**3. Basic modifications in the flow: numerical simulation results and comparison with experiment**

To demonstrate how the shock refraction on the interface with plasma triggers the chain of consecutive transformations in the flow, the model was numerically run for three types of plasma density distribution that would cover the most common ways of the plasma production. A relatively uniform distribution law is commonly observed in microwave discharges [37] or when the plasma spot was initiated at a considerable distance from the body shock. The exponential density distribution can be established during the exothermal expansion, for example, in large-area plasma sources created with internal low-inductance antenna units [38], detonation, or the ultra-intense laser-induced breakdown in a gas [39]. And the power law density distribution can be found in the heated layers/spheres in thermodynamic equilibrium (for both media and radiation) in the presence of radiative heat conduction, as, for example, on stellar surfaces where it is formed as the result of the combined action of gravity, thermal pressure, and radiant heat conduction [24]. On the laboratory scale, the examples of such systems include: the gas of radiative spherical cloud when simultaneously considering the mechanical equilibrium and radiative transfer, in the diffusion approximation; a planar problem of the sudden expansion into vacuum of a gas layer with a finite mass and constant initial gas distribution; a problem of sudden expansion of a spherical gas cloud into vacuum; isentropic flows for which there is a class of self-similar solutions, such as for a strong explosion on a solid surface due to other body impact with generation of vapor cloud expanding into vacuum [40]; vaporization of the anode needle of a pulsed x-ray tube caused by a strong electron discharge [41]; explosion of wires by electric currents in vacuum systems; spark discharge in air in the early stages; the gas area near the edge of a cooling wave; motion of gas under the action of an impulsive load; the spherical shock wave implosion; a problem of bubbles collapsing in a liquid [40]; in the gas behind a weak blast wave [42].

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

The main difference in the treatment imposed by the nonhomogeneous plasma parameter distributions is that the refracted shock velocity becomes time dependent and the system of Eqs. (1)–(3) must be substantially modified. While the plasma parameter distribution will vary, the cold gas parameters will be always considered as distributed homogeneously. For easier comparison, all the simulation results presented below will be obtained for the same incident shock strength, heating intensity, and the interface parameters, with *M*<sup>1</sup> = 1.9,*T*<sup>1</sup> = 293 K,*T*1/*T*<sup>2</sup> = 0.10, equal radii *Rs* = *Rb*, adiabatic index *k* = 1.4 (air), and smooth boundary type. The timely order in the sequence of the flow modification stages demonstrated below follows from the model logic and agrees well with that observed in experiments: the shock front distortion and its weakening; flow parameter redistribution and pressure drop in the post-shock flow, followed with the body drag reduction; vortex generation in the plasma volume; and finally deformation or collapse of the plasma bubble.

#### **3.1 Shock front distortion**

assumes back reflections off it and associated losses of the shock energy; thus, the ratio *M*2/*M*<sup>1</sup> < 1. For an interface of a "smooth" type, when parameters across its

reflection. Thus, *M*2/*M*<sup>1</sup> approaches the unit, and the refraction effects become stronger [33]. An intermediate boundary type, when the parameters across its thickness can change faster than 1/*x*<sup>2</sup> (an "extended" interface), has been considered in [34]. For a particular case of the exponential gradient in the parameter

change, Eq. (3) can be replaced with more general *V*2*x=V*<sup>1</sup> ¼ ð Þ *T*2ð Þ *x =T*<sup>1</sup>

*<sup>n</sup>*¼<sup>0</sup> *<sup>n</sup>α<sup>n</sup>* <sup>þ</sup> <sup>∑</sup>*<sup>m</sup> <sup>m</sup>α<sup>m</sup>* 

The numerical constants are valid within the limits for *M*<sup>1</sup> = (1.6–2.4), *T*2/*T*<sup>1</sup> = (0–75), and *k* = 1.4 (air). The power coefficients *m* in the second sum run the values *m* = 1/16, 1/8, ¼, and ½. In Eq. (5), the coefficient *β* approaches exactly ½ for a limiting case of a smooth boundary and a value much smaller than ½ for

**3. Basic modifications in the flow: numerical simulation results**

To demonstrate how the shock refraction on the interface with plasma triggers the chain of consecutive transformations in the flow, the model was numerically run for three types of plasma density distribution that would cover the most common ways of the plasma production. A relatively uniform distribution law is commonly observed in microwave discharges [37] or when the plasma spot was initiated at a considerable distance from the body shock. The exponential density distribution can be established during the exothermal expansion, for example, in large-area plasma sources created with internal low-inductance antenna units [38], detonation, or the ultra-intense laser-induced breakdown in a gas [39]. And the power law density distribution can be found in the heated layers/spheres in thermodynamic equilibrium (for both media and radiation) in the presence of radiative heat conduction, as, for example, on stellar surfaces where it is formed as the result of the combined action of gravity, thermal pressure, and radiant heat conduction [24]. On the laboratory scale, the examples of such systems include: the gas of radiative spherical cloud when simultaneously considering the mechanical equilibrium and radiative transfer, in the diffusion approximation; a planar problem of the sudden expansion into vacuum of a gas layer with a finite mass and constant initial gas distribution; a problem of sudden expansion of a spherical gas cloud into vacuum; isentropic flows for which there is a class of self-similar solutions, such as for a strong explosion on a solid surface due to other body impact with generation of vapor cloud expanding into vacuum [40]; vaporization of the anode needle of a pulsed x-ray tube caused by a strong electron discharge [41]; explosion of wires by electric currents in vacuum systems; spark discharge in air in the early stages; the gas area near the edge of a cooling wave; motion of gas under the action of an impulsive load; the spherical shock wave implosion; a problem of bubbles collapsing

calculated from the fit coefficients *α*<sup>n</sup> and *α*<sup>m</sup> in the equation:

, there are no losses associated with the shock

*β* , where

ð5Þ

¼ 53*:*58 is the numerical constant

thickness change slower than 1/*x*<sup>2</sup>

*Aerodynamics*

*<sup>β</sup>* <sup>¼</sup> <sup>1</sup>*=*<sup>2</sup> � <sup>1</sup>*=*<sup>Σ</sup> and *<sup>Σ</sup>* <sup>¼</sup> <sup>∑</sup><sup>10</sup>

another limit of a sharp boundary [34].

**and comparison with experiment**

in a liquid [40]; in the gas behind a weak blast wave [42].

**106**

When the temperature/density in plasma is distributed uniformly, the relations Eqs. (1)–(3) apply. Results of numerical simulation in **Figure 2** obtained for this case demonstrate the shock front distortion as it progresses through the hot plasma sphere, at different propagation times starting at *n* = 0.05 (the most left curve) through Δ*n* = 0.05 time intervals. To highlight the size of the interface effect, the results are plotted comparatively, with the upper part of the diagram corresponding to the smooth, lower, and sharp type of the interface.

The initially spherical front acquires a nearly conical shape stretched in the propagation direction, in good agreement with the experimental observations. The most central part of the front (near the longitudinal symmetry axis) is affected by significant stretching due to longer interaction times and smaller angle of incidence. The curvature sign changes from negative to positive, with the inflection point location tending the intermediate area off the axis. Comparison between upper and lower curves shows that both types of the boundary produce identically shaped fronts; however, the smooth boundary results in stronger effect.

#### **Figure 2.**

*The shock front modification in homogeneous plasma, at several interaction times. The shock is incident from left to right. The outside part of the shock remains spherical (not shown in the picture).*

#### *Aerodynamics*

In the case of exponential plasma density distribution, the refracted shock velocity becomes dependent on time. Such a problem was considered in [19] assuming that the plasma density is exponentially decreasing in the shock propagation direction only, *ρ* = *ρ*<sup>00</sup> *Exp* (�*x*/*z*0), starting from some finite value *ρ*<sup>00</sup> at the leftmost point of the interface (*z*<sup>0</sup> is the characteristic length). The approximation neglecting the density change in the transverse direction can be applied, for example, to the case of plasma created by a laser sheet, with its body extended in the longitudinal direction. The task is reduced to obtain two separate solutions to the problem that must be tailored at the interface. The first solution models the shock wave refraction on the interface giving its speed and direction, and the second one describes its propagation in the inhomogeneous medium after crossing the interface. Using the approach [19] modified here for the sphere-to-sphere problem geometry, the time-dependent system of equations for the front's surface coordinates can be derived.

$$X\_i = \sigma (t - t\_{0i} + t\_\lambda)^{2/5} + \varepsilon (t - t\_{0i} + t\_\lambda)^{4/5} - \ge\_0,\\ Y\_i = y\_i - \sigma (t - t\_{0i} + t\_\gamma)^{2/5} - \sigma t\_\gamma^{2/5} \tag{6}$$

where *x*<sup>0</sup> ¼ *σt* 2*=*5 *<sup>λ</sup>* þ *εt* 4*=*5 *<sup>λ</sup> , t*0*<sup>i</sup>* <sup>¼</sup> ð Þ *xi*þ*xb <sup>V</sup>*<sup>1</sup> *, t<sup>γ</sup>* ¼ 5*V*<sup>1</sup> cos *α<sup>i</sup>* sin *γ<sup>i</sup>* ffiffiffiffi *T*2 *T*1 � � q *=* 2*σ* cos *α<sup>i</sup>* � *γ<sup>i</sup>* f g ð Þ and the parameter *t*<sup>λ</sup> is the solution of the following equation

$$\frac{5}{2} \frac{V\_1 \cos a\_i \cos \beta\_i}{\sigma \cos \left(a\_i - \gamma\_i\right)} \sqrt{\frac{T\_2}{T\_1}} = t\_\lambda^{-3/5} + \frac{2\varepsilon}{\sigma} t\_\lambda^{-1/5} \tag{7}$$

When a plane shock wave propagates through a gas with the density that drops to zero according to a power law *<sup>ρ</sup>*�*xN*, the so-called energy cumulation effect takes place [44]. In the gas-dynamical approximation, a strong plane shock wave propagating in such a medium accelerates very quickly accumulating virtually infinite energy. This interesting phenomenon, as applied to the shock refraction problem, was considered in [35] assuming the plasma density as changing in the longitudinal (*x*-) direction only. Applying it to the present geometry, the refracted shock coor-

