A unified expression for the spectrum of turbulence is developed by asymptotically matching known expressions for small and large wave numbers, and a formula for the one-dimensional spectral function which depends on the turbulence Reynolds number Reλ is provided. In addition, formulas relating all the length scales of turbulence are provided. These relations also depend on Reynolds number.
The effects of free-stream turbulence on laminar heat transfer and pre-transitional flow in gas turbines are re-examined in light of these new expressions using our recent thoughts on an ‘effective’ frequency of turbulence and an ‘effective’ turbulence level. The results of this are that the frequency most effective for laminar heat transfer is about 1.3U/Le, where U is the free-stream velocity and Le is the length scale of the eddies containing the most turbulent energy, and the most effective frequency for producing pre-transitional boundary layer fluctuations is about 0.3U/η where η is Kolmogorov’s length scale. In addition, the role of turbulence Reynolds number on stagnation heat transfer and transition is discussed, and new expressions to account for its effect are provided.