2.1.3. Turbulent Flow

All of the above fluid flow papers assume laminar fluid flow, whereas turbulent flow has only been treated in a few publications. Turbulence is inherently time-dependent, but current works on topology optimisation of turbulent flow restrict themselves to the steady-state time-averaged approximation of turbulence, namely the Reynolds-Averaged Navier–Stokes (RANS) equations.

In the work by Othmer [21], turbulence is in the model, but the influence on the design sensitivities is neglected. Kontoleontos et al. [80] presented the first work on topology optimisation of turbulent flow, including the Spalart–Allmaras turbulence model in their continuous adjoint sensitivity analysis. For a shape optimisation example, they showed that the typical "frozen turbulence" assumption produces sensitivities of the incorrect sign in some cases. Yoon [81] presented a discrete adjoint approach to density-based optimisation of turbulent flow problems using the Spalart–Allmaras turbulence model and a modified wall equation. However, the meshes used are much too coarse to capture turbulence properly. Dilgen et al. [82] demonstrated the application of automatic differentiation for obtaining exact sensitivities for density-based topology optimisation of large scale two- and three-dimensional turbulent flow problems, using the one-equation Spalart–Allmaras model and the two-equation k-*ω* model. As Kontoleontos et al. [80] did for shape optimisation, they demonstrated that the "frozen turbulence" assumption gives inexact sensitivities for topology optimisation, even with the incorrect sign for some cases. Yoon [83] used a *k* − model, analogous to the model applied [82], to design 2D flow components minimising turbulent energy i.e., minimising noise.

#### 2.1.4. Non-Newtonian Fluids

In the preceding works, the fluids are all assumed to be Newtonian. However, treating more sophisticated fluids, including e.g., long polymer chains or blood cells, calls for implementation of non-Newtonian fluids, which have nonlinear behaviour of the viscosity. There are a wide variety of models that can be applied and a few have been implemented for use in a topology optimisation context.

Pingen and Maute [84] applied topology optimisation to non-Newtonian flows for the first time, using a density-based LBM formulation and a Carreau–Yasuda model for shear-thinning fluids. Ejlebjerg Jensen et al. [85] optimised viscoelastic rectifiers using a non-Newtonian fluid model based on dumbbells in a Newtonian solvent, which introduced a memory in the fluid. Jensen et al. [86] considered the bi-stability behaviour for a crossing between two viscoelastic fluids. Hyun et al. [87] suggested a density-based formulation for minimising wall shear stress by considering shear thinning non-Newtonian effects. Zhang and Liu [88] applied a level set based approach to minimise flow shear stress in arterial bypass graft designs, where the blood flow is modelled using a steady non-Newtonian modified Cross model. Zhang et al. [89] used an explicit boundary-tracking level set method with remeshing to optimise micropumps for non-Newtonian power-law fluids. Romero and Silva [90] extended their previous work [35] to cover non-Newtonian fluids and compare it to the Newtonian case. Dong and Liu [91] proposed a bi-objective formulation for the design of asymmetrical fixed-geometry microvalves for non-Newtonian flow.

#### *2.2. Species Transport*

In this section, the included papers are focused on a transport of matter or species due to the presence of a fluid. The transported matter does not necessarily need to be modelled itself, as long as the objective of the optimisation is related to the transport.

Okkels and Bruus [92] coupled a convection–reaction–diffusion equation to the fluid flow to model catalytic reactions, distributing the porous catalytic support to maximise the mean reaction rate of the microreactor. Andreasen et al. [93] used a convection–diffusion equation to model and optimise microfluidic mixers, in which well-known design elements such as herring-bones and slanted grooves appeared automatically. Gregersen et al. [94] applied topology optimisation to an electrokinetic model in order to maximise the net induced electroosmotic flow rate. Schäpper et al. [95] used more advanced reaction-kinetics using multiple convection-diffusion type equations to optimise microbioreactors. Kim and Sun [96] optimised the gas distribution channels in automotive fuel cells. Makhija et al. [97] optimised a passive micromixer using a porosity model for LBM. Deng et al. [98] used a physical model similar to [93] but omits the pressure constraint and use a quasi-Newton approach optimised three-dimensional and extruded two-dimensional microfluidic mixers. Makhija and Maute [99] introduced an explicit level set optimisation methodology using an X-FEM-based hydrodynamic Boltzmann model including transport. The ability to eliminate the spurious diffusion in void areas, especially dubious when modeling species concentration, is highlighted. Oh et al. [100] used the Navier–Stokes and a convection-diffusion equation to model and optimise the osmotic permeate flux over a membrane wall. Chen and Li [101] optimised micromixers under the assumption that reverse flow structures [8] inserted in a microchannel increases the mixing. Hyun et al. [102] designed repeating units for sorting particles using principles in deterministic lateral displacement. Andreasen [103] used a density-based framework to design dosing units of a secondary fluid utilising the inertia of the driving fluid. Yaji et al. [104] presented an optimisation of vanadium redox flow batteries by including a reaction term depending on the local flow speed and concentration level in a two-dimensional setting. Guo et al. [105] presented a methodology to model and optimise pure convection-dominated transport using a Lagrangian mapping method. Only the Navier–Stokes equations are approximated by FEM, while the Lagrangian transport is modelled cross-section-wise. Behrou et al. [106] presented a density-based approach for the design of proton exchange membrane fuel cells using a depth-averaged two-dimensional approximation of reactive porous media flow. Chen et al. [107] extended their previous work [104] to three-dimensional problems. Dugast et al. [108] used a level set method and the LBM to maximise the reaction in a square reactor and investigated the problem for a range of flow situations.

