Low Reynolds Number Airfoil Design Lecture Notes Michael S. Selig⁄ Department of Aerospace Engineering University of Illinois at Urbana{Champaign Urbana, Illinois 61801 USA VKI Lecture Series Sponsored by NATO Research and Technology Organization (RTO) Applied Vehicle Technology (AVT) Panel 24{28 November 2003 Abstract An approach to low Reynolds number airfoil design is described, and several example design cases are presented and discussed. The overall approach involves using the inverse method PROFOIL for design and XFOIL for analysis. Validation of these methods and the low Reynolds number airfoil design philosophy is supported by UIUC wind tunnel experiments. These notes derive largely from four prior publications of the author (see Refs. 1{4) and the contributions of the respective co-authors are gratefully acknowledged. 1 Introduction For over 100 years, airfoil design has continued to capture the interest of practitioners of applied aerodynamics. The fleld is fueled by the ever-growing combination of airfoil design requirements for unique applications, such as UAVs, and this state of afiairs is likely to continue. When one considers all possible permutations of the myriad of airfoil design re- quirements, it quickly becomes apparent that the number of unique sets of requirements far ⁄ Associate Professor, 306 Talbot Laboratory, 104 S. Wright St. email: [email protected]. Presented at the von Karman Institute for Fluid Dynamics, Lecture Series: Low Reynolds Number Aerodynamics on Aircraft including applications in Emerging UAV Technology, 24{28 November 2003. Copyright (cid:176)c 2003 by Michael S. Selig. Published by the von Karman Institute for Fluid Dynamics or NATO-RTO/AVT, with permission. 1 exceeds the collection of existing airfoils. For this reason, the advancement and use of meth- ods for airfoil design continues to be the economical solution. In contrast, the enrichment of airfoil \catalogs" for their own sake is felt to be of limited value. The objective of this lecture focuses on the design of airfoils for low Reynolds numbers, which has been the subject of considerable research as documented in several major con- ferences and books.5{9 First, various approaches to airfoil design are discussed, leading to the use of inverse methods being favored over the alternatives. Second, several sections of these notes outline the important in(cid:176)uence of laminar separation bubbles on low Reynolds number airfoils, leading to the need for an inverse method that has the unique capabilities of being able to more or less directly control the movement of transition with angle of at- tack. The desired behavior of transition forms the basis of a design philosophy that has been implemented in a methodology for inverse airfoil design. Finally, to illustrate the overall approach to low Reynolds number airfoil design, several example airfoils are presented in this lecture. In each case, state-of-the-art tools for airfoil design10,11 and analysis12{14 were used. Although these airfoils were each designed for speciflc applications, the systematic and parametric studies show useful performance trends and trade-ofis in airfoil design at low Reynolds numbers. As will be shown, the overall design process has been validated through wind tunnel tests, and these results are presented together with the predictions. 2 Various Approaches to Airfoil Design In this section, various approaches to airfoil design are brie(cid:176)y summarized. The alternative to our great legacy of airfoil design by geometric means guided by empirical study (Fig. 1a) is to use an inverse method, and there are certain advantages to be had by adopting the latter whilerealizingthatoftengeometricconstraintsmuststillbeachieved. Byadoptinganinverse approach, the degree to which the aerodynamic performance can be controlled has reached a high level of sophistication. Inverse design in the classic sense involves specifying a desired velocity distribution (Fig. 1b) based on boundary-layer and consequently performance con- siderations. Taking this one step further by directly prescribing the desired boundary-layer characteristics (Fig. 1c) is a step closer to controlling the desired outcome|the performance. Thus, employing an inverse boundary-layer-like approach can give the designer tremendous power in achieving the performance goals in the face of all the trade-ofis that one must consider in the process of airfoil design. Continuing the sequence in going from geometry based methods to inverse methods, the flnal step is one wherein the performance is speci- fled using an optimization scheme (Fig. 1d), but implementation of an e–cient optimization method has met with limited success relative to the other approaches. Nevertheless, each of these four