Swirling turbulent pipe flows: Inertial region and velocity–vorticity correlations
Introduction
Turbulent swirling flows have been studied in a variety of contexts due to its application in combustion chambers, industrial rotating flows, turbomachinery, transport of particles in swirling pipe flows, and others (Orlandi and Fatica, 1997, Eggels, 1994, Leclaire and Jacquin, 2012, Facciolo et al., 2007, Kitoh, 1991, Pierce and Moin, 1998, Zonta and Soldati, 2013). Swirling flow in a pipe is the canonical case representing these varied applications. The present work is partly motivated by a fundamental issue of the location of the inertial region in these flows. The inertial region corresponds to the region where viscous forces play a less important role in the flow dynamics, and the dominant balance is between the turbulent inertia and pressure forces.
In a non-swirling pipe flow, the different regions in the flow are typically divided into a near-wall, overlap (or logarithmic) and outer regions (e.g., Tennekes and Lumley, 1972). These regions are essentially identified by observing the mean streamwise velocity profiles (rather than based on any physical argument or equations), and therefore considerable uncertainty exists in locating where one region starts and ends. For example, based on this subdivision of flow regions, there is no consensus on the wall-normal location where the logarithmic (also synonymously called the inertial) region begins. More recently, however, starting from the Reynolds Averaged Navier–Stokes (RANS) equations there has been considerable progress made in defining different regions of wall-bounded turbulent flows (e.g., Klewicki, 2010). Specifically, it is now well established that in non-swirling canonical flows (such as in turbulent flat plate boundary layers, pipes and channels), the wall-normal position where inertial region begins, or essentially the start of the logarithmic behaviour in the mean streamwise velocity () profile, is located at (e.g., Klewicki, 2010, Marusic et al., 2013, Morrill-Winter and Klewicki, 2017). Here, the friction Reynolds number , where and are respectively the fluid kinematic viscosity, friction velocity (based on the wall-normal gradient of at the wall, ), whereas is the boundary layer thickness or pipe radius or channel half-width. Also, here we distinguish the streamwise friction velocity from the spanwise friction velocity , because in swirling pipe flows (which is the focus here) there is a non-zero azimuthal mean velocity component . The velocity in the radial direction is denoted as .
A recent analysis of swirling pipe flow results shows that for a fixed and for increasing swirl strength (to be defined below), the inertial or the log-region increases in extent (Chin and Philip, 2019). However, the effect of increasing on the start of the log-region in swirling pipe flows is still unknown, and this is one of the issues that we will consider in this paper.
Typically, the axial Reynolds-averaged mean momentum equation is employed to define the start of the inertial or the log-region. And this starting position is identified as the location where viscous force becomes relatively small compared to the other two forces, namely the pressure-gradient force and turbulent inertia (e.g., Klewicki, 2010, Wei et al., 2005). In the swirling flow, however, one has to consider simultaneously the mean axial as well as the azimuthal momentum equation, and ensure that the viscous forces in both the equations are sub-dominant to define the start of the inertial region (Chin and Philip, 2019). Pressure-gradient force becomes a constant once we fix and swirl strength, and it is the third term, i.e., the turbulent inertia — rising from the dynamics of the motion — that is the main contributor in the mean equations. In fact, the turbulent inertia term embodies in itself the effect of turbulence manifested in the force balance. In the streamwise (or the axial direction in a pipe), physically, the turbulent inertia represents the flux of streamwise turbulent momentum carried by the wall-normal turbulent fluctuations. Turbulent inertia is also the ‘unclosed’ term in the modelling of turbulence. Hence, there has been a consistent effort to accurately measure this term in experiments at higher Reynolds number (e.g., some recent attempts are Fernholz and Finley, 1996, Philip et al., 2013, Zimmerman et al., 2017, Baidya et al., 2019, Baidya et al., 2019), although at lower Reynolds numbers, Direct Numerical Simulation (DNS) of the Navier–Stokes and continuity equations has been more informative (e.g., Hoyas and Jiménez, 2006, Schlatter and Örlü, 2010, Lee and Moser, 2015). A fruitful approach in uncovering the underlying physics of the turbulent inertia (TI) has been to decompose the TI term into velocity–vorticity correlations (e.g., Tennekes and Lumley, 1972, Klewicki and Hirschi, 2004, Chin et al., 2014, Brown et al., 2015). This allows understanding the turbulence inertia as a force that is governed by two physically distinct mechanisms; one, the transport, and two, the stretching/re-orientation of the underlying vorticity field. Although there are differences between the velocity–vorticity correlations for the different wall-turbulent flows, all the studies (that we know of) are limited to non-swirling streamwise-aligned flows.
