Design and Numerical Analysis of a Plug Nozzle

: The purpose of this study is to simulate the flow in the plug nozzle. Further, the evolution of the parameters of this flow was studied. The mass of the nozzle and of the exhaust gases was calculated by the simulation. In addition, all these results are then compared with those obtained by the numerical calculations. In the second part of the study, the method of an approximate method was used to generate the profile of the plug nozzle for different values of heat capacity ratio γ and the number of exit Mach. Finally, our results are compared with those obtained numerically and experimentally (available in open literature).


Introduction
Most liquid propellant engines used on launchers are equipped with fixed geometry nozzles. These types of nozzles limit engine performance as they operate at optimum efficiency at a single point along the flight path. Conversely, a nozzle with a variable cross-section ratio optimizes the pulse delivered throughout the flight path: Several advanced nozzle concepts have been studied for this purpose [1]. Among these nozzles, (Aerospike) is a self-adaptive nozzle with altitudes for one-stage in orbit (SSTO) applications [2].
The full-length two-dimensional plug nozzle has a continuously expanding flow, while in off-design conditions, it has wave interactions in the central jet flow. On the contrary, the truncated plug nozzles present complexities of the base flow and its transition in addition to wave interactions in the flow of the central jet. Therefore, the design of plug nozzles with such complex flow physics continues to pose a challenge to the aerospace community [3]. It is in this context that the role of predictive tools such as empirical models and computational fluid dynamics (CFD) becomes important. To fully understand the aerospike nozzle, we compared two methods of designs of these nozzles for different ME and γ, and attempted to validate the CFD for complete length and truncated Aerospike nozzle flows with custom experiments for a range of PR (Chamber Pressure P0 / Ambient Pressure Pa).

The Literature Review
The plug nozzle concept was first developed by the Germans prior to World War II for aeronautical applications. Plug nozzles have a central body in the vicinity of the neck and the process of gas expansion is directly or indirectly regulated by ambient pressure, the gas flow is regulated by detent waves from the flow deviation due to the plug surface [1]. Based on weight/thrust ratio considerations, the cap is generally truncated, resulting in a very complex base flow. The use of these nozzles in the past was very rare, for example the Second World War fighter aircraft named Messerschmitt Me 262 was equipped with annular plug nozzle.
For the first time in 1950, Griffith of Rolls-Royce, Ltd., proposed the concept of plug nozzle for rocket propulsion in an American patent [4]. In 1959, Krase was the first to propose methods to designate ideal plug nozzle contours by simple approximate calculations [5]. In 1961, Berman and Cromp made studies on the modification of the end of plug and they found out that if half-cone angles at the end of the plug are used, a decrease in performance is of only 1 % [6]. In the same year, Rao discussed the use of plug contour optimization as the case in the conventional nozzle and obtained optimal contours [7]. In 1964, Angelino described an approximate method for axisymmetrical and two-dimensional plug nozzle design based on a simple technique [8]. Balasaygun studied the difference in performance between the plug nozzles and the conventional nozzles, and he found out that the nature of the flow in the plug nozzles is auto-adjustable, resulting in better performance when the nozzle is operated at a lower pressure ratio than the design [9]. In 1974, Johnson et al presented an optimization analysis for axial plug nozzles with variable input geometry [10].
The improvement in the efficiency of numerical simulation has led to the extensive study of this type of nozzle. The first numerical simulation was in 1997, Rommel et al. studied the development of the flow field as a function of ambient pressure variations using a computer study of a plug nozzle [11]. To minimize weight, McConnaughey conducted a numerical study of a three-dimensional aerospike and concluded that a 50 % truncation of the plug nozzle resulted in a 0.5 % reduction in performance only [12]. In 1998, Hagemann et al. carried out a numerical study based on the method of characteristics for the flow field simulations of plug nozzles, and they discussed the flow phenomena observed in experiments and numerical simulations of different adaptive plug nozzles in altitude [13]. In 2002, Ito et al. studied flow structures and thrust performance of axisymmetric truncated plug nozzles using a numerical simulation. They obtained a high gain from the plane plug nozzle (aerospike) of about 5 % to 6 % compared to the axisymmetric plug nozzle and for pressure ratios greater than the design ratio, the pressure distribution on the nozzle wall was not affected by the external flow [14]. Besnard et al. presented the manufacture, design and testing of a thrust engine equal to 1 000 lbf (4 448 N) of plug nozzle type. Their results showed that variations in heat capacity ratio led to a difference in thrust characteristics [15]. In 2006, using the Prandtl Meyer function for different specific heat ratios, Zebbiche plotted the profiles of the plug nozzle. Comparing the obtained performance with an MNL (Minimum Length Nozzle), he concluded that plug nozzles perform better [16]. In 2010, Shahrokhi and Noori used CFD to study the different flow properties of Aerospike nozzle [17]. In 2012, Karthikeyan et al. studied the effect of plug truncation of an aerospike nozzle on acoustic behavior [18]. In 2014, Chutkey et al. conducted a numerical and experimental study on the behavior of flow fields at truncated annular plug nozzle of different lengths [19].
In recent years, starting in the year 2015, Shanmuganathan et al. conducted a numerical comparative study on linear and annular plug nozzles and concluded that the annular nozzle was better than the linear nozzle [20]. In 2017, Kumar et al. compared the full length to the optimized plug nozzle models and discussed the aerospike nozzle design procedure and the parameters governing its design [21]. In 2019, Suryan et al. did a two-dimensional axisymmetric numerical analysis on flow through truncated conical plug nozzles with the introduction of a base bleed. They found that the introduction of the base bleed compensates for the loss of thrust due to the truncation of the conical supersonic plug nozzle [22]. In 2020, numerical simulations on an annular nozzle fed by a toroidal combustion chamber were carried out by Ferlauto et al. [23], in order to understand what are the main phenomena that affect the interaction between the primary flow and the injection of secondary flow on the truncated plug. In particular, the location of the injection orifice was studied. In 2021, Alshiya et al. did an experimental study on a plug aerospike nozzle in terms of pressure through an experimental setup of a jet flow apparatus with the assistance of CFD results in order to the reduction of potential core length. They also demonstrated that the spike configuration results in improvement of efficiency compared to C-D nozzle. The study was carried out on an aerospike nozzle with a 15° half-angle conical plug [24].

