Experimental/Numerical Investigation of the Influence of a Pressure Gradient on Acoustic Roughness Receptivity in the Boundary Layer of a 2-D Airfoil

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1 Experimental/Numerical Invetigation of the Influence of a Preure Gradient on Acoutic Roughne Receptivity in the Boundary Layer of a 2-D Airfoil S. Herr, A. Wörner, W. Würz, U. Rit, S. Wagner Univerität Stuttgart, Intitut für Aerodynamik und Gadynamik Pfaffenwaldring 21, D-755 Stuttgart, Germany Summary In the preent paper, the influence of a preure gradient on the acoutic roughne receptivity i invetigated by mean of wind tunnel experiment and Direct Numerical Simulation (DNS). The receptivity function for three different location of a 3-D urface non-uniformity on a 2-D airfoil i calculated from the experimental and numerical data and compared to each other. The combined effect of the roughne acoutic receptivity and the amplification of the generated diturbance are taken into account in order to evaluate the poition on the airfoil at which a urface non-uniformity i mot dangerou in provoking an early tranition. Introduction The problem of the boundary layer tranition from laminar to turbulent flow till attract much attention becaue of it fundamental and practical importance. It ha three main part. Firt, there i the laminar flow receptivity to external (acoutic or vortical) perturbation. The econd part i the region where the boundary layer intabilitie develop according to linear tability theory. And finally, there i the nonlinear flow breakdown to turbulence. Thi paper i devoted to experimental and numerical invetigation of the linear 3-D acoutic receptivity of a 2-D laminar boundary layer in the preence of a localized (3D) quai teady urface non-uniformity. Preure gradient trongly influence the amplification of diturbance in a tranitional boundary layer. The location of tranition i not only determined by the amplification of the diturbance, but alo by their initial amplitude. A the initial amplitude itelf i determined by the proce of receptivity, it hould be of interet whether a large cale preure gradient may have an influence on the receptivity of a boundary layer to freetream perturbation. Thi problem ha been poorly tudied o far. Mot tudie conider a Blaiu flow a bae flow [8], [6]. The 3-D vibrational receptivity of a 2-D elf-imilar boundary layer with an advere preure gradient wa invetigated experimentally in [5], and recently the influence of a favorable preure gradient wa tudied in [4]. A numerical invetigation on the influence of the preure gradient on boundary layer receptivity can be found in [1]. In the preent tudy, we would like to determine the role a large cale preure gradient play with regard to the acoutic receptivity of a boundary layer in the preence of a mall cale urface non-uniformity on an airfoil. In an earlier experiment and accompanying DNS, the acoutic receptivity function of a 3-D roughne element on an airfoil wa determined quantitatively [13]. The receptivity function wa defined a the ratio of the initial (i.e. at the urface non-uniformity) amplitude of the intability wave, generated in the boundary layer, to the amplitude of the external acoutic perturbation and the amplitude of the correponding (localized) urface non-uniformity

2 eg av (α β)= e B int S (α β) ea ac e A v (α β) for every panwie and treamwie wave number of the developing TS-wave train a propoed in [3]. The urface roughne wa located at the point of neutral tability of branch I for the invetigated frequency (188 Hz). It i located at max = 2% on the ymmetrical airfoil (XIS4MOD) [11], figure 1. The boundary layer profile at thi poition (H 12 = 2:58) i cloe to that of the Blaiu boundary layer figure 2, and therefore the accompanying DNS could be performed for the Blaiu flow [9]. In a preliminary invetigation, two additional poition were choen for the preent tudy. The poition max = 15% correpond to a favorable preure gradient (figure 1). The poition max = 3% lie in the region of the intability ramp of the airfoil figure 1. DNS were performed for Falkner-Skan elf-imilar boundary layer. For the poition max = 3% a Falkner-Skan profile with Hartree parameter β H = ;:5 wa ued and for poition max = 15% a Hartree parameter β H = :5 wa ued ee figure 2. Numerical Approach The DNS are baed on the velocity-vorticity formulation of the