2002_arch of comp meth engg_a review and comparative study of upwind biased schemes for compressible...

Arch. Comput. Meth. Engng.Vol. 9, 3, 207-256 (2002) Archives of Computational

Methods in EngineeringState of the art reviews

A Review and Comparative Study of Upwind Bia-sed Schemes for Compressible Flow Computation.Part III: Multidimensional Extension on Unstruc-tured Grids

P.R.M. LyraDepartamento de Engenharia MecanicaUniversidade Federal de PernambucoRecife/PE 50.740-530, Brasil

K. MorganCivil and Computational Engineering CentreSchool of EngineeringUniversity of WalesSwansea SA2 8PP, U.K.

Summary

The edge based Galerkin finite element formulation is used as the basic building block for the constructionof multidimensional generalizations, on unstructured grids, of several higher order upwind biased proceduresoriginally designed for the solution of the 1D compressible Euler system of equations. The use of a centraltype discretization for the viscous flux terms enables the simulation of multidimensional flows governedby the laminar compressible Navier Stokes equations. Numerical issues related to the development andimplementation of multidimensional solution algorithms are considered. A number of inviscid and viscousflow simulations, in different flow regimes, are analyzed to enable the reader to assess the performance ofthe surveyed formulations.

1 INTRODUCTION

In Parts I and II of this paper [91, 92], the numerical performance of different upwindbiased schemes for the solution of 1D compressible flows has been surveyed. In this paper,we consider a multidimensional extension, in which a Galerkin finite element formulationon an assembly of unstructured simplex elements is employed to obtain the discretization inspace. Time discretization will be achieved by the use of an explicit multi stage scheme. Thepresentation will be restricted to two dimensional problems, but the methods described arecapable of direct extension to enable the simulation of three dimensional applications [104,114, 115].

Methods based upon structured mesh methodologies remain the most widely used nu-merical tool of the computational fluid dynamicist and a large number of algorithms may beimplemented, in a fairly straightforward and computationally efficient manner, to producecomputer codes for multidimensional analysis. However, the large elapsed time necessaryto produce structured grids for extremely complex configurations, the difficulty in control-ling the quality of the elements or cells and the unstructured type overhead which ariseswhen adaptive meshing techniques are implemented represent the main disadvantages ofthe structured approach.

The use of unstructured mesh methods has been steadily increasing over the past twodecades, despite the associated increase in computational time and memory requirements.It is the possibility of computing geometrically complex designs, and the evident naturalenvironment for the incorporation of adaptivity which may be essential for resolving verysmall scale flow features, that stimulates many CFD practitioners to devote much atten-

c©2002 by CIMNE, Barcelona (Spain). ISSN: 1134–3060 Received: February 2002

208 P.R.M. Lyra and K. Morgan

tiontothedevelopmentanduseof unstructuredmeshmethodologies.Theenthusiasmoftheresearchcommunityisnowbeingmirroredbytheindustrial community, whoareattractedtotheopportunitiesaffordedbytheapproachinenablingtherapidanalysisofnewandmodifieddesigns[44 ]. Discretizationtechniqueswhicharebaseduponintegralformulations,suchasthefinitevolumeandfiniteelementmethods,arenaturalcandidatesforusewithunstructuredmeshes.Substantialprogresshasbeenmadeinthedevelopmentofunstructuredmeshsolutionalgorithmssothataccuracyhasbeenimprovedandcom-putationalcostshavebeenreduced.Significantaccomplishmentsincludethedevelopmentofdatastructuresandsearchingalgorithms,reducinggatherscatteroperationsandmem-oryrequirements[2, 10, 14 , 53, 71 , 72, 96 , 100 , 114 ]; thedevelopmentofmeshgeneratorsabletoaccuratelymodelcomplexgeometries,withhighqualitymeshessuitableforbothinviscidandviscoussimulation[9 , 31, 41 , 42, 57, 64, 82, 85 , 99, 103 , 117 , 139 , 143 , 148 ];thederivationofdifferentstrategiestoassessaccuracyandtoadaptthediscretization[5,7, 8, 21 , 22, 29, 32, 51, 60 , 77, 79 , 84, 117 , 132 , 152 , 153 ]; thedesignofalgorithmswhichmimicstructuredmeshalternatingdirectioni mplicittechniquesandthedesignofstrate-giestoefficientlytransferinformationbetweenunstructuredmeshesusedwithamultigridaccelerationprocedure[41 , 52, 63 , 66, 67, 73, 79, 100 , 101 , 115, 131 , 133 , 138]; theex-tensionof mostof theavailablemethodologiestodevelophighordermultidimensionalalgorithms, suchasswitchedartificialviscosity, MUSCLandmanyfluxlimitedorhybridschemes[10, 12, 13 , 23, 52, 53 , 68, 74, 79, 95 , 129 , 137 ];oruseofthegeneralizedGalerkin,orresidual, methods[4, 15, 18 , 24, 25 , 33, 43, 49 , 50, 56 , 58, 59, 61 , 75, 107 , 136 ]. ThesedevelopmentshaveensuredthatunstructuredmeshapproachesarenowtrulycompetitivesolutionproceduresinCFD.Furthermore,theexploitationofthevectornatureoftheop-erationsandnaturalorforcedparallelizationinvolvedinacomputationalimplementationofthesolutionalgorithmsontheavailablevectorandparallelcomputershascontributedtotheeffectivenessofthesecomputertoolsforfluiddynamicsdesign.

Theprincipalattentionof thisworkinvolves reviewingtheextensionofmanywellestablishedstructuredmeshproceduresin toalgorithmswhichmaybeimplementedonunstructureddiscretizations.TheedgebasedfiniteelementapproachofPeraireetal[114],whichhasfeaturesincommonwiththatadoptedbyBarth[10]andMavriplis[100]inthefinitevolumecontext, isemployedfortherepresentationofthetriangulargrid.Thisallowsadirectimplementationofdifferenttypesofstandard1Dupwindorcenteredshockcapturingmethodswithinafiniteelementunstructuredgridcontext[87, 93 , 94, 95 , 78, 97 , 105 , 113 ].Itisknownthattheuseofthisdatastructurehasadditionalbeneficialeffects,intermsofbothCPUtimeandmemoryrequirements.Thesewillbeofparticularimportancewhentheextensionofthemethodstothesolutionoflargescale3Dproblemsisenvisaged.

2 THE GOVERNING EQUATIONS

Followinganondimensionalization,baseduponrepresentativelength,andfreestreamvelocity, densityandviscosity, theequationswhichgoverntheunsteadylaminarflowofaNewtoniancompressibleviscousfluid,intheabsenceofexternal sourceterms,maybewrittenintheconservationform

∂U

∂t+

∂F j

∂xj+

∂Gj

∂xj= 0 in Ω × I for j = 1, . . . Nd (1)

where the summation convention is employed, xj is the Cartesian coordinate, Ω is thespatial domain, t is the time, I = (t0, T ) is the time interval t0 ≤ t ≤ T and the Ndindicates the number of spatial dimensions to be considered in the model. In equation (1),U is the vector of the conservative variables, while F j and Gj denote the inviscid and

A Review and Comparative Study of Upwind Schemes. Part III 209

viscous flux vectors in the direction xj , respectively. For two dimensional simulations, i.e.Nd = 2, these vectors can be written as

U =

u1

u2

ε

F j =

uj

u1uj + pδ1j

u2uj + pδ2j

(ε + p)uj

Gj = − 1R

0

τ1j

τ2j

uiτij +µ

Pr∂T

∂xj

(2)

Here , p, T and ε represent the fluid density, pressure, temperature and total specificenergy, respectively, and uj is the component of the velocity vector in direction xj . Inaddition, δij denotes the Kronecker delta, Pr the Prandtl number, µ the molecular dynamicviscosity and R the free stream Reynolds number. The shear stress tensor components, τij,and the heat flux vector components qj are defined by

τij = µ

(∂ui

∂xj+

∂uj

∂xi

)− 2

3µ

∂uk

∂xkδij qj = −

(µ

R Pr

)∂T

∂xj(3)

The equation set is closed by the addition of the perfect gas equations of state which, innon dimensional form, may be written as

p = (γ − 1) (ε − 0.5uiui) T =γp

(γ − 1)(4)

where γ = cp/cv, with cp and cv being the specific heats of the fluid at constant pressureand at constant volume respectively. Finally, Sutherland’s experimental law relating thedynamic coefficient of viscosity to the temperature has been adopted. In non dimensionalform, this is expressed as

µ =

[(γ − 1)M 2∞

]−1 + S

T + S

[(γ − 1)M 2

∞T]3/2

S =S0

M 2∞T∞(5)

where M∞ and T∞ are the free stream Mach number and temperature and S0 is a constantfor a given fluid [45]. The Prandtl number may be assumed constant for ideal gases atmoderate temperature [65]. It can be observed that, for any set of boundary and initialconditions, the flow is characterized by the dimensionless parameters γ, R, Pr, M∞, T∞and the angle of incidence, α, of the free stream.

