AEROSPATIALE missiles Engineer. yAEROSPATIALE missiles Engineer. zProfessor, Dept. of Mechanical and Aerospace Engineering. AIAA Associate Fellow.
AIAA 99-2108 Three Dimensional Optimization of Supersonic Inlets ´ Christophe Bourdeau, Gerald Carrier, Doyle Knight Department of Mechanical and Aerospace Engineering Center for Computational Design and Khaled Rasheed Department of Computer Science Center for Computational Design Rutgers University - The State University of New Jersey 98 Brett Road Piscataway, NJ 08854-8058
35st AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit June 20–23, 1999/Los Angeles, California For permission to copy or republish, contact the American Institute of Aeronautics and Astronautics 1801 Alexander Bell Drive, Suite 500, Reston, VA 22091
35th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit
Three Dimensional Optimization of Supersonic Inlets Christophe Bourdeau, Gerald Carriery, Doyle Knightz
Department of Mechanical and Aerospace Engineering Center for Computational Design and Khaled Rasheed x
Department of Computer Science Center for Computational Design Rutgers University - The State University of New Jersey 98 Brett Road Piscataway, NJ 08854-8058 An innovative design methodology has been implemented and used to address the intricate exercise of high speed inlets design. An ecient and robust process allows to optimize the aerodynamic performance of inlets. This optimization process links together an optimizer with a fast and accurate simulation tool into an automated optimization loop. This paper presents the implementation of these new design techniques and their application to a Mach 3 inlet case. The signi cant improvements obtained using two di erent optimizers are presented and compared. The results of these optimizations have been veri ed using a full Reynolds Averaged Navier-Stokes solver. All the following results are thoroughly analysed and placed into an industrial context.
1 Introduction
applied to various elds, including aerodynamic design of di erent air vehicles. Optimizations have been carried out successfully for several two-dimensional problems. Hussaini et al.1 and Borivikov et al.2 have addressed two-dimensional nozzles. Several works have dealt with two-dimensional supersonic and hypersonic inlets. Zha et al.3 and Shukla et al.4 addressed optimization of two-dimensional supersonic missile inlets. Moreover Blaize et al.5,6 performed both single
ight condition and mission optimizations on twodimensional supersonic missile inlets. Substantial improvements of the performance over the whole mission were demonstrated. Nevertheless, wind tunnel experiments as well as Reynolds Averaged NavierStokes (RANS) calculations have proved the ow eld through supersonic missile inlets to be highly three-dimensional,7 even for inlets with rectangular cross-section under symmetric free stream conditions. Moreover missile inlets are commonly facing highly three-dimensional ows with both angle of attack and side-slip e ects through the missile mission. Thus a methodology for optimization of three-dimensional inlets is clearly needed. The research presented in this paper intends to provide the aeronautical industries with an ecient and powerful tool to optimize the aerodynamic performance of a full three-dimensional supersonic inlet. The most important issue is the cost of the performance evaluation of such three-dimensional systems. If RANS simulations are now commonly used for manual design process which requires few di erent calculations, they are out of the scope of current automated design
The aerospace industry has greatly bene ted from the progress accomplished by numerical ow eld simulation tools. More and more accurate ow solvers are now available to the aerodynamic engineer. Wind tunnel experiments, which tend to be more expensive and time consuming than numerical simulations, are now typically required only for nal veri cations. Thus, aerodynamicists are o ered the possibility to explore more deeply the complex and highly multi-dimensional design space. However, the classical design process mainly relies on engineers' experience and on the limited human capability to cope with a large number of coupled parameters. As a consequence, the design process can be long, laborious and extremely expensive, without any guarantee to lead to the best performing design for the problem which is addressed. In that context, automated design process strategies are of rst interest when it comes to complex industrial problems. Actually, the association of ecient optimization algorithms and fast aerodynamic performance analysis tool has proved to be able to give a better design than classical design methods, while reducing the time cost. Such automated design strategies have already been AEROSPATIALE missiles Engineer y AEROSPATIALE missiles Engineer z Professor, Dept. of Mechanical and Aerospace Engineering.
AIAA Associate Fellow. x Research Associate, Dept. of Computer Science, Center for Computational Design Copyright c 1999 by C. Bourdeau, G. Carrier, D. Knight and K. Rasheed. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission.
1
35th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit process, which requires hundreds of ow eld simulations, each of them taking several hundreds CPU hours on a typical workstation. Therefore, a hybrid ow solver based upon an innovative combination of an Euler ow-solver and a one-dimensional subsonic di user model has been developed. The simulation time with this tool has been reduced to few CPU minutes, allowing the current automated design optimizations. After describing the problem addressed by the present study, the optimization methodology and the di erent element it involves are described. Then the results for a Mach 3 inlet optimization are analysed and ascertained using several RANS calculations.
be expressed as the search for the global optimum: Maximize (Geometry ) Geometry subject to constraints One of the constraints imposed for the inlet design is to achieve a sucient mass ow rate coecient ".
2.3 Three-dimensional Geometry Model for Generic Inlet Supersonic Part
The geometry investigated is a multi-ramp mixed compression inlet. Lateral compression is achieved by sidewalls allowed to open towards the incoming ow. These features yield a fully three-dimensional inlet geometry. The cross-section of the inlet can vary from rectangular to more complex shapes.
