Direct Numerical Simulation of Airfoil Separation Bubbles

Transcription

1 Direct Numerical Simulation of Airfoil Separation Bubbles Ulrich Maucher, Ulrich Rist and Siegfried Wagner 1 Abstract. In the present paper results of Direct Numerical Simulations of Laminar Separation Bubbles caused b streamwise increasing pressure gradient are shown. Small amplitude random disturbances are strongl amplified and therefore the flow shows unstead behaviour. The spatial growth of the disturbances and the corresponding velocit profiles derived b Fourier transform are compared with results of linear stabilit theor (so-called Tollmien-Schlichting waves (TS-waves)). Cases with and without artificial disturbances are considered. Two major time scales are observed. The first one is corresponding to the hdrodnamic instabilit (TS-waves), while the second, large one, is associated with an oscillation of the bubble and occurs in undisturbed cases. The damping influence of an upper boundar condition in the numerical domain with interaction between the viscous boundar laer and the inviscid outer flow (interaction model) on this large-scale effect is also shown. 1 INTRODUCTION When a laminar boundar laer is subjected to an adverse pressure gradient it will eventuall separate. As the separated boundar laer resembles a free shear laer which is known to be ver unstable with respect to a broad spectrum of disturbances the flow will most probabl become turbulent. Turbulence, in consequence, will entrain high-momentum fluid from the outer part of the separated flow towards the wall and will cause re-attachment in man cases. Such a phenomenon where a closed reverse-flow region eists at the wall is termed a Laminar Separation Bubble (LSB). The involved flow phsics are not et full understood. Since flight performance characteristics of laminar airfoils and compressor blades strongl degrade with the eistence of laminar separation bubbles there is great interest in further investigations. 2 NUMERICAL METHODS Here the computational techniques used for the present investigations are introduced. The calculations in this paper are based on two approaches, Direct Numerical Simulation 1 Universität Stuttgart, Institut für Aerodnamik und Gasdnamik, Pfaffenwaldring 21, 755 Stuttgart, German 1994 U. Maucher, U. Rist and S. Wagner ECCOMAS 94. Published in 1994 b John Wile & Sons, Ltd. (DNS), and Linear Stabilit Theor (LST) [6]. Both methods have alread been discussed in the literature. Therefore, onl the essentials are outlined here. To etract basic features of the DNS-results Fourier Transform is used. 2.1 Direct Numerical Simulation The calculations presented in this paper are performed with a numerical scheme originall developped b Fasel et al. [2], improved b Kloker [4] and more recentl b Kloker et al. [5]. The method is based on the complete Navier-Stokes equations in vorticit-velocit formulation for unstead, threedimensional, incompressible flow. Since the major changes of the code deal with features which are used to calculate the two-dimensional (2-D) component of the flow, this paper is restricted to 2-D flows. Consequentl, onl the equations for the 2-D components will be eplained in this section. The velocit components are denoted b u in streamwise () and v in wall-normal direction () (see figure 1). Figure 1. d.s. Integration domain for the DNS of a LSB; d.s.: disturbance strip. All variables are nondimensionalized with a reference length L and the free stream velocit U 1. The nondimensional variables relate to the corresponding dimensional ones (denoted b bars) as = L, = ȳ p Re, t = t U1 L L, u = ū, v = v p (1) U Re, Re = 1L U 1 U, 1 f

2 where is the kinematic viscosit and Re is the Renolds number. The vorticit is defined ; (2) In contrast to the disturbance-flow formulation used b Kloker [4][5] (i.e. the flow is split into a stead base-flow and an unstead disturbance-flow), now the total-flow formulation is used, since a solution for a stead base flow cannot be obtained due to the high amplification rates of random disturbances in the separated flow considered here. Thus the vorticit-transport equation (3), the Poisson-equation (4) and the equation for continuit (5) have to be solved to obtain!, v, and @ (v!) = 2! 2! 