Experiment and Large Eddy Simulation
Several heliostats are arranged on the field for concentrating beam solar radiation onto a receiver. The flow-induced dust deposition and vibration of heliostats will lead to failure of the receiver and the entire system in eventuality. In view of this, fluid flow analysis behind a heliostat model in a narrow channel of limited length is presented depicting a field-type condition. For this purpose, experiment and large eddy simulation (LES) are performed. Laser Doppler velocimetry (LDV) technique is used for horizontal velocity measurement around a heliostat with 25° angle of inclination at a Reynolds number of 60000 basing on the length of an inclined plate. The wall-resolved coarse and fine grids are utilized for LES with dynamic- Smagorinsky-Lilly and subgrid-scale kinetic energy transport equation based models.
Comparison between local instantaneous and statistical quantities, like horizontal velocity and its root-mean-square values are presented. The flow development in the wake region of heliostat model is described using the time-dependent and the statistical horizontal velocity. Based on the adopted LES approach the coherent structures are identified with the λci-criterion. The analysis of power spectral density and swirling strength isosurface reveal the responsible coherent structures in the inertial subrange for transport of energy to smaller structures. Interestingly, the presence of hairpin-type vortices is observed extending from the two sides of model heliostat using the λci-criterion. Further, instantaneous span-wise vorticity contour reveals their alignment with flow represented by streamlines. The provided insight to these flow features and their organization is expected to frame strategies for mitigating the unsteady flow-induced vibrations and dust depositions on heliostats.
Concentrated solar thermal (CST) based technologies are versatile, in view of their applications. One such novel application is solar convective furnace system, which was recently proposed by Patidar et al. (2015). In this system, heliostat based central receiver is used for generating hot air. For this purpose, an open volumetric air receiver was specially designed by Sharma et al. (2015a, 2015b). Such a system will be useful in desert regions, like Rajasthan having direct normal irradiance of 5-6 kWh/m2/day. However, a high wind-speed of nearly 47m/s, see e.g. Lakshmanan et al. (2009), may cause vibration of heliostat and even initiate the saltation process. This will lead to deposition of dust on heliostats and reduction in their reflectivity, see e.g. Niknia et al. (2012). The soil composition of Rajasthan has high fraction of fine sand particles (< 60μm) Gupta (1986). These are prone to saltation and deposition on the surface of heliostat. This was verified by SEM based analysis in Yadav et al. (2014a). The deposition of dust on an inclined square finite plate was investigated by Yadav et al. (2014a, 2014b) in a wind tunnel. They demonstrated using experiment and simulation that the deposition is affected by location and separation between the inclined finite flat plates at a given angle of attack. Thus, for a better understanding of the deposition process, both experimental and numerical studies on fluid flow around an inclined flat plate are reviewed in this section.
Nature of fluid flow behind a bluff body, such as flat plate or a circular cylinder, is very complex including separation of flow, unsteady vortex shedding and presence of small and large –scale turbulent structures. The pioneering experimental investigation by Fage and Johansen (1927) was conducted at 18 different angles of inclination ranging from 0.150 to 900over a wide range of horizontal velocity. They reported a relation between angle of inclination and vortex shedding frequency for infinite flat plate. The Strouhal number of 0.148 for an angle of attack between 30° to 90° was calculated based on the projected length of the inclined flat plate normal to the flow direction.
They also explained the mechanism of vortex shedding from the leading and trailing edges at the same rate. Roshko (1954b) extended this study to cylinder and wedge in addition to a flat plate. A wake based Strouhal number was defined and observed to be ubiquitous compared to the one based on the projected length. The free-streamline theory based notched-hodograph method by Roshko (1954a) was used to estimate the drag from vortex shedding frequency. This method was found by Wu (1961) to yield results in agreement with that of Fage and Johansen (1927) at angles of inclination greater than 300. A new model was proposed to understand the two-dimensional wake region behind an obstacle. This model accounted for coefficients of lift, drag and normal force as stated by Fage and Johansen (1927).
