On the Continuous Adjoint of Prominent Explicit Local Eddy Viscosity-based Large Eddy Simulation Approaches for Incompressible Flows

Abstract

The manuscript deals with continuous adjoint companions of prominent explicit Large Eddy Simulation (LES) methods grounding on the eddy viscosity assumption for incompressible fluids. The subgrid-scale approximations considered herein address the classic Smagorinsky-Lilly, the Wall-Adapting Local Eddy-Viscosity (WALE), and the Kinetic Energy Subgrid-Scale (KESS) model, whereby only static implementations, i.e., those without dynamically adjusted model parameters, are considered. The associated continuous adjoint systems and resulting shape sensitivity expressions are derived. Information on the consistent discrete implementation is provided that benefits from the self-adjoint primal discretization of convective and diffusive fluxes via unbiased, symmetric approximations, frequently performed in explicit LES studies to minimize numerical diffusion. Algebraic primal subgrid-scale models yield algebraic adjoint LES relationships that resemble additional adjoint momentum sources. The KESS one equation model introduces an additional adjoint equation, which enlarges the resulting continuous adjoint KESS system with potentially increased numerical stiffness. The different adjoint LES methods are tested and compared against each other on a flow around a circular cylinder at $\text{Re}_\text{D} = {140000\,}$ for a boundary (drag) and volume (deviation from target velocity distribution) based cost functional. Since all primal implementations predict similar flow fields, it is possible to swap the associated adjoint systems –i.e., applying an adjoint WALE method to a primal KESS result– and still obtain plausible adjoint results. Due to the LES’s inherent unsteady character, the adjoint solver requires the entire primal flow field over the cost-functional relevant time horizon. Even for the academic cases studied herein, the storage capacities are in the order of terabytes and refer to a practical bottleneck. However, in the case of suitable, time-averaged cost functional, the time-averaged primal flow field can be used directly in a steady adjoint solver, which results in a drastic effort reduction.

Appendices

A Standard Laminar Variations

A state perturbation and subsequent isolation is applied to:

• the continuity equation (1):

  $$\begin{aligned} \frac{1}{T} \int _{t^\textrm{s}}^{t^e} \int \hat{p} \frac{\partial \overline{v}_\textrm{k}^\prime }{\partial x_\textrm{k}} \textrm{d} \Omega \textrm{d} t = \frac{1}{T} \int _{t^\textrm{s}}^{t^e} \int \hat{p} \overline{v}_\textrm{k}^\prime n_\textrm{k} \textrm{d} \Gamma \textrm{d} t - \frac{1}{T} \int _{t^\textrm{s}}^{t^e} \int \overline{v}_\textrm{k}^\prime \frac{\partial \hat{p}}{\partial x_\textrm{k}} \textrm{d} \Omega \textrm{d} t \end{aligned}$$

  (70)

• the momentum equations (2)

  $$\begin{aligned} \frac{1}{T} \int _{t^\textrm{s}}^{t^e} \int \hat{v}_\textrm{i} \frac{\partial \rho \overline{v}_\textrm{i}^\prime }{\partial t} \textrm{d} \Omega \textrm{d} t&= \frac{1}{T} \bigg [ \int \hat{v}_\textrm{i} \rho \overline{v}_\textrm{i} \textrm{d} \Omega \bigg ]_{t_s}^{t_e} - \frac{1}{T} \int _{t^\textrm{s}}^{t^e} \int \overline{v}_\textrm{i}^\prime \frac{\partial \rho \hat{v}_\textrm{i}}{\partial t} \textrm{d} \Omega \textrm{d} t \end{aligned}$$

  (71)
  $$\begin{aligned} \frac{1}{T} \int _{t^\textrm{s}}^{t^e} \int \hat{v}_\textrm{i} \frac{\partial (\overline{v}_\textrm{k} \rho \overline{v}_\textrm{i})^\prime }{\partial x_\textrm{k}} \textrm{d} \Omega \textrm{d} t&= \frac{1}{T} \int _{t^\textrm{s}}^{t^e} \int \hat{v}_\textrm{i} \rho \left( \overline{v}_\textrm{k}^\prime \overline{v}_\textrm{i} + \overline{v}_\textrm{k} \overline{v}_\textrm{i}^\prime \right) n_\textrm{k} \textrm{d} \Gamma \textrm{d} t \nonumber \\&\qquad - \frac{1}{T} \int _{t^\textrm{s}}^{t^e} \int \overline{v}_\textrm{i}^\prime \left[ \rho \overline{v}_\textrm{k} \frac{\partial \hat{v}_\textrm{k}}{\partial x_\textrm{i}} + \frac{\partial \overline{v}_\textrm{k} \rho \hat{v}_\textrm{i}}{\partial x_\textrm{k}} \right] \textrm{d} \Omega \textrm{d} t \end{aligned}$$

  (72)
  $$\begin{aligned} \frac{1}{T} \int _{t^\textrm{s}}^{t^e} \int \hat{v}_\textrm{i} \frac{\partial p^\prime }{\partial x_\textrm{i}} \textrm{d} \Omega&= \frac{1}{T} \int _{t^\textrm{s}}^{t^e} \int \hat{v}_\textrm{i} p^\prime n_\textrm{i} - \int p^\prime \frac{\partial \hat{v}_\textrm{i}}{\partial x_\textrm{i}} \textrm{d} \Omega \textrm{d} t \end{aligned}$$

  (73)
  $$\begin{aligned} \frac{1}{T} \int _{t^\textrm{s}}^{t^e} \int \hat{v}_\textrm{i} \frac{\partial }{\partial x_\textrm{k}} \left[ 2 \left( {\mu ^\textrm{t}}^\prime S_\textrm{ik} + \mu ^e S_\textrm{ik}^\prime \right) \right] \textrm{d} \Omega \textrm{d} t&= \frac{1}{T} \int _{t^\textrm{s}}^{t^e} \int \hat{v}_\textrm{i} \left[ 2 \left( {\mu ^\textrm{t}}^\prime S_\textrm{ik} + \mu ^e S_\textrm{ik}^\prime \right) \right] n_\textrm{k} \nonumber \\&\qquad \qquad \qquad - \frac{\partial \hat{v}_\textrm{i}}{\partial x_\textrm{k}} \mu ^e \left( \overline{v}_\textrm{i}^\prime n_\textrm{k} + \overline{v}_\textrm{k}^\prime n_\textrm{i} \right) \textrm{d} \Gamma \textrm{d} t \nonumber \\&\qquad - \frac{1}{T} \int _{t^\textrm{s}}^{t^e} \int {\mu ^\textrm{t}}^\prime 2 S_\textrm{ik} \hat{S}_\textrm{ik} - \overline{v}_\textrm{i}^\prime \frac{\partial }{\partial x_\textrm{k}} \left[ 2 \mu ^e \hat{S}_\textrm{ik} \right] \textrm{d} \Omega \textrm{d} t \end{aligned}$$

  (74)

• the considered objectives (17):

  $$\begin{aligned} {J^f}^\prime&= \frac{1}{T} \int _{t^\textrm{s}}^{t^e} \int \left[ p^\prime \delta _\textrm{ik} - 2 {\mu ^\textrm{t}}^\prime S_\textrm{ik} - 2 \mu ^e S_\textrm{ik}^\prime \right] \textrm{d} \Gamma \textrm{d} t \end{aligned}$$

  (75)
  $$\begin{aligned} {J^I}^\prime&= \frac{1}{T} \int _{t^\textrm{s}}^{t^e} \int \overline{v}_\textrm{i}^\prime \left[ \overline{v}_\textrm{i} - \overline{v}_\textrm{i}^\textrm{tar}\right] \textrm{d} \Omega \textrm{d} t \end{aligned}$$

  (76)

Collecting all (non-objectives) variations yields

$$\begin{aligned} {L^{\text {lam}}}^\prime&= \frac{1}{T} \bigg [ \int \hat{v}_\textrm{i} \rho \overline{v}_\textrm{i} \textrm{d} \Omega \bigg ]_{t^\textrm{s}}^{t^e} \nonumber \\&+ \frac{1}{T} \int _{t^\textrm{s}}^{t^e} \int \overline{v}_\textrm{i}^\prime \big [ -\hat{p} n_\textrm{k} + \rho \overline{v}_\textrm{k} (\hat{v}_\textrm{k} n_\textrm{i} + \hat{v}_\textrm{i} n_\textrm{k}) + 2 \mu ^e \hat{S}_\textrm{ik} n_\textrm{k} \big ] + p^\prime \big [ \hat{v}_\textrm{i} n_\textrm{i} \big ] \nonumber \\ {}&\quad - {\mu ^\textrm{t}}^\prime \big [ \hat{v}_\textrm{i} 2 S_\textrm{ik} n_\textrm{k} \big ] + S_\textrm{ik}^\prime \big [ \hat{v}_\textrm{i} 2 \mu ^e n_\textrm{k} \big ] \textrm{d} \Gamma \textrm{d} t \nonumber \\&+ \frac{1}{T} \int _{t^\textrm{s}}^{t^e} \int \overline{v}_\textrm{i}^\prime \bigg [ -\frac{\partial \rho \hat{v}_\textrm{i}}{\partial t} - \rho \overline{v}_\textrm{k} \frac{\partial \hat{v}_\textrm{k}}{\partial x_\textrm{i}} - \frac{\partial \overline{v}_\textrm{k} \rho \hat{v}_\textrm{i}}{\partial x_\textrm{k}} + \frac{\partial \hat{p}}{\partial x_\textrm{i}} - \frac{\partial }{\partial x_\textrm{k}} \bigg [ 2 \mu ^e \hat{S}_\textrm{ik} \bigg ] \bigg ] - p^\prime \bigg [ \frac{\partial \hat{v}_\textrm{i}}{\partial x_\textrm{i}} \bigg ] \nonumber \\ {}&\quad + {\mu ^\textrm{t}}^\prime \bigg [ 2 S_\textrm{ik} \hat{S}_\textrm{ik} \bigg ] \textrm{d} \Omega \textrm{d} t \, . \end{aligned}$$

(77)

B Continuous Adjoint Smagorinsky-Lilly Model

The variation of the Smagorinsky-Lilly subgrid-scale viscosity is splitted in two contributions, viz.

• the subgrid length scale:

  $$\begin{aligned} \frac{1}{T} \int _{t^\textrm{s}}^{t^e} \int \hat{v}_\textrm{i} \frac{\partial }{\partial x_\textrm{k}} \bigg [ 4 \frac{\mu ^\textrm{t}}{l^\textrm{s}} {l^\textrm{s}}^\prime \overline{S}_\textrm{ik} \bigg ] \textrm{d} \Omega \textrm{d} t&= \frac{1}{T} \int _{t^\textrm{s}}^{t^e} \int \hat{v}_\textrm{i} 4 \frac{\mu ^\textrm{t}}{l^\textrm{s}} {l^\textrm{s}}^\prime \overline{S}_\textrm{ik} n_\textrm{k} \textrm{d} \Gamma \textrm{d} t \nonumber \\ {}&\quad - \frac{1}{T} \int _{t^\textrm{s}}^{t^e} \int 4 \frac{\mu ^\textrm{t}}{l^\textrm{s}} {l^\textrm{s}}^\prime \overline{S}_\textrm{ik} \hat{S}_\textrm{ik} \textrm{d} \Omega \textrm{d} t \end{aligned}$$

  (78)

• the velocity scale, cf. Eq. (33).

