Rapid supersonic performance estimation for a novel RBCC engine inlet

Aerospace Science and Technology 32 (2014) 51–59

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Rapid supersonic performance estimation for a novel RBCC engine inlet A. Murzionak ∗,1 , J. Etele 2 Carleton University, Ottawa, Ontario, K1S 5B6, Canada

a r t i c l e

i n f o

Article history: Received 4 June 2013 Received in revised form 28 November 2013 Accepted 11 December 2013 Available online 16 December 2013 Keywords: Rocket based combined cycle Detached shocks

a b s t r a c t A method to estimate the supersonic performance of an inlet designed for a rocket based combined cycle engine is developed. This method is to be used for exchange inlet geometry optimisation and as such is able to quickly predict properties that can be used in the design process such as total pressure loss and mass flow rate. The method is developed using solutions for shocks around sharp cones and estimations of shocks around 2D blunt wedges. The total pressure drop across the estimated shocks as well as the mass flow rate through the exchange inlet are calculated. The estimations for a selected range of freestream Mach numbers between 1.1 and 7 are compared against numerical finite volume simulations. The predicted mass flow rate does not differ from the numerical simulations by more than 5% for the tested conditions, while the total pressure is predicted to within 10% of finite volume simulations for most of the conditions considered. © 2013 Elsevier Masson SAS. All rights reserved.

1. Introduction One possible alternative to a pure rocket engine for space launch is the Rocket Based Combined Cycle (RBCC) engine. The RBCC engine combines within the same housing a rocket engine with an air breathing duct to form a common flow path. The engine operates in different modes depending on the flight condition to increase the mission averaged engine performance. The ejector mode is used to accelerate the vehicle from a static condition to supersonic speeds and relies on the pumping effect of the high speed rocket exhaust to entrain air. At low supersonic speeds the ramjet mode can be used, while at higher flight Mach numbers (Mach > 5) the scramjet (supersonic combustion ramjet) mode of operation can be employed. Once the vehicle leaves the atmosphere air-breathing ceases, and the engine operates as a pure rocket. Due to the air-breathing cycles the RBCC engine offers higher specific impulse over the flight profile as compared to a pure rocket. It has the potential to reduce the amount of propellant required making the vehicle lighter and thus reducing the cost of a space launch. From a theoretical perspective the RBCC engine offers better fuel performance than an equivalent rocket engine while still being capable of operation outside the atmosphere. From a practical

* 1 2

Corresponding author. E-mail address: [email protected] (A. Murzionak). MASc, Dept. of Mechanical and Aerospace Engineering. Associate Professor, Dept. of Mechanical and Aerospace Engineering.

1270-9638/$ – see front matter © 2013 Elsevier Masson SAS. All rights reserved. http://dx.doi.org/10.1016/j.ast.2013.12.007

perspective the RBCC engine adds more complexity over a rocket engine. This complexity comes from the need to combine different modes of operation and have seamless transition between these modes. The ejector mode of operation requires good mixing between the rocket and air streams. The simplest RBCC configuration with a single centred rocket nozzle results in long and heavy mixing duct [10]. Over the years a number of methods have been analysed to increase the mixing of the two streams and therefore reduce the weight of the mixing duct. The creation of large scale axial vortices has been shown to yield an improvement in the mixing characteristics within the engine. Several methods have been used to generate these vortices such as pulsing rocket stream through periodic rocket throttling [9] and the use of hypermixing nozzles [4,14,18] or forced mixer lobes [17,22]. Another means of improving the mixing performance is to increase the contact area between the rocket and air streams. This can be accomplished through the use of multiple thrusters [8,11], an effect employed with success to the Strutjet engine concept [6,20,21]. To obtain a similar increase in the shear layer area of the multiple thruster approach while reducing the number of parts, a method has been developed to design a nozzle that can create an annular exhaust profile starting from a single combustion chamber [7]. The increased radius of the annulus, as well as any breaks in the annulus itself, can significantly increase the contact area and thereby promote mixing within the remainder of the duct. However, the geometry of this nozzle is such that to maintain the flow within a set overall engine size the entrained airflow must pass within the annular rocket stream. This requires

