Roe Scheme in Generalized Coordinates; Part I- Formulations

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1 Roe Scheme in Generalized Coordinates; Part I- Formulations M.. Kermani. G. Plett Department Of Mechanical & Aerospace ngineering Carleton Universit, Ottawa, Canada Abstract. Most of the computations b the Roe scheme 98 to solve the fluid euations are performed in the phsical domain either with unstructured grids, like some triangular grids, or structured grids, like some uadrilateral grids, as for two-dimensional computations. With the choice of triangular grids some benefits of the Roe scheme, as a non-diffusive scheme in essence, are partiall lost due to the grid line obliueness w.r.t. the flow. In contrast, uadrilateral grids are more appropriate for the Roe scheme as the are better able to align with the flow. Besides, with the choice of uadrilateral grids it is also possible to convert the fluid euations from the phsical to the computational domain. To solve the fluid euations b the Roe scheme in the computational domain or generalized coordinates and to give a formula for the numerical flu b the Roe scheme in generalized coordinates is the main goal of this paper. In detail, the following subjects are addressed in this paper. A comprehensive knowledge of the grid-geometr 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 phsical domain to the computational domain. The governing euations of fluid motion in the phsical domain and those in the computational domain are put side b side term b term and a formula is obtained to give the numerical flu Roe in generalized coordinates. This numerical flu is written in terms of the grid-geometr metrics of transformation from the phsical domain to the computational domain and flow parameters. In part II of this publication the method is applied to some determining inviscid and viscous cases. Ph.D. Candidate, AIAA student member. Professor, Associate Fellow AIAA. Copright c 00 b M.. Kermani &. Plett. Published b the American Institute of Aeronautics and Astronautics, Inc. with permission. M in. 0 A M 4. 0 in B p 0 35 pa in T 300 k in Figure : Schematic picture of inviscid shear flow slipline. Introduction The most important characteristic of the Roe scheme as outlined b Roe [7] is that, it is in essence a non-diffusive scheme. An diffusive behavior that this scheme shows is due to the grid obliueness with respect to the flow. Appling the Roe scheme, [7], to some test cases in which the flow and grids are totall aligned can demonstrate this claim. As an eample, consider an inviscid shear flow with two parallel streams with Mach numbers and 4 as depicted in Fig.. Because these streams are assumed to be inviscid, there are no mechanisms b which one of the streams could diffuse to the other. Therefore, the same condition as at the inlet plane dominates the whole domain. The first order accurate scheme of Roe captures this discontinuit within two grids, usuall in a solution like the eact solution, as shown in Fig.. For more detail about this case the reader is referred to [3, 4]. When the same flow is analzed with an obliue grid, the discontinuit smears over the grid with smearing depending on the flow angle with the grid [6]. To use the non-diffusive propert of the Roe American Institute of Aeronautics and Astronautics Paper

2 XY ul 999 Iter 400 GRID 9* 8 P b ********** Flow Flow Y Triangular Grid Quadrilateral Grid H shape Figure 3: Comparison of triangular grid with uadrilateral H for the grid alignment purposes MACH Figure : Mach number distribution along the transverse direction at 0.3,0.0,0.3,0.6,0.9 after convergence is obtained. All the profiles overlap on a single profile. scheme, it is alwas desirable to align the grid with the flow as much as it is possible. To totall align the flow with the grid is usuall a non-feasible task depending on the compleit of the geometr. However, the tpe of grid chosen in a case of stud could help to achieve this goal. For -dimensional cases with triangular grid shapes, the grids cannot generall be aligned with the flow. Therefore, the non-diffusive