Investigation of Computational Flow Field over a Re-entry Capsule at High Speed

Manned space programme has become inevitable in the modern world. One important aspect in such endeavors is getting the crew back to home safe and sound. Re-entry capsules are widely used for bringing back astronauts back to the Earth. The re-entry mission is a very crucial one, with speeds up to 30 times than that of sound and temperatures hot enough to ionize air. The study and design of a re-entry capsule has to be done with utmost concern over the life of returning asronauts. In this paper, simulations carried out over a re-entry vehicle for various free-stream conditions such as subsonic, supersonic and hypersonic flow regimes. Results were obtained by solving compressible Navier-Stokes equation with k-omega turbulent model. Simulations were carried out for various inlet conditions corresponding to an altitude of a re-entry vehicle by using CFD code SU2. The geometrical parameters of a re-entry vehicle such as spherical nose radius and cone angle were varied and the effects of the aerodynamic phenomena such as shock wave, flow separation and wake formation over a downstream of a module were studied. In this paper, the results pertaining to the pressure distribution, the Mach contour and temperature over a module is presented.


INTRODUCTION
After the space mission, a re-entry capsule returns to the Earth atmosphere. At that time, the speed of the re-entry capsule is very high and the speed gradually decreases with altitude. The shape of the capsule is generally selected after aerodynamic and thermal consideration. The fore-body of the capsule has a spherical nose which is aerodynamically more stable than other shapes [1]. Due to the shape of the nose, it produces high aerodynamic drag which affects the capsule. The shoulder radius (Frontal area) and flare angle in capsule have a significant influence on the fore body and the position of flow separation point [2]. Allen and Cheng [10] carried out the numerical simulation in re-entry module and studied the flow separation in which is observed experimentally earlier.
From the investigations [6], it is evident that aerodynamic data generation is possible by the following methods like Analytical method, CFD codes and Wind tunnel testing In our present work, geometrical parameters of a capsule such as radius of the spherical nose and cone angle were varied. The effects over the capsule such as shock stand-off distance and the flow separated wake region behind the capsule were investigated.

MATHEMATICAL FORMULATION
The 3-D fluid flow is governed by 3-D Navier Stokes (NS) equations which comprise continuity, momentum and energy equations.
The integral formulations of the three conservation equations are given below. This equation represents the integral form of the continuity equation, which is based on the law of mass conservation [6].
where, ρ is the density of fluid, v is the velocity of the vector, n is the unit normal vector, dS is the elemental surface area and Ω is the control volume. The expression for the momentum conservation inside an arbitrary control volume Ω which is fixed in space and is given by where e f is the body force term, τ is the viscous stress tensor. The energy conservation equation is given below.
where H is the total enthalpy, E is the total energy per unit mass, k is the thermal diffusivity coefficient, T is the absolute temperature and h q is the heat transfer per unit mass.
Finite volume formulation is given in integral form to solve the equation with SU2 [6,8] The dynamic viscosity is defined as a sum of molecular viscosity ( ) 1 μ and turbulent viscosity. Therefore, Molecular viscosity μ i by using Sutherland law is given by, Turbulence Viscosity is given by, Static pressure downstream of shock wave is analytically determined by [9] ( ) where P 1 refers to pressure at upstream region of the shock and P 2 refers to pressure at downstream region of the shock.

GRIDS AND COMPUTATIONAL DOMAIN
The design of the re-entry vehicle was adapted from Mehta [1] and the design was varied to study the effects of the nose radius and cone angle over the entire vehicle. The vehicle is moving at an arbitrary angle of attack; instead of an axisymmetric body, a 2D cut section of the vehicle was considered. Grids were clustered adjacent to the vehicle model to capture the flow field accurately near the wall. Structured grids were used since it requires low computational time and also based on the design. The grid details are given in Table 1. To model the flow near the walls with accuracy, the first grid point from the surface should be within the viscous sub-layer. Fig 1 shows a zoomed view over a re-entry module. The first cell height in the normal direction of the body surface is fixed at 6 micron. The domain size is selected such as to avoid the transonic shock reflection back into the domain. Domain size was chosen as 75 times the diameter of the module for all simulated cases Re-entry module was specified as a wall and the whole domain was specified as a Far field boundary condition. Dimensional details of the simulated model are given in Table 2. where R represents the typical spherical radius and θ represents typical flare angle in a capsule. Simulations were carried out for three different Mach numbers ranging from 1.2, 4, 6 for corresponding altitudes of 13, 25 and 28.7 km respectively and the corresponding ambient conditions were applied.

