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Chaotic behavior of human thermal plumes in an aircraft cabin mockup
International Journal of Heat and Mass Transfer 119 (2018) 223–235
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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Chaotic behavior of human thermal plumes in an aircraft cabin mockup Congcong Wang a, Junjie Liu a,⇑, Jiayu Li a, Fei Li b a b
Tianjin Key Laboratory of Indoor Air Environmental Quality Control, School of Environmental Science and Engineering, Tianjin University, Tianjin 300072, China Department of HVAC, College of Urban Construction, Nanjing Tech University, Nanjing 210009, China
a r t i c l e
i n f o
Article history: Received 25 May 2017 Received in revised form 10 October 2017 Accepted 13 November 2017
Keywords: Human thermal plume Mini-PIV Gaussian mixture model Spectrum analysis Phase Space Reconstruction Chaotic behavior
a b s t r a c t The human thermal plume induced by body heat loss has a significantly impact on human thermal comfort, contaminant transport and indoor air quality. Few studies focused on the temporal unsteady characteristics of human thermal plume. In this study, the human thermal plume generated by a heated manikin was measured in a 7-row cabin mockup by mini particle image velocimetry (mini-PIV) system; and its unsteady and chaotic behavior was determined out of statistical and chaotic method. Probability density distributions of velocity time series of human thermal plumes presented Gaussian mixture models with two peaks, which substantiated the oscillating characteristics of human thermal plumes. The energy region of the human thermal plume was concentrated between 0.1 Hz and 10 Hz determined out of the power spectrum analysis, and the power spectrum exponent of the human thermal plume above the head ranged from 0.9 to 1.2. Evolution of phase space reconstruction of velocity time series from single-spindle to double-spindle revealed the human thermal plume presents obvious autocorrelation and oscillating behavior qualitatively. In addition, the fractal dimension of human thermal plumes overhead ranged from 6 and 12 without integers and Kolmogorov entropies of analyzed points were all larger than zero indicating the human thermal plume was kind chaotic airflow quantitatively. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction The human thermal plume induced by body heat loss has a significantly impact on human thermal comfort, contaminant transport and indoor air quality in enclosed environments. A large amount of studies have been conducted to investigate the airflow and temperature distributions of the human thermal plume and its influence on propagation of gaseous and particle. Melikov 2015 demonstrated thermal plumes above the head and the convective boundary layer near human body have impacts on the human thermal comfort and air quality in breathing zone [1]. To quantify the airflow characteristic of human thermal plumes, computational fluid dynamics (CFD) and experimental measurements were used extensively. Murakami et al. [2,3] studied the wind environment around heated human body and concluded the velocity of thermal plume can be up to 0.23 m/s. Their numerical simulation results were verified merely by the smoke tracer method qualitatively. Dan et al. [4] used numerical simulation to expose the thermal plume airflow and heat transfer of a seated human body and recommended low-Re turbulence model to determine the human thermal plume airflow and temperature. Liu et al. [5]
⇑ Corresponding author. E-mail address: [email protected] (J. Liu). https://doi.org/10.1016/j.ijheatmasstransfer.2017.11.059 0017-9310/Ó 2017 Elsevier Ltd. All rights reserved.
analyzed the human thermal plume by large eddy simulation (LES) and found unsteady and intermittent characteristics of the thermal plume. Unfortunately, Liu’s numerical results lacked of experimental verification. For experimental studies, Graven et al. [6] determined the airflow field of human thermal plume qualitatively and quantitatively by shlieren image and particle image velocimetry (PIV), respectively. However, the human model in Graven’s study was too simple to reflect real human surface temperature. Zukowska’s [7] results showed that the airflow distribution of the thermal plume is closely related to the geometry, the surface temperature and area of the human body. For different clothing, breathing and body posture of the heated manikin, its convective boundary layer was investigated by PIV and pseudo color visualization (PCV) in a chamber without ventilation [8]. They found personal factors such as clothing and breathing can affect the volume flux of thermal plume and cause the asymmetric velocity distribution [13,14]. Koelblen et al. [9] investigated the airflow and temperature pattern of the thermal plume and found the velocity field varies with body postures. Voelker et al. [10] measured an ascending plume above the head and found the human thermal plume expanded growing up from the lower part of the torso. Particularly, human thermal plumes play a more significant role in forming the indoor environment of aircraft cabin and have impact on airflow distributions and
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pollutant transport because of large heat density from passengers and narrow space of aircraft cabin [11–13]. While, the characteristics of pure thermal plume generated by passengers are not clearly. Study of the characteristics of human thermal plumes can provide theoretical support for the design of better airflow distribution to rapidly eliminate body heat dissipation and keep passengers thermal comfort. The schlieren video of human thermal plume recorded by Gary [6] and PCV measured results [8] qualitatively revealed that human thermal plumes present unsteady and oscillating behavior. From our literature review, the average velocity field of human thermal plumes was studied emphatically, while few studies focused on unsteady and turbulent characteristics. A number of researchers have studied the periodicity and oscillating characteristics of pure thermal plumes from heated flat and analyzed those airflows by nonlinearity and chaotic theory [14–18]. These chaotic method can explain the correlation between the turbulent characteristics of airflow and the human body draft sensation. The main purpose of this paper is to study the unsteady and chaotic behavior of the human thermal plume. A mini-PIV system was utilized to measure the human thermal plume in a 7-row cabin mockup. In order to further understand the airflow mechanism of human thermal plumes and provide experimental data support for human comfort analysis, indoor air quality and numerical simulation, statistical method, spectrum method and chaotic method are used to study the airflow mechanism of the human thermal plume and analyze the experimental data. The statistical analysis, spectrum analysis and phase space reconstruction methods are used to substantiate the unsteady characteristics of human thermal plume qualitatively. To further analyze the unsteady characteristics of thermal plume quantitatively, the chaos theory including fractal dimension denoting to the relation between phase space structure feature and embedding dimension and Kolmogorov entropy representing quantification of chaos of transformed velocity time series based on phase space reconstruction based on are employed. 2. Experiment setup A heated manikin with a thermal power of 75w sitting in a fullscale 7-row single-aisle aircraft cabin mockup was served as passenger generating thermal plume. The prototype of the aircraft cabin mockup is a section of Boeing 737–200. The geometry sizes of the aircraft cabin mockup are 5.85 m in length, 3.25 m in width and 2.15 m in height. Particularly, the luggage rack height is about 1.6 m. In order to facilitate the adoption of laser, the walls of the middle three rows were constructed by transparent acrylic. In order to guarantee the controllability of temperature boundary condition, the aircraft cabin mockup was set up in a thermostatic chamber. To meet the objectives of this investigation, the manikin was wrapped with electrical resistance, a total heat of 75 W. The manikin’s sitting height is 1.4 m with a total volume 0.055 m3. The surface area of each manikin is 1.3 m2. The geometry of the manikin is similar to a real person, according to Chinese adult body size and seating space [19]. A mini-PIV system was utilized to determine the human thermal plume composing of a continuous wave laser and a CCD camera with a Canon 35 mm lens. Tracing particles were generated by diethylhexyl sebacate (DEHS) through a spray generator. As shown in Fig. 1, three cross sections were measured above the shoulders and head with each field area is about 0.32 m 0.16 m. Due to the obstruction of the luggage rack, there is a certain deviation between the center of the human head and the center of measured cross section, i.e., CS2. In the measurement process, there was no ventilation in the aircraft cabin mockup and temperatures were controlled by the ther-
