Vortex behavior in a dynamically moving hypersonic cavity Open Access

In our previous paper,¹ we presented the detailed thermochemical non-equilibrium fluid dynamics when hypersonic flow interacts with static cavities with and without conjugate heat transfer. The current manuscript builds on this research and answers the following fundamental question: when exposed to thermochemical non-equilibrium hypersonic flow, what are the dominant processes that dictate the dynamics in a cavity with an increasing length-to-depth (L/D) ratio, that is, when the cavity depth decreases with time? The motivation for investigating this phenomenon arises from the need to understand buffeting and acoustic tones in the fuselage openings and weapon bays of aerospace vehicles.^2–4 All previous investigations of high-speed flow interactions with cavities, including our recent paper,¹ have focused on fixed cavity geometries of various length-to-depth (L/D) ratios.^5,6 In this paper, we present a computational analysis to understand the flow physics as a cavity that dynamically changes geometrically from an L/D of 2 to 9.8, is exposed to a freestream Mach number of 11.

To contextualize the configuration, imagine a scenario similar to that of the bay with a door underneath an aircraft. Upon opening the doors to release a payload, the vehicle experiences a plethora of rapid, transient, and unsteady flow behavior that can destabilize vehicle control. At high Mach numbers, this problem would be emphasized as rapid deceleration (with a more significant gradient than subsonic flow deceleration) of the incoming air would create substantial change in the moments on the vehicle. This could place the vehicle into a catastrophically unrecoverable state, such as an unrecoverable angle of attack, or create such significant loading in a manner for which the vehicle was not designed to bear such that parts break away.

Although a bay door problem is one such example of a real-world application of this study, broader conclusions can be drawn from the phenomenological behavior of mixing, mass ejection, vorticity, and aerodynamic effects in this flow scenario. Two other examples include challenges such as (1) internal cavity flame stabilization, where cavity motion could be used to create beneficial mixing to the species and flameholding once ignition begins, or (2) other dynamic structures where the freestream is suddenly exposed to an increase or rapid decrease in relative volume, this rapid change could be utilized to potentially reduce boundary layer growth and manipulate vehicle lift, drag, and moments. Structural impacts could weaken the vehicle in the mid-flight and cause a rapid change in its shape. Assuming that the projectile is moving at a speed similar to the freestream, this means the flow will encounter a rapid change in the vehicle, likely in the form of a cavity, although similarly likely not rectangular. Regardless of the application, understanding how such rapid changes affect hypersonic flows is necessary for future considerations in prototyping. Future numerical and experimental works could utilize this information to make improvements in flame stabilization in internal flows by cavity motion (although it is noted that the Mach numbers will not be as high for internal flows relevant to flameholding as those for external flows) and to test solutions to external flow challenges such as the bomb bay problem.

Computational analysis of the interactions between a moving fluid and moving structures, sometimes also referred to as flow-structure interactions (FSI), is a research topic that has been pondered for many years and has significant applications, both in engineering and medical science.^7–21 In recent years, interest in aerothermodynamics and combustion processes relevant to high-speed propulsion^22–28 has led to a surge in studies focused on FSI investigations.^29–31 New research in this field has covered experimental work on cantilevered panels^32–35 and edge-claimed panels.^36–46 The first of these cases focuses on the development of a high-speed flow boundary layer on a panel, and the second focuses on the impact of shock on the panel. From a numerical standpoint, these experiments are practical for validation and are designed to analyze conjugate heat transfer, structural dynamics, and thermochemical non-equilibrium. Several other studies following a similar setup can be found in the literature.^{32–34,36,39}

To aid in defining the problem at hand from a modeling point of view, FSI can be classified into two categories: (1) two-way coupled, where the effect of moving or deforming structure on the fluid and vice versa is modeled, and the deformation, and consequently the mesh motion is not known a priori, and (2) one-way coupled, where the solid boundaries are prescribed with a predefined motion and the effect of this change in geometry is investigated on the flow, thermodynamic, and chemical processes. In certain situations where the deformation is not significant, researchers also look at this problem from the structural dynamics lens where the effect of deformation on fluid dynamics is ignored and only structural dynamics is investigated. Although we acknowledge that two-way coupled frameworks more accurately represent the physical processes that would take place when high-enthalpy, high-speed flows interact with structures, as a first step we assume that the solid is rigid and changes shape based on prescribed motion (as opposed to due to aerothermal forces from the flow) in this paper. This will facilitate understanding of the effect of dynamic cavity motion on fundamental flow dynamics and provide useful information on the use of prescribed motion for the control of aerospace vehicles. In a subsequent manuscript, we will take into account the mutual effects of flow and structural dynamics. We should note that this is the first analysis of its kind to investigate the cavity dynamics as it compresses (or expands) when embedded in thermochemical non-equilibrium hypersonic flow.

The simulation of this complex flow behavior requires the solution of governing equations that dictate high-speed flow of gases in thermochemical non-equilibrium with a dynamically moving mesh. In the next section, we briefly discuss the governing equations and appropriate models that accurately describe the physical processes of interest and the implementation of the mesh motion. This will be followed by a discussion of the detailed fluid dynamic behaviors in the cavity as it compresses at a rate of 60 m/s. The last section draws major conclusions from this study.

We described the detailed governing equations and underlying models in our previous paper,¹ but for the sake of completeness, the conservation equations for mass, momentum, species and energy (total and vibrational) relevant to thermochemical non-equilibrium flows are summarized below. For more details, including the underlying assumptions, the readers are referred to Redding et al.¹ We solve these equations in two dimensions. Although we acknowledge the limitations of two-dimensional (2D) calculations, it is appropriate for the configuration under consideration. We will investigate the impact of the spanwise direction on flow physics in a future research effort.

• Mass conservation

  $∂∂tρ+∇·ρu=0,$

  (1)

  where $ρ$ is the density of the mixture and u is the velocity vector.

• Momentum conservation

  $∂∂tρu+∇·ρuu=−∇p−∇·{−μ(∇u+(∇u)T)+2/3μ(∇·u)δ},$

  (2)

  where $μ$ is the mixture averaged dynamic viscosity and p is the pressure.

As mentioned above, the details of various models in Eq. (5); $QT−Vi$⁠, $QV−Vi$⁠, $QChem−Vi$⁠, and $Qradi$ can be found in our previous paper.¹ The Sutherland law is used to take into account the temperature dependence of thermal conductivity and viscosity.

Although we presented a detailed grid sensitivity analysis and validation of the solver in our previous papers^1,28 where a double cone with half angles of 25°and 55° was exposed to a Mach number of 11, for completeness, Fig. 1 shows the comparison of heat flux and wall pressure predicted by our calculations with measurements of Holden.⁴⁸ As can be observed, our results are very close to the experimental observations. The calculations that form the basis of this manuscript are performed under very similar flow conditions, and this close comparison gives us confidence in the validity of the governing equations and models used for the present analysis.

Figure 2 shows a schematic of the flow configuration. The operating and initial conditions, and the corresponding physical properties listed in Table II are based on an altitude of 39 360 feet above sea level. Based on the Reynolds number, the estimated Kolmogorov scale, $η$ and Taylor microscale, $λ$ are 4.885  $×$ 10^–6 m and 2.315  $×$ 10^–4 m, respectively. In the current work, a direct numerical simulation (DNS) level grid is used. Despite using DNS resolution, it is worth verifying grid invariance. To illustrate this, two calculations were performed on a static cavity case, one with a grid resolution of $η$ and the other with a resolution of $2η$⁠. Figure 3 shows the velocity profile at an axial location of 0.02 m from the inflow boundary. Unsurprisingly, the two profiles are identical, thus satisfying our curiosity and further providing us confidence in the numerical predictions. The generally accepted grid resolution for DNS calculations is $π1.5η$⁠. However, to account for any change in scales due to chemical reactions, even if the production rates are small (as is the case in the current configuration–see Sec. IV D), and the motion of the boundary, we use $η$ as the grid size in the rest of this manuscript.

The computational domain is divided into four rectangular sections. Structured grids are used to ensure that the initial cell sizes can be directly controlled. Table III shows the four sections and the corresponding finite volumes to maintain a grid size of $η$⁠.

Determining the initial static grid is only part of the challenge. As the bottom boundary moves upward, the cells are compressed, leading to high-aspect ratio cells. If we avoid cells with aspect ratios greater than 1000, for DNS calculations, this would just mean we have an over-resolved solution. The primary concern when considering the effect of high aspect ratio cells is the accuracy and stability of the solver. Since the simulation is transient, the flow is limited by the Courant number. This number, which depends on the length of the cell in the flow direction, limits the upper limit of the time step so that the flow convects less than the cell width in one time step. If the high-speed flow is in the direction of the longer dimension of the cell, high aspect ratios are not as problematic, since the Courant–Friedrichs–Lewy (CFL) numbers can still be maintained at relatively higher values (but still within the stability limits of explicit numerical schemes). However, in cases with extremely low aspect ratios (tower cell instead of pancake-like cells), some significant reduction in CFL can occur, causing extremely long simulation times. Fortunately, since the L/D of our cavity varies from 2 to 9.8, starting with a grid cell aspect ratio of close to 1, the cells in the cavity become more pancake-like and thus lead to reasonable time steps.

The results section is organized into four subsections. In the first two subsections, we, respectively, discuss the dynamics of the waves and shear layer in the compressing cavity. This is followed by a detailed analysis of turbulence spectra in the cavity. The last subsection will detail species generation and mass flux in and out of the compressing cavity. However, before we begin this discussion, let us take a look at the overall flow behaviors. Figure 4 shows the contours of the time evolution of the density gradient as the cavity compresses from L/D of 2 to 9.8. In these contours, the shock structures are easily observable, with a distinct shock near the leading edge of the domain. As the shear layer oscillates, this shock can be seen deflecting and will be discussed at length in the next section. Within the cavity, high-density gradients show that there are wave propagations within the cavity and that rapid changes in thermodynamic behavior will occur. Higher density would likely result in a higher prediction of collision frequency between particles, and hence, more reactions should be observed near these gradients.

The interaction of the flow entering the cavity with the walls lead to two dominant wave motions within the cavity: one that propagates horizontally, caused by the forcing of the incoming fluid, and one that propagates vertically, caused by the reflection of the wave between the lower wall and the shear layer. This observation suggests that a strong enough shear layer can behave like a wall that partially reflects and partially transmits energy. The forcing resulting from the dynamic compression of the cavity maintains the pressure intensity while maintaining sinusoidal dilatational behavior. To understand this process in detail, Fig. 5 shows the time-averaged pressures on the leading, bottom, and trailing walls. Since the cavity height changes with time, each of these figures shows the non-dimensional wall distance as the abscissa. Note that while the bottom wall length does not change in time, near-wall physics is certainly different because of its upward motion. The time-averaged pressures on the front and bottom walls show a rise in pressure near the corners of the cavity. A higher rise is seen on the trailing edge of the cavity, which is consistent with previous literature⁴⁹ on static cavities. Although not completely unexpected, this phenomenon is still true in the present case–note that the moving cavity causes a change in the location of the shear layer. At all three walls, the pressure is highest at the edges and lowest at the center. While the presence of flow turning or impingement at the corners can explain these observations for the leading and trailing walls, to identify why the center of the bottom wall behaves differently from the edges, we look at the time evolution of pressure along the bottom wall.

Figures 6 and 7 show the time history of pressure along the bottom wall. The red dot indicates the maximum pressure at a given time instant. In each instant, a sharp gradient near the red dot indicates a shock wave that is traveling in the cavity as it compresses. Note that the time instances can be correlated with the L/D ratio of the cavity as indicated in the figures. This shock wave motion between the left and right edges of the cavity continues throughout the computation. If this wave were to oscillate in the absence of other flow characteristics, one would expect the peak pressure magnitude to attenuate over time. However, as Fig. 8 shows, while there is a decreasing trend in the pressure magnitude, there seems to be a periodicity in peaks. To identify the reasons for it, let us probe the peak at 4.6125  $μ$s. The bottom part of Fig. 8 shows the pressure contours in the cavity at three times (at the time instant of interest and just before and after it). These figures show the constructive interference between the horizontal wave that we identified earlier and the vertical fluid motion between the bottom wall and the shear layer; thus leading to a high pressure at each of the two corners that we observed in Fig. 5.

Before the cavity starts to compress, the shear layer impinges on the trailing wall in a cyclic manner, causing significant pressure waves and unsteady motion. As a side note, this is one of the reasons why an angled trailing wall is often utilized in engines. However, in this case, the shear layer is pushed away from the cavity trailing edge and, due to the compression rate of the cavity, is unable to return or impinge on the trailing wall again as long as the compression rate is maintained. This reduces the occurrence of pressure waves generated from the trailing edge of the cavity, just as one would expect to see in State II in our prior work. Instead, the horizontal and vertical pressure waves that we discussed in the previous section are allowed to continue their oscillation within the cavity (as long as the compression rate of the cavity is maintained and the forcing from the fluid remains). To elaborate, what is conveyed here is that since the shear layer does not impinge on the trailing edge, there are no additional pressure waves generated to destructively or constructively interact with the internal waves of the cavity; hence, the motion and structure of the two primary waves caused by the initial flow impingement are largely maintained. If the compressive motion of the cavity were to cease, it is expected that these waves would eventually dissipate.

The compression of the cavity and subsequent shear layer oscillations cause the shear layer to behave like a lifting surface to the incoming flow, causing deflections of the flow that are strong enough to create a shock whose angle is time-varying and dependent on the relative angle of the shear layer to the incoming flow direction. This deflection interacts with the primary shock structure in the domain, which is initiated near the boundary layer. This change in the shock angle of the primary shock would likely affect the macro-level parameters; such as a rapid change in the moment, lift, and drag of a vehicle. It also, however, hints at an optimum cavity wall motion rate to have a completely stable flow over a vehicle, with a reduced boundary layer. Such an optimum flow may include one that does not deflect the leading shocks, but also exhibits the vortical behavior as seen here which follows a predictable energy cascade in the inertial range. If this optimum or ideal behavior is hoped for, it likely lies at a compression rate that is between keeping the shear layer away from the trailing edge, while also avoiding shock deflection from too rapid deflection of the shear layer. There may be more significant room for stabilization, while also amplifying the benefits of cavity flow (or the necessity of cavity flow), in expansion or joint compression/expansion cavity motion in hypersonic flow—future work that is in post-processing will investigate the expansion case.

As shown in Fig. 9, as the flow enters the cavity, two counter-rotating vortices develop and as the cavity compresses, break down into multiple smaller vortices with much higher intensities and rotation rates. This is because of the interaction between the large vortices with the bottom wall and the shear layer. The size of the smaller vortices is dictated by the cavity size– single large vortices tend to optimize the space in the cavity, distributing their size, magnitude, and number to be more evenly distributed about the cavity. To quantify this behavior, we measure the area of vortical motion (defined by positive values of Q) as a function of time. Note that the shear layer is not accounted for during this measurement and only the interior of the cavity is considered.

Figure 10 shows that the total area of vorticity-dominated regions is relatively consistent throughout the motion of the cavity. This implies that as the cavity size decreases, the larger vortices breakup into smaller ones with increasing intensities, confirming the previous observation.

This observation that energy transfer takes place from large to smaller vortices (via splitting) is analogous to energy cascading in turbulence. To quantify this energy transfer and its rate, we calculate the energy spectrum by summing the squares of the magnitude of fast Fourier transforms (FFT) of the fluctuating velocity components at three spatial locations–the top left, top center, and top right of the cavity. Figures 11(c) and 12(c) show the FFT of the vertical velocity components at the top left and center locations, respectively. Both these locations are dominated by a frequency of 25 300 Hz, indicative of the shear layer oscillations near the leading edge of the cavity. We expected to observe the same frequency at the trailing edge of the cavity, but as shown in Fig. 13(c), the peak frequency there is significantly lower as compared to the other two locations.

This further confirms that the shear layer does not impinge on the trailing edge. As we will see later in Sec. IV D, this is likely because the vertical velocity of the mass that leaves the cavity fluctuates between the forced motion of the cavity and the shear layer (see Fig. 19).

Next, let us look at the energy cascade shown in Figures 11(b), 12(b), and 13(b). In addition to the energy spectrum, the figures also show two lines, one with a slope of −5/3 (in red) and the other with a slope of −1 (in green). In the region at the top of the cavity, no distinct cascade behavior is observed. A reverse cascade is also observed. More interestingly, for all three locations, the spectrum seems to be closer to the line with a slope of −1. This can have some intriguing implications if such behavior is specific to cavity flow where the distinct vortices are found. To check whether the turbulent flow in the cavity follows this behavior, we investigate the energy spectrum at the center of the cavity. Note that since the spatial location of the center changes with time, as shown in Fig. 14, we pick the center at each temporal location and then calculate the spectrum based on that dataset.

Figure 15 shows the energy spectrum at the (moving) center of the compressing cavity. The inertial range has a slope of −1. Acknowledging that the absolute spatial location where we extract velocity information changes in time (as compared to the traditional calculation of energy spectrum at a fixed spatial location), a −1 slope of energy cascade was reported by Feynman⁵⁰ in quantum experiments on liquid helium. It was found that a vortex ring can breakup into smaller rings by pinching, where two “sides” of the ring come closer to each other until their distance is of the order of atomic scales. Once this occurs, one larger (energy-containing) ring can breakup into smaller vortex rings. This is something that is observed in the flow behaviors of this dynamically compressing cavity.

The idea that quantum behavior can be observed at the macroscopic scale is not new by itself. Quantum behavior has been observed at the macroscale in superfluids, superconductors, and dilute quantum gases. It has never been seen, even in a comparative sense, in hypersonic flow over a cavity. Here we propose an analog: vortex rings within a hypersonic cavity exhibit quantum-like behavior when undergoing compression. These fields of course are often considered separate, and in the macroscale—we should still expect that the energy eventually dissipates as described by contemporary turbulence theory. However, due to the forcing of the cavity, the larger scale vortices are pinched off similarly to the discussion by Feynman.⁵⁰ A corollary of the similarity in the physical mechanism is perhaps the reason for the −1 slope. In quantum fluids, this type of turbulence is referred to as Vinen turbulence. Recently, Barenghi et al.⁵¹ argued the existence of physical mechanisms that prohibit the development of the classical Kolmogorov spectrum. The observations from our simulations may point to one of those situations where the forcing, resulting from the cavity motion, prevents Kolmogorov-type behavior.

It is noted that non-Kolmogorov-scale turbulence has been investigated before, typically following a steeper slope in other macroscale scenarios. Non-Kolmogorov turbulence has been observed in helicity-driven turbulence,^52,53 some atmospheric flows,^54,55 and in interfacial turbulence problems.^56,57 Blood flow^58–60 is another macro-scale flow that has seen temporary instances of following the −1 law. Given the non-Newtonian behavior of blood, some future efforts should focus on flow viscosity and the effect that would have on a similar problem. Nevertheless, in this scenario, it seems evident that the forcing and active compressing of the cavity along with the unique vortex behavior and pinching is the primary cause for this behavior.

In contrast, as shown in Fig. 16, when the cavity is static, the spectrum shows a closer alignment with the −5/3 turbulence cascade. One obvious conclusion from this observation is that the energy cascade as measured in the center of the cavity is affected by the motion of the cavity in terms of both the slope and the range of scales. Since all inflow parameters are constant, the slope change is purely due to the motion, and the local change in aerodynamic behavior at the cavity center is due to the motion. As pointed out before, it should be noted that the spectrum for the moving cavity case was calculated by tracking velocity at different spatial locations. This comparison also leads to a few questions, including (a) is the slope of the energy spectrum a function of the compression rate? and (b) what is the slope of the energy cascade for expanding cavities? For the latter question, it can be hypothesized that expanding cavities may result in steeper slopes because larger spatial volumes may lead to faster dissipation. However, these questions are topics for a follow-up study which will require a parametric set of experiments/computations at a range of compressive and expansive cavity motions.

Figure 17 shows the time evolution of Y_O in the cavity. The production takes place at the leading edge of the cavity, which then convects throughout the cavity following the vortical motions. Other species follow similar trends. The species concentration, irrespective of the small amount, is higher around the vortices as compared to their cores. This means that as the vortices move from larger-scale vortices to smaller-scale, the species generated gets dissipated faster and may be ejected downstream at a higher rate. This behavior is further exaggerated by the rate of change in the cavity area.

The observations that (1) chemistry production is low and (2) the species follow the flow path and vortical motion means that, in theory, the Damköhler number, $Da=τflowτchem$ should be low. The fastest reaction is that with the lowest bond energy, that is, that of O₂ dissociation (O₂ + M $→$ O + O + M). The constants for this dissociation reaction for an Arrhenius-type rate law, $k=CTn exp(−θkT)$ are A = 3.61  $×$ 10¹², n = −1 and C = 5.94  $×$ 10⁴. The time evolution of the Damköhler number is computed at the center of the cavity. Values much less than one are indicative of a transport-dominated flow and values close to or greater than one indicate reaction-dominated behavior. For the current case, as shown in Fig. 18, the Damköhler number is much smaller than 1. The locations where it peaks indicate the production/destruction of species due to chemical reactions.

Figure 19 shows the net mass flux (defined as positive if outflow is greater than inflow and vice versa) in the cavity as time evolves. Before the cavity starts to move at 100  $μ$s, there is a net inflow of air (indicated by the red color and negative mass flux values), which is obvious. At around 25  $μ$s, the shear layer starts to oscillate and at times, it leads to a positive outflow of gases from the cavity. After the cavity starts to compress, the net flow of gases is always positive, meaning more mass of the air is leaving the cavity than is staying. The gas itself may also compress, but it is not a significant enough compression to overcome the geometry compression, and mass flux is leaving the cavity at a higher rate than it is entering.

The point at which the cavity begins to move (100  $μ$s) is defined by the vertical line in this plot. As mentioned before, after this point, the mass flux is dominantly out of the cavity (blue region), seemingly increasing, but with noticeable fluctuations. It is self-evident that as long as there is no significant compression of the gas, the mass flux out of the cavity should be greater than that into the cavity once the motion begins. However, for it to be trending upward means that the rate change of the cavity volume has an increasingly significant effect on the mass flux of the cavity as it is transitioning to shallower and shallower cavity size: more flow gets ejected downstream faster as the cavity is changed. For practical applications, this could mean that a cavity with a moving wall might only be predictably controlled in deeper cavities, but this would be a design decision. It also suggests that other motions of the cavity walls should be tested and different rates should be considered given different starting points of L/D.

Given that the generation of products in this flow appears low and that the Damköhler number confirms the transport-dominated behavior, the species that are generated follow the dominant flow structures (in this case, moving at and around vortices and eventually ejected downstream). This implies that the motion of species can also provide an indication of flow ejection from the cavity and where it occurs. Figure 20 shows the mass fraction of NO along the top cross section of the cavity at all times. Although this is a busy figure, the intent here is to show where most species are ejected downstream. It was originally thought that the species would have its highest mass fraction leaving the cavity at the trailing edge, but it appears that the high pressure from the trailing edge along with the vortical motion in the cavity forces the species toward the center, then ejects it downstream once the vortices begin to break down into smaller and smaller size vortices with higher magnitude. To identify the temporal location of maximum mass ejection from the middle part of the cavity, Fig. 21 shows the mass flow rate of NO as a function of time. A rapid increase the rate is observed at $∼$ 1.7  $×$ 10^–4 s, which corresponds to L/D of 5.85–this is the critical L/D ratio at which ejection maximizes.

This study details the first-ever investigation of high-speed flow over a dynamically moving cavity using direct numerical simulations. It is found that the shear layer due to the mass flux out of the cavity is deflected and never reattaches with the trailing edge as long as the compression is present. The impingement of such a shear layer would typically cause significant pressure fluctuations in the cavity, but since it is not occurring here, two main pressure waves were observed. One of these waves is vertical in its motion and the other is horizontal, following a dilatational wave motion. The vertical wave oscillates between the shear layer and the bottom wall, implying that the shear layer has reflectivity and transmissivity. Singular time steps just before, during, and after the peak pressure on the bottom walls indicate that peak pressures are due to the interactions between the vertical and horizontal wave motion. The pressure reduction near the centers of each of these walls is not steady but is due to the vertical and horizontal waves becoming more out of phase. The high pressure is due to the constructive interference of these waves; most often, this occurs near the corners. The shear layer also interacts with the incoming boundary-generated shockwave, deflecting the angle of the shock downstream. Since shock waves in hypersonics can be used to generate compression lift on a vehicle, the rapidly changing shock angle due to the shear layer's deflection would certainly cause the lift to be affected. This means that cavity compression can be used to modify the lift in a hypersonic vehicle. The shear layer and wave dynamics can be summarized by the procedural logic as follows:

Cavity flow compression $→$ deflects the shear layer upward $→$ shock at the leading edge of the shear layer is deflected $→$ angle of the shock initiated early near the inflow boundary is deflected $→$ lift, moment, and drag of the vehicle are changed.

Next, it is found that the total area of the vortical regions is relatively constant throughout the simulation. This means that the “circular” vortices must become smaller and more in quantity to maintain the constant area. Another thing that could be reasoned from the transfer of energy from large to small-scale vortices is a turbulence energy cascade. It was found that at the top surface of the cavity, the cascade is highly non-linear but it seemingly follows a −1 slope. At the center of the cavity, however, the slope of energy transfer from large to small vortices is decidedly −1 caused by the cavity motion. This confirms speculations from other researchers in the literature that external physical mechanisms can lead to non-Kolmogorov behaviors. That said, it should be noted that the energy spectrum at the center of the cavity is calculated using velocity fluctuations at different spatial locations (corresponding to the center of the cavity at different times).

By analyzing the species concentrations and movement, it is found that most of the mass outflow happens from the middle of the cavity and not from the trailing edge, as one would expect. It is also found that beyond an L/D of 5.85, the mass is ejected at a significantly higher rate as compared to the 2 < L/D < 5.85.

Research was sponsored by the Army Research Laboratory and was accomplished under Cooperative Agreement Number W911NF-24-2-0152 and W911NF-22-2-0058. Luis Bravo was supported by the 6.1 basic research program in propulsion sciences. The authors gratefully acknowledge the High-Performance Computing Modernization Program (HPCMP) resources and support provided by the Department of Defense Supercomputing Resource Center (DSRC) as part of the 2022 Frontier Project, Large-Scale Integrated Simulations of Transient Aerothermodynamics in Gas Turbine Engines. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Laboratory or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein.

The authors have no conflicts to disclose.

Jeremy Redding: Conceptualization (lead); Data curation (lead); Formal analysis (lead); Investigation (lead); Methodology (lead); Validation (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (equal). Prashant Khare: Funding acquisition (equal); Investigation (supporting); Project administration (equal); Resources (equal); Supervision (lead). Luis Bravo: Funding acquisition (equal); Investigation (supporting); Project administration (lead); Supervision (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.