Journal of Wind Engineering & Industrial Aerodynamics 189 (2019) 1–10
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An application of quadrant and octant analysis to the atmospheric surface layer Xuebo Li a, b, Tianli Bo a, b, * a b
Key Laboratory of Mechanics on Western Disaster and Environment, Department of Mechanics, Lanzhou University, Lanzhou, 730000, PR China Department of Mechanics, Lanzhou University, Lanzhou, 730000, China
A R T I C L E I N F O
A B S T R A C T
Keywords: Quadrant analysis Octant analysis Atmospheric surface layer Reynolds stress
The quadrant and octant analysis methods are used to investigate the characteristic of the atmospheric surface layer (ASL) turbulent flow based on the field observation data obtained from the Qingtu Lake Observation Array (QLOA). The results indicate that a stronger thermal convection intensity can lead to a larger intensity ratio of Reynolds stress from each quadrant to the total Reynolds stress. In the near-neutral ASL, the time ratios of Reynolds stress from each quadrant against the total Reynolds stress at different heights and friction velocities are approximately constant, i.e., DQ10.19, DQ20.296, DQ30.2 and DQ40.314. And the turbulent kinetic energy (TKE) ratios of each quadrant against the total TKE (streamwise or vertical) for the ejection events and sweep events are pretty similar at different wall-normal heights. The flow characteristics tend to be isotropic with the increasing of the vertical distance. Moreover, a threshold value is also applied to separate the most important events from the less significant ones. Last, the contribution of Octant 2 and Octant 8 is dominant to the Reynolds stress, and the intensity ratios of Octant 2 and Octant 8 to the total Reynolds stress will be increased with the thermal convection intensity. The abundant dynamics behaviors revealed by the quadrant and octant analysis could benefit us a wider understanding of the ASL coherent structures.
1. Introduction Studies of wall-bounded turbulent flows have become a hot topic for debate in recent years. Fully developed channel flow and boundary layer flows are two typical wall-bounded turbulent flows (Shi et al., 2016). A large number of studies have found that there exist coherent structures in the wall-bounded turbulent flows, and turbulent fluctuations are not completely random. The coherent structures may dominate the momentum and scalar transport in the wall-bounded turbulent flows. Reynolds stress is an important statistic in wall-bounded turbulent flow. The Reynolds stress obtained by quadrant analysis can be divided into four categories: Q1(þu,þw), Q2(-u,þw), Q3(-u,-w) and Q4(þu,-w), where, u and w are streamwise and vertical velocity components, respectively. The Q2 and Q4 quadrant motions are called ejection events and sweep events respectively, and Q1 and Q3 quadrant motions are called outward interactions and inward interactions. That is, quadrant decomposition can be considered as a bridge between Reynolds stress and coherent structures. Detailed study of coherent structures in wall-bounded turbulent flows can be found in Robinson (1991a), Jimenez (2007), Lee and Wu
(2008), Smits et al. (2011), Jimenez and Lozano-Duran (2016). About 50 years ago, Kline et al. (1967) first observed that the flow near the wall of a boundary layer is organized into narrow streaks of high- and low-momentum fluid that are quite elongated in the streamwise direction and that are spaced, on average, at approximately 100 viscous lengths ν/uτ in the spanwise direction, where uτ is the friction velocity (Wallace, 2016). Corino and Brodkey (1969) observed the same visualization in the very near-wall region of a turbulent pipe flow by a high-speed movie camera. Then, Wallace et al. (1972) also determined the timescales for each of the quadrants from autocorrelations of the quadrant-conditioned signals, and they observed that the timescales for the ejection and sweep quadrants, Q2 and Q4, are significantly larger than those for the interaction quadrants, Q1 and Q3. In addition, many scholars (Kovasznay et al., 1970; Raupach and Thom, 1981; Raupach et al., 1991; Moriwaki and Kanda, 2006; Jin et al., 2015) used quadrant decomposition to study the momentum transport, heat flux and scalar transport (CO2, H2O, etc.). Robinson (1991b) used the turbulent boundary layer DNS database of Spalart (1988) to study the flow structure and its relationships to important transport properties of the flow.
