Raupach's model, however. In my model, the ob-

In the limit of large *p*

stacle's aerodynamic properties likely change as

λ(Φ)V

-λ(Φ)V

its orientation with the wind changes. Other-

1-

≈ exp

.

(17)

wise, the concept of streamlining would be

meaningless. Thus, *C*R 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-

1

es are additive. Mathematically, my hypothesis is

(18)

2

*

2

(13)

where *c*2 is another empirical constant. Hence,

using eq 5, 17 and 18 in eq 16, we get

where *C*Ri is the drag coefficient for a particular

face of the sastrugi-like obstacle, and *A*i is the sil-

2

τR (Φ) = λ(Φ)ρ*U*h ∑ 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

⋅

exp-*c*2 λ(Φ)

faces. When 0 ≤ Φ ≤ 90, the wind can see the

(19)

.

front triangular face; this has form drag coeffi-

*

cient *C*R1. When β ≤ Φ ≤ 180 β, the wind can see

the side ridge; this has drag coefficient *C*R2.

In eq 19, define

When 180 β ≤ Φ ≤ 180, the wind can see only

the rear silhouette; this has drag coefficient *C*R3.

^

.

(20)

As I mentioned, I found no study that evaluat-

^

For the four Φ regions identified earlier, CR fol-

reviewing several studies that treated somewhat

similar geometries (i.e., Arie and Rouse 1956;

lows.

Arya 1973, 1975; Banke et al. 1976, 1980; Taylor

1988), I estimated what I feel are representative

values for the three faces. Here, I use *C*R1 = 0.10,

^

(21a)

Continuing now with R92's model, if *p *rough-

ness elements cover a ground area *S*, the stress

they produce through form drag is

^

.

τR (Φ, *p*) = ρ*U*h ∑ CRi (Φ)Ai (Φ) 1 - . (14)

(21b)

Here, *V *is the volume sheltered by an individual

roughness element; the term raised to power *p*,

thus, represents a mutual sheltering effect.

^

(21c)

In my model, from eq 5

λ(Φ)

2γ

=

=

.

(15)

^

(21d)

Combining eq 11, 19 and 20 in eq 10, we get

Using this in eq 14 gives

λ(Φ) 2

2

τR (Φ, *p*) =

ρ*U*h ∑ CRi (Φ)Ai (Φ)

-*c*1λ(Φ) Uh

τ(Φ) =

ρ*U*h *C*Sh

exp

*

⋅ 1 - λ(Φ)*V * .

(16)

^ (Φ) exp-*c *λ(Φ) Uh .

+λ(Φ)CR

(22)

2

*

5

Integrated Publishing, Inc. |