Basics of Control of the Bow Shock Wave, Drag and Lift Forces, and Stability in a Steady Supersonic Flow Past an AD Body Using Permanently Operating Thermally Stratiﬁed Energy Deposition

: A new method of high-speed ﬂow control using permanently operating thermally stratiﬁed energy deposition is presented. The paper focuses on the analysis of the dependence of the characteristics of a steady supersonic ﬂow and an aerodynamic (AD) body on the temperature values in the layers of a stratiﬁed source and the possibility of making the transition from one steady ﬂow mode to another by changing the temperature in the layers. A detailed visualization of the dynamics of the ﬁelds of density, pressure, temperature, and local Mach number is presented during the controlled establishment of steady ﬂow modes. Multiple generation of the Richtmyer–Meshkov instability is shown. The sharp peaks accompanying the development of the Richtmyer–Meshkov instabilities were obtained, which remain in the steady ﬂow mode established under the action of a stratiﬁed energy source. Basic approaches for controlling the bow shock wave


Introduction
The problem of controlling a supersonic/hypersonic flow past an aerodynamic (AD) body due to energy supply at various points of the flow and the body surface has been studied since the second half of the last century and is now a well-developed area of aerospace engineering [1,2]. This is evidenced by numerous reviews devoted to various aspects of this topic [2][3][4][5][6][7][8][9]. In [2], a selected survey of aerodynamic drag reduction at high speeds is presented. The historical review on the idea of flow control by energy deposition to the supersonic flow past AD body, which appeared decades ago, is presented in [3]. The review of research on the direction of energy deposition for active and passive flow/flight control is presented in [4]. The studies on bow shock wave control and practical implementation of various devices for flow control are considered in [5] along with the physics-based mechanisms of their action.
A review of approaches to the organization of energy deposition in plasma aerodynamic applications is presented in [6] from the point of view of the physical mechanisms of interaction of various types of discharges with gas flows. Numerous examples of the use of discharge plasma to control high-speed flows are given. In a review [7], the author observes the experiments of the successful application of discharge-generated plasma for precise control of the structure and parameters of a supersonic air flow. The main attention is paid to the analysis of the specific properties of electric discharges. In [8], the results on the organization of the structure of plasma filaments using a powerful discharge are presented. A review of methods of flow control, including surface plasma actuators, laser energy deposition, and the impact of plasma generated by microwave impulses, is presented in [9]. In a review [10], the physical foundations of visualization of high-speed processes in gases as well as the basics of modern technologies for digital processing of flow images are considered.
An efficiency of energy deposition in the form of longitudinal filaments ("hot spikes") for reducing the aerodynamic drag was established in numerous studies [11][12][13]. In the experiments, the energy deposition to the supersonic flow was provided by MW impulse [14], laser [15][16][17], electrical discharge [18,19], MHD influence [20,21], etc. It was shown that the injection of the energy of microwave discharge to the supersonic flow past a blunt cylinder decreases the stagnation pressure and provides significant (up to 80%) decrease in a drag force. A vortex mechanism of these phenomena was established in the accompanied calculations [14]. In [18], the authors obtained the curvature of the shock wave under the action of a plasma zone formed by longitudinal pulsed discharge. Thermal fluctuations driven by a burst plasma discharge are experimentally investigated using a frequency analysis based on the Schlieren images in [19].
Theoretically, the possibility of obtaining self-sustained flow pulsations under the action of a homogeneous longitudinal energy source located symmetrically with respect to the streamlined body was demonstrated in [22][23][24]. The mechanisms of these phenomena are associated with the transfer of masses of a gas inside the shock layer and are different from the study carried out in present work considering the asymmetric location of the energy source relatively an AD body. The mechanism for the appearance of pulsations due to the asymmetric location of the energy source relative to the streamlined body was obtained in [25] and is associated with the dynamics of the generated asymmetric tripleshock configurations. Features of a supersonic flow containing an asymmetrically located extended homogeneous energy source were studied experimentally in [26] and theoretically in [27,28], where the possibility of producing a lift (pitch) force acting on an AD body at a zero angle of attack was shown.
Recently, much attention has been paid to experimental studies of a stratified plasma medium and the effect of such a medium on a shock wave. This interaction is distinguished by specific properties; in particular, blurring, distortion, or even complete disappearance of fragments of the shock wave front inside such a striped plasma region were obtained [29][30][31][32][33]. It has also been shown that the resulting plasma structures can enhance the mixing of gas mixtures [8]. The problem of the shock wave positioning control was considered in [30], where with the use of Mie scattering, the three-dimensional shape of the shock wave structure was obtained before and after electrical discharge organization, which gives possibilities for establishing plasma-based triggering mechanisms. In [33], ionization strata were obtained in the plasma zone created by a glow gas discharge, and the impact of the striation zone on a preliminary plane shock wave was investigated. In the experiments, the curvature, and in some cases, complete disappearance of the shock wave front was registered. In the accompanied numerical simulation, the multiple manifestation of the Richtmyer-Meshkov instability was shown, which destroyed the wave front due to the formation of many sharp peaks at the wave front, causing its practical disappearance in the density field.
The shock wave boundary layer interaction controlled by surface arc plasma actuators was studied [34,35]. In [34], an array of 16 surface arc plasma actuators was employed to control the interaction of the shock wave with the boundary layer in a flow with a Mach number of 2. In [35], the streamwise pulsed arc discharge array was used for the impact on the supersonic flow, in which five actuators were connected in a series with adjustable frequency. As the result of both the experiments, the disappearance of a fragment of the separation shock wave was established. In [36], localized arc filament plasma actuators and nanosecond dielectric-barrier discharge plasma actuators were discussed from the point of view of initiation of the Kelvin-Helmholtz instability. In [37,38], a manifestation of the Richtmyer-Meshkov instability as well as vortex lines generated by the Kelvin-Helmholtz instability were experimentally recorded during the interaction of a blast wave with an area of density and temperature inhomogeneities.
Theoretically, it was shown that the interaction of a shock wave with a temperaturestratified plasma medium leads to multiple manifestations of the Richtmyer-Meshkov instability, which causes sharp peaks in the region of the shock wave front. In this case, the wave front is distorted up to complete disappearance in the density field [33,[39][40][41][42]. It was shown that under the action of a stratified energy source, the redistribution of energy behind the shock wave front occurs in such a way that the vortex structures obtained behind the wave are characterized by sharp fluctuations of the energy parameters, the amplitude of which is up several times greater than the value for the homogeneous energy source with the same total energy [40][41][42].
In our recent works [43][44][45], the effect of stratified plasma regions on the bow shock wave and characteristics of an AD body was investigated. In [43], the impact of a combined energy source consisting of two thermal layers on the supersonic layer produced by an AD body was studied. A double-vortex mechanism of action on the body was obtained, which makes it possible to control the stagnation parameters and drag forces by changing the temperature in the layers of the combined source. In [44,45], the effect of a timelimited stratified impulse on the supersonic flow past a pointed body and the multi-vortex mechanism of action on the AD body were studied. The possibility of producing short-term lift (pitch) forces at zero angle of attack was shown.
The present paper studies the impact of a permanently operating a thermally stratified energy source on a steady supersonic flow past a pointed body. The study is based on the system of Navier-Stokes equations for a viscous heat-conducting gas (air); the freestream Mach number is 2. The paper focuses on the analysis of the dependence of the characteristics of a steady supersonic flow and an AD body on the temperature values in the layers of a stratified source and the possibility of making the transition from one steady flow mode to another by changing the values of temperature in the layers. The research is carried out on controlling the shape and position of the bow shock wave as well as the dynamics of the characteristics of an AD body, the parameters, and stability of a steady supersonic flow past the body. In addition, the possibility of the initiation and suppression of self-sustained flow pulsations and the formation of a steady flow with oppositely directed constantly acting lift (pitch) force at zero angle of attack by changing temperature in the layers of a thermally stratified energy source was established.

