Design Optimizationof the Intake of a Small-Scale Turbojet ...

Copyright c© 2007 Tech Science Press CMES, vol.18, no.1, pp.17-30, 2007

Design Optimization of the Intake of a Small-Scale Turbojet Engine

R. Amirante1, L.A. Catalano2, A. Dadone1 and V.S.E. Daloiso1

Abstract: This paper proposes a gradient-basedprogressive optimization technique, which can beefficiently combined with black-box simulationcodes. Its efficiency relies on the simultaneousconvergence of the flow solution, of the gradientevaluation, and of the design update, as well ason the use of progressively finer grids. The devel-oped numerical technique has general validity andis here applied to the fluid-dynamic design opti-mization of the intake of a small-size turbojet en-gine, at high load and zero flight speed. Two sim-plified design criteria are proposed, which avoidsimulating the flow in any turbojet componentsother than the intake itself. Using a geometricallyconstrained polynomial profile, both design op-timizations have been produced in less than theamount of computational work to perform nineflow analyses; moreover, both optimizations haveprovided almost coincident intake profiles. Neg-ligible performance improvements have been ob-tained by removing one geometrical constraint,at the price of almost tripling the CPU time re-quired. Finally, the original and the optimal pro-files have been mounted on the same small-scaleturbojet engine and experimentally tested, to as-sess the resulting improvements in terms of over-all performances. All numerical and experimentalachievements can be extended to the intake of amicroturbine for electricity generation.

Keyword: Design optimization, Finite-Difference Progressive optimization, microtur-bine intake.

1 Dipartimento di Ingegneria Meccanica e Gestionale, Po-litecnico di Bari, Bari, Italy

2 Contact author. DIMEG-Sezione Macchine ed Energet-ica, CEMeC, Politecnico di Bari, Bari, Italy. E-mail: [email protected]

1 Introduction

In the last decade, the interest in microturbinesand their industrial employment have significantlygrown up thanks to the very high power/weightratio, to the small environmental impact and tothe potentialities of achieving high efficiencies.In particular, this recent development responds tothe need for reliable and environmentally friendlyelectricity generators for the new markets pro-vided by the deregulation [Carno (1998)]. In-deed, applications involving electric power rang-ing from 20÷80 kW to 1 MW are consideredas routine applications requiring simple and inex-pensive plants and characterized by low pollutantemissions, especially low NOx emissions.

Meanwhile, the use of microturbines in bothcivil and military aeronautic and aerospace ap-plications has become more and more attractive:microturbines have been already employed forpower generation in small radio-controlled spotterplanes, in the spy-rockets and in the MIAI Abramstanks as auxiliary power generators. More re-cently, the development of unmanned aerial ve-hicles (UAVs) has increased the interest for smallturbojet engines [Carno (1998)] derived from tur-bocharger rotor components. A small-scale tur-bojet engine can also be employed as gas genera-tor core for small ramjet engines, powering super-sonic UAVs. For both applications, i.e., for smallportable power generation systems and for mini ormicro UAVs, the potentially very high power den-sity of the gas turbine allows a strong reductionin battery, and thus of the overall system weight[Decuyepere and Verstraete (2005), Fernandez-Pello (2005), Guidez, Ribaud, Dessornes, and Du-mand (2004), Hendrik, Verstraete and De Bruyn(2004)].

In order to be competitive with large-scale gasturbines and reciprocating engines, the microtur-

18 Copyright c© 2007 Tech Science Press CMES, vol.18, no.1, pp.17-30, 2007

bines should offer comparable thermal efficiency.The most common strategy to improve efficiencyis to preheat the inlet air to the combustor in arecuperator using the exhaust heat [Watts (1999),Hamilton (1999)]. Obviously, the reduced dimen-sions give rise to specific problems which havebeen addressed in the literature, where some cri-teria involved in the design of heat recuperatorscan be found [Jacobson (1998), Rodgers (1999)Rodgers (1997), McDonald (1996)]. Further-more, a careful design of microturbine compo-nents can significantly increase the thermal (orconversion) efficiency. Therefore, robust but rel-atively simple design methodologies are needed,in order to improve the competitiveness of thesesmall-scale equipments, while reducing their de-sign costs. In this context, the automatic de-sign of microturbines employing CFD and opti-mization techniques has attracted the interest ofmany researchers. This is also due to the possibil-ity of exploring the effects of the scale reductionwith respect to the standard turbine plants, e.g.,the combustion chamber is considered in Jacob-son (1998) and Waitz, Gauba and Tzeng, (1998),while Guidez, Dumand, Courvoisier and Orain(2005) explore the lower Reynolds number ef-fects, the high level of external heat losses andthe tip clearance controlling. Guidez, Dumand,Courvoisier and Orain (2005) also simulate thebehaviour of different parts of the entire plant, i.e.,the centrifugal compressor, the combustion cham-ber, the turbine.

