Aeronautics and Astronautics Part 5 doc

Physico - Chemical Modelling in Nonequilibrium Hypersonic Flow Around Blunt Bodies 25v v e O2 N2 without correctionwith correction Fig.. Temperature along stagnation line with weakly ion

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Physico - Chemical Modelling in Nonequilibrium Hypersonic Flow Around Blunt Bodies 25

v

v e

O2

N2

without correctionwith correction

Fig 9 Temperature along stagnation line with weakly ionized gas; M∞=18

149Physico - Chemical Modelling in Nonequilibrium Hypersonic Flow Around Blunt Bodies

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26 Aeronautics and Astronautics

Modif Dunn & Kang

With Gupta curve fit constants and Park CVD

Fig 10 Temperature proﬁle with Gupta curve ﬁt constants, M∞=25.9

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Dunn & Kang With Park CVD With Hansen CVD

With Gupta Curve fit Constants

Modif Dunn & Kang

M =25.9 ∞

Fig 11 Stagnation heat ﬂux, M∞=25.9

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VV

e

O2

N2

Fig 12 Temperatures distribution along the stagnation line, M∞=23.9

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Ο

Mach=25,9; H=71 KmExperiment, H=71 Km

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Dunn & Kang Dunn & Kang with Gupta

Fig 14 Dunn and Kang, and Modiﬁed Dunn and Kang Interferograms computed

Fig 15 Fringe patterns on 1 in diameter cylinder with Park(93) model and Exact equilibriumconstant

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Fig 16 Fringe patterns on 1 in diameter cylinder with Gardiner model and Exact

equilibrium constant

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S/L

0 100 200 300 400 500

Fig 17.Effect of chemical kinetics: a) Contours Mach number b) Surface pressure

The present study has shown that the prediction of hypersonic flowfield structures, shockshapes, and vehicle surface properties are very sensitive to the choice of the kinetic model.The large dispersion in the wall heat flux reaches 60 % as observed in the RAM-CII case Themanner in which the backward reaction rates are computed is quite important as indicated

by the interferograms that were obtained The model of Park (93) gives a better prediction

of hypersonic flowfield around blunt bodies Park(93) is identify as the model for hypersonicflow around blunt bodies with a confidence acceptable to a wide range of Mach number There

is also great sensitivity to the choice of chemical kinetics in flowfield around double-wedge.More numerical simulations compared with experiments need to be conducted to improvethe knowledge of the thermochemical model of air flow around double-wedge

7 References

[1] Park, C (1990) Nonequilibrium Hypersonic Aerothermodynamics New York, Wiley.

[2] Gupta, R N., Yos, J M., Thompson, R A., and Lee, K P (1990) A Review of ReactionRates and Thermodynamic and Transport Properties for an 11-Species Air Model for

Chemical and Thermal Nonequilibrium Calculations to 30 000K", NASA RP-1232 [3] Vincenti, W G., Jr.Kruger, C H (1965) Introduction to physical Gas Dynamics Krieger, FL [4] Gardiner, W C (1984) Combustion Chemistry, Springer-Verlag, Berlin.

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[5] Shinn, J L., Moss, J N., Simmonds, A L (1982) Viscous Shock Layer Heating Analysisfor the Shuttle Winward Plane Finite Recombination Rates, AIAA 82-0842

[6] Dunn,M G., Kang, S W (1973) Theoretical and Experimental Studies of re-entry Plasma,NASA CR-2232

[7] Park, C (1993) Review of Chemical-Kinetic Problems of Future NASA Mission, I: Earth

Entries", Journal of Thermophysics and Heat Transfer Vol 7, No 3, pp 385-398.

[8] Hansen, C F (1993) Vibrational Nonequilibrium Effect on Diatomic Dissociation Rates"

AIAA Journal, Vol 31, No 11, pp 2047-2051.

[9] Macrossan, N M (1990) Hypervelocity ﬂow of dissociating nitrogen donwnstream of a

blunt nose Journal of Fluid Mechanics, Vol 27, pp 167-202.

[10] Josyula, E (2001) Oxygen atoms effect on vibrational relaxation of nitrogen in blunt body

ﬂows Journal of Thermophysics and Heat Transfer, Vol 15, No 1, pp 106-115.

