Formula & Calculator
Turbulent Flat Plate Skin Friction Coefficient
Empirical local skin-friction coefficient for turbulent boundary-layer flow over a flat plate.
Interpretation
Turbulent flat plate skin friction coefficient: C_f = 0.074 / Re_x^(1/5), for fully turbulent flow (Re > 5e5). Example: Re_x=1e7 → C_f = 0.074 / 10^(1.4) = 0.074/25.12 ≈ 0.00295.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| C_f | Skin friction coefficient | |
| Re_x | Local Reynolds number |
What it means
This empirical correlation gives the local skin friction coefficient for a turbulent boundary layer on a flat plate, derived from the 1/7‑power law velocity profile. It is valid for smooth surfaces and for Re_x up to about 10⁷. The coefficient decreases slowly with Re_x, reflecting the fuller velocity profile and higher mixing. This formula is used to estimate turbulent skin friction drag, which is often the dominant component for aircraft and ships. It is also used in performance calculations and in drag polar construction. Understanding this relation is important for aerodynamic drag estimation and for design optimisation.
Worked example
Turbulent Skin Friction – Two Examples
Real‑World| Parameter | Value |
|---|---|
| Re_x | 1×10⁶ |
| Parameter | Value |
|---|---|
| Re_x | 1×10⁷ |
Common mistakes
- Turbulent flat plate skin friction coefficient: C_f = 0.074 / Re_x^(1/5).
- Re_x: Local Reynolds number.
- Valid for turbulent flow over a smooth flat plate.
- Applicable for Re_x up to ~10⁷.
- Exponent is 1/5 (or sometimes 1/6 for other correlations).
Applications
The turbulent flat plate skin friction coefficient, C_f = 0.074/Re_x^(1/5) (approximate, often used for smooth turbulent flow), is used to estimate turbulent friction drag. Turbulent flow is typical for most aircraft, except for small laminar‑flow regions. Engineers use this to compute drag for cruise and climb conditions, and to evaluate the effect of surface roughness. The exponent 1/5 is based on the 1/7 power law for velocity profile. By using this correlation, aerospace engineers can quickly estimate turbulent friction drag and assess the impact of design changes on overall drag and performance.
- Drag estimation for commercial and military aircraft
- Performance analysis at high Reynolds numbers
- Surface roughness and wing design trade‑offs
- Wind tunnel correlation and flight test validation
- Preliminary design of turbulent boundary layer profiles
Frequently Asked Questions
It is an empirical correlation for the local skin‑friction coefficient of a turbulent boundary layer over a flat plate. It is used to estimate turbulent drag.
Cf = local skin‑friction coefficient
Rex = Reynolds number based on distance x
Turbulent mixing transfers momentum more effectively, resulting in higher velocity gradients near the wall and thus higher shear stress.
It is valid for smooth plates with Rex between about 5×10⁵ and 10⁷. For higher Re, more refined formulas (e.g., Schlichting or Prandtl’s 1/7 power) are better.
- Using the 1/5‑power turbulent formula for Reynolds numbers above ~10⁷, where more refined (Schlichting) formulas are needed.
- Applying it to rough surfaces without correction.
- Confusing local and average values.
At x = 1 m, Rex = 10⁷. Cf = 0.074 / (10⁷)^(1/5) = 0.074 / 10^(1.4) = 0.074 / 25.12 ≈ 0.00295.
It decreases as Rex^(−1/5), which is a slower decrease than laminar (Re−1/2), so turbulent drag is relatively higher at high Re.
Roughness increases the turbulent skin friction, especially at lower Reynolds numbers. The formula is for smooth surfaces.
The average Cf for a fully turbulent plate is given by 0.074/ReL^(1/5) for the entire length, assuming transition at the leading edge.
For a wider Re range, Schlichting proposed Cf = 0.455 / (log10 Rex)2.58, which is more accurate for high Re.