complete Navier-Stoke equation for incompreible fluid. ω t ; rot(vω)= 1 Re ω with v =(u v w) and ω =(ω x ω y ω z ) All length are non-dimenionalized uing a reference length L and all velocitie uing the freetream velocity Ũ at the inflow boundary, where denote dimenional variable. A ketch of the integration domain i hown in figure 3. The flowfield i dicretized uing fourth-order accurate finite difference in treamwie (x-) and wall-normal (y-) direction, and a pectral method in panwie (z-) direction. The time integration i done uing a fourth order, four tep Runge- Kutta cheme. Detail concerning the numerical method can be found in [7]. A preure gradient i impoed on the boundary layer by precribing the treamwie velocity at the upper boundary of the integration domain. For the tudy hown here, thi i done uing the Falkner-Skan anatz for elf imilar boundary layer. Thi mean that the treamwie velocity at the upper boundary u e i et to: u e (x)=u x m with m = β H (2;β H ) If β H < the flow i decelerated, which mean that an advere preure gradient i impoed, wherea for β H > the flow i accelerated or a favorable preure gradient i impoed. The ound wave i modelled uing the analytical olution of the econd Stoke problem. Thi olution i added to the teady Falkner-Skan or, at zero preure gradient, Blaiu velocity profile, which i precribed at inflow. The mall roughne element within the equiditant carteian grid i modelled by extrapolating the velocity at the lowet row of grid point in uch a way a to fulfil the no-lip and the non-permeability condition at the urface of the roughne. Thi i done uing fifth order polynomial in y;direction which are conitent to the finite difference repreentation of the flowfield.

3 In contrat to the experiment, the interaction of a planar ound wave with a roughne element of one dicrete panwie wave number i calculated, and the complex receptivity function i evaluated for every panwie wave number eparately. For the 2-D cae, the evaluation of the receptivity function require four numerical imulation. In the firt imulation the teady flow over a flat plate with a roughne element located at a certain ditance from the leading edge i calculated uing the unteady Navier-Stoke equation tarting with the flow over a flat plate without roughne. In the econd imulation, the interaction of the ound wave with the roughne i calculated uing the previouly calculated teady flow a initial condition. The difficulty now i to extract the Tollmien-Schlichting(TS) wave, which i created by the interaction of the ound wave with the roughne, from the total olution. There are two problem. The firt one i that the ound wave itelf ha the ame frequency a the created TS-wave, and the econd i that there i a numerically created TS-wave reulting from an approximate inflow condition. Thee problem are olved uing a method uggeted by Crouch & Spalart [2]. Therefore, a third imulation i needed including the ound wave but no roughne. In thi imulation the TS-wave created at the inflow boundary i alo preent. So the TS-wave created at the roughne can be extracted from the total olution by ubtracting the reult of the third imulation from the reult of the econd one. Next, a Fourier analyi i ued to determine the amplitude and phae part of the TS-wave at every treamwie poition. The theoretical amplitude of the TS-wave at the center of the roughne element i found by matching the amplitude of the TS-wave created by the interaction with the amplitude of a TS-wave created in a fourth imulation (the o called reference imulation) by blowing and uction at the wall uptream of the roughne element. Thi can be een in figure 5, where the amplification curve of a TS-wave which i created by the interaction of a ound wave with a urface non-uniformity located at Re δ1 = 125 within an advere preure gradient region i hown together with it reference imulation. For the 3-D cae, the third imulation i not neceary becaue the TS-wave can be ditinguihed from the ound wave and the numerical TS-wave becaue of it different panwie wave number. Experimental Approach The hot-wire meaurement were carried out in the Laminar Wind Tunnel of the Intitut für Aerodynamik und Gadynamik [1]. The high contraction ratio of 1:1, a well a five creen and filter, reult in a very low turbulence level of le than 21 ;4. Downtream of the airfoil model a loudpeaker wa mounted in a treamline haped houing to inert