1 Introduction

An efficient mixing process is very important for the engineering implementation of an airbreathing propulsion system. Many mixing enhancement devices have been proposed in recent years, including transverse jet (Lee and Mitani, 2003; Huang et al., 2012b; 2012c), ramp (Alexander et al., 2006; Huang et al., 2013b), strut (Huang et al., 2011c; Sujith et al., 2013), pylon (Gruenig et al., 2000; Gruber et al., 2008; Takahashi et al., 2010; Lee, 2012; Pohlman and Greendyke, 2013), and cavity (Yu and Schadow, 1994; Huang et al., 2012a; 2013a), as well as double cavities in parallel or tandem (Huang et al., 2011b) and some combinations (Hsu et al., 2010; Grady et al., 2012). These devices can provide an axial vortex which has been proved to be responsible for improving mixing in supersonic flows.

Vergine et al. (2015) studied the influences of the imposed interaction and subsequent dynamics of a system of selected supersonic streamwise vortices on the reacting plume morphology and its evolution. The hydrogen plume issued from two pylon-type injectors. Huang and Yan (2013) gave a detailed review of transverse injection schemes in supersonic flows, and found that the mixing process in the transverse injection flow field is influenced by many geometric parameters, suggesting that it is a multi-objective problem. Vinogradov et al. (2007) provided an overview on the pre-injection scheme involved in a pylon, and concluded that the pylon is a useful tool to enhance the mixing process in a transverse injection flow field.

The unsteadiness in the flow field produced by emission of acoustic oscillations from wall-mounted cavities has proved to be an effective tool to enhance entrainment and fuel air mixing (Das et al., 2015). However, the mixing enhancement induced by a wall-mounted cavity has rarely been investigated, and the cavity is more often used to hold the flame in the scramjet engine. Yu and Schadow (1994) were the first to propose cavity based mixing phenomena. They observed that the spreading of a cavity actuated mixing layer was much higher than that of an ordinary mixing layer for the same velocity and density ratios. Handa et al. (2014) experimentally compared supersonic mixing fields in three ducts without any devices, with a rectangular cavity, or with the proposed cavity. They found that mixing, as well as jet penetration, was enhanced far more rapidly in the duct with the proposed cavity. This effect is induced by the large-amplitude jet fluctuation due to oscillatory secondary flows. However, detailed flow field structures and quantitative evaluation were lacking. This information should be explored further using a computational fluid dynamics approach. Recently, with the improvement in computer capability, this approach has become an efficient tool to support ground experimental tests.

In this study, three physical models were employed to investigate the supersonic mixing enhancement mechanism induced by the proposed cavity configuration, and grid discrepancy analysis was used to choose a suitable grid scale. A large eddy simulation and optimization design of these configurations, as well as a study of the combustion process, will be conducted in the near future, but this is beyond the scope of this paper.

2 Physical model and numerical approach

2.1 Physical model

The physical models have a height of 7.5 mm and a width of 42 mm at the entrance, and a section with a 3° divergence angle to avoid unstarting in the wind tunnel due to injection or boundary layer growth (Fig. 1). Fig. 1 shows the plan and symmetric views of the three physical models employed in this study, namely the conventional physical model, novel physical model I, and novel physical model II. The origin of the coordinate system is located in the centre of the middle injector. The middle injector is located in the symmetric plane of the model. For all three physical models, injection ports of 1.5 mm diameter are located 22 mm downstream of the entrance of the divergence section, and the distance between two injectors is 12 mm. These specifications are the same as those employed by Handa et al. (2014).

Fig. 1
figure 1

Plan and symmetric views of the three physical models employed in this study: (a) conventional physical model; (b) novel physical model I; (c) novel physical model II (unit: mm)

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The length and depth of the cavity mounted in novel physical models I and II are 10 mm and 5 mm, respectively, and the leading edge of the cavity is located at the entrance of the divergence section for novel physical models I and II. For novel physical models I and II, three parts of the rear face of the cavity are cut out, and the length and width of the cut out part are 10 mm and 6 mm, respectively.

