Skin friction drag

Skin friction drag is a type of aerodynamic or hydrodynamic drag, which is resistant force exerted on an object moving in a fluid. Skin friction drag is caused by the viscosity of fluids and is developed from laminar drag to turbulent drag as a fluid moves on the surface of an object. Skin friction drag is generally expressed in terms of the Reynolds number, which is the ratio between inertial force and viscous force.

Total drag can be decomposed into a skin friction drag component and a pressure drag component, where pressure drag includes all other sources of drag including lift-induced drag.^[1] In this conceptualisation, lift-induced drag is an artificial abstraction, part of the horizontal component of the aerodynamic reaction force. Alternatively, total drag can be decomposed into a parasitic drag component and a lift-induced drag component, where parasitic drag is all components of drag except lift-induced drag. In this conceptualisation, skin friction drag is a component of parasitic drag.

Flow and effect on skin friction drag

Laminar flow over a body occurs when layers of the fluid move smoothly past each other in parallel lines. In nature, this kind of flow is rare. As the fluid flows over an object, it applies frictional forces to the surface of the object which works to impede forward movement of the object; the result is called skin friction drag. Skin friction drag is often the major component of parasitic drag on objects in a flow.

The flow over a body may begin as laminar. As a fluid flows over a surface shear stresses within the fluid slow additional fluid particles causing the boundary layer to grow in thickness. At some point along the flow direction, the flow becomes unstable and becomes turbulent. Turbulent flow has a fluctuating and irregular pattern of flow which is made obvious by the formation of vortices. While the turbulent layer grows, the laminar layer thickness decreases. This results in a thinner laminar boundary layer which, relative to laminar flow, depreciates the magnitude of friction force as fluid flows over the object.

Skin friction coefficient

Definition

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.

Laminar flow

Blasius solution

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

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.

Transitional flow

The Computational Preston Tube Method (CPM)

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.

Turbulent flow

Prandtl's one-seventh-power law

${\displaystyle c_{f}={\frac {0.0576}{Re_{x}^{1/5}}}\ }$

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.

Skin friction drag

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

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

Relationship between skin friction and heat transfer

In the point of view of engineering, calculating skin friction is useful in estimating not only total frictional drag exerted on an object but also convectional heat transfer rate on its surface.^[8] This relationship is well developed in the concept of Reynolds analogy, which links two dimensionless parameters: skin friction coefficient (Cf), which is a dimensionless frictional stress, and Nusselt number (Nu), which indicates the magnitude of convectional heat transfer. Turbine blades, for example, require the analysis of heat transfer in their design process since they are imposed in high temperature gas, which can damage them with the heat. Here, engineers calculate skin friction on the surface of turbine blades to predict heat transfer occurred through the surface.

Effects of skin friction drag

A 1974 NASA study found that for subsonic aircraft, skin friction drag is the largest component of drag, causing about 45% of the total drag. For supersonic and hypersonic aircraft, the figures are 35% and 25% respectively.^[9]

A 1992 NATO study found that for a typical civil transport aircraft, skin friction drag accounted for almost 48% of total drag, followed by induced drag at 37%.^[10][11]

Reducing skin friction drag

There are two main techniques for reducing skin friction drag: delaying the boundary layer transition, and modifying the turbulence structures in a turbulent boundary layer.^[12]

One method to modify the turbulence structures in a turbulent boundary layer is the use of riblets.^[13][14] Riblets are small grooves in the surface of the aircraft, aligned with the direction of flow.^[15] Tests on an Airbus A320 found riblets caused a drag reduction of almost 2%.^[13] Another method is the use of large eddy break-up (LEBU) devices.^[13] However, some research into LEBU devices has found a slight increase in drag.^[16]

