Continuous Adjoint Methods for Turbulent Flows, Applied to Shape and Topology Optimization: Industrial Applications

Abstract

This article focuses on the formulation, validation and application of the continuous adjoint method for turbulent flows in aero/hydrodynamic optimization. Though discrete adjoint has been extensively used in the past to compute objective function gradients with respect to (w.r.t.) the design variables under turbulent flow conditions, the development of the continuous adjoint variant for these flows is not widespread in the literature, hindering, to an extend, the computation of exact sensitivity derivatives. The article initially presents a general formulation of the continuous adjoint method for incompressible flows, under the commonly used assumption of “frozen turbulence”. Then, the necessary addenda are presented in order to deal with the differentiation of both low- and high-Reynolds (with wall functions) number turbulence models; the latter requires the introduction of the so-called “adjoint wall functions”. An approach to dealing with distance variations is also presented. The developed methods are initially validated in $2D$ cases and then applied to industrial shape and topology optimization problems, originating from the automotive and hydraulic turbomachinery industries.

Discussion on the Total Pressure Losses and Fluid Power Dissipation as Objective Functions

In Sect. 7, the volume-averaged total pressure losses and the fluid power dissipation are used as the objective functions for problems related to internal aerodynamics.

The relation between Eqs. 78 and 113 for topology optimization problems is scrutinized in this “Appendix”. If term $\alpha v_i^2$ is neglected from Eq. 113 (along with the porous friction term, $T_{a,v}$, in Eq. 101b) the proof that follows holds for shape optimization problems as well.

The total kinetic energy of an incompressible fluid is

$$\begin{aligned} E_{kin} = \frac{1}{2}\int _{\Omega } v_i^2 d{\Omega } \end{aligned}$$

(120)

while its time derivative reads

$$\begin{aligned} \frac{\partial E_{kin}}{\partial t} = \int _{\Omega } v_i \frac{\partial v_i}{\partial t} d\Omega \end{aligned}$$

(121)

Using the momentum equations in which the porosity dependent term $\alpha v_i$ has been added, the integrand on the r.h.s. of Eq. 121 is written as

$$\begin{aligned} v_i \frac{\partial v_i}{\partial t}&= \underbrace{v_i v_j \frac{\partial v_i}{\partial x_j}}_{term 1} + \underbrace{v_i \frac{\partial p}{\partial x_i}}_{term 2} - \underbrace{v_i \frac{\partial }{\partial x_j} \left[ \left( \nu + \nu _t\right) s_{ij}\right] }_{term 3} + \alpha v_i^2 \end{aligned}$$

(122)

where the strain tensor, $s_{ij}$, is given by

$$\begin{aligned} s_{ij}&= \left( \frac{\partial v_i}{\partial x_j}+ \frac{\partial v_j}{\partial x_i}\right) \end{aligned}$$

(123)

By taking into account the continuity equation, the development of the terms appearing on the r.h.s. of Eq. 122 yields

$$\begin{aligned} term1&:~ v_i v_j \frac{\partial v_i}{\partial x_j} = v_j \frac{1}{2}\frac{\partial (v_i^2)}{\partial x_j}= \frac{1}{2} \frac{\partial (v_j v_i^2) }{\partial x_j} \end{aligned}$$

(124)

$$\begin{aligned} term2&:~ v_j \frac{\partial p}{\partial x_j} = \frac{\partial \left( v_j p\right) }{\partial x_j} \end{aligned}$$

(125)

$$\begin{aligned} term3&:~ v_i \frac{\partial }{\partial x_j} \left[ \left( \nu + \nu _t\right) s_{ij} \right] = \frac{\partial }{\partial x_j} \left[ \left( \nu + \nu _t\right) v_i s_{ij}\right] \nonumber \\&\quad \qquad \qquad \qquad \qquad \qquad \qquad - \left( \nu + \nu _t\right) s_{ij} \frac{\partial v_i}{\partial x_j} \end{aligned}$$

(126)