the ound wave with a frequency f ac = 188Hz a external perturbation. Thi ound wave had a ound preure level of approximately 1dB in the tet ection, that i 16dB higher than the natural ound preure level for the frequency range 1Hz; 5Hz of the wind tunnel itelf. The roughne wa modelled by the deflection of the membrane of a vibrating ource [12]. The active diameter (6mm) wa choen to be approximately one half of the TS-wavelength (at f ac ) to allow a ignificant amount of amplitude at higher panwie wave number. The membrane i vibrating with a very low frequency compared to the acoutic frequency f v : f ac = 1 : 64. Thi lead to an amplitude modulation of the reulting TS-wave, which i viible in the pectrum a dicrete idepeak at frequencie f TS1 2 = f ac f v. The TS-wave at thee combination frequencie reult from the cattering of the ound on the urface vibrator only. They are eparated from the TS-amplitude at f ac which i contaminated by ignal from different ource (acoutic, probe vibration etc.). On the other hand, the ratio between the acoutic frequency and the frequency of the urface vibration i big enough that the roughne can be aumed quai tationary and the reulting receptivity function can be compared to DNS for tationary roughne. The hape

4 of the membrane wa meaured and double Fourier tranformed in treamwie and panwie direction in order to determine the complex wave number pectrum of the urface vibration at the vibrational frequency: ea v (α β)= Z ; Z ; ea m ( z)e ;i(α+βz) ddz : Therefore, the reulting receptivity function i independent of the actual hape of the roughne. The ratio between the frequencie were choen a integer power of two ( f v : f ac : f amplerate = 1:64:8) and the frequencie were trictly phae locked becaue they were ubdivided from a ingle quartz baed clock. Therefore it wa poible to ue both the amplitude and the phae part of the Fat Fourier Tranform of the hot-wire ignal for further data proceing. The meaurement procedure wa controlled by a PC and the reult were monitored online. The complete etup i ketched in figure 4. Detail on the etup, the meaurement procedure and data evaluation can be found in [12]. Two main et of data were collected, coniting of 13 repectively 5 can in panwie direction which covered the whole width of the TS-wave train developing downtream of the roughne. The time ignal at every meaurement point were analyzed with a FFT and the complex amplitude ea TS 1=2 raw( i z)=a TS 1=2 raw( i z)e iφ TS 1=2 ( i z) formed by the amplitude part (A TS 1=2 raw) and the phae part (φ TS 1=2) for the combination frequencie wa ued for further proceing. The amplitude were normalized by the local freetream velocity U δ. The phae were corrected in treamwie and panwie direction by adding multiple of 36 o. The wave train at the right combination frequency developing downtream of the ource at max = 15% i plotted in figure 7 and 8. The complex value of the wave train in phyical pace were mapped for each panwie cut to wave number pectra by the complex Fourier tranform Z eb TS ( i β)= ea TS ( i z)e ;iβz dz : ; After thi decompoition, the downtream development of wave with different panwie wave number can be followed eparately. Similar to the procedure ued in the DNS the initial amplitude at the poition of the roughne i found by comparion to a reference amplification curve. The reference in the experimental cae i the amplification a it i predicted by the linear tability theory. Thi i ketched in figure 6 for the wave number β = :269 1 mm and the roughne located at max = 15%. In the ame figure, the phae part for the ame wave number i plotted. The phae for all wave number can be interpolated by a traight line. The gradient of thi line correpond to the treamwie wave number α r of the TS-wave. A a reult, a dependence of α r on β can be obtained which i called the diperion dependence. The propagation angle can be found a θ = arctan β α r. From the double Fourier decompoition of the hape of the membrane, the vibrational amplitude that generate eigenmode of the boundary layer can be found along the diperion function. Reult The receptivity function a a function of the propagation angle θ i hown in figure 9 for the three invetigated treamwie poition of the roughne element. The big ymbol mark the

