Skin-friction coefficient

Momentum boundary layer calculations are useful to estimate the skin friction on a number of objects, such as on a ship hull, airplane fuselage and wings, a water surface, and a terrestrial surface. Once we know the boundary layer thickness, occurring where the velocity is 99% of the free-stream velocity, skin friction coefficient and the skin friction drag on the solid surface can be calculated. Estimate the laminar boundary layer thickness of a 1-m-long, thin flat plate moving through a calm atmosphere at 20 m/s. 

Table E4.4.1 Boundary layer thickness and skin friction coefficient for aflat plate moving at 20 m/s in the atmosphere...

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. 

Nacht, S., Close, J., Yeung, D., and Gans, E.H., Skin friction coefficient changes induced by skin hydration and emollient application and correlation with perceived skin feel, J. Soc. Cosmet. Chem., 1981 32 55-65. 

Schlichting [1] has surveyed experimental measurements of friction coefficients for turbulent flow on flat plates. We present the results of that survey so that they may be employed in the calculation of turbulent heat transfer with the fluid-friction-heat-transfer analogy. The local skin-friction coefficient is given by... 

The form and skin friction coefficients are explained in books on Fluid Flow or Transport Phenomena (for example W. J. Seek et al. Transport Phenomena, Wiley 1999). We derived that for skin friction using data for rough tubes. 

Then the average local skin friction coefficient becomes... 

The Chilton-Colburn analogy has been obserx ed to hold quite well in laminar or turbulent flow over plane surfaces. But this is not always the case for internal flow and flow over irregular geometries, and in such cases specific relations developed should be used. When dealing with flow over blunt bodies, it is important to note that/in these relations is the skin friction coefficient, not the total drag coefficient, which also includes tlie pressure drag. 

The skin friction coefficient is calculated from the log-law relation in an implicit iteration loop... 

Note relationships exist for both laminar and turbulent flow defining the skin friction coefficient Cf as a function of Reynolds number (Re). 

The skin friction coefficient varies with Re, the Mach number, M, (when it involves incompressible fluids of local speeds less than the sonic speed, 1), and the character of the boundary layer. The momentum transferred between the air and the body surface appears as a velocity deficit in the viscous wake behind the body. 

The plot below shows how Re and the location of the transition from the laminar flow to the turbulent flow affect the skin friction coefficient. 

The Reynolds analogy, defined as the ratio of the Stanton number to the local skin friction coefficient St/(c//2) is a function of the Prandtl number and is extremely useful for estimating heat transfer. Pressure drop can be used to predict heat transfer in pipes, and the skin friction can be used to predict Stanton number for external flows. 

The average skin friction coefficient for the length of the plate up to x is defined as... 

Equation 6.17 indicates that on a flat plate the average skin friction coefficient is equal to twice the local skin friction coefficient at the trailing edge. 

The term (cfl2), represents the skin friction coefficient corresponding to a constant-property boundary layer at the same local length Reynolds number. Similarly, the Stanton number is given by... 

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 6.10 Local laminar boundary layer skin friction coefficient on a flat plate at uniform temperature, Pr = 0.725 and 0o = 3 [5]. 

FIGURE 6.11 Local skin friction coefficient for air flowing in a laminar boundary layer on a flat plate, Pr = 0.75, 0O = 0.505 [11],... 

Reference Enthalpy Method. The behavior of the skin friction coefficient indicated in Figs. 6.10 and 6.11 can be correlated to a very good approximation by the modified incompressible formula... 

A convenient form of the skin friction coefficient compatible with the Crocco and van Driest formulations is... 

TABLE 6.2 Comparison of Skin Friction Coefficients From Reference Enthalpy Methods... 

From Table 6.2 for air, it can be seen that Eq. 6.62 gives some improvement in comparison with the van Driest results over the older formulas. At the high speeds where air behaves as a real gas, Wilson [15] shows that equations equivalent to Eqs. 6.61 and 6.62 yield skin friction coefficients that agree with those found from numerical integrations of the boundary layer equations to within 5 percent for total enthalpies corresponding to free-stream speeds up to 25,000 ft/s (7620 m/s). Similar close agreement is achieved between the use of the Eckert reference enthalpy and results of Cohen [16]. 

The skin friction coefficient and Stanton number are defined under the conditions of similarity on axisymmetric and two-dimensional bodies as follows. The components of the shear stress in the xt direction are given by... 

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. 

Hence, the ordinate in Fig. 6.22 can also be used in conjunction with Eq. 6.107 or 6.109 to calculate the cross flow skin friction coefficient for cases of very small yaw angles (ts =1). Note that Iaw is equal to unity because the solution of Eq. 6.102 with Pr = 1 and an insulated surface is / = 1. Although the trends exhibited in Figs. 6.21 and 6.22 are generally similar, it must be cautioned that such large variations in the Reynolds analogy factor occur that the latter is no longer a useful concept. The heat transfer parameter for a cooled surface shows a rather small variation with Pp for Pp > Vi, a fact first utilized in Ref. 44 to obtain relatively simple expressions for the local heat flux to blunt bodies in hypersonic flow. 

Solutions of the momentum equation (Eq. 6.117) [45] yield velocity distributions generally similar to those of Fig. 6.19, and the skin friction parameter/" shown by the line labeled 1 in Fig. 6.21. The skin friction coefficient is given by... 

Given the momentum thickness at some location on a body with streamwise pressure gradients, the evaluation of the local skin friction coefficient from Eq. 6.157 requires knowledge of the shape factor H. Alternatively, the same input information can be used to evaluate both the skin friction coefficient and the shape factor by solving Eqs. 6.152-6.156 iteratively. 

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