The dimensionless pressure from the solid phase (ice crystals) defined in equation 26, is shown in Figure 4b. It is clear from this figure that the solid-phase pressure increases nearly linearly from the leading (x′ = 1) to the trailing edge (x′ = 0) where maximum compression is achieved; as the velocity increases, fair increases, and, thus the ice crystal's contribution to the total lift decreases.
The predictions shown in Figure 4 qualitatively agree with the experimental measurements in (17,19,20) for the compression of wind-packed snow, in their porous walled piston-cylinder apparatus. In these experiments, one measured the time for the air to escape through the porous side walls of the porous cylinder-piston apparatus and also measured the dynamic forces on the piston from the trapped air and the snow crystals as the air escaped. Although this is obviously not a realistic model for a ski or snowboard, it provides some guide for the forces that one can anticipate. Furthermore, the experiments suggest that for snowboarding on fine-grained less permeable (wind-packed) snow at velocities greater than 10 m·s−1, the lift forces generated by the trapped air in the compressed snow layer should provide a significant fraction of the total lift.
Friction (wind-packed snow, U = 20 m·s−1, η = 0.02, 0.04, 0.06 and 0.08).
The study of snowboard or skis as sliders on snow has a long history related to the more general study of friction. Skiing is unique among these studies. For alpine skiing or snowboarding, the skier seeks minimal sliding friction to increase their downhill speed, while for track and cross-country skiing one requires his or her skis to combine minimum friction on the forward glide and maximum friction or grab when the skier pushes off against the snow for acceleration or for travel uphill. To our knowledge, most of the literature on ski or snowboard friction is focused on the presence of a micron thick fluid film that forms beneath the running surface as a melt water lubrication layer (3-7). No one has ever investigated the associated lift mechanism due to the trapped air and the solid phase (ice crystals) for different values of friction coefficient.
In Figure 5a, we plot the centerline pore-pressure distribution beneath a snowboard surface for various values of friction coefficient, η, as one glides over a 10-cm wind-packed snow layer at a velocity, U = 20 m·s−1. The corresponding λ, k, xa′, xs′, fair and fsolid are listed in Table 1B. The values of η, 0.02 to 0.08, are chosen to span a broad range of thermal, bottom snowboard surface and ice crystal conditions (12). It is clear from this figure that a decrease in the sliding friction leads to an increase in the trapped air's contribution to the total lift and compression ratios at the leading and trailing edges, ë = h 2/h 0 and ë/k = h 1/h 0 increase. As η decreases from 0.08 to 0.02, fair increases from 24 to 63%, the compression ratio at the leading edge ë = h 2/h 0 increases from 0.55 to 0.77, and the compression ratio at the trailing edge, h 1/h 0 = ë/k increases from 0.54 to 0.68 providing for a much larger tilt angle. This behavior is further confirmed in Figure 5b, where one can readily see the pronounced increase in solid-phase lift force as η increases. These results indicate that for downhill skiing, a decrease in the sliding friction not only reduces the snow frictional drag force that the skier encounters as he or she glides down the slope at a given speed, but also increases the trapped air's contribution to the total lift. These results provide a new insight into the role of friction in ski and snowboard performance. For a smooth air cushioned glide one wants a low value of η. This can be obtained by waxing the bottom surface of the snowboard or ski.
Snowboard control and stability.
Shift of xc′. Figure 6 provides the critical insights for snowboard control and stability. In contrast to an airplane, where the center of mass is fixed and one controls the angle of attack and moments about the center of mass by flap and rudder control, a snowboarder can alter his/her center of mass by shifting their weight from the front to the rear foot. Under most conditions this shift in xc′ is small and varies between roughly 0.37 and 0.47. However as shown in Figures 6a and b, this has a dramatic impact on the distribution of the pressure loading from the air and solid (ice crystal) phases. When one places more of their weight on the rear foot xc′ decreases and approaches the lower limit of 0.37, whereas when one shifts their weight to the front foot xc′ increases and approaches the upper limit of 0.47. When xc′ = 0.37 the contribution of the trapped air pressure is both large and asymmetrically distributed with large pressures near the rear of the snowboard. In contrast, when xc′ = 0.47 the contribution of the trapped air is much smaller and the pressure distribution is symmetric. This change is accompanied by a transfer of lift forces from the air to the solid phase and a change in angle of attack of the snowboard, which is plotted in Figure 6c. The dashed lines in Figure 6a and b crossing the pressure profiles show this shift in xc′.