In this paper we are concerned with the intake ofmicroturbines for either electricity generation orpropulsion, because its improved design can in-fluence the fluid-dynamic behaviour of the othercomponents and the overall efficiency. In partic-ular, the intake of the Pegasus small-scale turbo-jet engine, produced by AMV (Amsterdam, TheNetherlands), has been here considered and nu-merically analyzed to verify the boundary layergrowth and the velocity profile at the inducer in-let. As previously underlined, the small scale andthe simplicity of the microturbine under investiga-tion do not justify complex and expensive designefforts. Therefore, the presence of the rotor bladesand the shaft rotation have been neglected, so that

the turbulent flow in an axisymmetric nozzle hasbeen computed, using the CFD code Fluent [Flu-ent 6.1 Inc. (2003)]. The computed flow-fieldshows a wide recirculation zone at the inducer in-let, spanning almost 8% of the inducer inlet sec-tion. This result suggests that the intake profilecan be more properly designed, with the aims ofreducing the fluid-dynamic losses, improving thevelocity profile at the inducer inlet and increasingthe flow rate.

A trial and error approach, often used in the in-dustrial environment, requires large amounts ofcomputer time and man-power, and may not leadto a full optimization. Many recent contribu-tions are available, which regard the develop-ment of design optimization methods for turbo-jet components and for entire aircraft configura-tions: P ’ascoa, Mendes, Gato and Elder (2004)propose a numerical procedure which allows torecover the blade geometry that gives the de-sired aerodynamic blade load distribution by re-lating the axial distribution of the mean tangen-tial velocity component through the cascade withthe blade camber-line angle. Morino, Bernar-dini and Mastroddi (2006) have recently devel-oped a multi-disciplinary optimization method forthe conceptual design of innovative aircraft con-figurations, based on the integrated modeling ofstructures, aerodynamics, and aeroelasticity. Bothmethods require the availability of the source filesof the analysis codes. For such a reason, an al-ternative, efficient, automatic optimization strat-egy that can be used in conjunction with a black-box commercial code is needed. Genetic al-gorithms are unaffordable, because they requiremany flow analyses, even using optimal meshpartitioning and float-encoded genetic algorithms[Rama Mohan Rao, Appa Rao and Dattaguru(2004)]. An alternative approach, that can becombined with black-box commercial codes, hasbeen recently proposed by Poloni, Giurgevich,Onesti, and Pediroda (2000): a genetic algorithmis used only for few generations to start the op-timization process; then, a gradient-based tech-nique is used, the sensitivity derivatives beingevaluated by means of a properly trained neuralnetwork. The proposed method is rather complex

Design Optimization of the Intake of a Small-Scale Turbojet Engine 19

and still expensive, because it requires the amountof computational time to perform many configu-ration simulations. Dadone and Grossman (2000)have recently proposed a more efficient technique,named progressive optimization, which is capableof performing a complete design optimization inthe amount of computational work to perform fewflow analyses. However, the adjoint method ofDadone and Grossman (2000) does not allow theuse of black-box analysis codes, since the sourcecode needed to derive the adjoint system is notavailable. Incidentally, even if the source codewere available, the derivation of the adjoint sys-tem would be very cumbersome for equations in-cluding turbulence.

In order to overcome these limits, the present pa-per introduces a gradient-based progressive op-timization procedure, which can be simply andefficiently combined with commercial black-boxsimulation codes. The efficiency of the proposedoptimization procedure relies on the simultaneousconvergence of the flow solution, of the gradientevaluation and of the design update, as well as onthe use of progressively finer grids. Hence, the so-lution accuracy increases while approaching theoptimum and the initial configurations can be op-timized at a very low computational cost.

The design optimization of the intake of the Pega-sus small-scale turbojet engine is approached as asingle-point optimization problem. Nevertheless,experimental results will demonstrate that simi-lar improvement performances are obtained overthe entire working range. The entire procedurecould be reformulated as a multi-point optimiza-tion problem by employing the auto-adjustingweighted object optimization proposed by Zhu,Liu, Wang and Yu (2004).