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Nomenclature A

area

Subscripts

ds

shock stand-off distance

B

EI

exchange inlet

˙ m

mass flow rate

M

Mach number

P

pressure

r

radial direction

R

radius

T

temperature

V

velocity

x

streamwise direction

β

shock angle

γ ε

ratio of specific heats

CB CW i ms n o r s sl t w

region after expansion wave but before the reflection shock centre body cowl cowl shock Mach stem normal stagnation reflected shock shock slipline tangential wedge free stream conditions

hyperbolic eccentricity

∞

θ

flow direction, surface angle

Superscript

τ

Mach wave angle

∗

chocked conditions

Fig. 2. Main shocks generated by the EI geometry. Fig. 1. Cutaway view of exchange inlet geometry: A – Centre body, B – Fairings, C – Cowl, D – Air passage, E – Rocket flow path, F – Rocket exhaust.

exchanging the traditional positions of the hot and cold streams in an axisymmetric RBCC engine. The generation of such flow field is accomplished through the use of the Exchange Inlet (EI) [13] shown in Fig. 1. The rocket flow path (E) expands the flow of hot gases from the combustion chamber into the mixing duct as an annular stream (this can be fully annular or segmented as shown) where it is then mixed with the air entrained through the air passages of the inlet (D). The geometry in Fig. 1 has been tailored for subsonic flight conditions by modifying the shapes of the cowl (C), fairings (B) and centre body (A). At these flight speeds maximum air entrainment is obtained using fairly rounded leading edges [23,12]. However, these blunted shapes can cause increased losses at supersonic flight conditions. Therefore, a rapid means of evaluating the supersonic performance of the inlet is required so that a balance can be found. The following describes a methodology where the both the entrained air mass flow and the total pressure through the EI can be rapidly evaluated for use in an optimisation exercise during the design process. Estimates of both shock positions and strengths over the various components of the EI are obtained and compared to three dimensional numerical simulations to evaluate the accuracy of the estimated performance.

2. Methodology 2.1. Overview The exchange inlet has three main components that generate shocks: the cowl, the fairings, and the centre body (Fig. 2). The shock from the centre body is approximated using a cone shock solution, while the shocks due to the other two components are approximated using a detached 2D shock solution. As seen from Fig. 2, the centre body shock is a cone, while the other two shocks are somewhat more elaborate. The estimation method is built around determining the shock shapes and finding the property changes across these shocks. The centre body generated shock is approximated using the Taylor–Maccoll equation for a conical shock solution [1]. The total pressure drop is obtained from the shock geometry and is the same along the entire shock surface. While the shock itself closely resembles a cone shock, the flow field behind the shock is affected by the curvature of the centre body which causes an acceleration of the flow. This acceleration is assumed to be isentropic and as such has no effect on total pressure but it will affect both the speed and the direction of the flow. Therefore, this acceleration still needs to be considered because the resulting Mach number has effect on the next shock. The acceleration along the surface of the centre body is calculated using a Prandtl–Meyer expansion

A. Murzionak, J. Etele / Aerospace Science and Technology 32 (2014) 51–59

53

Fig. 3. Shock generated by the fairing with shock plane shown.

