An unstructured Shock-Fitting solver for two ...

An unstructured Shock-Fitting solver for two-dimensional flows


Renato Paciorri , Aldo Bonfiglioli Dipartimento di Meccanica e Aeronautica, University of Rome, La Sapienza, Italy E-mail: [email protected]


Dipartimento di Ingegneria e Fisica dell’Ambiente, University of Basilicata, Italy E-mail: [email protected] Keywords: Computational Fluid Dynamics, Shock-Fitting, unstructured-grids. SUMMARY. A new floating Shock-Fitting technique featuring the explicit computation of shocks by means of the Rankine-Hugoniot relations has been implemented on unstructured grids. This paper illustrates the algorithmic features of this orginal technique and the results obtained in the computation of the hypersonic flow past a circular cylinder and a Mach reflection. 1 INTRODUCTION Shock waves occur very frequently in nature and technological applications. Their presence characterizes compressible flows not only in aeronautics and aerospace, but also in other areas of theoretical and applied physics. The extracorporeal shock wave lithotripsy, used for breaking kidney stones, and sonoluminescence, which is the emission of short bursts of light from imploding bubbles in a liquid excited by sound, are two such examples arising in different fields. In all flows where shock waves occur, these play an important role that affects the overall flow behavior. We could mention numerous examples testifying the importance of shock waves and their effects. Among these are: the flow around space vehicles re-entering the atmosphere at hypersonic speeds, the buffeting phenomenon induced by oscillating shock waves on transonic wings and turbomachinery blades, shock-induced noise and shock-induced boundary-layer separation that occurs inside highly over-expanded nozzles. It is therefore not surprising that computing shocks correctly has always been one of the most important issues in the numerical simulation of compressible flows. The numerical models used in the simulation of compressible flows can be cast into two main categories: Shock-Capturing (S-C) and Shock-Fitting (S-F). The technique for compressible flow computations known as S-F was proposed and developed by Gino Moretti [Moretti and Abbett, 1966, Moretti, 1987]. It consists in finding the precise shock location and computing the shock motion by means of the Rankine-Hugoniot (R-H) equations. Besides Moretti, other researchers applied this technique in the seventies and eighties [Salas, 1976, Marconi and Salas, 1973, Yamamoto et al., 1984] and in the early nineties [Hartwich, 1990, Dadone and Fortunato, 1994]. In those years, the S-F technique, used in conjunction with numerical schemes based on the quasi-linear equations, allowed to accurately compute flows with strong discontinuities using the modest computational resources available at that time. In the nineties, however, the development of modern S-C schemes based on the integral conservation equations along with the strong growth in computing resources diminished the interest in S-F schemes, which were considered too complex and not general enough as compared to S-C schemes. The development and the application of the S-F technique was pursued with good results by only one research team [Nasuti and Onofri, 1998]. Despite the widespread use of S-C codes, shock solutions obtained by means of S-C schemes are plagued by a number of problems pertaining to accuracy [Lee and Zhong, 1999, Carpenter and Casper, 1999, 1

