**5.1 Governing fluid equations**

As discussed above the base pressure measurements in the wind-tunnel testing are affected by presence of sting attached to model. The free-flight data depend on quality of the transmitted telemetry data. The fluid dynamic equations describing the flowfield around a space vehicle include equations of continuity, momentum, and total energy. A numerical simulation of unsteady, compressible, axisymmetric laminar Navier-Stokes equations is an alternative to the expensive experimental testing of the reentry vehicles. The governing fluid dynamics equations can be written in the following conservation form in order to capture shocks and discontinuities as

$$\frac{\partial U}{\partial t} + \frac{\partial F}{\partial \mathbf{x}} + \frac{1}{r} \frac{\partial (rG)}{\partial r} + \frac{H}{r} = 0 \tag{3}$$

Temperature *T* is related to pressure and density by the perfect gas equation of state. The ratio of the specific heats *γ* is assumed constant and is equal to 1.4. The coefficient of molecular viscosity is evaluated in the flow solver employing Sutherland's formula. The flow is assumed to be laminar, which is consistent with experimental results of Cassanto [37] and Bulmer [42].

### **5.2 Numerical technique**

To simplify the spatial discretization in numerical technique, Eq. (3) can be written in the integral form over a finite computational domain *Ω* with the boundary of the domain *Γ* as

$$\frac{d}{dt}\int\_{\varDelta} \mathbf{U} d\varDelta + \int\_{\varGamma} \{\mathbf{F} dr - \mathbf{G} dx\} + \int\_{\varDelta} \mathbf{H} d\varDelta = \mathbf{0} \tag{4}$$

The contour integration around the boundary of the cell is performed in anticlockwise sense in order to keep flux vectors normal to boundary of the cell. The computational domain *Ω* is having a finite number of non-overlapping quadrilateral cells. The conservation variables within the computational cell are represented by their average values at the cell centre.

The inviscid fluxes are computed at the centre of the cell resulting in flux balance. The summation is carried out over the four edges of the cell. The derivatives of primitive variables in the viscous flux are evaluated by using the method of lines. A system of ordinary differential equations in time is obtained after integrating Eq. (4) over a computational cell. In the cell-centered spatial discretization scheme is non-dissipative, therefore, artificial dissipation terms [55] are added by blending of second and fourth differences of the vector conserved variables. The blend of second and fourth differences provides third order back ground dissipation in smooth region of the flow and first-order dissipation in shock waves.

The spatial discretization described above reduces the integral equations to semi-discrete ordinary differential equations (ODE). The ODE is solved using multi-stage Runge-Kutta time stepping scheme of Jameson et al. [55]. The numerical algorithm is second-order accurate in space discretization and time integration. The scheme is stable for a Courant number ≤2. Local time steps are used to accelerate to a steady-state solution by setting the time step at each point to the maximum value allowed by the local Courant-Friedrichs-Lewy (CFL) condition.
