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An analytic expression for wind-velocity profile within the saltation layer
Geomorphology 60 (2004) 359 – 369 www.elsevier.com/locate/geomorph
An analytic expression for wind-velocity profile within the saltation layer Z.S. Li a,b,*, J.R. Ni a,b, C. Mendoza c b
a Department of Environmental Engineering, Peking University, Beijing, 100871, China The Key Laboratory of Water and Sediment Sciences, Ministry of Education, Beijing, 100871, China c Department of Civil Engineering, University of Missouri, Rolla, MO 65409-0030, USA
Received 19 March 2003; received in revised form 20 August 2003; accepted 28 August 2003 Available online 21 November 2003
Abstract The vertical wind-velocity profile within the saltation layer is investigated theoretically. New equations for the fluid shear stress distribution in the saltation layer and the velocity profile affected by saltation are derived. The dependence of the velocity profile on the threshold shear velocity, the vertical variations of sand grain velocity, the vertical sand-mass-flux distribution, and the fluid shear stress within the saltation are demonstrated. Velocity profiles with or without Bagnold’s focus are predicted. These velocity profiles are not very sensitive to the vertical distribution of sand-mass flux. Predicted velocity profiles are compared with profiles measured in the wind tunnel, and satisfactory agreement has been achieved. D 2003 Elsevier B.V. All rights reserved. Keywords: Wind-velocity profile; Saltation layer; Shear stress profile; Analytic expression
1. Introduction Data abound with evidence that wind-velocity profiles in the saltation layer are significantly affected by the presence of saltating sand grains and do not follow entirely the logarithmic law (e.g., Bagnold, 1941; Kawamura, 1951; Zingg, 1953; Spies et al., 1995; McKenna-Neuman and Maljaars, 1997). Owen (1964) suggested that the grain saltation layer acts on the flow above it in a way similar to that of solid * Corresponding author. Department of Environmental Engineering, Peking University, Beijing, 100871, China. Tel.: +86-106275-3962; fax: +86-10-6275-6526. E-mail address: [email protected] (Z.S. Li). 0169-555X/$ - see front matter D 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.geomorph.2003.08.008
roughness on a flow, and that the logarithmic law governs the flow outside of it. The validity of the logarithmic law outside the saltation layer has been repeatedly confirmed by many experiments (e.g., Kind, 1976; Rasmussen et al., 1985; Rasmussen and Mikkelsen, 1991; Sherman, 1992). In his analysis of the wind-velocity profile within the saltation layer, Owen (1964) advanced the complementarity of the fluid shear stress at height z, sa(z), and the grain-borne shear stress at height z, sp(z), under steady-state conditions. The sum of the fluid shear stress and the grain-borne shear stress remains constant with height and is equal to the fluid shear stress above the saltation layer. Further, Raupach (1991) exploited the similarity in momentum sink
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distribution from the fluid flow between vegetation canopies and saltation layers to study the wind field within the saltation layer. From imposing the same global constraints for the fluid shear stress as in the numerical models, he found h i2 sa ðzÞ ¼ qa U*2 1 ð1 U*t =U* Þez=Hs ð1Þ where qa is the air density, U* is the flow friction velocity outside the saltation layer, U*t is the threshold friction velocity, and Hs is the characteristic height at which the grain-borne stress is zero. McEwan (1993) adopted a triangular and a hyperbolic tangent stress distribution to investigate the wind-velocity profile within the saltation layer. The latter was favored due to its visual similarity to the predictions from the self-regulatory saltation model of McEwan and Willetts (1991) rather than from sound physical bases. Other researchers have computed, using numerical methodologies, the profile of the grain-borne shear stress and/or the wind velocity in the saltation layer from fairly complex sand grain saltation models. Sørensen (1985), Jensen and Sørensen (1986), Ungar and Haff (1987) and Werner (1990) investigated the steady-state case, whereas Anderson and Haff (1991) and most recently Shao and Li (1999) simulated the time evolution of the saltation process. Although their extensive calculations reproduced the overall features of measured wind-velocity profiles, they did not propose analytic expressions of the saltation-modified wind profile. In this paper, new analytical formulae for the fluid shear stress and the saltation modified velocity distributions in the saltation layer are developed. Also, new data were collected on wind-velocity profiles in the saltation layer making novel use of a hack tube in a wind tunnel to complement the available data in the literature and to validate the derived analytical formula for the wind-velocity distribution.
