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1 DNS of Point-Source Induced Transition in an Airfoil Boundary-Layer Flow C. Stemmer, M. Kloker, U. Rist, and S. Wagner Institut fur Aerodynamik und Gasdynamik, Universitat Stuttgart Stuttgart, Germany Abstract. Laminar-turbulent transition induced by a harmonic point source disturbance in a at-plate boundary layer with adverse pressure gradient is investigated by spatial Direct Numerical Simulation (DNS) based on the complete threedimensional Navier{Stokes equations for incompressible ow. A local disturbance is introduced into the two-dimensional (2-D) base ow at the wall by a harmonic point source. Thus Tollmien{Schlichting waves of a single frequency and a large number of obliqueness angles are stimulated and propagate downstream simultaneously, undergoing amplication by primary and subsequent instabilities, and eventually lead to breakdown of the laminar ow. The development of the wave train in the boundary layer is investigated by the spectral amplitude evolution and the vorticity/shearlayer dynamics. The computational aspects of this LAMTUR project are discussed in detail for runs on the NEC S-4 and the CRAY T3E supercomputers. 1 Introduction The investigation of laminar-turbulent transition has engaged many researchers for more than one century. During the last decades, DNS has become an increasingly reliable and powerful tool in transition research to supplement theory and experiment. This could be achieved by increased computer power and using advanced high-order numerics (for a basic review see [1]). To reduce the number of parameters, transition research has focused on so-called controlled transition with the excitation of only a few instability waves in the laminar boundary layer. A step further towards the understanding of more complex transition mechanisms is the involvementofnumerous oblique waves of, at rst, a single frequency. The harmonic point source generates by its spatial localiation a full spectrum of oblique disturbance waves with well-dened initial phase relations. Moreover, it can be considered as a model for the generation of disturbances by a localied surface roughness and a sound wave. The present study deals with a 2-D boundary layer present on an airfoil at considerable angle of attack including an extended region of adverse pressure gradient (APG). Studies on a Falkner-Skan boundary layer near separation ( h =-0.18) by Kloker [2] haveshown that, by exciting a 2-D and a pair of 3-D waves of identical frequency (so called K-type scenario), the instability waves periodically cause local separation cells in the boundary layer, precipitating ultimate breakdown of the APG laminar ow.

2 2 Stuttgart Transition Group The DNS results discussed in this paper were also compared with in- ight experiments carried out by four other German universitary groups (RWTH Aachen, TU Berlin, TH Darmstadt and University Erlangen) using wing gloves for a research motorglider [4]. For any details or results not reported in this paper see [3,4]. 2 Governing Equations The complete Navier-Stokes equations for incompressible, 3-D unsteady ow in the vorticity-velocity formulation are used (see [5,6,2]). The equations are solved in the so-called disturbance formulation. For this purpose, the system is split in a 2-D steady base ow (denoted by the index B) and the 3-D unsteady disturbance ow (denoted by a prime). Thus, all variables f=f B +f 0 are nondimensionalied using the reference values ~ L =0:065m, ~ U1 =38.9m/s in y-direction, a streching with p Re L =384.7 is performed. 3 Numerical Method The simulation is based on the spatial model with non-periodic inow-outow conditions in a rectangular integration domain (Figure 1) which includes a disturbance strip for the disturbance introduction. First, the steady 2-D base y u (x) e U ym δ(x) x0 x1 x2 x x3 disturbance strip xn = 0 = λ Fig. 1. Integration domain with disturbance strip ow is calculated using the Navier-Stokes equations with a prescribed streamwise velocity distribution at the upper boundary. Secondly, the disturbances are introduced and the 3-D disturbance ow is calculated. 3.1 Base Flow One steady transport equation for the spanwise vorticity component has to be solved together with two Poisson-type equations for the velocity components v B and u B (for more details see [3]). Fig. 2 summaries the relevant properties of the calculated base ow.