mostatic chamber with closed circulatory air condition system. Thermocouples were used to monitor temperature distributions of the manikin, aircraft cabin wall and ambient air. When the monitoring temperatures were stable, the experiment was started [20]. The constant temperature distributions in measurement process are listed in Table 1 with manikin divided into five parts. The temperatures of aircraft cabin wall and air surrounding manikin are 19.5 °C similarly. In order to accurately sample the turbulent characteristics of the human thermal plume, the sampling frequency of mini-PIV was set as 400 Hz and recording time was 100 s. 3. Result 3.1. Velocity and vorticity Fig. 2 shows the average velocity distributions of the human thermal plume with sampling time 100 s. In the figures, vectors and contour represent velocity directions and velocity magnitude, respectively. Human thermal plumes rise from two side shoulders and collide with each other above the head. Maximum velocity appearing at CS2 located above the head is up to 0.24 m/s. The overall thermal plume flows upward in the CS2 section while moving toward the manikin body in CS1 and CS3. More close to the manikin body, the velocity is larger in CS1 and CS3. Evidently, the thermal plume expands growing up upward via entrain ambient air in CS1 and CS3 driven by buoyancy. The airflow fields above the left and right shoulders are not completely symmetrical because of obstruction of luggage rack and asymmetrical heated manikin surface. In order to demonstrate the unsteady characteristics of human thermal plumes, instantaneous velocity distributions of CS2 section measured at 5th, 10th, 15th and 20th seconds are presented in Fig. 3. The airflow patterns of the four moments are basically different to each other and inconsistent with the average velocity distributions. The maximum velocity at 5th and 15th seconds is about 0.25 m/s while that is up to 0.3 m/s at 10th and 20th seconds. The velocity near the head is lower because of heat loss at that part is less simulating human with hair. Moreover, two strands thermal plumes rising from shoulders appear above head alternately. Summarily, the instantaneous velocity distributions of CS2 substantiate the asymmetrical and oscillating behavior of the human thermal plume. Vorticity is introduced to demonstrate the unsteady turbulent characteristic of human thermal plume which can be calculated as follows [21]:
w¼
@ v @u @x @y
ð1Þ
where v and u represent velocity components at x and y direction, respectively. Fig. 4 shows the instantaneous vorticity distribution of CS2 at 5th, 10th, 15th and 20th seconds. Irregular shape and random distribution of vorticity contours appear on the four moments inconsistently. The maximum vorticity can be up to 12 s1. At the center part of CS2, the sizes of vorticity are more obvious because of the collision of two thermal plumes rising from shoulders. The vorticity appearing on edge position of cross sections may cause by air entrainment. The obvious vorticity distributions also reveal high fluctuation and turbulent kinetic energy of thermal plume. The random and irregular vorticity contours reveal unsteady behavior of the human thermal plume. For the convenience of the researches on temporal unsteady and chaotic behavior of the human thermal plume, several points are selected on CS1, CS2 and CS3 as shown in Fig. 5. In order to focus on the study of the thermal plume over the head in detail, more points are selected.
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Fig. 1. Schematic diagram of mini-PIV measurement of human thermal plume in the aircraft cabin mockup.
Table 1 Temperature distributions of human manikin in measurement process. Head
Chest
Belly
Thigh
Leg
Wall
Ambient air
27 °C
28 °C
30 °C
30 °C
28 °C
19.5 °C
19.5 °C
In order to verify whether the unsteady thermal plumes are periodical, continuous instantaneous velocity of points CS1-B, CS2-B and CS3-B are presented in Fig. 6. It can be seen that the velocity time series fluctuate between 0.05 m/s and 0.4 m/s. Comparing the velocity of the three points, the velocity of CS2 thermal plume is not lager than others despite combined of the other two thermal plumes. Therefore, the observation substantiates the two plumes appearing overhead alternately. Velocity sequences of CS1 and CS3 are inconsistent in velocity magnitude and temporal trend. Hence, the thermal plumes rising from left and right side shoulders are asymmetric because of asymmetrical manikin surface temperature and geometric space for plume growing up, potentially. The temporal trend of each velocity sequence is analyzed in detail, and the periodic fluctuation is not found. But the peaks and troughs of velocity sequences appear alternately as shown in Fig. 6 reveal oscillating characteristic of human thermal plumes above shoulders and head. 3.2. Statistical analysis In order to analyze the unsteady behavior of the instantaneous velocity sequences of the human thermal plume, the frequency distribution and probability density distribution were introduced in this paper. The frequency distribution was determined out of velocity magnitude time series with 40,000 intervals of the measured thermal plume by the mini-PIV. The probability density distribution was calculated by the relative frequency of velocity magnitude and classified interval. According to study of Coleman and Steele [22], if the sampling signal is continuously stable, the frequency distribution of measured results is close to a normal distribution curve. Therefore, Gaussian model was used to fit the probability density distribution to analyze the unsteady characteristic of the human thermal plume.
Fig. 7 reflects the frequency distribution, probability density distribution and Gaussian fitted curves of three points of each measured cross section, i.e., A, B and C in CS1 and CS3. In the figure, relative frequency represents the frequency distribution determined out of 40,000 intervals velocity magnitudes. Meanwhile, the probability density distribution of velocity relative frequency is described by a cumulative fit curve. G1(x) and G2(x), as shown in Fig. 7, are represented Gaussian fitted curves. Apparently, the probability density distributions of the six points as shown in Fig. 7 represent mixed Gaussian model, i.e., there are two obvious peaks of relative frequency. As a consequence, the velocity magnitude time series are concentrated upon two median velocities, i.e., two peaks which means the human thermal plumes present unsteady oscillating characteristic. The magnitudes and frequency of the medians of the two Gaussian models at each point are different, respectively. At point C, the frequency of G2(x) is too small can be ignored. While, at point A, The frequency of low speed is dominant. In other words, the more close to the human body, the velocity is larger and the corresponding frequency is higher. Comparing CS1 and CS3, the major difference is at point A which reveals average velocity of CS3 is larger than CS1. Mixed Gaussian characteristics are more obvious at point A, i.e., the more obvious oscillating behavior of human thermal plume. The two frequency peaks with different velocity magnitude represent two different stream thermal plumes. Fig. 8 shows the probability density distributions of measured thermal plume overhead, i.e., the CS2. Detail positions of sampling points of CS2 are shown in Fig. 6. Obviously, the two frequency peaks of the fitted curves become increasingly evident from point A to F. In other words, Gaussian mixture models are more accurate representation of probability density. At point A, the G1(x) curve is corresponding to the cumulative fit curve and G2(x) can be ignored. Therefore, the velocity series of thermal plume closest to
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Fig. 2. Average velocity distributions of the human thermal plume.
the head tends to normal distribution. The relative frequency of G1 (x) is higher than G2(x) at points A, B and C with smaller velocity magnitudes. Conversely, the relative frequency of G1(x) is lower than G2(x) at points D, E and F far away from the head. Meanwhile, the median velocity and relative frequency of G2(x) grows larger as far away from the head. The relative frequency of two Gaussian fitted curves are similar, which reveals the probability of plume occurrence with different velocity magnitude is equal. But, at other positions, i.e., D, E and F the unsteady thermal plumes are driven by large velocity plumes with higher frequency. Comparing Gaussian mixture models of CS1, CS2 and CS3, the thermal plume far away from the manikin thermal boundary layer present obvious instability and oscillating airflows. The thermal plume above head
is a mixture of two strands of plumes rising from shoulders. Summarily, the human thermal plumes are unsteady and oscillating airflows with different velocity and different relative frequency. 3.3. Spectrum analysis The probability density distributions show the unsteady behavior of the human thermal plume qualitatively. While, in order to analyze the periodicity of the unsteady velocity series, the power spectrum was introduced in this paper. The power spectrums are determined out of fast Fourier Transform of the velocity series measured by mini-PIV with sampling frequency 400 Hz. Fig. 9
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Fig. 3. Instantaneous velocity distributions of CS2.
