common to have concentration dependencies for gas species other than those **in**volved **in** the reaction (Javed and Chakraborty, 2006). Numerical Schemes and Computational Grid Compressible Navier-Stokes equations, together **with** the energy and phase equations, are solved us**in**g OpenFOAM. OpenFOAM is a C++ code library of classes for writ**in**g CFD codes (Jasak, 2007). These equations are discretized us**in**g the f**in**ite volume method, where the doma**in** is divided **in**to cells. Integration of the dependent variables over each cell, together **with** the application of Gauss theorem, generates a set of discretized equations **with** the divergence terms represented as fluxes across the cell faces, are evaluated us**in**g appropriate **in**terpolation schemes; here a l**in**ear **in**terpolation is utilized. Time **in**tegration is carried out us**in**g SUNDIALS libraries (H**in**dmarsh v.d., 2005) developed at Lawrence Livermore National Laboratory. Robustness of the ODE solver allows tak**in**g large time steps. At least an order of magnitude speed up is observed when compared to built-**in** ODE solver libraries of OpenFOAM. This cuts computation time considerably. Follow**in**g the procedure of Rhie and Chow (1983), discretisation of the term is left; a Poisson equation is constructed, and the equation set solved sequentially us**in**g the result**in**g PISO (pressure implicit splitt**in**g operator) algorithm (Issa, 1986). Solution is performed implicitly by matrix **in**version us**in**g “**in**complete Cholesky conjugate gradient” methods. Figure 2. Computational Grid **in** the Vic**in**ity of the Cavity Region (Not to Scale). While simulat**in**g wave dom**in**ated high Mach number flows such as the one **in**vestigated here, it is necessary to have appropriate boundary conditions that do not reflect waves. Walls naturally reflect waves, while **in**lets and outlets should not be reflective. Therefore at the outlet a wave transmissive boundary condition is used **in** accordance **with** Po**in**sot and Lele (1992) and the farfield pressure is specified to be atmospheric. At the **in**lets temperature and total pressure are specified and velocities are computed from these specifications. This approach assures proper treatment of boundary conditions also assur**in**g conservation of mass **with****in** the computational doma**in** as well. For the turbulent k**in**etic energy boundary condition at the air **in**let turbulence **in**tensity is assumed. Inverse time scale of eddy dissipation is also chosen accord**in**gly **with** the characteristic length scale of turbulence at the **in**let. In order to ensure numerical stability, the solution of the 62 non-react**in**g flow is fed as the **in**itial condition to the react**in**g flow solver. As for the ma**in** fuel **in**jection location the upstream total pressure is specified high enough such that flow is choked at the **in**jection site as desired. Moreover, for the **cavity** backwall **in**jection site, the **in**jection velocity is specified as the boundary condition. In order to start the reaction, enthalpy of certa**in** ignition sites (two or three locations **in** the vic**in**ity of the **flame** hold**in**g **cavity**) are multiplied by a factor greater than one (typically 1.5), if the temperature at that ignition site is below 2000 K, for a certa**in** duration until the **flame** is self-susta**in****in**g. This approach emulates a spark at that ignition cite. Rais**in**g the temperature directly often results **in** errors **with** the JANAF thermo-dynamics database as the temperature can **in**stantaneously go above the upper limit for the curve-fitt**in**g polynomial, therefore enthalpy is used as a proxy. A total of 0.15 million grid po**in**ts were used for simulations **with** f**in**er mesh **in** the **cavity** and upstream fuel **in**jection locations. Computational grid is twodimensional and is sufficient to capture the essential features of the flowfield such as the flow re-circulation **in** the **cavity**, shock reflections **with****in** the **combustor** and the bow shock produced due to ma**in** fuel **in**jection. Grid structure at the vic**in**ity of the **cavity** is shown **in** Figure 2. Note the ref**in**ed mesh structure **in**side the **cavity**. NUMERICAL RESULTS Air **with** a mass flow rate of enters the isolator section at a Mach number of 1.4. Stagnation temperature of the air enter**in**g the **combustor** is . Flow condition **in**vestigated is tabulated **in** Table 3. Especially at low to moderate flight Mach numbers, such as the flow condition **in**vestigated here, the temperature of the **in**com**in**g air is not sufficient to ignite the fuel all by itself. With these conditions a preferential location **with** a favorable equivalence ratio, temperature, pressure and velocity must exist such that the fuel can ignite. A rich radical pool is conta**in**ed **in** this very region that is crucial **in** **flame** hold**in**g. Hydrogen (H2) is selected as the fuel due to its reduced characteristic **combustion** times. Fuel **in**jection is split **in**to two locations **in** order to aid **in** **flame** hold**in**g **in**side the **cavity**. Ma**in** fuel at a flow rate of is **in**jected sonically from the lower wall 44.5 mm upstream of the **cavity** lead**in**g edge. 33% of the total fuel flow is **in**jected at the **cavity** back wall to atta**in** a more desirable equivalance ratio **in**side the **cavity**. Fuel is **in**jected at a temperature of . Note that the flow at the upstream **in**jection port is choked. This is typically the case **in** many **ramjet**/sc**ramjet** **combustor**s as it easily allows to ma**in**ta**in** a constant fuel flow rate.

