A dynamic surface drag model for large-eddy simulation of turbulent boundary-layer flow over fractal-like roughness

Many flows especially in geophysics involve turbulent boundary layers forming over rough surfaces with multiscale height distribution. Such surfaces pose special challenges for large eddy simulation (LES) when the filter scale is such that only part of the roughness elements of the surface can be resolved. Here we consider LES of flows over rough surfaces with power-law height spectra Eh(k) ∼ kbeta s (-3 ≤ betas < -1), as often encountered in natural terrains. The surface is decomposed into resolved and subgrid-scale height contributions. The effects of the unresolved small-scale height fluctuations are modeled using a local equilibrium wall model (log-law or Monin-Obukhov similarity), but the required aerodynamic roughness length must be specified. It is expressed as the product of the subgrid-scale root-mean-square of the height distribution and an unknown dimensionless quantity, alpha, the roughness parameter. Instead of specifying this parameter in an ad-hoc empirical fashion, a dynamic methodology is proposed based on test-filtering the surface forces and requiring that the total drag force be independent of filter scale or resolution. This dynamic surface roughness model is inspired by the Germano identity traditionally used to determine model parameters for closing subgrid-scale stresses in the bulk of a turbulent flow. As a first step, a new method to simulate flow over horizontally resolved, but vertically unresolved surfaces is developed and tested. Then, a series of LES of fully developed flow over multiscale rough surfaces are performed, considering various types of multiscale surfaces: The first type is isotropic, built using random-phase Fourier modes with prescribed power-law spectra. The second type is highly anisotropic, namely evolved fluvial-like landscapes obtained from a numerical solution of the Kardar-Parisi-Zhang equation. Finally, a fluvial landscape from the Llano River catchment in Texas is considered. Results show that the dynamic model yields well-defined, rapidly converging, values of alpha. Effects of spatial resolution and spectral slopes are investigated. The accuracy of the dynamic model is tested by showing that predicted mean velocity profiles are approximately independent of resolution for the dynamically computed values of alpha, whereas resolution-dependent results are obtained when using other, incorrect, alpha values. Strong dependence of alpha on betas is found. The model is reported to be less accurate when the surface deviates from scale-invariance. In an appendix, some ideas for a similarly-posed dynamic approach for estimating fluxes of passive scalar into a turbulent boundary layer over a rough surface are presented; there arc some inherent challenges due to limitations of the Reynolds analogy, since inertial pressure force effects associated with turbulent momentum transport are not present for scalar transport.