Statement of the Problem and Methodology
The impact of a permanently acting, thermally stratified energy source on a supersonic flow past a sharpened plate is considered. The angle at the top of the pointed part of the body is 90 • (Figure 1). The numerical simulations are based on the Navier-Stokes equations for viscous heat conductive gas (air); the gas is supposed to be perfect with the ratio of specific heats γ = 1.4. The divergent form of the full Navier-Stokes system of equations for the dimensionless variables [46] is solved numerically: For the dependence of dynamic viscosity on temperature, the Sutherland's law in nondimensional form is used: (1 + )/( + ) s1 = 0.4096 (120 K). The coefficient of heat conductivity k is supposed to depend on temperature in nondimensional form as = . .
Here, when passing from dimensional to dimensionless parameters, the following scaling variables were used: where is the dimensionless value of D. The fields of gas parameters in a steady supersonic flow past the body at t = 0.6 were set as the initial conditions for studying the process of the interaction of the shock layer and the stratified energy source. At this time, the pressure and density at the top of the body differ from their theoretical values evaluated from the Bernoulli's relation by 1.81% and 1.75%, accordingly (relative to the theoretical values). The parameters of the gas on the wall are taken as the wall parameters. Boundary conditions on the body's boundaries mean the absence of according normal flows on the boundaries and the absence of the reflection in the directions normal to the exit boundaries of the computation domain. The numerical simulations are based on the Navier-Stokes equations for viscous heat conductive gas (air); the gas is supposed to be perfect with the ratio of specific heats γ = 1.4. The divergent form of the full Navier-Stokes system of equations for the dimensionless variables [46] is solved numerically:

∂U ∂t
For the dependence of dynamic viscosity on temperature, the Sutherland's law in nondimensional form is used: The coefficient of heat conductivity k is supposed to depend on temperature in nondimensional form as k = T 0.5 .
Here, when passing from dimensional to dimensionless parameters, the following scaling variables were used: The fields of gas parameters in a steady supersonic flow past the body at t = 0.6 were set as the initial conditions for studying the process of the interaction of the shock layer and the stratified energy source. At this time, the pressure and density at the top of the body differ from their theoretical values evaluated from the Bernoulli's relation by 1.81% and 1.75%, accordingly (relative to the theoretical values). The parameters of the gas on the wall are taken as the wall parameters. Boundary conditions on the body's boundaries mean the The constantly acting, thermally stratified energy deposition is modelled by a set of rarefied gas layers of the same width, located in front of the bow shock wave (see Figure 1). The gaps between the layers were set equal to half the layer width. Inside the layers (indicated by the index j), gas density is specified to be reduced, ρ j = α j ρ ∞ , α j < 1, j = 1 ÷ N es ; N es is a number of layers in the energy source (in these calculations, N es = 5). Pressure and velocity in the layers are set equal to their freestream values: p j = p ∞ , u j = u ∞ , v j = 0. Therefore, temperature inside the layers is increased compared to its freestream value, T j = α j −1 T ∞ . Thus, a thermally stratified energy source is determined by a set of the parameters of the gas rarefaction {α j } = (α 1 , α 2 , . . . α Nes ) in its layers. In the calculations, such an energy source is obtained via the boundary condition at the entrance boundary, at x = 0, where the parameters of the heated layers are set along with the parameters of the oncoming flow.
The energy source is supposed to be fixed rigidly relative to the streamlined body, and its axis of symmetry coincides with the axis of symmetry of the body. It is assumed that the energy source arises in the steady flow at time t i , moves along with the oncoming flow towards the bow shock wave, interacts with it, and then acts continuously throughout the entire interaction process. The defining characteristics of the flow, AD body, and stratified energy source used in the simulations are presented in Table 1. A numerical code based on the complex conservative difference schemes [47] was used in the simulations. When constructing the schemes, the systems of differential consequences for the spatial derivatives of system (1) were used, which provides these schemes with a second order of approximation in space and time. In the vicinity of the boundaries of the body, the schemes were constructed taking into account the location of the boundary inside the computational grid, which ensures the calculations to be conservative near the boundaries. The five-point stencil of the Lax's scheme was applied for the scheme construction, so in the calculations, the structural staggered orthogonal grids were used. The distance between the stencil nodes on the lower time layer is assumed to be 2h x , 2h y , where h x , h y are the spatial steps in the xand y-directions.

