Optimal design of supersonic nozzle contour for altitude test facility

Transcription

1 Journal of Mechanical Science and Technology 26 (8) (2012) 2589~ DOI /s x Optimal design of supersonic nozzle contour for altitude test facility Khin Oo Mon and Changjin Lee * Department of Aerospace Engineering, Konkuk University, Seoul, , Korea (Manuscript Received September 15, 2011; Revised January 3, 2012; Accepted March 22, 2012) Abstract This paper develops a robust and practical design for supersonic nozzles to be used in an altitude engine test facility. Although many studies have been conducted on nozzle design, none of these present a robust yet practical and simple method for designing supersonic nozzles. This research attempts to develop such design for supersonic nozzles by combining method of characteristics (MOC), optimization algorithm, and computational fluid dynamics analysis for design verification. Preliminary design optimal techniques were adopted to reduce nozzle length while keeping the exit area constant in the design. Optimization produced a smooth flow by generating a parallel and uniform flow at the exit. A two-dimensional model was initially used because of the axisymmetrical characteristic of the flow in this study. The optimal nozzle was designed for the operation of a test facility at Mach number 2.3 and altitude of 7 km. The optimal design produced a uniform and parallel flow at the given test condition. Keywords: Altitude engine test facility (AETF); Computational fluid dynamics (CFD); Method of characteristics (MOC); Optimal nozzle contour; Parallel and uniform flow Introduction Designing a supersonic aerodynamic nozzle that provides a uniform exit flow is a highly challenging aspect in the construction of an altitude engine test facility (AETF). Engine tests on the ground level cannot simulate the effects of flight altitude and speed on engine performance. Thus, the test facility is utilized to cover a wide range of simulated flight conditions. To reduce the costs and the time involved in development, altitude engine testing is generally scheduled to verify the performance and the reliability of the flight vehicle engine prior to the actual flight test. Test facilities adhere to safety requirements in measuring efficiency and performance in high altitude environments. Therefore, the successful set up of an AETF is highly related to nozzle design. Moreover, defining an optimal nozzle contour is necessary to ensure a uniform flow and to prevent losses and flow separation. A primary objective of the present work is to design a nozzle contour that delivers a uniform and parallel flow and that is particularly useful for supersonic altitude nozzles. Reducing the nozzle length by keeping the area ratio constant and by combining the three methods mentioned earlier is a secondary objective that aids in the achievement of a practical, robust, and compact nozzle design. * Corresponding author. Tel.: , Fax.: address: cjlee@konkuk.ac.kr Recommended by Associate Editor Jeonghoon Yoo KSME & Springer 2012 Designing the contour in basic nozzle designs is generally conducted according to the method of characteristics (MOC). Existing literature shows two basic nozzle configurations created through this method: rocket nozzles designed to produce maximum thrust and wind tunnel nozzles aimed at generating a uniform flow [1-5]. MOC is a classical method used for defining the characteristics of nozzle curvature. It consists of compatibility equations that are not used in the subsonic section. The compatibility relations that serve as important equations in computing are presented in detail in Refs. [6, 7]. Korte [8] designed the wind tunnel nozzle based on the inviscid flow assumption. The design assumes the thickness of the boundary layer to be small enough to be neglected compared with the characteristic length (nozzle radius). Instead, they adopted a real gas in this inviscid design. The quality of flow quality of the inviscid design was compared against ideal gas; computational fluid dynamic (CFD) calculation revealed no basic difference between the two. Thus, the inviscid flow assumption for an ideal gas is relevant to the design of the supersonic nozzle in the present study. An optimization algorithm is used to extend the preliminary contour to attain robustness and compactness for an altitude test facility. Recently, studies have shown that additional techniques can be applied to the classical MOC method to meet various design requirements. The optimization technique is a plausible method for improving design quality and flow characteristics. The design of a liquid rocket engine is an example of utiliz-