The curves in Figure 6 apply to a neutral stability condition in which the sum of moments about the CM vanishes (see equation 30). For an airplane one examines stability by changing the angle of attack from the neutral stability position and sees if the moment about the CM is restoring (stable) or produces a nonrestoring moment (unstable). A similar analysis can be performed for a ski or snowboard except that one wants to examine the moment created by a shift in xc′.
One can rewrite equation 30 as
Introducing dimensionless variables, equation 34a reduces to
where M′ = M/mgcosαh L. In Figure 6d we have plotted the equivalent moment (M′) or the pitching moment curves for a snowboard. For each set of curves in Figure 6a, b (same value of xc′) we ask how the moment would increase or decrease about the neutral equilibrium position if the angle of attack of the snowboard was fixed but xc′ shifted. This calculation leads to the moment profiles in Figure 6d. Each curve crosses the neutral position at M′ = 0 and all curves have a positive slope indicating that a forward shift of xc′ produces a clockwise moment and a backward shift a counterclockwise moment. If one shifts their weight (changes xc′) and does not change their angle of attack one must apply the moment indicated in this figure to maintain the same snowboard position. This requires a muscular input and for minimum effort one will want to go to a new neutral stability condition where no effort is required. This is shown in Figure 6d as the trajectory a-b-c. This requires that one change their angle of attack and move along the curve shown in Figure 6d where M′ = 0.
In the same way that one can shift their weight from the front to rear foot or the reverse, one can tilt the snowboard from one lateral edge to the other in a maneuver called edging. This maneuver requires a more difficult analysis since the lateral tilt changes the simple parabolic pressure distribution described by equation 12 in the transverse plane of the snowboard. Edging maneuvers, which are required for turning, are a subject for future study.
Shift of lc. The vertical displacement of the center of mass (CM) will also affect the snowboarder's stability, see equation 34. Similar to the shift of xc′ in Figure 6, if one raises or lowers his or her center of mass vertically through up-turn-down or down-turn-up body motion without changing their compression ratios at the leading and trailing edges, the initial neutral moment balance is broken and an unbalanced pitching moment M′ defined in equation 34b is generated. To maintain stability, one has to input a muscular moment or change the compression ratios of the snow layer to get back to a new neutral moment balance position. The latter requires no muscular input, and is accompanied by a transfer of lift forces between the trapped air and the solid ice crystals as well as changes of snow compression at the leading and trailing edges. We have plotted in Figure 7a the centerline pore-pressure distribution beneath a snowboard surface as one snowboards over a 10-cm-think wind-packed snow layer at U = 20 m·s−1. The air-pressure profiles for M′ = 0 for these different values of lc, lc = 0.8, 1.0, and 1.2 m are shown. The corresponding solid-phase pressure is plotted in Figure 7b and the results for U, λ, k, xa′, xs′, fair, and fsolid are summarized in the Table 1D. Table 1D shows that as one vertically lowers his or her CM from lc = 1.2 to 0.8 m, the trapped air's contribution to the total lift increases from 40 to 63% and the solid-phase lift contribution decreases from 60 to 37%, accordingly. Meanwhile, the required compression ratio at the leading edge ë = h 2/h 0 changes from 0.62 to 0.77, and the compression ratio at the trailing edge, ë/k = h 1/h 0 changes from 0.60 to 0.69. In the current study we choose the vertical displacement of the CM from crouching to standing position to be approximately 0.4 m (12).
