*Experimental Analysis of Waverider Lift-to-Drag Ratio Measurements in Rarefied… DOI: http://dx.doi.org/10.5772/intechopen.100328*

simulating the waverider with an incidence of 10° with conditions similar to those obtained experimentally with a Mach 4–8 Pa flow.

To achieve similarity between numerical and experimental curves, the accomodation coefficients which correspond to the specular reflection fraction and the absorbed fraction at surface are adapted. The sum of both fraction need to be equal to 1 for a consistent simulation. These coefficients are summarized in **Table 8** for each nozzle.


#### **Table 8.**

*Reflection and absorption accomodation parameters.*

**Figure 17.** *Comparison of experimental and DS3V results for Mach 2–8 Pa and Mach 4–8 Pa.*

**Figure 18.** *Comparison of experimental and DS3V results for Mach 4–2.6 Pa and Mach 4–71 Pa.*

For each of the experimental conditions i. e. nozles, the simulation of the flow around the wavereider was performed for the same incidences as those tested experimentally. Comparison between experimental and numerical results are presented on **Figures 17** and **18**. One can observe à good correlation between the numerical and the experimental results. Nevertheless, there are some differences in particular for the case Mach 2–8 Pa and Mach 4–2 Pa wich are the more rarefied conditions and presents small force values. The coefficient of the specular reflexion parameter seems to decrease with the rarefied parameter (Kn, *<sup>ψ</sup>* <sup>¼</sup> *<sup>M</sup>*ffiffiffiffiffi *Re* <sup>p</sup> or Re2). For low altitudes, typically the condition Mach 4–71 Pa, the reflection parameter tends to 1.

## **6. Discussion**

#### **6.1 Influence of rarefaction effects on fligth performance**

The optimal flight conditions are given by the maximum values of the drag-lift ratio. These values will depend on the effects of rarefaction as summarized in

*Experimental Analysis of Waverider Lift-to-Drag Ratio Measurements in Rarefied… DOI: http://dx.doi.org/10.5772/intechopen.100328*

**Table 9** in which are reported the maximum values of the ratio drag-lift, the angle of incidence corresponding for the various conditions of experiment studied in this work. Similarly, for each case, the values of the accommodation coefficient optimized in the framework of the DSMC numerical simulations are reported.

**Figure 19** shows the maximum value of the L/D ratio and the corresponding angle as a function of the Knudsen number. It can be seen that the variation of the optimum angle of incidence increases linearly with the number of Knudsen, while the value of the L/D ratio decreases linearly with the logarithm of the number of Knudsen and this without distinction of the Mach number.

The study of the evolution of the ratio L/D with the rarefaction parameter, shows the influence of the Mach number as illustrated in **Figure 20**. Indeed, both the maximum value of the ratio L/D and the corresponding angle of incidence follow a linear variation for the experimental conditions at Mach 4, while the values for Mach 2 do not follow this trend. Isobar results shows that lower Mach number needs higher angle of incidence to optimize de ratio L/D.

As shown in **Figure 21**, the reflection accommodation coefficient used for the DSMC numerical simulations also follows a linear trend with the Knudsen number and increases towards 1 with decreasing Knudsen number. The same conclusions are made with the DSMC parameter (reflection parameter). However, when the comparison is made with the rarefaction parameter, only the results obtained with the same number of mach follow a linear trend.

In conclusion, for each parameter, the Knudsen number regroup all nozzles on a same trend (linear or logarithmic. It is not the case for the rarefaction parameter but a study isomach seems possible. To valid these conclusions, studies in hypersonic Mach 20 will be realized.

### **6.2 Estimation of the skin friction contribution**

'The different effects of Mach number and geometry optimization on the aerodynamic performance of waveriders have been demonstrated. However, the effects of viscous drag, which have been little studied, can also have an impact on an elementary waverider geometry. For comparison purposes, and to complete our results, we consider the results presented by Rasmussen, carried out with a Mach number of 4 and with a geometry similar to the one used for the present study [48].