2 2590 K. O. Mon and C. Lee / Journal of Mechanical Science and Technology 26 (8) (2012) 2589~2594 ing the optimization method. The application of this method is one way to meet the demand for cost reduction in space missions [9]. Colonno et al. [10] studied the implementation of optimization in designing the CH4-LOX nozzle to maximize the vehicle velocity increment. They developed a multidisciplinary optimization method by incorporating a number of optimization methods. Their optimal results were compared with the results of an MOC-based preliminary design. The procedure targeted maximum system-level performance with manageable computational cost. The nozzle was designed to achieve maximum thrust for the liquid rocket engine [9]. Furthermore, a method that combines the preliminary MOC design with optimization has been introduced to produce a more uniform exit flow [11-13]. Adjusting the nozzle length by keeping the area ratio constant can efficiently reduce the engine weight and minimize the configuration dimensions, which are advantageous at high altitude. Motivated by previous studies, the present work adopts the optimization algorithm based on Sequential Quadratic Programming (SQP) under the gradient optimizer. In SQP, the selected design variables are repeatedly modified in a systematic model until the objective and constraint functions meet the prescribed conditions. Generally, short nozzles are designed to enable flow separation, as longer nozzles are likely to cause insufficient flow relaxation. Therefore, aerodynamic nozzles require design options that influence flow quality [14]. This study proposes a robust design method that combines three different techniques. Flow field in the nozzle was checked by a commercial CFD code, FLUENT, to measure the accuracy and the reliability of the design. The supersonic optimal nozzle is designed for the operation of a test facility at Mach 2.3 and altitude of 7 km. 2. Design of supersonic nozzle for an altitude test facility 2.1 Preliminary design (MOC) Various supersonic/hypersonic nozzle contours have been developed recently by combining MOC and optimization techniques to produce a uniform flow distribution [9-13]. In the present study, the preliminary nozzle contour is designed based on MOC and then optimized using gradient optimization algorithm to tune up performance. A nozzle consists of a subsonic and a supersonic section. MOC only investigates the supersonic part, which is from the throat to the exit of the nozzle, as the supersonic flow field is Fig. 1. Computational domain of a nozzle. independent of the upstream conditions of a sonic line. Fig. 1 presents a sketch of the nozzle. The simplest design for the supersonic section of the nozzle can be rendered through MOC, as MOC provides numerical data for designing supersonic nozzle contours. Some studies suggest a calculation procedure to outline the supersonic portion of nozzle contours [8, 9]. The results generally include characteristic lines with curvature that reflects changes in flow properties from one point of the flow to the next, as shown in Fig. 2. The supersonic region of the nozzle is further divided into the throat region and the straightening region. Flow accelerates from subsonic to supersonic speed in the throat region. The wall slopes decrease even when the cross-sectional area increases in the straightening region. Two different approaches are generally adopted in designing the straightening zone. The first approach assumes a vertical, straight sonic line at the throat, and the second one assumes a curved sonic. Multidimensionality of a converging subsonic flow causes the sonic line at the throat to form a slightly curved vertical line. Thus, the second approach is preferred in the current analysis. To produce design data for the aerodynamic nozzle in the preliminary design process, the following steps were conducted: Formulation of compatibility equations Discretization of compatibility equations Solving of numerically discretized equations The detailed procedure for constructing compatibility equations through MOC is described in Refs. [1-8]. The compatibility equations are derived from continuity, momentum, and energy equations for steady, adiabatic, and axisymmetric flows. The derived compatibility equation along the right and left running characteristics is given as follows: d( ν ± θ) = 1 dr M 2-1 ± cot( θ ) r dy = tan ± ( θ + μ). (2) dx Eqs. (1) and (2) represent the characteristic and compatibility equations, respectively. Both are valid on upward characteristics (C - ) and downward characteristics (C + ) that describe the variation of the flow properties along the characteristic lines (Mach lines). Here, M is the Mach number, θ is the expansion angle, γ is the specific heat ratio, µ is the Mach angle defined as arcsin (1/M), and ν is the Prandtl-Meyer function defined as follows: ( ) γ + 1 γ 1 ν arctan 2 1 arctan 2 1. γ 1 γ 1 M = M + (3) A continuous gradient of the Prandtl-Meyer angle ensures (1)

