15th. Annual (International) Conference on Mechanical Engineering-ISME2007 May 15-17, 2007, Amirkabir University of Technology, Tehran, Iran

XXXX should be replaced by paper number ISME2007-3056

Numerical Computation of Supersonic-Subsonic Ramjet Inlets; a Design Procedure

M. Akbarzadeh M. J. Kermani Graduate Student Assistant Professor mecmohsen2001@gmail.com mkermani@aut.ac.ir

Department of Mechanical Engineering

Amirkabir University of Technology (Tehran Polytechnic) Tehran, Iran, 15875-4413

multiple shock waves generated by the external-compression surfaces and the cowl and by the internal-compression surfaces from the cowl lip to the engine face. In this paper three different kinds of ramjet inlets are studied. These are inlet 1 as simple spike, inlet 2 as multi shocks external compression and inlet 3 as mixed compression. They are analyzed by Euler equations using Roe scheme with 3rd order accurate in space and second order accurate in time and also second order predictor-corrector MacCormack scheme (second order in both time and space), [1]. Most of the computations in high speed aerodynamics are performed in the supersonic combustion engine inlets. In contrast, in this study the flow of the interior part of engine inlet is directed to a subsonic combustion chamber and is much more complicated by forming a terminal normal shock. The effect of combustion in the present study modeled by imposing a high back pressure level associated with the combustion, [2]. The results are compared with each other and that of the commercial FLUENT software package. Governing Equations The governing equations of flow motion are 2D, unsteady, compressible and inviscid in full conservative form with no body forces which are given below,

0=+∂∂

+∂∂

+∂∂

HyG

xF

tQ

α (1)

where Q , F , G and H are: `

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

tevu

Q

ρρρρ

,

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

+=

0

2

uhuv

pu

u

F

ρρρ

ρ

,

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

+=

0

2

vhpv

uvv

G

ρρ

ρρ

,

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

=

0

21

vhv

uvv

rH

ρρ

ρρ

, (2)

and α is equal zero (2D analysis).

Abstract The performance of three different ramjet engine inletsis numerically considered in this study. The geometriesused are planar with the free stream Mach = 2.5. Inlet 1 is chosen from literature, a single oblique shockfollowed by a normal one, which given a low value oftotal pressure recovery. Inlets 2 and 3 are designed in away to produce a series of oblique shocks merging atthe cowl lip of the engine followed by a normal shock.The compression process in inlet 2 is completelyperformed in external part of the inlet. Instead inlet 3 isgiven a mixed internal and external compression.Inviscid Roe scheme with 3rd order accurate in spaceand second order accurate in time and second orderpredictor-corrector MacCormack scheme are applied inthis study and also commercial FLUENT softwarepackage. The results are in good agreement withliterature. Keywords: Supersonic Inlet, Multi Staging ExternalCompression, Mixed External and InternalCompression, Roe Scheme, MacCormack Scheme

Introduction The engine inlet is of prime importance for all air-breathing propulsion systems. Its major function is tocollect the atmospheric air at free stream Mach number,slow it down (probably involving a change of direction)and so compress it efficiently. In this role the inlet isperforming an essential part of the engine cycle and itsefficiency is directly reflected in the engineperformance. In addition, the inlet must present the air to the downstream component at the suitable velocitiesand with an acceptable degree of uniformity of velocityand pressure under any flight condition. Finally, theinlet has to achieve all this with minimum external dragand minimum disturbance to the external flow aroundaircraft. The supersonic inlet consists of a spike (center-body or fore-body) and an integrated duct, in which the initialcompression is being carried out by the spike. Theprinciple of staging a supersonic compression so as toreduce the loss of inlet total pressure is considered inthis study to modify inlet efficiency. By increasing theMach number, oblique shock numbers that needed tosave total pressure are increasing as well. The analysisof mixed-compression inlet flow is considered in thethird inlet and is complicated by the formation of

2

not exceed the maximum for shock attachment at a defined Mach number especially in second inlet, it limits numbers and angles of oblique shocks. 3.2 Inlet 2 The inlet is to be, is considered for Mach = 2.5 (according to the free stream speed of this study) and of wedge type (planar geometry) with shocks focused at the lip of engine cowl. Literature suggests that four shocks are needed and corresponding optimum flow turning angle is °5.31 , but with simple calculations can show that the external shock attachment limit is °30 . This could be observed assuming say, °3 lip vertex angle and using °27 cowl internal angle. With optimum flow turning the reverse angle of the duct would be °5.4 and a trial calculation for °5.9 , °5.10 and °5.11 wedge turns (first, second and third wedge angle respectively ) leads to a value 31.1 for the Mach number of the terminal normal shock. The calculations proceed as follows,

