= *p* < 0.05: significant difference by comparing the CR group to the PDNA group. ◦ = *p* < 0.05: significant difference by comparing the CR group to the PDA group.

0.84 ± 0.10 a,n

0.84 ± 0.10 a,m

1.18 ± 0.08 a,i,n,#,◦

1.02 ± 0.16 i,n,#

1.01 ± 0.16 i,n,◦ 1.04 ± 0.14 b,i,n,#,◦

0.89 ± 0.16 i,n,#

0.89 ± 0.14 i,n,◦

The stride length was affected by group (F (2, 24) = 4.415, *p* < 0.05), task (F (1, 24) = 41.213, *p* < 0.001) and direction (F (2, 48) = 132.916, *p* < 0.001). In particular, healthy controls walked with longer strides when compared to patients, whereas no differences were observed among the sides in the pathological group. All groups reduced their stride lengths during the dual task condition, and the 120◦ turn showed the lowest value compared with the other conditions.

Additionally, the walking velocity changed with group (F (2, 24) = 2.764, *p* < 0.05), task (F (1, 24) = 21.818, *p* < 0.001) and direction (F (2, 48) = 104.061, *p* < 0.001). In general, controls walked faster than patients, and the latter group showed similar values in both sides. Furthermore, the larger the turning the lower the velocity: the 120◦ turn showed the lowest velocity value compared to other directions.

The calculated LRI index significantly changed with group (F (2, 24) = 4.672, *p* < 0.05), task (F (1, 24) = 14.712, *p* < 0.001) and direction (F (2, 48) = 70.937, *p* < 0.001). Because of the lack of a significant difference, the data of the PDNA and PDA were pooled together. Our data confirmed that patients walked with a lower LRI compared to controls in all the investigated conditions because of their reduced walking speed, and such a pattern is more pronounced at turning 120◦ (Figure 4).

**Figure 4.** Comparison of locomotor rehabilitation index values between the control group (white bars) and the pathological group (black bars), for the forward direction (WF), turning 60◦ (T60) and turning 120◦ (T120), in both single (ST) and dual (DT) tasks. Average values ± SD have been reported; data of the PDNA and PDA were pooled together. A significant post-hoc test has been reported: \* = *p* < 0.05 by comparing the control group to the pathological one; c = *p* < 0.001: significant difference by comparing the ST to DT; f = *p* < 0.001: significant difference by comparing WF to T60; i = *p* < 0.001: significant difference by comparing T60 to T120; *n* = *p* < 0.001: significant difference by comparing WF to T120 (same legend as in Tables 2 and 3).

Finally, iso-velocity curves were calculated for each group and condition to understand the complex interaction between stride length and frequency in determining the walk velocity (Figure 5).

During the straight gait (Figure 5, panel a), in single tasking, the patients with PD used the same step frequency compared to controls, but they walked with a shorter stride length. As a consequence, their walking velocities were lower. During the dual task, all curves moved down and to the left: this implied a decrease in the walking speed for controls and patients, and the ratio between the stride length and step frequency was the same.

**Figure 5.** Stride length is plotted as a function of the step frequency (i.e., the iso-velocity curves): (**a**) walking forward (WF); (**b**) turning at 60◦ (T60); (**c**) turning at 120◦ (T120). The symbols refer to different groups as follows: CRs in the single task •; CRs in the dual task -; the PDNA in the single task -; PDNA in the dual task ; PDA in the single task ; PDA in the dual task Δ. For PDNA sometimes the symbols are not seen because they exactly coincide with the data of the other two groups.

This trend occurs similarly in turning (Figure 5, panels b and c), independently of group and task. Importantly, the most significant reduction in the walking velocity occurred while turning 120◦ and confirms that a larger angle of turning leads to a lower walking velocity.

#### **4. Discussion**

The present study focused on the effects of dual task and turning on the kinematic parameters in mild to moderate Parkinson's disease. We hypothesized that the simple cognitive and mechanical tasks will display similar effects in both populations, while the combination of these two stimuli will show a higher impact on patients. We found that: *(i)* the temporal walking parameters were affected by the mental task, as well as by the mechanical demand (turning), but no significant differences among populations were observed. On the contrary, the stride length and walking speed were lower in PD patients compared to controls; *(ii)* the turning task had the capability to alter the walking parameters, especially in people with PD, and the major changes in the walking strategy have been observed while turning at a larger angle (120◦); *(iii)* the combination of the cognitive and the mechanical task was challenging for patients. Indeed, their stride length and walking velocity showed significant alterations compared to controls in all walking conditions; finally, *(iv)* no significant differences were observed between the not affected/more affected side in all the investigated parameters, suggesting an equal symmetry between the right and left body side.

Taking them together, our results highlighted that a simple mental task alone is not sizeable enough to alter the walking strategy in patients with PD with mild to moderate impairments, whereas the combination of this cognitive task with a change of direction has the capability to modify the walking strategy, especially with a higher turning angle (120◦).

#### *4.1. E*ff*ects on Gait Variables: A Task Comparison*

In single tasking, our data confirmed that patients with mild to moderate PD walked with a reduced gait velocity and a shorter step length compared to controls. Therefore, patients showed a lower locomotor index rehabilitation if compared to controls with no evident differences between the not affected and more affected side. This finding suggests that the mild to moderate pathology compromises, in a similar way, the gait kinematics of the inferior limbs (e.g., it did not compromise the symmetry among body sides). Since, postural control and gait are linked to cognitive function both in healthy and pathological subjects [18,22,42], it is possible to assume that PD patients reduced their stride length in order to increase the time spent on the ground, increasing the walking stability.

As we hypothesized, a simple dual task condition (repeating the days of the week backwards) affected both groups similarly: all participants increased the ground contact (and the cycle time) and decreased their cadence and frequency [12,19]. This finding endorses that subjects tried to maintain their postural stability by spending more time on the ground, as this would prevent the risk of falling [18]. The lack of a difference among groups confirms that, by adding a cognitive load, a low disease severity could not play a major role in determining motor impairments [23]. In addition, the high focus on the additional task means a larger proportion of the attentional capacity is at the expense of walking performance: people walked even more slowly with much shorter steps [7,12,15,22,43–46]. This "compensatory strategy" could be useful in achieving a greater control of gait and balance disruption [20], and in counterbalancing the fluctuations of the center of mass.

Finally, the similar ratio between the stride length and step frequency means that subjects did not change their stride pattern. This latter outcome suggests that the simple cognitive task was probably not too demanding for all participants. More complex cognitive tasks (i.e., concurrent loads, mental tracking) would probably point out the gap among groups [19,20].

#### *4.2. E*ff*ects on Gait Variables: A Direction Comparison*

In the present study, we investigated two turning angles that are very common during daily activity [38]: the first (60◦) represents a simple turn and the other (120◦) a more difficult turn.

Our spatiotemporal data are in line with the previous literature [8,10,18,21]. With respect to directions, whereas healthy participants were impacted by the 120◦ turn, PD patients were also affected by the 60◦ turn, but only for the walking velocity. Additionally, in this case, no appreciable differences between the not affected and more affected side were observed.

Generally, the 120◦ turning is characterized by an increased support time and a reduced number of strides (i.e., cadence) and frequency, as well as a decreased speed and length. This angle requires additional attentional resources: in fact, it relies more heavily on proprioception (i.e., directed by the basal ganglia function) than both forward walking and the 60◦ turn [47]. By considering the mechanical approach, the impact of a larger turning could be explained by the kinetics of the movement. Indeed, to move into a larger angle of turn, the subjects decreased their speed more (just for a greater angle of turn) and, probably, used their turn foot as "pivot foot". When the turn angle increased, the ankle plantar flexion moment (and its peak) increased and the external rotators of the lower extremity played a much greater role than in straight walking [48–50]. These alterations of the plantar flexors have been accomplished with a reduced ankle power generated in the pre-swing phase (push-off power). Therefore, the larger the angle of turn the higher the mechanical demand. As a result, PD patients tended to reduce the step length and to increase the contact time primarily to improve the stability of the body. They showed an initial alteration of the walking pattern at a 60◦ turn, but only the walking velocity was affected. On the other hand, a 120◦ turn showed evident differences when compared to the straight direction in terms of both the temporal and spatial parameters; this finding supports the idea of a more challenging stimulus.