* Corresponding author. Key Laboratory of Mechanics on Western Disaster and Environment, Department of Mechanics, Lanzhou University, Lanzhou, 730000, PR China. E-mail address:
[email protected] (T. Bo). https://doi.org/10.1016/j.jweia.2019.03.013 Received 17 August 2018; Received in revised form 8 March 2019; Accepted 9 March 2019 0167-6105/© 2019 Elsevier Ltd. All rights reserved.
X. Li, T. Bo
Journal of Wind Engineering & Industrial Aerodynamics 189 (2019) 1–10
Although many scholars have applied the quadrant decomposition to the Reynolds stress in the ASL (Shaw et al., 1983; Maitani and Shaw, 1990; Katul et al., 1997; Boldes et al., 2007; Katsouvas et al., 2007; Fernandez-Caban and Masters, 2017), the data collection and results analysis are still limited for synchronous measurements with different heights and different stability regimes. Therefore, based on the three-dimensional velocities and temperature measurement data of 11 wall-normal heights obtained from the QLOA, the coherent structures at different shear flow and heat flux situations were analyzed by quadrant analysis and octant analysis. Compared with laboratory experiments and numerical simulations, the Reynolds number of ASL flow can reach to O (106), and there are more uncertain factors, such as wind directions, heat flux, flow stabilities and surface roughness. Therefore, the data is very rare which are suitable for analysis after pre-processing. The distribution of sweep and ejection events by way of quadrant analysis are analyzed at high Reynolds numbers in the present study. It should be noted that the different features were revealed from the different stability conditions. To a deep study of the main structures to the lower speed away from the wall (ejections events) and higher speed towards the wall (sweep event) reveal the features which contribute to a more complete description of the three-dimensional fluid-dynamic structures that characterize the wind in high Reynolds number and different stability regimes. Furthermore, using quadrant analysis and octant analysis to analyze the variations of physical quantities varying with the heights, friction velocities and thermal convection intensity can not only reveal the structural characteristics of the ASL, but also can be used to verify the flow characteristics of numerical simulation and laboratory simulation. Therefore, related researches on this subject were carried out in this manuscript. In Section 2, the experimental facility and data pretreatment were introduced. Section 3 gives the introduction of quadrant analysis and octant analysis. Section 4 gives the main results. The main conclusions are given in Section 5.
Suzuki et al. (1988) extended the quadrant analysis to the Octant analysis by using the addition of another scalar component θ' (temperature). For example, Volino and Simon (1994) discussed the Octant decomposition of the thermal turbulence. Vincont et al. (2000), Li et al. (2010) and Park et al. (2012) also have done some works on the Octant analysis. It should be noted that Wallace (2016) summarizes the history and development of quadrant analysis in turbulence studies over the past half-century in the Annual Review of Fluid Mechanics and pointed out that it has been used in an extraordinarily wide range of applications in turbulent shear flows. Quadrant decomposition can determine the intensity ratio and time ratio from each quadrant to the total Reynold stress. The ejection events and the sweep events (corresponding to Q2 and Q4) not only have an important contribution to Reynolds stress, but also to the burst time (Wallace et al., 1972; Katul et al., 1997; Katsouvas et al., 2007). The joint probability density function obtained by velocity fluctuation u’ and w’ can be used to determine the correlation coefficient between velocity fluctuations and estimate the correlation (Liepmann and Laufer, 1947; Wallace et al., 1977). However, none of these authors attempted to determine the quadrant contributions to this covariance (Wallace, 2016). Willmarth and Lu (1972) focused on intense Reynolds