Web Supplement: Evaluation of the CFD Model with Tree Foliage Parametrization
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Web Supplement: Evaluation of the CFD model with tree foliage parametrization
The ability of the 3-D CFD model (i.e., Sect. 2.1) to reproduce measured or direct numerical simulation (DNS) model profiles of spatially-averaged flow is ensured prior to its use to inform and parametrize the 1-D column model. Parametrization of drag and turbulent kinetic energy (TKE) generation and dissipation due to vegetation (i.e., Sect. 2.1) requires evaluation, despite being very similar to previous work (e.g., Liu et al. 1996; Foudhil et al. 2005; Dalpe and Masson 2009; Santiago et al. 2013). No measurements, DNS, or large-eddy simulation (LES) model results for canopies composed of both bluff (i.e. buildings) and porous (i.e. trees) obstacles are currently available, to the authors’ knowledge, that would be appropriate for evaluation of the CFD model; hence, it is tested against wind-tunnel measurements of flow through vegetation canopies. This is done for forest and forest-clearing scenarios; first, however, evaluation of the code for building-only arrays is addressed.
B.1 Flow through cubic arrays
Evaluation of CFD model flow through arrays of cubic obstacles, with a different computational package but identical model formulation, has been achieved by Santiago et al. (2007) against wind-tunnel results, by Santiago et al. (2008) against DNS model results, and by Santiago et al. (2010) against LES and scale model results. Evaluation of the current computational package has been conducted by Simon- Moral et al. (2014) for aligned arrays of cubes.
B.2 Continuous forest
Brunet et al. (1994) measured ‘steady state’ wind-tunnel flow profiles in and above a canopy of model vegetation sufficiently far downstream to avoid effects from the upwind edge. Measurement of flow profiles of mean horizontal wind speed, mean shear and standard deviations of horizontal and vertical wind was conducted downstream at x = 4 m. This dataset was used to test and tune the CFD model of Foudhil et al. (2005); further details are available in their article.
B.2.1 Simulation design
The non-dimensional quantity CDV LD h is preserved in the current CFD model implementation and canopy height (h) is increased to a realistic full-scale value of 10 m. To compensate, CDV is reduced to 2 -3 0.2 and LD to 0.159 m m , both realistic values. Furthermore, the x-axis is scaled identically to the canopy height , which increases from 0.047 m to 10 m. At the inlet boundary, the free-stream velocity is set to 10.2 m s−1, and TKE and its dissipation rate are estimated following Foudhil et al. (2005). Domain height is 60 m and there are 20 computational levels in the canopy. B.2.2 Results
The CFD model is found to be sensitive to parameter C5 in Eq. 4. Hence, results using ‘tuned’ values of this parameter (similar to Foudhil et al. 2005) are presented in addition to the value of 1.26 which is based on theoretical arguments (Sect. 2.1). Sensitivity to C5 is apparent in Fig. B1, where a modest decrease in C5, of ≈10-30%, yields noticeably different results. This parameter controls the destruction of dissipation due to the foliage – lower C5 yields a higher dissipation rate and smaller k in the canopy (and elsewhere depending on vertical transport). In general, the model captures the qualitative trend of the observations, and with C5 = 1.00 it further reproduces the wind-tunnel results in a quantitative sense, with the exception of the above-canopy k. The Foudhil et al. (2005) CFD model captures the wind-tunnel data well, which is not surprising because it uses a value of C5 that is tuned to the same data set. However, given that the foliage parametrization implemented in both our CFD model and that of Foudhil et al. (2005) does not represent the effects on the flow of ‘waving’, we do not expect the CFD model to compare well with the wind-tunnel results, especially the k profiles. Hence, a C5 parameter that is tuned to the results may not yield better results for more general cases.
B.3 Flow at the edge of a forest
Raupach et al. (1987) performed wind-tunnel experiments in several forest-clearing configurations. Velocity and turbulence statistics were measured. Their experiment with alternating patches of clearing and forest, each of length 21 h, is chosen to test the CFD model, where h is the canopy height (Fig. B2). This dataset has been used to test several other RANS and LES model implementations: the k-l model of Wilson and Flesch (1999), the k- model of Foudhil et al. (2005), and the LES models of Yang et al. (2006) and Dupont and Brunet (2008). The ability of the model to reproduce adjustment of flow profiles after a clearing-to-forest transition is evaluated. In one sense it is essential that the model be able to represent discontinuous foliage, because tree foliage distributions in cities tend to be patchy. On the other hand, the assumption made throughout the current contribution is that tree foliage is evenly-distributed across canopy spaces (Sect. 1.3), and hence foliage-no foliage transitions are less frequent, because foliage layer edges correspond to building/obstacle walls for foliage layers situated within the canopy. In this sense, the continuous forest evaluation (Sect. B.1 and B.2) is more appropriate. Nevertheless, foliage still presents vertical boundaries, and horizontal boundaries in two cases (see ‘Tree4’, ‘Tree5’ in the simulations in Sect. 3 and the Appendix; Figs. 2 & 3).
