AIAA 2002-4133 - Sapienza

AIAA 2002-4133 Optimal Shape Design of Supersonic, Mixed-Compression, FixedGeometry Air Intakes for SSTO Mission Profiles. Francesco Creta , Mauro Valorani

Riemann

Y

8

β

β2

β1

hyperbolic shock (vertex X0 )

βS S

M0

ξ

µ

sonic line M th =1

A0 X0

X standoff

mass flow continuity

.

m in

S

. m cowl lip

λS out

λB

38th Joint Propulsion Conference July 7–10 /Indianapolis, IN For permission to copy or republish, contact the American Institute of Aeronautics and Astronautics 1801 Alexander Bell Drive, Suite 500, Reston, VA 22091

Optimal Shape Design of Supersonic, Mixed-Compression, Fixed-Geometry Air Intakes for SSTO Mission Profiles. Francesco Creta∗, Mauro Valorani† The problem of maximizing the performance of a fixed-geometry air intake geometry of a vehicle accelerating over a wide range of flight Mach number is addressed. An extension of the Seddon-Goldsmith procedure is used to estimate the flow pattern involving a curved bowshock, a triple point interaction, and wall shock reflection which characterizes the subcritical regime of operations. The approximate model has been validated against detailed CFD calculations of the flowfield about the air intake. Finally, the approximate model is adopted to find the geometry which optimizes the fuel-to-mass ratio over a constant dynamic pressure trajectory.

Introduction The air intake is the most critical part of a supersonic/hypersonic airbreathing propulsion system. It must deliver air to combustion chamber for a wide range of flight Mach number at a desired rate and flow conditions. This delivery must be accomplished by as little losses and drag as possible. At high flight Mach number the compression process is accomplished by a succession of oblique shocks generated both ahead (external compression) and inside the converging part of the air intake (internal compression). The supersonic compression ends at a normal shock located at or downstream the intake throat and might be followed by further diffusion in the subsonic flow regime (ramjet). The intake geometry needs to be properly shaped to achieve an optimal performance. This task is relatively manageable if the intake is to be operated at nearly constant free stream conditions, as it happens for a cruiser type vehicle. However, a reusable airbreather launcher, such as a SSTO vehicle, attains orbital speeds and heights by a continuous acceleration (accelerator type vehicle), which forces the air intake to operate over a wide range of flight Mach numbers and altitudes.2, 6 To obtain an envelope of nearly optimal intake performance requires adopting a variable-geometry intake strategy,13 with a multipoint design selection along the ascent corridor. The air intake during off-design conditions displays several flow regimes, classified as subcritical or supercritical according to whether the flow Mach number is respectively below or above the design Mach number. The subcritical regime is prompted by the need ∗ Ph.D. Student; University of Rome “La Sapienza”, Italy, Department of Mechanical and Aeronautical Engineering, Via Eudossiana 18, 00184 Roma; e-mail address: [email protected] † Associate Professor, Ph.D.; Senior Member AIAA; University of Rome “La Sapienza”, Italy, Department of Mechanical and Aeronautical Engineering, Via Eudossiana 18, 00184 Roma; e-mail address: [email protected] c Copyright 2002 by F.Creta, M.Valorani. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission.

of reducing the massflow captured by the intake so as to match the reduced massflow requirement elaborated by the engine, reduction that is accomplished by spilling air over the intake cowl lip. Spilling might occur under both subsonic or supersonic flow conditions. Subsonic spillage involves the formation of a complex shock pattern ahead of the intake duct including a strong bow shock, whereas supersonic spillage can be accomplished by oblique shock waves alone. The off-design performance might be largely lower than at the design point. Before resorting to a wholly variable-geometry intake strategy, it is important to understand how and under what conditions an optimal accelerator performance for a vehicle flying over a range of variable free stream conditions could be achieved by adopting a fixed-geometry intake. Of special interest is to understand how to choose the design point for a fixed-geometry intake, its geometry and how its overall performance evaluated over a flight corridor can be predicted. Clearly, computational fluidynamics (CFD) can be used to obtain the flowfields about the intake at different free-stream conditions for several intake geometries, and next derive the overall performance prediction from this database.5 This procedure is very time consuming and at the design stage it is preferable to adopt a ”lighter” tool of analysis and to postpone the use of CFD tools in the verification stage of the design process. In this paper, we illustrate a procedure to attain an intake shape optimal over a flight corridor whose core tool is an approximate, algebraic, model of the shock pattern developed during the subcritical operation regime. This approximate shock model allows us to predict the intake off-design performance with a minimal computational effort thus allowing for a parametric investigation of very many intake shape designs and ultimately for an automatic design procedure to be implemented. Moreover, the simplification process followed to obtain the approximate shock model allows us to pinpoint the main mechanisms governing the interaction between the shock pattern ahead of the intake