The air flows into the combustor from left to right with a static pressure of 43882.48 Pa and total temperature of 297 K. Hydrogen was used as the injectant in this study, and was injected into the combustor with sonic velocity. Its static pressure was 166863.1 Pa, and its total temperature was 297 K. The parametric distributions at the exit of the wall orifice and at the entrance of the combustor are uniform. The effect of the flow field property in the inlet on combustor performance was not considered in this study, although the incoming boundary layer thickness has a great impact on the flow inside the cavity. This information will be obtained in future work.

2.2 Numerical method

Steady state computational data were obtained using FLUENT version 6.3.26, and mesh was generated using Gambit (Fluent Inc., 2006). A Dell workstation at the Science and Technology on Scramjet Laboratory, China, using up to 32 processors, provided a parallel computing environment for flow solutions.

For this study, the 3D Reynolds-averaged Navier-Stokes (RANS) equations were solved using a coupled, implicit, second-order upwind solver. Cell fluxes were computed using an AUSM scheme and the viscosity determined using mass-weighted-mixing-law. The operational fluid was air, treated as an ideal gas with no reactions modeled.

The k-ω shear stress transport (SST) model was used for turbulence modeling. The SST model combines the advantages of the k-ω model near solid surfaces with those of the k-ε model, which has good free-shear-flow properties, making it well suited for this flow. The SST model also shows better performance than either the k-ω or k-ε models in adverse pressure gradient flows (Freeborn et al., 2009). The SST model was utilized successfully in previous studies on transverse injection (Huang, 2014; 2015) and backward-facing step flows (Huang et al., 2014a). The Courant-Friedrichs-Levy (CFL) number remained at 0.5 with suitable under-relaxation factors to ensure stability. No-slip conditions (u=v=w =0) were assumed for wall boundaries. At the outflow, all the physical variables were extrapolated from the internal cells based on the flow being supersonic (Kim et al., 2004).

The solution was considered to have converged when most of the residuals reached their minimum values after falling more than three orders of magnitude, and the difference between the computed inflow and the outflow mass flux was required to drop below 0.005 kg/s. Fig. 2 shows the convergence history for novel physical model II, and it represents the residuals of the variable parameters, including the velocities, energy, k and ω for the turbulence model, and the species considered in the current study, namely air, H2, and O2. Three grid scales were employed to analyze grid independency for the conventional physical model, namely coarse grid (942703 cells), moderate grid (1200223 cells), and refined grid (1492543 cells). The grid was multi-blocked and highly concentrated close to the wall surfaces, the fuel injector, and the cavity to ensure the accuracy of the numerical simulation. Fig. 3 represents the grid system for novel physical model I, and Fig. 4 depicts a close-up view of the grid system around the cavity in novel physical model I. To show the configuration of the cavity in the flowpath, Fig. 5 shows details of the meshes at the bottom wall of the cavity. The height of the first row of cells is set at a distance of 0.001 mm for the walls, and the maximum wall y+ is less than 9.1. This proved to be suitable for turbulent mixing simulation of supersonic flows (Pudsey and Boyce, 2010), as well as combustion flow field simulation in a cavity-based combustor (Huang et al., 2014b).

Fig. 2
figure 2

Convergence history for novel physical model II

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Fig. 3
figure 3

Grid system for novel physical model I

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Fig. 4
figure 4

Close-up view of the grid system around the cavity in novel physical model I

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Fig. 5
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Close-up view of the grid system at the bottom wall of the cavity in novel physical model I

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3 Code validation and grid independency analysis

The numerical approach mentioned above has been utilized successfully to analyze the flow field in cases with a cavity (Huang et al., 2011a) and with transverse injection ports (Huang, 2014). For convenience, the code validation process is not provided in this section, and readers are referred to Huang et al. (2011a) and Huang (2014). A comparison between numerical results and published experimental data, as well as the grid independency analysis, has been provided by Huang et al. (2011a) and Huang (2014).

