2D Contour Design and Analysis of the Divergent Portion of a Satellite Thruster Nozzle using MOC

GRD Journals- Global Research and Development Journal for Engineering | Volume 02 | Issue 05 | April 2017 ISSN: 2455-5703

2D Contour Design and Analysis of the Divergent Portion of a Satellite Thruster Nozzle using MOC Veda Jose PG Scholar Department of Mechanical Engineering Mar Athanasius College of Engineering, Kerala, India Vinod Yeldho Baby Assistant Professor Department of Mechanical Engineering Mar Athanasius College of Engineering, Kerala, India

Aneesh K Johny PG Scholar Department of Mechanical Engineering Mar Athanasius College of Engineering, Kerala, India

Abstract This paper deals with the design of supersonic nozzles based on the theory of method of characteristics in two-dimensional regime.

A MATLAB program was developed and utilized to compute the minimum length of the supersonic nozzle at the diverging section of the nozzle for the optimum Mach number at the nozzle exit with uniform flow. Numerical analysis is done and the solution is established for steady, in viscid, irrotational supersonic flow in two dimensions. An arbitrary shape is assumed for the converging section of the supersonic nozzle because of the rational assumption, the flow holds the consistency in the converging section. The design is focused on the diverging section of the nozzle.

Keywords- C-D nozzle, Design, Method of Characteristics, Minimum Length, Supersonic Flow __________________________________________________________________________________________________

I. INTRODUCTION Nozzles for aerospace propulsion are designed for supersonic flows. Traditionally the supersonic nozzle is divided into two parts.

The converging section for sonic and subsonic flows and diverging section for supersonic flows. Both the sections have significance in the performance of the nozzle. So it is possible for design and study both the portions independently. This paper mainly focused on the design and study of the supersonic section of the nozzle. There are many methods for the design, but here the focus is mainly on the method of characteristic (MOC). The conditions of a two dimensional, steady, isentropic, irrotational flow expressed mathematically by nonlinear differential equation. The method of characteristics is a mathematical method which is used to find solutions for mentioned velocity potential, therefore satisfying given boundary conditions for which the governing partial differential equations (PDEs) become ordinary differential equations (ODEs). We consider nozzle giving a parallel and uniform flow at the exit section. This type of nozzle is named by Minimum Length Nozzle having centered expansion, compared to the other existing types it gives the minimal length.

A. Minimum Length Nozzle Convergent - divergent shape is more preferable in order to expand a flow through a duct from subsonic to supersonic speed. For an improper nozzle contour, chances for shock waves to occur inside section is high [2]. The method of characteristics provides a technique for designing the contour for a shock free supersonic nozzle, taking into account the multidimensional flow inside the duct. This section illustrate the method and how to do the design process.

Fig. 1: Straight Sonic Line Minimum Length Supersonic Nozzle [1]

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2D Contour Design and Analysis of the Divergent Portion of a Satellite Thruster Nozzle using MOC (GRDJE/ Volume 02 / Issue 05 / 028)

Rocket nozzles are very short in order to minimize weight. Also, in cases where rapid expansions are desirable, such as the non-equilibrium flow in modern gas dynamic lasers, the nozzle length is as short as possible In such minimum- length nozzles, the expansion section is shrunk to a point, and the expansion takes place through a centred Prandtl- Meyer wave emanating from a sharp-corner throat with an angle đ?œƒđ?‘Šđ?‘šđ?‘Žđ?‘Ľ, đ?‘€đ??ż as sketched [1]. The length of the supersonic nozzle, denoted as L in Fig 1 is the minimum value consistent with shock free, isentropic flow. If the contour is made shorter than L, shocks will develop inside the nozzle. For a fluid element moving along a streamline, constantly accelerated while passing through these multiple reflected waves, for the minimum-length nozzle, the expansion contour is a sharp corner at point a. There are no multiple reflections and the fluid element encounters only two systems of waves, the right-running waves emanating from point ‘a’ and the left-running waves emanating from point d.

B. Method of Characteristics Method of characteristics is a numerical procedure to solve certain classes of flow problems. This procedure enables solutions to be obtained for two dimensional, planar and axisymmetric, and three-dimensional supersonic flows. The procedure may also be employed for all one-dimensional unsteady flows. Method of characteristics has been used for many years to solve problems in various fields of study that are described by hyperbolic partial differential equations. When a characteristic of one family extends into a flow field, it can encounter (1) a characteristic of another family, (2) a boundary, (3) a free surface, or (4) a shock wave.

