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A Possible Mechanism for the Appearance of the Carbuncle Phenomenon in Aerodynamic Simulations

Marcus V. C. Ramalho1 Universidade de Brasília, Brasília, DF, 70919-970, Brazil

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João Luiz F. Azevedo2 Instituto de Aeronáutica e Espaço, São José dos Campos, SP, 12228-903, Brazil

The paper discusses a possible mechanism for the appearance of the carbuncle phenomenon in the numerical computation of supersonic flows. The hypothesis here explored considers that such mechanism may involve the interaction between shock waves and density inhomogeneities created within the nonphysical structure of the numerical shocks. The present proposal seems to be supported by two-dimensional results of the numerical simulation of steady flows around a blunt body and of the unsteady propagation of a plane shock.

I. Introduction It has been known for nearly 20 years that some shock-capturing methods can generate spurious solutions

when applied to seemingly simple problems such as the calculation of the flowfield containing a detached shock wave ahead of a blunt body in supersonic flow. The so-called carbuncle phenomenon, first reported by Peery and Imlay1, was initially associated with the use of Roe’s approximate Riemann solver in the spatial discretization of the convective terms in the Euler equations. However, it was later found to affect several other methods. In its most typical form, the carbuncle phenomenon consists of a protrusion ahead of the shock, which contains a region of circulating and possibly stagnated flow (see Fig. 1). The overall flowfield appears to satisfy the Euler equations, at least in a weak sense and in its discretized form, since solutions including the carbuncle satisfy usual convergence tests. The phenomenon was later observed by several authors, and a fairly extensive literature on its possible causes and cures was subsequently developed.2-9

It has been found that the carbuncle appears to occur only with shock-capturing schemes that are designed to preserve contact discontinuities.3 One explanation is that such schemes provide insufficient dissipation in the shock region, particularly in the direction parallel to the shock,2, 3, 12 thereby suffering from instabilities that may result in the formation of the carbuncle. Dumbser et al.4 analyzed the linear stability of several discretization schemes and concluded that those which are more carbuncle-prone are also likely to be inherently unstable under certain conditions. Quirk2 observed that, although adding dissipation in the direction parallel to the shock was a common method to suppress the carbuncle in contact discontinuity-preserving schemes, there were no physical or numerical grounds for resorting to such a “cure”. It was simply a convenient means to get rid of the problem.

A different type of explanation was attempted by Robinet et al.,5 who suggested that the origin of the carbuncle could lie in the physical instability of the surface of discontinuity itself, i.e., the shock wave. According to earlier research on the problem of shock wave stability (see, for example, Ref. 10), the plane shock wave formed in a polytropic gas is always stable. Robinet et al.,5 nevertheless, reported to have found a mode which could give rise to instability and that had been overlooked by previous researchers. Moreover, the characteristics of this mode seemed to agree with observations that had been made in connection with the carbuncle phenomenon. This discovery, however, was later contested by Coulombel et al.,11 who reaffirmed that a plane shock wave in a polytropic gas could not, in fact, be physically unstable, and showed that the derivation of Robinet et al.5 was incorrect in some respects.

1 Graduate Student, Instituto de Física, Caixa Postal 04455; E-mail: mvcramalho@gmail.com. AIAA Member. 2 Senior Research Engineer, Aerodynamics Division, Departamento de Ciência e Tecnologia Aeroespacial; E-mail: azevedo@iae.cta.br. Associate Fellow AIAA.

48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition4 - 7 January 2010, Orlando, Florida

AIAA 2010-872

Copyright © 2010 by M.V.C. Ramalho and J.L.F. Azevedo. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

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As pointed out by Roe et al., and Kitamura et al.,7,8 we still lack a satisfactory explanation for the occurrence

of the carbuncle phenomenon. Xu’s12 observation that numerical shock instabilities do not occur when unstructured meshes are used in the computation, because the shock would not be systematically aligned with the grid, is not borne out, for instance, by results obtained in the present paper. Kitamura et al.8 found that the carbuncle may affect schemes which incorporate “entropy corrections” or which provide dissipation of a multidimensional character. As mentioned above, Quirk2 noted that there are no physical reasons for the addition of extra doses of dissipation in critical directions in schemes that otherwise correctly preserve contact discontinuities. It seems certain, however, that the carbuncle only occurs in shock-capturing methods, since it has never been observed when the method of shock-fitting is used.