shear stress occurrences by determining the contribution to Reynolds shear stress of each quadrant from u' w' velocity fluctuation product values that are outside of what they called the hole, defined by the set of hyperbolas in each quadrant, u' w' H u' w' , where u' w' is the mean value of this product for a given distance from the wall. Shi et al. (2016) defined threshold Rc which is a case-determined value instead of a free parameter to propose to single out the larger Q2 and Q4 motions that directly contribute to the total Reynolds stress in an average sense. As for some wind tunnel studies, the research behind multiple windbreaks across-wind indicates that Reynolds stress is mostly contributed by the sweep and ejection events (Shiau, 1995). Shiau (1998, 1999) and Shiau and Hsieh (2002) were also employed to analyze stress component contribution to the Reynolds stress by the quadrant analysis technique. Results of the quadrant analysis above a two-dimensional trapezoidal shape of hill indicate that the sweep and ejection events are the major contributors to the Reynolds stress (Shiau and Hsu, 2003). Quadrant analysis technique was adopted to provided information on the turbulent bursting events contributing to the Reynolds shear stress at the near rigid boundary region in the water channel (Raushan et al., 2018). Since the Reynolds number similarity hypothesis (the turbulence beyond a few roughness heights from the wall is independent of the surface condition) of Townsend (1980) has been proposed, many scholars have analyzed the coherent structures in the smooth and rough surface boundary layer, and when the Reynolds number is high enough and the roughness scale is smaller than the boundary layer thickness, Townsend's hypothesis can be well verified (Raupach et al., 1991; Jimenez, 2004; Kunkel and Marusic, 2006; Schultz and Flack, 2007). The Q2 and Q4 events in quadrant decomposition are also used as a second-order turbulence statistics to test the effect of roughness on turbulence statistics analysis. For the quadrant analysis in the ASL, most scholars have analyzed the flow properties under different vegetation cover. For example, H€ ogstr€ om and Bergstr€ om (1996), Maitani and Shaw (1990), Bergstr€ om and H€ ogstr€ om (1989) and Boldes et al. (2003) analyzed the flow properties above the forest. Boppe et al. (1999), Boppe and Neu (1995) and Katsouvas et al. (2007) analyzed the flow properties above ocean surface. And the flow in the fields of paddy fields, maize and wheat cultivated surface and low vegetation plants were also analyzed (Maitani and Ohtaki, 1987; H€ ogstr€ om and Bergstr€ om, 1996; Katul et al., 1997; Shaw et al., 1983; Shaw, 1985; Shiau and Chen, 2002). In the ASL, the application of quadrant analysis and octant analysis to the Reynolds stress for sand bed and bare surface is relatively limited. Katul et al. (1997) analyzed the ratio of the quadrant component to the total momentum and heat flux on bare surface, but the results of the analysis for different heat flux cases are still lacking.
2. The experimental facility and data pretreatment The field observations were carried out at the Qingtu Lake Observation Array (QLOA) site. Details of the site and the arrangement of the facilities have been introduced in Wang and Zheng (2016) and Liu et al. (2017). Continuous measurements of all three components of velocity and temperature were acquired by a span-wise and vertical array of 17 sonic anemometers (Campbell Scientific CSAT3) in an ‘L’ shaped configuration. The schematic of the set-up can be found in Wang and Zheng (2016). The span-wise array covered an overall distance of 30 m with 6 anemometers placed 5 m apart and at the height of z ¼ 5 m. The vertical array consisted of 11 probes placed from z ¼ 0.9–30 m, and 11 anemometers were installed in a logarithmic manner for z ¼ 0.9, 1.71, 2.5, 3.49, 5, 7.15, 8.5, 10.24, 14.65, 20.96, 30 m. And the measurement array from the southeast view can be seen in Fig. 1. Not all of the data are suitable for analysis before pre-processing to adjust the wind direction, de-trending and the application of data selection criteria, details of which are provided in Hutchins et al. (2012) and Wang and Zheng (2016). It should be noted that the near-neutral data were selected at the condition of |z/L| 0; w' < 0; w' < 0; w' > 0; w'