Therefore, whereas the 60◦ turn could represent a not suitable training stimulus, the 120◦ may be a good challenge for people with PD. As expected, all gait alterations are exacerbated during walking with a combination of the two stimuli, especially in PD patients.

#### *4.3. E*ff*ects on Gait Variables: Cognitive Task and Mechanical Task*

The combined effects of dual (cognitive) and mechanical (turn) tasks represent an important training stimulus in people with PD. Indeed, our data shows that the concomitant presence of a simple mental and turning (60◦) task produced only a marginal effect on the main kinematics. In particular, the timing parameters showed no significant differences compared with the straight line, and the differences between the single and dual task are comparable to the ones obtained for healthy subjects. Furthermore, even the step length and the step frequency showed a similar trend.

On the contrary, the matching between the same mental task and a more complex mechanical demand (120◦ turn) played a greater "destructive" impact especially in patients with PD. Indeed, their temporal and spatial variables showed a much more marked gap than the controls' ones.

Our results were based on a relatively small heterogeneous sample of disabled patients, who lived independently in the community. Therefore, further researches are necessary to extend these findings to a larger sample (i.e., patients with more severe gait deficits or episodes of freezing), or to other conditions (i.e., OFF phase performance).

#### **5. Conclusions**

Our data showed that a mechanical task (i.e., turning) has the potential to modify gait strategy in people with Parkinson's disease, without changes in symmetry of the lower limbs. Of greatest interest, the concomitant presence of a mechanical task and a simple cognitive task did not produce a further impairment of this gait strategy. Therefore, using the investigated combined condition (turning and repeating the days of the week backwards) could represent a significant training stimulus in such patients. Indeed, the improvement of the mental and physical characteristics is very important in improving the functionality of patients at the early stages of their pathology.

**Author Contributions:** Conceptualization, F.N., F.B., A.M., M.B.; methodology, F.N., A.M.; software, F.N., E.B.; data curation, F.N., E.B., A.M.; writing—original draft preparation, F.N., A.M., M.B.; writing—review and editing, F.N., A.M., M.B. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Acknowledgments:** We would like to thank Alessandro Corsi, Gianluca Fedel and Davide Nisi for their help in data collection, and the subjects for participating in the study.

**Conflicts of Interest:** The authors acknowledge that there are no conflicts of interest pertaining to this manuscript.

#### **References**

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

### *Article* **Race Walking Ground Reaction Forces at Increasing Speeds: A Comparison with Walking and Running**

**Gaspare Pavei 1,\*, Dario Cazzola 2, Antonio La Torre 3 and Alberto E. Minetti 1**

Received: 10 June 2019; Accepted: 1 July 2019; Published: 3 July 2019

**Abstract:** Race walking has been theoretically described as a walking gait in which no flight time is allowed and high travelling speed, comparable to running (3.6–4.2 m s−1), is achieved. The aim of this study was to mechanically understand such a "hybrid gait" by analysing the ground reaction forces (GRFs) generated in a wide range of race walking speeds, while comparing them to running and walking. Fifteen athletes race-walked on an instrumented walkway (4 m) and three-dimensional GRFs were recorded at 1000 Hz. Subjects were asked to performed three self-selected speeds corresponding to a low, medium and high speed. Peak forces increased with speeds and medio-lateral and braking peaks were higher than in walking and running, whereas the vertical peaks were higher than walking but lower than running. Vertical GRF traces showed two characteristic patterns: one resembling the "M-shape" of walking and the second characterised by a first peak and a subsequent plateau. These different patterns were not related to the athletes' performance level. The analysis of the body centre of mass trajectory, which reaches its vertical minimum at mid-stance, showed that race walking should be considered a bouncing gait regardless of the presence or absence of a flight phase.

**Keywords:** body centre of mass; human gait; race walking; force plate

#### **1. Introduction**

Ground reaction forces (GRFs) have often been used in biomechanical studies to describe human locomotion [1,2] because they show that the forces exerted by the foot on the ground are a key determinant of the final gait kinematics. Ground reaction force analysis is nowadays used also for the detection of pathological gaits [3,4], gait asymmetry [5,6], injury prevention [7] and in the estimation of muscles force [8]. Moreover, the analysis of GRF peaks and the timing of peaks occurrence can explain how velocity is generated and increased, which could be important in sport activity such as running (e.g., [9]).

 In race walking, the BCoM trajectory can be correctly computed only by using a forward dynamics approach, whereas inverse dynamics computation has been shown to be biased [11]. Thus, the measurement and analysis of GRFs at increasing speed is key to investigating race walking mechanics, even more than in walking and running. Each animal or human gait has its own "locomotor signature", ultimately represented by the trajectory of the body centre of mass (BCoM), with its asymmetry and related energies [12], and race-walking trajectory has never been analysed in such a fashion due to the lack of consistent GRF datasets. Starting from the ground reaction forces recorded during a stride, it is also possible to represent the "locomotor signature" by showing the Lissajous contour [12] also for race walking gait.

Race walking is an Olympic discipline where athletes are required to complete the distance in the shortest time according to two constrains: no flight time can occur between steps and the knee has to be locked in extension from touch down to mid-stance. This rule induces race walking to manage a different kinematics compared with walking and running that causes, also, somewhat different ground reaction forces patterns [13–15]. However, race walking dynamics have been less studied when compared with walking and running [16], and often investigations have been focused on one speed only, missing potentially relevant information about velocity generation.

The aim of this study was to analyse the ground reaction forces and BCoM trajectory during race walking on a wide range of speeds and to compare the three components of GRFs (i.e., forward, lateral, vertical) with walking and running.

#### **2. Materials and Methods**

Fifteen male athletes (mean ± SD, 23.0 ± 5.5 years old, 1.78 ± 0.05 m height, 64.7 ± 5.2 kg body mass and with a 10,000 m personal best of 44:26 ± 3:34 min:s) participated in this study. All subjects gave their informed consent for inclusion before they participated in the study. The study was conducted in accordance with the Declaration of Helsinki, and the protocol was approved by the Ethics Committee of the University of Milan.

The "dynamometric corridor" was composed of five 3D strain gauge platforms (Bertec, USA) in order to obtain a 4 m long and 0.4 m wide footpath placed in the middle of a 40 m walkway where athletes could race walk at a constant speed. Athletes were asked to perform three trials at each self-chosen low, medium and high speed, hence, each subject completed 9 trials.

Ground reaction forces were recorded at 1000 Hz and normalised to athlete's body weight (BW) and as a percentage of stance time. Stance phase was defined using a 10 N threshold on vertical (FZ) force. The inversion between braking and propulsive on antero-posterior (FX) force was set when the force from negative (braking) became positive (propulsive). Speeds were clustered when within 3% of the target speed (2.78, 3.06, 3.33, 3.61, 3.89, 4.17 m s−1), similar to the cluster used in Nillson and Thorstensson [17]. Walking and running GRF values were also taken from the Nillson and Thorstensson [17] study for comparison purposes.

The BCoM position was computed by double integration of the 3D acceleration, obtained by the force signal, according to Cavagna [10]; the integration constants were calculated as described in the Appendix of Saibene and Minetti [18]. The obtained BCoM trajectory was transformed in local coordinates (as the sampling occurred over an instrumented treadmill). The resulting 3D contour included several consequent strides, each of which was forced to become a closed loop and centred on (0, 0, 0) by subtracting average 3D coordinates to allow a description based on a Fourier Series with 6 harmonics [12]. Walking and running BCoM data were extracted from our cumulated database on subjects matched for anthropometry and age (*n* = 10).

