Further applications of scheme for reducing numerical viscosity: 3D hypersonic flow

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1 Journal of Physics: Conference Series Further applications of schee for reducing nuerical viscosity: 3D hypersonic flow To cite this article: S A Elaskar et al 2009 J. Phys.: Conf. Ser View the article online for updates and enhanceents. Related content - TVD Schee for Adiabatic/Isotheral Dinshaw S. Balsara - An RKDG finite eleent ethod for the one-diensional inviscid copressible gas dynaics equations in a Lagrangian coordinate Zhao Guo-Zhong, Yu Xi-Jun and Zhang Rong-Pei - Hierarchical reconstruction for solving nonlinear conservation laws Yingjie Liu, Chi-Wang Shu and Zhiliang Xu Recent citations - Atospheric Reentry Dynaics of Conic Objects J. P. Saldia et al This content was downloaded fro IP address on 23/10/2018 at 02:34

2 Further applications of schee for reducing nuerical viscosity: 3D hypersonic flow S A Elaskar 1-2, O A Falcinelli 1, J P Taagno 1 and J P Saldía 1 1 Dpto. de Aeronáutica. Universidad Nacional de Córdoba. 2 CONICET Av. Velez Sarfield Córdoba (5000), Argentina. E-ail: selaskar@efn.uncor.edu Abstract. A Total Variation Diinishing (TVD) schee ipleented in a non structured finite volue forulation for solving the 3D Euler equations is presented. To siultaneously, achieve adequate accuracy in sooth flows, high resolution at flow discontinuities and to avoid spurious oscillations, different flux liiter functions are applied in a wave-to-wave basis. By appropriately using copressive liiter functions with waves of the faily two to four (linear waves), and diffusive liiter functions with waves of the faily one and five (non linear waves), iportant reductions of the nuerical viscosity can be achieved. This sort of adaptive schee has satisfactorily been applied to the shock tube proble and to the slip interface between two parallel flows. It has also been used when coputing the flow over a spherically blunted bi-conic body at free strea Mach nubers varying fro five to fifteen. The results obtained with the blunted body are presented here, and they show that the new adaptive schee besides being less diffusive, it does not lose robustness regarding other TVD ethods. 1. Introduction When solving the fluid echanic equations nuerically using finite volue techniques, the necessity of coputing convective fluxes arises. Traditionally, polynoial functions were used to approach the variations of variables. This approach gives excellent results when the variables undergo sooth variations but it has serious difficulties if the solution contains discontinuities. In these cases, the schees that use second or higher order approxiations present inconveniences during the convergence process and the solutions have oscillations next to the discontinuity. On the other hand, the schees that use first order approxiations generate solutions without oscillations but, the discontinuities ay poorly be resolved (schee highly diffusive). To deal with this proble, flux liiter functions were built as linear cobinations of first and second order approxiations [1,2]. These schees are known as TVD although, strictly speaking, the TVD condition has forally only been deonstrated for scalar convection equations. If in the linear cobination, the first order approach has ore weight than that of second order, the resultant schee is diffusive and reciprocally, if the second order approach has ore weight the schee is copressive. When extending these concepts fro systes of linear equations to non linear, the certainty that they continue being TVD is lost [3], and it becoes necessary to take decisions about as how to carry out such extension. By instance: To apply only one liiter function or so any as it is the nuber of equations. To apply the liiter functions according to the spectral decoposition of related variable jups [3]. What variable ust be kept in ind for the calculation of the scalar liiter function. For the ore robust schees the nuber of liiter functions is equal to the nuber of equations; and in addition, a spectral decoposition is used [3]. For the three-diensional Euler equations syste, the spectral decoposition akes that three lineally degenerate failies of waves appear [4]. Discontinuities associated with these eigenvectors c 2009 Ltd 1

