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## Journal of the Brazilian Society of Mechanical Sciences

##
*Print version* ISSN 0100-7386

### J. Braz. Soc. Mech. Sci. vol.24 no.3 Rio de Janeiro July 2002

#### http://dx.doi.org/10.1590/S0100-73862002000300002

**Numerical study of wedge supported oblique shock wave-oblique detonation wave transitions**

**C. A. R. Pimentel ^{I}; J. L. F. Azevedo^{II}; L.F. Figueira da Silva^{III}; B. Deshaies^{III}**

^{I}Centro Técnico Aeroespacial Instituto Tecnológico de Aeronáutica CTA/ITA/IEAA 12228-900 São José dos Campos, SP. Brazil

^{II}Centro Técnico Aeroespacial Instituto de Aeronáutica e Espaço CTA/IAE/ASE-N 12228-904 São José dos Campos, SP. Brazil

^{III}Laboratoire de Combustion et de Détonique; UPR 9028 du CNRS – ENSMA et Université de Poitiers 86960 Futuroscope. FRANCE

**ABSTRACT**

The results of a numerical study of premixed Hydrogen-air flows ignition by an oblique shock wave (OSW) stabilized by a wedge are presented, in situations when initial and boundary conditions are such that transition between the initial OSW and an oblique detonation wave (ODW) is observed. More precisely, the objectives of the paper are: (i) to identify the different possible structures of the transition region that exist between the initial OSW and the resulting ODW and (ii) to evidence the effect on the ODW of an abrupt decrease of the wedge angle in such a way that the final part of the wedge surface becomes parallel to the initial flow. For such a geometrical configuration and for the initial and boundary conditions considered, the overdriven detonation supported by the initial wedge angle is found to relax towards a Chapman-Jouguet detonation in the region where the wedge surface is parallel to the initial flow. Computations are performed using an adaptive, unstructured grid, finite volume computer code previously developed for the sake of the computations of high speed, compressible flows of reactive gas mixtures. Physico-chemical properties are functions of the local mixture composition, temperature and pressure, and they are computed using the CHEMKIN-II subroutines.

**Keywords:** Supersonic combustion, oblique shock waves, oblique detonation waves

**Introduction**

Supersonic flow of a reactive mixture around a wedge leads to the formation of an oblique shock wave (OSW) which may trigger combustion and eventually lead to the onset of an oblique detonation wave (ODW). This configuration, which is of interest to the propulsion of hypersonic vehicles (Dunlap, Brehm and Nicholls, 1958), has been studied both experimentally (Lehr, 1972, Dabora et al., 1991, Desbordes, Hamada and Guerraud, 1995) and numerically (Li, Kailasanath and Oran, 1994, Figueira da Silva and Deshaies, 1995, 2000, Deshaies, Figueira da Silva and René-Corail, 1997).

Previous comparisons between numerical and experimental results (Viguier et al., 1996) for wedge triggered supersonic combustion have shown a rather good agreement concerning the overall structure of the flow.

In particular, the change of the OSW/ODW transition length was shown to be directly connected to the characteristic time of the chemical process, both in the experiments and in the computations. Nevertheless, some important differences were observed concerning (i) the exact value of this transition length, which is always smaller in the experiments than in the calculations and (ii) the structure of the transition region. A possible reason for these discrepancies is the important difference that exists between the initial and boundary conditions of the experiments, where a gaseous wedge is used, and those of the computations, which consider a solid wedge. A second possible explanation for the observed discrepancies could be the lack of accuracy of the computation itself. Indeed, the experimental results used for the comparisons were obtained for physical and flow conditions for which the heat release time of the chemical process, which fixes the size of the elementary mesh, remains very small when compared to the induction time, which fixes the minimum size of the computational domain.

In the framework of the Cartesian, structured mesh computational code used previously, the number of mesh points required is incompatible with a reasonable computational time. In order to overcome this difficulty, an unstructured, upwind, finite volume method, together with an adaptive refinement of the computational mesh is used in the present work. This tool allows, in particular, a better definition of the OSW/ODW transition region. The computations consider flows with variable composition, finite rate chemistry and in which the physico-chemical properties are functions of the local mixture composition, temperature and pressure. Physico-chemical properties are computed using the CHEMKIN-II (Kee, Rupley and Miller, 1991) subroutines.