Statistical differences across speeds and multiple peaks were tested by a two-way ANOVA using a Bonferroni post-hoc test, whereas differences between the two main gait patterns (see Results) were tested using a *t*-test and the significant level was set at *p* < 0.05 (SPSS 19, IBM).

#### **3. Results**

Mean curves of vertical (Fz), antero-posterior (Fx) and medio-lateral (Fy) ground reaction forces during race walking stance phase are presented in Figure 1. Subjects were clustered in two different groups according to different Fz curves: (i) M shape (Figure 1) that was similar to walking, displaying two peaks and a valley and (ii) N shape (Figure 1) that showed a huge first peak followed by a plateau. The Fx curve showed a first negative braking peak and a smaller propulsive peak near the end of stance phase, without substantial differences on the Fx curve among the two groups. Braking and propulsive impulse (i.e., the area of the two phases delimited by the abscissa at 0 value) were always very similar denoting a substantially constant inter-strides speed. The Fy curves were averaged between right and left stance since they were specular with no significant differences. After a first medial small peak, the

force was lateral, medial at mid stance and lateral again at two-thirds of the stance. The two groups reached the first lateral peak with two different shapes, more consistently for the M shape group.

**Figure 1.** Ground reaction force traces (as a fraction of body weight, BW) in the three axes (vertical, antero-posterior and medio-lateral axis from top to bottom) at increasing speed (m s−1) are shown. Right and left columns represent the M shape and N shape vertical force patterns, respectively.

In Figure 2, vertical (Fz) peaks and valleys in race walking, walking and running [17] are presented. When analysing the M shape group, the first peak was higher than walking, whereas the second was comparable, and the valley instead was much higher in race walking than in walking without falling under the BW value. In the M shape group, the first Fz peak was always significantly higher (*p* < 0.001) than the second. The N shape Fz peak was slightly higher than the M shape, with significant difference only at 2.78 m s−1 (*p* < 0.01) and 3.61 m s−1 (*p* < 0.05). Running showed higher peaks than the other gaits. All peaks increased linearly with speed, with significant differences between 4.17 m s−1 and the other speeds in the N shape and the second M shape peak (*p* < 0.01). The first M peak increased with speed but with a less significant trend, whilst the valley was speed independent.

**Figure 2.** Ground reaction force vertical peaks (as a fraction of BW) in the three gaits (W, walking; R, running; RW, race walking) at increasing speed (m s−1) are shown. M and N refer to race walking M shape and N shape groups, respectively.

Figure 3 shows the antero-posterior (Fx) peaks in the three gaits. Braking and propulsive peaks increased linearly with speed and with the same values in walking and running, whereas race walking reported higher braking than propulsive peaks (*p* < 0.001) in both groups at each speed. The braking peak was significantly lower (*p* < 0.05) in the M shape than the N shape except for 3.33 m s−1 where the M shape peak was higher and at 3.89 m s−1 that was not different.

**Figure 3.** Antero-posterior peaks (as a fraction of BW) of ground reaction force in the three gaits (W, walking; R, running; RW, race walking) at increasing speed (m s−1) are shown. "Brake" is braking peak; "prop" represents propulsive peak. M and N refer to race walking M shape and N shape groups, respectively.

In Figure 4A, the difference between Fy medial and lateral peak are shown: walking and running Fy increased in a similar fashion linearly with speed, and race walking always showed higher values (significant differences among groups only at 3.89 m s−1 (*p* < 0.001)); in Figure 4B, the amplitude of medial and lateral peaks in race walking were almost speed independent. Also, across several speeds, the first lateral peak was significantly higher than the second one. In the N shape group, the medial peak was also greater than the lateral, whereas in the M shape group, the lateral and medial force peaks were comparable. The N shape group often showed significantly greater peak values than the M shape one.

**Figure 4.** (**A**) Ground reaction medio-lateral "delta" force (as a fraction of BW), expressed as the peak-to-peak force difference, for the three gaits (W, walking; R, running; RW, race walking) at increasing speed (m s−1) are shown. (**B**) Peak medial (med) and lateral (lat) force (as a fraction of BW) during race walking at increasing speed (m s−1). M and N refer to race walking M shape and N shape groups, respectively.

Figure 5A (M shape group) and b (N shape group) shows the timing of the peaks, valley and inversion of the vertical, antero-posterior and medio-lateral ground reaction forces in relation to the normalised stance phase for each speed tested. Most of the variables did not show speed dependency, since their relative timing across speeds did not change; however, in the N shape group, the propulsive peak at 4.17 m s−1 occurred significantly earlier (*p* < 0.05) than in other speeds. In the M shape group, the peak brake and peak lateral forces showed some timing variations, (*p* < 0.05). It was interesting to note that some peaks occurred together: Fx brake, Fy lateral and Fz; medial Fy and Fx inversion; and Fx propulsive and Fy lateral, without differences among groups. The M shape group showed a later inversion of antero-posterior force at speed < 3.89 m s−1 (*p* < 0.01), an earlier propulsive peak at some speeds (*p* < 0.01), an earlier lateral peak at speed < 3.61 m s−1 (*p* < 0.05) and an earlier medial peak at high speed (*p* < 0.05) than the N shape group.

**Figure 5.** Timing of race walking force peaks (%stance) at the different speeds (m s−1). (**A**) N shape group. (**B**) M shape group.

By comparing race walking peaks' timing with walking and running [17], braking and vertical peaks were anticipated in race walking, whereas propulsive peak timing was similar to walking and occurred later than running.

The 3D BCoM trajectory is presented in Figure 6 in comparison with running at the same speed and walking (1.94 m s−1). The race walking BCoM volume was smaller than in running and walking, with a narrower displacement in the medio-lateral direction and smaller vertical excursion. Walking and running showed a lower minimum of the BCoM vertical trajectory compared with race walking, whereas in the upper part, walking and race walking showed the same maximum, which was lower than in running. In race walking, BCoM was in the lowest part of the trajectory during stance, as in running, without showing the arc of circle characteristic of walking during the stance phase. The set of equations, based on the Fourier Series (truncated at the 6th harmonic), needed to describe the BCoM 3D trajectory of race walking (for example at 3.61 m s−1) is:

$$
\begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 5.017 \sin(2t - 0.609) + 0.296 \sin(4t - 2.219) + 0.191 \sin(6t + 1.380) \\ 5.080 \sin(t + 0.000) + 0.491 \sin(3t + 2.362) + 0.148 \sin(5t - 1.778) \\ 14.421 \sin(2t + 1.250) + 2.069 \sin(4t + 0.578) + 0.371 \sin(6t - 0.906) \\ + 1.865 \sin(t + 0.101) + 0.130 \sin(3t - 2.211) + 0.054 \sin(5t - 2.107) \\ + 0.203 \sin(2t - 0.610) + 0.052 \sin(4t - 1.232) + 0.025 \sin(6t - 2.281) \\ + 2.957 \sin(t - 2.110) + 0.554 \sin(3t + 2.859) + 0.113 \sin(5t + 2.357) \end{bmatrix}, t = 0 \dots 2 \pi
$$

where x, y, and z are the antero-posterior, medio-lateral and vertical axis, respectively. This kind of equation was used to represent the walking, running and race walking BCoM trajectory in Figure 6.

**Figure 6.** 3D representation using the Fourier Series of the Body Center of Mass (BCoM) trajectory of walking (1.94 m s−1), race walking and running (2.78 m s−1). The mean contours (in black) have been shifted in the antero-posterior and medio-lateral axes (x and y, respectively) to better appreciate the 2D projection (in grey) on each plane. Black arrows on the contour projections in the XZ plane represent the BCoM dynamic movement along the trajectory, which is counter-clockwise for walking and clockwise for race walking and running. Progression direction is shown by the black arrows on the antero-posterior (x) axis, and the axes thickness is 0.02 m.

