Skin friction drag

The skin friction coefficient is defined as:[2]

${\displaystyle c_{f}={\frac {\tau _{w}}{{\frac {1}{2}}\rho _{\infty }v_{\infty }^{2}}}}$

where:

• ${\displaystyle c_{f}}$ is the skin friction coefficient.
• ${\displaystyle {\rho _{\infty }}}$ is the density of the free stream (far from the body's surface).
• ${\displaystyle {v_{\infty }}}$ is the free stream speed, which is the velocity magnitude of the fluid in the free stream.
• ${\displaystyle {\tau _{w}}}$ is the skin shear stress on the surface.
• ${\displaystyle {{\frac {1}{2}}\rho _{\infty }v_{\infty }^{2}\equiv q_{\infty }}}$ is the dynamic pressure of the free stream.

The skin friction coefficient is a dimensionless skin shear stress which is nondimensionalized by the dynamic pressure of the free stream. The skin friction coefficient is defined at any point of a surface that is subjected to the free stream. It will vary at different positions. A fundamental fact in aerodynamics states that ${\displaystyle ({\tau _{w}})_{laminar}<({\tau _{w}})_{turbulent}}$.[3] This immediately implies that laminar skin friction drag is smaller than turbulent skin friction drag, for the same inflow.

The skin friction coefficient is a strong function of the Reynolds number ${\displaystyle Re}$, as ${\displaystyle Re}$ increases ${\displaystyle c_{f}}$ decreases.

${\displaystyle c_{f}={\frac {0.664}{\sqrt {\mathrm {Re} _{x}}}}\ }$

where:

• ${\displaystyle Re_{x}={\frac {\rho vx}{\mu }}}$, which is the Reynolds number.
• ${\displaystyle x}$ is the distance from the reference point at which a boundary layer starts to form.

The above relation derived from Blasius boundary layer, which assumes constant pressure throughout the boundary layer and a thin boundary layer.[4] The above relation shows that the skin friction coefficient decreases as the Reynolds number (${\displaystyle Re_{x}}$) increases.

CPM, suggested by Nitsche,[5] estimates the skin shear stress of transitional boundary layers by fitting the equation below to a velocity profile of a transitional boundary layer. ${\displaystyle K_{1}}$(Karman constant), and ${\displaystyle {\tau }_{w}}$(skin shear stress) are determined numerically during the fitting process.

${\displaystyle u^{+}=\int _{0}^{Y^{+}}{\frac {2(1+K_{3}y^{+})}{1+[1+4(K_{1}y^{+})^{2}(1+K_{3}y^{+})(1-exp(-y^{+}{\sqrt {1+K_{3}y^{+}}}/K_{2}))^{2}]^{0.5}}}\,dy^{+}}$

where:

• ${\displaystyle u^{+}={\frac {u}{u_{\tau }}},~u_{\tau }={\sqrt {\frac {{\tau }_{w}}{\rho }}},~y^{+}={\frac {u_{\tau }y}{\nu }}}$
• ${\displaystyle y}$ is a distance from the wall.
• ${\displaystyle u}$ is a speed of a flow at a given ${\displaystyle y}$.
• ${\displaystyle K_{1}}$ is the Karman constant, which is lower than 0.41, the value for turbulent boundary layers, in transitional boundary layers.
• ${\displaystyle K_{2}}$ is the Van Driest constant, which is set to 26 in both transitional and turbulent boundary layers.
• ${\displaystyle K_{3}}$ is a pressure parameter, which is equal to ${\displaystyle {\frac {\nu }{\rho }}{u_{\tau }}^{3}{\frac {dp}{dx}}}$ when ${\displaystyle p}$ is a pressure and ${\displaystyle x}$ is the coordinate along a surface where a boundary layer forms.

${\displaystyle c_{f}={\frac {0.027}{Re_{x}^{1/7}}}\ }$

The above equation, which is derived from Prandtl's one-seventh-power law,[6] provided a reasonable approximation of the drag coefficient of low-Reynolds-number turbulent boundary layers.[7] Compared to laminar flows, the skin friction coefficient of turbulent flows lowers more slowly as the Reynolds number increases.

A total skin friction drag force can be calculated by integrating skin shear stress on the surface of a body.

${\displaystyle F=\int \limits _{surface}c_{f}{\frac {\rho v^{2}}{2}}dA}$