Amplitude Modulation of Pressure in Turbulent Boundary Layer

The interaction between large and small scale motions from the point of pressure fluctuation is studied. Using the small pressure probe, both the static pressure and wall pressure fluctuations were measured inside the zeropressure gradient boundary layer at relatively high Reynolds numbers. How the large scales in outside affect the small scales near wall is analyzed by means of statistical method. High amplitude positive and negative pressure fluctuations are also analyzed which associate with coherent motions inside the boundary layer. Another interesting aspect is the amplitude modulations of pressure and this topic is reported in this paper. INTRODUCTION We have developed a small pressure probe and measured both static and wall pressure simultaneously in turbulent boundary layers up to Reynolds numbers based on the momentum thickness 21000. The statistical features were already reported in the previous studies [1,2]. Here, in this paper, we investigate the instantaneous feature of turbulence character, especially the large scale and small scale interaction of pressure fluctuations. In Fig.1, we plot the pressure intensity profile, which shows the logarithmic relation; p p rms B y A p     ) / log( ) ( 2  , (1) where p A and p B are constant 52 2. Ap  , 30 2. Bp  , but p B depends on flow field. Here, y is a distance from the wall,  is a boundary layer thickness and subscript + indicates the normalization by inner variables. This relation is similar with that observed in the intensity of stream-wise velocity component predicted by attached eddy model [3,4]; u u rms B y A u     ) / log( ) ( 2  , (2) where u A is constant 25 . 1  u A but u B depends on flow field. This velocity logarithmic relation was derived by Perry et al. [4] based on the Townsend’s attached eddy hypothesis. The basic Townsend’s idea said [3],“It is difficult to imagine how the presence of the wall could impose a dissipation length-scale proportional to distance from it unless the main eddies of the flow have diameters proportional to distance of their “centres” from the wall, because their motion is directly influenced by its presence. In other words, the velocity fields of the main eddies, regarded as persistent, organized flow patterns, extend to the wall and, in a sense, they are attached to the wall.” This idea was extended by Perry et al [4] that the distribution of eddies with a population density inversely proportional to distance from the wall. And in their model, they propose that 2 ) (  rms u is in proportion to the logarithmic of distance from the wall. From the recent high-Reynolds number experiments, it was found that large scale motions in overlap region have significant interaction with the motions close to the wall [5]. A series of researches by Melbourne university group have reported the detailed properties of this interaction. This process is conceptually expressed as “footprint”. We generally believe the attached eddy hypothesis relates with this footprint. The large-scale and small-scale interactions, or footprint, are characterized by statistical methods, such as space-time correlation, conditional sampling, joint-probability density functions etc. Among these, one possibility is the amplitude modulation method [6]. Although there are some discussions about the ability of this method [7], we apply this to the pressure fluctuation and discuss the similarity and difference with that of velocity fluctuations. Fig.1 Square of pressure intensity normalized by inner variables at distance  / y from the wall. The spatial resolutions of probes are corrected by PDF shapes. : 11260 Re   , : 16190 Re   ,  : 20940 Re   . Solid line is direct numerical simulation (DNS) data from Schlatter et al. (2010) at 4060 Re   . June 30 July 3, 2015 Melbourne, Australia 9 3A-3