Roy, 2003], stability [Pandolfi and D’Ambrosio, 2001] and, more in general, solution quality. Despite the continuous efforts made over the last 20 years, these shortcomings have not been completely overcome and appear to be particulary severe when unstructured-grids are used [Gnoffo and White, 2004]. The S-F technique, on the contrary, does not suffer from these numerical problems but, as it was said before, it has been abandoned over the last decades. We believe there exist today good reasons for reconsidering the S-F technique in the context of unstructured grids. During their historical development, Computational Fluid Dynamics (CFD) techniques have first been developed to use structured grids. Compared to unstructured-grid methods, the former offer a number of advantages, most notably a reduced algorithmic complexity which results in improved computational efficiency. Nevertheless, starting in the early nineties, the CFD community has shown an increasing interest in unstructured-grid techniques. This is mainly due to the following features, which are peculiar to unstructured grids: i) the ability to describe complex geometries and ii) the possibility of locally adapting the grid to follow the flow features. This latter capability makes them particularly well suited to simulate compressible flows, which may develop discontinuities such as shock waves and contact discontinuities. Moretti’s S-F technique was born at a time when CFD practitioners used solvers based only on structured meshes. The algorithmic complexity and the difficulties encountered in the implementation of the S-F methodology within structured-grid solvers has been one of the main reasons for the loss of interest in S-F. In the traditional implementation of the S-F technique, the discontinuities are made up of shock points connected by shock segments which move over and independently of the background structured grid. Moreover, new shock points may arise or disappear when shocks interact or weaken into Mach lines or when compression waves coalesce into a new shock. The algorithmic tools that can most effectively handle the motion of the shocks, their intersections, etc. are far from the IJK data structures used in structured-grid methods. On the contrary, the unstructured grid technology appears to be much better suited to handle the motion, insertion and removal of shock points as well as the local re-meshing that is required when the discontinuities move over a background mesh. To our knowledge, only a few attempts have been made to incorporate S-F ideas within S-C unstructured-grid solvers. These include the work by Van Rosendale [Rosendale, 1994], Parpia and Parikh [Parpia and Parikh, 1994], Tr´epanier [Trepanier et al., 1996] and co-workers and, more recently, H¨anel [Gloth et al., 2003] and co-workers. The strategies adopted in [Rosendale, 1994, Parpia and Parikh, 1994, Trepanier et al., 1996] are similar: rather than explicitly solving the R-H relations to compute the shock, use is made of the property of Roe’s approximate Riemann solver [Roe, 1981] to return the exact jump relations when the discontinuity is aligned with the mesh. Therefore, all three methods try to locally align the edges of the triangulation with the discontinuities by means of grid motion and/or adaptation. On the contrary, in [Gloth et al., 2003] the grid is stationary and made of arbitrary polygonal elements. The shock motion is computed using a level-set method. The local shock speed is computed by solving the R-H equations with values on both sides of the discontinuity obtained by one-sided extrapolation. A special treatment is also required to compute the flux integrals in those cells that are crossed by the shock front. The present paper tries to revive Moretti’s original floating S-F technique by formulating it as a local adaption algorithm on unstructured triangular grids and coupling it with an unstructured, vertex-centred, S-C code. The use of a S-C code, rather than a solver based on the quasi-linear equations, as was tipically done in the traditional S-F implementation, allows to use an hybrid technique in which, for example, the strongest shock is fitted and the other shocks are captured. This could be

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advantageous particularly in the three dimensional case, where fitting interacting shocks might be hard to handle analitically. In the proposed approach, the shock front is discretized as a polygonal curve and treated as an internal boundary for the CFD code: its local speed and downstream nodal values are computed by coupling the upstream moving wave predicted by the S-C code with the R-H relations. The discontinutity is allowed to move over and independently of a background triangular grid which is locally adapted to ensure that the edges of the shock are part of a constrained Delaunay triangulation (CDT) that covers the entire computational domain. Results are persented for the bow shock that is formed ahead of a circular cylinder moving at hypersonic speed and a Mach reflection. 2 SHOCK FITTING ALGORITHM The present S-F algorithm is able to compute single shocks occurring within a two-dimensional flow field discretized by an unstructured mesh made of triangles. In its current version, the algorithm does not include any procedure for shock detection. Therefore, the fitted shocks must be present within the flow domain at the beginning of the simulation. In order to illustrate the algorithmic features of the proposed methodology, let us consider a twodimensional domain and a shock front crossing this domain at time level , see Fig. 1a). The shock front is shown in bold by a polyline whose endpoints are the shock points, marked by squares. A background triangular mesh, whose nodes are denoted by circles, covers the entire computational domain and the position of the shock points is completely independent of the location of the nodes of the background grid. While each node of the background mesh is characterized by a single set of state variables, two sets of values, corresponding to the upstream and downstream states, are assigned to each shock point. We assume that at time the solution is known in all grid and shock points. The computation of 
 the subsequent time level can be split into several steps that will be described in detail in the following sub-sections. Shock

shock Upstream

Downstream d

Downstream

l

Upstream τ n

Cells enclosing phantom nodes

Phantom nodes

τ n Cells crossed by shock

a)

b)

Figure 1: a) shock front moving over the background triangular mesh at time : dashed lines mark the cells to be removed; dashed circles denote the phantom nodes; b) the triangulation around the shock has been rebuilt.