#### **4. Discussion**

This paper described and analysed the race walking ground reaction forces patterns in the three planes of motion at increasing speeds. The GRF peak values were comparable with the paired speeds of previous studies in literature [13–15,19], with a few discrepancies which are discussed below.

#### *4.1. Speed Adaptation*

The speed increased vertical peaks almost linearly and the second peak in the M shape race walking group was more affected than the first one (Figure 2). On the contrary, the valley results were speed independent (Figure 2). The peak propulsive and braking forces also increased with speed and the braking peak was always greater than the propulsive peak, whereas the timing of the inversion was not affected by speed (Figure 3). The braking peak is expected to be passive in race walking as explained by the "locked-knee" rule, but since the braking and propulsive impulses were the same (as they should for the constant speed), such a peak force asymmetry seems to suggest a strategy to minimize peak muscle involvement in the propulsive phase. Medio-lateral forces, both in absolute terms or expressed as peaks difference, were similar at increasing speed (Figure 4), and this suggests that the leg and trunk muscles do not need to increase their activity when speed increases, as the body centre of mass trajectory was not further laterally deviated. Such speculation is supported from a gait optimisation perspective, as a lateral deviation with respect to the progression direction has been shown as an avoidable and functionally ineffective feature in all animal gaits (apart from penguins [20]). Timing of peaks, when expressed as stance percentage, was speed independent except for few variations (Figure 5). This is a typical gait stereotype, which could be beneficial from a motor learning perspective [21], since, in this way, athletes do not have to change their pattern to gain speed but only perform it faster. In fact, when increasing speed, the contact time is reduced [22], and the gait events are anticipated in absolute timing. Since the majority of the peaks increased with speed, as occurred in walking and running [17], comparison within the same gait or with other gaits should be performed only at matched speed.

#### *4.2. Race Walking versus Walking and Running*

When comparing GRFs across race walking, walking and running, some differences were found. Vertical forces were higher in running, whereas race walking peaks seemed to increase with speed following the same trend of walking values at higher speeds (Figure 2). However, while in walking, the valley values dropped under body weight and decreased with speed; in race walking, the valley values were slightly higher than body weight and speed independent (Figure 2). In walking, the trough drops under body weight due to the "centrifugal reaction force" caused by the arc of circumference quickly travelled by BCoM during the stance phase [23,24]. In race walking, the BCoM during stance does not move along a circle with leg length as radius [11], but lowers down as in running. This could explain why the trough in vertical force signal does not drop under body weight; however, this hypothesis does not explain why an M shape pattern is exhibited, as in walking, whilst the BCoM pathway is similar to running. Also, both vertical peaks occurred earlier than walking with a timing similar to running (for the first). When considering the vertical forces, running is the most stressful gait due to the highest peaks. Race walking propulsive peaks were comparable to running, differently, the braking peaks were higher than running (Figure 3). The inversion between braking and propulsive in race walking occurred earlier than in running, and a shorter braking time involves a higher peak. The propulsive peak timing occurred as in walking, but later than running (Figure 5). This could be advantageous to avoid rapid changes in force production but required a constant average activation.

The medio-lateral forces (expressed as delta peaks) in race walking showed the lowest values as in the other gaits, but they were higher than walking and running (Figure 4). This is probably due to the kinematics of the pelvis, which shows a medio-lateral excursion in order to accept the straight knee from heel strike to midstance [22,25,26]. The first lateral peak was simultaneous with the vertical peak, as occurred in walking and running, and the medial peak, which was the greatest, was in line with braking–propulsive inversion.

The BCoM trajectory shows some interesting features of race walking compared with walking and running. At first, the volume was smaller than the other two gaits with less excursion in the medio-lateral direction, which is not expected when considering the great excursion of the pelvis, typical of this gait [22]. However, the BCoM is the weighted mean of all the segments, and a single segment could bring a remarkable bias in the estimated BCoM trajectory [11]. The vertical displacement showed a higher minimum than both walking and running. This can be explained by the race walking rule that requires the knee to be straight during the stance phase. In this phase, BCoM lowers its trajectory as in running, without the knee flexion, race walking BCoM is mechanically forced to stay on

a higher trajectory. In walking, when increasing speed, the minimum reaches smaller values. Walking and race walking, at slow speed (2.78 m s−1), show the same upper limit in BCoM trajectory. When speed is increased, flight time occurs also in race walking and the maximum is slightly higher, without approaching running values; whereas, the minimum is almost unchanged. The crossing point between the right and left part of the contour in race walking occurs in the upper part as in running, whereas in walking, it is located in the lower part. Also, the potential and kinetic mechanical energies are in phase as in running, whereas in walking, they are out of phase. The symmetry values on the three axes showed a behaviour similar to walking and running. However, a greater number of strides is necessary to give an appropriate description of the symmetry behaviour at increasing speed also in race walking. In conclusion, the BCoM contour of race walking resembles the running pattern, even when no flight time is present, with a smaller excursion.

#### *4.3. Di*ff*erent Vertical GRF Groups*

Besides the vertical peaks' difference, the clustering of M and N shape athletes showed other small differences in anterior-posterior and medio-lateral forces in the first part of the stance phase, before the braking–propulsion inversion. The braking peak was higher in the N than in the M shape group and the lateral force pattern was less homogeneous with a delayed peak in the N shape group.

Fenton [14] suggested that the magnitude and timing of the vertical peak was an index of smoothness and "fluidity" of the stride. Moreover, the absence of a second vertical peak would direct the force more in the progression direction, on the contrary, a vertical force would cause a vertical displacement that could end in flight phase [14]. In our data, both groups (M and N shape) showed the same first vertical peak, therefore, they should have the same stride "fluidity". As for the vertical displacement, the double integration of acceleration in both groups did not show any appreciable difference in the BCoM trajectory during stance and no difference in flight time.

The different vertical GRF patterns among athletes was already pointed out by Fenton [14], and it was also evident when comparing Fenton's with Cairns and colleagues' [13] data [13,16]. Fenton explained this difference with athletes' performance level: M shape athletes were the least, whereas N shape were the most trained. Despite our relatively small sample size (larger than in the Fenton paper [14]), eight athletes showed an N shape pattern and seven showed an M shape pattern: within groups, the performance level was very different; however, between groups, the level was the same (PB 10,000 m min:s M shape: 44:39; N shape 44:18). This allowed us to conclude that performance level should not be the trigger for different patterns. As shown in Figure 1, the pattern was well characterised and different across the whole range of speeds and no athlete changed it by increasing speed. A further explanation could be related to the different athletes' techniques learnt from different coaches. When analysing this aspect, the sample size decreases even more, with just a couple of athletes for each coach; however, we found that coach technique was not the determinant of the difference either.

#### **5. Conclusions**

The present comprehensive analysis of ground reaction forces shows that race walking is a gait that shares features with walking and running. Similarly, the increase in speed is achieved by increasing force peaks, which occur at the same relative instant of the stance phase, thus, a comparison among subjects or studies should be done only at the same speed. The peculiarity of race walking kinematics and dynamics features is remarkable, also in the ground reaction forces analysis, since, differently from walking and running, athletes showed two different vertical force patterns within the same gait. These different patterns do not change the 3D trajectory of the body centre of mass and related spatiotemporal parameters, and do not seem to be related to the athletes' performance level. Further investigations are needed to understand which biomechanical factors cause these patterns. The BCoM trajectory obtained by ground reaction forces showed that race walking has the same pattern of running, even at slow speed where no flight time is present, within a smaller volume.