Fresh snow (K 0 = 1.7 × 10−8 m2, φ0 = 0.8, d = 1.0 mm). Because the Darcy permeability of fresh snow is roughly 34 times larger than the wind-packed snow, air can not be trapped efficiently. At a given speed, one needs a much larger compression of the fresh snow layer and a larger contribution from the solid phase to generate the required lift, see Figure 8. Figure 8a shows the centerline pore-pressure distribution beneath a snowboard surface for the two different snow types. It is clear from this figure that, for a given speed, U = 20 m·s−1, the trapped air's contribution to the total lift is much larger in less permeable wind-packed snow than in highly permeable fresh snow (fair wind-packed snow = 54%, fair fresh snow = 18%). The values of h/h 0 at both the leading and trailing edge decrease by roughly a factor of two (for fresh snow, h 2/h 0 = 0.38, h 1/h 0=0.31; while for wind-packed snow, h 2/h 0 = 0.70, h 1/h 0 = 0.65). The maximum compression ratio that can be achieved for a given type of snow is obtained from the expression: h 1/h 0 min = 1 − φ0 + φmin. For fresh snow h 1/h 0 min = 0.26, for wind-packed snow h 1/h 0 min = 0.46. Figure 8a shows that for fresh powder snow, an increase in velocity U leads to an increase in fair and a decrease in k, the same behavior observed in Figure 4 for wind-packed snow.
Figure 8b shows the corresponding solid-phase pressure beneath the snowboard surface. Because the trapped air inside fresh snow is not nearly sufficient to support the weight of the snowboarder with his/her equipment, the lift force from the ice crystals carries much of the load (fsolid fresh snow = 82%). This is in sharp contrast to the case of wind-packed snow, where fsolid wind-packed snow = 46% at the same velocity. This behavior was previously predicted in (19,20) where the relaxation of the pore pressure in the piston-cylinder apparatus was too rapid to support the full weight of the piston when it was dropped from rest.
The primary difference in the lift mechanics of snowboarding and skiing is due to the dimensionless parameter ε. The width of a ski is one third that of a snowboard, and ε for a ski is roughly 1/16 that for a snowboard. Thus, solutions of equation 15 for a long slender planing surface (ski) differ greatly from those for snowboarding. In general, due to the large increase in the pore-pressure relaxation at the lateral edges, the required snow compression is larger in skiing than in snowboarding. In Figure 9a, we have compared the centerline pore-pressure distribution beneath a ski and snowboard for wind-packed snow for a velocity of 20 m·s−1. As shown in this figure, although the required compression at the leading edge is almost the same, the tilt angle γ of the planing surface (or the compression ratio from the leading to trailing edge, k = h 2/h 1) is much larger in skiing than in snowboarding (k skiing = 1.315, k snowboarding = 1.072), and thus the snow layer at the trailing edge of the ski is compressed more (h 1/h 0 skiing = 0.54, h 1/h 0 snowboarding = 0.65). For U = 20 m·s−1, 42% of the total lift force is generated by the trapped air for skiing on wind-packed snow, which is less than that for snowboarding, where fair = 54%. The difference in fair is not great, considering the large difference in ε and the pore-pressure profiles shown in Figure 9a., The skier's center of mass is closer to the trailing edge (x′c skiing = 0.4, x′c snowboarding = 0.45), and one needs to increase the tilt angle γ to satisfy the moment balance equation 30. By doing so, the center of pore air pressure shifts towards the trailing edge enhancing the pore pressure's contribution to the total lift and compensates in part for the pressure leakage at the lateral edges. The dependence of the pore-pressure distribution on the velocity U of the skier is similar to that for snowboarding. Here U is chosen as 10, 20, and 30 m·s−1 for skiing.
