Invariants for slightly heated decaying grid turbulence

The paper examines the validity of velocity and scalar invariants in slightly heated and approximately isotropic turbulence generated by passive conventional grids. By assuming that the variances $\langle {u}^{2} \rangle $ and $\langle {\theta }^{2} \rangle $ ($u$ and $\theta $ represent the longitudinal velocity and temperature fluctuations) decay along the streamwise direction $x$ according to power laws $\langle {u}^{2} \rangle \sim {(x- {x}_{0} )}^{{n}_{u} } $ and $\langle {\theta }^{2} \rangle \sim {(x- {x}_{0} )}^{{n}_{\theta } } $ (${x}_{0} $ is the virtual origin of the flow) and with the further assumption that the one-point energy and scalar variance budgets are represented closely by a balance between the rates of change of $\langle {u}^{2} \rangle $ and $\langle {\theta }^{2} \rangle $ and the corresponding mean energy dissipation rates, the products $\langle {u}^{2} \rangle { \lambda }_{u}^{- 2{n}_{u} } $ and $\langle {\theta }^{2} \rangle { \lambda }_{\theta }^{- 2{n}_{\theta } } $ must remain constant with respect to $x$. Here ${\lambda }_{u} $ and ${\lambda }_{\theta } $ are the Taylor and Corrsin microscales. This is unambiguously supported by previously available data, as well as new measurements of $u$ and $\theta $ made at small Reynolds numbers downstream of three different biplane grids. Implications for invariants based on measured integral length scales of $u$ and $\theta $ are also tested after confirming that the dimensionless energy and scalar variance dissipation rate parameters are approximately constant with $x$. Since the magnitudes of ${n}_{u} $ and ${n}_{\theta } $ vary from grid to grid and may also depend on the Reynolds number, the Saffman and Corrsin invariants which correspond to a value of $- 1. 2$ for ${n}_{u} $ and ${n}_{\theta } $ are unlikely to apply in general. The effect of the Reynolds number on ${n}_{u} $ is discussed in the context of published data for both passive and active grids.