Methodology
In this section, we will discuss the design process of two-dimensional plug nozzles discussed in reference [16]. The design method is based on the function of Prandtl Meyer. . As the gas is perfect, the velocity vector is tangent with a stream line, which will be regarded as the contour of Plug wall to require (the main idea of this method).
To have a Mach number required at the exit, the flow to the throat must be tilted at an angle θB (Flow deviation compared to the horizontal).
where N is the selected point number.
Once the number of Mach Mi in point i is known. In this case we can write: And for the point i + 1 we have: λ is the polar ray of a Mach wave with: α and β (angles respectively at tops A and i of the triangle connecting the points A, i and i + 1 of the Fig. 2). ϕ is the polar angle of Mach.

The Pressure
The pressure ratio is given by: , and represent static pressure, density and temperature, respectively, with 0 index for chamber condition.

The temperature
For the temperature ratio we have:

The Mass of a Plug
Let us suppose that the shape of the wall between two successive points is a straight line, if the number of points N (number of the discretization points of the nozzle wall) is very high. The sum of these lines gives the mass of a plug by: M − the density of a structural material of a plug nozzle, tM − the thickness of structural material of a plug nozzle, l − the unit of nozzle depth, x, y -Cartesian co-ordinates of a point.

Pressure Force on the Plug
The axial pressure force exerted on the plug is the sum of all axial pressure forces exerted on all panels. 1 1 Fx− axial pressure force exerted on the wall of the central body.

Gas Mass in the Divergent
The mass of gas in the divergent can be considered as the sum of the triangles placed adjacent to each other as shown in Fig. 1, and the uniform area between the Mach AE line and the horizontal.

Validations of Results with Simulation
In order to validate the calculation method for the plug nozzles [16], we performed a simulation of the flow around the profile of a plug nozzle generated by the reference method [16]. We compared several variables (mass, axial pressure force and gas mass) for ME = 1.5, 2.0 and 3.0. The results are presented in tabular form. Tabs 1-3 show the comparison between the simulation results and those presented in the reference [16]. The error in the last column of Tabs 1-3 is a relative error calculated using the following formula: By analyzing the values in Tabs 1-3, we can see that our simulation results are in good agreement with the numerical results of the reference [16]. The error rate does not exceed 0.19 % for the mass value, 0.2 % for the value of the axial pressure force, and 0.15 % for the value of the gas mass. This also confirms and demonstrates the ability of CFD (Simulation) solvers to accurately predict flows in the plug nozzle. We also note that the more it increases the number of design Mach ME, the higher the relative error is.

[ ] [ ]
Figs 3a, 3b and 3c show the wall pressure ratio comparison between the numerical method and the simulation for ME = 1.5, 2.0 and 3.0 in this order. The results show a good similarity. Fig. 3d shows the Iso-Pressure contours for a plug nozzle that works in the design Mach number obtained by our simulation. The figure shows the Prandtl-Meyer expansion fan around the lip. In addition, there are no pressure fluctuations or turbulence corresponding to a typical flow along the nozzle of a plug. Fig. 4 illustrates the mathematical-physical model, the boundary conditions and the mesh adopted for Plug nozzle.
For both ME = 1.5 and 2.0, the pressure ratio decreases continuously in the divergent before stabilizing at the end of the nozzle, approaching the value of the atmospheric pressure at the tip of the nozzle at a value of 2.5 and 1.25, respectively. On the other hand, for ME = 3.0, the pressure ratio first falls rapidly at the nozzle throat expansion region until it reaches a value of 0.8, then it continues to fall in the expansion part, and then it finally stabilizes at the tip of the nozzle. Regarding this slight drop in pressure at the expansion part, Kbab et al. [25] use a method to reduce the weight of the nozzle without significant impact on the thrust, This is done by truncating the nozzle in the expansion part at a point where the best compromise (weight/performance) is respected. They concluded that if an axisymmetric nozzle is truncated to 79 % of its ideal length, we get a nozzle with a weight gain of 20.85 % and a loss of thrust equal to only 0.987 %. During the non-viscous calculations, the ambient conditions around the nozzle were modeled by applying a computational domain of 30Rth in the x-direction by 20Rth in the y-direction. Rth represents the distance between point A and point B (see Fig. 1).
Figs 5a, 5b, and 5c represent the evolution of the Mach number along the wall of the plug nozzle. Fig. 5d shows the Iso-Mach contours for a plug nozzle that works in the design Mach number obtained by our simulation. It is noted that the flow increases from M = 1 in the col until M = design Mach.

