Physical criterion study on forward stagnation point heat flux CFD computations at hypersonic speeds

Transcription

1 Appl. Math. Mech. -Engl. Ed. 317, DOI /s c Shanghai University and Springer-Verlag Berlin Heidelberg 2010 Applied Mathematics and Mechanics English Edition Physical criterion study on forward stagnation point heat flux CFD computations at hypersonic speeds Bang-ming LI, Lin BAO, Bing-gang TONG Graduate University, Chinese Academy of Sciences, Beijing , P. R. China Communicated by Zhe-wei ZHOU Abstract In order to evaluate uncertainties in computational fluid dynamics CFD computations of the stagnation point heat flux, a physical criterion is developed. Based on a quasi-one-dimensional hypothesis along the stagnation line, a new stagnation flow model is applied to obtain the governing equations of the flow near the stagnation point at hypersonic speeds. From the above equations, the compatibility relations are given at the stagnation point and along the stagnation line, which consist of the physical criterion for checking the accuracy in the stagnation point heat flux computations. The verification of the criterion is made with various numerical results. Key words quasi-one-dimensional hypothesis, stagnation point heat flux, hypersonic flow, physical criterion Chinese Library Classification O Mathematics Subject Classification 76K05 1 Introduction At present, the computational fluid dynamics CFD offers a powerful approach for solving problems numerically in fluid mechanics. However, there exist numerous puzzled uncertainties in calculating the flow field gradient quantities such as the heat flux and the friction on the surface of a hypersonic flight vehicle, which brings difficulties in the verification and the validation of the numerical solutions. From our point of view, it is necessary to reveal the flow physics near the stagnation point at hypersonic speeds, which forms the physical criterion for checking the accuracy of CFD solutions. Affirming that it is a new idea, this research has made an original exploration. For the forward stagnation point heat transfer in high-speed flows, CFD solutions are greatly affected by the scheme and the grid spacing near the wall. The grid Reynolds number, which stands for the characteristic grid spacing near the wall, is defined as Re c = ρ u n, 1 μ Received May 5, 2010 / Revised Jun. 8, 2010 Project supported by the National Natural Science Foundation of China Nos , Corresponding author Bing-gang TONG, Professor, tongbg@gucas.ac.cn

2 840 Bang-ming LI, Lin BAO, and Bing-gang TONG where the characteristic length n is the grid normal height near the stagnation point. Klopfer and Yee [1] simulated the aerodynamic heating on a blunt body numerically and found that solutions strongly relied on Re c and numerical schemes. Moreover, Re c should be under three for getting a rational stagnation point heat flux. Hoffmann et al. [2 3] also mentioned that Re c distinctly affected numerical solutions, and some adjustment of it would greatly improve the stagnation point heat flux prediction. Lee and Rho [4] discussed the differences among several schemes on computing the hypersonic blunt-body stagnation point heat flux and found that the AUSM+ scheme was better than the others, yet the heat flux still depended on Re c. In summary, present difficulties of the heat flux computation originated from that the heat flux prediction accuracy strongly depends on Re c and the numerical schemes. However, there is no convincing principle about how to choose them. Zoby et al. [5] reviewed existing thermal environment prediction methods and believed that in current circumstances, people should continue to develop high-precision numerical schemes for a high resolution of the flow field, and another way of great importance was to unveil the specific physical characteristics of hypersonic flows. This inspires us that a careful theoretical analysis should be exploited to find the physical criteria for judging the reliability of the stagnation point heat flux computations. Among theoretical studies of the stagnation point heat transfer, based on the self-similar feature of a boundary-layer flow, many solutions have been published. As is well known, Lees [6] gave a good prediction of the heat flux on the blunt-body s surface, including the stagnation point and its downstream part. Famous Fay-Riddell formula derived by Fay and Riddell [7] obtained an excellent prediction of the stagnation point heat flux. Applying the perturbation theory, van Dyke [8] presented the high-order boundary-layer theory, which divided the flow field into an inviscid and a viscous region. Furthermore, both regions gave detailed descriptions. However, from our point of view, all that is not enough. Considering that the stagnation point heat flux is determined by the temperature distribution along the stagnation line, to make clear of the flow field s physical characteristic along the stagnation line is of remarkable importance. The above classical theories cannot give guidance to heat flux computations, which only reveal the flow feature tangential to the wall. Moreover, whether a boundary layer exists near the stagnation point is a question. Therefore, we decide to create a new flow model along the stagnation line. Belouaggadia et al. [9] theoretically and numerically studied the shock wave stand-off distance of a blunt body in hypersonic nonequilibrium flows, where along the stagnation line, the quasi-one-dimensional hypothesis was developed under the premise of an inviscid flow. Fortunately, this quasi-one-dimensional hypothesis can be adopted and expanded to the near wall viscous region. Therefore, a new stagnation flow model is established in the present paper. Originating from the N-S equations, governing equations of the flow near the stagnation point are developed and immediately reveal the heat transfer characteristic near the stagnation point. To obtain the physical criterion, eventually, the straightforward relations between flow variables at specific positions are given from the above equations. 2 Stagnation flow model As shown in Fig. 1, a two-dimensional cylinder in hypersonic uniform flows is studied as the simplified model of a blunt body. According to van Dyke s high-order boundary layer theory, the flow field is divided into two parts, in which the outer inviscid region is treated as Belouaggadia s work. For the inner viscous region, as discussed previously, a new approach would be presented later on. For convenience, a solar system is established, where u and v are defined as φ and r direction velocities, that is, the tangential and the normal velocities, respectively. L is the shock wave stand-off distance. D is the diameter of the cylinder. θ equals β φ + π/2, where β is the shock wave angle. In hypersonic flows, the stagnation enthalpy approximates to the kinetic energy of the