**Author Contributions:** Initial project discussion was conducted by all authors. Specific personal contributions were as follows: conceptualization, G.P.; methodology, G.P., D.C. and A.E.M.; software, G.P., D.C. and A.E.M.; formal analysis, G.P.; investigation, G.P., D.C. and A.L.T.; data curation, G.P.; writing—original draft preparation, G.P.; writing—review and editing, G.P., D.C., A.L.T. and A.E.M.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).


**Abstract**

Stable nitrogen and carbon isotopes were used to trace the interaction between atmospheric particles < 2.5 mm in diameter (PM2.5) with atmospheric physical variables and atmospheric chemical species in an urban environment. PM2.5 were collected daily at two sites in Mexico City during three dry seasons during two-week periods. PM2.5 varied between 10 and 70 μg/m3 , with the highest concentrations occurring during low-speed southerly winds and the lowest during high-speed easterly winds. Stable carbon isotope composition (δ 13C) showed that the main carbon source of PM2.5 included emissions from fossil fuel combustion, along with low-molecular-weight carbon emissions and suspended dust. Stable nitrogen isotope values (δ15N) in PM varied between −9.9 and 21.6‰. The most 15N-enriched particles generally occurred during low wind speeds and correlated significantly with hourly averaged ambient NOx and NO2 concentrations. Simultaneous samples from MER (commercial site) and XAL (industrial site) showed that PM2.5 mass concentration was generally lighter and the carbon and nitrogen isotopic compositions were heavier at the commercial site relative to the industrial site. The δ15N of PM2.5 increased with the %N in PM2.5 concordant with an isotopic fractionation during gas-to-particle condensation. Results suggest that wind speed, along with the nitrogen emission source, determines the nitrogen isotopic composition of PM2.5.

**Keywords:** PM2.5, δ 13C, δ 15N, Mexico City, ambient air


2. Governing equations of hypersonic non-equilibrium polyatomic gas flows

The Martian atmosphere is composed mostly of carbon dioxide (96%), nitrogen (1.9%), argon (1.9%), and others. Small admixtures of nitrogen (N2) and argon (Ar) in the Mars atmosphere do not play a significant role in the process of heat transfer to descent vehicles (at least, at moderate velocities of flight till the convective heat transfer prevails). It is possible to restrict ourselves by consideration of model atmosphere as the pure carbon dioxide. The pressure on the planet surface is taken equal to 6.0 mbar. It is 0.6% of Earth's mean sea level pressure. The atmosphere is quite dusty.

The conditions of a flow corresponding to the last stage of flight of space vehicles in an atmosphere of Mars (V∞ ≤ 6 km/s, r∞ > 105 kg/m3 , H < 60 km) were studied. Determining process at such velocities is a process of dissociation. Up to 75% of full gas flow energy can be spent on it.

The region where non-equilibrium physical and chemical processes realized is a significant part from all considered regions (Figure 2). Velocity of physical and chemical processes, as a rule, grows together with density of gas. For considered flow conditions, the degree of gas ionization is small and does not bring the appreciable contribution to internal gas energy. The translational degrees freedom becomes equilibrium on distances of several free path lengths of molecules behind front of a shock wave for considered altitude. The distribution of rotation energy also is established slightly later. Therefore, it is usually supposed that translational and vibration degrees of freedom of particles are in equilibrium. At high temperatures that observed in a shock layer, the characteristic times of a vibration energy relaxation of molecules and characteristic times of dissociation become one order. Thus the account of non-equilibrium excitation of vibration degrees of freedom of carbon dioxide molecules is necessary. The region of relaxation behind the bow shock wave has a specific structure that consists of sequential relaxation zones. The flow in shock and boundary layers are being especially non-equilibrium. It causes the energy redistribution of the internal energy.

Since the Martian atmosphere is strongly rarefied, non-equilibrium processes affect heat transfer at the more significant part of the descent trajectory. The most thermal-loaded part of the typical descent trajectory is the region of frozen chemical reactions and equilibrium-excited vibration degrees of freedom.

We consider the conditions typical for a high-temperature shock layer, while the translational and rotational relaxation are supposed to proceed fast as well as intra-mode VV-transitions in CO2, O2, CO and inter-mode VV-exchange between CO2 symmetric and bending modes. All other vibration energy transitions, dissociation, recombination, and exchange reactions are considered to be slower with relaxation times comparable with the mean time of the gas dynamic parameters variation. Such a relation between the characteristic times makes it possible to introduce vibration

Numerical Modeling of Hypersonic Aerodynamics and Heat Transfer Problems of the Martian Descent Modules

The existing experimental and theoretical data on relaxation times of different processes in mixtures containing CO2 molecules show that in a wide range of conditions the following

Here τtr, τrot are the characteristic times of translational and rotation relaxation; τVVm are the

to inter-mode transitions; τr is the characteristic time of chemical reactions; and θ is the mean

On the basis of the kinetic theory principles, the closed self-consistent three-temperature description of a flow in terms of densities of species, macroscopic velocity, gas temperature, and three vibration temperatures are obtained [7, 8]. The set of governing equations contains the conservation equations of mass, momentum, and total energy coupled to the equations of non-equilibrium three-temperature chemical kinetics as well as the relaxation equations for

1�2�3 < τVT3 << τr � θ, m ¼ 1, 2, 3 (4)

http://dx.doi.org/10.5772/intechopen.71666

dt þ r ∇ � v ¼ 0 (5)

dt þ ∇ � P ¼ 0 (6)

i ð Þ i ¼ CO2; CO; O2; O; C (8)

dt þ ∇ � q þ P : ∇v ¼ 0 (7)

k�m correspond

45

temperatures for the coupled (symmetric-bending) and asymmetric CO2 modes.

1�2 << τVT2 � τVV0

times of intra-mode VV exchanges; τVT2 , τVT3 are the times of VT transitions; τVV0

Under condition (5) the set of equations are obtained in the following form: dr

> r dv

The equations of non-equilibrium chemical kinetics written in the following form

Here ni is the number density of species i (1—CO2, 2—CO, 3—O2, 4—C, 5—O).

Ri t τtr < τrot < τVVm � τVV0

time of gas dynamic parameters change.

dni

tensor; and q is the heat flux.

Figure 2. (a) The region of different flow regimes of flow for MSRO vehicle (red line) and MARS EXPRESS vehicle (blue line); (b) Mars descent vehicles.

The thermo-chemical model of the processes occurring in the shock layer includes the chemical reactions, dissociation and recombination of CO2 molecules, dissociation and recombination of diatomic molecules, exchange reactions, processes of vibration energy exchange between various levels of molecules, influence of the vibration relaxation on the chemical reactions and vice versa (CVDV-processes), processes of excitation and deactivation of the electronic states of molecules, and spontaneous radiation processes for excited particles.

We consider a high-temperature flow of the mixture taking into account vibration excitation and the following exchange reactions, dissociation, and recombination:

$$\rm{CO}\_2 + M \leftrightarrow \rm{CO} + \rm{O} + M,\\
\rm{o} \rm{CO} + M \leftrightarrow \rm{C} + \rm{O} + M,\tag{1}$$

$$\rm O\_2 + M \leftrightarrow O + O + M,\\
\rm OO\_2 + O \leftrightarrow CO + O\_2.\tag{2}$$

$$\text{CO} + \text{CO} \leftrightarrow \text{CO}\_2 + \text{C}, \mathfrak{v}\text{CO} + \text{O} \leftrightarrow \text{O}\_2 + \text{C},\tag{3}$$

where M is a molecule or an atom [10].

As known molecule of СО2 have three vibration modes with different characteristic temperatures: symmetric, deformation (twice degenerate), and asymmetric. According to fast exchange of vibration energy between the different modes, it is assumed that molecules reach the Boltzmann distribution with a uniform temperature.