Together with the expansion of ${l^\textrm{s}}^\prime$ from Eq. (31), the varied Smagorinsky-Lilly Lagrangian reads

$$\begin{aligned} {L^{\text {smag}}}^\prime&= \frac{1}{T} \bigg [ \int \hat{v}_\textrm{i} \rho \overline{v}_\textrm{i} \textrm{d} \Omega \bigg ]_{t^\textrm{s}}^{t^e} \nonumber \\&+ \frac{1}{T} \int _{t^\textrm{s}}^{t^e} \int \overline{v}_\textrm{i}^\prime \big [ -\hat{p} n_\textrm{k} + \rho \overline{v}_\textrm{k} (\hat{v}_\textrm{k} n_\textrm{i} + \hat{v}_\textrm{i} n_\textrm{k}) + 2 \mu ^e \hat{S}_\textrm{ik} n_\textrm{k} \big ] + p^\prime \big [ \hat{v}_\textrm{i} n_\textrm{i} \big ] + \overline{S}_\textrm{ik}^\prime \big [ \hat{v}_\textrm{i} 2 \mu ^e n_\textrm{k} \big ] \nonumber \\&\qquad \quad - \hat{v}_\textrm{i} 4 \frac{\mu ^\textrm{t}}{l^\textrm{s}} {l^\textrm{s}}^\prime \overline{S}_\textrm{ik} n_\textrm{k} - \hat{v}_\textrm{i} \left[ 2 \mu ^\textrm{t} C_{qrik}^a \overline{S}_\textrm{qr}^\prime \right] n_\textrm{k} + \mu ^\textrm{t} C^\textrm{b} \overline{S}_\textrm{qr} \left( \overline{v}_\textrm{q}^\prime n_\textrm{r} + \overline{v}_\textrm{r}^\prime n_\textrm{q} \right) \textrm{d} \Gamma \textrm{d} t \end{aligned}$$

(79)

$$\begin{aligned}&+ \frac{1}{T} \int _{t^\textrm{s}}^{t^e} \int \overline{v}_\textrm{i}^\prime \bigg [ -\frac{\partial \rho \hat{v}_\textrm{i}}{\partial t} - \rho \overline{v}_\textrm{k} \frac{\partial \hat{v}_\textrm{k}}{\partial x_\textrm{i}} - \frac{\partial \overline{v}_\textrm{k} \rho \hat{v}_\textrm{i}}{\partial x_\textrm{k}} + \frac{\partial \hat{p}}{\partial x_\textrm{i}} - \frac{\partial }{\partial x_\textrm{k}} \bigg [ 2 \bigg ( \mu ^e \hat{S}_\textrm{ik} + \mu ^\textrm{t} C^\textrm{b} \overline{S}_\textrm{ik} \bigg ) \bigg ] \bigg ] \nonumber \\&\qquad \quad - p^\prime \bigg [ \frac{\partial \hat{v}_\textrm{i}}{\partial x_\textrm{i}} \bigg ] \nonumber \\&\qquad \quad + d^\prime \bigg [ \frac{\kappa }{2} \left( 1 - \text {sign}(\kappa d - C^\textrm{s} {{l^\textrm{g}}}) \right) \bigg ] \bigg [ 4 \frac{\mu ^\textrm{t}}{l^\textrm{s}} \overline{S}_\textrm{ik} \hat{S}_\textrm{ik} \bigg ] \nonumber \\&\qquad \quad + {C^\textrm{s}}^\prime \bigg [ \frac{{{l^\textrm{g}}}}{2} \left( 1 + \text {sign}(\kappa d - C^\textrm{s} {{l^\textrm{g}}}) \right) \bigg ] \bigg [ 4 \frac{\mu ^\textrm{t}}{l^\textrm{s}} \overline{S}_\textrm{ik} \hat{S}_\textrm{ik} \bigg ] \nonumber \\&\qquad \quad + {{l^\textrm{g}}}^\prime \bigg [ \frac{C^\textrm{s}}{2} \left( 1 + \text {sign}(\kappa d - C^\textrm{s} {{l^\textrm{g}}}) \right) \bigg ] \bigg [ 4 \frac{\mu ^\textrm{t}}{l^\textrm{s}} \overline{S}_\textrm{ik} \hat{S}_\textrm{ik} \bigg ] \textrm{d} \Omega \textrm{d} t \, . \end{aligned}$$

(80)

with $C_{qrik}^a = \overline{S}_\textrm{qr} \overline{S}_\textrm{ik} / (\overline{S}_\textrm{lm} \overline{S}_\textrm{lm})$ and $C^\textrm{b} = \hat{S}_\textrm{ik} \overline{S}_\textrm{ik} / (\overline{S}_\textrm{lm} \overline{S}_\textrm{lm})$. Assuming $d^\prime = 0$, ${C^\textrm{s}}^\prime = 0$, ${{l^\textrm{g}}}^\prime = 0$ leads to the following set of continuous adjoint Smagorinsky-Lilly equations

$$\begin{aligned}&- \frac{\partial \hat{v}_\textrm{k}}{\partial x_\textrm{k}} + -\frac{\partial j^\Omega }{\partial p} = R^{\hat{p}, \text {smag}} = 0 \quad \text {in} \quad \Omega ^\textrm{O} \end{aligned}$$

(81)

$$\begin{aligned}&- \frac{\partial \hat{v}_\textrm{k}}{\partial x_\textrm{k}} = R^{\hat{p}, \text {smag}} = 0 \quad \text {in} \quad \Omega \setminus \Omega ^\textrm{O} \end{aligned}$$

(82)

$$\begin{aligned}&-\frac{\partial \rho \hat{v}_\textrm{i}}{\partial t} - \frac{\partial \rho \overline{v}_\textrm{k} \hat{v}_\textrm{i}}{\partial x_\textrm{k}} - \rho \overline{v}_\textrm{k} \frac{\partial \hat{v}_\textrm{k}}{\partial x_\textrm{i}} + \frac{\partial }{\partial x_\textrm{k}} \left[ \hat{p} \delta _\textrm{ik} - 2 \bigg ( \mu ^e \hat{S}_\textrm{ik} + \mu ^\textrm{t} C^\textrm{b} \overline{S}_\textrm{ik} \bigg ) \right] + \frac{\partial j^\Omega }{\partial \overline{v}_\textrm{i}} = R^{\hat{v}_\textrm{i}, \text {smag}} = 0 \quad \text {in} \quad \Omega \end{aligned}$$

(83)

$$\begin{aligned}&-\frac{\partial \rho \hat{v}_\textrm{i}}{\partial t} - \frac{\partial \rho \overline{v}_\textrm{k} \hat{v}_\textrm{i}}{\partial x_\textrm{k}} - \rho \overline{v}_\textrm{k} \frac{\partial \hat{v}_\textrm{k}}{\partial x_\textrm{i}} + \frac{\partial }{\partial x_\textrm{k}} \left[ \hat{p} \delta _\textrm{ik} - 2 \bigg ( \mu ^e \hat{S}_\textrm{ik} + \mu ^\textrm{t} C^\textrm{b} \overline{S}_\textrm{ik} \bigg ) \right] = R^{\hat{v}_\textrm{i}, \text {smag}} = 0 \quad \text {in} \quad \Omega \setminus \Omega ^\textrm{O} \end{aligned}$$

(84)

$$\begin{aligned}&\hat{v}_\textrm{i} = 0, \frac{\partial \hat{p}}{\partial n} = 0 \quad \text {on} \quad \Gamma ^\textrm{in} \end{aligned}$$

(85)

$$\begin{aligned}&\bigg [ \hat{p} \delta _\textrm{ik} - 2 \bigg ( \mu ^e \hat{S}_\textrm{ik} + \mu ^\textrm{t} C^\textrm{b} \overline{S}_\textrm{ik} \bigg ) \bigg ] n_\textrm{k} n_\textrm{i} - \rho \overline{v}_\textrm{k} (\hat{v}_\textrm{k} n_\textrm{i} + \hat{v}_\textrm{i} n_\textrm{k}) = 0, \frac{\partial \hat{v}_\textrm{i}}{\partial n} = 0 \quad \text {on} \quad \Gamma ^\textrm{out} \end{aligned}$$

(86)

$$\begin{aligned}&\hat{v}_\textrm{i} + \frac{\partial j_\textrm{i}^\Omega }{\partial p} = 0, \frac{\partial \hat{p}}{\partial n} = 0 \quad \text {on} \quad \Gamma ^\textrm{O} \end{aligned}$$

(87)

$$\begin{aligned}&\hat{v}_\textrm{i} = 0, \frac{\partial \hat{p}}{\partial n} = 0 \quad \text {on} \quad \Gamma ^\textrm{wall} \setminus \Gamma ^\textrm{O} \end{aligned}$$

(88)

$$\begin{aligned}&\hat{v}_\textrm{i} n_\textrm{i} = 0, \bigg [ \hat{p} \delta _\textrm{ik} - 2 \bigg ( \mu ^e \hat{S}_\textrm{ik} + \mu ^\textrm{t} C^\textrm{b} \overline{S}_\textrm{ik} \bigg ) \bigg ] n_\textrm{k} t_\textrm{i} = 0, \frac{\partial \hat{v}_\textrm{i}}{\partial n} = 0 \quad \text {on} \quad \Gamma ^\textrm{symm} \, . \end{aligned}$$

(89)

C Further Subgrid Length-Scale Considerations

An improved near wall behavior can be achieved by manipulating the subgrid length scale in line with, e.g., a Van-Driest damping function, i.e.,

$$\begin{aligned} l^\textrm{s} = C^\textrm{s} {{l^\textrm{g}}} \bigg [1 - \left( \textrm{exp}^{\frac{-y^+}{A^+}}\right) ^{\gamma _1} \bigg ]^{\gamma _2} \qquad \qquad \textrm{with} \qquad \qquad y^+ = \frac{\rho d u^\mathrm {\tau }}{\mu } \quad \textrm{and} \quad u^\mathrm {\tau } = \frac{\tau ^\textrm{w}}{\rho }\, , \end{aligned}$$