behavior of the Roe scheme, the characteristic for which it was originall developed, is partiall lost. Although total alignment of the flow with all the grid lines is an impossible task, however, the tpe of grid chosen could partiall help in this matter. This is illustrated in Fig. 3, in which a triangular grid is compared with a uadrilateral H shape grid in terms of the grid obliueness with the flow. As shown in this figure, in general, the flow can be better aligned with uadrilateral H grids. To solve the governing euations of flow with the H grids, there are two major approaches: - To directl solve the euations on the phsical domain, or - to solve the euations on the computational domain after transforming the euations from the phsical domain to the computational domain. The second approach is to solve the governing euations in generalized coordinates. There are onl a few reports available in the literature on the formulation of the Roe scheme in generalized coordinates, and most of them are ver brief. To note a few, the reader is referred to [,, 8]. In this paper the numerical flu Roe in generalized coordinates are described based on the flow parameters and grid-geometr. Application for these formulations are given in part II of this publication as outlined in Ref. [5]. Grid Geometr To solve the governing euation of fluid motion, a comprehensive knowledge of the grid sstem is reuired b the flow solver. One wa to detail and to make known the configuration of the grid sstem to the flow solver is through the metrics of transformation from the phsical domain to the computational domain. In this approach the normal direction to each cell face and the area of the cell faces that the flues cross, are determined in terms of metrics of transformation. This knowledge is referred to as grid-geometr in this paper as in Ref. [3]. The gridgeometr in terms of the metrics of transformation are determined as follows. Metrics of Transformation Consider Fig. 4, in which the grid configuration in both the phsical and the computational domain are shown. In the transformation from the phsical domain to the computational domain, a one to one transformation is assumed. That is, a point in the phsical domain, sa point A, corresponds to one and onl one point in the computational domain, point B, and vice versa. Therefore, a point in the phsical domain and its correspondent in the computational domain could be denoted b the same inde j,k. In general, the phsical domain and the computational domain are related b:,, This relation could be determined in several was, -algebraicall, b solving elliptic, hperbolic or parabolic PD s, or 3 b a given set of data. For all of the above-cases, relation applies and must somehow be introduced to the solver. The metrics of transformation,,, and are the parameters that relate two domains in the following form, see for eample Ref. []:,, American Institute of Aeronautics and Astronautics Paper

3 Const. n Aj,k j j kkkk k Bj,k θ n β Const. kkkk k k k j j k k j j j j Bod: 0 Figure 4: Grid configuration in phsical domain left and computational domain right. 0 0,0,, where is the ratio of the volumes in the computational domain to that of the phsical domain also called the acobian of transformation:, 3 and,,, and are the inverse metrics of transformation which are computed b central differencing for internal nodes, and one-sided differencing for the boundaries. j+,k j,k, j+,k j,k, j,k+ j,k, j,k+ j,k. 4 Cosine Directions of the Grid ines Fig. 5 shows a bod fitted grid sstem, in which the bod is denoted b a constant line, sa 0. The net line almost parallel to the bod is denoted b assuming, and the following lines, 3, etc. The grid lines in the transverse direction are denoted b constant lines. The gradient of,, is a vector pointing the direction of maimum change of and it is normal to iso- lines. The cosine direction of the unit vector ˆn which is taken along could be determined as follows: ˆn cos θ î + sin θ ĵ, 5 where θ is the angle between the vector and the -ais, î + ĵ, and +. This Figure 5: Bod fitted grid sstem with the unit vectors normal to the grid lines. concludes: cos θ +, sinθ +. 