approaches to design have their respective strengths of allowing more or less direct control over particular characteristics of airfoils. Each of these characteristics could be considered as a design variable (Fig. 1d) that ideally should be incorporated into a single design methodology. Apart from the design variables of choice, a second consideration involves having the ability to control the performance over multiple operating points. Figure 3 illustrates this concept. Typically, airfoil design requirements include information regarding C (point l;max C in the flgure) as well as the operating range over which low drag is achieved (points A 2 to B). These requirements can be translated into speciflc characteristics to be embodied in the pressure distribution. For instance, low drag at points A and B requires extended runs of laminar (cid:176)ow on the lower and upper surfaces, respectively, while the high lift require- ment is achieved by limiting the leading edge suction peak behavior, each of which must be achieved at the corresponding design lift coe–cient. Collectively this approach is referred to as multipoint design, which is clearly a desired feature of any airfoil design method. 3 Laminar Separation Bubbles and Transition Low Reynolds number airfoil (cid:176)ows are principally distinguished by their associated laminar separation bubbles such as that depicted in Fig. 4. In past research, considerable attention has been focused on laminar separation bubbles because they are the leading culprit to the degradation in performance relative to airfoils at higher Reynolds numbers. When laminar separationbubblesdoappear, theyarecausedbytheinabiltyofthe(cid:176)owtomakeatransition to turbulent (cid:176)ow in the attached boundary layer on the surface of the airfoil. Instead, the laminar (cid:176)ow separates before transition. When this happens, transition occurs in the free shear layer, and the so-called laminar separation bubble is formed when the turbulent (cid:176)ow reattaches to the airfoil surface downstream of transition. For the most part, the resulting pressuredragovertheregionofthelaminarseparationbubbleisresponsiblefortherelatively high drag that can sometimes accompany airfoils at low Reynolds numbers. The existence of a laminar separation bubble and its extents can be deduced by examining surface oil (cid:176)ow visualization as will be discussed later in these notes. As shown by Drela,15 the drag contribution owning to a bubble can be approximated by considering the integral boundary layer equation d(cid:181) (cid:181) du e = ¡(2+H) +2C f d» u d» e from which the drag coe–cient can be determined using 2((cid:181) +(cid:181) ) us ls C = d c For the special purpose here of considering the drag due to a laminar separation bubble, it is more helpful to express the integral boundary layer equation as 1 d(‰u2(cid:181)) C H du e = f ¡ e ‰u2(cid:181) d» 2(cid:181) u d» e e Inside the laminar separation bubble, the skin friction is nearly zero and hence it can be assumed that C ’ 0. In this case the integral boundary layer equation becomes f ¢‰u2(cid:181) ¢u e ’ ¡H e ‰u2(cid:181) u e e or ¢‰u2(cid:181) ’ ¡‰u –⁄¢u e e e 3 Thus, the drag increment due to a laminar separation bubble is proportional to the ⁄ product of the average mass defect ‰u – and drop in the edge velocity ¢u . A graphical e e interpretation of this result is shown in Fig. 5. Three edge velocity distributions are shown together with the drag increment that grows downstream. For case 1, transition is assumed tohaveoccurredontheairfoilsurfacewithoutthemechanismofalaminarseparationbubble. Prior to transition, the drag increases slowly, while downstream of transition the growth is more rapid and consistent with a turbulent boundary layer. For case 2, laminar separation is shown to have occurred as indicated by the plateau in the velocity distribution, which is accompanied by no growth in the drag increment until transition occurs. At transition and subsequent reattachment, there is a jump in the drag that continues to grow consistent with the turbulent boundary layer behavior. For case 3, a longer laminar separation bubble is considered. In this case, the larger drop in the edge velocity at transition gives rise to a correspondingly larger jump in the drag increment as compared with case 2. Clearly, between the two limiting cases of transition without a bubble (case 1) and transition at the end of a long bubble (case 3), there is an optimum where the bubble drag increment can be minimized. For most airfoils, this minimum case in relation to the total airfoil drag is associated with a thin laminar separation bubble that has only a small drop in the edge velocity at transition. From this discussion in connection with Fig. 5 that shows the drag increment through a bubble, the location of transition dictates the size of the bubble and with it the drag of the bubble. Thus, an airfoil designer focuses on the questions: where is transition, where should it be, and how can it be controlled? Taken together answers to these questions should yield more knowledge of airfoil (cid:176)ows and help in the development of a design philosophy. As a step in this direction, the following section outlines an approach to flnding the transition location, which is the flrst step in low Reynolds number airfoil design. 