Surface Body-Force Model for Turbulent Flow over Rough Walls

DNS can be used with immersed boundary methods at low Reynolds numbers to resolve almost exactly the flow over surfaces of any specified roughness, and determine roughness modeling quantities like the equivalent sandgrain roughness $k_s$. What is needed to implement this information in RANS closures for general engineering applications is a technique for reproducing the effect of roughness when computed on a smooth-wall finite-volume mesh, using computed/modeled information only in the wall-adjacent cell, without outer-flow assumptions or restrictions. In this talk, we present a surface body-force model in which the effect of roughness is modeled as a source term in the $x$-momentum equation in the wall-adjacent cell, as the quadratic drag model $f_x = - \alpha_t \vert u_c \vert u_c$, together with the \textit{near-wall} function $u_c^+ = 1/\sqrt{\alpha_t\, \Delta y}$. Calibrations against reference pipe-flow data show that the roughness `wavenumber' $\alpha_t$ is a linear function of $k_s$ for small roughness, and depends on the cube of wall-cell height $\Delta y$. In preliminary computations within a $k$-$\epsilon$ closure, this roughness model appears to describe quite accurately the wall shear shear stress in flat-plate and adverse pressure-gradient boundary layers, provided the wall-cell height $\Delta y$ is sufficiently small.