Topology optimization method based on the Wray–Agarwal turbulence model

Abstract

One of the current challenges in fluid topology optimization is to address these turbulent flows such that industrial or more realistic fluid flow devices can be designed. Therefore, there is a need for considering turbulence models in more efficient ways into the topology optimization framework. From the three possible approaches (DNS, LES, and RANS), the RANS approach is less computationally expensive. However, when considering the RANS models that have already been considered in fluid topology optimization (Spalart–Allmaras, k–ε, and k–ω models), they all include the additional complexity of having at least two more topology optimization coefficients (normally chosen in a “trial and error” approach). Thus, in this work, the topology optimization method is formulated based on the Wray–Agarwal model (“WA2018”), which combines modeling advantages of the k–ε model (“freestream” modeling) and the k–ω model (“near-wall” modeling), and relies on the solution of a single equation, also not requiring the computation of the wall distance. Therefore, this model requires the selection of less topology optimization parameters, while also being less computationally demanding in a topology optimization iterative framework than previously considered turbulence models. A discrete design variable configuration from the TOBS approach is adopted, which enforces a binary variables solution through a linearization, making it possible to achieve clearly defined topologies (solid–fluid) (i.e., with clearly defined boundaries during the topology optimization iterations), while also lessening the dependency of the material model penalization in the optimization process (Souza et al. 2021) and possibly reducing the number of topology optimization iterations until convergence. The traditional pseudo-density material model for topology optimization is adopted with a nodal (instead of element-wise) design variable, which enables the use of a PDE-based (Helmholtz) pseudo-density filter alongside the TOBS approach. The formulation is presented for axisymmetric flows with rotation around an axis (“2D swirl flow model”). Numerical examples are presented for some turbulent 2D swirl flow configurations in order to illustrate the approach.

Appendices

Appendix A: Comparison of sensitivities with finite differences

A comparison of the computed sensitivities (using dolfin-adjoint) with finite differences is presented in this appendix. The comparison is performed for the optimized topology for the rotating nozzle for turbulent flow under 10 L/min and 2500 rpm (Sect. 6.1), by considering the simulations for laminar (0.06 L/min and 25 rpm) and turbulent (10 L/min and 2500 rpm) flows. The set of points selected for comparison with finite differences in the computational domain is shown in Fig. 25. The comparison is performed for the same configurations considered for laminar and turbulent flows in Sect. 6.1. For $\alpha = 1$ (fluid), the finite difference approximation is considered through the backward difference approximation: $\frac{\text{{d}}J}{\text{{d}}\alpha } = \frac{J(\alpha ) - J(\alpha - {\Delta }\alpha )}{{\Delta }\alpha }$, where $J = {\Phi }_{\text {rel}}$. For $\alpha = 0$ (solid), the finite difference approximation is considered through forward difference approximation: $\frac{\text{{d}}J}{\text{{d}}\alpha } = \frac{J(\alpha + {\Delta }\alpha ) - J(\alpha )}{{\Delta }\alpha }$. The computed sensitivities are shown in Fig. 26, for a step size of $10^{-3}$. As can be seen, the computed sensitivities for this work (by using dolfin-adjoint) and finite differences are close to each other. For a better insight about the differences between the two sensitivities, Fig. 27 depicts the relative differences as defined below, which resulted small. The computed relative difference values may be viewed in sight of the fact that smaller objective function values may hinder the computation of finite differences due to computational errors, as observed in Yoon (2020); Haftka and Gürdal (1991). Furthermore, since a discrete algorithm (not continuous) is being considered in this work, this amount of difference does not seem to pose a problem.

$$\begin{aligned} \left. r_d\right| _\text {laminar} = \left. \frac{\left. \frac{\text{{d}}J}{\text{{d}}\alpha }\right| _\text {FD} - \left. \frac{\text{{d}}J}{\text{{d}}\alpha }\right| _\text {p}}{\text {max}\left| \frac{\text{{d}}J}{\text{{d}}\alpha }\right| _\text {p,\ all\ points}}\right| _\text {laminar} \end{aligned}$$

(31)

$$\begin{aligned} \left. r_d\right| _\text {turbulent} = \left. \frac{\left. \frac{\text{{d}}J}{\text{{d}}\alpha }\right| _\text {FD} - \left. \frac{\text{{d}}J}{\text{{d}}\alpha }\right| _\text {p}}{\text {max}\left| \frac{\text{{d}}J}{\text{{d}}\alpha }\right| _\text {p,\ all\ points}}\right| _\text {turbulent} \end{aligned},$$

(32)

where the subscript “p” indicates the “present work” approach (by using dolfin-adjoint) and “FD” indicates “Finite Differences.” The relative differences values are higher in the turbulent case due to the higher non-linearity of the fluid flow problem.