We consider the conditions typical for a high-temperature shock layer, while the translational and rotational relaxation are supposed to proceed fast as well as intra-mode VV-transitions in CO2, O2, CO and inter-mode VV-exchange between CO2 symmetric and bending modes. All other vibration energy transitions, dissociation, recombination, and exchange reactions are considered to be slower with relaxation times comparable with the mean time of the gas dynamic parameters variation. Such a relation between the characteristic times makes it possible to introduce vibration temperatures for the coupled (symmetric-bending) and asymmetric CO2 modes.

The existing experimental and theoretical data on relaxation times of different processes in mixtures containing CO2 molecules show that in a wide range of conditions the following relations are valid:

$$
\tau\_{tr} < \tau\_{nt} < \tau\_{VV\_m} \sim \tau\_{VV\_{1-2}'} < \tau\_{VV\_2} \sim \tau\_{VV\_{1-2-3}'} < \tau\_{VV\_3} < \tau\_r \sim \theta, \quad m = 1, \ 2, \ 3 \tag{4}
$$

Here τtr, τrot are the characteristic times of translational and rotation relaxation; τVVm are the times of intra-mode VV exchanges; τVT2 , τVT3 are the times of VT transitions; τVV0 k�m correspond to inter-mode transitions; τr is the characteristic time of chemical reactions; and θ is the mean time of gas dynamic parameters change.

On the basis of the kinetic theory principles, the closed self-consistent three-temperature description of a flow in terms of densities of species, macroscopic velocity, gas temperature, and three vibration temperatures are obtained [7, 8]. The set of governing equations contains the conservation equations of mass, momentum, and total energy coupled to the equations of non-equilibrium three-temperature chemical kinetics as well as the relaxation equations for vibration temperatures.

Under condition (5) the set of equations are obtained in the following form:

The thermo-chemical model of the processes occurring in the shock layer includes the chemical reactions, dissociation and recombination of CO2 molecules, dissociation and recombination of diatomic molecules, exchange reactions, processes of vibration energy exchange between various levels of molecules, influence of the vibration relaxation on the chemical reactions and vice versa (CVDV-processes), processes of excitation and deactivation of the electronic

Figure 2. (a) The region of different flow regimes of flow for MSRO vehicle (red line) and MARS EXPRESS vehicle (blue

We consider a high-temperature flow of the mixture taking into account vibration excitation

As known molecule of СО2 have three vibration modes with different characteristic temperatures: symmetric, deformation (twice degenerate), and asymmetric. According to fast exchange of vibration energy between the different modes, it is assumed that molecules reach

CO2 þ M \$ CO þ O þ M, CO þ M \$ C þ O þ M, (1)

O2 þ M \$ O þ O þ M, CO2 þ O \$ CO þ O2, (2)

CO þ CO \$ CO2 þ C, CO þ O \$ O2 þ C, (3)

states of molecules, and spontaneous radiation processes for excited particles.

and the following exchange reactions, dissociation, and recombination:

where M is a molecule or an atom [10].

line); (b) Mars descent vehicles.

44 Advances in Some Hypersonic Vehicles Technologies

the Boltzmann distribution with a uniform temperature.

$$\frac{d\rho}{dt} + \rho \left| \nabla \cdot \mathbf{v} \right| = 0 \tag{5}$$

$$
\rho \frac{d\mathbf{v}}{dt} + \nabla \cdot \mathbf{P} = 0 \tag{6}
$$

$$
\rho \frac{de}{dt} + \nabla \cdot \mathbf{q} + \mathbf{P} : \nabla \mathbf{v} = 0 \tag{7}
$$

The equations of non-equilibrium chemical kinetics written in the following form

$$\frac{dn\_i}{dt} + n\_i \nabla \cdot \mathbf{v} + \nabla \cdot (n\_i \mathbf{V}\_i) = \mathbf{R}\_i^r \ (i = \mathbf{CO}\_2, \mathbf{CO}, \mathbf{O}\_2, \mathbf{O}, \mathbf{C}) \tag{8}$$

Here ni is the number density of species i (1—CO2, 2—CO, 3—O2, 4—C, 5—O). Values ri = mini is the species density, mi and ni are the species mass and number density, r ¼ Pri is the mixture density; v is the macroscopic gas velocity, e is the mixture total energy per unit mass; Ri t , R12, R3 are the production terms due to dissociation, recombination, exchange reactions and slow processes of CO2 vibration relaxation; Vi is the diffusion velocity; P is the pressure tensor; and q is the heat flux.

We consider the condition that corresponds to rapid translational and rotational relaxation, VVm is the vibration energy exchanges within modes and VV0 12 is the exchange between symmetric and bending CO2 modes. In this case the vibration CO2 distributions depend on the vibration temperatures T12 of the combined (symmetric + bending) mode, and T3 of the asymmetric mode. The vibration distributions of CO and O2 are supposed to be close to the thermal equilibrium; vibration spectra are simulated by the harmonic oscillator model. Values Evibr1(T12,T3) = E12(T12) + E3(T3), Evibr2(T), Evibr3(T) are the specific vibration energies of molecular species CO2, CO, and O2, respectively; thus E12(T12), E3(T3) are the specific vibration energies of non-equilibrium CO2 modes.

The equations of non-equilibrium vibration kinetics written in the following form

$$
\rho\_1 \frac{dE\_{12}}{dt} + \nabla \cdot \mathbf{q}\_{12} = R\_{12} - m\_1 E\_{12} R\_1' + E\_{12} \nabla \cdot (\rho\_1 \mathbf{V}\_1) \tag{9}
$$

derivation leads to the definition of two coefficients of viscosity: shear coefficient of viscosity and

Numerical Modeling of Hypersonic Aerodynamics and Heat Transfer Problems of the Martian Descent Modules

Transport coefficients (bulk and shear viscosity, heat conductivity, diffusion, pressure- and thermo-diffusion of multi-component gas mixture) are calculated according to the basic kinetic theory. The algorithms for thermal conductivity, vibration thermal conductivity, diffusion,

Here [A;B] are the bracket integrals depending on the cross sections of rapid processes (see for

The values of diffusion fluxes can be written down through thermodynamic forces (external

η ¼ kT B½ � ; B =10, ζ ¼ kT F½ � ; F , prel ¼ kT F½ � ; G (12)

∇ln T (13)

http://dx.doi.org/10.5772/intechopen.71666

47

ij, DTi are connected to the

; Ai � �=3n: (15)

cij, Bcij, Fcij, Fcij are found from the linear integral equa-

� �∇ln p: (14)

DTi

ij is the multi-component coefficients of diffusion, DTi is the coefficients of thermo-

nj n � cj

Here nj is the number of jth mole components, n is the common mole's number. Let us take the

kinetic theory methods. In order to define the multi-components diffusion coefficients, it is necessary to solve the system of the linear algebraic equations that in case of gas mixture have

; Di � �=3n, DTi ¼ Di

tions for the first-order correction to the distribution function. The relaxation pressure determine by the slow non-equilibrium processes. This quantity is basically supposed to be small

The total heat transfer of a multi-component mixture is defined by effects of heat conductivity

; Ai � � are the bracket integrals depending on the cross sections of rapid

ij and Dij, where Dij is the binary diffusion coefficients.

thermal diffusion, shear, and bulk viscosity coefficients computation are developed.

