DOCSLIB.ORG

Gas-Kinetic Bgk Method for Three-Dimensional Compressible Flows

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Gas-Kinetic Bgk Method for Three-Dimensional Compressible Flows AIAA-2002-0550 GAS-KINETIC BGK METHOD FOR THREE-DIMENSIONAL COMPRESSIBLE FLOWS by Yeefeng Ruan* and Antony Jameson+ Stanford University, CA 94303 Gas-kinetic BGK scheme with symmetric limited positive (SLIP) method, which was developed by Xu and Jameson, is modified and extended to three-dimensional compressible flows for the first time. BGK scheme is an approximate Riemann solver that uses collisional Boltzmann equation as the governing equation for flow evolutions. To demonstrate the robustness and accuracy of the scheme, it is applied to some typical shock flows ranging from 1D to 3D, and the computational results are compared with the exact results or experimental results. The simplified steady state BGK method is used as the flux solver of Jameson’s multi-grid, time-stepping FLO87 computer program for wing calculations. The results show that Gas-kinetic BGK scheme is an excellent shock-capturing method for subsonic and supersonic flows. the simplified Boltzmann equation. BGK-type schemes are different from other Boltzmann-type schemes, 1. INTRODUCTION which solve the collisionless Boltzmann equation by neglecting the collision term of the Boltzmann The development of numerical schemes based on the equation. Because of the inclusion of the collision term kinetic theory for compressible flow simulations started of the Boltzmann equation, BGK-type schemes take in 1960s. Chu’s method1, based on the gas-kinetic BGK into account the particle collisions in the whole gas model, with discretized velocity space, is one of the evolution process within a time step, from which a earliest kinetic methods used for shock tube time-dependent gas distribution function and the calculations. In the 1980’s and 90’s, many researchers resulting numerical fluxes at the cell interface are have contributed to gas-kinetic schemes. Pullin2 was the obtained. Moreover, due to its specific governing first to split the Maxwellian distribution into two parts equation, the BGK method gives Navier-Stokes and used the complete error function to obtain the solutions directly in smooth regions. In the numerical fluxes. The resulting scheme was named discontinuous regions, the scheme provides a delicate Equilibrium Flux Method (EFM). By applying upwind dissipative mechanism to get a stable and crisp shock technique directly to the collisionless Boltzmann transition. Since the gas evolution process is a equation, Mandal3 and Deshpande derived the same relaxation process from a non-equilibrium state to scheme, which is named Kinetic Flux Vector Splitting equilibrium one, the entropy condition is always (KFVS). Numerical schemes based on Boltzmann satisfied by the BGK method. Due to the dissipative equation are called Boltzmann-type schemes. nature of BGK methods, rarefaction shock, carbuncle phenomena or odd-even decoupling have never been A typical Godunov-type finite volume scheme has two observed, although they occasionally appear in the stages: reconstruction stage for constructing initial cell- other Godunov-type schemes. Finally, gas-kinetic BGK averaged values at cell interfaces and evolution stage schemes are positivity-preserving, which makes certain for computing flow variables at later times by solving that only positive densities and pressures are obtained physical governing equations. Gas-kinetic schemes from numerical computations. based on the Bhatnagar-Gross-Krook (BGK) model4 are new methods to model the gas evolution process by The development of gas-kinetic BGK methods for solving compressible flow simulations and other applications has attracted much attention and become matured in the 11 *Graduate Student, AIAA Student Member last decade. In 1993, Prendergast and Xu first +Professor, AIAA Member successfully used gas-kinetic BGK method to solve hydrodynamic equations. In 1995, Xu13,15,16,17 ã Copyright 2002 by the authors. Published by the American introduced Jameson’s SLIP formulation5,6 to construct Institute of Aeronautics and Astronautics, Inc., with high-order BGK schemes. In 1998, Kim8,9,10 and permission. Jameson developed the LED-BGK solver on 1 American Institute of Aeronautics and Astronautics AIAA-2002-0550 unstructured adaptive meshes. In 2000, Chae7 and Kim We end up with an equivalent reformulation of the proposed a modified gas-kinetic BGK scheme by conservation law. The first step is the variable introducing the Prantl number correction into flux reconstruction stage and the second step is the flux calculations. Lately Xu14 has used BGK schemes to evolution stage. solve shallow water equations, ideal MHD equations, and for chemical reactions. In this paper, the authors will present numerical results Reconstruction Stage of three-dimensional compressible flows with the gas- kinetic BGK method. This is the first time that the gas- In the reconstruction stage, to avoid generating spurious kinetic BGK scheme has been used for three- oscillations in the solutions, limiter is used to interpolate cell interface variables for higher order dimensional flows, and the results show that the BGK schemes. Jameson has developed a general symmetric scheme provides a very good tool for solving limited positive (SLIP) formulation construction of multidimensional flows, especially for flows with limiters: strong shock waves. The shock-capturing property of the BGK scheme makes it very attractive for subsonic q and supersonic flow simulations. 