In Figure 9b we plot the corresponding solid-phase pressure, which clearly demonstrates the significant contribution of the solid phase to the total lift in the case of skiing. Furthermore, due to the larger value of k = h 2/h 1, one observes a steeper increase in the local solid-phase pressure from the leading to trailing edge for skiing.
Having obtained the centerline pore-pressure distribution, we can plot two dimensional pore-pressure distributions beneath a snowboard or ski surface using equation 21. The results are shown in Figures 10a for snowboarding and 10b for skiing, where the snow is wind-packed, h 0 = 10 cm, and U = 20 m·s−1. This figure shows the effect of the dimensionless parameter, ε. One also notes the parabolic pore-pressure distribution in the transverse plane of a ski or snowboard. For skiing, due to the large tilt angle and prevailing continuous drainage in the lateral direction, the pore air pressure increases sharply to its peak value near the trailing edge and then decays rapidly to atmospheric pressure, in contrast to that observed for snowboarding, where the axial pressure relaxation at the trailing edge is much more gradual.
In this paper, we have developed a new theoretical analysis of the lift forces generated during downhill skiing or snowboarding, which incorporates the lift contribution from both the transiently trapped air and the compressed ice crystals. This study is an important practical application and extension of the lubrication theory for highly compressible porous media developed in F&W. The results presented herein agree with the more qualitative predictions in (17,19,20), where the pore pressures generated in snow were measured for the first time using a porous cylinder-piston apparatus.
The pore air pressure distribution was obtained using a consolidation theory based on Darcy's law, where the local change of Darcy permeability of snow was estimated using Shimizu's (15) classical empirical relationship. In our analysis we have neglected the motion of solid particles in the horizontal (x, y) plane and have assumed that the snow is compressed uniformly in the vertical direction. This assumption is appropriate for straight downhill skiing because one leaves a relatively clean track of compressed snow and the motion of the escaping snow at the lateral edges of the skis or snowboard is small compared with that of the escaping air. However, during edging and rapid turning maneuvers one has to consider the motion of the snow powder, and in particular its momentum, in determining the lateral forces, because under these conditions the snowboard or skis also act as a snowplow. The assumption of uniform vertical compression implies that the Darcy permeability K is primarily a function of x, or K does not change with the depth of the snow layer. This is reasonable because most skiing conditions involve a fresh snow layer on a packed base, which consists of recrystallized snow having a permeability that is 20-30 times smaller than fresh powder snow (11). Thus, the packed snow with much lower permeability acts as an impermeable base.
The lift force generated by the solid phase (ice crystals) was obtained from the experiments in (19,20). These experiments neglect the shear forces produced in the ice crystal structure at the lateral edges of the skis or snowboard. However, studies examining the deformation of the snow at the edge of a compression surface suggest that this force is small compared with the main compression force exerted by the snow directly beneath the planing surface (16).
The great enhancement in lift generated by a planing surface as it glides over a soft porous media is a new concept. The application to human skiing or snowboarding was first suggested in F&W. In a more recent study, Wu et al. (17,20) have applied this idea in the design of a future generation train that can glide on a soft porous track whose mechanical properties are similar to goose down pillows. The key insight in the latter application is that one could greatly enhance the lift and reduce the drag due to friction in the solid phase if the lateral loss of pore pressure at the side walls of the track could be eliminated. If this could be achieved, the authors predicted that it would be possible to support a 50-ton train car moving on a porous material with permeability properties that do not differ greatly from fresh snow at velocities greater than approximately 10 m·s−1.
This research was performed in partial fulfillment of the requirements for the Ph.D. degree from the City University of New York by Qianhong Wu. Dr. Wu was supported by NIH grant 19544 and the Mario Capelloni Dissertation Fellowship.
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Darcy Permeability of Snow beneath a Ski or Snowboard
The empirical relationship of Shimizu (15) for the Darcy permeability of snow is expressed as:
where ρS is the density of snow and d the mean diameter of the snow particles.