2. Velocity profile within the saltation layer This section introduces the derivation of a new equation for the wind-velocity distribution within the saltating layer that accounts for the presence of the hopping sand grains that alter the wind flow. The
analysis starts by considering, as in Owen (1964), that the total shear stress within the saltation layer (s) is s ¼ sa ðzÞ þ sp ðzÞ
ð2Þ
and that the velocity profile may be described, in accordance with Raupach (1991), McEwan (1993) and Anderson and Haff (1991), among others, in the form sffiffiffiffiffiffiffiffiffiffi dU 1 sa ðzÞ ¼ ð3Þ dz jz qa where U is the flow velocity at height z, and j is von Karman’s constant. It is easily seen from the combination of Eqs. (2) and (3) that since sa(z) depends on sp(z) for a given s, an estimation of sp(z) is required for the determination of the wind-velocity profile. Following Liu et al. (1996), sp(z) results from the sum of the differences in horizontal momentum (passing through a unit area per unit time) produced by (1) both the descending and ascending sand grains, (2) the descending grains, and (3) the ascending sand grains. The last two differences are far smaller than the first one, so, for an idealized case, they can be neglected. Therefore, for uniform saltating grains in equilibrium, sp(z) can be simplified as sp ¼ mn1 ðzÞw¯ 1 ðzÞ½¯u2 ðzÞ u¯ 1 ðzÞ
ð4Þ
where m is the mass of a single grain, n1(z) is the number of ascending grains at height z, w¯1,(z) is the mean vertical velocity of ascending grains at height z, u¯1(z) the mean horizontal velocity of ascending sand grains at height z, and u¯2(z) is the mean horizontal velocity of descending sand grains at height z. Eq. (4) will be used as the basis for further analysis of the grain-borne shear stress profiles. It is conveniently rewritten in the following dimensionless form sp ðzÞ mn1 ðzÞ w¯ 1 ðzÞ u¯ 2 ðzÞ u¯ 1 ðzÞ ¼ mn1 ð0Þ w¯ 1 ð0Þ u¯ 2 ð0Þ u¯ 1 ð0Þ sp ð0Þ
ð5Þ
where the argument 0 refers to parameters evaluated at the bed surface, i.e., z = 0. This equation will be
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subjected to a series of transformations before solving for sp(z). Firstly, the first term on the right-hand side is expressed as mn1 ðzÞ u¯ ð0Þ 2mn1 ðzÞ¯uðzÞ ¼ ð6Þ mn1 ð0Þ u¯ ðzÞ 2mn1 ð0Þ¯uð0Þ where u¯(0) and u¯(z) are defined approximately as u¯(0)i1/2[u¯1(0) + u¯2(0)] and u¯(z)i1/2[u¯1(z) + u¯2(z)], respectively. When the saltation layer is in equilibrium, the number of ascending particles equals the number of descending ones, i.e., 2mn1(z)u¯(z) = q(z) and 2mn1(0)u¯(0) = q(0), where q(z) and q(0) are the mass flux of grains at z and at the bed surface, respectively, and Eq. (6) is now written as mn1 ðzÞ qðzÞ u¯ ð0Þ ¼ mn1 ð0Þ qð0Þ u¯ ðzÞ
ð7Þ
Secondly, as the vertical component of the drag force on the saltating grains can be reasonably neglected (e.g., Kawamura, 1951), then 8 1 > 2 > < mw¯ 1 ð0Þ ¼ mg H¯ 2 ð8Þ > 1 1 > : mw¯ 21 ð0Þ ¼ mw¯ 21 ðzÞ þ mgz 2 2 results from the application of the energy conservation ¯ is the mean maximum height of law. In Eq. (8), H sand grain saltation, and g is the gravitational acceleration constant. It follows from Eq. (8) that w¯ 1 ðzÞ ¼ w¯ 1 ð0Þ
rffiffiffiffiffiffiffiffiffiffiffiffiffi z 1 H¯
ð9Þ
The substitution of Eqs. (9) and (7) into Eq. (5) yields rffiffiffiffiffiffiffiffiffiffiffiffiffi
sp ðzÞ qðzÞ z u¯ ð0Þ½¯u2 ðzÞ u¯ 1 ðzÞ 1 ¼ ð10Þ sp ð0Þ qð0Þ H¯ u¯ ðzÞ½¯u2 ðzÞ u¯ 1 ð0Þ Furthermore, the third term on the right-hand side of Eq. (10) becomes u¯ ð0Þ½¯u2 ðzÞ u¯ 1 ðzÞ ½¯u2 ð0Þ þ u¯ 1 ð0Þ½¯u2 ðzÞ u¯ 1 ðzÞ ¼ u¯ ðzÞ½¯u2 ð0Þ u¯ 1 ð0Þ ½¯u2 ð0Þ u¯ 1 ð0Þ½¯u2 ðzÞ þ u¯ 1 ðzÞ ð11Þ
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From the ballistics of individual sand grains, the velocities are u¯ 1 ðzÞ ¼ u¯ 1 ð0Þ þ at1 ; u¯ 2 ðzÞ ¼ u¯ 1 ð0Þ þ at2 ; u¯ 2 ð0Þ ¼ u¯ 1 ð0Þ þ at ð12a; b; cÞ where a is the acceleration of the particles, t = t1 + t2 = 2bw ¯ 1(0)/g (Liu, 1995; McEwan and Willetts, 1994) is the overall particles flight time, in which b is a coefficient, and t1 = bw ¯ 1(0)/g bw ¯ 1(z)/g and t 2 = bw ¯ 1(0)/g + bw ¯ 1(z)/g are the travel time of the ascending and descending particles, respectively. The substitution of Eq. (12a – c) into Eq. (11) yields u¯ ð0Þ½¯u2 ðzÞ u¯ 1 ðzÞ t2 t1 ¼ u¯ ðzÞ½¯u2 ð0Þ u¯ 1 ð0Þ t which can be further modified to the form rffiffiffiffiffiffiffiffiffiffiffiffiffi z u¯ ð0Þ½¯u2 ðzÞ u¯ 1 ðzÞ ¼ 1 u¯ ðzÞ½¯u2 ð0Þ u¯ 1 ð0Þ H¯
ð13Þ
ð14Þ
after using the expressions for t1, t2, t, and Eq. (9). Thirdly, since sa(0) = qaU*2 t, sp ð0Þ ¼ s sa ð0Þ ¼ qa U*2 qa U*2 t
ð15Þ
Combining (Eqs. (14), (15), (10) and (4) results in z qðzÞ sa ðzÞ ¼ qa U*2 1 ð1 U*2 t =U*2 Þ 1 qð0Þ H¯ ð16Þ Finally, the integration of Eqs. (16) and (3) together results in the following wind-velocity distribution within the saltation layer: U 1 ¼ U* j
Z
z z0
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ! u U2 1u z qðzÞ t1 1 *t dz 1 z qð0Þ U2 H¯ * ð17Þ
which accounts for the influence of the sand grain loading of the flow on the wind velocity in an explicit form. ¯ and q(z)/q(0) should be determined The terms H before Eq. (17) can be applied. Several investigators have found simple expressions for the mean saltation layer height. From theoretical calculations, Owen
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¯ = 0.82U*2/g whereas Nalpanis et al. (1980) proposed H ¯ = 1.4 –1.9U*2/g from the analysis of (1993) found H ¯ ~U 2/g. experimental data. It is clear then that H * Often, the studies of vertical distribution of the sand mass flux have yielded disparate results. From the fitting of wind tunnel data, Zingg (1953) proposed the modified power-law function: qðzÞ z 1=n1 ¼ 1þ ð18Þ qð0Þ r1 where r1 is a scale height and n1 is the exponent. The data in Kawamura (1951), Rasmussen and Mikkelsen (1991), Stout and Zobeck (1996), Sterk and Raats (1996), and Butterfield (1999) also follow a similar function. Other data displays an exponential dependence of mass flux with height (e.g., Williams, 1964; Greeley et al., 1996) of the form qðzÞ ¼ ez=r2 qð0Þ
ð19Þ
where r2 is a scale height. According to Nalpanis (1985), r2 = U*2/kg defines the characteristic height of the mass flux profile, since k can be assumed to be constant for a single experiment. The same relation resulted from White and Mounla (1991) and Nalpanis et al. (1993). Eqs. (18) and (19) suggest that q(z)/q(0) is a function of z/r, and the scale height r is the height at which q(z)/q(0) = 2 1/n in Eq. (18) but equals to 1/e in Eq. (19). Although r does not modify the structure of equations for q(z)/q(0), an appropriate selection of r can simplify the resulting wind-velocity distribution function and its application. Just as for the case of r2 in Eq. (19), r~U*2/g. ¯ and r to U*2/g permits The proportionality of H writing Eq. (17) in the functional form U 1 ¼ U* j
Z
z
1 z vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! ! !ffi u u U2 2gz 2gz dz t1 1 *2t 1 f BU 2 U2 U * * * ð20Þ z0
where B is a coefficient to be determined from observations.