[11] Peter, A G and Roop, N G and Judy, L S (1989) Conservation Equation and PhysicalModels for Hypersonic Air ﬂows in Thermal and Chemical Nonequilibrium NASA TP2867

[12] Knab, O and Fruaudf, HH and Messerschmid, EW (1995) Theory and validation of

the the physically consistent coupled vibration-chemistry-vibration model J Thermophys Heat Transfer, nˇr9, Vol.2, pp.219-226.

[13] Tchuen G., Burtschell Y., and Zeitoun E D (2008) Computation of non-equilibriumhypersonic ﬂow with Artiﬁcially Upstream Flux vector Splitting (AUFS) schemes

International Journal of Computational Fluid Dynamics, Vol 22, Nˇr4, pp 209 - 220.

[14] Tchuen G., and Zeitoun D E (2008) Computation of thermo-chemical nonequilibrium

weakly ionized air flow over sphere cones International journal of heat and fluid flow,

Vol.29, Issue 5, pp.1393 - 1401

[15] Tchuen G., and Zeitoun D E (2009) Effects of chemistry in nonequilibrium hypersonic

ﬂow around blunt bodies Journal of Thermophysics and Heat Transfer, Vol 23, Nˇr 3,

pp.433-442

[16] Burtschell Y., Tchuen G., and Zeitoun E D (2010) H2 injection and combustion in a Mach

5 air inlet through a Viscous Mach Interaction European Journal of Mechanics B/ﬂuid, Vol.

29, Issue 5, pp 351-356

[17] Lee Jong-Hun (1985) Basic Governing Equations for the Flight Regimes of Aeroassisted

Orbital Transfer Vehicles Thermal Design of Aeroassisted Orbital Transfer Vehicles, H.

F Nelson, ed., Volume 96 of progress in Astronautics and Aeronautics, American Inst of Aeronautics and Astronautics, Vol 96, pp 3-53.

[18] Appleton, J P and Bray, K N C (1964) The Conservation Equations for a

Nonequilibrium Plasma J Fluid Mech Vol 20, No 4, pp 659-672.

[19] Tchuen, G., Burtschell, Y., and Zeitoun, E D., 2005 Numerical study of nonequilibriumweakly ionized air flow past blunt body Int J of Numerical Methods for heat and fluidflow, 15 (6), 588 - 610

[20] Blottner, F G., Johnson, M., and Ellis, M., 1971 Chemically Reacting Viscous FlowProgram for Multi-Component Gas Mixtures Sandia Laboratories, Albuquerque, NM,Rept Sc-RR-70-754

[21] Wilke, C R., 1950 A viscosity Equation for Gas Mixture J of Chem Phys 18 (4), 517-519.[22] Ramsaw JD., and Chang CH., 1993 Ambipolar diffusion in two temperaturemulticomponent plasma Plasma Chem Plasma process 13 (3), 489-498

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[23] Masson, E A., and Monchick, 1962 Heat Conductivity of Polyatomic and Polar Gases.The Journal of Chemical 36 (6), 1622-1640

[24] Ahtye, W F., 1972 Thermal Conductivity in Vibrationnally Excited Gases Journal ofChemical Physics, 57, 5542-5555

[25] Candler, G V., and MacCormarck, R W., 1991 Computation of weakly ionizedhypersonic ﬂows in thermochemical nonequilibrium Journal of Thermophysics and heattransfer, 5 (3), 266-273

[26] Taylor R., Camac, M., and Feinberg, M., 1967 Measurement of vibration-vibrationcoupling in gas mixtures, In Proceeding of the 11th Intenational Symposium oncombustion, Pittsburg, PA, 49-65

[27] Sharma, S P., Huo, W M., and Park, C., 1988 The Rate Parameters for CoupledVibration-Dissociation in a Generalized SSH Approximation Flows AIAA-88-2714.[28] Shatalov, O P., and Losev, S A., "Modeling of diatomic molecules dissociation underquasistationary conditions", AIAA 97-2579, 1997

[29] Roe, P., 1983 Approximate Riemann Solvers, Parameters vectors and difference schemes.Journal of Computational Physics, Vol 43, 357-372

[30] Lobb, K., "Experimental measurement of shock detachment distance on sphere ﬁred in

air at hypervelocities", in The High Temperature Aspect of Hypersonic Flow ed Nelson W.