6 n. 6 shows the obliueness of iso- lines which are used throughout this paper and Ref. [3]. ikewise, the normal direction to iso- lines are determined from, where is a vector showing the direction of maimum change in. Therefore, ˆn cos β î + sin β ĵ, 7 where β is the angle between the vector and - ais, î + ĵ, +, and: cos β, sin β It should be noted that the formulations given here are not limited to orthogonal grids and also applies to non-orthogonal grids. Cosine Directions at the Cell Faces In the current stud, all the information including those of the grid geometr and flow parameters are all stored at nodes. In fact this approach is called the cell verte. Assuming the cell faces are located at the mid-point between the adjacent nodes, a simple interpolation can provide the grid geometr at the cell face. Fig. 6 shows the enlargement of the control volume associated with node A phsical-domain and B computational-domain. The boundar of this control volume is represented b dashed lines and denoted b east, W west, north, and S south. Consider the unit vector normal to the face. Its cosine directions are: 3 American Institute of Aeronautics and Astronautics Paper

4 Aj,k,K,K+ W ^ n β S θ,k +,K ^ n j,k j,k+ j,k Bj,k j+,k Figure 6: nlargement of control volume A in phsical-domain and B in computational-domain. ˆn cos θ î + sin θ ĵ 9 where θ is the angle between the normal direction to the face and the -ais. Different tpes of interpolation could be used to obtain the cosine direction at the mid point. The following arithmetic averaging is used here: and cos θ [cos θ j+,k + cos θ j,k ] [ ] j+,k + j,k sin θ [sin θ j+,k + sinθ j,k ] [ ] j+,k + j,k 0. Similarl at the north boundar of this control volume, i.e. at face, where: and ˆn cos β î + sin β ĵ cos β [cos β j,k+ + cos β j,k ] [ ] j,k+ + j,k sin β [sin β j,k+ + sinβ j,k ] [ ] j,k+ + j,k 3. 4 Cell Face Areas The boundaries of the control volume A consist of area segments S, S, etc. To obtain total mass total massmass-flu area, momentum and energ passing through these area s, we need to obtain these area s. Consider cell-face with the area S. Its projected area along the and aes are and, respectivel. Therefore, S +. 5 oting that S from the phsical domain corresponds to from the computational domain. Hence, [ ] S +, 6 or S [ + ], 7 and appling ns. we obtain: S +. 8 A similar euation for the cell face area of the north boundar, S, is obtained as follows: + S. 9 ns. 8 and 9 give the area of the cell faces, S and S, in terms of the metrics of transformation from the phsical domain to the computational domain. These euations are used throughout this paper and in Ref. [3]. Fluid uations in the Phsical Domain The inviscid compressible flow euations in full conservative form with no bod force can be written as: Q t + F + G 0 0 where Q, F, and G are: ρ Q ρu ρv t 4 American Institute of Aeronautics and Astronautics Paper

5 V Figure 7: Volume V surrounded b closed surface S. F G ρu p + ρu ρuv t + pu ρv ρuv p + ρv t + pv S ds where ρ, p, u, v, t are densit, pressure, - and - velocit component, and the total energ per unit volume, respectivel. t is the sum of the internal- and kinetic energ per unit volume given b: t ρe + V, where V u + v and e is the internal energ per unit mass. For a caloricall perfect gas, pressure is estimated b the euation of state: p ργ e, where γ is the ratio of specific heats, which is constant for perfect gases with a value of.4 here as applied to air. Consider n. 0 which is written in a compact form as follows: Q t + R 0 where R is the total flu vector, given b R F î+ G ĵ and î and ĵ are the unit vectors along the and aes, respectivel. Integrating n. over an arbitrar volume V gives: V Q t dv + V R dv 0. 