3. Comparisons with data Ni et al. (2002) carried out experiments in a straight-blowing type wind tunnel. The tunnel body has a total length of about 35 m and is capable of producing wind speeds from 3 to 25 m/s. The windvelocity profiles were measured for five wind speeds and for two test sands. The details of experiment set-up were described by [Ni et al. (2002)]. The results of the velocity measurements are shown in Fig. 1. In Fig. 1, two different parts are easily distinguished in each profile: the upper part follows the logarithmic law, to the wind-velocity profile outside the saltation layer, while the lower part shows a pronounced downward concavity indicative of the deviation from the log law, which is a well-known effect of the saltating grains, and represents the windvelocity profile within the saltation layer. Owen (1964) proposed a logarithmic equation for the wind-velocity profile outside the saltation layer U 1 2gz ¼ ln U* j C0 U 2 *
! ð21Þ
where C0 is an empirical constant. In Fig. 2, wind-velocity data by Bagnold (1941), Horikawa and Shen (1960), and McKenna-Neuman and Maljaars (1997) as well as data in Fig. 1 are plotted in a [U/U*, log(2gz/U*2 )]-plane, consistent with the structure of Eq. (20). In the upper part of the figure, the sampled data for the various freestream velocities collapse into a straight line, representative of Eq. (21) but into a single downward concave curve associated with Eq. (20) in the lower part. The data in the upper part in Fig. 2 are clearly from outside the saltation layer and in the lower part from within the saltation layer. The agreement of the data and the concave-downward segment of the curve in Fig. 2 resulted from calculations with Eqs. ¯ and an appropriate choice (20) and (19) with r2 = H of B. Several calculated wind-velocity profiles available in the literature (e.g., Sørensen, 1985; Shao and Li, 1999) have shown the concave-downward curves in a [U, log(z)]-plane as well. In contrast, other measured near-bed velocity profiles (e.g., Butterfield, 1999) seem to indicate that they still follow the log
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Fig. 1. Measured wind velocity profiles (Ni et al., 2002). Sand A: mean diameter = 0.17 mm, the sorting index (Folk and Ward, 1957) = 0.35. Sand B: mean diameter = 0.35 mm, the sorting index = 0.60.
Fig. 2. Comparison of predicted vertical wind-velocity distributions with Bagnold’s (1941), Horikawa and Shen’s (1960), McKenna-Neuman and Maljaars’ (1997), and Ni et al.’s (2002) data.
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law. Although this would represent a challenge to the notion that the saltating grains induce profound modifications of the wind flow within the saltation layer, this apparently contradictory result could be justified by the low concentration of sand grains in the air generated by the low wind friction speeds, ranging from 0.27 to 0.44 m/s, predominant in those experiments.
4. Discussion 4.1. Sensitivity of Eq. (20) to changes of q(z)/q(0) The results of the calculations obtained with Eq. (20) combined with Eqs. (18) and (19) and assuming ¯ are shown in Fig. 3. The closeness of that r1 = r2 = H the results demonstrates that Eq. (20), combined with Eq. (18) at n1 = 1.0, can represent the empirical data in Fig. 2 as well as Eq. (20) combined with Eq. (19) did.
It also indicates that Eq. (20) is not very sensitive to the form of the q(z)/q(0) equation. 4.2. Effect of the ratio B/C0 on Bagnold’s focus Because the results obtained with the wind-velocity profile equation are not very sensitive to the structure of the vertical sand-mass-flux profile, Eq. (19) is now taken as representative of q(z)/q(0) for further analysis of Eq. (20). Fig. 4 shows windvelocity profiles computed using Eqs. (20) and (21) with 12 different values of the (B,C0)-pair, for sand grains with a d = 0.25 mm. From Bagnold’s formula qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi U*t ¼ A gdðqp qa Þ=qa
ð22Þ
where U*t is the threshold friction velocity, A is a constant, and qp is the density of sand particles. Substitution of A = 0.08, g = 980 cm/s2, qp = 2.65 g/
Fig. 3. Results from Eq. (20) combined with Eqs. (18) and (19).