Abernathy (1962) extended the free-streamline theory of Roshko (1954b) for an inclined flat plate. Separation between free-vortex shear layers was found to be independent of flow constriction. This distance was used to devise a wake based Strouhal number. These wake based Strouhal numbers were mentioned besides the customary Strouhal number in Roshko (1954b) and Abernathy (1962). Perry and Seiner (1987) experimentally investigated turbulent wakes past nominal two-dimensional bluff bodies along their plane of symmetry. They concluded that the flow was still two-dimensional in view of the observed differences with two-dimension based theory. This was attributed to the waviness in vortex cores along the span-wise direction.
Kiya and Matsumura (1988) showed that the shearing stresses in the turbulent wake behind a normal thin beveled flat plate is mainly due to the random velocity fluctuations with half of the vortex shedding frequency. Chein and Chung (1988) determined the strength and location of vortices shed from an inclined flat plate using the Kutta condition. Their discrete vortex method predicted both macroscopic and microscopic features better than previous numerical practices. An experimental study for estimating Strouhal numbers with rectangular bluff bodies with at various aspect ratios was conducted by Knisely (1990). The authors investigated drag and lift characteristics with varying angles of attack. A rapid increase in Strouhal number was observed at low inclinations, which was associated with reattachment of separated shear layers. The dynamics of flow in the near-wake region behind a vertical flat plate was experimentally investigated by Leder (1991). Fluid mixing was reported in the separation region. The deficiency of using isotropic eddy-viscosity approaches in some of the modeling approaches was concluded. Systematic two and three -dimensional numerical simulations of flow over a flat plate are performed to investigate drag and vortex shedding.
For instance, Najjar and Vanka (1995a) used a range of Reynolds number (Re) flows in two-dimension to study the drag and wake characteristics including shedding frequencies and compare the results with previous studies. They stated that the flow become three-dimensional at a Re ~200.Later on it was established with three-dimensional direct numerical simulation (DNS) by Najjar and Vanka (1995b). They showed that the calculated drag agreed well with experiments. A study focusing on transition from two to three-dimensional flow behind various bluff bodies was made by Thompson et al. (2006).They identified two modes of instabilities having onset at Re ~107 and 125 for a normal flat plate. Thus, at a higher Re, say beyond 10000, extremely complex flow structures and the related instabilities are expected behind a flat plate. Najjar and Balanchandar (1998) employed three-dimensional computation to correlate fluctuating aerodynamic coefficients with vortex shedding frequencies.
Laser Doppler anemometry (LDA) investigation by Lam (1996) showed that the trailing edge vortices, around a plate inclined at an angle of 300 and Re ~30000, are more dominant than that of leading edge. Lam and Leung (2005) performed similar experimental investigations using particle image velocimetry (PIV) at 200, 250 and 300angle of inclinations at a Re ~5300. They reported that the vortex formation at the leading edge involves more complicated mechanism than the Kevin-Helmholtz instability of trailing edge. The trailing edge vortices showed to retain more energy than their leading edge counterparts in wake-region at a point.
Lam and Wei (2010) conducted unsteady Reynolds-averaged Navier-Stokes (URANS) simulations for analyzing these observations. Their analysis provided an insight to the differences in the vortex shedding processes at the leading and trailing edges. They also explored how vortex shedding induces fluctuating lift and drag on the plate as in Najjar and Balachandar (1998). Narasimhamurthy and Andersson (2009) performed DNS of the wake behind a normal flat plate to investigate turbulence characteristics. Statistics like mean flow and Reynolds stresses exhibited characteristics similar to that of a cylinder. A mean recirculation zone was found to extend up-to two-times the height of flat plate in the wake region. They also explored the relation between base pressure and vortex formation. They revealed that the reduction in base pressure increases the vortex shedding frequency with a shorter recirculation zone.