3 are very difficult to solve exactly except for schees that use higher copressive liiter functions, however, these schees are not very robust in solving discontinuities associated with the non linear wave failies [2]. In soe papers, looking for the best way to capture contact discontinuities without the characteristic searing linked to degenerate linear waves, different schees were proposed. One of the ost popular was introduced by Harten [5] using an Artificial Copression Method (ACM) that renders the local slope of the inod liiter steeper; however, if the field becoes genuinely nonlinear the liiter needed should not be so copressive [6]. The ACM ipleented on superbee liiter functions [7,8] was not utilized in solving the gasdynaics equations [6], probably because it was recognized that it was not a high-resolution gasdynaics shock-capturing technique [3]. The ACM was also ipleented on inod liiter functions to solve the scalar wave and Burgers equations [6], and in second order central difference schees for Euler equations [9]. More recent ACM applications are given in Ref. [10]. In this work an alternative schee is presented, which has the capacity to solve satisfactorily discontinuities associated with lineally degenerate wave failies and without losing robustness in dealing with gasdynaics shock waves. This new schee was ipleented in a three-diensional coputational code to solve the Euler equations using a non-structured finite volue technique. 2. Description of the schee The three-diensional Euler equations can be written as: U t + F = 0 (1) U is the vector of conservative variables, and F is the 3D vectorial flow. The teporal change of the conservative variables can be expressed as: lfaces n+ 1 n t * = l l l Vol l= 1 U U F n A (2) where the flux of the conservative variables F has been replaced by the nuerical flux tensor F. Equation 2 allows the use of a locally aligned syste of coordinates whose unit vector i coincides with the noral to the face l of the cell, and the unit vectors j and k are tangential directions. To achieve second order accuracy the nuerical flux at the interface between cells l and l+1 in the direction noral to the face l [11], is calculated by: * * fi + fi+ 1 1 i+ 1/2 = + Φi+ 1/2 ri+ 1/2 2 2 f (3) where f i and f i+1 are the physical fluxes noral to the face in each cells, eigenvector, and Φ i+ 1/2 is, in the original Harten-Yee schee [12-14], defined as: r i + 1/2 is the -th right i+ 1/2 gi gi+ 1 i+ 1/2 i+ 1/2 i+ 1/2 Φ = + λ +γ α (4) being: 2

4 g ( + 1/2 + 1/2 S 1/2 1/2 ) S = ax 0, in λ α, λ α 2 i i i i i ( i 1/2 ) S = sign λ + (6) (5) gi+ 1 gi si αi 1/2 0 + α i+ 1/2 γ i+ 1/2 = 0 si α = 0 i+ 1/2 (7) where α i+ 1/2 is the jup of the conserved variables across the interfaces between cells i and i+1, and λ i 1/2 is the -th eigenvalue of the Jacobian atrix. Since the local Rieann proble is solved with rotated data, the eigensyste is calculated in the locally aligned coordinate frae. The liiter function given in Eq. (5) is known as inod [1-4]. The inod selects the iniu possible value, so that the schee is TVD. The other end is the liiter function superbee that ponders the contribution of the high order flux [2]. The ipleentation of the superbee function leads to an excessively copressive schee which it is not very robust for general practical aerospace applications. In the nuerical solution of the three-diensional Euler equations five wave failies appear. If the five wave failies are enuerated in correspondence with their speed, being one the slowest and five the faster, it can be deonstrated that for waves of the failies two to four, the characteristic velocities at both sides of the discontinuity are the sae and equal to the velocity discontinuity [2,3]. This property akes very difficult to solve theses waves accurately, unless they are solved diffusely. In this work, it is explored the possibility of ipleenting different liiter functions for different wave failies. The objective is to iprove the nuerical resolution of the discontinuities associated with the failies two to four using copressive liiter functions (superbee), and without losing robustness due ainly to the use of diffusive liiter functions (inod) for the wave failies one and five. To introduce in the nuerical fluxes calculations the liiter function superbee the Eq. (5) is replaced by the following expression: g i 0 if α α < 0 = i+ 1/2 i 1/2 1 ax 0,in ( 2 r,1 ),in ( r,2 ) λi 1/2 αi 1/2 if αi+ 1/2 αi 1/2 0 2 (8) being: i+ 1/2 αi+ 1/2 λ r = λ α (9) i 1/2 i 1/2 To iprove the overall schee robustness, the ipleentation of different liiter functions is carried out only in those cells interfaces where the greater relative intensities of the discontinuities in 3