In all of the above mentioned previous papers, mostly semi-infinite wedges leading to overdriven ODW have been studied. In such a situation, the existence of a stationary ODW is subject to the presence of a supporting wedge. The use of an unstructured mesh technique allows the study of more realistic configurations, such as a wedge of finite length. To allow for combustion chambers of finite dimension, only finite dimension wedges can be used, thus leading to an interaction between the ODW and the expansion fan which occurs at the end of the wedge. This expansion fan weakens the ODW. As a consequence, the overdriven ODW may either relax toward a Chapman-Jouguet (CJ) ODW or be led to extinction, due to the quenching of the chemical reactions. The CJ ODW has, from the point of view of practical systems, the double advantage of being isolated from the downstream flow by a sonic surface and of leading to the minimum entropy increase among all possible detonation waves. It is important to note that this process could lead to a self-sustained CJ ODW, since obtaining a direct initiation of a Chapman-Jouguet ODW requires very high values of the inlet flow temperatures and pressures. Such values remain incompatible with the envisaged propulsive system (oblique detonation wave engine) for hypersonic trans-atmospheric vehicles (Figueira da Silva and Deshaies, 2000).

The present results evidence the influence of successive mesh adaptions on the structure of the computed flowfield, thus leading to an overall flow structure of the transition region which resembles the one observed experimentally. Finally, the first results which evidence the formation of a Chapman-Jouguet ODW over a wedge of finite dimensions are presented.

**Mathematical Formulation**

The supersonic reactive flow will be computed using the unsteady 2-D Euler equations, thus neglecting molecular transport. These balance equations of mass, momentum, energy and species mass fraction can be written as:

where

and

with *p*, *Y _{K}* and e given by

where

In these equations r is the mixture density, *u* and *v* are the Cartesian velocity components, *p* is the static pressure, *T* is the static temperature, e is the total energy per unit of mass, *e* is the internal energy, *R* is the universal gas constant. The internal energy, the standard-state enthalpy and the specific heat at constant pressure per unit of mass of species *k* are noted , and . and *WK* are the mass fraction, the molar production rate and the molecular weight of chemical species *k*, respectively.

**Kinetics Mechanism**

The chemical kinetics mechanism for the reactive mixture of Hydrogen and air is due to Balakrishnan and Williams (1994). This mechanism considers 9 species (H_{2}, O_{2}, H, O, OH, HO_{2}, H_{2}O_{2}, H_{2}O, N_{2}) and 21 elementary reactions, and it is given in Table 1.

The chemical production rates are given by the Arrhenius law:

The calculation of *k _{f}* and of the molar production rates are performed using the CHEMKIN-II package (Kee, Rupley and Miller, 1991). The thermodynamic properties are calculated according to the procedures developed by Kee, Rupley and Miller (1991).

**Numerical Method of Solution**

The system of partial differential equations (1) - (7) is discretized and solved numerically using a time-split, upwind, finite volume procedure, as indicated in previous papers (Figueira da Silva, Azevedo and Korzenowski, 1999, 2000). The time step splitting procedure used leads to a separate integration of the "fluid dynamic" and "chemical" evolutions of the flowfield. For the integration of the chemical source terms, the code uses an ordinary differential equations solver, VODE (Byrne and Dean, 1993), tailored for the solution of systems of equations that are mathematically stiff. The main differences between the algorithms used previously (Figueira da Silva and Deshaies, 1995, 2000) and the present one concern the integration of the "fluid dynamics" part, which uses an unstructured adaptive mesh technique. For time-march of the "fluid dynamics", a fully explicit, 2nd-order accurate, 5-stage Runge-Kutta time-stepping scheme (Mavriplis, 1988) is used. The algorithm used for the discretization of the fluxes found in Eq. (3) is the Liou AUSM^{+} flux vector splitting scheme (Liou, 1996), with a nominal second order MUSCL extrapolation and MINMOD limiter for the primitive variables (*p*, *u*, *v*, *T*, *Y _{k}*) (Hirsch, 1990).

