Design Methods For Axisymmetric Supersonic Nozzle Contours

Design Methods for Axisymmetric Supersonic Nozzle Contours Bholanath Behera 1 and K. Srinivasan2 1

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Air Breathing Propulsion Project, Vikram Sarabhai Space Centre Thiruvananthapuram - 695022, India Email: [email protected]

Department of Mechanical Engineering, Indian Institute of Technology Madras, Chennai – 600036, India Email: [email protected]

Abstract: Supersonic nozzles find application in many engineering systems. In several applications, minimum length nozzles are preferred for space and weight minimization. In this paper, two methods for design of axisymmetric supersonic nozzles have been compared. The methods examined are based on method of characteristics, namely, the analytical method, and the second-order method. Demonstrative results are presented for Mach 2.5 for various parameters. The results indicate that the nozzle length obtained by the second order accurate method is the least among all. Such methods hold promise in active nozzle contour control systems. Keywords: Supersonic Nozzle Design 1. Introduction Axisymmetric supersonic nozzles are inevitable in rocket nozzles, propulsive systems and mixing devices. Conventional applications of supersonic nozzles include experimental facilities like supersonic wind tunnels, missiles, variable area nozzles, gas dynamic lasers, molecular beams, air guns, re-entry vehicles etc. Challenging endeavours such as reusable launch vehicles require nozzles to operate in various flow regimes without compromise on their performance like thrust loss. Hypersonic air breathing vehicles are being vigorously pursued by several countries and space agencies. These applications would benefit from active modification of nozzle contours, since the aerodynamic range of operation is vast, and since the computational capabilities and possibilities are enormous. In all these applications, the shape of the nozzle contour is extremely important (see for instance [1]). Supersonic nozzles with gently curved expansion sections are normally used in wind tunnels where high quality uniform flow is desired in the test section. Hence wind tunnel nozzles are long with a relatively slow expansion. In contrast, in rockets and gas-dynamical lasers, smaller nozzle lengths are preferred to have rapid expansion and also to minimize weight in case of rocket nozzles. In such cases, the nozzles are called Minimum Length Nozzles (MLN) in which expansion section is shrunk to a point and the expansion takes place through a centered Prandtl-Meyer wave emanating from a sharp corner throat. Some of the aforementioned applications may demand active modifications to the nozzle contour, based on closed-loop feedback. Such active modifications would involve real-time computation of the nozzle contour, and would hence demand efficient nozzle design methods. Therefore, this paper focuses on design methods for supersonic nozzle contours. In particular, design methods for axisymmetric nozzles have been analysed using two methods based on the Method Of Characteristics (MOC). The methods chosen are:(i) Foelsh’s analytical method [2] and (ii) the second order accurate method outlined by Argrow and Emanuel [3]. The comparison between the two design methods is

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made using Mach 2.5 nozzle contour obtained by the two methods. The results indicate that the nozzle length obtained by the second order accurate method is the least. The following section describes the design procedure for Foelsch’s analytical method. 2. Foelsch’s Analytical Method Traditionally, the equations for the nozzle contour are derived by integration of the characteristic equations for axisymmetric flow. Since it is not possible to obtain a closed-form integral of these equations, the flow in a nozzle is approximated by the flow in a cone. This provides a good approximation for an analytical solution of the problem. That is, the coordinates of the contour of an axisymmetric nozzle, as well as the streamlines in the nozzles flow are determined from simple Eq. (2). The conical source flow emanating from the sonic section is converted into a parallel and uniform flow by a transition curve, as depicted in Fig. 1. This figure shows a section through the upper supersonic part of a three dimensional nozzle with axial symmetry. The flow in the nozzle is mainly divided into three regions. Region OAB: This is the starting of source flow. Region ABDC: Radial flow region. Region CDE: Transition region. Region EDX: In this region flow is fully parallel and uniform.

Fig. 1. Conversion of radial flow into parallel flow The computational procedure is as follows: Assuming a design Mach number of 2.5, the semi cone angle (ω) is taken as half of θmax. The ratio of radius at exit to the radius of critical arc (τ ), and the radius at throat are calculated as per reference [2]. Then the ν at every point is calculated taking maximum θpoint as ω. The Mach number at every point is then calculated using Prandtl-Meyer relation. The co-ordinates of transition curve and parallel flow section are calculated as per the equations given by Foelsch [2]. Results have been presented in Fig. 2. Figure 2 shows the wall contours obtained for Mach number 2.5 with semi cone angle = θmax/2 for various θ divisions. It is clear from the Fig. 2 that the results are almost coincident for θ values less than 2 degrees, confirming the grid independence.

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Fig. 2. Wall contour obtained by Foelsch’s analytical method for axisymmetric supersonic nozzle for Me = 2.5 for semi cone angle = θmax/2 for various θ divisions 3. Second Order Method For axisymmetric flows, there are two different types of minimum length nozzles (MLN’s). First one is straight sonic line MLN which has a straight sonic line at the throat and the wall at the throat generates a centered expansion. The other one is curved sonic line MLN which has a circular arc sonic line and is followed by a conical flow region with no centered expansion. In this section we discuss the wall contour of a supersonic, axisymmetric, minimum length nozzle with straight sonic line by using the second order accurate method of characteristics outlined by Argrow and Emanuel [3]. The computational characteristic grid for the design of a straight sonic line minimum length nozzle is shown in Fig. 3. The kernel computation starts at point A. For steady, supersonic, irrotational, axisymmetric flow of a perfect gas, the MOC equations are given by the following equations. For the right running C− characteristics, the MOC equations are –