4 Finding Transition As will be seen, having knowledge of the transition location is key to the development of the low Reynolds number airfoil design philosophy presented later. In what follows, two tools for determining transition are discussed|flrst through experiments and then through predictions. 4.1 Tool #1: Wind Tunnel Tests All experiments were conducted in the University of Illinois at Urbana{Champaign (UIUC) subsonic wind tunnel (Fig. 6), which has a nominal test section that is 2.81-ft high and 4-ft wide. The test set-up depicted in Figs. 7 and 8 was used for this study.16,17 As seen in Fig. 7, two 6-ft long Plexiglas splitter plates are inserted 2.8 ft apart into the test section to isolate the airfoil models from both the support hardware and the tunnel side wall boundary layers. The 1-ft chord airfoil models were inserted horizontally between the splitter plates with nominal gaps of 0.040{0.080 in. between the end of the airfoil model and the splitter plates. Performance data were taken at Reynolds numbers of 100,000, 200,000, 350,000 and 500,000. The lift was measured using a strain gauge load cell, and the drag was determined using 4 the momentum deflcit method.16 To account for spanwise drag variations at low Reynolds numbers,18 the drag was obtained from an average of eight equidistant wake surveys over the center of the model so that a 10.5-in. wide span was covered. The overall uncertainty in both the lift and drag measurements was estimated at 1.5%.16,17 All lift and drag measurements were corrected for wind tunnel interference and validated with data from the NASA Langley Low Turbulence Pressure Tunnel.4,16,18{20 As has been well documented, low Reynolds number airfoil (cid:176)ows are highly sensitive to the tunnel (cid:176)ow quality. Consequently, tunnel (cid:176)ow quality measurements were taken and documented in detail in Refs. 4 and 21. Only a subset of those results, in particular the turbulence intensity measurements, are included in these lecture notes. The turbulence intensity was measured using hot-wire anemometry. Speciflcally, the hot-wire system was a TSI Incorporated IFA 100 anemometer in conjunction with a TSI Model 1210-T1.5 hot- wire probe. The probe makes use of a 1.5-micron platinum-coated tungsten wire. The probe was mounted in the tunnel end-(cid:176)ow orientation with the wire perpendicular to the tunnel (cid:176)oor in order to measure the axial turbulence intensity. A PC equipped with a data acquisition card was used to log the signal from the anemometer. A HP 35665A Dynamic Signal Analyzer, which performed a FFT (Fast Fourier Transform) analysis, was employed to allow the turbulence spectrum to be monitored over a broad range of frequencies. More details of the method are given in Ref. 21. The turbulence intensity was calculated from data using a total of 50,000 samples with a sample frequency of 10,000 Hz. Figure 9 shows the resulting turbulence levels for both the tunnel empty case and with the full measurement apparatus installed. In general these levels are considered to be su–ciently low for taking low Reynolds number airfoil measurements. The surface oil (cid:176)ow visualization technique made use of a (cid:176)uorescent pigment (Kent- Moore 28431-1) suspended in a light, household-grade mineral oil that was sprayed onto the surface of the model using a Paasche Model VL double-action airbrush. The model was then subjected to 20{45 min of continuous wind-tunnel run time at a flxed Reynolds number and angle of attack. During this period, the oil moved in the direction of the local (cid:176)ow velocity at a rate dependent on the balance of forces dictated by the boundary-layer skin friction coe–cient C and surface tension of the oil. As a result, regions of the (cid:176)ow could be f identifled and compared with the NASA Langley Low-Turbulence Pressure Tunnel (LTPT) data.19,20 Figure 10 shows a photograph of the surface oil (cid:176)ow pattern made visible under (cid:176)uo- rescent light. Figure 11 conceptually illustrates the connection between the salient surface oil (cid:176)ow features and the skin friction distribution. Note that the skin friction distribution, though conceptual, is