3 K. O. Mon and C. Lee / Journal of Mechanical Science and Technology 26 (8) (2012) 2589~ Fig. 2. Notation of the characteristic. the continuous curvature of the contour along the nozzle axis. The new Mach number is calculated by applying the Prandtl- Meyer function. The notations are illustrated in Fig. 2. After designing the expansion section, as shown in Fig. 1, the slope of the wall contour is gradually straightened until it becomes parallel to the flow. New characteristics in this region can be calculated using the previous steps. The contour is finally obtained by numerically solving the compatibility equations. The MOC technique analyzes the contour within the required boundary conditions based on the design problem. The nozzle exit condition is specified in terms of the exit Mach number of M E = 2.3. The nozzle inlet and outlet boundary conditions are defined based on the altitude of 7 km. Assuming a uniform and parallel exit flow, the exit diameter is fixed at 564 mm in the present study because of the engine facility requirements. Based on the required data, the design of the twodimensional axisymmetric contour is numerically solved in inviscid flow, which assumes perfect gas. The nozzle curvature should be designed using an infinite number of contours depending on the specified boundary conditions. The throat diameter can then be estimated from the isentropic area ratio relation, as given in Eq. (4). γ A 1 2 γ γ 1. * = 2 + M A M γ Depending on the throat diameter and the boundary conditions, the throat characteristic (Fig. 1) permits the characteristic solution to fill almost the entire supersonic domain and spreads the contour design to cover the entire length from throat to exit. By solving the discretized compatibility equations numerically, a preliminary design of the aerodynamic nozzle is obtained. Notably, the MOC-based contour design does not proceed upstream of the throat. As a result, the nozzle contour containing the throat, initial expansion, and straightening regions can be produced with an exit diameter and nozzle length of 564 mm and 1230 mm, respectively. Fig. 4 presents the (4) Fig. 3. Summary of the nozzle design process including the design methods. preliminary design of the nozzle curvature using MOC technique vis-a-vis the optimal nozzle curvature. The entire procedure for developing application techniques is briefly demonstrated in Fig. 3. In this study, three sets of techniques are utilized in the design of an optimal nozzle contour. The nozzle design based on MOC is a classic and convenient method for constructing the nozzle contour. The optimization method presented above uses a simple model that has been shown in practice to be accurate in solving different problems on supersonic nozzle flows. In the final stage of the procedure, the nozzle design is verified through CFD solver using computational aerodynamics to complete the optimal nozzle design. Numerical calculation determines the velocity flow distribution inside the nozzle. The optimal nozzle shows that the inviscid assumption can effectively initiate the design of the supersonic flow configuration despite the exclusion of the viscous effect, as the problem can be assumed by the CFD solver to be a viscous flow, although the initial design is based on the inviscid flow. Accordingly, calculation time and effort are reduced while the accuracy and the reliability of design are confirmed. 2.2 Optimal design (optimization algorithm) In general, long nozzles may suffer substantial losses in total enthalpy, resulting in insufficient relaxation. Short nozzles may be inadequate for large flow separation. Hence, designing

4 2592 K. O. Mon and C. Lee / Journal of Mechanical Science and Technology 26 (8) (2012) 2589~2594 Table 1. Optimum design limits. Objective Minimize (L) (m) Constraints 0 M 2.3 Design variables 439,000 Pa P 0 600,000 Pa 477 K T K 22 N c 100 Table 2. Comparison of the design contour data. Variables can be prescribed by simple isentropic relations [6] considering the altitude change from ground level to 7 km. Based on the altitude and the fixed exit Mach number, the total pressure calculated from the isentropic relations at ground level and at an altitude of 7 km is 439,000 Pa and 600,000 Pa, respectively; the total temperature of these two conditions is 477 K and 522 K, respectively. A range of the number of characteristic lines