First, Section 1 provides a brief description of thePegasus small-scale turbojet engine and the nu-merical simulation of the flow field in the existingintake. Then, Section 2 describes the proposedgradient-based optimization procedure for black-box analysis codes, which is validated in Sec-tion 3, by means of an inverse design application.Furthermore, Section 3 presents the direct designoptimization of the intake profile by employingtwo different objective functions. The resulting

optimal profiles, some details of the correspond-ing flow fields, and a demonstration of the com-putational efficiency are provided. Finally, Sec-tion 4 describes the experimental rig employed totest the turbojet engine, and compares the over-all measured performances of the reference intakewith the optimized intake performances.

2 Flow analysis of the Pegasus turbojet en-gine intake

The Pegasus small-scale turbojet engine producedby AMV (Amsterdam, The Netherlands) is com-posed by the intake, a single-stage centrifugalcompressor, with a radial and an axial diffuser, anannular combustion chamber, a single-stage axialturbine and a convergent exhaust nozzle.

This paper considers only the first component ofthe Pegasus engine, with the aim of analyzing theinfluence of the nozzle profile on the boundarylayer and on the velocity profile at the rotor in-let, as well as of optimizing this profile. A sim-plified axisymmetric steady flow model has beendefined by neglecting the influence of the induceron the inlet flow; as well, the shaft rotation hasbeen neglected. Consequently, the axial part ofthe inlet nozzle has been extended three diameters(180mm) downstream of the rotor inlet section. Aconstant outlet pressure, pu = 0.93atm, has beenimposed at this section, far downstream of the sec-tion of interest (refer to Fig. 1). The imposedpressure value corresponds to high-load work-ing conditions of the Pegasus engine. The ex-ternal intake diameter is De = 57.3 mm, whereasthe shaft diameter is Di = 18 mm. The rotor-shaft blind nut is represented by an elliptical arcwith semi-axes a = 6mm and b = 9 mm. Theshaft has been considered as a fixed solid bound-ary. The computational domain has been extendedslightly upstream of the intake inlet, by empiri-cally defining an inlet surface with enforced nor-mal direction of the velocity vector and ambienttotal conditions (p0 = 1atm, T0 = 293K). Thisaxisymmetric geometry has been discretized bymeans of a structured grid with 8432 cells (referto Fig. 1). The compressible, turbulent, steady,axisymmetric flow has been computed by meansof the CFD code FLUENT c©, version 6.1.18 [Flu-


ent 6.1 Inc. (2003)], using a three-level full-multigrid technique to accelerate convergence tosteady state.

Figure 1: Computational structured grid for thesimplified intake geometry.

Fig. 2 shows the axial shear stress distribution onthe outer wall, computed using three well-knownturbulence models. The three shear stress distri-butions are very similar to each other: all of themshow that separation occurs immediately down-stream of the corner between the convergent ductand the axial one. The corresponding recircula-tion bubble is clearly shown in Fig. 3, which pro-vides the streamlines computed using the Spalart-Allmaras turbulence model.

Figure 2: Axial shear stress on the outer wall com-puted with different turbulence models.

Figure 3: Zoom of the streamlines (Spalart-Allmaras turbulence model).

Figure 4: Mach-number contours (Spalart-Allmaras turbulence model).

The re-attachment occurs sufficiently ahead of theinducer inlet section, denoted by a dashed line,thus proving the hypothesis of neglecting the in-fluence of the inducer blades to be realistic. Thelow total-pressure flow region downstream of therecirculation bubble significantly perturbs the ve-locity profile at the inducer inlet, as shown by theMach-number contours provided in Fig. 4. Corre-spondingly, the relative velocity profile becomesskew at the blade tip, where no additional twistinghas been provided by the manufacturer to com-pensate this effect. This low total-pressure flowaffects approximately 8% of the rotor inlet sec-tion, causing also a reduction of the mass flowrate. These arguments suggest to re-define theintake profile with the aim of eliminating the re-circulation bubble and of reducing the boundary-layer growth. This goal requires a numericaldesign optimization technique, since the simplerstrategy of choosing a reasonable smooth profiledoes not guarantee suitable operating conditions,as it will be shown in Section 3.1.