3 LAMTUR: Point-Source Induced Transition 3 Re δ DNS 3500 DNS + (0,0) 3000 BL- calculation H Fig. 2. Integral boundarylayer values Re 1 (based on Re δ1 4 displacement thickness) and shape parameter H 12 symbols: boundary layer cal- H 2 12 culation solid: steady 2-D DNS (base ow) dashed: unsteady 3-D DNS 3.2 Calculation of the Disturbance Flow For the solution of the unsteady Navier-Stokes equations, the set of three disturbance-vorticity transport 0 + (v0! 0 x ; u 0! 0 y + vb! 0 x ; ub! 0 (u0! 0 ; w 0! 0 x + ub! 0 + u 0! B )= ~! 0 x 0 ; (v0! 0 x ; u 0! 0 y + vb! 0 x ; ub! 0 (w0! 0 y ; v 0! 0 ; vb! 0 ; v 0! B )= ~! 0 y @x (u0! 0 ; w 0! 0 x + ub! 0 + u 0! B (w0! 0 y ; v 0! 0 ; vb! 0 ; v 0! B )= ~! 0 (3) have tobesolved. For the disturbance velocities u 0 v 0 w 0 three Poisson-type equations are 2 u 0 2 u 2 where the Laplacian ~ is dened as = ~v 0 x 2 w 2 w 0 0 y 2 v @ 2 2 (4) (5) (6) ~ = : (7)

4 4 Stuttgart Transition Group Boundary Conditions. At the upper boundary (y=y M ) in the potential ow, the disturbance vorticities vanish. For the disturbance velocity v 0 exponential decay at the upper boundary is prescribed. The no-slip condition for u 0 and w 0 is satised at the wall (y=y 0 ) v 0 is also ero except for the disturbance strip (Figure 3), where the disturbance function for the harmonic point source is prescribed. The function is designed such that, at any time step of the excitation cycle, no net mass ow isintroduced, see Figure 3. The disturbances are introduced well downstream of the inow boundary of the rectangular integration domain which permits to harmlessly force all disturbance variables to be ero at this boundary. As for the outow boundary conditions, the implementation of the welltested method of \articial relaminariation" [5] is applied, where basically the three disturbance vorticity components are slowly forced to a ero value in a damping domain ( x 3 x x 4, x 4 x N in Figure 1). 3.3 Computational Aspects In streamwise and wall-normal direction, fourth-order nite dierences are employed (alternating upwind/downwind/central for the streamwise convective terms), and the periodic spanwise spatial dimension is discretied using afourier spectral ansat with a number of (K+1) modes. The basic spanwise wavenumber for the fundamental Fourier mode (k=1) is related to the spanwise width of the integration domain through =2=, yielding =3.3 in the considered case. For saving computational eort, the symmetric initial/boundary conditions are exploited by calculating a symmetric ow eld with respect to =0. For the time integration, a fourth-order Runge-Kutta scheme is used (see, e.g. [2]). The spanwise Fourier ansat principally reduces the 3-D problem in physical space to a set of (K+1) 2-D problems in Fourier space thus enabling a largely parallel computation in Fourier space. However, the modes are coupled by the nonlinear convective terms of the vorticity transport equations amplitude U Z [λ ] Fig. 3. Normal velocity v 0 distribution across the disturbance strip over one spanwise wavelength

