SUMMARYIn this paper, we consider and examine alternate finite element computational strategies for time-dependent Navier-Stokes equations describing high-speed compressible flows with shocks in a viscous and conducting medium, with the ultimate objective of establishing the desired features of a general mathematical and computational framework for such initial value problems (IVP) in which: (a) the numerically computed solutions are in agreement with the physics of evolution described by the governing differential equations (GDEs) i.e. the IVP, (b) the solutions are admissible in the non-discretized form of the GDEs in the pointwise sense (i.e. anywhere and everywhere) in the entire space-time domain, and hence in the integrated sense as well, (c) the numerical approximations progressively approach the same global differentiability in space and time as the theoretical solutions, (d) it is possible to time march the solutions (this is essential for efficiency as well as ensuring desired accuracy of the computed solution for the current increment of time, i.e. to minimize the error build up in the time marching process), (e) the computational process is unconditionally stable and non-degenerate regardless of the choice of discretization, nature of approximations and their global differentiability and the dimensionless parameters influencing the physics of the process, (f) there are no issues of stability, CFL number limitations and (g) the mathematical and computational methodology is independent of the nature of the space-time differential operators.We consider one-dimensional compressible flow in a viscous and conducting medium with shocks as model problems to illustrate various features of the general mathematical and computational framework used here and to demonstrate that the proposed framework is general and is applicable to all IVP. The Riemann shock tube with a single diaphragm serves as a model problem. The specific details presented in the paper discuss: (1) Choice of the form of the GDEs, i.e. strong form or weak form. medium. Extension of this work to real gas models will be presented in a separate paper. It is worth noting that when the medium is viscous and conducting, the solutions of gas dynamics equations are analytic. (5) It is also significant to note that upwinding methods based on addition of artificial diffusion such as SUPG, SUPG/DC, SUPG/DC/LS and their many variations are neither needed nor used in this present work. (6) Numerical studies are aimed at resolving the localized details of the shock structure, i.e. shock relations, shock width, shock speed, etc. as well as the over all global behaviour of the solution in the entire space-time domain. (7) Numerical studies are presented for Riemann shock tube for high Mach number flows with special emphasis also on time accuracy of the evolution which is ensured by requiring that the approximations for each increment of time satisfy non-discretized form of the GDEs in the pointwise sense, and hence in the integrated sense as well. (8) Compari...

The public reporting burden for this collection of information is estimated to average t hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing the burden, to the Department of Defense, Executive Services and Communications Directorate (0704-0188). Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to any penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number. PLEASE DO NOT RETURN YOUR FORM TO THE ABOVE ORGANIZATION. REPORT DATE (DD-MM-YYYY)12. REPORT Distribution is unlimited AFRL-SR-AR-TR-07-0087 SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITOR'S ACRONYM(S)Air SUPPLEMENTARY NOTES ABSTRACTWork performed during the third year of the Grant: (i) Preliminary research on fluid-solid interaction, development of mathematical models, computational methodology in hpk framework.(ii) Preliminary research towards development of concepts for a priori and a posteriori error estimation in hpk mathematical and computational framework. (iii) Re-discretizations, moving meshes and solution mapping strategies and associated computational infrastructures for BVPs and IVPs in hpk framework (Appendix A contains technical report on this work). SUBJECT TERMShpk framework, variational consistency, variational inconsistency, higher order global differentiability, fluid-solid interaction, error estimation, re-discretization, moving meshes SummaryThis report summarizes the work completed during the third year of the three year AFOSR Grant no. F 49620-03-1-0298 to the University of Kansas, Lawrence, KS (K.S.Surana, PI) and Grant no. F 49620-03-1-0201 to Texas A & M University, College Station, TX (J.N.Reddy, PI). The development of hpk mathematical and computational framework for all boundary value problems (BVPs) and initial value problems (IVPs) regardless of their origin or fields of application has been the main thrust of this research. Previous two reports have presented the mathematical and computational developments for BVPs and IVPs in hpk framework in which the order of approximation space k defining global differentiability of order (k -1) has been pointed out as an independent computational parameter in addition to h and p. Successful applications of this new hpk framework have been presented in various areas of continuum mechanics to demonstrate the benefits of using this framework as apposed to h, p framework used currently in the finite element processes.The thrust of the work completed during the third year of these grants have been in the following areas:(I) Preliminary research towards development of mathematical models for fluid-solid interaction and associated computational infrastructure in hpk mat...

