Fluid flow in rocket nozzles

An analysis is presented to simulate the turbulent compressible flow in converging diverging nozzles using a time-marching finite difference scheme of MacCormack. The Navier-Stokes equations governing the physical domain are transformed to a rectangular computational domain and the resulting equations are cast into finite difference form. The effect of turbulence is incorporated by using the BaldwinLomax model. An efficient time saving mechanism is adopted where the continuity equation shows a steep oscillatory characteristics. A computer software is developed and the results thus obtained show an excellent agreement with the experimental results generated inhouse as well as those available in the literature Nomenclature AArea of nozzle cross section A, Nozzle throat area A'Constant in Eq (39) a Sonic velocity k Thermal conductivity of gas M Mach number P Static pressure PwWall pressure P0Stagnation pressure R Gas constant T Static temperature T0Stagnation temperature t Time u Axial velocity v Radial velocity x Axial coordinate in physical domain y_Radial coordinate in physical domain CC Constant due to turbulence y Ratio of specific heats At Time step size A£ Grid size in the axial direction Ar| Grid size in the radial direction e Conditon number C Axial coordinate in computational domain T| Radial cooridanate in computational domain * Scientist ** Scientist, Associate Fellow, AIAA Copyright ©1998 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. 0 Local wall angle 0' Local flow angle p Density of gas H First coefficient of viscosity A. Second coefficient of viscosity Subscripts i j Mesh point indices T Turbulent M Molecular Superscripts n Denotes time step Introduction Flow simulation in a rocket nozzle is essential to predict the performance of a rocket motor accurately. Traditional method of prediction of rocket motor performance couples the boundary layer solution with inviscid flow techniques. A number of studies such as Serra, Cline', Chakravarthy and Jones and Shukla for the development of inviscid codes in rocket nozzles have been reported hi the literature. Although the method has predicted satisfactorily the design parameters for subsonic condition, it has limited application hi the presence of strong inviscid/viscous boundary layer interaction. In the presence of strong interaction the solution of Navier-Stokes equations is required for the design of high performance rocket nozzles. Presently, such solutions are within the reach of the designer due to the advent of high speed computers. Many computer softwares are reported for viscous flow simulation hi rocket nozzles and other fluid flow problems. Cline 6J developed computer softwares christened as VNAP and VNAP2 for solving the two-dimensional axisymmetric tune dependent compressible Navier-Stokes equations by a second order accurate finite difference method. Swanson and Hasen solve the asymmetric and axisymmetric nozzle flows respectively. Mani, Tiwari and Drummond I0 included the chemistry for various species of discharged gases hi the flow simulation. Barber and Cox Jr. " presented a state of the art of CFD codes for hypersonic vehicle propulsion. However a detailed prediction of i American Institute of Aeronautics and Astronautics rocket motor performance involves the invocation of nozzle flow routine many times. These being the days of supercomptuers, the development of such a software appears to be well within the reach of a software scientist. With the above objective in mind an attempt is made to reduce the computation time of a nozzle flow software to a considerable extent. In this paper, an analysis is presented to simulate the steady viscous turbulent compressible flow in converging-diverging axisymmetric nozzles using a time marching scheme. The Navier-Stokes equations governing an axisymmetric flow are written for the physical domain and are transformed to a rectangular computational domain having a boundary fitted coordinate system12. Turbulence is handled with the Bladwin Lomax turbulence model . These equations are then cast into finite difference form in a variable mesh network. Interior mesh points are computed using the MacCormack's explicit predictorcorrector finite difference scheme 15. A backward difference analog is used for the predictor step and a forward difference analog for the corrector step. The inlet boundary as well as the subsonic region of the exit boundary are solved by two-step two independent variable characteristic scheme. The supersonic region of the exit boundary and the axis of symmetry are handled by extrapolation. While the free-slip wall is solved by the characteristic scheme, the continuity equation and the equation of state are used along with the wall boundary conditions to solve the no-slip wall. A computer software was developed on the above analysis. A close observation of the convergence pattern of the solution indicates that the continuity equaiton is too oscillatory at the boudary layer region. An isentropic flow relation is used locally to contain this oscillation. Roughly 30% reduction in computation time is achieved by the above modification. The software is applicable to axisymmetric as well as 2D nozles. Sample computations are carried out and the results thus obtained show an excellent agreement with the test results generated inhouse as well as those available in the literature.