°= 5.91δ °= 38.311β °= 107.22M °= 5.102δ °= 84.372β °= 72.13M °= 5.113δ °= 63.483β °= 31.14M

The values of βsinM are successively 10.1 , 06.1 and

99.0 which, while not equal, are not greatly disparate, so the Oswatich principle is considered. Intersection of the first shock with the bounding streamline (defined by the required engine mass flow) locates the intake lip and itself helps the engine to operate in its full mass flow capacity. By striking appropriate angles back from the lip the leading edges of the second and third wedges are located. Inside the duct following a °5.4 reverse angling, the shape is such as to give a subsonic diffuser terminating at the engine face. The external cowl line, following an initial °3 vertex angle, is an arbitrary shape and also terminating at the position of engine face. Inlets 2 and 3 are designed in a way to produce a series of oblique shocks merging at the cowl lip of the engines followed by a normal shock. The geometry and computational grid of inlet 2 are shown in figure 2 and 3, respectively.

Fig2. Geometry of inlet 2

The working fluid is assumed as an ideal gas with constantspecific heat value.

RTP ρ= (3)

Geometry and Boundary Conditions 3.1 Inlet 1 Inlet 1 is chosen from literature, [3]. The simplest form ofstaged compression is the two-shock inlet in which asingle angle wedge or cone projects forward of the duct.For the explanation of boundary condition, figure 1 isshown. Apart from the quantities of boundary conditionsof inlet 1 which will be given in next parts, the boundaryconditions at the inlet is set to 40 kPa static pressure value and Mach number = 2.5. The flow is also assumed to bearriving to the computational domain as normal. At theexit plan, static pressure boundary is used. In this study theflow of the interior part is directed to a subsoniccombustion chamber. The effect of combustion issimulated by imposing constant pressure levels associatedwith combustion of the exit of the engine inlet. All flow parameters are extrapolated to the top of the computationaldomain and it is taken far enough from the engine inlet, so the oblique shocks generated from the spike leading edgeand cowl lip can not reach this boundary. At the centerline (from inlet plan to the leading edge of the spike), asymmetry condition is enforced.

Fig1. Geometry, boundary condition and computational grid for inlet 1

For a two dimensional oblique shock of angle β the loss is that corresponding to the component Mach numbernormal to the shock. The decreasing rate of total pressureloss with increasing oblique shock numbers makes it possible to advise systems of supersonic compression bystages, yielding high pressure recovery overall. Thenumbers and type of stages used depend upon free streamMach number and some other factors. Corresponding toOswatitsch principle for two dimension systems,maximum shock pressure recovery is obtained when theoblique shocks are of equal strength, [4].

112211 sinsinsin −−=== nnMMM βββ L (4) Limit on numbers of oblique shock is that the flow inpassing through the compression system produced byspike is turned outwards from the inlet axis and requires tobe turned back to the axial direction within the subsonicdiffuser. With efficient compression the turning angles canbe quite large. Also the external angle of the cowl must

3

Fig5. Computational grid for inlet 3

domain 1 )30229( × , domain2 )41173( × Flow solvers The main goals of this study were developing a CFD solver code for solving the flow fields of different kinds of ramjet engine inlet diffusers which are designed in a way that are proper for supersonic speeds with subsonic inflow engines and also designing two inlets which have got almost suitable pressure recovery in supersonic speeds. The two schemes, used in this study are (1) Roe scheme with 3rd order accurate in space and second order accurate in time as an upwind scheme and (2) second order predictor-corrector MacCormack scheme (second order in both time and space)[5,6]. In the first case, using Roe scheme as an approximate Riemann solver (ARS) for solving linearized form of the fluid equations is followed by a penalty in which the entropy conditions are not satisfied, as the expansion shocks (physically impossible phenomena in nature) are valied solution of the linearized fluid equations. Therefore expansion shocks are captured in the same way as the expansion waves are captured. This problem is fixed by an entropy correction technique and the corresponding entropy correction formula is chosen from reference [7], which is suitable on a wide variety of test cases and can cause the expansion shock to totally disappear from the sonic-expansion regions. A first order upwind scheme is also applied using commercial FLUENT software package, so the results are compared with each other and with the results of FLUENT software too. Results The present computation is performed for the inlet geometries which are shown above, and some of resultsare given in this part. For the first case the quantities of boundary conditions are given in the following table and is considered for comparing to the literature (in fact a case study for the result validation).