5 central waves are registered, and using the conventional Harten-Yee TVD schee in all other cases. Notice that the coparison aong the intensity of the waves cannot be ade using directly the coefficients of the spectral decoposition ( α i +1/2 ) since these coefficients depend on the odule assigned to each eigenvector. In the local coordinate syste adopted for coputing the nuerical fluxes across each face, the corresponding eigenvectors are given by: u u a 0 0 u a + v r 1 = v r 2 = r 3 = 1 r 4 = 0 r5 = v w w 0 1 w u v w H u a + + v w H + u a 2 (10) 1 i+1/2 2 i+1/2 5 α i+1/2 where H is the stagnation enthalpy. It can be deduced fro Eq. (10) that α, α and 3 4 easure the density jup in the waves 1, 2 and 5 respectively and that α i+1/2 y α i+1/2 easure the oentu jup in waves three and four. To copare these jups it becoes necessary the select reference values for the density and velocity. Thus: αi+ 1/2 αi+ 1/2 αi+ 1/2 αi+ 1/2 αi+ 1/2 1 = 2 = 3 = 4 = 5 = ρref ρref ρref uref ρref uref ρref I, I, I, I, I (11) In this investigation is taken as density reference ρ ref = 0.5 (ρ i + ρ i+1 ) and as the velocity reference the average of the sound velocities at the cells u ref = 0.5 (c i + c i+1 ). Finally, if the axiu of I 1, I 5 is higher than the axiu of I 2, I 3, I 4 the conventional Harten- Yee TVD schee is used; otherwise, the values of g i 2, g i 3, g i 4 are calculated with the liiter function superbee and g i 1, g i 5 with the liiter function inod. 3. Ipleentation 3.1 The liiter functions evaluation For the evaluation of g i and g i + 1 in Eq. (4), it is necessary to calculate the spectral decopositions of the conservative variables increents at the interfaces i-1/2, i+1/2 and i+3/2. In the context of three-diensional not structured eshes of tetrahedrons, the identification of the cells i and i+1 is intuitive (they are two cells that share a face) but the deterination of the points i-1 and i+2 is not direct. If two tetrahedrons that share a face are analyzed, the nodes not belonging to the coon face can be used as representative points for i-1 and i+2. Then, these points can be used as iaginary cells. In this work these ideas have been ipleented, being the nodal values calculated as a pondered average of the conservative variables between all cells that are in contact with the nodes i-1 and i+2. Such pondered average is given by [15]: U node k = n d U cell i cell i i= 1 GCcell i -node k n d 1 i= 1 GC -node k (12) 4

6 where d GCcell i -node k is the distance that separates the gravity center of the cell i fro the node k, and n is the cell nuber in contact with the node k. 3.2 Boundary conditions The treatent of the boundary conditions is carried out through the iaginary cells technique [2,3]. Five different types of boundaries are considered: 1 Subsonic inlet, 2 - Supersonic inlet, 3 Subsonic exit, 4 - Supersonic exit, 5 - Non penetration (solid boundary and syetry). 4. Test cases In order to verify the accuracy and robustness of the adaptive schee three test cases were siulated. The first one is the flow inside of a shock tube. This exaple was used to explore the capacity of the adaptive schee which has been described, to odel contact discontinuities; the flow does not have velocity regarding the contact discontinuity (discontinuity in waves of the faily 2). The second test is the siulation of a slip interface layer between two flows with different velocities and densities, but equal pressures. This test was chosen to study the capacity of the schee to solve flows in which the discontinuity is in the velocity tangent to a given interface (discontinuity in waves of the failies 3 and 4). The third case is the flow around a spherically blunted bi-conic body at several supersonic Mach nubers. The object of this third case was to evaluate the robustness of the schee to capture strong shock waves (discontinuity in waves of the failies 1 and 5). Although the ain objective of this work is to present coputer results pertaining to the third case, soe brief coents on the other two cases are appropriate. A ore detailed description of test cases one and two ay be found in Ref [15]. 4.1 Flow inside a shock tube (ST) It is well known that an analytical solution for the flow inside a ST can be built considering three types of waves: shocks, expansion fans and contact discontinuities [2-4]. Figures 1 and 2 show analytical and coputed solutions in ters of the density plotted as a function of x/l (being x the distance along the tube and L its total length). In Figure 1, the analytical solution is copared with the results obtained applying both, the adaptive TVD schee here proposed and the conventional Harten-Yee TVD schee. It can be appreciated that the siulation of the non linear waves is equally good for both schees TVD, however, the capture of the contact discontinuity has been iproved notably. In Figure 2, the results obtained applying the adaptive schee is copared with those obtained using superbee liiter functions in all waves. It shall be rearked that the results obtained using the flux liiters superbee have a tendency to show oscillations, which are not present if the new schee is used. 4.2 Slip surface The second test case is depicted by an air layer flowing over another one. The velocities and densities in both flows are different, but the pressures are equal (no forces can be supported by the slip surface). The straightforward solution predicts the slip of a flow on another without interferences. However, due to the nuerical viscosity, the coputed solution produces an apparent ixture zone that gets wider downstrea the end of an assued splitter plane. The spreading of such unphysical ixing region can be interpreted as a easure of the accuracy of the nuerical ethod. In Figure 3 is presented the absolute value of the percentage of error incurred in the velocity calculated at a lateral plane of the coputational doain [15]. The results using the traditional Harten-Yee TVD schee, are shown in the upper part, those obtained with the new schee in the iddle, and in the lower part, are shown the results calculated using superbee liiter functions in all waves. 5

