**2. Numerical schemes**

Hypersonic calculations about complex geometries require large amounts of computer sources, but solution adaptive grids, convergence, acceleration, parallel processing may make run time manageable. The thin-layer approximation (Edwards 1992) is invoked, because viscous effects are generally present boundary layer near the wall. This

Computations of Flowfield over Reentry Modules at High Speed 353

complicated as compared to explicit code. The numerical solution can be accelerated

CFD-FASTRAN code (Liever et al., 2003) is based on compressible finite volume solver for thermo-chemical non-equilibrium flow analysis. Roe's flux difference upwind scheme is

The chemical behaviour of air depends on the trajectory points. Below about 2000 K, deviations from perfect gas behaviour are negligible. In the atmosphere, this temperature is first reached behind normal shocks at a Mach number of about six. Most of the test facilities are operated at low freestream temperatures and so real gas effects are difficult to obtain, even at high Mach numbers. The perfect gas equation of state is thus sufficient for many hypersonic speeds when the peak temperature is below the above limit. When the temperature in the flow is high enough to cause air to dissociate, the density of the fluid determines the character of the fluid/chemistry interaction. At relative high densities, the intermolecular collision rate is very high, and reactions proceed to equilibrium rapidly. The flow can be considered to be chemical equilibrium at all points in the flow field at all times as in the limit of infinite reaction rates. As the density is lowered, the chemical relaxation time increases and eventually becomes significant compared to the flow convection time. Air chemistry with numerical simulation is explained in details by (Edwards, 1992; and Gnoffo et al. 1989). (Rusanov, 1976) has reviewed fluid dynamics over reentry capsules. For a given thermodynamic state, the equilibrium composition of air is uniquely determined. Tables can be generated for the composition and properties of air in chemical equilibrium. These tables with a linear interpolation or store the coefficients are used with flow solver. The perfect gas equation of state is modified by an effective specific heats ratio that is evaluated by table interpolation or curve evaluation. In the case of non-equilibrium air chemistry, the concentration of all the species should be known at all points in the flow field determined the properties of air undergoing chemical reactions. This requires solving additional transport equations for each species present in the flow. A source terms presents in the transport equations, accounts for the net production of the species arising in the chemical reactions. This term depends on the forward and backward rates of the reactions being modelled, which will differ with the gas species and reaction set of the gas model. The non-equilibrium gas model requires solving an additional set of partial differential equations, the number of equations being dependent on the number of species in the model. In the frozen flow approximation (Hansen, 1959; Bailey 1967; & Srinivasan & Weilmuenster, 1987) individual species do not react with one another, but their effect is felt on the fluid dynamics because of variable thermodynamic properties and non-uniform concentration. Thus the species continuity equations are required. The computation time is saved due to not evaluating the production and destruction associated with chemical reactions. The flow is considered as a mixture of thermally perfect gases, with the specific heats and enthalpy of each species of the mixture determined as a function of temperature from a molecular data

A catalyst reduces the necessary activation energy of a reaction, hence more collisions level to reactions. The fully catalyst is equivalent to equilibrium of gases (thermal load). Poor

employing local time step, multi-grid, and preconditioning scheme.

**2.2 Air chemistry equations** 

base.

catalyst reduces the release of reaction.

used with the Van Leer flux limiter to provide higher-order spatial accuracy.

approximation neglects viscous derivatives in the directions tangent to the body surface, simplifying the equations somewhat alleviating grid resolution requirements. Numerical solution (Mehta et al., 1991) of the Navier-Stokes equations is obtained using Galerkin spatial discretiztion with multistage time integration. Table 2 depicts the various numerical schemes available to simulate the reentry flow field. Figure 3 shows the interlinking of computational fluid dynamics with the interaction of different discipline of sciences and engineering.

## **2.1 Algorithms**

In the flux-splitting approach, the flux vector is separated into the sum of two vectors, one associated with the positive eigenvalues of the flux Jacobian, and the other with the negative values. Then appropriate one-sides differences can be used on the split terms. The main requirement being that the eigenvalues of the split flux Jacobian be positive or negative definite. The split fluxes are discretized with a one-sided difference scheme. This is analogous to adding dissipation to a central difference scheme where the dissipation increases near flow discontinuities such as shock waves, yielding improved stability.