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Fig. 4. Wind-velocity profiles calculated with Eqs. (20) and (21).
cm3, and qa = 0.001226 g/cm3 into Eq. (22) yields 0.18 m/s for U*t. For a constant value of C0, the velocity profiles in the near-bed flow region diverge with increasing values of B; while for a constant B, the velocity profiles converge to form the well-recognized Bagnold’s focus with increasing values of C0. In general, two types of velocity profiles can be distinguished: one with Bagnold’s focus and the other without it. The focus is located at an identifiable height in some profiles, but not in others; for the latter case, the wind velocity is almost constant at the same height regardless of the magnitude of the friction velocity. The singular shape of the wind-velocity distribution for U* = 0.25 m/s throughout Fig. 4 may be related to the fact that the friction velocity exceeds
only slightly the threshold friction velocity in this case. The forms of the velocity profiles depicted in Fig. 4 can be immediately related to measured ones (Fig. 5): velocity profiles of Bagnold (1941) and Kawamura (1951) are similar to those in Fig. 4K and L; profiles of Horikawa and Shen (1960) and Kadib (1965) to those in Fig. 4E and I; profiles of Zingg (1953) to those in Fig. 4A and F; and profiles of Belly (1964), Gerety (1985), and Butterfield (1991) to those in Fig. 4B –D, G and H. It is noted from Fig. 4 that the ratio B/C0 controls the form of the wind-velocity profiles. Bagnold’s focus is more likely to form for decreasing values of the ratio. A value of B/C0 c 10 separates those velocity profiles with a focus from those without it.
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Fig. 5. Some measured wind-velocity profiles available in the literature.
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The magnitude of B/C0 is also related to the size of the saltating sand and its grain size distribution. The velocity profiles measured in the experiments with Sand A (finer sand) and Sand B (coarser sand) presented in Fig. 1 parallel those in Fig. 4A and E, respectively. It could be surmised then that the saltation modified wind-velocity profiles for finer and more uniform sand grains are associated with larger values of B/C0. If the view that Bagnold’s focus corresponds to the mean saltation height of uniformly sized grains (Bagnold, 1941) is accepted, then an apparent contradiction seems to emerge from the calculation from Eq. (20) in Fig. 4. On one hand, it is known that the mean saltation height is proportional to the magnitude of the friction velocity (Owen, 1980), and on the other hand, the figure does not reveal a direct relation between focuses and friction velocities. There may be two mechanisms contributing to the form of velocity profiles. With increasing wind speed, the increasing concentration of saltating
367
grains extracts energy from the flow and slows down the wind velocity (Chepil and Woodruff, 1963). Also, the saltating grains transport energy from the upper flow region to the region near the bed where the wind speed is lower than the grain speed at the same height (Zingg, 1953). The relative contribution of the two mechanisms would depend on the size of the particles and on the grain size distribution. Therefore, further study of the link between B/C0 and the grain size distribution is required in future research. 4.3. Comparison of Eqs. (16) and (1) It is instructive to compare the newly derived expression for the fluid shear stress profile within the saltation layer, Eqs. (16) and,Eq. (1) of Raupach (1991). It is clear from the derivation of Eq. (16) that ¯ ) and q(z)/q(0) on its the terms (1 U*2t/U*2), (1 z/H right-hand side reflect the effects of the threshold friction velocity, the variation of the grain velocity
Fig. 6. Comparison between Eqs. (16) and (1).
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with height, and the vertical distribution of sand-mass flux on the fluid shear distribution, respectively. In ¯ ). contrast, Eq. (1) does not have the term (1 z/H Further, the exponent of the ratio U*t/U* is 2 in Eq. (16) and 1 in Eq. (1). The term e z/Hs in Eq. (1) and the parallels in Eq. (19) may be associated with the vertical distribution of sand-mass flux. Fig. 6 depicts a comparison of results obtained ¯ from Eq. (16) with q(z)/q(0) = e z/H and Eq. (1) for four wind friction velocities and d = 0.25 mm, for which U*t is 0.18 m/s according to Bagnold’s (1941) U*t formula. The predictions are quite different. The ratio sa(z)/qaU*2 in Eq. (16) increases with increasing ¯ and reaches its maximum at z/H ¯ = 1, thus indicatz/H ing that, within the saltation layer, the fluid shear stress increases with height and equals the fluid shear stress outside of it at the effective height of the saltation layer. At this height the grain-borne shear stress is zero, in agreement with the current understanding of the fluid shear stress. Although for the results from Eq. (1) the ratio sa(z)/qaU*2 also increases with increasing values of z/Hs, the ratio does not reach its maximum at z/Hs = 1. This implies that the grainborne shear stress is not zero at the effective height of saltation layer, which disagrees with existing evidence. It appears that Eq. (16), derived from energy considerations of the saltating particle motion, is more reasonable than Eq. (1) obtained from invoking the analogy of the saltation layer and a vegetation forest as distributed momentum sinks.
5. Conclusions New equations for the fluid shear stress and windvelocity distribution within the saltation layer are derived. These equations state the dependence of both profiles on the threshold shear velocity, the vertical variations of sand grain velocity, the vertical sandmass-flux distribution, and the fluid shear stress within the saltation layer. There is a good agreement between the velocity data obtained during previous wind tunnel experiments and the derived equations. Velocity profiles in the saltation layer are concave downward curves when compared with the profiles outside the saltation layer. The new equation, in line with observations reported in the literature, is capable of generating wind-velocity profiles with and without
Bagnold’s focus. The equation is not very sensitive to the mathematical description of the vertical sand-mass flux.
Acknowledgements The financial support of this research is from the National Natural Science Foundation of China under Grant No. 49625101.