The solution of this set of equations is sought over the closed spatial domain Ω, withboundary surface Γ, and the time interval I. This initial/boundary value problem requiresadditionally boundary and initial conditions, which are taken here in the form

F n = njF j = Fn ; Gn = njGj = G

n at Γ × I (6)

andU(x, t0) = U

0(x) on Ω × t0 (7)

Here nj denotes the component, in direction xj, of the unit outward normal vector to Γ, x

is the vector (x1, x2) and Fn and G

n are the normal fluxes at the boundary. The exactform of F

n and Gn will depend upon the local solution and the boundary being simulated,

while U0 is assumed to be a known function.


3 A FINITE ELEMENT APPROACH

Although our ultimate goal is to simulate numerically the full system of Navier Stokesequations, we begin by considering the Euler equation set, which is obtained from theNavier Stokes equations given in (1) by neglecting all shear stress and heat conductionterms. i.e. Gj ≡ 0. The problem formulated in the previous section may be put into avariational form [58] by introducing a trial function space, T , and a weighting functionspace, W. These spaces are defined to consist of all suitably smooth functions and to besuch that

T = U(x, t)|U (x, t0) = U0(x) on Ω at t = t0 W = W (x) for x ∈ Ω) (8)

A weak variational formulation [58] of the problem is then to determine U ⊂ T , satisfyingthe problem boundary conditions, such that ∀ t > t0

∫Ω

∂U

∂tW dΩ =

2∑j=1

∫Ω

F j(U)∂W

∂xjdΩ −

∫Γ

FnW dΓ (9)

for every W in W. The fundamental lemma of the variational calculus [28] assures thatthe solution U of this problem is identical to the solution of the problem formulated in thestrong form in equations (1) to (7).

3.1 Approximate Variational Formulation

A discrete variational formulation corresponding to the weak variational formulation givenin equation (9) is now determined. Assuming that the spatial domain Ω is discretized intoan unstructured assembly of linear triangular elements, with the nodes numbered from 1to p, subspaces T (p) and W(p), of the trial and weighting function spaces T and W, aredefined by

T (p) = U(x, t)|U =p∑

J=1

UJ(t)NJ(x); UJ(t0) = U0(xJ) = U0

J

W(p) = W(x)|W =p∑

J=1

aJNJ(x)(10)

where NJ is the standard linear finite element shape function associated with node J ,located at x = xJ , UJ is the value of U at node J and a1, . . . , ap are constants. Thesespaces are conveniently defined for use with a Galerkin finite element formulation. Anapproximate weak variational formulation can then be stated as

find U ⊂ T (p), satisfying the problem boundary conditions, such that∀ t > t0∫Ω

∂U

∂tNI dΩ =

2∑j=1

∫Ω

F j(U )∂NI

∂xjdΩ −

∫Γ

FnNI dΓ

(11)

for each I = 1, 2, . . . , p. In equation (11), the generic weighting function W has been directlyreplaced by the nodal shape functions NI , as they form a basis for the space W(p). Theintegrals appearing here can be evaluated by summing individual element contributions,and the compact support of the shape function NI means that the variational statement


may be rewritten asfind U ⊂ T (p), satisfying the problem boundary conditions, such that ∀ t > t0∑E∈I

∫ΩE

∂U

∂tNI dΩ =

∑E∈I

2∑j=1

∫ΩE

F j(U)∂NI

∂xjdΩ −

∑B∈I

∫ΓB

FnNI dΓ

(12)

for each I = 1, . . . , p, where the summations just extend over those elements E and bound-ary edges B which contain node I.

To avoid the necessity for numerical integration, the inviscid flux F j is approximatedin a piecewise linear fashion in terms of its nodal values, i.e. we adopt the representation

F j(U) =p∑

J=1

F jJNJ(x); F j

J = F j(UJ) (13)

Alternatively, this is equivalent to adopting a Lobatto quadrature rule [48] over each el-ement E when evaluating numerically the integrals that appear in the right hand side ofequation (12). Inserting the assumed forms for U and F j(U), given in equations (10)and (13) respectively, into equation (12), all the integrals can be evaluated in closed form.The left hand side integral may be expressed as

∑E∈I

∫ΩE

∂U

∂tNI dΩ =

∑E∈I

[∫ΩE

NINJ dΩ]

dUJ

dt=

[M

dU

dt

]I

(14)

where M is the finite element consistent mass matrix. Using the fact that the shape functiongradients are constant on each element, the integral over the computational domain, on theright hand side of equation (12), can be expressed as

∑E∈I

2∑j=1

∫ΩE

F j(U)∂NI

∂xjdΩ =

∑E∈I

2∑j=1

[ΩE

3∂NI

∂xj

]E

(F jI + F j

J + F jK) (15)

where ΩE denotes the area of element E with nodes I, J and K. Similarly, the integralover the computational boundary on the right hand side of equation (12) can be expressedas ∑

B∈I

∫ΓB

FnNI dΓ =

∑B∈I

[ΓB

6(2F

nI + F

nJ )

](16)

where ΓB denotes the length of the boundary edge B with nodes I and J .The standard finite element data structure consists of the physical coordinates listed by

node number, a list of the nodal connectivity for each element and a list of the boundaryedge connectivities. With this geometrical and topological data, the integrals discussedabove can be performed by a loop over the elements and a loop over the boundary edges,with the element and edge contributions to the nodes being accumulated during the process.

3.2 Edge Based Formulation

An alternative to the element based data structure, in which nodal values are obtained bysumming element contributions as sketched in Figure 1(a), is to represent the unstructuredgrid in terms of an edge based data structure [10, 100, 114]. The nodal values are thenobtained by summing edge contributions, as illustrated in Figure 1(b). In this case, thephysical coordinates are listed by node number and a list of boundary edge connectivities is


(a)

I

I1

I2

I3

IIm

mEI

2E

1E

(b)

I

I1

I2

IIm

I3

S3

S 2

S1

SIm

Figure 1. Sketch of the triangles and edges sharing node I and different datastructure for a 2D unstructured grid. (a) element based data structure;(b) edge based data structure

also provided. However, in this case, the topology of the mesh in the interior of the domainis characterized in terms of the mesh edges and their connectivities.

For an interior node, such as node I in Figure 1(b), equation (15) can be rearranged as

∑E∈I

2∑j=1

∫ΩE

F j ∂NI

∂xjdΩ =

mI∑S=1

∑E∈IIS

2∑j=1

[ΩE

3∂NI

∂xj

]E

(F jI + F j

IS)

−

∑E∈I

2∑j=1

[ΩE

3∂NI

∂xj

]E

(F jI)

(17)

where mI is the number of edges in the mesh which are connected to node I and thesummation over E extends over those elements that contain the edge IIS. The secondterm on the right hand side of equation (17) is zero, as it is equivalent to the integration ofthe gradient of a constant function. In this way, expression (17) can be evaluated througha sweep over the edges and by adding the symmetric contribution to the vertices associatedwith each edge, as sketched in Figure 1(b). A similar, but somewhat more elaborate,algebraic manipulation can be undertaken for nodes which lie on the boundary in orderto retain the symmetric nature of the edge contributions. This process is described inAppendix A. In this way, the discrete equation (12) can be conveniently expressed as

[M

dU

dt

]I

= −mI∑S=1

2∑j=1

CjIIS

(F jI + F j

IS) + 〈

2∑f=1

Df (4FnI + 2F

nJf

+ F nI − F n

Jf)〉I (18)

where CjIIS

denotes the weight that must be applied to the sum of the fluxes in the xj

direction on the edge S, which joins nodes I and IS, to obtain the contribution madeby the edge to node I. The weight which is applied to the same quantity to obtain thecontribution made by the edge S to node IS will be denoted by Cj

ISI . In addition, Df

represents the boundary face correction coefficient, which is necessary for nodes I which lieon the boundary, and J1, J2 are the two boundary nodes which are connected to node I.


These weights can be readily computed, see Appendix A, as

CjIIS

= −∑

EεIIS

ΩE

3

[∂NI

∂xj

]E

+ 〈Γf

12nj

IIS〉IIS

Df = −Γf

12(19)

where the bracketed term is only non zero if IIS is a boundary edge. The quantity Γf

denotes the length of the boundary edge joining nodes I and IS and njIIS

is the componentin the xj direction of the unit normal to the edge IIS. It is readily verified, see Appendix A,that these weights satisfy the relations

mI∑S=1

CjIIS

− 〈2∑

f=1

3DfnjIIS

〉I = 0 for j = 1, 2

CjIIS

+ CjISI = 0 for j = 1, 2 and s = 1, . . . ,mI

(20)

With the weight coefficients defined in equation (19) determined in a preprocessing stage,the computer code for the flow simulation can be written so that equation (18) is formedby looping over each edge in the mesh and then sending edge contributions to the appro-priate nodes. A loop over the boundary edges is performed and the extra boundary edgecontributions added to the appropriate boundary nodes.

