_{e}

_{D}

_{DB}

_{DA}

_{R}

_{N}

_{T}

^{on}

^{off}

_{a}

_{ref}

_{a}

Aerodynamic analysis is crucial to aerospace vehicle design and development. Accurate prediction of the performance from an aerodynamics point of view provides an edge during the design process and simulations prior to flight. A classical approach for accurate prediction of the aerodynamic characteristics is aerodynamic wind tunnel experimentation. Another valuable source of data involves the use of computational fluid dynamics (CFD), since it is not reliant on expensive infrastructure and gives insight into phenomena that may be difficult to capture with wind tunnel experiments. In the case of rocket aerodynamics, both approaches have been widely applied.

The Grot sounding rocket’s main design objective was maximising the apogee; thus, accurate performance estimation was an important aspect of design and mission analysis. In order to obtain more realistic results, it was of interest to investigate the impact of the propulsion system operation on the overall aerodynamics.

In general, it is not possible to accurately obtain the net axial force acting on an in-flight rocket by simply adding the static thrust to the drag value calculated for a rocket with a non-operational propulsion system. This phenomenon is often related to a change in pressure acting on the rear of the rocket; this pressure creates a drag component commonly known as base drag [1].

In the analysis of base drag, several approaches and methodologies have been utilised in the literature to obtain valid results. Moore et al. [2] utilised wind tunnel experiments to investigate the impact of angle of attack, fin deflection, fin location and fin thickness on base drag, in an attempt to extend the Brazzel and Henerson [3] technique. Huffman [4], in his work, outlined a mathematical model for axial force coefficient calculation based on effects from overall pressure influence, base drag and friction drag. The mathematical model outlined the base drag for power-on and power-off conditions for differing Mach regimes. Al-Obaidi [5] performed drag prediction for bodies flying at supersonic speeds. The drag constituents taken into account were skin friction drag, base drag and wing-body interference drag. A coherent relationship between the base drag coefficient and the base pressure coefficient was established. The paper concluded that classical panel methods are not able to capture the effects of base drag; however, modified panel methods may be used. Kumar [6] outlined an improved method for prediction of motor-on base drag and overall drag of a rocket. Furthermore, the paper claimed expansion of a pre-existing Brazzel’s method to a high thrust coefficient and Mach number range. The classical method (Brazzel’s method) and the improved semi-empirical model were compared. It was concluded that they yield numerically similar results, supporting the use of CFD for accurate modelling of the physics behind the flow.

The purpose of this paper is to study the effect of engine operation on the aerodynamic performance of the Grot sounding rocket. This purpose is achieved by the use of CFD computations. Furthermore, a comparison of CFD results and an analytical model found in the literature is performed. It is expected that the models would not completely coincide, since the Grot sounding rocket has an unusual aft geometry and two stages with significantly different calibres.

The rocket used for the study is a sounding rocket developed by the Students’ Space Association, known as Grot. Grot is a 2-m-long dart-type rocket (Fig. 1), which comprises two stages. The first stage accelerates the dart (second stage) to an altitude of 2 km, where aerodynamic separation takes place and the motor (first stage) plunges back to the ground, while the dart continues flying to an altitude of 18.5 km. Key parameters describing the Grot sounding rocket can be found in Table 1.

Grot main dimensional parameters.

Parameter | Value [m] |
---|---|

Total rocket length | 2.000 |

First stage length | 0.983 |

Second stage length | 1.017 |

First stage diameter | 0.135 |

Second stage diameter | 0.045 |

Reference diameter | 0.135 |

Pre-flight distance of centre of mass from the rocket’s base | 0.979 |

Nozzle exit diameter | 0.084 |

Various types of nozzles have found use in rocketry; most configurations consist of converging and diverging sections, with the throat located at the minimal cross-sectional area. The variations between the types usually concern the diverging section, as it is determined to be the most crucial for achieving high performance.

The conical nozzle (Fig. 2) is one of the simplest and oldest geometries. One can define a theoretical correction factor

Large divergence angles enable light and short designs at the expense of performance, while small angles lead to the majority of the momentum being axial, which in turn improves the specific impulse, but the increased length influences the complexity of the design and the weight of the propulsion system and vehicle. The optimal setting is usually found to have a half-angle between 12° and 18°, depending on the application and mission profile. The Grot rocket uses a conical nozzle with a half-angle of 15°.

Thrust is a reaction force generated when an exhaust plume is ejected from a motor; this in turn applies a force pushing the motor in the opposite direction, as described by Newton’s third law. The inertial term of the thrust equation can be derived from Newton’s second law, as shown in Eq. (2):

An additional source of thrust is the pressure force acting on the nozzle exit. The full thrust force equation for a rocket motor can be seen in Eq. (3).