3 Progressive optimization technique

A general methodology to optimize the describedconfiguration, as well as other fluid-dynamiccomponents, is proposed in this section. Itsstarting point is the progressive optimizationtechnique presented in [Dadone and Grossman(2000)] and applied to the design of ducts, airfoilsand cascades. However, the adjoint method ofDadone and Grossman (2000) does not allow theuse of black-box analysis codes, since the sourcecode needed to derive the adjoint system is notavailable. Incidentally, even if the source codewere available, the derivation of the adjoint sys-tem would be very cumbersome for equations in-cluding turbulence. Thus, the adjoint formulationhas been here replaced by a finite-difference ap-proach. This increases the computational work re-quired to find out the optimal solution. The com-putational work is also affected by the numberof design variables. Indeed, for each global stepof the optimization process, one (not fully con-verged) computation of the flow field is requiredfor each perturbed design parameter in order toevaluate the corresponding sensitivity derivative.Nevertheless, the very high efficiency due to thesimultaneous convergence of the flow solutionand of the optimization process, as well as to theuse of different grid levels, is retained. Moreover,this approach can be combined with all commer-cial black-box codes currently used for the analy-sis of engineering systems and components.

After defining the objective function, I, its mini-mum value can be efficiently found as follows:

(1) fix an initial vector of n design variables,ξ 0 =

(ξ 0

1 , ... , ξ 0n

), which define the intake

profile;

(2) start the flow computations on a coarse grid;

(3) advance the flow solver for several iterationsand compute the objective function, I�;

(4) add a small increment, Δξ i, to the value ofeach design parameter and perturb the grid;using a restart solution, compute the flow withthe same convergence level, and evaluate theobjective function, I�

i ;

(5) compute the objective function gradient bymeans of finite differences:

∂ I∂ξi

=I�i − I�

Δξi, i = 1, . . .,n; (1)

(6) update the design variables according to therelation:

ξ �+1i = ξ �

i −ai∂ I∂ξi

, i = 1, . . .,n, (2)

where ai are positive parameters;

(7) repeat steps 3 to 6 until the norm of the objec-tive function gradient is sufficiently reduced;

(8) refine the mesh;

(9) repeat steps 3 to 6 until the norm of the objec-tive function gradient is further reduced;

(10) repeat steps 8 and 9 until the finest grid levelis reached;

(11) repeat steps 3 to 6 until the objective functiongradient becomes sufficiently small.

The coefficients ai in Eq. (2) are evaluated as:

ai =Δξc

k |∇iI|0max

ci, i = 1, . . .,n. (3)

In Eq. (3), Δξc is the typical change of the designvariables, kis a coefficient which can vary from 40to 100 accordingly to the considered application,and finally |∇i I|0max is the largest absolute valueof the sensitivity derivatives computed at the firstglobal step of the optimization process. The am-plification factor, ci, varies from a minimum valueequal to 1 to a maximum value, cM, which de-pends on ∇ I. Normalizing the gradient at the �th

iteration as

r = log|∇ I|0|∇ I|� , (4)

cM is set to 10 if r is less than 1, to 40 if r is greaterthan 2, whereas it varies linearly between 10 and40 for intermediate values of the considered ratio.At the beginning of the process ci is equal to 1;then it is increased by 50% if the corresponding


sensitivity derivative maintains its sign, otherwiseit is decreased by 50%. This approach performsbetter than the steepest descent method, corre-sponding to ci equal to 1. Indeed, large changesare applied to the design variables whose sensi-tivity derivates maintain their sign, while smallvariations are assigned to the design parameterswhose sensitivity derivatives are changing theirsign.

The flow solver convergence level decreases pro-gressively according to the mesh level and to ∇ I,so as to increase the accuracy of the objectivefunction computation, while approaching the op-timal solution. Thus the initial configurations,which differ considerably from the optimized one,are analyzed at a very low computational cost: in-deed, the sole aim of the first analyses is to findout an approximate direction in the design space,leading to the reduction of the objective function.

In order to further increase the efficiency of thepresented strategy, the mesh is generated com-pletely only a few times throughout the overalloptimization process. In all other cases, the posi-tion of the mesh nodes is updated using a grid per-turbation technique. Accordingly, the last com-puted flow solution can be used as the starting so-lution for the next flow computation. Taking intoaccount the distance,s j, between the mesh nodesand the nearest boundary node, the mesh nodeposition has been perturbed as follows [Marocco(1984), Medic, Mohammadi, Petruzzelli, Stanciuand Hecht (1999)]:

δ s j =1∫

Γdγsβ

j

∫Γ

δ sw

sβj

dγ , (5)

where both integrals are extended over the solidboundaries Γ, δ sw is the displacement of eachnode located at the wall and β ≥ 2.

