# Shock Equations and Jump Conditions for the 2D Adjoint Euler Equations

^{*}

## Abstract

**:**

## 1. Introduction

- The correct formulation of the inviscid adjoint equations at shocks must be addressed, which has to consider linearized perturbations to the shock location. This analysis was carried out in [3,4] for the quasi-1D adjoint Euler equations and in [5,6] for the 2D adjoint Euler equations. In both cases, it was shown that, for typical cost functions, the adjoint variables are continuous at shocks where they have to obey an adjoint boundary condition. The correct formulation and approximation of adjoint equations in flows with shocks has also been addressed in [7,8,9].
- The accuracy of discretized adjoint approximations with shocks must also be considered. For numerical computation, the adjoint shock boundary condition is usually not explicitly enforced, but the discretized adjoint equations (either continuous or discrete) yield the correct solution as long as adequate levels of smoothing are applied, in such a way that the shock is spread over an increasing number of grid points as the mesh spacing is reduced [10,11,12]
- The last point to address is the assessment of the impact of the shock equations on practical applications such as aerodynamic design or error estimation, among others. In numerical implementation, especially in 2D, such shock conditions and increased smoothing are usually ignored under the assumption that they have little impact on both the adjoint solution and the sensitivities. However, results in [6] show that explicitly accounting for the adjoint shock condition can have a significant impact on both the sensitivities and the optimization procedure. For adjoint-based error correction with shocks, it was shown in [2] with 1D examples that the key to obtaining meaningful results is the use of discrete solutions with a well-resolved viscous shock.

## 2. Adjoint Equations for Shocked Flows

## 3. Jump Conditions for the Adjoint Gradient across a Shock

## 4. Numerical Tests

^{6}nodes and 10.4 × 10

^{6}triangular elements throughout the flowfield. The near-wall distance is around 10

^{−5}chord lengths, which should be more than adequate to resolve the Euler flow and adjoint fields. Notice that this compares well to the current state of the art on these types of mesh-converged Euler computations [15,20]. The flow and drag-based adjoint solutions are computed with the SU2 direct and continuous adjoint solvers using a central scheme with JST artificial dissipation.

#### 4.1. Supersonic Flow

#### 4.2. Transonic Flow

#### 4.3. Normal Shock with Normal Extension

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Bardos, C.; Pironneau, O. Control of Shocks in CFD. In Computational Fluid Dynamics 2004; Groth, C., Zingg, D.W., Eds.; Springer: Berlin/Heidelberg, Germany, 2006; pp. 27–39. [Google Scholar] [CrossRef]
- Giles, M.; Pierce, N.; Süli, E. Progress in adjoint error correction for integral functionals. Comput. Vis. Sci.
**2004**, 6, 113–121. [Google Scholar] [CrossRef][Green Version] - Giles, M.; Pierce, N.A. On the Properties of Solutions of the Adjoint Euler Equations. In Numerical Methods for Fluid Dynamics VI; Baines, M., Ed.; IFCD: Oxford, UK, 1998; pp. 1–16. [Google Scholar]
- Giles, M.; Pierce, N. Analytic adjoint solutions for the quasi-one-dimensional Euler equations. J. Fluid Mech.
**2001**, 426, 327–345. [Google Scholar] [CrossRef][Green Version] - Giles, M.; Pierce, N. Adjoint Equations in CFD: Duality, Boundary Conditions and Solution Behavior, AIAA Paper 97–1850. In Proceedings of the 13th AIAA Computational Fluid Dynamics Conference, Snowmass Village, CO, USA, 29 June–2 July 1997. [Google Scholar]
- Baeza, A.; Castro, C.; Palacios, F.; Zuazua, E. 2-D Euler Shape Design on Nonregular Flows Using Adjoint Rankine-Hugoniot Relations. AIAA J.
**2009**, 47, 552–562. [Google Scholar] [CrossRef] - Bardos, C.; Pironneau, O. A Formalism for the Differentiation of Conservation Laws, C.R. Acad. Sci. Paris Série I
**2002**, 335, 839–845. [Google Scholar] [CrossRef] - Bardos, C.; Pironneau, O. Derivatives and Control in the Presence of Shocks. Comput. Fluid Dyn. J.
**2003**, 11, 383–392. [Google Scholar] - Alauzet, F.; Pironneau, O. Continuous and discrete adjoints to the Euler equations for fluids. Int. J. Numer. Meth. Fluids
**2012**, 70, 135–157. [Google Scholar] [CrossRef][Green Version] - Giles, M. Discrete adjoint approximations with shocks. In Hyperbolic Problems: Theory, Numerics, Applications; Hou, T.Y., Tadmor, E., Eds.; Springer: Berlin/Heidelberg, Germany, 2003; pp. 185–194. [Google Scholar] [CrossRef]
- Giles, M.; Ulbrich, S. Convergence of linearized and adjoint approximations for discontinuous solutions of conservation laws. Part 1: Linearized approximations and linearized output functionals. SIAM J. Numer. Anal.
**2010**, 48, 882–904. [Google Scholar] [CrossRef][Green Version] - Giles, M.; Ulbrich, S. Convergence of linearized and adjoint approximations for discontinuous solutions of conservation laws. Part 2: Adjoint approximations and extensions. SIAM J. Numer. Anal.
**2010**, 48, 905–921. [Google Scholar] [CrossRef][Green Version] - Lozano, C. Singular and Discontinuous Solutions of the Adjoint Euler Equations. AIAA J.
**2018**, 56, 4437–4452. [Google Scholar] [CrossRef] - Lozano, C. Entropy and Adjoint Methods. J. Sci. Comput.
**2019**, 81, 2447–2483. [Google Scholar] [CrossRef] - Peter, J.; Renac, F.; Labbé, C. Analysis of finite-volume discrete adjoint fields for two-dimensional compressible Euler flows. J. Comput. Phys.
**2022**, 449, 110811. [Google Scholar] [CrossRef] - Emanuel, G. Analytical Fluid Dynamics (Chapter 6, Chapter 19.3 and Appendix H); CRC Press: Boca Raton, FL, USA, 2000. [Google Scholar]
- Koren, B.; van der Maarel, E. On steady, inviscid shock waves at continuously curved, convex surfaces. Theor. Comput. Fluid Dyn.
**1993**, 4, 177–195. [Google Scholar] [CrossRef][Green Version] - Economon, T.; Palacios, F.; Copeland, S.; Lukaczyk, T.; Alonso, J. SU2: An Open-Source Suite for Multiphysics Simulation and Design. AIAA J.
**2016**, 54, 828–846. [Google Scholar] [CrossRef] - Unconstrained Shape Design of a Transonic Inviscid Airfoil at a Cte. AoA. In the SU2 Tutorial Collection. Available online: https://su2code.github.io/tutorials/Inviscid_2D_Unconstrained_NACA0012/ (accessed on 27 February 2023).
- Vassberg, J.; Jameson, A. In Pursuit of Grid Convergence for Two-Dimensional Euler Solutions. J. Aircr.
**2010**, 47, 1152–1166. [Google Scholar] [CrossRef][Green Version] - Lozano, C.; Ponsin, J. Analytic Adjoint Solutions for the 2D Incompressible Euler Equations Using the Green’s Function Approach. J. Fluid Mech.
**2022**, 943, A22. [Google Scholar] [CrossRef] - Sartor, F.; Mettot, C.; Sipp, D. Stability, Receptivity, and Sensitivity Analyses of Buffeting Transonic Flow over a Profile. AIAA J.
**2015**, 53, 1980–1993. [Google Scholar] [CrossRef]

