DESIGN OF A MULTIMODE AXISYMMETRIC NOZZLE FOR A HYPERSONIC WIND TUNNEL BY METHODS OF NUMERICAL OPTIMIZATION

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1 International Conference on Methods of Aerophysical Research, ICMAR 2008 DESIGN OF A MULTIMODE AXISYMMETRIC NOZZLE FOR A HYPERSONIC WIND TUNNEL BY METHODS OF NUMERICAL OPTIMIZATION S.M. Aulcheno 1, V.M. Galin 2, V.I. Zvegintsev 1, A.N. Shiplyu 1 1 Khristianovich Institute of Theoretical and Applied Mechanics SB RAS, Russia, , Novosibirs, Institutsaya Str., 4/1. 2 Toms Polytechnical University, Russia, , Toms, Pr. Lenina, 30. Introduction Despite significant achievements in the field of numerical simulations, which are primarily related to the improved computer performance, design of flying vehicle still cannot be performed without wind-tunnel (WT) experiments [1]. Experiments at high Mach numbers require rather sophisticated facilities from the engineering viewpoint to be developed [2]. If viscous effects are ignored, the supersonic segment of the WT contour should have a uniform characteristic at the exit and should be calculated in a manner similar to the method described in [3] to ensure a uniform flow with a prescribed Mach number M out in the test section. Obviously, a change in M out necessitates a change in the WT contour. To reduce the cost of creating supersonic segments with different values of M out, it was proposed [4] to mae the WT nozzle as a combination of a fixed segment and a replaceable segment adjacent to the cross section with the minimum area. A similar structure was used in [5], where the shape of the replaceable segment was calculated by direct methods, and the nozzle was approximated by splines. In [6], the replaceable segment was found among the set W(γ,M out,g) of supersonic contours with a uniform characteristic at the exit, where γ is the ratio of specific heats (in what follows, γ=1.4) and G is the ratio of the flow rate through the streamline based on which the nozzle contour is constructed to the flow rate of a nozzle with a corner point (Figs. 1 and 2). Y b Y b b' d a a' 0 c Fig. 1. Family of nozzles W(γ,M out,g). ab nozzle with a corner point; a'b' intermediate streamline for a flow rate G; ac characteristic closing the expansion fan; cb uniform characteristic; α=arcsin(1/m out ). α X a 2 a 1 e α1 X 0 c 1 c 2 Fig. 2. Multimode nozzle. a 1 d replaceable segment of the nozzle for M out1 ; a 2 d replaceable segment of the nozzle for M out2 ; db fixed segment of the nozzle; c 1 b uniform characteristic for M out1 ; c 2 b uniform characteristic for M out2 ; α 1=arcsin(1/M out1 ); α 2=arcsin(1/M out2 ). α2 M out and G used as parameters. The entire contour is not used; the study involves only the segment of the contour smoothly attached to the fixed segment. The fixed segment also belongs to the set of contours W(γ,M out,g). S.M. Aulcheno, V.M. Galin, V.I. Zvegintsev, A.N. Shiplyu,

