Law-of-the-wall analytical formulations for Type-A turbulent boundary layers

Abstract

In-depth analyses of existing direct numerical simulations (DNS) data from various sources supported a logical and important classification of generic turbulent boundary layers (TBL), namely Type-A, -B and -C TBL, based on distribution patterns of time-averaged wall-shear stress. Among these types, Type-A TBL and its related law, as represented by the DNS data of turbulence on a zero-pressure-gradient semi-infinite flat-plate, was investigated in terms of analytical formulations of velocity independent on Reynolds ( Re ) number. With reference to the analysis from von Karman in developing the conventional law-of-the-wall, the current study first physically distinguished the time-averaged local scale used by von Karman from the ensemble-averaged scale defined in the paper, and then derived the governing equations with the Re -independency under the ensemble-averaged scales. Based on indicator function (IDF) and TBL thickness, the sublayer partitions were rigorously defined. The analytical formulations for entire TBL, namely the complete law-of-the-wall, were established, including the formula in inner, buffer, semi-logarithmic (semi-log) and wake layer. The researches were featured by introducing the general damping and enhancing functions (GDF and GEF) and applying these functions to both linear and logarithmic coordinates. These law formulations were proved uniform and consistent in time-averaged local and ensemble-averaged scales, which were validated by the existing DNS and experiment data. Based on the similarity of relevant properly-scaled governing equations, the law formulations were logically reasoned being applicable to the temperature in Type-A thermal TBL. The findings advance the current understandings of the conventional TBL theory and its well-known foundations of law-of-the-wall.

Fluid Dynamics Equations based on Constitutive Relation of Symmetric Shearing Deformation

“…4(a) and (b), the streamwise velocities at these Res are in the good agreements with each other. Moreover, the near-wall behaviors of the velocity profiles present a high fidelity to the law-of-the-all formulations published in [12], which include the law relations in inner, buffer and semi-log sublayers. Further, Fig.…”

“…Firstly, the DNS data of channel turbulence used in the current study are carefully benchmarked by the existing studies in [10,11] and the law-of-the-wall formulation obtained in [12]. These comparisons are presented in Fig.…”

Accurate Method for Estimating Wall-Friction based on Analytical Wall-Law Model

“…The movement of the camera and laser along the flow direction was controlled by the displacement platform with a controlling accuracy of 0.03mm. In the experiment, a force balance method was applied to measure the ensemble-averaged (spatial-temporal mean) WSS of fully-developed TBL, since the ensemble-averaged WSS was pin-pointed as an important parameter in the wall-law development [5,7] . In order to cleanly eliminate the effects of laminar and transition regions upstream the measuring locations, a flat plate at a shorter length of 1.255m was separately applied and therefore, via the subtraction of two measurements, the ensemble-averaged net wall friction can be obtained accurately for the WSS in a fully-developed turbulence section.…”

“…Within the context, searching for the turbulence friction walllaws has been one of the well-known efforts ever since Prandtl [1] developed the turbulent boundary layer (TBL) theory and von Karman [2] established the law-of-the-wall for the TBL on a zero-pressure-gradient flat-plate (ZPGFP). From then on, the numerous studies on TBL wall-laws, such as the works in [3][4][5][6] , have provided the profound understandings on the Reynolds number (Re) independent relations between the WSS and the velocity profiles, the TBL velocity similarity and even the extension of the law model to the temperature field in thermal turbulence [7] . These fundamental researches firmly demonstrate and prove that the accurate and detailed knowledges on WSS are important for characterizing a wall-bounded turbulence and is vital for successfully exploring more precise wall-law relations based on the ever expanding experimental and computational (mainly the Direct Numerical Simulation, DNS) data of complex-geometry wall turbulence.…”

“…From the authors' point of view, in order to improve the accuracy of the traditional wall-law model, two key issues have to be addressed and resolved so that more accurate law formulations can be developed, which include 1) the TBL sublayers have to be quantitatively well-defined and 2) the analytical wall-law models have to be constructed for the sublayers with not only the continuity of analytical function of velocity, but also its derivative particularly at the joint locations of sublayers, since the WSS calculation needs to accurately evaluate the velocity derivative. Within this context, Wang et al [5] successfully developed a wall-law model satisfying these requirements and strictly validated the model using DNS data. On the other hand, the experimental fluid mechanics has developed the quite matured particle image velocimetry (PIV) [25,26] method, which is nowadays popularly applied as a nonintrusive means to measure the velocity vector in both two-dimensional and three-dimension field.…”