Characteristics Analysis of Dual Bell Nozzle using Computational Fluid Dynamics

Yeasir Mohammad Akib, Asif Kabir, Mahdi Hasan Department of Industrial and Production Engineering, Rajshahi University of Engineering and Technology, Rajshahi-6204, Bangladesh Department of Mechanical Engineering, Bangladesh University of Engineering & Technology, Dhaka1000, BangladeshDepartment of Mechanical Engineering, Chittagong University of Engineering & Technology, Chittagong-4349, Bangladesh yeasir.akib@gmail.com, 2* asifkabir121@gmail.com, mahdishuvo.bd@gmail.com


Introduction
Advanced nozzle configurations have been studied in recent years. Plug nozzles, either linear or axisymmetric, nozzles with extendable exit cones (EEC), and dual-bell nozzles are presently under consideration by space industries and agencies as possible main engine candidates for future launchers [1,2]. Especially Aerospace engineers began to focus on the dual bell nozzles concept as a means for equipping future aerospace vehicles. Currently, several research organizations (NASA, ONERA, etc.) and aviation and space industries (Boeing, Snecma Motors, Dassault, etc.) are working on the improvement of the performances and reliability of the supersonic rocket engine nozzles and space launchers nozzles [3].
The dual-bell concept was first initiated in literature in 1949 by F. Cowles and C. Foster, and was patented in the 1960s by Rocketdyne [4]. Due to the development of modern CFD capabilities, research activity was revived in the 1990s. In 1994, Tests at Rocketdyne conducted by Horn and Fisher and in Europe by the Future European Space Transportation Investigations Program (FESTIP) at the European Space Agency (ESA) investigated the influence of the extension contour geometry on the flow behavior in the first experimental study and confirmed the feasibility of this nozzle design. [4].Horn and Fisher found that a dual-bell nozzle could provide enough thrust to carry 12.1% more payload than a conventional nozzle of the same area ratio [4]. Since the early nineties, many studies, mostly numerical, have been made by Goel and Jensen [6], Hagemann et.al. [7], Immich and Caporicci [8] (within the FESTIP program) to understand and attempted to predict the behavior of this new nozzle concept. A numerical study of the feasibility was made by Karl and Hagemann [9]. Calculations were made to verify the transition duration and stability of the flow in Dual-Bell nozzles. Various parametrical studies have also been realized to understand the §ow phenomena and optimize the contour design [10,11]. P. Goel and R. Jensen performed the first numerical analysis of dual-bell nozzles, which was published in 1995 [12]. Throughout the 2000s, several numerical and experimental studies of dualbell nozzles were conducted in the United States and Europe [2]. A dual-bell nozzle has three characteristic geometric features: an inner base nozzle contour, a wall inflection, and an outer extension nozzle contour ( Fig.1, left). The contour inflection links the base nozzle to the inflection and provides two stable operation modes. At low altitude, the high ambient pressure forces the flow to separate at the inflection ( Fig.1, upper right). The separation is symmetrical and controlled, limiting the generation of high amplitude side loads. During the flight, as the ambient pressure decreases, the separation point leaves the inflection and moves abruptly in the extension down to the nozzle end. This flow transition leads to the high altitude mode (Fig.1, lower right): the extension is flowing full, offering a large area ratio for improved altitude performances. The area ratio limitation of conventional nozzles is circumvented for an overall performance gain [13].

Nozzle design methodology
A full-length dual-bell nozzle is created using MATLAB. After completing the meshing part using ANSYS, FLUENT software is used for the analysis of this model. Flow behavior along the nozzle is obtained. Air is taken as working medium for the nozzle. The area ratio of the dual-bell nozzle was determined using the isentropic flow relations. The value of the ratio of specific heats was assumed to be 1.23 due to the large assumed pressure and the high temperatures.