NUMERICAL SCHEME
CFD has been employed to analyze the flow over a reentry module to obtain optimum design parameters such as nose radius and cone angle. Numerical simulations were carried out with a CFD code "SU2" with Navierstokes k-ω SST turbulence model for various Mach numbers at an arbitrary altitude. For adding viscous effect Sutherland law was incorporated into the CFD solver. The results obtained from the numerical simulations were validated with analytical calculations.

RESULTS AND DISCUSSIONS
Numerical simulations have been carried out for different inlet Mach numbers to study the effects of design parameters on the flow characteristics. Fig.2 shows comparison of pressure behind the detached shock predicted using analytical and CFD methods. The results are validated with analytical calculation and also with the previous works published by Bruce Ralphin Rose et al [2]., which shows significant contributions for the validation of the current computational tool.
Initially the front spherical nose radius and flare cone angle were varied to study their effect over the entire module. At M=1.2, formation of vortices behind the capsule due to the wake flow field were observed and these vortices were found to diminish when the nose radius was lowered from D1 to D4 as shown in Fig-3.
The variation of Cp with the distance from the vehicle nose for various nose radii at M = 1.2 is highlighted in Fig 4. Initially the pressure was high due to stagnation and from Cp distribution it is clear that the increase in nose radius gradually decreased the pressure distribution at the nose cone and vice versa.
The flare angle also plays a major role in the downstream flow characteristics of a module. If the flare angle was increased, the length of the module decreased. As the flow turned around the shoulder of the re -entry module, it expanded rapidly and began to separate. Increase in flare angle increased the flow separation and the wake region formed downstream as shown in Figure 5.
At M=4, flow field around the forebody was dominated by a bow shock wave. Due to increase in nose radius the curvature of the bow shock increased and the pressure domination behind the bow shock decreased. Flow separation and formation of the vortices behind the capsule due to the wake flow field were less compared to the case with smaller nose radius as shown in Fig 6. The variation of Cp with the distance from the vehicle nose for various nose radii at M = 4 are shown in Fig 7. Initially the pressure was high due to stagnation, but the initial pressure domination reduced when the nose cone radius was increased. And from Cp distribution it is clear that increase of nose radius gradually decreased the pressure distribution at the nose cone and decrease in nose radius suddenly decreased the pressure level at the end of a nose cone. At M=4, flow field around the forebody was dominated by a bow shock wave. Due to increase in nose radius the curvature of the bow shock increased and the pressure domination behind the bow shock decreased. Flow separation and formation of the vortices behind the capsule due to the wake flow field were less compared to the case with smaller nose radius as shown in Fig 6. The variation of Cp with the distance from the vehicle nose for various nose radii at M = 4 are shown in Fig 7. Initially the pressure was high due to stagnation, but the initial pressure domination reduced when the nose cone radius was increased. And from Cp distribution it is clear that increase of nose radius gradually decreased the pressure distribution at the nose cone and decrease in nose radius suddenly decreased the pressure level at the end of a nose cone. The flare angle also plays a major role in the flow characteristics downstream of a module. If the flare angle was increased, at the fore body of a capsule the bow shock is formed and behind the bow shock the pressure domination will be higher for higher flare angle. When the flow turns around the shoulder of the capsule, it expands rapidly and began to separate. From Cp distribution it is clear that due to change of flare angle there is no significant change in pressure distribution over a re-entry module which can be seen in Fig 9.