For notational convenience, define

CIIS= (C1

IIS, C2

IIS) LIIS

= |CIIS| Sj

IIS=

CjIIS

|CIIS| (21)

and, using this notation, we can write equation (18) as

[M

dU

dt

]I

= −mI∑S=1

F SIIS︷︸︸︷

LIIS

2∑j=1

(F jISj

IIS+ F j

ISSj

IIS) + 〈

2∑f=1

Df (4FnI +2F

nJf

+F nI−F n

Jf)〉I (22)

From the asymmetry of the edge weights expressed in equation (20), the numerical dis-cretization scheme can be immediately observed to possess a conservation property, in thesense that the sum of the contributions made by any interior edge is zero. It is also appar-ent, by using the results of equation (20), that this is a central difference type discretizationfor the spatial derivatives. This means that, to construct practical solution algorithms forthe Euler equations, we will need to replace the actual flux function F S

IISin equation (22)

by a consistent numerical flux FIIS. By adopting different forms for this numerical flux

function, we are able to construct a number of different algorithms for the solution of thecompressible Euler equations [93, 94, 97, 114]. Some of these algorithms will be presentedin section 4.

3.3 Computation of the Viscous Fluxes

The high Reynolds number compressible flow problems of interest here are assumed to beinviscid dominated, in the sense that, apart from very thin regions close to solid walls,the flow field is governed by the inviscid portion of the Navier Stokes equations. As aresult, high resolution shock capturing schemes are needed to resolve the shock wavesand contact discontinuities present in the solution. Since the viscous terms in the Navier


Stokes equations are parabolic or elliptic in nature, the numerical discretization of theseterms is accomplished by a centered technique. This means that the numerical proceduresadopted here for compressible Navier Stokes flow simulations are built using the schemedeveloped in the previous section to discretize the inviscid part of the equations and a mixedfinite element formulation for the viscous terms. The inherent numerical dissipation of ashock capturing scheme has, in principle, a conflicting effect on the physical viscosity in theboundary layer region, where steep gradients of the flow variables exist. However, numericalevidence supports the suggested procedure, as the contamination of the boundary layer bynumerical diffusion can be contained within acceptable limits.

Following the procedure employed for the inviscid equations in section 3.1, the left handside of the weak approximate variational statement of equation (12) must be complementedby the addition of the viscous terms

−∑E∈I

2∑j=1

∫ΩE

Gj

(U ,

∂U

∂x1,

∂U

∂x2

)∂NI

∂xjdΩ +

∑B∈I

∫ΓB

GnNI dΓ (23)

for each I = 1, . . . , p, where the domain and the boundary integrals have been decomposedinto the sum of individual element and boundary face contributions. These integrals can beevaluated using different strategies. Here, the viscous terms are discretized by a Galerkinscheme, using a finite element mixed formulation [79, 113]. This is convenient when theedge based data structure is employed to represent the mesh. In this case, the nodal valuesof the viscous terms are directly evaluated as

GjI = Gj

(U I ,

∂U

∂x1

∣∣∣∣I,

∂U

∂x2

∣∣∣∣I

)(24)

which requires the nodal values of the gradients of the variables. When linear shape func-tions are adopted in the finite element method, the gradients over each element are con-stant, with multiple values defined at each nodal point. One possible way to compute acontinuous nodal value for the gradient of the solution is the use of the least squares re-construction [150, 152]. This procedure, when written using an edge based data structure,leads to the expression[

M∂U

∂xj

]I

=mI∑S=1

2∑j=1

CjIIS

(U I + U IS) − 〈

2∑f=1

DfnjIJf

(5U I + UJf)〉I (25)

This enables the gradients of U to be approximated at each nodal point I = 1, . . . , p andis similar to equation (18). The variation of the viscous flux Gj can then be representedusing a piecewise linear approximation of the form adopted for the inviscid flux F j inequation (13). In this way, the Navier Stokes equations may be discretized as

[M

dU

dt

]I

= RI ≡ −mI∑S=1

2∑j=1

CjIIS

(F j

I + F jIS

) − (GjI + Gj

IS)

+〈2∑

f=1

Df

(4F

nI + 2F

nJf

+ F nI − F n

Jf)

−(4GnI + 2G

nJf

+ GnI − Gn

Jf)〉I

(26)

which is in a form that is suitable for use in conjunction with an edge based data structure.


It must be noted here that the discretization of the viscous terms employing the mixedformulation involves information from two layers of points surrounding the point underconsideration, while adopting the consistent finite element computation of the gradientswould use information from those points directly connected to the point being considered.However, no adverse effect has been observed with the use of the mixed formulation forsome subsonic numerical experiments performed by Peraire et al [113]. Similar conclusionswere derived from some comparisons for supersonic and hypersonic regimes performed byLyra [79] and by Manzari [98]. Luo et al [78] also found practically identical results com-paring the mixed formulation with an alternative edge based finite element approach. Thedifficulty of evaluating the standard finite element expression using an edge based datastructure [110], the significant storage overhead required by the alternative edge based ap-proach and the similar results obtained with all three approaches, motivates the adoptionof the mixed formulation.

3.4 Time Discretization

Equation (26) represents the time evolution of the unknown vector U I(t) at node I of themesh. A practical solution algorithm is produced by discretizing the time dimension, usingan explicit hybrid multistage time stepping scheme [54, 115].

3.4.1 Mass lumping

The consistent finite element mass matrix M is replaced by the diagonal lumped massmatrix ML. This enables truly explicit time integration and does not alter the final steadystate solution, which is of primary concern here. Despite some possible loss of temporalaccuracy, this approximation was also adopted for the few transient computations reportedin the present work.

3.4.2 Multistage scheme

Multistage schemes represent an important family of time marching schemes in which eachtime step consists of K simple explicit stages. Assuming that the nodal values U I(t) andRI are known at time tn, the solution is advanced, over the time step ∆t to time tn+1,according to

U(0)I = Un

I...U

(k)I = Un

I + αk ∆t [ML]−1I R

(k−1)I k = 1, . . . ,K

...Un+1

I = U(K)I

(27)

where R(k−1)I represents the right hand side of equation (26) at stage (k − 1). The coef-

ficients αk are constants, which depend upon the number, K, of stages being employed.These coefficients are normally chosen to provide an enhancement of the stability rangewhen steady state computation is considered, and to ensure the highest accuracy for timedependent problems [45]. Here, the parameter values

k=3 stages ⇒ α1 = 3/5; α2 = 3/5; α3 = 1k=4 stages ⇒ α1 = 1/4; α2 = 1/3; α3 = 1/2; α4 = 1k=5 stages ⇒ α1 = 1/4; α2 = 1/6; α3 = 3/8; α4 = 1/2; α5 = 1

are adopted [46, 115]. To reduce the computational cost, a selective multistage scheme [46,55] can be used, where the dissipative term is only recalculated at certain stages and


remains frozen during the others. To increase the stability range of the explicit multistagetime stepping scheme of equation (27), and therefore increase the value of the local Courantnumber, C, a residual smoothing strategy [46, 53, 54, 115] can also be utilized. The simpleexplicit forward time integration is a particular technique which can be derived from theabove expression when the number of stages is set to one. In most high speed applicationsanalyzed here, the explicit one stage scheme is adopted, as the use of the multi stageprocedure was found to produce little increase in the size of the allowable value of theCourant number for such simulations. Further study is required in this area and the specialclass of multi stage TVD Runge Kutta type time discretizations presented by Shu [126] andShu et al [127] might lead to better performance.

3.4.3 Stability of explicit time integration

In the absence of a well grounded stability criterion for the full set of Navier Stokes equa-tions, a criterion derived by analogy with the advection diffusion equation is employed [129].The resultant expression is closely related to one that can be developed based upon thespectral radius of the viscous Jacobian matrices [133]. In this way, the local time step forviscous computations is determined by

∆tI = 2C[ML]I

mI∑S=1

LIIS|(λmax)IIS

| +4L2

IISµI max

(2;

γ

Pr

)Re∞I [ML]I

−1

(28)

where λmax represents the maximum eigenvalue of ASIIS

, i.e.