(90)

where $y^+$ and $u^\mathrm {\tau }$ refer to the dimensionless distance to the nearest wall as well as the friction velocity, respectively. The constants are usually assigned to $A^+ \approx 25$, $\gamma _1 = 3$, and $\gamma _2 = 0.5$. Its total variation can be expanded to

$$\begin{aligned} {l^\textrm{s}}^\prime&= {C^\textrm{s}}^\prime {{l^\textrm{g}}} \bigg [1 - \left( \textrm{exp}^{\frac{-y^+}{A^+}}\right) ^{\gamma _1} \bigg ]^{\gamma _2} + C^\textrm{s} {{l^\textrm{g}}}^\prime \bigg [1 - \left( \textrm{exp}^{\frac{-y^+}{A^+}}\right) ^{\gamma _1} \bigg ]^{\gamma _2} \end{aligned}$$

(91)

$$\begin{aligned}&\qquad + C^\textrm{s} {{l^\textrm{g}}} \gamma _2 \bigg [1 - \left( \textrm{exp}^{\frac{-y^+}{A^+}}\right) ^{\gamma _1} \bigg ]^{\gamma _2 - 1} \gamma _1 \left( \textrm{exp}^{\frac{-y^+}{A^+}}\right) ^{\gamma _1 - 1} \textrm{exp}^{\frac{-y^+}{A^+}} \frac{1}{A^+} \big [ \frac{\rho d^\prime u^\mathrm {\tau } + \rho d \frac{ {\tau ^\textrm{w}}^\prime }{\rho }}{\mu } \big ] \end{aligned}$$

(92)

$$\begin{aligned}&\qquad \qquad \text {with} \qquad \qquad {\tau ^\textrm{w}}^\prime = \mu \frac{\partial v_\textrm{i}^\prime }{\partial x_\textrm{k}} t_\textrm{i} n_\textrm{k} \bigg |^\textrm{w} \, , \end{aligned}$$

(93)

where $t_\textrm{i}$ refers to the flow tangential vector. In line with the primal implementation, the variation of the local length scale couples field quantities with the (dimensionless) nearest wall, which requires additional implementation effort.

D Continuous Adjoint Wall Adapting Local Eddy-Viscosity Model

The variation of the WALE subgrid turbulent viscosity is splitted in two contributions, viz.

• scaling with the filtered shear rate tensor $\overline{S}_\textrm{ik}^\prime$, cf. Eq. (49)

• scaling with its traceless symmetric companion ${\overline{S}_\textrm{ik}^\textrm{d}}^\prime$, cf. Eq. (8), that can be expressed in two ways. Variation of the first version results in six contributions that can be expanded as follows:

$$\begin{aligned} \frac{1}{T} \int _{t^\textrm{s}}^{t^e} \int \hat{v}_\textrm{i} \frac{\partial }{x_\textrm{k}} \bigg [ 2\mu ^\textrm{t} \alpha _\textrm{qr} \frac{1}{2} \frac{\partial \overline{v}_\textrm{q}^\prime }{\partial x_\textrm{m}} \frac{\partial \overline{v}_\textrm{m}}{\partial x_\textrm{r}} \overline{S}_\textrm{ik} \bigg ] \textrm{d} \Omega \textrm{d} t&= \frac{1}{T} \int _{t^\textrm{s}}^{t^e} \int \hat{v}_\textrm{q} \bigg [ \mu ^\textrm{t} \alpha _\textrm{ir} \frac{\partial \overline{v}_\textrm{i}^\prime }{\partial x_\textrm{k}} \frac{\partial \overline{v}_\textrm{k}}{\partial x_\textrm{r}} \overline{S}_\textrm{qm} \bigg ] n_\textrm{m} - \mu ^\textrm{t} \alpha _\textrm{ir} \overline{v}_\textrm{i}^\prime \frac{\partial \overline{v}_\textrm{k}}{\partial x_\textrm{r}} \overline{S}_\textrm{qm} \hat{S}_\textrm{qm} n_\textrm{k} \textrm{d} \Gamma \textrm{d} t \nonumber \\&+ \frac{1}{T} \int _{t^\textrm{s}}^{t^e} \int \overline{v}_\textrm{i}^\prime \frac{\partial }{\partial x_\textrm{k}} \bigg [ \mu ^\textrm{t} \alpha _\textrm{ir} \frac{\partial \overline{v}_\textrm{k}}{\partial x_\textrm{r}} \overline{S}_\textrm{qm} \hat{S}_\textrm{qm} \bigg ] \textrm{d} \Omega \textrm{d} t \end{aligned}$$

(94)

$$\begin{aligned} \frac{1}{T} \int _{t^\textrm{s}}^{t^e} \int \hat{v}_\textrm{i} \frac{\partial }{x_\textrm{k}} \bigg [ 2\mu ^\textrm{t} \alpha _\textrm{qr} \frac{1}{2} \frac{\partial \overline{v}_\textrm{q}}{\partial x_\textrm{m}} \frac{\partial \overline{v}_\textrm{m}^\prime }{\partial x_\textrm{r}} \overline{S}_\textrm{ik} \bigg ] \textrm{d} \Omega \textrm{d} t&= \frac{1}{T} \int _{t^\textrm{s}}^{t^e} \int \hat{v}_\textrm{m} \bigg [ \mu ^\textrm{t} \alpha _\textrm{qk} \frac{\partial \overline{v}_\textrm{q}}{\partial x_\textrm{i}} \frac{\partial \overline{v}_\textrm{i}^\prime }{\partial x_\textrm{k}} \overline{S}_\textrm{mi} \bigg ] n_\textrm{r} - \mu ^\textrm{t} \alpha _\textrm{qk} \frac{\partial \overline{v}_\textrm{q}}{\partial x_\textrm{i}} \overline{v}_\textrm{i}^\prime \overline{S}_\textrm{mi} \hat{S}_\textrm{mi} n_\textrm{k} \textrm{d} \Gamma \textrm{d} t \nonumber \\&+ \frac{1}{T} \int _{t^\textrm{s}}^{t^e} \int \overline{v}_\textrm{i}^\prime \frac{\partial }{\partial x_\textrm{k}} \bigg [ \mu ^\textrm{t} \alpha _\textrm{qk} \frac{\partial \overline{v}_\textrm{q}}{\partial x_\textrm{i}} \overline{S}_\textrm{mi} \hat{S}_\textrm{mi} \bigg ] \textrm{d} \Omega \textrm{d} t \end{aligned}$$

(95)

$$\begin{aligned} \frac{1}{T} \int _{t^\textrm{s}}^{t^e} \int \hat{v}_\textrm{i} \frac{\partial }{x_\textrm{k}} \bigg [ 2\mu ^\textrm{t} \alpha _\textrm{qr} \frac{1}{2} \frac{\partial \overline{v}_\textrm{r}^\prime }{\partial x_\textrm{m}} \frac{\partial \overline{v}_\textrm{m}}{\partial x_\textrm{q}} \overline{S}_\textrm{ik} \bigg ] \textrm{d} \Omega \textrm{d} t&= \frac{1}{T} \int _{t^\textrm{s}}^{t^e} \int \hat{v}_r \bigg [ \mu ^\textrm{t} \alpha _\textrm{qi} \frac{\partial \overline{v}_\textrm{i}^\prime }{\partial x_\textrm{k}} \frac{\partial \overline{v}_\textrm{k}}{\partial x_\textrm{q}} \overline{S}_\textrm{rm} \bigg ] n_\textrm{m} - \mu ^\textrm{t} \alpha _\textrm{qi} \overline{v}_\textrm{i}^\prime \frac{\partial \overline{v}_\textrm{k}}{\partial x_\textrm{q}} \overline{S}_\textrm{rm} \hat{S}_\textrm{rm} n_\textrm{k} \textrm{d} \Gamma \textrm{d} t \nonumber \\&+ \frac{1}{T} \int _{t^\textrm{s}}^{t^e} \int \overline{v}_\textrm{i}^\prime \frac{\partial }{\partial x_\textrm{k}} \bigg [\mu ^\textrm{t} \alpha _\textrm{qi} \frac{\partial \overline{v}_\textrm{k}}{\partial x_\textrm{q}} \overline{S}_\textrm{rm} \hat{S}_\textrm{rm} \bigg ] \textrm{d} \Omega \textrm{d} t \end{aligned}$$

(96)

$$\begin{aligned} \frac{1}{T} \int _{t^\textrm{s}}^{t^e} \int \hat{v}_\textrm{i} \frac{\partial }{x_\textrm{k}} \bigg [ 2\mu ^\textrm{t} \alpha _\textrm{qr} \frac{1}{2} \frac{\partial \overline{v}_\textrm{r}}{\partial x_\textrm{m}} \frac{\partial \overline{v}_\textrm{m}^\prime }{\partial x_\textrm{q}} \overline{S}_\textrm{ik} \bigg ] \textrm{d} \Omega \textrm{d} t&= \frac{1}{T} \int _{t^\textrm{s}}^{t^e} \int \hat{v}_\textrm{m} \bigg [ \mu ^\textrm{t} \alpha _\textrm{kr} \frac{\partial \overline{v}_\textrm{r}}{\partial x_\textrm{i}} \frac{\partial \overline{v}_\textrm{i}^\prime }{\partial x_\textrm{k}} \overline{S}_\textrm{mq} \bigg ] n_\textrm{q} - \mu ^\textrm{t} \alpha _\textrm{kr} \frac{\partial \overline{v}_\textrm{m}}{\partial x_\textrm{i}} \overline{v}_\textrm{i}^\prime \overline{S}_\textrm{mq} \hat{S}_\textrm{mq} n_\textrm{k} \textrm{d} \Gamma \textrm{d} t \nonumber \\&+ \frac{1}{T} \int _{t^\textrm{s}}^{t^e} \int \overline{v}_\textrm{i}^\prime \frac{\partial }{\partial x_\textrm{k}} \bigg [ \mu ^\textrm{t} \alpha _\textrm{kr} \frac{\partial \overline{v}_\textrm{r}}{\partial x_\textrm{i}} \overline{S}_\textrm{mq} \hat{S}_\textrm{mq} \bigg ] \textrm{d} \Omega \textrm{d} t \end{aligned}$$