In the more accurate integral form, Eq. (3) can be presented as Eq. (4):

For an ideally expanded nozzle, the second term tends to zero; nozzle expansions are illustrated in Fig. 3.

The Grot sounding rocket with the directions of the relevant axial forces is shown in Fig. 4. In the following discussions, it shall always be assumed that any component of drag is defined as positive when acting in the direction of the flow velocity

The axial forces present during the flight of the rocket vary depending on the Mach number and operation status of the propulsion system. Therefore, two cases were distinguished, the first being the non-operational propulsion system and the second being the operational propulsion system.

When the propulsion system is inactive, the only external force acting on the rocket is the drag. By definition, drag is assumed to be positive when acting in the direction of the airflow. The total axial forces in the case of a non-operational motor are presented in Eq. (5):

The common approach in analysis of aerodynamic characteristics is to use coefficients instead of the forces. The drag coefficient is defined in Eq. (7):

The coefficients of other discussed forces were computed analogously.

While the motor is in the operational state, the thrust force generated by the exhaust plume propels the rocket forward during flight and the drag force acts in the opposite direction. The total axial force, Fon, is expressed by Eq. (8):

The drag in the engine-on case can be expressed as follows:

In Eq. (9),

The difference

The National Aeronautics and Space Administration’s (NASA) computer programme Chemical Equilibrium with Applications (CEA) was designed to calculate the chemical equilibrium compositions and properties of complex mixtures [8]. For this study, the theoretical rocket performance mode was utilised, knowing the product concentrations from the set of reactants that made up the propellant in the form of fuel and oxidiser. The thermodynamic and transport properties can be obtained for the product mixture at the exhaust of the motor. The motor uses a typical AP+HTPB+AL composition, with the exact composition being confidential.

Since the rocket and its propulsion systems are essentially axisymmetric in nature, it was sufficient to perform 2D axisymmetric CFD simulations for the sake of this study (the only elements of the rocket that are not axisymmetric are the stabilising fins, which do not affect base drag in a meaningful way). To simplify the problem, it was assumed that the exhaust gases at the nozzle inlet are in a chemical equilibrium state (i.e. they are fully combusted). Moreover, Ansys® DesignModeler was used for preparing the geometry, Ansys® Mesher for meshing and defining the boundary conditions and Ansys® Fluent 20.2 for the numerical solution and post-processing.

The considered geometry was prepared on the basis of the computer-aided design (CAD) model of the Grot sounding rocket and technical drawings of the nozzle (Fig. 5). Given that this case study involved an axisymmetric model, only half of the geometry was modelled in Ansys® DesignModeler. The outline of the Grot sounding rocket without fins in the axisymmetric view can be seen in Fig. 6.

In addition, a domain study was conducted in order to minimise and preferably eliminate the influence of domain size on the results; the selected domain size can be seen in Fig. 7.

The overall quality of the mesh and the solver settings determine the accuracy of the results. However, the mesh used in a study may not require a large number of elements in order to obtain a fairly accurate solution. For this reason, a mesh independency study was conducted in order to determine the appropriate mesh settings so as to have minimal effect on the accuracy of results; as a result, the final mesh contained 626,458 elements. In addition, a structured mesh is favourable for increasing the efficiency of the calculations as it requires less memory. The nozzle and the plume region experience high gradients of pressure, temperature and Mach number; therefore, it was advantageous to utilise a dense-structured mesh in these areas; for this purpose, it was necessary for the fluid domain to be split into regions (Fig. 8).

Secondly, performing external aerodynamic CFD simulations requires the modelling of the boundary layer during meshing; this is done in the form of inflation layers. For this to be done correctly, one must estimate the boundary layer thickness formed in the range of the Reynold numbers experienced during the flight of the vehicle [9] and model the inflation layers to contain the approximated boundary layer thickness. It was decided to keep the 30 ≤ ^{+} ≤ 300 to implement the wall function approach for the shear stress transport (SST) ^{+} ≤ 1, while keeping the results relatively accurate. A similar choice of y+ has been observed in a previous paper [10]. For the nozzle, a bias factor was applied for capturing the boundary layer, followed with a dense mesh in the plume region. The mesh for the aforementioned domain regions was divided in the same manner (Figs 9 and 10).

By using the NASA CEA software, the physical properties of the exhaust gases were obtained as a function of motor operation duration. For this case study, the input chamber pressure was known beforehand. This case study made use of the species transport model, since it models the interaction between the exhaust gases and the free-stream air. The motor operating point chosen for the aerodynamic analysis is the one providing the highest value of the thrust force.