d) iso-Mach contours
We note that in the divergent part, the number of Mach increases until reaching the value of the nozzle designing Mach number at the outlet. We notice that the number of Mach at the outlet of the plug nozzle is M = 1.4 for nozzle of ME = 1.5 with a length of 11.35 mm. If we truncate the nozzle of ME = 3.0 to a length equal to 11.35 mm, we obtain a truncated nozzle with the number of Mach at the outlet equal to 1.78. Finally, we have two nozzles of the same length, but different Mach number (weight gain).

Design Method Validation
In this part the design method is validated by comparing the design method described in reference [16] with an approximate method [8] for different value of γ and ME.
Figs 6a, 6b, 6c, and 6d represent the contours of the plug nozzles generated by the design method described in reference [16] for different ME design Mach and different heat capacity ratio γ. A very good agreement is observed on the contours except for the last part of the contour, where there is a difference between the profiles. Fig. 7 and the following table show this difference.
Tab. 4 represents a comparison of the y-coordinates for each of the references [16] and [8] for the last part of the contour. This comparison was quite satisfactory since the results of both methods are very similar and the margin of error does not exceed 0.07 %.

Validations with Experimental Results
For the experimental comparison, we used the results of the future European space transport investigation program (FESTIP) [26]. We used two configurations: • a nozzle with design Mach number ME = 4.23 and with a truncation of 40 % of its ideal length, • a nozzle with design Mach number ME = 4.23 and with a truncation of 20 % of its ideal length. The analysis of flow for the Plug nozzle was done using ANSYS Fluent. The k-w SST model was used as the turbulence model. The baseline solver was selected as a double-precision density-based coupled solver with Implicit Time Integration. Least-square cell-based gradient is used for spatial discretization in which the solution was assumed to vary linearly was used and a second-order upwind scheme was used for interpolating the values of pressure, momentum, turbulent kinetic energy, specific dissipation rate, and energy. The computational analysis was conducted under steady conditions. The initialization for steady-state problem was done using full multigrid (FMG) initialization to get the initial solution, and the inlet boundary was provided to give the reference value. Sutherland equation is used for calculating the viscosity of air. 22       Very similar results were found for the different PR in both configurations, the error in all cases does not exceed 2.7 %. This difference may be justified by the difference between the approaches used. This proves the ability of CFD to accurately predict the flows in these types of nozzles. For a PR closer to the design condition, the pressure decreases along the plug as shown in Figs 13 and 14, while for a PR further from design conditions, a corrugated pressure distribution is observed due to the wave reflections. Note that for any value of PR, the flow remains attached to the surface of the plug. This leads to improved performance in plug nozzles, which is not the case in the conventional nozzle (there is a separation of the flow on the wall).

Conclusion
The plug nozzle is an improved rocket nozzle which includes a centerbody or plug around which the working fluid flows. The main characteristics of this nozzle are its interaction with the external environment, which avoids the phenomenon of separation of flows that affects a nozzle with a conventional contour. These advantages derive from the generation of a gas expansion at the nozzle lip and its influence on the behavior of the pressure along the wall. The present study is the first one devoted to the design of a profile of a plug nozzle using the method defined in reference [16]. The parameters of the flow (pressure and number of Mach) will then be evaluated and discussed for different Mach design and specific heat ratio γ. The validation of our results was done by comparing them with those obtained by the approximate method used in the reference [8]. This comparison was quite satisfactory since the results of both methods are very similar and the margin of error does not exceed 0.07 %.
Then a CFD study was conducted on this nozzle in both non-viscous and viscous cases using an appropriate turbulence (k-ω_sst) model and mesh. The results obtained were compared with those obtained by the numerical method but also with those of the experimental available in reference [26] for different truncation (20 % and 40 %) and different operating mode. A concordance of the results obtained was noted, the error in all cases did not exceed 2.7 %. This difference may be justified by the difference between the approaches used. This also confirms and demonstrates the ability of CFD (Simulation) solvers to accurately predict flows in this type of nozzles.
As far as prospective research is concerned, our study can be enriched by the addition of fluid injections in different position (throat, base of the nozzle) to see improved performance in different operating mode.