Fig. 4. Instantaneous vorticity distributions of CS2.
shows the power spectrum distributions of sampling points, i.e., A, B, C, D, E and F of CS2. In Fig. 9, vertical axis and abscissa axis represent power and discrete sampling frequency, respectively, by logarithmic coordinate axis. Because of fast Fourier Transformation is an even function, the figures shows symmetrical half curves. From the observation of the power spectrum, energy region, inertial sub-region and dissipation region can be quantified, the characteristics of these regions are concluded. The inertial sub-region of the power spectrum concentrate on area of curves with slope is minus 5/3. And the frequency of the inertial sub-region is about 5 Hz as shown in Fig. 9. Therefore, when the human thermal plume was studied by the CFD method with LES, the time step of simulation recommended should be less than 0.2 s [23]. And the sampling
frequency of instrument measured the human thermal plume should be larger than 20 Hz, in order to obtain the overall turbulence information. The dissipation region of the power spectrum located behind 20 Hz. While, the energy region of the power spectrum concentrates between 0.02 Hz and 1 Hz. Therefore, the turbulent airflows of the human thermal plume are mostly low frequency and large scale structures. Obviously, the turbulent kinetic energy distribution fluctuates strongly in the energy region. And there is no prominent frequency with highest energy which reveals the human thermal plume overhead fluctuate nonperiodically. Comparison of the six sampling points, the turbulent kinetic energy in the energy region increases with the distance far from the manikin head. Therefore, the airflows of the thermal
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Fig. 5. Positions of research points on CS1, CS2 and CS3.
plume furthest from the head, fluctuate most strongly and unsteadily which corresponding to the results of probability density distributions. In order to quantify impact of unsteady behavior of the thermal plume on human thermal comfort, the power spectrum exponent is defined as follows:
b¼
ln G ln f
ð2Þ
In practical calculation, b is the slope of curve fit of lnG and lnf by least square method. According to Zhu’s study [24], the power spectrum exponent of nature wind with most comfortable is between 1.1 and 2, while that of mechanical wind is less than 1.1. In Fig. 9, the red line is the fitted curves of lnG and lnf. The b values of human thermal plume overhead range from 0.9 to 1.2, which reflect the comfort of human thermal plumes are between nature wind and mechanical wind.
Fig. 10 shows the accumulation of power spectrum energy of sampling points, i.e., A, B, C, D, E and F at cross section 2. From the observation of Fig. 10, the energy accumulations increase rapidly between 0.1 Hz and 10 Hz, which represents the energy regions of sampling points range from 0.1 Hz to 10 Hz corresponding to the results of power spectrum distributions. Moreover, the energy accumulations reach up to 90% at frequency 10 Hz. Therefore, the turbulent airflows of human thermal plume mostly contain low frequency structures. After frequency 10 Hz, the energy of point A is larger than F, which reflects the human thermal plumes closest to the head contain higher frequency and small scale structures than that farthest from head.
3.4. Phase space reconstruction The unsteady behavior of human thermal plume reflects the chaotic airflows characteristic. To examine the chaotic behavior
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Fig. 6. Continuous instantaneous velocity variations of points CS1-B, CS2-B and CS3-B.
Fig. 7. Probability density distributions of sampling points of CS1 and CS3.
of the human thermal plume, the phase space reconstruction was introduced in this paper. We reconstructed the original onedimensional univariate time series to phase space reconstruction with Takens principle [25] as follows:
xi ¼ ðx1 ; x2 ; . . . ; xN Þ
X i ¼ xi ; xiþ1 ; . . . ; xiþðm1Þs
ð3Þ
ð4Þ
where xi is the one-dimensional univariate time series; Xi is the phase space series after reconstructing; m presents the embedding dimension; s denotes the delay time; N is the sample number of univariate time series; i ¼ 1; 2; . . . ; M; M ¼ N ðm 1Þs, M denotes
the number of phase points in Xi. In this study 40,000 intervals of velocity magnitude measured by mini-PIV are served as the onedimensional univariate time series. While, the two-dimensional phase space series (i.e. m = 2) are analyzed. And the six sampling points at CS2, i.e., A, B, C, D, E and F are analyzed in Fig. 11. For the one-dimensional univariate time series, the autocorrelation function method can be used to determine the delay time. When the autocorrelation function drops up to 1–1/e of initial value, the corresponding time s is the optimal delay time of reconstructed phase space. The two-dimensional phase space reconstruction reflects the correlation and fluctuation of signal on time scale. Therefore, two-dimensional phase space reconstructions are deter-
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Fig. 8. Probability density distributions of sampling points of CS2.
Fig. 9. Power spectrum distributions of sampling points in CS2.
mined out of the human thermal plume time series measured by mini-PIV as shown in Fig. 11. In Fig. 11, the x-coordinate and y-coordinate, represent original time series and reconstructed series, respectively. The gathered points shape reflects the correlation of velocity series. Literatures show the shape of two-dimensional phase space reconstructions of natural wind and mechanical wind present spindle and ellipse,
which can be concluded that the velocity series correlation of natural wind is stronger than that of mechanical wind [24]. From observation of the phase space reconstruction of the point A, the contour shape presents a spindle shape, which shows that there is a strong correlation of the velocity time series. And most of velocity phase points range from 0.1 m/s to 0.2 m/s. With the changing of point’s positions, the global correlation attenuates
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Fig. 10. The energy accumulation distributions of power spectrum of sampling points in CS2.
Fig. 11. Phase space reconstruction evoluation of sampling points in CS2.
and the range of velocity magnitudes of phase points expands. Moreover, the phase space reconstruction points gradually change from the spindle to the discrete shapes with the distance from the head farther and farther. At points E and F, the phase space reconstruction points gather two spindles with different velocity ranges
corresponding to the results of Gaussian mixture model. This phenomenon shows that there are two strands airflows combining the human thermal plume. And, the phase space reconstruction with two spindles reveals oscillating characteristics of the human thermal plume. At points E and F, the velocity range of the two spindles
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Fig. 12. The C(r,m) and r relationship, log-log profile.
Fig. 13. The relationship between the correlation dimension and the embedding dimension.
in the phase space separates each other, while at points C and D, the phase space ellipses overlap with each other. Therefore, at points C and D, the human thermal plume form two kinds superimposed velocity time series with an overlapping velocity range around 0.2 m/s. Summarily, from the observation of phase space reconstruction, the velocity time series of the human thermal plume above head present unsteady characteristic and oscillating behavior. And, the human thermal plume presents strong correlation between velocities at different times.
Fig. 14. Kolmogorov entropy distributions of the human thermal plume overhead.
3.5. Fractal dimension The probability density distribution, power spectrum and phase space reconstruction substantiate the unsteady characteristic of human thermal plume qualitatively, only. In order to determine the unsteady and chaotic behavior of the human thermal plume quantitatively, the fractal dimension is calculated by GrassbergerProcaccia algorithm [26] based on the embedding theorem and theory of phase space reconstruction. Firstly, for one-dimensional univariate time series the correlation integral can be defined as:
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C ðr; mÞ ¼
1 ðN ðm 1ÞsÞ
M X M X H r X i X j 2 i¼1 j¼1
M X M 1 X H r r ij ¼ 2 M i¼1 j¼1
ð5Þ
m-dimensional cubic boxes with side length is r. For an attractor for state space and a trajectory x(t) in the domain of attraction, s denote a small time interval. p(i0, i1, . . . , im) denote the joint probth ability of that system trajectory located at the ith 0 box and at im box at time t = ms. And Kolmogorov entropy can be defined as following formula [27] (9):
where H denote Heaviside step function:
H r r ij ¼
1; r r ij 0 0; r r ij 0
K ¼ lim lim lim ð6Þ
rij denote the distances of metric space; r is the critical scaling. Secondly, in the critical scaling, the connection of C(r,m) and r can be described as following formula (7):
C ðr; mÞ / r D2
ð7Þ
where D2 is the correlation dimension. Finally, the fractal dimension of a strange attractor from a one-dimensional univariate time series can be denoted as:
D¼
log C ðr; mÞ m!saturation r!0 log r lim
lim
ð8Þ
Fig. 12 shows the logarithmic curves of correlation integral and critical scaling determined out of 4000 intervals velocity time series of six points located at CS2 measured by mini-PIV. In the calculation, the embedding dimension ranges from 2 to 40 and critical scaling value is 0.0625–1, which are sufficient to examine the fractal dimension of the human thermal plume. From the observation of Fig. 12, when the embedding dimension m is small, the slopes of linear parts of the curves of lnG and lnf are smaller and the gaps between adjacent curves are larger. While, the slopes of linear parts gradually increase and the gap width become smaller with the increase of m. Finally, the linear parts of the curves tend to parallel to each other, which reveals the embedding dimension tends to saturation and the correlation dimension is stable. Summarily, stable correlation dimension reflects the velocity series of human thermal plume are chaotic and the complexity of their system attractors is similar to correlative system characteristic. The positions of the linear part are not the same, that is to say, the complexities of the chaotic systems are not the same. Fig. 13 shows the relationship between the correlation dimension and the embedding dimension determined out of the slope of the linear part of each curve fitted by the least square method. As shown in Fig. 13, when the embedding dimension m reaches saturation, the correlation dimension does not change obviously and reaches a stable value. And, the stable value of correlation dimension is fractal dimension. The correlation dimension can reach saturation, which indicates that the human thermal plume is not a completely random airflow but a chaotic airflow. It can be seen that the fractal dimension of the chaotic characteristics of the thermal plume at the six points is not an integer between 6 and 12, which indicates that the thermal plume is a kind of chaotic airflow with correlation structure. The correlation dimension is a measurement of the complexity of the attractor in the phase space. From the observation of Fig. 14, the saturation embedding dimension ranges from 18 to 30, which reflects the degree of freedom in a chaotic system of the thermal plume. And, the airflow of point A has the highest degree of freedom. Therefore, the velocity series of point A of thermal plume overhead has more complex structure with more factors affecting velocity changing.
