airflow in ramjet inlets

NUMERICAL SIMULATIONS OF INVISCID AIRFLOWS IN RAMJET INLETS

M. Akbarzadeh and M. J. KermaniDepartment of Mechanical Engineering, Amirkabir University of Technology (Tehran Polytechnic),

Tehran, Iran, 15875-4413

E-mail: [email protected]

Received August 2008, Accepted April 2009

No. 08-CSME-30, E.I.C. Accession 3068

ABSTRACT

The performances of three different ramjet inlets and an entire ramjet are numerically studiedin this paper. The fluid is assumed to be inviscid. Inlet 1 is a SCRAMJET inlet and is chosenfrom the literature. Inlets 2 and 3 are instead designed based on the Oswatitsch principle. Inlets2 and 3 produce a series of oblique shocks merging at the engine cowl lip followed by aterminating normal shock just downstream of the inlet throat. In ramjet, the combustion ismodeled using a non-uniform volumetric heat source distributed in the combustor area. Theposition of the terminating normal shock in Inlets 2 and 3 is controlled via the inlet’s backpressure. Instead, in ramjet it is bounded by the amount of heat rate added in combustor andthe exhaust nozzle throat area. For the numerical simulations, the Roe (1981) andMacCormack (1969) schemes are used. To prevent the spurious numerical oscillations in highresolution computations by Roe scheme the van Albada flux limiter (1982) is used, while inMacCormack scheme artificial viscosity terms are added to damp the oscillations. To doublecheck the accuracy of the computations, the Fluent software package has also been used.Comparisons show very good agreement.

LES SIMULATIONS NUMERIQUES DE COURANTS ATMOSPHERIQUES NON-VISQUEUX DANS DES ENTREES DE STATOREACTEURS

RESUME

Les performances de trois entrees de statoreacteurs differents et un statoreacteur entier sontete etudies dans une facon numerique dans cet article. Le fluide est ete suppose etre non-visqueux. L’entree 1 est une entree SCRAMJET et ete choisie de la litterature. Au contraire, lesentrees 2 et 3 ont ete dessinees fonde sur le principe d’Oswatitsch. Les entrees 2 et 3 produisentun enchaınement des chocs obliques fondant a la levre sur le capot moteur suivi par un choctermine normal en aval de la gorge de l’entree. Dans le statoreacteur, la combustion est eteimitee utilisant une source de chaleur volumetrique non-uniforme qui est ete distribuee dans lachambre de combustion. La position du choc termine normal dans les entrees 2 et 3 a eteconduite par la contre-pression de l’entree. Cependant, dans un statoreacteur, il est lie parl’amont de chaleur qui est ete ajoute dans la chambre de combustion et la tuyere. Pour lessimulations numeriques, les schemas de Roe (1982) et MacCormack (1969) ont ete utilises. Aempecher les oscillations numeriques fausses dans les calculs en haute resolution par le schemede Roe, le limiteur de fluidifiant de van Albada (1982) est ete utilise, pendant que dans le schemede MacCormack, les termes de viscosite artificielle ont ete ajoutes a s’amortir les oscillations. Averifier l’exactitude des calculs, le progiciel de Fluent est ete utilise. Les comparaisons montrentun tres bon accord.

Transactions of the Canadian Society for Mechanical Engineering, Vol. 33, No. 2, 2009 271

1. INTRODUCTION

Ramjet engines are very promising devices for supersonic air–breathing vehicles. Theseengines are mechanically much simpler than turbojet types, while they can operate moreefficiently at supersonic speeds. However, the design of these engines encounter some problemsas they are expected to cover flights at various operating conditions. Ramjet engines consist of(1) an inlet-diffuser (either conical or wedge shaped- center body), (2) an integrated combustionchamber and (3) an exhaust nozzle. The compression process is carried out by the inlet-diffuser,followed by the combustion chamber. Finally, in the exhaust nozzle, the potential energy of thehot and compressed gases converts to kinetic energy to provide the required thrust for theengine [1].

The role of engine inlet is very vital for all air-breathing propulsion systems; especially forramjets in which all the compression process is performed by the inlet. This is as opposed to,say, turbojets that contain compressors to enhance the compression processes.

A ramjet inlet is supposed to perform the following tasks: (1) collect the atmospheric air atthe designed supersonic speed (generally the inlet expected to do this task at any free streamMach number and any flight altitude during the wide spectrum of the ramjet flight), (2) slow itdown through a series of oblique shocks followed by a terminating normal shock (generally thisinvolves a change in the flow direction as the flow usually passes through a series of obliqueshocks), and (3) deliver the compressed air flow to the ramjet combustion chamber at Machnumber of about Mcc < 0.5, where Mcc is the Mach number at the combustion chamberentrance face (generally Mcc 5 0.5 is a suitable Mach number for subsonic combustors in orderto give stabilized combustion processes). In addition, the inlet is expected to deliver thecompressed airflow to the combustion chamber at an acceptable level of uniformity of thepressure and speed. These are to assure stable combustion processes in the ramjet combus-tion chambers. These tasks, as noted earlier, should be done under any flight condition. Finally,the inlet should achieve all of these with (a) minimum external drag and disturbance to theexternal flows around it, and (b) at highest possible pressure recovery (i.e. highest inletefficiency). As the ramjet inlet is the ‘‘one man show component’’ to do the compressionprocess, so its appropriate performance is highly important. In this role, the inlet is consideredan essential part of the engine cycle and its efficiency is directly translated to the engineperformance.

The supersonic inlet consists of a spike (center-body or fore-body) and an integratedduct. The initial compression is done by the spike. When designing a supersonic inlet, anincrease in the flight Mach number requires the increase in the number of oblique shocks inorder to save total pressure recovery. Therefore, the principle of staging a supersoniccompression like the Oswatitsch principle is used to reduce the inlet losses in a most efficientmanner.

The combined compression process takes place via multiple shock waves generated by theexternal compression surfaces and the cowl, and later by the internal compression surfaces fromthe cowl lip toward the inner part of the engine face [2,3].

Most of the computations reported in the literature are for supersonic combustionchamber inlets and scramjet inlets [4,5,6]. The geometry of these inlets is not very com-plicated as the flow in these inlets lack any normal shock. On the other hand, the flowfield through the ramjet inlets passes through several oblique shocks then it is con-verted to subsonic flow through a terminating normal shock. In the following, the flow isdelivered to the combustion chamber. Ramjet inlets are much more complicated due


to the formation of a terminal normal shock. Combustion processes are introduced to theramjet inlets using an elevated back pressure [7]. The problem definition is given in thefollowing.

