Abstract
The continuous adjoint approach is a technique for calculating the sensitivity of a flow to changes in input parameters, most commonly changes of geometry. Here we present for the first time the mathematical derivation of the adjoint system for multiphase flow modeled by the commonly used drift flux equations, together with the adjoint boundary conditions necessary to solve a generic multiphase flow problem. The objective function is defined for such a system, and specific examples derived for commonly used settling velocity formulations such as the Takacs and Dahl models. We also discuss the use of these equations for a complete optimisation process.
Highlights
The adjoint method is currently attracting significant interest as an optimization process in CFD.The objective of the adjoint approach is to calculate the sensitivity of the flow solution with respect to changes in the input parameters, most commonly changes in the geometry
Examples of the application of the continuous adjoint method for single phase flow can be found in a range of areas [1,2] such as automotive [3,4,5], aerospace [6,7] and turbomachinery [8,9,10], and implementations of the equations can be found in general purpose CFD codes such as STAR-CCM, ANSYS Fluent [11]
These equations no longer depend on the objective function, so when switching from one optimisation objective to another, they remain unchanged and only the boundary conditions have to be adapted to the specific objective function
Summary
The adjoint method is currently attracting significant interest as an optimization process in CFD. The slip (drift) between the phases is primarily due to the gravitational settling of the dispersed phase This might adequately describe solid particles in water or an emulsion of immiscible liquids, and in these cases a commonly used mathematical model is the drift flux model. A far more robust equation set has been produced and the computational resources required to solve the system have been reduced This makes it a very appropriate basis from which to develop an adjoint formulation suitable for applying to dispersed multiphase flows in this regime.
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