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A fully implicit unfactored algorithm for three-dimensional Euler equations is developed and tested on multi-block curvilinear meshes. The convective terms are discretized using an upwind TVD scheme. The large sparse linear system generated at each implicit time step is solved by GMRES* method combined with the block incomplete lower-upper preconditioner. In order to reduce the memory requirements and the matrix-vector operation counts, an approximate method is used to derive the Jacobian matrix, which only costs half of the computational efforts of the exact Jacobian calculation. The comparison between the numerical results and the experimental data shows good agreement, which demonstrates that the implicit algorithm presented is effective and efficient.

The interaction of shock wave and turbulence for transonic flow over a circular cylinder is investigated using detached-eddy simulation (DES). Several typical cases are calculated for free-stream Mach number M_∞ from 0.85 to 0.95, and the physical mechanisms relevant to the shock wave and turbulence interaction are discussed. Results show that there exist two flow states. One is unsteady flow state with moving shock waves interacting with turbulent flow for M_∞ < 0.9 approximately, and the other is quasi-steady flow with stationary shocks standing over the wake of the cylinder for M_∞ > 0.9, suppressing the vortex shedding from the cylinder. Moreover, local supersonic zones are identified in the wake of the cylinder and generated by two processes, i.e., reverse flow and shock wave distortion induced the supersonic zone. Turbulent shear layer instabilities are revealed and associated with moving shock wave and traveling pressure wave.

In this paper, we investigate the initial-boundary value problem of compressible Euler equations including friction and heating that model the transonic Fanno–Rayleigh flows through symmetric variable area nozzles. In particular, the case of contracting nozzles is considered. A new version of a generalized Glimm scheme (GGS) is presented for establishing the global existence of entropy solutions with bounded variation. Modified Riemann and boundary Riemann solutions are applied to design this GGS, which is constructed using the contraction matrices acting on the homogeneous Riemann (or boundary-Riemann) solutions. The extended Glimm–Goodman’s type of wave interaction estimates are investigated to determine the stability of the scheme and the positivity of gas velocity that results in the existence of the weak solution. The limit of approximation solutions serves as an entropy solution. Moreover, a quantitative relation between the shape of the nozzle, friction, and heat is proposed for the global existence result in the contracting nozzle. Numerical simulations of the contraction-expansion and expansion-contraction nozzles are presented to validate the scheme.

This paper supports quick and accurate prediction of the flutter onset speed of an F-16 Block 40/50 configured with external stores in the transonic flight regime. Current flutter prediction methods are reviewed and hypothesized mechanisms for limit cycle oscillation (LCO) are summarized. New efforts to correlate transonic small disturbance (TSD) theory methods with flight tests are outlined. Vibration analysis and structural optimization of an F-16 finite element model were used to match ground vibration testing results. Frequency tuning was found to be critical for accurate flutter speed predictions. Sensitivity to nonlinear aerodynamic effects and store modeling was examined.

We are concerned with the study of a bifurcation problem driven by a degenerate operator of Baouendi–Grushin type. Due to its degenerate structure, this differential operator has a mixed regime. Studying the combined effects generated by the absorption and the reaction terms, we establish the bifurcation behavior in two cases. First, if the absorption nonlinearity is dominating, then the problem admits solutions only for high perturbations of the reaction. In the case when the reaction dominates the absorption term, we prove that the problem admits nontrivial solutions for all the values of the parameter. The analysis developed in this paper is associated with patterns describing transonic flow restricted to subsonic regions.

2D and 3D transonic flows in a channel of variable cross-section are studied numerically using a solver based on the Reynolds-averaged Navier–Stokes equations. The flow velocity is supersonic at the inlet and outlet of the channel. Between the supersonic regions, there is a local subsonic region whose upstream boundary is a shock wave, whereas the downstream boundary is a sonic surface. The sonic surface gives rise to an instability of the shock wave position in the channel. Computations reveal a hysteresis in the shock position versus the inflow Mach number. A dependence of the hysteresis on the velocity profile given at the inlet is examined.

In the study of multidimensional systems of conservation laws people confront more difficulties than that for one-dimensional systems. The difficulties include characteristic boundary, free boundary associated with unknown nonlinear waves, various nonlinear wave structure, mixed type equations, strong singularities, etc. Most of them come from the complexity of characteristics. We will give a survey on the progress obtained in the study of this topics with the applications in various physical problems, and will also emphasize some crucial points for the further development of this theory in future.