Yang et al. (2012) conducted two and three -dimensional DNS at Re ~1000 and mentioned the reliability of the three-dimensional approach. They found unequal vortices shedding from leading and trailing edges behind the inclined plate. Saha (2007) reported a Hopf bifurcation using DNS in the separated near-wake behind a normal flat plate that instills unsteadiness at a Re ~ 30 – 35. Further, the steadiness of far-wake for a Re ~75 – 140 was reported to become unsteady at a Re = 145. This was attributed to large-scale vortices with low frequency. This is similar to that of the reported flow characteristics in Thompson et al. (2006). A tertiary vortex street was observed to appear in far-wake region at a Re = 150. Their frequency decreases with increasing Re. The drag and Strouhal numbers were reported to be nearly unaffected by these transitions.
Further investigation by Zhang et al. (2009) revealed various transition zones between steady and chaotic flow through Hopf and subsequent bifurcations. This was verified by calculating the largest Lyapunov exponent value, which is nearly zero for non-chaotic flows, else distinctly positive. Drabble at al. (1990) studied the stream-wise fluctuating forces on a flat plate normal to the flow direction with inlet velocity fluctuations. The response was defined through an admittance function, which increased with fluctuations in coherent flows and decreased for turbulent flows. This is relevant for further studies with varying inlet velocity to mimic wind gusts as in Lakshmanan et al. (2009). Indeed, DNS is limited only to low Re in view of computational requirements, see e.g. Chandra and Grötzbach (2007, 2008).
URANS, Detached Eddy Simulation (DES) and Large Eddy Simulation (LES) are some of the available options, especially, for high Re. Their degree of simplicity leads to uncertainties, which are being evaluated. Breuer et al. (1998, 2000) used various subgrid stress models for LES analysis of flow past a circular cylinder at Re~3900 and 140000. It was found that the importance of models increases with Re. Breuer and Jovicic (2001) conducted LES using the Smagorinsky (1963) model of flow around a flat plate inclined at 180 with Re ~20000. These results were consistent with the experimental findings by Lam (1996). They found that the Smagorinsky model has drawbacks in transitional flows in the near-wall region and for high Re. At high Re turbulent flows Smagorinsky coefficient should be local time dependent. In order to calculate this parameter as a function of space and time, Germano et al. (1991) proposed a dynamic approach.
The phenomenon of backscattering, which is found in transition flows, was addressed in this model. This serves as an extension to Smagorinsky model. Thus, the dynamic subgrid stress model proposed by Germano with Lilly’s modification Lilly (1992) and even with kinetic energy transport model as in Kim and Menon (1997) can be used to capture the physics of separated flows behind the heliostat plate. Breuer et al. (2003) employed DES, URANS in addition to LES on a coarser grid to evaluate their relative capabilities and found LES that adequately describes the flow characteristics. Afgan et al. (2013) carried out extensive LES based fluid-flow analysis over a flat plate using the Smagorinsky model with Van Driest wall-damping to represent the flow in near-wall region as in Moin and Kim (1982). These are summarized in Table I.
It may be noted that most of the experiments are performed for Re ≤ 20000. Indeed, a few wall-resolved LES based flow analysis is reported at a higher Re, mostly, with an inclined-isolated flat plate or around a cylinder. The current work aims at understanding flow around a heliostat that comprise of a finite square inclined flat plate, stand and base at a higher value of Re. Heliostats are radially located on the field in a staggered layout. The distance between these heliostats varies depending on their radial positions. Thus, the reported observations using an isolated infinite or finite flat plate only serves as a reference point. The flow features behind a heliostat model is expected to be more complex considering the presence of surrounding heliostats or walls. For analyzing flow features behind a heliostat, based on recommendations wall-resolved LES approach is adopted. In addition, experiments are performed for validating the adopted numerical approach. These are described subsequently.
Experiments are performed with one heliostat model to analyze fluid flow around this object. For instantaneous local measurement a two-dimensional laser Doppler velocimetry (LDV) technique is used, see Figure 1. LDV is a well-known non-intrusive in which the Doppler frequency of the seed particles crossing the formed interference fringe by un-shifted and shifted laser beam, as shown in this figure, is measured. The product of spacing between two adjacent fringes and the Doppler frequency provides the velocity component. This is in line with the plane formed due to this dual laser beam crossing.