Contour Design
Duel bell nozzle is defined by three section: converging part, throat and diverging part. Converging section and throat is designed using two circle equation having two different radii ( Figure 2). This nozzle feature a divergent section made of two parts. The first part is known as base or primary nozzle. The second bell ( Figure 3) starts with a slope angle higher than the first bell end, such to yield an attached flow with a centered expansion at the inflection point in under expanded regime and a separated flow in the second bell in strong over expanded regime [14].  The first Parabola length of the nozzle is determined by Where K is a percentage of the length of an equivalent 15% conical nozzle, ε is nozzle exit ratio, Rt is the throat radius, θe is nozzle exit angle. A coordinate system is defined with the axial (x) axis passing through the line of symmetry and the radial (y) axis centered at the throat in order to define the nozzle further. The first and second curves define the entrance and exit of the throat of the nozzle and are based on circular curves. The third curve is a parabola. Equation of first parabola dual-bell is The coefficients a, b and c are determined by the derivatives of the contour at the point where the circle from the throat meets the beginning of the parabola xN, and the length of the nozzle Ln where definition of xN is And also Slope of xN is Similarly, slope at exit is In matrix form, full system of equations for the parabolic coefficients for the first parabola is Full Length of dual-bell nozzle is determined by Similarly, full system of equations for the second parabola of the dual-bell nozzle Where a', b' and c' are the coefficients of the second curve. The final design parameters are listed in Table 1.

Modeling and Flow Analysis
Flow through the dual bell nozzle is analyzed with the ANSYS Fluent 17.2. At first, we meshed it in ANSYS Workbench mesh generator. Mapped face meshing is used for getting good mesh as it consists of square cells of equal size. Element size of the characteristic cells was 0.002m and the behavior was selected as soft. Our constructed model has an axial distance from -0.01184 m to 0.78266 m. Different boundary conditions are specified in different positions. At the inlet, stagnation pressure is 101,325 Pa and stagnation temperature is 300 K. The static pressure at the exit is 2026.5 Pa (PR 50) and 1013.25 Pa (PR 100). In the next step, we have used the Fluent solver. As the flow rocket nozzle is highly compressible, density based solver is used for calculation. The -epsilon turbulence model was selected for the simulation due to the focus on the flow structures within the nozzles as it is a commonly used model for flow analysis in nozzle. Hybrid initialization is used for solution initialization. And the last step is the contour and velocity vector generation for some major parameters like pressure, temperature and Mach number. We have done it for pressure ratio 50 and 100.

Mach Number
For the pressure ratio 50 and 100, the Mach number is subsonic at the inlet. At the starting of the second parabolic contour, the flow becomes supersonic and slightly hypersonic. The exit Mach number has some variations. In the boundary layer, the flow becomes supersonic but at the midpoint of the exit, the flow becomes subsonic to transonic. The observable difference PR (pressure ratio) 50 and 100 is the variation of the Mach number through the midpoint of the second parabola.

Total pressure
Total pressure at the inlet is very high for both figure 7(A) and 7(B). It drops significantly from the midpoint of the second curve and becomes low at the exit. Exit total pressure varies from 17.6 kPa to 0.731 kPa.

Total temperature
The total temperature for both figure 9(A) and 9(B), it seems high till the starting point of the second curvature. Then, the temperature varies significantly through the axial distance. At the exit, the temperature varies from 102-106 K as the exit gas meets the air.

Verification
In our result section we have already discussed about six parameters. For more consistency, our results are compared with numerical data calculating by the students of WORCESTER POLYTECHNIC INSTITUTE [15] only for Mach number. Compared data has given below with error percentage.

Conclusion
A critical appraisement of dual-bell nozzles was presented in this paper. This paper represents the design of a dual-bell nozzle profile and the study of the fluid parameters behavior like Mach number, pressure, temperature, velocity vectors under under-expanded conditions. A code in MATLAB was developed to draw contour design. CFD calculations were achieved within a numerical domain by applying boundary condition assuming isentropic environment. Then we have compared our result with ref [15] only for the Mach number contours. For future work, we can simulate some conditions for over-expansion and compare it with traditional convergent-divergent (CD) nozzle. Moreover creating a small scaled version of dual bell nozzle can help us for testing it in real conditions.