The first results using the adaptive refinement technique were presented in Rocha Pimentel et al. (1999) and Figueira da Silva, Azevedo and Korzenowski (2000). This technique is used in order to concentrate mesh points in regions of interest within the flowfield, thus making a more efficient use of the storage and CPU resources. The concept behind using an adaptive mesh strategy is to refine regions where large gradients of flow properties occur. In order to identify these regions, a sensor is defined (Figueira da Silva, Azevedo and Korzenowski, 2000) that is based on gradients of the primitive variables (*p*, *u*, *v*, *T*, *Y _{k}*). This sensor marks the triangles where the gradient exceeds a prescribed value. The mesh enrichment procedure introduces an additional node for each side of a triangle marked for refinement, thus making 4 new triangles. Each neighbor of a marked triangle is divided in two new triangles. Since only stationary flows are of interest here, convergence of the solution is ensured between each adaptive refinement of the mesh. This code was validated by comparisons with the numerical results of a previous structured fixed mesh computer program (Figueira da Silva, Azevedo and Korzenowski, 2000). More details concerning the solution algorithm may be found in Azevedo and Figueira da Silva (1997).

The computational domain is presented in Fig. 1. In all the cases considered here, the solid surface of the wedge is assumed non-catalytic and adiabatic where *n* is the local normal to the surface. Since the Euler equations are considered, boundary layer effects will not be accounted for and, thus, only slip conditions are used . The flow enters the computational domain parallel to the wedge symmetry plane with given temperature, pressure, Mach number and composition. Non-reflecting characteristic boundary conditions are used at the exit of the computational domain.

**Results and Discussion**

**Influence of the Mesh Adaption Procedure**

A typical result illustrating the effect of the mesh adaption procedure on the flowfield corresponding to the wedge configuration is shown in Figs. 2 and 3. In Fig. 2, the influence of the grid adaption procedure on the computational mesh and on the computed pressure field is shown. The corresponding fields of Mach number and OH mass fraction are given in Fig. 3. Important differences exist concerning the flow structure from the initial to the final mesh.

In particular, Fig. 2 shows that the mesh refinement leads to the appearance of a large pressure jump, located between the OSW and the wedge surface, and which is absent from the result of Fig. 2(a). Simultaneously, the OSW/ODW transition point is better defined as the mesh spacing is decreased. For these flow conditions, the adaption procedure evidences a transition point which eventually seems to be a triple point. The OSW, a slip line and two detonation waves are connected to this triple point. Each of the successive adaption passes leads to a better rendering of the details of the flowfield and only a few modifications are observed between the last two passes, thus indicating that there exists a limiting stationary state.

In particular, the OSW/ODW transition point moves downstream as the mesh is refined. A limiting, converged, position seems to exist from Fig. 2(c) to Fig. 2(d). Moreover, a quantitative change of the pressure level at the transition region is observed: the maximum values of pressure are 14.0 and 19.6 atm in Figs. 2(a) and (c), respectively. Furthermore, as it can be verified in Fig. 3, the increase of pressure which goes with the mesh refinement leads to an increase of the subsonic region downstream of the triple point. In this region, the downstream part of the flow may influence the upstream one.

It should be noted that the level of refinement currently allowed by the available computer resources remains insufficient for the cellular structure of the detonation wave, which is observed in the experiments (Dabora et al., 1991, Desbordes, Hamada and Guerraud, 1995), to be obtained. Nevertheless, the typical computational mesh size, 9µm, was chosen in order to remain smaller than the induction distance behind the leading shock of the ODW, 100µm and, as a consequence, smaller than the typical size of the elementary cell of the detonation wave, 1mm. A computation of the cellular structure of the detonation wave requires a specific study for such a geometry.