1 X
s!0 r!0 m!1 ms
pði0 ; i1 ; . . . ; im Þlog2 pði0 ; i1 ; . . . ; im Þ
while, order-q Renyi entropies are defined as follows:
K q ¼ lim lim lim
1
s!0 r!0 m!1 ms
X 1 log2 pq ði0 ; i1 ; . . . ; im Þ q1 i ...i 0
ð10Þ
m
To simplify the calculation of Kolmogorov entropy, Grassberger and Procaccia demonstrated K1is the Kolmogorov entropy and K2 is the best estimation of K1, i.e., the second order Renyi entropies [27] which can be defined as:
K 2 ¼ lim lim
1
s
r!0 m!1 n
log2
C m ðr Þ C nþm ðrÞ
ð11Þ
If the time series is chaotic, the Kolmogorov entropy is constant larger than zero [27]. And, if the Kolmogorov entropy is larger, the degree of the chaotic time series is larger with more information
3.6. Kolmogorov entropy To evaluate the chaotic human thermal plume quantitatively, Kolmogorov entropy was introduced as follows. The phase space of the m-dimensional dynamic system can be divided into several
ð9Þ
i0 ...im
Fig. 15. The detail surface temperature of heated manikin.
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loss. Fig. 14 shows the Kolmogorov entropy distribution of airflows of human thermal plume measured by mini-PIV. From the observation, the Kolmogorov entropy values are all larger than zero and increases with the distance far from the head, which reveals the velocity series of the six points are chaotic. And, the degree of chaotic of velocity series is largest at point F and lowest at point A. The human thermal plume becomes more chaotic with increase of distance from the head. 4. Discussion In this study, a heated manikin with a thermal power of 75 W was served as passenger generating thermal plume. While, the manikin surface temperature distribution has great influences on the unsteady characteristics of human thermal plume. The unsteady behavior of the human thermal plume was positively related to the temperature difference of human body surface. The velocity magnitudes of thermal plume under different surface temperature condition were different [28]. Therefore, the detail surface temperature distribution measured by the infrared thermal imager as shown in Fig. 15, which can be used as boundary condition of numerical and experimental research of human thermal plume. The sensitivity analysis of surface temperature distribution to airflow of human thermal plume can be evaluated qualitatively as following equation which described the velocity of a steady heated source [29].
U
@W @W 1 @ @ 2 w Þ u 2 þ gbDT þW ¼ ðr u w @r @z r @r @z
where U and W are the velocity of different directions, respectively. r and z are the coordinate of radial direction and vertical direction. u are fluctuating velocity. g denotes the gravitational acceleraand w tion. b represents the air thermal expansion coefficient. DT is the temperature difference between manikin surface and ambient air in this study. For a steady heat source, the velocity of a specific position is only related to DT. For the same experiment condition of human thermal plume, the velocity is linearly related to the temperature difference. 5. Conclusion In this paper, we focused on the unsteady and chaotic behavior of the human thermal plume determined by mini-PIV with frequency 400 Hz in a 7-row aircraft cabin mockup. Through the analysis of vorticity, probability density distribution, power spectrum and chaos theory of the measured thermal plume, we can draw conclusions as follows: 1. The maximum average velocity of the human thermal plume can be up to 0.24 m/s above the manikin head. And, the instantaneous velocity and vorticity distributions revealed the human thermal plume present unsteady and oscillating characteristic. 2. Probability density distributions of velocity time series of the human thermal plume form Gaussian mixture models with two peaks, which substantiated the oscillating characteristic of the human thermal plume. 3. The energy region of the human thermal plume concentrated between 0.1 Hz and 10 Hz determined out of the power spectrum analysis. The power spectrum exponent of the human thermal plume above the head ranges from 0.9 to 1.2 and the comfort level is between natural wind and mechanical wind. 4. The evolution of the phase space reconstruction of velocity time series from single-spindle to double-spindle revealed the human thermal plume presents obvious autocorrelation and oscillating behavior.