(λmax)IIS= |uIIS

· SIIS| + cIIS

(29)

with uIISand cIIS

denoting the edge values of the fluid velocity vector and the speedof sound respectively and SIIS

denoting the unit vector in the direction of the weightingcoefficient vector (21). These edge values are obtained by averaging the appropriate nodalvalues. In the limit of vanishing viscosity, this heuristic criterion reduces to the criteriondeduced by performing a stability analysis based upon the energy method [34]. It alsoreduces to the pure diffusive limit, either for the momentum equation or the energy equation,when the convective term is negligible, i.e. (λmax)IIS

→ 0.

3.4.4 Local time stepping

When a steady simulation is performed, local time stepping [113] is employed to acceleratethe convergence rate towards steady state, since the correct modelling of the transientdevelopment of the flow is not of interest. This is implemented by specifying a constantvalue for the Courant number through the mesh and evaluating the value of the time stepfor each node by using the relation given by equation (28). For true transient simulations,the minimum local time step, min(∆tI ) ∀ I, over the mesh is adopted.

3.5 Implementation of the Boundary Conditions3.5.1 Far field

For a node I located at the far field boundary, the flux Fn, is determined by employing

Roe’s [118] approximate Riemann solver to resolve the interface between the computedvalue U I and the free stream value U∞. This means that

FnI =

12F n(U I) + F n(U∞) − |An(U I ,U∞)| (U∞ − U I). (30)


where the Roe matrix An(U I ,U∞) is evaluated here in the direction normal to the bound-ary. For a node I located at the far field boundary, the flux G

n, is determined by employingthe free stream value at inflow, i.e. G

n = Gn∞, and the computed value at outflow, i.e.

Gn = Gn.

3.5.2 Solid wall

At a solid wall, the flux Fn is set equal to F n for inviscid flow computations. The normal

velocity at each node on the solid wall is set to zero after each stage, k, of the time steppingscheme, so that

u(k) · n = 0 at ΓW (31)

In general, this condition is not consistent with the initial free stream condition and anadditional computational problem might be encountered when attempting to simulate highspeed flows over obstacles. In fact, truly impulsive start of any mechanical system is notphysically possible owing to inertia and it is even mathematically inconsistent for incom-pressible fluid or solid mechanics [36]. Rapid start, or impulsive acceleration is quite legit-imate and can be implemented by imposing the free stream condition fully after a certainsmall time interval. Alternatively, one can impose the free stream condition directly, butrelax the imposition of the solid wall boundary condition. This is accomplished here byreplacing the condition of equation (31) by

u(k) · n = [u(k−1) · n](1 − κ) at ΓW (32)

The introduction of the parameter κ means that the solution is allowed to penetrate thewall at the start of the transient but, as time evolves, the normal velocity at the wall goesto zero. This procedure has been found to be very useful for the simulation of high speedflow past blunt bodies, where the value κ = 0.8 has been typically used [79]. For viscousflow computations, the velocity components are set to zero at a solid wall. However, therelaxation approach described in equation (32) is now employed for both the normal andtangential velocity components when simulating high speed flows past blunt bodies. Inaddition, isothermal or adiabatic conditions are set. At a point on an isothermal wall, thetemperature, or energy, is set to the prescribed value. At an adiabatic wall, the conditionof zero temperature gradient is weakly imposed through the discretization of the energyequation.

4 HIGH RESOLUTION SCHEMES FOR 2D PROBLEMS

Although a number of multidimensional upwind based schemes for the compressible Eulerequations have been developed over the last decade [19, 119, 123, 141], more research isrequired before a truly universal multidimensional upwind scheme can be identified. Theavailable theory is complicated and the implementation for practical applications is cur-rently expensive. For an arbitrary discretization, the design of multidimensional schemes isnormally accomplished by considering the propagation of information locally in the direc-tion normal to the cell faces. Here, with the use of the edge based data structure describedin section 3.2, we assume that the wave moves in the direction of the weight coefficient vec-tor CIIS

, which is the direction of the centered, or Galerkin, numerical flux F IIS. Despite

the mesh dependence of this approach, the errors involved reduce with the increase in theorder of the scheme, with the high resolution schemes based upon this approach workingsatisfactorily. This has already been confirmed by many researchers and will be demon-strated for a number of 2D applications. Part II [92] should be consulted for the backgroundof the LED condition for 1D problems and for a definition of the adopted notation.


4.1 Conditions to Ensure the LED Property in Multidimensions

The hope of obtaining multidimensional high order TVD schemes through the direct exten-sion of the 1D TVD requirements is dashed by the theorem of Goodman and LeVeque [35],which states that, except in certain trivial cases, any method that is TVD in two spacedimensions is at most first order accurate. However, for the scalar case the LED propertycan still be proved [52]. Following the introduction of the lumped mass matrix C and con-sidering an interior node so that Df ≡ 0), the scalar counterpart of the discrete formulationof equation (18) can be rewritten as

[ML

du

dt

]I

= −mI∑S=1

2∑j=1

CjIIS

(F jIS

− F jI ) (33)

where the property of equation (20) has been used. As already mentioned, we assume alocally one dimensional propagation of information in the direction SIIS

of the weightingcoefficient. If the wave speed aSIIS

is approximated as

aSIIS=

2∑j=1

(F jIS

− F jI )

uIS− uI

SjIIS

if uI+1 = uI

2∑j=1

∂F j

∂u

∣∣∣∣ISj

IISif uI+1 = uI

(34)

then the semi discrete scheme of equation (33) can be written in the equivalent form[ML

du

dt

]I

= −mI∑S=1

LIIS

[aSIIS

∆uIIS

](35)

Alternatively, this scheme can be expressed as[ML

du

dt

]I

=∑k

BkI(uk − uI) (36)

where BkI is a coefficient which depends on a compact stencil of points and is such that

BkI ≥ 0 ∀ k (37)

This condition is sufficient to guarantee that a local maximum cannot increase and a localminimum cannot decrease, so the scheme is LED [52, 92].

The scheme of equation (35) does not satisfy the positivity condition of equation (37)which is required for an LED scheme whenever aSIIS

> 0. Adding a generic dissipative term,defined using the dissipation coefficient αS

IIS, we produce

[ML

du

dt

]I

=mI∑S=1

LIIS

[(αS

IIS− aSIIS

)∆uIIS

](38)

together with an analogous expression for node IS. The requirement of equation (37) thenimplies that the condition

αSIIS

≥ |aSIIS| (39)


α L

βLα R

βR

ιSII

IIL

IS

IR

Figure 2. The edge IIS and the surrounding triangles

must hold if the scheme expressed in equation (38) it to be LED. However, this schemeis at most first order accurate and we should seek a limited higher order scheme followingthe procedure used in 1D [79, 92]. A possible dissipative high order numerical flux can bewritten as

FIIS= LIIS

αSIIS

[∆uIIS

− L(2)(∆u+IIS

, ∆u−IIS

)]

(40)

where the differences ∆u±IIS

are defined as [52]

∆u+IIS

= ∇+u · lIIS; ∆u−

IIS= ∇−u · lIIS

(41)

Here, lIISis the vector connecting the nodes I and IS and ∇±u are gradients of u evalu-

ated in the triangles into which and out of which lIISpoints. These triangles have nodes

(I, βL, αL) and (IS, βR, αR), as illustrated in Figure 2. These reconstructions based onthe adjacent triangles are not unique and further considerations and alternatives will bediscussed in the next section.

It is convenient to introduce the ratio of differences

r+IIS

=∆u+

IIS

∆uIIS

; r−IIS=

∆u−IIS

∆uIIS

; rIIS=

r+IIS

r−IIS

=∆u+

IIS

∆u−IIS

(42)

The semi discrete scheme of equation (35), using equation (40) as the dissipative term, maythen be expressed as[

MLdu

dt

]I

=mI∑S=1

LIIS

[(αS

IIS− aSIIS

)∆uIIS− αS

IISΦ(1)(rIIS

)∆u−IIS

](43)

where the one parameter function Φ(1) has been used to replace the notation for a limitedaverage function L(2) [92]. The difference ∆u−

IIScan be generically represented as

∆u−IIS

= εIαL(uI − uαL

) + εIβL(uI − uβL

) (44)


where the adjacent triangle, represented in Figure 2, is used to determine the desiredvariation. Inserting equation (44) into equation (43), we obtain

[ML

du

dt

]I

=mI∑S=1

LIIS

(αS

IIS− aSIIS

)(uIS− uI)

+ αSIIS

Φ(1)(rIIS) [εIαL

(uαL− uI) + εIβL

(uβL− uI)]

(45)

which is in the general form of equation (36). Apart from the positivity condition ofequation (39) for the lower order scheme and the non negative range of the limiter functions,0 ≤ Φ(1)(r) ≤ min(2, 2r) [92, 134], the coefficients εIαL

and εIβLmust be non negative in

order to guarantee the LED property of the scheme.One possible way to determine ∆u−

IISconsists in defining a fictitious node IL as shown

in Figure 2. Then, using linear interpolation or extrapolation of the values of u at thevertices αL, I and βL, we can compute uIL

. The linear interpolation is represented usingthe triangle shape functions, but without considering the compact support normally usedin the finite element method, i.e.

uIL= NαL

uαL+ NIuI + NβL

uβLwith NαL

+ NI + NβL= 1 (46)

The second equation here is required for consistency with a constant field u. Combiningthese expressions, we can write

uIL= NαL

uαL+ [1 − (NαL

+ NβL)]uI + NβL

uβL(47)

and it follows that the variation ∆u−IIS

can be approximated as

∆u−IIS

= ∆uILI = uI − uIL= NαL

(uI − uαL) + NβL

(uI − uβL) (48)

Comparing equations (44) and (48), we see that εIαL= NαL

and εIβL= NβL

. The linearinterpolation functions NαL

, NβLare, by definition, always larger than, or equal to, zero if IL

belongs to the dashed area to the left , including the boundaries, in Figure 2. The positivitycondition of equation (37) is then assured for the scheme expressed in equation (45) and soalso is the LED property.