Fig. 6 shows the effect of different grid scales on wall static pressure in the conventional physical model. The grid scale has only a slight impact on the wall static pressure distribution. The largest discrepancy occurs when x≈50 mm. The discrepancy between the results predicted by the moderate and refined grids is much smaller than that between the results predicted by the coarse and moderate grids. This may imply that the number of cells in the moderate grid system is sufficient for this investigation. The number of time steps increases with the number of grid cells to obtain a steady flow field. This is related to the parallel computing environment and the numerical method employed (Smirnov et al., 2014). The accumulation of stochastic error is proportional to the number of time steps and depends on the accuracy of the scheme and the approximation error (Smirnov and Nikitin, 2014). Fig. 7 depicts a comparison of the Mach number (Ma) contours on the symmetric and four cross-sectional planes of the conventional physical model with different grid scales. The grid scale makes a slight difference to the Mach number contours as well. The Mach disk and the bow shock wave upstream of the injector are clearly captured by all three grid scales, and the core of the counter-rotating vortex pair (CVP) moves towards the upper wall with the increasing horizontal distance. Thus, 2290350 and 2301429 cells were employed for the simulations of novel physical models I and II, respectively, in the following discussion.

Fig. 6
figure 6

Comparison of wall static pressure (Pw) resulting from different grid scales used in the conventional physical model

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Fig. 7
figure 7

Comparison of Mach number contours on the symmetric and four cross-sectional planes resulting from different grid scales used in the conventional physical model

(a) Coarse grid; (b) Moderate grid; (c) Refined grid

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4 Results and discussion

Fig. 8 depicts the comparison between the Mach number contours on the symmetric and four cross-sectional planes for the three physical models with a moderate grid scale. The subsonic area of the case with the cavity is larger than that of the case with only transverse injection ports (Figs. 8b and 8c). Further, the peach-shaped vortex (Huang et al., 2013b) at x=20 mm is broken up, especially for novel physical model I (Fig. 9). The mode of the flow field structure is symmetric at x=20, 40, 60, and 80 mm, and a larger separation region occurs on the top wall of the combustor. Fig. 9 compares the vortex structure at the cross-sectional plane x= 20 mm. The peach-shaped vortex generated in the core flow is clearly broken up. This may be induced by the strong interaction between the shear layer over the cavity and the jet. Also, the reflected oblique shock wave has a large impact on the augmentation of mixing of the transverse jet (Zare-Behtash et al., 2015). At the same time, the case with the cavity mounting upstream of the transverse injection ports shows stronger 3D effects.

Fig. 8
figure 8

Comparison of Mach number contours on the symmetric and four cross-sectional planes for three physical models with a moderate grid scale

(a) Conventional physical model; (b) Novel physical model I; (c) Novel physical model II

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Fig. 9
figure 9

Comparison of vortex structure at the cross-sectional planex=20 mm

(a) Conventional physical model; (b) Novel physical model I; (c) Novel physical model II

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Fig. 10 shows a comparison of the hydrogen mole fraction contours on the symmetric and four cross-sectional planes for the three physical models with a moderate grid scale. The vortex moves towards the top wall of the combustor due to the interaction of the shear layer over the cavity and the jet (Fig. 9). The peach-shaped vortex at the core flow is broken into small vortexes which move towards the top wall of the combustor. At the same time, the interactional effect is more serious for the middle vortex occurring on the symmetric plane (Figs. 10b and 10c). This implies that the case with the cavity is beneficial to jet penetration, especially for novel physical model II (Fig. 10c).

Fig. 10
figure 10

Comparison of hydrogen mole fraction contours on the symmetric and four cross-sectional planes for three physical models with a moderate grid scale

(a) Conventional physical model; (b) Novel physical model I; (c) Novel physical model II

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To evaluate the influence of the cavity mounted upstream of the transverse injection ports on the fuel spread process, the hydrogen mole fraction contour on the lower wall of the combustor was compared (Fig. 11). Fig. 11 represents a comparison of the hydrogen mole fraction contours on the lower wall for three physical models with a moderate grid scale. Clearly, the hydrogen mole fraction is the largest for the case with only the transverse injection ports (Fig. 11a). The hydrogen mole fraction for novel physical model I is the smallest (Fig. 11b). The hydrogen injected from the middle injector for novel physical model II is away from the lower wall of the combustor (Fig. 11c), and this may imply that its mixing process is more complete and its penetration depth is larger. More fuel was wrapped into the core flow and mixed with the air stream. This result is consistent with that observed in Fig. 10c.