1 of 11 American Institute of Aeronautics and Astronautics Paper 2002-4133

Reference Intake Geometry The hypersonic flight requirement of a deep integration between the vehicle aerodynamic structures and its propulsion system rules out high drag axissymmetric intake configurations in favour of twodimensional mixed-compression intakes with forebody precompression. Most of the recent hypersonic vehicle concepts, such as the Hyper-X7, 8 demonstrator, adopt this configuration which involves a shape design with several external and internal compression ramps. The reference intake geometry for our approximate model should therefore be able on one hand to reproduce the main flow features of a two-dimensional mixed-compression intake and on the other hand to display an off-design shock pattern involving a limited number of shock-shock and shock-wall interactions. In fact, the difficulty in dealing with complex phenomena such as multiple shock interactions and reflections5 without invoking CFD analysis tools, will force us to consider very simple intake configurations only. For this reason, the subcritical, off-design, shock pattern generated by the interaction between the bowshock and the external ramp oblique shocks should be kept at a minimum level of complexity thereby ruling out multi-ramp configurations in spite of their higher on-design efficiencies. A single ramp geometry was therefore chosen to accomplish external compression.5 Similar difficulties arise with multiple internal ramps when attempting to predict the successive shock reflections and interactions arising under off-design operations. Therefore, the internal compression will be accomplished by a Prandtl-Meyer single family isentropic compression profile in spite of its greater length.

σth MD θD

Aentry δramp

compressione esterna external compression

A th

A c =1

duct and the engine during off-design operations, thus enabling us to establish general intake design guidelines. The paper is organized as follows. First a reference intake geometry will be introduced; then the optimal performance at on-design conditions of the reference intake will be parametrically analyzed and discussed. Next, the approximate, algebraic, model of the shock pattern developed during the subcritical operation regime will be illustrated. The following section will present the results of the validation of the simplified model, validation which will be carried out by comparing the flow pattern generated by a given intake shape as predicted by the simple model and by an accurate CFD model. Finally, the off-design prediction of the air intake produced by the simplified model will be used to estimate the overall fuel mass fraction requirement of a vehicle flying along a constant dynamic pressure flight corridor. The main conclusion of this analysis is that the intake geometry, which is optimal over a flight corridor, differs from that optimal at the design point, and the reasons causing this circumstance can be clearly identified.

compr. P-M interna internal P−M compr.

Fig. 1 Reference intake geometry and definition of the three design parameters (MD , θD , σth ).

Such reference intake geometry can be uniquely identified by specifying the following three independent design parameters (Fig. 1): (i) the intake design Mach number MD , (ii) the oblique shock wave angle θD at the design Mach number and (iii) the flow deviation angle σth downstream of the throat. Note that the intake design Mach number MD is lower than the free stream Mach number M∞ , since the flow is decelerated across the oblique shocks generated by the ramps located along the vehicle forebody. Clearly, the wedge angle δramp is uniquely identified once the design Mach number and the oblique shock wave angle θD are given. The on-design capture area A0 is taken equal to the intake duct height Ac . The length of the external compression is such that the oblique shock wave hits the intake cowl lip at the design Mach number. Thus, each pair of values (θD , σth ) uniquely identifies a geometry within the family of intakes having a design Mach MD . By conventionally setting the intake duct height Ac equal to unity, both the external ramp length LN and the intake duct entry area Aentry can thus be found: LN =

1 tan θD

Aentry = (1 − LN tan δramp ) cos δramp The Prandtl-Meyer compression wave realizing the internal compression is centered at the intersection of the ramp wedge and a line stemming from the cowl lip and with a slope equal to the Mach line angle evaluated downstream the leading oblique shock. The intake duct geometry is found by tracing a streamline starting at the cowl lip through the Prandtl-Meyer compression. At the end of the compression, the intake cross section area defines the intake throat Ath where the flow Mach number is minimum Mth and, by construction, the flow angle is σth . The proposed intake configuration provides the simplest way to realize a mixed external-internal compression, and as such has little chances to represent a ”real world” intake. Moreover, no viscous effects will be accounted for in predicting the performance delivered, and the performance estimate will be overly optimistic


since no wave losses are associated to the PrandtlMeyer compression, whereas oblique shocks (at least one) always participate to the internal compression in a real intake. However, this simplified configuration will suffice to highlight the main phenomena affecting the choice and identification of an intake design optimal over a range of flight Mach number.