The results shown in Fig. 2(d) correspond to a total CPU time of 504 hours in a HP-PA 7200 processor running at 160 MHz. The structured, fixed mesh, computer code used previously (Figueira da Silva and Deshaies, 1995, 2000) would achieve steady state after the same CPU time on a mesh which has a resolution roughly equivalent to the finest regions of the grid obtained after two passes of adaptive refinement, Fig. 2(c). This clearly demonstrates the benefits of using an unstructured adaptive mesh technique. One way to increase the computational efficiency of the code, which is currently being pursued, consists in the implementation of a mesh coarsening technique. Indeed, as the mesh is progressively refined and the evolution of the flow variables gets steeper, Fig. 2 shows that several cells are found to lie in regions where gradients are small. The elimination of those no longer needed cells would decrease both the storage requirements and the execution time of the code. Another possibility that is currently being envisaged is the parallelization of the code by the use of domain decomposition strategies.

**Analysis of the Transition Region**

The overall structure of the flow, as well as the different types of OSW/ODW transitions are described in the literature (Dabora et al., 1991, Li, Kailasanath and Oran, 1994, Figueira da Silva and Deshaies, 1995, 2000, and Deshaies, Figueira da Silva and René-Corail, 1997). For the sake of clarity, the process which results in a stationary ODW is briefly recalled hereafter. As illustrated in Fig. 4, for the three different initial conditions given in Tab. 2, the flowfield may be divided in three regions:

I) An induction region, located between the OSW and the wedge surface, whose length is controlled by the induction time of the chemical process (*t _{i}*), and which eventually leads to;

II) A transition region. In this region, a large increase in temperature leads to the onset of pressure waves which modify the conditions downstream of the initial OSW, where a decrease of the chemical induction time occurs. Previous studies (Figueira da Silva and Deshaies, 2000) show that the type of transition obtained numerically is influenced by the ratio between the heat release time, *t _{q}*, and the induction time of the chemical kinetics,

*t*When

_{i}.*t*/

_{q}*t*is large, smooth, gradual, transitions are more likely obtained (see Fig. 4(a)) and, on the contrary, when

_{i}*t*/

_{q}*t*® 0, abrupt transitions are found to occur (see Fig. 4(b)). A decrease in

_{i}*t*/

_{q}*t*may be obtained either by an increase of the pressure or a decrease of the temperature of the shocked gases. The different flow structures observed for the transition region are discussed in detail subsequently.

_{i}III) Finally, the detonation is formed at the end of the transition region. A slip line separates combustion products of the ODW from the products issued from the transition region. The final angle of this detonation corresponds to the one which may be computed by considering only the initial and final states of the gases, i.e., using a shock and detonation polar technique (Rocha Pimentel et al., 1999, Figueira da Silva and Deshaies, 2000).

Previous structured, fixed mesh computations (Figueira da Silva and Deshaies, 2000) have only evidenced two types of transition. These two types of transition are also obtained with the present code, as shown in Figs. 4(a) and (b). In the first type, illustrated in Fig. 4(a), the OSW angle is gradually modified by the pressure waves, while, in the second one, shown in Fig. 4(b), an oblique detonation wave is formed in the transition region which leads to an abrupt change of the OSW angle. These two types of transitions are not observed in the experiments of Dabora et al. (1991), Desbordes, Hamada and Guerraud (1995) and Viguier et al. (1996), where the OSW/ODW transition appears to be an abrupt one, involving also a transverse detonation wave which takes place between the transition point and the wedge surface. This is illustrated in the photograph in Fig. 5, due to Viguier et al. (1996).

A first numerical evidence of such a transition was obtained by Rocha Pimentel et al. (1999) on the basis of the numerical procedure presented in the preceding section. Further comparisons show that, at least for the set of boundary and initial conditions used, this third kind of transition is obtained numerically for wedge angles close to the maximum one allowed for a plane ODW to exist, d_{max}. For instance, the transition presented in Fig. 4(c), corresponding to case 2, and which evidences a transverse wave, is obtained for the same wedge angle as the one used in case 1 (Fig. 4(a)), where a transition of the smooth type is obtained. The wedge angles corresponding to cases 1 and 2 are close to the maximum allowed for a plane ODW to exist d_{max} = 39 deg. for *M*_{0} = 7). The realization of a transverse wave in the transition region, starting from the conditions of case 1, required an increase of the initial pressure and, thus, a decrease of *t _{q}*/

*t*.