Analysis of the Grid Convergence
The grid convergence analysis was conducted using three difference grids ( Table 2). The flow dynamics during the steady-state establishing was studied ( Figure 2). In Figure 2a, the density fields in isochores are presented, which were calculated using these three difference grids. One can see that the shapes and the positions of the bow shock waves almost coincide. The dynamics of the parameters at the top of the body obtained using these three difference grids is shown in Figure 2b. In Table 2, the relative errors for the parameters at the top of the body at the steady state in comparison with the theoretically ones obtained from the Bernoulli's relation are presented. One can see that the values at the top of the body differ from their theoretical values less than by 1.9% (for Grid1). Hence, it can be stated that the grid convergence has taken place. For the calculations, Grid1 (h x = h y = 0.0005) was used for two computation domains of 9.5 × 10 6 nodes (with the coordinate of the axis of symmetry of the body y 0 = 0.7) and of 13 × 10 6 nodes (with y 0 = 0.9), accordingly (counting the middle point of the scheme stencil). Table 2. Characteristics of grids and relative errors at t = 0.6.

Grid
Steps of the axis of symmetry of the body y0 = 0.7) and of 13 × 10 6 nodes (with y0 = 0.9), accordingly (counting the middle point of the scheme stencil).

Analysis of the Flow Dynamics during the Interaction of an Energy Source with Equally Heated Layers with a Supersonic Flow Past an AD Body
The dynamics of the density fields during the interaction of a thermally stratified energy source with a supersonic flow past a body "a plate pointed by a wedge" is presented in Figure 3  3). First, the source approaches the steady bow shock wave (images for t = 0.8 and t = 1.0) and begins to interact with it. As a result of the interaction of the bow shock wave with the boundaries of thermal layers (which are contact discontinuities), the multiple Richtmyer-Meshkov instabilities develop, accompanied by the formation of vortex structures [39,44] (t = 1.2). These vortex structures determine a new multi-vortex mechanism of action on a streamlined body (t = 1.4-1.6) (see [44], where this mechanism is described for a time-limited stratified impulse).  The dynamics of the density fields during the interaction of a thermally stratified energy source with a supersonic flow past a body "a plate pointed by a wedge" is presented in Figure 3  First, the source approaches the steady bow shock wave (images for t = 0.8 and t = 1.0) and begins to interact with it. As a result of the interaction of the bow shock wave with the boundaries of thermal layers (which are contact discontinuities), the multiple Richtmyer-Meshkov instabilities develop, accompanied by the formation of vortex structures [39,44] (t = 1.2). These vortex structures determine a new multi-vortex mechanism of action on a streamlined body (t = 1.4-1.6) (see [44], where this mechanism is described for a time-limited stratified impulse). After that, the vortex structures are carried away from the computational domain, and the layered structure of the density field of the entire flow is established up to a new steady flow mode (t = 1.8-3.6). Note that the new steady flow mode established under the action of the stratified source differs significantly from the initial unperturbed steady flow mode (image for t = 1.8 vs. image for t = 3.6).
The corresponding dynamics of the density fields in a surface view are shown in Figure 4. It can be seen that the interaction is accompanied by almost complete destruction of the front of the bow shock wave in the zone of layers of the stratified energy source. In addition, sharp peaks are visible, indicating the occurrence of Richtmyer-Meshkov instabilities. After that, the vortex structures are carried away from the computational domain, and the layered structure of the density field of the entire flow is established up to a new steady flow mode (t = 1.8-3.6). Note that the new steady flow mode established under the action of the stratified source differs significantly from the initial unperturbed steady flow mode (image for t = 1.8 vs. image for t = 3.6).
The corresponding dynamics of the density fields in a surface view are shown in Figure 4. It can be seen that the interaction is accompanied by almost complete destruction of the front of the bow shock wave in the zone of layers of the stratified energy source. In addition, sharp peaks are visible, indicating the occurrence of Richtmyer-Meshkov instabilities.  The according dynamics of pressure, temperature fields and the fields of a local Mach number are presented in Figures 5-7. A significant change in the steady flow mode is clearly seen from the images of the pressure fields shown in Figure 5. One can see that, under the action of a stratified energy source, the pressure in the shock layer, as a whole, decreases. This is accompanied by a change in the shape of the front of the bow shock wave: in the region of heated layers, the wave front becomes wavy, and at the periphery, the bending of the front of the bow shock wave changes to the opposite. Note that since, by construction, the boundaries of the source layers are contact discontinuities, the source layers are not visible on the pressure field images. The according dynamics of pressure, temperature fields and the fields of a local Mach number are presented in Figures 5-7. A significant change in the steady flow mode is clearly seen from the images of the pressure fields shown in Figure 5. One can see that, under the action of a stratified energy source, the pressure in the shock layer, as a whole, decreases. This is accompanied by a change in the shape of the front of the bow shock wave: in the region of heated layers, the wave front becomes wavy, and at the periphery, the bending of the front of the bow shock wave changes to the opposite. Note that since, by construction, the boundaries of the source layers are contact discontinuities, the source layers are not visible on the pressure field images.  The temperature fields shown in Figure 6 are consistent with the density fields, and the resulting steady flow is characterized by a layered temperature structure. The fields of the local Mach number also become layered in the new steady flow mode ( Figure 7). In this case, the shape of the subsonic region between the body and the bow shock wave changes. It can also be seen that in the new steady flow mode, the sound line emanating from the corner point of the body is wavy, reflecting the layered structure of the flow in the shock layer.  