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2.1 Cell removal around the shock front The first step consists in removing the cells around the shock front. More precisely, all cells of the background mesh that are crossed by the shock line are removed along with the mesh points which are located too close to the shock front. These nodes are declared as “phantom” nodes and all cells having at least one phantom node among their vertices are also removed from the background triangulation, see dashed circles and cells in Fig. 1a). 2.2 Local re-meshing around the shock front Following the cell removal step, the remaining part of the background mesh is split into two regions separated by a hole containing the shock. The hole is then re-meshed using a CDT: both the shock segments and the edges belonging to the boundary of the hole are constrained to be part of the final triangulation, see Fig. 1b). No further mesh point is added during this triangulation process. At this stage, the computational domain is discretized by a modified mesh in which the shock points and the shock edges are part of the triangulation. This mesh differs from the background mesh only in the neighbourhood of the shock front. 2.3 Computation of the tangent and normal unit vectors The tangent and normal unit vectors along the shock front have to be computed in each shock point since this is required to properly compute the jump relations, see Fig. 1b). The computation of these vectors in a generic shock point is carried out through specific finite-difference formulas which involve the coordinates of the shock point itself and of its neighbouring shock points belonging to its domain of dependence. 2.4 Solution update using the capturing code
 The solution is updated at time level using the unstructured, S-C code and the modified mesh as input. The modified mesh is split into two non-communicating parts by the shock front which is treated by the code as if it were an internal boundary, see Fig. 2a). This is achieved by replacing each shock point by two superimposed mesh points: one belonging to the downstream region and the other to the upstream region. The downstream and upstream states are correspondingly assigned to each pair of shock nodes. A single time step calculation is performed by the unstructured
 solver which provides updated nodal values within all grid and shock points at time level . However, the downstream state of the shock points has not been correctly computed except for the following combination of variables:


 

  and  where  is the shock normal, 

 
  (1)        are the values of the acoustic and flow velocity of

the downstream state of the shock nodes computed by the unstructured solver. 2.5 Shock calculation Each shock point is characterized by an upstream state, a downstream state and a shock speed,  . The upstream state has been updated at time level
 in the previous step 2.4. Concerning
the downstream state, only the quantity has been correctly updated. The complete downstream state and the shock speed are computed through a system of five algebraic equations where four equations are the R-H relations written with respect to the shock point moving at the unknown speed

4

Shock point Upstream state

Downstream state

Phantom node

Cell containg the phantom node

a)

b)

Figure 2: a) the shock is treated as an internal boundary; b) interpolation of the phantom nodes.



. Finally, the last equation of the system is the variable combination of Eq.1 where the value has been computed in the previous step 2.4.




2.6 Interpolation of the phantom nodes
 In the previous steps all nodes of the modified mesh have been updated at time level . Nevertheless the phantom nodes, which belong to the background mesh but had been excluded from the modified mesh, have not been updated, see Fig. 2b). The update of the phantom nodes is performed by first locating each of these within the modified mesh (a search operation which can be efficiently performed on a Delaunay mesh) and then using linear interpolation. 2.7 Shock displacement
 is computed using the shock speed calculated The new position of each shock point at time in the previous step 2.5 by the following first order integration formula:




   




(2)

The new position of the shock line is obtained, see Fig. 3, by displacing all shock points according to Eq. 2. 2.8 Interpolation of the jumped nodes During the shock displacement step described in section 2.7, it may occur that the shock passes over several nodes of the background mesh. For instance, in Fig. 3 a mesh node whose position at time level was located in the upstream region, is jumped by the shock, so that its position at
 time is in the downstream region. The state of these “jumped” nodes has to be correctly re-computed. This is achieved through linear interpolation. More precisely, the interpolation uses the states of the two shock points belonging to the shock edge jumping the mesh point. In the case shown in the Fig. 3, the jumped node is re-computed by interpolating between the downstream states
of the shock points  and   (marked in red).