On the basis of this latter observation, the purpose of this paper is to suggest another hypothesis and to undertake a preliminary investigation thereof. The hypothesis is motivated by the occurrence of real physical “carbuncles” under certain conditions in experiments with supersonic flows. It is posited that, in numerical calculations, the carbuncle phenomenon may result from a physical instability – namely the Richtmyer-Meshkov instability – whose occurrence is made possible by the fact that the captured shock is not a perfect discontinuity, but rather contains intermediate non-physical states. The present rationale differs from the suggestion of Robinet et al.5, referred to above, in that the physical instability involved in our hypothesis is not that of the sharp and discontinuous shock wave itself, but, as will be explained later, that of the interaction between a shock wave and density inhomogeneities.

The paper is organized as follows. Section II briefly summarizes the governing equations and indicates the numerical methods used for their solution. Section III describes a test case involving the two-dimensional calculation of the supersonic flow around a blunt body. Section IV discusses some physical and numerical experiments previously reported in the literature where the formation of carbuncle-like flows was observed, although not identified by this name and not related to the shortcomings of numerical shock-capturing methods. The possible relation of these experimental results with the formation of carbuncles by means of baroclinic vorticity generation in numerical tests of the type described in section III is also investigated. Section V provides a brief description of the Richtmyer-Meshkov instability and its relation to baroclinic vorticity generation. Such a discussion provides the basis for the construction of a parameter that may be used to “sense” the possibility of the formation of carbuncles throughout the flowfield as a shock-capturing calculation evolves over time. This is done in several numerical tests in section VI. Section VII contains some concluding remarks.

II. Governing Equations and Numerical Procedure The flow is modeled by the two-dimensional Euler equations, which can be written in integral form as

(1)

Figure 1. A typical carbuncle.

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where V denotes a control volume, S represents its boundary and n is the outward normal to the S boundary. The vector of conserved quantities, Q, and the convective flux vectors, E and F, are given by

Q = [ρ, ρu, ρv, e] T (2)

E = [ρu, ρu2 + p, ρuv, (e + ρ)u] T, F = [ρu, ρuv, ρv2 + p, (e + ρ)v] T

In these definitions, ρ denotes the density, p is the pressure, u and v represent the Cartesian velocity components, and e is the total energy per unit volume.

The equations are discretized using a cell-centered based finite volume procedure, where the discrete vector of conserved variables, Qi, is defined as an average over the i-th control volume as

Qi = (3)

With this definition, Eq. (1) can be rewritten for the i-th control volume as

(4)

In the spatial discretization, the surface integral in Eq. (4) is approximated by

(5)

where k spans the n control volumes which are neighbors of the i-th volume, Eik and Fik are the numerical convective fluxes between the volumes i and k, and the terms ∆xik and ∆yik are calculated as

∆xik = yk2 – yk1 ∆yik = xk2 – xk1 where the points (xk1, yk1) and (xk2, yk2) are the vertices which define the interface between cells i and k.

The spatial discretization thus consists in (i) determining the geometry of the control volumes or cells (in other words, choosing a mesh); and (ii) determining the intercell (or numerical) fluxes Eik and Fik by means of an appropriate scheme. In the present paper, two types of flow are investigated: the supersonic steady flow around a blunt body (a circular cylinder) and the unsteady motion of a plane shock wave propagating down a duct. In the first case, an unstructured triangular mesh is employed, and the intercell numerical fluxes are calculated using four methods, as will be indicated throughout the paper: Liou’s first-order AUSM+ scheme,13 Jameson’s centered scheme,14 Van Leer’s first-order method15 and Roe’s original scheme.16 The approach adopted in the present work to extend the formulation of the first three methods to unstructured meshes follows Ref. 17 and consists in defining a local one-dimensional system normal to the interface considered. Roe’s scheme is implemented in a similar manner, following the formulas provided in Ref. 18. In the simulations of a plane shock wave propagating in a duct, a quadrilateral structured mesh is used, together with the original Roe scheme for the calculation of the numerical fluxes.

Once the convective operator, C(Qi), is calculated, Eq. (4) can be advanced in time. In the simulations of supersonic flow around a blunt body reported in this paper, a fully explicit, 5-stage, Runge-Kutta method is used as the time-stepping scheme.14 In the case of the plane shock propagating along a duct, a simple explicit formula of the type used in Ref. 18 (see, for instance, Equation 16.128 thereof) is employed.