70

60

50

Shock angle

60

40 Intake length 9 8.5 8 7.5 7 6.5 6 5.5 5 4.5 4 3.5 3 2.5 2 1.5 1

30

20

MD = 5.0

θD

10

Shock angle

50

On-Design Performance Analysis

0

-80

-60

-40

-20

0

20

40

Flow turning angle

ηC MD =5.0

40

30

Adiabatic compression efficiency 0.99 0.975 0.96 0.945 0.93 0.915 0.9 0.885 0.87 0.855 0.84 0.825 0.81 0.795 0.78

20

10

0

-80

-60

-40

-20

0

Flow turning angle

σth

20

40

Specific impulse map

70 65 60

MD = 5.0

55

θD

Isp MD =5.0

50

Shock angle

For a given design Mach number MD , the intake geometry design parameters (θD , σth ) will be subject to geometrical and gasdynamical constraints, which will define the domain of feasible intake geometries. For a given shock angle θD ∈ (θDmin , θDmax ) , with θDmin equal to the flow Mach angle (a function of MD ) and θDmax equal to the detatchment angle (a function of MD and δramp ), the deviation angle σth will lie between two extremes, one corresponding to a limiting isentropic compression yielding sonic throat conditions and the other corresponding to a zero strength compression, so that σth = δramp (MD , θD ). Any point chosen within the feasible domain corresponds to a physically admissible air intake geometry. Appropriate performance indices at on-design operation can be computed for such a geometry, these being (i) adiabatic compression efficiency ηc , defined as the ratio of the isentropic enthalpy change over the actual enthalpy change accomplished through the external oblique shock and a normal shock at the throat, and (ii) the engine specific impulse Isp , or equivalently, the specific thrust Tsp . The latter index is computed by carrying out a First Law analysis2 of the engine cycle with prescribed combustion and expansion efficiencies as illustrated in the Appendix. At this stage, performance maps of adiabatic compression efficiency and specific impulse can be constructed as a function of θD and σth for a given design Mach number MD . Fig. 2 shows one of such maps for the design Mach number MD = 5.0 intake family. Within each feasible two-dimensional domain in the (θD , σth ) plane, there exists a one-dimensional subdomain of intake geometries (identified by a dashed line in Fig.2) yielding a selection of configurations which are optimal in terms of highest adiabatic efficicency ηc , that is, for a given value of σth , there exists an optimal value of θD such that the configuration of the oblique shock plus the normal throat shock yields a minimum total pressure loss. The peak performance is found at the left side of this one-dimensional subdomain and suggests that low oblique shock angles intake geometries defined for the design Mach number MD = 5.0, by θD ≈ 12deg , with a very strong flow turning provided by the Prandtl-Meyer compression σth ≈ −70deg yield the maximum performance of ηc ≈ 0.99 and Isp ≈ 4600. Clearly, this class of geometries is unacceptable because they are too long, the flow turning too extreme and the predicted per-

LN Adiabatic compression efficiency map MD = 5.0

70

45

Isp 4600 4500 4400 4300 4200 4100 4000 3900 3800 3700 3600 3500 3400 3300 3200 3100 3000 2900 2800 2700 2600 2500 2400

40 35 30 25 20 15 10 5 0

-80

-60

-40

-20

0

Flow turning angle

σth

20

40

Fig. 2 On-design performance maps for intakes with design Mach number MD = 5.0

formance over-optimistic, since no viscous effects are accounted for, and no wave losses are associated to the Prandtl-Meyer compression.

An Approximate Shock Pattern Model In assessing the conclusions drawn from the inspection of the on-design performance maps, one has to consider that an optimal on-design geometry will, most likely, no longer be optimal under off-design operations, and especially so in the subcritical regime. From this follows the importance of devising a subcritical shock pattern prediction model capable of computing ’real time’ performance parameters along a given constant dynamic pressure trajectory. Such a model should require very little CPU time, so that it could be incorporated in an automatic optimization procedure. The model presented in this work is an extension of the model originally developed and illustrated by Seddon and Goldsmith (1985).1 Given an intake geometry and flight conditions, the model assumes that, during sub-critical operations, a curved bowshock stabilizes in front of the entry section thus prompting spillage of air over the cowl lip at


shock detachment angle δmax relative to the current Mach M0 . The local flow angle will therefore be λS = δmax . This conclusion is justified by the consideration that point S must act as a dividing point between the part of bowshock downstream of which the flow is subsonic, and supersonic respectively. Therefore, the oblique shock is of the strong type underneath point S and of the weak type above it. The maximum slope of a weak shock or the minimum slope of a strong shock will therefore occur where λS = δmax .

subsonic speed. The shock pattern is further complicated because of the ’triple point’ interaction occurring between the curved bowshock and the oblique shock emerging from the ramp. An oblique shock impinging on the intake ramp, and a contact discontinuity emerge past the triple point interaction (see Fig. 3). Riemann

Y

8

β

β2

β1

hyperbolic shock (vertex X0 )

βS S

M0

ξ

µ

4. The sonic line joining the sonic point and the cowl lip is considered to be straight. This assumption is somewhat oversimplified but it is nearly impossible to account for the dependence of the actual sonic line shape3 with the Mach number and the intake geometry and to keep the model simple as required. Therefore, assuming the sonic line straigth will imply a sistematic over/underestimation of the massflow crossing through it. This issue will require calibration of the model against comparison with CFD results.

sonic line M th =1

A0 X0

X standoff

mass flow continuity

.

m in

S

.

λS

m out cowl lip

λB

Fig. 3 An algebraic model of the intake subcritic flow pattern.

