An extension of Moretti's classical shock-fitting technique is proposed to solve complicated unsteady viscous flows. This version allows the automatic treatment of flow structures featuring triple points and shock interactions. A fitting of contact discontinuities has also been introduced for a number of problems. The fitting procedure is used with a Navier-Stokes solver based on the A scheme. Validation tests for selected cases are presented.
IntroductionT HE numerical treatment of shocks by floating shock-fitting techniques was introduced in the 1970s by Moretti. 1 ' 2 The method explicitly assumes the shocks to be discontinuities of the flowfield, governed by the Rankine-Hugoniot relations. It basically relies on a one-dimensional approach, with the shock strength computed following the procedure reported in Ref. 3. Therefore, it can easily handle shocks approximately oriented along one of the directions of the families of coordinate lines. As a consequence, initially the fitting was limited to these shocks, considering only their intersections with the other family of coordinate lines.Moretti introduced many improvements to his original version, including the ability to compute viscous flows, 4 and then proposed a generalized extension of the technique to fit shocks having any orientation with respect to the coordinate lines. 5 This approach was shown to be successful in many practical applications of supersonic flows. 5 " 7 Unfortunately, it requires a procedure to determine the local shock slope, which was not sufficiently robust, particularly when complicated shock interactions take place, or when many discontinuities are too close to one another. On the other hand, the quality of the results obtained prompted other attempts to be made to improve robustness. 8 ' 9 Although such efforts have not resulted yet in a satisfactory general version of the shock-fitting technique, renewed attention to possible improvements arose from its capability to compute accurately flow transients that are characterized by complex shock interactions. In such cases, the location and the speed of the moving shocks can be predicted precisely, without adding any computational cost, whereas to achieve solutions of comparable accuracy other methods need a specific handling of the discontinuities (like the use of very fine grids, or moving adaptive grids to be redefined at each integration step, or high-order schemes), which can eventually raise the computational time to impractical levels.The aim of this paper is to present a further development of the methodology to predict unsteady behavior of both viscous and inviscid supersonic flows. An algorithm that integrates the Navier-Stokes equations written in nonconservative form, coupled to the fitting of shocks and contact discontinuities, is proposed. This method can be seen as a natural evolution of Moretti's original technique. In particular, it has been carried out by introducing modifications to the procedure proposed by Moretti, 5 ' 10 with the extensions introduced to compute...