III. A Test Case The case that motivated the present research is the calculation of the two-dimensional inviscid flow of a

polytropic gas (γ = 1.4) around a cylinder at zero angle of attack using an unstructured triangular mesh. The geometry of the problem, together with the mesh, are shown in Fig. 2. A finite volume code is used in all computations. First-order accurate spatial discretization is implemented with Liou’s AUSM+ method.13

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Uniform freestream flow conditions are initially prescribed throughout the entire domain. As to boundary

conditions, freestream properties are imposed along the far field boundary, i.e., the left part of the computational domain in Fig. 2(a), and slip conditions are enforced on the cylinder surface. Zero-order extrapolation of all properties is employed on downstream boundaries, i.e., the right limits of the domain in Fig. 2(a). Figure 3 shows Mach number contours for a freestream Mach number (M∞) of 12.2. The carbuncle, although not as

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severe as that illustrated in Fig. 1, is clearly visible above the symmetry line, between the shock and the cylinder. It must be noted that the mesh itself is not symmetric. Figure 4 displays a detailed view of the carbuncle and shows velocity vectors forming a vortex-like pattern around a region of nearly stagnated flow. Figure 5 shows the evolution of the L2 norm of the density residue and confirms that it decreases by more than ten orders of magnitude, thereby attesting to the numerical convergence of the solution.

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IV. Shock-Vortex Interactions and Baroclinic Vorticity Generation It was noted by Roe et al.7 and Roe19 that numerically generated carbuncles resemble flows that may be

physically realized by the introduction of plates or spikes in front of a blunt body immersed in supersonic flow. This section discusses flows that were previously reported in the literature and that also resemble typical carbuncles. These flows do not result from modifications of the body itself, but rather involve the interaction between axial vortices and shock waves.

Kalkohran et al.20,21 and Cattafesta and Settles22 set up experiments in which tip axial vortices generated by a wedge impinged on detached shock waves formed in front of blunt bodies in supersonic flow. It is observed that, as a result of this interaction, the shock wave bulged forward in the upstream direction or acquired a conical shape, behind which a highly unsteady flow pattern develops. Figures 6 to 8, reproduced from Refs. 20 and 21, illustrate some of the flowfields obtained in these experiments. Thomer et al.23 conducted numerical simulations of the interaction of a plane shock with an axial vortex. The geometry of their simulation is reproduced in Fig. 9, with some of their results reproduced in Fig. 10.

These flows strongly resemble the typical carbuncle patterns formed in the numerical solution of the blunt body problem, such as illustrated in Fig. 1. The results of Thomer et al. reproduced in Fig. 10 are very similar, for example, to the flows obtained by Elling24 in his study of carbuncles as self-similar entropy solutions, here reproduced in Fig. 11. Thomer et al.23 describe their results in terms of vortex breakdown, whereupon “a free stagnation point is formed downstream from the shock, followed by a region of reversed flow with a bubble-like flow structure. (…) The shock remains normal near the axis of the vortex, but is curved further away from it. The flow downstream of the shock is slightly oscillating”. Such a description, although related to a physical phenomenon, could well be applied to patterns that have been recognized as carbuncles in the CFD literature.

In the context of the investigation of shock waves formed within plasmas, Kremeyer et al.25 sought to demonstrate that the phenomenon of “shock splitting” observed in some experiments should not be attributed to electromagnetic effects, but rather to the purely gas dynamic effect of the shock “curving and bowing as it passes through the different transverse density (or temperature) profiles”. In this connection, those authors observed that vorticity is generated at the shock by means of the so-called baroclinic mechanism. This corresponds to a source term in the vorticity evolution equation which has the following form:

(∂ω/∂t)B = grad(ρ) x grad(p) / ρ2 (6)

Such observation has prompted the authors of the present study to reproduce this phenomenon of shock bowing in the simulation of the test case described in Section II. It should first be noted that the transverse density gradients used to deform the shock in the simulations of Kremeyer et al.25 are also present in the interactions of the shock with axial vortices investigated in Refs. 20, 21, 22 and 23. The density profiles of the axial vortices simulated by Thomer et al. (see Fig. 2 of Ref. 23), in particular, have a form similar to the one shown in Fig. 12.

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Figure 6. Flowfield illustrating interaction of a tip vortex with the shock in front of a cylinder, Mach = 3.1. Reproduced from Ref. 20, Fig. 6, reprinted with permission of the American Institute of Aeronautics and Astronautics, Inc.

Figure 7. Shadowgraph of the flowfield during weak vortex-cylinder interaction, Mach = 2.49. Reproduced from Ref. 21, Fig. 5, reprinted with permission.