(97)

$$\begin{aligned} \frac{1}{T} \int _{t^\textrm{s}}^{t^e} \int \hat{v}_\textrm{i} \frac{\partial }{x_\textrm{k}} \bigg [ 2\mu ^\textrm{t} \alpha _\textrm{qr} \delta _\textrm{qr} \frac{1}{3} \frac{\partial \overline{v}_\textrm{n}^\prime }{\partial x_\textrm{m}} \frac{\partial \overline{v}_\textrm{m}}{\partial x_\textrm{n}} \overline{S}_\textrm{ik} \bigg ] \textrm{d} \Omega \textrm{d} t&= \frac{1}{T} \int _{t^\textrm{s}}^{t^e} \int \hat{v}_\textrm{n} \bigg [ \mu ^\textrm{t} \alpha _\textrm{qq} \frac{2}{3} \frac{\partial \overline{v}_\textrm{i}^\prime }{\partial x_\textrm{k}} \frac{\partial \overline{v}_\textrm{k}}{\partial x_\textrm{i}} \overline{S}_\textrm{nm} \bigg ] n_\textrm{m} - \mu ^\textrm{t} \alpha _\textrm{qq} \frac{2}{3} \overline{v}_\textrm{i}^\prime \frac{\partial \overline{v}_\textrm{k}}{\partial x_\textrm{i}} \overline{S}_\textrm{nm} \hat{S}_\textrm{nm} n_\textrm{k} \textrm{d} \Gamma \textrm{d} t \nonumber \\&+ \frac{1}{T} \int _{t^\textrm{s}}^{t^e} \int \overline{v}_\textrm{i}^\prime \frac{\partial }{\partial x_\textrm{k}} \bigg [ \mu ^\textrm{t} \alpha _\textrm{qq} \frac{2}{3} \frac{\partial \overline{v}_\textrm{k}}{\partial x_\textrm{i}} \overline{S}_\textrm{nm} \hat{S}_\textrm{nm} \bigg ] \textrm{d} \Omega \textrm{d} t \end{aligned}$$

(98)

$$\begin{aligned} \frac{1}{T} \int _{t^\textrm{s}}^{t^e} \int \hat{v}_\textrm{i} \frac{\partial }{x_\textrm{k}} \bigg [ 2\mu ^\textrm{t} \alpha _\textrm{qr} \delta _\textrm{qr} \frac{1}{3} \frac{\partial \overline{v}_\textrm{n}}{\partial x_\textrm{m}} \frac{\partial \overline{v}_\textrm{m}^\prime }{\partial x_\textrm{n}} \overline{S}_\textrm{ik} \bigg ] \textrm{d} \Omega \textrm{d} t&= \frac{1}{T} \int _{t^\textrm{s}}^{t^e} \int \hat{v}_\textrm{m} \bigg [ \mu ^\textrm{t} \alpha _\textrm{qq} \frac{2}{3} \frac{\partial \overline{v}_\textrm{k}}{\partial x_\textrm{i}} \frac{\partial \overline{v}_\textrm{i}^\prime }{\partial x_\textrm{k}} \overline{S}_\textrm{mn} \bigg ] n_\textrm{n} - \mu ^\textrm{t} \alpha _\textrm{qq} \frac{2}{3} \frac{\partial \overline{v}_\textrm{k}}{\partial x_\textrm{i}} \overline{v}_\textrm{i}^\prime \overline{S}_\textrm{mn} \hat{S}_\textrm{mn} n_\textrm{k} \textrm{d} \Gamma \textrm{d} t \nonumber \\&+ \frac{1}{T} \int _{t^\textrm{s}}^{t^e} \int \overline{v}_\textrm{i}^\prime \frac{\partial }{\partial x_\textrm{k}} \bigg [ \mu ^\textrm{t} \alpha _\textrm{qq} \frac{2}{3} \frac{\partial \overline{v}_\textrm{k}}{\partial x_\textrm{i}} \overline{S}_\textrm{mn} \hat{S}_\textrm{mn} \bigg ] \textrm{d} \Omega \textrm{d} t \, . \end{aligned}$$

(99)

Assembling all contributions yields

$$\begin{aligned} ... = \frac{1}{T} \int _{t^\textrm{s}}^{t^e} \int&\frac{\partial \overline{v}_\textrm{i}^\prime }{\partial x_\textrm{k}} \bigg [ \hat{v}_\textrm{q} \mu ^\textrm{t} \alpha _\textrm{ir} \frac{\partial \overline{v}_\textrm{k}}{\partial x_\textrm{r}} \overline{S}_\textrm{qm} n_\textrm{m} + \hat{v}_\textrm{m} \mu ^\textrm{t} \alpha _\textrm{qk} \frac{\partial \overline{v}_\textrm{q}}{\partial x_\textrm{i}} \overline{S}_\textrm{mi} n_\textrm{r} \nonumber \\ {}&\qquad + \hat{v}_r \mu ^\textrm{t} \alpha _\textrm{qi} \frac{\partial \overline{v}_\textrm{k}}{\partial x_\textrm{q}} \overline{S}_\textrm{rm} n_\textrm{m} + \hat{v}_\textrm{m} \mu ^\textrm{t} \alpha _\textrm{kr} \frac{\partial \overline{v}_\textrm{r}}{\partial x_\textrm{i}} \overline{S}_\textrm{mq} n_\textrm{q} \nonumber \\ {}&\qquad - \hat{v}_\textrm{n} \mu ^\textrm{t} \alpha _\textrm{qq} \frac{2}{3} \frac{\partial \overline{v}_\textrm{k}}{\partial x_\textrm{i}} \overline{S}_\textrm{nm} n_\textrm{m} - \hat{v}_\textrm{m} \mu ^\textrm{t} \alpha _\textrm{qq} \frac{2}{3} \frac{\partial \overline{v}_\textrm{k}}{\partial x_\textrm{i}} \overline{S}_\textrm{mn} n_\textrm{n} \bigg ] \nonumber \\&- \overline{v}_\textrm{i}^\prime \bigg [ \mu ^\textrm{t} \alpha _\textrm{ir} \frac{\partial \overline{v}_\textrm{k}}{\partial x_\textrm{r}} \overline{S}_\textrm{qm} \hat{S}_\textrm{qm} n_\textrm{k} + \mu ^\textrm{t} \alpha _\textrm{qk} \frac{\partial \overline{v}_\textrm{q}}{\partial x_\textrm{i}} \overline{S}_\textrm{mi} \hat{S}_\textrm{mi} n_\textrm{k} \nonumber \\ {}&\qquad + \mu ^\textrm{t} \alpha _\textrm{qi} \frac{\partial \overline{v}_\textrm{k}}{\partial x_\textrm{q}} \overline{S}_\textrm{rm} \hat{S}_\textrm{rm} n_\textrm{k} + \mu ^\textrm{t} \alpha _\textrm{kr} \frac{\partial \overline{v}_\textrm{m}}{\partial x_\textrm{i}} \overline{S}_\textrm{mq} \hat{S}_\textrm{mq} n_\textrm{k} \nonumber \\ {}&\qquad - \mu ^\textrm{t} \alpha _\textrm{qq} \frac{2}{3} \frac{\partial \overline{v}_\textrm{k}}{\partial x_\textrm{i}} \overline{S}_\textrm{nm} \hat{S}_\textrm{nm} n_\textrm{k} - \mu ^\textrm{t} \alpha _\textrm{qq} \frac{2}{3} \frac{\partial \overline{v}_\textrm{k}}{\partial x_\textrm{i}} \overline{S}_\textrm{mn} \hat{S}_\textrm{mn} n_\textrm{k} \bigg ] \textrm{d} \Gamma \textrm{d} t \nonumber \\ +\frac{1}{T} \int _{t^\textrm{s}}^{t^e} \int&\overline{v}_\textrm{i}^\prime \frac{\partial }{\partial x_\textrm{k}} \bigg [ \mu ^\textrm{t} \alpha _\textrm{ir} \frac{\partial \overline{v}_\textrm{k}}{\partial x_\textrm{r}} \overline{S}_\textrm{qm} \hat{S}_\textrm{qm} + \mu ^\textrm{t} \alpha _\textrm{qk} \frac{\partial \overline{v}_\textrm{q}}{\partial x_\textrm{i}} \overline{S}_\textrm{mi} \hat{S}_\textrm{mi} \bigg ] \nonumber \\&\qquad + \mu ^\textrm{t} \alpha _\textrm{qi} \frac{\partial \overline{v}_\textrm{k}}{\partial x_\textrm{q}} \overline{S}_\textrm{rm} \hat{S}_\textrm{rm} + \mu ^\textrm{t} \alpha _\textrm{kr} \frac{\partial \overline{v}_\textrm{r}}{\partial x_\textrm{i}} \overline{S}_\textrm{mq} \hat{S}_\textrm{mq} \bigg ] \nonumber \\ {}&\qquad - \mu ^\textrm{t} \alpha _\textrm{qq} \frac{2}{3} \frac{\partial \overline{v}_\textrm{k}}{\partial x_\textrm{i}} \overline{S}_\textrm{nm} \hat{S}_\textrm{nm} - \mu ^\textrm{t} \alpha _\textrm{qq} \frac{2}{3} \frac{\partial \overline{v}_\textrm{k}}{\partial x_\textrm{i}} \overline{S}_\textrm{mn} \hat{S}_\textrm{mn} \bigg ] \textrm{d} \Omega \textrm{d} t \, . \end{aligned}$$

(100)

Further permuting of indices

$$\begin{aligned} ... = \frac{1}{T} \int _{t^\textrm{s}}^{t^e} \int&\frac{\partial \overline{v}_\textrm{i}^\prime }{\partial x_\textrm{k}} \bigg [ \hat{v}_\textrm{q} \mu ^\textrm{t} \alpha _{im} \frac{\partial \overline{v}_\textrm{k}}{\partial x_\textrm{m}} \overline{S}_\textrm{qr} + \hat{v}_\textrm{q} \mu ^\textrm{t} \alpha _{mk} \frac{\partial \overline{v}_\textrm{m}}{\partial x_\textrm{i}} \overline{S}_\textrm{qr} \nonumber \\ {}&\qquad + \hat{v}_\textrm{q} \mu ^\textrm{t} \alpha _{mi} \frac{\partial \overline{v}_\textrm{k}}{\partial x_\textrm{m}} \overline{S}_\textrm{qr} + \hat{v}_\textrm{q} \mu ^\textrm{t} \alpha _{km} \frac{\partial \overline{v}_\textrm{m}}{\partial x_\textrm{i}} \overline{S}_\textrm{qr} \nonumber \\ {}&\qquad - \hat{v}_\textrm{q} \mu ^\textrm{t} \alpha _\textrm{nn} \frac{2}{3} \frac{\partial \overline{v}_\textrm{k}}{\partial x_\textrm{i}} \overline{S}_\textrm{qr} - \hat{v}_\textrm{q} \mu ^\textrm{t} \alpha _\textrm{nn} \frac{2}{3} \frac{\partial \overline{v}_\textrm{k}}{\partial x_\textrm{i}} \overline{S}_\textrm{qr} \bigg ] n_\textrm{r} \nonumber \\&- \overline{v}_\textrm{i}^\prime \bigg [ \mu ^\textrm{t} \alpha _{im} \frac{\partial \overline{v}_\textrm{k}}{\partial x_\textrm{m}} \overline{S}_\textrm{qr} \hat{S}_\textrm{qr} + \mu ^\textrm{t} \alpha _{mk} \frac{\partial \overline{v}_\textrm{m}}{\partial x_\textrm{i}} \overline{S}_\textrm{qr} \hat{S}_\textrm{qr} \nonumber \\ {}&\qquad + \mu ^\textrm{t} \alpha _{mi} \frac{\partial \overline{v}_\textrm{k}}{\partial x_\textrm{m}} \overline{S}_\textrm{qr} \hat{S}_\textrm{qr} + \mu ^\textrm{t} \alpha _{km} \frac{\partial \overline{v}_\textrm{m}}{\partial x_\textrm{i}} \overline{S}_\textrm{qr} \hat{S}_\textrm{qr} \nonumber \\ {}&\qquad - \mu ^\textrm{t} \alpha _\textrm{nn} \frac{2}{3} \frac{\partial \overline{v}_\textrm{k}}{\partial x_\textrm{i}} \overline{S}_\textrm{qr} \hat{S}_\textrm{qr} - \mu ^\textrm{t} \alpha _\textrm{nn} \frac{2}{3} \frac{\partial \overline{v}_\textrm{k}}{\partial x_\textrm{i}} \overline{S}_\textrm{qr} \hat{S}_\textrm{qr} \bigg ] n_\textrm{k} \textrm{d} \Gamma \textrm{d} t \end{aligned}$$

