A direct design approach using the Euler equations

Abstract

Recently, a fully coupled formulation of the surface shape design problem, called the direct design approach, has been proposed in which both the target surface pressure and the unknown nodal coordinates appear explicitly in the formulations. The proposed method is generally applicable, but in the past it has only been applied in the context of ideal fluid flows [Ashrafizadeh, A., Raithby, G.D. and Stubley, G.D. 2003, Direct design of ducts. Journal of Fluids Engineering, Transaction ASME, 125, 158–165; Ashrafizadeh, A., Raithby, G.D. and Stubley, G.D. 2002, Direct design of shape. Numerical Heat Transfer-Part B, 41, 501–510; Ashrafizadeh, A., Raithby, G.D. and Stubley, G.D. 2004, Direct design of airfoil shape with a prescribed surface pressure. Numerical Heat Transfer, Part B: Fundamentals, 46(6), 505–527]. The present article extends the application of this method to the design of ducts carrying flows governed by the nonlinear coupled Euler equations, using a cell-centered finite volume method to discretize the governing equations. The details of the linearization and the imposition of the target pressure are discussed in this article. Also, the validation test cases and the design examples are presented, which show the robustness of the method in handling complex geometries and complex physical phenomena (such as shock waves). It is also shown that the computational cost of a direct shape design solution is comparable to the solution of the corresponding analysis problem.