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de Goes Maciel E.S.,ITA Aeronautical Technological Institute
WSEAS Transactions on Fluid Mechanics | Year: 2011

In this work, numerical simulations involving supersonic and hypersonic flows on an unstructured context are analyzed. The Van Leer and the Radespiel and Kroll schemes are implemented on a finite volume formulation, using unstructured spatial discretization. The algorithms are implemented in their first and second order spatial accuracies. The second order spatial accuracy is obtained by a linear reconstruction procedure based on the work of Barth and Jespersen. Several non-linear limiters are studied, as well two types of linear interpolation, based on the works of Frink, Parikh and Pirzadeh and of Jacon and Knight. Two types of viscous calculation to the laminar case are compared. They are programmed considering the works of Long, Khan and Sharp and of Jacon and Knight. To the turbulent simulations, the Wilcox and Rubesin model is employed. The ramp problem to the inviscid simulations and the re-entry capsule problem to the hypersonic viscous simulations are studied. The results have demonstrated that the Van Leer algorithm yields the best results in terms of the prediction of the shock angle of the oblique shock wave in the ramp problem and the best value of the stagnation pressure at the configuration nose in the re-entry capsule problem. The convective time step is the best choice to accelerate the convergence of the numerical schemes, as reported by Maciel. In terms of turbulent results, the Wilcox and Rubesin model yields good results, proving the good capacity of this turbulence model in simulate high hypersonic flows. This paper is the first part of this work and is related to the theory and inviscid solutions. The second paper of this work is concerned with the laminar and turbulent viscous results.


de Goes Maciel E.S.,ITA Aeronautical Technological Institute
WSEAS Transactions on Fluid Mechanics | Year: 2011

In this work, numerical simulations involving supersonic and hypersonic flows on an unstructured context are analyzed. The Van Leer and the Radespiel and Kroll schemes are implemented on a finite volume formulation, using unstructured spatial discretization. The algorithms are implemented in their first and second order spatial accuracies. The second order spatial accuracy is obtained by a linear reconstruction procedure based on the work of Barth and Jespersen. Several non-linear limiters are studied, as well two types of linear interpolation, based on the works of Frink, Parikh and Pirzadeh and of Jacon and Knight. Two types of viscous calculation to the laminar case are compared. They are programmed considering the works of Long, Khan and Sharp and of Jacon and Knight. To the turbulent simulations, the Wilcox and Rubesin model is employed. The ramp problem for the inviscid supersonic simulations and the re-entry capsule for the viscous hypersonic simulations are considered. The results have demonstrated that the Van Leer algorithm yields the best results in terms of the prediction of the wall pressure distribution and the shock angle in the inviscid simulations and the best value of the stagnation pressure at the configuration nose in the viscous simulations. Moreover, the Van Leer algorithm in the SS case and using the Wilcox and Rubesin turbulence model predicts the best value of the lift aerodynamic coefficient. Hence, the Wilcox and Rubesin model yielded good results, proving its good capacity to predict high hypersonic flows. This paper is the second part of this work and is concerned with the laminar and turbulent viscous results.


Maciel E.S.G.,ITA Aeronautical Technological Institute | Pimenta A.P.,ITA Aeronautical Technological Institute
WSEAS Transactions on Applied and Theoretical Mechanics | Year: 2013

This work, second part of this study, presents a numerical tool implemented to simulate inviscid and viscous flows employing the reactive gas formulation of thermochemical non-equilibrium. The Euler and Navier-Stokes equations, employing a finite volume formulation, on the context of structured and unstructured spatial discretizations, are solved. These variants allow an effective comparison between the two types of spatial discretization aiming verify their potentialities: solution quality, convergence speed, computational cost, etc. The aerospace problems of the hypersonic "hot gas" flows around a double ellipse and around a reentry capsule, in two-dimensions, are simulated. The reactive simulations will involve an air chemical model of seven species: N, O, N2, O2, NO, NO+ and e-. Eighteen chemical reactions, involving dissociation, recombination and ionization, will be studied. The Arrhenius formula will be employed to determine the reaction rates and the law of mass action will be used to determine the source terms of each gas species equation.