*, Y*2*<sup>i</sup>* ¼ *yi* � ð Þ *V*2*=V*<sup>1</sup> sin *γ*½ � *nR* � ð Þ *xi* þ *xb* (8)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos <sup>2</sup>*α<sup>i</sup>* � ð Þ *<sup>T</sup>*2*=T*<sup>1</sup> ð Þ *<sup>M</sup>*2*<sup>n</sup>=M*1*<sup>n</sup>* <sup>2</sup> <sup>þ</sup> sin <sup>2</sup>*<sup>α</sup>*

½ � *t*0*<sup>i</sup>* � ð Þ ð Þ *xi* þ *xb =V*<sup>1</sup>

and the shock velocity ratio *V*2/*V*<sup>1</sup> is determined from Eq. (3). Here *a* and *ai* are the maximum (from the point *O* in **Figure 1**) and the local (from the boundary, at the periphery from the axis) distances to the zero density plane correspondingly, *t*0*<sup>i</sup>* is the local time of the shock front portion travel to the boundary, the constant *b* = 0.59 was determined in [44], and the constant *N* = 3.25 is taken from [40]. The system of Eqs. (8)–(10) was run for the distance to the zero density plane *a* = 3.0*R*b, at times starting at *n* = 0.025 through the equal increments *Δn* = 0.025 (note that the interaction times here are twice as short than in **Figure 3a**). The results presented in **Figure 3b** demonstrate considerably stronger stretching of the shock front per unit of time due to significant front acceleration in the plasma area. Thus, as seen from **Figures 2** and **3**, the most common type of the front deformation for all three types of the density distribution is its continuous stretching. To make more exact conclusion about the front distortion, the local front inclination angle was computed, for *M*<sup>1</sup> = 1.9,*T*2/*T*<sup>1</sup> = 10.0, and the smooth type of the

In **Figure 4a**, the angles are presented for the uniform law of distribution, at times starting at *n* = 0.05 through the intervals *Δn* = 0.05, and the radii of the incident shock front and the interface are *Rs* = *Rb* = 0.3 cm. The curves in **Figure 4b** are for the angles corresponding to the exponential law of distribution, at the same

*The shock front inclination angle φ vs. distance yi, at several interaction times, for the uniform (a), exponential (b), and power law (c) density distributions. The time sequence for the curves is from upper to lower.*

<sup>q</sup> <sup>þ</sup> ð Þ *xi* <sup>þ</sup> *xb*

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos <sup>2</sup>*α<sup>i</sup>* � ð Þ *<sup>T</sup>*2*=T*<sup>1</sup> ð Þ *<sup>M</sup>*2*<sup>n</sup>=M*1*<sup>n</sup>* <sup>2</sup> <sup>þ</sup> sin <sup>2</sup>*<sup>α</sup>*

, *ai* ¼ *a* � *Rb*ð Þ 1 � cos *α<sup>i</sup>* ,

*<sup>b</sup>*�<sup>1</sup> (9)

(10)

*V*<sup>1</sup>

dinates can be determined as

*V*<sup>1</sup> cos *γ<sup>i</sup>*

interface.

**Figure 4.**

**109**

*<sup>X</sup>*2*<sup>i</sup>* <sup>¼</sup> *ai* � ð Þ *Gi=<sup>b</sup>* ð Þ *<sup>t</sup>*0*<sup>i</sup>* � *<sup>t</sup> <sup>b</sup>*

*Gi* <sup>¼</sup> *<sup>V</sup>*<sup>1</sup> cos *<sup>γ</sup><sup>i</sup>*

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

*<sup>t</sup>*0*<sup>i</sup>* <sup>¼</sup> *b a*ð Þ � *Rb*ð Þ <sup>1</sup> � cos *<sup>α</sup><sup>i</sup>*

the shock velocity component *<sup>V</sup>*2*xi* <sup>¼</sup> *Gi*ð Þ *<sup>t</sup>*0*<sup>i</sup>* � *<sup>t</sup> <sup>b</sup>*�<sup>1</sup>

q

The parameters *σ* and *ε* in Eqs. (6) and (7) are related to the effective explosion energy *E* in the thermal spot and the scale of the density gradient *z0* as *σ* = *ϛ* (*E*/*ρ*00) 1/5 and *ε* = (*K*/*z*0)*σ*<sup>2</sup> , and the numerical parameters *ϛ* = 1.075 and *K* = 0.185 are borrowed from Ref. [43].

The simulation results presented in **Figure 3a** were obtained for the interaction times starting at *n* = 0.05 through *Δn* = 0.05 time intervals, *z*<sup>0</sup> = 0.225 cm, *ρ*1/*ρ*<sup>00</sup> = 10, *Rb* = 0.1 cm, and the parameters *α* = 7 and *β* = 402.79 correspond to the specific explosion energy *<sup>E</sup>=ρ*<sup>00</sup> <sup>¼</sup> <sup>11</sup>*:*<sup>707</sup> � <sup>10</sup>3J. The most striking result in this case is that such a density profile can generate virtually perfect conical shock fronts, exactly as observed in the experiment [13].