Fig. 11
figure 11

Comparison of hydrogen mole fraction contours on the lower wall for three physical models with a moderate grid scale

(a) Conventional physical model; (b) Novel physical model I; (c) Novel physical model II

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The injectant/air mixing efficiency was chosen to analyze the mixing enhancement induced by the wall-mounted cavity, and it was defined as (Segal, 2009)

$$\varphi = {{{{\dot m}_{{\rm{fuel}},{\rm{mixed}}}}} \over {{{\dot m}_{{\rm{fuel}},{\rm{total}}}}}} = {{\int {{\alpha _{{\rm{react}}}}\rho \mu {\rm{d}}A} } \over {\int {\alpha \rho \mu {\rm{d}}A} }},$$
(1)
$${\alpha _{{\rm{react}}}} = \left\{ {\begin{array}{*{20}c} {\alpha, \quad \quad \quad \quad \quad \quad \quad } & {\alpha \leq {\alpha _{{\rm{stoic}}}},}\\ {\alpha (1 - \alpha)/(1 - {\alpha _{{\rm{stoic}}}}),} & {\alpha > {\alpha _{{\rm{stoic}}}},} \end{array}} \right.$$
(2)

where α is the injectant mass fraction, αreact is the injectant fraction mixed in a proportion that reacts, αstoic is the injectant stoichiometric mass fraction, ṁfuel,mixed is the mixed injectant mass flow rate, and ṁfuel,total is the total injectant mass flow rate. ρ and μ are the local density and velocity, respectively, and A is the cross section of the axial station where mixing is evaluated. The value for the hydrogen stoichiometric mass fraction was 0.02831, and the cross section of the axial station varied due to the special properties of the flowpath. In this study, the mixing efficiencies at cross-sectional areas (x=20, 40, 60, and 80 mm) were evaluated and were all downstream of the jet orifice.

Fig. 12 depicts a comparison of the mixing efficiency for the physical models employed in this study. The mixing of air and injectant is enhanced by the wall-mounted cavity. Further, it is obvious that proposed novel physical model I was the best configuration for the flowpath design of the scramjet engine, and its mixing efficiency at x=40 mm was 100%. The conventional physical model without the wall-mounted cavity was the worst configuration, as the air and injectant were not mixed completely, even at the exit of the model; the mixing efficiency at x=80 mm was 98.64%, slightly lower than that of novel physical model I at x=20 mm. This further implies that the interaction between the shear layer over the cavity and the jet has a large impact on the mixing enhancement in supersonic flows.

Fig. 12
figure 12

Comparison of the mixing efficiency of the physical models employed in this study

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5 Conclusions

In this paper, the mixing process induced by a cavity mounted upstream of the transverse injection ports was investigated numerically, and grid independency analysis was conducted. The mixing efficiencies at four cross-sectional planes, namely x=20, 40, 60, and 80 mm, were compared for the physical models employed in this study. We have come to the following conclusions:

  1. 1.

    In the case with only the transverse injection ports the vortex was broken up by the strong interaction between the shear layer over the cavity and the jet. Accordingly, the area of the subsonic region was larger for the case with the cavity mounted upstream of the transverse injection ports.

  2. 2.

    Hydrogen penetrated deeper in the case with the cavity mounted upstream of the transverse injection ports, especially for novel physical model II. This is beneficial to the mixing process in supersonic flows.

  3. 3.

    Novel physical model I had the highest mixing efficiency. The strong interaction between the shear layer over the cavity and the jet has a large impact on the mixing enhancement in supersonic flows, and this is a fundamental step for the optimization of scramjet combustor configurations.