A new time-dependent analysis of the global and local fluctuating velocity signals in grid turbulence is conducted to assess the scaling laws for non-equilibrium turbulence. Experimental datasets of static- and active-grid turbulence with different Rossby numbers $R_o({=}U/\varOmega M$: $U$ is the mean velocity, $\varOmega$ is the mean rotation rate and $M$ is the grid mesh size) are considered. Although the global (long-time-averaged) non-dimensional dissipation rate $C_\varepsilon$ is independent of the Reynolds number $Re_\lambda$ based on the global Taylor microscale, the local (short-time-averaged) non-dimensional dissipation rate $\left \langle C_\varepsilon (t_i) \right \rangle$ ($t_i$ is the local time) both in the static- and active-grid turbulence clearly show the non-equilibrium scaling $\left \langle C_\varepsilon (t_i)\right \rangle / \sqrt {Re_0} \propto \left \langle Re_\lambda (t_i) \right \rangle ^{-1}$ ($\left \langle Re_\lambda (t_i) \right \rangle$ and $Re_0$ are the Reynolds numbers based on the local Taylor microscale $\lambda (t_i)$ and the global integral length scale, respectively), which has only been confirmed for global statistics in the near field of grid turbulence. The local value of $\left \langle L(t_i) / \lambda (t_i) \right \rangle$ ($L(t_i)$ is the local integral length scale) shifts from the equilibrium to non-equilibrium scaling as $\left \langle Re_\lambda (t_i) \right \rangle$ increases, further confirming that the non-equilibrium scalings are recovered for local statistics both in the static- and active-grid turbulence. The local values of $\left \langle C_\varepsilon (t_i) \right \rangle$ and $\left \langle L(t_i) / \lambda (t_i) \right \rangle$ follow the theoretical predictions for global statistics (Bos & Rubinstein, Phys. Rev. Fluids, vol. 2, 2017, 022601).

The decay of stably stratified turbulence generated by a towed rake of vertical plates is investigated by direct numerical simulations (DNS) of temporally evolving grid turbulence in a linearly stratified fluid. The Reynolds number $Re_M=U_0M/\nu$ is 5000 or 10 000 while the Froude number $Fr_M=U_0/MN$ is between 0.1 and 6 ($U_0$: towing speed; $M$: mesh size; $\nu$: kinematic viscosity; $N$: Brunt–Väisälä frequency). The DNS results are compared with the theory of stably stratified axisymmetric Saffman turbulence. Here, the theory is extended to a viscosity-affected stratified flow regime with low buoyancy Reynolds number $Re_b$, and power laws are derived for the temporal variations of the horizontal velocity scale ($U_H$) and the horizontal and vertical integral length scales ($L_H$ and $L_V$). Temporal grid turbulence initialized with the mean velocity deficit of wakes exhibits a $k^{2}$ energy spectrum at a low-wavenumber range and invariance of $U_H^2L_H^2L_V$, which are the signatures of axisymmetric Saffman turbulence. The decay of various quantities follows the power laws predicted for low-$Re_b$ Saffman turbulence when $Fr_M$ is sufficiently small. However, the decay of $U_H^2$ at $Fr_M=6$ is no longer expressed by a power law with a constant exponent. This behaviour is related to the scaling of kinetic energy dissipation rate $\varepsilon$, for which $\alpha =\varepsilon /(U_H^3/L_H)$ is constant during the decay for $Fr_M\leq 1$ while it varies with time for $Fr_M=6$. We also examine the experimental data of towed-grid experiments by Praud et al. (J. Fluid Mech., vol. 522, 2005, pp. 1–33), which is shown to agree with the theory of low-$Re_b$ Saffman turbulence.