Power Decay Law in Moderate Intensity Nearly Homogeneous Isotropic Turbulence

An experimental investigation of turbulence kinetic energy in the nearly homogeneous and isotropic region of a moderate intensity flow ($12\%$), with the Taylor Reynolds number varying between 175 and 540, generated by an active grid for nominal mean velocities of \Ams allows for an assessment of the applicability of the power decay law. A combination of single and cross hot-wires probes the flow to determine important flow quantities. The cross-wire in particular has a limited calibration range requiring the flow angles to fall within $\pm 33^{\circ}$. This occurs for, at the nominal velocities of 4, 6, 8 and 12 $m \, s^{-1}$, are at $x/M_U$ of 60, 75, 80 and 95, respectively. By comparing the velocity power spectrum in the downstream, the cross stream and in the vertical direction leads to conclusion that the turbulence is isotropic starting in pseudo inertial subrange. Using Kolmogorov (1941) normalization supports local isotropy with the collapse of the high wavenumber range. An additional power law is derived analytically for nearly homogenous, isotropic turbulence that relates the turbulence kinetic energy to the dissipation rate. The typical decay power law requires determining the location of the virtual origin via assumption or some other method. The selection of different virtual origins leads to differing decay exponents. The derived power law relationship allows for the determination of the decay exponent independently from the virtual origin. In deriving a power decay law in homogenous isotropic turbulence it is assumed that the normalized dissipation rate, $C_{\epsilon ,q}$, is constant. $C_{\epsilon ,q}$ is within $5\%$ of the asymptotic value for 6, 8 and 12 \ms from 75 to 142, 80 to 142 and 95 to 142 mesh lengths downstream of the active grid, respectively. $C_{\epsilon ,q}$ is within $15\%$, for 4 \ms, for the downstream locations of 60 to 110. The decay exponents obtained from the turbulence kinetic energy, the dissipation rate and the derived power law within $\pm 3 \%$ when the virtual origin is chosen such that the Taylor microscale squared is linear. Conversely, if the virtual origin is assumed to be zero, the decay exponents can vary as much as $50\%$. The ratio of the dissipation rate calculated from a temporal derivate over the dissipation rate calculated from the decay power law is found to be constant and within $\pm 10\%$ of unity with a virtual origin such that the Taylor microscale squared is linear. On the other hand, if the virtual origin is assumed to be zero, the ratio of dissipation rate calculated from a temporal derivate over the dissipation rate calculated from the decay power law is found to vary and have a slope that is 10 times lager then when the virtual origin is chosen such that the Taylor microscale squared is linear