Effect of the Granularity of Heterogeneous Forest Cover on the Drag Coefficient

Abstract

We investigated the effect of the granularity of heterogeneous forest cover on the momentum absorption by conducting a large-eddy simulation of the flow over a surface covered by square-shaped forest/non-forest patches ranging in size from 64 to 512 m. The modification of the drag coefficient by heterogeneity is analyzed using integral identity equations, which decompose the drag coefficient into contributions from the momentum fluxes, pressure, and forest form drag. From a macroscopic viewpoint, the drag coefficient significantly differs in each case, and its tendency is not monotonic with the granularity. The identity equation shows that the effect of the vertical momentum flux becomes persistent on the non-forest area with an increase in patch size, with the spatial development altering the drag contribution, assigning increasing importance to the forest area. The drag coefficient defined from a local viewpoint shows a strong overshoot at the front edge of the forest patch, with a sudden decrease away from the front edge. A distance of almost 10 forest heights is required to arrive at a nearly homogeneous condition. However, spatial development is still obvious in the mean advection and pressure deviation terms with spanwise heterogeneity inducing an overshoot of the drag coefficient by enhancing the secondary flow. As such, the effect of the edge remains in a transitional manner, even if a locally homogeneous region is observed in the downstream and at the centre of patches.

Appendix: Derivation of Identity Equations

Equations 13 and 17 are derived as follows. We write the Navier–Stokes equations of the mean streamwise velocity component $\overline{u}_{1}$ as

$$\begin{aligned} {{\overline{C_{for}}}}-{\displaystyle \frac{\partial {\overline{\tau ^{\star }}_{13}}}{\partial {x_{3}}}}=R+F_{1} \,, \end{aligned}$$

(22)

where $R=-A-T+P+D_{h}$ (see Eq. 15). A stress-free wall at $x_{3}/H=1$ is assumed. The momentum absorption by both forest and surface ($x_{3}/H=0$), $-\overline{\tau _{rgh}}/\rho \equiv \int _{0}^{H}{{\overline{C_{for}}}}{\hbox {d}}x_{3}-\overline{\tau _{1,3sfc}}/\rho$ is acquired by integrating Eq. 22 in the $x_{3}$-direction from 0 to H,

$$\begin{aligned} -\frac{\overline{\tau _{rgh}}}{\rho } = \int _{0}^{H}R {\hbox {d}}x_{3}+HF_{1}, \end{aligned}$$

(23)

which is substituted into Eq. 12. Further, Eq. 23 is horizontally averaged and substituted into Eq. 16. The drag coefficients of interest become

$$\begin{aligned} C_{d,l} = \frac{2}{\overline{u}^{2}_{b}}\left\{ \int _{0}^{H}R{\hbox {d}}x_{3}+HF_{1}\right\} \,\, {\mathrm{and}}\quad C_{d,g}=\frac{\mathrm{2}}{{{\langle {{u}}_{b}\rangle }}^\mathrm{2}}{} { HF}_\mathrm{1}\, . \end{aligned}$$

(24)

A double periodic condition and the boundary conditions of $\overline{u_{1}u_{3}}=0$ at $x_{3}=0$ and $x_{3}=H$ lead to $\langle R \rangle = 0$. The spatio-temporally constant driving force $F_{1}$ can be expressed as the properties of the momentum transport. Again, we integrate Eq. 22 in the $x_{3}$-direction from 0 to $x'_{3}$, where $0< x'_{3}/H < 1$, and subtract from Eq. 23 to give

$$\begin{aligned} \int _{x'_{3}}^{H}{{\overline{C_{for}}}}{\hbox {d}}x_{3}+\overline{\tau ^{\star }}_{13}(x'_{3}) = \int _{x'_{3}}^{H}R{\hbox {d}}x_{3}+(H-x'_{3})F_{1}, \end{aligned}$$

(25)

which is integrated again, first, in $x'_{3}$ from 0 to $x''_{3}$, and then in $x''_{3}$ from 0 to H, which leads to

$$\begin{aligned} \int _{0}^{H}\int _{0}^{x''_{3}}\left\{ \int _{x'_{3}}^{H} {{\overline{C_{for}}}}{\hbox {d}}x_{3}+\overline{\tau ^{\star }}_{13}\right\} {\hbox {d}}x'_{3}{\hbox {d}}x''_{3}= \int _{0}^{H}\int _{0}^{x''_{3}}\int _{x'_{3}}^{H}R{\hbox {d}}x_{3}{\hbox {d}}x'_{3}{\hbox {d}}x''_{3}+\frac{1}{3}H^{3}F_{1}.\nonumber \\ \end{aligned}$$

