Numerical Optimization and Wind-Tunnel Testing of Low

Numerical Optimization and WindTunnel Testing of Low ReynoldsNumber Airfoils y Th Lutz W Wurz and S Wagner University of Stuttgart D Stuttgart Germany Abstract A numerical optimization to ol has been applied to the design of low Reynolds number airfoils Re The aero dynamic mo del is based on the Eppler co de with ma jor extensions A new robust mo del for the calculation of short n transitional separation bubbles was implemented along with an e transition criterion and Drelas turbulent b oundarylayer pro cedure with a mo died shap efactor relation The metho d was coupled with a commercial hybrid optimizer and applied to p erform unconstrained high degree of freedom optimizations with ob jective to minimize the drag for a sp ecied lift range One resulting airfoil was tested in the Mo del WindTunnel MWT of the institute and compared to the classical RG airfoil In order to allow a realistic recalculation of exp eriments conducted in the MWT the limiting nfactor was evaluated for this facility A sp ecial airfoil featuring an extensive instability zone was designed for this purp ose The investigation was supplemented by detailed measurements of the turbulence level Toget more insight in design guidelines for very low Reynolds numb er airfoils the inuence of variations of the leadingedge geometry on the aero dynamic characteristics was studied exp erimentally at Re Intro duction For the Reynoldsnumber regime of aircraft wingsections Re sophisticated di rect and inverse metho ds for airfoil analysis and design are available In the hands of exp erienced users these metho ds allow a carefully directed airfoil design Usually sp ecic airfoils are b eing develop ed for any new aircraft in order to maximize the p erformance for the intended range of application Windtunnel tests mainly serve for verication of the predicted aero dynamic characteristics and are necessary means to determine the stall b ehav ior and to optimize turbulators or the ap settings Doubtless the total number of design lo ops was signicantly reduced due to the availability of reliable prediction metho ds This do es not hold for the airfoil design at low Reynolds numb ers Re In this regime transition usually takes place in a separated b oundarylayer if the clean airfoil is considered To predict the inuence of laminar separation of unsteady transitional separa tion bubbles and of the viscousinviscid interaction is still a challenge This is esp ecially true in the range of the critical Reynolds number where strong nonlinearities o ccur In general the uncertainty in the airfoil analysis increases with decreasing Re For this reason it is necessary to include the lessons learned from windtunnel tests in the design metho dology of low Reynolds number airfoils Extensive exp eriments physical interpretation of the results Research Assistant Institute for Aero dynamics und Gas Dynamics IAG Pfaenwaldring y Professor Head of Institute Institute for Aero dynamics und Gas Dynamics IAG Pfaenwaldring and the transfer into successively improved airfoil designs as p erformed e g by Selig et al is therefore the most promising approach Being aware of the ab ove mentioned problems an attempt was made to p erform direct nu merical shap e optimizations of low Reynolds numb er airfoils Re This approach is considered to be instructive for two reasons First the optimizer may nd unconven tional solutions which could inspire new ideas for an improved manual design Second weak points of the aero dynamic mo del can be identied if the optimized airfoils are examined in a windtunnel test This in turn may lead to improvements of the prediction metho ds The intention of the present optimizations was to p erform an unconstrained design of completely new airfoils rather than to mo dify existing ones This necessitates considering a large number of design variables in order to enable a detailed representation of the airfoil shap e and the pressure distribution Furthermore sto chastic optimization algorithms should b e applied to reduce the p ossibility of getting trapp ed in a lo cal optimum A high degree of freedom problem in combination with a sto chastic optimization strategy however requires a very large number of airfoils to be generated and analyzed until the optimization pro cess converges If the optimizations are to be p erformed with acceptable computation time a most ecient aero dynamic mo del is needed For the present investigations a p otentialow metho d coupled with an integral b oundarylayer pro cedure a simplied bubble mo del and n an e databasemetho d for transition prediction was applied Viscousinviscid interaction was neglected in order to minimize the computational eort which actually is roughly two orders of magnitude b elow