5

t

Pι+1

t+ ∆ t

jumped node t+ ∆ t

Pιt

jumping edge

P ι+1

wι

A n ∆t

t+ ∆ t

Pι

Figure 3: Mesh points jumped by the shock and their re-computation. 3 UNSTRUCTURED SOLVER As described in section 2, the proposed S-F algorithm provides a time dependent mesh in which the discontinuity is treated as an internal boundary of zero thickness. Communications with the CFD code are only required in step 2.4, when the modified grid containing the shock front is fed to the code which returns provisional values of the nodal variables at the subsequent time level. This reduced amount of communication allows to treat the CFD code as a black box, thus making the proposed S-F algorithm re-usable in conjunction with virtually any vertex-centred unstructured CFD code. It is however worth briefly describing the EULFS solver used to produce the numerical results presented in Sect. 4. A more detailed description can be found in [Bonfiglioli, 2000]. In the testcases presented in Sect. 4, which are restricted to two spatial dimensions, the governing conservation laws are the Euler equations for a non-viscous, non-conducting, perfect gas. The computational domain is partitioned into a set of non-overlapping triangular cells. Roe’s parameter vector [Roe, 1981] is chosen as dependent variable: it is stored in the vertices of the mesh and assumed to vary linearly and continuously in space, just as with iso-P1 linear Finite Elements (FE). The control volumes over which the system of conservation laws is discretized are the median dual cells. To enhance accuracy, the Euler equations are solved in preconditioned form, using the vanLeer, Lee, Roe matrix [van Leer et al., 1991]. This approach is often referred to as the Hyperbolic-Elliptic (HE) splitting as it allows to achieve full or partial (depending on the flow regime) diagonalisation of the original unsteady Euler system. In supersonic flows, this is re-cast into a set of four scalar equations describing convection of entropy and total enthalpy along the streamlines and convection of the acoustic variables along the Mach lines. In subsonic flows, the acoustic component is described by a coupled 2 2 Cauchy-Riemann system, while entropy and total enthalpy are convected along the streamlines, just as in supersonic flows. The spatial discretization relies upon the so-called Residual Distribution (RD) (or fluctuation splitting) schemes [Deconinck et al., 1993a] which share common features with both FE and Finite Volume (FV) techniques. RD schemes operate by evaluating the flux integrals (or cell residuals) over each triangular cell and distributing them among the forming vertices. The nodal residual, which is driven to zero as steady state is approached, is obtained by assembling the fractions of cell residuals scattered from the elements surrounding the node. A distinctive feature adopted in many RD codes is that the flux balance over a triangular cell in not computed using numerical quadrature, but using

6

the quasi-linear form of the equations and a multi-dimensional generalisation of Roe’s linearisation [Deconinck et al., 1993b]. Contrary to most FV schemes, no Riemann problems are involved in RD schemes since the functional representation of the dependent variables is continuous across the elements. The discretization of the viscous fluxes can be shown to be equivalent to a standard FE discretization, while the inviscid fluxes are distributed in an upwind-biased fashion. These upwinding directions are based upon the eigenstructure of the quasi-linear form of the equations, as described in details in [Bonfiglioli, 2000]. Concerning time integration, a simple Euler explicit scheme has been used and mass lumping applied to the time derivative term. 4 APPLICATIONS 4.1 Blunt body problem The first test case deals with the high speed flow past a circular cylinder at free-stream Mach

 . This and similar testcases have been used by other authors to assess the number, accuracy of high order S-C schemes within a structured grid setting. It has been observed in [Lee and Zhong, 1999, Carpenter and Casper, 1999, Roy, 2003] that, when captured shock waves are present, the order of accuracy of the computed solution downstream of the shock wave is reduced below the design order of the scheme, tipically to first order only. This is due to the fact that the high order scheme needs to be reduced to a first order scheme across the shock wave to capture the discontinuity without oscillations. The result is that the numerical solution is just first-order accurate downstream of the discontinuity, regardless of the design accuracy of the discretization used. In a S-F code this is not an issue since the discontinuity is explicitly computed by means of the R-H equations and elsewhere the flow is smooth so that higher order schemes can be used everywhere. The present calculations have been performed using the EULFS code both in S-C and S-F mode.






Table 1: Nodes and cells of the background and modified meshes. level

Background mesh cells nodes

coarse fine

610 2544

Initial shock nodes

351 1365

29 61

Final modified mesh cells nodes shock nodes 624 358 29 2580 1383 57

Two grids of increasing spatial resolution (see Tab. 1) have been created by specifying the distribution of the boundary nodes. These are evenly spaced and the distance between two adjacent boundary nodes of the coarse mesh equals R/12.5, R being the radius of the circular cylinder. The fine mesh is obtained by halving the distance between two adjacent boundary nodes and remeshing the interior of the computational domain. For this reason the number of cells and background grid nodes is approximately four times that of the coarse mesh, whereas the number of shock points is nearly twice. Different RD schemes have been selected when the code was used in S-C and S-F mode. In both cases, the entropy and total enthalpy scalar convection equations were discretized using the non-linear, monotone and second-order accurate, scalar PSI scheme. Concerning the two scalar equations describing convection along the Mach lines in supersonic regions and the 2 2 CauchyRiemann system in subsonic regions, different schemes have been used in capturing and fitting mode. 7