B.3.1 Simulation design
In the CFD model, a regular mesh of 840 x 2 x 200 cells is used. Only two cells are simulated in the y-direction, and 200 in the vertical. In the z direction, the vegetation canopy is resolved with 10 cells, and hence the domain top is at 20 h. Other simulations with higher domain tops (i.e., at 40h, 100h) are also simulated and similar results are obtained. Other mesh resolutions were also used (e.g., a canopy resolution of 5 cells), and results were again very similar. Hence, it is concluded that results are not very sensitive to the choice of grid layout within these ranges.
The forest is modelled using a leaf area index (LAI) of 2.0 (Dupont and Brunet, 2008), evenly distributed in the vertical for 0 < z ≤ h. The sectional drag coefficient due to vegetation foliage (CDV) is set to 0.2, such that CDV LAI = 0.4 as for previous modelling of these wind-tunnel measurements (Yang et al. 2006). The inlet profiles used are as follows,
u 骣z U = * ln 琪 k 桫z0 , (B1)
u2 k = * Cm , (B2)
u3 e = * k z , (B3)
where the values chosen for u*, κ, z0/h and C are 0.5, 0.42, 0.0032 and 0.09, respectively. Vegetation (tree) foliage is accounted for in the momentum, turbulent kinetic energy, and dissipation equations as discussed in Sect. 2.1.
B.3.2 Results
u Raupach et al. (1987) measured vertical profiles of mean horizontal velocity , standard deviation of horizontal velocity (u), standard deviation of vertical velocity (w) and momentum flux ( u' w ' ) at different locations (x/h = −8.5, 0.0, 2.1, 4.3, 6.4 and 10.6, as indicated by the dashed lines in Fig. B2). x/h = 0.0 indicates the upwind edge of the canopy. From these experimental data TKE (k) was computed assuming v =w (Wilson and Flesh, 1999).
Vertical profiles of u and k are compared between the CFD model and the wind tunnel. The streamwise velocity component u is normalized by mean streamwise velocity component at x/h = −8.5 2 and z/h = 2 (i.e., u) and k is normalized by u . Note that u is influenced to a small extent by the upwind vegetation canopy. CFD model results for several values of C5 are computed to explore the sensitivity of the results to this critical parameter. Results from an LES implementation (Dupont and Brunet, 2008) and another RANS CFD model (Foudhil et al. 2005) are included.
The wind speed generally compares favourably for all values of C5 and adjusts the profile appropriately relative to the upwind clearing (Fig. B3). The current CFD model performs similarly to, or perhaps better than, the Dupont and Brunet (2008) LES and Foudhil et al. (2005) k- models, with the exception of the superior performance of the LES model well above the canopy (as z/h goes to 2.0). It slightly underestimates wind speed above the canopy, and it slightly underestimates/overestimates the canopy streamwise velocity for C5 = 1.00/C5 = 1.26, respectively. Interestingly, all of the numerical models underestimate the streamwise velocity component immediately after entering the canopy at x/h = 2.1. The TKE k is less well-reproduced by the models (Fig. B4), and it is more sensitive to the vegetation parametrization (i.e., Eqs. 3 and 4). The current CFD model with C5 = 1.26 provides perhaps the best results of all models in the clearing (x/h = −8.5) and well downstream of the canopy edge (x/h = 10.6). Immediately downwind of the canopy edge (i.e., x/h = 2.1) none of the models perform well, although the LES model performs qualitatively better than the k- models. Overall, the current model with
C5 = 1.10 performs at least as well as the LES model and perhaps the best of all the models. C5 = 1.26, however, overestimates k, in particular within the canopy, downwind of the forest edge, presumably because larger C5 suppresses dissipation.
Overall, the current CFD model performs well with C5 = 1.10, and certainly no worse than the Foudhil et al. (2005) CFD model or the Dupont and Brunet (2008) LES model. Performance with both
C5 = 1.00 and C5 = 1.26 is reasonable. Canopy edge flow is a difficult process to simulate numerically, even with LES models (Yang et al. 2006; Dupont and Brunet, 2008).
B.4 Overall Assessment
The evaluation of the model against the “continuous forest” case yields an optimal C5 of ≈1.00, whereas that of the forest-clearing case gives a value of ≈1.10, both slightly lower than the theoretical value of 1.26 calculated based on Sanz (2003). A survey of the application of this tree foliage parametrization for CFD model flow through vegetation foliage (i.e. Sect. 2.1), unearths a wide range of values for C5 (0.4 – 1.5), as well as for other parameters involved in the parametrization of the impacts of foliage on k and (Green, 1992; Liu et al. 1996; Foudhil et al. 2005; Endalew et al. 2009; Dalpe and
Masson, 2009; Rosenfeld et al. 2010). As such, it appears likely that C5 is case dependent, or more correctly stated, the parametrization as a whole requires a suite of parameters that is case dependent. Clearly this is not optimal, but its rectification, such as the development of a new source term in the - equation to represent the spectral shortcut due to vegetation, is beyond the scope of the present work.
