High-order numerical methods have been considered and implemented in order to assess their applicability in a range of complex ows centering on shockinduced turbulent mixing. Speci cally, Weighted Essentially Non-Oscillatory (WENO) variable reconstruction schemes of fth and ninth order accuracy have been investigated within the context of a nite volume Godunov solver. In addition to this there have been further numerical developments to assess the HLLC Riemann solver and various quasi-conservative multi-component models in conjunction with the high-order methods. Understanding the physics of fundamental ow instabilities and turbulence is increasingly necessary to the development of a vast range of engineering applications with relation to uid dynamics. It is desirable to develop numerical methods that possess su cient accuracy to capture the detail of such ows while remaining robust and viable in terms of cost. The WENO schemes have been tested on a number of cases in comparison with more traditional second-order MUSCL schemes. These include two and three dimensional, single and multi mode Richtmyer-Meshkov instabilities with differing initial perturbations, a cube of homogeneous decaying turbulence and two hypersonic geometry cases were simulated. The results from this research were consistent. The higher-order methods provided measurably greater resolution of small scale uctuations. By conducting grid convergence studies it was seen that the e ect of the higher-order methods was comparable to the e ect of increasing the number of grid points. The cost analysis repeatedly showed that despite the additional cost of using a higher-order method they were much better value as they could resolve ow features on a signi cantly coarser grid. The high-order methods were not only validated for a range of ow problems but shown to o er great value for their additional cost; they could potentially help advance understanding and development in a wide range of elds much faster than is currently the case
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