5. The fractal dimension of human thermal overhead ranges from 6 and 12 without integers, which demonstrated the human thermal plume is kind chaotic airflow. The Kolmogorov entropies of analyzed points were all larger than zero. With more and more far from the head, the chaotic characteristics of the human thermal plume were more complex. Conflict of interest We declare that no conflict of interest exists in the submission of this manuscript, and manuscript is approved by all authors for publication. Acknowledgments The research presented in this paper was supported by the National Natural Science Foundation of China (NSFC) through Grant No. 51278332. We would also like to thank for the support of the National Basic Research Program of China (The 973 Program) through Grant No. 2012CB720100. References [1] A.K. Melikov, Human body micro-environment: the benefits of controlling airflow interaction, Build. Environ. 91 (2015) 70–77. [2] S. Murakami, J. Zeng, T. Hayashi, CFD analysis of wind environment around a human body, J. Wind Eng. Ind. Aerodyn. 83 (1999) 393–408. [3] S. Murakami, S. Kato, J. Zeng, Combined simulation of airflow, radiation and moisture transport for heat release from a human body, Build. Environ. 35 (2000) 489–500. [4] D.N. Sørensen, L.K. Voigt, Modelling flow and heat transfer around a seated human body by computational fluid dynamics, Build. Environ. 38 (2003) 753– 762. [5] Y. Liu, Z. Liu, J. Luo, Numerical investigation of the unsteady thermal plume around human body in closed space, Proc. Eng. 121 (2015) 1919–1926. [6] B.A. Craven, G.S. Settles, A computational and experimental investigation of the human thermal plume, J. Fluids Eng. 128 (2006) 1251–1258. [7] D. Zukowska, A. Melikov, Z. Popiolek, Thermal plume above a simulated sitting person with different complexity of body geometry, in: Proceedings of the 10th International Conference on Air Distribution in Rooms—Roomvent: Technical University of Denmark, 2007, pp. 191–198. [8] D. Licina, J. Pantelic, A. Melikov, C. Sekhar, K.W. Tham, Experimental investigation of the human convective boundary layer in a quiescent indoor environment, Build. Environ. 75 (2014) 79–91. [9] B. Koelblen, A. Bogdan, Impact of clothing, breathing and body posture on the shaping of a thermal plume above a human, Int. J. Ventil. 13 (2015) 397–410. [10] C. Voelker, S. Maempel, O. Kornadt, Measuring the human body’s microclimate using a thermal manikin, Indoor Air 24 (2014) 567–579. [11] Y. Yan, X. Li, J. Tu, Effects of passenger thermal plume on the transport and distribution characteristics of airborne particles in an airliner cabin section, Sci. Technol. Built Environ. 22 (2016) 153–163. [12] M. Li, B. Zhao, J. Tu, Y. Yan, Study on the Carbon Dioxide Lockup Phenomenon in Aircraft Cabin by Computational Fluid Dynamics, Tsinghua University Press, Building Simulation, 2015, pp. 431–441. [13] M. Kühn, J. Bosbach, C. Wagner, Experimental parametric study of forced and mixed convection in a passenger aircraft cabin mock-up, Build. Environ. 44 (2009) 961–970. [14] B. Cetegen, Y. Dong, M. Soteriou, Experiments on stability and oscillatory behavior of planar buoyant plumes, Phys. Fluids 10 (1998) 1658–1665. [15] J.M. Lopez, F. Marques, Instability of plumes driven by localized heating, J. Fluid Mech. 736 (2013) 616–640. [16] R. Chakravarthy, L. Lesshafft, P. Huerre, Local linear stability of laminar axisymmetric plumes, J. Fluid Mech. 780 (2015) 344–369. [17] K. Ichimiya, H. Saiki, Behavior of thermal plumes from two-heat sources in an enclosure, Int. J. Heat Mass Transfer 48 (2005) 3461–3468. [18] S. Guo, S. Zhou, L. Qu, X. Cen, Y. Lu, Evolution and statistics of thermal plumes in tilted turbulent convection, Int. J. Heat Mass Transfer 111 (2017) 933–942. [19] B.O.L. Protection, M.O. Labour, Human Dimensions of Chinese Adults, 1988. [20] J. Li, J. Liu, C. Wang, N. Jiang, X. Cao, PIV methods for quantifying human thermal plumes in a cabin environment without ventilation, J. Visual. 1–14 (2016). [21] P.A. Durbin, B.P. Reif, Statistical Theory and Modeling for Turbulent Flows, John Wiley & Sons, 2011. [22] H.W. Coleman, W.G. Steele, Experimentation, Validation, and Uncertainty Analysis for Engineers, John Wiley & Sons, 2009. [23] C. Wang, J. Liu, J. Li, Y. Guo, N. Jiang, Turbulence characterization of instantaneous airflow in an aisle of an aircraft cabin mockup, Build. Environ. 116 (2017) 207–217.
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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Chaotic behavior of human thermal plumes in an aircraft cabin mockup Congcong Wang a, Junjie Liu a,⇑, Jiayu Li a, Fei Li b a b
Tianjin Key Laboratory of Indoor Air Environmental Quality Control, School of Environmental Science and Engineering, Tianjin University, Tianjin 300072, China Department of HVAC, College of Urban Construction, Nanjing Tech University, Nanjing 210009, China
a r t i c l e
i n f o
Article history: Received 25 May 2017 Received in revised form 10 October 2017 Accepted 13 November 2017
Keywords: Human thermal plume Mini-PIV Gaussian mixture model Spectrum analysis Phase Space Reconstruction Chaotic behavior
a b s t r a c t The human thermal plume induced by body heat loss has a significantly impact on human thermal comfort, contaminant transport and indoor air quality. Few studies focused on the temporal unsteady characteristics of human thermal plume. In this study, the human thermal plume generated by a heated manikin was measured in a 7-row cabin mockup by mini particle image velocimetry (mini-PIV) system; and its unsteady and chaotic behavior was determined out of statistical and chaotic method. Probability density distributions of velocity time series of human thermal plumes presented Gaussian mixture models with two peaks, which substantiated the oscillating characteristics of human thermal plumes. The energy region of the human thermal plume was concentrated between 0.1 Hz and 10 Hz determined out of the power spectrum analysis, and the power spectrum exponent of the human thermal plume above the head ranged from 0.9 to 1.2. Evolution of phase space reconstruction of velocity time series from single-spindle to double-spindle revealed the human thermal plume presents obvious autocorrelation and oscillating behavior qualitatively. In addition, the fractal dimension of human thermal plumes overhead ranged from 6 and 12 without integers and Kolmogorov entropies of analyzed points were all larger than zero indicating the human thermal plume was kind chaotic airflow quantitatively. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction The human thermal plume induced by body heat loss has a significantly impact on human thermal comfort, contaminant transport and indoor air quality in enclosed environments. A large amount of studies have been conducted to investigate the airflow and temperature distributions of the human thermal plume and its influence on propagation of gaseous and particle. Melikov 2015 demonstrated thermal plumes above the head and the convective boundary layer near human body have impacts on the human thermal comfort and air quality in breathing zone [1]. To quantify the airflow characteristic of human thermal plumes, computational fluid dynamics (CFD) and experimental measurements were used extensively. Murakami et al. [2,3] studied the wind environment around heated human body and concluded the velocity of thermal plume can be up to 0.23 m/s. Their numerical simulation results were verified merely by the smoke tracer method qualitatively. Dan et al. [4] used numerical simulation to expose the thermal plume airflow and heat transfer of a seated human body and recommended low-Re turbulence model to determine the human thermal plume airflow and temperature. Liu et al. [5]
⇑ Corresponding author. E-mail address: [email protected] (J. Liu). https://doi.org/10.1016/j.ijheatmasstransfer.2017.11.059 0017-9310/Ó 2017 Elsevier Ltd. All rights reserved.
analyzed the human thermal plume by large eddy simulation (LES) and found unsteady and intermittent characteristics of the thermal plume. Unfortunately, Liu’s numerical results lacked of experimental verification. For experimental studies, Graven et al. [6] determined the airflow field of human thermal plume qualitatively and quantitatively by shlieren image and particle image velocimetry (PIV), respectively. However, the human model in Graven’s study was too simple to reflect real human surface temperature. Zukowska’s [7] results showed that the airflow distribution of the thermal plume is closely related to the geometry, the surface temperature and area of the human body. For different clothing, breathing and body posture of the heated manikin, its convective boundary layer was investigated by PIV and pseudo color visualization (PCV) in a chamber without ventilation [8]. They found personal factors such as clothing and breathing can affect the volume flux of thermal plume and cause the asymmetric velocity distribution [13,14]. Koelblen et al. [9] investigated the airflow and temperature pattern of the thermal plume and found the velocity field varies with body postures. Voelker et al. [10] measured an ascending plume above the head and found the human thermal plume expanded growing up from the lower part of the torso. Particularly, human thermal plumes play a more significant role in forming the indoor environment of aircraft cabin and have impact on airflow distributions and