3 Considering the first integral of n. 3 and assuming an kind of arbitrar variations for Q over V, see Fig. 7, an average Q could be defined as follows: Q V V QdV. The second integral in n. 3, is a volume integral over V which is converted to a surface integral through the Divergence Gauss theorem, i.e. R dv R ds, where S is the surface surrounding V and ds is the surface element vector, V S with the normal direction to the surface aiming outward, as shown in Fig. 7. Hence n. 3 becomes: V Q t + R ds 0 4 S n. 4 is the integral form of the fluid euation, i.e. n. 0, in the phsical domain. It represents the rate of change of the mean value of Q over the control volume V see Fig. 7 which varies b the net flu R crossing the closed surface S. n. 4 is epanded over the control volume A of Fig. 6: V Q t + R n S + R n S R Wn S W R Sn S S 0, 5 where R n, R n, etc. are the numerical flues in the direction normal to the cell faces in the phsical domain which must be determined. In n. 5, it should be noted that R ds R n S because the normal component of R is taken to be in the same direction as ds, so the dot product is positive. On the other hand, R ds W R Wn S W because the normal component of R W is in the opposite direction of ds W, so the dot product is negative. Fluid uations in Generalized Coordinates n. 0 is transformed to a generalized coordinates sstem b the chain rule, t τ, +, + 6 Viviand [0] and Vinokur [] separatel have shown that the resulting euations can be written in a strong conservation form similar to the original euations in the phsical domain. This allows that all the shock capturing techniues applied to the governing euations in phsical domain coordinates or phsical domain, can also be applied to the resulting euation in generalized coordinates. This is because the tpe of euation does not change in the transformation. For eample, if the original euation in the phsical domain is hperbolic, it will remain hperbolic in an other coordinate sstem. The fluid euations for the inviscid, unstead and compressible flow in full conservative form in generalized coordinates becomes: Q t + F + G American Institute of Aeronautics and Astronautics Paper

6 G R v V S F S u Figure 8: Schematics of R and its components. Figure 9: Velocit vector V and its components. where Q Q, F F + G, G F + G. 8 n. 7 is integrated over the control volume B as shown in Fig. 6. ote that the control volumes A and B are euivalent map of each other; the first one in the phsical domain, and the second one in the computational domain. That is each point of the control volume A corresponds to onl one point of control volume B and vice versa. Also each face of A, sa face, corresponds to the same face, face, of control volume B. Integrating n. 7 over B and appling the Gauss theorem gives, V ol. comp. Q t + F + G F W G S 0, 9 where V ol. comp., and F, G, etc. are the numerical flues in the computational domain. Relating uations of Phsical- and Computational-Domain Consider ns. 5 and 9, one written in Cartesian coordinates of the phsical domain, and the other in generalized coordinates in the computational domain. These euations are put side b side term b term in this section. First term: The volume of the control volumes A and B are related b the acobian of transformation, i.e. V ol.comp. V ol. ph.. 30 V On the other hand according to n. 8, one can write: Q Q/. Therefore, the first term of n. 9 becomes: V ol. comp. Q t V ol. comp. Q t V Q t, 3 which is eactl the same as the first term of n. 5. The other terms: Similarl, it can be shown that the other terms of ns. 5 and 9 are euivalent. This task is performed for the second term of the ns. 5 and 9, i.e. for the terms corresponding to the cell face. The same procedure could be etended to the other cell faces. Face of the control volume A has an area of S and it is assumed that all the flow parameters are known at this face. The flu vectors F and G can be determined according to n. and the total flu vector, R, at this cell face is written as: R F î + G ĵ. 3 The schematic picture of R and its two components F and G are shown in Fig. 8. ρu ρv R ρu + p ρuv î + ρuv ρv + p ĵ 33 ρuh ρvh or after combining terms F and G : ρuî + vĵ R ρuuî + vĵ + pî ρvuî + vĵ + pĵ ρhuî + vĵ, 34 u î + v ĵ is the total velocit at the cell face and is denoted b V. The velocit vector V and its components are shown in Fig. 9. Therefore n. 34 can be written in the compact form of: R ρ V ρu V + pî ρv V + pĵ ρh V American Institute of Aeronautics and Astronautics Paper