Fig. 3. Interaction of different area with CFD.

Riemann solvers are a class of schemes that propagate flow information according to solutions of Riemann problems. These schemes have the attributes of monotonicity, enabling sharp resolution of discontinuities, and robustness for capturing strong shocks. The one-dimensional inviscid problem can be solved exactly, but needs an expansive iterative approach, so most schemes are linearized, solve an approximate Riemann problem. The implicit numerical is allowed to have much higher time-step and having quadratic convergence. However the implicit algorithm needs computation of the Jacobian matrices of numerical fluxes, and having iterative solution of linear system. The implicit code is more complicated as compared to explicit code. The numerical solution can be accelerated employing local time step, multi-grid, and preconditioning scheme.

CFD-FASTRAN code (Liever et al., 2003) is based on compressible finite volume solver for thermo-chemical non-equilibrium flow analysis. Roe's flux difference upwind scheme is used with the Van Leer flux limiter to provide higher-order spatial accuracy.

#### **2.2 Air chemistry equations**

352 Computational Simulations and Applications

approximation neglects viscous derivatives in the directions tangent to the body surface, simplifying the equations somewhat alleviating grid resolution requirements. Numerical solution (Mehta et al., 1991) of the Navier-Stokes equations is obtained using Galerkin spatial discretiztion with multistage time integration. Table 2 depicts the various numerical schemes available to simulate the reentry flow field. Figure 3 shows the interlinking of computational fluid dynamics with the interaction of different discipline of sciences and

In the flux-splitting approach, the flux vector is separated into the sum of two vectors, one associated with the positive eigenvalues of the flux Jacobian, and the other with the negative values. Then appropriate one-sides differences can be used on the split terms. The main requirement being that the eigenvalues of the split flux Jacobian be positive or negative definite. The split fluxes are discretized with a one-sided difference scheme. This is analogous to adding dissipation to a central difference scheme where the dissipation

Riemann solvers are a class of schemes that propagate flow information according to solutions of Riemann problems. These schemes have the attributes of monotonicity, enabling sharp resolution of discontinuities, and robustness for capturing strong shocks. The one-dimensional inviscid problem can be solved exactly, but needs an expansive iterative approach, so most schemes are linearized, solve an approximate Riemann problem. The implicit numerical is allowed to have much higher time-step and having quadratic convergence. However the implicit algorithm needs computation of the Jacobian matrices of numerical fluxes, and having iterative solution of linear system. The implicit code is more

increases near flow discontinuities such as shock waves, yielding improved stability.

engineering.

**2.1 Algorithms** 

Fig. 3. Interaction of different area with CFD.

The chemical behaviour of air depends on the trajectory points. Below about 2000 K, deviations from perfect gas behaviour are negligible. In the atmosphere, this temperature is first reached behind normal shocks at a Mach number of about six. Most of the test facilities are operated at low freestream temperatures and so real gas effects are difficult to obtain, even at high Mach numbers. The perfect gas equation of state is thus sufficient for many hypersonic speeds when the peak temperature is below the above limit. When the temperature in the flow is high enough to cause air to dissociate, the density of the fluid determines the character of the fluid/chemistry interaction. At relative high densities, the intermolecular collision rate is very high, and reactions proceed to equilibrium rapidly. The flow can be considered to be chemical equilibrium at all points in the flow field at all times as in the limit of infinite reaction rates. As the density is lowered, the chemical relaxation time increases and eventually becomes significant compared to the flow convection time. Air chemistry with numerical simulation is explained in details by (Edwards, 1992; and Gnoffo et al. 1989). (Rusanov, 1976) has reviewed fluid dynamics over reentry capsules. For a given thermodynamic state, the equilibrium composition of air is uniquely determined. Tables can be generated for the composition and properties of air in chemical equilibrium.