Similar arguments can be used, and the same conclusions reached, if we consider thenode IS instead of I. Furthermore, any of the various one dimensional higher order schemesderived in Part II [92] can be adapted to the unstructured 2D scalar formulation in themanner employed for the scheme defined by equation (40). The extension to systems ofconservation laws may be accomplished by following one of the possible upwind general-izations presented in Part I [91] and Part II [92]. As for the non linear equation system in1D, no theoretical guarantee that the resultant scheme will be LED exists. Despite the adhoc nature of the multidimensional extension, and of the extension to systems of non linearequations, the numerical schemes designed with these assumptions remain second order ac-curate for smooth solutions and, in general, give non oscillatory and sharp solutions. Thiscan be observed in the large amount of successful applications performed by many CFDpractitioners and by the results presented here.

4.2 Reconstruction Step

In multidimensions, if only surrounding edges are to be used to construct the scheme, theresult will be at most first order accurate. Thus, the use of a larger support is necessary


when building higher order schemes. For limited schemes, this is required by the intro-duction of the limiting procedure and by the MUSCL reconstruction. For the switchedartificial viscosity schemes, a larger stencil is needed for the computation of the higher or-der background diffusive term and, also, to determine the discontinuity sensor or switch. Avariety of strategies are available for the calculation of the backward and forward differences∆U±

IISon multidimensional unstructured discretizations [6, 10, 52, 94, 114, 120, 144]. These

strategies are in general based on gradient reconstruction and/or interpolation techniques.

4.2.1 The use of gradient reconstruction

When linear shape functions are adopted in the finite element method, the element gradientsare piecewise constant. One possible way to compute a continuous nodal value for thegradient of the solution, ∇U I , is to use the least squares reconstruction of equation (25).Using Figure 2, the differences ∆U±

IIScan be determined if, for instance, we take

∇U+ = ∇U IS; ∇U− = ∇U I (49)

and substitute in equation (41). Alternatively, one can adopt a central difference approxi-mation for the gradients, resulting in the expressions

∆U+IIS

= 2∇U+ · lIIS− ∆U IIS

; ∆U−IIS

= 2∇U− · lIIS− ∆U IIS

(50)

The use of these gradient reconstructions for the determination of ∆U±IIS

is very attractive,as it requires little extra memory for the computation. However, the alternatives representedby equations (25) and (49) or (25) and (50) lead to some small oscillations close to shocks, oreven instability when they are used in the simulation of certain hypersonic flow applications.This lack of robustness [79, 97] has also been found in the study by Cabello et al [16].Different alternatives are possible [1, 6, 10] and might be considered in a future work,which searches for a more robust gradient reconstruction procedure.

4.2.2 The use of a linear interpolation

The procedure discussed when proving the non negativity of the coefficients εIαLand εIβL

in section 4.1 constitutes another option to obtain ∆U±IIS

. Firstly, we should introduce thedummy nodes IL and IR shown in Figure 3. For edge S, the dummy nodes IL and IR arelocated equidistantly along the line which contains the nodes I and IS.

Basically, two choices can be used to obtain the linearly interpolated values at thedummy nodes. The first option uses the nodal points that belong to the adjacent trianglesto perform the interpolation or extrapolation. For instance, the triangle αL, I, βL in Figure3 is adopted to compute the value U IL

. The second option considers the actual triangles inwhich the dummy nodes lie. In this case, the triangle αL, βL, γL of Figure 3 is adopted tocompute the value U IL

. Both options prove to work for the many problems which have beenanalyzed, with roughly the same final results. The use of extrapolation is frequently trou-blesome and bad behavior has been experienced for some hypersonic flow simulations whenthe adjacent triangle alternative was considered. If the situation represented in Figure 4occurs, oscillations might remain in the computed solution or instabilities can develop, asthe wrong information is used to obtain the variation ∆U ISIR

. For this configuration, theuse of the adjacent triangle will result in the addition of no diffusion after the limiting pro-cedure is applied. The use of the actual triangle which contains the dummy node IR mightadd enough dissipation to stabilize and damp possible oscillations at node IS . Actually,this prove to be true for most inviscid and viscous applications analyzed [87, 93, 94, 95, 97].As a result, the use of the correct triangle to obtain the information for the interpolation isadopted here. Despite the good numerical performance, we can no longer prove the LED


α L

βLα R

βR

γL

γR

S

IIL

ISIR

Figure 3. Location of the dummy nodes for edge S and determination of trianglesto be used for the linear interpolation

adjacent triangle actual

triangle

shockfront

IIS

IL

IR

Figure 4. Sketch of a possible shock configuration on a triangular grid

property, as it is not possible to write the scheme in the general form given in equation (36).However, robustness and numerical evidence favor the use of this option.

When a dummy node falls outside the computational domain, different forms of extrap-olation may be used. When one node of an edge falls on the boundary, at least one of thedummy nodes will fall outside the computational domain, as illustrated in Figure 5(a). Inthis case, either a constant or a linear extrapolation, using the values at the actual nodes ofthe edge, is adopted to obtain the desired values at the dummy nodes. Despite the local firstorder nature of the scheme, when a constant extrapolation is adopted, the overall accuracyof the scheme was not deteriorated for the numerical examples investigated. This choice isnormally preferred, as it proves to be more robust. The linear extrapolation procedure wasfound to lead to oscillations in certain instances. Even when a node does not lie on the


boundary, we may still have, in special situations, a dummy node outside the domain, e.g.see Figure 5(b). These special dummy nodes might occur frequently for viscous meshes,but can be reduced by proper mesh generation. For such exterior nodes, we adopt eitherconstant or linear extrapolation, or consider the triangle adjacent to the edge for the ex-trapolation. Other possibilities for dealing with such nodes can be devised and are worthyof careful investigation. Alternatively, higher order reconstruction, such as quadratic recon-struction based upon gradient and Hessian recovery [11, 20, 149], could also be attemptedto reduce the mesh dependency problem that results when using 1D upwind based schemes.These reconstructions will require extra computation and any improvements produced inthe solution must justify this extra overhead.

(a)

actual nodes

dummy nodes

(b)

actual nodes

dummy nodes

adjacenttriangle

Figure 5. Dummy nodes outside the computational domain. (a) edges sharing anode with the boundary; (b) edges without a node on the boundary

4.3 Unstructured Grid Solution Algorithms

The semi discrete solution algorithm of equation (22) for the Euler equations, with a lumpedmass matrix, may be written in the form

[ML

dU

dt

]I

= −mI∑S=1

FSIIS

+ 〈2∑

f=1

Df (4FnI + 2F

nJf

+ F nI − F n

Jf)〉I (51)

where F IISdenotes the inviscid numerical flux function for side IIS, which is computed

in the direction SIISof the weight coefficient vector, as shown in equation (21). The

superscript S denotes that the quantity is computed in the direction of the weight coefficientvector. Similarly, for the solution of problems governed by the Navier Stokes equations, thesemi discrete solution algorithm, corresponding to equation (26), may be expressed as[

MdU

dt

]I

= −mI∑S=1

FSIIS

+mI∑S=1

2∑j=1

CjIIS

(GjI + Gj

IS)

+ 〈2∑

f=1

Df

(4F

nI + 2F

nJf

+ F nI − F n

Jf)

−(4GnI + 2G

nJf

+ GnI − Gn

Jf)〉I

(52)


When the differences ∆U ILIS, ∆U IIS

and ∆U ISIRhave been computed, all high resolu-

tion schemes presented in Part II [92], either using a switched or a limited approach, canbe directly generalized for multidimensional solution of the Euler equations using equa-tion (51) or the Navier Stokes equations using equation (52). To clarify matters, we presenta summary of the inviscid numerical fluxes FS

IISthat are obtained when these schemes are

extended for use with 2D unstructured triangular discretizations.