7 θ G S R R S n θ F n^ θ v S ^ j ^i V θ u S u u ^ n Figure 0: Schematic of: S, R and its component along S, 3 unit vector ˆn along S. The surface area S is a vector normal to S aiming outward, as shown in Fig. 0. The component of R along S is called R n. R n is determined b forming the dot product between the vectors R and the unit vector ˆn the unit vector along S, see Fig. 0, i.e. and R n R ˆn ρ V ρu V + pî ρv V + pĵ ρh V ˆn 36 ˆn cos θ î + sin θ ĵ. 37 The component of V along S is denoted b u as depicted in Fig.. u is determined as follows: u V ˆn u cos θ + v sinθ. 38 From ns. 0 and, cosθ and sinθ can be substituted for in n. 38. It results in: u u + + v On the other hand, î ˆn is the component of ˆn along the -ais, i.e. cos θ. Similarl, ĵ ˆn sin θ. Hence n. 36 becomes: R n ρu ρuu + p cosθ ρvu + p sinθ ρu H, 40 This is the normal components of total flu at face. Mass, momentum and energ crossing this face Figure : Schematic of V and its components in directions either: - or parallel and normal. are represented b R n S and are determined as follows S is taken from n. 8: ρu ρv R n S ρ u + v + + u + v + p + + u + v + p + + ρ u + v H }{{} S n. 4 is simplified as follows: ρ u + v ρu u R n S + v + p ρv u + v + p ρ u + v H + + n. 4 is split in to two terms as follows: ρu ρu + p R n S ρuv + ρuh According to n. : [ R n S F + ρv ρuv ρv + p ρvh Finall according to n. 8: ] G. 44 R n S F American Institute of Aeronautics and Astronautics Paper

8 n. 45 is an important euation for two reasons. Firstl, it shows that the second terms of ns. 5 and 9 are identical. The same idea could be etended to the other terms of ns. 5 and 9 as well. In n. 45, R n is the numerical flu of mass, momentum and energ crossing the cell face of the phsical domain. When R n is multiplied b its corresponding area S, it gives the mass, momentum and energ crossing the face. On the other hand, F is the numerical flu crossing face in the computational domain and F is the mass, momentum and energ crossing the face in the computational domain. R n S F represents the mass, momentum and energ crossing the cell face in the phsical domain which in turn are euivalent to those crossing the corresponding face of the computational domain. umerical Flu Roe in Generalized Coordinates To obtain an appropriate form for the numerical flu Roe in the computational domain generalized coordinates, Roe s numerical flu in the phsical domain is assumed given, see Refs. [, 8], and the numerical flu Roe is obtained in generalized coordinates. Considering face again: [ R n R n + R R ] n 4 κ ˆλ κ δwκ κ ˆT, 46 where R n and R R n are the normal component of total flues at face obtained at inner and outer R conditions see Fig., ˆλ s are the eigenvalues of the acobian matri determined at Roe s averaged condition, δw s are the wave amplitudes and ˆT s are the eigenvectors corresponding to the eigenvalues ˆλ s determined at Roe s averaged conditions. The term R n on the.h.s. of n. 46 is Roe s numerical flu crossing the cell face i.e. crossing mass-, momentum-, or energ- per unit area of the cell face in the phsical domain. To obtain mass, momentum or energ crossing the face, n. 46 is multiplied b S : R n S [ ] R n S + R R n S 4 κ ˆλ κ δwκ κ ˆT S. 47 The.H.S. of n. 47 is replaced b F according to n. 45. The other terms of n. 47 are also described in terms of the generalized coordinate parameters as shown in the following, after which the numerical flu F can be obtained. j,k C D j,k+ j,k j,k B R A j+,k Grid ine Control Volume Figure : Inner and outer R values associated with a cell face. Grid lines are shown b solid lines and control volume boundaries with dashed-lines. j j j j j+ W Figure 3: The cell face value is determined b the first order upwinding j. Right Hand Side of n. 47 R n S or R R n S Before obtaining R n S or R R n S, it is appropriate to distinguish the difference between the terms R n S or R R n S with R n S. R n S is the actual mass, momentum and energ crossing the face predicted b the Roe scheme. In contrast, R n S or R R n S is a virtual nonreal value for the mass, momentum and energ determined at flow conditions or R. n. 45 is a general euation and applies for all the flow conditions including inner and outer conditions and R: j R n S F 48 R R n S F R 49 and R Flow Conditions The flow conditions and R at the cell face can be determined in accordance with the degree of accurac first order, second order, etc. and the R j+ 8 American Institute of Aeronautics and Astronautics Paper

