Effective forcing for direct numerical simulations of the shear layer of turbulent free shear flows

A numerically efficient configuration to simulate turbulent flows is to use triply periodic domains, with numerical forcing techniques to sustain turbulence. Previous homogeneous shear turbulence simulations considered only idealized homogeneous shear flows and not the statistically stationary shear turbulence observed in practical free shear flows. In contrast, the current study mathematically derives the complete forcing technique from the large scales of the turbulent free shear flows. Different statistically stationary free shear flows are considered in this study, namely, a nearly homogeneous shear turbulent flow, turbulent mixing layer, a turbulent planar jet, and a turbulent round jet. The simulations are performed on triply periodic, statistically homogeneous cubic domains in the vicinity of the shear layer in the self-similar region. An a priori analysis is performed to calculate the effects of the different forcing terms and to predict the expected turbulence quantities. The forcing technique is then used to perform direct numerical simulations at different Reynolds numbers. Numerical results for the different cases are discussed and compared with results from experiments and other simulations of free shear turbulent flows. Anisotropy is observed both in the components of velocity and vorticity, with stronger Reynolds number dependence in the anisotropy of vorticity. Energy spectra obtained from the present homogeneous shear turbulence agree well with the spectra from temporally evolving shear layers. The results also highlight the effects of the additional forcing terms that were neglected in previous studies and the role of shear convection and the associated splitting errors in the unbounded evolution of previous numerical simulations.

Figure 1

Different turbulent free shear flows considered for the current study with the computational domain chosen (red cube): (a) nearly homogeneous shear turbulence (NHST), (b) mixing layer (ML), (c) planar jet (PJ), (d) round jet (RJ).

Figure 2

(a) Integral length scale normalized by the domain width, (b) Reynolds shear stress $\langle u_x^{\prime }{u}_y^{\prime }\rangle$ normalized by turbulent kinetic energy, for the four DNS, and (c) comparison of Reynolds number dependence of Reynolds shear stress $\langle u_x^{\prime }{u}_y^{\prime }\rangle$ with other studies. Dashed lines corresponds to the averaged value obtained from all simulations in the current study.

Figure 3

(a) Turbulent kinetic energy normalized by its expected value [Eq. (39)]. (b) Energy dissipation rate normalized by its expected value [Eq. (40)]. (c) Taylor microscale Reynolds number, ${\text{Re}}_{\lambda }$, for DNS 3. Dashed line corresponds to ${\text{Re}}_{\lambda }^o=80$.

Figure 4

(a) Root-mean-square velocity components along $x,y$, and $z$, normalized by $u_{\text{rms}}$, for DNS 3 $({\text{Re}}_{\lambda }^o=80).$ (b) Mean values for velocity fluctuations normalized by $u_{\text{rms}}$ from other studies plotted versus ${\text{Re}}_{\lambda }$. Dashed lines correspond to the averaged values from the current simulations (1.24, 0.92, and 0.78, respectively). (c) Average values of rms vorticity components along different directions normalized by ${\omega }_{\text{rms}}$ plotted versus ${\text{Re}}_{\lambda }$. Dashed line corresponds to isotropic turbulence. (a) Anisotropy in velocity, (b) ${\mathrm{Re}}_{\lambda }$ dependence of anisotropy, and (c) Anisotropy in vorticity.

Figure 5

Comparison of one-dimensional energy spectra along $x$ and $z$ directions. R & M refers to the energy spectra published by Rogers and Moser [4].

Figure 6

(a) Energy spectra normalized by $\varepsilon$ and $\nu$. (b) Shear stress spectra normalized by $\varepsilon$ and $\nu$. The dashed line corresponds to turbulence scaling from literature, ${\kappa }^{-5/3}$ in (a) and ${\kappa }^{-7/3}$ in (b).

Figure 7

Normalized turbulent kinetic energy budget. The lines correspond to experimental results from Panchapakesan and Lumley [29]. Symbols correspond to different simulations: DNS3, circles; DNS 3a, triangles; DNS 3b, squares.

Figure 8

Ratio of production to dissipation of kinetic energy. The blue line corresponds to simulations in the current study and the three lines correspond to three simulations by Kasbaoui et al., with different initial conditions.

Figure 9

Joint pdf of the normalized velocity fluctuations in the $x$ and $y$ directions from simulation with (a) just the off-diagonal term and (b) linear and nonlinear terms.

Figure 10

Marginal pdf of the normalized velocity fluctuations in the (a) $x$ and (b) $y$ directions from simulation with just the off-diagonal term, and linear and nonlinear terms.

Figure 11

Evolution of turbulent kinetic energy normalized by the expected value [Eq. (39)] for different treatment of the shear convection term.