A different, and somewhat simpler, model is required when spilling is accomplished at supersonic speed, when the oblique shock stemming from the external ramp has a slope angle larger than the value attained at the design point. Main Assumptions

With reference to the sketch and the nomenclature shown in Fig. 3 , the main assumptions of the model are the following: 1. The curved bowshock is assumed to describe a hyperbolic curve with vertex x0 in a x, y frame of reference, where the x-axis is parallel to the free-stream direction. The choice of a hyperbolic form is physically justified by the fact that the shock should have zero strenght at infinity, and therefore one asymptote of the hyperbola will be set parallel to a Mach line of slope µ = arcsin M10 where M0 is the (current) Mach. 2. At the triple point the hyperbola has a local slope β∞ with respect to the horizontal; the slope β∞ is found by solving a two-dimensional Riemann problem generated by the interaction of the curved bowshock and the oblique shock stemming at the base of the external ramp. 3. The flow past the curved bowshock becomes subsonic and then reaccelerates to supersonic speeds when passing over the cowl lip. The sonic line intersects the curved bowshock at a point S. There the slope of the hyperbola βs equals the oblique

Solving the flow model problem

Once these assumptions are laid out, it is necessary to find a ”solution” for this simplified flow model, which involves finding the correct shock shape and location, the capture area, solving the triple point interaction, and assessing what type of shock reflection occurs at the ramp for the shock emerging out the triple point interaction, all of this for a given intake geometry and flight conditions. A first guess of the capture area A0 is initially set and fed as input to determine the bowshock’s hyperbola vertex x0 which ultimately defines the bowshock’s standoff distance and hence the shock pattern. Finding x0 involves an iterative procedure (Fig. 4) aimed at enforcing a continuity condition between the massflow m ˙ in passing through the capture area at the upstream flow conditions, and the massflow m ˙ out flowing through the sonic line extending between the intake cowl and the bowshock at the sonic flow conditions, as illustrated in the inset of Fig.3. At this point, the procedure must verify if the guessed value of capture area A0 , with the resulting shock pattern, produces sonic conditions at the throat. If this is not so the whole procedure is re-iterated until convergence is reached at which point the shock system found is compatible with the intake geometry and flight conditions. The model here presented extends the original Seddon-Goldsmith model, by allowing the triple point to fall both within the capture tube (sketch at the bottom of Fig 5) and outside of it (sketch at the top of Fig 5). The first instance is more complicated than the latter because the evaluation of the capture area cannot be carried out independently of the bowshock


INPUT Flight Mach M 0 Design Parameters M D

σth θD

Design Geometry Generator

SUPER SONIC Spillage Possible

OUTPUT

YES

?

Supersonic Spillage

NO Triple Point Riemann Solver

M>1 M0

Subsonic Spillage Bowshock Standoff Distance Estimation M=1 M0 Ri

Throat Choked

Updated Capture Area

YES

? NO

No Stationary Solutions

YES

?

Fig. 5 Model scheme for triple point inside and outside of capture tube.

Generalized Exact Riemann Solver

The Riemann problem arising at the triple point involves the interaction of a weak shock (the oblique wedge shock), and a locally strong shock (the curved subcritical shock). Such combination of shocks admits, among the emerging waves, a strong outer oblique shock, so that the flow past it is subsonic and an inner wall-impinging shock. This requires the extension of a standard exact Riemann solver to account for strong shocks. The Riemann solver yielded the results reported in Fig.6, where for a given wedge angle δramp and free-stream Mach M0 the local slopes β∞ of the outer portion of the bowshock and β2 of the inner portion (see upper inset of Fig.3) are displayed.

Inner shock angle

110 105 100 95 90 85 80 75 70 65 60 55 50 45 40

β2 β2

1

δ ramp δ=5 5

15 20 15 20

1.5

2

110

β

105

25 25

30 30

35 35

2.5 3 3.5 4 Free−stream Mach number

40 40

4.5

5

β inf

8 100

Outer shock angle β

standoff distance estimation as the average pressure losses for the captured streamtube are functions of the standoff distance itself. The capture area relative to the latter case, on the other hand, can be determined independently of the triple point, since the captured streamtube does not embrace the triple point and therefore the related pressure losses are solely due to the oblique ramp shock and the downstream wallimpinging shock. Allowing for the triple point to fall both outside and inside of the capture streamtube makes it possible to further estimate the flight conditions at which it actually crosses the dividing streamline (i.e. the streamline dividing the captured streamtube and the outer flow). This can be adopted as a criterion to predict the onset of buzzing oscillations (Ferri’s criterion4 ).

β2

Fig. 4 Flow chart of the iterative procedure to find the model solution.

8

Stop Iteration on Capture Area

95 90

δ=3

3

85

55

80

10 10 15 15

75

20 20

70

60

25

25

30

30

δramp

65 1

Fig. 6

1.5

2

35 35

2.5 3 3.5 4 Free−stream Mach number M0

40

40

4.5

5

Riemann solver results: (β2 , β∞ )M0 ,δramp .