5 reult of the DNS which are in excellent agreement with the experimental reult. The velocity fluctuation (B ints A ac ) are normalized with the freetream velocity U δ at the poition of the roughne, and all linear dimenion (A v dz d) are normalized with the diplacement thickne δ 1 of the boundary layer at the poition of the roughne element which i indicated in figure 9. A a reult of thi normalization, the receptivity function depend on a local value of the boundary layer, the diplacement thickne δ 1. In order to be able to tudy excluively the dependence of the acoutic receptivity on the external preure gradient, DNS were performed where all other parameter, particularly the Reynold number Re δ1, baed on the boundary layer diplacement thickne, were kept contant at the location of the urface non-uniformity. In figure 1 the receptivity function G av for a 2-D roughne element located at Re δ1 = 125 i plotted veru the wall hear tre, which i non-dimenionalized by the freetream velocity and the diplacement thickne at the location of the roughne element. It can be een that for the cae tudied the acoutic receptivity depend only weakly on the external preure gradient. The wall hear tre at the roughne lie between :54;:62 which lead to a change in the receptivity coefficient of about 15%. Similar to the reult for vibrational receptivity, a favorable preure gradient lead to an increae in the acoutic receptivity wherea an advere preure gradient reduce it. Thi mean that the acoutic receptivity behave in revere to the intability which i decreaed by a favorable preure gradient and increaed by an advere one. Thi can be explained a follow: The wall hear tre du dy i a function of the Hartree parameter β H and increae if the flow i decelerated. The roughne enter the receptivity mainly by the fact that the teady part of the treamwie velocity U at the lowet row of grid point cale with h du dy (ee alo [1]). So thi i the reaon why the receptivity function for a roughne located at a fixed Re δ1 cale with the local wall hear tre. For practical application, it i of great interet to find the poition on the airfoil where a roughne with contant height reult in the highet TS-amplitude far downtream. If we aume a diturbance with A ac A v = 1 δ contant for every propagation angle θ, we find the initial TSamplitude a B int S = G av A ac A v. We conider a control poition far downtream of the 1 roughne at max = 45:2%. The N-factor from the roughne poition to thi control poition, N c, can be found in figure 1, and the amplitude at the control poition B TS 45:2% = B int S e N c are plotted in figure 11. The relation between thee amplitude reflect the relation of the amplitude reulting from a roughne of contant dimenional height placed at different treamwie poition cloe to the intability point of the airfoil. Figure 11 how that a roughne element placed at branch I of the neutral tability curve ( max = 2%) lead to earliet tranition. Concluion We have hown experimental a well a numerical reult concerning the influence of a preure gradient on the acoutic roughne receptivity in the boundary layer of a 2-D airfoil. The receptivity function evaluated experimentally for three different poition of the urface nonuniformity on the airfoil and numerically for Falkner-Skan flow with three different Hartree parameter agreed very well. The value of the receptivity function didn t change much when the poition of the urface non-uniformity on the airfoil wa varied. Thi i alo due to the fact that the receptivity function a it i defined here depend on the length cale that i ued for nondimenionalization. Since here the local diplacement thickne at the location of the roughne wa choen, the influence of the preure gradient on the receptivity could not be completely eparated from the influence of the growth of the boundary layer.

6 Therefore, DNS were performed where Re δ1 at the location of the roughne a well a the dimenionle frequency F were kept contant o that we were able to tudy only the influence of the preure gradient. Thi tudy howed that a favorable preure gradient increae the acoutic roughne receptivity wherea an advere preure gradient lightly reduce it, which i in accordance with increaed and decreaed wall hear tre, repectively. Thi influence i contrary to the influence of the preure gradient on the intability of the boundary layer. If we take into account that a combination of both effect determine the amplitude of the boundary layer diturbance far downtream, it turn out that