2 Presentation of the Generic Inlet Optimization Problem
Subsonic Di user
A generic di user is added to the supersonic part previously described, respecting some aerodynamic based design rules for its shape. The upper surface of this di user is kept at. The lower surface is composed of three planes with increasing length and angle. The latter angle respect a 3 angle, in order to limit separation and loss in the subsonic di user. The width of the di user is assumed to be constant. A three-dimensional view of the generic inlet is presented on Figure 1. The geometry model nally requires nine parameters. The parameters used to describe the two-dimensional shape of the inlet are presented on Figure 2. This parametrization of the inlet has proven to allow the investigation of a large spectrum of three-dimensional shapes.
2.1 Overview
The problem is the design of three-dimensional supersonic missile inlet. The function of a supersonic inlet is to capture supersonic ow and eciently decelerate it in order to provide the engine with a sucient mass
ow rate of high total pressure subsonic ow. First, a set of oblique shocks take place in the supersonic part of the inlet and decelerates the supersonic incoming
ow. Then a terminal shock system (an approximate normal shock) occurs in the viscinity of the geometrical throat. Finally, the ow is further decelerated in a subsonic di user. To achieve high performance, the inlet design must be optimized for the ight condition according to a set of constraints imposed by manufacturing considerations and engine speci cations. Therefore the generic inlet problem can be de ned as the maximization of the inlet aerodynamic performance within a space of feasible designs whichs corresponds to the given set of constraints imposed on the inlet.
Y X
Z
2.2 The Objective Function for Optimization Two main coecients are used to assess the aerodynamic performance of the inlet. The total pressure recovery coecient is representative of the eciency of the ow deceleration performed by the inlet. It is de ned as follows: = P texit=P t0, where P texit and P t0 are respectively the averaged total pressure in the exit plane of the subsonic di user and the freestream total pressure. The mass ow rate coecient " is de ned by: Entering The Inlet " = MaximumMassFlow MassFlow Which Could Enter The Inlet and represents the amount of ow captured by the inlet. Therefore, the problem addressed in the following can
Fig. 1 3D view of the generic inlet
Due to this large number of coupled design parameters, an exhaustive search using any human-based design methodology is not possible. Therefore, automated design methodologies based on numerical simulations for performance estimation are of great interest for such complex problems.
3 Methods of Solution
2
The innovative three-dimensional automated optimization process takes ground on the development of
35th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit back a penalty for this candidate inlet based upon the amplitude of the constraint violation (see Figure 3). These features, which prevents from the evaluation of non-feasible inlets from a physical point of view, save a signi cant amount of time during the optimization process, especially at the very begining.
l xe
ac S
a
3
a2
a1 x1
3.1.2 Optimization Algorithms
x2
Two di erent optimization algorithms are implemented within the previous optimization loop: GADO (a Genetic Algorithm) and CFSQP (a gradient based search algorithm). These two algorithms are described in the following sections.
x3
Fig. 2 2D parameters of the inlet
several tools linked within a loop algorithm presented on Figure 3. The optimization software which has been developed and used for the present research can used either a Genetic Algorithm or a gradient-based method. The simulation part has implied the development of a solver fast enough to allow a large number of calls and accurate enough so as to correctly predict the trends between investigated con gurations. Outside of the loop, some veri cations are made using full Navier-Stokes simulations. Generates
Constraints
Optimizer
The rst optimizer included in the optimization loop is a Genetic Algorithm called GADO, which is the acronym for Genetic Algorithm for Design Optimization. This stochastic optimizer rst generates a random population of potential candidates. Then mutations and recombinations are applied to individuals of the population in order to make the population evolve towards better solutions. GADO was developed by Khaled Rasheed8 at the Department of Computer Sciences at Rutgers University. Compared to classical Genetic Algorithms, several improvements have been included that make the search more ecient and reliable for engineering problems. Each individual is represented by a real vector, which is particularly well adapted to the parametric description of the inlet. Several innovative crossover and mutation operators have been developed in order to make the search process fast and accurate, i.e. more likely to nd the global optimum. Depending on the number of iterations allowed for the search, the stage of the optimization process is taken into account. Thus, a guided crossover operator (which mimics a gradientbased method) is applied in the last part of the search, with the view to accelerate the convergence. The shape of the population is also checked to detect premature convergence and a reseeding of the population can be performed in order the avoid the search process to be stuck in a local optimum of the design space. Finally, the penalty function has been tailored by the used of a penalty coecient which increases during the process, so as to guide the search towards feasible regions. This algorithm has already been used successfully for several di erent engineering test cases.5,6,8 The advantage of this optimization tool is its ability to explore large parts of the design space and to reach the global optimum of topologicaly complex design space.