In the proposed design applications, one coarseand two locally refined grid levels have been used.In all cases, the two mesh refinements have beenperformed when the conditions r > 1 and r > 1.5,respectively, have been maintained for three con-secutive iterations of the optimization process.Furthermore, the objective function has been con-sidered sufficiently reduced, when r > 3.

4 Optimization results

First, the optimization strategy has been testedversus an inverse-design problem to assess itscomputational efficiency. A target flow solutionhas been generated by analyzing a known configu-ration. The optimization process has been startedusing a set of design variables significantly dif-ferent from the target one. Hence, the aim ofthe numerical test is to verify that the optimiza-tion technique is able to recover the target con-figuration in the amount of computational time toperform few flow analyses. Then the above tech-nique has been applied to the design optimizationof the intake, with the aims of cancelling the re-circulation bubble and of reducing the boundary-layer growth. Using a geometrically constrainedprofile, two applications have been first consid-ered, which differ in the definition of the objectivefunction. The first application aims at minimiz-ing the difference between the Mach number dis-tribution computed in the rotor inlet section andthe Mach number value computed using a one-dimensional isentropic flow assumption; the sec-ond application aims at maximizing the kineticenergy at the rotor inlet section, i.e., at reducingthe fluid-dynamic losses. In order to validate theemployed single-point optimization approach, thedesign procedure has been repeated in off-designflow conditions, namely by imposing a differentoutlet pressure, i.e., a different flow rate. Finally,the design optimization has been repeated after re-moving one geometrical constraint.

4.1 Validation test

The aim of this validation test is to check that theprocedure is able to drive an arbitrary initial pro-file to the (known) target one, in the amount ofcomputational time to perform few flow analysesof the reference shape, namely with the efficiencycharacterizing other procedures, i.e., the progres-sive adjoint procedure of Dadone and Grossman(2000).

The target shape, denoted by a solid line in Fig. 5,corresponds to predefined values of the design pa-rameters ξi, and is used to compute a target pres-sure distribution, p j, that must be matched by the


pressure distribution p j, corresponding to the cur-rent trial shape. Accordingly, the objective func-tion is defined as:

I(ξ ) =1

2N

N

∑j=1

(p j − p j)2 (6)

where N is the number of control points (N = 50)chosen on the control surface which defines theouter wall. Five design variables have been em-ployed, the intake profile y(x) being representedby the fifth-order polynomial

y = ξ1x5 +ξ2x4 +ξ3x3 +ξ4x2 +ξ5x (7)

Figure 5: Inverse design. Target, initial and opti-mal profiles.

The initial and target pressure distributions on theouter wall are provided in Fig. 6 as dashed andsolid lines, respectively. It is noteworthy that, de-spite the smoothness of this reasonable target pro-file, the target pressure distribution is far from rep-resenting a suitable intake pressure distribution,since the overexpansion at the corner and the finalrecompression are interposed by a weak recom-pression and a further weak expansion. This com-plex pressure distribution also makes the inversedesign optimization a difficult task.

Fig. 7 provides the convergence histories of thelogarithms of the objective function, log10(I/I0),

Figure 6: Inverse design. Target, initial and opti-mal pressure distributions.

and of the magnitude of the normalized gradientof the objective function, log10(|∇I|/|∇I|0). Theabscissa in Fig. 7 is the required work, with a unitof work considered as the computational time torun a single analysis solution to convergence onthe finest mesh.

Figure 7: Inverse design. Convergence histories.

A large step towards the optimal solution is per-formed at the coarser grid levels at a low cost.Indeed, the mesh has been refined at work ≈ 3and at work ≈ 4.7, whereas the entire optimiza-tion process has required about 18 work units.


Fig. 5 shows that the optimal shape (symbols) isperfectly superposed to the target configuration(solid line). Similarly, Fig. 6 proves that the op-timal pressure distribution accurately matches thetarget pressure distribution.

4.2 Intake profile optimization

This subsection aims at improving the turbojetperformance by optimizing the intake profile.