_{i}The transverse wave that appears in this third kind of transition clearly plays a different role in the flowfield organization than the one evidenced in the abrupt transition presented in Fig. 4(b). Due to their respective angles of attack, the latter deflects the flow away from the wedge surface, while the former deflects the flow towards the wedge. More precisely, even if the overall features of the flow associated with OSW/ODW transition remain similar in all the computational cases, each type of transition leads to a different flow pattern in the transition region.

This is further illustrated in Fig. 6 where the fields of Mach number and the streamlines corresponding to cases 1, 3 and 2 are presented, respectively. In the three cases the flow becomes parallel to the wedge surface immediately downstream the leading OSW. When combustion occurs, and due to the expansion of the gases, the flow is deflected away from the wedge surface, before becoming eventually parallel to it as *x *® ¥ . As illustrated in Fig.6, these successive deflections of the flow, that occur in the region separating the wedge surface from the OSW and the resulting ODW, can be achieved progressively or by the way of a discontinuity. In the smooth transition (Fig. 6(a)), both the deflection due to chemical heat release and the one imposed by the boundary conditions, that lead to a flow parallel to the wedge surface, are achieved progressively in a typical length scale of the order of the computational domain. Figure 6(b) shows that, in the case of the abrupt transition, the first flow deflection, which is directed away from the wedge surface, proceeds via a discontinuity, while the second one is achieved continuously. The discontinuous deflection of the flow is immediately followed by a region of intense combustion, thus indicating the ODW nature of this discontinuity.

In the case presented in Fig. 6(c), the flow initially is deflected progressively away from the wedge surface, as in the smooth transition of Fig. 6 (a), thus indicating that at least part of the chemical heat release occurs in this region. This region is bounded downstream by a discontinuity which occupies the space between the OSW and the wedge surface and through which the flow is deflected towards the wedge. Eventually, the flow becomes parallel to the surface. The chemical process is only partly achieved before this discontinuity, which is followed immedately by a region where the chemical process is completed. A more intense chemical reaction is observed in the vicinity of the OSW than nearby the wedge, where the discontinuity crosses a region which consist mostly of burned gases. Furthermore, Fig. 7(c) also shows that a large region of subsonic flow, when compared to the size of the transition region, is found to occur downstream of the triple point. This subsonic pocket is either absent or of negligible extent in the two other types of transition. This new type of flow structure is found to occur when the wedge angle is close to the maximum allowed for a plane ODW to exist, d * _{max}*, at least in the range of parameters explored. Note also that, as shown in Fig. 2, the use of an adaptive mesh technique is essential to determine the type of transition which occurs. Indeed, the results shown in Fig. 2(a) would suggest a gradual transition, while a transverse wave is actually obtained Fig. 2(d).

The discussion above was developed from a purely kinematic point of view. The complete understanding of all the phenomena controlling each type of transition can only be obtained via a dynamic, two-dimensional, stability analysis. To the best of the authors' knowledge, such an analysis remains an open problem even for shock waves which occur in inert flows.

**Towards a CJ ODW**

The computations previously performed (Li, Kailasanath and Oran, 1994, Figueira da Silva and Deshaies, 1995, 2000, Deshaies, Figueira da Silva and René-Corail, 1997) and those reported in the preceding section involve mostly overdriven ODW. These detonation waves are supported by the boundary conditions along the wedge surface and, therefore, (i) are susceptible to be influenced by perturbations coming from this surface and (ii) require "infinite" wedges to exist. On the contrary, Chapman-Jouguet (CJ) detonations are stand-alone waves which are isolated from downstream influence by a sonic surface. Thus, stabilizing a CJ ODW could be interesting from the standpoint of propulsion applications. However, the direct initiation of a CJ ODW requires initial conditions and extents of the computational domain which are out of the range of the present numerical computations (Figueira da Silva and Deshaies, 2000). Another possibility to obtain a CJ ODW is to initially trigger an overdriven ODW by a wedge and, then, to deflect the wedge surface until the angle corresponding to the CJ wave (d * _{CJ}*) is attained. However, d

*is a function of the freestream parameters. Therefore, it is necessary to examine the effect of the wedge surface deflections beyond this value (d < d*

