HIGH ORDER METHODS FOR SUPERSONIC REACTIVE FLOWS AFSOR GRANT FA9550-05-1-0123 FINAL REPORT

During the grant period, we have developed the high order methods, namely spectral methods, WENO-Z finite difference methods and the Hybrid spectral-WENO finite difference scheme for supersonic reactive and non-reactive flows. High order methods are needed in these kind of problems because the numerical solutions contain small scales that are important in the simulations of turbulence in interface instability and combustion in modeling combustors. Also since most of the relevant problems are not steady, long term integrations have to be carried out. It is well known that high order accuracy methods are needed in this situation. We consider the problem of supersonic reactive flow in recessed cavities and simulate it by spectral multidomain techniques. This is important as a flameholder mechanism in scramjets. We have developed a multi-domain spectral method with stable and conservative penalty interface conditions for the numerical simulation of supersonic reactive recessed cavity flows with homogeneous grid. The multi-domain hybrid Spectral-WENO finite difference method is introduced for the numerical solution of two dimensional nonlinear hyperbolic systems in a Cartesian physical domain which is partitioned into a grid of rectangular subdomains. The main idea of the Hybrid scheme is to conjugate the spectral and WENO methods for solving problems with shock or high gradients such that the scheme adapts its solver spatially and temporally depending on the smoothness of the solution in a 1 given subdomain. Built as a multi-domain method, an adaptive algorithm is used to keep the solutions parts exhibiting high gradients and discontinuities always inside WENO subdomains while the smooth parts of the solution are kept inside a spectral one, avoiding oscillations related to the well-known Gibbs phenomenon and increasing the numerical efficiency of the overall scheme. A higher order version of the multiresolution analysis proposed by Harten is used to determine the smoothness of the solution in each subdomain. The Hybrid method is applied to the two-dimensional Shock-Vortex Interaction and the Richtmyer-Meshkov Instability (RMI) problems. We also have developed a different approach in attempting to get more meaningful results is to model statistically those scales that can not be resolved. Methods for modeling those scales are being developed and applied. Numerical issues, as boundary conditions, are being addressed and new boundary procedure is being presented. 1 A weighted multi-domain spectral penalty method with inhomogeneous grid for supersonic injective cavity flows Spectral methods have been actively used in the computational fluid dynamics community in the last decades due to the merit of high order accuracy maintained for long time integration. Spectral methods also have been applied to highly complex fluid systems and have been proved to yield accurate solutions even with the stiff or discontinuous spatial gradients. These systems include the supersonic shock bubble interactions [12], the supersonic cavity flows [11], and etc. The difficulty of implementing the spectral method to these complex fluid systems is to deal with the stiff or discontinuous spatial gradients successfully. The discontinuous solution is commonly found in most high speed fluid mechanical systems. The spectral approximation of such solutions yields spurious oscillations near the discontinuity, known as the Gibbs phenomenon. These Gibbs oscillations destroy both the accuracy and stability in general. The essential methodology to deal with such oscillations in the spectral solution is the spectral viscosity or filtering methods [5,16,20,23]. The filtering which is mathematically equivalent to the spectral viscosity method but practically more efficient, is used to stabilize the flow fields over the time integration. The filtering reduces the high order oscillations by attenuating the high modes in the solution in a smooth manner. The filtering method can be applied either globally or locally. By applying the filtering locally more accurate solution is obtained in the smooth region and the spurious oscillations in the neighborhood of the discontinuity is considerably reduced. Thus it is desirable to separate the locally non-smooth regions from the global smooth region. A multi-domain spectral method has been developed to address this problem 2 [8,11,13,14,17–19]. Once the physical domain is split into multiple subdomains, the proper interface conditions should be imposed at the domain interfaces. The simplest condition is obtained using the averaging method. With the averaging method, the flow field at the domain interface is given by the average of the two adjacent solutions across the interface. Thus the continuity of the solution is ensured with the averaging method. Although this method is simple and efficient to be implemented, it may cause the generation of nonphysical solutions at