Figure 8. Shadowgraph of the flowfield during moderate vortex-cylinder interaction, Mach = 2.49. Reproduced from Ref. 21, Fig. 7, reprinted with permission.

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Figure 9. Geometry of simulation of normal shock-vortex interaction. Reproduced from Ref. 23, Fig. 1, reprinted with permission.

Figure 10. Mach number contours in the mid-plane of the computational domain, M∞ = 1.6. Reproduced from Ref. 23, Fig. 3, reprinted with permission.

Figure 11. A structure resembling carbuncles; x-velocity component contours, M∞ = 3.0. Reproduced from Ref. 24, Fig. 3, reprinted with permission.

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It was thought, therefore, that the superposition of a narrow region containing a transverse density gradient

upstream of the shock, obtained in the two-dimensional calculation described in Section II, should make the shock bulge forward as the calculation further advanced in time, at least during some transient period. Such a behavior was confirmed by simulations and a deformed shock was obtained not only with the use of the AUSM+ method, but also with a central difference scheme in which artificial dissipation is explicitly provided.14 It must be emphasized that a centered scheme, even if not particularly suited for the calculation of hypersonic flows, does not suffer from the carbuncle, as one can see in Fig. 13(b). Hence, Fig. 13 displays the converged solutions – velocity vectors – obtained with the AUSM+ scheme and with a centered scheme, for simulations performed as described in Section II. Hence, the velocity vectors shown in Fig. 13(a) correspond to the same simulation whose results are shown in Figs. 3 and 4. Figure 12 shows the density gradient superimposed upstream of the obtained shocks and used to restart the calculation from the previous converged solutions (in Fig. 13). Figure 14, then, presents detailed views of the velocity vectors in the region near the shock protrusion generated by the interaction of the shock with the transverse density gradient after 100 iterations of the restarted calculation.

On the basis of these results, it appears to be possible that carbuncles may be “physically” created via the baroclinic mechanism of vorticity generation, where strong pressure gradients induced by the shock interact with some transverse density gradients. In the experiments discussed in this section, however, the transverse density gradient is externally provided, either artificially or by means of axial vortices impinging on the shock. It would remain to be found, therefore, what might be the source of some “spurious” baroclinic interaction in the calculation of the flow by a shock-capturing method.

It is recalled at this point that carbuncles have never been found where a shock-fitting procedure is used. A major difference between shock fitting and shock capturing is that the former does not contain a numerical shock structure formed by artificial, non-physical states. In this regard, a possible source of the instability associated with the carbuncle could be found by investigating whether the intermediate non-physical states generated by shock-capturing schemes have the potential of inducing baroclinic vorticity generation, particularly in the transient stages of the calculation. It might be possible that actual physical instabilities associated with the baroclinic mechanism of vorticity generation could play a role in the calculation of flowfields which include non-physical states within the numerical shock, thereby disturbing the steady state solution.

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Figure 14. Velocity vectors at iteration 100, starting with upstream transverse density gradient superimposed to previous converged shock solutions. M∞ = 12.2. (a) Centered scheme; (b) AUSM+.

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V. The Richtmyer-Meshkov Instability and a “Tracking” Parameter In 1960, Richtmyer26 presented the linear stability analysis of a plane surface subject to a small sinusoidal

perturbation accelerated by a shock. He found that such a configuration would be unstable and his prediction was later confirmed in experiments reported by Meshkov.27 This phenomenon, which more generally involves the interaction of shock waves with density inhomogeneities,28 came to be known as the “Richtmyer-Meshkov instability”.

The deposition of circulation is the dominant fluid dynamical process in the early stages of the development of the Richtmyer-Meshkov instability.28 As a moving shock traverses an interface, a misalignment of pressure and density gradients leads to rapid vorticity deposition on the interface by means of the baroclinic mechanism indicated in Eq. (6). This may occur not only with an interface which is made non-planar because of a perturbation, but also in the case of a planar interface initially at an angle with the shock, as illustrated in Fig. 15. The figure illustrates finite portions of gases at different pressures and densities separated by interfaces, one of which is a shock. The problem investigated in Ref. 28 is the physical Richtmyer-Meshkov instability that arises as the shock crosses the oblique interface.