The ratio L/D can be expressed as:

$$L/D = \frac{L}{D\_0 + D\_f + D\_b} \tag{5}$$

where L is the lift force, *D*<sup>0</sup> is the invicid wave drag, *Df* is the friction drag and *Db* is the base drag. In the following we assume that the base pressure is equal to the freestream pressure so that the base drag can be negleted [29]. The friction drag can be written as follows:


$$D\_f = q \, \mathbb{S}\_w \, \mathbf{C}\_f \tag{6}$$

**Table 9.** *Reflection and absorption parameters.*

#### **Figure 19.**

*Value of maximum L/D ratio and value of the corresponding angle of incidence as a function of the Knudsen number.*

where q is the dynamic pressure, *Sw* is the wetted area of the waverider and *Cf* is the friction coefficient.

Friction effets are related to the development of the boundary layer, that can be influenced by many factors. The friction coefficient may depend of such factors as the Mach and Reynolds numbers, the wall temperature, and in continuum regime others physical properties such turbulence and flow

*Experimental Analysis of Waverider Lift-to-Drag Ratio Measurements in Rarefied… DOI: http://dx.doi.org/10.5772/intechopen.100328*

#### **Figure 20.**

*Value of maximum L/D ratio and value of the corresponding angle of incidence as a function of rarefaction parameter value.*

separation. This means that the definition of the friction coefficient can be a complex function that strongly depends on the flow properties and the objet model geometrie. For the current study, the waverider geometrie can be

**Figure 21.**

*Evolution of the DS3V reflection parameter as a function of the Knudsen number and the rarefaction parameter.*

assimilated to a sharp plate, without the formation of a bow shock at the leading edge. Under this assumption, the skin friction can be defined by the following equation:

*Experimental Analysis of Waverider Lift-to-Drag Ratio Measurements in Rarefied… DOI: http://dx.doi.org/10.5772/intechopen.100328*

$$C\_f = \frac{1.328f(M, T\_w/T\_\infty, Re)}{\sqrt{\frac{\rho\_\infty V\_\infty l}{\mu\_\infty}}} \tag{7}$$

Where *f M*ð Þ , *Tw=T*∞, *Re* is a function depending on Mach number, wall temperature and Reynolds number. For simplicity this function value can be assumed to be equal to 1 for *Tw=T*<sup>∞</sup> ¼ 1, wich is our case. For the present study, the subscribe inf corresponds to the values of the free stream noted with the subscribe 1 in **Tables 1**–**4** presenting our experimental conditions.

**Table 10** summarizes values of the friction coefficient for the experimental conditions presented by Rasmussen and those of the present experimental work as well as the maximum value of L/D ratio and the corresponding incidence angle of


#### **Table 10.**

*Sumarize of the parameters for the optimum L/D ratio corresponding to the experimental conditions of this work and those predicted by Rasmussen.*

**Figure 22.** *Evolution of the maximum value of L/D as a function of the volume ratio.*

the waverider. The value *V*2*=*<sup>3</sup> *=*ð Þ *S cos* Θ , corresponds to the volume ratio wich is a function of the incidence angle Θ.

**Figure 22** plots the Rasmussen data for the Mach 4 case and our experimental resuls which presents the linear evolution of the maximum L/D value with the

**Figure 23.** *Evolution of the volume angle as a function of the friction coefficient.*

**Figure 24.** *Evolution of the maximum value of L/D as a function of the friction coefficient.*

*Experimental Analysis of Waverider Lift-to-Drag Ratio Measurements in Rarefied… DOI: http://dx.doi.org/10.5772/intechopen.100328*

volume ratio. These results confirm the increase of the L/D maximum value as approaching continuous regime, the Rasmussen conditions presents lower friction coefficient values compared to those of our experimental conditions.

Moreover the evolution of the volume ratio with the friction coefficient presented on **Figure 23** seems to show that there is a limite value for volume ratio as rarefaction effects increases, showing that there is also a limite for the angle of attack to optimize the L/D ratio at high altitudes for a given geometrie. As presented in **Figure 24** the maximum L/D value decreases drastically with the increase of the friction coefficient which reflects the increase of the viscous effects.