2 Section III Let a nozzle a 1 b with a uniform characteristic c 1 b and a Mach number M out1 at the exit be constructed by the method of characteristics. The following procedure is proposed for constructing a nozzle in the form of a combination of a fixed segment db, replaceable segment a 1 d for a Mach number M out1, and replaceable segment a 2 d for a Mach number M out2 (M out2 <M out1 ) (Fig. 3). First, the fixed segment of the nozzle is determined for the Mach number M out2. For this purpose, a uniform characteristic c 2 b for the Mach number M out2 is constructed on the nozzle contour a 1 b. Then, between the nozzle contour a 1 b and the characteristic c 2 b, the flow parameters - are calculated (in the upstream direction) until the characteristic dc 2 outgoing from the axis is obtained. Obviously the fixed segment db has the minimum length. These manipulations allow obtaining the fixed segment db and the replaceable segment a 1 d for the Mach number M out1. Then the replaceable segment of nozzle a 2 d for the Mach number M out2 is constructed. The method of a 2 d construction is similar to the approach proposed in [7]. This approach was further developed in [8], where the contours from the family W(γ,M out,g) were used as the initial approximation for the replaceable segment, and the final result was obtained by a direct method. Requirements to hypersonic wind-tunnel nozzles include geometric and gas-dynamic restrictions: flow uniformity at the nozzle exit, sufficiently large core flow free from the influence of the boundary layer, nozzle length related to structural features of the facility, etc. Another principal issue in hypersonic nozzle design is the allowance for viscosity [9]. Therefore, the problem posed can be successfully solved only by using a comprehensive approach combining the traditionally used method of characteristics, which allows fairly simple calculation of supersonic nozzle with a uniform characteristic at the exit for an ideal inviscid heat-non-conducting gas, direct methods for calculating viscous flows, and methods of numerical optimization. As solving optimization problems for viscous flows requires large computational resources and is usually successful only if a good initial approximation is available, the first stage of nozzle design by the method of characteristics is an important element of the chosen strategy. Approximation of the nozzle contour As the effectiveness of direct methods largely depends on the choice of the nozzle-contour approximation, various methods of presentation of the sought contour geometry were compared: 1. monotonic interpolation on the basis of a cubic spline with irregular arrangement of nodes [5,8]; 2. monotonic approximation on the basis of expansion over the basis functions with the use of third-power B-splines [7]; 3. monotonic approximation on the basis of interpolants utilizing rational splines [7]; 4. approximation on the basis of expansion over the basis functions in the form of logarithm powers. The latter method ensured the best results if direct methods based on solving gas-dynamic equations were used for design. Interpolation on the basis of a cubic spline with irregular arrangement of nodes was used for the method based on viscous flow calculations. Results of design The initial approximation for design of a multimode nozzle was a nozzle constructed by the method of characteristics by the method [3] for a uniform output Mach number M out =15.7. At the first stage, the nozzle contour defined by a large number of points (up to 600) in a tabular form was approximated by a cubic spline with irregular arrangement of nodes, which was accurately built on the minimum number of reference points, because high-velocity flows are extremely sensitive even to minor perturbations of the boundary. Thus, for 10 reference nodes, the mean absolute and relative errors were equal to This high accuracy was needed to enable the contour to be varied in the course of optimization problems (see below) with a small number of parameters, which is very 2

3 International Conference on Methods of Aerophysical Research, ICMAR 2008 important, because the calculation of the nozzle characteristics for a viscous flow is rather timeconsuming. The viscous flow in this nozzle was computed by the Fluent software system for a perfect gas with the ratio of specific heats equal to 1.4 and the Reynolds numbers per meter Re 1 = Navier- Stoes equations with the SST--ω model of turbulence were solved. The computational grid had 1000 nodes along the nozzle axis and 40 nodes over the nozzle radius. The nodes were more frequent near the entrance cross section and in the region of the boundary layer. Test computations on finer grids (1000х60), (2000х40), and (2000х60) were performed. As the nozzle characteristics, which were further used to form the functional of the optimization problem, differed within fractions of a percent, the minimum grid was used for basic computations. Figure 3 shows the axial distribution of the Mach number. It is seen that the Mach numbr in the woring zone of the nozzle (2.5 m < x < 3.5 m) is 14, and there are Mach number oscillations, which were not observed for the inviscid flow. After that, the initial contour was modified by solving the optimization problem. The functional equal to the sum of the root-mean-square deviations of the Mach number from its mean value in a given flow region was minimized. Functional minimization was performed by gradient-free search methods with adaptation and an element of randomness. These methods were implemented in the form of a specialized software complex synthesized on the basis of modified methods of rotating coordinates, direction cone, and matrix descent. Restrictions were taen into account by the method of penalty functions. For this purpose, a composite functional is introduced. The estimates for the penalty coefficients have to guarantee the existence of a local extreme point of the composite functional in the region where the restrictions are satisfied with prescribed accuracy, which allows the ravine structure of the minimized function to be avoided, i.e., the problem of nonlinear programming is solved in the following statement: find min F( x), x X under the conditions E N X = { x : x i xi x i}, ϕ ( x) = 0, j = 1,, Q, ψ ( x) 0, = 1,, Q. j The composite functional has the form 2 2 Q Q ϕ j ψ ψ Ф c = Ф K j. + K 1 1 j= ε j = ε ε Here ε and ε are the prescribed values of accuracy of satisfying the restrictions; К and К are the penalty coefficients whose initial values are prescribed and then are refined in the course of program operation. Note that Ф 0 should be greater than unity, for instance, Ф 0 = (1 + Ф), where Ф is the objective functional. The resultant axial distribution of the Mach number for the optimized nozzle is plotted in Fig. 4. It is seen that the Mach number oscillations are substantially smaller. 3

