Flow conditions

Computations were performed for a real gas re-entry (M = 20.5, altitude = 67.3 km) of Apollo capsule AS- 202 [1] (hereafter referred as Apollo, also see Figure 1), corresponding to conditions present 4800s after launch. The freestream conditions have a Mach number of 20.5, ambient temperature and density of 227 K and 1.37e-4 kg/m3, respectively. The capsule orientation results in the free-stream approaching at an angle of 18.4o in the pitch plane with a 2.0o side slip (yaw). Heat transfer rates at the capsule surface are compared with flight measurement data [1] and with those computed using the DPLR code [2].

Geometry and grid details

The Apollo geometry is shown in Figure 1 [1]. Also shown are the calorimeters (a-s) which are used to measure the heat transfer occurring at those locations. Following the analysis presented in [1], a structured grid of 2 million cells was created on the half-geometry of the capsule. An initial analysis was carried out using the half model and further refined by considering the full model and the 2.0o yaw. Care was taken to cluster the mesh in the shock region upstream of the capsule. The first cell off the body was located at 104 m to ensure a cell Reynolds number of less than unity.

Figure 1: Schematic of the Apollo geometry with locations of calorimeters (Fig. taken from ref. [1]).

Problem set-up

Compressible Navier-Stokes equations with consideration of thermal and chemical non-equilibrium were solved, using CFD++ [ 3]. Thermal non-equilibrium was considered by adopting a two- temperature model, wherein an additional energy balance equation is solved for the transport of energy contained in the vibrational-electronic mode of atoms. The bulk viscosity and thermal conductivity of the mixture were directly computed using the Gupta-Yos collision cross-section-based species and bulk viscosity fits [4]. Park’s 5 species, 5-reaction chemical model [5] was deployed for reactions occurring in the shock layer. The translational-rotational temperature and vibrational-electronic temperature exponents for forward reactions were both kept as 0.5 and for backward reactions as 1 and 0, respectively.

For half domain computations (ignoring side slip) symmetry boundary condition was specified in the symmetry/pitch plane. The capsule surface was treated as a wall in radiative equilibrium with surface emissivity taken as 0.85. Furthermore, the wall was treated as a super-catalytic boundary which returns the species back to the freestream N2 and O2 compositions.

Results

Figure 2a shows the radiative equilibrium temperature in the aft-body region over-lapped with oil flow pattern and Figure 2b shows the stream-traces in the wake of the capsule, both obtained without considering the side-slip (similar to [1]). The temperature contours compare qualitatively well with those reported in [1]. Recirculating flow-field and its imprint on the surface temperature can be clearly seen in Figure 2b.

Figure 2: Radiative equilibrium temperature contour overlapped with oil flow pattern on the aft-body region (a) and stream traces in the wake of the Apollo capsule (b).

Quantitative comparison of heat release rate in the aft-body region is shown in Figure 3. The results shown in Figure 3a were obtained from the full mesh, considering side-slip. The calorimeter sensors are grouped as shoulder-sensors (h,i), windward-sensors (a,b,c,d,f,g), separation-line sensors (j,k,l,n) and leeward-sensors (e,m,o,p,q,r,s) following ref. [1]. The flight data limits were also taken from ref. [1] and were obtained by performing Fourier function-based least-square fit to the flight data and allowing for +/- 20% uncertainty as reported in the measurements.

Figure 3: Comparison of heat release rate at sensor locations with DPLR and flight test data (a) and the effect of side slip in heat transfer prediction (b).

In general, CFD++ predictions fall either within the flight data limits or adjoins these limits, including sensor ‘s’ where the DPLR [16] computation reports an under-prediction. Wright et al. [1] report that the reason for such under-prediction is not clear and there could be a possibility of a local vortical structure which was missed by their computations.

The effect of considering the side-slip (yaw) in the heat transfer predictions is shown inFigure 3b. The side-slip orients the on-coming flow-field away from most sensor locations on the aft-side of the capsule and as a result the heat transfer is generally lowered. This, in-turn, brings the predictions closer to the flight data limits at most sensor locations.

References

1. Wright M. J., Prabhu D. K. and Martinez E. R., Analysis of Apollo command module afterbody heating part I: AS-202, Journal of thermophysics and heat transfer, Vol. 20, No. 1, 2006, pp. 16-30.
2. Wright M., Candler G. and Bose D., Data-parallel line relaxation method for the Navier- Stokes equations, AIAA Journal, Vol. 36, No. 9, 1998, 1603-1609.
3. Chakravarthy, S., “A Unified-Grid Finite Volume Formulation for Computational Fluid Dynamics,” Int. J. Numer. Meth. Fluids 31: 309-323, 1999.
4. Gupta R. N., Yos J. M., Thompson R. A., A review of reaction rates and thermodynamics and transport properties for an 11-species air model for chemical and thermal non-equilibrium calculations to 30000 K, NASA report 1232, 1990.
5. Park C., On the convergence of computation of chemically reacting flows, AIAA paper 85-0275, 1985.