1.1. Problem DefinitionIn this paper three different ramjet inlets and one entire ramjet are numerically studied. The

flow is assumed to be inviscid. Viscous turbulent computations are underway and will come in aseparate study. Inlet 1 is a supersonic inlet (suitable for scramjets), and is chosen from theliterature [8] for validating the results of the present computations.

The supersonic portions of Inlet 2 and 3 are designed based on the Oswatitsch principle [2] inthe present study. Inlet 2 gives three oblique shocks in the external portion of the inlet. The flowturning through these oblique shocks is very high, hence the inlet produces a very high amountof cowl drag as compared to Inlet 3. The supersonic flow then enters the internal portion ofInlet 2, and passes through a terminating normal shock following the inlet throat. This post-shock subsonic flow continues its motion through the divergent portion of the inlet until theflow is delivered to the combustion chamber at Mach number of about 0.5. Inlet 3, instead,gives two external oblique shocks and one internal oblique shock followed by a terminatingnormal shock. Similar to Inlet 2, the subsonic flow further diffuses through the diffuser of Inlet3, and the flow is delivered to the combustion chamber at Mach number of about 0.5. Also inthe present study computations are performed for an entire ramjet engine. The effect ofcombustion through the ramjet combustion chamber is modelled by distributing a simple butrealistic volumetric heat source (VHS).

The position of normal shock in Inlets 2 and 3 are controlled by the inlet back pressures. Forthe entire ramjet, as no boundary condition is imposed at the entrance face of the combustionchamber, the position of normal shock is fixed by (1) the rate of heat release (Watts) throughthe combustion chamber, and the exhaust nozzle throat area. A set of parametric studies areperformed to assess the effects of VHS and the exhaust nozzle throat area on the position of thenormal shock. The dependence of cowl drag (the pressure drag), thrust and thrust specific fuelconsumption (TSFC) on the geometrical and operational parameters are discussed in thepresent paper.

The numerical simulations are performed using the flux difference splitting scheme of Roe [9]with various spatial orders of accuracy (including the third-order upwind biased algorithm) andthe predictor-corrector scheme of MacCormack [10]. The Roe scheme is an ApproximateRiemann Solver (ARS), so it is computationally much more efficient than the Exact RiemannSolvers (ERS) used in Godunov type schemes. This feature of the Roe scheme has made it oneof the most popular density-based schemes for compressible flows. The spurious numericaloscillations in high resolution computations using the Roe scheme are damped by van Albadaflux limiter (1982), while the artificial viscosity terms are added to the MacCormack scheme todamp the unwanted numerical oscillations. The commercial software package Fluent has alsobeen used in a few cases to check the accuracy of the computation. The computations of thesupersonic inlet (Inlet 1) are compared with those reported in the literature [8]. All comparisonsshow very good agreement.

2. GOVERNING EQUATIONS

The governing equations are solved in generalized coordinates in the present study. The fluidis assumed to be inviscid. The equations for the inviscid, unsteady and compressible flow in full


conservative form and in generalized coordinates with no body force for both planar andaxisymmetric cases are: (see Refs. [11,12] for example)

LQ1

Ltz

LF1

Ljz

LG1

LgzaH1~0 (1)

where Q1 is the conservative vector, F1 and G1 are respectively the inviscid flux vectors inj- and g-directions, and H1 is the axisymmetric space term in generalized coordinates. In Eqn. 1a 5 0 corresponds to the planar case and a 5 1 to axisymmetric flow condition. Theconservative and flux vector term in generalized coordinate are connected to those in physicalspace using,

Q1:JQ, F1:J jxFzjyG� �

G1:J gxFzgyG� �

H1:JH , (2)

where jx, jy, gx, gy, and J are respectively the metrics and Jacobian of transformation,and

Q~

rru

rv

ret

2664

3775, F~

ru

ru2zp

ruv

ruht

2664

3775, G~

rv

ruv

rv2zp

rvht

2664

3775, H~ 1

y

rv

ruv

rv2

rvht

2664

3775 :

Here, r is the density, u and v are respectively the x and y velocity components, et is thetotal energy, p is the static pressure, and ht is the total enthalpy. In the present study, forthe air as an ideal gas, the equation of state p 5 rRT is used, wherein R (5 287.0 J/kg.K

for air) is the gas constant. The internal energy, e, is determined by assuming a constantvalue for the specific heat at constant volume, e 5 cvT where cv 5 R/(c 2 1), with c 5 1.4 for theair.

3. TIME DISCRETIZATION

For the Roe scheme used in the present study, the following two-step predictor-correctorexplicit time integration from the Lax-Wendroff family is used [13]. The predictor step providesthe flow condition in an intermediate step n + 1/2.

Qnz1=21 {Qn

1

Dt=2z

LF1

Lj

� �n

zLG1

Lg

� �n

zaHn1~0 (3)

Equation 3 gives Qnz1=21 , from which all the primitive variables at the time step n + 1/2 can be

determined. The predictor step is followed by the corrector step, which completes theintegration. In the corrector step, a central differencing in time around n + 1/2 is implemented asfollows:

Qnz11 {Qn

1

Dtz

LF1

Lj

� �nz1=2

zLG1

Lg

� �nz1=2

zaHnz1=21 ~0: (4)

For the present explicit time integration the CFL stability criteria has been used [10].


4. SPACE DISCRETIZATION

For the spatial discretization of the Roe scheme the inner (L) and outer (R) (see Fig. 1) flowconditions are determined using either the first-, second-, or third-order upwind biasedalgorithm. The MUSCL strategy, [14], is used to extrapolate the primitive variables pressure (p),velocity components (u, v) and temperature (T) to the cell faces.

For example, at the east face of the control volume (E) the L and R flow conditions aredetermined as follows:

qLE~qi,jz

1

41{kð ÞDW qz 1zkð ÞDEq½ �,

qRE~qiz1,j~

1

41{kð ÞDEEqz 1zkð ÞDEq½ �:

(5)

where q represents either of the four primitive variables, i.e. q 2 {p, u, v, T}, k 5 21 and1/3 correspond, respectively, to the second order upwind and the third order upwind-biasedalgorithms. For the first order algorithm the L and R side values of the primitive variablesat the E-face (see Fig. 1) are determined from: qL

E~qi,j and qRE~qiz1,j. In Eqn. 5, DWq,

DEq and DEEq are the jumps of the primitive variables at the cell faces, i.e. DWq 5 qi,j 2 qi21,j,DEq 5 qi+1,j 2 qi,j and DEEq 5 qi+2,j 2 qi+1,j. A similar formula can be written for the inner(L) and outer (R) q values at the north face of the control volume (i.e. qL

N and qRN) shown

in Fig. 1.

