Development of an improved multi-dimensional upwind scheme for Euler/Navier-Stokes computations
Huang, Weicheng
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Permalink
https://hdl.handle.net/2142/22070
Description
Title
Development of an improved multi-dimensional upwind scheme for Euler/Navier-Stokes computations
Author(s)
Huang, Weicheng
Issue Date
1994
Doctoral Committee Chair(s)
Lee, Ki D.
Department of Study
Aeronautical and Astronautical Engineering
Discipline
Aeronautical and Astronautical Engineering
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Engineering, Aerospace
Language
eng
Abstract
Modern upwind schemes employ an upwind-biased stencil to solve convective terms in order to satisfy the domain of dependence of the hyperbolic system. They demonstrate a superior capability of capturing flow discontinuities in one-dimensional computations. However, in multi-dimensional computations, the high resolution of upwind schemes might be lost, due to the direct extension of one-dimensional analogy, which is not compatible with flow physics under such circumstances. The primary goal of present research is to improve the solution accuracy of conventional upwind schemes for multi-dimensional calculations.
The way the upwind schemes are applied to the multi-dimensional computations is responsible for the degrading of the solution resolution. That being said, most of the conventional upwind schemes are applied in a direction-split manner; specifically speaking, a one-dimensional solver is applied to each spatial dimension separately. This method not only ignores the interaction between different coordinate directions but also applies the upwind stencil to the grid-aligned direction, which violates the multi-dimensional physics when discontinuities are not aligned with the grid. This problem can be resolved by employing a more physically meaningful multi-dimensional upwind solver.
The present research employs a true multi-dimensional wave model to the flux-difference-splitting scheme. In addition, a multi-dimensional reconstruction is developed to formulate the left and right states of the cell face. These reconstructed states are then used to calculate various waves propagating along the dominant upwind direction.
The current approach is applied to the two-dimensional Euler calculations as well as to the Navier-Stokes computations. Results of the present study indicate that this approach is highly promising in respect to improve solution accuracy. Specifically, through the application of the current method, the resolution of shocks is improved over the first-order grid-aligned scheme, especially when the discontinuities are oblique to the grid. Moreover, the odd-even decoupling phenomena, such as those in the results of the Navier-Stokes computation of the transonic airfoil flow, are eliminated. In addition, the three-dimensional results demonstrate that although twice as many grids are used, the first-order grid-aligned scheme can not provide the same resolution as the present method.
However, since nonlinear feedback of the multi-dimensional wave model degrades the convergence, additional research must be conducted to control the nonlinear feedback of the wave model in order to accelerate the convergence. Furthermore, due to its oscillatory nature, multi-dimensional limiters are required for the development of the high order multi-dimensional wave model.
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