Skin friction coefficient values

FIGURE 32.3 Effect of lubricant cosmetic ingredient on skin friction coefficient. Amount applied of each material 2 mg/cm2. Reproduced from Nacht et al.14 (mean of five subjects but P value was not published). Time = — 1 is immediately prior to application Time = —0 is immediately after application. 

Thus, Eqs. 6.49 and 6.50 indicate that the skin friction coefficient and Stanton number remain equal to their constant-property values. In terms of these dimensionless transfer coefficients, the effects of the linear dependence of viscosity on temperature just cancel those of the perfect gas variation of the density. It should be noted, however, that the density variation itself still affects the boundary layer thickness. 

Figure 6.21 shows the wall shear parameter /" required to evaluate the local skin friction coefficients by Eq. 6.106 or 6.108. These curves apply for the case where ts = 1. The doublevalued nature of /" for a cooled surface (/ = 0) for PP near separation (fZ = 0) is evident. Generally, /" is more sensitive to variations in Pp for a hot surface. In fact, for cold wall conditions (Iw = 0), the variation of f" with PP for PP > 0 is quite modest. Also, a cooled surface tends to retard separation that is, /" = 0 at a smaller value of Pp. 

Heat Transfer. The Stanton number over a rough surface behaves similarly to the skin friction coefficient at sufficiently high roughness Reynolds numbers k+, the Stanton number becomes independent of the free-stream velocity. At a given Re or Res, roughness causes an increase in local Stanton number over the smooth-plate value. These effects are shown in Fig. 6.48 for five values of the free-stream velocity. The geometry of the rough surface used in these experiments was the densest array of spheres of radius r as shown in Fig. 

The variation of the skin-friction coefficient, Cf = Xg/(0.5pUg ), for the plane wall jet is shown in Figure 6. There seems to be a fair agreement between the predicted values and the ones given by Myer s expression (Rajaratnam, 1976). The measured values of the current four runs exhibit a significant amount of scatter and are not shown here. 

Real-time wall shear stress is difficult to monitor precisely because it varies in space and time. MEMS sensors provide high spatial resolution to resolve variations in shear stress in a 3D bifurcation model for small-scale hemodynamics. The application of MEMS sensors with backside wire bonding G ig. fib) captured the spatial variations in shear stress in a 3D bifurcation model (Fig. 8). The measured skin friction coefficients at various positions correlated well with values derived from the exact Navier-Stokes solution of the flow within the bifurcation [13]. Therefore, the development of MEMS sensors has enabled the precise measurements of spatial variations in shear stress for small-scale hemodynamics otherwise difficult with conventional technologies such as computed tomography (CT scan), magnetic resonance imaging (MRI), ultrasound, and laser Doppler velocimetry. 

Micro- and Nanoscale Anemometry Implication for Biomedical Applications, Fig. 8 (a) An array of MEMS sensors embedded in a 3D bifurcation model, (b) Computational fluid dynamics (CFD) solutions for skin friction coefficient (Cf) at a Reynolds number of 6.7. Cf represents local wall shear stress values normalized by the... 

The proportion of the total torque due to the radial surfaces is small, typically around 15%, but the estimate of the total torque is still sensitive to large errors in the estimate of the radial surface torque. Figure 7 shows a recalculated prediction assuming that the local skin friction coefficient for the radial surfaces was double the value for the cylindrical surfaces. It can be seen that the prediction differs significantly from the experimental data. 

A mean value of skin friction coefficient should be applied to both the radial and the cylindrical surfaces. The value suggested is = 3.16. 10" Re where Re is either a function of R for the cylindrical surface, or a variable function of r for the radial surface. These numerical values are mean values of those determined for the different gap ratios and have been tested and found to give good correlation. 

Micro- and Nanoscale Anemometry Implication for Biomedical AppiicaUens, Rgure 8 (a) An array of MEMS sensors embedded in a 3D bifurcation model, (b) Computational fluid dynamics (CFD) solutions for skin friction coefficient (Cf) at a R nolds number of 6.7. Cj represents local wall shear stress values normalized by the upstream dynamic pressure. Cj values are shown along the interior surface of bifurcation, (c) Comparison of the CFD On blue), experimental On green), and theoretical On red) skin friction for the 180° edge. x/D/cos(12.5) is the x distance normalized to the diameter of the inlet pipe and parallel to the centerline of the outlet pipes... 

The streamwise development of the skin-friction coefficient c/ is presented in Fig. 3(a). The c/-distribution for the pure RANS and the zonal RANS-LES results are in good agreement with the pure LES solution. Downstream of the LES inflow of the zonal RANS-LES the skin-friction coefficient does not drop but rather immediately converges to the pure LES values. The structures generated by the original inflow method of Jarrin et al. [10] would too strongly dissipate such that a much larger streamwise extent would be necessary for the LES to recover the correct c/-level. 

Table 2.1 contains numerical values of relevant observed skin friction coefficients, Cf. The size of the additional resistance for real and model surfaces has been reviewed by Montieth and Unsworth (1990) and is summarized as follows. For water vapor transfer to rigid rough surfaces the following expression applies ... 

Probe geometry and material influence the measured value of the friction coefficient because friction is a probe-skin interaction phenomenon. Few studies have examined probe effects El-Shimi2 studied probe roughness and Comaish and Bottoms3 probe roughness and material. 

Amonton s law, stipulating a constant value for the friction coefficient whatever the pressure applied to skin, is not verified for skin. 

Iwai and Suzuki [15] numerically investigated the effects of rarefaction and compressibility on heat transfer for a flow over a backward-facing step in a microchannel duct. They applied the velocity shp boundary condition to all the walls and considered tem-peratme jump at the heated wall. Skin friction was seen to reduce when the velocity shp was taken into account. It was further reduced if the accommodation coefficient takes smaller values, which results in larger slip velocities. They found that the compressibil-... 

On evaluation of the coefficients a, b, and c based on the van Driest values of skin friction in Fig. 6.11, it is found that they differ from those given in the earlier references. Therefore, the following formula is adopted ... 

For a given value of x and the same flow properties, the boundary layer on a cone (k = 1) is thinner by a factor of 1/V3, and the surface shear stress and heat transfer are larger by a factor of V3. The local skin friction and heat transfer coefficients are related similarly ... 

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