**1. Introduction**

The drag is one of the four major forces acting on an airplane in flight. Depending on the nature of the body-flow interaction and the phase of flight, it can be divided into three basic components. The pressure drag is due to pressure distribution over the vehicle surface and skin friction. The induced drag results from the pressure redistribution following the formation of the trailing vortex system during the lift production and thus becomes significant only during takeoff and landing times. When the free stream Mach number exceeds 0.8, the local flow velocities at some points on the airframe may become supersonic, and shock waves set in at the corresponding locations. The gas compression in a shock wave results in the pressure increase in the flow in front of the moving body. This is usually felt as an abrupt increase in local drag and reduction in local lift that may be accompanied by associated changes in trim, pitching moment, and the stability and control characteristics of the airplane. This portion of the drag, the so-called wave drag, is up to four times larger than its subsonic component and can result in the aerodynamic efficiency reduced by more than 50%. The wave drag is the reason for the strong pushback (shock stall) an aircraft experiences when, during acceleration through the transonic flight regime, the Mach number first reaches its critical value and the local speed passes through the sonic point ("sonic barrier"). The increase in the drag by 1% is equivalent to 5–10% of payload, and thus it imposes significant limitations on the speed, range, and payload carried by a vehicle [1].

The shock wave weakening achieved by altering the shock formation processes is one of the efficient methods developed to reduce the wave drag. Swept wing

surface minimizes the shock strength, thereby reducing wave drag. Sweeping beyond the Mach line converts the wing design problem into a transonic issue and also enables extension into supersonic speeds of "leading edge thrust" [1, 2]. Moving lift forward onto the fuselage allows extending the lifting line and reducing the drag due to lift [3]. This approach finally led to the so-called Coke bottle of the transonic fighter regime [4]. A method using an active or passive flow separation control at cruise has been suggested in [5].

and the geometry of the interaction zone. A model providing an insight on the origin of the chain of consecutive transformations in the flow leading to the wave drag reduction can be used to optimize thermal energy input and thus achieve fundamentally lower drag values than that of conventional approaches.

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

In the interaction between a shock wave and a plasma region produced in a discharge, the initially simply structured planar, bow, or oblique shock wave loses its stability and eventually evolves into a complicated system of distorted and secondary shocks with flow separation regions and formation of vortices in the flow behind the shock [16, 17]. The main changes in the "shock-plasma" system include the shock's acceleration and its front distortion gradually increasing with time, followed with its weakening until it becomes unidentifiable [18, 19]; motion of the shock away from the body in the presence of heating [20]; substantial gas/plasma parameter redistribution in the flow behind the shock, particularly considerable reduction of gas pressure; a vortex system forming in the aftershock flow [12]; remarkable positive dynamics in the vortex strength as the shock propagates well beyond the interface suggesting not only boundary but volume effects as well; the time delays in the effects on the flow relative to the discharge on-off times and a finite pressure rise time [21]; and finally a strong distortion or collapse of the plasma region [22]. Similar phenomena are commonly observed in other settings involving the shock interacting with an interface: in the front separation region control experiments [23], in combustion [22], and in impulsively loaded flows having place in astrophysics plasmas [24] and fusion research [25]. The shock-flame interaction, where the instability induced by the shock wave passage through a flame results in a sharp increase of the burning velocity [26, 27], is used in the supersonic combustion

The existing models used to describe the observed phenomena can be split into

two major approaches. The first one considering thermal heating as a possible reason includes Mach number decrease due to gas heating and the possibility of mean molecular weight and number density change caused by molecule dissociation and ionization [14]. The contribution of possible nonthermal mechanisms of the interaction is considered for flows involving atomic and molecular transitions, gas kinetics, electrical properties of plasmas, and non-equilibrium states as a result of fast evolving processes such as radiation or fast expansion. Among them are appearance of charged particles leading to upstream momentum transfer in the hypersonic flow [12, 28, 29]; the possibility of deflection of the incoming flow by plasma in front of the shock via electronic momentum transfer collisions [30]; and the release of heat into the shock layer by the exothermic reactions enhancing the shock layer temperature and thus reducing the pressure and the density behind the

