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ABSTRACT
Vegetated flows are typical in many aquatic systems such as natural and man-made wetlands, and therefore attract significant attention of researchers and engineers. This study first solves the Navier–Stokes–Forchheimer (NSF) equation for laminar vegetated flow and then modifies the obtained velocity distribution for turbulent flow. It demonstrates that (i) for flows through emergent and over submerged vegetation, the laminar velocity distributions are expressed by the Jacobi elliptic functions for which the parabolic law is recovered for zero vegetation; (ii) for flow through emergent vegetation, the laminar velocity distribution exhibits a typical boundary-layer profile, while its turbulent counterpart is simply uniform; and (iii) for flow over submerged vegetation, both laminar and turbulent velocity distributions are similar to those in conventional channel flows for the water layer, but both are approximated by hyperbolic sine laws for the vegetation layer. The laminar solutions meet the NSF equation and all boundary conditions; and the turbulent solutions agree with laboratory and field data.
Acknowledgements
The authors thank Dr Nina Nikora, University of Aberdeen, UK for providing experimental data for . They are also grateful to the Editor, Associate Editor and anonymous reviewers for constructive suggestions which improved clarity of the presentation.
Notation
| A, B | = | parameters (−) |
| Ai | = | stem area with i = 1, 2, … (m2) |
| a, b, c | = | interim parameters (−) |
| a1, b1 | = | parameters (−) |
| C, D | = | interim parameters (−) |
| CD | = | drag coefficient (−) |
| CE | = | Ergun constant (−) |
| E | = | E(φ|m) elliptic integral of the second kind (−) |
| Ei, E0 | = | values of E for φi and φ0, respectively (−) |
| F | = | F(φ|m) elliptic integral of the first kind (−) |
| Fi, F0 | = | values of F for φi and φ0, respectively (−) |
| Fd | = | vegetation drag (N) |
| g, g | = | gravity acceleration (m s–2) |
| H | = | dimensionless flow depth (−) |
| H, hv | = | flow depth and vegetation height, respectively (m) |
| K | = | K(m) complete elliptic integral of the first kind (−) |
| kp | = | intrinsic permeability (m2) |
| ks | = | Nikuradse's equivalent roughness (m) |
| m, mi | = | parameters of elliptic integrals or functions with j = 1 and 2 (−) |
| n | = | exponent (−) |
| p | = | pressure (Pa) |
| Q | = | discharge (m3s–1) |
| = | Reynolds number (−) | |
| So | = | bottom slope (−) |
| sn, cn | = | Jacobi elliptic functions (−) |
| t | = | time (s) |
| U, Uj | = | dimensionless velocities with j = 0, 1, 2, … 5 (−) |
| = | dimensionless average u (−) | |
| = | values of Ū for vegetation and water layers, respectively (−) | |
| u, u | = | time-volume-average velocity (m s–1) |
| uf, uf | = | pore-fluid velocity (m s–1) |
| ui | = | interface velocity (m s–1) |
| = | velocity gradient at interface (s–1) | |
| u1 | = | inflection velocity near bed (m s–1) |
| uj | = | characteristic velocities with j = 0, 1, 2, … ,5 (m s–1) |
| u* | = | interface shear velocity (m s–1) |
| V | = | time-volume-average velocity for 1D flow (m s–1) |
| Vf | = | pore-fluid velocity for 1D flow (m s–1) |
| W | = | dimensionless ω (−) |
| w | = | channel width (m) |
| x | = | flow direction (m) |
| Z | = | dimensionless z (−) |
| Zδ | = | dimensionless thickness of NBR (−) |
| z, z' | = | coordinates from bed and interface, respectively (m) |
| β | = | Forchheimer coefficient (m–1) |
| δ1 | = | thickness of NBR (m) |
| δ2 | = | velocity-dip position from interface (m) |
| η | = | porosity (−) |
| κ | = | von Kármán constant (−) |
| λ | = | interim parameter (m–3 s) |
| μ | = | dynamic water viscosity (Pa s) |
| υ | = | kinematic water viscosity (m2s–1) |
| ξ | = | interim parameter (−) |
| Π | = | Coles wake flow coefficient (−) |
| ρ | = | water density (kg m–3) |
| σ | = | stress tensor due to shear strains (Pa) |
| τ | = | shear stress (Pa) |
| τi, τ0 | = | interface and bed shear stresses, respectively (Pa) |
| φ | = | argument of elliptic integrals (−) |
| φi, φ0 | = | values of φ at interface and bed, respectively (−) |
| ω | = | imaginary part of u3 (m s–1) |