**Figure 3.**

*Shock front deformation at several interaction times, for the cases of exponential (a) and power law (b) density distributions. Note twice as short interaction times in graph (b) compared to graph (a).*

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

In the case of exponential plasma density distribution, the refracted shock velocity becomes dependent on time. Such a problem was considered in [19] assuming that the plasma density is exponentially decreasing in the shock propagation direction only, *ρ* = *ρ*<sup>00</sup> *Exp* (�*x*/*z*0), starting from some finite value *ρ*<sup>00</sup> at the leftmost point of the interface (*z*<sup>0</sup> is the characteristic length). The approximation neglecting the density change in the transverse direction can be applied, for example, to the case of plasma created by a laser sheet, with its body extended in the longitudinal direction. The task is reduced to obtain two separate solutions to the problem that must be tailored at the interface. The first solution models the shock wave refraction on the interface giving its speed and direction, and the second one describes its propagation in the inhomogeneous medium after crossing the interface. Using the approach [19] modified here for the sphere-to-sphere problem geometry, the time-dependent system of equations for the front's surface

<sup>4</sup>*=*<sup>5</sup> � *<sup>x</sup>*0*, Yi* <sup>¼</sup> *yi* � *<sup>σ</sup> <sup>t</sup>* � *<sup>t</sup>*0*<sup>i</sup>* <sup>þ</sup> *<sup>t</sup><sup>γ</sup>*

� � q

2*ε σ t* �1*=*5

, and the numerical parameters *ϛ* = 1.075 and *K* = 0.185

*<sup>V</sup>*<sup>1</sup> *, t<sup>γ</sup>* ¼ 5*V*<sup>1</sup> cos *α<sup>i</sup>* sin *γ<sup>i</sup>*

� �<sup>2</sup>*=*<sup>5</sup> � *<sup>σ</sup><sup>t</sup>*

ffiffiffiffi *T*2 *T*1

*=*

*<sup>λ</sup>* (7)

2*=*5 *γ* (6)

coordinates can be derived.

2*=*5 *<sup>λ</sup>* þ *εt*

1/5 and *ε* = (*K*/*z*0)*σ*<sup>2</sup>

are borrowed from Ref. [43].

<sup>2</sup>*=*<sup>5</sup> <sup>þ</sup> *<sup>ε</sup>*ð Þ *<sup>t</sup>* � *<sup>t</sup>*0*<sup>i</sup>* <sup>þ</sup> *<sup>t</sup><sup>λ</sup>*

*<sup>λ</sup> , t*0*<sup>i</sup>* <sup>¼</sup> ð Þ *xi*þ*xb*

*V*<sup>1</sup> cos *α<sup>i</sup>* cos *β<sup>i</sup> σ* cos *α<sup>i</sup>* � *γ<sup>i</sup>* ð Þ

2*σ* cos *α<sup>i</sup>* � *γ<sup>i</sup>* f g ð Þ and the parameter *t*<sup>λ</sup> is the solution of the following equation

energy *E* in the thermal spot and the scale of the density gradient *z0* as *σ* = *ϛ*

times starting at *n* = 0.05 through *Δn* = 0.05 time intervals, *z*<sup>0</sup> = 0.225 cm,

ffiffiffiffiffi *T*2 *T*1 r

The parameters *σ* and *ε* in Eqs. (6) and (7) are related to the effective explosion

The simulation results presented in **Figure 3a** were obtained for the interaction

*ρ*1/*ρ*<sup>00</sup> = 10, *Rb* = 0.1 cm, and the parameters *α* = 7 and *β* = 402.79 correspond to the specific explosion energy *<sup>E</sup>=ρ*<sup>00</sup> <sup>¼</sup> <sup>11</sup>*:*<sup>707</sup> � <sup>10</sup>3J. The most striking result in this case is that such a density profile can generate virtually perfect conical shock fronts,

*Shock front deformation at several interaction times, for the cases of exponential (a) and power law (b) density*

*distributions. Note twice as short interaction times in graph (b) compared to graph (a).*

¼ *t* �3*=*5 *<sup>λ</sup>* þ

4*=*5

5 2

exactly as observed in the experiment [13].

*Xi* ¼ *σ*ð Þ *t* � *t*0*<sup>i</sup>* þ *t<sup>λ</sup>*

*Aerodynamics*

where *x*<sup>0</sup> ¼ *σt*

(*E*/*ρ*00)

**Figure 3.**

**108**

When a plane shock wave propagates through a gas with the density that drops to zero according to a power law *<sup>ρ</sup>*�*xN*, the so-called energy cumulation effect takes place [44]. In the gas-dynamical approximation, a strong plane shock wave propagating in such a medium accelerates very quickly accumulating virtually infinite energy. This interesting phenomenon, as applied to the shock refraction problem, was considered in [35] assuming the plasma density as changing in the longitudinal (*x*-) direction only. Applying it to the present geometry, the refracted shock coordinates can be determined as

$$X\_{2i} = a\_i - (G\_i/b)(t\_{0i} - t)^b,\\ Y\_{2i} = y\_i - (V\_2/V\_1)\sin\gamma[nR - (\varkappa\_i + \varkappa\_b)]\tag{8}$$

the shock velocity component *<sup>V</sup>*2*xi* <sup>¼</sup> *Gi*ð Þ *<sup>t</sup>*0*<sup>i</sup>* � *<sup>t</sup> <sup>b</sup>*�<sup>1</sup> , *ai* ¼ *a* � *Rb*ð Þ 1 � cos *α<sup>i</sup>* ,