To damp the spurious numerical oscillation in high resolution computations (i.e. in thesecond and third order extrapolations), the van Albada flux limiter (1982) is applied. This limitsthe slope of extrapolation,

qLE~qi,jz

w

41{kð ÞDW qz 1zkð ÞDEq½ �

qRE~qiz1,j{

w

41{kð ÞDEEqz 1zkð ÞDEq½ �:

(6)

Fig. 1. Configuration of grids with the inner (L) and outer (R) values for the cell (i,j). Grid lines areshown by solid lines and control volume boundaries with dashed lines [12].


where w is the limiter function defined by

wi,j:2 DW qð Þ DEqð Þze0

DW qð Þ2z DEqð Þ2ze0

(7)

and e0 is a small number which prevents indeterminacy in regions of uniform flow, i.e. in andaround DWq 5 DEq 5 0.

To avoid expansion shocks in the regions where the eigenvalues vanish, an entropy correctionformula [15] is used throughout this paper,

llnew/ ll2ze2� �.

2e lj jve

e~4:0 max 0, ll{lL� �

, lR{ll� �h i

,

8<: (8)

where ll is the eigenvalue of the Jacobian flux matrix determined at Roe’s averaged condition,and lL and lR are the eigenvalues determined at the inner and outer flow conditions,respectively.

4.1. Roe’s Numerical Flux SchemeThe numerical flux Roe is obtained based on the formulae [12,16]:

East Face : FE~1

2FL

E zFRE

� �{

1

2

X4

k~1

llkð Þ

E

�� dwkð Þ

E TTkð Þ

E (9)

North Face : GN~1

2GL

NzGRN

� �{

1

2

X4

k~1

llkð Þ

N

�� dwkð Þ

N TTkð Þ

N (10)

where l is the eigenvalue of the Jacobian flux matrix, T is the corresponding eigenvector and dw

is the wave amplitude vector.

4.2. Roe’s AveragingThe numerical flux for the Roe scheme is calculated at the so-called Roe-averaged values [9],

which are obtained from the inner L and outer R state values at the sides of, say, E cell face:

rrE~

ffiffiffiffiffiffiffiffiffiffiffirL

ErRE

q(11)

uuE~

ffiffiffiffiffiffirL

E

puL

EzffiffiffiffiffiffirR

E

puR

EffiffiffiffiffiffirL

E

pz

ffiffiffiffiffiffirR

E

p , vvE~


E

pvL

EzffiffiffiffiffiffirR

E

pvR

EffiffiffiffiffiffirL

E

pz


E

p (12)

hhtE~


E

pht

LEz


E

pht

REffiffiffiffiffiffi

rLE

pz


E

p : (13)


Similar formulas can be written for the quantities on the north side of a cell face (rrN , uuN ,etc.).

5. MacCormack SCHEME

In the present study, for some cases the governing equations are also solved using thepredictor-corrector scheme of MacCormack, detailed for example in [10]. This predictor-corrector scheme is of second-order accuracy both in space and time:

Predictor Step : Qnz1

1i,j~Qn

1i,j{Dt

Fn1iz1,j

{Fn1i,j

� �Dj

zGn

1i,jz1{Gn

1i,j

� �Dg

zHn1i,j

24

35 (14)

Corrector Step : Qnz11i,j

~1

2Qn

1i,jzQ

nz1

1i,j

� �{

Dt

2|

Fnz1

1i,j{F

nz1

1i{1,j

� �Dj

zG

nz1

1i,j{G

nz1

1i,j{1

� �Dg

zHnz1

1i,j

24

35 (15)

Here, forward differences are used for all spatial derivatives in the predictor step, followed bybackward differences in the corrector step. These give the net scheme to be spatially secondorder. As the forward and backward differences are alternated between the predictor andcorrector steps, this eliminates any bias due to the one-sided differentiation [6]. However, thealternation in the order of spatial differentiation introduces a pseudo-unsteadiness to thesolution when marched in time. Therefore, the MacCormack scheme cannot reach to machineaccuracy [17].

For the MacCormack scheme in regions with large discontinuities such as shocks, an artificialviscosity term (AV) is added to stabilize the solution. To do so, in the present study, a fourth-order dissipation term, with two adjustable constants Cx and Cy, are employed. These terms areadded to the right-hand side of predictor and corrector steps. The artificial viscosity term (AV)at the (i, j) node is given by [10]:

AVi,j~Cx

piz1,j{2pi,jzpi{1,j

�� piz1,jz2pi,jzpi{1,j

Q1iz1,j{2Q1i,jzQ1i{1,j

� �zCy

pi,jz1{2pi,jzpi,j{1

�� pi,jz1z2pi,jzpi,j{1

Qii,jz1{2Q1i,jzQ1i,j{1

� � (16)

where Cx and Cy in Eqn. 16 are adjustable constants which are in the range of 0 to 0.5 [10]. Inthe present study, using a basic trial and error approach we have tunned Cx and Cy to 0.1.

6. GEOMETRIES AND BOUNDARY CONDITIONS

6.1. Inlet 1; Supersonic Flow to the Combustion ChamberInlet 1 chosen from the literature [8], is a supersonic inlet, and is suitable for supersonic

combustion ramjets (scramjets). It is considered for validating the results of the presentcomputations. The inlet is a multi-oblique-shock supersonic inlet that delivers the air flow at


supersonic speeds to the combustion chamber. The geometrical dimensions of the inlet, thecomputational domain, and the boundary conditions are depicted in Fig. 2. The compressioncorner shown in the figure is 10-degrees. For the boundary conditions, supersonic flow at flightMach 5 2, and altitude < 10 km (p‘ 5 26.5 kPa and T‘ 5 226 K) enters to the computationaldomain. At the exit, for the supersonic outflow all the flow parameters of p, T, u, and v areextrapolated from the internal nodes. The line AC (a line extended from the leading edge of thecowl to the computational domain at inlet plane) is assumed to be a straight streamline. This isa totally true assumption for the supersonic approaching flow. Therefore, the verticalcomponent of velocity is set to zero along this line. Pressure along the AC line is also determinedusing Lp/Ly 5 0. Due to the symmetry conditions along the centerline, only one side of thecenterline is computed here, along which the symmetry conditions are applied. Along thesurface of the wall, no-penetration through the solid wall is permitted, i.e. ~VV :nn~0, where nn is theunit vector normal to the wall. Along the straight-lined wall surfaces the radius of curvaturesare infinite, hence, pressure is determined using +p:nn~0.