Table 3. Flow Parameters for Air and Hydrogen. Air Hydrogen Hydrogen (Upstream) (Cavity) 3.4 0.002 0.001 702 300 300 1.4 1.0 0.5 0.8 3.9 3.9 The react**in**g flow field **in** the **ramjet** **combustor** has been simulated by solv**in**g two-dimensional Navier- 0.975 0.87 1.08 1.185 1.185 1.395 1.08 1.605 1.5 1.29 1.605 1.5 1.395 1.395 63 Stokes equations **with** shear stress transport (SST) turbulence model us**in**g open source CFD software OpenFOAM (Jasak, 2007). Other open source numerical libraries, such as SUNDIALS (H**in**dmarsh et al., 2005) for the **in**tegration of ord**in**ary differential equations (ODEs), are also used for the solution. Detailed chemistry **with** 9 species and 25 reaction steps is taken **in**to account for the **combustion** of hydrogen fuel. 1.29 1.185 1.08 1.29 -50 0 50 100 X (mm) Figure 3. Pressure Distribution near the Cavity Region (Pressures **in** MPa) Figure 3 demonstrates the qualitative features of the flow field through pressure distribution. Appearance of a bow shock due to upstream fuel **in**jection is also seen. Due to this bow shock upstream boundary layer separates from the lower wall and as a consequence of this separation a region **with****in** which the boundary layer fluid (air) mixes **with** the jet fluid (fuel) subsonically just near the **in**jection location is formed. It is also known that an oblique shock wave jet **in**teraction enhances mix**in**g between supersonic airflow and gaseous fuel through the vorticity generated by the barocl**in**ic torque. This **in** turn has immediate ramification on the jet spread**in**g rate and due to enhanced mix**in**g **combustion** efficiency is also enhanced as well. Formation of another shock wave at the **cavity** trail**in**g edge, where the flow beg**in**s expand**in**g, is also clearly visible from Figure 3. This shock wave at the trail**in**g edge is due to the flow reattachment and is therefore often referred to as the reattachment shock. In this particular case the reattachment shock is a rather strong one. In general while design**in**g **ramjet**/sc**ramjet** **combustor**s this re-attachment shock is sought to be made as weak as possible **in** order to obta**in** maximum thrust from the propulsion device. Shock reflections are also clearly seen from this figure. These are due to the supersonic nature of the flow. Surface pressure distribution at the top surface is plotted **in** Figure 4. Pressure distribution is demonstrated for both react**in**g and non-react**in**g cases **in** order to allow comparison between these cases. Solid l**in**e represents the react**in**g flow case whereas the dashed one represent**in**g the non-react**in**g case. Pressure rise due to **combustion** at the **cavity** lead**in**g edge (where the **flame** anchors) is evident from this plot. Pressure drop gradually towards the **combustor** exit after this 0.975 0.975 0.87 sudden rise. Flow enthalpy also rises due to heat added through **combustion**. This enthalpy obta**in**ed from **combustion** **in** a propulsion device is used to accelerate the flow **in** order to provide thrust, which is the ultimate goal. Pressure (MPa) 2.0 1.8 1.5 1.3 1.0 0.8 0.5 0.3 0.87 Non-react**in**g React**in**g 0.0 -400 -300 -200 -100 0 100 200 300 400 X (mm) Figure 4. Pressure Distribution Along the Top Surface. The flow situation under **in**vestigation **in** this **ramjet** **combustor** is a wave dom**in**ated flow. Not only it is a high Mach number flow, but there is also chemical reaction **in**volved. Shock and boundary layer **in**teractions as mentioned earlier are also quite important for a flow condition like the one that is under **in**vestigation. Therefore **in**vestigat**in**g Figure 3 and Figure 4 together it can be said that this numerical simulation tool has an acceptable level performance **in** captur**in**g the important aspects of such a flow condition. Obviously the specification of correct boundary conditions for pressure is necessary. These