9 j R j+ j R j+ j j j j j j j+ j j j+ W W Figure 4: The cell face value is determined b the second order upwinding j + W; W j j. tpe of scheme used central, upwind, etc.. In this stud onl upwind schemes are considered. The first order upwind algorithm upwinding suggests that: j,k 50 where represents a primitive variable, i.e. either the [ρ,p,u,v] or [T,p,u,v]. For the first order etrapolation a zero-order polnomial, i.e. a straight line with zero slope, is used to etrapolate the primitive variable at node j,k to the cell face. Fig. 3 shows the etrapolation of a tpical primitive variable b first order upwinding. Second order upwinding recommends: j,k + W 5 where W is the jump of a primitive variable at the west face of the control volume, i.e. W j,k j,k. For the second order etrapolation a first-order polnomial, i.e. a straight line, is used to etrapolate the primitive variable from nodes j,k and j,k to the cell face, as shown in Fig A third order upwind-biased algorithm proposes: j,k + 4 [ κ W + + κ ] 5 where κ 3 and is the jump of primitive variable at, i.e. j+,k j,k. For the third order etrapolation a second-order polnomial, i.e. a parabola, is used to determine the primitive variable at the cell face, as shown in Fig. 5. That is, a second order polnomial curve fit between the points j,k, j,k and j+,k is used to obtain. This task is performed via n. 5. The R side flow condition is determined in a similar manner as follows. For the first order: R j+,k, 53 Figure 5: The cell face value is determined b third order upwind-biased etrapolation. for the second order: R j+,k 54 where j+,k j+,k, and for the third order upwind-biased: R j+,k 4 [ κ ++κ ].55 In the current stud, a third order upwind-biased algorithm, i.e. ns. 5 and 55, are used to determine the flow conditions at and R sides of the cell faces. It is noted for the higher order, i.e. higher than first order accurate cases, the solution is not monotone and non-phsical oscillations are produced, which must be damped. In the current stud, the van Albada flu limiter, [9], is used to damp the numerical oscillations. For more detail, the reader is referred to [3, 5]. Roe s Averaging Once and R conditions at face are obtained, Roe s averaged condition is determined as follows: where ˆρ W W R 56 û W u + W R ur W + W R ˆv W v + W R vr W + W R Ĥ W H + W R HR W + W R W ρ, W R ρ R. 60 Therefore, other primitive variables, e.g. p, T, etc., corresponding to the and R flow conditions could be obtained. 9 American Institute of Aeronautics and Astronautics Paper

10 ˆλ, ˆT, and δw Other terms of n. 47, i.e. ˆλ, ˆT, and δw, are briefl described here in terms of the parameters of generalized coordinates. For more detail regarding obtaining these terms, the reader is referred to [3] or [8]. However, in order to give a somewhat complete presentation of the Roe scheme formulation in generalized coordinates, the description for ˆλ, ˆT, and δw are also provided here, δw ˆT ˆT ˆT 3 ˆT 4 ˆλ ˆλ ˆλ 3 ˆλ 4 δw δw δw 3 δw 4 û ĉ cos θ ˆv ĉ sin θ Ĥ û ĉ 0 sin θ cos θ û û ˆv û + ˆv û + ĉ cos θ ˆv + ĉ sin θ Ĥ + û ĉ û ĉ û û û + ĉ 6 δp ˆρ ĉ δu ĉ ˆρ δu δp ĉ δρ ĉ δp + ˆρ ĉ δu ĉ 6 63 where u is the velocit component parallel to face obtained from u V V u, ĉ is the speed of sound determined at Roe s averaged condition from: ĉ γ Ĥ û + ˆv, 64 δρ ρ j+,k ρ j,k, δp p j+,k p j,k, δu u j+,k u j,k and δv v j+,k v j,k. Roe s umerical Flues in Generalized Coordinates Finall the numerical flu Roe is given in generalized coordinates in this section. Considering ns. 45, 48 and 49 also replacing S from n. 8, n. 47 becomes: F [ F + F R ] 65 4 κ ˆλ κ δwκ κ ˆT [ + ], Once is canceled from