>0 >0 0; w' < 0; w' < 0; w' > 0; w'
> 0; θ' > 0; θ' > 0; θ' > 0; θ' < 0; θ' < 0; θ' < 0; θ' < 0; θ'
> 0ðhot outward interactionÞ > 0ðhot ejectionÞ < 0ðcold ejectionÞ < 0ðcold outward interactionÞ > 0ðhot sweepÞ > 0ðhot wallward interactionÞ < 0ðcold wallward interactionÞ < 0ðcold sweepÞ
(11)
Fig. 2 (a) shows the 4 sets of example data about normalized Reynolds stress along with the height under different thermal stratifications. It can be seen from Fig. 2(a) that the normalized Reynolds stress is constant in the near-ground region (z/δ < 7/150–8/150), which also verifies that the data obtained at QLOA can be considered as the wall-bounded turbulence flow data. Fig. 2(b) shows the normalized heat flux varying with the heights. It can be seen from Fig. 2(b) that the normalized heat flux of each set of data at different heights is basically the constant, and there exist differences in the normalized heat flux while z/L is different. Therefore, the data obtained from 5 m were used to the analysis as an example to investigate the potential temperature scale features. The variation of the normalized heat flux with the z/L is given in Fig. 2(c). From Fig. 2(c), it can be found that the more stable/unstable of the z/L, the larger absolute value of the potential temperature scale.
(7)
T→∞ T
(10)
4. Results
It defines a fifth area in the (u’, w')-plane for the conditional analysis and it is used to separate the most important events with large values of |u' w' |, from the less important ones. The quadrant decomposition was carried out using the hyperbolic hole size (H) method of Lu and Willmarth (1973). The contribution to |u'w'| from a given quadrant, Qi, can be expressed as:
>0 >0 0; θ'
3.3. Octant analysis
The potential temperature scale is defined as w' θ ' θ* ¼ u*
w' w' w' w'
Here, Q1 in the temperature transport represents ejections, Q2 is inward interactions, Q3 is sweeps, and Q4 represents outward interactions.
height of z ¼ 5 m. The turbulent sensible-heat flux (heat exchange in the form of turbulence between the atmosphere and the surface due to temperature changes) can be expressed as w' θ '
: : : :
4.1. Quadrant analysis (9)
4.1.1. Reynolds shear stress The ratio of the quadrant components to the total Reynolds stress can be reflected from the intensity and timescale, i.e., the intensity ratio
For the scalar transport (w'θ' and w'q’, θ is sonic virtual temperature and q is the specific humidity), it can also be decomposed as same as quadrant analysis for momentum transport (see also Katul et al., 1997).
Ratiointensity ¼ u' w' Qi =u' w' , and time ratio Ratiotime ¼ DQi =Dtotal . Ratiotime means the ratio of the time in the Qi event to the total time in all four events. They represent the mean value of each quadrant Reynolds stress against the total average Reynolds stress and occupying time in each quadrant to the total time for all four quadrants, respectively. Fig. 3 reveals the Ratiointensity of each quadrant varying with the 4
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Journal of Wind Engineering & Industrial Aerodynamics 189 (2019) 1–10
larger than that of Q1 and Q3, which also shows that the ejection and sweep events play a dominant role in the atmospheric surface-layer turbulence. Compared with low-Reynolds-number results, u' w' Q =u' w' < 1, where, Reτ ¼ 187 by Wallace et al. (1972), Reτ ¼ 180 by i Kim et al. (1987) and Reτ ¼ 590 by Moser et al. (1999), the intensity ratio of the four quadrants measured under the ASL is larger. The reason for this could be the higher Reynolds number in the ASL. Accordingly, the turbulence of the flow is stronger, and then the fluctuation intensity is larger. In addition, the variation of the intensity ratio of the quadrant components to the Reynolds stress with z/L at 5 m shows that the larger the heat flux, the greater the average