Table1. Boundary conditions of inlet 1

M Inflow Static

pressure (Pa)

Static back pressure

(Pa)

Inflow static temp.

(K)

3.0 2815 39694 113 Figures 6 and 7 show the Mach number and static pressure contours, respectively.

Fig3. Computational grid for inlet 2;

domain 1 )25200( × , domain2 )48154( × 3.3 Inlet 3 The outward turning of flow that goes with externalcompression leads to the use of outward angles on the inletcowl, which even with attached shocks, result insignificant drag. Mixed compression implies the use ofboth external and internal compression, in appropriatedegrees, in order to relieve the external drag problem ofthe former whilst avoiding excessive boundary layer orother disadvantages from the later. The third inlet is shown in figure 4. The first two compressions are external andgive o20 total turn. The shocks are focused at the lipwhere a reverse o5.11 turn is made by means of theinternal cowl angle. The second wedge is continued until itmeats the reversed shock. A o5.11 change of direction ofthe surface at this interaction cancels shock reflection andallows the normal shock to be positioned. This type ofinlet diffuser may be termed 50/50 external/internal. In all above inlets, the computational domain is divided in to 2regions and computational grid is produced by solvingLaplace's equations in each of them. The geometry andcomputational grid of inlet 2 are shown in figure 4 and 5,respectively.

Fig4. Geometry of inlet 3

4

(b)

Fig 8. Contours of Mach for inlet 2 (Roe scheme); (a) Pback = 450000 Pa, (b) Pback = 495000 Pa

(a)

(b)

Fig 9. Contours of static pressure for inlet 2 (Roe scheme); (a) Pback = 495000 Pa,

(b) Pback = 450000 Pa The most comprehensive study is performed in third case (and the most efficient inlet in this study, as will be shown in the next parts). Figures 10 and 11 show the contour of Mach number for two different conditions; first the supercritical condition and second near critical condition in which the normal shock wave is placed near the throat of engine inlet. Static pressure counters for inlet 3, are shown in figure 12. Like inlet 2, the flow enters to the engine at Mach number of about 0.5.

Fig 6. Mach number contours (critical condition) by

Roe scheme

Fig 7. Static pressure contours (critical condition) by

Roe scheme The obtained results are in very good agreements withliterature and critical condition is obtained at the sameback pressure that is presented by literature (this case isstudied for validating the obtained results) [3]. The contours of Mach number and static pressure for inlet2 are shown in figures 8 and 9, respectively. The flow enters to the engine at Mach number of about 0.5 (that is usually recommended for subsonic combustion jetengines) [4]. The boundary conditions of inlets 2 and 3 are given in thefollowing table in which a wide range of back pressure is studied, until achieving critical conditions.

Table2. Boundary conditions of inlets 2 and 3

M Inflow Static

pressure (Pa)

Static back pressure

(Pa)

Inflow static temp.

(K) 2.5 40000 - 250

(a)

5

(a)

(b)

Fig 12. Static pressure contours (inlet 3), (a) Roe scheme Pback = 530000 Pa, (b) MacCormack

scheme Pback = 500000 Pa The Static back pressure corresponding to critical condition for each inlet is given in table 3.

Table3. Static back pressure at critical conditions Roe Scheme

(Pa) MacCormack Scheme (Pa)

Inlet 2 525000 500500 Inlet 3 533000 506500

The static back pressures predicted by MacCormack scheme are a bit lower than the other (Roe Scheme). it might be because of more instability in flow field especially near critical condition which is an almost instable situation and because of the most diffusion that there is in MacCormack scheme and the pseudo unsteadiness that is entered to the flow field by this scheme. Figure 13 shows the configuration of total temperature contours for inlet 1.