both sides, the numerical flu F crossing the face of the computational cell, see Fig. 6 Right, is obtained: [ ] F F + F R 66 4 κ ˆλ κ δwκ κ ˆT [ + n. 66 represents the numerical flu Roe in generalized coordinates. The obtainment of Roe s numerical flu in generalized coordinates has been the main message of this paper, which is completed b deriving n. 66. Similarl the numerical flu G at the north face of the control volume B, as shown in Fig. 6, could be determined as follows: [ ] G G + G R 67 4 κ ˆλ κ δwκ κ ˆT [ + where the superscript and R correspond to inner and outer states at the cell face, and ˆλ û ĉ ˆλ ˆλ 3 û û 68 ˆλ 4 û + ĉ also δw ˆT ˆT δw δw δw 3 δw 4 û ĉ cos β ˆv ĉ sinβ Ĥ û 0 sin β cos β û δp ˆρ ĉ δu ĉ ˆρ δu δp ĉ δρ ĉ δp + ˆρ ĉ δu ĉ ĉ ] ]., 69 0 American Institute of Aeronautics and Astronautics Paper

11 ˆT 3 ˆT 4 û ˆv û + ˆv û + ĉ cos β ˆv + ĉ sinβ Ĥ + û ĉ 70 where all hat states denote Roe s averaged condition determined at the face, cos β and sinβ are given b ns. 3 and 4, δρ ρ j,k+ ρ j,k, δp p j,k+ p j,k, δu u j,k+ u j,k, δv v j,k+ v j,k and ĉ γ Ĥ û + ˆv. Discussion The direct application of Roe s scheme in the phsical domain has been reported in the literature, see Refs. [, 6]. However, onl a few reports are given for the application of Roe scheme in generalized coordinates. When, this scheme is applied over the uadrilateral grids, there is no advantage of directl solving the fluid euations vs. that of solving the fluid euations in the transformed domain of computation, i.e. generalized coordinates. On the other hand, the uadrilateral grids are more suitable for the Roe scheme. Because, this kind of grid is better able to be aligned with the flow and less numerical diffusion results. This is as opposed to triangular grids which are inevitabl almost obliue to the whole flow field and produce unnecessar numerical diffusion, which lessens the non-diffusive benefits of the Roe s scheme. Similar forms for n. 66 are reported in a few places in the literature, see for eample Ref. [8]. However, to the knowledge of the authors, none has described the numerical flu Roe as clearl as it is given here b n. 66. Conclusion A simple formulation for the numerical flu Roe in generalized coordinates has been developed. The method has been applied to several inviscid and viscous test cases which is given in Part II of this publication. Acknowledgement The financial support for the course of this stud is provided b the Ministr of Culture and Higher ducation of I.R. of Iran and Carleton Universit. The are gratefull acknowledged. References [] Hirsch, C., umerical Computation of Internal and ternal Flows, Vol., 990. [] Hoffmann, K. A. and Chiang, S. T., Computational Fluid Dnamics for ngineers, Vol. II, A Publication of ngineering ducation Sstems, Wichita, Kansas, USA, 993. [3] Kermani, M.., Simulation of the Viscous- Turbulent and multi-dimensional Gasdnamics effects on Flows in Inlet Diffusers of Supersonic Vehicles, Ph.D. thesis, Department of Mechanical & Aerospace ngineering, Carleton Universit, Ottawa, Canada, 00. [4] Kermani, M.. and Plett,. G. Modified ntrop Correction Formula for the Roe Scheme, AIAA [5] Kermani, M.. and Plett,. G. Roe Scheme in Generalized Coordinates; Part II- Application to Inviscid and Viscous Flows, AIAA [6] Manna, M., A Three Dimensional High Resolution Upwind Finite Volume uler Solver, von Karman Institute for Fluid Dnamics, Technical ote 80, April 99. [7] Roe, P.., Approimate Riemann Solvers, Parameter Vectors and Difference Schemes. Comput. Phs., Vol. 43, pp , 98. [8] Tannehill,. C., Anderson, D. A., Pletcher, R. H., Computational Fluid Mechanics and Heat Transfer, Second dition, 997. [9] van Albada, G. D., van eer, B., and Roberts, W. W., A Comparative Stud of Computational Methods in Cosmic Gas Dnamics, Astron. Astrophs., Vol. 08, pp 76-84, 98. [0] Viviand, H., Conservative Forms of Gas Dnamic uations, Rech. Aerosp. o. 974-, pp , 974. [] Vinokur, M., Conservation euations of gas- Dnamics in Curvilinear Coordinates Sstems,. Comput. Phs. Vol. 4, pp. 05-5, 974. American Institute of Aeronautics and Astronautics Paper