Evaluation of some proposed forms of Lagrangian velocity correlation coefficient K. Manomaiphiboon, A.G. Russell * School of Civil and Environmental Engineering, Georgia Institute of Technology, 311 Ferst Drive, Atlanta, GA 30332-0512, USA Received 21 November 2002; accepted 9 April 2003 Abstract This work evaluates four different forms of Lagrangian velocity correlation coefficient for stationary homogeneous turbulence at very large Reynolds numbers through consideration of simple mathematical and physical requirements. It is shown that some of them do not comply well with the requirements and may not be appropriate for use. Ó 2003 Elsevier Inc. All rights reserved. Keywords: Turbulence; Lagrangian velocity correlation coefficient; Inertial subrange theory 1. Introduction One of the most fundamental statistics of a turbulent flow is the Lagrangian velocity correlation coefficient (shortly, correlation coefficient). In stationary homoge- neous turbulence, its definition is given by R L ðsÞ¼ huðtÞuðt þ sÞi hu 2 i ; ð1Þ where R L is the correlation coefficient, s is the time lag, uðtÞ is the Lagrangian velocity of a fluid element at time t, and hi denotes an ensemble average (that is equiva- lent to a time average for stationary turbulence) of a quantity. The objective of this work is to evaluate four different forms of R L proposed in the literature for sta- tionary homogeneous turbulence at very large Reynolds numbers through consideration of essential mathemati- cal and physical requirements. The first is the classical exponential form given by Taylor (1921). This form has been discussed to a large extent in Tennekes (1979). It is included here for comparison. The others are two forms given by Frenkiel (1953) and a recent proposal of Al- tinsoy and Tu! grul (2002). Their expressions are given in the next section. It is important that a proper form of R L should comply with the following requirements: I: R L isevenaroundtheorigin s ¼ 0with jR L ðsÞj 6 1 ¼ R L ð0Þ. Also, it vanishes fast as jsj!1 such that its integral over s holds, i.e. lim jsj!1 R L ðsÞ¼ 0 and R 1 0 jR L ðsÞj ds < 1. II: R L is smooth over s. At the origin, dR L =ds ¼ 0 and d 2 R L =ds 2 < 0. III: As a result, the Lagrangian integral time scale T L , defined by T L ¼ Z 1 0 R L ðsÞ ds; ð2Þ is bounded or well defined. IV: In addition, let E L denote the Lagrangian turbulent energy spectrum. Mathematically, R L and E L can be expressed as the Fourier transform pairs: R L ðsÞ¼ 1 hu 2 i Z 1 0 E L ðxÞ cosðxsÞ dx; ð3Þ and E L ðxÞ¼ 2hu 2 i p Z 1 0 R L ðsÞ cosðxsÞ ds; ð4Þ where x is the turbulence frequency. The Fourier cosine transforms are used in the above relations due to the evenness of both R L and E L . According to the inertial subrange theory (K41) (Kolmogorov, 1941), E L can be expressed by E L ðxÞ¼ k" ex 2 ðor / x 2 Þ ð5Þ * Corresponding author. Tel.: +1-404-894-3079; fax: +1-404-894- 8266. E-mail address: trussell@themis.ce.gatech.edu (A.G. Russell). 0142-727X/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0142-727X(03)00065-1 International Journal of Heat and Fluid Flow 24 (2003) 709–712 www.elsevier.com/locate/ijhff