These transport coefficients are defined by functions Bcij, Fcij, Gcij:

Ji ¼ �ri

dj ¼ ∇

The basic way of finding the independent transfer coefficients D∗

Dij ¼ Di

cij , Að Þ3

cij , Að Þ c;1

of various kinds of energy and diffusion. The heat flux is given by the formula

X j D∗ ijdj � ri

nj n � � þ

coefficient of bulk viscosity.

instance [7, 8] for definition).

mass forces it is neglected):

difference between D∗

; Di � �, Di

processes. Functions Acij, Að Þ 12

compared to p, and by value prel is neglected.

the next form

Here Di

diffusion, dj is the diffusion driving forces:

Here D∗

$$
\rho\_1 \frac{dE\_3}{dt} + \nabla \cdot \mathbf{q}\_3 = R\_3 - m\_1 E\_3 R\_1' + E\_3 \nabla \cdot (\rho\_1 \mathbf{V}\_1) \tag{10}
$$

xi is the molar fraction of species i (1—CO2, 2—CO, 3—O2, 4—C, 5—O). Values q12, q3 are the fluxes of vibration energy in the combined and asymmetric modes, respectively.

The vibration relaxation of molecules time of СО2 molecules is calculated under the usual formulas by approximation of theoretical and experimental data for particles of different types.

### 3. Transport and source terms

The transport theory of polyatomic gas mixtures taking into account internal molecular structure, different rates of vibration transitions, and unharmonicity has been developed for a five component CO2/O2/CO/O/C mixture taking into account vibration excitation of diatomic molecules.

The transport properties in the viscous gas approximation are determined by the first-order distribution functions. The zero-order and the first-order distribution functions are known and express by means different gradients, the diffusive driving forces, etc. [7, 8].

Pressure tensor, diffusion velocity, heat flux, and vibration energy fluxes are expressed in terms of macroscopic parameters gradients and transport coefficients.

The pressure tensor is obtained in the form:

$$\mathbf{P} = \begin{pmatrix} p - p\_{rel} \end{pmatrix} \text{ I} - 2\eta \text{ S} - \zeta \nabla \cdot \mathbf{v} \text{I}. \tag{11}$$

Here S is the strain rate tensor, I is the unit tensor, η is the shear viscosity coefficients, and ζ and prel are the bulk viscosity coefficient and relaxation pressure appearing in the diagonal elements of the pressure tensor due to rapid inelastic non-resonant processes. The Navier-Stokes equation derivation leads to the definition of two coefficients of viscosity: shear coefficient of viscosity and coefficient of bulk viscosity.

Transport coefficients (bulk and shear viscosity, heat conductivity, diffusion, pressure- and thermo-diffusion of multi-component gas mixture) are calculated according to the basic kinetic theory. The algorithms for thermal conductivity, vibration thermal conductivity, diffusion, thermal diffusion, shear, and bulk viscosity coefficients computation are developed.

These transport coefficients are defined by functions Bcij, Fcij, Gcij:

We consider the condition that corresponds to rapid translational and rotational relaxation,

symmetric and bending CO2 modes. In this case the vibration CO2 distributions depend on the vibration temperatures T12 of the combined (symmetric + bending) mode, and T3 of the asymmetric mode. The vibration distributions of CO and O2 are supposed to be close to the thermal equilibrium; vibration spectra are simulated by the harmonic oscillator model. Values Evibr1(T12,T3) = E12(T12) + E3(T3), Evibr2(T), Evibr3(T) are the specific vibration energies of molecular species CO2, CO, and O2, respectively; thus E12(T12), E3(T3) are the specific vibration

The equations of non-equilibrium vibration kinetics written in the following form

dt þ ∇ � q12 ¼ R12 � m1E12Rr

dt þ ∇ � q3 ¼ R3 � m1E3Rr

fluxes of vibration energy in the combined and asymmetric modes, respectively.

express by means different gradients, the diffusive driving forces, etc. [7, 8].

terms of macroscopic parameters gradients and transport coefficients.

P ¼ p � prel

xi is the molar fraction of species i (1—CO2, 2—CO, 3—O2, 4—C, 5—O). Values q12, q3 are the

The vibration relaxation of molecules time of СО2 molecules is calculated under the usual formulas by approximation of theoretical and experimental data for particles of different types.

The transport theory of polyatomic gas mixtures taking into account internal molecular structure, different rates of vibration transitions, and unharmonicity has been developed for a five component CO2/O2/CO/O/C mixture taking into account vibration excitation of diatomic molecules. The transport properties in the viscous gas approximation are determined by the first-order distribution functions. The zero-order and the first-order distribution functions are known and

Pressure tensor, diffusion velocity, heat flux, and vibration energy fluxes are expressed in

Here S is the strain rate tensor, I is the unit tensor, η is the shear viscosity coefficients, and ζ and prel are the bulk viscosity coefficient and relaxation pressure appearing in the diagonal elements of the pressure tensor due to rapid inelastic non-resonant processes. The Navier-Stokes equation

I � 2η S � ζ∇ � vI: (11)

12 is the exchange between

1 þ E12∇ � ð Þ r1V1 (9)

1 þ E3∇ � ð Þ r1V1 (10)

VVm is the vibration energy exchanges within modes and VV0

energies of non-equilibrium CO2 modes.

46 Advances in Some Hypersonic Vehicles Technologies

3. Transport and source terms

The pressure tensor is obtained in the form:

r1 dE12

> r1 dE3

$$
\eta = kT[B,B]/10,\ \zeta = kT[F,F].\ p\_{rel} = kT[F,G] \tag{12}
$$

Here [A;B] are the bracket integrals depending on the cross sections of rapid processes (see for instance [7, 8] for definition).

The values of diffusion fluxes can be written down through thermodynamic forces (external mass forces it is neglected):

$$\mathbf{J}\_i = -\rho\_i \sum\_{\vec{j}} D\_{i\vec{j}}^\* \mathbf{d}\_{\vec{j}} - \rho\_i D\_{T\_i} \nabla \ln T \tag{13}$$

Here D∗ ij is the multi-component coefficients of diffusion, DTi is the coefficients of thermodiffusion, dj is the diffusion driving forces:

$$\mathbf{d}\_{\circ} = \nabla \left( \frac{n\_{\circ}}{n} \right) + \left( \frac{n\_{\circ}}{n} - c\_{\circ} \right) \nabla \ln p. \tag{14}$$

Here nj is the number of jth mole components, n is the common mole's number. Let us take the difference between D∗ ij and Dij, where Dij is the binary diffusion coefficients.

The basic way of finding the independent transfer coefficients D∗ ij, DTi are connected to the kinetic theory methods. In order to define the multi-components diffusion coefficients, it is necessary to solve the system of the linear algebraic equations that in case of gas mixture have the next form

$$D\_{\vec{\eta}} = \left[\mathbf{D}^{\dot{\mathbf{i}}}, \mathbf{D}^{\dot{\mathbf{i}}}\right] / \mathfrak{Im}, \ D\_{\overrightarrow{\mathrm{Ti}}} = \left[\mathbf{D}^{\dot{\mathbf{i}}}, \mathbf{A}^{\dot{\mathbf{i}}}\right] / \mathfrak{Im}. \tag{15}$$

Here Di ; Di � �, Di ; Ai � � are the bracket integrals depending on the cross sections of rapid processes. Functions Acij, Að Þ 12 cij , Að Þ3 cij , Að Þ c;1 cij, Bcij, Fcij, Fcij are found from the linear integral equations for the first-order correction to the distribution function. The relaxation pressure determine by the slow non-equilibrium processes. This quantity is basically supposed to be small compared to p, and by value prel is neglected.

The total heat transfer of a multi-component mixture is defined by effects of heat conductivity of various kinds of energy and diffusion. The heat flux is given by the formula

$$\mathbf{q} = -\lambda^{\prime}\nabla T - \lambda\_{12}\nabla T\_{12} - \lambda\_{3}\nabla T\_{3} - p\sum\_{i=1}^{5}D\_{T\_{i}}\mathbf{d}\_{i} + \sum\_{i=1}^{5}\rho\_{i}h\_{i}\mathbf{V}\_{i} \tag{16}$$

theoretically in the literature [18] for weak and strong non-equilibrium conditions but up to

Numerical Modeling of Hypersonic Aerodynamics and Heat Transfer Problems of the Martian Descent Modules

The experiments and kinetic theory show that bulk viscosity can significantly influence shock wave structure for polyatomic molecules. In polyatomic gases of the deviation from local equilibrium effects itself as bulk viscosity. From the Chapman-Enskog's theory, it can be proved that for any perfect monoatomic gas, the coefficient of bulk viscosity is equal to zero.