4.3.1 Switched artificial viscosity approaches

The switched artificial viscosity solution algorithm for multidimensional simulations, usingequations (51) or (52), employs the numerical flux function

FSIIS

=2∑

j=1

CjIIS

(F jI + F j

IS) − LIIS

αSIIS

[ε(2)IIS

∆U IIS+ ε

(4)IIS

(∆U IIS

−(

hdU

dl

)IIS

)](53)

where the parameters ε(2)IIS

, ε(4)IIS

have the definitions presented in Part II [92]. The valuesfor the edge gradients are obtained as the average of the nodal values and are given by

dU

dl

∣∣∣∣IIS

=12

∂U

∂xj

∣∣∣∣∣I

+∂U

∂xj

∣∣∣∣∣IS

dxj

dl(54)

where lIISgives the local coordinate at edge IIS , as indicated in Figure 2. The pressure

switch is computed by

ΥI =|pIS

− 2pI + pIL|

(1 − θ)(|pIS− pI | + |pI − pIL

|) + θ(pIS+ 2pI + pIL

) + ε(55)

with the dummy value of the pressure pILdetermined by using one of the procedures

discussed in the last section. For the scaling factor, we can use the scalar coefficient αSIIS

=LIIS

(λmax)IIS, with (λmax)IIS

computed by equation (29). Alternatively, we can use amatrix coefficient αS

IIS= |LIIS

AS(U I , U IS)|, where the standard Roe matrix, between the

states U I and U IS, is evaluated in the direction of CIIS

, i.e.

ASIIS

= AS(U IIS) =

2∑j=1

ddU

F jSj

IIS

∣∣∣∣IIS

=2∑

j=1

SjIIS

AjIIS

(56)

The CUSP flux splitting artificial viscosity scheme [79, 92] is readily extended for 2D ap-plications in a similar fashion, but this has not been attempted here.

4.3.2 Algebraic approaches

The schemes discussed in section 4.1 are directly generalized with the inviscid numerical fluxfunctions now computed in the direction of CIIS

. Employing the variables and parametersdefined for the 1D counterparts, see section 4.1 and Part II [92] for details, we can producethe following algorithms:Taylor–Galerkin LED Schemes [79, 94, 95]

FSIIS

=2∑

j=1

CjIIS

(F jI + F j

IS)

− LIISRS

IIS[λ∗(ΛS

IIS)2∆W

SIIS

+ |ΛSIIS

|(∆W SIIS

− ∆WSIIS

)]

(57)


where λ∗ = ∆tS/LS, LS is the length of the edge S and ∆tS = (∆I + ∆IS)/2 denotes the

local time step for the edge S.Second Order Limited Positive Schemes [66, 67, 79]

FSIIS

=2∑

j=1

CjIIS

(F jI + F j

IS)

− LIISRS

IIS|ΛS

IIS|[∆W S

IIS− sign(AS

IIS)min

(|AS

IIS|, βIIS

BSIIS

)]

(58)

Galerkin/Osher LED Schemes [79, 90]

FSIIS

=2∑

j=1

CjIIS

(F jI + F j

IS) − LIIS

[∫ U IS

U I

|AS(U)| dU − ∆FSIIS

] (59)

where LIIS∆F

SIIS

denotes the limited flux variation defined by equation (67) in Part II [92].

4.3.3 Geometric approaches (MUSCL)

A straightforward extension of the 1D MUSCL schemes discussed in Part II [92] can beaccomplished for any of the first order upwind schemes summarized in Part I [92]. This isachieved by considering the limited interface values UL, UR and the direction CIIS

.Roe/MUSCL Schemes [79, 87, 97]

FSIIS

=2∑

j=1

CjIIS

[F j(UL) + F j(UR)

]−LIIS

|AS(UL, UR)| (UR − UL) (60)

Osher/MUSCL Schemes [79, 90]

FSIIS

=2∑

j=1

[F j(UL)Sj

IIS+ F j(UR)Sj

IIS

]−LIIS

[∫ UR

UL

|AS(U)|dU (61)

Liou–Steffen AUSM/MUSCL Schemes [79, 93]

F IIS= LIIS

MS

L/R[cLΘL + cRΘR]

−|MSL/R|[cRΘR − cLΘL] + 2pL/R [S1

L/R∆1 + S2L/R∆2]

(62)

where cLΘL, cRΘR are evaluated using the limited interface values UL and UR respectively.The interface Mach number MS

L/R and pressure pL/R are computed using the equationspresented in section 4.4 of Part I [91], but using now the limited interface values of the statevariables. A detailed description of the unstructured mesh implementation of the AUSMflux splitting scheme, which was originally presented in a structured mesh context [70], canbe found in Lyra et al [93].

5 OTHER IMPORTANT ISSUES

It is appropriate to make some remarks about practical issues related to the implementationof these schemes. The importance of the strategies to be discussed is such that they can beresponsible for the success or failure of the analysis. This is particularly true for hypersonicflow simulation.


5.1 Enhancement of Stability and Convergence Rate

Certain parameters and aspects of the methods analyzed can influence the stability andconvergence rate of the computation. Some of these parameters have little or negligibleeffect for low Mach number simulation. However, they can drastically affect the robustnessof the scheme when high speed flow simulation is attempted.

5.1.1 Positivity of thermodynamic variables

Negative pressure and density can appear during the time integration path, because ofthe presence of large gradients or quasi rarefaction zones or the impulsive start from freestream conditions. To prevent the appearance of local spurious negative values of the ther-modynamic variables and p during the convergence process, the pressure and density areupdated using a relaxation process which is such that they always remain positive ([138]).In this process, the pressure is updated as

pn+1 = pn + ∆p[1 + η(α + |∆p

pn|)]−1 (63)

whenever ∆p/pn≤α, where η = 2 and α = −0.2. The use of this form of update for thepressure, and a similar procedure for the density, leads to an increase in the robustnessfor calculations at high Mach numbers and during the initial transient stages of a calcu-lation following an impulsive start from free stream. When the Venkatakrishnan/Thomaslimiter [92] is adopted with MUSCL schemes, the limited pressure interface value, which isnormally computed as

pLIIS

= pI +12L(2)(pIS

− pI , pI − pIL) (64)

is, instead, computed using

pLIIS

= pI + pIL(2)

(pIS

− pI

pIS+ pI

,pI − pIL

pI + pIL

)(65)

A similar modification could be used with other limiters, but this has not been attemptedin this work. In addition, for some particular cases, normally at the start of the tran-sient, negative values of the square of the sound speed appear when solving the Riemannproblem for the dummy edges. A possible explanation for this behavior could be the ex-tra approximation involved in the computation of the dummy node values of the variablesand the possibility for creating non physical states at the downwind dummy node. Tocircumvent such a problem, we eliminate these particular dummy edges from the limitingprocedure. This proves to be of particular importance when a symmetric stencil of pointsis used as support for the limiting procedure. Finally, it is well known that negative valuesfor the square of the average sound speed, c2

IIS, may be produced when Roe’s scheme is

adopted [26, 27, 38], as the average values may lie outside the interval between c2I and

c2IS

. The fix proposed by Einfeldt et al [27] might be necessary and has been found to beparticularly important for equilibrium real gas simulation by Yee [146].

5.1.2 Background diffusion

The use of the central difference, or Galerkin, type second order scheme when building thehigh resolution flux function leads to a procedure which has no background diffusion in thesmooth areas of the flow. On the contrary, the Taylor Galerkin LED schemes or the SLIP(2)

schemes [92] incorporate a form of background dissipation, which proves to be important


in assisting the damping of high frequency modes in the solution. This results in improvedconvergence rates in steady state simulations. Of course, such additional diffusion has thedrawback of smearing discontinuities and should be well controlled. When the SLIP(2)

approach is adopted, it has been found [79] that the parameter value β = 0.6 gives goodconvergence behavior without damaging the solution accuracy.

5.1.3 Freezing the non linear artificial diffusion

The lack of background dissipation and the non linear nature of the limiting procedure arebelieved to be two of the main reasons for the poor convergence rate observed for certainapplications. It seems that the limiter reacts to small scale oscillations in smooth regionsand introduces too much non linearity. It has been suggested that freezing the non linearnumerical diffusion when the solution approaches steady state can help to drop the residualtowards machine zero and this has been confirmed in numerical experiments [79, 95]. Forthe flux limited approach, fully described in section 4 of Part II [92], the freezing strategy isimplemented by preventing the update of the limited corrective flux FC

IISterm. No signifi-

cant differences were found in the solutions produced with and without the implementationof the freezing strategy. For the inviscid computations included here, the freezing strategywas normally imposed when the L2 norm of the density residual had dropped by three or-ders of magnitude. No limiter freezing was employed for the viscous computations reportedhere.