The overall dimensions of the original intake havebeen retained, i.e., its length and external radiushave been kept constant. At this stage, also thecorner between the convergent duct and the axialone is maintained in the original position. Thiscondition reduces the number of design parame-ters, thus modifying Eq. (7) as follows:

y = ξ1x5 +ξ2x4 +ξ3x3 +a4x2 +ξ5x (8)

a4 being the value which allows to enforce theoutlined geometrical constraint.

Two different definitions of the objective functionhave been considered, both of them aiming at re-ducing, or even cancelling, the recirculation bub-ble presented in Fig. 3, as well as at improving thevelocity profile at the inducer inlet. In particular,the first definition aims at minimizing the differ-ence between the computed Mach number distri-bution in the inducer inlet section and the Machnumber value that would be computed using aone-dimensional isentropic flow assumption. Thecorresponding objective function is defined as:

I =1

2 ·NN

∑j=1

(Mj −Mis

)2(9)

In Eq. (9),N is the number of nodes (N = 69)on the inducer inlet section, Mj is the computedMach number in the jth node, and Mis is theone-dimensional isentropic Mach number (Mis =0.324). The boundary conditions used to computethe results plotted in Figs. 3 and 4 have been used.Moreover, the Spalart-Allmaras turbulence modelhas been used to compute the turbulent flow in theintake.

The original profile, denoted by a dot-dashed linein Fig. 8, is taken as initial profile of this de-sign optimization, as well as of the following one.

Fig. 9 shows the Mach number contours com-puted with the optimized intake profile, providedin Fig. 8 as dashed line. This figure proves thatthe separation bubble has been eliminated and thatat the outer wall a gradual recompression takesplace, as demonstrated by the reduced boundarylayer growth. The mass flow rate increases 2%with respect to the original intake.

Figure 8: Comparison between the two optimalprofiles with fixed corner.

Figure 9: First design optimization. Mach num-ber contours of the flow for the optimal intakeconfiguration.

Fig. 10 shows the convergence histories of the ob-


jective function and of its gradient. The two gridrefinements have been performed at work ≈ 0.14and at work ≈ 4 respectively whereas the entireoptimization process has required approximately9 work units.

Figure 10: First design optimization. Conver-gence histories.

The second design optimization aims at reducingthe fluid-dynamic losses, i.e., at maximizing theflow kinetic energy at the inducer inlet section.The corresponding objective function to be maxi-mized is:

I =12

N−1

∑j=1

mav2j (10)

where v j and ma are the velocity and the massflow rate in the center of the jth element of theinducer inlet section.

Fig. 8 also shows the optimal intake profile (solidline) resulting from this second application, whichpractically coincides with the previous optimizedprofile. Correspondingly, also the Mach numbercontours are very similar to the previous contoursand thus they are omitted for brevity. On thecontrary, both optimal profiles significantly differfrom the original one. Fig. 11 shows the con-vergence histories of the objective function and ofits gradient. The two grid refinements have been

performed at work ≈ 0.2 and work ≈ 2, respec-tively whereas the entire optimization process hasrequired approximately 9 work units.

Figure 11: Second design optimization. Conver-gence histories.

Figs. 12 and 13 provide the Mach-number distri-butions and the total pressure distributions at theinducer inlet section corresponding to the two op-timal profiles and to the original one. The two op-timal distributions are superposed and are denotedby solid lines, whereas the dashed line refers tothe original intake. Both figures remark the flowrate increase and the loss reduction. It is note-worthy that the optimal inlet velocity profile isslightly flatter than the initial one. In the designoptimizations to follow, only the second objec-tive function and the corresponding optimal pro-file and flow solution will be considered.

In order to check the off-design behaviour of theoptimal intake profile, a new single-point designoptimization has been performed, with a lowerdownstream pressure, pu = 0.88atm. Note thatthe pressure reduction implies higher values of theengine rotation and of the flow rate. The intakeprofile optimized for the new operating conditions(which are even out of the operating range of theturbojet engine) is plotted in Fig. 15 (solid line)together with the previous optimal intake (dashedline) and the original intake profile (dot-dashed


Figure 12: Design optimization with fixed cor-ner. Mach-number distributions at the inducer in-let section.

Figure 13: Design optimization with fixed cor-ner. Total pressure distributions at the inducer in-let section.

line).

The two optimal profiles are very close, thusdemonstrating that the design optimization is notvery sensitive to a variation of the operation con-ditions. This conclusion proves that this designapplication can be afforded by the simple single-point design strategy here employed. Similar re-sults have been obtained enforcing a higher down-stream pressure, pu = 0.96atm, i.e., for lower val-ues of the engine regime and of the flow rate.