$$\mathbf{G}\_{i} = \frac{V\_{1}\cos\gamma\_{i}\sqrt{\cos^{2}a\_{i}\cdot(T\_{2}/T\_{1})(M\_{2n}/M\_{1n})^{2} + \sin^{2}a}}{\left[t\_{0i} - ((\varkappa\_{i} + \varkappa\_{b})/V\_{1})\right]^{b-1}}\tag{9}$$

$$t\_{0i} = \frac{b(a - R\_b(1 - \cos a\_i))}{V\_1 \cos \gamma\_i \sqrt{\cos^2 a\_i \cdot (T\_2/T\_1)(M\_{2n}/M\_{1n})^2 + \sin^2 a}} + \frac{(\varkappa\_i + \varkappa\_b)}{V\_1} \tag{10}$$

and the shock velocity ratio *V*2/*V*<sup>1</sup> is determined from Eq. (3). Here *a* and *ai* are the maximum (from the point *O* in **Figure 1**) and the local (from the boundary, at the periphery from the axis) distances to the zero density plane correspondingly, *t*0*<sup>i</sup>* is the local time of the shock front portion travel to the boundary, the constant *b* = 0.59 was determined in [44], and the constant *N* = 3.25 is taken from [40].

The system of Eqs. (8)–(10) was run for the distance to the zero density plane *a* = 3.0*R*b, at times starting at *n* = 0.025 through the equal increments *Δn* = 0.025 (note that the interaction times here are twice as short than in **Figure 3a**). The results presented in **Figure 3b** demonstrate considerably stronger stretching of the shock front per unit of time due to significant front acceleration in the plasma area.

Thus, as seen from **Figures 2** and **3**, the most common type of the front deformation for all three types of the density distribution is its continuous stretching. To make more exact conclusion about the front distortion, the local front inclination angle was computed, for *M*<sup>1</sup> = 1.9,*T*2/*T*<sup>1</sup> = 10.0, and the smooth type of the interface.

In **Figure 4a**, the angles are presented for the uniform law of distribution, at times starting at *n* = 0.05 through the intervals *Δn* = 0.05, and the radii of the incident shock front and the interface are *Rs* = *Rb* = 0.3 cm. The curves in **Figure 4b** are for the angles corresponding to the exponential law of distribution, at the same

**Figure 4.**

*The shock front inclination angle φ vs. distance yi, at several interaction times, for the uniform (a), exponential (b), and power law (c) density distributions. The time sequence for the curves is from upper to lower.*

times as in **Figure 4a**, for *z*<sup>0</sup> = 0.225 cm and *σ* = 7, and that in **Figure 4c** are for the power law of the distribution at shorter times, *n* = 0.025 through the intervals *Δn* = 0.025. The time sequence for the curves in all graphs is from upper to lower. The minimum angle corresponding to a location next to the longitudinal symmetry axis is common for all three types of the distribution. The angle increase to 90° as *y*<sup>i</sup> > 0 is due to normal incidence at this point.

distribution quite closely follows the inclination angle change suggesting that this

For the exponentially distributed density, the Mach number

*<sup>M</sup>*2*x*ðÞ¼ *<sup>t</sup> <sup>σ</sup>*ð Þ *<sup>t</sup>* � *<sup>t</sup>*0*<sup>i</sup>* <sup>þ</sup> *<sup>t</sup><sup>λ</sup>* �3*=*<sup>5</sup> <sup>þ</sup> *<sup>ε</sup>*ð Þ *<sup>t</sup>* � *<sup>t</sup>*0*<sup>i</sup>* <sup>þ</sup> *<sup>t</sup><sup>λ</sup>* �1*=*<sup>5</sup> h i*<sup>=</sup>* <sup>20</sup>*:*<sup>043</sup>

and almost zero pressure at the stagnation point is typical.

shock modification occurred, confirms these conclusions.

the shock front surface as

**111**

On the axis (*yi* = 0), the pressure ratio stays the same as for the incident shock due to normal position of the front relative to the propagation direction (*φ* = 90°).

becomes time dependent, and the front deformation develops with an acceleration. The results for *p*21, obtained for times starting at *n* = 0.05 through *Δn* = 0.05 increments, *z*<sup>0</sup> = 0.225 cm, *σ* = 7, and *Rs* = 0.1 cm, are presented in **Figure 5b**. For the most part of the shock front, the pressure significantly drops, with the distribution closely matching the inclination angle change (**Figure 4b**). The pressure ratio levels are in good agreement with the observations of cone-shaped fronts [13] where the reduction by three to four times in the pressure in the central area of the shock front