function. The flow field between the shock and the centre body surface is estimated using a linear interpolation of the data from these two surfaces. This provides a Mach number field behind the centre body shock which is used to calculate the shock on the fairing geometry. Based on this Mach number field shocks are fitted around the fairings. For the purpose of the shock fitting the fairing is represented as an infinite 2D blunt body. Multiple shocks are fitted along the leading edge of the fairing at different radii. This takes into account slight variations in the geometry of the leading edge of the fairing as well as the difference in the length of the shocks at different radii. This procedure results in a 3D shock surface. The flow field from the previous step is projected onto this three dimensional shock surface, and property changes across the entire shock are calculated as follows. The total pressure drop is averaged behind each 2D shock segment at a given radial location (an average is required as the incoming Mach number is not uniform across this shock and the angle of the fairing shock changes along its length). This results in a radial variation in total pressure which is then projected onto the cowl shock. As the Mach number in the region between the fairing and cowl shocks changes considerably due to both the curvature of the fairing and the centre body, additional considerations have to be undertaken to properly account for this effect. These considerations are expanded further in the following sections. The cowl is assumed to be an infinite 2D blunt object similar to the fairings and thus allows a similar curve fit approach to be used. Once the shape of this shock is found, the calculated flow field from the previous shocks are used to obtain the properties of the flow entering the cowl shock. The cowl shock is then adjusted for a Mach stem, if one exists, as determined from the shape of the EI and the flow properties behind the cowl shock. The flow properties are then averaged in the radial direction to obtain a single value for the total pressure drop across the EI. In addition to the total pressure at the exit of the EI, the entrained air mass flow rate is also estimated. 2.2. Fairing shock estimation Although the solution of the leading shock around the centre body follows the Taylor–Maccoll solution, the remaining shocks require some additional theory to properly estimate their position and strength. Fig. 3 shows a cut of the EI geometry and the faring shock geometry. The calculations for the fairing shock are performed for a number of shock planes located at different radii along the leading edge of the fairing. In each plane an infinite 2D blunt wedge is assumed and the shock shape around this body is found following the approach of Billig [5]. The proposed method requires the determination of the stand-off distance of the shock (d s ) and the overall shape of the shock (Fig. 4). Both R and θ w are properties of the fairing geometry, while β w , d s , and R s are

Fig. 4. Variables used to determine detached shock geometries.

values that must be determined. The stand-off distance is not actually required for the pressure loss estimations across a single shock, however, when the intersections between the fairing and cowl shocks become important this distance is required. The shock shape is defined by Eq. (1) with n = 1.7,

(x − xo + a + ds )n an

−

yn bn

=1

(1)

The location of the leading edge of the object generating the shock is specified by xo , while the parameters a and b are related to the shock angle β w through

b a

= tan(β w )

(2)

where the value of β w is found from an oblique shock solution based on the known wedge angle θ w and incoming Mach number. As per the method of Billig [5] the shock stand-off distance is a function of the physical radius of the object and the incoming Mach number (Eq. (3)),

 d s / R = 0.386 exp

4.67

 (3)

M2

Although an analytical solution for this distance is applicable for Mach numbers greater than five [16] the expression used here can be applied to lower Mach numbers. From geometry the remaining unknown in Eq. (1), a, can be related to both R s and the eccentricity of the hyperbola. The eccentricity can be found from β w (which is equal to the angle of the hyperbola asymptote), while the radius of the lower portion of the shock is found from a curve fit of multiple 2D detached shock simulations as

 R s / R = 0.1589 exp

6.3366 M

 + 1.9187

(4)

The results of this shock fitting procedure are shown in Fig. 5. As can be seen, the comparison of the estimated shock shape and that calculated using 2D numerical simulations is reasonable both at low and high Mach numbers. The shock shape matches well both near the leading edge of the object as well as in the far field where the shock solution approaches the oblique shock solution. In cases where the cone shock generated by the centre body intersects the shock shape fitted around the fairing, the free stream Mach number is applied to the fairing shock above the point of intersection. In all other cases the Mach field behind the cone shock is applied at each radial location to a fitted fairing shock to calculate the property changes across the shock. With both the shock angle and the incoming Mach number, enough information is known to calculate the total pressure drop across the fairing shock. However, due to the curvature of the fairing, the flow can

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Fig. 7. Side view of cowl shock structure (Point I – shock/expansion wave intersection; Point T – triple point).