3 Forward stagnation point heat flux CFD computations at hypersonic speeds 841 Fig. 1 Sketch map of stagnation flow structure incoming flows. Thus, the dimensionless expression is shown as follows: p = p, ρ = ρ, ū = u, v = p s ρ s u h = h, T = T, r = h s T s 2r D 1 L. v u, Here, the subscript s stands for the flow stagnation variables and for the incoming. The dimensionless coordinate r is set to be zero at the wall and one just behind the bow shock wave. L= 2L D, h s=c p T s = u Discussions of the flow variables along the stagnation line As mentioned above, the inner viscous layer still applies the quasi-one-dimensional flow hypothesis. Let us discuss flow variables under the above hypothesis afterward. First, for the symmetry of the flow, the flow variables v, p, ρ, andt satisfy the relations as the following equation, which coincides with the hypothesis [v, p, ρ, T] 2 =0. 3 However, the tangential velocity u does fit the symmetry condition like equation 3. It is u, not the partial derivative of u with respect to φ, that is zero along the stagnation line. Therefore, a discussion about the tangential velocity and momentum equation is necessary. From the exact solution of the classical incompressible stagnation flow [10], the tangential momentum equation cannot simply be omitted although its every part is zero because it has the limit of 0/0 term by dividing the convective scale. Moreover, from Fay-Riddell s work [7], we know that the tangential velocity can be written as u = u 0φ. Thus, a new flow variable u u is brought in. Using instead of u as the flow variable, the limit form of the tangential momentum equation exists by dividing φ. More significantly, u u remains constant in the φ direction. Thus, satisfies equation 4, which also coincides with the quasi-one-dimensional flow hypothesis u =0. 4 All this shows that the flow variables along the stagnation line change little in the φ direction. Therefore, the flow near the stagnation line can be treated as a quasi-one-dimensional flow.

4 842 Bang-ming LI, Lin BAO, and Bing-gang TONG 4 Governing equations of a compressible stagnation flow 4.1 Governing equations of the quasi-one-dimensional flow Foremost, the N-S equations in the solar system are given as ρrv + ρu =0, 5a ρ v v + u v r u2 = 1 rτrr + τ rφ τ φφ r r r, 5b ρ v u + u u r + uv = 1 rτrφ + τ φφ + τ rφ r r r, 5c ρ v e + u e v u = τ rr r + τ φφ r + v u + τ rφ r + v r u r + 1 r kr T + k T r, 5d where τ rr, τ rφ,andτ φφ are components of the stress tensor, and k and e are the heat transfer coefficient and the specific internal energy, respectively. Here, e= c v T, h= c p T,andc p /c v =γ. Besides, as the perfect gas is concerned, the gas specific heat ratio γ is a constant. Next, equations 5 are nondimensionalized according to equations 2. Suppose that the, v, T, ρ, andp as flow variables. The nondimensional equations along the stagnation line are finally obtained as equations 6 for more details, refer to Appendix A, and for simplicity, the dimensionless variables are no longer marked with a superscript afterward. flow is quasi-one-dimensional and arrange u v ρ ρ + v + L u 1+rL + v ρv v γ 1 p + 2γ = 1 2 Re 3 + 2L 1+Lr μ v ρv u + L 1+rL ρ u = 1 u μ Re 1 T ρv γ + γ 1 v γ + L 1+rL 1 = k ReP rγ 1 T + Lk 1+rL + + 2μ 3Reγ 1 =0, 6a μ v L 1+rL μ L 1+rL u + v + L 1+rL v + u u + v, 6b γ 1 1 p 2γ φ, 6c v 2 L 1+rL u + v v T u + v L rl 2 u + v 2. 6d The above equations have been greatly simplified by utilizing the quasi-one-dimensional hypothesis and the flow symmetry condition. Especially, for the tangential pressure gradient in equation 6c, we individually discussed it as follows.