(101)

$$\begin{aligned} \frac{1}{T} \int _{t^\textrm{s}}^{t^e} \int&\overline{v}_\textrm{i}^\prime \frac{\partial }{\partial x_\textrm{k}} \bigg [ \mu ^\textrm{t} \alpha _{im} \frac{\partial \overline{v}_\textrm{k}}{\partial x_\textrm{m}} \overline{S}_\textrm{qr} \hat{S}_\textrm{qr} + \mu ^\textrm{t} \alpha _{mk} \frac{\partial \overline{v}_\textrm{m}}{\partial x_\textrm{i}} \overline{S}_\textrm{qr} \hat{S}_\textrm{qr} \bigg ] \nonumber \\ {}&\qquad + \mu ^\textrm{t} \alpha _{mi} \frac{\partial \overline{v}_\textrm{k}}{\partial x_\textrm{m}} \overline{S}_\textrm{qr} \hat{S}_\textrm{qr} + \mu ^\textrm{t} \alpha _{km} \frac{\partial \overline{v}_\textrm{m}}{\partial x_\textrm{i}} \overline{S}_\textrm{qr} \hat{S}_\textrm{qr} \bigg ] \nonumber \\&\qquad - \mu ^\textrm{t} \alpha _\textrm{nn} \frac{2}{3} \frac{\partial \overline{v}_\textrm{k}}{\partial x_\textrm{i}} \overline{S}_\textrm{qr} \hat{S}_\textrm{qr} - \mu ^\textrm{t} \alpha _\textrm{nn} \frac{2}{3} \frac{\partial \overline{v}_\textrm{k}}{\partial x_\textrm{i}} \overline{S}_\textrm{qr} \hat{S}_\textrm{qr} \bigg ] \textrm{d} \Omega \textrm{d} t \end{aligned}$$

(102)

followed by isolation of $\overline{S}_\textrm{qr}$ (with $\alpha _{mi} = \alpha _{im}$) provides

$$\begin{aligned} ... = \frac{1}{T} \int _{t^\textrm{s}}^{t^e} \int&\frac{\partial \overline{v}_\textrm{i}^\prime }{\partial x_\textrm{k}} \bigg [ \hat{v}_\textrm{q} 2 \mu ^\textrm{t} \overline{S}_\textrm{qr} \bigg ( \alpha _{im} \frac{\partial \overline{v}_\textrm{k}}{\partial x_\textrm{m}} + \alpha _{mk} \frac{\partial \overline{v}_\textrm{m}}{\partial x_\textrm{i}} - \alpha _\textrm{nn} \frac{2}{3} \frac{\partial \overline{v}_\textrm{k}}{\partial x_\textrm{i}} \bigg ) n_\textrm{r} \bigg ] \nonumber \\&- \overline{v}_\textrm{i}^\prime \bigg [ 2 \mu ^\textrm{t} \overline{S}_\textrm{qr} \hat{S}_\textrm{qr} \bigg ( \alpha _{im} \frac{\partial \overline{v}_\textrm{k}}{\partial x_\textrm{m}} + \alpha _{mk} \frac{\partial \overline{v}_\textrm{m}}{\partial x_\textrm{i}} - \alpha _\textrm{nn} \frac{2}{3} \frac{\partial \overline{v}_\textrm{k}}{\partial x_\textrm{i}} \bigg ) n_\textrm{k} \bigg ] \textrm{d} \Gamma \textrm{d} t \end{aligned}$$

(103)

$$\begin{aligned} \frac{1}{T} \int _{t^\textrm{s}}^{t^e} \int&\overline{v}_\textrm{i}^\prime \frac{\partial }{\partial x_\textrm{k}} \bigg [ 2 \mu ^\textrm{t} \overline{S}_\textrm{qr} \hat{S}_\textrm{qr} \bigg ( \alpha _{im} \frac{\partial \overline{v}_\textrm{k}}{\partial x_\textrm{m}} + \alpha _{mk} \frac{\partial \overline{v}_\textrm{m}}{\partial x_\textrm{i}} - \alpha _\textrm{nn} \frac{2}{3} \frac{\partial \overline{v}_\textrm{k}}{\partial x_\textrm{i}} \bigg ) \bigg ] \textrm{d} \Omega \textrm{d} t \, . \end{aligned}$$

(104)

Adding these –as well as those from Eq. (49)– WALE-related variational contributions to the laminar system (77) yields

$$\begin{aligned} {L^{\text {wale}}}^\prime&= \frac{1}{T} \bigg [ \int \hat{v}_\textrm{i} \rho \overline{v}_\textrm{i} \textrm{d} \Omega \bigg ]_{t^\textrm{s}}^{t^e} \nonumber \\&+ \frac{1}{T} \int _{t^\textrm{s}}^{t^e} \int \overline{v}_\textrm{i}^\prime \big [ -\hat{p} n_\textrm{k} + \rho \overline{v}_\textrm{k} (\hat{v}_\textrm{k} n_\textrm{i} + \hat{v}_\textrm{i} n_\textrm{k}) + 2 \mu ^e \hat{S}_\textrm{ik} n_\textrm{k} \big ] + p^\prime \big [ \hat{v}_\textrm{i} n_\textrm{i} \big ] + S_\textrm{ik}^\prime \big [ \hat{v}_\textrm{i} 2 \mu ^e n_\textrm{k} \big ] \nonumber \\&\qquad \quad + \overline{v}_\textrm{i}^\prime \bigg [ 2 \mu ^\textrm{t} \overline{S}_\textrm{qr} \hat{S}_\textrm{qr} C_\textrm{ik}^c n_\textrm{k} \bigg ] - \frac{\partial \overline{v}_\textrm{i}^\prime }{\partial x_\textrm{k}} \bigg [ \hat{v}_\textrm{q} 2 \mu ^\textrm{t} \overline{S}_\textrm{qr} C_\textrm{ik}^c n_\textrm{r} \bigg ] \nonumber \\&\qquad \quad + {C^\textrm{w}}^\prime \left[ \hat{v}_\textrm{i} 4 \frac{\mu ^\textrm{t}}{l^\textrm{s}} \overline{S}_\textrm{ik} n_\textrm{k} \right] + {{l^\textrm{g}}}^\prime \left[ \hat{v}_\textrm{i} 4 \frac{\mu ^\textrm{t}}{l^\textrm{s}} \overline{S}_\textrm{ik} n_\textrm{k} \right] \textrm{d} \Gamma \textrm{d} t \end{aligned}$$

(105)

$$\begin{aligned}&+ \frac{1}{T} \int _{t^\textrm{s}}^{t^e} \int \overline{v}_\textrm{i}^\prime \bigg [ -\frac{\partial \rho \hat{v}_\textrm{i}}{\partial t} - \rho \overline{v}_\textrm{k} \frac{\partial \hat{v}_\textrm{k}}{\partial x_\textrm{i}} - \frac{\partial \overline{v}_\textrm{k} \rho \hat{v}_\textrm{i}}{\partial x_\textrm{k}} + \frac{\partial \hat{p}}{\partial x_\textrm{i}} - \frac{\partial }{\partial x_\textrm{k}} \bigg [ 2 \bigg ( \mu ^e \hat{S}_\textrm{ik} + \mu ^\textrm{t} C_\textrm{ik}^c \bigg ) \bigg ] \bigg ] \nonumber \\&\qquad \quad - p^\prime \bigg [ \frac{\partial \hat{v}_\textrm{i}}{\partial x_\textrm{i}} \bigg ] \nonumber \\&\qquad \quad + {C^\textrm{w}}^\prime \bigg [ 4 \frac{\mu ^\textrm{t}}{l^\textrm{s}} \overline{S}_\textrm{ik} \hat{S}_\textrm{ik} \bigg ] \nonumber \\&\qquad \quad + {{l^\textrm{g}}}^\prime \bigg [ 4 \frac{\mu ^\textrm{t}}{l^\textrm{s}} \overline{S}_\textrm{ik} \hat{S}_\textrm{ik} \bigg ] \textrm{d} \Omega \textrm{d} t \, , \end{aligned}$$

(106)

with

$$\begin{aligned} C_\textrm{ik}^c = \overline{S}_\textrm{qr} \hat{S}_\textrm{qr} \bigg ( \alpha _{mi} \frac{\partial \overline{v}_\textrm{k}}{\partial x_\textrm{m}} + \alpha _{mk} \frac{\partial \overline{v}_\textrm{m}}{\partial x_\textrm{i}} - \alpha _\textrm{nn} \frac{2}{3} \frac{\partial \overline{v}_\textrm{k}}{\partial x_\textrm{i}} - \beta _\textrm{ik} \bigg ) \, . \end{aligned}$$

(107)

Assuming the contributions from ${l^\textrm{s}}^\prime$ to vanish leads to the following set of continuous adjoint WALE equations

$$\begin{aligned}&- \frac{\partial \hat{v}_\textrm{k}}{\partial x_\textrm{k}} + -\frac{\partial j^\Omega }{\partial p} = R^{\hat{p}, \text {wale}} = 0 \quad \text {in} \quad \Omega ^\textrm{O} \end{aligned}$$

(108)

$$\begin{aligned}&- \frac{\partial \hat{v}_\textrm{k}}{\partial x_\textrm{k}} = R^{\hat{p}, \text {wale}} = 0 \quad \text {in} \quad \Omega \setminus \Omega ^\textrm{O} \end{aligned}$$

(109)

$$\begin{aligned}&-\frac{\partial \rho \hat{v}_\textrm{i}}{\partial t} - \frac{\partial \rho \overline{v}_\textrm{k} \hat{v}_\textrm{i}}{\partial x_\textrm{k}} - \rho \overline{v}_\textrm{k} \frac{\partial \hat{v}_\textrm{k}}{\partial x_\textrm{i}} + \frac{\partial }{\partial x_\textrm{k}} \left[ \hat{p} \delta _\textrm{ik} - 2 \bigg ( \mu ^e \hat{S}_\textrm{ik} + \mu ^\textrm{t} C_\textrm{ik}^c\bigg ) \right] + \frac{\partial j^\Omega }{\partial \overline{v}_\textrm{i}} = R^{\hat{v}_\textrm{i}, \text {wale}} = 0 \quad \text {in} \quad \Omega \end{aligned}$$