2 (3) 2 v 2 v 2 @ (5) For discretisation of the governing equations in streamwise and wall-normal direction, central finite-differences of fourth-order-accurac are used. The integration in time is performed with a fourth-order accurate Runge-Kutta scheme. The four stages per time step of this scheme are coupled with a centered-upwind-downwind-(and vice versa)-centered discretisation for the -convection term. Optionall, the flow can be disturbed b time-periodic suction and blowing within a so called Disturbance Strip on the surface of the plate (figure 1). In that wa, TS-waves with prescribed frequenc and amplitude can be generated (see [5] for details). In -direction the integration domain etends from to f and covers a certain region of the boundar laer flow over an airfoil including a laminar separation bubble, while in -direction the domain tpicall covers approimatel 1 displacement thicknesses at the inflow boundar. At the free-stream boundar a streamwise pressure gradient is imposed b prescribing the free-stream velocit distribution u e() of the eternal flow, assuming inviscid flow. For the simulations discussed in this paper u e() is chosen to approimate the potential flow distribution from s/c=.215 to s/c=.578 on the upper side of an FX66-S-196 airfoil at 9 degrees angle of attack (see figure 2a), as obtained from an eperiment. The vorticit is set to zero. At the inflow boundar, stead flow is assumed. An approimated Hartree-parameter H calculated from the local slope of the u e velocit at the inflow boundar is used to calculate Falkner-Skan profiles. At the outflow boundar a new condition based on the so-called Relaminarisation Zone technique, a special buffer domain developed and etensivel tested b Kloker et al. [5] is introduced. Now, the total 2-D vorticit is artificiall damped to b multipling it with a function which smoothl decas 1.8 u e 1.4 f,ia a c f 1.5 u w. a c Figure 2. a) Velocit distribution (u e) at the free-stream boundar. a denotes the beginning of the region with acceleration at the wall and c constant u velocit. The dashed line shows the data with interaction (ia) at the upper boundar, crosses mark eperimental data. b) Velocit distribution (u w) at the wall. from 1 to near the outflow boundar. In addition, the freestream u-velocit is set to a constant value and the velocit at the wall is accelerated to its free stream value (see figure 2b). This leads to uniform and irrotational flow throughout the boundar laer. This new method has been rigorousl tested for spatiall evolving TS waves in Blasius flow ( 3), its applicabilit for laminar separation bubbles is shown in the present paper ( 4). To take the displacement effect of the separation bubble on the potential flow into account, a model for viscous-inviscid interaction developed b Gruber [3] based on thin-airfoil theor can be used optionall. The v-velocit v e at the free-stream boundar is split into an inviscid component V and a part v c related to displacement v c = v e ; V = ve + e du d, (6) where V has been replaced b the prescribed potential flow distribution U using the continuit equation (5). U in the present case coarsl approimates the potential flow over an FX66-S-196 airfoil (figure 2a). Now u e can be updated from Z r u e = U + 1 v c d, (7) = l ; where l, and r are the left, and the right boundar, respectivel, where displacement effects of the boundar laer and especiall the bubble are considered. The code neglects curvature of the airfoil since the curvature radius is ver large compared to the boundar laer thickness. To start the simulation, initial values for the whole flowfield are needed. These are taken either from the solution of a boundar-laer calculation, or from the averaged values of a previous simulation with the same boundar-conditions. After a period of transient behaviour the appearance of a quasi-periodic state of the flow is observed. a) b) 2 U. Maucher, U. Rist and S. Wagner

3 2.2 Linear Stabilit Theor In the present paper unstead flows are investigated. The major interest focuses on the development and the profiles of the disturbances. For the validation of the DNS-results comparisons with results of Linear Stabilit Theor (LST) can be used. For the LST calculations local velocit profiles of a stead base flow are needed. As for the simulations discussed in this paper no stead base flow eists, the stabilit calculations are performed for the mean base flow profiles generated b averaging the DNS velocit components over a long time. The local streamwise variation of the base flow is neglected in the theor. Thus uniform stead 2-D base flow is assumed locall and the profiles depend on onl. Disturbances of the form v (,,t) =A()e ;i cos (r ; t ; Θ()) (8) are considered, where v, A, i, r,, and Θ denote a velocit-component of the wave, amplitude, spatial amplification rate, streamwise wave number, frequenc, and phase, respectivel. Negative i means amplification in -direction. This ansatz is inserted into the Navier-Stokes equation. Linearisation then ields the so-called Orr-Sommerfeld equation. From the solution of the Orr-Sommerfeld equation eigenfunctions (disturbance velocit profiles) and amplification rates for given base-flow velocit profiles, Renolds number, and frequenc, are obtained. 