For the S-F solutions, the linear, second order accurate, and therefore not necessarily monotonicitypreserving, system LDA scheme, could be used since the shock is fitted and the flow-field is smooth elsewhere. In the S-C solutions, a non-linear system scheme (B scheme) has been used which is a blending of two linear system schemes: the first order, monotone N scheme and the system LDA scheme. Non-linearity is brought into the scheme by the blending function [Abgrall, 2001] which is able to switch between the N scheme in the neighbourhood of a discontinuity and the LDA scheme in smooth regions of the flow field. The solution computed by the unstructured code in S-C mode was used to initialize the flow-field and to determine the initial position of the shock front. In the S-F simulation, the initial upstream state in the shock points is set equal to the freestream conditions, while the initial downstream state is computed from the upstream state and the shock slope assuming zero shock speed. The coarse background mesh is made of 351 nodes and 29 shock points are preset at the beginning of the simulation. Afterwards, the flow-field and the shock position are integrated in time until steady state is reached. During the computation the number of nodes of the modified mesh may vary with respect to that of the background mesh. Indeed, the modified mesh used in the last time iteration is made of 358 gridpoints, 29 of which are shock points. The re-meshing technique does not significantly increase the number of points and cells with respect to the background mesh, since the addition of the shock nodes in the modified mesh is partly balanced by the removal of the phantom nodes. Figures 4a) and 4b) show the comparisons between the solutions computed by the EULFS code working in S-C and S-F mode on the coarse and fine meshes, respectively. Figures 4a) and 4b) also include a detailed view of the stagnation point region showing the modifications introduced by the S-F algorithm to the background mesh in order to take into account the shock line. The 2D flow around a circular cylinder at M ∞=20 pressure isolines (Coarse mesh)

3

1

2

2D flow around a circular cylinder at M ∞=20 pressure isolines (Fine mesh)

3

1

2

fitted shock

-1

0.5

1

0

0

Y

Y

0

Y

Y

fitted shock 0.5

1

-1

-0.5

-0.5

captured shock

-2

0

captured shock

-2

-1 0.5

1

1.5

2

-1

2.5

0.5

1

1.5

X

0

1

2

3

4

5

0

1

2

X

3

4

5

2.5

6

X

a) Figure 4: Blunt body at b) fine grid.

2

X

6






b)

 . Comparison between the S-C and S-F solutions: a) coarse grid,

improvements in solution quality due to the S-F technique are evident in the aforementioned figures; particularly when comparing the coarse grid solutions, see Fig. 4a). More specifically, the coarse grid solution computed in S-C mode is characterized by a very large shock thickness and by strong

8

spurious disturbances affecting the shock layer region. These disturbances originate from the shock and are caused by the mis-alignment between the mesh and the shock. On the contrary, the S-F solution shows a very neat field. As far as the fine grid solutions are concerned, see Fig. 4b), the one computed in S-F mode appears to be very similar to that obtained on the corse mesh, whereas the S-C solution still shows the presence of spurious oscillation even if the shock thickness has been halved. Also in this case, the S-F solution shows a significant gain in terms of solution quality with respect the S-C solution. Of course, the resolution of the S-C solutions could be significantly improved if local grid-refinement were used in the shock region, see e.g. [Gnoffo and White, 2004]. However, the results presented in [Yamaleev and Carpenter, 2002] seem to indicate that even grid redistribution and local grid refinement do not significantly improve the numerical solution accuracy compared to that obtained on a uniform grid. A quantitative analysis of the S-F solutions was carried out by comparing the estimates of the shock position and the pressure distribution computed on both grid levels with the reference solution 
 computed by [Lyubimov and Rusanov, 1973]. In Figures 5a) and 5b) the distance ( ) between the shock and the cylinder wall and the normalized pressure at the wall are plotted against the azimuthal angle (  ). This analysis clearly proves that the coarse grid solution is grid-independent despite the extremely limited number of cells enclosed in the shock layer. A comparison between the S-F and the

500

1.4

Y

400

d sh

0

0.8

0

Θ

1

2

300

3

p/p∞

1

1

dsh

X

0.6

200

0.4

Coarse Fine Lyubimov et al.