**Figure 6.**Flow past a NACA0012 airfoil at M = 1.5 and α = 0°. Contour lines for the adjoint x−momentum variable ψ

_{2}(

**a**) and y−momentum variable ψ

_{3}(

**b**). The bow and fish-tail shocks are indicated for reference.

**Figure 7.**Flow past a NACA0012 airfoil at M = 1.5 and α = 0°. Contour plot for the adjoint y−momentum variable ψ

_{3}near the bow shock. The bow shock and the cutting line are indicated for reference.

**Figure 8.**Flow past a NACA0012 airfoil at M = 1.5 and α = 0°. Adjoint variables along the cutting line indicated in Figure 7. Mach number and tangential velocity are also shown for reference.

**Figure 9.**Flow past a NACA0012 airfoil at M = 1.5 and α = 0°. Adjoint normal derivatives and shock relations across the bow shock.

**Figure 10.**Mach contours and cutting line for flow past a NACA0012 airfoil at M = 0.8 and α = 1.25°.

**Figure 11.**Flow past a NACA0012 airfoil at M = 0.8 and α = 1.25°. Contour lines for the x−momentum variable ψ

_{2}(

**a**) and y−momentum variable ψ

_{3}(

**b**). The shock position is indicated for reference.

**Figure 12.**Flow past a NACA0012 airfoil at M = 0.8 and α = 1.25°. Contour plot for the adjoint y−momentum variable ψ

_{3}near the shock. The shock and the cutting line are indicated for reference.

**Figure 13.**Adjoint variables along the cutting line indicated in Figure 12. Mach number and tangential velocity are also shown for reference.

**Figure 14.**Flow past a NACA0012 airfoil at M = 0.8 and α = 1.25°. Adjoint normal derivatives and shock relations across the shock.

**Figure 15.**(

**a**) Mach contours and cutting line and (

**b**) surface Mach distribution for flow past a NACA0012 airfoil at M = 0.8 and α = 5.794°.

**Figure 16.**Adjoint variables along the cutting line indicated in Figure 15. Mach number and tangential velocity are also shown for reference.

**Figure 17.**Flow past a NACA0012 airfoil at M = 0.8 and α = 5.794°. Adjoint normal derivatives across the shock.

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**MDPI and ACS Style**

Lozano, C.; Ponsin, J.
Shock Equations and Jump Conditions for the 2D Adjoint Euler Equations. *Aerospace* **2023**, *10*, 267.
https://doi.org/10.3390/aerospace10030267

**AMA Style**

Lozano C, Ponsin J.
Shock Equations and Jump Conditions for the 2D Adjoint Euler Equations. *Aerospace*. 2023; 10(3):267.
https://doi.org/10.3390/aerospace10030267

**Chicago/Turabian Style**

Lozano, Carlos, and Jorge Ponsin.
2023. "Shock Equations and Jump Conditions for the 2D Adjoint Euler Equations" *Aerospace* 10, no. 3: 267.
https://doi.org/10.3390/aerospace10030267