(110)

$$\begin{aligned}&-\frac{\partial \rho \hat{v}_\textrm{i}}{\partial t} - \frac{\partial \rho \overline{v}_\textrm{k} \hat{v}_\textrm{i}}{\partial x_\textrm{k}} - \rho \overline{v}_\textrm{k} \frac{\partial \hat{v}_\textrm{k}}{\partial x_\textrm{i}} + \frac{\partial }{\partial x_\textrm{k}} \left[ \hat{p} \delta _\textrm{ik} - 2 \bigg ( \mu ^e \hat{S}_\textrm{ik} + \mu ^\textrm{t} C_\textrm{ik}^c \bigg ) \right] = R^{\hat{v}_\textrm{i}, \text {wale}} = 0 \quad \text {in} \quad \Omega \setminus \Omega ^\textrm{O} \end{aligned}$$

(111)

$$\begin{aligned}&\hat{v}_\textrm{i} = 0, \frac{\partial \hat{p}}{\partial n} = 0 \quad \text {on} \quad \Gamma ^\textrm{in} \end{aligned}$$

(112)

$$\begin{aligned}&\bigg [ \hat{p} \delta _\textrm{ik} - 2 \bigg ( \mu ^e \hat{S}_\textrm{ik} + \mu ^\textrm{t} C_\textrm{ik}^c \bigg ) \bigg ] n_\textrm{k} n_\textrm{i} - \rho \overline{v}_\textrm{k} (\hat{v}_\textrm{k} n_\textrm{i} + \hat{v}_\textrm{i} n_\textrm{k}) = 0, \frac{\partial \hat{v}_\textrm{i}}{\partial n} = 0 \quad \text {on} \quad \Gamma ^\textrm{out} \end{aligned}$$

(113)

$$\begin{aligned}&\hat{v}_\textrm{i} + \frac{\partial j_\textrm{i}^\Omega }{\partial p} = 0, \frac{\partial \hat{p}}{\partial n} = 0 \quad \text {on} \quad \Gamma ^\textrm{O} \end{aligned}$$

(114)

$$\begin{aligned}&\hat{v}_\textrm{i} = 0, \frac{\partial \hat{p}}{\partial n} = 0 \quad \text {on} \quad \Gamma ^\textrm{wall} \setminus \Gamma ^\textrm{O} \end{aligned}$$

(115)

$$\begin{aligned}&\hat{v}_\textrm{i} n_\textrm{i} = 0, \bigg [ \hat{p} \delta _\textrm{ik} - 2 \bigg ( \mu ^e \hat{S}_\textrm{ik} + \mu ^\textrm{t} C_\textrm{ik}^c \bigg ) \bigg ] n_\textrm{k} t_\textrm{i} = 0, \frac{\partial \hat{v}_\textrm{i}}{\partial n}&= 0 \quad \text {on} \quad \Gamma ^\textrm{symm} \, . \end{aligned}$$

(116)

1.1 D.1 Alternative Derivation

Starting with the second option to express $S_\textrm{qr}^d$. A perturbation expands to

$$\begin{aligned} {S_\textrm{qr}^d}^\prime = \underbrace{ \overline{S}_\textrm{ql}^\prime \overline{S}_\textrm{lr} + \overline{S}_\textrm{ql} \overline{S}_\textrm{lr}^\prime + \overline{W}_\textrm{ql}^\prime \overline{W}_\textrm{lr} + \overline{W}_\textrm{ql} \overline{W}_\textrm{lr}^\prime }_{A_\textrm{qr}} - \frac{1}{3} \delta _\textrm{qr} \bigg [ \underbrace{2 \overline{S}_\textrm{lm} \overline{S}_\textrm{lm}^\prime - 2 \overline{W}_\textrm{lm} \overline{W}_\textrm{lm}^\prime }_{B} \bigg ] \, . \end{aligned}$$

(117)

The calculus is divided into two parts, where the first part $A_\textrm{qr}$ is further expanded to

$$\begin{aligned} A_\textrm{qr}&= \frac{1}{2} \left( \frac{\partial \overline{v}_\textrm{q}^\prime }{\partial x_\textrm{l}} + \frac{\partial \overline{v}_\textrm{l}^\prime }{\partial x_\textrm{q}}\right) \overline{S}_\textrm{lr} + \frac{1}{2} \left( \frac{\partial \overline{v}_\textrm{l}^\prime }{\partial x_\textrm{r}} + \frac{\partial \overline{v}_\textrm{r}^\prime }{\partial x_\textrm{l}}\right) \overline{S}_\textrm{ql} + \frac{1}{2} \left( \frac{\partial \overline{v}_\textrm{q}^\prime }{\partial x_\textrm{l}} - \frac{\partial \overline{v}_\textrm{l}^\prime }{\partial x_\textrm{q}}\right) \overline{W}_\textrm{lr} + \frac{1}{2} \left( \frac{\partial \overline{v}_\textrm{l}^\prime }{\partial x_\textrm{r}} - \frac{\partial \overline{v}_\textrm{r}^\prime }{\partial x_\textrm{l}}\right) \overline{W}_\textrm{ql} \end{aligned}$$

(118)

$$\begin{aligned}&= \frac{1}{2} \left[ \frac{\partial \overline{v}_\textrm{q}^\prime }{\partial x_\textrm{l}} \left( \overline{S}_\textrm{lr} + \overline{W}_\textrm{lr} \right) + \frac{\partial \overline{v}_\textrm{l}^\prime }{\partial x_\textrm{q}} \left( \overline{S}_\textrm{lr} - \overline{W}_\textrm{lr} \right) + \frac{\partial \overline{v}_\textrm{l}^\prime }{\partial x_\textrm{r}} \left( \overline{S}_\textrm{ql} + \overline{W}_\textrm{ql}\right) + \frac{\partial \overline{v}_\textrm{r}^\prime }{\partial x_\textrm{l}} \left( \overline{S}_\textrm{ql} - \overline{W}_\textrm{ql} \right) \right]. \end{aligned}$$

(119)

Contracting with $\alpha _\textrm{qr}$ (cf. Eq. (44)) allows for further simplifications, viz.

$$\begin{aligned} \alpha _\textrm{qr} A_\textrm{qr}&= \frac{1}{2} \left[ \frac{\partial \overline{v}_\textrm{q}^\prime }{\partial x_\textrm{l}} \alpha _\textrm{qr} \left( \overline{S}_\textrm{lr} + \overline{W}_\textrm{lr} \right) + \frac{\partial \overline{v}_\textrm{l}^\prime }{\partial x_\textrm{q}} \alpha _\textrm{qr} \left( \overline{S}_\textrm{lr} - \overline{W}_\textrm{lr} \right) + \frac{\partial \overline{v}_\textrm{l}^\prime }{\partial x_\textrm{r}} \alpha _\textrm{qr}\left( \overline{S}_\textrm{ql} + \overline{W}_\textrm{ql}\right) + \frac{\partial \overline{v}_\textrm{r}^\prime }{\partial x_\textrm{l}} \alpha _\textrm{qr} \left( \overline{S}_\textrm{ql} - \overline{W}_\textrm{ql} \right) \right] \end{aligned}$$

(120)

$$\begin{aligned}&= \frac{1}{2} \bigg [ \alpha _\textrm{iq} \left( \overline{S}_\textrm{kq} + \overline{W}_\textrm{kq} \right) + \alpha _\textrm{kq} \left( \overline{S}_\textrm{iq} - \overline{W}_\textrm{iq} \right) + \alpha _\textrm{qk} \left( \overline{S}_\textrm{qi} + \overline{W}_\textrm{qi}\right) + \alpha _\textrm{qi} \left( \overline{S}_\textrm{qk} - \overline{W}_\textrm{qk} \right) \bigg ] \frac{\partial \overline{v}_\textrm{i}^\prime }{\partial x_\textrm{k}} \, . \end{aligned}$$

(121)

Raising symmetry arguments ($\alpha _\textrm{qr} = \alpha _\textrm{qr}$, $S_\textrm{qr} = S_\textrm{rq}$, $W_\textrm{qr} = - W_\textrm{rq}$) finally provides

$$\begin{aligned} \alpha _\textrm{qr} A_\textrm{qr}&= \bigg [ \alpha _\textrm{iq} \left( \overline{S}_\textrm{kq} + \overline{W}_\textrm{kq} \right) + \alpha _\textrm{kq} \left( \overline{S}_\textrm{iq} - \overline{W}_\textrm{iq} \right) \bigg ] \frac{\partial \overline{v}_\textrm{i}^\prime }{\partial x_\textrm{k}} \, . \end{aligned}$$

(122)

The second part is considered now, i.e.,

$$\begin{aligned} B&= \overline{S}_\textrm{lm} \left( \frac{\partial \overline{v}_\textrm{l}^\prime }{\partial x_\textrm{m}} + \frac{\partial \overline{v}_\textrm{m}^\prime }{\partial x_\textrm{l}} \right) - \overline{W}_\textrm{lm} \left( \frac{\partial \overline{v}_\textrm{l}^\prime }{\partial x_\textrm{m}} - \frac{\partial \overline{v}_\textrm{m}^\prime }{\partial x_\textrm{l}} \right) \end{aligned}$$

(123)

$$\begin{aligned}&= \frac{\partial \overline{v}_\textrm{l}^\prime }{\partial x_\textrm{m}} \left( 2 \overline{S}_\textrm{lm} - 2 \overline{W}_\textrm{lm} \right) \, . \end{aligned}$$

(124)

Employing symmetry conditions finally yields

$$\begin{aligned} \alpha _\textrm{qr} \frac{1}{3} \delta _\textrm{qr} B = \alpha _\textrm{qq} \frac{2}{3} \left( \overline{S}_\textrm{lm} - \overline{W}_\textrm{lm} \right) \frac{\partial \overline{v}_\textrm{l}^\prime }{\partial x_\textrm{m}} \, . \end{aligned}$$

(125)

Hence, the required expression $\alpha _\textrm{qr} {S_\textrm{qr}^d}^\prime$ reads as follows

$$\begin{aligned} \alpha _\textrm{qr} {S_\textrm{qr}^d}^\prime&= \alpha _\textrm{qr} ( A_\textrm{qr} - \frac{1}{3} \delta _\textrm{qr} B ) \end{aligned}$$

(126)

$$\begin{aligned}&= \bigg [ \underbrace{\alpha _\textrm{iq} \left( \overline{S}_\textrm{kq} + \overline{W}_\textrm{kq} \right) + \alpha _\textrm{kq} \left( \overline{S}_\textrm{iq} - \overline{W}_\textrm{iq} \right) - \alpha _\textrm{qq} \frac{2}{3} \left( \overline{S}_\textrm{ik} - \overline{W}_\textrm{ik} \right) }_{\gamma _\textrm{ik}} \bigg ] \frac{\partial \overline{v}_\textrm{i}^\prime }{\partial x_\textrm{k}} \, . \end{aligned}$$