can be estimated by defining the various values until the required Mach number is satisfied. Finally, the range of 22 lines to 100 lines of characteristics is considered adequate without causing any additional errors. In executing the optimization problem, the gradient optimizer systematically assigns values to the design variables until the objective is satisfied. The optimizer executes the model several times to calculate the gradients. As described in Table 1, the optimization process can be carried out using the MOC-based code to produce an optimal nozzle contour. When the design variables approach the convergent optimum, the optimal design is satisfied. Fig. 4 compares the preliminary and the optimal nozzle curvatures. The x-axis presents the length of the nozzle, and the y-axis presents the nozzle height. In Table 2, output parameters of the preliminary and the optimal designs are compared. Based on the isentropic relation, the exit diameter is calculated from Eq. (4) as 563 mm. The preliminary design based on MOC and the optimization algorithm yield an exit diameter of 565 mm and 564 mm, respectively. Table 2 compares the exit Mach number, throat Mach number, and exit diameter in the three cases. Both MOC-based and optimization-based nozzles satisfy the required exit Mach number and fixed exit diameter but produce different nozzle lengths. The optimal nozzle presented in this study is designed using the MOC technique, with the initial contour as basis for optimization. Optimization was performed on the supersonic nozzle using a slightly different nozzle length. Modification of the basic contour by MOC is necessary because the oscillatory characteristics of the Mach number that appears in the optimal design are better than those in the preliminary design. An op- Isentropicbased data MOC-based data Optimizationbased data M E M * D E (mm) aerodynamic nozzles that yield uniform flow is hinged on the choice of design; flow quality is also influenced by many factors [14]. The optimization technique is considered because it produces an optimal nozzle design with an appropriate nozzle length. SQP [10, 11], a commercially available optimization code, is adopted in the optimized design because it facilitates the definition of parameters, constraints and, design objective. The nozzle design is optimized using the MOC-based code and the optimizer to improve the uniform velocity distribution. The objective function seeks to provide a better uniform flow through appropriate nozzle length, thereby creating a practical and simple nozzle design for AETF. The choice of the objective function generally depends on the characteristics of the problem [8-13]. Here, nozzle length is selected as the design objective to be minimized, and the optimization algorithm is added until the required value is satisfied. The design variables and constraints as well as the design objective are described in Table 1; L in the objective represents nozzle length, and M in the constraints represents the axial Mach number. The design variables P 0, T 0, and N c denote total pressure, total temperature, and number of characteristic lines, respectively. The design objective of this problem is to minimize the nozzle length for a given test condition. Maximizing the length in the objective is not a suitable option for satisfying the required design conditions to obtain the appropriate nozzle length. The design variables are the parameters used in defining the boundary conditions or quantities related to the specified altitude. In the altitude nozzle design, the altitude is based on the test facility requirements, and the boundary conditions are determined according to the specified altitude. In this work, the altitude is specified as 7 km, and the required exit Mach number is fixed to 2.3. Constraints are defined by selecting particular variables and described using the values in Table 1. The constraints are within the limitations to prevent producing output outside the allowable region of design interests. The upper value for Mach is the specified Mach number, with zero as its lower value. The design variables influence the optimizer to generate a new set of design values until the objectives are satisfied. Defining the design variables within a set of limitations can prevent physically empty values and restrict the design space. Moreover, the design variables are intended to satisfy all the constraints prescribed in the optimization study. Thus, the appropriate choice of boundaries can significantly improve the process of optimization. The range of design variables for pressure and temperature Fig. 4. Comparison of nozzle contours with MOC and optimization.