The oblique shock emerging from the triple point and impinging the ramp can reflect at the wall following three distinct modalities: (i) regular oblique shock reflection, (ii) a Mach reflection or (iii) no reflection at all. In this latter case the oblique shock is strong and bends so as to impinge the ramp wall at a right


angle. Fig. 8 displays the domains of existence of the three reflection modes as a function of the ramp angle δramp. and flight Mach M0 . A CFD validation procedure has confirmed that intake geometries featuring the Mach reflection mode during subcritical operation provide very unfavourable flow patterns as illustrated in Fig.7. These are originated by the very strong velocity gradients across the vortex sheet emanating from the triple point which ultimately generate large recirculation regions within the intake’s convergent duct leading, among other things, to a drastic reduction of the captured mass-flow.

temperature rise and total pressure drop across the intake.

Fig. 9 Shock pattern as predicted by the simple model and by CFD for a MD = 3.5 θD = 25deg σth = −10deg intake. Mach refl.

Fig. 7 Mach contours and streamlines at flight Mach 4.0 illustrating recirculation for a MD = 5.0 θD = 42deg σth = +10deg intake (close up).

In contrast, intakes displaying no shock reflection at the ramp wall are preferable as the flowfields they originate are far more uniform and recirculation regions far smaller. 6 Regular reflection

5.5 5

Cruise Mach

4.5 4

Mach reflection

3.5 3 2.5

No regular refl.

2

δδmax ΜΑΞ

1.5 1

0

5

10

15

20 25 30 Ramp angle

35

40

45

50

Fig. 8 Parametric maps classifying the type of shock reflection at the ramp wall as a function of intake ramp angle (deg) and cruise Mach.

Model Validation Once the procedure has converged, the proposed model problem delivers the following information: the capture area, all main flow states (assumed piecewise constant within each flow portion enclosed by shocks or wall boundaries), shocks’ locations, structures, and strengths. Derived quantities will include overall static

Fig. 10 Triple point path as Mach is varied, as predicted by CFD and by model for a MD = 3.5 θD = 25deg σth = −10deg intake.

The model was subject to an extensive validation procedure through comparison with inviscid CFD results. The CFD solver used is based on a shock-fitting technique,14 which allows to sharply measure shock


Md=3.5

1 44

INTAKE B

3.5 3.5 0.9

CFD ALGEBRAIC

33 0.8

A0 yy

2.5 2.5

0.7 22 0.6 1.5 1.5 0.51 1 0.5 0.4 0.5 1 1

0

2 2

2

3 2.5 x3

4

5

4 3

5

3.5

Mach x

Frame 001  19 Sep 2000  Valori medi sulla presa d’aria

1 0.9 0.8

1 0.9 0.8

0.7

0.7

0.6

0.6

0.5

p0/p0∞

Total Pressure Drop

angles and location so as to improve the accuracy of the comparison. For those intakes which display a normal incident bowshock in subcritical operation, the comparison campaign has allowed for a thorough verification of the simplifying assumptions made and for error estimation. In particular the straight sonic line assumption was extensively investigated and the model calibrated accordingly. Fig.9 shows a direct comparison between the CFD generated flowfield and the algebraic model for a MD = 3.5 θD = 25deg σth = −10deg intake at flight Mach 2. The comparison shows that the triple point location is well captured although the bowshock shape is somewhat in disagreement as the shock moves into the farfield. For the same intake geometry, Fig.10 shows the trajectory followed by the triple point as the Mach is lowered in sub-critical operation as computed by CFD and by the algebraic model. Note that the curved ascending-descending path of the triple point obtained through CFD can be properly described by the simplified algebraic model. The algebraic model also allows us to predict the Mach number (transition Mach) below which air is spilled subsonically and above which supersonically. Fig.11 shows that for a given design Mach and oblique shock angle, the range of supersonic spillage, defined by the difference between the design Mach and the transition Mach, decreases for higher absolute values of the flow deviation, i.e. for stronger PrandtlMeyer internal compressions. This is because for stronger on-design internal compressions the design throat Mach approaches unity thus leaving less margin before chocking conditions arise as the flight Mach is lowered.

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

0

INTAKE B

ALGEBRAIC CFD

3

4

2

2.5 x

3

5

3.5

Mach

Fig. 12 Results of the comparison with CFD data: capture area A0 and total pressure losses as a function of the flight Mach number for an intake geometry with design parameters MD = 3.5 θD = 25deg σth = −10deg..

3.5

Transition Mach number

supersonic spillage 3

θD 18 19 20 21

θ=30

2.5

θ=25 subsonic spillage

2

θ=20

1.5 1 −40

−30

−20 −10 Flow deviation angle

0

σth from − 50 to − 6 from − 40 to − 5 from − 20 to − 4 from − 11 to − 3

Table 1 Ranges of design parameters (in deg) examined for an intake designed at MD = 3.5.