Each of these encounters or interactions leads to a set of computational rules that are called unit processes. Application of the unit processes allow the solutions to be gradually advanced into the flow field. Therefore, the procedure is generally called the Pointto-Point Method [1]. We now consider the first type of interaction. We begin by considering the calculation at an interior field point. Suppose that, along a curve LM, we know the flow conditions-that is, the velocity and its direction, pressure, temperature, Mach number, and so on. This curve is called the initial value line and must not be a characteristic curve or else the problem is ill posed. Now select points A and B on the initial value curve as shown in figure 2.

Fig. 2: The representation of an interior field point [1]

With velocity (magnitude and direction) and Mach number known at points A and B and ν (M) (Prandlt-Meyer function) are known, so the Riemann invariants đ??ś1 and đ??ś11 can be calculated from the equations (1) and (2). Characteristic Ó? from point A intersects with Characterestic II from point B at point P, as shown in the figure 2. Since ν+Îą = constant, along a characteristic of Type I and ν-Îą = constant, along a characteristic of Type II, it follows that, at point P, in Fig 3. đ??śđ??ź = đ?œˆđ??´ + đ?›źđ??´ =đ?œˆđ?‘ƒ + đ?›źđ?‘ƒ (1) đ??śđ??źđ??ź = đ?œˆđ??ľâˆ’ đ?›źđ??ľ = đ?œˆđ?‘ƒâˆ’ đ?›źđ?‘ƒ (2) So, we have two equations with two unknowns, đ?œˆđ?‘ƒ đ?‘Žđ?‘›đ?‘‘ đ?›źđ?‘ƒ , which may be readily solved to obtain đ??ś1 + đ??ś2 đ?œˆđ?‘? = (3) 2 đ??ś1 − đ??ś2 đ?›źđ?‘? = (4) 2

Fig. 3: The straight line approximation of the characteristics [1]

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2D Contour Design and Analysis of the Divergent Portion of a Satellite Thruster Nozzle using MOC (GRDJE/ Volume 02 / Issue 05 / 028)

Since, from �+1

đ?›žâˆ’1

đ?œˆđ?‘? = {√ tan−1 [√ (đ?‘€đ?‘? 2 − 1)] − tan−1 (√đ?‘€đ?‘? 2 − 1)} đ?›žâˆ’1 đ?›ž+1

(5)

ν =ν(M), the Mach number, at P can be determined. Moreover, the direction of the velocity at P is known from equation.

For isentropic flow, the Mach number can be used to determine the values of the stagnation-to-static-temperature and pressure ratios at point P. Since đ?‘‡đ?‘‚ đ?‘Žđ?‘›đ?‘‘ đ?‘ƒđ?‘œ are constants for this flow, the static temperature and pressure at P can be calculated by working outward from the initial value curve LM, it is then possible to build a net or mesh, with points connected by the characteristics. Of course, to complete the process, we need to determine the direction of the characteristics. Since Mach number varies throughout the flow field, the characteristic lines AP and BP are not necessarily straight, but rather may be curved. However, if the size of the mesh is kept small enough, the lines AP and BP can be approximated by straight lines inclined at the Mach angle to the flow direction. Naturally, the accuracy of this procedure depends on the mesh size. The smaller the mesh, the more accurate the solution will be.

To find the direction of the characteristics, we employ đ?‘‘đ?‘Ś ( ) = tan(đ?›ź − đ?œ‡) (6) đ?‘‘đ?‘Ľ 1 đ?‘‘đ?‘Ś đ?‘‘đ?‘Ľ 2

( ) = tan(đ?›ź + đ?œ‡)

(7)

As we have seen, these equations tell us that at a given point the characteristics of family I are inclined to the x-axis at an angle of Îą-Îź, whereas at a given point the characteristics of family II are inclined to the x-axis at an angle of Îą+Îź.To solve the forgoing pair of equations, we may use the explicit Euler method that was used to solve ordinary differential equations. However, this situation is somewhat different because we must solve the pair of equations simultaneously. Thus, instead of selecting a set value of the difference Δx, here we write the finite-difference versions of the equations as follows. đ?‘‘đ?‘Ś đ?‘Śđ?‘ƒ − đ?‘Śđ??´ ( ) = = tan(đ?›ź − đ?œ‡)đ??´ = đ?‘š1 (8) đ?‘‘đ?‘Ľ 1 đ?‘Ľđ?‘ƒ − đ?‘Ľđ??´ đ?‘‘đ?‘Ś đ?‘Ś −đ?‘Ś ( ) = đ?‘ƒ đ??ľ = tan(đ?›ź − đ?œ‡)đ??ľ = đ?‘š2 (9) đ?‘‘đ?‘Ľ 2