Compressibility effects are prevalent when the fluid accelerates beyond Mach 0.3; thus, it was necessary to use the density-based solver in this study. As mentioned in Subsection 3.2, the SST _{a}_{a}

For qualitative analysis of the obtained flow fields, the Mach number and static pressure, the contour plots for the operational and non-operational propulsion systems were analysed for the Mach number 1.2. The shock waves developed on the rocket appeared dimmer in the Mach number contour plot for the active motor than for the inactive motor (see Figs 11 and 12). This was due to the exhaust plume travelling at a Mach number three times greater than the free-stream air. Figures 13 and 14 present the static pressure for the same Mach number. Figures 15 and 16 show the closed-up static pressure contour plots for the aforementioned contour plots. It is important to note that the static pressure near the aft end (as defined by the red lines in Fig. 17) is lower in the engine-off case.

To evaluate the thrust force, it was necessary to define the nozzle exit surface, as the calculations were made for an axisymmetric model. The exit surface of the nozzle is represented by a black line in Fig. 17.

The approach utilises the Ansys® Fluent post-processing surface integral reports and Eq. (3) presented in Subsection 2.3. However, to be able to use such an approach, one needs to define the inertial term from Eq. (3). For this purpose, a variable,

The raw data for the total drag coefficients for active and inactive propulsion systems, along with the difference between them, are presented in Table 2 and Figs 18 and 20.

Computed drag coefficients.

Δ_{D} | |||
---|---|---|---|

0.3 | 0.460 | 0.453 | -0.007 |

0.6 | 0.378 | 0.457 | 0.079 |

0.8 | 0.359 | 0.462 | 0.100 |

1.0 | 0.494 | 0.668 | 0.170 |

1.2 | 0.628 | 0.744 | 0.120 |

1.6 | 0.577 | 0.659 | 0.080 |

2.0 | 0.506 | 0.576 | 0.070 |

2.5 | 0.435 | 0.495 | 0.060 |

3.0 | 0.383 | 0.429 | 0.046 |

_{D}

The total drag coefficients for both computational cases (active and inactive propulsion systems) are shown in Fig. 18. As expected, they are not equal; the difference is presented in Fig. 20. Further discussion of the results necessitates the introduction of some additional coefficients. The coefficient of drag acting on the nozzle exit (already defined in Eq. (6)) shall be denoted as

In Eq. (11), the pressure acting on the nozzle exit is not taken into account, since it is already a component of thrust, as defined by Eq. (4). The _{D,B}

The established Δ_{D}

Indeed, the plot of both sides of the approximate equality in Fig. 21 proves that they are indeed almost identical. Consequently, it is confirmed that the influence of propulsion system operation on drag can be fully explained by the pressure acting on the aft end of the rocket.

Since the atmospheric parameters change with flight altitude, it is worth considering whether they affect the previously shown results. Additional numerical simulation for the flight altitude of 5 km was performed. For this altitude, the static pressure of 54,050 Pa and static temperature of 255.7 K (−17.45°C) were assumed. Figure 22 presents the comparison of Δ_{D}_{D}_{D}

Two series of axisymmetrical flow simulations of the Grot rocket were performed: one, with the propulsion system being inactive; and in the other, with the propulsion system operating with full thrust. The goal was to study and quantify the effect of motor operation on the rocket’s drag coefficient. It has been demonstrated that drag is decreased during the thrusting phase. The dependence of the observed difference in drag coefficient on the Mach number has been found to be very non-linear and rather hard to approximate by a simple regressive model. For the sake of accurate flight simulation, it seems necessary to simply use interpolated CFD data. It has been shown that the analytical models of base drag found in a previous work [4] are in good agreement with the CFD results obtained in the supersonic domain. An important result is the proof that for all considered Mach numbers, the change in Grot’s drag coefficient due to motor operation is fully described by the change of pressure forces acting on the nozzle exit and aft end regions. This conclusion holds despite Grot’s non-standard geometry, with the nozzle extending noticeably from the aft end of the rocket. Therefore, it seems advisable to continue research on a general physical model for base drag prediction by attempting to link the base pressure with the rocket’s velocity and basic propulsive parameters.

#### Computed drag coefficients.

Δ_{D} |
|||
---|---|---|---|

0.3 | 0.460 | 0.453 | -0.007 |

0.6 | 0.378 | 0.457 | 0.079 |

0.8 | 0.359 | 0.462 | 0.100 |

1.0 | 0.494 | 0.668 | 0.170 |

1.2 | 0.628 | 0.744 | 0.120 |

1.6 | 0.577 | 0.659 | 0.080 |

2.0 | 0.506 | 0.576 | 0.070 |

2.5 | 0.435 | 0.495 | 0.060 |

3.0 | 0.383 | 0.429 | 0.046 |

#### Grot main dimensional parameters.

Parameter | Value [m] |
---|---|

Total rocket length | 2.000 |

First stage length | 0.983 |

Second stage length | 1.017 |

First stage diameter | 0.135 |

Second stage diameter | 0.045 |

Reference diameter | 0.135 |

Pre-flight distance of centre of mass from the rocket’s base | 0.979 |

Nozzle exit diameter | 0.084 |