|FIG. 1. (a) Image of beam crossing, showing fringes, idealized seed particle and intensity signal at photo detector and (b) two-dimensional LDV setup at IIT Jodhpur|
Geometry and Conditions
In the reported flow measurement experiment a single heliostat is considered. Heliostat model comprise of three distinct structures, namely, inclined square flat plate (P), stand (S) and base (B). The model is placed at an inclination of 250 with respect to the horizontal plane. The plate is located at 7 mm above the stand inside the test section of a sub-sonic wind-tunnel as shown in Figure 2. The angle of inclination is comparable to the latitude of Jodhpur and serves as the reference point for further investigations. Indeed the angle of inclination will vary throughout the day in order to concentrate solar radiation to a fixed absorber. Preliminary analysis revealed its variation from 25 to 65°. The dimensions of experiment heliostat model are given in Table II.
TABLE II. Dimensions (in mm) of heliostat model
|Components||Dimensions (length × width× height/thickness)|
|Plate (P)||60 × 60 × 3|
|Stand (S)||18 × 6 × 50|
|Base (B)||74 × 68 × 20|
|FIG. 2.(a) Experimental heliostat model with (b) two-dimensional LDV system|
A schematic of the measurement section at the symmetry plane is shown in Figure 3(a). In the performed experiment a diffuser with 5° divergence angle is attached to this section having a length of about 10 times the length of inclined plate. The constant free-stream average inlet air velocity at the wind tunnel is set to U0 = 16 m/s. The required length scale (D) for calculating ReD, length (L) and height (H) of the wind-tunnel are given in Table III. The heliostat model is placed inside the wind-tunnel with its center at a height Hc = 82.5 mm. In the measurement section, the domain size along horizontal (z), vertical (y) and span-wise (x) direction is 8.3D, 2.5D and 2.5D, respectively. The domain length along the horizontal direction is comparable to Breuer and Jovicic (2001).
Moreover, the selected length of test section represents separation between heliostats along radial direction at some azimuthal angle in a field-layout. The narrow channel, especially in the span-wise direction, depicts the presence of neighboring heliostats with stand as wall. Thus, the selected domain size is an attempt to simulate flow around a single heliostat in the simplest possible way. The outcome will serve as the starting point for further detailed investigations with multiple heliostats. Time dependent horizontal velocity component is measured along the vertical direction at a distance of 5 mm, upstream of the leading edge and 5, 10, 30 and 200 mm downstream of the trailing edge of heliostat model.
The Reynolds number, over a time step Δt with spatial discretized length of Δx. Subscript “i” means center of element with Δx/2 length on each side, “r,” “l,” “t” and “b” mean right, left, top and bottom. Least squares cell-based gradient evaluation assumes that the solution varies linearly between adjoining grid elements. Suppose a vector ri joins the cell c0 to adjoining ci, then the gradient of a variable ϕ can be represented as:
To obtain the cell gradient, let . Each of these components is calculated as; here,Wi0 is a weight factor and is determined by Gram-Schmidt orthogonalization, Anderson and Bonhus (1994). The simulations have been performed for a total of 17 flow through times considering Afgan et al. (2013). Out of this 13 flow through times based LES analysed data is used for instantaneous and statistical analysis.
DISCUSSION AND RESULTS
The measured and LES based time-dependent horizontal velocity is compared at 5 mm downstream of the trailing edge of the inclined plate. This serves as the first test of the adopted LES approach. The measured horizontal velocity consists of more than 1000 LDV bursts over 90 seconds following Lam (1996).
|(a) Experiment – full data set||(b) Experiment – partial data set|
|(c) Coarse grid –k equation model||(d) Coarse grid – DSGS model|
|(e) Fine grid – k equation model||(f) Fine grid – DSGS model|
|FIG. 9. (a), (b) Measured and (c), (d), (e) (f) LES analyzed horizontal velocity at 5mm downstream of the trailing edge|
Figure 9(a) clearly depicts that the flow is, practically, independent of time. For an elaborated view the measured horizontal velocity for 10 seconds is shown in Figure 9(b). The random oscillations are, mostly, within ±6-9 m/s. Furthermore, LES analyzed time-dependent horizontal velocity at the same location is shown for the selected models and grids in Figures 9(c) to 9(f). These are analyzed over 13 flow-through times. As in the experiment, the LES analyzed oscillations are bounded within ±6-10 m/s. For a reference, dotted lines are provided at ±6 m/s. Thus the applicability of LES approach to capture flow features behind a heliostat model is inferred. The time resolution in LES is much better than that of the experiment. This is evident even from visual inspection of Figures 9(a) to 9(f). Therefore, LES is expected to provide an acceptable estimate of St, if any, for shedding of vortices having the same or different characteristic frequencies.
