#### 5.1. Cycle Time ${t}_{cyc}$

The cycle time,

${t}_{cyc}$, plays a significant role in high-speed dynamic soaring. This relates to cases of very small

${t}_{cyc}$ values which cause difficulties in flying and controlling the vehicle. Small

${t}_{cyc}$ values are a characteristic feature of high-speed dynamic soaring [

5,

6,

17]. There are

${t}_{cyc}$ values of such small a magnitude in high-speed dynamic soaring that they are regarded critical. This is because cycle times of around 2 s, or smaller, are considered as difficult to fly in practice.

The cycle time is basically given by the relation

For expanding this expression, reference is made to the load factor relation in curved flight

Using this relation and accounting for Equation (21b), the cycle time dependent on the average speed

${\overline{V}}_{inert}$ and the lift coefficient

${C}_{L}$ can be expressed as

The optimal lift coefficient for achieving the maximum speed

${V}_{inert,max}$ is the one associated with

${({C}_{L}/{C}_{D})}_{max}$. Denoting this lift coefficient by

${C}_{L}^{*}$, and referring to Equations (24) and (26), the following result for the optimal cycle time dependent on the maximum speed

${V}_{inert,max}$ is obtained, yielding

The relationship between

${t}_{cyc}$ and

${V}_{inert,max}$ is graphically addressed in

Figure 13. There is a decrease of

${t}_{cyc}$ with

${V}_{inert,max}$ until a value of about

${V}_{inert,max}=260\mathrm{m}/\mathrm{s}$is reached. Thereafter, an increase of

${t}_{cyc}$ occurs. With reference to the

${V}_{inert,max}=260\mathrm{m}/\mathrm{s}$ value and

Figure 11, it can be concluded that the

${V}_{inert,max}$ region involving the

${t}_{cyc}$ decrease relates to the incompressible regime, while the

${V}_{inert,max}$ region showing the

${t}_{cyc}$ increase is associated with the compressible regime.

The reason why

${t}_{cyc}$ shows an increase in the upper

${V}_{inert,max}$ region (rather than a continuation of the decrease) is due to

${C}_{L}^{*}$ and its dependence on the Mach number

$Ma$. The relationship between

${C}_{L}^{*}$ and

$Ma$ is shown in

Figure 14, based on an examination of the drag polar presented in

Figure 3 with regard to

${({C}_{L}/{C}_{D})}_{max}$ and

${C}_{L}^{*}$, respectively. This relationship is such that

${C}_{L}^{*}$ is constant in the incompressible

$Ma$ region whereas there is a decrease of

${C}_{L}^{*}$ in the compressible

$Ma$ region. That decrease of

${C}_{L}^{*}$ is so strong that it outweighs the opposing effect of the increase of

${V}_{inert,max}$ in the denominator of Equation (30), with the result that

${t}_{cyc}$ is increased.

The results presented in

Figure 13 show that the decrease of

${t}_{cyc}$ is stopped and even reversed, occurring in the upper

${V}_{inert,max}$ range where the

${t}_{cyc}$ level is lowest. This can be considered a favorable effect, especially if

${t}_{cyc}$ is very small and of a magnitude regarded as critical.

#### 5.2. Load Factor $n$

The loops performed in high-speed dynamic soaring involve rapid turning at a large speed. The centrifugal force associated with this flight condition leads to high loads acting on the vehicle, yielding load factors of an order of magnitude as large as 100 [

5]. Accordingly, large loads are typical for high-speed dynamic soaring [

6].

The load factor, dependent on the average speed

${\overline{V}}_{inert}$ and the lift coefficient

${C}_{L}$, is given by Equation (28). The lift coefficient for achieving the maximum speed

${V}_{inert,max}$ is

${C}_{L}^{*}$. Referring to Equations (24) and (26), the following result for the load factor flying at the maximum speed

${V}_{inert,max}$ is obtained, yielding

The load factor

$n$ dependent on

${V}_{inert,max}$ is presented in

Figure 15. According to Equation (31), there is a quadratic increase of

$n$ with

${V}_{inert,max}$ until a value of about

${V}_{inert,max}=260\mathrm{m}/\mathrm{s}$is reached. Thereafter,

$n$ shows a decrease. The two

${V}_{inert,max}$ regions, one showing a decrease and the other an increase of

$n$, relate to the incompressible and compressible regimes, respectively. The reason for these differences in the

$n$ effects is again due to

${C}_{L}^{*}$ and its dependence on the Mach number

$Ma$, as shown in

Figure 14.