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(Eds.), Proceedings of International Workshop on the Physics of Blown Sand. Aarhus University, Denmark, pp. 301 – 325. Raupach, M.R., 1991. Saltation layers, vegetation canopies and roughness lengths. Acta Mech. Suppl. 1, 83 – 96. Shao, Y.P., Li, A., 1999. Numerical modelling of saltation in the atmospheric surface layer. Boundary-Layer Meteorol. 91, 199 – 225. Sherman, D.J., 1992. An equilibrium relationship for shear velocity and apparent roughness length in aeolian saltation. Geomorphology 5, 419 – 431. Sørensen, M., 1985. Estimation of some aeolian saltation transport parameters from transport rate profiles. In: Barndorff-Nielsen, O.E., Møller, J.T., Rasmussen, K.R., Willetts, B.B. (Eds.), Proceedings of International Workshop on the Physics of Blown Sand. Aarhus University, Denmark, pp. 141 – 190. Spies, P., McWan, I.K., Butterfield, R.B., 1995. On wind velocity profile measurements taken in wind tunnels with saltating grains. Sedimentology 42, 515 – 521. Sterk, G., Raats, P.A.C., 1996. Comparison of models describing the vertical distribution of wind-eroded sediment. J. Soil Sci. Soc. Am. 60, 1914 – 1919. Stout, J.E., Zobeck, T.M., 1996. The Wolfforth field experiment: a wind erosion study. Soil Sci. 161, 616 – 632. Ungar, J.E., Haff, P.K., 1987. Steady state saltation in air. Sedimentology 24, 289 – 299. Werner, B.T., 1990. A steady-state model of wind-blown sand transport. J. Geol. 98, 1 – 17. White, B.R., Mounla, H., 1991. An experimental study of Froude number effect on wind-tunnel saltation. Acta Mech. Suppl. 1, 145 – 157. Williams, G., 1964. Some aspects of the eolian saltation load. Sedimentology 3, 257 – 287. Zingg, A.W., 1953. Wind tunnel studies of the movement of sedimentary material. Proc. 5th Hydraulic Conf. Inst. of Hydraulics, Iowa City, pp. 111 – 135.
An analytic expression for wind-velocity profile within the saltation layer Z.S. Li a,b,*, J.R. Ni a,b, C. Mendoza c b
a Department of Environmental Engineering, Peking University, Beijing, 100871, China The Key Laboratory of Water and Sediment Sciences, Ministry of Education, Beijing, 100871, China c Department of Civil Engineering, University of Missouri, Rolla, MO 65409-0030, USA
Received 19 March 2003; received in revised form 20 August 2003; accepted 28 August 2003 Available online 21 November 2003
Abstract The vertical wind-velocity profile within the saltation layer is investigated theoretically. New equations for the fluid shear stress distribution in the saltation layer and the velocity profile affected by saltation are derived. The dependence of the velocity profile on the threshold shear velocity, the vertical variations of sand grain velocity, the vertical sand-mass-flux distribution, and the fluid shear stress within the saltation are demonstrated. Velocity profiles with or without Bagnold’s focus are predicted. These velocity profiles are not very sensitive to the vertical distribution of sand-mass flux. Predicted velocity profiles are compared with profiles measured in the wind tunnel, and satisfactory agreement has been achieved. D 2003 Elsevier B.V. All rights reserved. Keywords: Wind-velocity profile; Saltation layer; Shear stress profile; Analytic expression
1. Introduction Data abound with evidence that wind-velocity profiles in the saltation layer are significantly affected by the presence of saltating sand grains and do not follow entirely the logarithmic law (e.g., Bagnold, 1941; Kawamura, 1951; Zingg, 1953; Spies et al., 1995; McKenna-Neuman and Maljaars, 1997). Owen (1964) suggested that the grain saltation layer acts on the flow above it in a way similar to that of solid * Corresponding author. Department of Environmental Engineering, Peking University, Beijing, 100871, China. Tel.: +86-106275-3962; fax: +86-10-6275-6526. E-mail address: [email protected] (Z.S. Li). 0169-555X/$ - see front matter D 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.geomorph.2003.08.008
roughness on a flow, and that the logarithmic law governs the flow outside of it. The validity of the logarithmic law outside the saltation layer has been repeatedly confirmed by many experiments (e.g., Kind, 1976; Rasmussen et al., 1985; Rasmussen and Mikkelsen, 1991; Sherman, 1992). In his analysis of the wind-velocity profile within the saltation layer, Owen (1964) advanced the complementarity of the fluid shear stress at height z, sa(z), and the grain-borne shear stress at height z, sp(z), under steady-state conditions. The sum of the fluid shear stress and the grain-borne shear stress remains constant with height and is equal to the fluid shear stress above the saltation layer. Further, Raupach (1991) exploited the similarity in momentum sink
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distribution from the fluid flow between vegetation canopies and saltation layers to study the wind field within the saltation layer. From imposing the same global constraints for the fluid shear stress as in the numerical models, he found h i2 sa ðzÞ ¼ qa U*2 1 ð1 U*t =U* Þez=Hs ð1Þ where qa is the air density, U* is the flow friction velocity outside the saltation layer, U*t is the threshold friction velocity, and Hs is the characteristic height at which the grain-borne stress is zero. McEwan (1993) adopted a triangular and a hyperbolic tangent stress distribution to investigate the wind-velocity profile within the saltation layer. The latter was favored due to its visual similarity to the predictions from the self-regulatory saltation model of McEwan and Willetts (1991) rather than from sound physical bases. Other researchers have computed, using numerical methodologies, the profile of the grain-borne shear stress and/or the wind velocity in the saltation layer from fairly complex sand grain saltation models. Sørensen (1985), Jensen and Sørensen (1986), Ungar and Haff (1987) and Werner (1990) investigated the steady-state case, whereas Anderson and Haff (1991) and most recently Shao and Li (1999) simulated the time evolution of the saltation process. Although their extensive calculations reproduced the overall features of measured wind-velocity profiles, they did not propose analytic expressions of the saltation-modified wind profile. In this paper, new analytical formulae for the fluid shear stress and the saltation modified velocity distributions in the saltation layer are developed. Also, new data were collected on wind-velocity profiles in the saltation layer making novel use of a hack tube in a wind tunnel to complement the available data in the literature and to validate the derived analytical formula for the wind-velocity distribution.
