Comparison of Several Turbulence Models as Applied to Hypersonic Flows in 2D Part I

Transcription

1 Comparison of Several Turbulence odels as Applied to Hpersonic Flows in D Part I EDISSON SÁVIO DE GÓES ACIEL and AILCA POTO PIENTA Aeronautical Engineering Division Aeronautical Technological Institute (ITA) Pça. al. do Ar Eduardo Gomes, 50 Vila das Acácias São José dos Campos SP BAZIL edisavio@edissonsavio.eng.br, amilcar@ita.br Abstract: - In the present work, the Van Leer flux vector splitting scheme is implemented, on a conservative and finite-volume context. The -D Favre-averaged Navier-Stokes equations are solved using an upwind discretization on a structured mesh. The Cebeci and Smith and Granville algebraic models and the Jones and Launder and Coakle two-equation models are used in order to close the problem. The phsical problem under stud is the cold gas hpersonic flow around a reentr capsule configuration. The implemented scheme is first order accurate in space. The time integration uses a unge-kutta method of five stages and is second order accurate. The algorithm is accelerated to the stead state solution using a spatiall variable time step. This technique has proved excellent gains in terms of convergence rate as reported in aciel works. The results have demonstrated that the Granville model has ielded more critical pressure field than the others models. The aerodnamic coefficient of lift is better predicted b the Coakle turbulence model. Finall, the stagnation pressure ahead of the reentr capsule configuration is better predicted b the Granville turbulence model. This paper treats the numerical results obtained from these simulations. Ke-Words: Cebeci and Smith model, Jones and Launder model, Coakle model, Granville model, Favreaveraged Navier-Stokes equations, Van Leer algorithm, First order scheme. 1 Introduction Conventional non-upwind algorithms have been used extensivel to solve a wide variet of problems ([1]). Conventional algorithms are somewhat unreliable in the sense that for ever different problem (and sometimes, ever different case in the same class of problems) artificial dissipation terms must be speciall tuned and judiciall chosen for convergence. Also, complex problems with shocks and steep compression and expansion gradients ma def solution altogether. First order upwind schemes are in general more robust but are also more involved in their derivation and application. Some upwind schemes that have been applied to the Euler equations are, for example, [-3]. Some comments about these methods are reported below: [] suggested an upwind scheme based on the flux vector splitting concept. This scheme considered the fact that the convective flux vector components could be written as flow ach number polnomial functions, as main characteristic. Such polnomials presented the particularit of having the minor possible degree and the scheme had to satisf seven basic properties to form such polnomials. This scheme was presented to the Euler equations in Cartesian coordinates and threedimensions. [3] emphasized that the [4] scheme had low computational complexit and low numerical diffusion when compared to other methods. The also mentioned that the original method had several deficiencies. It ielded pressure oscillations in the proximit of shock waves. Problems with adverse mesh and with flow alignment were also reported. [3] proposed a hbrid flux vector splitting approach which alternated between the [4] scheme and the [] scheme, at the shock-wave regions. This strateg assured that strength shock resolution was clearl and well defined. There is a practical necessit in the aeronautical industr and in other fields of the capabilit of calculating separated turbulent compressible flows. With the available numerical methods, researches seem able to analze several separated flows, threedimensional in general, if an appropriated turbulence model is emploed. Simple methods as the algebraic turbulence models of [5-6] suppl satisfactor results with low computational cost and allow that the main features of the turbulent flow be detected. This work describes four turbulence models applied to hpersonic flows in two-dimensions. The ISBN:

2 [] scheme, in its first-order version, is implemented to accomplish the numerical simulations. The Favreaveraged Navier-Stokes equations, on a finite volume context and emploing structured spatial discretization, are applied to solve the cold gas hpersonic flow around a reentr capsule in twodimensions. Turbulence models are applied to close the sstem, namel: [5; 7-9]. The convergence process is accelerated to the stead state condition through a spatiall variable time step procedure, which has proved effective gains in terms of computational acceleration (see [10-11]). The results have shown that the [8] scheme ields the best results in terms of the prediction of the lift aerodnamic coefficient; however, the [9] turbulence model predicts the best value of the stagnation pressure. oreover, the [9] scheme also predicted the most severe pressure fields. Navier-Stokes Equations The -D flow is modeled b the Navier-Stokes equations, which express the conservation of mass and energ as well as the momentum variation of a viscous, heat conducting and compressible media, in the absence of external forces. The Navier-Stokes equations are presented in their two-equation turbulence model formulation. For the algebraic models, these two-equations are neglected and the [] algorithm is applied onl to the original four conservation equations. The integral form of these equations ma be represented b: E E n F F n ds GdV 0 t QdV e v x e v V S, V (1) where Q is written for a Cartesian sstem, V is the cell volume, n x and n are components of the unit vector normal to the cell boundar, S is the flux area, E e and F e are the components of the convective, or Euler, flux vector, E v and F v are the components of the viscous, or diffusive, flux vector and G is the source term of the two-equation models. The vectors Q, E e, F e, E v and F v are, incorporating a k- formulation, represented b: ρ ρu ρv ρu ρu p ρuv t ρv ρuv ρv p t Q, Ee, Fe, Ev e e pu e pv ρk ρku ρkv ρs ρsu ρsv xx x 0 τ xx τ x ; fx x x (a) 0 0 t x τ x 0 t τ 0 F v, and G, (b) f 0 G k G s where the components of the viscous stress tensor are defined as: t t xx u x 3 u x v e ; t x u v x e v 3 u x v ; (3) e. The components of the turbulent stress tensor (enolds stress tensor) are described b the following expressions: xx T u x 3 T u x v e 3k ; x T u v x e ; (4) v 3 u x v e 3k. T T Expressions to f x and f are given bellow: f f x t xx xx u t x x v q x t x x u t v q ; (5), (6) where q x and q are the Fourier heat flux components and are given b: q q x PrL T PrT e i x ; Pr Pr e e (7) e. (8) L The diffusion terms related to the k- equations are defined as: T T k x ; T k T s x ; s x 1 e k e ; (9) 1 k x 1 e s e. (10) 1 T s In the above equations, is the fluid densit; u and v are Cartesian components of the velocit vector in the x and directions, respectivel; e is the total energ per unit volume; p is the static pressure; k is the turbulence kinetic energ; s is the second turbulent variable, which is the rate of dissipation of T i ISBN:

3 the turbulence kinetic energ (k- model) for this work; the t s are viscous stress components; s are the enolds stress components; the q s are the Fourier heat flux components; G k takes into account the production and the dissipation terms of k; G s takes into account the production and the dissipation terms of s; and T are the molecular and the turbulent viscosities, respectivel; Pr L and Pr T are the laminar and the turbulent Prandtl numbers, respectivel; k and s are turbulence coefficients; is the ratio of specific heats; e is the laminar enolds number, defined b: e V EF l EF, (11) where V EF is a characteristic flow velocit and l EF is a configuration characteristic length. The internal energ of the fluid, e i, is defined as: e i e ρ 0.5 u v. (1) The molecular viscosit is estimated b the empiric Sutherland formula: bt 1 1S T, (13) where T is the absolute temperature (K), b = 1.458x10-6 Kg/(m.s.K 1/ ) and S = K, to the atmospheric air in the standard atmospheric conditions ([1]). The Navier-Stokes equations are dimensionless in relation to the freestream densit,, the freestream speed of sound, a, and the freestream molecular viscosit,. The sstem is closed b the state equation for a perfect gas: p (γ 1) e0.5ρ u v ρk, (14) considering the ideal gas hpothesis. The total enthalp is given b H e p. 3 Van Leer Algorithm, Turbulence odels, Spatiall Variable Time Step, Initial and Boundar Conditions The space approximation of the integral Equation (1) ields an ordinar differential equation sstem given b: V dq dt, (15) with representing the net flux (residual) of the conservation of mass, conservation of momentum and conservation of energ in the volume V. The residual is calculated as: with, (16) 1/ i1/,j i1/,j c i1/,j 1/ d i1/,j i1/,j, where the superscripts c and d are related to convective and diffusive contributions, respectivel. The cell volume is given b: V i, j 0.5 x xi1, j i1, j1 x i1, j xi1, j1 x i1, j1 xi, j i1, j x i, j xi1, j1 1 x i1, j1 xi, j1 x 1 xi, j i1, j (17) The convective discrete flux calculated b the AUS scheme (Advection Upstream Splitting ethod) can be understood as a sum of the arithmetical average between the right () and the left (L) states of the cell face (i+½,j), involving volumes (i+1,j) and (), respectivel, multiplied b the interface ach number, plus a scalar dissipative term, as shown in [4]. Hence, a a au au 1 av av, j S i1/, j i1/, j ah ah ak ak as as L a a 0 au au Sxp av av S p i1/, j, (18) ah ah 0 ak ak 0 as as 0 L i1/, j i1/ 1 S S S defines the normal where t i 1/,j x i1/, j area vector for the surface (i+½,j). The normal area components S x and S to each flux interface are given in Tab. 1. Figure 1 exhibits the computational cell adopted for the simulations, as well its respective nodes and flux interfaces. The quantit a represents the speed of sound, which is defined as: p k 0. 5 a. (19) ISBN:

