High-order Analysis And Application Of The Curved Shock Theory

The most distinctive features of supersonic flows are shock waves.However,curved shock theory is still in the process of development.A high-order curved shock theory is proposed which covers theoretical derivation,analysis and applications on inverse design method of supersonic flow field,Mach reflection,and the integration method for dual-waverider and inlet are emphasize.The main research works that have been done in this paper include:(1)The development of curved shock theory and its application in supersonic flow are summarized and classified systematically.The various design methods based on the shock theory and the future research and development trends are also analyzed.(2)Second-order curved shock theory is developed and applied to planar and axisymmetric curved shocks.Explicit equations are given in an influence coefficient format,relating the second-order gradients of pre-shock and post-shock flow parameters to shock curvature gradients.Theoritical analyses of curved shock properties are presented in planar and axis symmetry flows.(3)Three inverse design methods are demonstrated.First,the post-shock flow fields behind known curved shocks are solved using the second-order curved shock equations.Second,the second-order theory is applied to capture the curved shock shape with limited flow field information.In terms of the residual sum of squares of the curved shock,the second-order curved shock equations give a value one order of magnitude better than those given by the Rankine-Hugoniot equations and the first-order equations.This improved accuracy makes the second-order theory a good candidate for solving shock capture problems in computational fluid dynamics algorithms.Third,method of curved-shock characteristic is developed based on the curved shock theory and applied to supersonic flowfield calculation and inverse design in planar/axisymmetry,external/internal,uniform/nonuniform flow.With acquired derivatives,the flow parameters of post-shock flowfield can be quickly solved.Compared with the Method of Characteristics,the information of gradients helps the method of curved-shock characteristic increase not only the computational efficiency but also the accuracy.This makes the method of curved-shock characteristic more effective,accurate and robust than the method of characteristic.Several supersonic flow-fields are solved using the method of curved-shock characteristic.The results show that the method of curvedshock characteristic can reduce an order of magnitude of the computation resources with an improved accuracy than the method of characteristic.(4)The method is also applied on the inverse design of internal flows to analyze the post-curved-shock internal flowfield.A series of planar and axisymmetrical flowfields with centerbodies are solved under the condition that the shock curves are given.Afterwards,the Mach-reflection flow fields induced by curved shock(CMR)in steady,planar/axis symmetry flows are analyzed to supplement with the well studied phenomena caused by oblique shock(OMR).It is found that the incident shock curvature has a great effect on the Mach reflection patterns.The algebraic solution from von Neumann’s three-shock theory is not equivalent to the CMR.An improved four shock theory characterized with incident shock,reflected shock,Mach stem and expansion/no/compression waves is presented.A pressure equilibrium criterion to identify whether there are expansion/compression or no waves is proposed.On the basis of the four-shock theory,an analytical model for predicting the Mach-stem in the CMR is established.It is also available for the OMR by neglecting the shock curvature.Predictions of the Mach stem geometric shape and shock structure based on the model are found to have a better agreement with numerical results than the previous models.(5)A design method of the dual waverider with given pressure distribution based on the dual waverider theory is presented to analyze the post-curved-shock external flowfield.A dual-waverider with given pressure distribution is generated to combine the outward and inward flowfield.Numerical results show that the lateral pressure gradient will disappear with the given pressure distribution,which is beneficial to the vehicle performance.Furthermore,three-dimensional numerical simulations were performed for a test dual waverider at different bluntness radii to investigate the leading-edge curved-shock impact on aerodynamic performance as well as the heat flux.For the configuration without duct,the lift-drag ratio reduces when the leading-edge radius increases.By contrast,with duct installed,there is a positive association between the lift-drag ratio and the bluntness radius.As for heat flux,the real maximum value in both external and internal parts can be reduced effectively when the leading edge of the vehicle is blunter.Furthermore,since the whole curved-shock structure varies with the bluntness radius of the leading edge,the local maximum value of the heat flux on the nose region is affected as well as those values on the external and internal parts of the configuration and the upper surface of the duct.We first develop the curved shock theory to the high-order area.The method of curved-shock characteristics is proposed to solve aerodynamic parameters and their gradients in supersonic flowfields.Afterwards,based on these gradients,the typical post-curved-shock flows in the inlet and the dual-waverider are analyzed.The fourshock theory is further improved,and the geometry expression of the Mach stem is given considering the curvature impact.The Mach-stem shape in curved-shock Mach reflections can be captured accurately by using the geometry equation.These studies further promote the development of the curved shock theory and provide a theoretical basis for the design of supersonic vehicles.