C., Pergamon Press, Macmillan Co., New York, 1964

[31] Rose, P H., Stankevics, J O., "Stagnation-Point Heat Transfer Measurements in Partially

Ionized Air" AIAA Journal, Vol 1, No 12, 1963, pp 2752-2763.

[32] Hornung, H G., "Non-equilibrium dissociating nitrogen ﬂow over spheres and circular

cylinders" Journal of Fluid Mechanics, Vol 53, 1972, pp 149-176.

[33] Joly, V., Coquel, F., Marmignon, C., Aretz, W., Metz, S., and Wilhelmi, H., "Numerical

modelling of heat transfer and relaxation in nonequilibrium air at hypersonic speeds", La Recherche Aérospatiale, Vol.3, 1994, pp 219-234.

[34] Séror, S., Schall, E., and Zeitoun, E D., "Comparison between coupled euler/defectboundary-layer and navier-stokes computations for nonequilibrium hypersonic ﬂows,

Computers & Fluids, Vol.27, 1998, pp 381-406.

[35] Fay, J A., Riddell, F R., "Theory of stagnation point heat transfer in dissociated air" J Aero Sciences, Vol 25, 1958, pp 73-85.

[36] Walpot, L M., "Development and Application of a Hypersonic Flow Solver" PhD thesis,

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A Frequency-Domain Linearized Euler Model

for Noise Radiation

Andrea Iob, Roberto Della Ratta Rinaldi and Renzo Arina

Politecnico di Torino, DIASP, 10129 Torino

Italy

1 Introduction

Aeroacoustics is the branch of Fluid Mechanics studying the mechanism of generation

of noise by fluid motions and its propagation The noise generation is associated withturbulent and unsteady vortical flows, including the effects of any solid boundary in the flow.Experimental studies in this field are very difficult, requiring anechoic wind tunnels and verysensitive instruments able to capture high frequency, low amplitude, pressure fluctuations.Computational Aeroacoustics (CAA) can be a powerful tool to simulate the aerodynamicnoise associated to complex turbulent flow fields As sound production represents only avery minute fraction of the energy associated to the flow motion, CAA methods for acousticpropagation have to be more accurate compared to the solution schemes normally used inComputational Fluid Dynamics (CFD)

A direct approach to aeroacoustic problems would imply to solve numerically the fullNavier-Stokes equations for three-dimensional, unsteady, compressible flows Soundwould then be that part of the flow field which dominates at large distances from theregion characterized by intense hydrodynamic fluctuations, propagating at the local soundspeed The direct noise simulation is hardly achievable in practice, except for very simpleconfigurations of academic interest, and in a limited region of space Even if the smallamplitude of the fluctuations allow a linearization, the equations of acoustic disturbances

on an arbitrary base ﬂow are very complicated and their solution is not straightforward

To solve aeroacoustic problems of practical interest some simplifying approximations arenecessary One way to obtain realistic solutions is provided by the hybrid approach Itdecouples the computation of the flow field from the computation of the acoustic field Whenusing a hybrid method the aeroacoustic problem is solved in two steps: in the first step, thehydrodynamic flow field is solved using a CFD method, then the noise sources are identifiedand the acoustic field is obtained Extraction of noise sources from the fluid dynamic fieldcan be done using an aeroacoustic theory such as the Lighthill’s analogy [Crighton (1975);Goldstein (1976)] A hybrid approach is based on the fundamental assumption that there is

a one-way coupling of mean flow and sound, i.e., the unsteady mean flow generates soundand modifies its propagation, but sound waves do not affect the mean flow in any significantway This assumption is not so restrictive, because acoustic feedback is possible only whenthe mechanical energy in the unsteady mean flow is weak enough to be influenced by acousticdisturbances This occurs principally in the vicinity of a starting point for flow instability (forinstance, upstream edges of cavities or initial areas of shear layers) Since the fluid-dynamic