intensity ratio of the quadrant components to the total Reynolds stress. When the stratification stability changes from the unstable to the neutral condition, the intensity ratio gradually decrease. It indicates that the intensity ratio of the quadrant component and the stratification stability have a good correlation. The increase of the heat flux intensity makes the turbulence motion more “turbulent”, and intensity of quadrant component increase. Katul et al. (1997) pointed out that the time ratio of the Q2 and Q4 quadrants over tall natural grass and bare soil surfaces in the ASL are Deject Dsweep0.29 (D2D4 0.29) at z ¼ 2.65 m. Our data show that Deject0.296 and Dsweep0.314 at the height of 5 m in the near-neutral ASL, which have no obvious difference with Katul et al. (1997). In addition, Katul et al. (1997) points out that the time ratio of the ejection event is a constant, but the time ratio of the sweep event depends on the thermal stability. It can be seen from Fig. 4 (b) that the time ratio has a significant change with z/L, especially with the heat flux intensity increasing, the time ratio of Q2 and Q4 quadrants decreases. When the height is below 30 m, the time ratio of each quadrant component is similar in the nearneutral condition. As shown in Fig. 4(c), the relative error is less than 5%. That is, DQ10.19, DQ20.296, DQ30.2 and DQ40.314. At the same time, the results from Fig. 4(c) also show that DQ4> DQ2> DQ3> DQ1. At the near surface of the ASL, not only the intensity ratio of Q2 and Q4 quadrants is larger, but also the time ratio is larger.
Fig. 3. Distributions of quadrant contributions to the Reynolds shear stress varying with the heights.
heights. The solid point is the results of the unstable ASL, and the solid line is the results of near-neutral ASL. Also, we compared the results to the low and moderate Reynolds numbers from Wallace et al. (1972) (Reτ ¼ 187) and Nagano and Tagawa (1988) (Red ¼ 40,000). It can be seen from Fig. 3 that the absolute intensity ratio of Q1, Q2, Q3 and Q4 increases slightly along with the heights. In order to demonstrate the effect of the heat flux on the intensity ratio and the time ratio, the intensity ratio and the time ratio at the height of 5 m varying with z/L and height were given in Fig. 4. In Fig. 4(a), we take the consideration to the near-neutral condition (0.1 < z/L < 0.1) that the intensity ratio of the four quadrants are Q1, -1.47 0.32; Q3, -1.34 0.26; Q2, 2.86 0.27; Q4, 2.23 0.34 at the heights of 5 m. In the near neutral ASL, the intensity ratio of Q2 and Q4 is
Fig. 4. (a) and (b) are the variations of the intensity ratio and the time ratio at the height of 5 m with z/L, respectively. (c) the variations of the time ratio with heights averaged by all near-neutral ASL data. 5
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Journal of Wind Engineering & Industrial Aerodynamics 189 (2019) 1–10
4.1.2. Joint probability density functions of velocity fluctuations Wallace et al. (1977) analyzed the channel flow data obtained by Brodkey et al. (1974) at Reτ ¼ 194, yþ ¼ 45, to obtain such JPDFs for the Reynolds shear stress across the channel from the viscous sublayer to the centerline. Maitani and Shaw (1990) obtained the velocity fluctuation component u’ and w’ above the surface of the atmosphere and the vegetation cover, and the aggregation of Q2 and Q4 were revealed. Nolan et al. (2010) analyzed the JPDF from the wall surface to the boundary layer in the wind tunnel and found the increasing law of the JPDF along the height. An extension of quadrant analysis was carried out by Wallace et al. (1977), who made a detailed analysis of the Reynolds shear stress in a turbulent channel flow by determining its joint probability density function (JPDF). Z u' w' ¼
∞
þ∞
u' w' P u' ; w' du' dw'