6.2. The Oswatitsch PrincipleThe design of the supersonic portion of Inlets 2 and 3 is performed using the Oswatitsch

principle. The principle states that for a two-dimensional supersonic duct, the maximum shock-pressure-recovery can be obtained if the strength of the train of oblique shocks are equal [2].That is, the components of Mach numbers in directions normal to the shocks are equal (seeFig. 3-(Left)),

M1 sin b1~M2 sin b2~M3 sin b3~ . . . ~Mn{1 sin bn{1 (17)

where n is the number of shock waves (n 2 1 oblique shocks and one terminating normalshock), Mi (1 # i # n 2 1) is the Mach number upstream of the ith shock and b is the shock

Fig. 2. Geometry and boundary conditions for Inlet 1 (all dimensions in meter).


angle, as shown in Fig. 3-(Left). In Eqn. 17, Mi sin bi ~Mnið Þ is the component of Mach

number normal to the ith shock. In fact Eqn. 17 implies that Mniremains constant throughout

the train of oblique shocks for minimized shock losses.

6.3. Inlet 2; External CompressionInlet 2 is designed based on the Oswatitsch principle (see Section 6.2 for detail). The inlet

receives flow at Mach 5 2.5, and is designed to give the supersonic compression solely in theexternal portion of the inlet. The inlet is of planar geometry with shocks focused at its cowl lip.

According to the Oswatitsch principle, for the optimum supersonic compression, thecomponent of Mach numbers normal to the train of shocks should be same. This leads to rampangles of d1 5 9.5u, d2 5 10.5u and d3 5 11.5u (see Table 1). These flow turning angles (d1 to d3)determine the Mach number upstream of the terminating normal shock, i.e. M4 5 1.30. Theresults of the calculations are summarized in Table 1. As shown in this table, the value ofMi sin bi~Mni

is within the range of (1.29, 1.30), so Mniis almost constant. That is, the

Oswatitsch principle is fairly well adhered to.For an inlet, the captured air flow is determined from the spacing between the engine

centerline and the so-called bounding streamline. This spacing is shown by A‘/2 in Fig. 3-(Right). Here, the bounding streamline is a streamline (or streamtube in axisymmetric cases)originating from the engine cowl and extended to freestream, as shown in Fig. 3-(Right). Hence,the location of cowl lip is determined from the intersection of the first shock with the boundingstreamline, see Fig. 3-(Right). This fixes the spike apex angle d1 (see Fig. 3-(Right) or Table 1).

Fig. 3. Procedures for inlet design: (Left)- Shock angles b and flow turning angles d for a typicalsupersonic inlet, and (Right)- the configuration of the cowl lip position and the bounding streamline.

Table 1 Flow turning angles, the corresponding shock wave angles, and the Mach number sequencesfor Inlet 2

d1 5 9.5u b1 5 31.38u M2 5 2.11 M1 sin b1 5 1.30d2 5 10.5u b2 5 37.84u M3 5 1.72 M2 sin b2 5 1.29d3 5 11.5u b3 5 48.63u M4 5 1.30 M3 sin b3 5 1.29


With M1 5 2.5, and d1 and b1 values given in Table 1, the post-shock Mach number is obtainedas M2 5 2.11. Also with known turning angle d2 5 10.5u, the ramp location of the secondoblique shock that will merge with the first shock at the cowl lip can be determined. The sameprocedure can be repeated until the last shock. Complete information including the flow turningangles for Inlet 2 is given in Table 1.

The geometry and boundary conditions for Inlet 2 are shown in Fig. 4-(Left). For thesupersonic inflow and outflow, the boundary conditions are exactly the same as those of theInlet 1. For the subsonic outflow (where the flow is delivered to the engine combustionchamber) the back pressure is specified, and other three parameters T, u, and v are extrapolatedfrom the interior nodes.

6.4. Inlet 3; Combined Internal and External CompressionIn the case of Inlet 2, in which the compression process is solely performed in the external

portion of the inlet, the outward turning of flow (i.e. diversion of flow away from the enginecenterline) due to the train of oblique shocks requires the inlet cowl to highly bend inward (i.e.turn toward the centerline). This bend of the cowl lip significantly raises the engine pressuredrag, and is called the cowl drag. Alternatively, combined compression, i.e. supersoniccompressions in both external and internal portions of the inlet, is used to reduce the cowl drag.

The schematic of Inlet 3 that possesses combined internal and external compression isillustrated in Fig. 4-(Right). Here, the first two compression processes are performed in theexternal portion of the inlet, and give 20u total turn. That is, 9.5u flow turning by Ramp 1,followed by 10.5u flow turning by Ramp 2 (see Fig. 4-(Right)). These shocks are focused at thecowl lip, where an inward bend (turning toward the centerline) angle of 11.5u is made at thecowl lip. On the surface of the spike, Ramp 2 is continued just before it meets the reflectedshock from the cowl lip. An inward diversion of the spike surface (turning toward thecenterline) at the intersection point eliminates the shock reflection and allows the normal shockto be formed due to the elevated back pressure. The boundary conditions specified for Inlet 3are identical to those of Inlet 2.

Fig. 4. Schematic of geometry and boundary conditions for: (Left)- Inlet 2 giving externalcompression solely (all dimensions in meter), and (Right)- Inlet 3 with combined external andinternal compressions (all dimensions in meter).


6.5. The Entire RamjetThe last case studied in this paper is the entire ramjet engine with axisymmetric flow and

geometry. The engine consists of inlet-diffuser, a subsonic combustion chamber followed by anexhaust nozzle. The flight conditions are Mach 5 3 at altitude of about 10 km (p‘ 5 26.5kPa

and T‘ 5 226K). The inlet of the engine is designed based on the Oswatitsch principle describedin Sec. 6.2. The designed inlet gives two oblique shocks merging at the engine cowl lip. Theseshocks are followed by a terminating normal shock located immediately downstream of theinlet throat. Then, the subsonic flow is delivered to the combustion chamber, where the effect ofcombustion is simplified by a volumetric heat source (VHS) function that is devised to give afairly linear distribution of the total temperature along the combustion chamber. The VHS isdefined as,

V H S~ru�cpDTt=L units : Watt

m3 �

(18)

where u* is defined as u* 5 max(u, 0), in which u is the axial velocity component, DTt is the totaltemperature rise throughout the combustion chamber (~Tto

{Tti), Tti

and Ttoare respectively

the total temperature at the entrance and exit of the combustion chamber, L is the combustionchamber axial length and cp is the specific heat at constant pressure (5 cR/(c 2 1)). In thefollowing, we explain why such a simplified volumetric heat source (VHS), given by Eqn. 18,can properly predict the rate of heat generation through a combustion chamber [18]. It is notedthat the present computation is aimed only for inviscid cases, but it is shown below that theVHS function in Eqn. 18 is suitable for both viscous and inviscid cases.