2. Velocity profile within the saltation layer This section introduces the derivation of a new equation for the wind-velocity distribution within the saltating layer that accounts for the presence of the hopping sand grains that alter the wind flow. The
analysis starts by considering, as in Owen (1964), that the total shear stress within the saltation layer (s) is s ¼ sa ðzÞ þ sp ðzÞ
ð2Þ
and that the velocity profile may be described, in accordance with Raupach (1991), McEwan (1993) and Anderson and Haff (1991), among others, in the form sffiffiffiffiffiffiffiffiffiffi dU 1 sa ðzÞ ¼ ð3Þ dz jz qa where U is the flow velocity at height z, and j is von Karman’s constant. It is easily seen from the combination of Eqs. (2) and (3) that since sa(z) depends on sp(z) for a given s, an estimation of sp(z) is required for the determination of the wind-velocity profile. Following Liu et al. (1996), sp(z) results from the sum of the differences in horizontal momentum (passing through a unit area per unit time) produced by (1) both the descending and ascending sand grains, (2) the descending grains, and (3) the ascending sand grains. The last two differences are far smaller than the first one, so, for an idealized case, they can be neglected. Therefore, for uniform saltating grains in equilibrium, sp(z) can be simplified as sp ¼ mn1 ðzÞw¯ 1 ðzÞ½¯u2 ðzÞ u¯ 1 ðzÞ
ð4Þ
where m is the mass of a single grain, n1(z) is the number of ascending grains at height z, w¯1,(z) is the mean vertical velocity of ascending grains at height z, u¯1(z) the mean horizontal velocity of ascending sand grains at height z, and u¯2(z) is the mean horizontal velocity of descending sand grains at height z. Eq. (4) will be used as the basis for further analysis of the grain-borne shear stress profiles. It is conveniently rewritten in the following dimensionless form sp ðzÞ mn1 ðzÞ w¯ 1 ðzÞ u¯ 2 ðzÞ u¯ 1 ðzÞ ¼ mn1 ð0Þ w¯ 1 ð0Þ u¯ 2 ð0Þ u¯ 1 ð0Þ sp ð0Þ
ð5Þ
where the argument 0 refers to parameters evaluated at the bed surface, i.e., z = 0. This equation will be
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subjected to a series of transformations before solving for sp(z). Firstly, the first term on the right-hand side is expressed as mn1 ðzÞ u¯ ð0Þ 2mn1 ðzÞ¯uðzÞ ¼ ð6Þ mn1 ð0Þ u¯ ðzÞ 2mn1 ð0Þ¯uð0Þ where u¯(0) and u¯(z) are defined approximately as u¯(0)i1/2[u¯1(0) + u¯2(0)] and u¯(z)i1/2[u¯1(z) + u¯2(z)], respectively. When the saltation layer is in equilibrium, the number of ascending particles equals the number of descending ones, i.e., 2mn1(z)u¯(z) = q(z) and 2mn1(0)u¯(0) = q(0), where q(z) and q(0) are the mass flux of grains at z and at the bed surface, respectively, and Eq. (6) is now written as mn1 ðzÞ qðzÞ u¯ ð0Þ ¼ mn1 ð0Þ qð0Þ u¯ ðzÞ
ð7Þ
Secondly, as the vertical component of the drag force on the saltating grains can be reasonably neglected (e.g., Kawamura, 1951), then 8 1 > 2 > < mw¯ 1 ð0Þ ¼ mg H¯ 2 ð8Þ > 1 1 > : mw¯ 21 ð0Þ ¼ mw¯ 21 ðzÞ þ mgz 2 2 results from the application of the energy conservation ¯ is the mean maximum height of law. In Eq. (8), H sand grain saltation, and g is the gravitational acceleration constant. It follows from Eq. (8) that w¯ 1 ðzÞ ¼ w¯ 1 ð0Þ
rffiffiffiffiffiffiffiffiffiffiffiffiffi z 1 H¯
ð9Þ
The substitution of Eqs. (9) and (7) into Eq. (5) yields rffiffiffiffiffiffiffiffiffiffiffiffiffi
sp ðzÞ qðzÞ z u¯ ð0Þ½¯u2 ðzÞ u¯ 1 ðzÞ 1 ¼ ð10Þ sp ð0Þ qð0Þ H¯ u¯ ðzÞ½¯u2 ðzÞ u¯ 1 ð0Þ Furthermore, the third term on the right-hand side of Eq. (10) becomes u¯ ð0Þ½¯u2 ðzÞ u¯ 1 ðzÞ ½¯u2 ð0Þ þ u¯ 1 ð0Þ½¯u2 ðzÞ u¯ 1 ðzÞ ¼ u¯ ðzÞ½¯u2 ð0Þ u¯ 1 ð0Þ ½¯u2 ð0Þ u¯ 1 ð0Þ½¯u2 ðzÞ þ u¯ 1 ðzÞ ð11Þ
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From the ballistics of individual sand grains, the velocities are u¯ 1 ðzÞ ¼ u¯ 1 ð0Þ þ at1 ; u¯ 2 ðzÞ ¼ u¯ 1 ð0Þ þ at2 ; u¯ 2 ð0Þ ¼ u¯ 1 ð0Þ þ at ð12a; b; cÞ where a is the acceleration of the particles, t = t1 + t2 = 2bw ¯ 1(0)/g (Liu, 1995; McEwan and Willetts, 1994) is the overall particles flight time, in which b is a coefficient, and t1 = bw ¯ 1(0)/g bw ¯ 1(z)/g and t 2 = bw ¯ 1(0)/g + bw ¯ 1(z)/g are the travel time of the ascending and descending particles, respectively. The substitution of Eq. (12a – c) into Eq. (11) yields u¯ ð0Þ½¯u2 ðzÞ u¯ 1 ðzÞ t2 t1 ¼ u¯ ðzÞ½¯u2 ð0Þ u¯ 1 ð0Þ t which can be further modified to the form rffiffiffiffiffiffiffiffiffiffiffiffiffi z u¯ ð0Þ½¯u2 ðzÞ u¯ 1 ðzÞ ¼ 1 u¯ ðzÞ½¯u2 ð0Þ u¯ 1 ð0Þ H¯
ð13Þ
ð14Þ
after using the expressions for t1, t2, t, and Eq. (9). Thirdly, since sa(0) = qaU*2 t, sp ð0Þ ¼ s sa ð0Þ ¼ qa U*2 qa U*2 t
ð15Þ
Combining (Eqs. (14), (15), (10) and (4) results in z qðzÞ sa ðzÞ ¼ qa U*2 1 ð1 U*2 t =U*2 Þ 1 qð0Þ H¯ ð16Þ Finally, the integration of Eqs. (16) and (3) together results in the following wind-velocity distribution within the saltation layer: U 1 ¼ U* j
Z
z z0
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ! u U2 1u z qðzÞ t1 1 *t dz 1 z qð0Þ U2 H¯ * ð17Þ
which accounts for the influence of the sand grain loading of the flow on the wind velocity in an explicit form. ¯ and q(z)/q(0) should be determined The terms H before Eq. (17) can be applied. Several investigators have found simple expressions for the mean saltation layer height. From theoretical calculations, Owen