1 u − v L(u,v) = (u + v)(1− ) 2 u + v 2. NUMERICAL SCHEMES FOR • For q = 1 , the MinMod limiter is obtained CONSERVATION LAWS • For q = 2 , Van Leer's limiter is obtained To illustrate our approach we consider the one- • For q = 3 or larger, the higher-order SLIP method is dimensional hyperbolic system of conservation laws obtained + = ut f (u) x 0 = ∆ ∆ Let x j j x be a uniform mesh and x the mesh size. W , W are cell-averaged variables and where u is velocity and f is flux. j j+1 W 1 ,W 1 are the interpolated values at cell j+ j− In constructing a finite volume scheme, we proceed in 2 2 two steps: First, we introduce a small spatial scale, ∆x , interface. To second order accuracy, the interpolated and we consider the corresponding average u(x,t) values at the left and right state of the interface (i = 1/ 2) can be written as inside each cell: ∆x ∆x L = + 1 ∆ ∆ x − ≤ x ≤ x + to be W 1 W j L( W 1 , W 1 ) i 2 i 2 j+ 2 j+ j− 2 2 2 R = − 1 ∆ ∆ + W 1 W j+1 L( W 3 , W 1 ) ut (x,t) j+ 2 j+ j+ 2 2 2 1 é ∆x ∆x ù ê f (u(x + ,t)) − f (u(x − ,t))ú = 0 ∆x ë 2 2 û where L(u,v) is the nonlinear limiter. Next, we introduce a small time-step, ∆t , and integrate over the slab t ≤ τ ≤ t + ∆t , Hence Evolution Stage The evolution stage is basically to solve a Riemann 1 u(x,t + ∆t) = u(x,t) − problem. This area has been well developed in the past ∆ x decades: some well-known Riemann solvers for the cell-interface fluxes estimations are: é t+∆t ∆x t+∆t ∆x ù êòòf (u(x + ,τ )dτ − f (u(x − ,τ )dτ ú ë ττ=t 2 =t 2 û • The exact iterative solver 2 American Institute of Aeronautics and Astronautics AIAA-2002-0550 • Roe's approximate solver BGK Model of the Boltzmann Equation • The Harten-Lax-VanLeer Einfeldt (HLL) solver One of the main functions of the particle collision is to drive the gas distribution function f back to the The exact Riemann solver is too costly, and Roe's equilibrium state g corresponding to the local values approximate solver permits unphysical solutions such of ρ , ρU and ρε . The collision theory assumes that as expansion shock waves. In this study, a robust and τ accurate high-order evolution method based on gas- during a time dt , a fraction of dt / of molecules in a kinetic theory is used to construct the numerical fluxes given small volume undergoes collision, where τ is the across a cell interface. This gas-kinetic BGK scheme is average time interval between successive particle not just a simple alternative to the Riemann solver or collisions for the same particle. The collision term in another upwinding method. It has abundant physical the BGK model alters the velocity-distribution function basis to describe the numerical fluid. It provides better from f to g . This is equivalent to assuming that the solutions for compressible flows, especially for very rate of change ( df /)dt of f due to collisions is strong shock flows and supersonic flows. −( f − g) /τ . So the Boltzmann equation becomes ∂f ∂f f − g + u = − 3. GAS-KINETIC BGK METHOD ∂t ∂x τ Boltzmann Equation At the same time, due to the mass, momentum and energy conservation in particle collision, the collision Due to the unique form of the equilibrium distribution term (g − f ) /τ satisfies the compatibility condition, function g in classical statistical physics, at each point in space and time, there is a one to one correspondence g − f ψ dudξ = 0 between g and the macroscopic densities, e.g. mass, ò τ α momentum and energy. So, from macroscopic flow variables at any point in space and time, we can The BGK model coincides in form with the equations construct a unique equilibrium state. However, in the in the theory of relaxation processes and is therefore real physical situation, the gas does not necessarily stay sometimes called the relaxation model. in the local thermodynamic equilibrium state, such as gas inside a shock or boundary layer, even though we If τ is a local constant, the solution may be written as can construct a local equilibrium state there from the corresponding macroscopic flow variables. Usually, we 1 t − − ' τ do not know the explicit form of the gas distribution f (x,t,u,ξ ) = g(x − u(t − t ), t ' , u,ξ )e (t t ) / dt ' τ ò 0 function f in an extremely dissipative flow region, t0 such as that inside a strong shock wave.
