**4. The analysis of shadowgraphs and calculation of helium density field at a hypersonic flow over a cone**

As it is known from the theory of the Schlieren technique [27], relative change of light intensity Δ*I=I*<sup>0</sup> in geometrical optics approach in case of small deflection angles of the light *ε* is directly proportional to the magnitude of the deflection angle of the light rays in the optical inhomogeneity and can be defined by a ratio

$$\frac{\Delta I}{I\_0} = \frac{\epsilon f}{d}; \quad \Delta I = I - I\_0,\tag{6}$$

where *d* = 0:16 mm is a width of a slit, *f* = 1 000 mm is a focal length of optical system, *I*<sup>0</sup> is light intensity in the absence of an optical inhomogeneity, *I* is light intensity in the presence of an optical inhomogeneity. Since the streamlined body has an axial symmetry, when processing shadow pictures the assumption of

axisymmetric structure of the flow was used. The coordinate system is chosen in such a way that the x-axis is perpendicular to the edge of the imaging knife, the zaxis is directed along the probing light beam of the shadow device, and the y-axis is along the symmetry axis of the cone and the flowing stream. The origin is set at the apex of the cone. In this coordinate system, the Abel equation can be written in the form (where R is the radius of the inhomogeneity) [21, 27, 28]:

$$\varepsilon(\mathbf{x}) = 2 \int\_{\mathbf{x}}^{\mathbb{R}} \frac{d \ln n(r)}{dr} \, \frac{\mathbf{x} dr}{\sqrt{r^2 - \mathbf{x}^2}}. \tag{7}$$

For calculations we use Abel equation transformation where *d* ln *n=dr* is expressed explicitly:

$$\frac{d\ln n}{dr} = -\frac{1}{\pi} \frac{d}{dr} \int\_{r}^{\mathbb{R}} \frac{e(\mathbf{x}) d\mathbf{x}}{\sqrt{r^2 - \mathbf{x}^2}}. \tag{8}$$

After repeated integration [21, 27, 28] the distribution of refraction index is received:

$$\ln \frac{n(r)}{n\_0} \approx \frac{n(r) - n\_0}{n\_0} = \frac{\Delta n}{n\_0} = \frac{1}{\pi} \int\_{r}^{\mathbb{R}} \frac{\varepsilon(\mathbf{x}) d\mathbf{x}}{\sqrt{\mathbf{x}^2 - r^2}},\tag{9}$$

where *n*<sup>0</sup> is known in advance value of refraction index on the line of integration, for example in the region of undisturbed flow *n*<sup>0</sup> ¼ *n R*ð Þffi 1.

The flow area was divided into annular zones, the number of which was selected in accordance with the resolution of the video camera matrix. The deflection angle within the annular zone *ri* ½ � ;*ri*þ<sup>1</sup> is assumed to be constant. Then the expression (9) can be written as a sum of elementary integrals:

$$\frac{\Delta n\left(r\_{j}\right)}{n\_{0}} = \frac{1}{\pi} \sum\_{i=j}^{N-1} \varepsilon(r\_{i}) \int\_{r\_{i}}^{r\_{i+1}} \frac{dr}{\sqrt{r^{2} - r\_{j}^{2}}}.\tag{10}$$

The distribution of the refractive index in the flow is calculated as a result of independent integration at each section separately. With such processing of shadow, characteristic distortions arise in the distribution of the refractive index, which are associated with a high level of noise in the experimentally measured light intensity along the y axis. To eliminate random noise, a preliminary smoothing of the luminous intensity function along the y-axis was carried out using the least squares method [29–32]:

$$\int\_{\mathcal{Y}\_{\min}}^{\mathcal{Y}\_{\max}} I''(y^2) dy \to \min. \tag{11}$$

At the same time the smoothed function values should differ from the experimental function values not more than the standard deviation *δ* of the probe light intensity *I*<sup>0</sup> noise. Calculation of helium density *ρ* was carried out with use of Gladstone-Dale Equation [21, 28].

$$n - 1 = K\_{H \epsilon \rho},\tag{12}$$

where *KHe* <sup>¼</sup> <sup>0</sup>*:*19607 cm*=*g<sup>3</sup> is Gladstone-Dale constant for helium. The value for the given gas and given wavelength is considered to be a constant in the wide *Investigation of Hypersonic Conic Flows Generated by Magnetoplasma Light-Gas Gun Equipped… DOI: http://dx.doi.org/10.5772/intechopen.99457*

**Figure 5.** *Results of density field calculations (hypersonic helium flow past cone with the half-angle τ*<sup>1</sup> ¼ 3°*).*

**Figure 6.** *Results of density field calculations (hypersonic helium flow past cone with the half-angle τ*<sup>2</sup> ¼ 12°*)*

pressure range. Its value was calculated for gas under normal conditions with use of tabular values of density and refraction index [21]. As far as the initial values *n*<sup>0</sup> and *ρ*<sup>0</sup> are arbitrarily chosen values, we can calculate Δ*ρ* from the following equation:

$$
\Delta n = K\_{He} \Delta \rho. \tag{13}
$$

Calculation results for helium density fields at a hypersonic flow over the cones with a half-angle *τ*<sup>1</sup> ¼ 3° and *τ*<sup>2</sup> ¼ 12° are shown in **Figures 5** and **6**.
