This paper presents a mathematical model that was developed to study instabilities (primarily thermoacoustic oscillations) experienced inside a channel (with a rectangular cross section) heated symmetrically (from its top and bottom.) The heated channel is configured to simulate a combustion chamber of a rocket hybrid rocket motor and is connected to a converging–diverging nozzle in the downstream and to a plenum with a flow straightener in the upstream side. The working fluid is supplied from a pressurized storage tank to the upstream plenum through a throttle valve. A multi-component approach is used to model this test apparatus. In this integrated component model, the unsteady flow through the throttle valve and the nozzle is assumed to be one-dimensional and isentropic where as the flow in the forward plenum and the heated channel is assumed to be a two-dimensional, unsteady, compressible, turbulent, and subsonic. The physics based mathematical model of the flow in the channel consists of conservation of mass, momentum (two-dimensional Navier-Stokes) and energy equations subject to appropriate boundary conditions as defined by the physical problem stated above. The working fluid is assumed to be compressible where the density of the fluid is related to the pressure and temperature of the fluid through a simple ideal gas relation. The governing equations are discretized using second order accurate central differencing for spatial derivatives and second order accurate (based on Taylor expansion) finite difference approximations for temporal derivatives. The resulting nonlinear equations are then linearized using Newton’s linearization method. The set of algebraic equations that result from this process are then put into a matrix form and solved using a Coupled Modified Strongly Implicit Procedure (CMSIP) for the unknowns (primitive variables, i.e., pressure, temperature, and the velocity field) of the problem. The turbulence model equations and the unsteady flow equation for the throttle valve are solved using a second order accurate explicit finite difference technique. Convergence and grid independence studies were done to determine the optimum mesh size and computational time increment. Furthermore, two benchmark cases (unsteady driven cavity and laminar channel flows) were simulated using the developed numerical model to verify the accuracy of the proposed solution procedure. Numerical experiments were then carried out to simulate the thermoacoustic oscillations inside rectangular channels with various aspect ratios ranging from 5 to 20 for various operating conditions (i.e., for Re numbers between 102 and 106) and to determine the flow regions where these oscillations are sustained. The numerical simulation results indicate that the mathematical model for the gas flow in the heated channel predicts the expected unsteady temperature and pressure distributions, and the velocity field, successfully. Furthermore, it is concluded that the proposed integrated component model is successful in generating the characteristics of the instabilities associated with thermal, hydrodynamic, and thermoacoustic oscillations in heated channels.
Specific features of mathematical modeling of flows with detonation waves on unstructured computational grids
