Flow over natural or engineered surfaces: an adjoint homogenization perspective

Natural and eng (i) neered surfaces are ne (v) er smooth, but (i) rregular, rough at d (i) fferent scales, compl (i) ant, poss (i) bly porous, l (i) qu (i) d (i) mpregnated or superhydrophob (i) c. The correct numer (i) cal modell (i) ng of flu (i) d flow (i) ng through and around them (i) s (i) mportant but poses problems. For med (i) a character (i) zed by a per (i) od (i) c or quas (i) -per (i) od (i) c m (i) crostructure of character (i) st (i) c d (i) mens (i) ons smaller than the rele (v) ant scales of the flow, mult (i) scale homogen (i) zat (i) on can be used to study the effect of the surface, a (v) o (i) d (i) ng the numer (i) cal resolut (i) on of small deta (i) ls. Here, we re (v) (i) s (i) t the homogen (i) zat (i) on strategy us (i) ng adjo (i) nt (v) ar (i) ables to model the (i) nteract (i) on between a flu (i) d (i) n mot (i) on and regularly m (i) cro-textured, permeable or (i) mpermeable walls. The approach descr (i) bed allows for the easy der (i) (v) at (i) on of aux (i) l (i) ary/adjo (i) nt systems of equat (i) ons wh (i) ch, after a (v) erag (i) ng, y (i) eld macroscop (i) c tensor (i) al propert (i) es, such as permeab (i) l (i) ty, elast (i) c (i) ty, sl (i) p, transp (i) rat (i) on, etc. When the flu (i) d (i) n the ne (i) ghbourhood of the m (i) crostructure (i) s (i) n the Stokes reg (i) me, class (i) cal results are reco (v) ered. Adjo (i) nt homogen (i) zat (i) on, howe (v) er, perm (i) ts s (i) mple extens (i) on of the analys (i) s to the case (i) n wh (i) ch the flow d (i) splays nonl (i) near effects. Then, the propert (i) es extracted from the aux (i) l (i) ary systems take the name of effect (i) (v) e propert (i) es and do not depend only on the geometr (i) cal deta (i) ls of the med (i) um, but also on the m (i) croscop (i) c character (i) st (i) cs of the flu (i) d mot (i) on. Examples are shown to demonstrate the usefulness of adjo (i) nt homogen (i) zat (i) on to extract effect (i) (v) e tensor propert (i) es w (i) thout the need for < (i) tal (i) c>ad hoc parameters. In part (i) cular, notable results reported here (i) n (i) nclude:( (i) ) an or (i) g (i) nal formulat (i) on to descr (i) be f (i) ltrat (i) on (i) n porous med (i) a (i) n the presence of (i) nert (i) al effects; ( (i) (i) ) the m (i) croscop (i) c and macroscop (i) c equat (i) ons needed to character (i) ze flows through poroelast (i) c med (i) a; ( (i) (i) (i) ) an extended Na (v) (i) er's cond (i) t (i) on to be employed at the boundary between a flu (i) d and an (i) mpermeable rough wall, w (i) th roughness elements wh (i) ch can be e (i) ther r (i) g (i) d or l (i) nearly elast (i) c; ( (i) (v) ) the m (i) croscop (i) c problems needed to def (i) ne the rele (v) ant parameters for a Saffman-l (i) ke cond (i) t (i) on at the (i) nterface between a flu (i) d and a porous substrate; and ( (v) ) the macroscop (i) c equat (i) ons wh (i) ch hold at the d (i) (v) (i) d (i) ng surface between a free-flu (i) d reg (i) on and a flu (i) d-saturated poroelast (i) c doma (i) n.