0.2 0

Coarse Fine Lyubimov et al.

2

1.2

0

30

Θ

60

100

0

90

a)




0

30

Θ

60

90



b)



. Comparison between the S-F and reference solutions: a) shockFigure 5: Blunt body at wall distance, b) normalized pressure at wall S-C solutions is displayed in Fig. 6 where the normalized pressure distributions along the line at 45 degrees is plotted. The S-C solutions are characterized by a finite shock thickness that significantly affects the distribution even on the fine grid. Nevertheless, significant differences between the S-C solutions and the reference one are also visible in regions far from the shock. On the contrary, the differences between the shock fitting solutions and the reference one are very small and the fine grid solution is nearly superimposed to the reference one.

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4.2 Mach reflection

The second test-case deals with the Mach reflection that occurs when a uniform flow at 

is deflected by an agle of    degrees. This case gives the opportunity to verify the hybrid solution technique in which some of the shock are fitted while others are captured. Three solutions have been computed by: i) capturing all shocks, ii) fitting all shocks, iii) capturing the incident shocks and fitting the reflected shock and the Mach stem. In all cases the contact discontinuity has not been fitted, but captured. Concerning the RD schemes used for the spatial discretisation, the entropy and total enthalpy transport equations have been discretized using the PSI scheme, while the system B scheme has been used for the remaining two equations. All three solutions are displayed in Figure 7, which focuses on the Mach stem region. Observe, in particular, that fitting the Mach stem removes the oscillations that are observed in the subsonic region downstream of the captured Mach stem (top of Figure 7a).



5 CONCLUSIONS We believe to have shown that fourty years [Moretti, 2002] after it was first proposed, Moretti’s S-F technique can still be effectively used to accurately compute flows with shocks on relatively coarse meshes. Its use in conjunction with unstructured grids, which is the original contribution proposed in the present paper, allows to overcome many of the algorithmic difficulties that have contributed to the demise of S-F algorithms over the last decades. The use of a S-C code to compute the smooth regions of the flow provides additional flexibility in the sense that it allows to use an hybrid technique in which some of the shocks are fitted while others are captured. This could be an useful feature for extending the methodology to three spatial dimensions. References [Abgrall, 2001] Abgrall, R. (2001). Towards the Ultimate Conservative Scheme: Following the Quest. Journal of Computational Physics, 167(2):277–315. [Bonfiglioli, 2000] Bonfiglioli, A. (2000). Fluctuation splitting schemes for the compressible and incompressible Euler and Navier-Stokes equations. Int. J. Computational Fluid Dynamics, 14:21– 39. [Carpenter and Casper, 1999] Carpenter, M. H. and Casper, J. H. (1999). Accuracy of Shock Capturing in Two Spatial Dimensions. AIAA Journal, 37(9):1072–1079. [Dadone and Fortunato, 1994] Dadone, A. and Fortunato, B. (1994). Three-dimensional flow computations with shock fitting. Comp. and Fluids, 23:539–50. [Deconinck et al., 1993a] Deconinck, H., Paill`ere, H., Struijs, R., and Roe, P. (1993a). Multidimensional upwind schemes based on fluctuation-splitting for systems of conservation laws. Computational Mechanics, 11(5/6):323–340. [Deconinck et al., 1993b] Deconinck, H., Roe, P., and Struijs, R. (1993b). A Multi-dimensional Generalization of Roe’s Flux Difference Splitter for the Euler Equations. Journal of Computers and Fluids, 22(2/3):215–222. [Gloth et al., 2003] Gloth, O., H¨anel, D., Tran, L., and Vilsmeier, R. (2003). A front tracking method on unstructured grids. Comp. & Fluids, 32:547–570. [Gnoffo and White, 2004] Gnoffo, P. and White, J. A. (2004). Computational Aerothermodynamic Simulation Issues on Unstructured Grids. AIAA-CP-2004-2371. [Hartwich, 1990] Hartwich, P. M. (1990). Fresh look at floating shock fitting. AIAA J., 29:1084–91. [Lee and Zhong, 1999] Lee, T. K. and Zhong, X. (1999). Spurious numerical oscillations in simulation of supersonic flows using shock-capturing schemes. AIAA Journal, 37(3):313–319. 10