Figure 15, however, could also be regarded as representing an enlarged view of a group of cells at one stage in the numerical solution of a flow containing a shock wave with a shock-capturing method, as, for example, the two-dimensional problem of a plane shock wave propagating in a tube containing gas at rest. The portion with properties pb and ρb could represent one cell in the yet undisturbed region of the gas; the portion with properties p1 and ρ1 would represent one cell in the already shocked region; and the domain with properties p0 and ρ0 could be regarded as a cell associated with the artificial, intermediate state generated by the shock-capturing method to connect the states upstream and downstream of the shock.

Thus, the same environment that is set up just prior to the physical occurrence of the Richtmyer-Meshkov instability can be realized on a smaller scale as a result of the spatial discretization of the flowfield, with the assumption of uniform, averaged flow properties within each cell, and the application of a shock-capturing method, which produces intermediate states within the numerical shock. If the cells and their interfaces could be regarded as physical regions of gas at different states separated by physical interfaces, then the portions of the numerical grid, where pressure and density gradients are of such magnitude and direction as to produce strong vorticity deposition via the baroclinic mechanism, could be susceptible to instability in roughly the same way as the physical configuration shown in Fig. 15 is susceptible to the Richtmyer-Meshkov instability.

This observation motivated the definition of a parameter based on Eq. (6) which seeks to measure the intensity of baroclinic vorticity generation within, and immediately upstream of, the numerical shock. As the flow solver advances the computation in time, the maximum value of this parameter throughout the flow field is calculated according to the following algorithm:

1) At each iteration, find the cells in which the density is larger than the freestream density, using some practical threshold value, e.g., ρ ≥ 1.01* ρ∞, and which have at least one neighbor cell whose density is approximately equal to or smaller than that of the undisturbed flowfield, e.g., ρ ≤ 1.01* ρ∞. These cells, which will be named “0-cells”, should correspond to “intermediate” states created by the shock-capturing method. Store the pressure and density of these cells as p0 and ρ0, respectively; 2) For each 0-cell found in (1), identify the neighbor cell for which the pressure is largest among the neighbors. Call this neighbor a “1-cell”. Store its pressure as p1; 3) For each 0-cell found in (1), identify the neighbor whose density is approximately equal to that of the undisturbed flow field. Call this neighbor a “b-cell”. Store its density as ρb;

Figure 15. Geometry of the Richmyer-Meshkov instability with a planar interface.28

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4) For each 0-cell found in (1), find the angle between the normal direction to the interface between the 0-cell and the 1-cell and the normal direction to the interface between the 0-cell and the b-cell. Call this angle θ; 5) For each 0-cell found in (1), calculate the quantity

k = │p1 - p0│*│ ρb - ρ0│*│sin θ│ / ρ02 ;

6) Identify the 0-cell where k is a maximum at the given iteration. Store the value of k and the position of this 0-cell. Advance the solution to the next time step and return to (1).

This algorithm is applied in some numerical tests described in the next section. As will be seen, the location of the points where the k parameter attains its largest values, as well as the magnitude of the parameter, seem to correlate with the occurrence, or the absence, of carbuncles in the computational tests.

VI. Numerical Tests The first application consisted in using the procedure introduced in the previous section to the test case

described in Section II. The expectation was that, given the geometry of the mesh and the fact that the carbuncle structure was located above the symmetry line, the location of the points where the k parameter reached its maximum values as the calculation evolved in time should also point to that same region of the domain. One should observe that higher values of k would be associated with larger production of vorticity via the baroclinic mechanism.

First, though, it would be appropriate to look at how the maximum value of the k parameter over the flowfield evolved as the iterations proceeded. Figure 16 shows the evolution of the maximum k for the test case of Section II for the first 2000 iterations, using the first-order AUSM+ scheme for the simulations. The CFL number is 0.2 and the mesh is the same one illustrated in Fig. 2 (here called GA1).

As can be seen in Fig. 16, the maximum value of the k parameter over the flowfield attains a series of peaks

as the calculation evolves from the initial freestream flow imposed over the entire domain to a shock wave that detaches itself away from the surface of the cylinder. A criterion was devised to select the highest values of k among those peaks as follows: take all local maxima of k that are greater than or equal to half of the overall maximum value of k. In Fig. 16, these values are indicated by squares. The spatial coordinates of the volumes (cell centers) in which these selected local maxima occur are, then, plotted in the computational domain. If the hypothesis advanced in the previous section is correct, the geometric location of the cells where k attains its highest values, as the computation evolves over time, should trace the generation and subsequent motion of the carbuncle in space.