Fig. 5. Shock shape produced by a blunt body at different Mach numbers (R = 0.1 m).

Fig. 6. Flow acceleration around the fairing.

re-accelerate which would have an effect on the following cowl shock. Starting from point A in Figs. 3 and 6 which is located at the point where the fitted fairing shock at the given radial location intersects the symmetry plane (i.e., where this shock would meet its counterpart from an adjacent fairing), one can draw a Mach wave back to the fairing surface (point B, Fig. 6). With the surface angle at point B as a reference value, the subsequent supersonic turning of the flow along the fairing surface is calculated assuming isentropic flow through a series of expansion waves. This process is repeated for all the fitted shock planes over the range of radial positions considered resulting in an x–r distribution for the Mach number. It should be noted that this method can be used only if the Mach number at point A is larger than or equal to one. It is possible for the resultant Mach number after the fairing shock to be lower than one (e.g., at lower M ∞ ). In such cases the reacceleration of the flow is not computed. As a consequence, this also means that the cowl shock is not computed since the flow is subsonic after the fairing shock. 2.3. Cowl shock estimation The initial shock shape around the cowl is fitted in the same manner as for the shock generated by the fairings. However, since past the cowl the flow becomes completely internal a Mach stem may be required to satisfy both continuity and momentum (especially at higher Mach numbers). This generates a non-uniform flow field in that the flow passing through the Mach stem becomes subsonic, while above the Mach stem the flow can be either sub-

sonic or supersonic (and thus may or may not encounter reflected shocks). For a finite sharp wedge geometry and a known back pressure Ben-Dor [3] describes a method for calculating the size of the required Mach stem. Although in this case the back pressure is not known, the internal EI geometry is set and thus a similar iterative approach can be used to establish the correct size of the Mach stem as follows. The initial fitted cowl shock is estimated based on the Mach number at the leading edge of the cowl (M 1 (x, r ), Fig. 7) as found from the consideration of the flow through the fairing shock. From the point of contact of this shock with the centre body, increasing sizes of Mach stem are assumed. Treating the Mach stem as a normal shock and knowing the shape of the fitted cowl shock, the conditions both behind the cowl shock and the Mach stem (regions 2 and 4 respectively in Fig. 7) can be determined. If the flow in region 2 is still supersonic a reflected shock will be formed at the triple point in order to satisfy the condition of equal pressures across the slipline. Enforcing this conditions allows the angle of the reflected shock to be found as



βr = M 2

2γ M 12 − (γ − 1) 2 2γ M 1n − (γ − 1)

 1 (γ + 1) + (γ − 1)

2γ

(5)

where once this value is known, both the flow angle (θr ) and the Mach number M 3 can be found from the usual shock relations. The angle of the slipline (θsl ) is then the difference between the flow direction after the cowl shock and that leaving the reflected shock,

θsl = θi − θr

(6)

This angle creates a converging area of subsonic flow behind the Mach stem as bounded by the centre body and the slipline which will accelerate to sonic conditions at some point downstream. If one assumes the slipline remains linear until this choke point then the isentropic area required to choke the flow can be found and thus the location downstream of the Mach stem at which this area is obtained (x∗ ) can be calculated. As this location depends on the size of the initially assumed Mach stem, its value can be used to validate the assumed Mach stem size as follows. At the choke location the flow pressure is known, while the Mach number is unity and its direction must be parallel with the centre body surface. Above the slipline the flow at condition 3 is not aligned with the centre body, but everywhere along this line the pressure must match that below including at the choke point. Therefore, if one assumes isentropic flow between regions 3 and 5 in Fig. 7, then knowing both the pressure (p ∗ = p 5 ) and the angle through which the flow must be turned at the choke location (θr − θCB ) the Mach number M 5 can be reduced to

