**2. Model of the interaction**

It will be assumed here that a thermal spot (plasma region) has been created via a discharge at a distance in front of a stationary solid body and the spot's continuous expanding ceases by the time of interaction. This could be a case of a relatively slow evolving plasma bubble or when thermal energy is deposited far enough from the body, so the system had enough time to achieve a thermal equilibrium state. The plasma region is allowed to move with the hypersonic flow toward the bow shock formed in front of the body. The interaction starts when the plasma region arrives at the shock location, and that is when the time *t* in the model relations starts to be counted. For blunt bodies that are common in the experiments, a spherical shock wave and interface geometry can be an adequate approximation.

In **Figure 1**, the spherical interface (dashed curve) separates the hot plasma (medium 2) and the surrounding cold gas (medium 1). Both media will be treated as ideal gases with initially equal pressures on both sides of the interface.

The cold gas temperature *T*<sup>1</sup> will be always distributed homogeneously, and *T*<sup>2</sup> is the plasma temperature right behind the interface with varied parameter distribution along the shock motion direction. The gas temperature can change across the interface abruptly (stepwise) or smoothly (distributed over a distance) [34], and *T*<sup>2</sup> > *T*1. The radii *Rb* and *Rs* are for the plasma boundary and the shock front correspondingly, and other parameters are shown in the figure. In the reference frame stationary for the plasma region, the shock wave is incident on the interface from left to right, center to center, with constant horizontal velocity *V*1. If the hotter medium (plasma) is uniform, a portion of the shock front crossing the interface accelerates to velocity *V*<sup>2</sup> in a steplike manner, and after this its velocity remains constant. Due to the acceleration, this portion advances faster compared to the

#### **Figure 1.**

*Shock-plasma interaction diagram in the vertical plane of symmetry. As the initially spherical shock progresses through the spherical interface (dashed curve), its shape gradually deforms (green curve). Only the upper part of the diagram is shown.*

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

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

It will be assumed here that a thermal spot (plasma region) has been created via a discharge at a distance in front of a stationary solid body and the spot's continuous expanding ceases by the time of interaction. This could be a case of a relatively slow evolving plasma bubble or when thermal energy is deposited far enough from the body, so the system had enough time to achieve a thermal equilibrium state. The plasma region is allowed to move with the hypersonic flow toward the bow shock formed in front of the body. The interaction starts when the plasma region arrives at the shock location, and that is when the time *t* in the model relations starts to be counted. For blunt bodies that are common in the experiments, a spherical shock

In **Figure 1**, the spherical interface (dashed curve) separates the hot plasma (medium 2) and the surrounding cold gas (medium 1). Both media will be treated as

The cold gas temperature *T*<sup>1</sup> will be always distributed homogeneously, and *T*<sup>2</sup> is the plasma temperature right behind the interface with varied parameter distribution along the shock motion direction. The gas temperature can change across the interface abruptly (stepwise) or smoothly (distributed over a distance) [34], and *T*<sup>2</sup> > *T*1. The radii *Rb* and *Rs* are for the plasma boundary and the shock front correspondingly, and other parameters are shown in the figure. In the reference frame stationary for the plasma region, the shock wave is incident on the interface from left to right, center to center, with constant horizontal velocity *V*1. If the hotter medium (plasma) is uniform, a portion of the shock front crossing the interface accelerates to velocity *V*<sup>2</sup> in a steplike manner, and after this its velocity remains constant. Due to the acceleration, this portion advances faster compared to the

*Shock-plasma interaction diagram in the vertical plane of symmetry. As the initially spherical shock progresses through the spherical interface (dashed curve), its shape gradually deforms (green curve). Only the upper part*

wave and interface geometry can be an adequate approximation.

ideal gases with initially equal pressures on both sides of the interface.

adequate timing order.

*Aerodynamics*

**Figure 1.**

**104**

*of the diagram is shown.*

**2. Model of the interaction**

reminder that is still propagating in the colder media thus resulting in the continuously increasing front stretching toward the hotter medium [18].

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 dynamics.

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

$$\overline{X\_i} = (\overline{\nu}\cos\chi - \mathbf{1})(n - (\overline{\mathbf{x\_i}} + \overline{\mathbf{x\_b}})) + \Delta\overline{\mathbf{x\_i}} \\ \overline{Y\_i} = \overline{\gamma\_i} - \overline{\nu}\sin\chi \bullet (n - (\overline{\mathbf{x\_i}} + \overline{\mathbf{x\_b}})) \tag{1}$$

Here *n* = *t*/*τ* is the dimensionless time, 0 < *n* < 2, *α* is the local incidence angle at the interaction point (**Figure 1**), *xb* ¼ 1 � cos *α*, Δ*x* ¼ ð Þ *Rs=Rb* ð Þ cos *β* � *η* , the parameter

$$\eta = \frac{2(R\_\sharp + R\_b)(R\_\sharp - nR\_b) + n^2 R\_b^2}{2R\_\sharp[(R\_\sharp + R\_b) - nR\_b]} \tag{2}$$

and the bar over the variable means its dimensionless equivalent. The dimensionless shock velocity

$$
\overline{v} = V\_2 / V\_1 = \sqrt{\overline{T} \left(\overline{M}\right)^2 \cos^2 a + \sin^2 a} \tag{3}
$$

and the refraction angle *<sup>γ</sup>* <sup>¼</sup> *<sup>α</sup>* � tan �<sup>1</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *TM* tan *<sup>α</sup>* <sup>p</sup> <sup>Þ</sup> � are determined by the 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 reflections off the interface.

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> changes across the interface (a "sharp" interface)

$$\frac{1}{M\_{1n}(k-1)}\left\{\left[2kM\_{1n}^2 - (k-1)\right]\left[(k-1)M\_{1n}^2 + 2\right]\right\}^{\frac{1}{2}} \times \left\{\left[\frac{2kM\_{2n}^2 - (k-1)}{2kM\_{1n}^2 - (k-1)}\right]^{\frac{k-1}{2n}} - 1\right\}.$$

$$= M\_{1n}\left(1 - \frac{1}{M\_{1n}^2}\right) - M\_{2n}\left(1 - \frac{1}{M\_{2n}^2}\right)\left(\frac{T\_2}{T\_1}\right)^{\frac{1}{2}}$$

and the normal and tangential components of the Mach numbers are related as *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>* ffiffiffiffiffiffiffiffiffiffiffiffiffi *T*1*=T*<sup>2</sup> p . The "sharp" interface

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 thickness change slower than 1/*x*<sup>2</sup> , there are no losses associated with the shock 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> *β* , where *<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> *<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>* <sup>¼</sup> <sup>53</sup>*:*58 is the numerical constant calculated from the fit coefficients *α*<sup>n</sup> and *α*<sup>m</sup> in the equation:

$$\frac{T\_2}{T\_1} = \sum\_{n=0}^{10} \alpha\_n (\frac{M\_1}{M\_2})^n + \sum\_{\text{w}} \alpha\_\text{w} (\frac{M\_1}{M\_2})^n \tag{5}$$

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.

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

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

*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).*

**3.1 Shock front distortion**

**Figure 2.**

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to the smooth, lower, and sharp type of the interface.

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

fronts; however, the smooth boundary results in stronger effect.

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 another limit of a sharp boundary [34].
