3.7 Optimization Procedure
4.1.6 Adjoint Two-Phase Reynolds-Averaged Navier-Stokes System
All functional derivatives including volume integrals are summarized in a compact form
δpL∗Ω·δp∗ =
where the superscript (Ω) indicates the volumetric part of the differentiated Lagrangian.
The integration limits of the spatial cost functional are conceptually extended to the entire volume. Contributions of the cost functional are active in areas of Ω∗O ∩Ω∗ ̸= ∅ only.
Assuming temporal independence of the cost functional allows a combination of both integrants in each differentiated direction. Demanding first-order optimality conditions
finally yields the adjoint field equations, viz.
R∗ˆp =−∂vˆ∗k
∂x∗k +∂ j∗Ω
∂ p∗ = 0 (4.16)
R∗ˆc =−∂ˆc
∂ t∗ −vk∗∂ˆc∗
∂ x∗k −ρ∗∆
[ ˆ
vi∗gi∗−ˆvi∗vk∗∂ vi∗
∂ x∗k − ∂ µ∗t
∂ ρ∗ Sik∗ ∂vˆ∗i
∂ x∗k ]
+µ∗∆ [
2Sik
∂vˆ∗i
∂ x∗k ]
+ ∂ j∗Ω
∂ c = ∂
∂ x∗k [
ν∗c∂ˆc∗
∂ x∗k ]
(4.17) R∗iˆvi =−ρ∗
[∂vˆ∗i
∂ t∗ +v∗k∂vˆi∗
∂ x∗k ]
+ ∂
∂ x∗k [
ˆ
p∗δik−2µ∗effSˆik∗ ]
+ ˆc∗ ∂ c
∂ x∗i +ρ∗ˆvk∗∂ v∗k
∂ x∗i +∂ j∗Ω
∂ vi∗ = 0. (4.18) The adjoint equations are similar to their primal companions. However, additional advec-tion and cross-coupling terms occur. Above all, the adjoint concentraadvec-tion equaadvec-tion contains significantly more terms that scale with the two fluids’ density or viscosity difference. The term on the r.h.s. of (4.17) is of particular importance since this additional diffusivity bridges the gap between an adjoint sharp vs. an under-resolved diffusive interface formu-lation. The latter is illustrated utilizing a plane Couette flow example discussed in Sec.
4.3.2.
Interpretation of Primal vs. Dual Time Horizon
Originating from integration by parts, the adjoint time integration is inverted, e.g. declin-ing w.r.t. the physical time. The latter is why the complete primal solution has to be stored during the (forward) integration to be available to the adjoint solver for its subsequent (backward) integration. Several memory-reduced methods such as one-shot (piggy-bag) methods (Gherman and Schulz [2005], Kuruvila et al. [1994], ¨Ozkaya and Gauger [2009]) for pseudo-time stepping simulations or check-pointing approaches (Giering and Kaminski [1998], Griewank and Walther [2000], Hinze and Sternberg [2005], Hinze et al. [2006]) were suggested. They reduce the memory requirements at the expense of enhanced (partially rerunning) computing efforts, cf. Kapellos et al. [2019]. The formulation of the present adjoint problem is pseudo-time dependent. Although the primal flow problem is formerly unsteady, the steady state in terms of, e.g. the wave field att∗ → ∞(∆t∗ → ∞) is sought.
4.1.7 Boundary Conditions
The adjoint boundary conditions result from the boundary integrals of Eqns. (4.4), (4.6), (4.12) which likewise have to disappear for all variations (cf. Tab. 4.1). They are
summa-rized in a compact form, viz.
where the superscript (Γ) indicates the boundary part of the differentiated Lagrangian.
The adjoint boundary conditions follow from the first-order optimality conditions for the boundary Lagrangian and also consider the variation of primal boundary conditions de-noted in Tab. 3.1. The respective companions of the primal boundary conditions are denoted below. Separate formulations are employed to describe conditions for boundaries which contain cost functional contributions.
