DNS of compressible pipe flow exiting into a coflow
Highlights
► DNS conducted of turbulent pipe flow exiting into a coflow. ► Varied Mach number and coflow velocity. ► Statistics near nozzle collapse using wall-shear stress and density from upstream. ► Asymptotic theory can be used to qualitatively predict near-nozzle behaviour. ► Self-similarity of jets only found for small coflow magnitudes.
Introduction
Jet noise research over several decades has shown that several different sources contribute to the overall sound radiation from subsonic jets. These sources are: (i) large scale structures mainly occurring close to the potential core region, (ii) the breakdown of large scale structures into fine-scale turbulence near the end of the potential core, (iii) fine-scale turbulence within the initial shear layers of fully turbulent jets, and (iv) trailing-edge noise resulting from the interaction between the flow and the solid wall at the nozzle exit.
The ultimate goal of this research is to perform direct noise computations, i.e. simulating directly both hydrodynamic and acoustic fields, that include all these sound sources. This requires the inclusion of the nozzle in the simulations, with a turbulent flow inside the nozzle. Furthermore, a large computational domain is necessary to capture the acoustic farfield. Due to limitations of available computational power, previous direct noise simulations of jets have not been able to include all possible noise generation mechanisms. In the only existing direct numerical simulation (DNS) of jet flow (Freund, 2001), only the free part of the jet was simulated and the acoustic field was then calculated using an acoustic analogy, thus entirely omitting the nozzle trailing-edge noise mechanism. Only in more recent work using large eddy simulation (LES) has the nozzle been included and the effect of perturbations on initially laminar shear layers has been investigated (e.g. Bogey and Bailly, 2010). These studies have in common the observation of considerable vortex-pairing noise being generated by the breakdown of the initially laminar shear layers and the sound originating from the turbulence in the nozzle shear layers is not represented. Therefore, for the current study, we require the flow exiting the nozzle to be fully turbulent. To simplify the problem we use a round pipe as a canonical nozzle, since a spatially developing pipe flow with sufficient streamwise length results in well defined turbulent upstream conditions. Spatially-developing pipe simulations in Sandberg et al. (2011) found that the structure of the flow changed considerably when approaching the nozzle exit and the focus of the current paper is to investigate this near-nozzle behaviour, in particular the effect of the nozzle lip on the turbulent flow.
At the trailing edge of the nozzle, classical boundary-layer theory breaks down because streamwise gradients occur on a length-scale similar to the boundary layer thickness. The asymptotic solutions provided in the literature (e.g. Stewartson, 1968) were derived for laminar boundary layers. In Sandberg and Sandham (2008), it was shown that the asymptotic scalings could produce results with reasonable accuracy for turbulent boundary layers when the eddy viscosity of the turbulence is accounted for. Here, the applicability of these results is evaluated for turbulent pipe flow exiting the nozzle. However, the behaviour of a turbulent flow in the vicinity of a round trailing edge at moderate Reynolds number is unclear and will be investigated in the present paper.
For round turbulent jets without coflow streamwise similarity of jets at various flow speeds was shown in Crow and Champagne (1971). However, bearing in mind the application of jets in flight conditions, a coflow with varying magnitude was specified for all DNS presented in the current work. Bradbury and Riley (1967) found for plane jets issuing into a coflow that the spread of jets with varying ratios of jet exit velocity to freestream velocity could be collapsed when accounting for the effective origins. In a study of coflowing round jets, Nickels and Perry (1996) showed that the excess velocity of the jet scales as the inverse of the streamwise coordinate. This behaviour was found upstream of the region where the asymptotic limit of the excess velocity being much smaller than the coflow magnitude is reached. All their cases with different coflow values collapsed if the streamwise coordinate was scaled with the momentum radius. More recently, Uddin and Pollard (2007) revisited the concept of a virtual origin in the context of round jets in coflow. It was found that the velocity distribution at the jet inlet could be used to predict the location of the virtual origin and that virtual-origin corrected data from jets with different jet velocity to coflow ratios could be collapsed. In the present contribution, we evaluate whether the above observations still hold for larger coflow magnitudes as considered in the current DNS study.
Section snippets
Numerical approach
The compressible Navier–Stokes equations for the conservative variables are solved in cylindrical coordinates using a newly developed finite-difference DNS code. For the spatial discretization in the radial and streamwise directions a 4th-order standard finite-difference scheme with Carpenter boundary stencils is applied. A spectral method using the FFTW3 library is used in the azimuthal direction, enabling an axis treatment that exploits parity conditions of individual Fourier modes. Time
Results
Four DNS were conducted with the flow parameters listed in Table 2. The simulations were run for at least 200 nondimensional time units (based on radius and bulk velocity inside the pipe) to allow the initial transients to leave the domain. All four cases were then continued for a further 600–800 time units to achieve statistical convergence.
In Fig. 1 (bottom), contours of the azimuthal vorticity component are shown for case M84-c2 to illustrate qualitatively the fully developed turbulent pipe
Conclusions
Direct numerical simulations of fully turbulent pipe flow exiting a nozzle and developing a turbulent jet were conducted. The jet Mach number and the coflow magnitude were varied at a constant target Reynolds number of Rejet = 7,500. It was shown that turbulence statistics at the nozzle exit could be collapsed with fully developed turbulent pipe flow profiles by using the wall shear-stress, and in the case of higher Mach number cases also the wall density, from the fully developed flow region
Acknowledgments
This project was partly supported by the Royal Academy of Engineering/EPSRC research fellowship (EP/E504035/1) and computing time from EPSRC Grant EP/G069581/1.
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