The VHS function in Eqn. 18 has the units of W/m3, and the rate of total heat generationwithin the combustion chamber (cc) can be determined from:

Qcc:ð

cc

V H S:dv~

ðcc

ru�cp

DTt

L

� �:dv, (19)

where dv is the volume element within the combustion chamber. In a real combustion pro-cess, the small values of fluid velocities close to the walls degrade the mixing process betweenthe air and fuel, then a smaller amount of heat release is expected in wall regions. Thismatter is incorporated in the formula given by Eqn. 18, where the VHS function decreasesin the vicinity of walls since u reduces close to the walls in viscous flows due to no-slipconditions at walls. On the other hand, the viscous dissipation term, mW, appearing in theenergy equation:

mW:mLu

Lyz

Lv

Lx

� �2

z2Lu

Lx

� �2

zLv

Ly

� �2" #

zlLu

Lxz

Lv

Ly

� �2( )

units : Watt

m3 �

(20)

increases close to the walls due to large gradients of velocity in the wall regions, where m is theviscosity, and l 5 2(2/3)m. The mW term in Eqn. 20 is in fact the rate at which the mechanicalwork is irreversibly converted to thermal energy due to viscous effects. Hence, in fundamentalnature, mW behaves like a volumetric heat source. The increase in mW close to the wallscompensates the reduction of VHS in these regions. Therefore, in the net, a fairly uniformprofile for the rate of heat generation per unit volume (i.e. VHS + mW < Const.) is achieved intransverse directions for viscous flows. This is in fair agreement with real combustion processes.For the computations of inviscid flows (as is the case in the present study), mW 5 0, and in the


absence of viscous effects there is no means to reduce u close to the walls. Therefore, the VHSfunction gives a fairly uniform profile similar to a real combustion process.

Following the heat injection through the combustion chamber, the pressurized hot gasesenter the exhaust nozzle and produce the propulsive thrust.

The geometry and boundary conditions for the ramjet engine are shown in Fig. 5.

6.6. The Performance AnalysisTo make performance analysis of the inlets and the entire ramjet, we consider an entire

engine. Consider a jet engine surrounded by a control volume around the engine as shown inFig. 6. Applying the linear momentum equation [19] for the engine:

SF!

~LLt

ðC:V :

V!

rdvð Þzð

C:S:

V!

rV!:d A!� �

: (21)

Simplifying this equation along the x-axis for steady state condition, we obtain:

FzFcowlzFf { Pe{P?ð ÞAe~ _mm Ve{V?ð Þ, (22)

where F is the engine thrust, Fcowl is the cowl drag defined by [2]

Fcowl:ð

Ap

p{p?ð ÞdA (23)

Fig. 5. Geometry and boundary conditions for the entire ramjet.

Fig. 6. Configuration of a control volume surrounding the ramjet engine.


where Ap is the circumferential projected area of the inlet or engine (i.e. the cross-sectional areaof the cowl surface perpendicular to the inflow direction). Ff in Eqn. 22 is the skin friction drag(which is zero in the case of inviscid flow computations). Solving Eqn. 22 for the thrust force,we obtain:

F~ _mm Ve{V?ð Þ|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}momentum thrust

z Pe{P?ð ÞAe|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}pressure thrust

{Fcowl{Ff , (24)

where _mm is the total mass flow captured by the engine (5 r‘V‘A‘), Ve and V‘ are respectivelythe velocity leaving and entering the control volume, and Ae is the nozzle exit area. In Eqn. 24,_mm Ve{V?ð Þ is the momentum thrust, (Pe 2 P‘)Ae is the pressure thrust. In the case of

incomplete expansion to ambient pressure, pressure thrust will exist for the price of some loss inmomentum thrust. The most suitable position for the normal shock is right at the inlet throat,as it gives the weakest normal shock. The rise of inlet back pressure (i.e. the pressure at theentrance face of the combustion chamber) pushes the terminating normal shock toward the inletthroat. However, the inlet throat is an unstable position for the normal shock, as slightlyexcessive inlet back pressure will make the inlet disgorge the normal shock and give a hugeamount of pressure drag (cowl drag). So the inlet back pressure should be properly tuned forthe sake of better performance of the inlet. There are two important downstream issues thataffect the inlet back pressure, namely the rate of heat release within the combustion chamber(Qcc as given by Eqn. 19) and the exhaust nozzle throat area. The increase of Qcc and decreaseof the exhaust nozzle throat area both raise the inlet back pressure and push the normal shockupstream.

The inlet or the engine cowl drag (as illustrated by Fcowl in Fig. 6) is a type of pressure dragand is used to determine the cowl drag coefficient,

CDcowl~

Fcowl

12r? cRT?ð ÞM2

?Ap

, (25)

The total pressure recovery of the inlet, g, is defined as the ratio of total pressure atcombustion chamber entrance face, Pt,f, to that of the free stream, Pt,‘:

g~Pt,f

Pt,?(26)

For the entire engine, the thrust and the energy consumption or conversion values areusually used to represent the engine performance. Specifically, the propulsion efficiency (gp),the kinetic energy conversion efficiency (ge), the overall efficiency (go) and the thrust speci-fic fuel consumption (TSFC) are used for these reasons. These parameters are given asfollows:

gp~2

1zVj

V?

; ge~_mm V 2

j {V 2?

� �.2

_mmf Qnet,p; go~

F V?

_mmf Qnet,p; TSFC~

_mmf

F: (27)

Here V‘ and Vj are the free stream and exhaust nozzle jet velocities, respectively, and is thecaptured air mass flow rate. Qnet,p in Eqn. 27 is the heating value of a typical hydrocarbon fuel(5 43100 kJ/kgfuel [20]) and _mmf Qnet,p is heat rate (kW) of the fuel. For the present computation


in which the combustion chamber is modelled by distribution of a volumetric heat source(VHS), an equivalent fuel consumption can be obtained by using Eqn. 19:

_mmf Qnet,p~Qcc [ _mmf ~Qcc

Qnet,p: (28)

Using _mmf Qnet,p~Qcc, Eqn. 27 gives:

go~FV?