A. Murzionak, J. Etele / Aerospace Science and Technology 32 (2014) 51–59

 M5 =

2 + (γ − 1) M 32 2 + (γ − 1) M 42

 (γ + 1) − 2

1

γ −1

55

1/2 (7)

The angle of the Mach wave in this region is then simply sin(τ ) = 1/ M 5 and so the location of this wave can be traced back starting from the choke location to where it intersects the reflected shock. Wave interaction theory can be used to calculate the change in the angle of this Mach wave through the shock/expansion interaction and thus find τ B . The linear approximation of this wave within region 2 allows to find the location along the cowl from which this wave emanates. Since at this location the flow direction must be parallel to the cowl surface this requires that θ B = θC W . However, knowing both the Mach number (M 2 ) and the angle of this expansion wave (τ B ), the resulting flow angle can be calculated independently and thus compared to the cowl surface angle to see if the above condition is satisfied. If it is not, then the assumed size of the Mach stem must be enlarged until the angles become consistent. Once this process has been iterated to completion the cowl shock shape is finalised as being a combination of the fitted solution about most of the cowl along with a Mach stem portion near the centre body. Then the property changes across this composite shock can be calculated and averaged in the radial direction to provide a measure of the EI performance. 3. Results 3.1. 3D simulation setup To validate estimation method 3D numerical simulations are performed using Ansys-CFX [2] software, which is a time accurate, finite volume, Reynolds averaged Navier–Stokes solver. Due to the symmetry of the EI only one eighth of the EI is modelled. Fig. 8(a) shows the section being modelled, while Fig. 8(b) shows the entire computational domain and the conditions applied along each boundary. For the outlets no conditions are specified (therefore engine unstart conditions are not considered), while along the free surface and the inlet the pressure is set to the values listed in Table 1 depending on the case under consideration (the velocity and temperature are also specified along the inlet). Two simulations are done at Mach 2 and 7 to test the effects of using a free slip and no slip condition along the geometry walls. At both the low and high flight speeds the impact of the no slip calculations (i.e., those which include the boundary layer) on both the total pressure ratio and mass flow through the engine are found to be small and thus the remaining conditions are performed using free slip wall boundaries to improve the calculation and convergence times. Although the flow fields are steady, to improve convergence all simulations are performed using a transient analysis with air modelled as a perfect gas. A second order “high resolution” advection scheme with double precision is used along with the Shear Stress Transport turbulence model with the turbulence intensity set to 5%. The simulations are run until no change is seen in either total pressure or Mach number at which point a steady state has been achieved. To estimate the grid dependence of the results three different meshes of increasing size (∼1.5, 2.8, and 5.4 million nodes) are used and the Grid Convergence Index (GCI) [19,15] is calculated for the mass flow, mass flow averaged Mach number, and mass flow averaged total pressure, at the exit of the EI. Using a safety factor of 1.25 the expected error for the finest mesh is 0.05%, 0.09% ˙ P o(exit ) , and M exit respectively. On the coars, and 0.48% for m, est mesh these values increase to 0.27%, 0.07%, and 0.60%. Given the relative computational time between the coarse and fine mesh,

Fig. 8. Numerical simulation setup. Table 1 Flight conditions. Mach

Altitude (km)

Speed (m/s)

Static pressure (kPa)

Total pressure (kPa)

Static temperature (K)

Total temperature (K)