2.3 Fourier Transform Fourier analsis ields the comple amplitudes of the frequenc spectrum of a periodic signal [1]. Since all DNS cases presented in this paper are unstead, an important attempt to understand and validate results is Fourier Analsis. The following discrete ansatz definesthe Fourier Transform (FT) that is used X L;1 n F(,, L t )= t k= ;i 2nk f (,,t 1 + k t)e L, (9) where f, t, t 1, F, L, n denote the waveform to be decomposed into a sum of sinusoids, the time step, the initial point of the analzed time window, the comple Fourier amplitude, the number of discrete points (in time) and an integer counting the frequenc as follows: = 2n L t. (1) If a function is not periodic or if the arbitrarl chosentime window L t does not consist of an integer multiple of the period, noise is introduced into the spectrum b the FT. This is particularl apparent if there is a large discrepanc between the starting and end point of the interval. Especiall in strong unperiodic DNS cases it is often difficult to determine a time window in such a wa, that results of FT are meaningful and independent of the choice of the time window. Using a Hanning window to modif the time signal before appling FT ma help to reduce such undesirable effects: f h (t)=f (t) h.5 ;.5 cos 2 t ; t1 t 2 ; t 1 i. (11) 3 VERIFICATION OF THE PARALLEL-OUTFLOW CONDITION To validate the parallel-outflow condition, test calculations for the well known Blasius flow with artificiall ecited disturbances have been performed. The base flow is described b Re =1 5, U 1 =3 m m2 s, = 1.5 1;5 s, L =.5m. The frequenc of the disturbances is t = 11. The spatial development of the ecited small-amplitude TS-waves is compared with results of LST in figure 3. The amplification rates for the u-velocit maimum normal to the wall derived from DNS and the respective results from LST are shown. Ecellent agreement between both approaches is observed. There is almost no upstream influence of the parallelization zone. Even at the beginning of the acceleration of u at the wall, the amplification rates agree well. Since amplification rates are ver sensitive even to slight variations of the base flow, the results demonstrate the suitabilit of the parallel-outflow boundar condition. 2. i. ;2. Figure 3. d.s. DNS LST Amplification rates ln (u ma) compared with LST for Blasius flow. 4 RESULTS FOR FLOWS WITH SEPARATION So far, onl 2-D simulations for flows with strong adverse pressure gradient and a separation bubble have been performed with the new code. The velocit distribution u e prescribed at the free-stream boundar is taken from an eperiment for a FX 66-S-196 airfoil at 9 degrees angle of attack and a chord-renolds number of Re= (see figure 2a). A case with fied velocit u e at the upper boundar and without artificial disturbances is investigated etensivel. Furtheron the influence of the IA-model at the upper boundar and the development of artificial disturbances is briefl presented. All DNS are performed using Re=1.51 5, U 1 =3.28 m s, ;5 m2 = s, L =.743m. The integration domain etends from = (c/s=.215) to f = (c/s=.578; in a 3 U. Maucher, U. Rist and S. Wagner

4 cases with interaction f = 4.977, c/s=.498) with 722 (562) grid points and from = to =48.19 with 97 grid points ( = ;3, =.52). In the case of artificiall ecited disturbances the frequenc of the disturbances is chosen to be t = Results for fied u e, self-ecited disturbances The simulation was started using the averaged flow fluid data from a previous simulation as initial condition. The initial transient stage of development was terminated at approimatel t=16.76=t tr. Fort>t tr a quasi-periodic behaviour of the flow field was observed, as discussed subsequentl. An overall impression of the flow-field at t ; t tr=2.72 can be obtained from figure 4, where lines of constant velocit component u are shown. Although stead inflow and freestream boundar conditions are used, the flow ehibits largeamplitude unstead disturbances and remains unstead for all times. Due to the parallel-outflow condition at the end of the integration domain, all fluctuations of the u-component disappear. A distinct separation bubble is visible through the reverse flow region. Downstream of the bubble, strong variations of u in streamwise direction are apparent. Since there are no artificial disturbances, these fluctuations indicate the appearance of self-ecited wave packets, possibl due to the strong instabilit of the separated flow Figure a Fied u e: u = -.4, -.2 (dashed), (dotted),.2,.4,..., 1.6 (solid). In figure 5 the instantaneous time signals for the vorticit at the wall at different downstream positions are plotted versus t ; t tr. In addition to the high-frequenc disturbances corresponding to the wave packets in figure 4, a low frequenc oscillation appears, especiall in the region near the inflow boundar (i.e. at =3. and =3.5 in figure 5). The low frequenc oscillation lets the separation bubble periodicall appear and disappear (note that negative vorticit at the wall corresponds to reverse flow, i.e., separation). The highfrequenc oscillations are compared with results from LST in the following section.! wall.5 ;.5.1 ;.1 1. ;1. 