4.1.3. Quadrant-hole analysis Willmarth and Lu (1972) extended quadrant analysis by focusing on intense Reynolds stress occurrences. They did this by determining the contribution to the Reynolds stress of each quadrant from u'w’ velocity fluctuation product values that are outside of what they called the hole, defined by the set of hyperbolas in each quadrant, u' w' ¼ H u' w' where u' w' is the mean value of this product for a given distance from the wall. One set of data at height of 5 m from 9:00 to 10:00 on March 27, 2014 was taken as an example to introduce our results. Fig. 6 shows the variation of the Reynolds stress ratio with the threshold H at each quadrant at the height of z ¼ 5 m. It can be seen from Fig. 6 that with the increase of the threshold H, the decay rate of the ratio in Q1 and Q3 Reynolds stress is larger than from Q2 and Q4. In particular, when H ¼ 15, the ratio between Q1 and Q3 is close to zero, but the ratio between Q2 and Q4 still has a certain intensity value. It indicated that not only the intensity of Q2 and Q4 is greater than that of Q1 and Q3, and for the contribution to the Reynolds stress, Q2 and Q4 contains more fluctuation information whose intensity is larger. Based on the data of 11 heights from 9:00 to 10:00 on March 27, 2014, the variation of the Reynolds stress ratio at different heights with the hyperbolic threshold H are given in Fig. 7. The plot consists of two superimposed contour plots; a continuous color scale contour plot and discrete isolines. The continuous color scale contours show the stress
(12)
where, P(u',w’) is the joint probability density function (JPDF) of the velocity fluctuations u’ and w'. Fig. 5 shows quadrant maps for the fluctuating component of velocity u’ and w’ of the JPDF at the 9 different heights from 0.9 m to 30 m in ASL. Due to almost all the experiment data have a similar trend along with the heights, so we take one set of the data as an example to introduce our results. The concentrated areas of the solid lines indicate the concentrated of the fluctuation component. The experimental data at all heights show that there are obvious the concentrated areas of the solid lines when u’ and w’ approaches 0. It demonstrated that the most of fluctuation components are smaller. Although the height measured in this paper is less than the thickness of the boundary layer, it is still possible to find a variation trend based on the variation of JPDF with heights. That is, with the height increasing, the JPDF pattern changes from the ellipse of the streamwise long axis to the ellipse of the vertical long axis, and more and more “round”, which means that the higher the height is, the better the symmetry of the JPDF pattern, and the wind velocity tends to isotropic. At the same time, it can be found from the figures that JPDF has a very good aggregation in Q2 and Q4 at all altitudes, which proves that the ejection events and sweep events have a dominated role in the ASL. The JPDF, given by the quadrant decomposition of Reynolds stress along with the heights, not only can be used to the prediction of the velocity component distribution of Reynolds stress at any height, but also can be used to verify the change of the Reynolds stress with height in numerical simulation.
Fig. 6. Contributions to Reynolds stress from truncated quadrants of u, w plane and percentage time spent in ‘hole’ as a function of ‘hole’ size.
Fig. 5. Joint probability density function at different heights. z/L ¼ 0.29, u* ¼ 0.27 m/s (9:00 a.m. to 10:00 a.m., March 27, 2014). Contour levels are from P(u',w') ¼ 0.1 to 0.6 in increments of 0.1. (a) to (i) indicates the measurement height at z/δ ¼ 0.9/150, 1.71/150, 2.5/150, 5/150, 7.15/150, 8.5/ 150, 14.65/150, 20.96/150 and 30/150, respectively.
Fig. 7. Contours of stress fraction for each quadrant with isolines of duration fraction superimposed. Contour levels are from u' w' Qi =u' w' ¼ -0.5 to 0 in increments of 0.05 for (a) and (c), from u' w' Qi =u' w' ¼ 0 to 1 in increments of 0.1 for (b) and (d). 6
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Journal of Wind Engineering & Industrial Aerodynamics 189 (2019) 1–10
Fig. 8. (a) and (b) represents the ratios of Q2 events to streamwise and vertical turbulent kinetic energy at different heights, respectively. Solid line represents ratio(u2þ) ¼ 0.355, dotted line represents ratio(w2þ) ¼ 0.376.
