**5. Results**

#### **5.1 Measurements of lift and drag forces**

The lift-to-drag ratio is of most important consideration when designing spacecraft and others space vehicles to reach higher lift-to-drag ratios. In addition to their shape, the incidence angle plays a significant role over the lift-to-drag ratio. The incidence angle is defined as the angle at which the leading edge of the vehicle is set in relation to the flowing air, and has a direct influence on how far the vehicle will glide for a given altitude. At high Mach numbers or high altitudes because of viscous effects, the skin friction will increase the viscous drag, decreasing L/D ratios.

Drag and lift forces have been measured for each one of the nozzles and for several angle of attack covering negative and positive angles. The objective is to study the behavior of the waverider in gliding phase, maximizing the Lift-to-Drag ratio (positive angles) and in the landing phase, reaching a target as quickly as possible (negative angles). For nozzles N1 and N2, the attack angles range between �25° and 25° while for N3 and N4 angles are ranged between �50° and 25°. It was not possible to achieve angles below �25° with nozzles N1 and N2 because the vacuum pumps are not powerful enough to achieve the required pressure *P*1. **Figures 11** and **12** plots the lift and drag forces measured for each operating

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

**Figure 11.** *Lift and drag forces: Mach 2–8 Pa and Mach 4–8 Pa.*

condition. Whatever the experimental condition, the lift force behaves linearly with the angle of attack Θ, while the drag force follows a quadratic behavior at order 2. Lift forces can be described with an equation at first order as:

$$L\sharp\sharp = a\Theta - b \tag{3}$$

The fitting coefficients a and b are sumarized on **Table 6**.

Concerning the drag forces, an equation in second order can describe their evolution with the angle of attack, as follows:

$$Drag = c(\Theta)^2 + d\Theta + \varepsilon \tag{4}$$

**Table 7** presents the corresponding coefficients c,d and e.

The y-intercept is negative due to the shape of the waverider, the asymmetry between the two surfaces creates a negative lift at 0° angle of attack. Negative lift values means that the force acts in the same direction as the gravity force.

**Figure 12.** *Lift and drag forces: Mach 4–2 Pa and Mach 4–71 Pa.*


#### **Table 6.**

*Fitting coefficients for lift forces.*

Comparisons of lift and drag forces show that those measured at the operating condition Mach 4–71 Pa are the highest and on the contrary, the lowest forces are obtained for Mach 2–8 Pa and Mach 4–2 Pa which are close in terms of values and shapes. One of the major parameters to design a space craft is the lift-to-drag ratio value. This value may determine the gliding ability of the waverider. The L/D ratio is a dimensionless number that can be determined directly from the aerodynamic


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

**Table 7.**

*Fitting coefficients for drag forces.*

**Figure 13.** *Comparison of L/D ratio for all experimental conditions.*

forces but also from the lift and drag coefficients. It does not follow a linear or quadratic function but has a more complex form. **Figure 13** presents the evolution of the ratio lift-to-drag with the angle of attack Θ for the overall experimental conditions. As can be seen, the L/D ratio will increase to reach a maximum and then decrease and converge to 0.5–0.6. This evolution occurs for both positive and negative angles and presents a quasi symmetry around the 0° angle. This symmetry is possible because the thickness of the waverider is small, 6.6 mm, compared to its length, 100 mm, so that the waverider can be compared to a flat beveled plate. The waverider can be rotated 180° lengthwise and the sliding performance will be similar or even better if the fitting curves are realistic. Negative angle behavior shows that the optimal landing angle is between 5° and 15°, depending on the ambient operating pressure of the nozzle used.

In the following, only the results of positive angles that illustrate the ability of the waverider to glide and maximize the travel distance before landing will be analyzed. The analysis of the results shows that the angle for which the value of the L/D ratio is maximum changes as a function of the operating conditions of the nozzle i.e. as a function of the static pressure *P*<sup>1</sup> and the Mach number. By relating the pressure *P*<sup>1</sup> of the flow to the atmospheric pressure for a given flight altitude, it is possible to associate to each nozzle an equivalent flight altitude, 80 km for Mach 4–2 Pa and Mach 2–8 Pa, 70 km for Mach 4–8 Pa and 50 km for Mach 4–71 Pa. This means that the angle of the maximum value of the L/D ratio decreases with altitude while the value itself increases. This is due to the fact that when the pressure decreases and the speed remains constant, the waverider must adapt its incidence in order to maintain an optimal L/D ratio and travel as far as possible. The influence of pressure in the isomach condition is shown in **Figure 14**. The values of the L/D ratio vary from 2.8 and 1.2 respectively for angles varying from 5° and 15°.