1. ;1. Figure 5. t ; t tr Fied u e: Time signals at different downstream positions (=3. (top), =3.5, =4., =4.5) Comparison with LST Results from a Fourier analsis of the wall-vorticit time signals as plotted in figure 5 are shown in figure 6. Starting at =3. a remarkable amplitude is observed onl for ver low frequencies. However, the frequencies about =23 and =3 eceed the surrounding amplitudes. Further downstream (=3.5) a frequenc hillock appears between 2 and 32 with amplitudes that are more than two magnitudes larger than other frequencies (ecept the ver low ones). The large-amplitude frequenc-band strongl grows in downstream direction and as it gains large amplitudes, other frequencies are also strongl amplified (=3.75, =4., =5.). log 1! wall ;2 ;4 ;6 ;8 ;1 3 6 Figure 6. =3. =3.5 =3.75 =4. =5. Fied u e: u amplitude-spectrum. The amplification of each frequenc component can be determined b calculating its amplification rate u ln u e which in turn is comparable with LST. In the present case, however, due to the self-ecited nature of the disturbances, these amplification rates show large-amplitude random oscillations, and a comparison of i with LST is difficult. We therefore show a comparison of u -velocit profiles in figure 7. The time window used to generate the u -profiles as well as the av- 4 U. Maucher, U. Rist and S. Wagner

5 eraged base flow for LST etends from t ;t tr=2.94 to The comparison is performed for the frequenc =18. Initiall (=3.) the disturbance amplitude is ver small (O(1 ;8 )) and the DNS profile resembles more to the base flow profile than to the LST-eigenfunction. With increasing, however, the disturbance profile continuousl changes from its initial shape towards the shape of a TS-wave. Especiall at =4. the agreement with LST is ver good. Further downstream, the profiles start to differ, but this can be attributed to nonlinear effects due to the large disturbance amplitude of the waves in the DNS which in addition to changing the disturbance profiles feed energ into higher harmonics (waves with a multiple of the fundamental frequenc). 1 A =18 =18,LST =3. = Figure 7. base flow =3.5 =5. Fied u e, self-ecited disturbances: u -profiles. u A denotes. u,ma The phase Θ of the frequenc =3 of the vorticit at the wall versus is plotted in figure 8. The analzed time interval corresponds to the interval shown in figure 5. Θ ; f Figure 8. Fied u e, self-ecited disturbances: Phase Θ of the disturbance mode =3 of the vorticit at the wall. There are three different zones. Near the inflow boundar there are onl slight variations of the phase with a large wavelength. This can be eplained with the low frequenc oscillation of the bubble. Figure 5 shows that the phase of the low-frequenc oscillations differs at different -positions (=3. and =3.5). In the domain where TS-waves are dominant the phase has the tpical shape for propagating waves (continuousl decreasing). Near the outflow boundar the TSwaves are strongl damped and consequentl the phase approaches a constant value, which is different from the value at the inflow boundar. Therefore, pressure disturbances which possibl could be generated b the outflow-boundar and then propagate upstream with infinite velocit (in incompressible flow i.e. constant phase) seem not to have remarkable impact on the development of disturbances in the present code Low frequenc bubble oscillation In the region between =2.8 and =3.7 the flow periodicall separates and reattaches. The condition with fied u-velocit at the upper boundar is not able to take displacement effects caused b the bubble into account. In real flows the displacement effect of a growing bubble leads to an acceleration of the outer flow and reduction of the adverse pressure gradient and consequentl to a decrease of the bubble. On the other hand this decelerates the outer flow and promotes separation. Both adverse effects stabilize each other. Regarding the long-time behaviour it seems that the bubble oscillations slowl deca. The simulation has been stopped at t ; t tr=23.4 and it is not certain if the low frequenc oscillations would completel disappear for larger simulation time. Effects comparable to the bubble oscillation in the present simulation have been observed in previous 2-D simulations of flat-plate separation bubbles [3] or backward facing steps in channel flow. Possibl there is no solution for the stead base flow for some 2-D problems. 