the influence of the preure gradient on the acoutic roughne receptivity i much weaker than it influence on the boundary layer tability. Therefore the mot dangerou poition for a urface non-uniformity on an airfoil with repect to tranition i the location of branch I of the neutral tability curve. Acknowledgement Thi work wa performed under grant of the German reearch council (DFG) and i a contribution to the reearch program Tranition. Reference [1] Choudhari M., Lian N., Street C.L.: Acoutic receptivity due to weak urface inhomogeneitie in advere preure gradient boundary layer; NASA Technical Memorandum 4577, 1995 [2] Crouch J.D., Spalart P.R.: A tudy of non-parallel and nonlinear effect on the localized receptivity of boundary layer; J. Fluid Mech., pp , Vol. 29, 1995 [3] Gater M., Grant I.: An experimental invetigation of the formation and development of a wave packet in a laminar boundary layer; Proc. R. Soc. of London A; 347, pp , 1975 [4] Ivanov A.V., Kachanov Y.S., Bake S., Neemann K.: Influence of favorable preure gradient on 3-D vibrational receptivity of boundary layer; ICMAR Proc. Part 2, ITAM, Novoibirk, pp , 2 [5] Kachanov Y.S., Koptev D.B., Smorodkiy B.V.:3-D tability and receptivity of two-dimenional elfimilar boundary layer with advere preure gradient; Laminar-Turbulent Tranition, Proceeding IUTAM Sympoium, Ed. Fael H., Saric W., Berlin Springer-Verlag 1999 [6] Koorygin V.S., Radetztky R.H., Saric W.S.: Laminar boundary layer ound receptivity and control; Laminar-Turbulent Tranition Vol. IV, Ed. Kobayahi R., Springer 1995 [7] Rit U., Fael H.: Direct numerical imulation of controlled tranition in a flat-plate boundary layer J. Fluid Mech., pp , Vol. 298, 1995 [8] Saric W.S., Radetzky J.A., Hoo R.H. jr: Boundary layer receptivity of ound with roughne; Boundary Layer Stability and Tranition, FED-Vol. 114, Ed. Reda, Reed, Kobayahi, ASME, 1991 [9] Wörner A., Rit U., Herr S., Würz W., Wagner S., Kachanov Y.S.: Study of the acoutic receptivity of a Blaiu boundary layer in the preence of a urface non-uniformity; ECCOMAS 2, Barcelona [1] Wortmann F.X., Althau D.: Der Laminarwindkanal de Intitut für Aerodynamik und Gadynamik der Technichen Hochchule Stuttgart;Zeitchrift für Flugwienchaften Nr.12 Heft 4, 1964 [11] Würz W.: Hitzdrahtmeungen zum laminar-turbulenten Strömungumchlag in anliegenden Grenzchichten und Ablöeblaen owie Vergleich mit der linearen Stabilitättheorie und empirichen Umchlagkriterien; Diertation, Univerität Stuttgart, 1995 [12] Würz W., Herr S., Wagner S., Kachanov Y.S.: Experimental invetigation on 3-D acoutic receptivity of a laminar boundary layer in the preence of urface non-uniformitie; New Reult in Numerical and Experimental FLuid Mechanic 2, Ed. Nitche W., Heinemann H.-J., Hilbig R., Vieweg 1999 [13] Würz W., Herr S., Wörner A., Rit U., Wagner S., Kachanov Y.S.: Study of 3-D wall roughne acoutic receptivity on an airfoil; Laminar-Turbulent Tranition, Proceeding IUTAM Sympoium, Ed. Fael H., Saric W., Berlin Springer-Verlag 1999

7 U/U roughne control N-factor poition Re max = 1.2 * meaurement 1. poition / max 3 f [Hz] y/δ roughne poition / max = 15 % 2 % 3 % Hz α i U/U 1. δ Figure: 2 Boundary layer profile at the choen treamwie poition of the roughne / 1. max Figure: 1 Velocity ditribution and poition of the roughne in tability diagram y x z roughne element U Figure: 3 Integration domain for the DNS Figure: 4 Experimental etup at the Laminar Wind Tunnel log u max B ints -5.5 Figure: 5 imulation β H = -.5 reference imulation x roughne Amplification a calculated by DNS ln(b TS /A ac ) [-] Φ [ ] roughne [mm] 25 Figure: 6 Downtream development of amplitude and phae for β = :269 1 mm θ = 25o

8 z [mm] 6 amplitude right idepeak A TS,2 /U δ [%] z [mm] 6 phae right idepeak Φ TS,2 [ ] panwide can -4 panwie can [mm] [mm] Figure: 7 Amplitude part of the wave train roughne at max = 15% Figure: 8 Phae part of the wave train roughne at max = 15% G av [-].5 / max = 15 % δ 1 =.31 mm G av [-].5 / max = 2 % δ 1 =.356 mm G av [-].5 / max = 3 % δ 1 =.484 mm.4 Re δ1 = 66.4 Re δ1 =85.4 Re δ1 = DNS Θ [ ] Θ [ ] Θ [ ] Figure: 9 Receptivity function for roughne at different treamwie poition G av β H < β H > du/dy B TS(45.2%) roughne at / max : % 2. % 7 3. % Θ [ ] Figure: 1 Receptivity function for different Hartree parameter β H Figure: 11 TS-amplitude at control poition for roughne at different treamwie poition