The baroclinic effect is one of the mechanisms that can be responsible for vorticity generation on an interface when there is nonalignment of pressure and density gradients in the hot gas region [32]. Richtmyer-Meshkov instability (RMI) takes place when two gases of different densities are accelerated by a passage of a shock wave. The acceleration causes growth of small perturbations of the interface, followed by a nonlinear regime with bubbles appearing in the case of lighter gas penetrating a heavier gas and with spikes appearing in the case of a heavier gas moving into the lighter one. The vortex sheet rolls up and accumulates into periodic

The previous research on this topic was quite successful in explaining some separate features of the interaction. However, the experiments show that each of the processes is not an isolated phenomenon and their coexistence and specific time sequence in their development cannot be covered by the existing models. The model described below is capable of explaining the full set of the features observed

for increased chemical reaction rates.

vortex cores in the post-shock flow [21].

shock wave [31].

**103**

The shock wave interaction with other components of the aircraft (centerbodynacelle, nacelle-wing, body-wing, body-body, and body/wing-propulsion exhaust efflux) [1, 4] can be used to obtain favorable wave interference leading to a weakened shock system. Up to 25% reduction of the wave drag and increase in the lift-todrag ratio were reported [6].

One of the approaches in which the shock energy is effectively redirected into the thrust production uses the interaction between the forebody shock with a wing raised above the body. The shock reflection onto the afterbody results in pressure increase and an increment of thrust. Though these methods were proven to be effective in a number of localized problems such as centerbody inlets and nacelle-wing interactions, significant difficulties were encountered in other applications [1].

The local passive porosity method [7] utilizes high transonic flow sensitivity to local geometry changes resulting in the replacement of strong shocks with a system of weaker ones (*λ*-shock system). Localized wall deformations/bumps were shown to weaken the shock wave through the generation of nearly isentropic waves upstream [8]. The trailing edge methods use truncation of the region or morphing trailing edge to alter the airfoil shape [9].

The "physical spikes" representing an artificial sharpening of the front region of a blunt body can reduce the drag up to 30–45% [10]. It works through the replacement of nearly normal shock wave with a weaker, oblique shock structure. The upstream inert gas jets or liquid/solid particles injected in front of the body [11] is another method of the same type; however, its effect is weakened by injection thrust in the drag direction.

Thermal energy deposited in the flow upstream of the body is another spike-like working approach reporting 20–30% drag reduction rates in experiments [5, 12, 13] and 40–96% (with 65-fold "return" of the invested energy) predicted in numerical studies [14]. The energy is delivered into the flow via RF, microwave, or optical discharge, with focused electron beam or localized combustion combined with the spike. Pulsed energy deposition methods have been shown to be more efficient than continuous if an appropriate repetition rate is chosen [15]. A combination of methods, such as simultaneous addition of the heat at the tip of the physical spike, leads to flow reattachment on the nose and can further reduce the drag up to 75% [1, 5]. The study showed that, under a number of simplifying assumptions, a thermally created phantom body enveloping the airplane results in finite rise-time signatures that can theoretically eliminate the shock wave, but practically this would require a power input approximately equivalent to twice that necessary to sustain the airplane flight. Thus, the idea of the thermal energy deposition in the flow can be practically realized if highly effective ways of the energy deposition and the shock-thermal area interaction are found.

