Production and dissipation of kinetic energy in grid turbulence

In grid turbulence, not so far behind the grid, an average flow can be observed with a close to sinusoidal velocity profile, corresponding to the wakes behind the grid bars. The kinetic energy of this mean flow decays rapidly and a close to isotropic flow is observed further downstream. We show how these wakes behind the grid bars influence the downstream turbulence. In particular, we investigate the decay rate of kinetic energy, the behavior of the normalized dissipation rate, and the sensitivity of the flow on initial conditions. We show that the initial value of the ratio of the length scale of the turbulence to the mesh-size determines the precise decay of the mean-flow and the generation of the turbulent kinetic energy. We further show how a simple turbulence model can estimate the degree of nonequilibrium and inhomogeneity of grid turbulence and how this model can be extended to take into account disequilibrium observed in direct numerical simulations of decaying isotropic turbulence.

Figure 1

Sketch of the geometry considered in the present investigation. A uniform incoming flow $U_0{\mathit{e}}_x$ is perturbed by a grid. The resulting flow is decomposed into a sinusoidal mean profile $U\left(y\right)=\overline{U_x}-U_0$ and a fluctuating part.

Figure 2

Temporal evolution of the kinetic energy of the turbulence for values of the lenghtscale ratio $0.01\le {\alpha }_L\left(0\right)\le 10$. (a) Initial intensity $I_k=0.1$; (b) $I_k=0.01$. (c) Same results as in (b) but time is normalized by $t_{\text{peak}}$ and kinetic energy by $k_{\text{peak}}$. (d) Local power-law decay exponent of the kinetic energy $n$.

Figure 3

Temporal evolution of (a) the kinetic energy of the mean field and (b) the dissipation of the turbulence, for values of the lenghtscale ratio $0.01\le {\alpha }_L\left(0\right)\le 10$; initial intensity $I_k=0.01$.

Figure 4

(a) Normalized dissipation rate for values of the lenght-scale ratio $0.01\le {\alpha }_L\left(0\right)\le 10$ and initial intensity $I_k=0.01$. (b) Production $p$ of turbulent kinetic energy.

Figure 5

Inhomogeneity of the kinetic energy induced by the periodic profile of the turbulence production term. (a) The maximum ($k^{+}=k+A_k$; solid lines) and minimum values ($k^{-}=k-A_k$; dashed lines) of the kinetic energy are shown for values of the lenghtscale ratio $0.01\le {\alpha }_L\left(0\right)\le 10$ and initial intensity $I_k=0.01$. Line types as in the previous figures. (b) The peak of the kinetic energy $k^{+}$ at the cross-stream ($y$) position where the kinetic energy is maximum, divided by its peak at the $y$ position where the kinetic energy is minimum.

Figure 6

(a) Temporal evolution of the kinetic energy of the turbulence for values of the length-scale ratio $0.01\le {\alpha }_L\left(0\right)\le 10$ and $I_k=0.01$. (b) Evolution of the dissipation rate. Also shown in (a), (b) are DNS results of flow through an array of cylinders [42]. (c) Kinetic energy at a cross-stream position where the production of $k$ is minimum, compared with data obtained from a fractal square grid (FSG) and a regular square grid (RG) [27]. (d) Nonequilibrium of the flow as measured by the normalized dissipation, compared to experimental data-points [18].

Figure 7

(a) Peak values of the kinetic energy and (b) associated peak-times for $0.001\le {\alpha }_L\left(0\right)\le 10$ and $I_k=0.01$.

Figure 8

Temporal evolution of (a) the lengthscale ratio and (b) the turbulent timescale $\tau =k/\epsilon$ for $0.01\le {\alpha }_L\left(0\right)\le 10$ and $I_k=0.01$.

Figure 9

Results from the two-scale model Eqs. (49, 50, 51, 52). (a) Temporal evolution of the kinetic energy $k$, energy flux $f$ and dissipation rate $\epsilon$. (b) Evolution of the normalized dissipation rate.

Figure 10

Analytical solution for the temporal evolution of the turbulent kinetic energy for freely decaying turbulence for different initial values of the ratio $t_0=k\left(0\right)/\epsilon \left(0\right)$.