Returns Aerodynamic Performance
If constraints violated
Candidate inlet
The Genetic Algorithm GADO
verifications
Mesh generation
Simulation
Analysis Objective
Performance Analysis Module
Fig. 3 Automated Optimization Loop 3.1 Optimization Methodology 3.1.1 Automated Process Description
The heart of the optimization loop implemented for designing supersonic inlet is called the optimizer. The optimizer generates candidate inlet designs decribed by nine geometrical parameters (see Section x2.3). These parameters are passed to the analysis module which returns the aerodynamic performance of this candidate inlet. Based on these simulation results, the objective function and some aerodynamic constraints are calculated. In order to reduce the computational cost of the design process, some constraints (mainly based on geometrical considerations) are also implemented and checked before running the analysis module. If any of those constraints is violated, no evaluation is undertaken and the optimizer is simply given
The Gradient-Based Algorithm CFSQP
The second optimizer included in the optimization loop is a gradient-based algorithm called CFSQP, 3
35th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit formance evaluation is based on the di erent stages the inlet (see Figure 4). Thus, 2es3d rst uses a Euler simulation to account for the supersonic compression occuring above the inlet ramps. Then corrections are applied for the loss through the terminal shock system and for the viscous losses through the subsonic di user. Figure 5 summarizes the di erent elements of the analysis process performed by 2es3d.
which is the acronym for C code for Feasible Sequential Quadratic Programming. The main advantage of CFSQP is that it requires less evaluations to reach an optimum and so can be an alternative to the use of stochastic optimizer when the design space is more or less continuous and the evaluation cost is expensive. CFSQP has been designed to solve nonlinear problems. It seeks to minimize the objective function with a set of nonlinear constraints. A sequential quadratic programming method is tted to the actual nonlinear problem. Then, the former is solved and a minimization is performed along the line de ned by the current point and the minimum of the quadratic problem. This process is repeated until convergence. This algorithm has already been applied to engineering problems.9 However, the shape of the design space and the starting con guration can have an important in uence on the nal optimum which remains, by definition of this search process, a local one. To correct this feature, a multi-start CFSQP strategy has been investigated to make the search more global and more likely to reach the global optimum.
Supersonic flow
Subsonic flow Shocks
Supersonic diffuser
Throat section
Subsonic diffuser
Fig. 4 Critical operating regime of a supersonic inlet Geometry
ε Mass flow rate coefficient Grid generation
η Total pressure recovery coefficient
3.2 Performance Analysis Methodology The accuracy and usability of any automated design process is mainly grounded on its performance estimation module. Indeed, the analysis part of the process must be accurate enough to predict the trend between candidate inlets and fast enough for the several hundred of three-dimensional inlet analyses needed by the optimization to be performed within an acceptable time frame. To answer these requirements a hybrid ow solver, called 2es3d (an acronym for Euler + Semi-Empirical Simulation 3-D), has been implemented to be used inside the optimzation loop and validated.10
Euler Supersonic Simulation
Fig. 5
2es3d
Terminal shock correction
Subsonic diffuser correction
ε η
automated simulation process
3.2.2 Simulation Models Euler Calculation
To address the supersonic compression which occurs above the ramps and the associated losses through the di erent oblique shocks, a Euler calculation is performed using GASP 11,12 version 3.2 by AeroSoft, Inc. as a Euler ow solver. In this respect, a grid of the inlet has to be generated. This grid is created with GridPro developed by Program Development Corporation. The Euler simulation uses a third order accurate upwind scheme (Van Leer scheme) to compute the inviscid uxes. A Jacobi scheme with inner iterations is used for relaxation. A tangencial velocity boundary condition is applied on all the inlet walls and a supersonic out ow condition is used for the inlet exit plane. GASP as well as GridPro are run automatically without any user intervention, using the possibility, both softwares have, to be run in batch mode.
3.2.1 Simulation Overview
The compression which occurs in the inlet must be achieved with the minimum total pressure loss. Moreover, the engine requires a minimumamount of ow to be captured by the inlet to work in optimal conditions. Figure 4 describes the ow eld in the inlet working in critical operating regime, i.e. in the regime which leads to the maximum eciency for the compression. First, a supersonic compression is performed through a series of oblique shocks. The ow remains supersonic until the throat region, where a shock system close to a normal shock occurs, downstream of the geometrical throat. After this shock system, the ow is subsonic and is compressed along a diverging duct which acts as a subsonic di user. Basically, total pressure losses occur across the various shocks and through the viscous e ects in the subsonic di user. The methodology developed for inlet aerodynamic per-
Virtual Terminal Shock Model (VTS)
For the case of supersonic inlets, the main total pressure loss occurs through the terminal shock system previously described. At the critical operating regime, which is the one to be considered for optimization, the terminal shock is located at the aerody4
35th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit namical throat which is a position known to be close to the geometrical throat or above the boundary layer bleed. This shock is modeled by taking into consideration each individual cell of the mesh at the chosen shock position (see Section x5.5) and by applying the Rankine-Hugoniot formulae for this cell. An averaged value of the total pressure is computed behind the vertical shock. Then the information about the ow just behind the vertical shock are passed to the onedimensional subsonic di user model.