This observation shows that the waverider needs a high angle of attack at the beginning of its flight at high altitude, (low pressure) and must decrease its incidence to have an optimal L/D ratio when it decreases its flight attitude. A non-optimized L/D ratio will decrease the range of its flight. For example, if the waverider is launched at 80 km altitude with an incidence of 10° the L/D ratio is not optimal and for a flight altitude between 80 km and 50 km the waverider will be able to travel 33 km. Indeed between 80 km and 70 km, by referring to the curve for the Mach 2-8 Pa nozzle the value of L/D is of 0.89 and between 70 km and 50 km the M4 8Pa curve gives a L/D value of 1.21. However, by varying the angle of incidence the optimal L/D ratio between 70 km and 80 km is 1.25 and 1.64 for an

**Figure 14.** *Pressure influence on L/D ratio for isomach experimental conditions(Mach 4).*

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

altitude ranged between 70 km and 50 km. With these data the new maximum flight distance that the waverider will be able to cover is 43 km. There is a variation of 10 km, which is not much, but during the last 50 km the L/D values could increase significantly due to the increase of the atmospheric pressure. The variation of the distance flown between the optimized and non-optimized flight can be greater, in the order of 50 km to 100 km. During the re-entry into the atmosphere, the speed of the waverider will vary and increase while decreasing the altitude. In this work, Mach 2 and Mach 4 nozzles operating with a static pressure of 8 Pa were used to observe the influence of the speed on the aerodynamic performance of the waverider. The L/D ratio is presented in **Figure 15**. As observed, the L/D ratio increases with Mach number as well as its optimal angle of attack to a lesser extent. For the Ma 2 8 Pa curve, the maximum is reached at about 25° while for the Ma 4 8 Pa curve the optimal angle is 15°. For the same pressure, when the speed increases, the value of the L/D ratio also increases. The maximum value of the L/D ratio can reach higher values for smaller angles, so to optimize the L/D ratio of waveriders, their speed must be increased when they fly at high altitude.

A first conclusion can be drawn from these experimental results. To optimize the performance of the waverider, the L/D ratio must be maintained at its optimal value by increasing the Mach number at high altitude and by increasing the angle of incidence by decreasing the flight altitude, in order to glide over the greatest possible distance. For the test model of this work the incidence will be between 20° and 25° at 80 km and will gradually decrease to 8° at 50 km. In terms of performance, the waverider will travel 1.2 km for every vertical kilometer lost at 80 km altitude and at 50 km, the waverider will travel more than 3 km. In fact, a waverider geometry is optimized for a specific flying speed and pressure. The geometry of this waverider is optimized to fligh at Mach 10 and altitudes ranging between 40 km and 50 km as Rolim presented in his paper, giving higest L/D values.

#### **5.2 DSMC simulations and comparison with experimental results**

Numerical simulations have been carried out using the direct Monte Carlo method, a stochastic method, solving the flow fields in transitional regime i.e. from continuum low density regime to free molecule regime. For this purpose, the

**Figure 15.** *Mach number influence on L/D ratio for isobar experimental conditions (P*1*=8 Pa).*

well-known DS3V software developed by Prof. Graeme A. Bird have been used [43, 44]. The main objective is to determine the accommodation parameters between the gaz flow and the waverider surface for our experimental conditions and analyze how they are related to rarefaction effects.

The comparison criterion between the numerical and experimental results are the drag and lift forces with the purpose to obtain numerical results as close as possible to those of the experiments.

We have selected the DS3V program by many reasons. The first one is the avaibility as it is free to download from the Bird website (http://www.gab.com.au). This program was designed to be able to run on personal computers with Microsoft windows. Finally, the use of DS3V is quite simple for a beginner as it uses a set of menus which creates files containing all the simulation information for post processing. However, one disadvantages associated with the DS3V codes is the creation/importation of the 3D geometry. Indeed, DS3V does not have an integrated geometry package and does not accept basic 3D CAD files. The geometry file have to be a Raw Triangle files with a series of x, y and z coordinates which form a 'triangle' and when put together they form a mesh.

To design the waverider, we have used the software Rhinoceros 7, because it can create triangular mesh with coordinates associated to each triangle apex. It is possible to select gas-surface interaction models among two models: the diffuse model or the CLL model [45–47]. The diffuse reflection model was adopted for this study. However, the reflection and absorption parameters characterizing the interaction with the waverider surface were adapted to the experimental conditions to be simulated. As the DSMC simulation is a probabilistic calculation, we considered that a flow time of 1 ms was sufficient to obtain valid results, and in particular this time is sufficient for the aerodynamic forces to be stabilized. Simulation were realized allowing 200 Mb memory. **Figure 16** shows the result obtained with the DS3V code