6

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2 Will-be-set-by-IN-TECH

field and the acoustic field are computed separately, numerical accuracy for the mean flowsimulations used as an input of hybrid methods is less critical than in direct computation.Simpler, more flexible and lower-resolution schemes are applicable provided that numericaldissipation is carefully controlled to prevent the artificial damping of high-frequency sourcecomponents Incompressible flow solutions can be adequate for evaluating acoustic sourceterms based on the low Mach numbers approximation Time-accurate turbulence simulationapproaches such as DNS, LES, DES and unsteady RANS methods can be used to compute thespace-time history of the flow field, from which acoustic sources are extracted Because of thehigh computational cost of the time-accurate simulations, there have been efforts to use steadyRANS calculations in conjunction with a statistical model to generate the turbulent acousticterms

Once the acoustics sources have been evaluated, the generated noise has to be propagated

in the surrounding region with linearized propagation models The main focus of the presentchapter is the description of a computational method for noise propagation in turbomachineryapplications In the next Section a linearized model is presented Section 3 describes thenumerical algorithm based on a Discontinuous Galerkin approximation on unstructuredgrids, and in Section 4 several applications are presented

2 Governing equations

In principle the propagation of acoustic waves could be directly studied using the equations ofthe fluid motion, i.e the Navier Stokes equations However, it is possible to introduce someapproximations in the Navier Stokes equations in order to obtain equations more suitablefor aeroacoustics At frequencies of most practical interest, viscous effects are negligible inthe acoustic field because the pressure represents a far greater stress field than the viscousstresses Moreover, these disturbances are always small, also for very loudly acoustic waves.The threshold of pain, i.e the maximum Sound Pressure Level (SPL) which a human canendure for a very short period of time without the risk of permanent ear damage, is equal to

140 dB, which corresponds to pressure ﬂuctuations of amplitude equal to

A=√ 2pref10(SPL/20) ≈90 Pa , (1)

where prefis the reference pressure corresponding to the threshold of hearing at 1 kHz for a

typical human hear For sound propagating in gases it is equal to pref = 2×10−5Pa The

atmospheric pressure of the standard air is equal to p0 = 101325 Pa, which is 103 greaterthan the pressure variation associated with an acoustic wave at the threshold of pain, i.e.,

p /p0 = O

10−3

, where the superscript (.) denotes acoustic quantities and the subscript(.)0denotes mean ﬂow quantities The corresponding density ﬂuctuations of a progressiveplane wave are

a small perturbation of the mean ﬂow ﬁeld Therefore it is possible to linearize the equations

of motion Considering acoustic waves as a perturbation of the mean flow field, defining

p =p − p0, ρ =ρ − ρ0, v =v − v0, (3)

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for Noise Radiation 3

and assuming small perturbations, it is possible to obtain the equations for the propagation ofthe sound waves, i.e., the Linearized Euler Equations (LEE) For a two-dimensional problemand a steady mean ﬂow ﬁeld, LEE are formulated as

is the acoustic perturbation vector,F xandF yare the ﬂuxes along

x and y directions respectively, H contains the mean ﬂow derivatives and S represents the

acoustic sources The ﬂuxes,F xandF y, and the termH have the following expressions

whereu= [ρ , u, v , w , p ]Tis the acoustic perturbation vector expressed in the cylindrical

coordinate system, i.e u , v , and w are the velocity components in (z, r, θ) directionsrespectively FAX

z ,FAX

r ,FAX

θ are the ﬂuxes along z, r, and θ directions respectively, HAX

contains the terms due to the cylindrical reference frame and to the mean ﬂow derivativesand SAX represents the acoustic sources In Section 4.3 it will be shown that a genericturbomachinery tonal wave can be expanded in a sum of complex duct modes, having the

form ˆf(z, r, t ) ·exp(I mθ)where m is an integer number which identiﬁes the azimuthal mode

andI the imaginary unit Therefore, using the dependence of the acoustic ﬁeld onθ, the

problem can be reduced, from a three-dimensional problem, to a two-dimensional one in(r, z).For a single duct-mode the LEE become

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2.1 Frequency domain approach