5 LAMTUR: Point-Source Induced Transition 5 and are transformed to physical space ("pseudospectral method" with dealiasing procedure) for the calculation of the nonlinear vorticity terms, which in turn are parallelied in streamwise direction. The uniform equidistant grid for the point source problem contains about (K=20-93) points in (x y )-directions (N, NY, 2K). The problem has been run on the supercomputers of the hww GmbH, Stuttgart, the NEC S-4/32 (PVP-type, 32 processors, 8 GB RAM) and the CRAY T3E-512 (MPP-type, 512 processors with 128 MB RAM each) for a high-resolution simulation with K+1=94 Fourier modes in spanwise direction. The code performance on a single processor on the NEC S-4/32 is about 3.5 s per gridpoint and time step on the CRAY T3E it is about 107 s. Hence, 30 processors of the MPP-machine have to make up for the vectoriation advantage of the NEC S-4/32 to reach the same computation speed. Here, the average vector length was 163. As for the K=20 problem on the NEC S-4 (1.75 GB memory requirement), the serial code reached 900 MFLOPS (of 2 GFLOPS theoretical peak performance) at a vector operation ratio of 98.9% after adapting the code to the specic features of the NEC S-4. The NEC S-4 parallel version reached a speed-up of over 9 (8236 concurrent MFLOPS) when computing on 11 processors parallel in everyday operation. The performance, however, depends heavily on the load of the computer. About 90% of the overall time was spent in parallel execution and only 10% of the total computation time for I/O, system-calls and serial program parts. An optimal adaptation of the code to the distributed-memory architecture of the CRAY T3E enforces modulo (N, K+1)=0 and a maximum x-ydomain sie requiring less memory than the RAM available on one processor (128 MB at present). Finally, a grid has been used for the high-resolution simulation, corresponding to a total used memory of 7.5 GB. We remark here that the memory overhead to match the distributed-memory architecture roughly is 80% in our case, i.e., 1.8 times the memory sie is required compared to a shared-memory machine. The speed-up of the parallel code on the CRAY T3E was 88.5 for the concurrent use of 94 Processors (including I/O, etc.). It is noted that the x-discretisation is the crucial point in resolution requirements. In the high-resolution case, the grid spacing was x=0164, y= and =105. The time step in this computation had to be reduced compared to the rst case due to a ner resolution in x. The number of time steps per fundamental disturbance period was 1000 in the nal simulation stages, and 700 for the standard case (the inow boundary for the high-resolution simulation has been shifted downstream compared to a rst run, using unsteady boundary conditions in the second run at the inow boundary extracted from the rst computation within the domain thus the number of grid points in x-direction is smaller despite the ner resolution).

6 6 Stuttgart Transition Group 4 Numerical Results The discrete waves will be presented in the frequency-spanwise wave number spectrum (h k), where the rst index gives multiples of the fundamental frequency, and the second of the fundamental spanwise wave number. Mode (1,0) corresponds to a 2-D wave, mode (1,6) has an obliqueness angle of approx. 45 and (1,20) represents a 3-D wave with an approximate angle of 77. The downstream amplitude development (u 0 -max over y) of the dierent modes for the basic frequency is shown in Figure 4 in a logarithmic scale. Fig. 4. Amplitude development of dierent modes in the frequency { spanwise wavenumber spectrum 4.1 DNS Data Analysis The behaviour of the wave train can be studied by means of Figure 5 which shows lines of corresponding phases of uctuations! 0 for a single (disturbance) frequency at the wall. The time signal was Fourier analysed over one disturbance cycle to obtain the Fourier phases of the uctuations. The left margin of the picture marks the downstream end of the disturbance strip. The boomerang-shaped phase distribution just after the introduction of the disturbances has qualitatively been observed, for instance, by Gilyov et al., Seifert & Wygnanski, both experimentally, and Mack & Herbert theoretically for ZPG ow (see [3]). Curved wave fronts with a straightened middle part are formed. 3-D waves with high wavenumbers are responsible for the regular pattern in spanwise direction beside the point source. These waves decline in amplitude due to the stability properties of the base ow (see [3]). Downstream of x =6:5, the typical wave train has fully developed and exhibits an half opening angle with respect to the centerline of 12, which is similar to the value observed in Blasius ow and is found in the joint experiments as well. The center region of the wave train straightens downstream of x 7:5.

LAMTUR: Point-Source Induced Transition 7 0.5 Z -0.5 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0-5.0 5.0 phase Fig. 5. Phaselines for the disturbance frequency =9:45 of!