5 K. O. Mon and C. Lee / Journal of Mechanical Science and Technology 26 (8) (2012) 2589~ Table 3. Nozzle boundary conditions. Total pressure (P 0 ) 513,038 Pa Total temperature (T 0 ) 500 K Exit Mach number (M E ) 2.3 Specific heat ratio (γ) 1.4 Table 4. Comparison of design accuracies for M E = 2.3. Variables Isentropic-based data Optimal design with CFD-checked data m = ρav (kg/s) M E T 0 (K) Fig. 5. Nozzle geometry used in CFD calculation. (a) timal nozzle with appropriate length can promote a uniform and smooth flow. 2.3 Accuracy of optimal design (CFD) As the optimum nozzle generates the contour only from the throat to the exit, the contraction part is added to verify the accuracy of the flow field in the supersonic section. In the contraction zone, the flow is entirely subsonic, but it accelerates to sonic speed. A schematic geometry of the AETF nozzle used in CFD calculation is presented in Fig. 5. The typical process for designing an optimal nozzle contour was described in the previous section. To visualize the flow field for the nozzle using the commercial CFD code, the contraction section of the nozzle is constructed based on the ISO method [14]. This construction specifies the geometry and the method for designing nozzles and Venturi nozzles. A two-dimensional model is used because of the axisymmetrical characteristics of the flow in this study. The computational domain is meshed using a regular and a structured grid of quadrilateral elements for M E = 2.3. The grid consists of points for calculating the contraction zone and points for calculating the region from the throat to the exit. The governing equations include the physical laws on conservation of mass, momentum, and energy. The famous k- ε model was selected as the turbulent model. The parameters required for boundary conditions are described in Table 3. A compressible CFD solver computes the flow field until a convergent solution is obtained within the optimal design conditions. Accuracy of the optimal nozzle design is estimated by computing the flow field. The optimal nozzle flow field with uniform flow produced by the CFD results is presented in Fig. 6. In proposing a simple nozzle design for AETF, a typical procedure for M E = 2.3 was presented in the previous sections. To confirm that the procedure is applicable to any supersonic (b) Fig. 6. (a) Numerical grids of optimal contour; (b) Velocity distribution along the nozzle. AETF nozzle contour, the optimal nozzle at M E = 2.3 was compared against the required conditions. This detailed comparison (Table 4) confirms the accuracy and the reliability of the procedure. The design based on optimization shows the nozzle exit diameter to be 564 mm. Therefore, the mass flow rate should be kg/s, as calculated from the mass flow rate equation. Given the isentropic exit diameter, the mass flow rate becomes kg/s compared with kg/s using the CFD optimized design. Inlet and outlet mass flow rates are identical. Combining optimization and CFD in contour design is an interesting approach to achieving a compact design for the optimal nozzle. Moreover, the internal profiles can be viewed based on the CFD calculation. The commercial CFD code helps in verifying the accuracy of the optimal nozzle design, thus saving time and effort. The purpose of presenting distribution profiles and accuracy tables is to confirm that the flow field satisfies the given requirements and that the design process is reliable. 3. Conclusions A number of studies on nozzle design have been discussed. However, most of the existing research focuses on rocket nozzles, wind tunnel nozzles, and so on. There are no studies that