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¯ = 0.82U*2/g whereas Nalpanis et al. (1980) proposed H ¯ = 1.4 –1.9U*2/g from the analysis of (1993) found H ¯ ~U 2/g. experimental data. It is clear then that H * Often, the studies of vertical distribution of the sand mass flux have yielded disparate results. From the fitting of wind tunnel data, Zingg (1953) proposed the modified power-law function: qðzÞ z 1=n1 ¼ 1þ ð18Þ qð0Þ r1 where r1 is a scale height and n1 is the exponent. The data in Kawamura (1951), Rasmussen and Mikkelsen (1991), Stout and Zobeck (1996), Sterk and Raats (1996), and Butterfield (1999) also follow a similar function. Other data displays an exponential dependence of mass flux with height (e.g., Williams, 1964; Greeley et al., 1996) of the form qðzÞ ¼ ez=r2 qð0Þ
ð19Þ
where r2 is a scale height. According to Nalpanis (1985), r2 = U*2/kg defines the characteristic height of the mass flux profile, since k can be assumed to be constant for a single experiment. The same relation resulted from White and Mounla (1991) and Nalpanis et al. (1993). Eqs. (18) and (19) suggest that q(z)/q(0) is a function of z/r, and the scale height r is the height at which q(z)/q(0) = 2 1/n in Eq. (18) but equals to 1/e in Eq. (19). Although r does not modify the structure of equations for q(z)/q(0), an appropriate selection of r can simplify the resulting wind-velocity distribution function and its application. Just as for the case of r2 in Eq. (19), r~U*2/g. ¯ and r to U*2/g permits The proportionality of H writing Eq. (17) in the functional form U 1 ¼ U* j
Z
z
1 z vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! ! !ffi u u U2 2gz 2gz dz t1 1 *2t 1 f BU 2 U2 U * * * ð20Þ z0
where B is a coefficient to be determined from observations.
3. Comparisons with data Ni et al. (2002) carried out experiments in a straight-blowing type wind tunnel. The tunnel body has a total length of about 35 m and is capable of producing wind speeds from 3 to 25 m/s. The windvelocity profiles were measured for five wind speeds and for two test sands. The details of experiment set-up were described by [Ni et al. (2002)]. The results of the velocity measurements are shown in Fig. 1. In Fig. 1, two different parts are easily distinguished in each profile: the upper part follows the logarithmic law, to the wind-velocity profile outside the saltation layer, while the lower part shows a pronounced downward concavity indicative of the deviation from the log law, which is a well-known effect of the saltating grains, and represents the windvelocity profile within the saltation layer. Owen (1964) proposed a logarithmic equation for the wind-velocity profile outside the saltation layer U 1 2gz ¼ ln U* j C0 U 2 *
! ð21Þ
where C0 is an empirical constant. In Fig. 2, wind-velocity data by Bagnold (1941), Horikawa and Shen (1960), and McKenna-Neuman and Maljaars (1997) as well as data in Fig. 1 are plotted in a [U/U*, log(2gz/U*2 )]-plane, consistent with the structure of Eq. (20). In the upper part of the figure, the sampled data for the various freestream velocities collapse into a straight line, representative of Eq. (21) but into a single downward concave curve associated with Eq. (20) in the lower part. The data in the upper part in Fig. 2 are clearly from outside the saltation layer and in the lower part from within the saltation layer. The agreement of the data and the concave-downward segment of the curve in Fig. 2 resulted from calculations with Eqs. ¯ and an appropriate choice (20) and (19) with r2 = H of B. Several calculated wind-velocity profiles available in the literature (e.g., Sørensen, 1985; Shao and Li, 1999) have shown the concave-downward curves in a [U, log(z)]-plane as well. In contrast, other measured near-bed velocity profiles (e.g., Butterfield, 1999) seem to indicate that they still follow the log
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Fig. 1. Measured wind velocity profiles (Ni et al., 2002). Sand A: mean diameter = 0.17 mm, the sorting index (Folk and Ward, 1957) = 0.35. Sand B: mean diameter = 0.35 mm, the sorting index = 0.60.
Fig. 2. Comparison of predicted vertical wind-velocity distributions with Bagnold’s (1941), Horikawa and Shen’s (1960), McKenna-Neuman and Maljaars’ (1997), and Ni et al.’s (2002) data.
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law. Although this would represent a challenge to the notion that the saltating grains induce profound modifications of the wind flow within the saltation layer, this apparently contradictory result could be justified by the low concentration of sand grains in the air generated by the low wind friction speeds, ranging from 0.27 to 0.44 m/s, predominant in those experiments.
4. Discussion 4.1. Sensitivity of Eq. (20) to changes of q(z)/q(0) The results of the calculations obtained with Eq. (20) combined with Eqs. (18) and (19) and assuming ¯ are shown in Fig. 3. The closeness of that r1 = r2 = H the results demonstrates that Eq. (20), combined with Eq. (18) at n1 = 1.0, can represent the empirical data in Fig. 2 as well as Eq. (20) combined with Eq. (19) did.
It also indicates that Eq. (20) is not very sensitive to the form of the q(z)/q(0) equation. 4.2. Effect of the ratio B/C0 on Bagnold’s focus Because the results obtained with the wind-velocity profile equation are not very sensitive to the structure of the vertical sand-mass-flux profile, Eq. (19) is now taken as representative of q(z)/q(0) for further analysis of Eq. (20). Fig. 4 shows windvelocity profiles computed using Eqs. (20) and (21) with 12 different values of the (B,C0)-pair, for sand grains with a d = 0.25 mm. From Bagnold’s formula qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi U*t ¼ A gdðqp qa Þ=qa
ð22Þ
where U*t is the threshold friction velocity, A is a constant, and qp is the density of sand particles. Substitution of A = 0.08, g = 980 cm/s2, qp = 2.65 g/
Fig. 3. Results from Eq. (20) combined with Eqs. (18) and (19).