7 LAMTUR: Point-Source Induced Transition Z phase Fig. 5. Phaselines for the disturbance frequency =9:45 of! 0 at the wall As the development becomes increasingly non-linear, this shape deformation becomes more pronounced (x >8:0). The sharply dened wave front shape dissolves for x>8:5 where the region of rapid nal breakdown begins (recall the increasing APG, Figure 2). The saturation of the amplitudes of the disturbance waves (Figure 4) can be observed at the same downstream location. To investigate the spanwise development of the disturbance intensity, the u 0 -rms amplitudes are shown in Figure 6. The development ofu 0 -rms on the centerline (=0) is surpassed by that at =95 and =0.19. The centerline u 0 uctuations are not the most intensive uctuations in the area of transition (x >8:0). The transitional regime (non-linear wave development), where disturbance waves of large obliqueness angles reach amplitudes beyond 1% u 0 =U 1 and detailed structures can be visualied experimentally, is limited to about one or two TS wavelengths. For the comparison with the experimental results, patterns of the instantaneous wall vorticity! distribution from the high resolution simulation with K=93 (which can be compared to hot-lm data from the measurements) are evaluated, delivering footprints of the structures in the boundary-layer ow (Figure 7). Very early (x 8:0), regions with negative vorticity (white areas) at and close to the wall appear indicating downstream traveling local separation ones. The areas of local separation periodically alternate with areas of very high vorticity (black areas) at the wall. The at center-part of the wave front deforms in the transition region (Figure 7a) and two areas of high shear o centerline (x=8.1, =0:25) accelerate compared to the surrounding structures at the wall (x=8.35 in Figure 7b). These highshear areas divide the negative shear area into a centerpart (the remainders Fig. 6. u 0 -rms (max. over y) for dierent spanwise positions ( =0:95b= =2)

$at the wall ( wall shear) a) at t = t 0 and b) at t = t 0 + T=2 of which can be found one time period later at x = 8:6 in Figure 7a, however, largely disintegrated) and two \legs" (white \streaks",$ x=8.4 and =0:25). These streaks can be associated with structures inside the boundary layer to be described below.

x=8.4 and =0:25). These streaks can be associated with structures inside the boundary layer to be described below.

8 8 Stuttgart Transition Group a) 0.5 b) ! x x -8 Fig. 7. Vorticity! at the wall ( wall shear) a) at t = t 0 and b) at t = t 0 + T=2 of which can be found one time period later at x = 8:6 in Figure 7a, however, largely disintegrated) and two \legs" (white \streaks", x=8.4 and =0:25). These streaks can be associated with structures inside the boundary layer to be described below. They fall back compared to the faster centerpart and persist for a surprisingly long time embedded in high-shear areas. The outer ends of the streaks (x=8.4 and =0:35, Figure 7b) remain visible in the ow pattern for one time period longer than the aforementioned structures. The rapid breakdown of the wave train leads to periodically generated turbulent spots rather than to an instant \broad" breakdown region in. The high-shear area expands in spanwise direction downstream of x = 8:5 but the anks align with the streamwise direction for x 8:8 and do not extend across the entire width of the integration domain. In conclusion, the harmonic point source can be regarded as a turbulent spot generator with well-dened initial disturbance conditions. A streamwise cut at the centerline reveals no distinctive structures associated with breakdown (mind the stretching of the y-coordinate by a factor of 8). Two shear layers are observed at x=8.4 and x=8.5 in Figure 8a. These layers (dark areas at an inclination of 45 ) intensify on their way downstream, get ejected towards outer regions of the ow and rapid breakdown occurs (compare Figure 9a half a fundamental time-period later at x = 8:5-8:6). The development o-center at =0.19 (= =0.1) reveals further details (Figure 9b). At this spanwise location, a high-shear layer develops much earlier and can be clearly distinguished at x=8.4 above an area of high-shear at the wall, the latter induced by ow reversal (early development can be seen as soon as x=8.25 in Figure 8b). It extends towards outer regions of the

LAMTUR: Point-Source Induced Transition 9 a) Y b) Y 1 5.0 1 5.0 8.0 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.0 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8-0.

contours at t = t 0 in the x-y plane a) at = 0 and b) o-center =0:19 boundary layer (y=6.

The virulent breakdown of this high-shear layer (Figure 8b, x=8.6) triggers the breakdown to a turbulent-spot train.