consistent with the results of many computational studies.22{27 The authorsbelievethattheuniqueshapeoftheC distribution, inparticularthestrongnegative f C spike, has yet to be experimentally verifled (as no experimental data could be found); f however, the oil (cid:176)ow patterns observed seem to conflrm the validity of the negative C spike f concept. Several important (cid:176)ow features can be identifled and related to the underlying skin friction and surface tension forces. In Fig. 10, laminar (cid:176)ow is seen to exist from the leading edge to approximately 0:40c. The oil streaks are characteristically smooth in this region until laminar separation occurs, which is identifled in Fig. 11 as the point where C = 0. f (Note again that the (cid:176)ow shown in Fig. 11 is conceptual, and it is not intended to match 5 Fig. 10 in detail.) Downstream of the point of laminar separation, the original airbrushed \orange-peel" texture that existed prior to running the tunnel still exists, indicating that the (cid:176)ow is stagnant in this region. This stagnant (cid:176)ow is consistent with the known behavior of (cid:176)ow in the interior leading-edge region of a laminar separation bubble. As sketched, the C magnitude in this region is quite small due to the low (cid:176)ow speed and negative in sign f due to reverse (cid:176)ow at the surface. In the presence of a laminar separation bubble, transition takes place in the free shear layer above the surface of the airfoil. Downstream of this point, reattachment occurs in a process that is known to be unsteady as vortices are periodically generated and impinge on the airfoil surface.27,28 These unsteady vortices colliding with the surface lead to a relatively high shear stress that tends to scour away the oil at the mean reattachment point, pushing oil upstream or downstream of the reattachment point. As seen in Fig. 11, the reattachment line is less distinct because the bulk of the oil has been pushed away revealing the underlying black airfoil surface. In Fig. 10, the tunnel run time was long enough that the reattachment line at 0:58c is even harder to see than in Fig. 11. In the original high-resolution color photographs that were archived, this feature is clear and easily quantiflable. Downstream of reattachment the boundary layer is turbulent. The high skin friction in this area relative to the laminar boundary layer upstream tends to clear away more oil, again making the black surface downstream more visible than in the upstream region. The remaining visible feature of the (cid:176)ow is a line where the oil tends to pool, termed here the \oil accumulation line." This intrinsic feature of the oil (cid:176)ow has no direct connection to laminar(cid:176)ow, reverse(cid:176)owinthebubble, ortheensuingturbulent(cid:176)owdownstream. However, it does indicate a relatively important feature of the (cid:176)ow with regard to the nature of the skin friction in the vicinity of reattachment. The negative C spike shown in predictions and f sketched conceptually in Fig. 11 is most likely responsible for generating the oil accumulation line. Assuming that this is the case, the (cid:176)uctuating high skin friction that is generated over the unsteady reattachment zone will tend to push the oil upstream ahead of the mean reattachment point. At some location on the airfoil, however, the oil moving upstream will experience a balance of forces between the rapidly weakening skin friction force and that of the surface tension and oil adhesion that is retarding its motion. At the location where these two forces balance, the oil accumulates into a line that becomes the most distinguishable feature of the oil (cid:176)ow. Consequently, it is speculated that this (cid:176)ow feature is sometimes mislabeled as \reattachment" as will be discussed below. 4.2 Tool #2: XFOIL Predictions In the data presented here, XFOIL14 has been used as a post-design viscous/inviscid analysis tool. A linear-vorticity second-order accurate panel method is used for inviscid analysis in XFOIL. This panel method is coupled with an integral boundary-layer method and an en- type transition ampliflcation formulation using a global Newton method to compute the inviscid/viscous coupling, requiring less than a minute of elapse time per polar on modern desktop computers. For the current work n was set to the default value of 9, which crit is typical for a smooth wing surface in a low-turbulence environment. Each airfoil was represented in XFOIL using 230 panels distributed using XFOIL’s default paneling routine. XFOIL has proven to be well suited for the analysis of subcritical airfoils even in the presence 6 of signiflcant laminar separation bubbles. 