#### 5.3. Trajectory Extensions and Loop Radius ${R}_{cyc}$

High-speed dynamic soaring involves a closed-loop trajectory showing extensions in the longitudinal and lateral directions. A question is whether or not the extensions of the trajectory depend on the maximum speed

${V}_{inert,max}$. In case that the trajectory extensions are not independent of

${V}_{inert,max}$, the trajectory has to be adapted for a change in

${V}_{inert,max}$. This implies that a change in the trajectory extensions is necessary if

${V}_{w}$ changes, because

${V}_{inert,max}$ is dependent on the wind speed

${V}_{w}$ (

Figure 11). The loop radius is considered a quantity that can be used to describe the relationship between the extensions of the trajectory and

${V}_{inert,max}$.

With reference to Equations (21b) and (28), the radius of the loop in high-speed dynamic soaring can be expressed as

Since the lift coefficient for achieving the maximum speed

${V}_{inert,max}$ is

${C}_{L}^{*}$, the relation for the optimal loop radius is obtained as

This relation shows that there is no direct dependence of

${R}_{cyc}$ on

${V}_{inert,max}$ or

${V}_{w}$. The only effect is due to

${C}_{L}^{*}$ in case that

${C}_{L}^{*}$ depends on the Mach number

$Ma$ (

Figure 14). Thus,

${R}_{cyc}$ is constant in the incompressible

$Ma$ region since

${C}_{L}^{*}$ is constant here. In the compressible

$Ma$ region,

${R}_{cyc}$ shows changes inversely proportional to

${C}_{L}^{*}$, to the effect that

${R}_{cyc}$ increases with

${V}_{inert,max}$ because of the decrease of

${C}_{L}^{*}$ with an increase of

$Ma$.

The dependence of

${R}_{cyc}$ on

${V}_{inert,max}$ is presented in

Figure 16. The loop radius is constant up to a

${V}_{inert,max}$ value where compressibility becomes effective. Thereafter, a continual increase of

${R}_{cyc}$ takes place. The results from the trajectory optimization (Chapter 2) are determined using the average of the greatest extensions in the

x_{i} nd

y_{i} directions.

#### 5.4. Effects of Altitude $h$

There are different aspects concerning the effects of the altitude on high-speed dynamic soaring. Basically, these effects are due to the dependencies of the air density and the speed of sound on the altitude $h$, i.e., $\rho =\rho \left(h\right)$ and $a=a\left(h\right)$. This relates to all quantities treated above, including ${V}_{inert,max}$ as well as ${t}_{cyc}$, $n$ and ${R}_{cyc}.$

A main aspect is how the maximum speed performance in terms of

${V}_{inert,max}$ is influenced by the altitude. The relation of

${V}_{inert,max}$ given in Equation (26) shows that there is no term involving an explicit dependence on the altitude

$h$. However, there is an altitude effect which is associated with the dependence of

${\left({C}_{L}/{C}_{D}\right)}_{max}$ on

$Ma$ (

Figure 12). This refers to the following relation between

$Ma$ and

$h$:

Since this is relevant only if compressibility becomes effective for

${\left({C}_{L}/{C}_{D}\right)}_{max}$, it can be distinguished between the incompressible and the compressible

$Ma$ regions regarding the effect of

$h$ on

${V}_{inert,max}$. In the incompressible

$Ma$ region, where

${\left({C}_{L}/{C}_{D}\right)}_{max}$ is constant, there is no altitude effect so that

${V}_{inert,max}$ is independent of

$h$. In the compressible

$Ma$ region, an altitude effect exists due to the dependence of

${\left({C}_{L}/{C}_{D}\right)}_{max}$ on

$Ma$. As the speed of sound,

$a$, shows comparatively little changes with

$h$ (e.g., 3% between 0 and 3000 m), the relation for

${V}_{inert,max},$ Equation (26), together with the dependence of

${\left({C}_{L}/{C}_{D}\right)}_{max}$ on

$Ma$,

Figure 12, suggests that the effect of

$h$ on

${V}_{inert,max}$ is mall.