10

Fig. 11 Transition Mach as a function of intake design parameters

Fig.12 demonstrates the excellent agreement found for the capture area A0 and the total pressure recovery as functions of Mach for an intake geometry with design parameters MD = 3.5 θD = 25deg σth = −10deg.

Off-Design Performance Analysis The approximate algebraic model proposed will enable us to carry out an extensive off-design performance campaign relative to a great number of air

intake shapes, along given flight path trajectories. We chose to examine intake geometries for the design Mach number MD = 3.5, and different combinations of (θD , σth ) as detailed in Table 1. As a global performance parameter we selected the fuel consumption needed to accelerate a reference vehicle from an initial Mach, Mi = 1.6, to a final Mach, Mf equal to the intake geometry design Mach number MD = 3.5. The vehicle flies along a constant pitch and constant dynamic pressure q0 = 72.5kN/m2 acceleration trajectory. The vehicle is assumed to have variable,


where η0 is the total airbreathing engine efficiency as calculated through a simple First Law analysis (see Appendix) of the thermodynamic engine cycle, r is the radial distance from the center of the Earth, and hP R is the propellant heat of reaction. The numerical integration of Eq.(1) yields the end-of-mission Fuel to Mass Fraction (FMF) index Πf defined as: Πf =

mi − m f mi

(2)

We will define an intake geometry optimal if it will minimize the final FMF over an accelerating flight path. Note that Eq.(1) and hence Πf ultimately depends upon the following group of terms: De (3) η0 1 − F which can be identifiable as an effective overall efficiency of the engine. It is therefore clear how, everything being equal, in order to save fuel it is essential to adopt an intake design that during the mission, and especially during its off-design operation, minimizes the drag to thrust ratio De /F . q=72.5kN/m2 trajectory

θD = 20

0.3

η0

SIGMA=−10 σth −15

0.1

−20

0.05 −25 1.5

−20

0.06

0.04

−15 −10

0.02 SIGMA=−20 σth SP.No DRAG=0 Drag

{

1

1.5

2

2.5 flight Mach

3

3.5

4

Fig. 14 Instantaneous Fuel Mass Fraction along trajectory as a function of Mach for several intake design parameters σth and fixed θD = 20deg.

Fig.14 displays the instantaneous FMF, Πf (M ), as a function of the increasing flight Mach along the trajectory, as it results from the integration of Eq.(1). Fuel consumption is clearly faster during the subsonic spillage regime because of high drag and slower during supersonic spillage. Intakes with increasing absolute values of flow deviation, |σth |, because of their wider subsonic spillage ranges and higher spillage drag values, will tend to yield higher end-of-mission FMF values, Πf (MD ). Overall performance on a q=72.5kN/m2 trajectory

ON DESIGN GLOBAL EFFICIENCY

1

SIGMA=−25 σth

0.04

0.2 0.15

θD = 20 θ D=20

0.08

0

θD=20

0.25

0

q=72.5kN/m2 trajectory 0.1

2

2.5 flight Mach

3

3.5

4

FINAL Fuel Mass Fraction

Effective Overall Efficiency

0.35

reported in Tab.1. A sudden efficiency drop can be noticed in correspondence to the transition between the supersonic and subsonic spillage regimes due to the onset of the bowshock and the accompanying drastic increase in drag. Moreover, an inversion of performance occurs between the two regimes. It was concluded that this behaviour is due to the fact that intakes with the best performance both at design point and during supersonic spillage (when spillage drag is negligibly small), simultaneously show the highest De /F ratios during the subsonic spillage regime and hence the worst performance in such regime.

Instantaneous FMF (with Drag)

lumped-mass body subject to weight, thrust F and spillage drag De . The spillage drag is solely due to the intake and can be easily evaluated on the basis of the model output data as illustrated in the Appendix. No other drag sources have been accounted for, so as to highlight the effects of the intake off-design operations on the global performance of the vehicle. Manipulation of the equation of motion yields a relation for the instantaneous vehicle mass:  2  d V2 + gdr dm  = − (1) m η0 hP R 1 − DFe

Fig. 13 Effective overall efficiency along trajectory as a function of Mach for several intake design parameters σth and fixed θD = 20deg.

Fig.13 displays the global overall efficiency relative to intakes having θD = 20deg and different σth as

0.035

θTHETA D 0.03 FMF with Drag FMF No Drag 0.025

18 19 20 21

0.02

0.015 −30

−25

−20 −15 −10 −5 Flow turning angle SIGMA σth

0

Fig. 15 End-of-mission Fuel Mass Fraction as a function of intake design parameters.


Engine modelling can be kept at a minimum level of complexity through a First Law Analysis which views the propulsion plant as a thermodynamic closed cycle. In so doing, engine global efficiency can be defined as the ratio of thrust power to input thermal power, and can be expressed (see Fig. 16) as the product of a combustion efficiency ηb , a propulsive efficiency ηp and the thermodynamic cycle’s efficiency ηtc : . η0 =

F V0 m ˙ f hP R

=

ηb

ηtc

ηp

(4)

where hP R is the heat of reaction, m ˙ f is the fuel mass flow and where V0 is the flight velocity. ηb

combustion inefficiencies unburnt fuel poor mixing

ηtc

heat lost

ηp

lost flux k.e.

