| The 5th order weighted essentially non-oscillatory scheme((WENO))with free-stream preserving was developed and it was applied in computation of combustion flow-field induced by premixed mixture of methane and air in this thesis.The related works were summarized as follows:(1)development of 5th order WENO scheme which preserves free-stream properties and computation of premixed combustion flow-field induced by cylinders with different diameters.The geometry was transformed for body-fitted grid generation.The chemical reactions related to combustion were described by three elementary reaction models.Numerical flux was obtained by the WENO interpolation of solutions which was basically different from the standard WENO scheme.Then,numerical tests were carried to calibrate the accuracy and order of such a WENO scheme.Eulerian equation system with multi-species was solved by this developed 5th-order WENO scheme in the body fitted grids.In contrast to the standard WENO schemes in non-smooth grids,the developed WENO scheme demonstrated good priorities in preserving free freestream properties and avoiding discontinuous grid derivatives.Contours and snapshots of pressure,temperature and species mass fractions as well as their distribution along a symmetry passing through stand-off points to show the ignition delay were obtained in the premixed combustion flow-field.With increasing the cylindrical diameter,the standoff distances increase and induction length decrease either for the ignition delay.The bow shock and flame front are both pushed upstream but they are still remained in sequence.In contrast to reduced reaction models of 16species 41 steps and 15species 19steps,the model 53spesies 325steps provides good accuracy and ignition delay covers a wide range of pressure and temperature.Meanwhile,grid independence was also checked for the obtained results.The results demonstrate that a shock front and a flame front are both captured accurately by the coarse grids.The distance between the flame front and shock wave corresponds to ignition delay for the specified chemistry model.The results show good agreement with those from the reference.The distance between the flame front and shock wave decreases with increasing the cylinder radius.In contrast to second order TVD scheme,almost one fourth grids were used by the WENO scheme for almost the same numerical results.(2)A high order finite difference WENO method with free-stream preserving property is developed for the coupling of ideal MHD equation and CT equation in curvilinear coordinate system CT equation;which belongs to class of H-J equation,is introduced to control the divergence error.It can be proved that the magnetic potential is a linear function of spatial variable(x,y)in Cartesian coordinate system for free-stream.Therefore,in order to preserve the free-stream property,it is necessary to maintain the numerical solution of the magnetic potential linear to(x,y)when solving the CT equation.However,in general curvilinear coordinate,it is impossible to preserve this property when using the standard finite difference WENO method to solve CT equation.In this paper,a new finite-difference WENO method for solving H-J equations in 'curvilinear coordinate(?,?)is developed based on the idea of finite-difference method on unstructured mesh.Theoretical derivation and numerical results show that,this method can preserve the numerical solution linear to(x,y).Further more,combining with the free-stream preserving method for conservation law,a high order finite difference WENO method that has free-stream preserving property and divergence free property is obtained.Numerical results indicate that the method can preserve free-stream property of the ideal MHD equations,and reduce errors effectively comparing with the standard finite difference WENO method in static and moving curvilinear grids. |
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