The temperature fields shown in Figure 6 are consistent with the density fields, and the resulting steady flow is characterized by a layered temperature structure. The fields of the local Mach number also become layered in the new steady flow mode ( Figure 7). In this case, the shape of the subsonic region between the body and the bow shock wave changes. It can also be seen that in the new steady flow mode, the sound line emanating from the corner point of the body is wavy, reflecting the layered structure of the flow in the shock layer.    The temperature fields shown in Figure 6 are consistent with the density fields, and the resulting steady flow is characterized by a layered temperature structure. The fields of the local Mach number also become layered in the new steady flow mode (Figure 7). In this case, the shape of the subsonic region between the body and the bow shock wave changes. It can also be seen that in the new steady flow mode, the sound line emanating from the corner point of the body is wavy, reflecting the layered structure of the flow in the shock layer.
In Figure 8,  Figure 9 illustrates the flow analysis for different sets {α j } with equal values of α j (and values of temperature) in the source layers. In Figure 9a, the pressure fields are presented for the steady flow mode. Here, the front of the bow shock is wavy reflecting the stratification of the energy source. The bow shock wave is located more far from the body for the smaller α j (greater temperatures) in the layers (green curve), which, however, was expected. In addition, it can be seen that the more the angle of inclination of the bow shock wave changes, the smaller the values α j in the layers. The dynamics of pressure, density, and temperature at the top of the body as well as those of the average front surface temperature T a and the drag force F during the steady flow establishing for different sets {α j } with equal values α j are presented in Figure 9b,c. Here, where T k and p k are the temperature and pressure values at the k-point on the front surface of the body, N is a number of grid points on the front surface, and y 0 is the y-coordinate of the axis of symmetry of the body. It can be seen that in the steady flow mode, the pressure and density at the top of the body and the drag force of the front surface are lower for sets {α j } with smaller α j (green curves).    Figure 9a, the pressure fields are presented for the steady flow mode. Here, the front of the bow shock is wavy reflecting the stratification of the energy source. The bow shock wave is located more far from the body for the smaller αj (greater temperatures) in the layers (green curve), which, however, was expected. In addition, it can be seen that the more the angle of inclination of the bow shock wave changes, the smaller the values αj in the layers. The dynamics of pressure, density, and temperature at the top of the body as well as those of the average front surface temperature Ta and the drag force F during the steady flow establishing for different sets {αj} with equal values αj are presented in Figure 9b,c. Here, where Tk and pk are the temperature and pressure values at the k-point on the front surface of the body, N is a number of grid points on the front surface, and y0 is the y-coordinate of the axis of symmetry of the body. It can be seen that in the steady flow mode, the pressure and density at the top of the body and the drag force of the front surface are lower for sets {αj} with smaller αj (green curves). The bow shock wave is located more far from the body for the smaller α j (greater temperatures) in the layers (green curve), and accordingly, for this set, the temperature at the top and the average temperature are higher at the steady state (Figure 9b,c). Thus, by changing the temperature in the layers of the stratified energy source, it is possible to control the parameters at the top of the streamlined body in the stationary flow regime as well as the drag force of its surfaces and the surface temperature. The bow shock wave is located more far from the body for the smaller αj (greater temperatures) in the layers (green curve), and accordingly, for this set, the temperature at the top and the average temperature are higher at the steady state (Figure 9b,c). Thus, by changing the temperature in the layers of the stratified energy source, it is possible to control the parameters at the top of the streamlined body in the stationary flow regime as well as the drag force of its surfaces and the surface temperature. source during the new steady flow modes established are presented in a surface view. It can be seen that the magnitude of the sharp peaks accompanying the Richtmyer-Meshkov instabilities is greater when there are several hotter layers nearby in the source. At the same time, these sharp peaks are preserved in the flow patterns, which are characterized by the establishment of steady values of the flow parameters at the top of the body (the first and second lines of the images). For one heated layer, the peaks are much smaller in magnitude since it is located close to the less-heated layers. At the same time, these peaks are smoothed out to the time the steady flow is established, and only some surface nonmonotonicities can be seen there (the third line of images). In Figure 11, the flow analysis for these symmetrical sets in the stratified energy is illustrated. Figure 11a demonstrates the pressure fields for the steady flow mode. Here, the front of the bow shock is also wavy, reflecting the number of layers in the energy source. The front of the bow shock wave undergoes a significant modification, and the wave inclination angles also change. For one heated layer α 3 in the set {α j }, the shape of the bow shock practically repeats the shape of the pointed part of the body. curve). Thus, by leaving the pressure at the top and the surface drag force unchanged and changing the temperature in the layers of the stratified energy source, it is possible to control the temperature on the surface of the streamlined body.

Analysis of the Dynamics of the Flow Field Parameters during the Interaction of an Energy Source with Different Temperatures in the Layers with a Supersonic Flow Past an AD Body
Note that for symmetrical sets {αj} characterizing a stratified energy source, lift (pitch) forces are not initiated in contrast to energy sources with asymmetrical sets {αj}, which, as will be shown below, produce, in addition to surface drag forces, lift (pitch) forces as well (at zero angle of attack).