Fig. 25

Topology considered for the finite differences comparison

Fig. 26

Sensitivity values computed with the approach of the present work (from FEniCS/dolfin-adjoint) and from finite differences, for laminar and turbulent flows

Fig. 27

Relative differences for the cases shown in Fig. 26

Appendix B: 2D double pipe

The design of a 2D double pipe is performed in this Appendix, in order to show the Wray–Agarwal model (2018) being considered for topology optimization in a 2D case for a different initial guess configuration for the design variable. The topology optimization for this type of problem has been previously evaluated by Borrvall and Petersson (2003), for laminar Stokes flow. In this work, the inlets are set to be larger, which is reflected in setting the specified fluid volume fraction (f) as 50%, in order to make possible the formation of straight channels connecting the inlets to the outlets. Also, the outlet flow boundary condition is set as “stress free,” which is more generic (Hasund 2017) with respect to imposing fixed outlet velocity profiles as Borrvall and Petersson (2003). The design domain is shown in Fig. 28.

Fig. 28

Design domain for the 2D double pipe

The mesh consists of 16,181 nodes and 32,000 elements (i.e., 100 horizontal and 80 vertical rectangular partitions of crossed triangular elements, see Fig. 29). The input parameters, geometric dimensions, and material model parameters that are considered for the design are shown in Table 5. The maximum inlet Reynolds number is 0.375 (laminar flow case) and $2.8{\times }10^4$ (turbulent flow case). In order to consider a different configuration for the initial guess with respect to the other examples, the initial guess is chosen as shown in Fig. 30, where $d = 37.5$ mm and $r_c = 3.75$ mm. The TOBS approach is considered for $\varepsilon _\text {relax} = 0.1$ and $\beta _\text {flip\ limit} = 0.1$ (laminar flow case), and for $\varepsilon _\text {relax} = 0.05$ and $\beta _\text {flip\ limit} = 0.05$ (turbulent flow case).

Fig. 29

Mesh used in the design of the 2D double pipe

Fig. 30

Initial guess used in the design of the 2D double pipe

Table 5 Parameters used for the topology optimization of the 2D double pipe

The optimized topologies are shown in Fig. 31, where the maximum local Reynolds number is given as 0.46 (laminar flow case) and $2.9{\times }10^4$ (turbulent flow case). As can be noticed, both optimized topologies are essentially different, where both channels join in the middle of the design domain for the laminar flow (similarly to the optimized results obtained by Borrvall and Petersson (2003)), but are kept separated for the turbulent flow. This difference in the optimized topologies shown in Fig. 31 can be viewed from the fact that the high inlet fluid flow velocity from the turbulent flow case does not allow creating a bend in the channel without dissipating significantly more energy, while the energy expended for that in the laminar flow case is minimal. This is also shown in the energy dissipation values from Table 6, where it can be seen that the optimized topologies perform better for their respective fluid flow regimes (43% better in the laminar flow case, and 46% better in the turbulent flow case).

Fig. 31

Optimized topologies for the 2D double pipes ($f =$ 50%)

Table 6 Energy dissipation computed for the laminar and turbulent flow optimized topologies operating under each others’ flow configurations for the 2D double pipes

Appendix C: 2D U-bend channel

The design of a 2D U-bend channel is performed in this Appendix, in order to show the Wray–Agarwal model (2018) being considered for topology optimization in a 2D case. This topology optimization problem has been previously evaluated by Dilgen et al. (2018), for the Spalart–Allmaras and k-$\omega$ models considering the MMA (Method of Moving Asymptotes) algorithm. The 2D U-bend channel consists of an inlet and an outlet next to each other, with a “rod”-like structure in the middle of the channel that forces the optimized channel to go around it (see Fig. 32). The inlet and outlet zones, as well as the “rod”-like structure are kept as “non-optimizable.”