#### *2.3. Conjugate Heat Transfer*

Conjugate heat transfer is when the coupled heat transfer between a solid and the surrounding fluid is considered, with the temperature field of both of interest. Thus, in order to model conjugate heat transfer, it is necessary to build the thermal transport on top of the fluid flow model. Conjugate heat transfer is generally divided into groups based on the heat transfer mechanism in the fluid.

Figure 2 shows the three main heat transfer mechanisms: forced convection, where the flow is actively driven by a pump, fan or pressure-gradient; natural convection, where the flow happens passively from the natural density variations due to temperature differences; and diffusion where heat is transferred through a stagnant fluid through diffusion. Only the first two are considered in this review, since they include fluid motion modelled through fluid flow equations.

(**a**) Forced convection (**b**) Natural convection (**c**) Diffusion

**Figure 2.** Illustration of a metallic block subjected to different heat transfer mechanism in the surrounding fluid. (**a**) shows forced convection with a cold flow entering at the left-hand side; (**b**,**c**) show natural convection and pure diffusion, respectively, due to cold upper and side walls. Reproduced with permission from Alexandersen et al. [109].

#### 2.3.1. Forced Convection

Dede [110] and Yoon [111] presented the first works on topology optimisation of forced convection at almost the same time. Dede [110] used the commercial finite element analysis (FEA) software COMSOL to optimise both conduction and conjugate heat transfer problems. Yoon [111] presented a two-dimensional formulation, treating heat sink problems, as well as flow focusing in order to cool specific points. Thereafter, Dede [112] applied topology optimisation to design multipass branching microchannel heat sinks for electronics cooling. McConnell and Pingen [113] presented a two-layer pseudo-3D topology optimisation formulation based on the lattice Boltzmann method (LBM). Kontoleontos et al. [80] presented a continuous adjoint formulation for fluid heat transfer using the Spalart–Allmaras turbulence model and the impermeability directly as the design variable. However, the presented examples are not true conjugate heat transfer problems, since the solid temperature is predefined and enforced through a penalty approach, rather than modelling the solid temperature alongside the fluid temperature. Matsumori et al. [114] presented topology optimisation for forced convection heat sinks under constant input power. They interpolated the heat source to only be active in the solid and investigated both temperature-independent and -dependent sources. Marck et al. [115] investigated a multiobjective optimisation problem considering both fluid and thermal objectives using a finite volume-based discrete adjoint approach. Koga et al. [116] presented the development of an active cooling heat sink device using topology optimisation, which was manufactured and experimentally tested. However, they used a two-dimensional Stokes flow model to optimise for a three-dimensional turbulent application. In 2015, Yaji et al. [117] published three-dimensional results for forced liquid-cooled heat sinks using an Ersatz-material level set approach. Next, Yaji et al. [118] presented a topology optimisation method using the Lattice Boltzmann Method (LBM) incorporating a special sensitivity analysis based on the discrete velocity Boltzmann equation. Łaniewski Wołłk and Rokicki [119] treated large three-dimensional problems using a discrete adjoint formulation for the LBM implemented for multi-GPU architectures. Qian and Dede [120] introduced a constraint on the tangential thermal gradient around discrete heat sources with the goal of reducing thermal stress due to non-uniform expansion. Yoshimura et al. [47] proposed a gradient-free approach using a genetic algorithm and a Kriging surrogate model coupled to an immersed method known as the Building-Cube Method. Haertel and Nellis [121] developed a plane two-dimensional fully-developed flow model for topology optimisation of air-cooled heat sinks. Pietropaoli et al. [122] used the impermeability as the design variable to optimise internal channels.