đ?‘Ľđ?‘ƒ −đ?‘Ľđ??ľ

Because characteristics can be curved, there are other ways to evaluate the right hand side of equations. For example, we could base the inclination angle on the average value between the initial value line and the target point. This approach leads to an improved calculation of the location of the target point. Applying this method to equations yields (đ?›ź − đ?œ‡)đ??´ + (đ?›ź − đ?œ‡)đ?‘ƒ đ?‘‘đ?‘Ś đ?‘Śđ?‘ƒ − đ?‘Śđ??´ ( ) = = tan [ ] = đ?‘š1 (10) đ?‘‘đ?‘Ľ 1 đ?‘Ľđ?‘ƒ − đ?‘Ľđ??´ 2 (đ?›ź − đ?œ‡)đ??ľ + (đ?›ź − đ?œ‡)đ?‘ƒ đ?‘‘đ?‘Ś đ?‘Śđ?‘ƒ − đ?‘Śđ??ľ ( ) = = tan [ ] = đ?‘š2 (11) đ?‘‘đ?‘Ľ 2 đ?‘Ľđ?‘ƒ − đ?‘Ľđ??ľ 2 These variations will result in a slightly different characteristic and will cause the target point P to appear at a different location from that shown in figure đ?‘Śđ?‘ƒ = đ?‘Śđ??´ + đ?‘š1 (đ?‘Ľđ?‘ƒ − đ?‘Ľđ??´ ) (12) đ?‘Śđ?‘ƒ = đ?‘Śđ??ľ + đ?‘š2 (đ?‘Ľđ?‘ƒ − đ?‘Ľđ??ľ ) (13) Solving this pair of equations simultaneously yields đ?‘Ľđ?‘ƒ , the x location of point P. đ?‘Śđ??´ − đ?‘Śđ??ľ + đ?‘š2 đ?‘Ľđ??ľ − đ?‘š1 đ?‘Ľđ??´ đ?‘Ľđ?‘ƒ = (14) đ?‘š2 − đ?‘š1 These coordinates are used to plot the points and hence the profile of the nozzle.

II. MATLABS The design procedure using the MOC is little bit complex. So the manual calculation is much difficult and time consuming, So by writing this procedure in programming languages, the properties of the grid points can be easily find out even for a unskilled person. This is done using MATLAB programming. MATLAB is one of the prominent mathematical simulation software used by researchers and students across the wold. The programming using MATLAB is very simple. The program can be easily loaded to different simulation platforms like, ANSYS-Fluent, CFX etc.

III. DESIGN The design of the nozzle profile was done by solving the equations mentioned in the section II. For that we use a MATLAB program. The coordinates for the nozzle profile will be obtained by running the MATLAB program. The table 1 gives the boundary coordinates for the nozzle profile.

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2D Contour Design and Analysis of the Divergent Portion of a Satellite Thruster Nozzle using MOC (GRDJE/ Volume 02 / Issue 05 / 028)

Table 1: Nozzle profile coordinates obtained from the MATLAB program X coordinates Y coordinates 0

1

0.8652

1.1863

2.1077

1.4071

3.1391 3.7467

1.5155 1.5367

By using the coordinates in the table 1, the nozzle is designed using ANSYS-Design modeler 16.2, the figure 4 represents the CAD drawing of the nozzle profile. The throat area ratio obtained using MOC was (A/A*) = 1.5367, and actual area ration corresponding to Me =1.8 is (A/A*) =1.4389.

Fig. 4: Design of nozzle section created using the coordinates obtained from MOC

IV. MATHEMATICAL MODELLING AND NUMERICAL SIMULATION A. Mathematical Formulation The Cartesian coordinate system is used for the modelling, the basic governing equations for the compressible flow are [5], [6], Conservation of mass: đ?œ•đ?œŒ + ∇. (đ?œŒđ?‘˘) = 0 đ?œ•đ?‘Ą

(15)

đ??ˇđ?‘˘ ∇đ?‘? =đ??šâˆ’ đ??ˇđ?‘Ą đ?œŒ

(16)

Conservation of momentum: Conservation of energy: đ??ˇâ„Ž đ??ˇđ?‘? = + ∇. (đ?‘˜âˆ‡đ?‘‡) + ∅ đ??ˇđ?‘Ą đ??ˇđ?‘Ą đ?œ• đ?œ• ∇= + đ?œ•đ?‘Ľ đ?œ•đ?‘Ś đ?œŒ

(17) (18)

Where, Ď - Density of the fluid ÎŚ represents the viscous dissipation function h - Total enthalpy k - Thermal conductivity p - Pressure of the fluid T - Temperature of the fluid t - Time F - Total force acting By solving these equations we will be able to find out different parameters of flow inside and outside of the nozzle. All these calculations are done using different computer software due to complex numerical and computational procedures. The main software used are ANSYS-Fluent, CFX like proprietary software and open source software like open FOAM, REEF3D, etc. Here we use ANSYS-Fluent version 14 as the simulation software.