5 Forward stagnation point heat flux CFD computations at hypersonic speeds 843 From the Newton theory in hypersonic flows, the wall pressure distribution is definitely expressed as p p =ρ u 2 cos 2 φ. Thus, the dimensionless stagnation pressure gradient is 1 φ p s = 4 ρ ρ s ρ s h s p s. 7 Therefore, due to the tangential pressure gradient being constant, equation 6c remains an ordinary differential equation. However, the right-hand sides of equations 6b and 6d remain complicated, and needs to be simplified further. Van Dyke pointed out that the viscous layer near the stagnation point was a very thin layer the dimensionless thickness approximates to Re 1/2 for a large Reynolds number flow. Therefore, equations 6 can be simplified by the technique of order analysis. 4.2 Governing equations of the flow near the stagnation point at a large Reynolds number The dimensionless viscous region thickness is the order of Re 1/2, which is defined as the small parameter here. Thus, at the outer boundary of the viscous region, the orders of flow variables give the following equations: r = Oε, u = O1, T = O1, ρ = O1. 8 Through the continuity equation 6a, the order of the normal velocity gives v = Oε. Thus, by using of above relations, equations 6 can be simplified to the governing equations of flow near the stagnation point as follows: v ρ ρ + v + L u =0, p =0, ρv u + Lρ 1 T ρv γ + γ 1 γ u 2 γ 1 1 p + L 2γ φ = 1 Re v + L u 1 = ReP rγ 1 [ μ k T u 9a 9b ], 9c. 9d The normal momentum equation 9b shows that the pressure is constant in a normal direction in the viscous layer, which accords with the normal pressure gradient characteristic of a boundary layer flow. It is well known that, in the singular perturbation theory, a partial differential equation with a small parameter multiplied in the highest order derivatives whose degradation form makes it reduce order, which often results in a boundary layer type solution [11]. Hence, the tangential momentum equation 9c indicates that the distribution of u is a boundary the layer type and the same as the following energy equation 9d. Moreover, the dissipation term in the energy equation is neglected, which directly affects the temperature distribution along the stagnation line and results in a nonpeak temperature distribution. Thus, the energy equation 9d only consists of three terms that denote the heat convection, the compression work, and heat conduction. Near the stagnation point and in the condition of a large Reynolds number incoming flow, the dissipation term remains in Fay-Riddell s work [7], which is different from equations 9 derived above. However, since the kinetic energy in a boundary layer near the stagnation point is very small in comparison with the total enthalpy of the incoming flow, the dissipation term can also be omitted in Fay-Riddell s work, which should lead to the same result as in our derivations.

6 844 Bang-ming LI, Lin BAO, and Bing-gang TONG 5 Compatibility relations at the stagnation point and along the stagnation line From equations 9, we get the simplified relations namely, the compatibility relations at the stagnation point and along the stagnation line. For the stagnation point, the nonslip and the nonpenetration conditions are u =0, v =0. 10 Substitute equation 10 to equations 9. Then, the compatibility relations at the stagnation point give v =0, 11a p =0, 11b L γ 1 1 p 2γ φ = 1 μ u, 11c Re k T =0. 11d These compatibility relations can be interpreted as follows. The continuity equation 11a says that the divergence of the velocity vector at the wall is zero. The normal momentum equation 11b remains the same as equation 9b. The tangential momentum equation 11c only leaves the balance of the tangential pressure gradient and the viscous force. The energy equation 11d appears that the heat conduction term is equal to be zero due to no convection and no compression work done at the stagnation point. Specifically, equation 11c means the balance of two terms, and the degradation form of it indicates that the flow is a boundary layer type flow. To guarantee the computational accuracy of a boundary layer flow, the tangential width of the grid should be much larger than the normal height in the viscous layer for details, see Appendix B. That is, equation 11c requires a specific grid shape, so here, we no longer put it into the physical criterion. In all, the compatibility relations at the stagnation point 11a, 11b, and 11d reveal the constraint relations among flow variables and should be treated as the physical criteria for judging the reliability of heat flux computations. Besides, the stagnation line compatibility relation near the stagnation point is shown below. Based on the normal momentum equation 9b and the flow symmetry relation 3, we know that the pressure in the viscous region is a constant, so that the state equation of a perfect gas can be rewritten as ρt =1. 12 In order to combine the convection term and the compression work in the energy equation 9d, first, we make use of the continuity equation 9a to yield v + L u = v ρ ρ. 13 Then, substituting the state equation 12 into the right-hand side of equation 13 gives v ρ T = ρv ρ. 14