Load more

Recommended publications
  • Experimental Investigation Into Shock Waves Formation and Development Process in Transonic Ow
    Scientia Iranica B (2017) 24(5), 2457{2465 Sharif University of Technology Scientia Iranica Transactions B: Mechanical Engineering www.scientiairanica.com Experimental investigation into shock waves formation and development process in transonic ow M. Farahani and A. Jaberi Department of Aerospace Engineering, Sharif University of Technology, Tehran, Iran. Received 11 May 2016; received in revised form 12 August 2016; accepted 8 October 2016 KEYWORDS Abstract. An extensive experimental investigation was performed to explore the shock waves formation and development process in transonic ow. Shadowgraph visualization Shock waves technique was employed to provide visual description of the ow eld features. Based on formation process; the visualization, the formation process was categorized into two intrinsically di erent Transonic ow; phases, namely, subsonic and supersonic. The characteristics of subsonic phase are well Shadowgraphy known; however, those of the supersonic one are far less studied. The supersonic phase visualization; itself is made up of two consecutive phases, namely, approaching and sweeping. The e ects Splitting of shock of each phase on the ow eld characteristics and on shaping the supersonic regime were waves; studied in details. In order to generalize the results, three di erent models were tested. A Zeh shock. special terminology is suggested by authors to ease the process description and pave a way for future studies. Above all, as the transition from transonic regime to the supersonic one is a vague concept in terms of physical reasoning, a new explanation is proposed that can be used as a criterion for distinguishing between transonic and supersonic regimes. © 2017 Sharif University of Technology.
    [Show full text]
  • Bifurcating Mach Shock Reflections with Application to Detonation Structure
    BIFURCATING MACH SHOCK REFLECTIONS WITH APPLICATION TO DETONATION STRUCTURE Philip Mach A thesis submitted to the Faculty of Graduate and Postdoctoral Studies in partial fulfillment of the requirements for the degree of MASTER OF APPLIED SCIENCE in Mechanical Engineering Ottawa-Carleton Institute for Mechanical and Aerospace Engineering University of Ottawa Ottawa, Canada August 2011 © Philip Mach, Ottawa, Canada, 2011 ii Abstract Numerical simulations of Mach shock reflections have shown that the Mach stem can bifurcate as a result of the slip line jetting forward. Numerical simulations were conducted in this study which determined that these bifurcations occur when the Mach number is high, the ramp angle is high, and specific heat ratio is low. It was clarified that the bifurcation is a result of a sufficiently large velocity difference across the slip line which drives the jet. This bifurcation phenomenon has also been observed after triple point collisions in detonation simulations. A triple point reflection was modelled as an inert shock reflecting off a wedge, and the accuracy of the model at early times after reflection indicates that bifurcations in detonations are a result of the shock reflection process. Further investigations revealed that bifurcations likely contribute to the irregular structure observed in certain detonations. iii Acknowledgements Firstly, I would like to thank my supervisor, Matei Radulescu, who guided me through the complicated task of learning about the detonation phenomenon, provided me with most of the scripts for the numerical simulations, answered my questions (no matter how dumb they were), and was patient with my progress. I would have obviously been lost without him.
