Forcing frequency effects on turbulence dynamics in pulsatile pipe flow
Introduction
Pulsatile flows are encountered in a wide range of engineering applications and physical systems. Examples include biological flows, e.g. pulmonary ventilation and haemodynamics (Varghese, Frankel, Fischer, 2007, Xiao, Zhang, 2009, Huang, Yang, Lan, 2010), environmental flows, e.g. flow over ocean beds and sediment transport in coastal flows (Van Rijn, et al., 1990, Yan, 2011) and reciprocating flow in internal combustion engines (Semlitsch et al., 2014). Furthermore, the challenges associated with predicting unsteady turbulent flows using commercial computational fluid dynamics (CFD) software, e.g. computer codes that solve the Reynolds-averaged Navier-Stokes (RANS) have also been highlighted in a past review by Scotti and Piomelli (2002). Ultimately, improved modelling capabilities can only be achieved through improved physical understanding. Therefore, obtaining a first-principles understanding of the physics that governs pulsatile turbulent flows is of great practical interest. However, as noted in past work by Akhavan et al. (1991) and several others, the mean-squared effect of unsteadiness greatly complicates the statistical analysis of the instantaneous flow, relative to a traditional Reynolds-averaged approach, e.g. see Reynolds and Hussain (1972) for details. Hence, the fluid dynamic properties of pulsatile turbulent flows continue to be an active area of experimental and numerical research.
The instantaneous fluid motion in a pulsatile turbulent flow can be split into three separate parts: (i) a mean component; (ii) an oscillatory component and (iii) a turbulent fluctuation. In the absence of (i) and (iii), the flow reduces to an oscillatory laminar flow (with zero mass flux) which forms the basis for Stokes’ First and Second Problems (Stokes, 1851). Particularly relevant to this study are the past works of Womersley (1955) and Uchida (1956), who derived the theoretical solution for pulsatile laminar flow in a straight circular pipe. A basic theoretical outcome of these past studies is that the thickness of the oscillatory shear layer, ls, and the applied forcing frequency, ω, are related through the formula where ν is the kinematic viscosity of the fluid. The impact of varying the forcing frequency, and, hence, the forcing length-scale, upon pulsatile turbulent pipe flows has been considered in several past experimental (Ramaprian, Tu, 1980, Tu, Ramaprian, 1983, Shemer, Kit, 1984, Shemer, Wygnanski, Kit, 1985, Mao, Hanratty, 1986, Brereton, Reynolds, Jayaraman, 1990, Brereton, Reynolds, 1991, Brereton, Hwang, 1994, Lodahl, Sumer, Fredsøe, 1998, He, Jackson, 2009) and computational (Manna, Vacca, 2008, Manna, Vacca, Verzicco, 2012, Papadopoulos, Vouros, 2016) studies. In addition, the impact of unsteadiness upon turbulent channel flow has been examined in detail by Tardu et al. (1994), Binder et al. (1995), Scotti and Piomelli (2001) and Weng et al. (2016) among others. Some of these studies are discussed in further detail below.
Tu and Ramaprian (1983) conducted experiments in a fully-developed turbulent pipe flow, focusing on how pulsation affects the time-averaged turbulence intensity. Their work considered two separate forcing frequencies: one was comparable to the bursting frequency of near-wall cycle, whereas the other was far lower. Considerable differences between the mean turbulence intensity acquired under pulsatile and non-pulsatile conditions were observed for low frequency forcing, while negligible differences were observed for the higher forcing frequency. A later experimental study by Brereton and Hwang (1994) investigated how the instantaneous turbulence structures responded to pulsation by computing phase-averaged streak spacing over a range of forcing frequencies, ultimately yielding a scaling law based using rapid-distortion theory. He and Jackson (2009) experimentally studied phase-averaged turbulence intensities in a pulsatile turbulent pipe flow, noting that turbulence structure in the core region was not affected by the higher frequency forcing conditions, which they referred to as the “frozen” state. Manna et al. (2012) used direct numerical simulation (DNS) to investigate the near-wall region of pulsatile turbulent pipe flow, focusing on penetration depth of disturbances from the wall in the context of time- and space-averaged statistics of the first- and second-order moments, including vorticity fluctuations and Reynolds stress budgets. More recently, Papadopoulos and Vouros (2016) conducted a DNS study covering a wide range of high and very high frequencies. Following an analysis of the mean velocity and turbulence intensities, they also noted that phase-averaged turbulent quantities exhibit independence with respect to pulsation once the forcing frequency exceeds a certain level. In addition, Papadopoulos and Vouros (2016) also noted that the upper limit of high frequency forcing remains somewhat ambiguous.
In addition to studying how pulsation affects turbulence dynamics and flow structures, past studies have proposed several classification schemes for forcing conditions encountered in pulsatile turbulent flows. Early classifications were based on the relative magnitude of the applied forcing frequency and the bursting frequency measured in the near-wall region under non-pulsatile conditions. Detailed descriptions of the bursting process and associated flow events can be found in the work of Kline et al. (1967) and later studies by Corino and Brodkey (1969) and Bogard and Tiederman (1986). One of the first classifications was discussed by Mizushina et al. (1974), where the forcing conditions in pulsatile turbulent pipe flow were categorised based on the relative magnitude between the applied forcing frequency and the maximum value of the bursting period. Later work by Ramaprian and Tu (1983) introduced a classification scheme based on a turbulent Stokes number, representing the ratio of the pipe radius to the turbulent diffusion in one oscillation period. Additional classifications based on variants of the bursting frequency (Tardu and Binder, 1993) and the so-called turbulent Stokes layer thickness — which accounts for the summed effect of molecular and eddy diffusivities (Scotti and Piomelli, 2001) — have also been put proposed in past research.
