Transonic Navier-Stokes wing solution using a zonal approach. Part 1: Solution methodology and code validation
Abstract : A fast diagonalized Beam Warming algorithm is coupled with a zonal approach to solve the three dimensional Euler/Navier Stokes equations. The computer code, called Transonic Navier Stokes (TNS), uses a total of four zones for wing configurations (or can be extended to complete aircraft configurations by adding zones). In the inner blocks near the wing surface, the thin layer Navier Stokes equations are solved, while in the outer two blocks the Euler equations are solved. The diagonal algorithm yields a speedup of as much as a factor of 40 over the original algorithm/zonal method coded. The TNS code, in addition, has the capability to model wind tunnel walls. Transonic viscous solutions are obtained on a 150,000 point mesh for a NACA 0012 wing. A three order of magnitude drop in the L2 norm of the residual requires approximately 500 iterations, which takes about 45 min of CPU time on a Cray-XMP processor. Simulations are also conducted for a different geometrical wing called WING C. All cases show good agreement with experimental data.
Transonic Navier-Stokes solutions for a fighter-like configuration
The Transonic Navier-Stokes wing code is extended to a 16-zone TNS wing-fuselage code and used to solve the transonic viscous flow over a modified F-16A. The computer code, called Transonic Navier-Stokes Wing/Fuselage uses a zonal approach to solve the three-dimensional Euler and Navier-Stokes equations. With the zonal implementation, clustering suitable for viscous calculations is achieved on all solid surfaces. The transonic case has flow conditions of free-stream M = 0.9, alpha = 4.12 deg, and a Reynolds number based on root chord of 4.5 million. This case required about 3,000 iterations to reduce the L2-norm of the residual by three orders, which takes about 15 hr of cpu time on the Cray X-MP/48 processor. Pressure distributions, as well as separaton patterns, compare favorably with experiment for this transonic case.
A Historical Introduction to the Law of Obligations
1 PROLOGUE: THE PREHISTORY OF THE ENGLISH LAW OF OBLIGATIONS 2 STRUCTURAL FOUNDATIONS 3 UNITY AND FRAGMENTATION OF THE MEDIAEVAL LAW OF CONTRACT 4 TRESPASS, TRESPASS ON THE CASE, AND THE MEDIAEVAL LAW OF TORT 5 THE SUBSTANTIVE LAW OF TORTS 6 THE SUBSTANTIVE LAW OF CONTRACT PART 2 THE TRIUMPH OF TRESPASS ON THE CASE 7 TORT, PROPERTY, AND REPUTATION: THE EXPANSION OF THE ACTION ON THE CASE 8 THE RISE OF THE ACTION OF ASSUMPSIT PART 3 THE MODERN LAW OF TORT AND CONTRACT 9 TRESPASS, CASE, AND THE MORAL BASIS OF LIABILITY 10 THE LAW OF TORTS IN THE NINETEENTH CENTURY: THE RISE OF THE TORT OF NEGLIGENCE 11 THE LAW OF TORTS IN THE TWENTIETH CENTURY: EXPANSION AND COLLAPSE OF THE TORT OF NEGLIGENCE 12 FOUNDATIONS OF THE MODERN LAW OF CONTRACT 13 THE RISE OF THE WILL THEORY THE WILL THEORY AND THE CLASSICAL MODEL OF CONTRACT 14 THE DECLINE OF THE WILL THEORY: LEGAL REGULATION AND CONTRACTUAL FAIRNESS PART 4 UNJUST ENRICHMENT 15 UNJUST ENRICHMENT 16 LEGAL CHANGE AND LEGAL CONTINUITY