**4. Conclusion and controlling the drag**

The sequence of interconnected stages of the flow modification described here, in fact, is a complex way of the shock flow instability development that is triggered via the single mechanism of the shock refraction on an initially disturbed (curved) interface. At the moment of exiting the interface disturbance, the shock wave reaches a new stable state characterized by a new front structure and the gas parameter distribution behind it. The instability starts to develop in the form of a wavelike shock front stretching into the lower density plasma, and the global pressure drop is the consequence of the weakened shock wave. Pressure perturbations caused by the shock stretching result in the loss of stability of the flow behind it that eventually organizes into an intense clockwise rotating vortex structure. If the plasma density is nonuniform, the transition in the form of front stretching exhibits the pattern of motion that prevails the principle of exchange of stabilities, so the instability sets in as a secondary flow. Regardless of the interaction geometry, the instability mode is aperiodical and unconditional, and thus either a transition to another stable state or continuous development as a secondary flow is possible. The marginal state condition {*T*2/*T*<sup>1</sup> > (*M*1/*M*2) 2 , |*χ*| > 0} is the only requirement to trigger the instability [17], where the minimum heating requirement accounts for losses due to shock reflections off the interface. Independence of the instability locus on the plasma density distribution identifies the interface conditions as the sole triggering factor, though the density gradient can discriminate between qualitatively different outcomes at later stages of the instability evolution. With *T*<sup>21</sup> fixed, in the uniform density case, the perturbation growth rates are determined solely by the interface perturbation curvature *χ* that, for some geometries, can be replaced with the *relative* curvature between the shock front and the interface. The specific wave nature of the instability dissipation, when the overstretched shock

perturbation decays via the degeneration into an acoustic wave, allows the shock instability to decay even though the viscous damping mechanisms are not available in the flow (except the shock width layer). The initial state shock wave energy, in this case, converts into rotational energy of the flow thus continuously supporting the developing vortex structure until the perturbed shock vanishes.

In optimizing the ratio *Rs*/*Rb*, it should be noted that the interaction can be less significant if the size of the thermal spot becomes small compared to the shock front dimensions. It is not only because of a relatively smaller plasma volume or deposited energy but also due to smaller relative curvature of the shock front as seen from the plasma boundary (approaching a nearly flat shape as *Rs=Rb* ! 0). This results in shorter interaction time and consequently weaker effect. Higher temperature/density step across the interface and the smooth interface type can be also used to

In discussing validity of the model for a particular application, the following factors must be taken into consideration. First, the model assumes that the plasma region is not expanding, i.e., that the gas is in an equilibrium state. With the interaction times on the order of 0.1 μs and in the temperature range typical for many thermal energy deposition experiments, the approximation works well, for example, for diatomic gases whose vibrational relaxation times are on the order of tens of microseconds or longer (at atmospheric pressure). However, for considerably elevated gas temperatures or pressures and in specific gases, the relaxation times may become comparable to the interaction time, and the model would need a

Second, the model does not assume various degree deformation of the plasma region during the interaction time and thus should be considered as a valid approximation only when the coupling between the shock flow and the interface instability can be neglected. The model also assumes the ideal gas conditions. For the pressures typically developed in the shocks, the gas state equation can be still valid. However, the parameter jump relations across the shock in real gases may depart from their ideal equivalents significantly, and Eq. (4) must be corrected. An analysis of such effects must be done, and the criteria of ideality have to be applied and tested in the experiments used for comparison. For most energy deposition conditions though, the approach developed here appears to be quite adequate as, within the time frame of the interaction, the model predictions in the form of consistent time and dimen-

The shock refraction on an interface was considered here as the main mechanism of the wave drag reduction. The contribution of other thermal or nonthermal secondary mechanisms should be accounted for as well. In addition to those mentioned in the introduction, these can include the mechanisms participating in the formation of the thermal equilibrium and definite parameter distribution in the plasma during rapid expansion of the thermal spot. A blast wave produced at the early stages of the plasma creation can result in the pressure increase on the body at the first moment followed by the main pressure and the drag drop [48]. This may interfere with the shock refraction effects and overlap with the processes described in the model. The mechanisms controlling the effectiveness of the energy transfer to the gas during its deposition to the flow should be taken into account for better

The results of this work can be found useful in a number of other major applications. With optimized energy deposition, decreased pressure with an extended rise time on the ground can be used to alleviate the sonic boom or reduce the heating of a space craft at atmospheric entry. Other potential areas of application include magnetohydrodynamics (MHD), plasma chemical reactors, and molecular lasers for shock wave structure control; rocket plume optimization; shock waveassisted combustion, where a shock wave can be used to affect the ignition conditions in the gas; plasma-based piloting and combustion enhancement that can be used toward the design of efficient combustors for hypersonic propulsion systems, combustors for gas turbine, diesel engines, environmentally clean combustors, and

spark inhibition; localized flow problems, such as the Edney type IV shock

sion relationship are clearly visible in most of the experiments [12, 22].

strengthen the effect.

optimization.