an XFOIL analysis The complete airfoil optimization to ol has b een proven successful for the design of high Reynolds number Re NLF airfoils and is routinely applied for various design tasks In the following sections the ingredients of the metho d will briey be describ ed There after optimization examples of low Reynolds numb er airfoils including windtunnel verica tion will b e discussed Supplementary to this theoretical approach to airfoil design fundamental windtunnel tests for very low Reynoldsnumber airfoils Re will b e presented First detailed exp erimental investigations of airfoil characteristics for this Reynoldsnumber regime were published by Schmitz He proved that thin airfoils featuring a sharp leading edge have a lower critical Reynolds number than conventional airfoil designs In the exp eriments of Schmitz the cambered plate Go a shows the best p erformance up to Re The problem of how to provoke b oundarylayer transition for Re in order to avoid extensive laminar separation was examined by Hamma Among other designs a thick symmetrical airfoil with a sharp leading edge was tested in a water tunnel and showed promising results However very complex ow phenomena were observed For example laminar separation with laminar reattachment partly followed by a transitional separation bubble o ccurred for a sp ecic and Reregime Furthermore the separation angle abruptly increased for a distinct angle of attack Such complex ow phenomena cannot be predicted reliably with available theoretical mo dels The present authors therefore prefer an exp erimentbased approach with the intention to establish guidelines for the design of very low Reynoldsnumber airfoils In a recent test campaign the inuence of relevant geometric parameters like the leadingedge radius on the aero dynamic p erformance has been examined First results have been obtained and will b e discussed Aero dynamic Mo del For the present investigations the determination of the airfoil characteristics during the optimization pro cess is based on the airfoil design and analysis metho d of Eppler Ma jor extensions with resp ect to b oundarylayer computation were added to this to ol In the direct mo de the inviscid outerow is evaluated by means of a higherorder panel metho d whereas an inverse conformal mapping pro cedure is used in the design mo de Based on the inviscid velo city distribution the b oundarylayer development is calculated with a rstorder integral pro cedure The metho d is based on the numerical integration of von Karmans integral momentum equation and Wieghardts energy equation with the momen tum thickness and the energy thickness as dep endent variables For attached laminar ow algebraic closure relations are used which were derived from FalknerSkan selfsimilar proles in regions with adverse pressure gradient whereas solutions for the at plate with suction serve as a basis in accelerated ow regimes For the calculation of turbulent b oundarylayers the metho d prop osed by Drela was implemented in the Eppler co de with a new shap efactor relation H H Re With this approach Greens lagequation is intro duced as a third governing equation in order to account for history eects in case of nonequilibrium ows To close the problem Drela pro p osed an algebraic shap efactor relation whichwas derived from an evaluation of Swaords twoparameter b oundarylayer proles For very low lo cal Reynolds numbers Re and large values of H this relation yields H This b ehavior however should only o c cur with tangential blowing which is not considered in the present investigations Therefore an alternative empirical relation was derived which ensures H for any Re and furthermore satises the asymptotic condition H H for a rectangular velo city prole Reliable transition prediction is of essential imp ortance for successful design and opti mization of natural laminar ow airfoils For the present investigations the semiempirical n e metho d according to van Ingen and Smith Gamb eroni was implemented This approach takes advantage of the fact that the transition pro cess in subsonic Dows is typi cally asso ciated with the growth and breakdown of TollmienSchlichting waves TSwaves As long as the disturbance amplitude A is small enough the amplication can b e predicted n by means of linear stability theory The basic idea of the e metho d is that transition may be assumed when the most amplied frequency reaches a certain critical amplication factor n ln A A The value of n dep ends on freestream conditions the cr it cr it initial cr it receptivity mechanism and on the denition of the transition p oint Therefore n has cr it to be correlated individually for each windtunnel or for freeight conditions taking the numerical mo del used for the

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