Similar to the results of the ejection events, it can be seen from Fig. 9 that the contribution of Q4 events to the streamwise and vertical turbulent kinetic energy is also approximately constant at different heights and different friction velocity, ratio(u2þ) ¼ 0.355 and ratio(w2þ) ¼ 0.318, respectively. Figs. 8 and 9 also show the ratio of the velocity vector components in the ejection events and sweep events to the streamwise and vertical turbulent kinetic energy in the near neutral ASL, respectively. It can be seen from the figures that the contribution of the Q2 event to the turbulent kinetic energy is 0.355, and the ratio of the Q4 event is substantially equal to that of the vertical turbulent kinetic energy. Q2 and Q4 event are almost near to a total of 70% of the turbulent kinetic energy. That is, the contribution of the ejection and sweep events to the produce of turbulent kinetic energy plays a leading role. Compared with the streamwise turbulent kinetic energy, the ratio of vertical turbulent kinetic energy in the ejection and sweep events to the total vertical turbulent kinetic energy at different heights of different friction velocities is basically the same. Although the total contribution of the Q2 and Q4 events to the vertical turbulent kinetic energy is about 70%, the ratio of Q2 is 0.376, which is significantly larger than the 0.318 in Q4. This may be related to the 68 sets of near neutral data in this section. The vertical heat flux of the 47 sets of data is larger than 0 (w'T'>0), and the vertical heat flux of the 21 sets of data is less than 0 (w'T' 0, the ratio of the Q2 and Q4 quadrant components is positive, and Q2 and Q4 have a dominant role in the vertical heat transport. The reason causing this difference may be the vertical heat flux direction is changed (stable or unstable), corresponding to the different stratification stability. For the temperature scalar and Reynolds stress decomposition, the data can be decomposed into three two-dimensional components, i.e., u'w’, u'θ' and w'θ'. Using the relationship of three components, their contribution to the octant decomposition can be obtained. Fig. 11 (a), (b) and (c) give the contribution of the three two-dimensional components u'w’, u'θ' and w'θ' obtained from the data (20140327 11–12) from each octant value to total value. It can be seen from Fig. 10 that the results are similar to those of Li et al. (2010) that the mean gradient transport is dominant in the logarithmic region. Compared with the results of this set of data, the other data obtained in this paper have the same law. It indicates that O2 and O8 are the two most important events in the decomposition of the ASL. Volino and Simon (1994) points out that a turbulent “parcel” of fluid moving away from the wall (w'>0) would most likely be of relatively low velocity (u' 0) (corresponding to the heat flux transportation from the top to the bottom). And the two components of the vertical heat flux w’ and θ0 can also be analyzed by quadrant analysis, similar to the
Fig. 10. The ratio of the quadrant components to the vertical heat flux varying with the heights.
Fig. 11. (a), (b) and (c) are the octant opponents proportion of the u'w’, u'θ' and w'θ', (d) respects the ratio of the O2 and O8 octant Reynolds stress to the total Reynolds stress varying with z/L. 8
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Journal of Wind Engineering & Industrial Aerodynamics 189 (2019) 1–10
Table 2 The octant decomposition corresponding to the shear flow and the thermal turbulence. Octant
O1
O2
O3
O4
O5
O6
O7
O8
u'w’ w’θ’
Outward interaction Ejection
Ejection Ejection
Ejection Outward interaction
Outward interaction Outward interaction
Wall ward interaction Wall ward interaction
Sweep Wall ward interaction
Wall ward interaction Sweep
Sweep Sweep
Fig. 12. Quadrant decomposition and conceptual drawing of eddies in each octant.
