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A modified equilibrium distribution function for the Bhatnagar--Gross--Krook-type modeled Boltzmann equation has recently been proposed. The function was deduced using acoustics scaling to normalize the equation and allowed a correct recovery of similarly normalized Euler equations. It is a combination of a Maxwellian distribution plus three other terms that are moments of particle velocity. The lattice counterpart of the modified equilibrium distribution function also led to an exact recovery of the Euler equations; therefore, there is no Mach number limitation in the entire approach. This lattice counterpart was able to replicate aeroacoustics problems involving vorticity--acousfic and entropy--acoustic interactions correctly, and the simulations were carried out using a finite difference lattice Boltzmann method employing only a two-dimensional, nine-velocity lattice. Thus formulated, the numerical scheme has no arbitrary constants and all calculations were carried out using one single relaxation time and a set of constants derived from the analysis. This paper investigates the validity and extent of the formulation to capture shocks and resolve contact discontinuity and expansion waves in one- and two-dimensional Riemann problems. The simulations are carried out using the same two-dimensional, nine-velocity lattice, and identical set of constants and relaxation time; they are compared with theoretical results and those obtained by solving the Euler equations directly using Harten's first-order numerical scheme. Good agreement is obtained for all test cases. However, the modified equilibrium distribution function is not suitable for shock structure simulation; for that, an exact recovery of the Navier--Stokes equations is required.

Aerodynamic sensitivity analysis is performed for the Navier-Stokes equations, coupled with two-equation turbulence models using a discrete adjoint method and a direct differentiation method, respectively. Like the mean flow equations, the turbulence model equations are also hand differentiated to calculate accurately the sensitivity derivatives of flow quantities with respect to design variables in turbulent viscous flows. Both the direct differentiation code and the adjoint variable code adopt the same time integration scheme with the flow solver to solve the differentiated equations efficiently. The sensitivity codes are then compared with the flow solver in terms of solution accuracy, computing time, and computer memory requirements. The sensitivity derivatives obtained from the sensitivity codes with different turbulence models are compared with each other. Using two-equation turbulence models, it is observed that a usual assumption of constant turbulent eddy viscosity in adjoint methods may lead to inaccurate results in a case of turbulent flows involving strong shocks. The capability of the present sensitivity codes to treat complex geometry is successfully demonstrated by analyzing the flows over multielement airfoils on chimera overlaid grid systems.

For a low-Reynolds-number turbulence model, the boundary layer is solved accurately next to the solid boundaries. Generally, it is required that there is a sufficient number of computational points in a viscous sublayer. In this work the effect of the height of the first cell next to the solid surfaces is studied. Modifications to the momentum equation, to the energy equation, and also to the k-[element of] model that reduces the number of grid points needed in the viscous sublayer for an accurate result are suggested. It is shown that the accuracy of the simulation is enhanced when the proposed modifications are applied. For a coarse grid the stability is increased as well.

A numerical study has been carried out to investigate modeling issues on trailing-edge vortex shedding. The vortex shedding from a circular cylinder and a VKI turbine blade is calculated using a two-dimensional unsteady multi-block Navier-Stokes solver. The unsteady stresses are calculated from the unsteady solutions. The distributions of the unsteady stresses are analyzed and compared for the cylinder case and the cascade case, respectively. The time-averaged equations are then solved, and the effectiveness of the unsteady stresses in suppressing trailing-edge vortex shedding is checked. Finally, the time-independent solution produced by solving the time-averaged equations is compared with the time-averaged solution obtained by integrating the unsteady solutions. The numerical results have demonstrated that a time-independent vortex shedding solution can be achieved by solving the Navier-Stokes equations with the unsteady stresses and the time-averaged effects of the vortex shedding can be included.