3.2.3 Validation of 2es3d
A complete validation of three-dimensional inlet performance prediction using this method has been conducted.10 The 2es3d software has been proven to be able to predict the trend between inlet con gurations, which is the most important feature for its use inside an optimization loop. Its reliability with regards to the mesh re nement for the Euler simulation has been addressed and has shown a good agreement between ne, medium and coarse meshes. The accuracy of the Virtual Terminal Shock model for di erent positions of the terminal shock was also tested against Euler calculations with back-pressure. The nal result is an ecient simulation tool for inlet performance which can accurately predict the trend between inlet con gurations and allows important reduction in the computational resources needed for optimizations.
Subsonic Di user Model
The losses which occur in the subsonic di user are due to viscous e ects. As boundary layer develops with adverse pressure gradient due to compression, separation is likely to occur in the di user13 , leading to strong reduction of the inlet performance. A onedimensional model of the subsonic di user has been developed based upon a preliminary study done at Stanford University.14,15 The in ow conditions used to initialise the computation are the averaged ow conditions found immediately behind the terminal shock. The ow eld along the di user is calculated using a space marching strategy. A \weak-strong" method is applied when separation is detected to take into account the interactions between boundary layers and the non-viscous core ow. The velocity pro le of ColesVan Driest, corrected when boundary layer separate, is used to calculate the stess along the wall. Finally, the geometry of the di user is taken into consideration through the de nition of its di erent wall-elements. This model, which is able to predict separation accurately,15 has turned out to be more accurate than empirical formulae for subsonic di users.
3.3 Reynolds Averaged Navier-Stokes Veri cations Once the optimization process nds an optimum inlet con guration, a boundary layer bleed is added to this con guration and a Navier-Stokes grid is generated using GridPro. Then, the optimal con guration is ascertained using a Reynolds Averaged Navier-Stockes (RANS) computations performed with GASP . These RANS simulations have been proven to predict the aerodynamic performance of the real inlet with a good accuracy, i.e. within 2% of the experiments.7 The Navier-Stokes veri cations use a k-" two equations model for turbulence modeling, a third order accurate upwind scheme (Van Leer scheme) for inviscid uxes and a central di erencing scheme for viscous
uxes. Time integration is performed with a Jacobi relaxation scheme with inner iterations. The grid used for these computations presents a re nement near the walls and near the cowl leading edge which allow the boundary layer to be correctly resolved. Moreover a boundary layer bleed is added to the original duct. Noslip boundary conditions are applied at all inlet walls while a imposed back-pressure is imposed at the inlet exit plane. In order to reach the critical operating regime of the inlet, the back pressure is progressively increased, moving the terminal shock system upstream in the diffuser. The critical operating regime is reached when the terminal shock is located in the viscinity of the geometrical throat, close to the bleed slot.
Bleed Implementation
Supersonic inlets usually include a boundary layer bleed located in the throat region. The bleed role is two fold. Basically, it aims at removing the low energy
uid from the boundary layer which has developed on the supersonic compression ramps in order to prevent separation in the subsonic di user. Furthermore, it also helps to stabilize the terminal shock system close to the throat section (see Figure 4). In the present study, the bleed has not been taken into account for optimization, i.e. in 2es3d, since no boundary layer is computed in the Euler calculation performed by 2es3d. This strategy takes ground on the assumption that, given an optimal inlet found with the present method, it is possible, a posteriori, to add to the optimal inlet a bleed which removes a sucient amount of ow, so as to lead to an excellent nal design. This assumption has been validated by performing RANS simulations of the optimal inlet design on which two di erent boundary layer bleeds have been added (see Section x5).
4 Results of Optimizations for the Mach 3-Cruise conditions Inlet
5
Optimizations have been performed using both optimization algorithms presented in Section x 3.1.2. The objective was to compare the behavior of these two dif-
35th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit ferent methods relatively to this speci c problem. In this respect, the results of the optimizations are thoroughly analysed and the eciency and the reliability of the two methods are compared. Moreover several optimization parameters have been addressed and investigated through the di erent optimizations.
Entrance Cowl Section Area)=(Throat Section Area), is compared to the maximum contraction ratio which would allow a Pitot type inlet to self-start at the Mach number value calculated previously (Mach number at the the cowl entrance). This simpli ed criterion, which has been used succesfully in previous two-dimensional optimizations16 has been chosen to avoid an additional costly ow eld calculation. During all optimizations, the inlet was required to be self-started at Mach 2:5. (
4.1 Flight Condition of Optimization One ight condition has been investigated in this study. The inlet has been optimized for cruise condition. These condition was chosen to be representative of the cruise stage a typical missile is required to y consuming as little fuel as possible. The aerodynamic conditions are summarized in Table 1. The Mach number is 3, for an approximative altitude of 52000 feet. Moreover the inlet faces neither angle of attack nor side-slip angle. These conditions have been kept for all performed optimizations. Flight Condition Cruise Side-slip angle 0 Angle of Attack 0 Total Pressure 378000 Pa Total temperature 607 K
Constraint on the Mass Flow Rate
The inlet has to be adaptated to a given engine. The mass ow captured by the inlet is so required to be large enough to feed this engine. A constraint is consequently applied on the mass ow rate crossing the inlet. The mass ow rate captured by the inlet is measured through the mass ow rate coecient, ". Typically, this coecient is calculated using the mass
ow rate going through the capture area of the inlet as reference. However, since this reference is linked with the geometry of the inlet, its value changes with the candidate inlet during the optimization. That is why a constant mass ow rate reference has been used for the present optimization, regardless of the candidate inlet capture area. This strategy is nally consistent with the mass ow rate requirement.