The linearized Euler equations, beside acoustic waves, support also instability waves that,for a mean flow with shear-layers, are the well-known Kelvin-Helmholtz instabilities Inthe complete physical problem this instabilities are limited and modified by non-linearand viscous effects Indeed, in the linearized Euler equations, these two effects arenot present Therefore when solving LEE in presence of a shear-layer type mean flow,Kelvin-Helmholtz instabilities can grow indefinitely as they propagate down-stream fromthe point of introduction and the acoustic solution may be obscured by the non-physicalinstabilities [Agarwal et al (2004); Özyörük (2009)] By using a Fourier decomposition ofthe acoustics sources and solving the linearized Euler equations in the frequency domain onecan, in principle, avoid the unbounded growth of the shear-layer type instability, since theacoustic and instability modes correspond to different values of complex frequency [Rao &Morris (2006)] However, this could be accomplished in practice only if the discretized form

of the equations is solved using a direct solver The use of iterative techniques to solve theresulting global matrix has been discussed by Agarwal et al (2004) It is proved that the use ofany iterative technique to solve the global matrix is equivalent to a pseudo-time marchingmethod, and hence, produces an instability wave solution Therefore, the solution of theglobal matrix needs to be sought by using direct methods such as Gaussian elimination or

LU decomposition techniques

2.2 GTS-like approximation

In order to reduce computational time and memory requirements, the pressure gradients inthe momentum equations are neglected A similar approximation, termed Gradient TermsSuppression (GTS), is often used to overcome instability problems that prevent convergence

of time domain algorithms for the LEE [Tester et al (2008); Zhang et al (2003)] Whilethe original GTS approximation suppresses all mean-flow gradients, which are likely to besmall in the considered subsonic flows, in the present case, being interested in reducingthe computational time, only the density mean-flow gradients in momentum equations areneglected This allows to decouple the continuity equation and to solve only momentum

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and energy equations For an axial-symmetric problem the number of total unknowns is thusreduced by a factorT =5/4, whereas the non-zero terms in the coefﬁcient matrix of the linearsystem associated with the discretized form of Eqs (8) is reduced by a factorT2 Indeed, asmaller linear system can be solved faster, and, more important, its resolution requires lessmemory

Rigid Walls

If walls are assumed impermeable and acoustically rigid, no flow passes through theboundary and acoustic waves are totally reflected Assuming that the mean flow satisfiesthe slip flow boundary condition, an analogous slip flow condition must be imposed on thevelocity fluctuations

whereu is the acoustic velocity andn is the normal vector to the wall To apply this condition,

Eq (11) is used to express one of the velocity components in terms of the others

Axial symmetry

When dealing with axial-symmetric problems, the equations could be solved only for r ≥0

if an appropriate boundary condition is applied on the symmetry axis Along that boundary,

the acoustic velocity should be aligned with the r =0 axis, this can be achieved applying awall type boundary condition

Far-ﬁeld boundary

One of the major issues in CAA is to truncate the far-field domain preserving a physicallymeaning solution This leads to the necessity to have accurate and robust non-reflectingfar-field boundary conditions A large number of families of non-reflecting boundaryconditions has been derived in literature The most widely used for the Euler equationsare the characteristics-based boundary conditions [Giles (1990); Thompson (1990)] Thesemethods are derived applying the one-dimensional characteristic-variable splitting in theboundary-normal direction This technique is usually efficient and robust The maindrawback is that reflections are prevented only for waves that are traveling in theboundary-normal direction Not negligible reflections can be seen for waves that hit theboundary with other angles Another class of non-reflecting boundary conditions is based

on the asymptotic solutions of the wave equation [Bayliss & Turkel (1980); Tam & Webb(1993)] In this case, the governing equations are replaced in the far ﬁeld by an analyticsolution obtained imposing an asymptotic behavior to the system These conditions can

be very accurate Unfortunately, the asymptotic solution can be achieved only in a limitednumber of cases, reducing the applicability of this model to test cases

Another family of non-reflective boundary conditions is composed by the buffer zonetechnique [Bodony (2006); Hu (2004)] In this case, an extra zone is added to damp thereflected waves The damping can be introduced as a low-pass filter, grid stretching oraccelerating the mean flow to supersonic speed The main drawback of these techniques is

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A Frequency-Domain Linearized Euler Model for Noise Radiation