Analysis of LES data shows repeated occurrence of maxima and minima in horizontal velocity as indicated in Figure 9(d) by vertical dotted lines. However, periodicity is not very obvious. This may be associated with transport and interaction of multiple large-scale structures at the measurement location having different frequencies. This is expected in view of different structures in the modeled geometry viz. P, S and B as in Figure 2. In view of lack of any apriori estimation phase-averaged values are not adopted for analysis. More detailed analysis is provided for analyzing the underlying phenomena. The range of fluctuations is summarized in Table VI. A comparative qualitative and quantitative assessment reveals that LES analyzed values are well within the measured data and thus suitable for further analysis.
|TABLE VI. Velocity fluctuations in simulations and experiment|
|Mesh & Model||Velocity Fluctuations (m/s)|
|Coarse Grid – k equation model||±6-9|
|Coarse Grid–DSGS model||±6-9|
|Fine Grid –k equation model||±6-9|
|Fine Grid–DSGS model||±6-10|
Further qualitative and quantitative evaluation of the adopted LES approaches is performed using the statistical horizontal velocity and the corresponding RMS values at various horizontal positions along the centerline of heliostat model. These locations are depicted in Figure 10(a) and numbered 1 to 7. Their unequal spacing can be inferred from the indicated relative positions. The first point indicated by 1 is located at 5 mm and the last point indicated by 7 is located at 200 mm downstream of the trailing edge of inclined plate. As explained, the locations 1 and 7 are in near and far -wake region, respectively.
For statistical analysis of LES analyzed data thirteen flow-through times are considered. A comparison between LES and experimentally analyzed statistical horizontal velocity is shown in Figures 10(b) and (c), which depict a qualitative agreement. The coarse and fine-grid based LES analyzed data over-predicts and under-predicts the measured statistical horizontal velocity values, respectively. The increasing trend of statistical horizontal velocity is predicted by LES. The lowest value of horizontal velocity is a signature of highest level of fluctuations and vice versa. Mean flow-field development downstream of the heliostat model is inferred from increasing statistical horizontal velocity. This behavior is captured by LES and experiment. The extent of fluctuations indicate turbulent nature of the flow past heliostat and is quantified using statistical RMS values of horizontal velocity at the considered horizontal positions. The standard deviation in the measured horizontal velocity and its RMS values are estimated using eight temporal datasets including 100, 200, 300 up-to 800 measurements. The relative standard deviation of mean and RMS values allowed us underlining the measurement uncertainty. The maximum standard deviation or uncertainty is about 9% for the mean horizontal velocity at 5 mm and about 10% for the RMS of the horizontal velocity at 200mm downstream of the trailing edge of inclined plate.
|FIG. 10.(a) Measurement locations along center-line downstream of model; measured and LES analyzed statistical (b)horizontal velocity and (c)RMS of horizontal velocity at these locations|
The comparison between measured and LES analyzed statistical RMS values within 3.5-4 m/s of horizontal velocity clearly depict both the qualitative and quantitative predictive capability of the modeling approaches. High values of statistical RMS horizontal velocity are observed up-to 30 mm and the smallest value is obtained at 200 mm from the trailing edge of inclined plate. Thus, it is inferred that indeed the strong flow disturbance dissipates from near to far-wake region. The under-prediction of RMS values with LES beyond 30mm, within uncertainty bound, may be attributed to the applied atmospheric pressure outlet condition considering a realistic separation between heliostats in a field-layout. The LES analyzed statistical and RMS horizontal-velocities are practically independent of the selected sgs-models for the fine grid. This observation is interesting in view of DNS in which models are not expected to influence the flow features. Furthermore, the LES index of quality (LES_IQ) as explained in e.g. Geurts and Froehlich (2002), Celik et al. (2005) is computed using the following expression:
here, is the Kolmogoroff microscale of length, which is obtained using the characteristic length-scale of the test-section (H=150 mm) and h indicates the smallest grid-size near the wall. The computed value of and is acceptable for a LES. Using the suggested values of results in and is an indicator of a good LES. For DNS this parameter should be greater than 0.95. However, the value is beyond 0.8 for LES, which suggests that ca. 80% of the energy is resolved see e.g. Pope (2009) and even a step closer to DNS, as observed.