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pollutant transport because of large heat density from passengers and narrow space of aircraft cabin [11–13]. While, the characteristics of pure thermal plume generated by passengers are not clearly. Study of the characteristics of human thermal plumes can provide theoretical support for the design of better airflow distribution to rapidly eliminate body heat dissipation and keep passengers thermal comfort. The schlieren video of human thermal plume recorded by Gary [6] and PCV measured results [8] qualitatively revealed that human thermal plumes present unsteady and oscillating behavior. From our literature review, the average velocity field of human thermal plumes was studied emphatically, while few studies focused on unsteady and turbulent characteristics. A number of researchers have studied the periodicity and oscillating characteristics of pure thermal plumes from heated flat and analyzed those airflows by nonlinearity and chaotic theory [14–18]. These chaotic method can explain the correlation between the turbulent characteristics of airflow and the human body draft sensation. The main purpose of this paper is to study the unsteady and chaotic behavior of the human thermal plume. A mini-PIV system was utilized to measure the human thermal plume in a 7-row cabin mockup. In order to further understand the airflow mechanism of human thermal plumes and provide experimental data support for human comfort analysis, indoor air quality and numerical simulation, statistical method, spectrum method and chaotic method are used to study the airflow mechanism of the human thermal plume and analyze the experimental data. The statistical analysis, spectrum analysis and phase space reconstruction methods are used to substantiate the unsteady characteristics of human thermal plume qualitatively. To further analyze the unsteady characteristics of thermal plume quantitatively, the chaos theory including fractal dimension denoting to the relation between phase space structure feature and embedding dimension and Kolmogorov entropy representing quantification of chaos of transformed velocity time series based on phase space reconstruction based on are employed. 2. Experiment setup A heated manikin with a thermal power of 75w sitting in a fullscale 7-row single-aisle aircraft cabin mockup was served as passenger generating thermal plume. The prototype of the aircraft cabin mockup is a section of Boeing 737–200. The geometry sizes of the aircraft cabin mockup are 5.85 m in length, 3.25 m in width and 2.15 m in height. Particularly, the luggage rack height is about 1.6 m. In order to facilitate the adoption of laser, the walls of the middle three rows were constructed by transparent acrylic. In order to guarantee the controllability of temperature boundary condition, the aircraft cabin mockup was set up in a thermostatic chamber. To meet the objectives of this investigation, the manikin was wrapped with electrical resistance, a total heat of 75 W. The manikin’s sitting height is 1.4 m with a total volume 0.055 m3. The surface area of each manikin is 1.3 m2. The geometry of the manikin is similar to a real person, according to Chinese adult body size and seating space [19]. A mini-PIV system was utilized to determine the human thermal plume composing of a continuous wave laser and a CCD camera with a Canon 35 mm lens. Tracing particles were generated by diethylhexyl sebacate (DEHS) through a spray generator. As shown in Fig. 1, three cross sections were measured above the shoulders and head with each field area is about 0.32 m 0.16 m. Due to the obstruction of the luggage rack, there is a certain deviation between the center of the human head and the center of measured cross section, i.e., CS2. In the measurement process, there was no ventilation in the aircraft cabin mockup and temperatures were controlled by the ther-
mostatic chamber with closed circulatory air condition system. Thermocouples were used to monitor temperature distributions of the manikin, aircraft cabin wall and ambient air. When the monitoring temperatures were stable, the experiment was started [20]. The constant temperature distributions in measurement process are listed in Table 1 with manikin divided into five parts. The temperatures of aircraft cabin wall and air surrounding manikin are 19.5 °C similarly. In order to accurately sample the turbulent characteristics of the human thermal plume, the sampling frequency of mini-PIV was set as 400 Hz and recording time was 100 s. 3. Result 3.1. Velocity and vorticity Fig. 2 shows the average velocity distributions of the human thermal plume with sampling time 100 s. In the figures, vectors and contour represent velocity directions and velocity magnitude, respectively. Human thermal plumes rise from two side shoulders and collide with each other above the head. Maximum velocity appearing at CS2 located above the head is up to 0.24 m/s. The overall thermal plume flows upward in the CS2 section while moving toward the manikin body in CS1 and CS3. More close to the manikin body, the velocity is larger in CS1 and CS3. Evidently, the thermal plume expands growing up upward via entrain ambient air in CS1 and CS3 driven by buoyancy. The airflow fields above the left and right shoulders are not completely symmetrical because of obstruction of luggage rack and asymmetrical heated manikin surface. In order to demonstrate the unsteady characteristics of human thermal plumes, instantaneous velocity distributions of CS2 section measured at 5th, 10th, 15th and 20th seconds are presented in Fig. 3. The airflow patterns of the four moments are basically different to each other and inconsistent with the average velocity distributions. The maximum velocity at 5th and 15th seconds is about 0.25 m/s while that is up to 0.3 m/s at 10th and 20th seconds. The velocity near the head is lower because of heat loss at that part is less simulating human with hair. Moreover, two strands thermal plumes rising from shoulders appear above head alternately. Summarily, the instantaneous velocity distributions of CS2 substantiate the asymmetrical and oscillating behavior of the human thermal plume. Vorticity is introduced to demonstrate the unsteady turbulent characteristic of human thermal plume which can be calculated as follows [21]:
w¼
@ v @u @x @y
ð1Þ
where v and u represent velocity components at x and y direction, respectively. Fig. 4 shows the instantaneous vorticity distribution of CS2 at 5th, 10th, 15th and 20th seconds. Irregular shape and random distribution of vorticity contours appear on the four moments inconsistently. The maximum vorticity can be up to 12 s1. At the center part of CS2, the sizes of vorticity are more obvious because of the collision of two thermal plumes rising from shoulders. The vorticity appearing on edge position of cross sections may cause by air entrainment. The obvious vorticity distributions also reveal high fluctuation and turbulent kinetic energy of thermal plume. The random and irregular vorticity contours reveal unsteady behavior of the human thermal plume. For the convenience of the researches on temporal unsteady and chaotic behavior of the human thermal plume, several points are selected on CS1, CS2 and CS3 as shown in Fig. 5. In order to focus on the study of the thermal plume over the head in detail, more points are selected.
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Fig. 1. Schematic diagram of mini-PIV measurement of human thermal plume in the aircraft cabin mockup.
Table 1 Temperature distributions of human manikin in measurement process. Head
Chest
Belly
Thigh
Leg
Wall
Ambient air
27 °C
28 °C
30 °C
30 °C
28 °C
19.5 °C
19.5 °C
In order to verify whether the unsteady thermal plumes are periodical, continuous instantaneous velocity of points CS1-B, CS2-B and CS3-B are presented in Fig. 6. It can be seen that the velocity time series fluctuate between 0.05 m/s and 0.4 m/s. Comparing the velocity of the three points, the velocity of CS2 thermal plume is not lager than others despite combined of the other two thermal plumes. Therefore, the observation substantiates the two plumes appearing overhead alternately. Velocity sequences of CS1 and CS3 are inconsistent in velocity magnitude and temporal trend. Hence, the thermal plumes rising from left and right side shoulders are asymmetric because of asymmetrical manikin surface temperature and geometric space for plume growing up, potentially. The temporal trend of each velocity sequence is analyzed in detail, and the periodic fluctuation is not found. But the peaks and troughs of velocity sequences appear alternately as shown in Fig. 6 reveal oscillating characteristic of human thermal plumes above shoulders and head. 3.2. Statistical analysis In order to analyze the unsteady behavior of the instantaneous velocity sequences of the human thermal plume, the frequency distribution and probability density distribution were introduced in this paper. The frequency distribution was determined out of velocity magnitude time series with 40,000 intervals of the measured thermal plume by the mini-PIV. The probability density distribution was calculated by the relative frequency of velocity magnitude and classified interval. According to study of Coleman and Steele [22], if the sampling signal is continuously stable, the frequency distribution of measured results is close to a normal distribution curve. Therefore, Gaussian model was used to fit the probability density distribution to analyze the unsteady characteristic of the human thermal plume.
