“…Hence, this two-dimensional solution should be valid for lifting and non-lifting flows alike, contrary to what was claimed in Lozano & Ponsin (2021a), where the validity of the solution was restricted to non-lifting flows based on far-field behaviour. In 2-D, incompressible flow past a lifting airfoil is still irrotational.…”
Section: Analytic Adjoint Solutionsmentioning
confidence: 60%
“…Figure 16 shows the analytic drag-based adjoint solution. This solution has been compared to numerical solutions obtained by discretizing the adjoint equation directly in Lozano & Ponsin (2021 a , 2022). Figure 16.Analytic drag-based adjoint solution on the airfoil profile for incompressible, inviscid flow at α = 0° past a van de Vooren airfoil with trailing-edge angle and 12 % thickness.…”
Section: Sample Solutionsmentioning
confidence: 99%
“…where Γ 0 is the circulation of the base flow, so if δu, δv tend to zero at the far-field (which is a reasonable assumption as otherwise the free stream state would be perturbed), then the above term is at least O(1/r 2 ) and the far-field integral vanishes, thus guaranteeing that ψ drag is also a valid solution for lifting flows. The discrepancies observed in Lozano & Ponsin (2021a) can then be attributed to the dependence of the numerical solution on the distance from the body to the far-field (20 chord lengths for the meshes used in Lozano & Ponsin 2021a). However, for lift, we get the result…”
Section: Analytic Adjoint Solutionsmentioning
confidence: 99%
“…Figure 16 shows the analytic drag-based adjoint solution. This solution has been compared to numerical solutions obtained by discretizing the adjoint equation directly in Lozano & Ponsin (2021a.…”
Section: Airfoil With Finite Trailing-edge Anglementioning
confidence: 99%
“…In inviscid 2-D/3-D flows, the entropy variables provide another exact solution for an output measuring net entropy flux across boundaries (Fidkowski & Roe 2010;Lozano 2019a). A closely related solution, corresponding to a nearfield computation of aerodynamic drag, has been recently discovered by the authors (Lozano & Ponsin 2021a). Finally, an exact solution for the adjoint Navier-Stokes equations corresponding to a Blasius laminar boundary layer has been derived by Kühl, Müller & Rung (2021).…”
The Green's function approach of Giles and Pierce (J. Fluid Mech., vol. 426, 2001, pp. 327–345) is used to build the lift and drag based analytic adjoint solutions for the two-dimensional incompressible Euler equations around irrotational base flows. The drag-based adjoint solution turns out to have a very simple closed form in terms of the flow variables and is smooth throughout the flow domain, while the lift-based solution is singular at rear stagnation points and sharp trailing edges owing to the Kutta condition. This singularity is propagated to the whole dividing streamline (which includes the incoming stagnation streamline and the wall) upstream of the rear singularity (trailing edge or rear stagnation point) by the sensitivity of the Kutta condition to changes in the stagnation pressure.
“…Hence, this two-dimensional solution should be valid for lifting and non-lifting flows alike, contrary to what was claimed in Lozano & Ponsin (2021a), where the validity of the solution was restricted to non-lifting flows based on far-field behaviour. In 2-D, incompressible flow past a lifting airfoil is still irrotational.…”
Section: Analytic Adjoint Solutionsmentioning
confidence: 60%
“…Figure 16 shows the analytic drag-based adjoint solution. This solution has been compared to numerical solutions obtained by discretizing the adjoint equation directly in Lozano & Ponsin (2021 a , 2022). Figure 16.Analytic drag-based adjoint solution on the airfoil profile for incompressible, inviscid flow at α = 0° past a van de Vooren airfoil with trailing-edge angle and 12 % thickness.…”
Section: Sample Solutionsmentioning
confidence: 99%
“…where Γ 0 is the circulation of the base flow, so if δu, δv tend to zero at the far-field (which is a reasonable assumption as otherwise the free stream state would be perturbed), then the above term is at least O(1/r 2 ) and the far-field integral vanishes, thus guaranteeing that ψ drag is also a valid solution for lifting flows. The discrepancies observed in Lozano & Ponsin (2021a) can then be attributed to the dependence of the numerical solution on the distance from the body to the far-field (20 chord lengths for the meshes used in Lozano & Ponsin 2021a). However, for lift, we get the result…”
Section: Analytic Adjoint Solutionsmentioning
confidence: 99%
“…Figure 16 shows the analytic drag-based adjoint solution. This solution has been compared to numerical solutions obtained by discretizing the adjoint equation directly in Lozano & Ponsin (2021a.…”
Section: Airfoil With Finite Trailing-edge Anglementioning
confidence: 99%
“…In inviscid 2-D/3-D flows, the entropy variables provide another exact solution for an output measuring net entropy flux across boundaries (Fidkowski & Roe 2010;Lozano 2019a). A closely related solution, corresponding to a nearfield computation of aerodynamic drag, has been recently discovered by the authors (Lozano & Ponsin 2021a). Finally, an exact solution for the adjoint Navier-Stokes equations corresponding to a Blasius laminar boundary layer has been derived by Kühl, Müller & Rung (2021).…”
The Green's function approach of Giles and Pierce (J. Fluid Mech., vol. 426, 2001, pp. 327–345) is used to build the lift and drag based analytic adjoint solutions for the two-dimensional incompressible Euler equations around irrotational base flows. The drag-based adjoint solution turns out to have a very simple closed form in terms of the flow variables and is smooth throughout the flow domain, while the lift-based solution is singular at rear stagnation points and sharp trailing edges owing to the Kutta condition. This singularity is propagated to the whole dividing streamline (which includes the incoming stagnation streamline and the wall) upstream of the rear singularity (trailing edge or rear stagnation point) by the sensitivity of the Kutta condition to changes in the stagnation pressure.
Numerical solutions to the adjoint Euler equations have been found to diverge with mesh refinement near walls for a variety of flow conditions and geometry configurations. The issue is reviewed, and an explanation is provided by comparing a numerical incompressible adjoint solution with an analytic adjoint solution, showing that the anomaly observed in numerical computations is caused by a divergence of the analytic solution at the wall. The singularity causing this divergence is of the same type as the well-known singularity along the incoming stagnation streamline, and both originate at the adjoint singularity at the trailing edge. The argument is extended to cover the fully compressible case, in subcritical flow conditions, by presenting an analytic solution that follows the same structure as the incompressible one.
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