A multi-objective optimization problem in mixed and natural convection for a vertical channel asymmetrically heated

Abstract

This paper deals with a multi-objective topology optimization problem in an asymmetrically heated channel, based on both pressure drop minimization and heat transfer maximization. The problem is modeled by assuming steady-state laminar natural convection flow. The incompressible Navier-Stokes equations coupled with the convection-diffusion equation, under the Boussinesq approximation, are employed and are solved with the finite volume method. In this paper, we discuss some limits of classical pressure drop cost function for buoyancy driven flow and, we then propose two new expressions of objective functions: the first one takes into account work of pressure forces and contributes to the loss of mechanical power while the second one is related to thermal power and is linked to the maximization of heat exchanges. We use the adjoint method to compute the gradient of the cost functions. The topology optimization problem is first solved for a Richardson (Ri) number and Reynolds number (Re) set respectively to Ri ∈ {100, 200, 400} and Re = 400. All these configuration are investigated next in order to demonstrate the efficiency of the new expressions of cost functions. We compare two types of interpolation functions for both the design variable field and the effective diffusivity. Both interpolation techniques have pros and cons and give slightly the same results. We notice that we obtain less isolated solid elements with the sigmoid-type interpolation functions. Then, we choose to work with the sigmoid and solve the topology optimization problem in case of pure natural convection, by setting Rayleigh number to {3 × 10 3 , 4 × 10 4 , 5 × 10 5 }. In all considered

A Review of Topology Optimisation for Fluid-Based Problems

“…Pizzolato et al [158] extended their previous work to maximise the performance of multi-tube latent heat thermal energy storage systems, investigating many different working conditions. Ramalingom et al [159] applied their previous method to multi-objective optimisation of mixed and natural convection in a asymmetrically-heated vertical channel. Pollini et al [160] extended the work of Asmussen et al [157] to large-scale three-dimensional problems, producing results comparable to those of Alexandersen et al [151] with a computational time reduction of 80-95% in terms of core-hours.…”

“…PM is the least used method with only a single paper [79]. Surprisingly, FVM is the second least used method with only 12 papers [21,47,52,80,82,115,129,130,137,144,159,188] or 7%, despite the fact that FVM for many years has been the preferred discretisation method for computational fluid dynamics. This can probably be explained by several factors: topology optimisation originates from solid mechanics where FEM is the preferred method; discrete adjoint approaches are easier using FEM than FVM; stabilised FEM has grown to be a mature and accurate method [193,194].…”

The Boussinesq System with Nonsmooth Boundary Conditions: Existence, Relaxation, and Topology Optimization

“…[5,24]). A focus on non-smooth outflow conditions when the temperature appears can be found in [13,24,43,45].…”

“…These directional boundary conditions tries to capture this phenomenon, while limiting the reflection. It is worth noting that other boundary conditions can be used, namely the so-called local/global Bernouilli boundary conditions [13,24,45]. The latter implies the do-nothing boundary condition is satisfied for exiting fluid and that both the normal velocity gradient and the total pressure vanish for re-entering fluid.…”

“…In this paper, we choose to locate the solid thanks to a penalization term added in the unsteady Navier-Stokes equations, as exposed in [4]. However, the binary function introduced in [4] is usually replaced by a smooth approximation, referred as interpolation function [45], in order to be used in gradient-based optimization algorithms. We refer to the review papers [1,23] for many references that deal with numerical resolution of TO problems applied to several different physical settings.…”