(26)

We apply the partial-integral rule to the double- and triple-integral terms in Eq. 26,

$$\begin{aligned} \int _{0}^{H}\int _{0}^{x''_{3}} X(x'_{3}) {\hbox {d}}x'_{3}{\hbox {d}}x''_{3}= & {} \int _{0}^{H}(H-x_{3})X(x_{3}){\hbox {d}}x_{3}, \nonumber \\ \int _{0}^{H}\int _{0}^{x''_{3}}\int _{x'_{3}}^{H}X(x_{3}){\hbox {d}}x_{3}{\hbox {d}}x'_{3}{\hbox {d}}x''_{3}= & {} -\frac{1}{2}\int _{0}^{H}\{(H-x_{3})^{2}-H^{2}\}X(x_{3}){\hbox {d}}x_{3}. \end{aligned}$$

(27)

Then, the non-dimensional driving force $2HF_{1}/u_{ref}^{2}$, which is to be substituted into Eq. 24, becomes

$$\begin{aligned} \frac{2}{u_{ref}^{2}}HF_{1}=-h(R)+h({{\overline{C_{for}}}})-i(\overline{\tau ^{\star }}_{13}). \end{aligned}$$

(28)

It is noted that $-i(\overline{\tau ^{\star }}_{13})$ consists of the effect of the Reynolds number $6/(u_{ref}H/\nu )$, and subgrid-scale stress in the vertical direction $i(S_{v})$. The substitution of Eq. 28 with $u_{ref}=\overline{u}_{b}$ into Eq. 24 yields Eq. 13. The derivation of $C_{d,g}$ in Eq. 17 follows the same approach above. In order to relate the spatially-averaged property of the momentum transport, we consider the horizontal average of Eq. 26 with $\langle P\rangle$,$\langle D_{h}\rangle$, and horizontal advection $\langle A \rangle$ and $\langle T \rangle$ being zero,

$$\begin{aligned} \frac{2}{u_{ref}^{2}}HF_{1}=-\frac{6}{u_{ref}^2 H^{2}} \left\langle \int _{0}^{H}\int _{0}^{x''_{3}}{{u_{1}u_{3}}}{\hbox {d}}x'_{3}{\hbox {d}}x''_{3}\right\rangle +\langle h(C_{for})\rangle -\langle i({{\tau ^{\star }}}_{13})\rangle . \end{aligned}$$

(29)

Again, we may apply Eq. 27 to the first term on the r.h.s., and substitute it into $C_{d,g}$ in Eq. 24, with $u_{ref}={{\langle {{u}}_{b}\rangle }}$ to acquire Eq. 17. It is also noted that the advection in the vertical direction is compatible in the following way,

$$\begin{aligned} g\left( {\displaystyle \frac{\partial {\overline{u_{1}u_{3}}}}{\partial {x_{3}}}}\right)= & {} h\left( {\displaystyle \frac{\partial {\overline{u_{1}u_{3}}}}{\partial {x_{3}}}}\right) \nonumber \\= & {} \frac{6}{u_{ref}^{2}H^{2}}\int _{0}^{H}\int _{0}^{x''_{3}}\int _{x'_{3}}^{H}{\displaystyle \frac{\partial {\overline{u_{1}u_{3}}}}{\partial {x_{3}}}}{\hbox {d}}x_{3}{\hbox {d}}x'_{3}{\hbox {d}}x''_{3} \nonumber \\= & {} -\frac{6}{u_{ref}^{2}H^{2}}\int _{0}^{H}\int _{0}^{x''_{3}}\overline{u_{1}u_{3}}{\hbox {d}}x'_{3}{\hbox {d}}x''_{3} \nonumber \\= & {} i(\overline{u_{1}u_{3}}). \end{aligned}$$

(30)

We used the boundary conditions at $x_{3}=0$ and $x_{3}=H$ of $\overline{u_{1}u_{3}} = 0$, and this relation is used in a description of Fig. 7.