Fig. 16

Thrust work

Available work

A simplified model for the subcritical behaviour of a supersonic fixed-geometry, mixed-compression air intake was presented. The algebraic nature of the model was validated against detailed CFD calculations of the flowfield about the intake. For geometries which do not manifest shock reflection at the ramp, and hence have widely uniform flowfields, the model delivers a satisfactory agreement with CFD data. The model was successful in pinpointing the main mechanisms governing the relationships between intake geometry, subcritical shock patterns and ultimately compression efficiency. Given the low computing power requirements, it was possible to carry out an extensive off-design engine performance analysis so that the parametrical influence of intake design parameters could be clearly highlighted. Optimal geometries were found for which fuel consumption is minimal over a given accelerating trajectory. In particular it was found that such optimal geometries arise because intakes delivering the best performance both at design point and during supersonic spillage, simultaneously show the highest drag/thrust ratios during the subsonic spillage regime and hence the worst performance in such regime. A compromise geometry will therefore sacrifice part of its on-design efficiency in favour of lower drag values during the subsonic spillage regime. The subsonic spillage, or unstarted, operation is clearly a highly unfavourable flight condition and certainly one to be avoided during cruise and acceleration. To pursue this aim a well established solution is the internal bleeding of part of the captured air flow which causes the bowshock to be swallowed and started conditions to be reached. The proposed model can indeed yield a proper estimate of the amount of bleeding required. Since its main outcome is a relationship between the bowshock standoff, the captured mass flow and throat area at given flight conditions, then

Appendix First Law Analysis

Available heat

Conclusions

the amount of bleeding is simply found by enforcing that the standoff distance be zero. Under this limiting condition, the bowshock, at the inlet duct, becomes unstable, is swallowed and the intake is started. Analysis of the bleeding requirements to achieve supersonic spillage operation along the whole mission profile will be the subject of a forthcoming research activity.

Fuel chemical energy

Fig.15 displays the end-of-mission FMF, Πf (MD ). The importance of intake spillage drag appears clearly whenever it is neglected in the computation of the final FMF. We observe the emergence of optimal values for σth for given values of θD which minimize the final FMF. Considering intake geometries with increasing absolute values of |σth |, from right to left in the figure, we initially observe that an increase in η0 has a beneficial effect on FMF (in spite of increasingly wider subsonic spillage ranges) until we reach an optimum beyond which the deleterious effect of increasing De /F (and yet wider subsonic spillage ranges) prevails. This result clearly underlines that optimal, fixedgeometry, intake design geometries exist for mission profiles involving a continuos acceleration flight path and such geometries are different from those found optimal at constant Mach fligth path.

Global engine efficiency

With reference to the nomenclature of Fig.17 such efficiencies are defined as h4 − h 3 f hP R

(5)

(h4 − h3 ) − (h10 − h0 ) h4 − h 3

(6)

ηb = ηtc = ηp =

F V0 ' [(h4 − h3 ) − (h10 − h0 )]m ˙0

'

F V0 2

(m ˙ f +m ˙ 0 ) V10 ˙ 0 V20 2 −m

2

(7)

where m ˙ 0 is the captured air mass flow and f = m ˙ f /m ˙ 0 is the fuel to air ratio. Compression, combustion and expansion submodels can then be implemented in series, each representing, with its own efficiency, an energy transformation of the cycle. By assigning proper constant values to the combustion efficiency and the nozzle adiabatic expansion efficiency ηe , the effects on the global engine efficiency of the intake’s adiabatic compression efficiency ηc can be adequately singled out.


With reference to Fig.17, adiabatic compression efficiency is defined as γ−1 γ 1 ψ − πc . h3 − h X ηc (ψ, πc ) = = (8) h3 − h 0 ψ−1 . . where ψ = T3 /T0 is the temperature rise and πc = pt3 /pt0 the total pressure drop across the intake. Both ψ and πc are directly extracted as output of the intake model as a function of flight conditions. Carrying out the complete cycle analysis, yields a relationship between thermodynamic cycle efficiency and adiabatic compression efficiency: C p T0 C p T0 T10 (9) +1 − ηtc = 1 − ψ ηb f hP R T4 ηb f hP R where 1 T10 (10) = 1 − η e 1 − 1 − ηc 1 − T4 ψ . with ηe = (h4 − h10 )/(h4 − hY ) being the adiabatic expansion efficiency. ηb

n sio res

Co

Y X

0

c

p1

jection Heat re essure const. pr

Q OUT

a

c

A1

Ac A0 pw3

(b) Ac A0

Fig. 18

Spillage Drag

where A0 is the capture area, Ac the intake entry area, p0 and p1 the pre- and post-shock static pressures. For the subsonic spillage regime, the case in which the dividing streamline is below the triple point (see Fig. 18.2a) yields the following expression: De = (p1 − p0 ) (Ac − A0 ) +

10

De = (pw3 − p0 ) (Ac − A0 ) . _ Q IN Q OUT

(12)

(13)

References

Q IN

Thermodynamic cycle nomenclature

Once the engine global efficiency is computed, the unistalled thrust can be computed as F = . η0 m ˙ f hP R /V0 and specific impulse as Isp = F/(m ˙ f g0 ) = η0 hP R /(g0 V0 ).