After substituting Eqs. 124–126 into Eq. 122, we get

$$\begin{aligned} v_i \frac{\partial v_i}{\partial t}&= \frac{\partial }{\partial x_j} \left[ v_j \left( \frac{1}{2} v^2 +p\right) \right] -\frac{\partial }{\partial x_j} \left[ \left( \nu + \nu _t\right) v_i s_{ij} \right] \nonumber \\&\quad + \left( \nu + \nu _t\right) s_{ij} \frac{\partial v_{i}}{\partial x_j} + \alpha v_i^2 \end{aligned}$$

(127)

The Frobenius inner product of the strain tensor with the velocity gradient can be written as

$$\begin{aligned} \frac{\partial v_i}{\partial x_{j}} s_{ij}&= \frac{\partial v_i}{\partial x_{j}} \left( \frac{\partial v_i}{\partial x_{j}} + \frac{\partial v_j}{\partial x_{i}} \right) \nonumber \\&= \frac{1}{2}\frac{\partial v_i}{\partial x_{j}} \frac{\partial v_i}{\partial x_{j}} + \frac{1}{2}\frac{\partial v_j}{\partial x_{i}} \frac{\partial v_j}{\partial x_{i}} + \frac{\partial v_i}{\partial x_{j}}\frac{\partial v_j}{\partial x_{i}}\nonumber \\&= \frac{1}{2} \left( \frac{\partial v_i}{\partial x_{j}} + \frac{\partial v_j}{\partial x_{i}} \right) ^2 = \frac{1}{2} s_{ij}^2 \end{aligned}$$

(128)

Substituting Eq. 128 into Eq. 127, we get

$$\begin{aligned} v_i \frac{\partial v_i}{\partial t}&= \frac{\partial }{\partial x_j} \left[ v_j \left( p + \frac{1}{2} v_i^2 \right) \right] -\frac{\partial }{\partial x_j} \left[ \left( \nu + \nu _t\right) v_i s_{ij} \right] \nonumber \\&\quad + \frac{\left( \nu + \nu _t\right) }{2} s_{ij}^2 + \alpha v_i^2 \end{aligned}$$

(129)

For a steady state problem, the time derivative on the l.h.s. of Eq. 129 is zero. Taking this into consideration and using the Green–Gauss theorem for the conservative terms, the integration of Eq. 129 over $\Omega$ yields

$$\begin{aligned} -\int _{S} \left[ \left( p + \frac{1}{2} v_i^2 \right) \right] v_j n_j dS&= \int _{\Omega } \left[ \frac{\left( \nu \!+\!\nu _t \right) }{2} s_{ij}^2 + \alpha v_i^2 \right] d\Omega \nonumber \\&\quad - \int _{S} \left( \nu + \nu _t\right) v_i s_{ij} n_j dS \end{aligned}$$

(130)

The boundary integrals in Eq. 130 are zero along $S_W$ due to the no-slip velocity boundary condition. Thus, Eq. 130 becomes

$$\begin{aligned}&-\int _{S_{I,O}} \left[ \left( p + \frac{1}{2} v_i^2 \right) \right] v_j n_j dS\nonumber \\&\quad = \int _{\Omega } \left[ \frac{\left( \nu +\nu _t \right) }{2} s_{ij}^2 + \alpha v_i^2 \right] d\Omega - \int _{S_{I,O}}\!\! \left( \nu + \nu _t\right) v_i s_{ij} n_j dS \end{aligned}$$

(131)

If the flow at $S_I$ and $S_O$ is sufficiently free from intense flow gradients, it can be assumed that last integral on the r.h.s. of Eq. 130 is negligible.² Under this assumption and after taking into consideration Eq. 123, we get

$$\begin{aligned} - \int _{S_{IO}} \left[ \left( p + \frac{1}{2}v^2 \right) v_in_i\right] dS&\approx \int _{\Omega } \left[ \frac{\left( \nu + \nu _t\right) }{2} s_{ij}^2 +\alpha v^2_i \right] d\Omega \end{aligned}$$

(132)

or

$$\begin{aligned} F_{p_t} \approx F_{PL} \end{aligned}$$

(133)