CHL - A Finite Element Scheme for Shock Capturing_1

ộ dốc xt sốc cho thấy tốc độ gây sốc. Bất kỳ uốn sẽ chỉ ra rằng tốc độ thay đổi theo thời gian, không phải là trường hợp. Độ cao trên chính xác là 0,2 m, đó là chính xác. Có không vượt qua nhảy, mặc dù có một số undershoot khi C, nhỏ hơn 1. Cs là sản phẩm của tốc độ sốc phân tích và tỷ lệ của chiều dài thời gian bước chiều dài phần tử
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CHL - A Finite Element Scheme for Shock Capturing_1Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.comSimpo PDF Merge and Split Unregistered Version - http://www.simpopdf.comSimpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com Technical Report HL-93-12 In-House Laboratory Independent August 1993 Research Program A Finite Element Scheme for Shock Capturing by R. C. Berger, Jr. Hydraulics Laboratory U.S. Army Corps of Engineers Waterways Experiment Station 3909 Halls Ferry Road Vicksburg, MS 39180-6199 Final report Approved for public release; distribution is unlimited Assistant Secretary of the Army (R&D) Prepared for Washington, DC 20315Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com Waterways Experiment Station Cataloging-in-Publication Data Berger, Rutherford C. A finite element scheme for shock capturing / by R.C. Berger, Jr., ; prepared for Assistant Secretary of the Army (R&D). 61 p. : ill. ; 28 cm. - (Technical report ; HL-93-12) Includes bibliographical references. 1. Hydraulic jump - Mathematical models. 2, Hydrodynamics. 3. Shock (Mechanics) - Mathematical models. 4. Finite element method. I. United States. Assistant Secretary of the Army (Research, Development and Acquisi- tion) 11. U.S. Army Engineer Waterways Experiment Station. Ill. In-house Labo- ratory Independent Research Program (U.S. Army Engineer Waterways Experiment Station) IV. Title. V. Series: Technical report (U.S. Army Engineer Waterways Experiment Station) ; HL-93-12. TA7 W34 no.HL-93-12Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv Preface Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1 Basic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2 Shock equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 . Shock relations in 2-D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13 2-Numerical Approach Advective Dominated Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 The Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Petrov-Galerkin formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14 Shock Capturing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .20 Case 1: Analytic Shock Speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Case 2: Dam Break . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Case 3 : 2-D Lateral Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .43 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .55 ReferencesSimpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com Preface T his report is the product of research conducted from January 1992 through April 1993 in the Estuaries Division (ED), Hydraulics Laboratory (HL), U.S. Army Engineer Waterways Experiment Station (WES), under the In-House Laboratory Independent Research (ILIR) Program. The funding was providing by ILIR work unit Finite Element Scheme for Shock Capturing. Dr. R. C. Berger, Jr., ED, performed the work and prepared this report under the general supervision of Messrs. F. A. Herrmann, Jr., Director, HL; R. A. Sager, Assistant Director, HL; and W. H. McAnally, Chief, ED. Mr. Richard Stockstill of the Hydraulic Structures Division, HL, performed the test on supercritical contraction. At the time of publication of this report, Director of WES was Dr. Robert W. Whalin. Commander was COL Bruce K. Howard, EN.Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com ntroduction Background Shocks in fluids result from fluid flow that is more rapid than the speed of a compression wave. Thus there is no means for the flow to adjust gradually. Pressure, velocity, and temperatures change abruptly, causing severe fatigue and component destruction in military aircraft and engine turbines. This problem is not limited to supersonic aircraft; many parts of subsonic craft are supersonic. For example, the rotors of helicopters have supersonic regions as do the blades of the turbine engines used on many crafts. The shock is formed ...