Adjoint Treatment of Flow Turbulence

Γ^∗D

µ^∗eff∂ v_i^∗

∂ n^∗ [

−∂ˆv_i^∗

∂ n^∗ ]

dΓ^∗dt^∗ → s^∗ =−µ^∗eff∂ v_i^∗

∂ n^∗

∂ˆv_i^∗

∂ 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.

4.1.8 Adjoint Treatment of Flow Turbulence

In this thesis, primal flow turbulence modeling refers to Reynolds-averaging strategies (cf.

Sec. 2.1.6) and employs BVM based on differential transport equations for two turbu-lence parameters. The influence of the variation of the turbuturbu-lence parameters is an open discussion (Marta and Shankaran [2013]), and optimizations of complex engineering flow using entirely consistent, differentiated turbulence transport models are rare (Papoutsis-Kiachagias and Giannakoglou [2016], Kapellos et al. [2019]) due to the significant increase of complexity. Primal turbulence transport models inhere multiple nonlinearities and inter-parameter couplings that significantly hamper the robustness and the efficiency of a con-sistent adjoint framework and hinder their utilization in engineering applications.

The continuous adjoint framework gives access to dedicated adjoint turbulence modelling at a lower level of adjoint consistency. One research question of the present effort is to investigate the potential of an algebraic adjoint turbulence treatment that offers the algorithmic benefits of a frozen turbulence approach. In contrast to former studies, the present effort originates from analysing the adjoint complement to a simple unidirectional turbulent shear flow. This canonical flow is the foundation of virtually all wall function-based boundary conditions and grounds on the mixing-length hypothesis, cf. Prandtl [1925], Pope [2001]. With attention given to a plane 2D b.-l., e.g. the lower half of a channel illustrated in Fig. 2.15 of Sec. 2.3.3, an unconstrained optimization problem for the drag functional (3.7) reads

min J^∗ =−µ^∗effdv₁^∗ where the index (·)_W denotes a wall value. The integral in streamwise (x^∗₁) direction has been neglected, since the channel is assumed to be fully developed and only the wall normal direction (x^∗₂) is of relevance. The adjoint velocity is dimensionless for the considered drag objective. Using ρ^∗ν^∗ =µ^∗ together with a constant density ρ^∗, the total variation of the The frozen turbulence assumption neglects the variation of the turbulent viscosity, i.e.

δν^∗t = 0→δν^∗eff = 0. An isolation ofδv₁^∗ allows the formulation of first-order optimality Here, x^∗₂ = ∆^∗ marks the position of the outer boundary. The adjoint equation to (2.149) follows from the integral expression in (4.47) and reads

Rˆ^∗₁^v,CH,F : d

The superscript (F) indicates the reference to the frozen turbulence assumption. The boundary conditions along the wall as well as the outer boundary arise from the remaining terms, viz. Following the general derivation from above, the adjoint velocity is simply zero at the wall for other cost functionals.

A consistent approach also considers the variation of the turbulent viscosity as sug-gested by the mixing length hypothesis. As outlined in Sec. 2.1.6, the turbulent viscosity exclusively depends on the tangential mean velocity next to the wall distance, and the related variation reads δν^∗t = (κx^∗₂)²(dδv^∗₁/dx^∗₂). The latter augments (4.47) towards a Interestingly, (4.51) resembles (4.47) by doubling the turbulent contribution. Hence, the consistent adjoint to (2.149) reads

Rˆ^∗₁^v,CH,C : d

The superscript (C) serves to separate the adjoint formulation based on the consistent alge-braic closure model from the frozen turbulence framework. Necessary boundary conditions follow again from the boundary parts in (4.51) and agree with Eqns. (4.49)-(4.50).

A sensitivity rule of the objective w.r.t. a general control variable depends on the defini-tion as well as on the nature of the control. For example the reladefini-tionδv^∗₁ = 0 (4.50) along the channel wall holds as long as the wall is not subjected to control, cf. (3.24). However, if the wall is examined for its optimization potential, further variational contributions follow from the concept of material derivative (3.11) and are available based on a linear develop-ment of the local flow w.r.t. a perturbation in wall normal directionδv₁^∗ =−(dv₁^∗/dx^∗₂)δx^∗₂, cf. Sec. 3.2. The latter yields a shape sensitivity derivative expression

δuL^∗·δu^∗ =−(ν^∗+β ν^∗t)d ˆv1 The coefficientβ = 1 [β = 2] accounts for a frozen [consistent] algebraic formulation and β= 1 is in line with the general shape sensitivity (4.44). The simple algebraic manipulation of the adjoint turbulent viscosity also supports more general primal BVM.

Adjoint Two-Equation Wall Functions

The study refers to the baseline k^∗ −ε^∗ model from Sec. 2.1.6, c.f. Jones and Launder [1972]. The employed wall boundary conditions are of significance. In line with Sec. 2.2.8, they refer to standard approaches and employ prescribed shear stress and pressure loads on the wall face of the wall-adjacent elements to close the primal momentum equations.