1 .1 1 .3 1 .5 2 .0 2 .5 3 .0 4 .0 5 .0 7 .0

8 .1 10.3 12.1 15.8 18.6 20.9 24.6 27.4 32.0

338.7 387.8 442.6 590.2 737.7 885.3 1180.4 1499.6 2162.4

35.4 25.4 19.0 10.7 6.9 4.8 2.7 1.7 0.87

75.6 70.3 60.9 83.8 117.2 174.9 406.6 906.9 3620.6

236 221 217 217 217 217 217 224 237

293 296 314 390 487 606 910 1343 2565

the accuracy of the coarse mesh is considered sufficient for validating the estimation method and is thus used for all the results presented. 3.2. Shock geometry Fig. 9 shows the Mach number contours along symmetry plane 1 (shown in Fig. 8) calculated using the numerical simulations along with the estimated shock geometries overlaid as thick lines. The three primary shocks to be compared consist of the cone shock produced by the centre body (A), the projection of the detached bow shock produced by the fairing onto this symmetry plane (B), and the detached cowl shock (C) (with Mach stems (F) visible in some cases close to the centre body). In terms of the shock generated by the centre body (A) the estimations and numerical simulations agree reasonably well. With an increase of the free stream Mach number the centre body shock

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A. Murzionak, J. Etele / Aerospace Science and Technology 32 (2014) 51–59

Fig. 9. Mach contours along symmetry plane 1 with estimated shocks overlaid (A – Centre body shock, B – Fairing shock, C – Cowl shock, D – Acceleration of the flow due to centre body before fairing shock, E – Acceleration of the flow due to the fairing and centre body curvature before the cowl shock, F – Mach stem, G – Region of high Mach number after cowl shock).

becomes sharper as expected over the entire Mach number range considered. At the highest Mach number condition this shock falls within the EI (i.e., it terminates below the cowl) and as such the cowl sees the free stream Mach number, an effect accounted for in the estimation method. Overall, this level of accuracy is expected given the relative simplicity of the flow in this region.

The projection of the fairing shock also agrees reasonably well between the estimation method and the numerical simulations over the considered range of Mach numbers. Fig. 10 shows the Mach number contours on selected planes normal to the fairing to better illustrate the source of the estimation line B along the symmetry plane. Since the estimation produces numerous fairing shock shapes which depend on the radial position, their distance from symmetry plane shown in Fig. 9 changes. Connecting the intersection points between these fairing shocks and the symmetry plane as shown in Fig. 10 yields the single fairing shock illustrated in Fig. 9. As seen in Fig. 10(a) for M ∞ = 1.3 the location of the estimated fairing shock differs slightly from that calculated in the numerical simulations with the estimation method yielding a smaller standoff distance than that seen in the numerical simulations. However, both methods predict a fairly straight shock around the leading edge of the fairing at this Mach number (the numerical results appear curved at the middle and highest planes in Fig. 10(a) only because the plane is taken normal to the fairing which means the radial distance increases as one moves away from the fairing). By Mach 2 the fairing shock is highly curved and the stand-off distance has decreased significantly, features which are captured by the estimation method in Figs. 10(b)–10(d). This is also shown in Figs. 9(b)–9(d) where the projection of the fairing shock onto the symmetry plane matches well between the estimation method and the sharp change in Mach contours of the numerical simulations. In the case of the cowl shock, at the lowest Mach number considered (Fig. 9(a)) there is a large discrepancy between the estimation method and the numerical results in that there is no estimated cowl shock. This is because the estimation method predicts fully subsonic flow behind the fairing shock and thus no further shocks are considered. Although the numerical simulations also show the flow as fully subsonic immediately behind the fairing shock, it is re-accelerated by the shape of the centre body and again reaches supersonic velocities ahead of the cowl. This results in a weak cowl shock in the numerical simulations which is not captured by the estimation method. As the Mach number is increased this discrepancy is eliminated because the flow does not decelerate to subsonic velocities in this region, and thus the cowl shock is estimated (see Fig. 9(b)). At M = 2 the estimated cowl shock matches well with the numerical simulation near the cowl, but does not predict any Mach stem. However, in the numerical results the flow acceleration in region E near the centre body is enough to generate a Mach stem. This result is consistent with or without boundary layers included in the numerical simulations as shown in Fig. 11. The most noticeable effect of the no slip condition is the boundary layer along the centre body surface downstream of the cowl shock. Although a boundary layer is computed on all the surfaces, its effect as observed by the Mach contours is small outside of this region. In each case, both with and without the boundary layer, the numerical simulation predicts a Mach stem that is not predicted in the estimation method. Despite the non-uniformity of the flow below the cowl in the no slip simulations (Fig. 11(a)), as the numerical shock positions are relatively unchanged the total pressure through the inlet as compared to Fig. 11(b) is within 1.3% (see also Fig. 12). As well, given the axisymmetric nature of the geometry the slower moving boundary layer flow along the centre body represents a much smaller flow area than the 2D symmetry plane would suggest leading to a difference in the computed mass flows of less than 2.4% (see Fig. 14). As the Mach number is further increased to M = 3 (Fig. 9(c)) both the estimation method and the numerical simulations predict the existence of a Mach stem. However, as seen the estimation method predicts a smaller Mach stem further downstream than