As was previously mentioned, a recent DNS study by Papadopoulos and Vouros (2016) noted that the upper limit of high frequency forcing in a pulsatile turbulent pipe flow remains somewhat ambiguous. In that work, attention was directed towards the phase-averaged response of single-point turbulence statistics, namely, profiles of mean velocity and axial turbulence intensity, across a systematic range of forcing frequencies. The current study extends the past work of Papadopoulos and Vouros (2016) by performing DNS of pulsatile turbulent pipe flow across a wide range of forcing frequencies — including high, very-high and a new “ultra-high” regime. In addition, whilst the majority of past work regarding pulsatile turbulent pipe flow studies have focussed on single-point data, we extend our analysis to two-point statistics in both the frequency, wave-number and physical domains, with a particular emphasis on the Reynolds shear stress (RSS). Finally, this work introduces a new classification procedure to categorise forcing frequency conditions in pulsatile turbulent pipe flow based on the relative magnitude of the applied forcing frequency and the frequency co-spectra of the instantaneous RSS fluctuations.
This document is organised into four sections. The computational aspects of this study are described in Section 2, which includes details of the numerical method, averaging procedures and the validation tests. A list of the simulations considered in this study is also provided in Section 2, along with the classification of each case using the RSS co-spectra approach. Section 3 contains the key results of this study, where a detailed analysis of phase-averaged single- and two-point RSS statistics is performed. Finally, in Section 4, the conclusions of this work are given and recommendations for future research are made.
Section snippets
Computational details
This section describes the computational details and is divided into three parts. First, the governing equations and details of the simulation setup are provided. Second, the statistical averaging procedures are described and a validation of the current numerical algorithm is presented. Finally, a classification of the forcing conditions is given and a list of the cases considered in this study is provided.
Results
This section includes the key results of this work and is divided into four parts. First, the influence of pulsation upon first-order velocity statistics is presented in the context of phase-averaged and oscillatory velocity profiles. Second, the phase-averaged response of the RSS and its frequency spectra are analysed. Third, a quadrant analysis of the phase-averaged RSS is performed in the context of weighted joint probability density function along with an inspection of the instantaneous RSS
Summary & conclusions
DNS of pulsatile turbulent pipe flow were performed at a mean friction Reynolds number of 180 across a range of forcing frequencies. In order to classify the forcing conditions considered in this work, a categorisation procedure based on the applied forcing frequency and the frequency co-spectra of RSS was devised (Fig. 6). The impact of unsteady forcing upon the turbulence dynamics was investigated in the context of phase-average velocity profiles (Fig. 10), oscillatory velocity profiles (
CRediT authorship contribution statement
Z. Cheng: Conceptualization, Software, Validation, Formal analysis, Investigation, Resources, Writing - original draft, Writing - review & editing. T.O. Jelly: Conceptualization, Writing - original draft, Writing - review & editing, Supervision. S.J. Illingworth: Supervision. I. Marusic: Supervision. A.S.H. Ooi: Supervision, Funding acquisition.
Declaration of Competing Interest
We wish to confirm that there are no known conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its outcome. We confirm that the manuscript has been read and approved by all named authors and that there are no other persons who satisfied the criteria for authorship but are not listed. We further confirm that the order of authors listed in the manuscript has been approved by all of us. We confirm that
Acknowledgements
The authors gratefully acknowledge the support of the Australian Research Council. This research was undertaken using the LIEF HPC-GPGPU Facility hosted at the University of Melbourne. This Facility was established with the assistance of LIEF grant LE170100200. This work was also supported by resources provided by the Pawsey Supercomputing Centre with funding from the Australian Government and the Government of Western Australia.
References (50)
- et al.
Highly energy-conservative finite difference method for the cylindrical coordinate system
J. Comput Phys
(2002) - et al.
An experimental study of pulsating turbulent flow in a pipe
Eur. J. Mech. B/Fluids
(2009) - et al.
Spectral dynamic of pulsating turbulent pipe flow
Comput Fluids
(2008) - et al.
High Reynolds number effects in wall turbulence
Int. J. Heat Fluid Flow
(2010) - et al.
Pulsating turbulent pipe flow in the current dominated regime at high and very-high frequencies
Int. J. Heat Fluid Flow
(2016) - et al.
Flow effects due to pulsation in an internal combustion engine exhaust port
Energy Convers. Manag.
(2014) - et al.
Response of the streaks, the active and passive eddies in an unsteady channel flow
Int. J. Heat Fluid Flow
(2004) Analysis of laminar to turbulent transition of pulsating flow in ocean environment with energy gradient method
Ann. nucl energy
(2011)- et al.
An investigation of transition to turbulence in bounded oscillatory stokes flows Part 1. Experiments
J. Fluid Mech
(1991) - et al.
Cyclic modulation of reynolds stresses and length scales in pulsed turbulent channel flow
Proceed. Roy. Soc. Lond. Ser. A
(1995)