Zero wall normal gradients for the TKEk^∗ and a prescribed near-wall value of the energy dissipationε^∗, including the assurance of the local turbulence equilibrium P^∗k=ε^∗ in the wall-adjacent cell, serve to close the primal turbulence model equations, cf. Wilcox [1998].

The study resembles the investigation from before by directly imposing either the primal low-Re or the high-Re formulation. Again, an exemplary objective refers to the fluid flow-induced shear force in an internal formulation, cf. Sec. 3.2. The algorithmic structure for the low- and the high-Re situation is identical. The only difference refers to assigned specific values for the wall shear in line with either of the two solutions (2.152). The near-wall value of ϵ^∗ accommodates to the low- (ϵ^∗ = 2ν^∗k^∗/x^∗₂²) or the high-Re situation (ϵ^∗ = V_1,τ^∗3/(κx^∗₂) = (√

C^µk^∗)^3/2/(κx^∗₂)), cf. Sec. 4.2.6. As regards the adjoint approach, only the adjoint momentum is considered. The wall value ˆv^∗₁ does not differ for the low-and the high-Re situation. However, when attention is given to high-Re simulations, the resolution of ˆv₁^∗ in the very near-wall region is deemed computationally expensive and it is more convenient to follow the same implementation strategy as for the primal flow.

Employing a low-Re approach, one frequently imposes k^∗ = 0, ϵ^∗ = 2ν^∗ k^∗

x^∗₂² → ν^∗t = 0 (4.54)

for the turbulent quantities in the very near-wall regime. This again allows for the con-struction of a Lagrangian, viz.

L^∗ =− A variation is performed in App. D.2 and reveals the following relations at the inner and outer b.-l. position, viz. This again confirms the findings of Eqns. (4.49)-(4.50). Eqn. (4.57) is fulfilled if either

δv^∗₁ = 0 or δv^∗₁ =−dv₁^∗ holds that allows for a low-Re (LR) shape derivative expression (cf. Eqn. (4.53)) if a linear development of the local flow w.r.t. a perturbation in wall normal direction is applied.

Employing a high-Re k^∗−ϵ^∗ formulation, one frequently imposes k^∗ = V_1τ^∗2

√C^µ, ϵ^∗ = V_1τ^∗3

κ x^∗₂ → ν^∗t =C^µk^∗2

ϵ^∗ =V_1τ^∗ κ x^∗₂ = (κ x^∗₂)²dv^∗₁

dx^∗₂. (4.59) Hence, a possible Lagrangian, that is valid within the logarithmic layer [within in the first cell] from a continuous [discrete] perspective, reads

L^∗ =−

Substituting ϵ^∗ = V_1τ^∗3/(κ x^∗₂) = (k^∗√ A variation is performed in App. D.3. The results can be collected in a compact form and subsequently eliminated. The respective contributions at the inner and outer b.-l. position read Similar to the low-Re formulation, the relation is again fulfilled if either

δv₁^∗ = 0 or δv^∗₁ =−dv^∗₁ holds that allows for a high-Re (HR) shape derivative expression based on twice the tur-bulent viscosity (cf. Eqn. (4.53)). This is beneficial from a discrete perspective, since in line with the primal implementation, robust stress conditions that prescribe ˆτ^∗eff and ˆp^∗ (cf. Sec. 4.2.6) instead of a simple Dirichlet condition ˆv1(x2 = 0) = ˆv₁^W can be employed.

The latter requires the derivation of an adjoint complement to the universal LoW (4.171), which is outlined in Sec. 4.3.3.

One can conclude that there is no need for adjoint turbulent transport equations in the range of validity of the adjoint LoW, if the above-presented wall function-based two-equation closure is employed. An exception refers to volumetric objectives that explicitly depend on turbulent quantities and – of course – the wake regime of the b.-l. or other more complex wall-detached zones of interest. Nonetheless, the benefit can be substantial and comes at a negligible cost, which is particularly appreciated. The reason is attributed to the algebraic scaling of all mean flow and turbulence parameters with the friction velocity V_1,τ^∗ within the logarithmic layer. It should be noted that the adjoint equations possibly experience twice the primal turbulent viscosity since β = 2 [β = 1] is chosen in the

consistent [frozen] case. Since the suggested approach is only consistent in the sub-layer and the logarithmic region, it can only be hypothesized that the consistency improves compared to the frozen turbulence approach for other applications. However, shape optimization problems are by definition interested in the primal / adjoint near-wall flow. Hence a consistent adjoint formulation is particularly relevant in this region. Using a two-equation model, the consistency is restricted to the momentum equation and assumes the eddy-viscosity distribution to agree with the mixing-length results, e.g. 2µ^∗effSˆ_ik^∗ → 2 (µ^∗ + β µ^∗t) ˆS_ik^∗ in (4.18). Moreover, the robustness of the adjoint numerical procedure benefits from an augmented viscosity.