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Infinite Swept-Wing Reynolds-Averaged Navier-Stokes Computations with Full En Transition Criterion

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Infinite Swept-Wing Reynolds-Averaged Navier-Stokes Computations with Full En Transition Criterion 27TH INTERNATIONAL CONGRESS OF THE AERONAUTICAL SCIENCES INFINITE SWEPT-WING REYNOLDS-AVERAGED NAVIER-STOKES COMPUTATIONS WITH FULL EN TRANSITION CRITERION Zhang Kun, Song Wenping National Key Laboratory of Science and Technology on Aerodynamic Design and Research, School of Aeronautics, Northwestern Polytechnical University, Xi’an, China [email protected] [email protected] Keywords: full en method, transition, three- dimensional RANS, swept wing, linear stability Abstract accuracy of boundary layer heat conductivity n will reduce at least 25% [1]. A full e transition prediction method is coupled On the other hand, the method which can to the three-dimensional Reynolds-Averaged accurately predict the transition location is one Navier-Stokes (RANS) solver to predict the of the key technologies for designing the natural transition point automatically during the laminar flow wing. In order to improve the simulation of the flow around the infinite swept performance of the aircraft and to reduce air wings. The three-dimensional linear stability pollution during the cruise, the cruise drag equations are solved using the Cebeci- needs to be reduced. In general, for a typical Stewartson eigenvalue formulation. The swept-winged transport aircraft at cruise, the locations of the calculated transition points are frictional drag accounts for about 35% of the validated by the experimental results. With the total drag [2], so among the various drag reliable transition information, the accuracy of reduction technologies, the laminar flows drag the infinite swept wing’s aerodynamic reduction is one of the most promising performance calculated by the RANS solver has technologies. However, the design of natural been improved. laminar flow wing on which a wide range of laminar flow can be maintained must be based 1 Introduction on the reliable transition prediction method. Because of the complexity of the transition from It is well known that if the transition the laminar flow to turbulence flow, yet we information of the boundary layer flow is not cannot make a complete explanation of its included, the wing's aerodynamic characteristics mechanism. However, after half a century’s calculated by the RANS solver will not be theoretical and experimental research, there has accurate enough. The characteristics between been quite in-depth understanding of the the laminar flow and turbulent flow are very transition mechanism and a lot of methods for different, for example, the wall friction caused predicting the boundary layer transition point by turbulence flow is several times higher than have been developed. Among those methods, that caused by laminar flow. Hence, without the en method proposed by Smith, Gamberoni [3] taking into account the boundary layer flow's and Van Ingen [4], which based on the linear transition point, or miscalculating the transition stability theory, has been widely used in point, the calculated wing’s aerodynamic industry. In view of the en method has been characteristics, especially the drag characteristic successfully applied in two-dimensional will be a far cry from the experimental value. At boundary layer transition determination, Malik the same time, ignoring the boundary layer [5], Mack[6], Arnal[7], Cebeci[8] and other flow's transition information, the calculation investigators introduced this method into determining the three-dimensional boundary 1 ZHANG KUN, SONG WENPING layer’s transition. Especially during the past this reason, the wall pressure distribution from decade, the National Aeronautics and Space RANS solutions is used as the outer boundary Administration (NASA), the German Aerospace condition for the boundary-layer solution. Center (DLR), the French Aerospace Center (ONERA) and the other research institutes have 2.2 Three-Dimensional Laminar Boundary been carrying out a method of coupling the Layer Solver simplified en method to the three-dimensional As we all know that the prediction of transition Reynolds-averaged Navier-Stokes solver in n order to increase the accuracy of solving in the flows around wings with the e prediction aerodynamic performance for the wing, the high transition method requires the specification of lift devices and the aircraft. The simplified en velocity and temperature profiles of the laminar method owing to its rapid response speed has boundary layers. Generally there are two ways been widely used, but as the computer to obtain the velocity and temperature profiles, performance significantly improved, the use of a either the solutions of the RANS or the full en approach is also feasible. In this paper, boundary- layer equations. In our paper, we use we couple the full en method to the three- the latter method to get the velocity and dimensional Reynolds-averaged Navier-Stokes temperature profiles because the former method solver to increase the solver’s accuracy. need a lot of grid point in the region of The computational methods for