Figure 17 shows results for the test case of Section II, as well as for other 3 cases, namely: AUSM+ scheme, zero angle of attack, mesh GA2, which is an asymmetric triangular mesh in which the triangles have the opposite orientation of the ones of mesh GA1; AUSM+ scheme, angle of attack of 10 degrees and mesh GA1;

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and the original Roe scheme, zero angle of attack and a symmetric triangular mesh (here called GS). Figure 17 also displays the geometrical location of the centroids of the cells where k attains its highest values during the first 2000 iterations with a CFL number of 0.2 in all cases. These are shown alongside Mach number contours of the final converged solutions. The convergence histories for each of the 4 cases are shown in Fig. 18.

Figure 17. Mach number contours and location of cell centers where k reaches local maxima. M∞ = 12.2. (a) AUSM+, GA1 mesh; (b) AUSM+, GA2 mesh; (c) AUSM+, GA1 mesh, angle of attack = 10 degrees; (d) Roe, GS mesh.

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The carbuncles are clearly visible in all 4 cases illustrated in Fig. 17. Moreover, the location of the cell centers of maximum k appears to point to the location of the final, converged carbuncles. This could indicate that carbuncles appear where the baroclinic vorticity generation, as roughly measured by the k parameter, attains its maximum values as the flowfield evolves over time from its initial condition.

It was also determined, in each of the 4 cases illustrated in Fig. 17, at which iteration and at which point in the domain the velocity component parallel to the freestream direction first became negative. This was thought to give an indication of the first stage in the development of a vortex characteristic for the carbuncles. In all cases, this behavior occurred just a few iterations after the first time the k parameter reached its maximum value, as shown in Table 1. Furthermore, negative velocity components first appeared invariably in cells that are adjacent to the cell where k had attained its maximum value a few iterations earlier. This seems to corroborate the idea that there may be a connection between the “early stages” of the carbuncle and the misalignment of pressure and density gradients as measured by k. Figure 19 presents enlarged views of the plots of the locations of maximum k shown in Fig. 17, this time together with the points where the velocity component in the freestream direction first becomes negative (indicated by circles).

Table 1. Iterations of maximum k and first negative velocity component in the freestream direction (CFL = 0.2 in all cases).

Iteration at which k is maximum

Iteration in which velocity component in the freestream

direction first becomes negative AUSM+, GA1 mesh 54 57 AUSM+, GA2 mesh 54 57

AUSM+, GA1 mesh, α = 10 deg 53 57 Roe, GS mesh 74 82

Figure 18. Convergence histories for the other 3 solutions displayed on the left-hand side of Fig. 17. (Convergence history for case (a) is already shown in Fig. 5).

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

x 104

-12

-10

-8

-6

-4

-2

0

2AUSM+, Mesh GA2, Alpha 0 deg, CFL=0.2

Iteration

Log1

0 of

Den

sity

L2

norm

0 1 2 3 4 5 6 7

x 104

-12

-10

-8

-6

-4

-2

0

2

Iteration

Log1

0 of

Den

sity

L2

norm

AUSM+, Mesh GA1, Alpha 10 deg, CFL=0.2

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

x 104

-12

-10

-8

-6

-4

-2

0

2

Iteration

Log1

0 of

Den

sity

L2

norm

Roe, Mesh GS, Alpha 0 deg, CFL = 0.2

16

Finally, it must be noted that the previous discussion is not restricted to any specific family of shock-capturing schemes. The k parameter could be calculated in the course of a solution generated by any shock-capturing scheme, including those which are not likely to generate carbuncles. This was done for 2 discretization methods: Van Leer’s first-order flux vector splitting and Jameson’s second-order centered schemes. Other conditions were the same as in the test case described in Section II. As expected, the solutions converge and no carbuncles appear, as illustrated in Figs. 20 and 21. Figure 22 displays the evolution of the maximum value of the k parameter for these two test cases, together with that for the AUSM+ and Roe schemes. As can be seen from the graphs, the intensity of baroclinic vorticity generation immediately ahead of the shock, as roughly measured by k, is considerably lower for schemes that are known not to produce any carbuncles.

Figure 19. Points of maximum k, together with location of points where velocity component in the freestream direction first becomes negative (indicated by circles). (a) AUSM+, GA1 mesh; (b) AUSM+, GA2 mesh; (c) AUSM+, GA1 mesh, angle of attack = 10 degrees; (d) Roe, GS mesh.