4.3 Results XFOIL was used to predict the viscous pressure distributions on the E387 airfoil as shown in Figs. 12a{c. The characteristic pressure distribution caused by the presence of a laminar separation bubble is clearly seen. Oil (cid:176)ow visualization was performed to document the locations of the major surface (cid:176)ow features as presented in Figs. 13, 14, 15, and 16. These results were then compared with benchmark data and also XFOIL predictions. Figures 17 and 18 show the previously described (cid:176)ow features compared with benchmark data obtained at the NASA Langley LTPT. In the low drag range between ¡2 deg and 7 deg angle of attack, the agreement in the laminar separation line between the NASA LTPT and UIUC data sets is mostly within 0.01c to 0.02c, which is very near the uncertainty of the method. As previously discussed, the next feature to appear is the oil accumulation line. The UIUC oil accumulation line agrees fairly well with the \reattachment" line identifled in the NASA experiment. It is believed, however, that based on the previous reasoning this label given in the original reference20 is a misnomer. Had the UIUC tests been performed for a longer duration, the reattachment zone would be scoured clean with no remaining distinguishing feature, leaving only the oil accumulation line to be labeled as the \reattach- ment line," knowing that one must exist. Hence, it is speculated here and in prior UIUC work17 that such a scenario took place in the NASA study, i.e. the oil-accumulation line was misinterpreted as the reattachment line. Guided by this working assumption, the two results again are in good agreement. It must be stated, however, that the oil accumulation line might change slightly from one facility to the next since it is dictated by a force balance that depends on the skin friction forces of the boundary layer relative to the adhesion forces of the particular oil used. The predictions, however, show that the negative C region has a sharp upstream edge, which is most likely f where the oil accumulates regardless of the surface tension characteristics. Difierences in the oil accumulation line due to difierences in the type of oil used are therefore believed to be small. The good comparisons between UIUC and Langley data tend to support this assumption. Moving further downstream, the UIUC reattachment data is plotted, but unfortunately no direct comparison can be made because of the ambiguity with respect to the reattachment data reported in the NASA study. However, close inspection of the data suggests that at a Reynolds number of 300,000 and between 5 and 7 deg angle of attack, the LTPT line merges with the UIUC reattachment line. Perhaps in this case, the measurements at Langley were indeed the reattachment points. The conclusion to be drawn from this comparison of the oil (cid:176)ow visualization results is that the two facilities produce airfoil (cid:176)ows that are in close agreement. Moreover, if the arguments regarding the oil accumulation line are correct, then the agreement can be considered excellent and within the uncertainty of the measurements. To make comparisons with predictions, the upper-surface (cid:176)ow features for each airfoil were computed at Reynolds numbers of 200,000, 350,000, and 500,000 and at angles of attack from ¡2 to 18 deg using XFOIL14 (version 6.94, http://raphael.mit.edu/xfoil/). These data were then used to determine the point of laminar separation and reattachment, which are 7 bothdeflnedbyC = 0asillustratedinFig. 11. Acomparisonbetweenthesepredictionsand f experimental measurements are shown in Figs. 19, 20, and 21. The agreement is quite good, and this lends confldence to using XFOIL as an integral tool for analysis of low Reynolds number airfoils. 5 Design Philosophy As was discussed in Section 3, the performance of low Reynolds number airfoils is strongly dependent on the location of transition as that sets the length of the laminar separation bubble and consequently the magnitude of the drag rise attributable to the bubble. Thus, controlling transition is a key step towards mitigating the adverse efiects of laminar separa- tion bubbles on low Reynolds number airfoils. One common approach to controlling transition, in particular, promoting transition, is to employ an instability region in the pressure gradient or, as it is commonly called, a transition ramp. A general discussion of transition ramps can be found in Refs. 13 and 29, and additional details speciflc to low Reynolds number airfoils are discussed in Refs. 15, 30, 31, and 32. Figure. 