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Fig. 4. Wind-velocity profiles calculated with Eqs. (20) and (21).
cm3, and qa = 0.001226 g/cm3 into Eq. (22) yields 0.18 m/s for U*t. For a constant value of C0, the velocity profiles in the near-bed flow region diverge with increasing values of B; while for a constant B, the velocity profiles converge to form the well-recognized Bagnold’s focus with increasing values of C0. In general, two types of velocity profiles can be distinguished: one with Bagnold’s focus and the other without it. The focus is located at an identifiable height in some profiles, but not in others; for the latter case, the wind velocity is almost constant at the same height regardless of the magnitude of the friction velocity. The singular shape of the wind-velocity distribution for U* = 0.25 m/s throughout Fig. 4 may be related to the fact that the friction velocity exceeds
only slightly the threshold friction velocity in this case. The forms of the velocity profiles depicted in Fig. 4 can be immediately related to measured ones (Fig. 5): velocity profiles of Bagnold (1941) and Kawamura (1951) are similar to those in Fig. 4K and L; profiles of Horikawa and Shen (1960) and Kadib (1965) to those in Fig. 4E and I; profiles of Zingg (1953) to those in Fig. 4A and F; and profiles of Belly (1964), Gerety (1985), and Butterfield (1991) to those in Fig. 4B –D, G and H. It is noted from Fig. 4 that the ratio B/C0 controls the form of the wind-velocity profiles. Bagnold’s focus is more likely to form for decreasing values of the ratio. A value of B/C0 c 10 separates those velocity profiles with a focus from those without it.
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Fig. 5. Some measured wind-velocity profiles available in the literature.
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The magnitude of B/C0 is also related to the size of the saltating sand and its grain size distribution. The velocity profiles measured in the experiments with Sand A (finer sand) and Sand B (coarser sand) presented in Fig. 1 parallel those in Fig. 4A and E, respectively. It could be surmised then that the saltation modified wind-velocity profiles for finer and more uniform sand grains are associated with larger values of B/C0. If the view that Bagnold’s focus corresponds to the mean saltation height of uniformly sized grains (Bagnold, 1941) is accepted, then an apparent contradiction seems to emerge from the calculation from Eq. (20) in Fig. 4. On one hand, it is known that the mean saltation height is proportional to the magnitude of the friction velocity (Owen, 1980), and on the other hand, the figure does not reveal a direct relation between focuses and friction velocities. There may be two mechanisms contributing to the form of velocity profiles. With increasing wind speed, the increasing concentration of saltating
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grains extracts energy from the flow and slows down the wind velocity (Chepil and Woodruff, 1963). Also, the saltating grains transport energy from the upper flow region to the region near the bed where the wind speed is lower than the grain speed at the same height (Zingg, 1953). The relative contribution of the two mechanisms would depend on the size of the particles and on the grain size distribution. Therefore, further study of the link between B/C0 and the grain size distribution is required in future research. 4.3. Comparison of Eqs. (16) and (1) It is instructive to compare the newly derived expression for the fluid shear stress profile within the saltation layer, Eqs. (16) and,Eq. (1) of Raupach (1991). It is clear from the derivation of Eq. (16) that ¯ ) and q(z)/q(0) on its the terms (1 U*2t/U*2), (1 z/H right-hand side reflect the effects of the threshold friction velocity, the variation of the grain velocity
Fig. 6. Comparison between Eqs. (16) and (1).
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with height, and the vertical distribution of sand-mass flux on the fluid shear distribution, respectively. In ¯ ). contrast, Eq. (1) does not have the term (1 z/H Further, the exponent of the ratio U*t/U* is 2 in Eq. (16) and 1 in Eq. (1). The term e z/Hs in Eq. (1) and the parallels in Eq. (19) may be associated with the vertical distribution of sand-mass flux. Fig. 6 depicts a comparison of results obtained ¯ from Eq. (16) with q(z)/q(0) = e z/H and Eq. (1) for four wind friction velocities and d = 0.25 mm, for which U*t is 0.18 m/s according to Bagnold’s (1941) U*t formula. The predictions are quite different. The ratio sa(z)/qaU*2 in Eq. (16) increases with increasing ¯ and reaches its maximum at z/H ¯ = 1, thus indicatz/H ing that, within the saltation layer, the fluid shear stress increases with height and equals the fluid shear stress outside of it at the effective height of the saltation layer. At this height the grain-borne shear stress is zero, in agreement with the current understanding of the fluid shear stress. Although for the results from Eq. (1) the ratio sa(z)/qaU*2 also increases with increasing values of z/Hs, the ratio does not reach its maximum at z/Hs = 1. This implies that the grainborne shear stress is not zero at the effective height of saltation layer, which disagrees with existing evidence. It appears that Eq. (16), derived from energy considerations of the saltating particle motion, is more reasonable than Eq. (1) obtained from invoking the analogy of the saltation layer and a vegetation forest as distributed momentum sinks.
5. Conclusions New equations for the fluid shear stress and windvelocity distribution within the saltation layer are derived. These equations state the dependence of both profiles on the threshold shear velocity, the vertical variations of sand grain velocity, the vertical sandmass-flux distribution, and the fluid shear stress within the saltation layer. There is a good agreement between the velocity data obtained during previous wind tunnel experiments and the derived equations. Velocity profiles in the saltation layer are concave downward curves when compared with the profiles outside the saltation layer. The new equation, in line with observations reported in the literature, is capable of generating wind-velocity profiles with and without
Bagnold’s focus. The equation is not very sensitive to the mathematical description of the vertical sand-mass flux.
Acknowledgements The financial support of this research is from the National Natural Science Foundation of China under Grant No. 49625101.
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