[Lyubimov and Rusanov, 1973] Lyubimov, A. N. and Rusanov, V. V. (1973). Gas flows past blunt bodies, part ii, table of the gasdynamic functions. NASA TT F-715. [Marconi and Salas, 1973] Marconi, F. and Salas, M. (1973). Computation of three dimensional flows about aircraft configurations. Comp. & Fluids, 1:185–195. [Moretti, 1987] Moretti, G. (1987). Computation of flows with shocks. Ann. Rev. Fluid Mechanics, 19:313–7. [Moretti, 2002] Moretti, G. (2002). Thirty-six years of shock fitting. Computers and Fluids, 31:719–723(5). [Moretti and Abbett, 1966] Moretti, G. and Abbett, M. (1966). A time-dependent computational method for blunt body flows. AIAA J., 4:2136–41. [Nasuti and Onofri, 1998] Nasuti, F. and Onofri, M. (1998). Viscous and inviscid vortex generation during startup of rocket nozzles. AIAA J., 36:809–15. [Pandolfi and D’Ambrosio, 2001] Pandolfi, M. and D’Ambrosio, D. (2001). Numerical Instabilities in Upwind Methods: Analysis and Cures for the ”Carbuncle” Phenomenon. Journal of Computational Physics, 166(2):271–301. [Parpia and Parikh, 1994] Parpia, I. H. and Parikh, P. (1994). A solution-adaptive mesh generation method with cell-face orientation control. In AIAA, Aerospace Sciences Meeting and Exhibit, 32nd, Reno, NV; UNITED STATES; 10-13 Jan., number AIAA-1994-416. [Roe, 1981] Roe, P. L. (1981). Approximate riemann solvers, parameter vectors and difference schemes. J. of Comp. Physics, 43:357–372. [Rosendale, 1994] Rosendale, J. V. (1994). Floating shock fitting via lagrangian adaptive meshes. Technical report, Institute for Computer Applications in Science and Engineering, NASA Langley Research Center, Hampton, VA 23681-0001. NASA Contractor Report 194997; ICASE Report No. 94-89. [Roy, 2003] Roy, C. J. (2003). Grid Convergence Error Analysis for Mixed-Order Numerical Schemes. AIAA Journal, 41(4):595–604. [Salas, 1976] Salas, M. D. (1976). Shock fitting method for complicated two-dimensional supersonic flows. AIAA J., 14:583–8. [Trepanier et al., 1996] Trepanier, J.-Y., Paraschivoiu, M., Reggio, M., and Camarero, R. (1996). A conservative shock fitting method on unstructured grids. J. comp. Physics, 126:421–433. [van Leer et al., 1991] van Leer, B., Lee, W.-T., and Roe, P. (1991). Characteristic Time-Stepping or Local Preconditioning of the Euler equations. In AIAA 10th Computational Fluid Dynamics Conference. AIAA-91-1552-CP. [Yamaleev and Carpenter, 2002] Yamaleev, N. K. and Carpenter, M. H. (2002). On accuracy of adaptive grid methods for captured shocks. J. Comput. Phys., 181(1):280–316. [Yamamoto et al., 1984] Yamamoto, O., Anderson, D. A., and Salas, M. (1984). Numerical calculations of complex mach reflection. Technical Report 84–1976.

11

body

3

shock

400 2

Y

shock 45° 1

r

300

body 0

0

1

2

p/p

∞

X

200

Ref. [6] S-C coarse S-C fine S-F coarse S-F fine

100

0

1

1.2

1.4

1.6

1.8

r








. Normalized pressure along the 45 degrees line: comparison Figure 6: Blunt body at among the computed solutions and the reference solution.

Iso-Mach contours

Iso-Mach contours 2.75

2.75

2.5

The incident shock is captured; all other shocks are fitted

2.5

All shocks are captured

2.25

2.25

M = 1.5

2

Y

Y

M = 1.5

All shocks are fitted

1.75

2

All shocks are fitted

1.75

1.5

1.5

1.25

1.25 1

1.5

2

1

1.5

X

2

X

a) Figure 7: Mach reflection with




b)




and 

12





: detail of the Mach stem region.