Raupach's model, however. In my model, the ob-
In the limit of large p
stacle's aerodynamic properties likely change as
its orientation with the wind changes. Other-
wise, the concept of streamlining would be
meaningless. Thus, CR and A must depend on Φ.
But because I could find no study in the literature
Finally, following R92, for m and n greater than 1,
that reported the aerodynamic properties of an
I estimate the sheltered volume to be
obstacle like that depicted in Figure 3, I assume
that the drag contributions from the various fac-
es are additive. Mathematically, my hypothesis is
FD (Φ) = ρUh ∑ CRi (Φ)Ai (Φ)
where c2 is another empirical constant. Hence,
using eq 5, 17 and 18 in eq 16, we get
where CRi is the drag coefficient for a particular
face of the sastrugi-like obstacle, and Ai is the sil-
τR (Φ) = λ(Φ)ρUh ∑ CRi (Φ) i
houette area of that face. The sum is over all faces
presented to the wind at angle Φ.
The obstacle in Figure 3 presents only three
faces. When 0 ≤ Φ ≤ 90, the wind can see the
front triangular face; this has form drag coeffi-
cient CR1. When β ≤ Φ ≤ 180 β, the wind can see
the side ridge; this has drag coefficient CR2.
In eq 19, define
When 180 β ≤ Φ ≤ 180, the wind can see only
the rear silhouette; this has drag coefficient CR3.
CR (Φ) ≡ ∑ CRi (Φ) i
As I mentioned, I found no study that evaluat-
ed CR for the obstacle geometry I use here. But on
For the four Φ regions identified earlier, CR fol-
reviewing several studies that treated somewhat
similar geometries (i.e., Arie and Rouse 1956;
Arya 1973, 1975; Banke et al. 1976, 1980; Taylor
Region I: 0 ≤ Φ ≤ β
1988), I estimated what I feel are representative
values for the three faces. Here, I use CR1 = 0.10,
CRI = CR1.
CR2 = 0.30, and CR3 = 0.30.
Continuing now with R92's model, if p rough-
Region II: β ≤ Φ ≤ 90
ness elements cover a ground area S, the stress
they produce through form drag is
C m cos Φ + CR2[n sin Φ - (m / 2) cos Φ]
CRII = R1
τR (Φ, p) = ρUh ∑ CRi (Φ)Ai (Φ) 1 - . (14)
n sin Φ + (m / 2) cos Φ
Here, V is the volume sheltered by an individual
Region III: 90 ≤ Φ ≤ 180 β
roughness element; the term raised to power p,
thus, represents a mutual sheltering effect.
CRIII = CR2 .
In my model, from eq 5
Region IV: 180 - β ≤ Φ ≤ 180
S mnh2 A(Φ)
CRIV = CR3 .
Combining eq 11, 19 and 20 in eq 10, we get
Using this in eq 14 gives
τR (Φ, p) =
ρUh ∑ CRi (Φ)Ai (Φ)
⋅ 1 - λ(Φ)V .
^ (Φ) exp-c λ(Φ) Uh .