In general, as a fresh snow layer is compressed by a planar surface, the density of the snow layer changes due to the motion of the upper boundary. In the present analysis we assume that a fresh snow layer of the depth h 0 either rests on the ground or over a previously packed snow base whose permeability is much smaller than the new snow layer. If one assumes that the new snow layer is uniformly compacted in the vertical direction, whereas in the horizontal plane there is no crystal movement, the deformation-dependent average snow density, ρS(h), where h is the local instantaneous height of the new snow layer, is expressed as
if the snow layer has an initial uniform density ρS0 at its initial height h 0. If the mean diameter of the ice crystal, d, remains the same, and the initial Darcy permeability of snow is K 0, one obtains from equations A1 and A2,
In the current application shown in Figure 1b, one assumes that the bottom surface of the ski or snowboard is planar and that there is no lateral tilt or edging. Thus, the local thickness, h, of the fresh snow layer beneath the planing surface is a linear function of x:
where h 2 is the thickness of the snow layer at the leading edge and γ is tilt angle of the ski or snowboard. The slope of the ski relative to the ground or snow base is given by
where k = h 2/h 1 is the compression ratio. Substituting equations A4 and A5 into equation 3, we have
where K 2 is the Darcy permeability of the snow beneath the leading edge of the planing surface, which can be determined from equation A3 for given values of undeformed Darcy permeability, K 0, undeformed snow layer thickness, h 0, and the value of h 2, as shown in Figure 1b.
Equations A1 and A6 are described in the main text as equations 1 and 2a, respectively.
Pore-Pressure Distribution beneath a Ski or Snowboard without Lateral Tilt or Edging
From the analysis in (18,20) and Figure 13 in F&W, we know that the pressure distribution in the y direction is parabolic. Thus, one assumes a parabolic pressure profile in the cross-sectional plane of a ski or snowboard,
where E(x), F(x) are unknown functions and Pc(x) is the centerline pressure corresponding to the cross section at the location x. From symmetry P(x, y) = P(x, −y), it is evident from equation A7 that F(x) = 0. Thus, equation A7 can be written as:
E(x) is determined by applying a simplified boundary condition P∣y = ±W/2 = P 0 at the lateral edges of the ski or snowboard,
P 0 is very close to the atmospheric pressure because the porosity of the compressed snow changes abruptly at the lateral edges where the compression ends and the hydrodynamic resistance falls precipitously. Substituting equation A9 into A8, one obtains
Equation A10 is described in the main text as equation 12.
Centerline Pressure Distribution beneath a Ski Surface (ε << 1)
The governing equation for the centerline pressure distribution beneath a ski surface is given by:
where θW = (μA∣x = L/P 0) · (W 2/K 2). In equation A11 the small dimensionless parameter ε multiplies both the highest derivative term d 2 p′c(x′)/dx′2 and the nonlinear term (1/K′(x′))(dK′(x′)/dx′)(dp′c(x′)/dx′). Equation A11 is readily solved using singular perturbation methods.
The outer solution of equation A11 is obtained by neglecting the first two terms on the lefthand side,
Near the trailing edge, x′ = 0, one expects a sharp increase of pressure in a narrow region of thickness O(ε1/2 L). Both K′(x′) and dK′(x′)/dx′ remain nearly constant (see Fig. 2) within this layer and can be approximated by their values at x′ = 0. Introducing σ = x′/√ε, we obtain the inner solution at x′ = 0 by requiring
Similarly, at the leading edge, one introduces ζ = (1 − x′)/√ε to stretch the pressure boundary layer, and obtains the inner solution at x′ = 1
Thus the composite solution for the centerline pressure distribution beneath a ski surface is given by
Equations A11 and A15 are described in the main text as equations 19 and 20, respectively.
Keywords:©2006The American College of Sports Medicine
POROUS MEDIA; LIFT GENERATION; PORE PRESSURE; DARCY PERMEABILITY; DARCY'S LAW; SNOW