    [Show full text]
  • Investigation of Shock Wave Interactions Involving Stationary
    Investigation of Shock Wave Interactions involving Stationary and Moving Wedges Pradeep Kumar Seshadri and Ashoke De* Department of Aerospace Engineering, Indian Institute of Technology Kanpur, Kanpur, 208016, India. *Corresponding Author: [email protected] Abstract The present study investigates the shock wave interactions involving stationary and moving wedges using a sharp interface immersed boundary method combined with a fifth-order weighted essentially non- oscillatory (WENO) scheme. Inspired by Schardin’s problem, which involves moving shock interaction with a finite triangular wedge, we study influences of incident shock Mach number and corner angle on the resulting flow physics in both stationary and moving conditions. The present study involves three incident shock Mach numbers (1.3, 1.9, 2.5) and three corner angles (60°, 90°, 120°), while its impact on the vorticity production is investigated using vorticity transport equation, circulation, and rate of circulation production. Further, the results yield that the generation of the vorticity due to the viscous effects are quite dominant compared to baroclinic or compressibility effects. The moving cases presented involve shock driven wedge problem. The fluid and wedge structure dynamics are coupled using the Newtonian equation. These shock driven wedge cases show that wedge acceleration due to the shock results in a change in reflected wave configuration from Single Mach Reflection (SMR) to Double Mach Reflection (DMR). The intermediary state between them, the Transition Mach Reflection (TMR), is also observed in the process. The effect of shock Mach number and corner angle on Triple Point (TP) trajectory, as well as on the drag coefficient, is analyzed in this study.
    [Show full text]
  • Numerical Study of Interaction of a Vortical Density Inhomogeneity with Shock and Expansion Waves
    22/ NASA/CR- 1998-206918 ICASE Report No. 98-10 Numerical Study of Interaction of a Vortical Density Inhomogeneity with Shock and Expansion Waves A. Povitsky ICASE D. Ofengeim University of Manchester Institute of Science and Technology Institute for Computer Applications in Science and Engineering NASA Langley Research Center Hampton, VA Operated by Universities Space Research Association National Aeronautics and Space Administration Langley Research Center Prepared for Langley Research Center under Contract NAS 1-97046 Hampton, Virginia 23681-2199 February 1998 Available from the following: NASA Center for AeroSpace Information (CASI) National Technical Information Service (NTIS) 800 Elkridge Landing Road 5285 Port Royal Road Linthicum Heights, MD 21090-2934 Springfield, VA 22161-2171 (301) 621-0390 (703) 487-4650 NUMERICAL STUDY OF INTERACTION OF A VORTICAL DENSITY INHOMOGENEITY WITH SHOCK AND EXPANSION WAVES A. POVITSKY * AND D. OFENGEIM t Abstract. We studied the interaction of a vortical density in_homogeneity (VDI) with shock and expan- sion waves. We call the VDI the region of concentrated vorticity (vortex) with a density different from that of ambiance. Non-parallel directions of the density gradient normal to the VDI surface and the pressure gradient across a shock wave results in an additional vorticity. The roll-up of the initial round VDI towards a non-symmetrical shape is studied numerically. Numerical modeling of this interaction is performed by a 2-D Euler code. The use of an adaptive unstructured numerical grid makes it possible to obtain high accuracy and capture regions of induced vorticity with a moderate overall number of mesh points. For the validation of the code, the computational results are compared with available experimental results and good agreement is obtained.