Initiation and Suppression of Flow Pulsations by Changing the Temperature in the Layers of a Stratified Energy Source
The dynamics of the density fields during the interaction of a thermally stratified energy source with {αj} = (0.3,0.3,0.8,0.8,0.8) with the supersonic flow past a body are presented in Figure 12. The corresponding pressure fields are presented in Figure 13. The source is moving with the flow towards the steady bow shock wave and begins to interact with it (t = 0.8, 1.0). The vortex structure that accompanies multiple Richtmyer-Meshkov instabilities develops near the lower surface of the body (t = 1.2) and begins to affect this surface (t = 1.4). As the result of this impact, the separation bubble is originated near the lower surface, which gives the origin to the pulsation process of the whole flow (t = 1.8).
Flow pulsations are regulated by a mechanism similar to that described in [25] for one longitudinal energy source located asymmetrically towards an AD body. At the beginning of the action of the energy source, the front of the bow shock wave begins to move away from the body, and bow shock wave diffraction occurs with the formation of a tripleshock configuration (t = 1.4) Note that the front of the reflected shock wave in this triple The dynamics of pressure, density, and temperature at the top of the body, along with the average front surface temperature T a and drag force F during the steady flow establishing for these different symmetrical sets {α j }, are presented in Figure 11b,c, accordingly. It can be seen that in the steady flow mode, the pressure at the top and the drag force of the front surface are practically the same for all the considered sets {α j }. Therefore, the defining role in the establishing pressure values at the top of the body and the drag force play the central values α j . Thus, in order to obtain a drop in the drag force that is the same as in a set with all equally heated layers, there is no need to heat up all the layers; it is enough to heat several central layers. This fact can contribute to an increase in the efficiency of the impact of a stratified energy source. the steady density value is greater; consequently, the steady temperature in the top is smaller, and the averaged surface temperature is smaller, too (orange curve). Thus, by leaving the pressure at the top and the surface drag force unchanged and changing the temperature in the layers of the stratified energy source, it is possible to control the temperature on the surface of the streamlined body.
Note that for symmetrical sets {α j } characterizing a stratified energy source, lift (pitch) forces are not initiated in contrast to energy sources with asymmetrical sets {α j }, which, as will be shown below, produce, in addition to surface drag forces, lift (pitch) forces as well (at zero angle of attack).