the interface if the two adjacent subdomains have different order of approximation or different domain length, i.e. if the grid system is inhomogeneous. We define the grid inhomogeneity as the grid configuration such that the grid resolutions between the adjacent subdomains across the domain interface are different. Such difference can be obtained by having each domain have different order of polynomials or different physical domain length. If the distribution of the subdomain grids is inhomogeneous, the stable interface conditions derived for the homogeneous grid system are not enough and one needs to find the conditions with which the spatial inhomogeneity can be addressed properly. At the domain interface of two adjacent subdomains which have the degree of polynomials, N1 and N2 in x−direction, the ratio of the grid spacing ∆x1 to ∆x2 is approximately given by ∆x2 ∆x1 = ∆ ∆I · N 1 N 2 , (1) where ∆ and ∆ are the domain lengths of the two subdomains. If the grid spacing ratio ∆x2 ∆x1 is different and far from unity, we consider it as the inhomogeneous grid. If ∆x2 ∆x1 = 1, i.e. if the grid is homogeneous, the averaging method can play an efficient role as a stable interface condition. However, if the ratio is far from unity, the simple averaging interface condition can cause the growth at the domain interface. In the real computation, the values of N1, N2 and ∆ I , ∆ are chosen such that ∆x2 ∆x1 becomes close to unity but in general ∆x2 ∆x1 6= 1. The current work centers around the development of the method dealing with the solution in inhomogeneous grid, i.e. when ∆x2 ∆x1 6= 1. In [11], we performed a 2D direct numerical simulation (DNS) of the recessed cavity with the multi-domain spectral penalty method under the condition that the grid is homogeneous. For the homogeneous grid system, the size and the number of collocation points in each subdomain are the same in each dimension. In this study, we extend the previous work to the inhomogeneous grid system to consider the injectorcavity system with the local hydrogen fuel injector. The crucial part of the DNS of the injector-cavity system is to resolve the hydrogen jet injector without causing any instability or nonphysical growing modes at the domain interfaces. The ratio of the injector to the cavity length scale is about O(10). We use a smaller subdomain with higher order polynomials to resolve the narrow jet. In Figure 1 the local do3 main configuration is given for the cavity flameholder with (left figure) and without (right figure) the injector. The local domain configuration shown in the right figure is the typical domain system used in [11] for which grid system is homogeneous. The grid system in the left figure is inhomogeneous as the local injector is in the narrow domain. In [11], the stability analysis has been done with the assumption that each subdomain has the same length but can have different polynomial orders. With different polynomial orders, the stability is still maintained. In this work, we further show that the stability can be also maintained with the different domain length. For the 2D DNS, we use the grid inhomogeneity mainly due to the different domain length with the same polynomial order in each domain. The stability conditions can be derived with the inhomogeneous grid, but these are only necessary conditions. Consequently there can be a non-reflecting mode at the domain interface which can yield a growth in time. A weighted spectral penalty method is proposed in order to minimize such nonphysical reflecting modes at the inhomogeneous grid interfaces. We note that in [8] the multidomain spectral method has been also used for the localized incompressible stratified turbulence flows in which the strong adaptive averaging method has been used with the spectral filtering technique. In our work, we observe that the averaging method with the filtering technique does not yield a smooth profile across the inhomogeneous grid interface for the compressible supersonic flows. This is due to the fact that the averaging does not guarantee the stability for the inhomogeneous grid as shown in the following section. The characteristic decomposition type interface conditions also fail to provide the smooth solution across the inhomogeneous grid interfaces. The major developments of the current work are as follows. 1) A generalized conservative and stable penalty conditions are derived for the inhomogeneous grid. 2) The weighted spectral penalty method is developed to minimize the non-physical growth modes at the inhomogeneous domain interfaces. Using the proposed weighted penalty method, we carry out the 2D DNS of the injector-cavity system with various recessed angles. Cavity is an efficient flame-holder of scramjet engine as it generates the self-sustained recirculation region. The hot radicals from the chemical reactions residing in the recirculation region reduce the induction time and consequently maintain the auto-ignition. For the continuous autoignition and better fuel efficiency,