    [Show full text]
  • A Numerical Study of Postshock Oscillations in Slowly Moving Shock Waves
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector An lntemattonnl Journal computers & mathematics withapplIcatIona PERGAMON Computers and Mathematics with Applications 46 (2003) 719-739 www.elsevier.com/locate/camwa A Numerical Study of Postshock Oscillations in Slowly Moving Shock Waves Y. STIRIBA* CERFACS/CFD Team, 42 Av. Gaspard Coriolis, 31057 Toulouse Cedex 1, France youssef.stiriba&erfacs.fr R. DONAT+ Dept. de Matem&ica Aplicada, Universidad de Valencia Dr. Moliner 50, 46100 Burjassot (Valencia), Spain donat@uv. es (Received May 2001; revised and accepted October 200.2) Abstract-Godunov-type methods and other shock capturing schemes can display pathological behavior in certain flow situations. This paper discusses the numerical anomaly associated to slowly moving shocks. We present a series of numerical experiments that illustrate the formation and propagation of this pathology, and allows us to establish some conclusions tid question some previous conjectures for the source of the numerical noise. A simple diagnosis on an explicit Steger-Warming scheme shows that some intermediate states in the first time steps deviate from the true direction and contaminate the flow structure. A remedy is presented in the form of a new flux split method with an entropy intermediate state that dissipates the oscillations to a numerically acceptable level, and fix or reduce a variety of numerical pathologies. @ 2003 Elsevier Ltd. All rights reserved. Keywords-Nonlinear systems of conservation laws, Shock capturing schemes, Flux split meth- ods, Slowly moving shocks, Compressible flows. 1. INTRODUCTION The phenomenon of postshock oscillations in slowly moving shock waves is a numerical anomaly that has attracted much attention in recent times [l-6].
    [Show full text]
  • Chapter Five NORMAL SHOCK WAVES
    Mech 448 QUEEN'S UNIVERSITY Faculty of Engineering and Applied Science Department of Mechanical Engineering MECH 448 Chapter Five COMPRESSIBLE FLUID FLOW NORMAL SHOCK WAVES 2011 NORMAL SHOCK WAVES Mech 448 Mech 448 The changes that occur through a normal shock wave, i.e., SHOCK WAVES: It has been found experimentally that, under a shock wave which is straight with the flow at right angles to the some circumstances, it is possible for an almost spontaneous wave, is shown in the following figure: change to occur in a flow, the velocity decreasing and the pressure increasing through this region of sharp change. The possibility that such a change can occur actually follows from the analysis given below. It has been found experimentally, and it also follows from the analysis given below, that such regions of sharp change can only occur if the initial flow is supersonic. The extremely thin region in which the transition from the supersonic velocity, relatively low pressure state to the state that involves a relatively low velocity and high pressure is termed a shock wave. Mech 448 Mech 448 A shock wave is extremely thin, the shock wave normally A photograph of a normal shock wave is shown in the only being a few mean free paths thick. A shock-wave is following figure: analogous in many ways to a “hydraulic-jump” that occurs in free-surface liquid flows, a hydraulic jump being shown schematically below. A hydraulic jump occurs, for example, in the flow downstream of a weir. 1 Mech 448 Mech 448 In the case of a normal shock wave, the velocities both A complete shock wave may be effectively normal in ahead (i.e.
    [Show full text]
  • Shock Wave Numerical Structure and the Carbuncle Phenomenon
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Open Archive Toulouse Archive Ouverte Shock wave numerical structure and the carbuncle phenomenon Y. Chauvat, J.-M. Moschetta and J. Gressier∗ Department Models for Aerodynamics and Energetics { ONERA Ecole´ Nationale Sup´erieure de l'A´eronautique et de l'Espace 31400 Toulouse, France SUMMARY Since the development of shock-capturing methods, the carbuncle phenomenon has been reported to be a spurious solution produced by almost all currrently available contact-preserving methods. The present analysis indicates that the onset of carbuncle phenomenon is actually strongly related to the shock wave numerical structure. A matrix-based stability analysis as well as Euler finite volume computations are compared to illustrate the importance of the internal shock structure to trigger the carbuncle phenomenon. key words: carbuncle phenomenon, shock-capturing schemes, supersonic flows, Riemann solvers 1. INTRODUCTION Shock-capturing upwind methods developed since the 1980's can be classified into two distinct categories: 1. upwind methods which exactly preserve contact discontinuities (which might be referred to as contact-preserving methods), 2. upwind methods which introduce spurious diffusivity in the resolution of contact discontinuities. Schemes belonging to the second category never produce the carbuncle phenomenon but are hardly suitable for Navier-Stokes computations (e.g. Van Leer's method, see Fig. 3(a)) since they artificially broaden boundary layer profiles. On the other hand, schemes taken from the first category are attractive for viscous computations but turn out to be sensitive to the carbuncle phenomenon at various degrees [1] [2] with very few exceptions [3].