Initiation and Suppression of Flow Pulsations by Changing the Temperature in the Layers of a Stratified Energy Source
The dynamics of the density fields during the interaction of a thermally stratified energy source with {α j } = (0.3,0.3,0.8,0.8,0.8) with the supersonic flow past a body are presented in Figure 12. The corresponding pressure fields are presented in Figure 13. The source is moving with the flow towards the steady bow shock wave and begins to interact with it (t = 0.8, 1.0). The vortex structure that accompanies multiple Richtmyer-Meshkov instabilities develops near the lower surface of the body (t = 1.2) and begins to affect this surface (t = 1.4). As the result of this impact, the separation bubble is originated near the lower surface, which gives the origin to the pulsation process of the whole flow (t = 1.8).
Flow pulsations are regulated by a mechanism similar to that described in [25] for one longitudinal energy source located asymmetrically towards an AD body. At the beginning of the action of the energy source, the front of the bow shock wave begins to move away from the body, and bow shock wave diffraction occurs with the formation of a tripleshock configuration (t = 1.4) Note that the front of the reflected shock wave in this triple configuration acquires a wavy shape under the action of the source stratification (t = 3.0). As the distance from the body increases, the triple point moves up (t = 1.4-2.6), and dimensions of the triple configuration increase. In this case, the zone of increased pressure behind the reflected shock wave moves up along the upper surface of the wedge, pushing the bubble boundary above the top (t = 1.8-2.6). The pressure at the top drops (t = 2.2). Then, the bow shock wave stops and starts moving towards the body. The triple point starts to move down, and the dimensions of the triple configuration decrease (t = 2.6-4.2). The highpressure zone captures the top (t = 4.2) and pushes the boundary of the separation bubble away from the top. In this case, the bubble collapses, and the flow becomes unseparated It can be seen that pressure, density, and temperature at the top of the body (green, orange, and blue curves) (Figure 14a,b) have consistent behavior along with the average front surface temperature, drag, and lift forces (Figure 14b,c). The upper surface of the wedge and the upper horizontal surface are the least affected by the vortex structure created by the energy source; therefore, the oscillatory mode of the dynamics of parameters on these surfaces is also the least pronounced (brown curve) (Figure 14b).    force acting on the bottom surface). It can be seen that pressure, density, and temperature at the top of the body (green, orange, and blue curves) (Figure 14a,b) have consistent behavior along with the average front surface temperature, drag, and lift forces ( Figure  14b,c). The upper surface of the wedge and the upper horizontal surface are the least affected by the vortex structure created by the energy source; therefore, the oscillatory mode of the dynamics of parameters on these surfaces is also the least pronounced (brown curve) (Figure 14b). Note, however, that the entire flow oscillates along with the bow shock wave. Indeed, we can see this in Figure 15, where the fields of density, pressure, temperature, and local Note, however, that the entire flow oscillates along with the bow shock wave. Indeed, we can see this in Figure 15, where the fields of density, pressure, temperature, and local Mach number are presented for {α j } = (0.3,0.3,0.8,0.8,0.8) for the time instant of maximum value of pressure at the top (left row of images) and for the time instant of minimum value of pressure at the top (right row of images). When the pressure at the top is maximal, the dimensions of the triple configuration are the smallest, the front of the bow shock wave is at the closest distance from the body, and the separation bubble is just emerging; separation is almost absent, and the flow near the lower boundary is weakly subsonic (left row of images). When the pressure at the top is minimal, the dimensions of the triple configuration are the largest, the standoff distance of the bow shock wave from the body is maximal, the flow is separated with the formation of the separation bubble, and the flow is strongly subsonic near the lower boundary (right row of images). Therefore, it should be underlined that the standoff distance of the bow shock wave from the body is pulsing, too.
Thus, under the action of a stratified source with asymmetrically located hotter layers, a self-sustained pulsating flow mode can be initiated. This mode is undesirable for the flight of aircraft because it can lead to a negative impact on the process of control up to loss of controllability of an AD body. Such a flow mode can be characterized as a pulsating instability of the flow. Figure 16 shows how to suppress these flow pulsations by changing the temperature in the source layers. Here, the dynamics of pressure, density, and temperature at the top of the body are presented (Figure 16a,b) along with of the average front surface temperature, drag forces, and lift forces (Figure 16b,c). It can be seen that by successively decreasing the temperature (or increasing α j ) in the more heated layers, it is possible to partially or completely suppress these oscillations and obtain a stable steady flow mode (green and brown curves). This result is in agreement with [25], where the flow pulsations for one heated layer were obtained. Additionally, we can conclude that pressure and density at the top of the body are practically determined by their values in the layers closest to the axis of symmetry of the body, while the temperatures differ: for a source with more heated layers, the steady value of temperature and average temperature are lower. When the pressure at the top is maximal, the dimensions of the triple configuration are the smallest, the front of the bow shock wave is at the closest distance from the body, and the separation bubble is just emerging; separation is almost absent, and the flow near the lower boundary is weakly subsonic (left row of images). When the pressure at the top is minimal, the dimensions of the triple configuration are the largest, the standoff distance of the bow shock wave from the body is maximal, the flow is separated with the formation of the separation bubble, and the flow is strongly subsonic near the lower boundary (right row of images). Therefore, it should be underlined that the standoff distance of the bow shock wave from the body is pulsing, too. Thus, under the action of a stratified source with asymmetrically located hotter layers, a self-sustained pulsating flow mode can be initiated. This mode is undesirable for the flight of aircraft because it can lead to a negative impact on the process of control up to loss of controllability of an AD body. Such a flow mode can be characterized as a pulsating instability of the flow. Figure 16 shows how to suppress these flow pulsations by changing the temperature in the source layers. Here, the dynamics of pressure, density, and temperature at the top of the body are presented (Figure 16a,b) along with of the average front surface temperature, drag forces, and lift forces (Figure 16b,c). It can be seen that by successively decreasing the temperature (or increasing αj) in the more heated layers, it is possible to partially or completely suppress these oscillations and obtain a stable steady flow mode (green and brown curves). This result is in agreement with [25], where the flow pulsations for one heated layer were obtained. Additionally, we can conclude that pressure and density at the top of the body are practically determined by their values in the layers closest to the axis of symmetry of the body, while the temperatures differ: for a source with more heated layers, the steady value of temperature and average temperature are lower.  The corresponding pressure fields in isolines are shown in Figure 16d. In addition, the pressure fields are presented for {α j } = (0.3,0.3,0.8,0.8,0.8) for the time instant of maximum and minimum values of pressure at the top (blue and orange curves, respectively). It can be seen that the position of the bow shock wave is also determined by the temperature of the layers closest to the center: for the green and brown curves, the positions of the bow shock wave practically coincide. In this case, the front of the bow shock wave is located between the fronts for the time instants of maximum and minimum values of pressure at the top (blue and orange curves). In addition, it can be seen that in the region below the symmetry axis of the body, the bow shock wave front becomes rectilinear, and at the periphery, the wave front coincides for the green and brown curves. This means that far from the body, the determining parameters of the stratified source (which also determine the parameters of the bow shock) are the heated layers closest to the body's symmetry axis.   Thus, it is possible to suppress pulsations by alternately setting lower temperatures in the layers of a permanently operating stratified energy source. In this case, it is also possible to control the flow parameters and the bow shock wave when a steady flow mode is established.
Another approach to suppressing pulsations (orange curves) and converting the flow regime into a stable steady mode (green curves) is demonstrated in Figure 17. Here, we present an example of switching flow modes by changing at a certain moment the parameters of a permanently operating stratified energy source.
tions of the bow shock wave practically coincide. In this case, the front of the bow shock wave is located between the fronts for the time instants of maximum and minimum values of pressure at the top (blue and orange curves). In addition, it can be seen that in the region below the symmetry axis of the body, the bow shock wave front becomes rectilinear, and at the periphery, the wave front coincides for the green and brown curves. This means that far from the body, the determining parameters of the stratified source (which also determine the parameters of the bow shock) are the heated layers closest to the body's symmetry axis.
Thus, it is possible to suppress pulsations by alternately setting lower temperatures in the layers of a permanently operating stratified energy source. In this case, it is also possible to control the flow parameters and the bow shock wave when a steady flow mode is established.
Another approach to suppressing pulsations (orange curves) and converting the flow regime into a stable steady mode (green curves) is demonstrated in Figure 17. Here, we present an example of switching flow modes by changing at a certain moment the parameters of a permanently operating stratified energy source.  Figure 17a,b, along with of the average front surface temperature, drag forces and lift forces (Figure 17b,c). It can be seen that when replacing at t = 5.4 the set of values {α j } 1 in the stratified source with the set {α j } 2 , the flow after several oscillations comes to a steady state (green curves). In this case, in the new steady flow mode, pressure, density, and temperature at the top of the body as well as the drag force of the front surface are set at the maximum level of these parameters in the previous oscillatory mode, and the average surface temperature is set at the minimum values. At the same time, the absolute value of the total lift force decreases.
The corresponding pressure fields in isolines are shown in Figure 17d. In addition, the pressure fields are presented for {α j } = (0.3,0.3,0.8,0.8,0.8) for the time instants of maximum and minimum values of pressure at the top (blue and orange curves, respectively). It can be seen that the position of the bow shock wave is closer to the position at the moment of t corresponding to the maximum pressure at the top. At the same time, the deflection angle of the rectilinear bow shock wave at the periphery (green curve) exceeds the angles established in the oscillatory mode (orange and blue curves).
Thus, it is possible to suppress pulsations and to control the flow parameters and the bow shock wave by switching the sets {α j } in a permanently operating stratified energy source from one to another with lower temperatures in the layers.  (Figure 18). It can be seen that at different times, the flow patterns are exactly opposite (compare the images in the left and right rows). This is due to the assumption that the axis of symmetry of the stratified source coincides with the axis of symmetry of an AD body; i.e., the source is rigidly fixed relative to the body.  Figure 17a,b, along with of the average front surface temperature, drag forces and lift forces ( Figure  17b,c). It can be seen that when replacing at t = 5.4 the set of values {αj} 1 in the stratified source with the set {αj} 2 , the flow after several oscillations comes to a steady state (green curves). In this case, in the new steady flow mode, pressure, density, and temperature at the top of the body as well as the drag force of the front surface are set at the maximum level of these parameters in the previous oscillatory mode, and the average surface temperature is set at the minimum values. At the same time, the absolute value of the total lift force decreases.

