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 Graduate Student
[email protected]
M. J. Kermani Assistant Professor
[email protected]
Department of Mechanical Engineering Amirkabir University of Technology (Tehran Polytechnic) Tehran, Iran, 15875-4413
Abstract The performance of three different ramjet engine inlets is numerically considered in this study. The geometries used are planar with the free stream Mach = 2.5. Inlet 1 is chosen from literature, a single oblique shock followed by a normal one, which given a low value of total pressure recovery. Inlets 2 and 3 are designed in a way to produce a series of oblique shocks merging at the cowl lip of the engine followed by a normal shock. The compression process in inlet 2 is completely performed in external part of the inlet. Instead inlet 3 is given a mixed internal and external compression. Inviscid Roe scheme with 3rd order accurate in space and second order accurate in time and second order predictor-corrector MacCormack scheme are applied in this study and also commercial FLUENT software package. The results are in good agreement with literature. Keywords: Supersonic Inlet, Multi Staging External Compression, Mixed External and Internal Compression, Roe Scheme, MacCormack Scheme Introduction The engine inlet is of prime importance for all airbreathing propulsion systems. Its major function is to collect 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 is performing an essential part of the engine cycle and its efficiency is directly reflected in the engine performance. In addition, the inlet must present the air to the downstream component at the suitable velocities and with an acceptable degree of uniformity of velocity and pressure under any flight condition. Finally, the inlet has to achieve all this with minimum external drag and minimum disturbance to the external flow around aircraft. The supersonic inlet consists of a spike (center-body or fore-body) and an integrated duct, in which the initial compression is being carried out by the spike. The principle of staging a supersonic compression so as to reduce the loss of inlet total pressure is considered in this study to modify inlet efficiency. By increasing the Mach number, oblique shock numbers that needed to save total pressure are increasing as well. The analysis of mixed-compression inlet flow is considered in the third inlet and is complicated by the formation of
multiple shock waves generated by the externalcompression surfaces and the cowl and by the internalcompression 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, ∂Q ∂F ∂G + + + αH = 0 ∂t ∂x ∂y where Q , F , G and H are: ` ⎡ ρu ⎡ ρv ⎤ ⎡ρ ⎤ ⎢ 2 ⎢ ρuv ⎥ ⎢ ρu ⎥ ρu + p ⎥ , G=⎢ 2 Q=⎢ ⎥, F=⎢ ⎢ ρuv ⎢ ρv + ⎥ ⎢ ρv ⎥ ⎢ ⎢ ⎥ ⎢ ⎥ ⎣ ρet ⎦ ⎣⎢ ρuh0 ⎦⎥ ⎣⎢ ρvh0 ⎡ ρv ⎤ ⎢ ⎥ 1 ρuv H = ⎢ 2 ⎥, r ⎢ ρv ⎥ ⎢ ⎥ ⎣⎢ ρvh0 ⎦⎥
and α is equal zero (2D analysis).
(1)
⎤ ⎥ ⎥, p⎥ ⎥ ⎦⎥
(2)
The working fluid is assumed as an ideal gas with constant specific heat value. P = ρRT
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) 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 31.5° , 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 4.5° and a trial calculation for 9.5° , 10.5° and 11.5° wedge turns (first, second and third wedge angle respectively ) leads to a value 1.31 for the Mach number of the terminal normal shock. The calculations proceed as follows,
Geometry and Boundary Conditions 3.1 Inlet 1 Inlet 1 is chosen from literature, [3]. The simplest form of staged compression is the two-shock inlet in which a single angle wedge or cone projects forward of the duct. For the explanation of boundary condition, figure 1 is shown. Apart from the quantities of boundary conditions of inlet 1 which will be given in next parts, the boundary conditions at the inlet is set to 40 kPa static pressure value and Mach number = 2.5. The flow is also assumed to be arriving to the computational domain as normal. At the exit plan, static pressure boundary is used. In this study the flow of the interior part is directed to a subsonic combustion chamber. The effect of combustion is simulated by imposing constant pressure levels associated with combustion of the exit of the engine inlet. All flow parameters are extrapolated to the top of the computational domain and it is taken far enough from the engine inlet, so the oblique shocks generated from the spike leading edge and cowl lip can not reach this boundary. At the center line (from inlet plan to the leading edge of the spike), a symmetry condition is enforced.
δ 1 = 9.5° δ 2 = 10.5° δ 3 = 11.5°
For a two dimensional oblique shock of angle β the loss is that corresponding to the component Mach number normal to the shock. The decreasing rate of total pressure loss with increasing oblique shock numbers makes it possible to advise systems of supersonic compression by stages, yielding high pressure recovery overall. The numbers and type of stages used depend upon free stream Mach number and some other factors. Corresponding to Oswatitsch principle for two dimension systems, maximum shock pressure recovery is obtained when the oblique shocks are of equal strength, [4]. 1
sin β 1 = M
2
sin β
2
= L = M
n −1
sin β
n −1
M 2 = 2.107° M 3 = 1.72° M 4 = 1.31°
The values of M sin β are successively 1.10 , 1.06 and 0.99 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 4.5° 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.
Fig1. Geometry, boundary condition and computational grid for inlet 1
M
β 1 = 31.38° β 2 = 37.84° β 3 = 48.63°
(4) Fig2. Geometry of inlet 2
Limit on numbers of oblique shock is that the flow in passing through the compression system produced by spike is turned outwards from the inlet axis and requires to be turned back to the axial direction within the subsonic diffuser. With efficient compression the turning angles can be quite large. Also the external angle of the cowl must
2
Fig5. Computational grid for inlet 3 domain 1 ( 229 × 30) , domain2 (173 × 41) 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 sonicexpansion 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.
