Optimal mixing in two-dimensional stratified plane Poiseuille flow at finite Péclet and Richardson numbers

Abstract

We consider the nonlinear optimisation of irreversible mixing induced by an initial finite amplitude perturbation of a statically stable density-stratified fluid with kinematic viscosity ν and density diffusivity κ. The initial diffusive error function density distribution varies continuously so that ρ ∈ [ρ− 1 2 ρ 0 ,ρ+ 1 2 ρ 0 ]. A constant pressure gradient is imposed in a plane two-dimensional channel of depth 2h. We consider flows with a finite Péclet number Pe = U m h/κ = 500 and Prandtl number Pr = ν/κ = 1, and a range of bulk Richardson numbers Ri b = gρ 0 h/(ρU 2 ) ∈ [0, 1] where U m is the maximum flow speed of the laminar parallel flow, and g is the gravitational acceleration. We use the constrained variational direct-adjoint-looping (DAL) method to solve two optimization problems, extending the optimal mixing results of Foures et al. (2014) to stratified flows, where the irreversible mixing of the active scalar density leads to a conversion of kinetic energy into potential energy. We identify initial perturbations of fixed finite kinetic energy which maximize the time-averaged perturbation kinetic energy developed by the perturbations over a finite time interval, and initial perturbations that minimise the value (at a target time, chosen to be T = 10) of a 'mix-norm' as first introduced by Mathew, Mezic & Petzold (2005), further discussed by Thiffeault (2012) and shown by Foures et al. (2014) to be a computationally efficient and robust proxy for identifying perturbations that minimise the long-time variance of a scalar distribution. We demonstrate, for all bulk Richardson numbers considered, that the time-averaged-kineticenergy-maximising perturbations are significantly suboptimal at mixing compared to the mix-norm-minimising perturbations, and also that minimising the mix-norm remains (for density-stratified flows) a good proxy for identifying perturbations which minimise the variance at long times. Although increasing stratification reduces the mixing in general, mix-norm-minimising optimal perturbations can still trigger substantial mixing for Ri b 0.3. By considering the time evolution of the kinetic energy and potential energy reservoirs, we find that such perturbations lead to a flow which, through Taylor dispersion, very effectively converts perturbation kinetic energy into 'available potential energy', which in turn leads rapidly and irreversibly to thorough and efficient mixing, with little energy returned to the kinetic energy reservoirs.

Optimal unstirred state of a passive scalar

“…Given the initial distribution f 0 = f (x, y, 0) of a passive scalar field f , the variational method searches for the optimal velocity field u with respect to an augmented Lagrangian L. In fluid mixing and transport studies, the Lagrangian is commonly defined by a measure of homogenisation and other constraints, such as the Navier-Stokes equation and the normalisation condition of a seed perturbation field. Although it would be intuitive to use the Dirichlet seminorm to quantify homogenisation, a 'mix-norm' |∇ −1 f T | 2 has been shown to be numerically robust and efficient (Thiffeault 2012;Marcotte & Caulfield 2018). We therefore define a generic Lagrangian applicable to two measures of homogenisation as…”

“…To better quantify the homogenisation of the passive tracer, several other norms have also been used in the literature, including a class of ‘mix-norms’ (Mathew, Mezić & Petzold 2005; Thiffeault 2012). This type of norm has recently been applied to the optimisation problems of mixing in 2-D plane Poiseuille flow (Foures, Caulfield & Schmid 2014) and stratified plane Poiseuille flow (Marcotte & Caulfield 2018) using variational methods. From this perspective, our unstirring problem may be considered as an interesting test of the state-of-the-art of the methods used in this branch of fluid dynamics.…”

Mixing enhancement in binary fluids using optimised stirring strategies

“…2011; Foures etal. 2014; Marcotte & Caulfield 2018) and the choice of proposed by Mathew etal. (2005) based on advection-based mixing.…”

“…Throughout this paper we will be optimising with respect to this quantity, but we stress again that other norms can be employed without conceptual changes in the optimisation procedures. For our study, we select the mix-norm with an exponent k of −2/3, a value between the common choice k = −1 (see Lin et al 2011;Foures et al 2014;Marcotte & Caulfield 2018) and the choice of k = −1/2 proposed by Mathew et al (2005) based on advection-based mixing. In general, the exponent governs the relative emphasis of larger over smaller scales in the definition of mixedness, as can be seen from a Fourier transform of the integrand in the expression above.…”

“…Using wall-mounted blowing/suction control in a channel, improved mixing could be accomplished by directly targeting a mixing measure, rather than a flow instability (Foures, Caulfield & Schmid 2014). Further studies (Marcotte & Caulfield 2018; Vermach & Caulfield 2018) have extended this approach to higher dimensions and stratified flows.…”