Using different approaches in the heat design, some authors found little or no effect in the drag reduction. It is the main conclusion of many researchers that, for further progress, a careful design of the energy deposition must be applied. While most of the previous research was concentrated on finding optimal gas parameter values and the amount of energy, this work will point at the exact mechanism of the shock-plasma interaction. Recommendations in the form of adjustment parameters will be done based on closer attention to the effect of the gas parameter distribution

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

surface minimizes the shock strength, thereby reducing wave drag. Sweeping beyond the Mach line converts the wing design problem into a transonic issue and also enables extension into supersonic speeds of "leading edge thrust" [1, 2]. Moving lift forward onto the fuselage allows extending the lifting line and reducing the drag due to lift [3]. This approach finally led to the so-called Coke bottle of the transonic fighter regime [4]. A method using an active or passive flow separation

The shock wave interaction with other components of the aircraft (centerbodynacelle, nacelle-wing, body-wing, body-body, and body/wing-propulsion exhaust efflux) [1, 4] can be used to obtain favorable wave interference leading to a weakened shock system. Up to 25% reduction of the wave drag and increase in the lift-to-

One of the approaches in which the shock energy is effectively redirected into the thrust production uses the interaction between the forebody shock with a wing raised above the body. The shock reflection onto the afterbody results in pressure increase and an increment of thrust. Though these methods were proven to be effective in a number of localized problems such as centerbody inlets and nacelle-wing interac-

The local passive porosity method [7] utilizes high transonic flow sensitivity to local geometry changes resulting in the replacement of strong shocks with a system of weaker ones (*λ*-shock system). Localized wall deformations/bumps were shown to weaken the shock wave through the generation of nearly isentropic waves upstream [8]. The trailing edge methods use truncation of the region or morphing

The "physical spikes" representing an artificial sharpening of the front region of a blunt body can reduce the drag up to 30–45% [10]. It works through the replacement of nearly normal shock wave with a weaker, oblique shock structure. The upstream inert gas jets or liquid/solid particles injected in front of the body [11] is another method of the same type; however, its effect is weakened by injection

Thermal energy deposited in the flow upstream of the body is another spike-like working approach reporting 20–30% drag reduction rates in experiments [5, 12, 13] and 40–96% (with 65-fold "return" of the invested energy) predicted in numerical studies [14]. The energy is delivered into the flow via RF, microwave, or optical discharge, with focused electron beam or localized combustion combined with the spike. Pulsed energy deposition methods have been shown to be more efficient than continuous if an appropriate repetition rate is chosen [15]. A combination of methods, such as simultaneous addition of the heat at the tip of the physical spike, leads to flow reattachment on the nose and can further reduce the drag up to 75% [1, 5]. The study showed that, under a number of simplifying assumptions, a thermally created phantom body enveloping the airplane results in finite rise-time signatures that can theoretically eliminate the shock wave, but practically this would require a power input approximately equivalent to twice that necessary to sustain the airplane flight. Thus, the idea of the thermal energy deposition in the flow can be practically realized if highly effective ways of the energy deposition and

Using different approaches in the heat design, some authors found little or no effect in the drag reduction. It is the main conclusion of many researchers that, for further progress, a careful design of the energy deposition must be applied. While most of the previous research was concentrated on finding optimal gas parameter values and the amount of energy, this work will point at the exact mechanism of the shock-plasma interaction. Recommendations in the form of adjustment parameters will be done based on closer attention to the effect of the gas parameter distribution

tions, significant difficulties were encountered in other applications [1].

control at cruise has been suggested in [5].

trailing edge to alter the airfoil shape [9].

the shock-thermal area interaction are found.

**102**

thrust in the drag direction.

drag ratio were reported [6].

*Aerodynamics*

and the geometry of the interaction zone. A model providing an insight on the origin of the chain of consecutive transformations in the flow leading to the wave drag reduction can be used to optimize thermal energy input and thus achieve fundamentally lower drag values than that of conventional approaches.