Fig3. Computational grid for inlet 2; domain 1 ( 200 × 25) , domain2 (154 × 48) 3.3 Inlet 3 The outward turning of flow that goes with external compression leads to the use of outward angles on the inlet cowl, which even with attached shocks, result in significant drag. Mixed compression implies the use of both external and internal compression, in appropriate degrees, in order to relieve the external drag problem of the former whilst avoiding excessive boundary layer or other disadvantages from the later. The third inlet is shown in figure 4. The first two compressions are external and o give 20 total turn. The shocks are focused at the lip o where a reverse 11 . 5 turn is made by means of the internal cowl angle. The second wedge is continued until it o meats the reversed shock. A 11.5 change of direction of the surface at this interaction cancels shock reflection and allows the normal shock to be positioned. This type of inlet diffuser may be termed 50/50 external/internal. In all above inlets, the computational domain is divided in to 2 regions and computational grid is produced by solving Laplace's equations in each of them. The geometry and computational grid of inlet 2 are shown in figure 4 and 5, respectively.
Results The present computation is performed for the inlet geometries which are shown above, and some of results are 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 Inflow Static back Inflow M Static pressure static temp. pressure (Pa) (K) (Pa) 3.0 2815 39694 113
Fig4. Geometry of inlet 3
Figures 6 and 7 show the Mach number and static pressure contours, respectively.
3
Fig 6. Mach number contours (critical condition) by Roe scheme
(b) Fig 8. Contours of Mach for inlet 2 (Roe scheme); (a) Pback = 450000 Pa, (b) Pback = 495000 Pa
Fig 7. Static pressure contours (critical condition) by Roe scheme (a)
The obtained results are in very good agreements with literature and critical condition is obtained at the same back pressure that is presented by literature (this case is studied for validating the obtained results) [3]. The contours of Mach number and static pressure for inlet 2 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 jet engines) [4]. The boundary conditions of inlets 2 and 3 are given in the following table in which a wide range of back pressure is studied, until achieving critical conditions.
(b) Fig 9. Contours of static pressure for inlet 2 (Roe scheme); (a) Pback = 495000 Pa, (b) Pback = 450000 Pa
Table2. Boundary conditions of inlets 2 and 3 Inflow Static Static back Inflow M pressure pressure static temp. (Pa) (Pa) (K) 2.5 40000 250
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.
(a)
4
(a)
(a)
(b) Fig 12. Static pressure contours (inlet 3), (a) Roe scheme Pback = 530000 Pa, (b) MacCormack scheme Pback = 500000 Pa
(b) Fig 10. Mach number contours, (MacCormack scheme), (a) Pback = 480000 Pa, (b) 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 MacCormack (Pa) 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)
(b) Fig 11. Mach number contours, (Roe scheme), (a) Pback = 500000 Pa, (b) Pback = 530000 Pa
(a)
5
(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 FLUENT software package. A good agreement shown in the results.
(b) Fig 13. configuration of total temperature contours (a) Roe scheme, (b) Fluent software As expected for inviscid flow, total temperature remains constant throughout the computational domain. This is also a self-consistency check for obtained results. Engine cowl pressure drag force and total pressure recovery at the end of the inlet diffusers are given in table 4. As mentioned already, the cowl drag force of the second inlet is more than the third inlet and that is because of more cowl lip angle that forms stronger oblique shock started at the apex of cowl engine Whereas, there is not obvious difference between their pressure recovery so, with our assumptions (inviscid analysis) the third inlet is the most efficient.
List of Symbols
β δ
p p back
Table 4. Engine cowl pressure drag force and total pressure recovery (at critical conditions) Cowl Total Total pressure pressure pressure drag force recovery recovery (N) (Roe (Fluent scheme) software) Inlet 1 0.44 0.456 Inlet 2 36934.055 0.861 0.854 Inlet 3 11568.173 0.858 0.839
h0
α et
Oblique shock angle Wedge angle Static pressure Static back pressure (at the end of inlet) Total enthalpy H vector multiplier Total energy
References [1] Kermani, M. J., and Plett, E. G., 2001 “Roe Scheme in Generalized Coordinates: Part IFormulations”, 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.
Comparing the three recent inlets; first, second and third inlet, the following results are noticeable: (1) apart from the first inlet which is not efficient and is considered as a validation for produced CFD codes (and we do not consider it more), the sequence of compression Mach numbers are the same in two last inlets and so is the throat size (assuming the same mass flow capacity) and so is the pressure recovery. (2) The shortest possible supersonic section (defined by the distance from fore body apex to normal shock) has been used in each case, consistent with the particular design philosophy, this section is nevertheless longer in third than in second and because of it and increasing enclosure of shock wave, boundary layer (which is neglected in current study) might be expected to 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 drag corresponding to the external shape of the cowl is more than the third. So there is a compromise between the more efficient internal flow of the former and lower pressure drag of the latter.
[5]
[6]
[7]
6
Kermani, M. J., and Plett, E. G., 2001 “Roe Scheme in Generalized Coordinates: Part IIApplication to Inviscid and viscous Flows, AIAA Paper # 2001-0087 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 Kermani, M. J., and Plett, E. G., 2001 “Modified Entropy Correction Formula for the Roe Scheme, AIAA Paper # 2001-0083