1

Seddon, J., Goldsmith, E. L., ”Intake Aerodynamics”, Collins Professional and Technical Books, 1985.

2

Heiser, W., Pratt, T., ”Hypersonic Airbreathing Propulsion”, AIAA Education Series, 1994.

3

Rusanov, V. V., Lyubimov, A. N., ”Gas Flows Past Blunt Bodies”, Nauka Press, Moscow 1970, NASA Washington D.C. Feb. 1973.

4

Ferri, A., ”Elements of Aerodynamics of Supersonic Flows”, Mac Millan N.Y., 1949.

5

Valorani, M., Nasuti, F., Onofri, M., Buongiorno, C., ”Optimal shape design of air intakes for Air Collection Engines (ACE)”, Acta Astronautical Journal, Vol. 45, No. 12, pp. 729–745, 1999.

6

Czysz P., Bruno C., Kanerori K., ”Interaction of the Propulsion System and System Parameters Determines the Design Space Available for Solution”. Copyright 2001 by Paul A. Czysz, Published by the American Institute of Aeronautics and Astronautics, Inc. with permission. Released to the AIAA to publish in all forms.

7

Freeman, D., Reubush, D., McClinton, C., Rausch, V., Crawford, J. L., ”The NASA Hyper-X Program”, NASA TR 1997.

8

Huebner, L. D., Rock, K. E., Witte, D. W., Ruf, E. G., ”Hyper-X Engine Testing in the NASA Langley 8-Foot High Temperature Tunnel”, AIAA 20003605, 36th AIAA/ASME/SAE/ASEE Joint Propulsion Conference, July 17-19 2000, Huntsville AL.

Spillage Drag

A common definition of intake drag, consistent with the definition of uninstalled thrust, generally includes a cowl drag and a pre-entry drag contribution. The former includes viscous and suction effects on the cowl while the latter accounts for the resulting pressure forces on the captured streamtube. Spillage drag, on the other hand, is defined as the drag increase as the capture area is lowered below full-flow conditions. For supersonic intakes in subcritical operation, neglecting cowl suction (thin wall assumption) and viscous effects, spillage drag can be assumed equal to pre-entry drag. An expression for supersonic subcritical spillage drag can be derived by extending the Fraenkel1 (1950) expression valid for a Pitot intake to single-wedge intake. For the supersonic spillage case (see Fig. 18.1) the following expression was used: De = (p1 − p0 ) (Ac − A0 )

Ac

whereas the case in which the dividing streamline is above the triple point (Fig. 18.2b) yields

ηe

η = tc

(a)

w p1

A0

S

Fig. 17

pw2

+ (pw2 − p0 ) (Ac − A1 )

Q IN

mp

T

nsion

ηc

n ditio re t ad Hea pressu t. cons

2)

b

4 Expa

3

1)

(11) 10 of 11

American Institute of Aeronautics and Astronautics Paper 2002-4133

9

Valorani, M., Di Giacinto, M., Buongiorno, C., ”A Method to Predict the Performance of Oblique Detonation Wave Engines (ODWE)”, AIAA 98-S.5.09, 49th International Astronautical Congress, Sept 28Oct 2, 1998, Melbourne Australia.

10

Ikawa, I., ”Rapid Methodology for Design and Performance Prediction of Integrated SCRAMJET/Hypersonic Vehicle”, AIAA-89-2682, AIAA/ASME/SAE/ASEE 25th Joint Propulsion Conference, Monterey CA, July 10-12 1989.

11

Shneider, A., Koshel W., ”Detailed Analysis of a Mixed Compression Hypersonic Intake”, ISABE Paper No. 99-7036, 14th International Symposium on Air Breathing Engines, Florence, Italy Sept. 5-10 1999.

12

Soares, C., Rasmussen M., ”Integration of Scramjets with Waverider Configurations”, AIAA 892675, AIAA/ASME/SAE/ASEE 25th Joint Propulsion Conference, Monterey CA July 10-12 1989.

13

VonEggers, Rudd L., Rankins, F., Pines D. J., ”Moveable Cowl Control for Increased Hypersonic Performance”, AIAA 98-1575, AIAA 8th International Space Planes and Hypersonics and Tecnologies Conference, April 27-30 1998, Norfolk VA.

14

Nasuti, F., Onofri, M., ”Analysis of Unsteady Supersonic Viscous Flows by a Shock-Fitting Technique”, AIAA Journal Vol. 34 July 1996.