Table 1 Aerodynamic Conditions 4.2 Objective Function
4.4 GADO Optimizations Four optimizations have been performed using the Genetic Algorithm GADO as the optimizer. Di erent settings were used for the optimizer. The results of these optimizations are presented and analysed in the following sections.
Both of the optimizers used for this research aim at minimizing the objective function. Since the targeted objective was to optimize the total pressure recovery of the inlet, the objective function passed to the optimizer is ?, the negative of the total pressure recovery coecient.
4.4.1 Description of the Four Optimizations
4.3 Constraints
The four optimizations performed with GADO and its nal results are described in Table 2. Some GADO settings have been investigated through these optimisations such as the total number of iterations allowed to the optimizer and the size of the active population of GADO. Run Iterations Population Random max seed GADO 1 1200 60 23 0.69 GADO 2 1200 60 17 0.71 GADO 3 2400 90 23 0.71 GADO 4 2500 80 37 0.75
Geometrical Constraints
Prior to the analysis code call, some geometrical constraints are checked, based on the nine geometry parameters given by the optimizer. These constraints allow to eliminate, early in the process, unfeasible geometries according to manufacturing or physical considerations. This feature saves the time of a performance analysis with 2es3d. Unstart of the Inlet
Since the inlet is required to be self-started at a lower Mach number than the cruise one, a unstart criterion is applied, prior to the analysis code, to predict if the candidate inlet design is started in the required condition. The unstart constraint is handled using an approximate criterion. First, the Mach number value at the cowl entrance is computed using Rankine-Hugoniot formulae and taking into account only the two-dimensional geometry of the inlet. Then, the actual contraction ratio, de ne as
Table 2 GADO optimizations
The random seed is the number used to initialize the random process used for generating GADO initial population.
6
35th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit have been tried and two methods for passing the geometry parameters to CFSQP compared.
4.4.2 Results Analysis
The convergence history of all optimizations are presented in Figure 6. This graph shows that a total pressure recovery of = 0:60 is quickly reached by GADO during the rst 50 call to 2es3d. After this rst stage, the search process reveals several at zones for . These at zones are related to the exploration of new regions in the design space, which correspond to families of good designs. Each time a new best design is found, GADO further investigates the designs close to this point. The maximum total recovery reached during these four GADO optimizations are relatively close, and range between = 0:69 and = 0:75.
4.5.1 Description of the Five Optimizations
CFSQP, as a gradient-based optimizer, requires a starting point to initialise the search process. Then, prior to CFSQP optimizations, a random probe search was performed on the same design problem so as to nd feasible points. The rst two feasible designs found were used to start the gradient-based search. They are refered as rp1 and rp2 in the following. Their total pressure recovery are presented in Figure 3. Design rp1 0.44 rp2 0.48
0.75
Total pressure recovery
0.7
Table 3 Random probe designs used as CFSQP starting point
0.65 0.6
Figure 8 shows that these two designs are geometricaly very close. The rst one, rp1 will be used as the reference to compare the designs obtained with CFSQP.
GADO 1 0.55
GADO 2 GADO 3
0.5
GADO 4
Random probe1 Random probe 2
0.45 0.4
0
500 1000 Calls to the objective function
1500
Fig. 6 GADO convergence history
The optimal geometries found by GADO all have an non zero sidewalls opening angle, demonstrating the interest of taking into consideration the threedimensional feature of the inlet. Figure 7 shows the two-dimensional shape of the four optimal designs. In three cases out of four, the designs reached by GADO are similar. The design found by the second GADO optimization appears to belong to a di erent family, pointing out that several families of design can have similar level of performance. It also underlines the capability of GADO to nd the di erent designs leading to the same optimum performance, if such designs exist. From the short range of total pressure recovery and considering the uncertainty of the performance analysis, these four optimal designs may be considered as equaly good.
Fig. 8 Random probe design geometry
The perturbation used by CFSQP in approximating the gradients by nite di erence is called udelta. Different values of udelta have been investigated, showing its important in uence on the optimization behaviour. Moreover, udelta equally a ects the nine design variables used to parametrize the inlet geometry which are passed to CFSQP, whether they are angle or distance. That is why the use of a non-dimensional vector of variables has also been investigated. 4.5.2 Results Analysis
The description of the ve CFSQP optimizations is provided in Table 4. The setting parameters and the total pressure recovery are also described in this Table. Depending on the runs, from 9 to 25 line searches were performed during these optimizations. Convergence history is presented in Figure 9 for the ve di erent runs. The run CFSQP 1 has been performed with dimensional design variables and a low value for udelta. It has been unable to achieve a significant improvement of the total pressure recovery and the evolution between the initial geometry rp1 and the nal one is very limited, as can be seen on Figure 10. With this low value of udelta, CFSQP has explored a very limited region of the design space around the
Fig. 7 GADO four optimal geometries 4.5 CFSQP Optimizations Five optimizations were performed with the gradientbased optimizer CFSQP. Di erent settings of CFSQP 7
35th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit Run CFSQP 1 CFSQP 2 CFSQP 3 CFSQP 4 CFSQP 5
Start. Dimensional udelta point rp1 Yes 0.001 rp1 Yes 0.01 rp1 No 0.05 rp1 No 0.1 rp2 No 0.1
the number of call to the performance analysis code which has been required (and consequently, the computational cost) was less important. Figure 12 shows the two di erent geometries obtained with runs CFSQP 2 and 3, compared with the geometry of rp1. Larger variations for the angular parameters have been achieved with these two runs. Finally, the use of nondimensional parameters has proved to be more consistent with the gradient-based search and has leaded to better results in term of time needed to achieve a targeted improvement.