    [Show full text]
  • Supersonic and Subsonic Projectile Overtaking Problems in Muzzle Gun Applications
    AJCPP 2008 March 6-8, 2008, Gyeongju, Korea Supersonic and Subsonic Projectile Overtaking Problems in Muzzle Gun Applications Rajesh Gopalapillai, Suryakant Nagdewe and Heuy Dong Kim School of Mechanical Engineering, Andong National University, 388, Songchun-dong, Andong, Korea [email protected] Toshiaki Setoguchi Department of Mechanical Engineering, Saga University, 1, Honjo, Saga 840-8502, Japan Keywords: Blast Wave, Projectile, Ballistic Range, Overtaking, Chimera Mesh Abstract A projectile when passes through a moving shock ambient gas that has early been perturbed by the blast wave, experiences drastic changes in the aerodynamic wave. Behind the contact surface, an unsteady jet is forces as it moves from a high-pressure region to a developed by the discharge of the compressed gas. low pressure region. These sudden changes in the This unsteady jet may often be accompanied by a forces are attributed to the wave structures produced shock wave, the strength of which is again dependent by the projectile-flow field interaction, and are on the projectile acceleration4. responsible for destabilizing the trajectory of the Following the unsteady jet, the projectile is projectile. These flow fields are usually encountered discharged, consequently leading to strong interaction in the vicinity of the launch tube exit of a ballistic between the projectile and the unsteady jet when the range facility, thrusters, retro-rocket firings, silo former passes through the preceding unsteady jet. The injections, missile firing ballistics, etc. In earlier effects of this interaction are usually reflected in the works, projectile was assumed in a steady flow field unsteady fluctuating forces that act on the projectile when the computations start and the blast wave itself5.
    [Show full text]
  • Shock Jump Relations for a Dusty Gas Atmosphere R K Anand
    Shock jump relations for a dusty gas atmosphere R K Anand Shock jump relations for a dusty gas atmosphere R. K. Anand Department of Physics, University of Allahabad, Allahabad-211002, India E-mail: [email protected] Abstract This paper presents generalized forms of jump relations for one dimensional shock waves propagating in a dusty gas. The dusty gas is assumed to be a mixture of a perfect gas and spherically small solid particles, in which solid particle are continuously distributed. The generalized jump relations reduce to the Rankine-Hugoniot conditions for shocks in an idea gas when the mass fraction (concentration) of solid particles in the mixture becomes zero. The jump relations for pressure, density, temperature, particle velocity, and change-in-entropy across the shock front are derived in terms of upstream Mach number. Finally, the useful forms of the shock jump relations for weak and strong shocks, respectively, are obtained in terms of the initial volume fraction of the solid particles. The computations have been performed for various values of mass concentration of the solid particles and for the ratio of density of solid particles to the constant initial density of gas. Tables and graphs of numerical results are presented and discussed. Key words Shock waves . Dusty gas . Solid particles . Shock jump relations . Mach number 1. Introduction Understanding the influence of solid particles on the propagation phenomena of shock waves and on the resulting flow field is of importance for solving many engineering problems in the field of astrophysics and space science research. When a shock wave is propagated through a gas which contains an appreciable amount of dust, the pressure, the temperature and the entropy change across the shock, and the other features of the flow differ greatly from those which arise when the shock passes through a dust-free gas.