4.2 Interaction at the upper boundar, self-ecited disturbances In order to suppress the low-frequenc bubble oscillation the effect of an alternate upper boundar condition, which considers displacement effects b the bubble was investigated. A condition satisfing all epectations could not be found, et. The interaction model b Gruber [3] calculates the u e component b integration in a large -domain (see eq. (7) ) and therefore allows for upstream influence of disturbances which are strongl damped in real flow in upstream direction. This leads to an ecitation of disturbances with moderate amplitude at the upper boundar. Nevertheless, calculations using an interaction model show encouragingresults since the low-frequenc oscillation of the LSB is damped. Figure 9 shows the amplitude spectrum of the vorticit at the wall. When the interaction model is used, a broad spectrum of disturbances appears in the flow near the inflow boundar (=3.). In downstream direction some frequencies are amplified and a hillock appears in the spectrum for frequencies 15 <<32 at =3.5. Higher frequencies are still not amplified in this region. As the fundamental disturbances grow to ver large values all frequencies are strongl amplified and gain large amplitudes (=3.75 and =4.). 4.3 Interaction, artificiall disturbed In order to check the accurac of the present DNS b further comparisons with LST, a case with artificial disturbances has 5 U. Maucher, U. Rist and S. Wagner

6 log 1! wall ;2 ;4 ;6 ;8 ; Figure 9. =3. =3.5 =3.75 =4. Interaction, self-ecited disturbances:!wall amplitude-spectrum. profiles differ even more from the LST profiles. The disturbance amplitudes upstream of the position where the oscillating bubble originall appeared are now larger since the initial amplitude is comparabl high. The propagation of energ towards the wall caused b the artificiall ecited disturbances obviousl supresses the process leading to the oscillating bubble. The spectrum (see figure 11) shows the epected shape with strong peaks at the ecited frequenc and its higher harmonics. On the other hand, it is surprising that at an -position the amplitudes of the higher harmonics eceed other neighbouring frequencies b far, even when the amplitude of the fundamental wave is comparabl low and amplification of the higher harmonics b nonlinear interaction does not take place (=3.). Since the unstead effects are dominated b the ecited frequenc and its higher harmonics for all it is ver likel that, for the present case, the IA-model propagates discrete frequencies (here the higher harmonics) in upstream-direction. been computed. An artificial disturbance with a frequenc of = 3 and an amplitude of u ma=1 ;4 was ecited at the disturbance strip located at A base flow =3 =6 b.f. =2.66 = Figure 1. =9 =12 b.f. b.f. =3.16 =3,LST Interaction, artificiall disturbed: u -profiles. u A denotes. u =3,ma =4.17 Velocit profiles for the u -component are plotted in figure 1, and compared with the eigenfunctions obtained from LST-analsis for the time-averaged mean-flow profiles at four different -positions. Ever plot contains the profiles of the disturbance wave, its higher harmonics (=6, 9, 12), the LST-profile, and the local base flow. The first position (=2.66) is located in a region where the flow is still slightl accelerated and the LST and the DNS profiles fit ver well. At the second position (=3.16) the are ver similar with little differences onl. These are supposed to be due to the strong non-parallelism of the base flow caused b the deceleration (which is neglected b LST due to the assumption of a locall parallel base flow). In addition, a nonlinear evolution of the disturbance has started: the higher harmonic wave =6 is amplified and visible in the picture with ver low amplitude. Moving further downstream (=3.66, =4.17), the DNS log 1! wall ;2 ;4 ;6 ;8 ; =3. =3.5 =3.75 =4. Figure CONCLUSIONS Interaction, artificiall disturbed:!wall amplitude-spectrum. The results presented in this paper demonstrate that the numerical method can be applied to boundar laer flows with strong adverse pressure gradients including laminar separation bubbles, which, for eample, appear on laminar airfoils. The initial 2-D development of TS-waves can be predicted ver well. Nevertheless, 2-D simulations neglect a lot of basic features of separated or strongl decelerated flows. The growth of 3-D disturbances has major impact on these flows. Therefore 3-D calculations will be necessar to get better insight into the flow phsics of transitional airfoil flows. 6 U. Maucher, U. Rist and S. Wagner

7 ACKNOWLEDGEMENTS This research is supported b the Deutsche Forschungsgemeinschaft (DFG), Bonn-Bad Godesberg, FRG, under contract Ri 68/1-1. REFERENCES [1] E. Oran Brigham, THE FAST FOURIER TRANSFORM, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1 edn., [2] Hermann Fasel, Ulrich Rist, and Uwe Konzelmann. Numerical investigation of the three-dimensional development in boundar laer transition, 199. [3] Karl Gruber. Numerische Untersuchungen zum Problem der Grenzschichtablösung, Dissertation, Universität Stuttgart. [4] Markus Kloker. Direkte Numerische Simulation des laminarturbulenten Strömungsumschlages in einer stark verzögerten Grenzschicht. Dissertation, Universität Stuttgart, [5] Markus Kloker, Uwe Konzelmann, and Hermann Fasel. Outflow boundar conditions for spatial Navier-Stokes simulations of transition boundar laers, [6] L. M. Mack. Boundar-laer stabilit theor, U. Maucher, U. Rist and S. Wagner