Compared to the two previous cases, the effect of the power law density distribution is stronger (**Figure 5c**): the front becomes deformed to a larger degree and moves considerably faster (note twice as short interaction times, starting at *n* = 0.025, with *Δn* = 0.025 increments). The pressure ratio drops to the levels between 0.25 and 0, and the compression in the shock becomes weaker as it approaches the 0 density point. This is possible despite the sharp increase in the velocity because the density ahead of the wave decreases faster, similarly to the case

of a plane shock propagating through an unbounded plasma medium [40]. Since the gas pressure in front of the moving body becomes significantly lowered, the decrease in the drag follows. In the spherically symmetrical case, the drag can be numerically estimated by summing the longitudinal pressure component over the body's surface *d* ¼ ∑*<sup>i</sup>* 2*πripi* cos *α<sup>i</sup>* � *db*, where *α*<sup>i</sup> is the local angle of incidence on the body and *d*<sup>b</sup> is the summing result for the back body surface. While the total drag experienced by the body is dependent on its shape, it is still linearly proportional to the ambient pressure and will decrease accordingly. The drag reduction about 2–2.5 times recorded in experiments, at the times when the

**3.3 The parameter redistribution in the flow and generation of vorticity**

local refraction angle *γi*, and thus the velocity acquires an additional, locationdependent *y*-component. Then the velocity components for the flow behind the shock

*<sup>ω</sup><sup>i</sup>* <sup>¼</sup> *<sup>ω</sup>i*ð Þ¼ *Xi,Yi <sup>∂</sup>vx=∂<sup>y</sup>* � *<sup>∂</sup>vy=∂<sup>x</sup>* � �

As a result of refraction, the initially horizontal shock velocity vector rotates at

*vx* ¼ *vn* sin *φ* þ *V*<sup>2</sup> cosð Þ *φ* � *γ* cos *φ, vy* ¼ *vn* cos *φ* � *V*<sup>2</sup> cosð Þ *φ* � *γ* sin *φ* (13)

result in non-zero vorticity [45] that can be calculated at specific locations along

Thus, in addition to the rotation of the shock velocity vector at an angle *γ*, there is a turn for the flow velocity relative to its initial propagation direction, at the angle

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 10*T*<sup>1</sup> exp f g *X*2*i*ð Þ*t =z*<sup>0</sup>

(12)

� � q

*<sup>x</sup>*¼*Xi, <sup>y</sup>*¼*Yi* (14)

angle is the key parameter in the phenomenon.

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

### **3.2 Pressure drop and drag reduction**

In experiments, at the time when the gas pressure around the body significantly drops, recordings also show that the image of the front part of the shock wave becomes blurred or invisible and the deflection signals appear weaker and widened with time (shock "dispersion" and "disappearance") [10, 12]. As was seen above, continuous stretching of the shock front during its advancement through the plasma transforms it from being nearly normal into an inclined one. Due to the inclination, its progressively weakened intensity results in vanishing compression across the shock. As a result, the gas pressure in the flow in front of the body is lowered. If this continues, the shock wave degenerates in an ordinary pressure wave that becomes unidentifiable in the experiments.

The effect of the shock deformation on the pressure drop can be estimated through the change in the Mach number. For the homogeneous density distribution in plasma, the local inclination angle of the front φ, as defined through the tangent line, can be used to estimate the local parameter jump across the front. For this, the expressions for gas parameter variation across the normal shock front can be corrected, with the total Mach number replaced with its normal to the front component *M*2n = *M*<sup>2</sup> *sin*(φ-γ). Then the pressure jump across the refracted shock *p*<sup>21</sup> (ref) normalized to the one across the incident shock *p*<sup>21</sup> (inc):

$$p\_{21} = \frac{p\_{21}^{(ref)}}{p\_{21}^{(inc)}} = \frac{(2k/(k+1))M\_{2\mu}^2 - (k-1)/(k+1)}{(2k/(k+1))M\_1^2 - (k-1)/(k+1)}\tag{11}$$

In defining this parameter, the incident shock was taken as normal. This approximation works best for the area of the most strong interaction (near the symmetry axis area) or when the thermal spot dimensions (*Rb*) are considerably smaller compared to the incident shock radius *R*s.

In **Figure 5a**, the simulation results for pressure *p*<sup>21</sup> in the uniform density distribution case is presented, at times starting at *n* = 0.05 through *Δn* = 0.05 increments. Matching with the data in **Figures 2a** and **4a**, it is seen that the pressure

**Figure 5.**

*Pressure ratio p21 vs. coordinate yi, for the following plasma density distribution laws: (a) uniform, (b) exponential, and (c) power law. The time sequence for the curves is from upper to lower.*

times as in **Figure 4a**, for *z*<sup>0</sup> = 0.225 cm and *σ* = 7, and that in **Figure 4c** are for the power law of the distribution at shorter times, *n* = 0.025 through the intervals *Δn* = 0.025. The time sequence for the curves in all graphs is from upper to lower. The minimum angle corresponding to a location next to the longitudinal symmetry axis is common for all three types of the distribution. The angle increase to 90° as

In experiments, at the time when the gas pressure around the body significantly

drops, recordings also show that the image of the front part of the shock wave becomes blurred or invisible and the deflection signals appear weaker and widened with time (shock "dispersion" and "disappearance") [10, 12]. As was seen above, continuous stretching of the shock front during its advancement through the plasma transforms it from being nearly normal into an inclined one. Due to the inclination, its progressively weakened intensity results in vanishing compression across the shock. As a result, the gas pressure in the flow in front of the body is lowered. If this continues, the shock wave degenerates in an ordinary pressure wave