4 Section III Fig. 3 Axial distribution of the Mach number for the initial nozzle. Fig. 4 Axial distribution of the Mach number for the optimized nozzle. Based on the contour obtained, a multimode nozzle with output Mach numbers M out = 8, 10, 12, and 14 was constructed. For this purpose, an optimization problem was solved, where the nozzle throat diameter and the geometry of the initial segment 329 mm long were varied for a fixed nozzle length of 3 m, nozzle exit diameter of 410 mm, and M out. The geometry of the initial segment was defined by a cubic spline on a nonuniform grid and was smoothly conjugated with the fixed segment of the nozzle corresponding to the optimal nozzle for M out =14. The Fluent software system was again used to compute the viscous flow during nozzle optimization and design, but this system was integrated with optimization codes. Figures 5a and 5b show the distributions of the Mach numbers in the resultant nozzle for all computed regimes in the cross sections x = 2.5 and 3 m, and Fig. 5c shows the axial distributions of the Mach number in the constructed nozzles in the interval 2.5 m < x < 3.5 m. a Mach number distribution in the cross section x=2.5 m b Fig.5 Mach number distribution in the cross section x=3.0 m Axial distribution of the Mach number in the interval m. c All nozzles were designed for the middle values of the Reynolds number Re mid, based on possible parameters of the gas in the settling chamber (Table 1). The values of the root-mean-square deviations from the mean Mach number in two cross sections and in the indicated interval on the axis are given in Table 2. For each regime in the woring region of the multimode nozzle, the flow is fairly uniform. Though the basic length of the nozzle was chosen to be 3 m, and the root-meansquare deviations of the Mach number from its mean value were minimized in this cross section, the nozzle flow was computed to a greater length (4 m) to estimate the degree of flow uniformity in the core flow downstream of the nozzle exit cross section. 4

5 International Conference on Methods of Aerophysical Research, ICMAR 2008 Table 1. Range of Reynolds numbers for several Mach numbers at the nozzle exit M out Re Re min Re mid Re max Table 2. Root-mean-square deviation from the mean Mach number in two cross sections and in the interval on the axis (in %) for several Mach numbers. M out х=2.5 m х=3.0 m m It is important for practice to now the characteristics of designed nozzles for the minimum and maximum values of the Reynolds number in each range. The results of these computations are summarized in Tables 3-5 together with the characteristics of the basic variants. Table 3. Root-mean-square deviation from the mean Mach number (in %) and the mean Mach number in two cross sections and in the interval on the axis of the nozzle designed for М out = 8 and for different Reynolds numbers. Reynolds number Re min Re mid Re max х=2.5 m х=3.0 m m Table 4. Root-mean-square deviation from the mean Mach number (in %) and the mean Mach number in two cross sections and in the interval on the axis of the nozzle designed for М out = 10 and for different Reynolds numbers. Reynolds number Re min Re mid Re max х=2.5 m х=3.0 m m Table 5. Root-mean-square deviation from the mean Mach number (in %) and the mean Mach number in two cross sections and in the interval on the axis of the nozzle designed for М out = 12 and for different Reynolds numbers. Reynolds number Re min Re mid Re max х=2.5 m х=3.0 m m Figure 6 shows the shapes of the replaceable segment 320 mm long for М out =8, 10, and 12. 5