Fig. 7 reflects the frequency distribution, probability density distribution and Gaussian fitted curves of three points of each measured cross section, i.e., A, B and C in CS1 and CS3. In the figure, relative frequency represents the frequency distribution determined out of 40,000 intervals velocity magnitudes. Meanwhile, the probability density distribution of velocity relative frequency is described by a cumulative fit curve. G1(x) and G2(x), as shown in Fig. 7, are represented Gaussian fitted curves. Apparently, the probability density distributions of the six points as shown in Fig. 7 represent mixed Gaussian model, i.e., there are two obvious peaks of relative frequency. As a consequence, the velocity magnitude time series are concentrated upon two median velocities, i.e., two peaks which means the human thermal plumes present unsteady oscillating characteristic. The magnitudes and frequency of the medians of the two Gaussian models at each point are different, respectively. At point C, the frequency of G2(x) is too small can be ignored. While, at point A, The frequency of low speed is dominant. In other words, the more close to the human body, the velocity is larger and the corresponding frequency is higher. Comparing CS1 and CS3, the major difference is at point A which reveals average velocity of CS3 is larger than CS1. Mixed Gaussian characteristics are more obvious at point A, i.e., the more obvious oscillating behavior of human thermal plume. The two frequency peaks with different velocity magnitude represent two different stream thermal plumes. Fig. 8 shows the probability density distributions of measured thermal plume overhead, i.e., the CS2. Detail positions of sampling points of CS2 are shown in Fig. 6. Obviously, the two frequency peaks of the fitted curves become increasingly evident from point A to F. In other words, Gaussian mixture models are more accurate representation of probability density. At point A, the G1(x) curve is corresponding to the cumulative fit curve and G2(x) can be ignored. Therefore, the velocity series of thermal plume closest to
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Fig. 2. Average velocity distributions of the human thermal plume.
the head tends to normal distribution. The relative frequency of G1 (x) is higher than G2(x) at points A, B and C with smaller velocity magnitudes. Conversely, the relative frequency of G1(x) is lower than G2(x) at points D, E and F far away from the head. Meanwhile, the median velocity and relative frequency of G2(x) grows larger as far away from the head. The relative frequency of two Gaussian fitted curves are similar, which reveals the probability of plume occurrence with different velocity magnitude is equal. But, at other positions, i.e., D, E and F the unsteady thermal plumes are driven by large velocity plumes with higher frequency. Comparing Gaussian mixture models of CS1, CS2 and CS3, the thermal plume far away from the manikin thermal boundary layer present obvious instability and oscillating airflows. The thermal plume above head
is a mixture of two strands of plumes rising from shoulders. Summarily, the human thermal plumes are unsteady and oscillating airflows with different velocity and different relative frequency. 3.3. Spectrum analysis The probability density distributions show the unsteady behavior of the human thermal plume qualitatively. While, in order to analyze the periodicity of the unsteady velocity series, the power spectrum was introduced in this paper. The power spectrums are determined out of fast Fourier Transform of the velocity series measured by mini-PIV with sampling frequency 400 Hz. Fig. 9
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Fig. 3. Instantaneous velocity distributions of CS2.
Fig. 4. Instantaneous vorticity distributions of CS2.
shows the power spectrum distributions of sampling points, i.e., A, B, C, D, E and F of CS2. In Fig. 9, vertical axis and abscissa axis represent power and discrete sampling frequency, respectively, by logarithmic coordinate axis. Because of fast Fourier Transformation is an even function, the figures shows symmetrical half curves. From the observation of the power spectrum, energy region, inertial sub-region and dissipation region can be quantified, the characteristics of these regions are concluded. The inertial sub-region of the power spectrum concentrate on area of curves with slope is minus 5/3. And the frequency of the inertial sub-region is about 5 Hz as shown in Fig. 9. Therefore, when the human thermal plume was studied by the CFD method with LES, the time step of simulation recommended should be less than 0.2 s [23]. And the sampling
frequency of instrument measured the human thermal plume should be larger than 20 Hz, in order to obtain the overall turbulence information. The dissipation region of the power spectrum located behind 20 Hz. While, the energy region of the power spectrum concentrates between 0.02 Hz and 1 Hz. Therefore, the turbulent airflows of the human thermal plume are mostly low frequency and large scale structures. Obviously, the turbulent kinetic energy distribution fluctuates strongly in the energy region. And there is no prominent frequency with highest energy which reveals the human thermal plume overhead fluctuate nonperiodically. Comparison of the six sampling points, the turbulent kinetic energy in the energy region increases with the distance far from the manikin head. Therefore, the airflows of the thermal
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Fig. 5. Positions of research points on CS1, CS2 and CS3.
plume furthest from the head, fluctuate most strongly and unsteadily which corresponding to the results of probability density distributions. In order to quantify impact of unsteady behavior of the thermal plume on human thermal comfort, the power spectrum exponent is defined as follows:
b¼
ln G ln f
ð2Þ
In practical calculation, b is the slope of curve fit of lnG and lnf by least square method. According to Zhu’s study [24], the power spectrum exponent of nature wind with most comfortable is between 1.1 and 2, while that of mechanical wind is less than 1.1. In Fig. 9, the red line is the fitted curves of lnG and lnf. The b values of human thermal plume overhead range from 0.9 to 1.2, which reflect the comfort of human thermal plumes are between nature wind and mechanical wind.
Fig. 10 shows the accumulation of power spectrum energy of sampling points, i.e., A, B, C, D, E and F at cross section 2. From the observation of Fig. 10, the energy accumulations increase rapidly between 0.1 Hz and 10 Hz, which represents the energy regions of sampling points range from 0.1 Hz to 10 Hz corresponding to the results of power spectrum distributions. Moreover, the energy accumulations reach up to 90% at frequency 10 Hz. Therefore, the turbulent airflows of human thermal plume mostly contain low frequency structures. After frequency 10 Hz, the energy of point A is larger than F, which reflects the human thermal plumes closest to the head contain higher frequency and small scale structures than that farthest from head.
3.4. Phase space reconstruction The unsteady behavior of human thermal plume reflects the chaotic airflows characteristic. To examine the chaotic behavior
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Fig. 6. Continuous instantaneous velocity variations of points CS1-B, CS2-B and CS3-B.
Fig. 7. Probability density distributions of sampling points of CS1 and CS3.
of the human thermal plume, the phase space reconstruction was introduced in this paper. We reconstructed the original onedimensional univariate time series to phase space reconstruction with Takens principle [25] as follows:
xi ¼ ðx1 ; x2 ; . . . ; xN Þ
X i ¼ xi ; xiþ1 ; . . . ; xiþðm1Þs
ð3Þ
ð4Þ
where xi is the one-dimensional univariate time series; Xi is the phase space series after reconstructing; m presents the embedding dimension; s denotes the delay time; N is the sample number of univariate time series; i ¼ 1; 2; . . . ; M; M ¼ N ðm 1Þs, M denotes
the number of phase points in Xi. In this study 40,000 intervals of velocity magnitude measured by mini-PIV are served as the onedimensional univariate time series. While, the two-dimensional phase space series (i.e. m = 2) are analyzed. And the six sampling points at CS2, i.e., A, B, C, D, E and F are analyzed in Fig. 11. For the one-dimensional univariate time series, the autocorrelation function method can be used to determine the delay time. When the autocorrelation function drops up to 1–1/e of initial value, the corresponding time s is the optimal delay time of reconstructed phase space. The two-dimensional phase space reconstruction reflects the correlation and fluctuation of signal on time scale. Therefore, two-dimensional phase space reconstructions are deter-
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Fig. 8. Probability density distributions of sampling points of CS2.
Fig. 9. Power spectrum distributions of sampling points in CS2.
mined out of the human thermal plume time series measured by mini-PIV as shown in Fig. 11. In Fig. 11, the x-coordinate and y-coordinate, represent original time series and reconstructed series, respectively. The gathered points shape reflects the correlation of velocity series. Literatures show the shape of two-dimensional phase space reconstructions of natural wind and mechanical wind present spindle and ellipse,
which can be concluded that the velocity series correlation of natural wind is stronger than that of mechanical wind [24]. From observation of the phase space reconstruction of the point A, the contour shape presents a spindle shape, which shows that there is a strong correlation of the velocity time series. And most of velocity phase points range from 0.1 m/s to 0.2 m/s. With the changing of point’s positions, the global correlation attenuates
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Fig. 10. The energy accumulation distributions of power spectrum of sampling points in CS2.
Fig. 11. Phase space reconstruction evoluation of sampling points in CS2.
and the range of velocity magnitudes of phase points expands. Moreover, the phase space reconstruction points gradually change from the spindle to the discrete shapes with the distance from the head farther and farther. At points E and F, the phase space reconstruction points gather two spindles with different velocity ranges
corresponding to the results of Gaussian mixture model. This phenomenon shows that there are two strands airflows combining the human thermal plume. And, the phase space reconstruction with two spindles reveals oscillating characteristics of the human thermal plume. At points E and F, the velocity range of the two spindles
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Fig. 12. The C(r,m) and r relationship, log-log profile.