(a)

(a)

(b)

Fig 10. Mach number contours, (MacCormack scheme), (a) Pback = 480000 Pa, (b) Pback = 500000 Pa

(a)

(b)

Fig 11. Mach number contours, (Roe scheme), (a) Pback = 500000 Pa, (b) Pback = 530000 Pa

6

(3) Design of the subsonic diffuser poses different problems in studied cases, however because it is greatly dependent on the engine installation situation, no generalization is attempted.

Conclusion Supersonic inlet design is a difficult task due to the complex flow physics and computational limitations. Three different ramjet engine inlets studied in this paper which the first one was chosen from literature (for validating the obtained results). The two last inlets designed in such a way to produce three oblique shocks merging at the engine cowl lip leads to staging the compression process to avoid high total pressure losses. Flux difference Roe scheme and MacCormack scheme are applied to the above inlets and the results compared with each other and that of the commercial FLUENTsoftware package. A good agreement shown in the results. List of Symbols

β Oblique shock angle δ Wedge angle p Static pressure

backp Static back pressure (at the end of inlet)

0h Total enthalpy α H vector multiplier te Total energy

References [1] Kermani, M. J., and Plett, E. G., 2001 “Roe

Scheme in Generalized Coordinates: Part I-Formulations”, AIAA Paper # 2001-0086

[2] Duncan B., Thomas S., 1992, “Computational Analysis of Ramjet Engine Inlet Interaction”, SAE, ASME, and ASEE, Joint Propulsion Conference and Exhibit, 28th, Nashville, TN, July 6-8, 12 p.

[3] Gossiping P., Lesage F., Champlain Aide., 1998, “The Application of Computational Fluid Dynamics (CFD) For the Design of Rectangular Supersonic Intakes”, Proceedings of CFD 98 Conference.

[4] Goldsmith E.L., Sedona J., 1999, Intake Aerodynamics, Blackwell Science, Second Edition.

[5] Kermani, M. J., and Plett, E. G., 2001 “Roe Scheme in Generalized Coordinates: Part II-Application to Inviscid and viscous Flows, AIAA Paper # 2001-0087

[6] Kermani, M. J., ”Simulation of the Viscous Turbulent and multi-Dimensional Gasdynamics effects on Flows in Inlet Diffusers of Supersonic Vehicles", Ph.D. thesis, Department of Mechanical & Aerospace Engineering, Carleton University, Canada, 2001

[7] Kermani, M. J., and Plett, E. G., 2001 “Modified Entropy Correction Formula for the Roe Scheme, AIAA Paper # 2001-0083

(b)

Fig 13. configuration of total temperature contours (a) Roe scheme, (b) Fluent software

As expected for inviscid flow, total temperature remainsconstant throughout the computational domain. This isalso a self-consistency check for obtained results. Engine cowl pressure drag force and total pressurerecovery at the end of the inlet diffusers are given in table 4. As mentioned already, the cowl drag force of the secondinlet is more than the third inlet and that is because ofmore cowl lip angle that forms stronger oblique shockstarted at the apex of cowl engine Whereas, there is notobvious difference between their pressure recovery so,with our assumptions (inviscid analysis) the third inlet isthe most efficient.

Table 4. Engine cowl pressure drag force and total pressure recovery (at critical conditions) Cowl

pressure drag force

(N)

Total pressure recovery

(Roe scheme)

Total pressure recovery (Fluent

software) Inlet 1 - 0.44 0.456 Inlet 2 36934.055 0.861 0.854 Inlet 3 11568.173 0.858 0.839

Comparing the three recent inlets; first, second and thirdinlet, the following results are noticeable:

(1) apart from the first inlet which is not efficientand is considered as a validation for producedCFD codes (and we do not consider it more), thesequence of compression Mach numbers are thesame in two last inlets and so is the throat size(assuming the same mass flow capacity) and sois the pressure recovery.

(2) The shortest possible supersonic section (defined

by the distance from fore body apex to normalshock) has been used in each case, consistentwith the particular design philosophy, thissection is nevertheless longer in third than insecond and because of it and increasingenclosure of shock wave, boundary layer (whichis neglected in current study) might be expectedto be more adverse in third than in the second. On the other hand because of more cowl angle of second inlet than the third, pressure dragcorresponding to the external shape of the cowlis more than the third. So there is a compromisebetween the more efficient internal flow of theformer and lower pressure drag of the latter.