(127)

The condensed variable $\gamma _\textrm{ik}$ already equals the first part on the r.h.s. of Eq. (107). However, the derivation is continued and (127) plugged into the remaining part of Eq. (47), viz.

$$\begin{aligned} \frac{1}{T} \int _{t^\textrm{s}}^{t^e} \int \hat{v}_\textrm{q} \frac{\partial }{\partial x_\textrm{r}} \left[ 2 \mu ^\textrm{t} \gamma _\textrm{ik} \frac{\partial \overline{v}_\textrm{i}^\prime }{\partial x_\textrm{k}} \overline{S}_\textrm{qr} \right] \textrm{d} \Omega \textrm{d} t&= \frac{1}{T} \int _{t^\textrm{s}}^{t^e} \int \hat{v}_\textrm{q} \left[ 2 \mu ^\textrm{t} \gamma _\textrm{ik} \frac{\partial \overline{v}_\textrm{i}^\prime }{\partial x_\textrm{k}} \overline{S}_\textrm{qr} \right] n_\textrm{r} - 2 \mu ^\textrm{t} \hat{S}_\textrm{qr} \overline{S}_\textrm{qr} \gamma _\textrm{ik} \overline{v}_\textrm{i}^\prime n_\textrm{k} \textrm{d} \Gamma \textrm{d} t \end{aligned}$$

(128)

$$\begin{aligned}&\quad + \frac{1}{T} \int _{t^\textrm{s}}^{t^e} \int \overline{v}_\textrm{i}^\prime \frac{\partial }{\partial x_\textrm{k}} \left[ 2 \mu ^\textrm{t}\overline{S}_\textrm{qr} \hat{S}_\textrm{qr} \gamma _\textrm{ik} \right] \textrm{d} \Omega \textrm{d} t \, . \end{aligned}$$

(129)

Accordingly, the contribution to the adjoint momentum equation reads

$$\begin{aligned} R^{\hat{v}_\textrm{i}, \text {wale}} = R^{\hat{v}_\textrm{i}, \text {lam.}} - \frac{\partial }{\partial x_\textrm{k}} \bigg [ 2 \mu ^\textrm{t} \underbrace{ \overline{S}_\textrm{qr} \hat{S}_\textrm{qr} \left( \gamma _\textrm{ik} - \beta _\textrm{ik} \right) }_{= C_\textrm{ik}^c \text {cf. Eq. (107)}} \bigg ] \, , \end{aligned}$$

(130)

and proves the formal correctness of the former derivation, since

$$\begin{aligned} C_\textrm{ik}^c = \overline{S}_\textrm{qr} \hat{S}_\textrm{qr} \left( \gamma _\textrm{ik} - \beta _\textrm{ik} \right) \qquad \text {with} \qquad \gamma _\textrm{ik}&= \alpha _\textrm{iq} \left( \overline{S}_\textrm{kq} + \overline{W}_\textrm{kq} \right) + \alpha _\textrm{kq} \left( \overline{S}_\textrm{iq} - \overline{W}_\textrm{iq} \right) - \alpha _\textrm{qq} \frac{2}{3} \left( \overline{S}_\textrm{ik} - \overline{W}_\textrm{ik} \right) \end{aligned}$$

(131)

$$\begin{aligned}&= \alpha _\textrm{qi} \frac{\partial \overline{v}_\textrm{k}}{\partial x_\textrm{q}} + \alpha _\textrm{qk} \frac{\partial \overline{v}_\textrm{q}}{\partial x_\textrm{i}} - \alpha _\textrm{qq} \frac{2}{3} \frac{\partial \overline{v}_\textrm{k}}{\partial x_\textrm{i}} \, . \end{aligned}$$

(132)

Continuous Adjoint Kinetic Energy Subgrid-Scale Model

The variations of the KESS equation (11) are expanded and isolated, viz.

$$\begin{aligned} \frac{1}{T} \int _{t^\textrm{s}}^{t^e} \int \hat{k} \frac{\partial \rho k^\prime }{\partial t} \textrm{d} \Omega \textrm{d} t&= \bigg [ \int \rho \hat{k} k \textrm{d} \Omega \bigg ]_{t_s}^{t_e} - \frac{1}{T} \int _{t^\textrm{s}}^{t^e} \int k^\prime \frac{\partial \rho \hat{k}}{\partial t} \textrm{d} \Omega \textrm{d} t \end{aligned}$$

(133)

$$\begin{aligned} \frac{1}{T} \int _{t^\textrm{s}}^{t^e} \int \hat{k} \frac{\partial \rho (\overline{v}_\textrm{k}^\prime k + \overline{v}_\textrm{k} k^\prime )}{\partial x_\textrm{k}} \textrm{d} \Omega \textrm{d} t&= \frac{1}{T} \int _{t^\textrm{s}}^{t^e} \int \hat{k} \rho \left( \overline{v}_\textrm{k}^\prime k + \overline{v}_\textrm{k} k^\prime \right) n_\textrm{k} d \Gamma \textrm{d} t \nonumber \\&\quad - \frac{1}{T} \int _{t^\textrm{s}}^{t^e} \int \overline{v}_\textrm{k}^\prime \rho k \frac{\partial \hat{k}}{\partial x_\textrm{k}} + k^\prime \frac{\partial \overline{v}_\textrm{k} \rho \hat{k}}{\partial x_\textrm{k}} \textrm{d} \Omega \textrm{d} t \end{aligned}$$

(134)

$$\begin{aligned} \frac{1}{T} \int _{t^\textrm{s}}^{t^e} \int \hat{k} \left[ 2 {\mu ^\textrm{t}}^\prime S_\textrm{ik}^2 + 4 \mu ^\textrm{t} S_\textrm{ik} S_\textrm{ik}^\prime \right] \textrm{d} \Omega \textrm{d} t&= \frac{1}{T} \int _{t^\textrm{s}}^{t^e} \int \hat{k} 4 \mu ^\textrm{t} S_\textrm{ik} \left( \overline{v}_\textrm{i}^\prime n_\textrm{k} + \overline{v}_\textrm{k}^\prime n_\textrm{i} \right) \textrm{d} \Gamma \textrm{d} t \end{aligned}$$

(135)

$$\begin{aligned}&\quad - \frac{1}{T} \int _{t^\textrm{s}}^{t^e} \int \overline{v}_\textrm{i}^\prime \frac{\partial }{\partial x_\textrm{k}} \left[ \hat{k} 8 \mu ^\textrm{t} S_\textrm{ik} \right] - \frac{\mu ^\textrm{t}}{k} {k}^\prime \hat{k} S_\textrm{mn}^2 \textrm{d} \Omega \textrm{d} t \end{aligned}$$

(136)

$$\begin{aligned} \frac{1}{T} \int _{t^\textrm{s}}^{t^e} \int \hat{k} C_\varepsilon \rho \left( \frac{k^{3/2}}{{{l^\textrm{g}}}} \right) ^\prime \textrm{d} \Omega \textrm{d} t&= \frac{1}{T} \int _{t^\textrm{s}}^{t^e} \int \hat{k} C_\varepsilon \rho \left( \frac{3}{2} \frac{\sqrt{k}}{{{l^\textrm{g}}}} k^\prime - \frac{k^{3/2}}{{{l^\textrm{g}}}^2} {{l^\textrm{g}}}^\prime \right) \textrm{d} \Omega \textrm{d} t \end{aligned}$$

(137)

$$\begin{aligned} \frac{1}{T} \int _{t^\textrm{s}}^{t^e} \int \hat{k} \frac{\partial }{\partial x_\textrm{k}} \left[ \frac{{\mu ^\textrm{t}}^\prime }{\sigma } \frac{\partial k}{\partial x_\textrm{k}} + \frac{\mu ^\textrm{t}}{\sigma } \frac{\partial k^\prime }{\partial x_\textrm{k}} \right] \textrm{d} \Omega \textrm{d} t&= \frac{1}{T} \int _{t^\textrm{s}}^{t^e} \int \hat{k} \left[ \frac{{k}^\prime }{k} \frac{\mu ^\textrm{t}}{\sigma } \frac{\partial k}{\partial x_\textrm{k}} + \frac{\mu ^\textrm{t}}{\sigma } \frac{\partial k^\prime }{\partial x_\textrm{k}} \right] n_\textrm{k} - \frac{\mu ^\textrm{t}}{\sigma } \frac{\partial \hat{k}}{\partial x_\textrm{k}} k^\prime n_\textrm{k} \textrm{d} \Gamma \textrm{d} t \end{aligned}$$

(138)

$$\begin{aligned}&\quad - \frac{1}{T} \int _{t^\textrm{s}}^{t^e} \int \frac{{k}^\prime }{k} \frac{\mu ^\textrm{t}}{\sigma } \frac{\partial \hat{k}}{\partial x_\textrm{k}} \frac{\partial k}{\partial x_\textrm{k}} - k^\prime \frac{\partial }{\partial x_\textrm{k}} \left[ \frac{\mu ^\textrm{t}}{\sigma } \frac{\partial \hat{k}}{\partial x_\textrm{k}} \right] \textrm{d} \Omega \textrm{d} t .\end{aligned}$$

(139)