7 Forward stagnation point heat flux CFD computations at hypersonic speeds 845 Thus, the convection term is in direct proportion to the compression work. Merging them as one, finally, we get the simplified energy equation ρv T = 1 ReP rγ 1 k T. 15 There is no dissipation term, and it only has the convection and the conduction terms in equation 15. Computational solutions should coincide with this physical rule. Thus, equation 15 is also listed in the physical criterion. For simplicity, the energy equation 15 may be rewritten in the following form: R cc = ReP rγ 1ρv T k T = To sum up, we have obtained the stagnation point compatibility relations 11a, 11b, and 11d and the stagnation line compatibility relation 16, which in total consists of the physical criterion for the stagnation point heat flux computations. 6 Verification of the physical criterion as a reliability judgment for CFD computations of the stagnation point heat flux Here, let us begin with the concept of uncertainty for our judgment. Because the real stagnation point heat flux value is unknown, the CFD numerical solution and the prediction by the Fay-Riddell formula are compared in practice. The difference between them and the ratio of this difference to the Fay-Riddell solution are defined as absolute uncertainty and relative uncertainty, respectively. CFD results show that the low relative uncertainty is considered as a reliable one. As mentioned in Section 2, the CFD results of the stagnation point heat flux value are sensitive to Re c, i.e., solutions with large Re c have high uncertainties. However, there is no convincing principle about how to choose Re c to reduce the uncertainties. In this section, it will be proved that the computation of the stagnation point heat flux is reliable only if its results obey the physical criterion, which is theoretically obtained above. First, the deviation errors from the physical criterion in CFD computations will be shown. The deviation error is characterized by the differences between the theoretical and the computational values of the formulas from the compatibility relations 11a, 11b, 11d, and 16. Obviously, smaller deviation errors mean better resolutions in the computations for flow structures and temperature field nearby the stagnation zone, which could lead directly to the more accurate prediction of the stagnation point heat flux because the heat flux is determined by the slope of temperature profile. Second, the uncertainties of the stagnation point heat flux value in CFD computations will be calculated. Finally, it will be revealed that the small deviation error from the physical criteria is the necessary and sufficient condition to obtain low uncertainties in heat flux computations, so that the physical criterion can be proved as a reliability judgment for CFD computations of the stagnation point heat flux. The CFD code is described briefly. In the FVM framework employed here, the inviscid terms are discretized by AUSM+UP scheme, and the viscous term is approximated using a second-order central difference scheme. Though heat flux values in CFD computations are strongly dependent on the scheme and the grids, we just choose the AUSM scheme, which has better resolutions in heat computations [12], and discuss only the grid dependency. Two groups of examples are designed in verification. The first group includes six cases with varying normal grid height Re c and fixed tangential grid width Δφ, while the second group consists of five cases with Re c remaining unchanged and tangential grid width varying. All examples are in