The LES analyzed instantaneous swirling strength iso surface or λci-cretiron as in Zhou et al. (1999) corresponding to 900Hz using the DSGS model for coarse and fine grids is shown in Figure 11 at 0.34 second. This criterion identifies similar structures as Q or λ2, Green et al. (2007). The relationship between these criteria is explained by Charaborty et al. (2005). Swirling strength, given by λci, is a measure of the local swirl rate inside the vortex. In other words, the time period for completing one revolution of the streamline is given by 2π/λci. The identified coherent structures appear to be in the inertial subrange, which is inferred by analyzing power spectral density in Figure 14.
Such structures are responsible for transport of turbulent energy to smaller scales, as investigated by Terashima et al. (2015) for a planer jet. The presence of classical horseshoe vortex is clearly visible in front of base of the heliostat model, see Figure 11(a). It is evident that owing to a higher spatial resolution much smaller grid-resolved structures are captured by the fine grid as shown in Figure 11(b). This is attributed to lower values of grid-based local Courant number for the fine-grid in comparison to the coarse-grid. In the case of fine-grid, except in the wall-layer, this is found to be less than 1.
The aligned coherent structures with the flow direction are visible, especially near the lower-wall of test section, for LES with fine grid. Both the coarse and fine grids based swirling strength isosurface show the emanating coherent structures from the two sides of finite inclined flat plate. Their nonlinear interaction is leading to eventual formation of smaller structures. The smaller structures are clearly visible in Figure 11(b). It may be safely concluded that the overall flow-field is captured by both the grid-based LES simulations. Thus, for a clearer insight to the flow-field and on vortex shedding the coarse grid based wall-resolved LES is used.
|FIG. 11. Swirling strength isosurface (λci-criterion)for (a) coarse and (b) fine grids with DSGS at 0.34 second.|
In Figure 12, the vortices shed from the trailing edge of the plate are shown for the DSGS model at an interval of 0.01 second. The selected time-period corresponds to the estimated vortex-shedding frequency of about 109Hz from an inclined plate. Figure 12(b) and 12(c) indeed shows the detachment of coherent structures from the trailing edge during 0.32 to 0.33 second. Finally, at a later instant the attached structure on the trailing edge of plate reappears in Figure 12(c).Thus the entire series of event leads to continuous vortex shedding from the trailing edge.
As expected, the flow-field around heliostat is asymmetric along vertical direction, which is also inferred from the measured mean horizontal velocity, say, at a distance of 5 mm downstream of the leading edge in Figure 4. For completeness the fine grid based LES analyzed instantaneous swirling strength isosurface is shown in Figure 12(d). As in the coarse grid, at this instant of time, attached structures on the trailing edge of the finite inclined flat plate is observed.
Further, the vortex path is depicted from the leading edge into the wake region and is explained with more details. As a result of higher grid resolutions, the resolved smaller structures are captured. These vortices will exert dynamic loading on a structure placed in the wake of heliostat, Lam et al. (2005). It was found in Lam (1996) that the trailing edge vortices are stronger in sense that they carry more energy than that of leading edge. However, the current study involves a square plate, stand and base leading to even more complex interactions and resulting flow structures.
