Non-equilibrium turbulence scalings and self-similarity in turbulent planar jets

Abstract

We study the self-similarity and dissipation scalings of a turbulent planar jet and the theoretically implied mean flow scalings. Unlike turbulent wakes where such studies have already been carried out (Dairay et al. 2015 J. Fluid Mech . 781 , 166–198. ( doi:10.1017/jfm.2015.493 ); Obligado et al. 2016 Phys. Rev. Fluids 1 , 044409. ( doi:10.1103/PhysRevFluids.1. 044409)), this is a boundary-free turbulent shear flow where the local Reynolds number increases with distance from inlet. The Townsend–George theory revised by (Dairay et al. 2015 J. Fluid Mech . 781 , 166–198. ( doi:10.1017/jfm.2015.493 )) is applied to turbulent planar jets. Only a few profiles need to be self-similar in this theory. The self-similarity of mean flow, turbulence dissipation, turbulent kinetic energy and Reynolds stress profiles is supported by our experimental results from 18 to at least 54 nozzle sizes, the furthermost location investigated in this work. Furthermore, the non-equilibrium dissipation scaling found in turbulent wakes, decaying grid-generated turbulence, various instances of periodic turbulence and turbulent boundary layers (Dairay et al. 2015 J. Fluid Mech . 781 , 166–198. ( doi:10.1017/jfm.2015.493 ); Vassilicos 2015 Annu. Rev. Fluid Mech . 95 , 114. ( doi:10.1146/annurev-fluid-010814-014637 ); Goto & Vassilicos 2015 Phys. Lett. A 3790 , 1144–1148. ( doi:10.1016/j.physleta.2015.02.025 ); Nedic et al. 2017 Phys. Rev. Fluids 2 , 032601. ( doi:10.1103/PhysRevFluids.2.032601 )) is also observed in the present turbulent planar jet and in the turbulent planar jet of (Antonia et al. 1980 Phys. Fluids 23 , 863055. ( doi:10.1063/1.863055 )). Given these observations, the theory implies new mean flow and jet width scalings which are found to be consistent with our data and the data of (Antonia et al. 1980 Phys. Fluids 23 , 863055. ( doi:10.1063/1.863055 )). In particular, it implies a hitherto unknown entrainment behaviour: the ratio of characteristic cross-stream to centreline streamwise mean flow velocities decays as the −1/3 power of streamwise distance in the region, where the non-equilibrium dissipation scaling holds.

Turbulence dissipation and the role of coherent structures in the near wake of a square prism

“…In the following section we study the dependence of C on Re λ in the near-field region x 1 /d 10 where the turbulence is demonstrably non-Gaussian and has energy spectra with close to −5/3 frequency scalings at x 1 /d 2. The turbulence dissipation scaling (1) has been reported in axisymmetric wakes in the range 10 x 1 /d O(100) [10,11] and planar jets in the range 20 [13], where d is the size of the wake generator in the case of the wake and the size of the nozzle exit in the case of the jet. The turbulence dissipation scaling is being investigated in the very near field x 1 /d 10 where the turbulence fluctuations are clearly documented to be very non-Gaussian.…”

“…(2), ε is the mean turbulent dissipation rate while L and U are, respectively, length and velocity scales associated with the largest turbulent eddies. The scaling (1) has been found in important regions of various turbulent flows which extend over a number of turnover times and where well-defined −5/3 energy spectra exist: grid-generated turbulence (both fractal or multiscale and regular grids) [3][4][5][6][7], turbulent boundary layers [8], axisymmetric wakes [9][10][11], round and planar jets [12,13], and periodic turbulence, both forced and decaying [14]. In some of these flows, specifically grid-generated turbulence and decaying periodic turbulence, C ε has been seen to become constant quite abruptly far enough downstream, but the direct numerical simulation (DNS) of [15] has shown that this constant C ε is not a reflection of Kolmogorov equilibrium (in relation to which a constant C ε is typically established) but of a balanced nonequilibrium.…”

The non-equilibrium part of the inertial range in decaying homogeneous turbulence

“…Both (δu) 2 and (δu) 3 increasingly deviate away from their Kolmogorov equilibrium prediction as r increases in the inertial range. The matched asymptotic expansions (5) and (6) imply that (δu) 2 and (δu) 3 tend towards C(εr) 2/3 and − 4 5 εr, respectively, as Re λ → ∞ for fixed r/λ but not for fixed r/L. For fixed r/L smaller that 1, they tend to (δu) 2 = C(εr) 2/3 [1 − A 2 (r/L) 2/3 ]…”

The impact of upstream turbulence on a plane jet

“…This nozzle geometry resulted in a planar jet with exit height h = 20 mm, and an aspect ratio of 30; the latter was shown by Deo et al (2007a) to be sufficient to produce approximately 2D flow. Plywood sidewalls were mounted on the two short sides of the planar jet in order to reduce spreading in this dimension, thus promoting 2-dimensionality (Deo et al 2007b, c;Cafiero and Vassilicos 2019). The sidewalls extended 2.4 m = 120h downstream of the jet nozzle.…”

“…The plane jet is a canonical flow of both historical and contemporary significance. It is a relatively simple flow with well understood initial and boundary conditions, allowing for the development of analytical expressions for its evolution that can be validated against experiments, c.f., George (1989); Pope (2000); Cafiero and Vassilicos (2019). It is also a simplification of many practical flows of engineering interest, e.g., air curtains (Maurel et al 2000), building ventilation (Törnström et al 2001), flow control actuators (Gutmark and Grinstein 1999).…”