Figure 14: First design optimization. Optimal in-take profiles under different operating conditions.

Figure 15: Comparison between the optimal pro-files with fixed and variable starting abscissa.

In the previous design applications, the corner be-tween the convergent duct and the axial part of theduct has been preserved in the original location.In the final design optimization here considered,the corner point is allowed to move axially; cor-respondingly, its abscissa xc can assume non-zerovalues. Thus, the intake profile is represented by


the following equation:

y = ξ1(x−xc)5 +ξ2(x−xc)4 ++ξ3(x−xc)3

+a4(x−xc)2 +ξ5(x−xc) (11)

In Eq. (11), ξ1, ξ2, ξ3, ξ5, and xc are the fivedesign parameters. The previous reference op-erating conditions have been imposed, i.e., p0 =1atm, T0 = 293K at inlet, pu = 0.93atm at outlet.The computed optimal intake profile with variableinitial abscissa of the convergent duct, xc, is repre-sented in Fig. 15, together with the previous opti-mal profile corresponding to fixed initial abscissa,xc = 0. The computed optimal initial abscissa isxc = 1.56. The resulting profiles appear signifi-cantly different. Nevertheless, the present optimalintake determines a negligible further increase ofthe mass flow-rate, i.e., 0.1% for pu = 0.93atmand 0.3% for pu = 0.88atm.

The results computed for different operating con-ditions are summarized in Tab. 1. The conver-gence histories of the present design optimizationare plotted in Fig. 16, which underlines that thevariation of the initial abscissa, xc, determines areduction of the optimization efficiency. Indeed,25 work units are required to decrease the gradi-ent of the objective function three orders of mag-nitude. The two grid refinements have been per-formed at work ≈ 0.2 and work ≈ 1.4, respec-tively.

Table 1: Mass flow-rate for the original and theoptimal profiles for different operating conditions.

Mass flow rate [g/s]downstreampressurepu[atm]

PEGASUSMk3 origi-nal profile

optimalprofile(xc = 0)

optimalprofile(xc �= 0)

0.88 354.9 363.1 364.10.93 279.8 285.9 286.20.96 215.3 220.0 220.0

5 Experimental validation

Fig. 17 shows an overall view of the experimentalrig which has been set up to test the small-scaleturbojet engine. The microturbine assembly has

Figure 16: Design optimization with variable ini-tial abscissa, xc. Convergence histories.

undergone some modifications to allow controland data acquisition by means of an external PC.In particular, the original fuel tank has been sub-stituted by a new tank (3), which has been hangedup in order to measure the fuel flow rate by meansof a load cell (6). The fuel consumption is evalu-ated as the difference of two weightings within a 1minute period of steady operating conditions, themeasurement range of the load cell being 0÷30N, and the measuring error being 1% of the fullscale. The turbojet is suspended on a parallelo-gram (5) to measure the axial and the lateral com-ponents of the thrust. In particular, the axial thrustis measured by means of two load cells (4) witha measurement range of 0÷120 N, whereas theundesired lateral thrust is measured by means ofa load cell (2) with a range of 0÷60 N. The rota-tional speed is measured by a pick-up (1) mountedon the compressor outer case and facing the in-ducer blades. A data acquisition system allows todisplay and to record all measured quantities.

First, the Pegasus turbojet engine has beenequipped with the original intake. Then, the opti-mal profile provided by the third design optimiza-tion has been tested. Both configurations havebeen experimentally tested under the same ambi-ent conditions, to assess the real effects of the pro-


Figure 17: Overall view of the Pegasus turbojettest bed: 1) pick up, 2) side load cell, 3) fuel tank,4) one of the two rear load cells, 5) parallelogram,6) fuel load cell.

posed simplified design optimization of the intakeon the overall performances of the turbojet en-gine. Fig. 18 shows the thrust versus the fuel flowrate for the two configurations. The performanceimprovement due to the optimal intake profile isalso plotted. Fig. 18 underlines that optimal pro-file provides a performance improvement rangingfrom 3% to 4%. This performance improvementcan be considered fully satisfactory, since onlyone turbojet component has been analyzed andoptimized.

