SUMMARYThree preconditioners proposed by Eriksson, Choi and Merkel, and Turkel are implemented in a 2D upwind Euler flow solver on unstructured meshes. The mathematical formulations of these preconditioning schemes for different sets of primitive variables are drawn, and their eigenvalues and eigenvectors are compared with each other. For this purpose, these preconditioning schemes are expressed in a unified formulation. A cellcentered finite volume Roe's method is used for the discretization of the preconditioned Euler equations. The accuracy and performance of these preconditioning schemes are examined by computing steady low Mach number flows over a NACA0012 airfoil and a two-element NACA4412-4415 airfoil for different conditions. The study shows that these preconditioning schemes greatly enhance the accuracy and convergence rate of the solution of low Mach number flows. The study indicates that the preconditioning methods implemented provide nearly the same results in accuracy; however, they give different performances in convergence rate. It is demonstrated that although the convergence rate of steady solutions is almost independent of the choice of primitive variables and the structure of eigenvectors and their orthogonality, the condition number of the system of equations plays an important role, and it determines the convergence characteristics of solutions.
A new optimization techniques based on the adjoint lattice Boltzmann method is derived for airfoil inverse design in laminar compressible flows. In this study, the developed adjoint lattice Boltzmann scheme based on the circular function (CF) is extended for airfoil inverse design problems in laminar incompressible and compressible flows. New mathematical derivation based on compressible lattice Boltzmann equations (LBE) is developed which can find target shape of an airfoil with available desired pressure distribution. The adjoint lattice Boltzmann method is extended for both the incompressible and compressible flows by selecting the circular function idea for calculating the equilibrium distribution functions. So, the adjoint equation is also expanded based on CF idea for calculation of objective function gradient vector. The steepest decent technique is utilized as gradient optimizer. Also, a novel solution is presented to remove singularity problem of the adjoint boundary condition. In order to validate the developed optimization algorithm, results are presented for both incompressible and compressible inverse problem in steady and unsteady flow and accurate results are obtained.
Three-dimensional unsteady flow field around a finite circular cylinder standing in a flat-plate boundary layer is studied. For this purpose, two different numerical turbulence approaches as wall adapted local eddyviscosity LES (LES-WALE) and the zonal hybrid RANS-LES approach of Detached-Eddy Simulation (Zonal-DES) are used. Analysis is carried out for a finite circular cylinder with diameter of D = 3 mm and length-to-diameter ratio of L/D=6 which leads to the Reynolds number 2×10 4. Numerical simulation has been performed based on the LES-WALE and Zonal-DES turbulence models using coarse and fine grids. Ability and accuracy of two models in capturing the complex physics of present phenomenon are investigated by comparing their results with each other and validated experimental results. Also, effect of several important parameters such as time-averaged pressure coefficient, velocity, vortex shedding frequency and performance of the LES-WALE and Zonal-DES turbulence models are studied.
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