_{CJ}*), for instance, when it becomes parallel to the incoming flow (d = 0). This wedge geometry is of practical interest for propulsion applications. The conditions under which an overdriven ODW may relax to a CJ ODW may be examined by the numerical procedure used in the present work, due to the flexibility introduced by the unstructured meshes. Indeed, this study requires a larger and more complex computational domain involving different regions of mesh refinement.*

_{CJ}As illustrated in Fig. 7, an abrupt deflection of the wedge surface generates an expansion fan. This expansion fan interacts with the overdriven ODW and decreases both the pressure and the temperature downstream of the ODW, leading to an increase of the Mach number in the burned gases region and to a decrease of the wedge-ODW angle. This modification is also responsible for an increase of the induction length of the chemical kinetics, *l _{i}*, downstream of the ODW. For a numerically relevant solution to be obtained, it is clear that

*l*must remain smaller than the longitudinal dimension of the computational domain. When this condition is satisfied, the ODW becomes isolated from the downstream flow by a sonic surface.

_{i}In Fig. 8, the results of a computation for which a CJ ODW is obtained are presented. In this figure, the fields of temperature, pressure (together with some streamlines), and of the angles w^{±} = arctan (*v*/*u*) ± arcsin (1/*M*) are plotted. The w^{±} angles limit the Mach cone, which is the region of the flow that may be influenced by perturbations which occur at its vertex. In this figure, one can clearly see the OSW/ODW transition that occurs at the beginning of the wedge and that leads to the formation of an overdriven ODW. The expansion waves that emanate from the deflection of the wedge intercept this overdriven ODW, leading to a progressive bending of the ODW. In the case shown in Fig. 8, the progressive bending of the ODW seems to stop after the ODW reaches an angle of attack of 35 deg.This value is quite close to the one predicted by the detonation polar technique for a CJ ODW, q * _{CJ}* = 37 deg.

It should be noted that the length of the computational domain, 1 mm, remains too small to be relevant for practical applications. This domain size is chosen such that the OSW/ODW transition may be followed by a large region where the expansion fan interacts with the overdriven ODW, yet allowing for a reasonable CPU time for the computation, of the order of 100 hours in a HP-PA 7200 processor. Nevertheless, this very small computational domain remains sufficiently large when compared to the typical induction length of the chemical kinetics, at least for the case presented in Fig. 8. Indeed, the peak of pressure and the temperature immediately downstream the overdriven ODW and the CJ wave are (54 atm, 3700 K) and (16 atm, 3000 K), respectively, that leads to induction lengths for the chemical kinetics, *l _{i}*, of 20 µm and 100 µm. Moreover, most of the Mach cone angle lines w

^{+}> 40 deg. have overtaken the ODW. This is an indication that the normal Mach number downstream of the ODW has reached the sonic condition.

**Concluding Remarks**

The above analysis seems to indicate that a Chapman-Jouguet ODW is indeed obtained as the result of the interaction between an expansion fan and an overdriven ODW, at least for the computational conditions chosen here. A parametric analysis of such a configuration is needed involving larger computational domains, such that situations of practical interest may be investigated. Nevertheless, an increase of the computational domain would require modifications of the computer code. In particular, an adaptive refinement procedure involving not only a local decrease of the elementary mesh size in regions of large gradients, as in present the case, but also an increase of mesh size in regions of small gradients should be implemented. Since the steady solution is found at the outcome of an unsteady process involving the progressive buildup of the initial OSW, of the overdriven ODW and, eventually, of the CJ ODW, whose positions are unknown *a priori*, adaptive mesh refinement and coarsening are utterly important. Indeed, regions of small mesh size may be needed during the OSW/ODW/CJ ODW formation, but not in the final steady solution. In the framework of the unstructured mesh computations here performed, increasing the size of the elementary cell is not a straightforward task. This point is currently under investigation.

**Acknowledgments**

This work was accomplished in the framework of an international cooperation agreement between Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) and Centre National de la Recherche Scientifique (CNRS), while the first author was a graduate student with a scholarship from CAPES (Coordenação de Aperfeiçoamento de Pessoal de Nível Superior), Brazil. Additional partial support was provided by CNPq under the Integrated Project Research Grant No. 522.413/96-0.

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