Derivation of a realistic forcing term to reproduce the turbulent characteristics of round jets on the centerline

Turbulence forcing techniques are often required in the numerical simulation of statistically stationary turbulent flows. However, the existing forcing techniques are not based on physics, but rather arbitrary numerical methods that sustain the turbulent kinetic energy. In this work, a forcing technique is devised to reproduce the centerline turbulent characteristics of round jets in a triply periodic box. It is derived from the Navier-Stokes equations by applying a Reynolds decomposition with the mean velocity of the axisymmetric jet. The result is an anisotropic linear forcing term, which is intended to be used in a three-dimensional box to create turbulence. Four direct numerical simulations with different ${\text{Re}}_{\lambda }$ have been performed with these forcing terms. The budget of the terms in the kinetic energy equation is very close to the experimental measurement on the centerline. The anisotropy, kinetic energy $k$, and dissipation rate $\varepsilon$ of the simulations are also comparable to experimental values. Finally, the kinetic energy spectrum in the axial direction, $\varphi \left({\kappa }_1\right)$, is presented. With appropriate normalizations, the spectrum agrees well with the round jet spectrum on its centerline.

Figure 1

Temporal fluctuations of turbulent parameters: (a) ratio of the integral length scale, $l$, to the domain width, $L$; (b) anisotropy $\langle u_x^{*2}\rangle /\langle {\mathit{u}}^{*2}\rangle$; (c) the volume-averaged kinetic energy, $\langle k^{*}\rangle$, with respect to the expected value, $k_o$; (d) the volume-averaged dissipation rate, $\langle {\varepsilon }^{*}\rangle$, with respect to the expected value, ${\varepsilon }_o$. The parameters are plotted as a function of the time normalized by the a priori eddy time scale, ${\tau }_o$.

Figure 2

Energy budget comparison. Each term is normalized by $U_c^3/r_{1/2}$. The lines are experimental data from Ref. [19], and the points are the estimated values from Eqs. (48, 49, 50, 51, 52, 53).

Figure 3

Single-point values comparison: (a) anisotropy $\frac{\langle u_x^{\prime 2}\rangle }{\langle {\mathit{u}}^{\prime \mathit{2}}\rangle }$; (b) normalized kinetic energy $\frac{k}{U_c^2}$; (c) normalized dissipation rate $\frac{{\varepsilon }_k}{U_c^3/x_o}$. Boersma(98) [34] and Boersma(04) [35] are DNS. P&Li [36], P&Lu [19], Antonia [33], Romano [28], Xu [18], W&F [25], Burattini [17], and Darisse [22] are experiments.

Figure 4

Examination of the origins of the forcing term: (a) anisotropy $\frac{\langle {u_x^{*}}^2\rangle }{\langle {{\mathit{u}}^{*}}^2\rangle }$; (b) normalized kinetic energy $\frac{k}{U_c^2}$. The parameters are plotted as a function of the time normalized by the a priori eddy timescale, ${\tau }_o$. All (black) refers to the original forcing term; Mean (blue), the forcing term from $\nabla \overline{\mathit{u}}$ only; and Mean+Adv (red), the forcing term from $\nabla \overline{\mathit{u}}$ and advection normalization.

Figure 5

Comparison of normalized kinetic energy spectra in the longitudinal direction. $u_k={\left(\nu \varepsilon \right)}^{1/4}$ is the Kolmogorov velocity scale.