Scalar mixing efficiency in pulsatile channel flow

Pulsatile channel flow exhibits destabilising effects at certain frequencies with potentially significant non-modal transient growths. We explore the mixing efficiency of oscillatory channel flows at Reynolds number Re=7500 as they act on various distributions of passive scalars. The flows are controlled by a pulsating pressure gradient (defined with mass flow rate Q(t) = Q(1 + Q̃sin(Ωt)) and Womersley number Wo = h(Ω/ν)^(1/2)). Distinctive subharmonic eigenvalue orbits<span style="font-size:10.8333px"> (that form due to the non-normality of the Orr-Sommerfeld operator) can be tracked and identified. The eigenfunctions associated with the unstable eigenvalues can then be used as initial perturbations of Direct Numerical Simulations. We report that although instabilities and non-normal growths can be observed for a range of Womersley numbers, the most significant mixing events still occur within the ‘ballistic’ regime with relatively low pulsation frequency and large amplitudes. For larger Womersley numbers, although non-normal growths are also observed, the mixing efficiency is much lower due to the high pulsation frequency reducing the thickness of the oscillatory boundary layers. Therefore, we conjecture that it is more appropriate to employ the variational 'direct-adjoint-looping' method to locate the optimal pulsation scheme via the minimization of scalar variance and/or mix-norms instead of maximizing perturbation energy growth.