Cross Correlation and Length Scales in Turbulent Flows near Surfaces

Abstract

Two kinds of length scales are used in turbulent flows; ‘functional length scales’ such as mixing length, dissipation length L ∈ , etc., and ‘flow-field length scales’ derived from cross correlations of velocity, pressure, etc. in the flow. Some connection between these scales are derived here. We first consider the cross correlation \( {\hat R_{\nu \nu }}\left( {y,{y_1}} \right)\) of the normal components u at two heights y, y 1 above a rigid surface, normalised by the velocity y 1 (> y). For shear-free boundary layers it is found theoretically, and in field and numerical experiments that \( {\hat R_{\nu \nu }}{\text{ }} \simeq {\text{ }}y/{y_1} \). For shear layers it is also found that \( {\hat R_{\nu \nu }}{\text{ }} \simeq {\text{ }}f\left( {y/{y_1}} \right) \leqslant y,{y_1} \). This function f differs slightly between low Reynolds number numerical simulations and field experiments. The lateral structure defined by \( {\hat R_{\nu \nu }}\left( {y,{r_3};{y_1},0} \right) \) is also self similar and shows that the eddies centred at about y 1 appear to have constant lateral width a 3 above and below y l, where a \( a_3^ + \simeq 7 + 1/\left( {1.4d{U^ + }/d{y^ + }} \right) \), when normalised on u * and v, where U is the mean velocity. Results for \( L_ \in ^{ - 1} \) from direct numerical simulation are found to compare well with the formula \( L_ \in ^{ - 1} = {A_B}/y + {A_S}dU/dy/\sqrt {{\nu ^2}} \), for unidirectional and reversing turbulent boundary layers and channel flow, except near where \( dU/dy \simeq 0 \). The conclusion is that the large-scale eddy structure and length scales in these flows are determined by a combination of shear and blocking, and that the vertical component of turbulence has a self-similar structure in both kinds of boundary layer.