For harmonic oscillators, rapid inelastic VV and VV0

therefore do not give contribution to the coefficient ζ.

Coefficients fi,mn are the solutions of the system

11 β2�3

11 β3�3

11 β4�3

11 β5�3

0110 β1�3

0110 β2�3

0110 β3�3

Here matrix B is composed of the bracket integrals β

11 β2�4

11 β3�4

11 β4�4

11 β5�4

0110 β1�4

0110 β2�4

0110 β3�4

x1ctr x2ctr x3ctr x4ctr x5ctr x1crot, 1 x2cint, 2 x3cint,3

11 β2�1

11 β3�1

11 β4�1

11 β5�1

0110 β1�1

0110 0 β2�2

1001 β2�2

1001 β3�2

1001 β4�2

1001 β5�2

0110 0 0 β3�3

0011 0 0

Matrix B, vectors f and s are given above. Here сtr, crot, сint are specific heats of translation, rotational, internal degrees of freedom and cu is the total specific heat. The bracket integrals

11 β2�5

11 β3�5

11 β4�5

11 β5�5

0110 β1�5

0110 β2�5

0110 β3�5

The bulk viscosity coefficient can be obtained in the form [18]:

gas.

4—C, 5—O).

β2�1 11 β2�2

0

BBBBBBBBBBBBBBBBBBB@

β3�1 11 β3�2

β4�1 11 β4�2

β5�1 11 β5�2

β1�1 0110 β1�2

β2�1 0110 β2�2

β3�1 0110 β3�2

B ¼

Here p is the pressure, hi is the specific enthalpy of species i, λ 0 ¼ λtr þ λr þ λν is the thermal conductivity coefficient of all degrees of freedom which deviate weakly from local thermal equilibrium. They include the translational and rotational modes as well as CO and O2 vibration degrees of freedom. Thus, the coefficient λv = λv,CO + λr,O2. Coefficients λ12, λ3 correspond to the thermal conductivity of strongly non-equilibrium modes: combined (symmetric + bending) and asymmetric ones.

The fluxes of vibration energy in the combined and asymmetric CO2 modes in the harmonic oscillator approach depend only on the gradient of corresponding vibration temperature:

$$\mathbf{q}\_{12} = -\lambda\_{12}\nabla T\_{12} \quad \mathbf{q}\_3 = -\lambda\_3 \nabla T\_3 \tag{17}$$

The thermal conductivity coefficients are expressed in terms of bracket integrals

$$\boldsymbol{k}\lambda = k[\mathbf{A}, \mathbf{A}]/3, \ \lambda\_{12} = k\left[\mathbf{A}^{(12)}, \mathbf{A}^{(12)}\right]/3, \lambda\_3 = k\left[\mathbf{A}^{(3)}, \mathbf{A}^{(3)}\right]/3 \tag{18}$$

The algorithms of transport coefficients calculation are similar for various multi-temperature models and consist of the following steps:

For example, let us consider the bulk viscosity coefficient. In Navier-Stokes equations, the terms involving bulk viscosity multiplied by divergence of velocity and can play a significant role in flow fields with substantial dilatation. The bulk viscosity coefficient was discussed theoretically in the literature [18] for weak and strong non-equilibrium conditions but up to now it was not evaluated in real gas flows.

The experiments and kinetic theory show that bulk viscosity can significantly influence shock wave structure for polyatomic molecules. In polyatomic gases of the deviation from local equilibrium effects itself as bulk viscosity. From the Chapman-Enskog's theory, it can be proved that for any perfect monoatomic gas, the coefficient of bulk viscosity is equal to zero. Bulk viscosity results from contributions from the several internal degrees of freedom of the gas.

The bulk viscosity coefficient is defined by rotational energy transitions of all molecular species and VT vibration energy transfer in CO and O2 molecules and can be written as follows

$$
\zeta = \zeta\_r + \zeta\_{v, \text{CO}} + \zeta\_{v, O\_2} \tag{19}
$$

For harmonic oscillators, rapid inelastic VV and VV0 1�2 exchanges occur to be resonant and therefore do not give contribution to the coefficient ζ.

The bulk viscosity coefficient can be obtained in the form [18]:

$$\mathcal{L} = -kT\Big(\mathbf{x}\_1 f\_{1,10} + \mathbf{x}\_2 f\_{2,10} + \mathbf{x}\_3 f\_{1,10} + \mathbf{x}\_4 f\_{4,1} + \mathbf{x}\_5 f\_{5,1}\Big)\_{\prime} \tag{20}$$

where k is the Boltzmann constant, xi is the molar fraction of species i (1—CO2, 2—CO, 3—O2, 4—C, 5—O).

Coefficients fi,mn are the solutions of the system

q ¼ �λ0

48 Advances in Some Hypersonic Vehicles Technologies

models and consist of the following steps:

bracket integrals as coefficients.

1. Unknown functions Acij, Að Þ 12

tion function.

ing) and asymmetric ones.

∇T � λ12∇T12 � λ3∇T3 � p

The thermal conductivity coefficients are expressed in terms of bracket integrals

cij , Að Þ c;1

λ ¼ k½ � A; A =3, λ12 ¼ k Að Þ 12 ; Að Þ 12 h i

cij , Að Þ3

2. Transport coefficients are expressed in terms of expansion coefficients.

conductivity coefficient of all degrees of freedom which deviate weakly from local thermal equilibrium. They include the translational and rotational modes as well as CO and O2 vibration degrees of freedom. Thus, the coefficient λv = λv,CO + λr,O2. Coefficients λ12, λ3 correspond to the thermal conductivity of strongly non-equilibrium modes: combined (symmetric + bend-

The fluxes of vibration energy in the combined and asymmetric CO2 modes in the harmonic oscillator approach depend only on the gradient of corresponding vibration temperature:

The algorithms of transport coefficients calculation are similar for various multi-temperature

3. Integral equations are reduced to the linear systems of algebraic equations involving

4. Bracket integrals are simplified on the basis of some assumptions about cross sections of

5. Transport coefficients are found as solutions of transport linear systems using some numerical algorithms (for instance, the Gauss method or new iterative procedures). For example, let us consider the bulk viscosity coefficient. In Navier-Stokes equations, the terms involving bulk viscosity multiplied by divergence of velocity and can play a significant role in flow fields with substantial dilatation. The bulk viscosity coefficient was discussed

calculated for particular models of inter-molecular interaction potentials. In the present study, the Lennard-Jones potential is used for low and moderate temperatures whereas in

rapid processes; finally they are expressed in terms of the standard Ωð Þ l;r

and relaxation times which can be measured experimentally.  Here сtr, crot, сint are specific heats of translation, rotational, internal degrees of freedom and cu is the total specific heat. The bracket integrals β ij mm0 nn0 required for the evaluation of the bulk viscosity coefficient are determined by the energy variation in inelastic processes. The system (22) in this form has a unique solution.

The rates of vibration energy transitions are expressed in terms of corresponding relaxation times. The rate coefficients for non-equilibrium CO2 dissociation were calculated using the expressions proposed in Ref. [8] as an extension of the Treanor-Marrone's model [20] for threeatomic molecules. For the recombination rate coefficients, the detailed balance principle is used. For the rate coefficients of exchange reactions and dissociation of diatomic molecules, the Arrhenius formulas are applied. The vibration relaxation of molecules time of СО2 molecules is calculated under the usual formulas by approximation of theoretical and experimental data for particles of different types [10].