In the context of swirling flows, there is another turbulent inertia term corresponding to the azimuthal momentum equation. To our knowledge, a decomposition of the azimuthal turbulent inertia term into velocity–vorticity correlations has not been investigated before. As such, another aim of this paper is to present such a decomposition and to study its characteristics.
Here we would employ DNS of the Navier–Stokes and continuity equations to study the swirling flow in a pipe. The governing equations and the velocity–vorticity correlations will be presented in §2, whereas the computational methodology will be elaborate in §3. We would present four pipe flow cases, with two each at fixed and swirl strengths. We will first present the basic flow features and the mean and Reynolds stress statistics in §4, and then focus on the inertial region (in §5). Subsequently, we study the velocity–vorticity correlations (in §6) before the final conclusion is presented in §7.
At this point we should mention that typically swirling flow in pipes are studied by either rotating the pipe walls (e.g., Orlandi and Fatica, 1997, Eggels, 1994, Leclaire and Jacquin, 2012, Facciolo et al., 2007) or by keeping the wall fixed while imposing an azimuthal flow by swirl generators or radial jets (e.g., Kitoh, 1991, Pierce and Moin, 1998, Zonta and Soldati, 2013). We follow the latter methodology, primarily due to its direct application in the swirling particle transport industry that originally motivated our study. In particular, to simplify the problem we impose an azimuthal as well as an axial body force (equivalent to a mean pressure gradient in the corresponding directions) with stationary pipe wall to generate a swirling pipe flow (Pierce and Moin, 1998, Chin and Philip, 2019). Some relevant details of the numerical procedure are presented in §3, and a fuller description is presented elsewhere (Chin and Philip, 2019).
Section snippets
Governing equations and some pertinent observations
The governing equations are continuity and the incompressible Navier–Stokes equations in a cylindrical co-ordinate system for a pipe, where we employ () as the axial, radial and azimuthal directions. Here we will first focus on the mean equations and then on the velocity–vorticity correlations.
Numerical method and flow parameters
The numerical scheme employed for the swirling pipe flow simulations is that described by Blackburn and Sherwin (2004) and employed previously by us in a swirling flow simulation (Chin and Philip, 2019). It is based on a cylindrical-coordinate spectral element/Fourier spatial discretisation. The meridional semi-plane is a 11th order Gauss–Lobatto–Legendre nodal-based expansion in each element and a Fourier discretisation in the azimuthal direction. The length of the pipe is with a
Flow features and basic statistics
Here we will first present some flow visualisation results before presenting the mean velocity and Reynolds stress statistics.
Inertial region
Inertial region, as the name suggests, is dominated by the turbulent inertia (TI) force. Since there are three terms in both the x and momentum equations, respectively, (5), (10), the TI term is balanced by the pressure gradient (PG) term. Therefore, the viscous term is subdominant in the inertial region. The y-location where the viscous force (VF) term become smaller than the TI and PG term is considered as the start of the inertial region. Now, since the VF term (although in a different
Velocity–vorticity correlations
Before presenting results of velocity–vorticity correlations, we show data that will test the approximations we derived for the ‘components’ of these correlation. These velocity-velocity correlations and derivative, which are parts of the turbulent inertia term are for distances ‘far’ from the wall, or within the inertial region. The four relations were derived in section §2.1, and presented in (6) and (11) related to x- and -momentum equations respectively. Fig. 8(a) and (b) show the two
Conclusion
We performed direct numerical simulations of swirling pipe flows at low and moderate Reynolds numbers and compared them to non-swirling pipe flows. The introduction of the swirl promotes self-organisation of the flow in the near-wall region into high and low momentum zones. Swirling the flow increases drag as evident by a lower mean axial velocity profile. The azimuthal mean velocity profiles appear to scale well with the azimuthal friction velocity. There is also an increased turbulence
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
This work was supported with supercomputing resources provided by the Phoenix HPC service at the University of Adelaide. The authors acknowledge the financial support of the Australian Research Council.
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