9 LAMTUR: Point-Source Induced Transition 9 a) Y b) Y *10 ω *10 ω -1 Fig. 8. Instantaneous! contours at t = t 0 in the x-y plane a) at = 0 and b) o-center =0:19 boundary layer (y=6.5) and inuences nearby spanwise locations as can be seen in Figure 9a at x=8.4. The virulent breakdown of this high-shear layer (Figure 8b, x=8.6) triggers the breakdown to a turbulent-spot train. 5 Conclusions a) Y b) Y *10 ω *10 ω -1 The downstream development of a harmonic point-source disturbance in an adverse pressure-gradient boundary-layer has been investigated up to late stages of the breakdown by direct numerical simulations. Using the NEC S-4 (or CRAY T3E) and 15 (38) Mega grid points, a computing performance of 8 (2.6) GFLOPS has been achieved using 11 (94) processors with a paral- Fig. 9. Instantaneous! contours at t=t 0 + T=2inthex-y plane a) at =0 and b) o-center =0.19

10 10 Stuttgart Transition Group leliation scaling factor of 0.82 (0.94). The memory requirement amounted to 1.75 (7.5) GB, with 80% overhead for the run on the CRAY T3E. The amplitude development of the disturbance waves composing the wave train and the detailed investigation of the vorticity structures indicate that the breakdown is not of the fundamental type (K-type). Rather, a form of multiple oblique breakdown is observed that shows a kind of peak/valley splitting with the valley plane at the centerline of the wave train. Velocity uctuations are larger for spanwise positions just o-center than in the centerline. The major development of vorticity structures takes place o the centerline. Maxima and minima (local ow reversal) of the spanwise vorticity! appear at the wall. A clear-cut M-shaped coherent structure can be observed in nal stages related to! at the wall. Inside the boundary layer close to the boundary layer edge, areas of high shear develop where the shear at the wall is at a maximum. The nal stage is rapid due to the strong adverse pressure gradient. Acknowledgments The authors wish to express their deepest gratitude to Dr. Horst Bestek who initiated this work and gave many helpful suggestions along its path of development. He was Head of the Transition and Turbulence Research Group at the IAG up to his death in September This research has been nancially supported by the German Research Council (DFG) under contract number Be 1192/7-4. References 1. L. Kleiser, T. Zang, Numerical simulation of transition in wall-bounded shear ows. Ann. Rev. Fluid Mechanics, 23, 495{537, M. Kloker and H. Fasel, Direct numerical simulation of boundary-layer transition with strong adverse pressure gradient. In R. Kobayashi (ed.), Laminar-Turbulent Transition, IUTAM-Symposium, Sendai, Japan, pp. 481{ 488. Springer, Berlin, Heidelberg, C. Stemmer, M. Kloker and S. Wagner, DNS of harmonic point source disturbances in an airfoil boundary-layer ow. AIAA , 29th AIAA Fluid Dynamics Conference, Albuquerque, New Mexico. 4. J. Suttan, M. Baumann, S. Fuhling, P. Erb, S. Becker and C. Stemmer, In- ight research on boundary layer transition { works of the DFG-University Research Group. In H. Korner and R. Hilbig (eds.), Notes on Numerical Fluid Mechanics, vol. 60, pp. 343{350. Vieweg Verlag, 1997, 10 th Stab Symposium 96, Braunschweig. 5. M. Kloker, U. Konelmann and H. Fasel, Outow boundary conditions for Spatial Navier-Stokes Simulations of Transition Boundary Layers. AIAA J., 31, 620{628, U. Rist and H. Fasel, Direct numerical simulation of controlled transition in a at-plate boundary layer. J. Fluid Mech., 298, 211{248, 1995.

UNSTEADY DISTURBANCE GENERATION AND AMPLIFICATION IN THE BOUNDARY-LAYER FLOW BEHIND A MEDIUM-SIZED ROUGHNESS ELEMENT

UNSTEADY DISTURBANCE GENERATION AND AMPLIFICATION IN THE BOUNDARY-LAYER FLOW BEHIND A MEDIUM-SIZED ROUGHNESS ELEMENT Ulrich Rist and Anke Jäger Institut für Aerodynamik und Gasdynamik, Universität Stuttgart,