The paper presents mathematical details of the finite element processes using the Galerkin method with weak form and least square processes for 1-D Helmholtz equation with Robin boundary condition in h,p,k framework. The concepts of variational consistency (VC) and variational inconsistency (VIC) are discussed. It is shown that Galerkin method with or without weak form is variationally inconsistent (VIC). The VIC of the Galerkin method is due to non-zero wave number (κ term) as well as Robin boundary condition. These two aspects of VIC of the integral forms and their consequences are investigated individually as well as jointly. It is shown that the integral forms yield non-symmetric functional B(. , .) and hence the possibility of complex basis for the coefficient matrix for some choices of h, p and k, which may lead to spurious or oscillatory non-physical numerical solutions. It is demonstrated that the variational consistency of the Galerkin method or galerkin method with weak form cannot be restored through any mathematically justifiable means, hence spuriousness in the computed solution is inevitable for those ranges of physical and computational parameters for which basis of the coefficient matrix becomes complex, though may possibly be minimized using different approaches reported in the literature. Least square processes, on the other hand, are always variationally consistent for any wave number and any choice of computational parameters. Hence, the resulting coefficient matrices are always symmetric, positive definite and have real basis and thus are free of spurious numerical solutions. Periodic theoretical solutions of most model problems, including those used here, are of higher order global differentiability and hence necessitate use of higher order spaces for local approximations. Mathematical details as well as numerical studies are presented to illustrate various features of the proposed approach.

An implicit space-marching finite-difference procedure is described for solving the compressible form of the steady, two-dimensional Navier-Stokes equations in body-fitted curvilinear coordinates. The coupled system of equations is solved for the primitive variables of velocity and pressure by making multiple sweeps of the space-marching procedure. A new pressure correction method has been developed which significantly accelerates convergence of the iterative process. The scheme is used to compute incompressible flows by taking the low Mach number limit of the compressible formulation.Computed results are compared with other numerical predictions for low Reynolds number channel inlet flow, flow over a rearward-facing step in a channel, and external flow over a cylinder.

SUMMARYThis paper considers numerical simulation of time-dependent non-linear partial differential equation resulting from a single non-linear conservation law in h, p, k mathematical and computational framework in which k = (k 1 , k 2 ) are the orders of the approximation spaces in space and time yielding global differentiability of orders (k 1 −1) and (k 2 −1) in space and time (hence k-version of finite element method) using space-time marching process. Time-dependent viscous Burgers equation is used as a specific model problem that has physical mechanism for viscous dissipation and its theoretical solutions are analytic. The inviscid form, on the other hand, assumes zero viscosity and as a consequence its solutions are non-analytic as well as non-unique (Russ. Math. Surv. 1962 Math. 1965; 18:697-715) in which artificial viscosity is a function of spatial discretization, which diminishes with progressively refined discretizations. The thrust of the present work is to point out that: (1) viscous form of the Burgers equation already has the essential mechanism of viscosity (which is physical), (2) with progressively increasing Reynolds (Re) number (thereby progressively reduced viscosity) the solutions approach that of the inviscid form, (3) it is possible to compute numerical solutions for any Re number (finite) within hpk framework and space-time least-squares processes, (4) the spacetime residual functional converges monotonically and that it is possible to achieve the desired accuracy, (5) mechanism for approaching the solutions of inviscid form with progressively increasing Re. Numerical studies are presented and the computed solutions are compared with published work.