8 846 Bang-ming LI, Lin BAO, and Bing-gang TONG the following conditions: Ma=6, flight altitude H=20 km, cylinder radius R=0.01 m, and wall temperature T w =300 K. Let us examine group 1 first. Equation 16 is satisfied very well in all cases, as shown in Fig. 2. However, in Table 1, we can see that in the case of Re c =400, the deviation error from equation 11d exceeds the order of one, which means that this criterion is violated. On the other hand, Fig. 3 demonstrates that all cases except Re c =400 have given the reliable values of the heat flux by the relative uncertainties lower than 20%. Thus, we can conclude that small deviation error will result in low uncertainty. Fig. 2 Deviation errors from the compatibility relation along the stagnation line Remark 1 The axis variable η in Fig. 2 is the length along the stagnation line normalized by the boundary layer thickness. η is set to be zero at the stagnation point and one at the outer edge of the boundary layer. In order to avoid the 0/0 term singularity in equation 16, here, η is between 0 exclusive and 0.8. According to equation 16, R cc exactly equals one, and all numerical results fit it well. Table 1 Re c Deviation errors from the compatibility relations at the stagnation point data shown is the dimensionless value of v, etc., of which the exact solution is zero v p k T Remark 2 From Table 1, Equations 11a and 11b are not sensitive to Re c and the deviations remain small; however, equation 11d is strongly dependent on Re c. When Re c is less than 200, deviation errors are far less than one. However, the results of Re c = 400 do not. In the same way, group 2 is then investigated. As shown in Fig. 4 and Table 2, in the first three cases, which have small aspect ratio grids, the deviation errors from all compatibility relations are very small, but for the rest, with large aspect ratio grids, the maximum deviations in equations 11a and 16, both exceed the order of one. Table 3 also shows that relative uncertainties of the first three examples are all less than 20%. On the contrary, relative uncertainties of the rest are much larger than 20%, which indicates that these simulations have failed. The conclusion can be made in a similar manner that small deviations in the physical criterion do result in low uncertainties in computations.

9 Forward stagnation point heat flux CFD computations at hypersonic speeds 847 Fig. 3 Relations between q w/q wf-r and Re c Remark 3 In Fig. 3, relative uncertainties below 20% are chosen as a reliable prediction of the heat flux in practice. When Re c is above 300, the relative uncertainty is over 20%. That is, the results with Re c >300 are unreliable. Fig. 4 Deviation errors from the compatibility relation along the stagnation line Remark 4 In Fig. 4, as equation 16 shows, R cc =1 is the exact solution. The last two cases do not fit equation 16 well, while the others fit it very well. In order to avoid the 0/0 term singularity in equation 16, here, η is between 0 exclusive and 0.8. According to Appendix B, the reason is that the grid aspect ratio of the last two cases is not small enough. Table 2 Deviation errors from the compatibility relations at the stagnation point data shown is the dimensionless value of v, etc., of which the exact solution is zero v p k T Tangential grid width Remark 5 Table 2 shows that the deviation errors in the top three cases are all below the order of one. Meanwhile, the rest are not. This implies that the flow fields are not correctly computed and lead to the failure of temperature profile numerical prediction.

10 848 Bang-ming LI, Lin BAO, and Bing-gang TONG Table 3 Relative uncertainties of the stagnation point heat flux computation with different grid width Tangential grid width Relative uncertainties Remark 6 In Table 3, except the last two cases, relative uncertainties are small. The reason is the same as discussed in Fig. 4. In summary, the numerical results with different grid scales illustrate that the solution will be reliable only if all the compatibility relations are fulfilled in the CFD computations of the stagnation point heat flux. In other words, we can use the physical criterion as a judgment on the reliability of CFD computations. 7 Conclusions A theoretical analysis of the stagnation point heat flux on the blunt-body in hypersonic flows is presented, and a series of physical criteria have been deduced for decreasing the uncertainties in CFD computations of the heat flux; see details as follows: i A quasi-one-dimensional flow model along the stagnation line is developed. Originating from the N-S equations, utilizing the flow symmetry characteristic and following the order analysis technique, we get the governing equations of flow near the stagnation point, of which the solution is the same as Fay-Riddell s. ii Based on the above mentioned theoretical analysis, the compatibility relations on the stagnation point and along the stagnation line in the viscous layer have been derived, which are used as the physical criterion for checking the reliability of CFD heat flux computations. iii Through illustrations of the numerical examples for hypersonic stagnation point flow, the deviation errors from the physical criterion and the relative uncertainties of the stagnation point heat flux in CFD computations are enumerated, and it has been shown that small deviation errors below the order of one lead to low uncertainties less than 20%. Hence, it is proved to be true that we are able to obtain reliable solutions in numerical simulation only if the physical criterion is fulfilled, namely, the physical criterion provides a theoretical basis to evaluate the reliability of CFD computations of the stagnation point heat flux and to evaluate and improve grid system and discretization scheme. Acknowledgements The authors would like to express their thanks to Professor Gui-qing JIANG of CAAA and Professor Li-xian ZHUANG of USTC for their very beneficial helps. References [1] Klopfer, G. H. and Yee, H. C. Viscous hypersonic shock on shock interaction-on-blunt cowl lips. AIAA Paper [2] Hoffmann, K. A., Siddiqui, M. S., and Chiang, S. T. Difficulties associated with the heat flux computations of high speed flows by the Navier-Stokes equations. AIAA Paper [3] Hoffmann, K. A., Papadakis, M., and Suzen, Y. B. Aeroheating and skin friction computations for a blunt body at high speeds. AIAA Paper [4] Lee, J. H. and Rho, O. H. Accuracy of AUSM + scheme in hypersonic blunt body flow calculation. AIAA Paper [5] Zoby, E. V., Thompson, R. A., and Wurster, K. E. Aeroheating design issues for reusable launch vehicles a perspective. AIAA Paper [6] Lees, L. Laminar heat transfer over blunt nosed bodies at hypersonic flight speeds. Jet Propulsion 264, [7] Fay, J. A. and Riddell, F. R. Theory of stagnation point heat transfer in dissociated air. Journal of Aeronautical Science 252,