The octant decomposition corresponding to the shear flow and the thermal turbulence (in the unstable condition) were given in Table 2. Under the unstable condition, octant O2 corresponds to the ejection events in Reynolds stress decomposition and heat flux decomposition, that is, low-speed motions on the fluid “particle” and the high temperature away from the wall. When the two cases together the role is to promote each other to strengthen the process, the decomposition of octant, corresponding to the contribution of O2 is dominated, similarly in O8. Therefore, for the enhancement of the intensity of the vertical thermal convection in Fig. 11(d), the contribution of the O2 and O8 components to the total amount is also enhanced. Flow features that correspond to the different types of quadrants and octants can be seen in Fig. 12 (in the unstable condition). For the unstable case (hot wall), the O2 and O8 should be the most significant, corresponding to a turbulent ejection would be classified as a hot ejection and a turbulent sweep would be classified as a cold sweep. Similar to the stable condition, but for the scalar decomposition, the ejection and sweep events should correspond to the Q2 and Q4.
different friction velocity in the near neutral ASL is approximately constant, i.e., DQ10.19, DQ20.296, DQ30.2 and DQ40.314. Based on the joint probability density functions of the Reynolds stress component at different heights, it is discovered that the flow characteristic tends to be isotropic varying with the increasing height. The results of the hyperbolic analysis indicate that the H threshold is not the same as the intensity ratio of the Reynolds stress, which shows the higher height indicates the greater value of the velocity fluctuation component. In the near neutral ASL, the ratio of the turbulent kinetic energy (streamwise or vertical) to the total turbulent kinetic energy at different heights for different friction velocities is approximately constant. Furthermore, our results show that the contribution of O2 and O8 to the Reynolds stress decomposition and vertical heat flux decomposition is the largest, which reveals that the larger heat flux intensity corresponds to the larger ratio of intensity in O2 and O8. This indicates that the low-speed motions on the fluid “particle” and the high temperature tends to be away from the wall. The two cases promote each other to strengthen the process, so that the octant decomposition of the Reynold stress corresponded to the contribution of O2 is dominated, similarly in O8. In summary, the phenomena and laws revealed by the quadrant analysis and octant analysis in the ASL can promote the better understanding and prediction of the turbulent flow structures. Also, it can be used to verify the Reynolds stress characteristics in the numerical simulation and laboratory boundary layer simulation.
5. Conclusions Although the quadrant decomposition method has been employed in only a half-century, there are a great number of related results have been obtained in the characteristics of the turbulence structure at the atmosphere surface layer. However, the studies in the features of the turbulence structure at the sand surface are still limited, especially with the unprecedented dataset collected under different stratification stability conditions. As a consequence, most of the data measured in this work arranged in the near neutral (weak unstable condition) and unstable stratification due to the low Specific Heat Capacity of the sand as well the strong heat absorption and dissipation on the ground. In this work, the intensity ratio and the time ratio of the Reynolds stress component against the total Reynolds stress are determined by quadrant analysis. Our results indicate that the ejection events and the sweep events have a dominant role in Reynolds stress, and the intensity ratio of the quadrant components to the total Reynolds stress become stronger with the increasing heat flux. Moreover, the time ratio of the quadrant components to the total Reynolds stress at different heights for
Acknowledgements This research was supported by a grant from the National Natural Science Foundation of China (Nos. 11490551 and 11421062), the Fundamental Research Funds for the Central Universities (No. lzujbky2017-it66), the authors express their sincere appreciation to the support. References Bergstr€ om, H., H€ ogstr€ om, U., 1989. Turbulent exchange above a pine forest II. Organized structures. Boundary-Layer Meteorol. 49 (3), 231–263. Boldes, U., Scarabino, A., Di Leo, J.M., Colman, J., Gravenhorst, G., 2003. Characteristics of some organised structures in the turbulent wind above and within a spruce forest from field measurements. J. Wind Eng. Ind. Aerod. 91 (10), 1253–1269.
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