In the interaction between a shock wave and a plasma region produced in a discharge, the initially simply structured planar, bow, or oblique shock wave loses its stability and eventually evolves into a complicated system of distorted and secondary shocks with flow separation regions and formation of vortices in the flow behind the shock [16, 17]. The main changes in the "shock-plasma" system include the shock's acceleration and its front distortion gradually increasing with time, followed with its weakening until it becomes unidentifiable [18, 19]; motion of the shock away from the body in the presence of heating [20]; substantial gas/plasma parameter redistribution in the flow behind the shock, particularly considerable reduction of gas pressure; a vortex system forming in the aftershock flow [12]; remarkable positive dynamics in the vortex strength as the shock propagates well beyond the interface suggesting not only boundary but volume effects as well; the time delays in the effects on the flow relative to the discharge on-off times and a finite pressure rise time [21]; and finally a strong distortion or collapse of the plasma region [22]. Similar phenomena are commonly observed in other settings involving the shock interacting with an interface: in the front separation region control experiments [23], in combustion [22], and in impulsively loaded flows having place in astrophysics plasmas [24] and fusion research [25]. The shock-flame interaction, where the instability induced by the shock wave passage through a flame results in a sharp increase of the burning velocity [26, 27], is used in the supersonic combustion for increased chemical reaction rates.

The existing models used to describe the observed phenomena can be split into two major approaches. The first one considering thermal heating as a possible reason includes Mach number decrease due to gas heating and the possibility of mean molecular weight and number density change caused by molecule dissociation and ionization [14]. The contribution of possible nonthermal mechanisms of the interaction is considered for flows involving atomic and molecular transitions, gas kinetics, electrical properties of plasmas, and non-equilibrium states as a result of fast evolving processes such as radiation or fast expansion. Among them are appearance of charged particles leading to upstream momentum transfer in the hypersonic flow [12, 28, 29]; the possibility of deflection of the incoming flow by plasma in front of the shock via electronic momentum transfer collisions [30]; and the release of heat into the shock layer by the exothermic reactions enhancing the shock layer temperature and thus reducing the pressure and the density behind the shock wave [31].

The baroclinic effect is one of the mechanisms that can be responsible for vorticity generation on an interface when there is nonalignment of pressure and density gradients in the hot gas region [32]. Richtmyer-Meshkov instability (RMI) takes place when two gases of different densities are accelerated by a passage of a shock wave. The acceleration causes growth of small perturbations of the interface, followed by a nonlinear regime with bubbles appearing in the case of lighter gas penetrating a heavier gas and with spikes appearing in the case of a heavier gas moving into the lighter one. The vortex sheet rolls up and accumulates into periodic vortex cores in the post-shock flow [21].

The previous research on this topic was quite successful in explaining some separate features of the interaction. However, the experiments show that each of the processes is not an isolated phenomenon and their coexistence and specific time sequence in their development cannot be covered by the existing models. The model described below is capable of explaining the full set of the features observed

in experiments and thus fills the gap in understanding of this phenomenon. The shock refraction on an interface will be considered there as a mechanism [33] that triggers the chain of subsequent flow transformations leading to the wave drag reduction. The model has an advantage of pointing at the origin of the complex phenomena and describing each of the consecutive stages of its development in adequate timing order.

reminder that is still propagating in the colder media thus resulting in the continu-

The shock front development proceeds in two stages, first being affected by the conditions on the interface and second in the plasma volume. The shock refraction resulting in an increase of the absolute value of the shock velocity along with its vector rotation (at refraction angle *γ*) occurs at the moment when the shock front crosses the plasma interface. As the refracted shock continues to propagate in hotter medium, its dynamics is determined by the parameter distribution in the plasma volume [17, 19, 33, 35]. Even though the changes in the shock structure become visible only during this time, they are the still consequences of the interaction at both stages: the conditions on the interface are necessary to trigger the front instability, and the gas volume effects provide the means necessary for its positive

The relationship between the incident (*x*i, *y*i) and refracted (*X*i, *Y*i) shock front coordinates at a point of the interaction *i* has been derived in [33]. To recast it in a dimensionless form, the coordinates can be scaled with the plasma sphere radius *Rb*, the shock velocity with *V*1, gas temperature with *T*1, Mach number with *M*1*n*, and