max
0.48 0.65 0.63 0.64 0.66
Table 4 CFSQP optimizations
starting point, which explains the poor result obtained in term of total pressure recovery. 0.7
Total pressure recovery
0.65
0.6 CFSQP 1 CFSQP 2 CFSQP 3 CFSQP 4 CFSQP 5
0.55
Fig. 12 Optimal geometries CFSQP 3 and 4 compared to rp1 geometry
The run CFSQP 5 has been performed using rp2 as starting point and non-dimensional design variables. It has leaded to an optimal inlet whose total pressure recovery is close to runs 2, 3 and 4. The geometry of the optimal inlet presented in Figure 13 is also very similar to those from the three previous runs. Moreover, the two last runs showed that, starting from close point rp1 and rp2, CFSQP has explored the same optimal region.
0.5
0.45
0.4
0
50 100 150 Calls to the objective function
200
Fig. 9 CFSQP convergence history
Fig. 10 Optimal geometry CFSQP 1 compared to rp1 geometry
Fig. 13 Optimal geometries CFSQP 5 compared to rp1 geometry 4.6 Comparison Between Optimizers
The run CFSQP 2 used the same settings as run CFSQP 1, except for the parameter udelta which was increased by one order of magnitude. The improvements achieved with these settings are much better, compared to run CFSQ 1. However the evolution of angular design variables have been much slower than the distance based variable, due to the use of dimensional design variable (the range of the angle are much lower than the one of distance). According to Figure 9, the search process has met a large plateau with poor improvements between the 30th and 110th calls of the performance analysis code.
From the convergence history curves in Figures 6 and 9, it appears that CFSQP, even used with a udelta parameter adaptated to the design space of the problem and with non-dimensional variable is not likely to reach a targeted value of total pressure recovery faster than GADO. For instance, CFSQP is not faster than GADO to reach a target value of total pressure recovery of = 0:6, as proves Figure 14. Moreover CFSQP then stucks in a local optimum, when GADO continues to nd better performing inlets. Indeed CFSQP stops at = 0:61 when GADO continues to achieve improvement and nally reaches total pressure recovery values between = 0:69 and = 0:75. From the present results, a multi-start strategy using CFSQP does not seem to be a interesting alternative to the use of GADO, if the goal is achieving a given value of total pressure recovery whitin the smallest amount of time. Furthermore, GADO proved to be more likely
Fig. 11 Optimal geometry CFSQP 2 compared to rp1 geometry
Runs 3 and 4 of CFSQP have been performed using non-dimensional design variables. If the improvement achieved for if comparable to the run CFSQP 2, 8
35th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit boundary layer to the total pressure evolution in the duct mesh re nements have been applied on the inside duct. The parameter describing these re nements are provided in Table 5. Mean Min Max St. Dev. + y 1:0962 0:1852 5:8052 0:9104
Total pressure recovery
0.7
0.6
Table 5 Grid parameters for RANS simulations 5.2 De nition of the Boundary Layer Bleed
CFSQP 5 GADO 4 0.5
0.4
0
50
100 150 200 Calls to the objective function
250
No boundary layer bleed has been considered during optimization, since non-viscous modelization is used. For RANS computations, a bleed must be added so as to limit the separation which could occur in the subsonic di user. The bleed aims at removing the low energy ow from the the boundary layer, before it enters the subsonic di user. Current bleeds were designed to remove 5:8% of the capture ow. Two di erent bondary layer bleeds have been de ned, leading to two RANS calculations. The twodimensional shape of bleeds is described in Figure 16. It includes a slot, making the connection with the duct. The plenum is terminated by a sonic throat, in order to control the exiting mass ow rate. The two geometries di er mainly by the lenght of the slot: the slot of the second bleed is smaller (see Figure 17).
300
Fig. 14 Comparaison between the best GADO run and the best CFSQP run to nd a design close to the global optimum than CFSQP.
5 Results of Reynolds Averaged Navier-Stokes Veri cations
In order to ascertain the quality of the results obtained with the automated design process which was based on the empirical ow solver 2es3d, several Reynolds Averaged Navier-Stokes computations have been performed on one of the optimal geometries achieved during the several runs done with the two di erent optimizers (run GADO 1). RANS computations have been performed for both unstart and cruise conditions used during the optimization.