5.2 Improvement of Accuracy5.2.1 Adaptivity

A major attraction of unstructured grid methodologies is the possibility of efficiently im-plementing adaptive mesh strategies. Adaptivity reduces vulnerability to human errorsand assures more confidence in the numerical results and this is extremely important forhigh risk applications. The unstructured triangulations used for the 2D computations wereobtained by making use of the advancing front technique of Peraire et al [117]. An adaptivemesh enrichment procedure was used to improve the accuracy of computed steady statesolutions. The error estimates that are employed are based upon concepts from interpo-lation theory. In addition, to improve the efficiency of the refined mesh for the resolutionof directional features, such as shocks, the procedure adopted also provides a directionalindication of the error. The new spatial distribution of the grid spacing in the directionof each edge is determined by using an average of the values of the directional error indi-cators at the two nodes of the edge. As a result, new nodes and elements are introducedfor each side for which the calculated error exceeds a certain proportion of the maximumerror [76, 105]. It should be noted that the use of an edge based error indication and re-finement strategy is also advantageous as it leads to better grid alignment with the shocks,which assists the schemes proposed here, with their local 1D basis. Alternatively, adap-tive remeshing can be used using isotropic error analysis [79, 117, 18] or directional errorprocedures [3, 5, 83, 84, 109].

5.3 The Computational Implementation

The initial grids and the refined grids are defined according to the conventional elementbased data structure. This means that a preprocessing of the grid must be undertaken beforeit can be used with an edge based flow analysis algorithm. After the preprocessing stage,the element based data structure is discarded. The preprocessor stage consists basically ofthe following steps: (a) arrays defining the grid and boundary topology are constructed;these take the form of lists of edges and boundary faces with their respective connectivities;(b) the weight coefficients for the edges and the boundary faces are computed and stored;


(c) the information necessary for the use of the dummy nodes is determined and stored;(d) a colouring algorithm [30, 109] is employed to group the edges and boundary faces insuch a way that no repetition in the node numbering occurs amongst items of the samegroup. It should be noted that the information required to describe an unstructured meshis minimal when an edge based data structure is employed. The three nodes of the trianglethat contains the dummy node, and two shape functions evaluated at the dummy node forthe interpolation step, are kept in memory for each of the two dummy nodes that belong toeach side. This procedure represents a memory overhead of ten times the total number ofsides in a 2D computation. The alternating digital tree [14, 109] algorithm is adopted forthe searching operations required. When using the gradient reconstruction techniques, thedummy node information is not necessary. The colouring algorithm is adopted to preventrecurrence inside the loops in the algorithmic process and, therefore, allows implementationon vector and parallel machines [79, 96]. The operations performed inside the loops overthe edges and boundary faces, which take place in the edge based algorithm are: gatherinformation from the nodes of each edge; operate on this information; scatter it back tothe nodes of the edges and add it to the nodal quantities. These typical loops are entirelyvectorizable provided each group of edges, or boundary faces, is executed separately and acompiler directive instructing the computer is inserted before the vectorizable inner loop.The inner loops can also be distributed to multiple processors using the autotasking facilitiesavailable in certain shared memory computers or by inserting a compiler directive to forcethe tasking of the loop [96].

5.4 Wall Coefficients

The wall distribution of quantities such as the pressure coefficient, the heat transfer andthe skin friction are normally required in Navier Stokes simulations. The wall pressure, pw,is directly obtained using the flow variables and the equation of state (4). The heat flux atthe wall, qw, and the wall shear stress. τw, are obtained as [39]

qw =∂T

∂n= ∇T · n (66)

andτw = t2n1 − t1n2 (67)

Here, t1 and t2 denote the surface tractions, which are evaluated as

t1 = τ11n1 + τ12n2

t2 = τ21n1 + τ22n2

(68)

using the shear stress tensor components τij defined in equation (3). Making use of equa-tion (25) for the calculation of the gradients involved in the computation of equations (66)and (67), we can determine directly the heat transfer and the skin friction coefficient.

5.5 Further Considerations

Other factors which can affect stability, convergence rate, accuracy and efficiency, mainlyin high Mach number cases are now mentioned.

5.5.1 The variables limited

It was mentioned in Part II [92] that the implementation of the MUSCL formulations canbe extended, for systems of equations, by applying the limiting to the primitive, the con-servative or the characteristic variables. In a preliminary study, the use of the primitive


variables led to improved behavior compared with the use of the conservative variables orwith the mixing of primitive and conservative variables. It was observed that, dependingon the limiter adopted, the choice of the primitive variables produced good oscillation freesolutions. As the MUSCL approach requires extra computation during the reconstructionstage, the use of the characteristic variables during the limiting procedure will make iteven more expensive. This fact, and the numerical evidence, supports the choice of lim-iting the primitive variables that is adopted in this work. However, Yee et al [146, 147]report that the choice of characteristic variables plays an important role in the stabilityand convergence rate as the Mach number increases, so that the use of primitive variablesmight lead to failure in certain future applications. It was found to be important to applythe limiting procedure in the weighting coefficient direction and in the direction normal tothis coefficient, when the primitive or conservative variables are limited. The velocities, ormomenta, are projected onto the weighting coefficient direction and the normal direction tothis coefficient and then the limiter is applied, followed by another projection back to the x1

and x2 directions. For the algebraic limited schemes, the adoption of any set of variables,other than the characteristic variables, leads to solutions which contain oscillations, evenfor the 1D subsonic shock tube application.

5.5.2 Entropy parameter

The Roe approximate Riemann solver requires the implementation of an entropy fix [91].The value employed for the free parameter involved in this fix, and/or the different al-ternatives for scaling this fix, affect the stability and convergence rate and can influencethe accuracy of the results. In most of the computations presented here, the parameter δ1

of the Harten and Hyman entropy fix [37] is set to 0.1, with satisfactory behavior of thescheme [91]. For hypersonic blunt body calculations, Yee et al [146, 147] and Muller etal [108] claim that the use of a more elaborate entropy fix is required.

5.5.3 Implicit formulation

Since the correct modelling of the transient development of the flow is not the main concernhere, the use of an implicit formulation for the time discretization might be very attractive.Due to the scale of the problems normally faced in CFD, iterative solution approaches arethen essential. An approach that is well suited for use with unstructured meshes is the linerelaxation technique proposed by Hassan et al [39, 41]. This allows the direct incorporationof implicit formulations for the upwind schemes that have been considered, though this hasonly been attempted for 1D problems [79, 91]. Other possibilities include the use of moreadvanced iterative approaches, such as GMRES [124, 125, 135].

5.5.4 Viscous grid generation

The development of unstructured mesh generation and adaptivity procedures for viscousflow simulations is not addressed in detail, but it is mentioned because of its importance.Meshes suitable for viscous flow simulations may be obtained by generating a structuredgrid in the immediate vicinity of solid surfaces and completing the discretization of thecomputational domain by using the unstructured mesh approach. This approach has beensuccessfully utilized for many realistic viscous flow simulations [39, 112]. However, themethod is not very flexible for the inclusion of adaptivity or for the extension to generalcomplex 3D configurations. As an alternative, fully unstructured meshes may be generatedusing the advancing layers technique [40, 41] and the extension of these ideas to deal withhybrid unstructured meshes may also be contemplated [82, 86, 130].


6 NUMERICAL APPLICATIONS

In this section, the numerical solution of some inviscid and viscous compressible flow prob-lems is considered, with the objective of comparing the numerical performance of the algo-rithms that have been described. Most of the applications involve supersonic or hypersonicflows and these represent severe challenges. The results presented were obtained duringthe development of this research [79, 87, 88, 89, 90, 93, 94, 95] and so may represent thetypical performance that can be expected from a scheme rather than the best possible. Theviscous flow examples involve simple geometries, which means that structured triangularmeshes can be adopted. This also allows for direct comparison with the results produced bystructured mesh flow solvers on closely related grids [121, 122]. Available theoretical andexperimental data is also used. Preliminary simulations using fully unstructured grids andincorporating adaptive remeshing have already been undertaken [86] and more complex 2Dand 3D applications using these schemes have been addressed [104, 105, 115, 116].

6.1 Inviscid Flows6.1.1 Subsonic shock tube problem

The time dependent simulation of the subsonic shock tube problem is adopted as the firstapplication. This problem is described and extensively analysed in Parts I and II [91,92]. A 2D simulation of this 1D problem was performed using the G/LED and limitedMUSCL approaches with Roe’s approximate Riemann solver. The mesh employed is shownin Figure 6.