A. Murzionak, J. Etele / Aerospace Science and Technology 32 (2014) 51–59

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Fig. 11. Mach contours along symmetry plane 1 with estimated shocks overlaid for Mach 2 (A - Centre body shock, B – Fairing shock, C – Cowl shock, D – Acceleration of the flow due to centre body before fairing shock, E – Acceleration of the flow due to the fairing and centre body curvature before the cowl shock, F – Mach stem, G – Region of high Mach number after cowl shock).

Fig. 12. Total pressure ratio at the exit plane of the EI.

Fig. 10. Mach distribution around the fairing shock on selected planes: A – Fairing shock, B – Cowl shock, C – Estimated fairing shock and its projection onto the symmetry plane 1 (thick lines).

observed in the numerical simulations. At a Mach number of 7 (Fig. 9(d)) the agreement between the estimation method and the numerical simulations is the greatest. Along the symmetry plane

the estimated cowl shock and Mach stem locations are in close agreement, while the fairing and centre body shocks are also well placed in the estimation method as compared to the numerical results. Therefore, since the estimation method is able to capture the existence (or lack thereof) of the Mach stem and is also able to model the location of all the main shock structures with reasonable accuracy over a wide range of flight conditions, it is possible to use these shocks to calculate the performance across the entire EI.

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Fig. 13. Shock structure for M ∞ = 7: A – Centre body shock; B1 – Fairing shock; B2 – Fairing shock exposed to free stream; C – Cowl shock; D – Centre body and fairing shock intersection; E – Cowl and fairing shock intersection; F – Cowl and centre body shock intersection.

method is slightly conservative in that it over-predicts the loss in total pressure at most Mach numbers between 1 and 7. At the lowest Mach numbers the total pressure ratio is near unity as expected. Despite the lack of an estimated cowl shock at M = 1.3 the total pressure ratio still compares well to the numerical results and is within 2% indicating that when the estimation method does not predict a cowl shock, the numerical simulations predict only a weak one. At M = 2, although the estimation method does not predict a Mach stem, the overall pressure ratio still compares well to the numerical results with the estimated total pressure ratio approximately 4% lower than that predicted by the numerical simulations. As the Mach number is increased up to M = 5 the estimation method is never more than 10% away from the total pressure ratio predicted by the numerical simulations (with an average difference of 3%). At the highest Mach number considered, M = 7, the two data points in Fig. 12 still compare reasonably well on the scale shown, although due to the small absolute value of the total pressure ratio at this Mach number the percentage difference is approximately 30%. However, this difference is not simply a matter of the chosen scale as there are significant flow features which start to develop at the higher Mach number range that are not accounted for in the estimation method. Fig. 13(a) shows the shock structure around the EI at the Mach 7 flight condition. As can be seen, at this flight speed the centre body shock remains very close to the centre body itself while the fairing shock is also pushed very close to the surface. The result is that the shocks start to intersect each other. The centre body shock is so shallow that it intersects both the fairing shock (line D) and the cowl shock (line F). There is also a significant intersection between the fairing and cowl shocks (line E). Since the estimation method is based on two dimensional methods, the effect of these interactions is not easily incorporated into the methodology. As seen in Fig. 13(b) these interactions create a very three dimensional shock structure. Although the shock locations on the symmetry plane match quite well as seen in Fig. 9(d), due to the complex three dimensional nature of the flow at this condition the symmetry plane is not as representative of the overall flow as it is at the lower flight Mach number conditions. This is also why the no slip simulation at Mach 7 still predicts a total pressure ratio within 1.3% of the free slip simulations in Fig. 12 as the effects of any boundary layers, although slightly greater than those seen in Fig. 11(a), are overwhelmed by the effects of the three dimensional shock interactions (which are largely independent of the boundary layer within the cowl section). 3.4. Mass flow rate