Third ramp
Bleed slot
Bleed plenum Sonic throat
5.1 Grid Generation
Fig. 16 Bleed geometry
A grid of the optimal inlet given by the run GADO 1 has rst been generated using the same meshgenerator as for optimization (GridPro from Program Development Corporation). The resulting grid is composed of height domains for the inlet duct, and one other domain for the boundary layer bleed. The geometry of the bleed is described in Section x5.2. The inlet duct has 247428 cells. A three-dimensional view of this grid is presented in Figure 15.
According to this de nition, grids of the two boundary layer bleeds were created with GridPro. These two grids are presented in Figure 17 and are both composed of 31360 grid cells. The bleed grid was added to the duct grid using the non-overlapping feature of GASP . Bleed Geometry 1
Bleed Geometry 2
Fig. 17 Three-dimensional mesh for the boundary layer bleed 5.3 Naviers-Stokes computations settings Fig. 15 Three-dimensional grid for RANS computations In order to take into account the contribution of the
All RANS computations has been performed using the
ow solver GASP . A Van Leer third order accurate upwind scheme was used for the inviscid uxes and a 9
35th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit central di erencing for the viscous uxes. The time relaxation has been performed using a Jacobi scheme, with inner iterations and local time step. A two equations k-" model developed for low Reynolds numbers has been used (Chien model) for turbulence modeling. The methodology used to simulate the critical regime of the inlet is the following. First a fully supersonic computation of the inlet, using a supersonic out ow boundary condition at the exit plane of the inlet was converged. Then, this boundary condition was replaced by a sequence of back pressures and the solution reconverged until reaching the desired regime which is detected as the one leading to the maximumtotal pressure recovery. These RANS computations were performed at NCSA on the SGI O2000 system, taking advantage of the parallel capability of GASP .
RANS simulation
Fig. 19 Pressure contours from RANS computation at unstart Mach number = 2 5 M
5.4 Veri cation of the Unstart An approximate criterion was used during optimization process to verify if the inlet could self-start at the desired Mach number. To examine the accuracy of this criterion two simulations of the optimal inlet were performed at the unstart Mach number. The rst one is an Euler computation and does not take into consideration the bleed. The second one is a full RANS computation performed with the rst boundary layer bleed presented in section x 5.2. Both calculations give the optimal inlet to be unstarted at the desired Mach number, as can be seen on Figures 18 and 19. Those two gures show the pressure contour plots in the symmetry plane of the inlet. A strong normal shock is revealed along the last compression ramp, proving the unstart of the inlet.
5.5 Veri cations of the Inlet in Cruise Condition Total Pressure Recovery
Two RANS computations were performed using the two di erent boundary layer bleeds. First, the computations with the rst bleed geometry was converged in critical operating regime, using the methodology presented in Section x5.3. Seven di erent back-pressures were successively applied at the inlet exit to reach the critical regime. These back-pressures go from 170000 Pa to 225000 Pa. The targeted regime was reached with a 220000 Pa back-pressure. The total pressure recovery obtained from this rst computation (bleed 1) is max = 0:616. Then the same methodology was applied for the second computation which has been performed with the second bleed geometry, taking advantage of the fully supersonic solution of the rst computation. This second RANS computation gave a result within 1% of the rst RANS result and the critical regime was obtained with the same back-pressure. These RANS results proves the low in uence of the boundary layer bleed slot size on the optimal performance of the inlet in term of total pressure recovery. Nevertheless, the second bleed geometry seemed to lead to a more stable state of the computation at critical regime. For the same inlet, 2es3d predicted a total pressure recovery of 2es3d = 0:69. The discrepancy between the RANS results and 2es3d result is thus 12%. It
Euler simulation
Fig. 18 Pressure contours from Euler computation at unstart Mach number = 2 5 M
:
the physics and does account neither for the lateral compression performed by the opening side-walls nor for the boundary layer developement along the supersonic ramps. Both these two e ects are known to lead to a higher self-start Mach number of the inlet. A better unstart criterion based on a threedimensional Euler computation is currently being tested. It will allow to consider the fully threedimensional features of the compresion. Moreover a one-dimensional boundary layer calculation, based upon the results of the previous Euler computation could also help to build a more accurate unstart criterion to be used during the optimization process.