As shown in Section 4, the optimal intake is char-acterized by a mass flow rate 2-3% higher thanthe mass flow rate of the original intake. Alter-natively, for the same mass flow rate, the opti-mal profile is characterized by a higher pressureat the inducer inlet with respect to the original in-take, i.e., by a higher global compression ratio andaccordingly by an increased thermal (conversion)efficiency. Moreover, the improved inlet velocityprofile limits the pressure losses at the tip of theinducer blades. Fig. 18 also underlines that thebenefits of using a re-designed intake profile van-ish at low-load conditions. Indeed, at small mass

Figure 18: Thrust measurements for the existingand the optimal nozzles.

flow rate and rotational speed, the intake final re-compression is weak and the recirculation bubblecharacterizing the original intake disappears.

6 Conclusions

A small-scale Pegasus turbojet engine has beenanalyzed. A simplified, rotor-free, numerical sim-ulation of the flow in the intake has revealeda boundary-layer separation immediately down-stream of the sharp corner between the convergentintake portion and the axial duct. The presenceof a recirculation bubble reduces the inducer in-let velocity in 8% of the rotor inlet section anddetermines a 2-3% reduction of the mass flowrate. Further numerical computations have shownthat the choice of a reasonable smooth profile isnot sufficient to guarantee suitable intake opera-tion conditions. Therefore, a gradient-based pro-gressive optimization procedure appropriate to beused in combination with black-box simulationcodes has been developed to re-design the intakeprofile. The efficiency of the proposed optimiza-tion technique comes from the simultaneous con-vergence of the flow solution and of the optimiza-tion process, as well as from the use of nested gridlevels. Moreover, the finite-difference sensitivi-ties are computed by means of partially convergedflow solutions. The proposed design optimizationstrategy can be straightforwardly applied to all en-


gineering fields which employ commercial black-box codes as simulation tools.

An inverse design application with known opti-mal solution has been preliminarily considered tovalidate the robustness and the efficiency of theoptimization technique. In this application, theproposed optimization strategy has reduced themagnitude of the gradient of the objective func-tion by three orders of magnitude in the amountof computational work to perform 18 flow anal-yses. Then, two single-point optimizations suit-able for the design of the rotor-free intake havebeen defined. The first one is an inverse designwith prescribed Mach number at the rotor inletsection, whereas the second optimization requiresthe maximization of the kinetic energy at the in-ducer inlet section. Both design optimizationsaim at reducing the pressure losses, namely atcancelling the recirculation bubble and at reduc-ing the boundary-layer growth. During these opti-mizations, the location of the point connecting theconvergent and the axial portions of the intake hasbeen maintained fixed. The two resulting optimalintake profiles are almost coincident and producea 2% increase of the mass flow rate, together witha significant improvement of the velocity profileat the inducer inlet section. Note that these im-provements are due to the elimination of the recir-culation bubble. Both design optimizations havebeen produced in the amount of computationalwork to perform 9 flow analyses. Almost coin-cident intake profiles and performance improve-ments have been obtained by re-designing the in-take profile for higher (+25%) and lower (-25%)mass flow rates, thus suggesting that the designoptimization of a small-scale turbojet intake doesnot require a multi-point optimization formula-tion.

The last design optimization of the intake hasbeen performed by allowing the abscissa of thepoint connecting the two parts of the intake tochange. The resulting intake profile is signifi-cantly different from the previous ones but per-forms only slightly better. Indeed, the profileswith fixed connecting point have already suc-ceeded in cancelling the recirculation bubble, andthe unique possible improvement is the reduction

of the axial extension of the final recompressionregion to limit the boundary-layer growth aheadof the rotor inlet. This is achieved by shifting theconnecting point closer to the inducer. The nega-tive outcome is the increase of the computationaltime to the amount required to perform 25 flowanalyses.

Finally, an experimental rig has been set up totest the performance improvements of the turbo-jet engine, obtained by re-designing the intake.For most of the operating range, the optimal pro-file provides a performance improvement rangingfrom 3 to 4%, on account of the slightly increasedglobal compression ratio and of the improved inletvelocity profile, which limits the pressure lossesat the tip of the inducer blades. This performanceimprovement can be considered fully satisfactory,since only one component has been analyzed andoptimized.

We can conclude that the proposed progressiveoptimization technique for black-box simulationcodes is capable of producing a rather complexdesign optimization in an amount of computa-tional work which is acceptable for engineeringapplications. Moreover, the experimental tests al-low to state that the design optimization of the in-take of a small-scale microturbine can be affordedusing simplified geometry and flow model.

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