7 superbee liiter function in waves 2, 3 and 4 and inod liiter function in waves 1 and 5 inod liiter function in all waves exact solution Density [Kg/^3] x/t [] Figure 1. Shock tube results. Coparison between analytical solution, Harten-Yee TVD and proposed schee Density [kg/3] SUPERBEE - SUPERBEE - SUPERBEE - SUPERBEE - SUPERBEE NEW SCHEME x/t [] Figure 2. Shock tube results. Coparison between schee using superbee liiter function for all waves and the new schee. 6

8 Figure 3. Slip surface results. Error: white 0% e 2%, green 2% e 5%, yellow 5% e 10%, light blue 10% e 20%, red 20% e. As it can be deduced fro Figure 3, the proposed adaptive schee is notably less diffusive that the conventional Harten-Yee ethod, although ore diffusive than the schee using superbee in all five waves of the Euler equations. 4.3 Spherically blunted bi-conic body at supersonic speeds It was analyzed the flow around the spherically blunted bi-conic configuration shown in Figure 4. The nose radius is of 0.15, the aft part of the nose is conical with an angle of 10º and a length of 0.3, and there after, there is another conical part with an angle of 20º and 0.2 of length. Two Mach nubers, 5 and 15 were considered for nuerical siulations. By eans of these nuerical studies, the following properties of the coputer progra were intended to be analyzed: The quality of the results describing the flow on the spherical nose. The ability for capturing accurately the shock wave between the first and second cone. The preservation of the axial syetry of the flow despite using a 3D code. The potential of the new schee for capturing strong shock waves. It is realized nevertheless, that the results at M = 15 can be questioned because real gas effects are not accounted for. The esh, shown in Figure 5, has cells and nodes. The boundary conditions are: (1) supersonic flow enters through the control volue front; (2) supersonic flow exits through the back plane and (3) non penetration on the body and in the planes y-x and z-x. The flow that enters is air and has a density of 1.225Kg/3, and a pressure of Pa. The free strea coponents in the directions of the z- and y-axis are null, and in the direction of the x-axis are /s for M = 5, and /s for M = 15. 7

9 R X Meeting on Recent Advances in the Physics of Fluids and their Applications Figure 4. Spherically blunted bi-conic body Y Z X Y X Figure 5. Mesh used in the blunted bi-conic body case. In Figures 6 and 7 pressure distributions on the body for M = 5, are presented. Figure 6 shows the pressure distribution on the body, however due to the picture scale only the nose shock appears clearly defined. In Figure 7 the pressures plotted are of all the cells in contact with the body (fro 0 o to 90 o ). It can clearly be perceived the axial syetry of the results and the accurate capture of shock waves. The distributions of pressure agree satisfactorily if they are copared with results presented by other authors [16]. In Figures 8 and 9, the pressure distributions are plotted for M = 15. In Figure 8, again is shown the pressure distribution produced by the priary or nose shock. The resolution of the discontinuities is satisfactory, oscillations are not present and the axial syetry of the flow is aintained appropriately. 8

10 Figure 6. Pressure distribution over axial-syetric body. M = 5. Priary shock P/Pinfinito Distance fro de ipact point [] Figure 7. Pressure distribution over axial-syetric body. M = 5 external flow. It can be appreciated fro Figures 7 and 9 that the spread of the pressure results, due the cells arrangeents on the body at each x-station, is very sall copared with related local values. Figure 10 shows the shock in front of the blunt nose of the body at M = 15 as it is deterined by the new adaptive schee and by the traditional Harten-Yee TVD technique. It can be observed that the two figures look very uch alike. Consequently, the new schee captures strong shock waves as well as it does the traditional Harten-Yee. Furtherore, Figure 11 which is a close up of the blunt body nose region, shows that the shock is captured between two grid cells only. 9

11 Figure 8. Pressure distribution over axial-syetric body. M = 15. Priary shock P/Pinf Distance fro the ipact point [] Figure 9. Pressure distribution over axial-syetric body. M = 15 external flow. It is iportant to notice that a nuerical schee which uses only superbee flux liiters to siulate blunted bodies proble, introduces oscillations, produces negative pressures near discontinuities and it ends up being unstable. To ephasize this last point, two cases with different ways of introducing initial conditions were considered. In the first one, starting constant values typical of a free strea flow were used, and in the second one, the siulation was started fro a steady-state results obtained after using the new adaptive schee. In both cases, the schee using superbee functions becae unstable. 10