Maciel E.S.G.,ITA Aeronautical Technological Institute | Pimenta A.P.,ITA Aeronautical Technological Institute
WSEAS Transactions on Applied and Theoretical Mechanics | Year: 2013

This work, first part of this study, describes a numerical tool to perform thermochemical nonequilibrium simulations of reactive flow in two-dimensions. The Van Leer and Liou and Steffen Jr. schemes, in their first- and second-order versions, are implemented to accomplish the numerical simulations. The Euler and Navier-Stokes equations, on a finite volume context and employing structured and unstructured spatial discretizations, are applied to solve the "hot gas" hypersonic flows around a blunt body, around a double ellipse, and around an entry capsule, in two-dimensions. The second-order version of the Van Leer and Liou and Steffen Jr. schemes are obtained from a "MUSCL" extrapolation procedure in a context of structured spatial discretization. In the unstructured context, only first-order solutions are obtained. The convergence process is accelerated to the steady state condition through a spatially variable time step procedure, which has proved effective gains in terms of computational acceleration. The reactive simulations involve a Mars atmosphere chemical model of nine species: N, O, N2, O2, NO, CO2, C, CO, and CN, based on the work of Kay and Netterfield. Fifty-three chemical reactions, involving dissociation and recombination, are simulated by the proposed model. The Arrhenius formula is employed to determine the reaction rates and the law of mass action is used to determine the source terms of each gas species equation. The results have indicated the Van Leer TVD scheme as the most accurate one, both inviscid and viscous cases. In this paper is presented the blunt body results. In Part II is presented the results with the double ellipse and the entry capsule.


De Goes Maciel E.S.,ITA Aeronautical Technological Institute | Pimenta A.P.,ITA Aeronautical Technological Institute
WSEAS Transactions on Applied and Theoretical Mechanics | Year: 2012

This work presents a numerical tool implemented to simulate inviscid and viscous flows employing the reactive gas formulation of thermochemical non-equilibrium. The Euler and Navier-Stokes equations, employing a finite volume formulation, on the context of structured and unstructured spatial discretizations, are solved. These variants allow an effective comparison between the two types of spatial discretization aiming verify their potentialities: solution quality, convergence speed, computational cost, etc. The aerospace problem involving the hypersonic flow around a blunt body, in two-dimensions, is simulated. The reactive simulations will involve an air chemical model of seven species: N, O, N2, O2, NO, NO+ and e-. Eighteen chemical reactions, involving dissociation, recombination and ionization, will be simulated by the proposed model. This model was suggested by Blottner. The Arrhenius formula will be employed to determine the reaction rates and the law of mass action will be used to determine the source terms of each gas species equation.


De Goes Maciel E.S.,ITA Aeronautical Technological Institute
WSEAS Transactions on Applied and Theoretical Mechanics | Year: 2012

The present work compares the TVD schemes of Roe, of Van Leer, of Yee,Warming and Harten, of Harten, of Yee and Kutler and of Hughson and Beran applied to the solution of an aeronautical problem. Only the Van Leer scheme is a flux vector splitting one. The others are of flux difference splitting type. The Roe and Van Leer schemes reach second order accuracy and TVD properties by the use of a MUSCL approach, which employs five different types of nonlinear limiters, that assures TVD properties, being them: Van Leer limiter, Van Albada limiter, minmod limiter, Super Bee limiter and β-limiter. The other schemes are based on the Harten's ideas of the construction of a modified flux function to obtain second order accuracy and TVD characteristics. The implicit schemes employ an ADI ("Alternating Direction Implicit") approximate factorization to solve implicitly the Euler equations, whereas in the explicit case a time splitting method is used. Explicit and implicit results are compared trying to emphasize the advantages and disadvantages of each formulation. The Euler equations in conservative form, employing a finite volume formulation and a structured spatial discretization, are solved in two-dimensions. The steady state physical problem of the supersonic flow along a compression corner is studied. A spatially variable time step procedure is employed aiming to accelerate the convergence of the numerical schemes to the steady state condition. This technique has proved an excellent behavior in terms of convergence gains, as shown in Maciel. The results have demonstrated that the most accurate solutions are provided by the Roe TVD scheme in its Super Bee variant.


De Goes Maciel E.S.,ITA Aeronautical Technological Institute
WSEAS Transactions on Applied and Theoretical Mechanics | Year: 2012

In the present work, the Harten and Osher TVD/ENO and the Yee TVD symmetric schemes are implemented, on a finite volume context and using a structured spatial discretization, to solve the laminar/turbulent Navier-Stokes equations in the three-dimensional space. The Harten and Osher TVD/ENO schemes are flux difference splitting type, whereas the Yee TVD scheme is a symmetric one, which incorporates TVD properties due to the appropriated definition of a limited dissipation function. All three schemes are second order accurate in space. Turbulence is taken into account considering two algebraic models, namely: the Cebeci and Smith and the Baldwin and Lomax ones. A spatially variable time step procedure is also implemented aiming to accelerate the convergence of the algorithms to the steady state solution. The gains in convergence with this procedure were demonstrated in Maciel. The schemes are applied to the solution of the physical problem of the low supersonic flow along a ramp. The results have demonstrated that the most accurate results are obtained with the Harten and Osher ENO scheme. This paper is the third part of this work, TURBULENT RESULTS, considering the description of the turbulence models and the solutions obtained with them and compared with the laminar results.