**115**

correction for non-equilibrium effects.

*Wave Drag Modification in the Presence of Discharges DOI: http://dx.doi.org/10.5772/intechopen.86858*

The advantage of the model presented here is that it provides useful insight on the phenomenological origin of the drag reduction and its connection to the series of other phenomena that accompany the interaction. Its simple analytical solutions can be used for quick estimates and thus avoid the complexity of more exact numerical calculations. The dimensionless form of the equations and similarity law found in the spatial and temporal evolution of the flow perturbations with respect to the interface curvature [17] allow conclusions to be applied to any scale of the interaction. Though the model here was specifically developed for the spherical or sphereto-flat (as a limiting case) geometry of the interaction, the equations can be readily modified for other configurations. The shock reflections off the interface can be accounted in the model to its various degrees with the Mach number ratio *M*2n/*M*1n different from the unit [34], thus accommodating various types of the interface. Note that even plasma as a medium inside the heated area was considered, the nature of the effect is purely thermal and hence the model can be applied to conditions where only thermal heating in a neutral gas is present.

It was shown here that gas parameter redistribution in the flow behind the refracted shock front, including the global pressure drop, is the cause for the vorticity generation rather than its consequence. Thus, the pressure drop and the vortical perturbations in the post-shock flow can be considered rather competing because of the locally developing secondary flows resulting in an effective mixing. This eventually equalizes pressure in different parts of the volume and thus quenches (at least partially) the pressure drop and large-scale sucking effect on the interface. Thus, if pressure lowering is thought, the vortex generation must be minimized. Then the parameters of the interaction must be chosen in such a way that the shock front would be stretched considerably and as possible evenly, without sharp bends causing intensive vorticity. Spherical geometry and the exponential type of the plasma parameter distribution would be the best fit for this purpose as it was shown to generate vorticity of either micro-size and/or micro-intensity. Thus, the choice of the plasma parameter distribution alone can offer the advantage of depositing less energy of a smaller volume and closer to the body.

Other applications where the vorticity development must be avoided include magnetized target fusion experiments, for example, where, during the implosion of an inertial confinement fusion target, the hot shell materials surrounding the cold D-T fuel layer are shock-accelerated [25]. Mixing of the shell material and fuel is not desired in this case, and efforts should be made to minimize any abrupt imperfections or irregularities on the shock front. Considering a sharp type of the interface can be helpful in such cases since the refraction effect can be up to 40% weaker [18].

Contrary, in the applications where vorticity is beneficial for better mixing in the flow, sharp distortions spanning a larger portion of the shock front are necessary, and the uniform and the power law density distributions could be more desirable in this case. The examples include combustion where the burning speed is controlled by introducing turbulence in the flow [22]. A supersonic combustion in a scramjet may also benefit from vorticity in the flow as the fuel-oxidant interface is enhanced by the breakup of the fuel into finer droplets. Studies of deflagration to detonation transition (DDT) processes show that introduction of vorticity can result in an increase in burning velocity and detonation. In this case, introducing small perturbations of the interface that will cause sharp distortions on the shock front can be suggested [17].

### *Wave Drag Modification in the Presence of Discharges DOI: http://dx.doi.org/10.5772/intechopen.86858*

perturbation decays via the degeneration into an acoustic wave, allows the shock instability to decay even though the viscous damping mechanisms are not available in the flow (except the shock width layer). The initial state shock wave energy, in this case, converts into rotational energy of the flow thus continuously supporting

The advantage of the model presented here is that it provides useful insight on the phenomenological origin of the drag reduction and its connection to the series of other phenomena that accompany the interaction. Its simple analytical solutions can be used for quick estimates and thus avoid the complexity of more exact numerical calculations. The dimensionless form of the equations and similarity law found in the spatial and temporal evolution of the flow perturbations with respect to the interface curvature [17] allow conclusions to be applied to any scale of the interaction. Though the model here was specifically developed for the spherical or sphereto-flat (as a limiting case) geometry of the interaction, the equations can be readily modified for other configurations. The shock reflections off the interface can be accounted in the model to its various degrees with the Mach number ratio *M*2n/*M*1n different from the unit [34], thus accommodating various types of the interface. Note that even plasma as a medium inside the heated area was considered, the nature of the effect is purely thermal and hence the model can be applied to

the developing vortex structure until the perturbed shock vanishes.

*Aerodynamics*

conditions where only thermal heating in a neutral gas is present.

depositing less energy of a smaller volume and closer to the body.

front can be suggested [17].