The effect of the shock deformation on the pressure drop can be estimated through the change in the Mach number. For the homogeneous density distribution in plasma, the local inclination angle of the front φ, as defined through the tangent line, can be used to estimate the local parameter jump across the front. For this, the expressions for gas parameter variation across the normal shock front can be corrected, with the total Mach number replaced with its normal to the front component *M*2n = *M*<sup>2</sup> *sin*(φ-γ). Then the pressure jump across the refracted shock *p*<sup>21</sup>

<sup>¼</sup> ð Þ <sup>2</sup>*k=*ð Þ *<sup>k</sup>* <sup>þ</sup> <sup>1</sup> *<sup>M</sup>*<sup>2</sup>

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

In defining this parameter, the incident shock was taken as normal. This approximation works best for the area of the most strong interaction (near the symmetry axis area) or when the thermal spot dimensions (*Rb*) are considerably

In **Figure 5a**, the simulation results for pressure *p*<sup>21</sup> in the uniform density distribution case is presented, at times starting at *n* = 0.05 through *Δn* = 0.05 increments. Matching with the data in **Figures 2a** and **4a**, it is seen that the pressure

*Pressure ratio p21 vs. coordinate yi, for the following plasma density distribution laws: (a) uniform, (b) exponential, and (c) power law. The time sequence for the curves is from upper to lower.*

(inc):

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

<sup>1</sup> � ð Þ *<sup>k</sup>* � <sup>1</sup> *<sup>=</sup>*ð Þ *<sup>k</sup>* <sup>þ</sup> <sup>1</sup> (11)

(ref)

*y*<sup>i</sup> > 0 is due to normal incidence at this point.

that becomes unidentifiable in the experiments.

normalized to the one across the incident shock *p*<sup>21</sup>

*<sup>p</sup>*<sup>21</sup> <sup>¼</sup> *<sup>p</sup>*ð Þ *ref* 21 *p*ð Þ *inc* 21

smaller compared to the incident shock radius *R*s.

**Figure 5.**

**110**

**3.2 Pressure drop and drag reduction**

*Aerodynamics*

distribution quite closely follows the inclination angle change suggesting that this angle is the key parameter in the phenomenon.

On the axis (*yi* = 0), the pressure ratio stays the same as for the incident shock due to normal position of the front relative to the propagation direction (*φ* = 90°).

For the exponentially distributed density, the Mach number

$$M\_{2\mathbf{x}}(t) = \left[\sigma(t - t\_{0i} + t\_k)^{-3/5} + \varepsilon(t - t\_{0i} + t\_k)^{-1/5}\right] / \left(20.043\sqrt{10T\_1 \exp\left\{X\_{2i}(t)/z\_0\right\}}\right) \tag{12}$$

becomes time dependent, and the front deformation develops with an acceleration. The results for *p*21, obtained for times starting at *n* = 0.05 through *Δn* = 0.05 increments, *z*<sup>0</sup> = 0.225 cm, *σ* = 7, and *Rs* = 0.1 cm, are presented in **Figure 5b**. For the most part of the shock front, the pressure significantly drops, with the distribution closely matching the inclination angle change (**Figure 4b**). The pressure ratio levels are in good agreement with the observations of cone-shaped fronts [13] where the reduction by three to four times in the pressure in the central area of the shock front and almost zero pressure at the stagnation point is typical.

Compared to the two previous cases, the effect of the power law density distribution is stronger (**Figure 5c**): the front becomes deformed to a larger degree and moves considerably faster (note twice as short interaction times, starting at *n* = 0.025, with *Δn* = 0.025 increments). The pressure ratio drops to the levels between 0.25 and 0, and the compression in the shock becomes weaker as it approaches the 0 density point. This is possible despite the sharp increase in the velocity because the density ahead of the wave decreases faster, similarly to the case of a plane shock propagating through an unbounded plasma medium [40].

Since the gas pressure in front of the moving body becomes significantly lowered, the decrease in the drag follows. In the spherically symmetrical case, the drag can be numerically estimated by summing the longitudinal pressure component over the body's surface *d* ¼ ∑*<sup>i</sup>* 2*πripi* cos *α<sup>i</sup>* � *db*, where *α*<sup>i</sup> is the local angle of incidence on the body and *d*<sup>b</sup> is the summing result for the back body surface. While the total drag experienced by the body is dependent on its shape, it is still linearly proportional to the ambient pressure and will decrease accordingly. The drag reduction about 2–2.5 times recorded in experiments, at the times when the shock modification occurred, confirms these conclusions.

#### **3.3 The parameter redistribution in the flow and generation of vorticity**

As a result of refraction, the initially horizontal shock velocity vector rotates at local refraction angle *γi*, and thus the velocity acquires an additional, locationdependent *y*-component. Then the velocity components for the flow behind the shock

$$\nu\_{\mathbf{x}} = \nu\_{\pi} \sin \varphi + V\_2 \cos \left(\varphi - \chi\right) \cos \varphi,\\ \nu\_{\mathbf{y}} = \nu\_{\pi} \cos \varphi - V\_2 \cos \left(\varphi - \chi\right) \sin \varphi \tag{13}$$

result in non-zero vorticity [45] that can be calculated at specific locations along the shock front surface as

$$\alpha\_{i} = a \imath\_{i}(X\_{i,}Y\_{i}) = \left(\partial v\_{\mathbf{x}}/\partial \jmath - \partial v\_{\mathbf{y}}/\partial \mathbf{x}\right)\_{\mathbf{x} = X\_{i,}\jmath = Y\_{i}}\tag{14}$$