-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2AUSM+, Mesh GA1, Alpha 0 deg

a)

-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2AUSM+, Mesh GA2, Alpha 0 deg

b)

-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2AUSM+, Mesh GA1, Alpha 10 deg

c)

-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2Roe, Mesh GS, Alpha 0 deg

d)

17

Figure 20. Mach number contours obtained with Van Leer’s and Jameson’s schemes (M∞ = 12.2, α = 0 deg., GA1 mesh).

-1 -0.5 0 0.5 1 1.5

-1

-0.5

0

0.5

1

Mach contour lines, ideal gas - Van Leer

0.5

1

1.5

2

2.5

3

3.5

4

-1 -0.5 0 0.5 1 1.5

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Mach contour lines, ideal gas - Jameson

0.5

1

1.5

2

2.5

3

3.5

4

Figure 21. Convergence histories for the solutions illustrated in Fig. 20.

0 0.5 1 1.5 2 2.5 3 3.5 4

x 104

-12

-10

-8

-6

-4

-2

0

2

Iteration

Log1

0 of

Den

sity

L2

norm

Van Leer, CFL = 0.2

0 0.5 1 1.5 2 2.5 3 3.5

x 104

-12

-10

-8

-6

-4

-2

0

2

Iteration

Log1

0 of

Den

sity

L2

norm

Jameson, CFL = 0.2

0 200 400 600 800 1000 1200 1400 1600 1800 20000

10

20

30

40

50

60

Iteration

Max

imum

k

AUSM+RoeVan LeerJameson

Figure 22. Comparison of the evolution of the k parameter for different discretization schemes. CFL = 0.2 for all cases. GA1 mesh (AUSM+, van Leer and centered schemes) and GS mesh (Roe).

18

A second type of numerical test is the so-called “Quirk’s test”,2 which consists in the simulation of a plane shock propagating down a duct with one grid line in the x-direction slightly perturbed. The simulations reported here were run with a finite volume code which employed a 800x20 node structured grid. Reflecting (inviscid) wall boundary conditions were used in the bottom and top boundaries of the simulated duct, with initial conditions, as well as properties imposed at the entrance and exit, set so as to correspond to a Mach 6 shock initially located at x = 50.0 (the length of the duct is 800.0 and its width is 20.0). The grid points originally located at y = 11.0 were displaced in an odd-even pattern by δ = 10-3. Spatial discretization is performed with the first-order Roe scheme.

Figure 23(a) shows density contours at time t = 0.17 in the case where no perturbation of the grid lines is

introduced. As expected, the result is quite satisfactory. The use of the slightly modified grid line, however, leads to a distortion of the shock as it propagates down the duct. Figure 24(a) displays density contours at successive instants of time, showing the growth of a carbuncle as the solution evolves. Figure 24(b) superimposes, to the previous density contours plots, the spatial location of the cells where the k parameter attained its maximum values as the calculation evolved. Figure 24(c) shows the value of the k parameter for those cells against the x-coordinate of their spatial location. Figure 24(d) presents the same information as Fig. 24(c) with the value of maximum k shown in a logarithmic scale. It is important to observe the difference in scale when comparing the values of maximum k in Figs. 23(b) and 24(c).

In the calculation of k for the present test case, for each computational cell, the density difference was always taken in the “j”-direction (nominally y-direction), whereas the pressure difference was calculated in the “i”-direction (nominally x-direction). The θ angle, previously defined, therefore, was always 90 degrees, or very close to this value in the case of cells with one of their sides slightly misaligned with the horizontal direction when the grid lines were perturbed.

Again, there is considerable correlation between, on the one hand, the location of the points where the misalignment of density gradients, inside the numerical shock, with pressure gradients, across the shock, may lead to maximum baroclinic vorticity generation, and, on the other hand, the location of carbuncles. Note that the spatial location of the points where k attains its maximum values also corresponds, during the initial phases of the shock motion, to the region where the grid was initially distorted. This behavior is confirmed by an additional simulation in which the grid points originally located at y = 4.0 are displaced in a similar fashion as in the previous test case. Results are shown in Fig. 25. When the grid is not distorted, and no carbuncles develop, the values of maximum k are practically zero throughout the entire simulation and computational domain (Fig. 23(b)). This seems to indicate that, in Quirk’s test, distorting the grid leads to misalignments of pressure and density gradients from very early stages of the calculation, and hence to nonzero values of maximum k already at those stages, as shown in Fig. 24(c). Eventually, the interaction of the plane shock with these artificial density gradients may lead to a breakdown of the shock structure.