    [Show full text]
  • Fluid Dynamics Android Application: an Efficient Semi-Implicit Solver For
    Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 27 October 2019 doi:10.20944/preprints201910.0309.v1 Fluid Dynamics Android Application: An Efficient Semi-implicit Solver for Compressible Flows Shivam Singhaly1, Yayati Gupta2, and Ashish Garg∗y3 1Department of Mechanical Engineering, Indian Institute of Technology Kharagpur 2Department of Computer Science & Engineering, Indian Institute of Information Technology Dharwad 3GATE Aerospace Forum Educational Services, Delhi-110059 Abstract: The computing power of smartphones has not received considerable attention in the mainstream education system. Most of the education-oriented smartphone applications (apps) are limited to general purpose services like Massive Open Online Courses (MOOCs), language learning, and calculators (performing basic mathematical calculations). Greater potential of smartphones lies in educators and researchers developing their customized apps for learners in highly specific domains. In line with this, we present Fluid Dynamics, a highly accurate Android application for measuring flow properties in compressible flows. This app can determine properties across the stationary normal and oblique shock, moving normal shock and across Prandtl − Meyer expansion fan. This app can also measure isentropic flows, Fanno flows, and Rayleigh flows. The functionality of this app is also extended to calculate properties in the atmosphere by assuming the International Standard Atmosphere (ISA) relations and also flows across the Pitot tube. Such measurements are complicated and time-consuming since the relations are implicit and hence require the use of numerical methods, which give rise to repetitive calculations. The app is an efficient semi-implicit solver for gas dynamics formulations and uses underlying numerical methods for the computations in a graphical user interface (GUI), thereby easing and quickening the learning of concerned users.
    [Show full text]
  • Numerical Simulations of Shock Wave Refraction at Inclined Gas Contact Discontinuity
    INTERNATIONAL JOURNAL OF ENVIRONMENTAL & SCIENCE EDUCATION 2016, VOL. 11, NO. 16, 9026-9038 OPEN ACCESS Numerical Simulations of Shock Wave Refraction at Inclined Gas Contact Discontinuity Pavel V. Bulata,b and Konstantin N. Volkovc aNational Research University of Information Technologies, Saint Petersburg, RUSSIA; bMechanics and Optics University ITMO, Saint Petersburg, RUSSIA; cKingston University, London, UNITED KINGDOM ABSTRACT When a shock wave interacts with a contact discontinuity, there may appear a reflected rarefaction wave, a deflected contact discontinuity and a refracted supersonic shock. The numerical simulation of shock wave refraction at a plane contact discontinuity separating gases with different densities is performed. Euler equations describing inviscid compressible flow were discretized using the finite volume method on unstructured meshes and WENO schemes. Time integration was performed using a third-order Runge– Kutta method. The wave structure resulting from regular shock refraction is determined allowing its properties to be explored. In order to visualize and interpret the results of numerical calculations, a procedure for identifying and classifying gas-dynamic discontinuities was applied. The procedure employed dynamic consistency conditions and digital image processing methods to determine flow structure and its quantitative characteristics. The results of the numerical and experimental visualizations were compared (shadow patterns, schlieren images, interferograms). The results computed are in an agreement
    [Show full text]
  • One Dimensional Waves
    • DECEMBER 2019 One Dimensional Waves Application of Shock-Expansion Theory – Lesson 4 Intro Into 1D Wave Motion • So far, we have examined shock and expansion waves generated by flows over objects. These waves are stationary in the object’s reference frame, and steady-state analysis can be applied to derive their relations. • In this lesson we will discuss propagation of transient shocks and expansion waves in one dimension. • We will also briefly outline propagation of small amplitude acoustic waves. • Finally, we will combine these elements of the transient wave theory in the classical Sod shock tube problem. 2 Moving Normal Shock Waves Stationary shock • The governing equations of a moving normal shock can be readily derived from those obtained for the stationary Gas motion shock using the following variables: 푢 푢 Gas motion downstream of 2 1 ‐ 푊 – velocity of the gas ahead of the shock relative to the upstream of shock, shock shock 2 1 ‐ 푊 − 푢푝 – velocity of the gas behind the shock relative to the shock Shock moving with velocity 푊 휌1푊 = 휌2(푊 − 푢푝) 2 Motion induced Stagnant gas 푝 + 휌 푊2 = 푝 + 휌 푊 − 푢 푢 > 0 푊 1 1 2 2 푝 by moving shock 푝 ahead of 2 shock ℎ2 + 1Τ2 푣 = ℎ1 + 1Τ2 푊 − 푢푝 2 1 • The Mach number of the shock is given by: 2 푀1 = 푊Τ푎1 where the speed of sound is: 푎1 = (푑푝Τ푑휌)1 3 Moving Normal Shock Waves • The “jump” relations developed for stationary normal shocks can be easily rewritten for the moving shock using the transformation equation from the previous slide.
    [Show full text]