de Goes Maciel E.S.,ITA Aeronautical Technological Institute
WSEAS Transactions on Applied and Theoretical Mechanics | Year: 2012

In the present work, the Steger and Warming, the Van Leer, the Liou and Steffen Jr. and the Radespiel and Kroll schemes are implemented, on a finite volume context and using a structured spatial discretization, to solve the Euler and the Navier-Stokes equations in three-dimensions. A MUSCL ("Monotone Upstream-centered Schemes for Conservation Laws") approach is implemented in these schemes aiming to obtain second order spatial accuracy and TVD ("Total Variation Diminishing") high resolution properties. An implicit formulation is employed to the Euler equations, whereas the Navier-Stokes equations use an explicit formulation. The algebraic turbulence models of Cebeci and Smith and of Baldwin and Lomax are implemented. The problems of the supersonic flow along a compression corner (inviscid case), and of the supersonic flow along a ramp (viscous case) are solved. The results have demonstrated that the most severe and most accurate results are obtained with the Liou and Steffen Jr. TVD scheme. The first paper of this work treats the inviscid and laminar viscous results. In this paper, the second of this series, the turbulent results are presented.


De Goes Maciel E.S.,ITA Aeronautical Technological Institute
WSEAS Transactions on Fluid Mechanics | Year: 2013

In this work, second part of this study, the high resolution numerical schemes of Yee and Harten, of Yang second order, of Yang third order, and of Yang and Hsu are applied to the solution of the Euler and Navier-Stokes equations in three-dimensions. All schemes are flux difference splitting algorithms. The Yee and Harten is a TVD ("Total Variation Diminishing") second order accurate in space and first order accurate in time algorithm. The Yang second order is a TVD/ENO ("Essentially Nonoscillatory") second order accurate in space and first order accurate in time algorithm. The Yang third order is a TVD/ENO third order accurate in space and first order accurate in time algorithm. Finally, the Yang and Hsu is a UNO (Uniformly Nonoscillatory) third order accurate in space and first order accurate in time algorithm. The Euler and Navier-Stokes equations, written in a conservative and integral form, are solved, according to a finite volume and structured formulations. A spatially variable time step procedure is employed aiming to accelerate the convergence of the numerical schemes to the steady state condition. It has proved excellent gains in terms of convergence acceleration as reported by Maciel. The physical problems of the supersonic flows along a compression corner and along a ramp are solved, in the inviscid case. For the viscous case, the supersonic flow along a ramp is again solved. In the inviscid case, an implicit formulation is employed to marching in time, whereas in the viscous case, a time splitting or Strang approaches are used. The results have demonstrated that the Yang and Hsu UNO third order accurate algorithm has presented the best solutions in the problems studied herein. Moreover, it is also the best as comparing with the numerical schemes of Part I of this study.


De Goes Maciel E.S.,ITA Aeronautical Technological Institute
WSEAS Transactions on Fluid Mechanics | Year: 2012

In this work, the Frink, Parikh and Pirzadeh and the Liou and Steffen Jr. schemes are implemented following a MUSCL approach, aiming to guarantee second order accuracy and to achieve TVD properties, and employing an implicit formulation to solve the Euler equations in the two-dimensional space. These schemes are implemented according to a finite volume formulation and using a structured spatial discretization. The former scheme is a flux difference splitting one, whereas the latter is a flux vector splitting scheme. The MUSCL approach employs five different types of nonlinear limiters, which assure TVD properties, namely: Van Leer limiter, Van Albada limiter, minmod limiter, Super Bee limiter and β-limiter. All variants of the MUSCL approach are second order accurate in space. The implicit schemes employ an ADI approximate factorization to solve implicitly the Euler equations. Explicit and implicit results are compared, as also the computational costs, trying to emphasize the advantages and disadvantages of each formulation. The schemes are accelerated to the steady state solution using a spatially variable time step, which has demonstrated effective gains in terms of convergence rate according to Maciel. The algorithms are applied to the solution of the physical problem of the moderate supersonic flow along a compression corner. The results have demonstrated that the most accurate solutions are obtained with the Frink, Parikh and Pirzadeh TVD scheme using the Van Leer and Super Bee nonlinear limiters, when implemented in its explicit version.

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