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represented by the increase of the computational cost, as the thickness of the buffer zone could

be important to achieve a good level of accuracy

More recently, the Perfectly Matched Layer (PML) technique has been developed as a newclass of non-reﬂective boundary conditions The basic idea of the PML approach is to modifythe governing equations in order to absorb the out-going waves in the buffer region Theadvantage of this technique is that the absorbing layer is theoretically capable to damp waves

of any direction and frequency, resulting in thinner layers with respect to other buffer zoneapproaches, with benefits on the efficiency and the accuracy of the solution Originallyproposed by Berenger (1994) for the solution of the Maxwell equations, the PML techniquewas extended to CAA applying the split physical variable formulation to the linearizedEuler equations with uniform mean flow [Hu (1996)] It was shown that the PML absorbingzone is theoretically reflectionless to the acoustic, vorticity and entropy waves Nonetheless,numerical instability arises in this formulation, and in Tam et al (1998) the presence ofinstability waves is demonstrated In Hu (2001), it was shown that the instability of the splitformulation is due to an inconsistency of the phase and group velocities of the acoustic waves

in presence of a mean ﬂow, and a stable PML formulation for the linearized Euler equationwas proposed, based on an unsplit physical variable formulation

The PML technique can be seen as a change of variable in the frequency domain, for example,considering the vertical layer, this change of variable can be written as

x − → x+ω i x

x0

whereσ x > 0 is the absorption coefﬁcient and x0 is the location of the PML/LEE interface

To avoid instabilities, a proper space-time transformation must be used before applying thePML change of variable, so that in the transformed coordinates all linear waves supported

by the LEE have consistent phase and group velocities Assuming that the mean ﬂow in the

absorbing layer is uniform and parallel to the x axis, the proper space-time transform involves

a transformation in time of the form [Hu (2001)]

iω

and ˜F x, ˜F y, and ˜H are the terms of Eq (5) under the

assumption that the mean ﬂow is uniform and parallel to the x axis The damping constants

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σ xandσ yhave the following expressions

conditions are needed except those that are necessary to maintain the numerical stability ofthe scheme For this reason at the external boundary of the absorbing layer wall boundaryconditions are applied

Applying the same PML formulation to the LEE written for the turbomachinery duct modes,the following system is obtained

whereα zandα r are deﬁned as in the two-dimensional case and ˜F z, ˜F r, ˜F θ, and ˜H are the

terms of Eq (9) and Eq (10) under the assumption that the mean ﬂow is uniform and parallel

to the z axis.

Acoustic inlet

The PML formulation is also used to impose incoming waves at acoustic inlet boundaries Onthose boundaries incoming waves should be specified, but at the same time outgoing wavesshould leave the computational domain without reflections This can be achieved applyingthe PML equations to the reflected wave,ure[Özyörük (2009)], which can be expressed as thetotal acoustic field,u, minus the incoming prescribed acoustic wave, uin

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A Frequency-Domain Linearized Euler Model for Noise Radiation

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3 Numerical methods

The numerical solution of the LEE requires highly accurate and efﬁcient algorithms able

to mimic the non-dispersive and non-diffusive nature of the acoustic waves propagatingover long distances One of the most popular numerical scheme in CAA is the DispersionRelation Preserving (DRP) algorithm originally proposed by Tam & Webb (1993) The DRPscheme is designed for Cartesian or highly regular curvilinear coordinates However, inmany practical applications, complex geometries must be considered and unstructured gridsmay be necessary One of the most promising numerical scheme able to fulﬁll all the aboverequirements is the Discontinuous Galerkin method (DGM or DG method)

The DGM was ﬁrstly proposed in the early seventies by Reed and Hill in the frame of theneutron transport [Reed & Hill (1973)] Since then, the method has found its use in manydifferent computational models In the last years, in the context of CFD, DGM has gained

an increasing popularity because of its superior properties with respect to more traditionalschemes in terms of accuracy and intrinsic stability [Cockburn et al (2000)]

The DG method displays many interesting properties It is compact: regardless of the order

of the element, data are only exchanged between neighboring elements It is well suited forcomplex geometries because the expected dispersion and dissipation properties are retainedalso on unstructured grids Furthermore in the framework of DGM it is straightforward

to implement the boundary conditions, since only the flux needs to be specified at theboundary The main disadvantage of the DGM is its computational cost Because of thediscontinuous character, there are extra degrees of freedom at cell boundaries in comparison tothe continuous finite elements, demanding more computational resources This drawback can

be partially reduced with a static condensation technique and with a parallel implementation

of the algorithm, operations which are made easier by the compactness of the scheme[Bernacki et al (2006)]