0 50 100 150 200 250 300 350 400 4500

5

10

15

0 50 100 150 200 250 300 350 400 450-1

-0.5

0

0.5

1

Max

(k)

x-distance

Figure 23. Test case without any grid line perturbation: (a) Density contours of a propagating plane Mach 6 shock wave at t = 0.17, initially located at x = 50 (first-order Roe scheme, no perturbation introduced to the grid lines); (b) Maximum value of the k parameter plotted against the x-coordinate of the points of maximum k.

19

0 50 100 150 200 250 300 350 400 4500

10

0 50 100 150 200 250 300 350 400 4500

10

0 50 100 150 200 250 300 350 400 4500

1

2x 10

5

Max

(k)

0 50 100 150 200 250 300 350 400 450-20246

x-distance

Log1

0(M

ax(k

))

Figure 24. Test case with grid line perturbation: (a) Density contours of a propagating plane Mach 6 shock wave initially located at x = 50 at 5 subsequent different instants of time (first-order Roe scheme, odd-even perturbation introduced to the grid lines at y = 11); (b) Points of maximum k superimposed to the previous density contours; (c) Maximum value of the k parameter plotted against the x-coordinate of the points of maximum k; (d) Same as (c), with maximum k shown on a logarithmic scale.

0 50 100 150 200 250 300 350 4000

10

0 50 100 150 200 250 300 350 4000

10

0 50 100 150 200 250 300 350 4000

5

10

x 104

Max

(k)

0 50 100 150 200 250 300 350 400-20246

Log1

0(M

ax(k

))

x-distance

Figure 25. (a) Density contours of a propagating plane Mach 6 shock wave, initially located at x = 50, at its final position (first-order Roe scheme, odd-even perturbation introduced to the grid lines at y = 4); (b) Points of maximum k superimposed to the previous density contours; (c) Maximum value of the k parameter plotted against the x-coordinate of the points of maximum k; (d) Same as (c), with maximum k shown on a logarithmic scale.

20

VII. Conclusions The paper presents a preliminary investigation of a hypothesis for the generation of “carbuncles” in flow

solutions calculated by some shock-capturing schemes. In light of experimental results involving the interactions of shocks with longitudinal vortices, it is suggested that the origin of the carbuncles may be associated with the vorticity generated by the misalignment of pressure gradients across the shock with density gradients artificially created within the unphysical numerical shock structure. By introducing such unphysical states in the solution, shock-capturing schemes could create the conditions for the development of an instability that is physically and inherently related to the interaction of shock waves with density discontinuities, the Richtmyer-Meshkov instability.

A parameter (here named k) is used to roughly “track” the spots of the domain where the misalignment between pressure and density gradients immediately ahead of the shock could lead to maximum baroclinic vorticity generation as the solution evolves over time. It is found that these spots correlate well with the spatial location of the carbuncles which appear in different solutions to the problem of the two-dimensional supersonic (M∞ = 12.2) inviscid flow over a circular cylinder, as well as in the case of a Mach 6 plane shock propagating down a duct. Moreover, among the discretization methods tested in this study, the values of this parameter are higher for the schemes that effectively generated carbuncles than for methods that are known not to produce them.

In a sense, the thrust of the present paper is consistent with Roe’s hypothesis19 that spurious vortical structures can be introduced into a computation by discretization errors. Carbuncles tend to be produced by discretization methods which are the most accurate. The more accurate a computational method is designed to be from a physical standpoint, however, the more likely it is that the physical consequences of spatial discretization are realized. Godunov-type methods, in particular, take into account the direction of characteristics as well as the nonlinear interaction of waves by regarding each interface of the discretization as the physical interface of a Riemann problem. Such methods may highlight undesirable physical consequences of having intermediate states within the numerical shock.

Some steps in the calculation of the parameter k are somewhat arbitrary and one would, probably, need more accurate means of probing the density profiles within numerical shock structures so as to further investigate whether they effectively interact with high pressure gradients generated by the shock and, eventually, result in carbuncles. Nevertheless, the results of the present paper seem to lend support to such hypothesis.

Acknowledgments The authors gratefully acknowledge the partial support for the present work provided by Conselho Nacional

de Desenvolvimento Científico e Tecnológico, CNPq, under the Research Project Grant No. 312064/2006-3. The work is also supported by Fundação de Amparo à Pesquisa do Estado de São Paulo, FAPESP, under the Process No. 2004/16064-9.

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