Organization of the Opposite Effect on the Front Surfaces of the Body and the Initiation of Oppositely Directed Lift (Pitch) Forces
The corresponding pressure fields in isolines are shown in Figure 17d. In addition, the pressure fields are presented for {αj} = (0.3,0.3,0.8,0.8,0.8) for the time instants of maximum and minimum values of pressure at the top (blue and orange curves, respectively). It can be seen that the position of the bow shock wave is closer to the position at the moment of t corresponding to the maximum pressure at the top. At the same time, the deflection angle of the rectilinear bow shock wave at the periphery (green curve) exceeds the angles established in the oscillatory mode (orange and blue curves).
Thus, it is possible to suppress pulsations and to control the flow parameters and the bow shock wave by switching the sets {αj} in a permanently operating stratified energy source from one to another with lower temperatures in the layers.  (Figure 18). It can be seen that at different times, the flow patterns are exactly opposite (compare the images in the left and right rows). This is due to the assumption that the axis of symmetry of the stratified source coincides with the axis of symmetry of an AD body; i.e., the source is rigidly fixed relative to the body.  Figure 19. Dynamics of pressure, density, and temperature at the top of the body are presented in Figure 19a,b (green, orange and blue curves). One can see that the dynamic of these parameters is quite the same as for {αj} = (0.3,0.3,0.8,0.8,0.8) (compare with Figure 14a,b). The same can be said about the dynamics of the average temperature of the frontal surface and the drag force Fdrag (purple and green curves in Figure 19b Figure 19. Dynamics of pressure, density, and temperature at the top of the body are presented in Figure 19a,b (green, orange and blue curves). One can see that the dynamic of these parameters is quite the same as for {α j } = (0.3,0.3,0.8,0.8,0.8) (compare with Figure 14a,b). The same can be said about the dynamics of the average temperature of the frontal surface and the drag force Fdrag (purple and green curves in Figure 19b). However, forces  Figure 19. Dynamics of pressure, density, and temperature at the top of the body are presented in Figure 19a,b (green, orange and blue curves). One can see that the dynamic of these parameters is quite the same as for {αj} = (0.3,0.3,0.8,0.8,0.8) (compare with Figure 14a,b). The same can be said about the dynamics of the average temperature of the frontal surface and the drag force Fdrag (purple and green curves in Figure 19b  Thus, a stratified source with a reverse set {αj} R = (α5,α4,α3,α2,α1) produces the same drag force Fdrag and the parameters at the top as well as the average front surface temperature as in the case of {αj} = (α1,α2,α3,α4,α5) but an oppositely directed lift (pitch) force (at zero angle of attack).

Organization of the Opposite Effect on the Front Surfaces of the Body and the Initiation of Oppositely Directed Lift (Pitch) Forces
A natural continuation of the analysis of flow characteristics under the influence of a stratified source with a reversible set {αj} R in its layers becomes the approach with switching the steady flow modes with obtaining oppositely directed lift (pitch) forces. This approach is illustrated by Figure 20.  Figure 20d. It can be seen that the flow patterns are exactly symmetrical, which is to be expected given that the symmetry axis of the stratified source coincides with the symmetry axis of the body.
Thus, one steady flow mode can be changed to another steady mode characterized by an oppositely directed lift (pitch) force by switching the set {αj} = (α1,α2,α3,α4,α5) in the stratified source to the reverse one {αj} R = (α5,α4,α3,α2,α1). In this case, the drag force Fdrag and the parameters at the top as well as the average front surface temperature do not change. Thus, a stratified source with a reverse set {α j } R = (α 5 ,α 4 ,α 3 ,α 2 ,α 1 ) produces the same drag force Fdrag and the parameters at the top as well as the average front surface temperature as in the case of {α j } = (α 1 ,α 2 ,α 3 ,α 4 ,α 5 ) but an oppositely directed lift (pitch) force (at zero angle of attack).
A natural continuation of the analysis of flow characteristics under the influence of a stratified source with a reversible set {α j } R in its layers becomes the approach with switching the steady flow modes with obtaining oppositely directed lift (pitch) forces. This approach is illustrated by Figure 20. First, a steady flow mode is reached for the set {α j } = (0.4,0.4,0.8,0.8,0.8), t = 5.0 (orange curves). Then, the set {α j } = (0.4,0.4,0.8,0.8,0.8) in the stratified source is replaced by the set {α j } R = (0.8,0.8,0.8,0.4,0.4) (green curves). It can be seen that after several oscillations, a new steady flow mode is established with the same flow characteristics but with an oppositely directed total lift force (Figure 20a-c). The pressure fields in the isolines are shown in Figure 20d. It can be seen that the flow patterns are exactly symmetrical, which is to be expected given that the symmetry axis of the stratified source coincides with the symmetry axis of the body.