Fig. 14. Total pressure and mass flow rate comparison.

3.3. Total pressure Fig. 12 compares the averaged total pressure ratio over the EI between the estimation method and the numerical simulations for the Mach numbers considered. With the shapes and locations of all the shocks generated by the EI, it is possible to calculate the loss in total pressure across each shock and thus the cumulative effect from inlet to exit. As can be seen, the results compare reasonably well over the entire range of Mach numbers. The estimation

In addition to estimating the total pressure ratio across the EI, the methodology is also able to calculate the mass flow ingested through the EI. Physically, there is a minimum area within the EI as defined by the geometry of the cowl, fairings, and centre body, which will restrict the maximum amount of air capable of being ingested into the engine even under isentropic conditions. Therefore, in calculating the mass flow through the EI the estimation method compares the value as calculated based on the streamline geometry to this limiting isentropic value. Whichever result is lower is used as the value of mass flow through the EI. The results are shown in Fig. 14 compared to the mass flow predicted by the numerical simulations (the results using a no slip condition are also included at Mach 2 and 7). In general the estimation method over-predicts the mass flow through the EI below M ≈ 2 by an average of 5% and underpredicts the mass flow through the EI by the same amount at Mach numbers above this value. It should be noted that for the

A. Murzionak, J. Etele / Aerospace Science and Technology 32 (2014) 51–59

cases where M > 3 the mass flow through the inlet is much lower than that predicted by the isentropic limit. However, the geometry chosen for this work is optimised for a subsonic flight profile and so although the mass flow or total pressure ratio may not be ideal, it is the ability of the estimation method to accurately predict these values which is important. For all the results presented here the average calculation time for the estimation method to determine the shock positions and calculate the performance properties through the inlet is approximately 3 s per flight condition on a 3.0 GHz processor. This compares to an average numerical simulation time of several days on an 8 core system with each core running at 2.4 GHz. 4. Conclusions A method to rapidly assess the performance of the Exchange Inlet (EI) design for use with rocket based combined cycle engines at supersonic conditions is developed. The results of this method are compared to numerical simulations obtained using a commercially available finite volume Navier–Stokes solver (Ansys-CFX). The EI geometry produces three sets of shocks which are approximated by a cone shock and a series of detached bow shocks. The estimation method uses these shock approximations to define the shock locations, and based on these, to calculate the total pressure drop and mass flow across the entire inlet. A comparison of the estimated shock locations to the numerical simulations shows good agreement over a wide range of supersonic Mach numbers (1.3  M  7). The total pressure ratio at the exit plane of the EI differs on average by less than 3% for Mach numbers below 5.0. Good agreement is also seen for the mass flow rate through the EI where on average there is less than a 5% difference between the estimation and the numerical results and the maximum difference does not exceed 10% for any of the tested conditions up to Mach 7.0. At Mach 7 three dimension effects become more significant and the differences between the estimated total pressure ratio and the numerical simulations increases indicating there is an upper limit to the applicability of the proposed methodology. The estimation methodology takes approximately 3 seconds to evaluate a given flight condition which is orders of magnitude faster that the full three dimensional numerical simulations. This is advantageous for a rapid assessment of a proposed design and makes the methodology well suited for use in optimisation algorithms.

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