22 shows a concept sketch of a transition ramp on the upper surface velocity distribution. A ramp can be used on either surface of the airfoil, but in these lecture notesattentionwillbefocusedontheuppersurface. Asecondapproachistouseaturbulator or trip on the airfoil surface, such a zigzag trip like that shown in Fig. 23. Various types of boundary layer trips have been used in the past, but for greatest e–ciency three-dimensional trips are believed to be more efiective.33 The remainder of these lecture notes is concerned with the former approach to promoting transition|that of using a transition ramp. Of critical importance in the design of low Reynolds number airfoils is the upper surface pressure distribution. The tendency of the (cid:176)ow to form a laminar separation bubble can lead to a signiflcant degradation in performance owing to the high bubble drag. To mitigate these adverse efiects, a transition ramp in the pressure distribution is often employed to gradually bring the (cid:176)ow to transition in a thin bubble without a large pressure rise and high drag associated with an otherwise thick bubble. Illustrating a transition ramp by means of showing the pressure distribution (Fig. 22) is instructive, but the essence of a transition ramp can be more precisely deflned by examining the movement of transition with angle of attack. Figure 24 shows a concept sketch of a polar and in relation to this the transition locations on both surfaces of the airfoil. As can be seen, over the low drag range, transition on the upper surface is in the vicinity of the midchord of the airfoil. As the angle of attack is increased, transition moves forward as the upper surface pressure gradient becomes more adverse. In these notes, the transition ramp as indicated in Fig. 24 will be deflned as the chordwise extent over which transition moves while the airfoil operates in the low drag range of the polar. Given this approach to deflning a transition ramp, consider the Wortmann FX 63-137 airfoil shown in Fig. 25 together with its inviscid pressure distribution. As can be seen, on the upper surface there is a gradual adverse pressure gradient that will promote transition as has been described. Figure 26 shows the measured locations of the major (cid:176)ow features with angle of attack, and in this form the extents of the ramp are more clearly deflned. As can be deduced from the flgure, transition (which occurs near the point of reattachment) 8 moves from nearly 80% of chord at low angles of attack to near 20% of chord at an angle of attack of near ’10 deg. The corresponding performance is shown in Fig. 27. The low drag range of the polar extends from C = 0:5 to C ’ 1:6 which correspond to angles of attack l l of ¡2 deg to ’10 deg, respectively. In contrast to the FX 63-137 airfoil, which was designed for low Reynolds numbers, the high Reynolds number design NASA NLF(1)-0414F airfoil was also examined. At high Reynolds numbers, the need for a relatively long transition ramp is diminished. Thus, the pressure recovery region can occur rather abruptly so long as transition occurs before recovery. For this reason, a relatively short transition ramp is often employed. On the NASA NLF(1)-0414F airfoil this ramp occurs at ’70% chord on the upper surface (see Fig. 28). At low Reynolds numbers with a steep pressure recovery occurring this far aft, a long bubble can be expected to occur. Indeed as shown in Fig. 29 laminar separation occurs relatively far aft, and at the lowest Reynolds number of 200,000 reattachment does not occur. For the two higher Reynolds numbers of 350,000 and 500,000, reattachment does occur on the airfoil. However, the resulting long bubble leads to a signiflcant degradation in performance as show in Fig. 30. The contrast between the FX 63-137 and NASA NLF(1)-0414F airfoils illustrates large performance difierences resulting from correspondingly large difierences in their respective transition ramps. Even small changes in the shape of the transition ramp, however, can have important efiects on performance as follows. The efiect of subtle changes in the transition ramp is demonstrated using two example airfoils A and B. Figure 31 shows a comparison of the geometries and inviscid velocity distributions. These airfoils were designed to each have a difierent transition ramp that is re(cid:176)ected in a difierent shape for the transition curve (C -x =c curve) on the upper surface. l tr The two airfoils were analyzed using XFOIL, and Fig. 32 shows the drag polars and upper- surfacetransitioncurvesforaReynoldsnumberof200,000. Aswasmentioned, thetransition ramp is deflned here as the region over which the bubble moves gradually as deflned by the transition curve. (In this context, the transition ramp might be more aptly called a \bubble ramp."31) From Fig. 32, it can be seen that airfoil A has lower drag than airfoil B at lift coe–cients from around 0:3 to around 0:7, above which value airfoil B has lower drag. Also noticeable is the correlation between the drag