|(a) t = 0.32 second||(b) t = 0.33 second|
|(c) t = 0.34 second||(d) Fine Grid at t = 0.34second|
|FIG 12. Vortex shedding from the trailing edge analyzed using coarse (a, b, c) and fine (d) grid with DSGS model|
Interestingly, amidst these multiple vortices, hairpin-type structures are visible behind the square inclined finite flat plate in the LES analyzed data at 0.34secondin Figure 13(a) and 13(b). Similar structure with flow past a sphere was reported by Johnson and Patel (1999). The hairpin-type vortex extends from the two-sides of the inclined square finite plate rolls up-to the center forming a closed structure as depicted with swirling strength isosurface corresponding to 650 Hz in Figures 13(a) and 13(b). Indeed the λci-criterion identifies such structures as expected, Zhou et al. (1999).The velocity vector with horizontal-vorticity contour is shown in Figure 13(c) and 13(d) at a distance of z/D = 1.0 and 2.0 from the trailing edge of inclined plate. The observed counter-rotating vortices on these planes are regarded as a signature of hairpin vortex as explained by Adrian et al. (2000). It appears that these structures ensemble forming larger structures as inferred in Pope (2009). It is further observed that their angle of inclination with horizontal direction increases with distance from the trailing edge. Apparently, these structures are transported with the flow enclosing smaller structures and are extending up-to 3 times the length of plate i.e. z/D=3.0. Non-linear interaction between the vortex structures at the trailing edge as in Figure 11 and hairpin-type structures as in Figure 13 may lead to (a) generation of smaller structure, (b) excludes the presence of a unique Strouhal number and (c) the energy redistribution among structures with different frequencies. This aspect will be analyzed using frequency domain analysis of time-dependent velocity.
To understand the transport of coherent structures and the related energy dissipation process the power spectral density (PSD) is used. It employs the fast Fourier transform (FFT) to analyze the temporal data on a frequency domain. This may provide any prevailing dominant frequency at a particular location, which is essentially an indicator of vortex shedding. This is defined as,
Where, N is the sample size, u(tk) is the horizontal velocity at time tk and fs as the sampling frequency.The PSD of LES analyzed horizontal velocity using one-equation model and fine-grid is shown in Figure 14.These are performed at different horizontal positions, viz. 5, 10, 30 and 200 mm downstream of the inclined square flat-plate along its centre line as shown in Figure 10(a). The energy is distributed over a wide-range of frequencies up-to 200 Hz. The LES analyzed energy spectrum of horizontal velocity with a slope of about f-1/2in this region is attributed to the presence of very-large scale motions Katul et al. (2012), which is unlike f-1 scaling as in Perry and Marusic (1995). The experimental specific kinetic energy up-to 100Hz based on PSD is under-predicted by about 15% than that of LES analyzed values.
As expected, f-5/3 depicts that the inertial subrange and is captured with LES as in Calaf et al. (2013) in between 200 – 1000 Hz. Furthermore, the selected grid-resolution resolves structures beyond this region. Thus, this is an extremely well resolved LES; at this position the spatial resolution is near to a DNS. Unusual is that the inertial subrange is rather short in terms of frequency range. This poses a question that whether such a fluid flow can be termed as turbulent.
The second argument would be that such a scale, say beyond 1000Hz, is affected by the diffusive nature of these sgs-models and numerical approach. Indeed energy levels decrease with increasing distance along the flow direction, which is inferred from PSD at the selected locations along the centre-line of model. This indicates non-linear interactions leading to generation of smaller structures, which eventually dissipate the energy, Tennekes and Lumley (1972).
As expected, PSD of LES analyzed turbulent kinetic energy provided the same slopes and is not included avoiding repetitions. Apparently, at the considered locations along the centre-line dominant energy containing structure is not observed and thus, a unique St is not expected.
|FIG. 14. PSD of LES analyzed horizontal velocity with fine-grid and k-equation model at 5, 10, 30 & 200mm downstream of heliostat model.|
In order to understand the vortex dynamics in the wake-region LES analyzed positive and negative span-wise vorticity contours are shown in Figure 15 at certain instants of time. These are shown at the mid-plane of heliostat model. The considered instants account for the one flow-through time. The flow is indicated by instantaneous streamlines colored by vorticity. The vortices are found to be aligned with the streamlines and are thus inferred as convected downstream in the wake region of the model as shown for all instants of time. This is indicated by dotted arrows and is consistent with findings using PIV based measurement by Lam and Leung (2005).