In 2018, Zhao et al. [123] used a Darcy flow model for topology optimisation of cooling channels. Qian et al. [124] optimised active cooling flow channels for cooling an active phased array antenna with

many discrete heat sources. Sato et al. [125] used an adaptive weighting scheme for the multiobjective topology optimisation of active heat sinks. Yaji et al. [126] applied a local-in-time approximate transient adjoint method for topology optimisation of large scale problems with oscillating inlet flow. Haertel et al. [127] presented a pseudo-3D model for extruded forced convection heat sinks, actually considering the chip temperature by coupling a thermofluid design layer to a conductive base plate layer. Almost simultaneously, Zeng et al. [128] published a similar two-layer model for an air-cooled mini-channel heat sink, where the connection between the layers is tuned using full three-dimensional simulations of a reference heat sink design. Furthermore, Zeng et al. [128] manufactured and experimentally validated the performance of their optimised designs. Dilgen et al. [129] presented the first full conjugate heat transfer model for density-based topology optimisation of turbulent systems. In contrast to Kontoleontos et al. [80], the temperature field of the solid is modelled, thus rendering it true conjugate heat transfer. Furthermore, Dilgen et al. [129] treat large-scale three-dimensional problems comparing their thermal performance to equivalent two-dimensional designs. Ramalingom et al. [130] proposed a sigmoid interpolation function for mixed convection problems. Dugast et al. [131] applied a level set based approach in combination with the LBM to a variety of thermal control problems. Santhanakrishnan et al. [132] performed a comparison of density-based and Ersats-material level set topology optimisation for three-dimensional heat sink design using the commerical FEA software COMSOL. However, the designs, both density and level set based, show clear signs of being unconverged with unphysical designs and, thus, the study must be rendered inconclusive. Sun et al. [133] used density-based topology optimisation to generate guiding channels for an enhanced air-side heat transfer geometry in fin and tube heat exchangers. Lv and Liu [134] applied a density-based method to the design of a bifurcation micro-channel heat sink, comparing them to reference designs.

In 2019, Pietropaoli et al. [135] extended their previous work to three-dimensional internal coolant systems. Makhija and Beran [136] presented a concurrent optimisation method using a shape parametrisation for the external shape and a density-based parametrisation for the internal geometry. Subramaniam et al. [137] investigated the inherent competition between heat transfer and pressure drop. Yu et al. [138] applied a geometry projection method called moving morphable components (MMC) to the design of two-dimensional problems allowing for explicit feature size contol. Zhang and Gao [139] presented a density-based approach for optimising non-Newtonian fluid based thermal devices. Kobayashi et al. [140] used topology optimisation to design extruded winglets for fin-and-tube heat exchangers. Zeng and Lee [141] extended their previous work to the design of liquid-cooled microchannel heat sinks with in-depth numerical and experimental investigations. Jahan et al. [142] designed conformal cooling channels for plastic injection molds using a two-dimensional simplification. Yan et al. [143] developed a two-layer plane model based on analytical derivations and assumptions of the out-of-plane distribution for optimising microchannel heat sinks. Tawk et al. [144] proposed a density-based approach for optimising heat exchangers with two seperate fluids and a solid. Lundgaard et al. [145] presented a density-based methodology for distributing sand and rocks in thermal energy storage systems modelled by a transient Darcy's law coupled to heat transfer. Li et al. [146] applied a multi-objective density-based method to the design of liquid-cooled heat sinks, presented both extensive numerical and experimental comparisons to reference designs. Dong and Liu [147] applied topology optimisation to air-cooled microchannel heat sinks with discrete heat sources. Yaji et al. [148] suggested a multifidelity approximation framework to optimise turbulent heat transfer problems using a low-fidelity laminar flow model as the driver. Hu et al. [149] applied a density-based approach to optimisation of a microchannel heatsink with an in-depth comparison to a reference design with straight channels.

#### 2.3.2. Natural Convection

Alexandersen et al. [109] presented the first work on topology optimisation of natural convection problems, using a density-based approach for optimising both heat sinks and buoyancy-driven micropumps. On the contrary, Coffin and Maute [150] used an explicit level set method combined with the extended finite element method (X-FEM) for both steady-state and transient natural convection cooling problems. Alexandersen et al. [151] extended their initial paper to large-scale three-dimensional heat sink problems using a parallel framework allowing for the optimisation of problems with up to 330 million DOFs. Pizzolato et al. [152] applied topology optimisation to the design of fins in shell-and-tube latent heat thermal energy storage, including the temperature-dependent latent heat coupled with natural convection using a time-dependent formulation. Alexandersen et al. [153] applied their previously developed framework to optimise the design of passive coolers for light-emitting diode (LED) lamps showing superior performance compared to reference lattice and pin fin designs. Ramalingom et al. [130] proposed a sigmoid interpolation function for mixed convection problems. Lazarov et al. [154] performed an experimental validation of the optimised designs from Alexandersen et al. [153] using additive manufacturing in aluminium, showing good agreement with numerical results and highlighting the superiority of topology-optimised designs. Lei et al. [155] continued this work and used investment casting to experimentally investigate a larger array of heat sink designs comparing them to optimised pin fin designs. Saglietti et al. [156] presented topology optimisation of heat sinks in a square differentially heated cavity using a spectral element method. In order to reduce the computational cost, Asmussen et al. [157] suggested an approximate flow model to that originally presented by Alexandersen et al. [109], by neglecting intertia and viscous boundary layers. 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.