6 Section III Fig.6 Shapes of the replaceable segment of the nozzle Figure 7 shows the computed (by the Navier-Stoes equations) and experimental distributions of the Mach number in the nozzle exit section in the regimes with М out =8 and 10. The computed data are plotted by the red curve, and the experimental values are indicated by symbols. In both variants, the results are in good qualitative and quantitative agreement. М 9,0 8,0 7,0 6,0 5,0 4,0 3,0 2,0 1,0 0,0-0,20-0,15-0,10-0,05 0,00 0,05 0,10 0,15 0,20 R, м Fig. 7 Comparison of computed and experimental results in the exit cross section of the nozzle with the replaceable segment for М out =8. P 0 = 58 atm. Comparison of computed and experimental results in the exit cross section of the nozzle with the replaceable segment for М out =10. P 0 =72 atm. A multimode nozzle for the range М out =14 20 was also designed. The method of characteristics was used at the first stage. The axial distributions of the Mach number for inviscid contours for М out =14 and 20 obtained by flow calculations within the viscous gas model are plotted in Fig. 8. The design value М out =20 is not reached in the initial contour because of considerable viscous effects. At the next stage, a multimode nozzle for this range is designed by the above-described technique with the initial contour used as the basis for optimization. The root-mean-square deviations of the Mach number at the nozzle exit are smaller than 1%. 6

7 International Conference on Methods of Aerophysical Research, ICMAR 2008 Axial distribution of the Mach number in the nozzle with the replaceable segment for М out =14. Fig. 8 Axial distribution of the Mach number in the nozzle with the replaceable segment for М out =20 Conclusions A computational technology is developed to solve problems of design and optimization of supersonic and hypersonic wind-tunnel nozzles. The technology involves methods for solving direct and inverse problems, various models of the medium, and numerical methods for integration of equations of the gas flow. Classes of functions that define the nozzle contour geometry and the dimension of the space of varied parameters are chosen. Restrictions on functionals used and methods for solving the optimization problem are formulated. Examples of multimode nozzles designed for Mach number ranges М = 8-14 and М = demonstrate that it is possible to obtain meaningful solutions of problems of design of high-accuracy nozzles for wind tunnels in a wide range of imposed requirements. This research was supported by ISTC (Moscow) within the project 3550, which made mentioned results possible. REFERENCES 1. Kharitonov A.M. Techniques and Methods of Aerophysical Experiment. Wind Tunnels and Gas-Dynamic Facilities, Pt. 1. Published by the Novosibirs State Technical University, Novosibirs, pp. 2. Zvegintsev V. I., Kharitonov A. M., Chirasheno V. F., Chibisov S. V., Fletcher D., Paris S. Hyperboloid flare tests in the AT-303 hypersonic wind tunnel // 12 International Conference on the Methods of Aerophysical Research, Novosibirs, Pt 4: Proceedings. P Katsova O.N. Calculation of Equilibrium Gas Flows in Supersonic Nozzles, Comp. Center, Acad. of Sci. of the USSR, Moscow, Byrin A.P., Verhovsii V.P. Supersonic axisymmetric contoured wind-tunnel nozzle. Patent RF No dated Aulcheno S.M., Zamuraev V.P. Design of a multi-regime nozzle by means of a direct method of numerical optimization // 12 International Conference on the Methods of Aerophysical Research, Novosibirs, Pt. 1: Proceedings. P

8 Section III 6. Galin V.M., Zvegintsev V.I. Constructing a multimode axisymmetric nozzle for a hypersonic wind tunnel // in: Fundamental and Applied Problems of Modern Mechanics. Conf. Proc. Toms University, Toms, P Volov Yu.S., Galin V.M. Choosing approximations in direct problems of nozzle construction // Zh. Vych. Mat. Mat. Fiz Vol.47. No.5. P Volov Yu.S., Galin V.M. Optimization of a multimode wind tunnel // in: Fundamental and Applied Problems of Modern Mechanics. Conf. Proc. Toms University, Toms, P Korte J. Aerodynamic design of axisymmetric hypersonic wind-tunnel nozzle using a least-squares/parabolized Navier-Stoes procedure // J. of Spacecraft and Rocets Vol.29. No. 5. P