Thus, in addition to the rotation of the shock velocity vector at an angle *γ*, there is a turn for the flow velocity relative to its initial propagation direction, at the angle *g* = *tan*�<sup>1</sup> (*vy*/*vx*). The shock wave and the flow behind it will diverge at a relative angle *ξ* = *tan*�<sup>1</sup> (*vy*/*vx*)�*γ* (double refraction) giving rise to a non-zero circulation in the hot gas volume.

integrated over the whole plasma volume becomes considerably higher compared to the uniform density case because of the contribution from the regions located further from the axis. The same strong correlation between shock deformation, vortices of similar size, rotational direction, and topology with pressure/wave drag drop evolving in the same sequence and within the same time frames was con-

The existence of phenomenological connection between the shock's and the interface stability makes the chain of the transformations in the flow to continue. Since the flow instabilities take place upstream from the interface, the perturbations to the flow parameters propagating downstream, toward the interface, will disturb it. The overall pressure drop behind the refracted shock continuously mounting with time is responsible for the sucking effect resulting in the large-scale interface perturbation, moving it closer to the shock. The positive and essentially nonlinear dynamic in the pressure perturbation evolution [17] will support amplification of this global perturbation to the interface and thus determines the pattern in the interface instability structure. With increasing distortion of the interface, Kelvin-Helmholtz (KH) shearing instability may start to contribute resulting in the characteristic mushroom shapes of the interface perturbations [47]. The initial instability pattern associated with the global pressure drop behind the refracted shock will be of a larger scale, and the KH instability turning on at later stages would

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

2

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

, |*χ*| > 0} is the only requirement to

firmed in [46].

**3.4 The interface stability problem**

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

finally determine the smaller characteristic structure.

**4. Conclusion and controlling the drag**

marginal state condition {*T*2/*T*<sup>1</sup> > (*M*1/*M*2)

**113**

Numerical results for dimensionless vorticity *ω* ¼ *ωi=*ð Þ *V*1*=Rb* developing in the flow with the homogeneous plasma parameter distribution vs. the coordinate *Yi* are presented in **Figure 6a**, for times between *n* = 0.18 and 0.42 with *Δn* = 0.03 increments, and the last curve corresponds to *n* = 0.44 chosen for better resolution on the graph. The time sequence for the curves is in the up-direction. Only the upper half of the picture is shown; thus, the results on the graph correspond to a vortex sheet with rotations in both halves of the picture in opposite directions (in cylindrical geometry) or a toroidal vortex ring if considered in the spherical geometry.

Due to a small size of the structure, the *Y*-coordinate was scaled with the factor of 10<sup>2</sup> , and thus the picture is greatly zoomed in on the narrow region next to the symmetry axis. The spike in the vorticity intensity observed in close proximity to the symmetry axis exactly corresponds to the location where abrupt change in the shock front structure occurs (**Figures 2a** and **4a**). This trend is pointing at the fact that the local rate of change of the front inclination angle *φ* is the key factor in the origin of the vorticity. The nonlinear growth of the vortex intensity (**Figure 6**) demonstrates the possibility of a strong positive dynamics in its development.

The specific effect of the exponential density distribution has been studied for the interaction times between *n* = 0.05 and 0.40 through *Δn* = 0.05 increments, and the last two curves correspond to *n* = 0.43 and 0.45. (**Figure 6b**). Due to practically no areas on the shock front where its shape would change sharply enough (**Figures 2b** and **4b**), except a very narrow region near the symmetry axis, significant vorticity develops mostly in this region corresponding to the very tips of the fronts. The maximum vorticity intensity is about two orders of magnitude less compared to the levels found in the uniform density distribution case. Considerably weaker vorticity and a very small size of the structure found here can probably explain why sometimes the vortex system "does not develop" in experiments even though noticeable changes in the shock structure and pressure/drag reduction are present.

In the presence of the energy accumulation effect (power law density distribution), with more stretched fronts and softer distribution of the front inclination angle *φ*, the vorticity develops slower (**Figure 6c**), but still, its intensity steadily grows with time at all locations on the front. The curves on the graph correspond to times *n* = 0.025–0.250 with 0.025 increments and the distance *a* = 3.0*R*b. The vorticity originates further from the symmetry axis confirming a larger size of the vortex structure. As in other density distribution cases, the maxima of vorticity (centers) shift toward each other as the shock advances through the volume, following the same trend for the sharpest bending on the front and the shock and flow rotation angles to move closer to the axis [45]. The total vorticity intensity

**Figure 6.**

*Dimensionless vorticity ϖ vs. Yi, for the uniform (a), exponential (b), and power law (c) density distributions. The time sequence is from lower curve to upper.*

integrated over the whole plasma volume becomes considerably higher compared to the uniform density case because of the contribution from the regions located further from the axis. The same strong correlation between shock deformation, vortices of similar size, rotational direction, and topology with pressure/wave drag drop evolving in the same sequence and within the same time frames was confirmed in [46].