The described relationship between

${V}_{inert,max}$ and

$h$ is graphically addressed in

Figure 17 which shows

${V}_{inert,max}$ dependent on

${V}_{w}$ for

$h=0$ and

$h=3000\mathrm{m}$, thus including

${V}_{inert,max}$ for the altitude range between these values. The left part of the

${V}_{inert,max}$ curve which is associated with the incompressible Ma region holds for both

$h$ values, whereas the right curve part associated with the compressible Ma region shows a small reduction at

$h=3000\mathrm{m}$ compared to

$h=0$.

In regard to the quantities

${t}_{cyc}$,

$n$ and

${R}_{cyc}$, there are effects of the altitude

$h$ associated with the air density

$\rho $ and the speed of sound

$a$. The expressions given in Equations (30), (31) and (33) show that the effect of

$\rho $ manifests explicitly in a dependence that is linear in

$\rho $ or inversely proportional to

$\rho $. The effect of

$a$ relates to the dependencies of

${C}_{L}^{*}$ and

${\left({C}_{L}/{C}_{D}\right)}_{max}$ on

$Ma$ (

Figure 12 and

Figure 14) and the relation between

$Ma$ and

$a\left(h\right)$ as given by Equation (34). Accordingly, it can be distinguished between the incompressible and the compressible

$Ma$ regions regarding the effects of

$h$. In the incompressible

$Ma$ region, there is no altitude effect on

${C}_{L}^{*}$ and

${\left({C}_{L}/{C}_{D}\right)}_{max}$ so that each of

${t}_{cyc}$,

$n$ and

${R}_{cyc}$ is only dependent on

$\rho $as described by Equations (30), (31) and (33). In the compressible

$Ma$ region, an additional altitude effect exists due to the dependencies of

${C}_{L}^{*}$ and

${\left({C}_{L}/{C}_{D}\right)}_{max}$ on

$Ma$.

Concerning the optimal cycle time, the effect of the altitude is presented in

Figure 18 which shows

${t}_{cyc}$for two altitude cases. Basically,

${t}_{cyc}$increases with an increase of

$h$. This is a favorable effect, especially in the case of very small

${t}_{cyc}$ values. Furthermore, the difference between the two

${t}_{cyc}$ curves is significant. For the incompressible

$Ma$ region in

Figure 18, where

${t}_{cyc}$ is only dependent on

$\rho $, the following result on the relation between

${t}_{cyc}$ and

$h$ can be obtained from Equation (30) to yield

with

${\rho}_{0}$ and

${t}_{cyc,0}$ denoting the values at

$h=0$.

The effect of the altitude on the load factor is presented in

Figure 19 which shows

$n$for two altitude cases. As a basic result,

$n$decreases with the increase of

$h$. This can be considered a favorable effect as the load on the vehicle is reduced. For the incompressible Ma region, where

$n$ depends only on

$\rho $, the relation between

$n$ and

$h$ can be expressed as

where

${n}_{0}$ denotes the load factor at

$h=0$.

The effect of altitude on the optimal loop radius

${R}_{cyc}$is presented in

Figure 20, where

${R}_{cyc}$ is plotted for the two altitude cases under consideration. Basically,

${R}_{cyc}$ increases with the altitude

$h$ and the shape of the

${R}_{cyc}$ curve involving a constant part for a large

${V}_{inert,max}$ range is retained. The constant part of

${R}_{cyc}$ relates to the incompressible

$Ma$ region, where

${R}_{cyc}$ is dependent only on

$\rho $. Here, the effect of the altitude can be expressed as

where

${R}_{cyc,0}$ denotes the optimal loop radius at

$h=0$. In the compressible Mach number region, there are additional altitude effects due to the dependencies of

${C}_{L}^{*}$ and

${\left({C}_{L}/{C}_{D}\right)}_{max}$ on

$Ma$.