Analytical formulas for verification of aerodynamic force and moment computations

• Published in: Journal of computational physics Vol. 466; p. 111408
• Main Author: Nishikawa, Hiroaki
• Format: Journal Article
• Language: English
• Published: Cambridge Elsevier Science Ltd 01-10-2022
• Subjects: Aerodynamic forces; Algorithms; Circular cylinders; Computational fluid dynamics; Computational physics; Drag coefficients; Mathematical analysis; Pitching moments; Solvers; Stresses; Unstructured grids (mathematics); Velocity gradient; Verification; Yaw; Yawing moments
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2022.111408

In this paper, we derive analytical formulas for the drag, lift, and moment coefficients of a circular cylinder exposed to a fictitious flow defined by analytical functions in two and three dimensions, and demonstrate that these formulas serve as a useful tool for quickly verifying the implementation of force and moment computation algorithms in computational fluid dynamics solvers. Focusing on a typical second-order unstructured-grid solver, we will show that the lift and drag coefficients consist of second-order accurate pressure and first-order accurate viscous force coefficients, and thus it is not always possible to uniquely determine their orders of accuracy although all will be first-order accurate on fine enough grids. For the circular cylinder geometry, the pitching moment coefficient depends only on the viscous stresses and is therefore first-order accurate. In three dimensions, rolling and yaw moments have both pressure and viscous contributions, and therefore they are asymptotically first-order accurate. Viscous contributions to the force and moment coefficients are first-order accurate even for regular grids because the gradient stencil is not symmetric typically at a boundary and any linearly-exact gradient algorithm can be first-order accurate at best; hence, the velocity gradients used to compute the viscous stresses at a wall are first-order accurate. For verification, therefore, it is necessary to verify the pressure and viscous contributions separately; it can be performed rigorously with the analytical formulas presented in this paper.