4 Table 1 Values of S x and S. Surface S x S x x -1/ i 1,j i 1,j i+1/,j i1,j 1 i1,j x i1,j xi1,j 1 +1/ 1 i1,j 1 x i1,j 1 x 1 i-1/,j x x 1 1 with p+/- denoting the pressure separation and due to []: p p p, 0.5p 0, 0, 0.5p 1 p, 1,, if 1; if 1; if 1; if 1; if 1; (4) if 1. The definition of a dissipative term determines the particular formulation of the convective fluxes. The following choice corresponds to the [] scheme, according to [3]: Figure 1. Computational Cell. i+½,j defines the advective ach number at the (i+½,j) face, which is calculated according to [4]: i1/,j L, (0) where the separated ach numbers are defined b []:, 0.5 0, 0, 0.5 1, if 1; 1, if 1; if 1; if 1;, if 1; if 1. (1) L and represent the ach numbers associated with the left and the right states, respectivel. The advection ach number is defined b: S u S v a S. () x The pressure at the face (i+½,j), related to the cell (), is calculated b a similar formula: i1/,j L p p p, (3) i 1/, j VL i1/, j i i i 1/, j 1/ 1/, j,, j L 1, 1, if if 0 if 1 i1/ i1/ i1/, j, j, j 1; 1; 0. (5) The above equations clearl show that to a supersonic cell face ach number, the [] scheme represents a discretization purel upwind, using either the left state or the right state to the convective terms and to the pressure, depending of the ach number signal. This [] scheme is first order accurate in space. The time integration is performed using an explicit unge-kutta method of five stages, second order accurate, and can be represented in generalized form b: Q Q Q (0) (k) (n1) Q Q (n) (0) Q t (k) k (k1) (k1) Q V GQ, (6) with k = 1,...,5; 1 = 1/4, = 1/6, 3 = 3/8, 4 = 1/ and 5 = 1. The gradients of the primitive variables are calculated using the Green theorem, which considers that the gradient of a primitive variable is constant at the volume and that the volume integral which defines the gradient is replaced b a surface integral ([13]). The turbulence models studied herein are described in [14-15]. The interested reader is encouraged to read these references to become aware of the implemented turbulence models in this stud. [14-15] give full description of the turbulence models studied herein. The spatiall variable time step is described in details in [14] and the initial and ISBN:

5 boundar conditions to the present problem are discussed in [14-15]. Again, the reader is encouraged to read these references to become aware of the numerical tools emploed in this work. 4 esults Tests were performed in a Dual Core processor of.3ghz and.0gbtes of A microcomputer. Three orders of reduction of the maximum residual in the field were considered to obtain a converged solution. The residual was defined as the value of the discretized conservation equation. The entrance or attack angle was adopted equal to zero. The ratio of specific heats,, assumed the value 1.4. Table. Initial Conditions. Altitude L e o 40,000m 3.0m 1.66x Cebeci and Smith esults Figure 3 shows the pressure contours obtained b the [] scheme as using the [5] turbulence model. The pressure contours are smmetrical and homogeneous. No pre-shock oscillations are observable. The pressure peak is 43,85 unities. Figure 4 exhibits the ach number contours obtained b the [] scheme as using the [5] turbulence model. The contours are smmetrical and the shock is well captured. The ach number peak is A region of intense energ exchange is observable close to the trailing edge. Figure 1. eentr capsule configuration. Figure 3. Pressure contours (CS). Figure. eentr capsule viscous mesh. Figure 1 shows the reentr capsule configuration. It is composed of 5,040 rectangular cells and 5,185 nodes, which is equivalent to a mesh of 85x61 nodes on a finite difference context. The exponential stretching is of 7.5%. Detail of this mesh is shown in Fig.. The initial condition is defined in Tab.. The enols number was estimated based on [1]. Figure 4. ach number contours (CS). This region is better observable in Fig. 5, where the contours of temperature are shown. In this figure, it is possible to note at the trailing edge a region of ISBN:

6 intense energ exchange, suggesting the formation of a pair of circulation bubbles, The contours are smmetrical and homogeneous. The temperature peak is close to, K. Figure 5. Temperature contours (CS). Figure 6 presents detail of the region at the trailing edge of the reentr capsule. As can be seen, a pair of circulation bubbles is formed close to the trailing edge. This formation is due to boundar laer detachment at the upper and lower surfaces of the reentr capsule at the configuration end. This phenomenon was captured b the [5] turbulence model and proves it capacit of detect flow nonlinearit. Figure 7 shows the T contours obtained b the [] scheme as using the [5] turbulence model. The contours are smmetrical. The values of T are close to the expected values to be obtained. 4. Jones and Launder esults Figure 8 shows the pressure contours obtained b the [] scheme as using the [7] turbulence model. The pressure contours have not been developed. The solution present smmetr properties, but the pressure field is not formed. The pressure peak is 3.95, well below the same parameter as measured b the [5] turbulence model. Figure 6. Circulation bubble formation (CS). Figure 8. Pressure contours (JL). Figure 7. T contours (CS). Figure 9. ach number contours (JL). ISBN:

7 Figure 9 exhibits the ach number contours obtained b the [1] scheme as using the [7] turbulence model. The field is again not developed. The ach number peak is 7., a reasonable value. Good smmetrical properties are observable. Figure 10 presents the temperature field obtained b the [] scheme as using the [7] turbulence model. The temperature peak is 1,566.1 K and appears at the trailing edge region, suggesting a zone of high energ dissipation and, consequentl, the region of formation of circulation bubbles. in [16-17]. It seems that the limit of good results is restricted to the supersonic regime, as seen in [14-15; 18-19]. It is a severe penalization of the k- model to hpersonic regime. It does not happen with k-, k- or k- 1/. It is also important to note that the T values are ver small and it can influence the turbulence model performance. Figure 1. T contours (JL). Figure 10. Temperature contours (JL). 4.3 Coakle esults Figure 13 presents the pressure contours obtained b the [] scheme as using the [8] turbulence model. The contours are smmetrical and homogeneous. The pressure peak is unities. The shock is well captured. Figure 11. Velocit vector field (JL). Figure 11 shows that this region of high dissipation not corresponding to a zone of circulation bubble formation. As can be seen, the lines are attached to the surface and not separation occurs. Figure 1 exhibits the T contours obtained b the [] scheme as using the [7] turbulence model. Good smmetr properties are observable. The field is again not developed. It seems that the k- model did not present good behavior in hpersonic flow regimes. This same flow characteristic is observable Figure 13. Pressure contours (C). Figure 14 shows the ach number contours obtained b the [] scheme as using the [8] turbulence model. The contours are clean, without pre-shock oscillations. The ach number peak is 6.57, which is a reasonable value. Figure 15 exhibits the temperature contours obtained b the [] scheme as using the [8] turbulence model. The temperature ISBN:

8 peak is, K. The region at the trailing edge is a zone of high energ dissipation and energ exchange, with the consequent formation of circulation bubbles, as registered in Fig. 16, where a pair of vortices is formed. Figure 17 presents the T contours generated b the [] scheme as using the [8] turbulence model. Despite the low values of T, the turbulence effects were considered in its majorit. Although the maximum T should be close to the bod, the turbulent model captured the main flow features. Figure 14. ach number contours (C). Figure 17. T contours (C). 4.4 Granville esults Figure 18 shows the pressure contours obtained b the [] scheme as using the [9] turbulence model. Good smmetr properties are observed. The contours are homogeneous. The pressure peak is unities. Figure 15. Temperature contours (C). Figure 18. Pressure contours (G). Figure 16. Circulation bubble formation (C). Figure 19 exhibits the ach number contours generated b the [] scheme as using the [9] turbulence model. The contours are homogeneous and clean, free of pre-shock oscillations. The ach number peak is 6.56, which is an acceptable value. The region at the trailing edge seems a zone of circulation bubble formation. ISBN:

9 this region, high dissipation occurs with the consequent formation of circulation bubbles. A pair is observed in Fig. 1, as consequence of the energ exchange between zones and dissipation energ. Figure 19. ach number contours (G). Figure. T contours (G). Figure 0. Temperature contours (G). Figure 1. Circulation bubble formation (G). Figure 0 presents the temperature contours generated b the [] scheme as using the [9] turbulence model. The temperature peak is close to, K and happens at the trailing edge zone. In Figure shows the T contours generated b the [] scheme as using the [9] turbulence model. As interesting analsis, onl the algebraic models assured the high values to T close to the bod. oreover, in the entire field, the [9] turbulence model assured high values to the turbulent viscosit. As conclusion of this qualit analsis of the solutions, it is possible to highlight the [5] and the [9] turbulence models as the best, reproducing the expected behavior for these models. The [8] turbulence model has onl presented bad values to the turbulent viscosit. Finall, the [7] turbulence model, the k- model, has presented the bad performance in hpersonic flows, as expected from the references [16-17]. The good emploment of this model is restricted to transonic and supersonic flows, as reported in [14-15; 18-19]. 4.5 Quantitative Analsis Table 3 shows the lift and drag aerodnamic coefficients calculated b the [] scheme in the turbulent cases. As the geometr is smmetrical and an attack angle of zero value was adopted in the simulations, the lift coefficient should have a zero value. The most correct value to the lift coefficient is due to the [8] turbulence model. The second best is due to the [9] turbulence model once that the [7] turbulence model ields an incorrect solution. Another possibilit to quantitative comparison of the laminar and turbulent cases is the determination of the stagnation pressure ahead of the configuration. [0] presents a table of normal shock ISBN:

10 wave properties in its B Appendix. This table permits the determination of some shock wave properties as function of the freestream ach number. In front of the VLS configuration, the shock wave presents a normal shock behavior, which permits the determination of the stagnation pressure, behind the shock wave, from the tables encountered in [0]. So it is possible to determine the ratio pr 0 pr from [0], where pr 0 is the stagnation pressure in front of the configuration and pr is the freestream pressure (equals to 1/ to the present dimensionless). Table 3. Aerodnamic coefficients of lift and drag. Turbulence odel: c L : c D : [5] turbulence model 5.88x [7] turbulence model 6.31x [8] turbulence model.57x [9] turbulence model -.6x Hence, to this problem, = 7.0 corresponds to pr 0 pr = and remembering that pr = 0.714, it is possible to conclude that pr 0 = Values of the stagnation pressure to the turbulent cases and respective percentage errors are shown in Tab. 4. The are obtained from Figures 3, 8, 13, and 18. As can be observed, the [9] turbulence model has presented the best result, with a percentage error of 1.06%. Table 4. Values of the stagnation pressure and respective percentage errors. Turbulence odel: pr 0 : Error (%): [5] turbulence model [7] turbulence model [8] turbulence model [9] turbulence model Finall, Table 5 exhibits the computational data of the present simulations. It can be noted that the most efficient is the [5] turbulence model, considering that the [7] turbulence model have presented wrong solution. As final conclusion of this stud, the [9] turbulence model was the best when comparing these four turbulence models: [5; 7-9]. This choice is based on the second best estimative of the lift aerodnamic coefficient, considering the right results, and the best estimative to the stagnation pressure. Table 5. Computational data. Turbulence odel: CFL: Iterations: [5] turbulence model ,358 [7] turbulence model ,14 [8] turbulence model ,461 [9] turbulence model ,67 5 Conclusion This work describes four turbulence models applied to hpersonic flows in two-dimensions. The [] scheme, in its first-order version, is implemented to accomplish the numerical simulations. The Favreaveraged Navier-Stokes equations, on a finite volume context and emploing structured spatial discretization, are applied to solve the cold gas hpersonic flow around a reentr capsule in twodimensions. Turbulence models are applied to close the sstem, namel: [5; 7-9]. The convergence process is accelerated to the stead state condition through a spatiall variable time step procedure, which has proved effective gains in terms of computational acceleration (see [10-11]). The results have shown that the [8] turbulence model ields the best results in terms of the prediction of the lift aerodnamic coefficient; however, the [9] turbulence model predicts the best value of the stagnation pressure. oreover, the [9] scheme also predicted the most severe pressure fields. Hence, the best choice is the [9] turbulence model. It is important to highlight the bad behavior of the k- turbulence model to hpersonic flows. As in [16-17], this model presents problems to this flow regime. Good results are obtained in the supersonic regime, [14-15; 18-19], but disappoint behavior is observed in the hpersonic regime. 6 Acknowledgment The first author thanks the financial support conceded b the CAPES under the form of a scholarship. The authors would like also to thank the infrastructure of ITA that allowed the realization of this work. eferences: [1] P. Kutler, Computation of Three-Dimensional, Inviscid Supersonic Flows. Lecture Notes in Phsics, Vol. 41, pp , [] B. Van Leer, Flux-Vector Splitting for the Euler Equations. Proceedings of the 8th International Conference on Numerical ISBN:

11 ethods in Fluid Dnamics, E. Krause, Editor, Lecture Notes in Phsics, Vol. 170, pp , Springer-Verlag, Berlin, 198. [3]. adespiel, and N. Kroll, Accurate Flux Vector Splitting for Shocks and Shear Laers, Journal of Computational Phsics, Vol. 11, pp , [4]. Liou, and C. J. Steffen Jr., A New Flux Splitting Scheme, Journal of Computational Phsics, Vol. 107, pp. 3-39, [5] T. Cebeci, and A.. O. Smith, A Finite- Difference ethod for Calculating Compressible Laminar and Turbulent Boundar Laers, Journal of Basic Engineering, Trans. ASE, Series B, Vol. 9, No. 3, pp , [6] B. S. Baldwin, and H. Lomax, Thin Laer Approximation and Algebraic odel for Separated Turbulent Flows, AIAA Paper 78-57, [7] W. P. Jones, and B. E. Launder, The Prediction of Laminarization with a Two-Equation odel of Turbulence, International Journal of Heat and ass Transfer, Vol. 15, pp , 197. [8] T. J. Coakle, Turbulence odeling ethods for the Compressible Navier-Stokes Equations, AIAA Paper , [9] P. S. Granville, Baldwin Lomax Factors for Turbulent Boundar Laers in Pressure Gradients, AIAA Journal, Vol. 5, pp , [10] E. S. G. aciel, Analsis of Convergence Acceleration Techniques Used in Unstructured Algorithms in the Solution of Aeronautical Problems Part I, Proceedings of the XVIII International Congress of echanical Engineering (XVIII COBE), Ouro Preto, G, Brazil, 005. [CD-O] [11] aciel, E. S. G., Analsis of Convergence Acceleration Techniques Used in Unstructured Algorithms in the Solution of Aerospace Problems Part II, Proceedings of the XII Brazilian Congress of Thermal Engineering and Sciences (XII ENCIT), Belo Horizonte, G, Brazil, 008. [CD-O] [1]. W. Fox, and A. T. cdonald, Introdução à ecânica dos Fluidos, Ed. Guanabara Koogan, io de Janeiro, J, Brazil, [13] L. N. Long,.. S. Khan, and H. T. Sharp, assivel Parallel Three-Dimensional Euler/Navier-Stokes ethod, AIAA Journal, Vol. 9, No. 5, pp , [14] E. S. G. aciel, A. P. Pimenta, and N. E. astorakis, Assessment of Several Turbulence odels as Applied to Supersonic Flows in D Part I, Submitted to the International Journal of athematical odels and ethods in Applied Sciences (under review). [15] E. S. G. aciel, A. P. Pimenta, and N. E. astorakis, Assessment of Several Turbulence odels as Applied to Supersonic Flows in D Part II, Submitted to the International Journal of athematical odels and ethods in Applied Sciences (under review). [16] E. S. G. aciel, and A. P. Pimenta, Comparison of Several Turbulence odels as Applied to Hpersonic Flows in D Theor, Submitted to the 1 st Pan American Congress on Computational echanics (PANAC 015) (under review). [17] E. S. G. aciel, and A. P. Pimenta, Comparison of Several Turbulence odels as Applied to Hpersonic Flows in D esults, Submitted to the 1 st Pan American Congress on Computational echanics (PANAC 015) (under review). [18] E. S. G. aciel, A. P. Pimenta, and N. E. astorakis, Assessment of Several Turbulence odels as Applied to Supersonic Flows in D Part III, Submitted to the International Journal of athematical odels and ethods in Applied Sciences (under review). [19] E. S. G. aciel, A. P. Pimenta, and N. E. astorakis, Assessment of Several Turbulence odels as Applied to Supersonic Flows in D Part IV, Submitted to the International Journal of athematical odels and ethods in Applied Sciences (under review). [0] J. D. Anderson Jr., Fundamentals of Aerodnamics, cgraw-hill, Inc., EUA, 1008p, 005. ISBN: