Numerical Analysis of Shock Tube Problem by using TVD and ACM Schemes
|
|
- Joel Wright
- 6 years ago
- Views:
Transcription
1 Numerical Analysis of Shock Tube Problem by using TVD and Schemes Dr. Mukkarum Husain, Dr. M. Nauman Qureshi, Syed Zaid Hasany IST Karachi, Abstract Computational Fluid Dynamics (CFD) has grown from a mathematical curiosity to become an essential tool in almost every branch of fluid dynamics. Now, CFD as a computational technology is eminently suited to develop the concept of numerical test rig. Yee, Sandham and Djomehri developed a low dissipative high order artificial compression method () using characteristic based filters. The aim of these schemes is to minimize numerical dissipation for high speed compressible viscous flows containing shocks, shears and turbulence. In present studies, different shock tube test cases, namely, SOD, D LAX, and D inverse shock are simulated by using TVD and low dissipative high resolution artificial compression methods (). Comparison is made between numerical and analytical results. Computation shows that low dissipative high order is reasonably good for predicting shock waves, contact wave, and expansion fan. Key Words TVD, Low Dissipative High Order Artificial Compression Method, Shock Tube, SOD, D LAX, D Inverse Shock. Introduction TVD schemes are stable for nonlinear hyperbolic conservation laws. TVD schemes being able to capture discontinuities with ease and high resolution [-5]. While TVD formulations are very reliable, versatile, and quite accurate, they do have certain inherent accuracy limitations. At extrema points of the solution, TVD schemes automatically reduce to being first-order accurate discretization locally, while away from extrema they can be constructed to be of higher-order accuracy. Yee, Sandham and Djomehri [6-8] developed low dissipative high order shock capturing method using characteristic based filters. The aim of these schemes is to minimize numerical dissipation for high speed compressible viscous flows containing shocks, shears and turbulence. An artificial compression method [9] () of Harten is employed to identify non-smooth behavior and control the quantity of numerical dissipation to be added. Yee et al. utilized in an entirely different context than Harten originally intended. The fundamental idea of schemes consists of two steps, the first step is the high order temporal and spatial base scheme and the second step is the appropriate filtering for stability, shocks and for fine scale capturing. The shock tube problem [] is a general test case for the accuracy of computational fluid dynamics codes. It is often used as a test case for validation of numerical codes, because analytical solutions are available. In present work, numerical and analytical results of shock tube problem are computed and compared to analyze the capability of TVD scheme and low dissipative high order. Three different test cases are solved. Shock tube problem is sketched in figure and descriptions of test cases are given in Table. Table : Description of D Test Cases SOD LAX Inverse Shock p..57. ρ.5.5. v p ρ v
2 Driver Section High Region (4) Driven Section Low Region () P 4, ρ 4, V 4 P, ρ, V Diaphragm P 4 P Distance Fig : Initial Conditions in a Shock Tube Governing Equations The Navier-Stokes equations are used in present study. It can be written in conservation form under Cartesian coordinate system as Q F F + = v t x x () T Here, Q = ( ρ, ρu, E), T F = ρu ρu + p,( E + p) u () ( ) T Fv =, τ xx, uτ xx + k x the total energy E = p + ρ u (3) γ and kinematic viscosity μ is calculated using Sutherland formula as.5 μ T T + C = (4) μ T T + C where 5 μ =.7894 kg ( m s), T = 88.5K, C =.4K The thermal conductivity k is calculated by μc p k = Pr (5) Viscous stress tensor is calculated by following relations: 4 u τ xx = μ (6) 3 x Perfect gas assumption is used, which gives (7) Numerical Analysis Numerical analysis has been carried out using the following two different methods:. Second order [-3] scheme for convective part and second order central difference approximation for diffusive part of the governing equations T
3 . Forth order [6] for convective part and forth order central difference approximation for diffusive part of the governing equations Roe s approximate average state is used to calculate eigenvalues and eigenvector matrix [4]. Scheme The convective flux for scheme can be cast into the form F = Fi+ Fi x Δx (7) Here, _ F i + = [ F i + F i + + R Φ i + i + ] (8) where m m m m m m m Φ = σ a ( g j+ + g j) ψ a + γ α j+ j+ j+ j+ j+ (9) m =,, 3 z z δ ψ ( z ) = ( z + δ ) δ z < δ () ( U + U ) α () = R j j j+ j+ ( j+ j) m m m m g g α For α j+ j+ m () γ = j+ m For α = j+ σ ( z ) = ψ ( z ) + λβ ( θ ) z (3) For steady state calculation and/or implicit method, we have σ ( z ) = ψ ( z ) (4) limiter used for Eq. () in the present study is given as, m m m g j = min mod α, α (5) j j+ Low Dissipative High Order Method The fundamental idea of these shock-capturing schemes consists of two steps. The first step is a high-order spatial and temporal base scheme. Various standard high-order non dissipative or low dissipative base schemes fit in the current frame work. The second step is the appropriate filter for stability, shocks, contact discontinuities, and fine scale flow structure capturing. Various TVD, positive, WENO, and ENO dissipations, after a slight modification, are appropriate candidates as filters.. The Base Scheme Fourth order central differencing used for both convective and diffusion parts of Navier-Stokes equations is given below: F = Fi+, j, k 8Fi+, j, k+ 8Fi, j, k Fi, j, k (6) x Δx F 9 v = Fv Fv Fv 3 Fv 3 (7) x 8Δx i+,, j k i,, j k 4 x i+,, j k i,, j k Δ Other types of base schemes can also be formulated in the same manner.
4 . The Numerical Flux Filtering Scheme Non linear dissipation term R i + Φ i + of scheme in combination with Harten s switch applied to all characteristic waves as the filter numerical fluxes, * F% = kθ i + Ri i i + Φ + (8) + The parameter k is a problem dependent. The function θ i + is the Harten switch. For a general m+ points base scheme, Harten recommended θ = max ( θ j m j m ) i + +,....., θ (9) + p α α j+ j θ () = α + α j+ j Instead of varying k for the particular physics, one can vary p. The higher the p, the less the amount of numerical dissipation is added. Note that by varying the p one can essentially increase the order of accuracy of the dissipation term. The order of the dissipation depends on the value of p. For all the numerical examples, we use, θ = max ( θ, ) + j θ () i j + The new time level is defined as * * * Lf ( F% ) = F% F% () Δx i+, j, k i, j, k ) n+ n+ * * * Q = Q + Δt L F%, G%, H% (3) f ( ) Results and Discussion Figures -4 depict, respectively, the numerical results computed for SOD, D lax and D inverse shock problems and their comparisons with the respective analytical solutions. It is seen that the resolution for the shock wave is captured better than contact discontinuity for different schemes. The shock wave is a compressive wave, the characteristic lines of the conservation laws are convergent and so the dissipation near to the shock wave is controlled in a small level in the time marching steps regardless of the numerical dissipation. Near the contact wave the discrepancy of the curves by different schemes are noticeable. The contact discontinuity is a kind of linear wave in the theory of the hyperbolic conservation laws and the characteristic lines are parallel to each other so the dissipation can not be restrained. During the time marching the dissipation is accumulated, and the contact wave may span more and more grid points. Especially for the schemes that owned larger numerical dissipation, the jump is weakened more seriously. It is very difficult to resolve contact wave accurately by any scheme and it can also be seen from the results. The method is better than the TVD scheme especially in the vicinity of the contact wave. At the head/tail of the expansion fan, the method also gives better results than the TVD scheme..8.4 Density.8.4 a) Comparison of profile b) Comparison of Density Profile
5 .8 Mach a) Comparison of Mach profile Fig : Numerical Results of SOD Problem. 3 Density.8.4 a) Comparison of profile b) Comparison of Density profile.8 Mach.4 c) Comparison of Mach profile Fig 3: Numerical Results for D LAX Problem 3 Density 4 a) Comparison of profile b) Comparison of Density profile
6 Mach 6 4 c) Comparison of Mach profile Fig 4: Numerical Results for D Inverse Shock Problem Conclusion In present work, shock tube problem have been solved to study the applicability and accuracy of two different numerical methods, namely, TVD and low dissipative high order. Three test cases composed of SOD, D LAX, and D inverse shock are simulated for comparative purpose. Comparison is made between numerical and analytical results. Computation shows that low dissipative high order is reasonably good for predicting shock waves, contact wave, and expansion fan. References [] John D. Anderson, Jr., Computational Fluid Dynamics, the Basics with Application. McGraw-Hill Series in Mechanical Engineering, 995 [].J. C. Tannehill, D. A. Anderson, R. H. Pletcher, Computational Fluid Mechanics and Heat Transfer. Second Edition, January 997. [3] Toro EF, Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction [M]. Springer, 997. [4] Yee, H.C., Upwind and Symmetric Shock-Capturing Schemes. NASA TM-89464, May 987. [5] Yee, H.C., A Class of High-Resolution Explicit and Implicit Shock-Capturing Methods. VKI Lecture Series Mar.6-, 989, also NASA TM-88, Feb.989. [6] Yee, H.C., N. D. Sandham, and M. J. Djomehri, Low Dissipative High Order Shock Capturing Methods Using Characteristic-Based Filters. Journal of Computational Physics, 5, pages99-38, 999. [7] Mukkarum Husain, Lee Chun Xuan, Comparative Study of Modern Shock Capturing Schemes. ICFP9, Hangzhu, China, 9 [8] Kamran R. Q., Numerical Simulation of High Speed Vehicle using High Resolution, High-Order, and Low Dissipative Scheme. PhD thesis, Beihang University, Beijing, China, 8. [9] Harten A., The Artificial Compression Method for Computation of Shocks and Contact Discontinuities, III Self-Adjusting Hybrid Schemes. Math Comp, 3: , 978. [] Anderson, J. D., Jr., Modern Compressible Flow: With Historical Perspective. McGraw-Hill Book Co., New York, 98. [] Yee, H.C., G. H. Klopfer, and J. -L. Montagne, High-Resolution Shock-Capturing Schemes for Inviscid and Viscous Hypersonic Flows. Journal of Computational Physics, 88, pages 3-6, 99. [] A Harten, High Resolution Schemes for Hyperbolic Conservation Laws. J. Computational Phys.49, pages , 983. [3] Mukkarum Husain, Lee Chun Xuan, Comparative Study of Limiters for Scheme. IBCAST, Islamabad, Pakistan, 9. [4] P. L. Roe, Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes. Journal of Computational Physics, 43, 357(98)..
Debojyoti Ghosh. Adviser: Dr. James Baeder Alfred Gessow Rotorcraft Center Department of Aerospace Engineering
Debojyoti Ghosh Adviser: Dr. James Baeder Alfred Gessow Rotorcraft Center Department of Aerospace Engineering To study the Dynamic Stalling of rotor blade cross-sections Unsteady Aerodynamics: Time varying
More informationFaculty of Mechanical and Manufacturing Engineering, University Tun Hussein Onn Malaysia (UTHM), Parit Raja, Batu Pahat, Johor, Malaysia
Applied Mechanics and Materials Vol. 393 (2013) pp 305-310 (2013) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/amm.393.305 The Implementation of Cell-Centred Finite Volume Method
More informationA Review on the Numerical Solution of the 1D Euler Equations. Hudson, Justin. MIMS EPrint:
A Review on the Numerical Solution of the D Euler Equations Hudson, Justin 6 MIMS EPrint: 6.9 Manchester Institute for Mathematical Sciences School of Mathematics The University of Manchester Reports available
More informationNIA CFD Seminar, October 4, 2011 Hyperbolic Seminar, NASA Langley, October 17, 2011
NIA CFD Seminar, October 4, 2011 Hyperbolic Seminar, NASA Langley, October 17, 2011 First-Order Hyperbolic System Method If you have a CFD book for hyperbolic problems, you have a CFD book for all problems.
More informationOn the high order FV schemes for compressible flows
Applied and Computational Mechanics 1 (2007) 453-460 On the high order FV schemes for compressible flows J. Fürst a, a Faculty of Mechanical Engineering, CTU in Prague, Karlovo nám. 13, 121 35 Praha, Czech
More informationThe Development of a Navier-Stokes Flow Solver with Preconditioning Method on Unstructured Grids
Proceedings of the International MultiConference of Engineers and Computer Scientists 213 Vol II, IMECS 213, March 13-15, 213, Hong Kong The Development of a Navier-Stokes Flow Solver with Preconditioning
More informationEXPLICIT AND IMPLICIT TVD AND ENO HIGH RESOLUTION ALGORITHMS APPLIED TO THE EULER AND NAVIER-STOKES EQUATIONS IN THREE-DIMENSIONS RESULTS
EXPLICIT AND IMPLICIT TVD AND ENO HIGH RESOLUTION ALGORITHMS APPLIED TO THE EULER AND NAVIER-STOKES EQUATIONS IN THREE-DIMENSIONS RESULTS Edisson Sávio de Góes Maciel, edissonsavio@yahoo.com.br Mechanical
More informationModule 1: Introduction to Finite Difference Method and Fundamentals of CFD Lecture 13: The Lecture deals with:
The Lecture deals with: Some more Suggestions for Improvement of Discretization Schemes Some Non-Trivial Problems with Discretized Equations file:///d /chitra/nptel_phase2/mechanical/cfd/lecture13/13_1.htm[6/20/2012
More informationMid-Year Report. Discontinuous Galerkin Euler Equation Solver. Friday, December 14, Andrey Andreyev. Advisor: Dr.
Mid-Year Report Discontinuous Galerkin Euler Equation Solver Friday, December 14, 2012 Andrey Andreyev Advisor: Dr. James Baeder Abstract: The focus of this effort is to produce a two dimensional inviscid,
More informationEVALUATE SHOCK CAPTURING CAPABILITY WITH THE NUMERICAL METHODS IN OpenFOAM
THERMAL SCIENCE: Year 2013, Vol. 17, No. 4, pp. 1255-1260 1255 Open forum EVALUATE SHOCK CAPTURING CAPABILITY WITH THE NUMERICAL METHODS IN OpenFOAM by Reza KHODADADI AZADBONI a*, Mohammad Rahim MALEKBALA
More informationModeling & Simulation of Supersonic Flow Using McCormack s Technique
Modeling & Simulation of Supersonic Flow Using McCormack s Technique M. Saif Ullah Khalid*, Afzaal M. Malik** Abstract In this work, two-dimensional inviscid supersonic flow around a wedge has been investigated
More informationA HYBRID SEMI-PRIMITIVE SHOCK CAPTURING SCHEME FOR CONSERVATION LAWS
Eighth Mississippi State - UAB Conference on Differential Equations and Computational Simulations. Electronic Journal of Differential Equations, Conf. 9 (), pp. 65 73. ISSN: 7-669. URL: http://ejde.math.tstate.edu
More informationEuler Equations Lab AA Computer Project 2
Euler Equations Lab AA 543 - Computer Project 2 Mishaal Aleem February 26, 2015 Contents 1 Introduction.............................................. 1 2 Algorithms...............................................
More informationNumerical and theoretical analysis of shock waves interaction and reflection
Fluid Structure Interaction and Moving Boundary Problems IV 299 Numerical and theoretical analysis of shock waves interaction and reflection K. Alhussan Space Research Institute, King Abdulaziz City for
More informationCS205b/CME306. Lecture 9
CS205b/CME306 Lecture 9 1 Convection Supplementary Reading: Osher and Fedkiw, Sections 3.3 and 3.5; Leveque, Sections 6.7, 8.3, 10.2, 10.4. For a reference on Newton polynomial interpolation via divided
More informationComputational Fluid Dynamics for Engineers
Tuncer Cebeci Jian P. Shao Fassi Kafyeke Eric Laurendeau Computational Fluid Dynamics for Engineers From Panel to Navier-Stokes Methods with Computer Programs With 152 Figures, 19 Tables, 84 Problems and
More informationStudy on the Numerical Accuracy for the CFD
Study on the Numerical Accuracy for the CFD T.Yamanashi 1, H.Uchida, and M.Morita 1 Department of Mathematics, Master s Research Course of Faculty of Science, Tokyo University of Science,1-3 Kagurazaka,
More informationPartial Differential Equations
Simulation in Computer Graphics Partial Differential Equations Matthias Teschner Computer Science Department University of Freiburg Motivation various dynamic effects and physical processes are described
More informationExample 13 - Shock Tube
Example 13 - Shock Tube Summary This famous experiment is interesting for observing the shock-wave propagation. Moreover, this case uses the representation of perfect gas and compares the different formulations:
More informationFinal Report. Discontinuous Galerkin Compressible Euler Equation Solver. May 14, Andrey Andreyev. Adviser: Dr. James Baeder
Final Report Discontinuous Galerkin Compressible Euler Equation Solver May 14, 2013 Andrey Andreyev Adviser: Dr. James Baeder Abstract: In this work a Discontinuous Galerkin Method is developed for compressible
More informationOn the order of accuracy and numerical performance of two classes of finite volume WENO schemes
On the order of accuracy and numerical performance of two classes of finite volume WENO schemes Rui Zhang, Mengping Zhang and Chi-Wang Shu November 29, 29 Abstract In this paper we consider two commonly
More informationIMPROVING THE NUMERICAL ACCURACY OF HYDROTHERMAL RESERVOIR SIMULATIONS USING THE CIP SCHEME WITH THIRD-ORDER ACCURACY
PROCEEDINGS, Thirty-Seventh Workshop on Geothermal Reservoir Engineering Stanford University, Stanford, California, January 30 - February 1, 2012 SGP-TR-194 IMPROVING THE NUMERICAL ACCURACY OF HYDROTHERMAL
More informationStudies of the Continuous and Discrete Adjoint Approaches to Viscous Automatic Aerodynamic Shape Optimization
Studies of the Continuous and Discrete Adjoint Approaches to Viscous Automatic Aerodynamic Shape Optimization Siva Nadarajah Antony Jameson Stanford University 15th AIAA Computational Fluid Dynamics Conference
More informationProgram: Advanced Certificate Program
Program: Advanced Certificate Program Course: CFD-Vehicle Aerodynamics Directorate of Training and Lifelong Learning #470-P, Peenya Industrial Area, 4th Phase Peenya, Bengaluru 560 058 www.msruas.ac.in
More informationA fully implicit Navier-Stokes algorithm for unstructured grids incorporating a two-equation turbulence model
Copyright 1996, American Institute of Aeronautics and Astronautics, Inc. AIAA Meeting Papers on Disc, January 1996 A9618376, AIAA Paper 96-0414 A fully implicit Navier-Stokes algorithm for unstructured
More informationNUMERICAL VISCOSITY. Convergent Science White Paper. COPYRIGHT 2017 CONVERGENT SCIENCE. All rights reserved.
Convergent Science White Paper COPYRIGHT 2017 CONVERGENT SCIENCE. All rights reserved. This document contains information that is proprietary to Convergent Science. Public dissemination of this document
More informationDevelopment of a Maxwell Equation Solver for Application to Two Fluid Plasma Models. C. Aberle, A. Hakim, and U. Shumlak
Development of a Maxwell Equation Solver for Application to Two Fluid Plasma Models C. Aberle, A. Hakim, and U. Shumlak Aerospace and Astronautics University of Washington, Seattle American Physical Society
More informationHigh-Order CENO Reconstruction Scheme For Three-Dimensional Unstructured Mesh
High-Order CENO Reconstruction Scheme For Three-Dimensional Unstructured Mesh by Al-Amin Aziz A thesis submitted in conformity with the requirements for the degree of Masters of Applied Science Graduate
More informationLax-Wendroff and McCormack Schemes for Numerical Simulation of Unsteady Gradually and Rapidly Varied Open Channel Flow
Archives of Hydro-Engineering and Environmental Mechanics Vol. 60 (2013), No. 1 4, pp. 51 62 DOI: 10.2478/heem-2013-0008 IBW PAN, ISSN 1231 3726 Lax-Wendroff and McCormack Schemes for Numerical Simulation
More informationMath 690N - Final Report
Math 690N - Final Report Yuanhong Li May 05, 008 Accurate tracking of a discontinuous, thin and evolving turbulent flame front has been a challenging subject in modelling a premixed turbulent combustion.
More informationThe WENO Method in the Context of Earlier Methods To approximate, in a physically correct way, [3] the solution to a conservation law of the form u t
An implicit WENO scheme for steady-state computation of scalar hyperbolic equations Sigal Gottlieb Mathematics Department University of Massachusetts at Dartmouth 85 Old Westport Road North Dartmouth,
More informationTVD and ENO Schemes for Multidimensional Steady and Unsteady Flows A Comparative Analysis
TVD and ENO Schemes for Multidimensional Steady and Unsteady Flows A Comparative Analysis Philippe Angot IRPHE équipe MNM, UMR CNRS 38, Universit Aix-Marseille I. & II., 2, Av. Général Leclerc, 30 03 Marseille,
More informationImplementing third order compressible flow solver for hexahedral meshes in OpenFoam
Tutorial/Report in OpenFoam Course 8 Implementing third order compressible flow solver for hexahedral meshes in OpenFoam Martin Olausson, Chalmers University of Technology, SE-1 9 Gothenburg, Sweden Abstract
More informationExpress Introductory Training in ANSYS Fluent Workshop 04 Fluid Flow Around the NACA0012 Airfoil
Express Introductory Training in ANSYS Fluent Workshop 04 Fluid Flow Around the NACA0012 Airfoil Dimitrios Sofialidis Technical Manager, SimTec Ltd. Mechanical Engineer, PhD PRACE Autumn School 2013 -
More informationAxisymmetric Viscous Flow Modeling for Meridional Flow Calculation in Aerodynamic Design of Half-Ducted Blade Rows
Memoirs of the Faculty of Engineering, Kyushu University, Vol.67, No.4, December 2007 Axisymmetric Viscous Flow Modeling for Meridional Flow alculation in Aerodynamic Design of Half-Ducted Blade Rows by
More informationNumerical Methods for Hyperbolic and Kinetic Equations
Numerical Methods for Hyperbolic and Kinetic Equations Organizer: G. Puppo Phenomena characterized by conservation (or balance laws) of physical quantities are modelled by hyperbolic and kinetic equations.
More informationThree dimensional meshless point generation technique for complex geometry
Three dimensional meshless point generation technique for complex geometry *Jae-Sang Rhee 1), Jinyoung Huh 2), Kyu Hong Kim 3), Suk Young Jung 4) 1),2) Department of Mechanical & Aerospace Engineering,
More informationThe Immersed Interface Method
The Immersed Interface Method Numerical Solutions of PDEs Involving Interfaces and Irregular Domains Zhiiin Li Kazufumi Ito North Carolina State University Raleigh, North Carolina Society for Industrial
More informationCOMPUTATIONAL AND EXPERIMENTAL INTERFEROMETRIC ANALYSIS OF A CONE-CYLINDER-FLARE BODY. Abstract. I. Introduction
COMPUTATIONAL AND EXPERIMENTAL INTERFEROMETRIC ANALYSIS OF A CONE-CYLINDER-FLARE BODY John R. Cipolla 709 West Homeway Loop, Citrus Springs FL 34434 Abstract A series of computational fluid dynamic (CFD)
More informationUse of CFD in Design and Development of R404A Reciprocating Compressor
Purdue University Purdue e-pubs International Compressor Engineering Conference School of Mechanical Engineering 2006 Use of CFD in Design and Development of R404A Reciprocating Compressor Yogesh V. Birari
More informationA-posteriori Diffusion Analysis of Numerical Schemes in Wavenumber Domain
2th Annual CFD Symposium, August 9-1, 218, Bangalore A-posteriori Diffusion Analysis of Numerical Schemes in Wavenumber Domain S. M. Joshi & A. Chatterjee Department of Aerospace Engineering Indian Institute
More informationEVALUATION OF A GENERAL CFD-SOLVER FOR A MICRO-SCALE URBAN FLOW
EVALATION OF A GENERAL CFD-SOLVER FOR A MICRO-SCALE RBAN FLOW Jarkko Saloranta and Antti Hellsten Helsinki niversity of Technology, Laboratory of Aerodynamics, Finland INTRODCTION In this work we study
More informationPossibility of Implicit LES for Two-Dimensional Incompressible Lid-Driven Cavity Flow Based on COMSOL Multiphysics
Possibility of Implicit LES for Two-Dimensional Incompressible Lid-Driven Cavity Flow Based on COMSOL Multiphysics Masanori Hashiguchi 1 1 Keisoku Engineering System Co., Ltd. 1-9-5 Uchikanda, Chiyoda-ku,
More informationi.e. variable extrapolation along the characteristic propagation directions. This leads to a family of rst and second-order accurate schemes with an i
Cell-centered Genuinely Multidimensional Upwind Algorithms and Structured Meshes P. Van Ransbeeck, Ch. Hirsch Department of Fluid Mechanics Vrije Universiteit Brussel Brussels, Belgium A family of cell-centered
More information1.2 Numerical Solutions of Flow Problems
1.2 Numerical Solutions of Flow Problems DIFFERENTIAL EQUATIONS OF MOTION FOR A SIMPLIFIED FLOW PROBLEM Continuity equation for incompressible flow: 0 Momentum (Navier-Stokes) equations for a Newtonian
More informationNumerical Methods for (Time-Dependent) HJ PDEs
Numerical Methods for (Time-Dependent) HJ PDEs Ian Mitchell Department of Computer Science The University of British Columbia research supported by National Science and Engineering Research Council of
More informationThis is an author-deposited version published in: Eprints ID: 4362
This is an author-deposited version published in: http://oatao.univ-toulouse.fr/ Eprints ID: 4362 To cite this document: CHIKHAOUI Oussama, GRESSIER Jérémie, GRONDIN Gilles. Assessment of the Spectral
More informationA MULTI-DOMAIN ALE ALGORITHM FOR SIMULATING FLOWS INSIDE FREE-PISTON DRIVEN HYPERSONIC TEST FACILITIES
A MULTI-DOMAIN ALE ALGORITHM FOR SIMULATING FLOWS INSIDE FREE-PISTON DRIVEN HYPERSONIC TEST FACILITIES Khalil Bensassi, and Herman Deconinck Von Karman Institute for Fluid Dynamics Aeronautics & Aerospace
More informationModeling External Compressible Flow
Tutorial 3. Modeling External Compressible Flow Introduction The purpose of this tutorial is to compute the turbulent flow past a transonic airfoil at a nonzero angle of attack. You will use the Spalart-Allmaras
More information2DNS Documentation. Release Sayop Kim
2DNS Documentation Release 0.0.1 Sayop Kim September 25, 2014 Contents 1 Project description 3 1.1 Given task................................................ 3 1.2 Governing Equations...........................................
More informationA Hybrid Cartesian Grid and Gridless Method for Compressible Flows
rd AIAA Aerospace Sciences Meeting and Exhibit,, January 5, Reno, Nevada A Hybrid Cartesian Grid and Gridless Method for Compressible Flows Hong Luo and Joseph D. Baum Science Applications International
More informationEFFICIENT SOLUTION ALGORITHMS FOR HIGH-ACCURACY CENTRAL DIFFERENCE CFD SCHEMES
EFFICIENT SOLUTION ALGORITHMS FOR HIGH-ACCURACY CENTRAL DIFFERENCE CFD SCHEMES B. Treidler, J.A. Ekaterineris and R.E. Childs Nielsen Engineering & Research, Inc. Mountain View, CA, 94043 Abstract Preliminary
More informationTechniques for Using the Method of Manufactured Solutions for Verification and Uncertainty Quantification of CFD Simulations Having Discontinuities
Techniques for Using the Method of Manufactured Solutions for Verification and Uncertainty Quantification of CFD Simulations Having Discontinuities Ben Grier Clemson University Richard Figliola, Larry
More informationTVD Flux Vector Splitting Algorithms Applied to the Solution of the Euler and Navier-Stokes Equations in Three-Dimensions Part II
TVD Flux Vector Splitting Algorithms Applied to the Solution of the Euler and Navier-Stokes Equations in Three-Dimensions Part II EDISSON SÁVIO DE GÓES MACIEL IEA- Aeronautical Engineering Division ITA
More informationTwo-dimensional laminar shock wave / boundary layer interaction
Two-dimensional laminar shock wave / boundary layer interaction J.-Ch. Robinet (), V. Daru (,) and Ch. Tenaud () () SINUMEF Laboratory, ENSAM-PARIS () LIMSI-CNRS 5, Bd. de l Hôpital, PARIS 753, France
More informationValidation of Tools to Accelerate High-Speed CFD Simulations Using OpenFOAM
Validation of Tools to Accelerate High-Speed CFD Simulations Using OpenFOAM Daniel E. R. Espinoza, Thomas J. Scanlon and Richard E. Brown Centre for Future Air-Space Transportation Technology, University
More informationMathematical modeling of fluid flow using the numerical scheme with artificial viscosity
Mathematical modeling of fluid flow using the numerical scheme with artificial viscosity Ing. Tomáš Sommer, Ing. Martin Helmich Thesis supervised by: Doc. Svatomír Slavík, CSc., Doc. Luboš Janko, CSc.
More informationHigh Order Fixed-Point Sweeping WENO Methods for Steady State of Hyperbolic Conservation Laws and Its Convergence Study
Commun. Comput. Phys. doi:.48/cicp.375.6a Vol., No. 4, pp. 835-869 October 6 High Order Fixed-Point Sweeping WENO Methods for Steady State of Hyperbolic Conservation Laws and Its Convergence Study Liang
More informationNonoscillatory Central Schemes on Unstructured Triangulations for Hyperbolic Systems of Conservation Laws
Nonoscillatory Central Schemes on Unstructured Triangulations for Hyperbolic Systems of Conservation Laws Ivan Christov Bojan Popov Department of Mathematics, Texas A&M University, College Station, Texas
More informationA Novel Multi-Dimensional Limiter for High-Order Finite Volume Methods on Unstructured Grids
Commun. Comput. Phys. doi:.428/cicp.oa-27-39 Vol. xx, No. x, pp. -28 xxx 2x A Novel Multi-Dimensional Limiter for High-Order Finite Volume Methods on Unstructured Grids Yilang Liu, Weiwei Zhang, and Chunna
More informationExplicit and Implicit TVD and ENO High Resolution Algorithms Applied to the Euler and Navier-Stokes Equations in Three-Dimensions Turbulent Results
Explicit and Implicit TVD and ENO High Resolution Algorithms Applied to the Euler and Navier-Stokes Equations in Three-Dimensions Turbulent Results EDISSON SÁVIO DE GÓES MACIEL IEA Aeronautical Engineering
More informationNon-Newtonian Transitional Flow in an Eccentric Annulus
Tutorial 8. Non-Newtonian Transitional Flow in an Eccentric Annulus Introduction The purpose of this tutorial is to illustrate the setup and solution of a 3D, turbulent flow of a non-newtonian fluid. Turbulent
More informationA STUDY ON THE UNSTEADY AERODYNAMICS OF PROJECTILES IN OVERTAKING BLAST FLOWFIELDS
HEFAT2012 9 th International Conference on Heat Transfer, Fluid Mechanics and Thermodynamics 16 18 July 2012 Malta A STUDY ON THE UNSTEADY AERODYNAMICS OF PROJECTILES IN OVERTAKING BLAST FLOWFIELDS Muthukumaran.C.K.
More informationOn the Resolution Necessary to Capture Dynamics of Unsteady Detonation
On the Resolution Necessary to Capture Dynamics of Unsteady Detonation Christopher M. Romick, University of Notre Dame, Notre Dame, IN Tariq D. Aslam, Los Alamos National Laboratory, Los Alamos, NM and
More informationLab 9: FLUENT: Transient Natural Convection Between Concentric Cylinders
Lab 9: FLUENT: Transient Natural Convection Between Concentric Cylinders Objective: The objective of this laboratory is to introduce how to use FLUENT to solve both transient and natural convection problems.
More informationComputing & Verifying Compressible Fluid Dynamics:! The Good, The Bad and The Ugly!
: LA- UR 11-05852 Computing & Verifying Compressible Fluid Dynamics:! The Good, The Bad and The Ugly! Tariq Aslam! Los Alamos National Laboratory! WX-9: Shock and Detonation Physics! Slide 1 Background:!
More informationA Flow Feature Extraction Method for Shock-Fitting Computation
Tenth International Conference on Computational Fluid Dynamics (ICCFD10), Barcelona, Spain, July 9-13, 2018 ICCFD10-2018-015 A Flow Feature Extraction Method for Shock-Fitting Computation Zedong Chen,
More informationA Novel Approach to High Speed Collision
A Novel Approach to High Speed Collision Avril Slone University of Greenwich Motivation High Speed Impact Currently a very active research area. Generic projectile- target collision 11 th September 2001.
More informationSolving Partial Differential Equations on Overlapping Grids
**FULL TITLE** ASP Conference Series, Vol. **VOLUME**, **YEAR OF PUBLICATION** **NAMES OF EDITORS** Solving Partial Differential Equations on Overlapping Grids William D. Henshaw Centre for Applied Scientific
More informationCFD-1. Introduction: What is CFD? T. J. Craft. Msc CFD-1. CFD: Computational Fluid Dynamics
School of Mechanical Aerospace and Civil Engineering CFD-1 T. J. Craft George Begg Building, C41 Msc CFD-1 Reading: J. Ferziger, M. Peric, Computational Methods for Fluid Dynamics H.K. Versteeg, W. Malalasekara,
More informationSecond order blended multidimensional upwind residual distribution scheme for steady and unsteady computations
Journal of Computational and Applied Mathematics 25 (28) 378 389 www.elsevier.com/locate/cam Second order blended multidimensional upwind residual distribution scheme for steady and unsteady computations
More informationGrid. Apr 09, 1998 FLUENT 5.0 (2d, segregated, lam) Grid. Jul 31, 1998 FLUENT 5.0 (2d, segregated, lam)
Tutorial 2. Around an Airfoil Transonic Turbulent Flow Introduction: The purpose of this tutorial is to compute the turbulent flow past a transonic airfoil at a non-zero angle of attack. You will use the
More informationISSN(PRINT): ,(ONLINE): ,VOLUME-1,ISSUE-1,
NUMERICAL ANALYSIS OF THE TUBE BANK PRESSURE DROP OF A SHELL AND TUBE HEAT EXCHANGER Kartik Ajugia, Kunal Bhavsar Lecturer, Mechanical Department, SJCET Mumbai University, Maharashtra Assistant Professor,
More informationFOURTH ORDER COMPACT FORMULATION OF STEADY NAVIER-STOKES EQUATIONS ON NON-UNIFORM GRIDS
International Journal of Mechanical Engineering and Technology (IJMET Volume 9 Issue 10 October 2018 pp. 179 189 Article ID: IJMET_09_10_11 Available online at http://www.iaeme.com/ijmet/issues.asp?jtypeijmet&vtype9&itype10
More informationHigh-order, conservative, finite difference schemes for computational MHD
High-order, conservative, finite difference schemes for computational MHD A. Mignone 1, P. Tzeferacos 1 and G. Bodo 2 [1] Dipartimento di Fisica Generale, Turin University, ITALY [2] INAF Astronomic Observatory
More informationInviscid Flows. Introduction. T. J. Craft George Begg Building, C41. The Euler Equations. 3rd Year Fluid Mechanics
Contents: Navier-Stokes equations Inviscid flows Boundary layers Transition, Reynolds averaging Mixing-length models of turbulence Turbulent kinetic energy equation One- and Two-equation models Flow management
More informationLaminar and Turbulent Simulations of Several TVD Schemes in Two-Dimensions Part II
Laminar and Turbulent Simulations of Several TVD Schemes in Two-Dimensions Part II Edisson S. G. Maciel Abstract This work, second of this study, describes four numerical tools to perform perfect gas simulations
More informationOn the Construction, Comparison, and Local Characteristic Decomposition for High-Order Central WENO Schemes
Journal of Computational Physics 8, 87 09 (00) doi:0.006/jcph.00.79 On the Construction, Comparison, and Local Characteristic Decomposition for High-Order Central WENO Schemes Jianxian Qiu, and Chi-Wang
More informationNUMERICAL SOLUTION OF INVISCID AND VISCOUS FLOWS USING MODERN SCHEMES AND QUADRILATERAL OR TRIANGULAR MESH
Proceedings of the Czech Japanese Seminar in Applied Mathematics 24 Czech Technical University in Prague, August 4-7, 24 pp. 8 NUMERICAL SOLUTION OF INVISCID AND VISCOUS FLOWS USING MODERN SCHEMES AND
More informationInterface and Boundary Schemes for High-Order Methods
19th AIAA Computational Fluid Dynamics 22-25 June 29, San Antonio, Texas AIAA 29-3658 Interface and Boundary Schemes for High-Order Methods Xun Huan, Jason E. Hicken, and David W. Zingg Institute for Aerospace
More informationA DETAILED COMPARISON OF WENO AND SFV HIGH-ORDER METHODS FOR INVISCID AERODYNAMIC APPLICATIONS
Blucher Mechanical Engineering Proceedings May 2014, vol. 1, num. 1 www.proceedings.blucher.com.br/evento/10wccm A DETAILED COMPARISON OF WENO AND SFV HIGH-ORDER METHODS FOR INVISCID AERODYNAMIC APPLICATIONS
More informationHierarchical Reconstruction for Spectral Volume Method on Unstructured Grids
Hierarchical Reconstruction for Spectral Volume Method on Unstructured Grids Zhiliang Xu, Yingjie Liu and Chi-Wang Shu April 4, 2009 Abstract The hierarchical reconstruction (HR) [, 24] is applied to a
More informationRotorcraft Noise Prediction with Multi-disciplinary Coupling Methods. Yi Liu NIA CFD Seminar, April 10, 2012
Rotorcraft Noise Prediction with Multi-disciplinary Coupling Methods Yi Liu NIA CFD Seminar, April 10, 2012 Outline Introduction and Background Multi-disciplinary Analysis Approaches Computational Fluid
More informationSimulation of Flow Development in a Pipe
Tutorial 4. Simulation of Flow Development in a Pipe Introduction The purpose of this tutorial is to illustrate the setup and solution of a 3D turbulent fluid flow in a pipe. The pipe networks are common
More informationCase 3.1: Turbulent Flow over a 2D Multi-Element Airfoil. Summary of Results. Marco Ceze
Case 3.1: Turbulent Flow over a 2D Multi-Element Airfoil Summary of Results Marco Ceze (mceze@umich.edu) 2 nd International Workshop on High-Order CFD Methods, May 27-28, Cologne, Germany C3.1 1/14 Case
More informationNew Rules of ME Ph.D. Qualifying Exams 1/8
New Rules of ME Ph.D. Qualifying Exams 1/8 Qualifying Examination The student must pass a Qualifying Examination before the Dissertation Director, the Interdisciplinary Committee, and the courses for the
More informationImaging of flow in porous media - from optimal transport to prediction
Imaging of flow in porous media - from optimal transport to prediction Eldad Haber Dept of EOS and Mathematics, UBC October 15, 2013 With Rowan Lars Jenn Cocket Ruthotto Fohring Outline Prediction is very
More informationUnstructured Mesh Generation for Implicit Moving Geometries and Level Set Applications
Unstructured Mesh Generation for Implicit Moving Geometries and Level Set Applications Per-Olof Persson (persson@mit.edu) Department of Mathematics Massachusetts Institute of Technology http://www.mit.edu/
More informationIntroduction to ANSYS CFX
Workshop 03 Fluid flow around the NACA0012 Airfoil 16.0 Release Introduction to ANSYS CFX 2015 ANSYS, Inc. March 13, 2015 1 Release 16.0 Workshop Description: The flow simulated is an external aerodynamics
More informationA DRAG PREDICTION VALIDATION STUDY FOR AIRCRAFT AERODYNAMIC ANALYSIS
A DRAG PREDICTION VALIDATION STUDY FOR AIRCRAFT AERODYNAMIC ANALYSIS Akio OCHI, Eiji SHIMA Kawasaki Heavy Industries, ltd Keywords: CFD, Drag prediction, Validation Abstract A CFD drag prediction validation
More informationA new class of central compact schemes with spectral-like resolution II: Hybrid weighted nonlinear schemes. Abstract
A new class of central compact schemes with spectral-like resolution II: Hybrid weighted nonlinear schemes Xuliang Liu 1, Shuhai Zhang, Hanxin Zhang 3 and Chi-Wang Shu 4 Abstract In this paper, we develop
More informationReproducibility of Complex Turbulent Flow Using Commercially-Available CFD Software
Reports of Research Institute for Applied Mechanics, Kyushu University No.150 (71 83) March 2016 Reproducibility of Complex Turbulent Flow Using Commercially-Available CFD Software Report 3: For the Case
More informationSupersonic Flow Over a Wedge
SPC 407 Supersonic & Hypersonic Fluid Dynamics Ansys Fluent Tutorial 2 Supersonic Flow Over a Wedge Ahmed M Nagib Elmekawy, PhD, P.E. Problem Specification A uniform supersonic stream encounters a wedge
More informationContinued Investigation of Small-Scale Air-Sea Coupled Dynamics Using CBLAST Data
Continued Investigation of Small-Scale Air-Sea Coupled Dynamics Using CBLAST Data Dick K.P. Yue Center for Ocean Engineering Department of Mechanical Engineering Massachusetts Institute of Technology Cambridge,
More informationA new multidimensional-type reconstruction and limiting procedure for unstructured (cell-centered) FVs solving hyperbolic conservation laws
HYP 2012, Padova A new multidimensional-type reconstruction and limiting procedure for unstructured (cell-centered) FVs solving hyperbolic conservation laws Argiris I. Delis & Ioannis K. Nikolos (TUC)
More informationUsing a Single Rotating Reference Frame
Tutorial 9. Using a Single Rotating Reference Frame Introduction This tutorial considers the flow within a 2D, axisymmetric, co-rotating disk cavity system. Understanding the behavior of such flows is
More information(LSS Erlangen, Simon Bogner, Ulrich Rüde, Thomas Pohl, Nils Thürey in collaboration with many more
Parallel Free-Surface Extension of the Lattice-Boltzmann Method A Lattice-Boltzmann Approach for Simulation of Two-Phase Flows Stefan Donath (LSS Erlangen, stefan.donath@informatik.uni-erlangen.de) Simon
More informationTotal Variation Diminishing (TVD) methods
Total Variation Diminishing (TVD) methods 1 Wiggles (Spurious oscillations) High order methods such as QUICK, Central, 2 nd order Upwind and so on give minor undershoots and overshoots, specially near
More informationAil implicit finite volume nodal point scheme for the solution of two-dimensional compressible Navier-Stokes equations
Ail implicit finite volume nodal point scheme for the solution of two-dimensional compressible Navier-Stokes equations Vimala Dutta Computational and Theoretical Fluid Dynamics Division National Aerospace
More informationHigh Order Schemes for CFD: A Review. Juan Cheng 1. Institute of Applied Physics and Computational Mathematics, Beijing , China.
High Order Schemes for CFD: A Review Juan Cheng 1 Institute of Applied Physics and Computational Mathematics, Beijing 100088, China and Chi-Wang Shu 2 Division of Applied Mathematics, Brown University,
More information