:
However, the unstart problem is very complex and linked with unsteady phenomenon, which are not fully addressed by the present computations. Thus, the initial condition (a uniform ow eld in freestream conditions) may have in uenced the nal steady ow obtained with these computations. Nevertheless, these two computations show the limit of the unstart criterion used during optimization. Indeed, this criterion is based on a two-dimensional non-viscous analysis of 10
35th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit can be explained by the location of the terminal shock was Q_ 2es3d = 1:08 kg/s, whereas RANS computasystem. Indeed, it is to be noticed that the termi- tions predict a mass ow rate at the inlet exit of nal shock predicted by RANS computations is located Q_ RANS = 0:975 kg/s and a mass ow rate through at the downstream lip of the bleed, as presented on the bleed of Q_ bleed = 0:087 kg/s. The total mass ow Figure 20. In 2es3d, the Virtual Terminal Shock is ingested by the inlet is so Q_ tot = 1:062 kg/s, which assumed to be located at the geometical throat of the is 1.7% lower than the predicted mass ow rate by inlet, which roughly corresponds to the upstream lip 2es3d. of the bleed. Then by applying the VTS model at the downstream lip of the boundary layer bleed, the 6 Conclusion and Perspectives new prediction of 2es3d was 22es3d = 0:61, which is within 2% of the RANS values. This is in keeping with An automated optimaldesign process have been implethe previous work10 showing that, if applied at the cor- mented for the optimization of fully three-dimensional rect location, the VTS model discrepancy was no more supersonic inlets. Two di erent optimization algothan 5%. Finally, the RANS solutions analysis showed rithms have been used and compared through several the presence of a subsonic zone under the cowl of the optimizations. Compared to human decision based inlet, which can in uence the terminal shock location design cycle currently used in industries, this innovaat the critical regime. Therefore, in order to improve tive optimization strategy allows to investigate a larger the VTS position, a new criterion was successfully im- number of con gurations, while looking for maximum plemented: instead of assuming the VTS location to be aerodynamic performance of the inlet under speci c at the geometrical throat, a sweeping from the down- constraints. The use of arti cial intelligence techstream position in the inlet is performed, looking for niques guides the search process toward high interest the furthest point where the ow is fully supersonic. regions of the design space, minimizing the number These new methodology leaded to the following re- of computations required to reach a high performance sults: 23es3d = 0:635, within 3% of RANS results. All level. The results of the optimizations performed during this study proved the automated design process this results are gathered is Table 6. to be a robust and e ective tool, ready for industrial applications. Moreover the implementation of the automated design process is highly recon gurable so as to be easily adapted to new design problems. Taking advantage of these features, multi-objective optimizations for a whole missile mission and multi-disciplinary optimizations could be performed in the near future. RANS computations
Optimized Inlet with Bleed 1
Back pressure 220 000 kPa : Pressure contours
RANS computations
Acknowledgments
Optimized Inlet with Bleed 2
Technical and nancial support was provided by AEROSPATIALE missiles, France. Computational resources were provided by NCSA under Grant #DDM980001N. We would like to thank Michael Blaize, Xavier Montazel, Yan Kergaravat and Donald Smith for their invaluable assistance in this research.
Fig. 20 Pressure contours at critical conditions given by RANS computations Method Disc. RANS with bleed 1 0:616 { RANS with bleed 2 0:621 1% VTS at geometrical throat 0:69 12% VTS at downstream bleed lip 0:61 2% VTS at last position for M > 1 0:635 3%
References
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Table 6 Total pressure recovery predicted by the di erent methods Mass Flow Rate
During the optimizations, the mass ow rate was considered as a constraint and was required to be larger than Q_ ref = 1:08 kg/s. For the current optimal inlet, the mass ow rate predicted by 2es3d 11
35th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit 5 Blaize, M. and Knight, D., \Automated Optimization of TwoDimensional High Speed Missile Inlets," 36th AIAA Aerospace Sciences Meeting, January 1998, AIAA Paper 98-0950. 6 Blaize, M., Knight, D., Rasheed, K., and Kergaravat, Y., \Optimal Missile Inlet Design by Means of Automated Numerical Optimization," 82nd AGARD Fluid Dynamics Panel Symposium on Missile Aerodynamics, 1998. 7 Montazel, X., Kergaravat, Y., and Blaize, M., \Navier-Stokes Simulation Methodology for Supersonic Missiles Inlets," 13 th ISABE , Sept. 7-12 1997, AIAA-97-3147. 8 Rasheed, K., GADO : A Genetic Algorithm for Continuous Design Optimization , Ph.D. thesis, Department of Computer Science, Rutgers University, January 1998. 9 Lawrence, C., Zhou, J., and Tits, A., A C Code for Solving Constrained Nonlinear Optimization Problems Generating Iterates Satisfying all Inequality Constraints , ElectricalEngineering Department and Institute for System Research, University of Maryland. 10 Bourdeau, C., Blaize, M., and Knight, D., \Performance Analysis for an Automated Optimal Design of High Speed Missile Inlets," 37th AIAA Aerospace Science Meeting , January 11-14 1999, AIAA Paper 99-0611. 11 Aerosoft, Inc., \GASP , General Aerodynamic Simulation Program Version 3 User's Manual," Aerosoft, Inc., May 1996. 12 Aerosoft, Inc., \GASP Version 3.2, The General Aerodynamic SimulationProgram, User's Manual Addendum," Aerosoft, Inc., Feb. 1998. 13 Kline, S., Abbott, D., and Fox, R., \Optimum Design of Straight-Walled Di users," Journal of Basic Engineering , September 1959, pp. 321{331. 14 Childs, R. and Ferziger, J., \A Computational Method for Subsonic Compressible Flow in Di users," 21th AIAA Aerospace Sciences Meeting , January 10-13 1983, AIAA paper 83-0505. 15 Johnston, J., \Di user Design and Performance Analysis by Uni ed Integral Methods," 33rd AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit , July 6-9 1997, AIAA paper 97-2733. 16 Blaize, M., Automated Optimization Loop for TwoDimensional High Speed Inlet ver 5.6 , Aerospatiales Missiles, 1998.
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