Qcc

: (29)

As a consequence of the inviscid flow assumption, the drag is of pressure-type only. Thispressure drag is divided into cowl drag and drag force on the spike. The spike drag in the firstobservation of Eqn. 24, seems to have no contribution on the calculation of propulsive thrust asit is within the control volume of Fig. 6. But in fact the spike effect plays its influence on theinternal drag of the engine, and reduces the net momentum thrust (exhaust jet velocity), seeEqn. 24.

Also it is worthwhile to mention that the captured mass flow ( _mmf ) is independent of inlet backpressure (i.e. Qcc and the exhaust nozzle throat area) as long as the terminating normal shockremains swallowed by the engine. That is, the flow enters the engine supersonically. Also thecowl drag remains unchanged. These issues are discussed later in Sec. 7.

7. RESULTS AND DISCUSSIONS

The present computations are performed on Inlets 1 – 3, and the entire ramjet geometriesdescribed in Section 6. The results are given in this section.

7.1. Grid ConfigurationsIn this study, the computational domain is divided into two subdomains (except for Inlet 1

which has a single computational domain), and an algebraic method is employed to generatestructured quadrilateral grids, [21,22], in each subdomain. Fig. 7-Left shows sample of gridconfigurations for Subdomains 1 and 2 for Inlet 3.

To determine adequate grid sizes for each geometry, grid independency checks are conductedin the present study. A sample of results that shows the variation of pressure coefficient Cp

along the spike surface of Inlet 3 with various grid sizes are shown in Fig. 7-Right, where

Cp~p{p?

1=2r?V 2?: (30)

The grid sizes for grid independent computations and for all the geometries used in thepresent study are given in Table 2.

7.2. Inlet 1Inlet 1 is a scramjet inlet that is considered for validating the results of the present

computations. The flow supersonically enters the computational domain of the inlet and itcompresses over the 10-degree corner at point C (see Figs. 2 and 8 (a)). Figure 8 (a) also showsthe geometry of the inlet, over which for clarity purposes the pressure contours and streamlinesare superimposed. The Mach number contours are also shown in Fig. 8 (b). All the shocks and


expansion fans are clearly illustrated in this figure. Figure 8 (c) shows the distribution ofpressure coefficient Cp for Inlet 1 along the inner part of the top wall. The figure (Fig. 8 (c))shows the results of the present computations using the 1st, 2nd and 3rd order scheme of Roe, theMacCormack scheme, Fluent and the results from the literature [8]. As shown in this figure verygood agreement are achieved in all computations, except for the 1st order computations by Roescheme. The discrepancy is due to the diffusive behavior of the first-order upwind scheme.

The oblique shock, originated from the compression corner at C, say at the bottom wall, isreflected from the opposite wall (the top wall) then impinges on the bottom wall again. On theother hand, when the flow passes over the convex corner over the bottom wall a series ofisentropic expansion fans form (see Fig. 8 (b)). Due to these expansion fans, nonuniform Machprofiles are formed upstream of the shock originated from the opposite wall. Therefore, eachstreamline passing through the oblique shock will experience a shock with different strength.This produces a non-isentropic post-shock condition as shown in Fig. 8 (d). As the consequenceand according to the Crocco’s theorem [23,24] a rotational flow in this region is formed (see Fig.8 (e)).

Fig. 7. Computation for Inlet 3: (Left)- Computational grid and the corresponding subdomains forInlet 3; Subdomain 1: (229 6 31), Subdomain 2: (173 6 37), and (Right)- Grid independency test;distribution of pressure coefficient on the spike wall with different grid sizes for Inlet 3. Grid 1:Subdomain 1 (349 6 71) and Subdomain 2 (275 6 51); Grid 2: Subdomain 1 (401 6 103) andSubdomain 2 (301 6 103).

Table 2 The grid independent mesh system used in the present computation.

Inlet 1 Inlet 2 Inlet 3 Ramjet Engine

Subdomain 1 151 6 51 349 6 51 351 6 51 991 6 51Subdomain 2 - 275 6 51 275 6 51 451 6 41


Fig. 8. Geometry and boundary conditions for Inlet 1 with supersonic outflow delivered to thecombustion chamber; (a) Iso-bar lines with the streamline configuration; (b) Mach numbercontours; (c) Pressure coefficient along the internal wall; (d) The entropy contour; (e) The vorticitycontours.


7.3. Inlets 2 and 3The flows supersonically enter the computational domains assigned to Inlets 2 and 3 in

parallel directions to their symmetry lines. For Inlet 2, the flow passes over three ramps (seeFigs. 4-(Left) and 9), so three oblique shocks are originated from the lips of the ramps andmerge at the engine cowl lip that eliminate any extra reflection from the cowl lip. Then thesupersonic flow converts to subsonic flow in Inlet 2, via a terminating normal shock generatedby the high value of inlet back pressure imposed at the inlet’s exit plane. On the other hand, inInlet 3 the flow enters the internal diffuser after passing through two consecutive externaloblique shocks generated by two ramps (see Figs. 4-(Right) and 10). These two external obliqueshocks are followed by an internal oblique shock generated by the cowl internal angle in Inlet 3.Then similar to Inlet 2, the flow in Inlet 3 converts to subsonic flow via a terminating normalshock generated by the high value of inlet back pressure.

Figure 9 (a), (b), (c), and (f) show, respectively, the contours of Mach number, totaltemperature, the entropy and velocity vectors in and around the exit plane of Inlet 2. Theentropy at any point in Fig. 9-(c) is calculated w.r.t the inflow entropy using:

s~cplnT

T?{Rln

p

p?, (31)

in which T‘ and p‘ are respectively the static temperature and pressure at the inlet boundary.The line AA9 in Fig. 9 corresponds to the cowl lip section at x 5 0. Also shown in the figure isthe pressure coefficient Cp along the spike surface obtained at various inlet back pressures (seeFig. 9 (d)). The subsonic flow leaves Inlet 2 and enters the engine’s combustion chamber –notcomputed here– at Mach number Mcc < 0.5, where Mcc is the Mach number at the combustionchamber entrance face. Mcc < 0.5 is usually a suitable speed required by the subsoniccombustion chambers [2]. Figure 9 (e) shows the distribution of Mach number, Mcc, at the Inlet2 exit plane (i.e. the combustion chamber entrance face). As shown in this figure, Mach numberis just slightly less than 0.5 at the exit plane of the interior part of the inlet, where the flow isdelivered to the combustion chamber.