Together with the expansion in (55) and the laminar system Eq. (77), the varied KESS model’s Lagrangian reads

$$\begin{aligned} {L^{\text {kess}}}^\prime&= \frac{1}{T} \bigg [ \int \hat{v}_\textrm{i} \rho \overline{v}_\textrm{i} + \hat{k} \rho k \textrm{d} \Omega \bigg ]_{t^\textrm{s}}^{t^e} \nonumber \\&+ \frac{1}{T} \int _{t^\textrm{s}}^{t^e} \int \overline{v}_\textrm{i}^\prime \big [ -\hat{p} n_\textrm{i} + \rho \overline{v}_\textrm{k} (\hat{v}_\textrm{k} n_\textrm{i} + \hat{v}_\textrm{i} n_\textrm{k}) + 2 \mu ^e \hat{S}_\textrm{ik} n_\textrm{k} + \hat{k} \rho k n_\textrm{i} - \hat{k} 8 \mu ^\textrm{t} S_\textrm{ik} n_\textrm{k} \big ] + p^\prime \big [ \hat{v}_\textrm{i} n_\textrm{i} \big ] + S_\textrm{ik}^\prime \big [ \hat{v}_\textrm{i} 2 \mu ^e n_\textrm{k} \big ] \nonumber \\&\qquad \quad + {k}^\prime \bigg [ \hat{v}_\textrm{i} \frac{\mu ^\textrm{t}}{k} \overline{S}_\textrm{ik} n_\textrm{k} + \hat{k} \rho \overline{v}_\textrm{k} n_\textrm{k} - \frac{\mu ^\textrm{t} \hat{k}}{\sigma ^\textrm{k} k} \frac{\partial k}{\partial x_\textrm{k}} n_\textrm{k} + \frac{\mu ^\textrm{t}}{\sigma } \frac{\partial \hat{k}}{\partial x_\textrm{k}} \bigg ] - \frac{\partial k^\prime }{\partial x_\textrm{k}} \bigg [ \hat{k} \frac{\mu ^\textrm{t}}{\sigma } n_\textrm{k} \bigg ] \nonumber \\&\qquad \quad + {{l^\textrm{g}}}^\prime \bigg [ \hat{v}_\textrm{i} 2 C^\textrm{k} \rho \sqrt{k} \overline{S}_\textrm{ik} n_\textrm{k} \bigg ] \textrm{d} \Gamma \textrm{d} t \nonumber \\&+ \frac{1}{T} \int _{t^\textrm{s}}^{t^e} \int \overline{v}_\textrm{i}^\prime \bigg [ -\frac{\partial \rho \hat{v}_\textrm{i}}{\partial t} - \rho \overline{v}_\textrm{k} \frac{\partial \hat{v}_\textrm{k}}{\partial x_\textrm{i}} - \frac{\partial \overline{v}_\textrm{k} \rho \hat{v}_\textrm{i}}{\partial x_\textrm{k}} + \frac{\partial \hat{p}}{\partial x_\textrm{i}} - \frac{\partial }{\partial x_\textrm{k}} \bigg [ 2 \mu ^e \hat{S}_\textrm{ik} \bigg ] + \frac{\partial }{\partial x_\textrm{k}} \bigg [ \hat{k} 8 \mu ^\textrm{t} S_\textrm{ik} \bigg ] \bigg ] \nonumber \\&\qquad \quad - p^\prime \bigg [ \frac{\partial \hat{v}_\textrm{i}}{\partial x_\textrm{i}} \bigg ] \end{aligned}$$

(140)

$$\begin{aligned}&\qquad \quad + {k}^\prime \bigg [ -\frac{\partial \rho \hat{k}}{\partial t} - \rho k \frac{\partial \hat{k}}{\partial x_\textrm{k}} - \frac{\partial \overline{v}_\textrm{k} \rho \hat{k}}{\partial x_\textrm{k}} + \frac{\mu ^\textrm{t}}{k} \overline{S}_\textrm{ik} \hat{S}_\textrm{ik} - \frac{\mu ^\textrm{t}}{k} \hat{k} S_\textrm{mn}^2 +\hat{k} C_\varepsilon \rho \frac{3}{2} \frac{\sqrt{k}}{{{l^\textrm{g}}}} + \frac{\mu ^\textrm{t}}{\sigma ^\textrm{k} k} \frac{\partial \hat{k}}{\partial x_\textrm{k}} \frac{\partial k}{\partial x_\textrm{k}} - \frac{\partial }{\partial x_\textrm{k}} \left[ \frac{\mu ^\textrm{t}}{\sigma } \frac{\partial \hat{k}}{\partial x_\textrm{k}} \right] \bigg ] \nonumber \\&\qquad \quad + {{l^\textrm{g}}}^\prime \bigg [ 2 C^\textrm{k} \rho \sqrt{k} \overline{S}_\textrm{ik} \hat{S}_\textrm{ik} - \hat{k} C_\varepsilon \rho \frac{k^{3/2}}{{{l^\textrm{g}}}^2} \bigg ] \textrm{d} \Omega \textrm{d} t \, . \end{aligned}$$

(141)

Assuming ${{l^\textrm{g}}}^\prime = 0$ leads to the following set of continuous adjoint KESS equations

$$\begin{aligned} - \frac{\partial \hat{v}_\textrm{k}}{\partial x_\textrm{k}} + -\frac{\partial j^\Omega }{\partial p} = R^{\hat{p}, \text {kess}}&= 0 \quad{} & {} \text {in} \quad \Omega ^\textrm{O} \end{aligned}$$

(142)

$$\begin{aligned} - \frac{\partial \hat{v}_\textrm{k}}{\partial x_\textrm{k}} = R^{\hat{p}, \text {kess}}&= 0 \quad{} & {} \text {in} \quad \Omega \setminus \Omega ^\textrm{O} \end{aligned}$$

(143)

$$\begin{aligned} -\frac{\partial \rho \hat{v}_\textrm{i}}{\partial t} - \frac{\partial \rho \overline{v}_\textrm{k} \hat{v}_\textrm{i}}{\partial x_\textrm{k}} - \rho \overline{v}_\textrm{k} \frac{\partial \hat{v}_\textrm{k}}{\partial x_\textrm{i}} + \frac{\partial }{\partial x_\textrm{k}} \left[ \hat{p} \delta _\textrm{ik} - 2 \mu ^\textrm{e} \hat{S}_\textrm{ik} \right] \qquad \qquad \qquad&\nonumber \\ - \rho k \frac{\partial \hat{k}}{\partial x_\textrm{i}} + \frac{\partial }{\partial x_\textrm{k}} \left[ \hat{k} 8 \mu ^\textrm{t} S_\textrm{ik} \right] + \frac{\partial j^\Omega }{\partial \overline{v}_\textrm{i}} = R^{\hat{v}_\textrm{i}, \text {kess}}&= 0 \quad{} & {} \text {in} \quad \Omega \end{aligned}$$

(144)

$$\begin{aligned} -\frac{\partial \rho \hat{v}_\textrm{i}}{\partial t} - \frac{\partial \overline{v}_\textrm{k} \rho \hat{v}_\textrm{i}}{\partial x_\textrm{k}} - \rho \overline{v}_\textrm{k} \frac{\partial \hat{v}_\textrm{k}}{\partial x_\textrm{i}} + \frac{\partial }{\partial x_\textrm{k}} \left[ \hat{p} \delta _\textrm{ik} - 2 \mu ^\textrm{e} \hat{S}_\textrm{ik} \right]&\nonumber \\ - k \frac{\partial \hat{k}}{\partial x_\textrm{i}} + \frac{\partial }{\partial x_\textrm{k}} \left[ \hat{k} 8 \mu ^\textrm{t} S_\textrm{ik} \right] = R^{\hat{v}_\textrm{i}, \text {kess}}&= 0 \quad{} & {} \text {in} \quad \Omega \setminus \Omega ^\textrm{O} \end{aligned}$$

(145)

$$\begin{aligned} - \frac{\partial \hat{k}}{\partial t} - \frac{\partial \overline{v}_\textrm{k} \rho \hat{k}}{\partial x_\textrm{k}} - \frac{\partial }{\partial x_\textrm{k}}\left[ \frac{\mu ^\textrm{t}}{\sigma ^\textrm{k}} \frac{\partial \hat{k}}{\partial x_\textrm{k}} \right] + \frac{\mu ^\textrm{t}}{\sigma ^\textrm{k} k} \frac{\partial \hat{k}}{\partial x_\textrm{k}} \frac{\partial k}{\partial x_\textrm{k}} - \frac{\mu ^\textrm{t} \hat{k}}{k} S_\textrm{ik}^2&\nonumber \\ + C_\varepsilon \rho \frac{3}{2} \frac{\sqrt{k}}{{{l^\textrm{g}}}} \hat{k} + \frac{\mu ^\textrm{t}}{k} S_\textrm{ik} \hat{S}_\textrm{ik} + \frac{\partial j^\Omega }{\partial k} = R^{\hat{k}, \text {kess}}&= 0 \quad{} & {} \text {in} \quad \Omega \end{aligned}$$

(146)

$$\begin{aligned} - \frac{\partial \hat{k}}{\partial t} - \frac{\partial \overline{v}_\textrm{k} \rho \hat{k}}{\partial x_\textrm{k}} - \frac{\partial }{\partial x_\textrm{k}}\left[ \frac{\mu ^\textrm{t}}{\sigma ^\textrm{k}} \frac{\partial \hat{k}}{\partial x_\textrm{k}} \right] +\frac{\mu ^\textrm{t}}{\sigma ^\textrm{k} k} \frac{\partial \hat{k}}{\partial x_\textrm{k}} \frac{\partial k}{\partial x_\textrm{k}} - \frac{\mu ^\textrm{t} \hat{k}}{k} S_\textrm{ik}^2&\nonumber \\ + C_\varepsilon \rho \frac{3}{2} \frac{\sqrt{k}}{{{l^\textrm{g}}}} \hat{k} + \frac{\mu ^\textrm{t}}{k} S_\textrm{ik} \hat{S}_\textrm{ik} = R^{\hat{k}, \text {kess}}&= 0 \quad{} & {} \text {in} \quad \Omega \setminus \Omega ^\textrm{O} \end{aligned}$$

(147)

$$\begin{aligned}{} & {} {}{} & {} {}&\nonumber \\ \hat{v}_\textrm{i} = 0, \frac{\partial \hat{p}}{\partial n} = 0, \hat{k}&= 0 \quad{} & {} \text {on} \quad \Gamma ^\textrm{in} \end{aligned}$$

(148)

$$\begin{aligned} \bigg [ \hat{p} \delta _\textrm{ik} - 2 \mu ^\textrm{e} \hat{S}_\textrm{ik} \bigg ] n_\textrm{k} n_\textrm{i} - \rho \overline{v}_\textrm{k} \hat{v}_\textrm{i} n_\textrm{k} n_\textrm{i} = 0, \frac{\partial \hat{v}_\textrm{i}}{\partial n} = 0, \frac{\partial \hat{k}}{\partial n}&= 0 \quad{} & {} \text {on} \quad \Gamma ^\textrm{out} \end{aligned}$$

(149)

$$\begin{aligned} \hat{v}_\textrm{i} + \frac{\partial j_\textrm{i}^\Omega }{\partial p} = 0, \frac{\partial \hat{p}}{\partial n} = 0, \hat{k}&= 0 \quad{} & {} \text {on} \quad \Gamma ^\textrm{O} \end{aligned}$$

(150)

$$\begin{aligned} \hat{v}_\textrm{i} = 0, \frac{\partial \hat{p}}{\partial n} = 0, \hat{k}&= 0 \quad{} & {} \text {on} \quad \Gamma ^\textrm{wall} \setminus \Gamma ^\textrm{O} \end{aligned}$$

(151)

$$\begin{aligned} \hat{v}_\textrm{i} n_\textrm{i} = 0, \bigg [ \hat{p} \delta _\textrm{ik} - 2 \mu ^\textrm{e} \hat{S}_\textrm{ik} \bigg ] n_\textrm{k} t_\textrm{i} = 0, \frac{\partial \hat{v}_\textrm{i}}{\partial n}, \frac{\partial \hat{k}}{\partial n}&= 0 \quad{} & {} \text {on} \quad \Gamma ^\textrm{symm} \, . \end{aligned}$$

(152)