11 Forward stagnation point heat flux CFD computations at hypersonic speeds 849 [8] Van Dyke, M. Higher approximations in boundary-layer theory, part 1: general analysis. Journal of Fluid Mechanics 142, [9] Belouaggadia, N., Oliver, H., and Brun, R. Numerical and theoretical study of the shock stand-off distance in non-equilibrium flows. Journal of Fluid Mechanics 6071, [10] Batchelor, G. K. An Introduction of Fluid Mechanics, China Machine Press, Beijing 2004 [11] Li, J. C. and Zhou, X. Asymptotic Method in Mathematical Physics in Chinese, Science Press, Beijing 2002 [12] Yan, C., Yu, J. J., and Li, J. Z. Scheme effect and grid dependency in CFD computations of heat transfer in Chinese. Acta Aerodynamica Sinica 241, Appendix A Derivation of quasi-one-dimensional flow equations along stagnation line Stress terms, in the dimensional N-S equations at solar system 5 in subsection 4.1, are expressed as follows: τ rr = p + λ v+2μ v, A1 u τ φφ = p + λ v+2μ r + v, A2 r u τ rφ = μ + v r u, A3 r where v = rv r + u r,andλ = 2 μ is assumed for the Stokes fluid. 3 After the nondimensionalization presented in equations 2 in Section 2, we get the dimensionless N-S equations: v ρ ρ r + v r + L v 1+L r + L ū 1+L r =0, ρ v v r + L v 1+L r ρū ū + γ 1 p 2γ r = 1 2 μ v Re 3 r L 1+L r μ v + ū + L ū μ 1+L r r φ + L v 1+L r ū φ + 2L v 1+L r μ r L ū 1+L r + v, A5 ρ v ū + L r 1+L r ρ ū ū + v + L γ 1 1 p 1+L r 2γ φ = 1 ū μ + L v Re r r 1+L r ū L 2 v 1+L r 3 μ r 2L ū 1+L r + v + 2L 1+L r μ ū r φ + L v 1+L r ū φ, A6 = 1 T ρ v γ r + γ 1 γ v p r + 1 T k ReP rγ 1 r r L 1+L r v ū r + v + L 1+L r + L k 1+L r where Re=ρ su L/μ s, Pr=c pμ s/k s. ū + v T + r 2 μ 3Reγ 1 v 2 r A4 L L r 2 ū + v 2, A7

12 850 Bang-ming LI, Lin BAO, and Bing-gang TONG Then, according to the quasi-one-dimensional hypothesis, omitting terms related to v, p, ρ, T and, finally, we obtain quasi-one-dimensional flow equations 6 the along stagnation line. u Appendix B Order estimation of the discretization error in computation and the optimization of grid aspect ratio Originating from the tangential momentum equation 9c and listing its truncation error, we can get the optimization of grid aspect ratio. Details are as follows: where ρv u + R 1 + Lρ u 2 + R2 + L γ 1 2γ 1 p φ = 1 Re R 1 =Δr ρv 2 u, R 2 2 =Δφ 2 ρ u 2 μ u + R 3, B1. B2 u 2 From order analysis of subsection 4.2, we know that in the boundary layer, there exist v = Oε, 2 u = Oε 2, 2 u = O1, 2 u = O1. B3 2 In order to minimize the computation error, the truncation error should be in the same order. That means R 1/R 2=O1. Thus, we get Δr Δφ = εδφ. Due to ε being a small parameter, the grid aspect ratio in the boundary layer should be small as well. B4