These tables with a linear interpolation or store the coefficients are used with flow solver. The perfect gas equation of state is modified by an effective specific heats ratio that is evaluated by table interpolation or curve evaluation. In the case of non-equilibrium air chemistry, the concentration of all the species should be known at all points in the flow field determined the properties of air undergoing chemical reactions. This requires solving additional transport equations for each species present in the flow. A source terms presents in the transport equations, accounts for the net production of the species arising in the chemical reactions. This term depends on the forward and backward rates of the reactions being modelled, which will differ with the gas species and reaction set of the gas model. The non-equilibrium gas model requires solving an additional set of partial differential equations, the number of equations being dependent on the number of species in the model. In the frozen flow approximation (Hansen, 1959; Bailey 1967; & Srinivasan & Weilmuenster, 1987) individual species do not react with one another, but their effect is felt on the fluid dynamics because of variable thermodynamic properties and non-uniform concentration. Thus the species continuity equations are required. The computation time is saved due to not evaluating the production and destruction associated with chemical reactions. The flow is considered as a mixture of thermally perfect gases, with the specific heats and enthalpy of each species of the mixture determined as a function of temperature from a molecular data base.

A catalyst reduces the necessary activation energy of a reaction, hence more collisions level to reactions. The fully catalyst is equivalent to equilibrium of gases (thermal load). Poor catalyst reduces the release of reaction.

Computations of Flowfield over Reentry Modules at High Speed 355

anticlockwise sense. The computational domain is divided into a finite number of nonoverlapping quadrilateral grids. The conservation variables within the computational cell are represented by their average values at the cell centre. When the integral governing equation is applied separately to each cell in the computational domain, we obtain a set of

The convective fluxes are calculated at the centre of the grid, resulting in cell-centre flux balances. The contour integration of the inviscid flux vector is approximated at the side of

The viscous terms are defined at the centre of the cell. The above procedure is repeated to approximate the contour integration of the viscous flux vector. The inclusion of diffusive transport terms requires a choice of the locations at which the derivatives of the primitive variables should be integrated. The grid used for gradient evaluation forms a second grid that is shifted a half cell in the axial and radial directions in relation to the original computational grid. The viscous flux vectors are approximated in such a manner to preserve cell conservations and maintain algorithm simplicity. The derivatives of primitive variables at the vertices of the cell are evaluated by discrete application of the divergence theorem (Peyret & Vivind, 1993) to the auxiliary cell of the secondary grid. The net convective flux is

In the cell-centered spatial discretization schemes, such as the one described above which is non-dissipative, therefore, artificial dissipation terms are added. The approach of (Jameson et al., 1981) is adopted to construct the dissipative terms consisting of blend of second and fourth differences of the vector conserved variables. Fourth differences are added everywhere in the flow domain where the solution is smooth, but are 'switched off' in the region of shock waves. A term involving second differences is then 'switched on' to damp oscillations near shock waves. This switching is achieved by means of a shock sensor based on the local second differences of pressure. Since the computational domain is structured. The scaling quantity confirms the inclusion of the cell volume in the dependent variable. The blend of second and fourth differences provides third-order back ground dissipation in

The spatial discretization described above reduces the governing flow equations to semidiscrete ordinary differential equations. The integration is performed employing an efficient multistage scheme (Jameson et al., 1981). The following three-stage, timestepping method is adopted. A conservative choice of the Courant-Friedritchs-Lewy number (1.4) was made to achieve a stable numerical solution. The numerical algorithm is second-order accurate in space and time discretization. A global time step was used rather than the grid-varying time step to simulate time-accurate solution and is calculated as

the computational grid. The summation is carried out over the four edges of the grid.

coupled ordinary differential equation.

computed by summing the inviscid and viscous terms.

smooth region of the flow and first-order dissipation in shock waves.

**3.1.1 Inviscid fluxes** 

**3.1.2 Viscous fluxes** 

**3.1.3 Artificial dissipation** 

**3.1.4 Time marching scheme** 

mentioned (Mehta, 2001).


Table 2. Numerical method for Fluid Dynamics Equations.
