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Abstract1 To avoid un-realistic solutions like expansion shocks from appearing as a part of a solution it is necessary to satisfy the entropy condition for the Roe scheme. A variety of entropy fix formulae for the Roe scheme have been addressed in the literature. Three of the most famous are due to Harten-Hyman and Hoffmann-Chiang. These formulations have been assessed in this paper by applying them to the inviscid Burgers' equation and shock tube problem. These entropy fix formulations are unable to totally diminish the expansion shock in the vicinity of sonic expansions. Moreover, they are not universal, i.e. a single formulation is not adequate for the scalar Burgers' equation, shock tube problem and multi-dimensional cases and different formulations were suggested for each case. A simple modification to the Harten formulation is presented in this paper. This modification basically enlarges the band over which the entropy fix condition is enforced. The resulting formulation is able to totally remove the non-physical expansion shocks from the region of sonic expansion without affecting the rest of the computational domain. Comparison among the exact solution, and the entropy correction formulae of HartenHyman and Hoffmann-Chiang and the currently modified formula are shown here. The modified entropy fix formulation can totally diffuse the expansion shock. Moreover, the current formula does not affect the solution in the rest of the computational domain. Besides the modified formula, as a single formulation, can universally be applied over a wide range of applications from scalar equations to the governing equation of fluid motion. Finally in this paper, the following test cases are performed to assess the accuracy of the modified entropy formulation: inviscid shear flow, transonic flow over a bump, and transonic flow in a Laval Nozzle. Very accurate results are obtained. In the current study a second order upwind scheme of Roe with minmod flux limiter is applied.

1 Most of the computations by Roe scheme (1981) to solve the fluid equations is performed in the physical domain either over unstructured grids, like some triangular grids, or structured grids, like some quadrilateral grids, say for two-dimensional computations. With the choice of triangular grids some benefits of the Roe scheme, as a non-diffusive scheme in essence, is partially lost due to the grid line obliqueness w.r.t. flow. In contrast, quadrilateral grids are more appropriate for the Roe scheme as they are better able to align with the flow. Besides, with the choice of quadrilateral grids it is also possible to convert the fluid equations from physical to computational domain. There are only very few reports and articles on the application of the Roe scheme in generalized coordinates. To solve the fluid equations by Roe scheme in computational domain (or generalized coordinates) and to give a formula for the numerical flux Roe in generalized coordinates is the main goal of this paper. In detail, the following subjects are addressed in this paper. (1) A comprehensive knowledge of the grid-geometry is obtained. That is, the cosine direction of the grid lines, the cosine direction of the control volume faces and the area of the control volume faces are described in terms of the metrics of transformation from the physical domain to that of the computational domain. (2) The governing equations of fluid motion in physical domain and those in computational domain are put side by side term by term and a formula is obtained to give the numerical flux Roe in generalized coordinates. This numerical flux is written in terms of grid-geometry (metrics of transformation from the physical domain to the computational domain) and flow parameter. In part II of this publication the method is applied to some determining inviscid and viscous cases.

1 The recently developed formulation for the numerical flux resulting from use of the Roe scheme in generalized coordinates as outlined in Part I of this publication, is now applied to several inviscid and viscous test cases. The test cases studied are as follows: inviscid transonic flow over a bump and in a Laval nozzle (the results of which are not detailed in this paper), Blasius flow, plane Poiseuille flow, and laminar boundary layer interaction with an oblique shock over a flat plate. In this paper, the primitive variables are extrapolated to the cell faces by the MUSCL idea using the third order upwind biased scheme. The van Albada flux limiter is used to prevent spurious numerical oscillations. The viscous terms are centrally differenced.

/npsi/ctrl?lang=en http://nparc.cisti-icist.nrc-cnrc.gc.ca/npsi/ctrl?lang=fr Access and use of this website and the material on it are subject to the Terms and Conditions set forth at http://nparc.cisti-icist.nrc-cnrc.gc.ca/npsi/jsp/nparc_cp.jsp?lang=en NRC Publications Archive Archives des publications du CNRCThis publication could be one of several versions: author's original, accepted manuscript or the publisher's version. / La version de cette publication peut être l'une des suivantes : la version prépublication de l'auteur, la version acceptée du manuscrit ou la version de l'éditeur. Engineering, 110, pp. 74-81, 1988-05 Thermal performance representation and testing of air solar collectors Bernier, M. A.; Plett, E. G. Journal of Solar Energy