Fig. 32

Design domain for the 2D U-bend channel

Since there is a “non-optimizable” zone inside the computational domain (see Fig. 32), when considering the Helmholtz pseudo-density filter, the filter is applied over the whole computational domain (${\Omega }$); however, the design variable value outside the design domain (${\Omega }\setminus {\Omega }_\alpha$) is enforced as the previous value ($\alpha$). Therefore, the variable ${\alpha }_{f,\text {new}}$ is used in the equations in function of the position (${\varvec{s}}$):

$${\alpha }_{f,\text {new}} ({\varvec{s}}) = \left\{\begin{array}{ll} \alpha_f, & \quad \text {if}\; {\varvec{s}} \in {\Omega }_{\alpha}\\ \alpha, & \quad \text{if} \;{\varvec{s}} \in {\Omega} \setminus {\Omega}_{\alpha}\end{array} \right.$$

(33)

The local Reynolds number from Eq. (25) is redefined for this case, where the characteristic length is set as the width of the inlet (L, from Fig. 32) in the place of the external diameter (“$2 r_{{ext}}$”). As in the other numerical examples, the fluid is being considered as water.

The mesh consists of 92,431 nodes and 184,000 elements (i.e., 230 horizontal and 200 vertical rectangular partitions of crossed triangular elements, see Fig. 33). The input parameters, geometric dimensions, and material model parameters that are considered for the design are shown in Table 7. The maximum inlet Reynolds number is $5.4{\times }10^4$. In this work, in order to facilitate convergence, the initial guess is given from the reference topology from Fig. 34. The specified fluid volume fraction (f) is set as 30%. The TOBS approach is considered for $\varepsilon _\text {relax} = 0.2$ and $\beta _\text {flip\ limit} = 0.001$.

Fig. 33

Mesh used in the design of the 2D U-bend channel

Fig. 34

Reference topology for the 2D U-bend channel

Table 7 Parameters used for the topology optimization of the 2D U-bend channel

The optimized topology is shown in Fig. 35, with the corresponding sensitivities’ distribution shown in Fig. 36, where the maximum local Reynolds number is given as $2.7{\times }10^6$. As can be noticed, the optimized channel is expanded outwards with respect to the “rod”-like structure. Also, the optimized topology features a wider curve than the reference topology (Fig. 34). As a comparison, the optimized topology resulted in 35% less energy dissipation ($1.10{\times }10^2$ W/m vs. $1.70{\times }10^2$ W/m), and 11% ($2.72{\times }10^{-2}$ m vs. $3.04{\times }10^{-2}$ m) less head loss than the reference topology.

Fig. 35

Optimized topology for the 2D U-bend channel

Fig. 36

Sensitivities in the last optimization iteration, for the optimized 2D U-bend channel

The simulation of the optimized design is shown in Fig. 37. From it, it can be noticed that the small “bumps” of the optimized topology near the inlet and outlet of the design domain do not significantly change the velocity field of the fluid flow simulation, meaning that their contribution to the energy dissipation should be small. Also, the curve in the optimized channel reduces the bending necessary for the fluid to head towards the outlet, reducing energy dissipation and head loss.

Fig. 37

Optimized topology and variables for the optimized 2D U-bend channel

Figure 38 shows a comparison of streamlines between the simulations for the modeled solid material, for the Helmholtz filter ($r_H =$ 1.0 mm) and for the post-processed topology. As can be noticed, some differences arise between the simulations. First, the streamlines for the modeled solid simulation show an acceleration of the fluid flow after the curve, which is not present in the simulation of the post-processed topology. This effect can also be observed in Dilgen et al. (2018) (Figs. 10c and 14b of Dilgen et al. (2018)). This seems to be a drawback of the modeled material used in fluid topology optimization (inverse permeability), since this modeled material intrinsically implies that there will be fluid flow inside the solid material, even at extremely small values, which may possibly lead to small changes in the characteristics of the solid boundaries and may affect the turbulent flow modeling. When considering the Helmholtz filter, the boundaries are blurred, which has the effect of leading the topology optimization to focus on the main flow, giving less emphasis to local effects that may possibly destabilize the fluid flow (such as some specific inclusions in the middle of the channel) or lead the topology optimization to a relatively worse local minimum.

Fig. 38

Streamlines in the 2D U-bend channel optimized topology, by considering the simulation for the modeled solid material, the simulation when including the Helmholtz filter ($r_H =$ 1.0 mm) and the simulation when considering the post-processed topology