**114**

It was shown here that gas parameter redistribution in the flow behind the refracted shock front, including the global pressure drop, is the cause for the vorticity generation rather than its consequence. Thus, the pressure drop and the vortical perturbations in the post-shock flow can be considered rather competing because of the locally developing secondary flows resulting in an effective mixing. This eventually equalizes pressure in different parts of the volume and thus

quenches (at least partially) the pressure drop and large-scale sucking effect on the interface. Thus, if pressure lowering is thought, the vortex generation must be minimized. Then the parameters of the interaction must be chosen in such a way that the shock front would be stretched considerably and as possible evenly, without sharp bends causing intensive vorticity. Spherical geometry and the exponential type of the plasma parameter distribution would be the best fit for this purpose as it was shown to generate vorticity of either micro-size and/or micro-intensity. Thus, the choice of the plasma parameter distribution alone can offer the advantage of

Other applications where the vorticity development must be avoided include magnetized target fusion experiments, for example, where, during the implosion of an inertial confinement fusion target, the hot shell materials surrounding the cold D-T fuel layer are shock-accelerated [25]. Mixing of the shell material and fuel is not desired in this case, and efforts should be made to minimize any abrupt imperfections or irregularities on the shock front. Considering a sharp type of the interface can be helpful in such cases since the refraction effect can be up to 40% weaker [18].

Contrary, in the applications where vorticity is beneficial for better mixing in the flow, sharp distortions spanning a larger portion of the shock front are necessary, and the uniform and the power law density distributions could be more desirable in this case. The examples include combustion where the burning speed is controlled by introducing turbulence in the flow [22]. A supersonic combustion in a scramjet may also benefit from vorticity in the flow as the fuel-oxidant interface is enhanced by the breakup of the fuel into finer droplets. Studies of deflagration to detonation transition (DDT) processes show that introduction of vorticity can result in an increase in burning velocity and detonation. In this case, introducing small perturbations of the interface that will cause sharp distortions on the shock

In optimizing the ratio *Rs*/*Rb*, it should be noted that the interaction can be less significant if the size of the thermal spot becomes small compared to the shock front dimensions. It is not only because of a relatively smaller plasma volume or deposited energy but also due to smaller relative curvature of the shock front as seen from the plasma boundary (approaching a nearly flat shape as *Rs=Rb* ! 0). This results in shorter interaction time and consequently weaker effect. Higher temperature/density step across the interface and the smooth interface type can be also used to strengthen the effect.

In discussing validity of the model for a particular application, the following factors must be taken into consideration. First, the model assumes that the plasma region is not expanding, i.e., that the gas is in an equilibrium state. With the interaction times on the order of 0.1 μs and in the temperature range typical for many thermal energy deposition experiments, the approximation works well, for example, for diatomic gases whose vibrational relaxation times are on the order of tens of microseconds or longer (at atmospheric pressure). However, for considerably elevated gas temperatures or pressures and in specific gases, the relaxation times may become comparable to the interaction time, and the model would need a correction for non-equilibrium effects.

Second, the model does not assume various degree deformation of the plasma region during the interaction time and thus should be considered as a valid approximation only when the coupling between the shock flow and the interface instability can be neglected. The model also assumes the ideal gas conditions. For the pressures typically developed in the shocks, the gas state equation can be still valid. However, the parameter jump relations across the shock in real gases may depart from their ideal equivalents significantly, and Eq. (4) must be corrected. An analysis of such effects must be done, and the criteria of ideality have to be applied and tested in the experiments used for comparison. For most energy deposition conditions though, the approach developed here appears to be quite adequate as, within the time frame of the interaction, the model predictions in the form of consistent time and dimension relationship are clearly visible in most of the experiments [12, 22].

The shock refraction on an interface was considered here as the main mechanism of the wave drag reduction. The contribution of other thermal or nonthermal secondary mechanisms should be accounted for as well. In addition to those mentioned in the introduction, these can include the mechanisms participating in the formation of the thermal equilibrium and definite parameter distribution in the plasma during rapid expansion of the thermal spot. A blast wave produced at the early stages of the plasma creation can result in the pressure increase on the body at the first moment followed by the main pressure and the drag drop [48]. This may interfere with the shock refraction effects and overlap with the processes described in the model. The mechanisms controlling the effectiveness of the energy transfer to the gas during its deposition to the flow should be taken into account for better optimization.

The results of this work can be found useful in a number of other major applications. With optimized energy deposition, decreased pressure with an extended rise time on the ground can be used to alleviate the sonic boom or reduce the heating of a space craft at atmospheric entry. Other potential areas of application include magnetohydrodynamics (MHD), plasma chemical reactors, and molecular lasers for shock wave structure control; rocket plume optimization; shock waveassisted combustion, where a shock wave can be used to affect the ignition conditions in the gas; plasma-based piloting and combustion enhancement that can be used toward the design of efficient combustors for hypersonic propulsion systems, combustors for gas turbine, diesel engines, environmentally clean combustors, and spark inhibition; localized flow problems, such as the Edney type IV shock

interaction and the control of shock structures formed in the high-speed engine inlet; and astrophysics for understanding the dynamics of shock waves generated in the stellar interior.

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