• Along the Inlet, both the primal velocity and the concentration field are fixed and their variationsδvi∗ = 0 andδc= 0 vanish, cf. Sec. 3.4. The pressure is extrapolated with zero gradient, hence ∂δp∗/∂n∗ = 0. Inserting these relations in (4.19)-(4.21) yields the remaining parts of the derivatives of the Lagrangian along the inlet
δpL∗Γ ·δp∗ = According to (4.22) and (4.24), first-order optimality conditions demand a zero ad-joint velocity ˆvi∗ = 0. A condition results for the CH-VoF adjoint concentration only (ν∗c ̸= 0) and requires a vanishing Dirichlet value ˆc∗ = 0. VoF methods neglect the apparent concentration viscosity (ν∗c = 0) and therefore support any boundary condition. In the present work, the same Dirichlet condition is prescribed for the VoF and the CH-VoF formulation for consistency reasons. The adjoint pressure does not appear explicitly in the derivatives. In line with the primal formulation, a zero gradient is employed.
Further variational contributions arise if the external force functional is active, cf.
(3.22), viz. δJ∗Γ,E = [2µ∗effδSik∗ nk − δp∗ni]ri and the derivatives for the force-objective based Lagrangian are augmented
δpL∗Γ,E ·δp∗ = Therefore, the adjoint velocity is modified w.r.t. the (non-dimensional) force direc-tion to be minimized, viz. ˆvi = ri. Boundary conditions for adjoint pressure and concentration are not affected.
• A hydrostatic pressure is prescribed along theOutlet. As a consequence, its variation δp∗ = 0 is zero which directly fulfills the derivative of the boundary-based Lagrangian in pressure direction (4.19). All other quantities are extrapolated with zero gradients, e.g. ∂ δvi∗/∂n∗ = 0 and ∂ δc/∂n∗ = 0. An outlet far away from a fluid flow exposed geometry motivates the neglect of Sik∗ → 0 (δSik∗ = 0) and thus also k∗ → 0. As a consequence, the boundary-based Lagrangian simplifies to
δcL∗Γ·δc=
The adjoint velocity and the adjoint pressure are coupled via relation (4.29). In line with the primal formulation, a zero gradient is employed for the adjoint velocity (∂vˆi∗/∂ n∗ = 0) and the adjoint pressure is explicitly fixed via a Dirichlet condition to
ˆ
p∗out =[
vk∗ρ∗vˆ∗i +µ∗effSˆik∗ nk]
ni. (4.30)
VoF approaches follow fromν∗c = 0 which gives rise to a zero adjoint concentration value, i.e. ˆc∗ = 0, from (4.28). For the CH-VoF case, it is assumed that the ratio of convective to diffusive concentration transport along the outlet justifies the neglect of the latter and also ˆc∗ = 0 is employed.
In line with the inlet boundary, further variational contributions arise if the external force functional is active, viz. δJ∗Γ,E = −[δvi∗ρ∗vk∗ +vi∗(ρ∗∆δc vk∗+ρ∗δvk∗)]nkri, cf.
(3.22). Hence, both remaining derivatives can be simplified to
The adjoint concentration experiences a contribution from the variation of the con-vective momentum transport and the adjoint pressure-velocity relationship picks up the direction of the force to be minimized, viz.
ˆ
• Symmetry Planes and Slip Walls experience zero normal gradients for primal pressure as well as primal concentration, viz. ∂ δp∗/∂ n∗ = 0 and ∂ δc/∂ n∗ = 0.
The velocity field has a zero Dirichlet normal as well as zero Neumann tangential component, e.g., δvi∗ni = 0 and (∂ δvi∗/∂ n∗)ti= 0. The respective derivatives of the
The adjoint velocity field experiences the same distinction compared to the primal field, i.e. no adjoint flux normal to the symmetry plane (ˆv∗i ni = 0) and no normal gradient of its tangential component, viz. (∂ˆvi∗/∂ n∗)ti. The latter follows from a decomposition into wall normal and wall tangential components of the remaining parts of Eqn. (4.36). A zero gradient condition is employed for the adjoint pressure and concentration.
