increasing inflow turbulence intensities. The results pertaining to the lower turbulence intensities are reported in (a) and those for the higher free-stream turbulence levels in (b) with the curve obtained at lowest Tu repeated for comparison. Initiall, 1, the instabilit