12 Figure 10. Shock wave over axial-syetric body at M = 15. Right figure: Traditional Harten-Yee. Left figure: adaptive schee Figure 11. Close up of the blunt body nose region. (1) p/ p > 272; (2) 224 < p/ p < 272 ; (3) 176 < p/ p < 224 ; (4) 128 < p/ p < 176 ; (5) 80 < p/ p < 128 ; (6) 33 < p/ p < 80 11

13 5. Conclusions In previous test cases, it has been proven the capacity that the adaptive TVD schee here described has to odel contact discontinuities (ST test case), and to siulate interfaces between two flows with different velocities and densities (slip surface test case). Now, its potential to capture strong pressure discontinuities (shock waves) in supersonic flows has been tested. After coparing results obtained with the proposed TVD schee, with those obtained using the conventional TVD of Harten-Yee and with other solver that uses superbee functions as liiters for all nuerical fluxes, the following conclusions can be written. Fro the Shock Tube test case: The adaptive TVD schee diinishes the nuerical viscosity with regards to the conventional Harten-Yee TVD and it does not introduce oscillations. Fro the Slippery Surface test case: The adaptive TVD schee works ore efficiently than the conventional Harte-Yee TVD. However, the schee that only uses superbee liiters functions produces less nueric diffusion than the other two. Fro the Blunted Body test case: The adaptive TVD schee proposed capture strong shocks as well as it does the conventional TVD of Harten-Yee. The uch higher resolution of discontinuities (the shock in the nose region is captured between two grid cells, only), does not affect its robustness. No difficulties have ever been experiented in coputing the flow around a blunted bi-conic body in the free strea Mach nuber range of 5 to 15. Attepts to apply the nuerical schee that only uses superbee liiters functions failed (generate oscillations, produces negative pressure and it finishes being unstable). Acknowledgeent This work has been supported by eans of grants PIP No 5692 of the Consejo Nacional de Ciencia y Tecnología (CONICET) of Argentina, PICTO2005-UNRC: of the Secretaría de Ciencia, Tecnología e Innovación Productiva (SECYT) of Argentina, and of the Universidad Nacional de Córdoba. References [1] Sweeby P 1984 High resolution schees using flux liiters for hyperbolic conservation laws SIAM Journal on Nuerical Analysis [2] Hirsch C 1992 Nuerical Coputation of Internal and External Flows, Vol.2 Coputational Methods for Inviscid and Viscous Flows (London: John Wiley & Sons Ltd.). [3] Toro E 1999 Rieann Solvers and Nuerical Methods for Fluid Dynaics (Berlin: Springer- Verlag). [4] Leveque R 1992 Nuerical Methods for Conservation Law (Basel: Birkhäuser Verlag). [5] Harten A 1978 The artificial copression ethod for coputation of shocks and contact discontinuities: III self adjusting hybrid schees Matheatics of Coputation [6] Rider W 1992 The design of high-resolution upwind shock-capturing ethods Los Alaos National Laboratory. [7] Rider W and Woodruff S 1991 High-order solute tracking in two-phase theral hydraulics LA- UR Los Alaos National Laboratory. [8] RELAP5-3D Code Developent Tea 2005 RELAP5-3D code anual. Volue 1: code structure, syste odels, and solution ethods Idaho National Engineering and Environental Laboratory. [9] Lie K and Noelle S 2003 On the artificial copression ethod for second-order nonoscillatory central difference schees for systes of conservation laws SIAM Journal of Scientific Coputation [10] Lo S Blaisdell G and Lyrintzis A 2007 High-order shock capturing schees for turbulence calculations AIAA Paper Nº [11] Udrea B 1999 An advanced iplicit solver for MHD PhD Thesis, University of Washington. 12

14 [12] Harten A 1982 On to class of high resolution total-variation-stable finite difference schees Technical Report NYU. [13] Harten A 1983 High resolution schees for hyperbolic conservation laws Journal of Coputational Physics [14] Yee H Waring R and Harten A 1985 Iplicit total variation diinishing (TVD) schees for steady-state calculations Journal of Coputational Physics [15] Falcinelli O Elaskar S and Taagno J 2008 Reducing the nuerical viscosity in non structured three-diensional finite volues coputations Journal of Spacecraft and Rocket, AIAA [16] Pirolo J and Wardlaw Jr. A 1991 High teperature effects for issile type bodies using the Euler solver ZEUS AIAA Paper Nº