Discussion
It should be noted that in this work, symmetrical and asymmetric vortex structures (mushroom structures) initiated by the Richtmyer-Meshkov instability are not clearly visible (Figures 3, 6, 12, and 18; t = 1.2, 1.4). This is due to the fact that since the source is set on the left boundary (x = 0) and moves towards the bow shock wave for some time, the boundaries of the initial layers have time to undergo the dissipation due to the action of physical viscosity and thermal conductivity before interaction with the bow shock (as would happen in a real situation when using such an energy source). Therefore, the boundaries of the layers (which are contact discontinuities in the inviscid approach) are somewhat smeared, which contributes to the weakening of the vortices in the vortex structures emerging as the result of the interaction of the layers with the bow shock wave.
To confirm the conclusion about the manifestation of the Richtmyer-Meshkov instability, let us demonstrate the density fields in the case that when at the initial moment, the

Discussion
It should be noted that in this work, symmetrical and asymmetric vortex structures (mushroom structures) initiated by the Richtmyer-Meshkov instability are not clearly visible ( Figure 3, Figure 6, Figure 12, and Figure 18; t = 1.2, 1.4). This is due to the fact that since the source is set on the left boundary (x = 0) and moves towards the bow shock wave for some time, the boundaries of the initial layers have time to undergo the dissipation due to the action of physical viscosity and thermal conductivity before interaction with the bow shock (as would happen in a real situation when using such an energy source). Therefore, the boundaries of the layers (which are contact discontinuities in the inviscid approach) are somewhat smeared, which contributes to the weakening of the vortices in the vortex structures emerging as the result of the interaction of the layers with the bow shock wave.
To confirm the conclusion about the manifestation of the Richtmyer-Meshkov instability, let us demonstrate the density fields in the case that when at the initial moment, the stratified source is located in the immediate vicinity of the bow shock ( Figure 21). In this case, it does not have time to be strongly affected by the dissipation (Figure 21a), and the manifestation of the Richtmyer-Meshkov instability is noticeably clearer (Figure 21b). An enlarged image of the flow density field is presented in Figure 21c and in surface view in Figure 21d. The mushroom structures accompanying the Richtmyer-Meshkov instability causing by the baroclinic effects and generation of vorticity on the boundary can be seen [39]. In addition, a multiple structure of sharp peaks is visible (Figure 21d), which is formed in the points of manifestation of the instability (red dots in Figure 21c) and directly indicates the generation of this instability. stratified source is located in the immediate vicinity of the bow shock ( Figure 21). In this case, it does not have time to be strongly affected by the dissipation (Figure 21a), and the manifestation of the Richtmyer-Meshkov instability is noticeably clearer (Figure 21b). An enlarged image of the flow density field is presented in Figure 21c and in surface view in Figure 21d. The mushroom structures accompanying the Richtmyer-Meshkov instability causing by the baroclinic effects and generation of vorticity on the boundary can be seen [39]. In addition, a multiple structure of sharp peaks is visible (Figure 21d), which is formed in the points of manifestation of the instability (red dots in Figure 21c) and directly indicates the generation of this instability.

Conclusions
The basic approaches for controlling the bow shock wave, drag and lift (pitch) forces, and stability in a steady supersonic flow past an AD body using permanently operating

Conclusions
The basic approaches for controlling the bow shock wave, drag and lift (pitch) forces, and stability in a steady supersonic flow past an AD body using permanently operating thermally stratified energy deposition were herein developed. A detailed visualization of the dynamics of the fields of density, pressure, temperature, and local Mach number is presented during the steady flow mode establishing. The axis of symmetry of the stratified source was supposed to coincide with the axis of symmetry of an AD body. It was shown that a new steady flow mode established under the action of the stratified source differs significantly from the initial unperturbed one. The new results obtained on the influence of a permanently acting stratified energy source on a steady flow past an AD body are as follows: -Multiple generations of the Richtmyer-Meshkov instability were shown. A new steady flow mode was established, in which the front of the bow shock wave can be completely destroyed in the zone of the source layers. The sharp peaks accompanying the development of the Richtmyer-Meshkov instabilities were obtained, which remain in the steady flow mode established under the action of a stratified energy source. The magnitudes of the picks are greater for smaller α j (greater temperatures) in the layers of the energy source. -It was shown that the front of the bow shock becomes wavy, reflecting the stratification of the energy source, and the bow shock wave standoff distance is greater for the smaller α j (greater temperatures) in the layers. In addition, the more the angle of inclination of the bow shock wave changes, the smaller the values α j in the layers. The layered nature of the fields of density, temperature, and local Mach number was also shown up to the establishment of a new steady flow mode. -It was shown that by changing the temperature in the layers of the stratified energy sources given by symmetrical sets {α j }, it is possible to control the defining flow parameters. For a source with the same temperatures in the layers, the pressure and density at the top of the body and the drag force of the front surface are lower for sets {α j } with smaller α j (greater temperatures). The bow shock wave is located more far from the body for the smaller α j in the layers, and the temperature at the top and the average front surface temperature are higher. It was also shown that central layers of the stratified source play a decisive role in the formation of the parameters at the top and the drag forces. In this case (for the symmetrical sets {α j }), lift forces do not arise. -It was established that by setting asymmetric sets {α j } in a permanently operating stratified energy source, it is possible to provoke a pulsating flow mode. The mechanism of the self-sustained flow pulsations was described. The possibility to suppress the pulsations by alternately setting lower temperatures in the layers was shown. In this case, it was shown that one can control the flow parameters and the bow shock wave during the steady flow mode establishing. In addition, it is possible to suppress the pulsations and to control the flow parameters by switching at some time the set {α j } to another one with lower temperatures in the layers. -It was obtained that for the set {α j } = (α 1 ,α 2 ,α 3 ,α 4 ,α 5 ) and for the reverse set {α j } R = (α 5 ,α 4 ,α 3 ,α 2 ,α 1 ) in a permanently operating stratified energy source, the flow patterns are exactly opposite due to the assumption that the axis of symmetry of the stratified source coincides with the axis of symmetry of an AD body. -It was shown that a stratified source with a reverse set {α j } R (in comparison with the source with {α j }) produces equal, in absolute value, and oppositely directed lift (pitch) forces constantly acting on the body (at zero angle of attack) and the same drag force Fdrag, the parameters at the top, and the average front surface temperature. It was also shown that a steady flow mode can be changed to another one characterized by an oppositely directed lift (pitch) force by switching at some time the set {α j } to the reverse one {α j } R . In this case, the drag force Fdrag and the parameters at the top as well as the average front surface temperature do not change.
Funding: This research received no external funding.

Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: All obtained data underlying the conclusions made can be provided by the author upon request.

Conflicts of Interest:
The author declares no conflict of interest.