6 2594 K. O. Mon and C. Lee / Journal of Mechanical Science and Technology 26 (8) (2012) 2589~2594 include practical considerations in designing a high-altitude nozzle; nozzle designs for altitude test facilities have not been found in the past. An altitude optimal nozzle is designed in the current research, incorporating aerodynamic approaches to early nozzle design applications, particularly those that promote a uniform exit flow. A computational technique is developed in designing an optimal nozzle for a supersonic altitude test facility. A step-by-step description of the process is provided to allow its application in practical situations. The proposed technology involves methods for designing the preliminary nozzle and a numerical method for compatibility equations that define the geometry of the nozzle contour. In defining the objective function for optimal design, design constraints and variables similar to the MOC preliminary design were used. CFD calculation of the supersonic nozzle designed for M E = 2.3 confirmed the accuracy of the solutions. The result is a convenient, robust, and efficient method for designing the altitude nozzle, as described in this paper. A block diagram integrating the entire design procedure is shown, demonstrating the application methods to design an optimal nozzle. Combining these methods yields an appropriate optimal nozzle contour. The process presented in this study can be applied in generating suitable and optimal nozzle designs. Any supersonic nozzle can be designed using a wide range of imposed altitude requirements by the technique developed in this study [7] A. Mccabe, Design of a supersonic nozzle, Thesis and Dissertation of the Mechanics of Fluids Department, University of Manchester (3) (1964). [8] J. J. Korte, Inviscid design of hypersonic wind tunnel nozzles for a real gas, AIAA Paper (2000). [9] G. Cai, J. Fang, X. Xu, M. Liu, Performance prediction and optimization for liquid rocket engine nozzle, Aerospace Science and Technology, 11 (2007) [10] M. R, Colonno, E. Van der Weide and J. J. Alonso, The optimum vacuum nozzle: and MDO approach, AIAA Paper (2008). [11] S. M. Aulchenko, V. M. Galkin, V. I. Zvegintsev and A. N. Shiplyuk, Design of multimode axisymmetric hypersonic nozzles with the use of optimization methods, Journal of Engineering Physics and Thermophysics, 82 (6) (2009). [12] S. M. Aulchenko, V. M. Galkin, V. I. Zvegintsev and A. N. Shiplyuk, Numerical design of multimodal axisymmetric hypersonic nozzles for wind tunnels, Journal of Applied Mechanics and Technical Physics, 51 (2) (2010) [13] L. Shope, Frederick, Contour design techniques for super/hypersonic wind tunnel nozzles, AIAA Paper (2006). [14] ISO , Measurement of fluid flow by means of pressure differential devices inserted in circular cross-section conduits running full, Nozzles and Venturi Nozzles, Part 3 (2003). Acknowledgment This work was made possible by the financial support from the 2011 research grants (Grant No: ) of the National Research Foundation of Korea. References [1] H. W. Liepmann and A. Roshko, Elements of gasdynamics, Seventh Ed. John Wiley & Sons, Inc, New York (1996) [2] M. Gőing, Nozzle design optimization by method-of- characteristics, AIAA paper (1990). [3] G. V. R. Rao, J. E. Beck and T. E. Booth, Nozzle optimization for space-based vehilcles, AIAA paper A (1999). [4] G. V. R. Rao and J. E. Beck, Use of discontinuous exit flows to reduce rocket nozzle length, AIAA paper (1994). [5] V. I Plyashechnik, A. P. Byrkin and V. P. Verkhovsky, Gasdynamic design of shaped nozzles for supersonic wind tunnels, allowing for viscosity, West-East High Speed Flow Field Conference, Moscow, Russia, 11 (2007) [6] D. Anderson, Jr, Modern compressible flow with historical perspective, Third Ed. McGraw-Hill, New York (2004) Khin Oo Mon received her B.E. from Mandalay Technological University (MTU, Myanmar) in 2004 and M.E. in Mechanical Engineering from Yangon Technological University (YTU, Myanmar) in She is currently a Ph.D candidate for Aerospace Information Engineering at Konkuk University in Seoul, Korea. Her research interests are in the area of computational fluid dynamics of nozzle flow and LES analysis of flow instabilities in hybrid rocket. Changjin Lee received his B.S. and M.S. in Aeronautical Engineering from Seoul National University in 1983 and 1985, respectively. He received his Ph.D from the University of Illinois in Urbana-Champaign in Dr. Lee is currently a Professor at the Department of Aerospace Engineering in Konkuk University in Seoul, Korea. His research interests are in the area of combustion instabilities of rocket and jet propulsions.