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2D Contour Design and Analysis of the Divergent Portion of a Satellite Thruster Nozzle using MOC (GRDJE/ Volume 02 / Issue 05 / 028)

B. Numerical Simulation The simulation is done using ASNYS-CFX 16.2, which is a high performance computational fluid dynamics software tool. The operation using the CFX software is very simple compare to other packages like Fluent. The reliability and accuracy compared to other software tools are more and the operation is quick and robust across a wide range of multi-physics applications. CFX is mainly recognized for its robustness and speed [7]. Figure 5 represents the meshed geometry by the ANSYS-Mesh tool. A fine meshing scheme is selected.

Fig. 5: The meshed geometry

V. RESULTS AND DISCUSSION The simulation is completed with a reference pressure and temperature. The simulation is done to verify the conditions given for the design. The pressure, velocity and Mach contours are plotted. Figure 6 represents the pressure contour, the pressure is decreasing along the length of the nozzle. The maximum pressure is indicated by red colour, which is the reference pressure 25 bar.

Fig. 6: Pressure contour

Figure 7 represents the velocity contour, along the length of the nozzle the velocity shows increase. The velocity increases when the flow moves from throat to exit. Along the nozzle surface, we can clearly see that there is a low velocity region, indicated by blue colour. This shows the existence of a boundary layer in the nozzle surface. For a numerical comparison, the velocity at the exit from the contour can be found as 523.4 m/s and from numerical calculations the exit velocity is around 493.22 m/s.

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2D Contour Design and Analysis of the Divergent Portion of a Satellite Thruster Nozzle using MOC (GRDJE/ Volume 02 / Issue 05 / 028)

Fig. 7: Velocity contour

From the Mach number contour, represented in the figure 8, it is clear that the Mach number is increasing, meaning the flow is supersonic in nature at the exit of the nozzle. Sonic condition is existing at the throat section of the nozzle and exit Mach number is found to be 1.7 and the maximum Mach number is 1.8, just before exit. The Mach number value from the analysis shows a 5% variation from the design Mach number.

Fig. 8: The Mach number Contour

VI. CONCLUSION The point to point method is considered for the design of the nozzle due to its simplicity. For the design of the nozzle profile, method of characteristics is used. A special program is created for doing the design process, which is done using MATLAB. This gives the characteristics and coordinates of nozzle profile. The input data to the program was exit Mach number and values of Îł.

The coordinates were generated for an exit Mach number of 1.8 and Îł is 1.2. The MATLAB programme generated almost same results as calculated by manually. The coordinates then used to create the diverging section of the nozzle. The validation of the obtained nozzle profile was done by using ANSYS-CFX 16.2 software. The results obtained from the computational fluid dynamics software shows, the designed nozzle profile gives almost the same results as compared with the design parameter, Mach number.

The CFD analysis gives a promising results, and the result only have a variation of 5% from the design value. The exit Mach

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2D Contour Design and Analysis of the Divergent Portion of a Satellite Thruster Nozzle using MOC (GRDJE/ Volume 02 / Issue 05 / 028)

number obtained from the analysis was 1.7. Since the analysis shows a promising result, the MATLAB program can be utilized for future design application of nozzles. This program will be beneficial for the future research in space science and engineering.

ACKNOWLEDGEMENT The author wish to thank all the staff members and students of PG studies, Department of Mechanical Engineering, Mar Athanasius College of Engineering, India. All the equations are taken from the textbook for Gas Dynamics (2010) [1] by John and Keith, Modern Compressible Flow with Historical Perspective (2003) [2] by Anderson J D.

REFERENCE Books [1] [2] [3] [4] [5]

J. John and T. Keith, Gas Dynamics, 3rd edition, Pearson, India Anderson, J. D, Fundamentals of Aerodynamics, 4 th edition, McGraw-Hill, 2007 Anderson, J. D, Modern Compressible Flow with Historical Perspective, McGraw-Hill, 2003 P. H. Oosthuizen and W. E. Carscallen, Introduction to Compressible Fluid Flow, 2nd edition, 2009 P. Niyogi, S. K. Chakrabartty, M. K. Laha, Introduction to Computational Fluid Dynamics, Pearson, 2005

Website [6] www.en.wikipedia.org/wiki/Fluid_dynamics [7] www.ansys.com/Products/Fluids/ANSYS-CFX

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