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One of the key issues in a reentry risk analysis is the calculation of the aerodynamic coefficients. This paper presents a methodology to obtain these coefficients and couple it to a code that computes re-entry trajectories considering six degrees of freedom. To evaluate the different flight conditions encountered during the natural re-entry of conical objects, the Euler Equations for gasdynamics flows are used. A new scheme TVD (Total Variation Diminishing) is incorporated to a finite volume unstructured cell-centred formulation, for application to three-dimensional Euler flows. Finally, numerical results are obtained for a conical body at different attack angles and Mach. With these results, the calculation of the trajectories during atmospheric re-entry is completed.

In the case of natural reentries (non-controlled), the orbital evolution of an object can only be monitored, with no or limited ability to control risks. The time window for reentry of a satellite is usually provided with a standard error of ±10% to ±20% of the remaining orbital lifetime. For the controlled reentries, it is required to simulate the different scenarios until the right window for the mission is found, being that the total or partial disintegration, or the landing on a safe place.

In this paper, the main objective is to conduct numerical simulations of the supersonic flow regime on a conical body, thereby using a code developed at the Department of Aeronautics of the National University of Córdoba, Argentina [

The drag and lift coefficients,

The three-dimensional Euler equations can be written as

The temporal change of the conservative variables can be expressed as

Equation (

_{i}_{i+1} are the physical fluxes normal to the face in each cells,

The limiter function given in (

In the numerical solution of the three-dimensional Euler equations, five wave families appear. If the five wave families are enumerated in correspondence with their speed, being one the slowest and five the fastest, it can be demonstrated that for waves of the families two to four, the characteristic velocities at both sides of the discontinuity are the same and equal to the velocity discontinuity [

In this work, the possibility of implementing different limiter functions for different wave families is explored. The objective is to improve the numerical resolution of the discontinuities associated with the families two to four using compressive limiter functions (superbee), and without losing robustness mainly due to the use of diffusive limiter functions (minmod) for the wave families one and five. This technique implements the utilization of the superbee limiter function only in linear degenerate waves and the minmod function in nondegenerated nonlinear waves [

To introduce in the numerical fluxes calculations the limiter function superbee, (

To improve the overall scheme robustness, the implementation of different limiter functions is carried out only in those cells interfaces where the greater relative intensities of the discontinuities in central waves are registered, and using the conventional Harten-Yee TVD scheme in all other cases. Notice that the comparison among the intensity of the waves cannot be made using directly the coefficients of the spectral decomposition

In the local coordinate system adopted for computing the numerical fluxes across each face, the corresponding eigenvectors are given by [

In this investigation, _{i}

Finally, if the maximum of

For the evaluation of

The treatment of the boundary conditions is carried out through the imaginary cells technique [

The choice of a suitable set of coordinates and parameters of the trajectory to describe the movement of an object in atmospheric reentry is inherent to any investigation of guided spacecraft. To analyze a reentry trajectory it is appropriated to describe the motion of the center of mass using a set of elements known as Flight Coordinates [

Thus, the flight coordinates are described by the six orbital elements: magnitude of the position vector,

The gravitational force is always present. For nonpowered flight, the propulsion force is zero, while for flights outside the atmosphere, the aerodynamic force vanishes.

To derive the equations of motion, we must use an Earth-fixed reference system. The kinematics equations of motion are [

It is desirable to separate the aerodynamic force into two components and define the tangential component of the nongravitational force,

The force equations are [

Knowing the attitude of a space object means knowing the orientation of an axis system connected to the vehicle related to a vertical reference system. To specify the orientation of a rigid body in space, three independent parameters are needed. These parameters are commonly known as roll, pitch and yaw, the Euler angles.

However, the use of the Euler angles to compute the attitude evolution of a spacecraft is limited: the equations of motion in attitude have singularities for certain values of the pitch angle, namely

The quaternions are denoted as

Kinematics equations of motion have the following form [

The attitude dynamic equations of motion express the temporal dependence of the angular velocity related to the applied torques:

If the vehicle under consideration operates on a symmetry condition, the velocity vector defines this plane of symmetry. So the attitude of the object is properly described by the attack angle,

The aerodynamic force is decomposed into two components: the force opposite to the direction of motion, called Drag -and part of the non-gravitational force

The first phase of the calculation is to identify the object’s surface areas. So, the nodes of the tetrahedrons, whose faces form the surface of the object, are identified in the code input file.

The forces acting on each face have a magnitude equal to the pressure divided by the area. The direction is the incoming normal to the surface, and the point of application corresponds to the geometric center of the face. In this way, the resultant force on the body is calculated as the vector sum of all forces acting on the discretized surface. Finally, the resulting force components in directions parallel and perpendicular to the flow velocity vector are projected, in order to obtain the drag and lift coefficients, respectively.

The numerical simulations were performed using as a core calculation code the one developed at UNC [

GID has been developed as an interface for geometric modelling, meshing, income data, and display results of all types of numerical simulation programs. The different menus can be modified according to specific user needs. The graphical interface adapted to the code is designed to allow the entrance of initial conditions, to mark the object's surface for the calculation of forces, to define the number of iterations, and other parameters inherent to the code. The ultimate objective of the use of GID is writing the data file that enters the UNC's code and the subsequent display of results, from reading the output file written by the code.