predicting the boundary layer to obtain accurate viscous-layer transition point based on the Cebeci-Stewartson results which will cost huge compute time. This eigenvalue formulation [8] is described in was confirmed by Stock, H. W. [9]. In order to section 2. Section 3 answers the question: how solve the boundary-layer on arbitrary wings, we to couple the RANS solver with the transition utilize a non-orthogonal coordinate system for prediction method? In section 4, we validate the defining the wings. Keller’s box method is used reliability of the transition prediction method to discrete the three-dimensional laminar described in this paper by comparing the boundary-layer equations, and then, using the calculated results with the experimental results, Newton method to linearize above nonlinear and the result of the comparison approves the equations, finally, the Block-Elimination reliability of above method. method is used to solve the linear system. 2.3 Transition Prediction Solver 2 Computational Method Unlike the simplified en method, such as the en – As mentioned above, it is an iterative process to database method or the envelope method, which n determine the transition point using the e do not need to solve the linear stability equation method during solving the three dimensional for detecting the transition location, the present Reynolds-averaged Navier-Stokes (RANS) full method uses the Cebeci-Stewartson equations. Each iterative process contains three eigenvalue formulation which is based on the programs which will be described in detail in spatial amplification to solve the Orr- the following sections. Sommerfeld equation for three-dimensional flows. 2.1 Three-Dimensional RANS Solver In the solution of three-dimensional linear stability equation, the eigenvalue formulation The three-dimensional, unsteady, compressible requires a relationship between the two wave RANS equations in our solver are solved by numbers α and β . In the Cebeci-Stewartson means of a finite volume approach using a LU- SGS time-stepping method with multi-grid eigenvalue formulation the relationship is acceleration, and the SA turbulence model is computed by making use of concepts based on applied. In boundary-layer theory, the pressure group velocity using the saddle-point discussed gradient is nearly zero along the wall normal by Cebeci and Stewartson[8] and Nayfeh[10]. direction inside the boundary-layer region. For According to the saddle point method the 2 INFINITE SWEPT-WING REYNOLDS-AVERAGED NAVIER-STOKES COMPUTATIONS WITH FULL EN TRANSITION CRITERION formulation (∂∂αβ) is real and related by equations can be solved. Then, we use the linear ω,R stability code to analyze the laminar boundary’s the disturbance angle φ through velocity and temperature profiles applied by the ⎛⎞∂α =−tanφ (1) boundary layer solver, and find out the ⎜⎟ transition point with the full en method. If there ⎝⎠∂β ω,R does not find transition point by using the full en The Eq. (1) provides a relationship between the method, setting the laminar separation point as two wave numbers as needed in the eigenvalue the transition point approximately. Finally, we problem. n returned the transition information to the The procedure through using the full e method solution of RANS equations. Repeat the above to determine the three dimensional boundary process, the flow transition point was detected layer flows’ transition points contains two steps. automatically during the ongoing RANS The first step is to calculate the absolute neutral computation. curve which was named as “zarf” by Cebeci. The zarf passes through the critical points in Input α,,,βωR space at which R = Rcr and is of cp significant importance in transition point RANS Solver Laminar BL prediction for three dimensional flows. The Solver second step is to calculate the amplification rate Velocity Profiles Transition Position Iteration Temperature Profiles for different dimensional frequencies beginning Iteration on the lower branch of the zarf. The The Full en amplification rate Γ is: Transition xtr Prediction Method ⎛⎞∂α Γ=−αβii + ⎜⎟ (2) ⎝⎠∂β ω,R Solution The onset of transition may be evaluated by solving the integral Fig.1 Sketch of the coupling the RANS solver with the transition prediction method x Ndx=Γmax⎡⎤ max ()()φ (3) f ⎣⎦⎢⎥∫x0 φ Where f is the dimensional frequencyω* 2π ,ω* 4 Results is the dimensional radian frequency, x0 We studied two infinite wing configurations by corresponds to the x-location where the comparing their calculated transition locations amplification rate is zero on the zarf. Once the and the measured transition locations to validate amplification factor N is greater than a limiting our transition prediction method. The two factor Nlimit, then the transition happens. infinite wing sections normal to the leading edge are NACA 642A015 and NLF(2)-0415 airfoils respectively. 3 Calculation Procedures A Reynolds-averaged Navier-Stokes code, a n 4.1 Infinite Swept Wing with NACA 642 A015 laminar boundary code and a full e prediction Profile transition method are coupled (see Fig. 1). First, the flow simulation begins with full turbulence The infinite swept-wing with the NACA 642 model by the RANS solver.
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