Consider Fig. 4-(Right) again that shows the geometry of Inlet 3. Figure 10 (a), (b), (c), and(d) show, respectively, the contours of Mach numbers, total temperature, the entropy, anddistributions of Cp along the spike surface of Inlet 3 that are obtained at various inlet backpressures. Similar to the AA9 line in Fig. 9 the line AA9 in Fig. 10 corresponds to the cowl lipsection at x 5 0. As shown in these figures, total temperature remains constant throughout thecomputational domain. Also shown in this figure (see Fig. 10 (e)) is the distribution of Machnumber at the interior and exterior portions of Inlet 3 at the exit plane. Like Inlet 2, the Machnumber at the entrance face of the combustion chamber, Mcc, is computed < 0.5 (Fig. 10 (e)).The velocity vectors in and around the exit plane of Inlet 3 are also shown in Fig. 10 (f).Computations of Inlet 3 are also performed at various back pressures of the inlet to monitor theinfluences on the positions of normal shock. The results are shown in Fig. 11 (a) to (f). Forcomparisons of the results between various numerical solutions for Inlet 3, the flow fieldcomputation is performed using the MacCormack scheme as well, and the Mach numbercontours are shown in Fig. 12. Comparison between Fig. 11 (e) and Fig. 12 shows excellentagreement for the Mach number contours.

Based on the designs made in this paper, the throat for Inlet 2 is located at the cowl face,whereas in Inlet 3 the supersonic flow continues into the inner part of the diffuser until the flowmeets the throat a bit downstream of the intersection of the third oblique shock and the spike


Fig. 9. Computations for Inlet 2; (a) Mach number contours; (b) Total temperature contours; (c)Entropy contours; (d) Pressure coefficient along the spike surface at various inlet back pressures; (e)Profile of Mach numbers at the exit plane of the inlet for both interior and exterior parts; (f) Velocityvectors around the exit plane.


Fig. 10. Computations for Inlet 3; (a) Mach number contours; (b) Total temperature contours; (c)Entropy contours; (d) Pressure coefficient along the spike surface at various inlet back pressures; (e)Profile of Mach numbers at the exit plane of the inlet for both interior and exterior parts; (f) Velocityvectors around the exit plane.


surface. Inlet 3 has several advantages from different aspects. It delivers the flow to thecombustion chamber at a reasonable level of uniformity. Flow uniformity at the entrance faceof the combustion chamber (say uniform profiles of Mach number, pressure and temperature)

Fig. 11. Mach number contours for Inlet 3 at various back pressure values computed by the 3rd

order Roe scheme.


leads to more stable combustion processes in the engine combustion chamber. Inlet 3 also has abetter efficiency than Inlet 2. Performance analysis of Inlets 2 and 3 are discussed as follows.

The increase of inlet’s back pressure pushes the normal shock toward the inlet throat.Excessive increase of back pressure to values above the on-design value will make the inletdisgorge the normal shock and give huge amount of pressure drag. On the other hand, a backpressure less than an on-design value will suck the normal shock to deeper sections of the inlet,i.e. into larger sections of the diffuser duct. This will generate a stronger terminating normalshock, increase the inlet losses and degrade the performance of the inlet. Hence, there exists anoptimized inlet’s back pressure, the so-called ‘‘on-design back pressure’’. Table 3 summarizesthe performances of the inlets for the on-design back pressures (optimized condition) of Inlets 2and 3.

Inlet 1 is a supersonic inlet and is suitable for scramjet engines. It has the highest efficiency asit lacks any normal shock. On the other hand, Inlets 2 and 3 are suitable for ramjet engines. Thecowl drag of Inlet 2 (the inlet with external compression only) is problematic, as compared withthat of Inlet 3 (the inlet with combined internal and external compressions). As noted inTable 3, Inlet 2 has a cowl drag of about 5.5 times larger than that of Inlet 3. This is due to thelarge cowl lip angle of Inlet 2, which is bent inward as shown in Fig. 4-(Left). The inward bent isrequired to cover the flow-turnings (i.e. the flow diversions from the centerline directions)

Fig. 12. Mach number contours for Inlet 3 at on-design back pressure computed by MacCormackscheme.

Table 3 Summary of the computation for Inlets 1–3 for on-design (optimized) condition. Here: (1)Fcowl is the cowl drag (see Fig. 6 and Eqn. 24), (2) CDcowl

is the drag coefficient of the cowl (see Eqn.25), (3) _mm the mass flow rate (per unit depth) captured by the inlet.

Fcowl(1)(kN) CDCowl

(2) g (%)

_mm (3)(kg/s/m) Comments

Inlet 1 - - 94.99 246.12 Supersonic outflow, suitable for scramjetInlet 2 17.10 0.54 85.64 158.86 Subsonic outflow, suitable for ramjetInlet 3 3.11 0.16 89.36 167.48 Subsonic outflow, suitable for ramjet


through three external oblique shocks generated at the ramp lips. It is noted that the cowl dragcoefficient (CDcowl

) of Inlet 2 is only 3.3 times larger than that of Inlet 3. The difference is due thedifferences in their corresponding Ap. As noted in Table 3, Inlet 3 has a slightly better pressurerecovery efficiency (g) as compared to Inlet 2.

7.4. The Entire RamjetThe last case analyzed is the entire axisymmetric ramjet engine. For the geometry and

specifics of the engine refer to Fig. 5. With a designated inlet-diffuser for the ramjet, a series ofparametric studies are performed by varying two parameters and the influences on theperformance of the engine are studied. The parameters varied are the rate of heat releasethrough the combustion chamber (Qcc), and the exhaust nozzle throat diameter (DNozzle).