Given this situation, we opt to retain the original theoretical value of C5 = 1.26 for the combined building-tree foliage simulations in Sect. 3 and in the Appendix. There is no indication that another dataset would not yield another ‘tuned’ value of C5 entirely. As a precaution, the analysis is re-run for select scenarios with values of C5 of 1.10 and 1.00 in Sect. 3 and in the Appendix in order to assess the robustness of the results obtained using C5 = 1.26. Moreover, the primary aim of this contribution is to present a methodology for the development of an urban canopy parametrization for urban canopies with trees, and the final results are subject to the many simplifications and assumptions inherent in the use of a CFD model with highly simplified urban configuration, and in particular, the adoption of the now ‘standard’ vegetation source and sink terms in the k and equations (i.e., Eqs. 3 and 4).
B.5 The standard k- parametrization of vegetation canopy turbulence
A significant issue with the parametrization of the effects of foliage on the k and budgets (i.e., Eqs. 3 and 4) is that the flow, especially k and , are highly sensitive to these source and sink terms, in particular to C5 (Foudhil et al. 2005). In our tests, for example, a 10% reduction of C5 doubled the peak of (not shown), whereas a 30% reduction virtually eliminated all turbulence in the vegetation canopy space. One approach is to use C5 as a tuning parameter, as in Foudhil et al. (2005). However, its tuning to a particular dataset may not yield a set of coefficients that are capable of reproducing flows through vegetation more generally, in particular if the tuning is conducted against wind-tunnel measurements with artificial ‘foliage’ (Liu et al. 1996; Foudhil et al. 2005). A recent study by Silva Lopes et al. (2013) calibrated the coefficients in Eqs. 3 and 4 against LES models for both homogeneous forest and forest- clearing scenarios and determined the following optimal values: p = 0.0, d = 4.0, C4 = 0.0, C5 = 0.9. These values suggest that wake turbulence due to flow around leaves is converted rapidly to heat via viscous effects, and hence only the short-circuit effect (diminishment of larger eddies) needs to be accounted for in terms of the k and budgets. Although foliage wake production was included in the present CFD model formulation, it is notable that it played a very minor role in the reproduction of the spatially-averaged profiles. Conversely, the enhanced dissipation term was significant.
Introduction of the d k U term in Eq. 3, to represent the rapid dissipation of wake-scale turbulence generated by foliage, is not an optimal approach. Its introduction implies a corresponding term in the -equation (C5d U), which includes the coefficient C5 to which the model is very sensitive. A more parsimonious approach would not modify the k-equation source term (i.e., retain only the wake production term), but simply modify the existing source term in the -equation (i.e. C4). Or for greater flexibility, one additional source term could be added to the -equation (just as d k U is added to the k- equation in the ‘standard’ parametrization), to represent the enhanced dissipation of the wake turbulence 3 generated by the LD CDV U term in Eq. 3. This latter approach would reduce by one the number of terms derived heuristically. Such a development is beyond the scope of the present work; therefore, the ‘standard’ approach as described in Eq. 3 and 4 (Green, 1992; Liu et al. 1996; Sanz, 2003) is followed, with careful attention paid to the value of C5. More generally, the single-band (bulk spectral) approach taken here may not have a set of optimal parameters p, d, C4, C5, or they may be dependent on the scale and distribution of the foliage elements, for example. Transfer of energy from mean to turbulent flow cannot be differentiated between building drag and foliage drag, despite the large differences in scale. The spectral shortcut from large to small turbulent scales by foliage elements can only be represented as enhanced dissipation of turbulent kinetic energy. Two or greater band approaches are a possible alternative (e.g., Wilson, 1988), and they assume a clear separation of turbulent scales. Figure B1: Comparison of CFD modeled mean streamwise velocity component (), TKE () and Reynolds stress () profiles for a ‘continuous forest’ with wind tunnel results from Brunet et al. (1984) and CFD model results from
Foudhil et al. (2005). TKE is normalized using the free stream friction velocity (u*). h is tree foliage canopy height. Figure B2: Side view (i.e. x-z plane) of the forest-clearing model configuration in the wind tunnel (Raupach et al. 1987) and in the CFD model. Green areas indicate modeled foliage, and vertical dashed lines indicate locations of profile comparisons. h is tree foliage canopy height. Figure B3: Comparison of modelled streamwise velocity component profiles upwind and downwind of the upwind edge of the forest (x / h = 0.0) against wind-tunnel results from Raupach et al. (1987) for three values of the parameter C5. Also included are k- CFD model results from Foudhil et al. (2005) and LES model results from
Dupont and Brunet (2008) for the same case. The dashed red line indicates the C5 = 1.26 wind profile at x / h = −8.5
(i.e., approximately the center of upwind clearing). Streamwise velocities are normalized by u, the streamwise velocity component at z / h = 2.0 and x / h = −8.5, for each case. Figure B4: Comparison of modelled TKE profiles upwind and downwind of the upwind edge of the forest (x / H = * 0.0) with wind tunnel results from Raupach et al. (1987), for three values of the parameter C5. Also included are k- results from Foudhil et al. (2005) and LES model results from Dupont and Brunet (2008)* for the same case. TKE 2 (k) is normalized by the square of the wind speed at z / H = 2.0 and x / H = −8.5 (i.e., u ) for each case.
* TKE (k) is calculated as , assuming v = w.