ANALYSIS OF PRESSURE-STRAIN CORRELATIONS IN A SUPERSONIC PIPE, NOZZLE AND DIFFUSER USING GREEN’S FUNCTIONS

Abstract

Pressure-strain correlations along with the turbulent dissipation rate are important terms that need to be modelled in second-order turbulence closures. In this paper, we provide insights into the pressure-strain correlations in a supersonic pipe, nozzle and diffuser by performing Green’s function analyses based on DNS and LES data. The relative importance of the rapid and slow parts of pressure-strain correlations for the axial, azimuthal and radial pressurestrain correlations is presented and it is demonstrated that properly performed LES replicates the trends found in DNS and may be used to develop models for these correlations. INTRODUCTION DNS studies of supersonic channel flows with isothermal walls (Colemanet al., 1995; Foysiet al., 2004) have revealed that compressibility effects manifest themselves as mean density and temperature variations in the near-wall region. This leads to a reduction of pressure-strain correlations at supersonic Mach numbers and in turn to an increase in Reynolds stress anisotropy (Foysi et al., 2004). These observations were also made in DNS of supersonic pipe flow with isothermal wall (Ghoshet al., 2010). Effects of mean dilatation and extra rates of strain add further complications to supersonic flows and lead to changes in the turbulence structure which cannot be explained only by mean property variations. Such effects were described by Bradshaw (1974) and observed in LES and DNS of canonical supersonic nozzle and diffuser flows where fully developed supersonic pipe flow serves as inflow (Ghosh et al., 2008; Ghosh & Friedrich, 2014). It was observed that the Reynolds stresses decrease dramatically in the nozzle and increase in the diffuser. The pressure-strain correlations were found to play a pivotal role in changing the Reynolds stresses in these flows. Hence, it is important to gain insight into the behaviour of pressure-strain correlations in these flows and an elegant way of doing this is a Green’s function analysis based on DNS data. Foysi et al. (2004) used Green’s function to analyse pressure-strain correlations using supersonic channel flow DNS data and found the contribution of the slow terms to be greater than that of the rapid terms. Ghoshet al. (2010) carried out a similar study in cylindrical coordinates with DNS data of a supersonic pipe flow with isothermal wall. Recently Ghosh & Friedrich (2014) extended the Green’s function analysis to a supersonic nozzle and diffuser with isothermal walls using DNS data. In this paper we analyse LES data of supersonic pipe, nozzle and diffuser flow and compare the results with those obtained with DNS. Such a Green’s function analysis with LES data will enable us to easily gain insight into flows for which only LES is possible. MATHEMATICAL AND COMPUTATIONAL DETAILS We use modified Bessel functions to construct the Green’s functions in cylindrical coordinates. The effect of axial non-periodicity in the nozzle and diffuser is taken care of by using a series expansion involving cosine functions. The procedure is detailed in Ghosh et al. (2010); Ghosh & Friedrich (2014) and is not repeated here due to lack of space. It is used here to analyse DNS and LES data of supersonic pipe, nozzle and diffuser flows. 5th order lowdissipation compact upwind schemes have been used in the DNS for the convective terms and 6th order central schemes for the molecular transport terms. The LES uses 6th order compact central schemes for all terms (Ghosh et al., 2008). The pipe flow has a centerline Mach number of 1 .5 and a friction Reynolds number of 245. 256 × 128× 91 points have been used for the DNS in the axial, spanwise and radial directions respectively where the domain size is 10R× 2πR×R. This pipe flow also acts as inflow to the nozzle simulation which is also similarly discretized and has a domain length of 10 R and a ratio of nozzle to pipe radius of 1.58 at the exit. The incoming pipe flow for the diffuser simulation has a friction Reynolds number of 300 and a centerline Mach number of 1 .8. The diffuser domain 1 June 30 July 3, 2015 Melbourne, Australia 9 5C-1