Figure 13-(Left) shows a sample of computations for the entire ramjet at Qcc 5 16.0 MW

(DTt 5 1105 K) and DNozzle 5 0.22 m. The flight conditions are M‘ 5 3, P‘ 5 26.5 kPa, and T‘

5 226 K. The geometry of the entire ramjet and the contours of Mach number and totaltemperature in and around the entire engine are shown in Fig. 13-(Left). As noted in Fig. 13-(Left) (c) total temperature rises just within the combustion chamber, and it remains constantdownstream and upstream of the combustion chamber as well as outside the engine.Computations of the entire ramjet at three different levels of Qcc are shown in Fig. 13-(Right).The figure shows the Mach number contours. As Qcc increases the normal shock is pushed

Fig. 13. Computation for the entire ramjet: (Left)- Computation at the combustor heat release valueof Qcc 5 16 MW and the nozzle throat size of DNozzle 5 0.22 m, Fig-(a) the ramjet geometry, (b) theMach number contours and (c) the total temperature contours, and (Right)- Mach number contoursand normal shock movement toward the inlet throat versus different heat source values (a)_QQcc~7:5 MW (DTt 5 485K) (b) _QQcc~10 MW (DTt 5 690K) (c) _QQcc~16 MW (DTt 5 1105K).


upstream. The position of normal shock for each Qcc is also illustrated in Fig. 13-(Right).Table 4 summarizes the parametric study on the effect of Qcc and DNozzle on the position ofinlet-diffuser normal shock. As noted from this table, any decrease in DNozzle or any increase inQcc pushes the normal shock in upstream direction. These data are graphically pictured as notedin Fig. 14-(Left), for a given DNozzle. Figure 14-(Right) shows the increase of total temperaturealong the combustion chamber centerline. As expected, a linear profile for the total temperaturedistribution along the combustion chamber is obtained.

Table 5 summarizes a parametric study on the entire ramjet at various Qcc. As noted fromthis table, Fcowl, CDcowl

and _mm do not change with Qcc, since the normal shock for all of the casesremains swallowed inside the inlet-diffuser of the engine although it is pushed ahead when Qcc

increases (see Fig. 13-(Right)). Therefore, Qcc cannot affect the flow configurations outside ofthe engine. Hence, as long as the normal shock remains inside the inlet, the parameters Fcowl,CDcowl

and _mm will not change with Qcc. Similar effects are obtained when DNozzle changes (theseresults are not shown here), i.e. as long as the normal shock remains inside the inlet, theparameters Fcowl, CDcowl

and _mm will not be affected with DNozzle. The propulsion efficiency gp

reduces with Qcc as shown in Table 5, since the jet velocity enhances at higher heat rates of thecombustion chamber. On the other hand ge of the engine increases with Qcc, since the change in

Fig. 14. Computation for the entire ramjet: (Left)- Graphical representation of the data in Table 4;parametric study on the effect of nozzle diameter and heat source level on the position of normalshock, and (Right)- The effect of increase of total temperature along the centerline of the ramjet’scombustion chamber (DNozzle 5 0.22 m).

Table 4 Parametric study on the ramjet performance; the effect of Qcc and DNozzle values on thenormal shock location.

DNozzle 5 0.20 m DNozzle 5 0.22 m DNozzle 5 0.24 m

Qcc 5 7.5 MW (DTt 5 485K) Xs 5 0.61 m Xs 5 0.72 m Xs 5 0.85 mQcc 5 10 MW (DTt 5 690K) Xs 5 0.47 m Xs 5 0.67 m Xs 5 0.79 m

Qcc 5 16 MW (DTt 5 1105K) Xs 5 0.37 m Xs 5 0.52 m Xs 5 0.72 m


kinetic energy increases with the combustion chamber heat rate. As noted from Table 5, theoverall efficiency go of the engine gradually increases with Qcc. This is due to the dominanteffect of the engine thrust F in Eqn. 29. The TSFC reduces with Qcc since the increase in F isagain dominant as compared to the increase in _mmfuel (see Eqn. 27).

8. CONCLUSIONS AND FUTURE PLANS

In this paper three different inlets (Inlets 1 – 3) and an entire axisymmetric ramjet werenumerically studied. The highlights of the present study are given here.

1. Inlet 1, which is suitable for supersonic combustion ramjets, was chosen from the literatureand is used to validate the results of the present computations. Very good agreements wereachieved in all comparisons.

2. Inlets 2 and 3 were designed in this paper in a way to produce multiple oblique shocksmerging at the inlet cowl lip. Inlet 2 gives a solely external compressions via three shocks,while Inlet 3 produces combined external and internal compressions through two and oneoblique shocks, respectively. Very large amount of cowl drag is developed in Inlet 2 (about5.5 times more than that in Inlet 3), which is attributed to large turnings of cowl angle ofInlet 2.

3. The present study has concentrated on the design of supersonic portions of the inlets. To doso we used the Oswatitsch principle. No attempts were made on the design of subsonicportions of the inlet. This task will be done in a separate study.

4. The position of normal shock on the performance of inlet is very important. Generally it iscontrolled by the inlet back pressure. In the case of the entire ramjet, the combustionchamber heat rate (Qcc) and the exhaust nozzle throat opening (DNozzle) fix the position ofnormal shock in the ramjet’s inlet-diffuser.

5. A simplified but realistic volumetric heat source (VHS) is provided in the present study tosimulate the rate of heat release (Qcc) through the engine combustion chamber.

6. The computations performed here are inviscid and were carried out using (i) the fluxdifference splitting scheme of Roe (1981) with various orders of accuracy (including thethird-order upwind biased algorithm), and (ii) the second-order predictor-corrector schemeof MacCormack (1969). The commercial software package Fluent has also been used in

Table 5 Specifications of the ramjet engine at various Qcc. Here: (1) Fcowl is the cowl drag (see Fig. 6and Eqn. 24), (2) CDcowl

is the drag coefficient of the cowl (see Eqn. 25), (3) Fspike is the pressuredrag over the spike, (4) F is the thrust (see Fig. 6 and Eqn. 24), (5) _mm is the air mass flow ratecaptured by the engine, (6) gp is the propulsion efficiency (see Eqn. 27), (7) ge is the kineticenergy conversion efficiency (see Eqn. 27), (8) go overall efficiency (see Eqn. 27), and (9) TSFCis the thrust specific fuel consumption (see Eqn. 27).

Qcc Fcowl (1) CDcowl(2) Fspike (3) F (4) _mm (5) gp (6) ge (7) go (8) TSFC (9)

(MW) (kN) – (kN) (kN) (kg/s) (%) (%) (%)

kgfuel

kN:s

0 2.65 0.38 3.48 20.90 15.16 99.19 - - -7.5 2.65 0.38 29.1 2.50 15.16 92.56 29.51 29.79 70.3710 2.65 0.38 211.69 4.13 15.16 88.38 35.62 34.78 60.2816 2.65 0.38 215.97 7.06 15.16 81.88 40.60 37.64 55.70


some cases to check the accuracy of computations. Very good agreements were achieved inall comparisons. Viscous turbulent modelling of the problem studied in the present paper isunderway.

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airflow in ramjet inlets

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