• Along No-Slip Walls, zero normal gradients are prescribed for (effective) primal pressure as well as primal concentration, viz. ∂ δp∗/∂ n∗ = 0 and∂ δc/∂ n∗ = 0, and
the boundary-based Lagrangian takes the following simplified form
Therefore, the adjoint velocity is simply zero (ˆv∗i = 0) and the concentration variation requires a zero adjoint concentration gradient ∂ˆc∗/∂ n∗ = 0. Following the primal model, the adjoint pressure gradient is likewise set to zero, i.e. ∂pˆ∗/∂ n∗ = 0.
Further variational contributions occur if an internal force objective is active along a no-slip wall, viz. δJ∗Γ,I = [δp∗δik−2δµ∗effSik∗ −2µ∗effδSik∗]nkri, cf. (3.17), which
The –now dimensionless– adjoint velocity reads ˆvi = −ri and boundary conditions for adjoint concentration and pressure are not affected.
If the actual no-slip wall is not released for design, the variation of the velocity vanishes (δvi∗ = 0) and all optimality conditions are satisfied. However, if the wall is to be controlled, further contributions arise that have been discussed in Sec. 3.4, e.g.
a linear development of the flow reveals δv∗i = −δu∗(∂ v∗i/∂ n∗) where δu∗ =δn∗ = δx∗knk along ΓD, cf. (3.26). Since all respective optimality conditions are satisfied, sensitivity information can only be obtained from the final optimality condition in terms of a derivative of the Lagrangian in the direction of the control, i.e.
δuL∗ ·δu∗ =
boundary type vˆ∗i pˆ∗ cˆ∗ inlet vˆi∗ = 0 ∂ n∂pˆ∗∗ = 0 cˆ∗ = 0 outlet ∂ n∂vˆi∗∗ = 0 pˆ∗ = ˆp∗out cˆ∗ = 0 symmetry ˆvi∗ni = 0, ∂ n∂ˆv∗i∗ti = 0 ∂ n∂pˆ∗∗ = 0 ∂ n∂cˆ∗∗ = 0 wall (slip) ˆvi∗ni = 0, ∂ n∂ˆv∗i∗ti = 0 ∂ n∂pˆ∗∗ = 0 ∂ n∂cˆ∗∗ = 0 wall (no-slip) vˆi∗ = 0 ∂ n∂pˆ∗∗ = 0 ∂ n∂cˆ∗∗ = 0 inlet (Γ⊂ΓO,E) vˆi =ri ∂pˆ∗
∂ n∗ = 0 cˆ∗ = 0 outlet (Γ⊂ΓO,E) ∂ n∂vˆi∗∗ = 0 pˆ∗ = ˆp∗out,E ˆc∗ = ˆc∗out,E wall (no-slip, Γ⊂ΓO,I) vˆi =−ri ∂pˆ∗
∂ n∗ = 0 ∂∂ncˆ∗∗ = 0
Table 4.1: Boundary conditions for the adjoint equations where ti [ni] refer to the local boundary tangential [normal]. The first five lines refer to the general boundary conditions in the absence of any active boundary-based cost functional.
The latter gives rise to the desired shape sensitivity derivative s∗, which follows from (∂ vi∗/∂ n∗)ni= 0 and reads
δuL∗ =
∫ ∫
Γ∗D
µ∗eff∂ vi∗
∂ n∗ [
−∂ˆvi∗
∂ n∗ ]
dΓ∗dt∗ → s∗ =−µ∗eff∂ vi∗
∂ n∗
∂ˆvi∗
∂ n∗. (4.44) The relation will be further explored below in terms of a wall function-based turbulent flow description.
All adjoint boundary conditions are summarized in Tab. 4.1. Mind that adjoint flow turbulence has been neglected, but the subsequent Sec. 4.1.8 is concerned with an adjoint LoW. It is assumed, that a consistent adjoint treatment close to design boundaries is of particular importance. Therefore, the study analyses the adjoint complement to a simple unidirectional turbulent shear, which is the foundation of virtually all wall function-based primal boundary conditions using a classical mixing-length hypothesis, cf. Prandtl [1925], Pope [2001]. Note that the vast majority of primal boundary condition implementations resembles this generic flow model and assumes negligible curvature to split the boundary forces into normal and tangential traction, cf. Sec. 2.2.8. The following study sugests a similar strategy for the adjoint velocity boundary condition and complies with all adjoint velocity boundary conditions outlined in Tab. 4.1.