(M

)

−1 ⎡ γ −1 2 ⎤ ⎢1 + 2 M ⎥ ⎣ ⎦ 2

dM M

+ dθ

−

tan θ tan θ

(M

2

)

−1 −1

dr r

=

0

(1)

and

⎛ dr ⎞ ⎜ ⎟ ⎝ dx ⎠ char

⎡ ⎛ 1 = tan ⎢θ − sin −1 ⎜ ⎝M ⎣

⎞⎤ ⎟⎥ ⎠⎦

(2)

For the left running C+ characteristics, the MOC equations are –

(M

)

−1 ⎡ γ −1 2 ⎤ ⎢1 + 2 M ⎥ ⎣ ⎦ 2

dM M

− dθ

−

tan θ

tan θ

(M

2

)

−1 +1

dr r

=

0

(3)

and

⎛ dr ⎞ ⎜ ⎟ ⎝ dx ⎠ char

⎡ ⎛ 1 = tan ⎢θ + sin −1 ⎜ ⎝M ⎣

⎞⎤ ⎟⎥ ⎠⎦

(4)

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The differenced form of these equations are easily solved for x, r, M and θ variables, as outlined by Argrow and Emanuel [3].

Fig. 3. Regions in the Design of Minimum Length Nozzle The sweep procedure for the kernel and transition regions are explained in Argrow and Emanuel [3] and hence not being repeated here. Whenever the wall contour is poorly represented, it is corrected by the grid compression scheme i.e., by inserting the characteristics which takes more computational time. For this, a power law is used which is given as M ⎛ c ⎞ c ⎟ (5) υc = ⎜ ∆υ ⎜N ⎟ ⎝ c⎠ where, c represents the characteristic from 1 to Nc i.e. for Nc=5, c = 1, 2, 3, 4, and 5; Nc is total number of inserted characteristics, and Mc is the fixed integer exponent for the power law. For the same Nc, when Mc is varied, the wall contours would be different. The greater the Mc, more characteristics start closer to the sonic line resulting a more satisfactory wall contour. The throat coordinates are x = 0 and r = 0.1. Assuming 5 characteristics to produce the required acceleration, we get a total of 15 kernel region points. All the parameters of kernel region points are calculated first. The computations are based on unit processes 1, 2 and 3 described in [3]. When all the parameters of kernel region points are calculated, the parameters of transition region points are calculated. Lastly the wall contour points are determined. In all the results presented, the exit Mach number is 2.5 and γ= 1.4. Figure 4 shows the variation of kernel region points, transition region points and wall contour obtained with no grid compression, moderate grid compression and high grid compression for various characteristics. From these figures, it is clear that grid compression is more beneficial than introduction of more characteristics. This is made clear in Fig. 5 wherein the nozzle contours are shown for high grid compression for various number of characteristics. 4. Conclusions

The wall contours obtained using the various techniques used in the present study and the computational times have been compared in this paper. Figure 6 compares the nozzle contours

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obtained using Foelsch’s analytical method and the second order accurate method of Argrow and Emanuel [3]. The results show that the second order accurate method results in the shortest nozzle. These computations were performed on a personal computer (Intel Pentium II, 333 MHz) running on Linux operating system 6.0. The programs were written in ’C’ and the compiler used was EGCS Version 1.1.2. The results show that the computational time increased by 19%.

No. of characteristics = 18, No grid compression

No. of characteristics = 18, high grid compression

No. of characteristics = 5, Moderate grid compression

No. of characteristics = 5, high grid compression

Fig. 4. Wall contour with kernel and transition region points for Me=2.5 nozzle, for various number of characteristics and grid compressions from no grid compression to highest grid compression, for 18 characteristics. Lastly, we can conclude that methods which use the Prandtl-Meyer function in an indirect form should be preferred to gain computational advantage and accuracy. In this regard, the second order method holds immense potential in active nozzle modification applications.

Fig. 5. Wall contour for Me=2.5 for various characteristics with high grid compression (Nc=5 and Mc=5) when γ=1.4.

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Fig. 6. Nozzle length and wall contour of axisymmetric Laval nozzle by analytical method and axisymmetric MLN by second order accurate method, 1: Axisymmetric Laval nozzle by analytical method, 2: Axisymmetric MLN by second order accurate method

Nomenclature

x* r* A ν νe Me µ µe θ β θw θwmax θmax ω V γ

x co-ordinate of throat r co-ordinate of throat throat area Prandtl-Meyer function Prandtl-Meyer function corresponding to exit Mach number exit Mach number or design Mach number Mach angle Mach angle corresponding to exit Mach number deflection angle shock angle angle of the duct wall with respect to nozzle axis maximum angle of the duct wall with respect to nozzle axis expansion angle of the wall downstream of the throat semi cone angle velocity of the flow specific heat ratio

References

[1] George P. Sutton, “Stepped nozzle,” U.S. Patent 5, 779, 151 July 14, 1998. [2] Foelsch, K., “The Analytical Design of an Axially Symmetric LavalNozzle for a Parallel and Uniform Jet,” Journal of the Aeronautical Sciences, Volume 16, 1948, pp.161-166. [3] Emanuel, G. and Argrow, B. M., “Comparison of Minimum Length Nozzles,” Journal of Fluid Engineering, Trans. ASME, Volume 110, 1988, pp.283-288.

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