polar and the shape of the upper-surface transition curve. FortheC -rangefrom0:3{0:7, whereairfoilAhaslowerdrag, thetransitioncurveforairfoilA l isshallowerthanforairfoilB;thatis, thereisalargerchangeinthevalueofx =cforairfoilA tr than for B. For values of C from 0:7{1:2 where airfoil B has lower drag, the transition curve l for airfoil B is shallower than for A. This flgure shows that the steepness of the transition curve is a direct indication of the bubble drag. By adjusting the shape of this curve, it is therefore possible to tailor the drag polar of an airfoil at low Reynolds numbers. Figure 32 also includes an overlay of the variation of bubble size (x ¡x ) with C . The r s l size of the bubble for each C was obtained by determining the chordwise extent over which l the skin-friction C , as predicted by XFOIL, was less than or equal to zero. Studying the f bubble-size variation for the two airfoils further illustrates the connection between the shape of the transition curve and the bubble drag. The bubble is larger when the transition curve is steeper. Figure 33 shows the inviscid velocity distributions for airfoil A at C values of 0.5 and 1.0 l 9 with the upper-surface bubble location marked in bold. A similar plot for airfoil B is shown in Fig. 34. Comparing the velocity drops across the bubble for the four cases, it can be seen that while airfoil A has a smaller velocity drop than airfoil B at C = 0.5, the situation is l reversed for C = 1.0. Since the pressure drag due to the bubble increases with increasing l velocity drop across the bubble, airfoil A has smaller bubble drag at the low C and larger l bubble drag at the higher C . Thus, a steeper transition curve results in a larger bubble and l also a larger velocity drop across the bubble causing an increase in bubble drag. The connection between the transition ramp as deflned by the C -x =c transition curve l tr and the resulting performance forms the basis of the present low Reynolds number airfoil design philosophy. The philosophy is to prescribed the C -x =c transition curve realizing l tr that at most it can extend from the trailing edge to the leading edge. Between these two points (or simply the extent of the ramp) the airfoil performance can be further controlled by prescribingtheslopeofthecurve|theshallowertheslope, thelowerthedrag, andviceversa. There are however tradeofis that must be addressed as was illustrated in the prior example. The next step involves the implementation of this philosophy into a design methodology. 6 Methodology In this section, the philosophy of prescribing the C -x =c transition curve (which in efiect l tr is the transition ramp) is conveniently implemented in the inverse design code PROFOIL. First, however, general background on the method and its capabilities are presented, and following this its use in low Reynolds number airfoil design is described. The PROFOIL code10,11 embodies an inverse airfoil design method and an integral boundary-layer method for rapid analysis at the design points. The method draws on the pioneering work of Eppler12,13,34,35 in inverse airfoil design and analysis through conformal mapping (outlined in Figs. 35, 36 and 37) and integral boundary-layer techniques, respec- tively. PROFOIL difiers from the Eppler code in that laminar and turbulent boundary-layer developments can be directly prescribed through iteration on the velocity distribution. The method also allows for control over certain geometric constraints, such as the local geome- try, maximum thickness, thickness distribution, etc. Additional difierences are discussed in Refs. 10, 11, and 36. Both the boundary-layer and thickness-constraint capabilities are used in the examples presented in these lecture notes. A web-based version of PROFOIL and further discussion is available on the web.37 The general capabilities of PROFOIL are illustrated in Figs. 38, 39, and 40. In Fig. 38, three segments of an airfoil (to be designed) are specifled to include a velocity distribution, n-development (as used in the en method for transition prediction), and H development, 12 each to be achieved at difierent conditions. It should be noted that this is merely an example and is not intended to be use for any practical application. Figure 39 shows the solution that includes the relevant characteristics as prescribed, while Fig. 40 shows the flnal airfoil shape. For the low Reynolds number airfoil design problem, the design variable of choice is not the geometry nor the velocity, but instead it is the C -x =c transition curve as illustrated in l tr Figs. 31{34. In coping with this problem of in efiect prescribing the C -x =c transition curve, l tr a useful approach derives from an inherent feature of the Eppler theory for inverse airfoil 10