Such interaction between shear and eddies lead to generation of turbulent stresses and are consistent with observations by Breuer and Jovicic (2001), Lam (1996).The negative values of span-wise vorticity correspond to clockwise rotating vortices. The positive values of span-wise vorticity indicate counter clockwise rotating vortices. Figures 15(a) to (f) show the attached vortex on the trailing edge is rotating in the clockwise direction and that of leading edge along the counter clockwise direction.
Careful observations at the trailing edge show the stretching of clockwise rotating vortices, detachment, formation and transport of localized vortices from 0.31 to 0.34 second along the streamline. Behind the stand on this plane the nonlinear interaction of vortices leading to formation of smaller structures is inferred. The presence of clockwise rotating vortices is observed behind the base. Furthermore, the analysis of positive span-wise vorticity at 0.32 second indicates the presence of horseshoe vortex in front of the base. Dust particle-vortex interaction will be analysed at the next step and reported.
(Scale: 60mm )
|(b) t = 0.31 second|
|(c) t = 0.32 second||(d) t = 0.32 second|
|(e) t = 0.34 second||(f) t = 0.34 second|
|FIG. 15.LES analyzed positive (a, c, e) and negative (b, d, f) span-wise vorticity contours with streamlines on the mid-plane of model at 0.31, 0.32 and 0.34 seconds.|
The LES analyzed Reynolds stresses viz. cross-correlation of velocity components are computed using twelve flow-through times. These are depicted in Figures 16(a) to (c). These clearly depict the influence of vortices, as described in Figure 15, especially in the component. The other two components are localized in view of the affected region behind the heliostat model. The positive values of Reynolds stresses depicts that the velocity components are, more frequently, in the same direction. The negative values depict the opposite effect. Analyzing these quantities allows us to conclude that indeed the dominating will lead to shear stress on the wake-affected heliostat or mirror surface and its erosion with wind and dust. Thus, mitigating the same would be beneficial for a durable coating or mirror surface.
|FIG. 16. LES analyzed specific Reynolds stresses; (a)(b) and (c)|
Unsteady flow features behind a single heliostat in view of a field-layout is analyzed using the laser Doppler velocimetry and the large eddy simulation (LES). For this purpose a narrow channel of a limited length is considered indicating the presence of neighboring heliostats in a field-layout. The local measurement of the horizontal velocity depicts strong fluctuations at 5mm from the trailing edge of heliostat model. The decreasing level of oscillations from near-to-far wake region is inferred from the measured and LES analyzed root-mean-square values of horizontal velocity. Thus, dissipation of energy is inferred downstream of heliostat in the wake region.
Analysis of swirling strength iso surface or λci-criterion reveals that LES with fine-grid resolves smaller structures owing to lower values of Courant number with the same time-resolution. The observed hairpin-like structure in the wake behind heliostat is consistent with the presence of velocity vector at two different vertical planes showing the presence of counter rotating vortices. These are transported along the wake and may interact with heliostat in this region. Such coherent structures are regarded as responsible for energy transfer from large to smaller structures. The intensity of which may lead to its vibration. The PSD of LES analyzed horizontal velocity reveals a slope of about f-1/2in the low frequency region up to 200Hz infers the presence of very large scale motions.
An indication of which is found in local time-dependent horizontal velocity with time-scale of the order of flow-through time. Further analysis of span-wise vorticity reveals the alignment and transport of counter and clock-wise rotating vortices along the streamlines in the wake region. Thus, a heliostat in the wake will experience such unsteady flow features.
Further analysis with multiple heliostats in a field-layout will be performed to verify the observed flow-features and to establish their influence on heliostats. However, it may be safely concluded, while designing a heliostat field, flow based parameters are to be established. Such parameters will hold the key to mitigate flow-induced dust deposition and vibration effect on heliostat. This will, in turn, enhance the performance of concentrated solar thermal system based on heliostat in near future.
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