*Xi* ¼ ð Þ *v* cos *γ* � 1 ð*n* � ð Þ *xi* þ *xb* Þ þ Δ*x, Yi* ¼ *yi* � *v* sin *γ* ∙ð Þ *n* � ð Þ *xi* þ *xb* (1)

the interaction point (**Figure 1**), *xb* ¼ 1 � cos *α*, Δ*x* ¼ ð Þ *Rs=Rb* ð Þ cos *β* � *η* , the

*<sup>η</sup>* <sup>¼</sup> <sup>2</sup>ð Þ *Rs* <sup>þ</sup> *Rb* ð Þþ *Rs* � *nRb <sup>n</sup>*<sup>2</sup>*R*<sup>2</sup>

2*Rs*½ � ð Þ� *Rs* þ *Rb nRb*

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

*TM* tan *<sup>α</sup>* <sup>p</sup> <sup>Þ</sup>

2 �

*M*<sup>2</sup> 2*n*

ffiffiffiffiffiffiffiffiffiffiffiffiffi *T*1*=T*<sup>2</sup>

! *<sup>T</sup>*<sup>2</sup>

� *<sup>M</sup>*2*<sup>n</sup>* <sup>1</sup> � <sup>1</sup>

and the normal and tangential components of the Mach numbers are related as

8 < :

2*kM*<sup>2</sup>

2*kM*<sup>2</sup>

*T*1 � �<sup>1</sup> 2

p . The "sharp" interface

cos <sup>2</sup>*α* þ sin <sup>2</sup>*α*

and the bar over the variable means its dimensionless equivalent. The dimen-

*T M* � �<sup>2</sup>

�

problem geometry, heating intensity *T* ¼ *T*2/*T*1, and the ratio of normal components of Mach numbers in the two media *M* ¼ *M*2n/*M*1n that account for the shock

The Mach number ratio for normal incidence can be obtained using the refraction equation from [36] that was derived assuming steplike temperature *T*2/*T*<sup>1</sup>

q

*v* ¼ *V*2*=V*<sup>1</sup> ¼

and the refraction angle *<sup>γ</sup>* <sup>¼</sup> *<sup>α</sup>* � tan �<sup>1</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

changes across the interface (a "sharp" interface)

*M*1*<sup>n</sup>* ¼ *M*<sup>1</sup> cos *α*, *M*1*<sup>t</sup>* ¼ *M*<sup>1</sup> sin *α*, and *M*2*<sup>t</sup>* ¼ *M*1*<sup>t</sup>*

<sup>1</sup>*<sup>n</sup>* � ð Þ *<sup>k</sup>* � <sup>1</sup> � � ð Þ *<sup>k</sup>* � <sup>1</sup> *<sup>M</sup>*<sup>2</sup> <sup>1</sup>*<sup>n</sup>* <sup>þ</sup> <sup>2</sup> � � � � <sup>1</sup>

<sup>¼</sup> *<sup>M</sup>*1*<sup>n</sup>* <sup>1</sup> � <sup>1</sup>

*M*<sup>2</sup> 1*n*

!

*b*

(2)

(3)

are determined by the

<sup>2</sup>*<sup>n</sup>* � ð Þ *k* � 1

2*k* � 1

9 = ;

(4)

" #*<sup>k</sup>*�<sup>1</sup>

<sup>1</sup>*<sup>n</sup>* � ð Þ *k* � 1

Here *n* = *t*/*τ* is the dimensionless time, 0 < *n* < 2, *α* is the local incidence angle at

ously increasing front stretching toward the hotter medium [18].

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

time *t* with the characteristic time *τ* = *Rb*/*V*1:

dynamics.

parameter

sionless shock velocity

reflections off the interface.

2*kM*<sup>2</sup>

1 *M*1*<sup>n</sup>*ð Þ *k* � 1

**105**