3.1 Discontinuous Galerkin formulation

The DGM will be initially presented for the scalar problem of ﬁnding the solution u of the

hyperbolic conservation equation

The discontinuous Galerkin formulation is based on the idea of discretizing the domainΩ

into a set of E non-overlapping elementsΩe Introducing the notations

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To obtain an expression which explicitly contains the ﬂux at the element interfaces, thedivergence term in Eq (26) is integrated by parts

∂Ω w F(u ) · n=0 , (27)wheren is the outward-pointing normal versor referred to each element edge For interfaces

on the domain borders, the normal flux vector is evaluated using appropriate boundaryconditions In the general case a boundary condition defines the normal flux asF(u ) · n =

FBC(u) +GBC On internal interfaces,F(u ) · n is evaluated from the values of u In order for

the formulation to be consistent, the normal flux vector evaluated on right side of an internalinterface must be equal to minus the normal flux vector evaluated on the left side of the sameinterface Since one of the key feature of the DGM is the discontinuity of the solution amongthe elements, the consistency is not automatically guaranteed by the formulation Thereforethe normal fluxF(u ) · n is replaced by a numerical flux F R(u)which is uniquely defined nomatter of the side on which it is evaluated (see section 3.2) For ease of notation it is convenient

to introduce the following deﬁnition

whereF ∂+G ∂is equal toFBC+GBCfor interfaces on the domain borders and is equal to

toFRfor internal ones Furthermore, assuming that the ﬂux vector is a linear function of theunknown, yields

where A and A ∂ are two matrices representing the Jacobian of the physical ﬂux and the

Jacobian of the numerical ﬂux respectively Using Eq (28) and Eqs (29), the weak formulation

∂Ω w A ∂ u+G ∂

Given Eq (30), the discontinuous Galerkin approximation is obtained considering a ﬁnite

element space, W h , to approximate W On each element, a set of points called nodes or degrees

of freedom is identiﬁed The number and the position of the nodes depend on the type ofapproximation used The set of nodes is chosen to be the same on each element, in this way, onelement’s borders, there is a direct correspondence among the nodes deﬁned on neighboring

elements The nodes are numbered globally using the index jglob=1, 2, , nglobwith nglob

being the global number of degrees of freedom Beside the global numbering, there is a local

numbering On each element the nodes are identiﬁed using the index j eloc = 1, 2, , nloce where n e

loc is the number of degrees of freedom of the e-th element The correspondence

between local node numbers and global node numbers can be expressed through a matrixcalled connectivity matrix

jglob= C e, j e

The nodes of the discretization are used to deﬁne the ﬁnite element space W h: the vector space

W his generated by the Lagrangian polynomials deﬁned on the nodes of the discretization The

variable u ∈ W is therefore approximated in the W h space with an interpolation of its nodalvalues

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where u j(t) is the value of u in the j-th global node x j , y j

at the time t andΦj is the

Lagrangian polynomial deﬁned on the j-th global node with the property

Φi x j , y j

Although in this work Lagrangian interpolation functions are used, other types of

interpolation are possible Considering the vector space W h, the discrete weak form of

problem (23) consists in ﬁnding u h ∈ W hsuch that

∂Ω w h A ∂ u − G ∂

=0 ∀ w h ∈ W h (34)Substituting Eq (32) into the discrete weak form (34) leads to

This equation must hold for every admissible choice of weight functions w h, therefore it is

sufﬁcient to test it for the ngloblinearly independent functions of a base of W h In this way it

is possible to obtain nglob independent algebraic equations to solve for the nglobunknowns

u j The vector space W h is deﬁned as the space formed by the Lagrangian polynomials

Φi, therefore the functions Φi form a base for W h The i-th algebraic equation is obtained substituting w h=Φiinto Eq (35)

where ˆ(.)(l) j is the l-th component of the Fourier transform of(.),ω (l)is the angular frequency

of the l-th Fourier mode and Iis the imaginary unit Equation (37) represents the weak-formdiscontinuous Galerkin model for a scalar hyperbolic problem in the frequency domain Itcan also be written in matrix notation as

Tiêu đề	Physico-Chemical Modelling in Nonequilibrium Hypersonic Flow Around Blunt Bodies
Trường học	Not specified
Chuyên ngành	Aerospace Engineering
Thể loại	Research Paper

Định dạng
Số trang	40
Dung lượng	4,02 MB