See also

• Parasitic drag
• Pressure drag

References

1. Gowree, Erwin Ricky (20 May 2014). Influence of Attachment Line Flow on Form Drag (doctoral). p. 18. Retrieved 22 March 2022.
2. Anderson Jr., John D. (2011). Fundamentals of Aerodynamics (5th edition) Textbook. pp. 25–26.
3. Anderson Jr., John D. (2011). Fundamentals of Aerodynamics (5th edition) Textbook. p. 75.
4. White, Frank (2011). Fluid Mechanics. New York City, NY: McGraw-Hill. pp. 477–478. ISBN 9780071311212.
5. Nitsche, W.; Thünker, R.; Haberland, C. (1985). A computational Preston tube method. Turbulent shear flows, 4. pp. 261–276.
6. Prandtl, L. (1925). "Bericht uber Untersuchungen zur ausgebildeten Turbulenz". Zeitschrift für Angewandte Mathematik und Mechanik. 5 (2): 136–139. Bibcode:1925ZaMM....5..136P. doi:10.1002/zamm.19250050212.
7. White, Frank (2011). Fluid Mechanics. New York City, NY: McGraw-Hill. pp. 484–485. ISBN 9780071311212.
8. Incropera, Frank; Bergman, Theodore; Lavine, Adrienne (2013). Foundations of Heat Transfer. Hoboken, NJ: Wiley. pp. 402–404. ISBN 9780470646168.
9. Fischer, Michael C.; Ash, Robert L. (March 1974). "A general review of concepts for reducing skin friction, including recommendations for future studies. NASA Technical Memorandum TM X-2894" (PDF). Retrieved 22 March 2022. {{cite journal}}: Cite journal requires |journal= (help)
10. Robert, JP (March 1992). Cousteix, J (ed.). "Drag reduction: an industrial challenge". Special Course on Skin Friction Drag Reduction. AGARD Report 786. AGARD: 2-13.
11. Coustols, Eric (1996). Meier, GEA; Schnerr, GH (eds.). "Control of Turbulent Flows for Skin Friction Drag Reduction". Control of Flow Instabilities and Unsteady Flows: 156. ISBN 9783709126882. Retrieved 24 March 2022.
12. Duan, Lian; Choudhari, Meelan M. "Effects of Riblets on Skin Friction in High-Speed Turbulent Boundary Layers". Retrieved 22 March 2022. {{cite journal}}: Cite journal requires |journal= (help)
13. Viswanath, P. R (1 August 2002). "Aircraft viscous drag reduction using riblets". Progress in Aerospace Sciences. 38 (6): 571–600. Bibcode:2002PrAeS..38..571V. doi:10.1016/S0376-0421(02)00048-9. ISSN 0376-0421. Retrieved 22 March 2022.
14. Nieuwstadt, F. T. M.; Wolthers, W.; Leijdens, H.; Krishna Prasad, K.; Schwarz-van Manen, A. (1 June 1993). "The reduction of skin friction by riblets under the influence of an adverse pressure gradient". Experiments in Fluids. 15 (1): 17–26. Bibcode:1993ExFl...15...17N. doi:10.1007/BF00195591. ISSN 1432-1114. S2CID 122304080. Retrieved 22 March 2022.
15. García-mayoral, Ricardo; Jiménez, Javier (2011). "Drag reduction by riblets". Philosophical Transactions: Mathematical, Physical and Engineering Sciences. 369 (1940): 1412–1427. Bibcode:2011RSPTA.369.1412G. doi:10.1098/rsta.2010.0359. ISSN 1364-503X. JSTOR 41061598. PMID 21382822. S2CID 2785024. Retrieved 22 March 2022.
16. Alfredsson, P. Henrik; Örlü, Ramis (1 June 2018). "Large-Eddy BreakUp Devices – a 40 Years Perspective from a Stockholm Horizon". Flow, Turbulence and Combustion. 100 (4): 877–888. doi:10.1007/s10494-018-9908-4. ISSN 1573-1987. PMC 6044242. PMID 30069144.

Fundamentals of Flight by Richard Shepard Shevell