Asymptotic matching arguments and DNS for transpired turbulent flows

Abstract

The work discusses fundamental aspects of local solutions and asymptotic matching for transpired turbulent boundary layers with support from DNS simulations and previous analytical formulations validated through experimental data. The DNS data are obtained for porous wall pipe flows, for six different transpiration rates ($v_w/u_\tau = -\,0.063, -\,0.015$, 0.0, 0.015, 0.053 and 0.095) and moderate Reynolds number ($\mathrm{Re}_{{\tau }}$ = $u_\tau D/\nu$), varying from 340 to 460. In terms of the classical global parameters, the Reynolds number (Re = $UD/\nu$) range is 5700–10,500. Wall fluid injection or suction is applied continuously and throughout the axial pipe direction, meaning that the flow accelerates through the computational domain. The impossibility of application of cyclic boundary conditions at the inlet and outlet pipe sections is circumvented with the use of convective conditions. The work particularly discusses similarity of the mean and turbulent quantities in terms of the transpiration parameters $u/u_\tau$ ($u_\tau$ = friction velocity), $(u_\tau y)/\nu$, $(v_w u)/u_\tau ^2$ ($v_w$ = injection velocity), $(v_w y)/\nu$, $(u_\tau ^2 y)/(\nu w^*)$ (with $w^* = 2.3 u_\tau (1 + 9 v_w^+)$, $v_w^+$ = $v_w/u_\tau$), $u/w^*$, $u/u_c$ [with $u_c$ defined as in Guimaraes et al. (Int J Heat Fluid Flow 78:108436, 2019)] and ${yu_c}/{\nu }$. A discussion based on the arguments of Millikan and on the matched asymptotic expansions method shows that bilogarithmic mean velocity solutions match in a common domain provided certain asymptotic identities are satisfied. A parametrization scheme based on the classical similarity velocity, the friction velocity, is developed to correlated the points of maximum value for the elements of the Reynolds stress tensor with the transpiration rate.