Fig. 13. The relationship between the correlation dimension and the embedding dimension.
in the phase space separates each other, while at points C and D, the phase space ellipses overlap with each other. Therefore, at points C and D, the human thermal plume form two kinds superimposed velocity time series with an overlapping velocity range around 0.2 m/s. Summarily, from the observation of phase space reconstruction, the velocity time series of the human thermal plume above head present unsteady characteristic and oscillating behavior. And, the human thermal plume presents strong correlation between velocities at different times.
Fig. 14. Kolmogorov entropy distributions of the human thermal plume overhead.
3.5. Fractal dimension The probability density distribution, power spectrum and phase space reconstruction substantiate the unsteady characteristic of human thermal plume qualitatively, only. In order to determine the unsteady and chaotic behavior of the human thermal plume quantitatively, the fractal dimension is calculated by GrassbergerProcaccia algorithm [26] based on the embedding theorem and theory of phase space reconstruction. Firstly, for one-dimensional univariate time series the correlation integral can be defined as:
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C ðr; mÞ ¼
1 ðN ðm 1ÞsÞ
M X M X H r X i X j 2 i¼1 j¼1
M X M 1 X H r r ij ¼ 2 M i¼1 j¼1
ð5Þ
m-dimensional cubic boxes with side length is r. For an attractor for state space and a trajectory x(t) in the domain of attraction, s denote a small time interval. p(i0, i1, . . . , im) denote the joint probth ability of that system trajectory located at the ith 0 box and at im box at time t = ms. And Kolmogorov entropy can be defined as following formula [27] (9):
where H denote Heaviside step function:
H r r ij ¼
1; r r ij 0 0; r r ij 0
K ¼ lim lim lim ð6Þ
rij denote the distances of metric space; r is the critical scaling. Secondly, in the critical scaling, the connection of C(r,m) and r can be described as following formula (7):
C ðr; mÞ / r D2
ð7Þ
where D2 is the correlation dimension. Finally, the fractal dimension of a strange attractor from a one-dimensional univariate time series can be denoted as:
D¼
log C ðr; mÞ m!saturation r!0 log r lim
lim
ð8Þ
Fig. 12 shows the logarithmic curves of correlation integral and critical scaling determined out of 4000 intervals velocity time series of six points located at CS2 measured by mini-PIV. In the calculation, the embedding dimension ranges from 2 to 40 and critical scaling value is 0.0625–1, which are sufficient to examine the fractal dimension of the human thermal plume. From the observation of Fig. 12, when the embedding dimension m is small, the slopes of linear parts of the curves of lnG and lnf are smaller and the gaps between adjacent curves are larger. While, the slopes of linear parts gradually increase and the gap width become smaller with the increase of m. Finally, the linear parts of the curves tend to parallel to each other, which reveals the embedding dimension tends to saturation and the correlation dimension is stable. Summarily, stable correlation dimension reflects the velocity series of human thermal plume are chaotic and the complexity of their system attractors is similar to correlative system characteristic. The positions of the linear part are not the same, that is to say, the complexities of the chaotic systems are not the same. Fig. 13 shows the relationship between the correlation dimension and the embedding dimension determined out of the slope of the linear part of each curve fitted by the least square method. As shown in Fig. 13, when the embedding dimension m reaches saturation, the correlation dimension does not change obviously and reaches a stable value. And, the stable value of correlation dimension is fractal dimension. The correlation dimension can reach saturation, which indicates that the human thermal plume is not a completely random airflow but a chaotic airflow. It can be seen that the fractal dimension of the chaotic characteristics of the thermal plume at the six points is not an integer between 6 and 12, which indicates that the thermal plume is a kind of chaotic airflow with correlation structure. The correlation dimension is a measurement of the complexity of the attractor in the phase space. From the observation of Fig. 14, the saturation embedding dimension ranges from 18 to 30, which reflects the degree of freedom in a chaotic system of the thermal plume. And, the airflow of point A has the highest degree of freedom. Therefore, the velocity series of point A of thermal plume overhead has more complex structure with more factors affecting velocity changing.
1 X
s!0 r!0 m!1 ms
pði0 ; i1 ; . . . ; im Þlog2 pði0 ; i1 ; . . . ; im Þ
while, order-q Renyi entropies are defined as follows:
K q ¼ lim lim lim
1
s!0 r!0 m!1 ms
X 1 log2 pq ði0 ; i1 ; . . . ; im Þ q1 i ...i 0
ð10Þ
m
To simplify the calculation of Kolmogorov entropy, Grassberger and Procaccia demonstrated K1is the Kolmogorov entropy and K2 is the best estimation of K1, i.e., the second order Renyi entropies [27] which can be defined as:
K 2 ¼ lim lim
1
s
r!0 m!1 n
log2
C m ðr Þ C nþm ðrÞ
ð11Þ
If the time series is chaotic, the Kolmogorov entropy is constant larger than zero [27]. And, if the Kolmogorov entropy is larger, the degree of the chaotic time series is larger with more information
3.6. Kolmogorov entropy To evaluate the chaotic human thermal plume quantitatively, Kolmogorov entropy was introduced as follows. The phase space of the m-dimensional dynamic system can be divided into several
ð9Þ
i0 ...im
Fig. 15. The detail surface temperature of heated manikin.
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loss. Fig. 14 shows the Kolmogorov entropy distribution of airflows of human thermal plume measured by mini-PIV. From the observation, the Kolmogorov entropy values are all larger than zero and increases with the distance far from the head, which reveals the velocity series of the six points are chaotic. And, the degree of chaotic of velocity series is largest at point F and lowest at point A. The human thermal plume becomes more chaotic with increase of distance from the head. 4. Discussion In this study, a heated manikin with a thermal power of 75 W was served as passenger generating thermal plume. While, the manikin surface temperature distribution has great influences on the unsteady characteristics of human thermal plume. The unsteady behavior of the human thermal plume was positively related to the temperature difference of human body surface. The velocity magnitudes of thermal plume under different surface temperature condition were different [28]. Therefore, the detail surface temperature distribution measured by the infrared thermal imager as shown in Fig. 15, which can be used as boundary condition of numerical and experimental research of human thermal plume. The sensitivity analysis of surface temperature distribution to airflow of human thermal plume can be evaluated qualitatively as following equation which described the velocity of a steady heated source [29].
U
@W @W 1 @ @ 2 w Þ u 2 þ gbDT þW ¼ ðr u w @r @z r @r @z
where U and W are the velocity of different directions, respectively. r and z are the coordinate of radial direction and vertical direction. u are fluctuating velocity. g denotes the gravitational acceleraand w tion. b represents the air thermal expansion coefficient. DT is the temperature difference between manikin surface and ambient air in this study. For a steady heat source, the velocity of a specific position is only related to DT. For the same experiment condition of human thermal plume, the velocity is linearly related to the temperature difference. 5. Conclusion In this paper, we focused on the unsteady and chaotic behavior of the human thermal plume determined by mini-PIV with frequency 400 Hz in a 7-row aircraft cabin mockup. Through the analysis of vorticity, probability density distribution, power spectrum and chaos theory of the measured thermal plume, we can draw conclusions as follows: 1. The maximum average velocity of the human thermal plume can be up to 0.24 m/s above the manikin head. And, the instantaneous velocity and vorticity distributions revealed the human thermal plume present unsteady and oscillating characteristic. 2. Probability density distributions of velocity time series of the human thermal plume form Gaussian mixture models with two peaks, which substantiated the oscillating characteristic of the human thermal plume. 3. The energy region of the human thermal plume concentrated between 0.1 Hz and 10 Hz determined out of the power spectrum analysis. The power spectrum exponent of the human thermal plume above the head ranges from 0.9 to 1.2 and the comfort level is between natural wind and mechanical wind. 4. The evolution of the phase space reconstruction of velocity time series from single-spindle to double-spindle revealed the human thermal plume presents obvious autocorrelation and oscillating behavior.
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