The limit of iterations in the code can be determined by the number of steps, or the limit time independently. When it comes to any of these, the program completes its implementation.

The motivation of the simulations performed is to obtain the main aerodynamic characteristics (drag and lift coefficients) of a conical body. They are used in the calculation of the trajectory during reentry into the atmosphere [

The angle of the cone (10°) was chosen arbitrarily, in order to guarantee that this value is small. The cone's length (1 m) was chosen to facilitate the calculations.

It is known that the aerodynamic characteristics of a body with the given geometry depend, in the case of nonviscous flow, only on the Mach number and the incidence angle of the free flow, without considering heat transfer or changes in the properties of air. That's the reason why these variables are used as independent ones.

Two sets of simulations were performed. The first case is for a cone-shaped object that enters in the atmosphere by sharp (narrow) side forward, while the second one was calculated in the assumption that the vehicle moves by wide (obtuse) side forward. Both calculations were made for several attack angles and Mach numbers.

It is important to note that the pressure distribution around the body depends only on the free flow Mach number. Then to obtain the

The dimensions of the volume meshing are length in direction

The interface with GID requested that a new mesh was drawn every time a new calculation was made, because it needed a new data file for every case considered. For this reason, and because a nonstructured mesh was used, every case had a slightly different number of elements, due to arbitraries processes during meshing.

The criterion used for the meshing is based on

using the symmetry of the flow on a plan to reduce the volume of mesh, and on two planes in the case of zero attack angle,

concentration of all elements near the surface of the cone, in particular areas considered critical for the calculation: impact point and base,

significant reduction of elements in remote areas of the body, to reduce computational cost.

Figures

Pressure distribution on the object: (a) Mach = 1, attack angle = 0°; (b) Mach = 4, a.a. = 15°.

The implemented numerical scheme has the capacity to simulate compressible flows in subsonic, transonic, and supersonic regimens. From Figure

Figure

Sharp Side Forward: Drag coefficient as a function Mach for different angles of attack.

Sharp Side Forward: Lift coefficient as a function Mach for different angles of attack.

Note from Figure

Figure

Sharp Side Forward: (a) Attack angles depending on the time; (b) Altitude. Initial attack angles: 0° (blue), 15° (red) and 30° (green).

We can notice the moment when the objects sense the presence of the atmosphere (around 1500 seconds) and begin to experience high variation of the attack angle during their descent. The first trend is to decrease the initial attack angle. However, although the atmosphere promotes an oscillatory movement around the zero attack angle condition, this is very unstable for objects reentering sharp side forward, and it is possible that for certain initial conditions, the atmosphere turns the object and reverses its attitude.

In what concerns the time evolution of the altitude, it could be seen that it is not affected until the height of 150 km. This is a limitation of the atmospheric model that has information on the atmospheric density just until this height. Since the aerodynamic forces depend directly on this parameter, it is assumed that the atmosphere is so faint over 150 km that it does not affect the trajectory. Anyway, under this limit, all trajectories show bouncing movements, indicating that the attitude variation affects the objects dynamics.

In this case the characteristics of the mesh are number of tetrahedrons around 160,000 and number of nodes around 32,000. An additional criterion used for the meshing is based on the concentration of all elements over the cone’s surface, in particular areas considered critical for the calculation: impact point and base. Also smaller elements were used in the zones nearby the cone base and along the axis of the cone in order to determine more precisely detached shock waves. Apart from that, the mesh was made thinner over the sides of the cone compared to the aerodynamic shadow in order to more accurately determine the expansion waves.

To simulate the atmospheric reentry of a cone shaped object by wide side forward, it is takes into account that this object has an attack angle around 180

Wide Side Forward: Drag coefficient as a function Mach for different angles of attack.

Wide Side Forward: Lift coefficient as a function Mach for different angles of attack.

Figure

The result of these differences can be noticed in Figures

Wide Side Forward: (a) Attack angles depending on the time; (b) Altitude. Initial attack angles: 180° (dark blue), 175° (dark red) and 150° (dark green).

The most important consequence of the wide side forward reentry is over the altitude evolution. It can be seen in Figure

In this paper, a series of numerical simulations have been conducted using a code developed at UNC [

The goal was the calculation of aerodynamic characteristics of a cone under the effects of different flight conditions, to attach the results to other code that calculates the dynamic of atmospheric reentry [

It was possible to perform simulations of reentry trajectories for a specific cone under the influence of various aerodynamic effects for a variety of initial attack angles. Simulations show, as expected, that the trajectories are more affected when the object has initially a sharp side forward configuration.

From the obtained results, was compared the numerical slope of the normal force coefficient with analytical results. In none of the cases the errors exceed 6%. This good accuracy of the numerical results permits to induce that the error by not considering viscous effects in the calculation of the aerodynamic coefficients is low, and the obtained trajectories and attack angles evolution during the reentry are reliable.

Although the methodology implemented has been shown to be suitable for calculating the reentry trajectories inside the terrestrial atmosphere, it will be improved with the inclusion of viscous effects in the simulation of the aerodynamic flow in future works.