Figure 6. Mesh used for a 2D simulation of the shock tube problem:subsonic regime

To allow for comparison with the 1D results presented in Part II [92], the same mixtureof limiters is used for the G/LED scheme, i.e. upwind biased superbee for the linear fieldand symmetric MUSCL for the non linear fields, with the van Albada limiter adopted forthe MUSCL scheme. It should be noted that the results obtained with the MUSCL schemewere produced by limiting the primitive variables and not the characteristic variables usedfor the corresponding 1D simulation. This is for consistency with the remainder of theapplications considered here.

The computed density distributions are shown in Figure 7 and it can be observed thatthe corners of the expansion fan are slightly more rounded in the 2D results. Apart from this,the results are very similar and demonstrate the excellent performance of these unstructuredhigh resolution algorithms for this problem.

6.1.2 Regular shock reflection at a flat plate

In this example, the flow impinging on a flat plate at a Mach number of 2.0 and at angle ofattack of either −100, flow condition A, or +1900, flow condition B, is considered. ProblemsA and B have analytical solutions consisting of two regions of uniform flow separated by astationary shock at an angle of 29.30 and 209.30 respectively. A regular triangular mesh with800 elements and 441 nodes is used to discretize the rectangular computational domain. Thevalue 0.5 is adopted for the CFL number. The mesh was constructed so as to provide almostperfect grid alignment with the shock for flow condition A and skewed, almost orthogonal,grid alignment for flow condition B. It is well known that the solutions produced in 2D,by schemes based upon the extension of standard 1D upwind methods, frequently exhibit


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(b)

0

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-0.6 -0.4 -0.2 0 0.2 0.4 0.6

Figure 7. Density distributions for the shock tube problem: subsonic regime com-puted using Roe’s approximate Riemann solver. (a) G/LED with mixedlimiters and (b) limited MUSCL scheme with the van Albada limiter

(a)

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FLOW "A"FLOW "B"

ρ

x2

Figure 8. Density profiles at the centre of the computational domain for the pro-blem of regular shock reflection on a flat plate using Roe’s approximateRiemann solver. (a) first order upwind scheme; (b) G/LED MUSCLscheme with the limiter computed using an upwind biased stencil

strong mesh dependence [79, 95]. The loss of accuracy is much smaller when high resolutionschemes are used and this can be appreciated by comparing the computed density profiles forthis problem, shown in Figure 8(b), with the corresponding profiles, shown in Figure 8(a),obtained with a first order upwind scheme.

This simple test case allows us to gain an insight into the performance of the differenthigh resolution schemes for two dimensional simulations. Adopting the flow condition withangle of attack equal to −100, the results computed using schemes from all three classes ofhigh resolution approaches discussed in this work are presented in Figure 9. The algebraicapproaches use the MUSCL limiter computed using an upwind biased stencil, while thegeometric approaches use the Venkatakrishnan/Thomas limiter with the modifications fordensity and pressure given in equation (65). The parameter β for the USLIP(2) schemewas set to 0.6 for improved convergence behavior, but this led to results that were slightlymore smeared than those of the other algebraic schemes. The performance of all fourdifferent Riemann solvers together with the limited MUSCL approach is very similar. Any


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AV-SCALARAV-MATRIX

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MUSCL-ROE MUSCL-OSHERMUSCL-AUSM MUSCL-WPS

2x

ρ

Figure 9. Density profiles at the centre of the computational domain for the problem ofregular shock reflection on a flat plate obtained using different high resolutionschemes. (a) switched artificial viscosity approaches; (b) algebraic approaches;(c) geometric approaches

of these algebraic or geometric schemes achieves a good performance, with the number oftransition points varying from two to four for this particular problem configuration. Forboth switched artificial viscosity schemes, the weighting coefficient θ in the pressure switchis set to zero and the parameter µ(4) is set to 0.1. The parameter µ(2) was assigned thevalues 0.8 and 1.6 for the scalar and matrix scaled schemes respectively. No attempt hasbeen made to improve the solution quality by varying these parameter values. The resultsshown in Figure 9(a) are typical and demonstrate the improved accuracy obtained with theuse of a matrix scaling factor, producing a solution which is similar to that obtained withthe other high resolution approaches.

Previous studies [79] have shown that the MUSCL scheme using the Roe approximateRiemann solver and van Albada limiter, produces results with a small ripple ahead of theshock when the limiting is applied to either the primitive or the conservative variables.A monotonic solution was obtained when the characteristic variables were limited, butthe convergence behavior was worst with this option. The convergence history, of the L2

norm of the density residual, R, for the computations using the first order Roe upwindscheme, the G/LED scheme and the TG/LED scheme are presented in Figure 10(a), wherea logarithmic scaling is employed on the vertical axis. The convergence behavior was foundto depend strongly on the choice of the limiter. The results show that the background


(a)

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1st-UPW ==>

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<== G-LED (freezing)

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(b)

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Non-freezingFreezing

ρ

x2

Figure 10. Convergence behaviour of schemes used to solve the problem of regular shockreflection at a flat plate, with the MUSCL limiter computed using an upwindbiased stencil. (a) variation of the L2 norm of the density residuals; (b) solutionsat the centre of the computational domain obtained using the G/LED schemewith and without limiter freezing

dissipation present in the TG/LED scheme damps some high frequency modes, helpingthe convergence behavior of the algorithm, but it is not always enough to damp the nonlinear effects introduced by the limiting procedure. This means that, as with the G/LEDalgorithm, the residual does not drop beyond a certain level. Freezing the limiter valueswhen the solution approaches steady state ensures that the residual eventually reachesmachine zero. This effect is illustrated in Figure 10(a) for the case in which freezing ofthe limiters was instigated when the L2 norm of the residual of the conservative variablesdropped by three orders of magnitude, with the G/LED scheme. The solution valuesobtained with and without limiter freezing are seen to be indistinguishable in the plot ofFigure 10(b). It should also be noted that lowering the time step also reduced the nonlinear effects and the residual goes to machine zero for the G/LED scheme without anyfreezing of the limiters. However, this is achieved at the expense of slowing considerablythe convergence rate towards steady state and this is, in general, not desirable.

6.1.3 Supersonic flow past a circular cylinder

The third example consists of a steady flow past a circular cylinder, at a free streamMach number of 3 [79, 95]. A minmod symmetric limiter was adopted in this analysis.The presence of sonic points and stagnation and rarefaction zones makes this problemchallenging in terms of stability behavior. The entropy fix discussed in Part I [91] was foundto exert an important role. The adoption of a very small value for the adjustable parameterδk resulted in bad convergence behavior [79, 95], with the L2 norm of the density residualonly dropping around two orders of magnitude. The final mesh employed, following onelevel of adaptation, and the corresponding computed Mach number contours are shown inFigure 11. This mesh has 24,979 elements and 12,651 nodes. Note that both the bow shockand the quasi rarefaction zone behind the cylinder are well represented, with the circulationand the weak shocks captured. The form of the initial mesh was also found to be extremelyimportant, as very coarse triangulations at the back of the cylinder lead to non physicalsolutions and computational instability. A reasonable mesh, which enables the capture ofthe main flow features, is also important to allow the error indicator to detect regions wherethe discrete model must be improved during the adaptation. The variation of the computed


(a) (b)

Figure 11. Steady flow past a cylinder at a Mach number of 3. (a) final mesh;(b) corresponding computed distribution of the Mach number contours

Mach number and of the pressure coefficient along the horizontal symmetry line and overthe cylinder are presented in Figure 12. The sharp capture of the discontinuities is clearlyapparent, despite the use of a rather diffusive limiter. These results are in good agreementto those produced in other simulations of this problem [105]. The implementation of theprocedures to prevent the appearance of negative values of the thermodynamic variablesand the relaxation of the solid wall boundary condition are of paramount importance in thesuccess of this analysis during the initial stage of the transient, with the simulation startingfrom the free stream flow condition.

6.1.4 Hypersonic shock interaction

The previous examples involve computations in a relatively low supersonic regime. The nextcomputation considers a flow at high hypersonic Mach number, in which an oblique shockinteracts with the bow shock on a circular cylindrical leading edge [79, 95, 97]. The com-putation starts with the appropriate free stream and oblique shock boundary conditions.The undisturbed free Mach number is 15.0 and the disturbed flow has a Mach number of10.596 with 60 angle of attack. The problem definition is specified in Figure 13(a). This isan application with practical interest to the designers of hypersonic vehicles [105], as theflow field is typical of what may be experienced by the inlet cowl of the vehicles. The valuesCFL = 0.4, δk = 0.1 are adopted and the G/LED scheme applied, with the Roe Riemannsolver and the symmetric minmod limiter. Starting from an initial mesh of 3,217 elements,


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Figure 12. Steady flow past a cylinder at Mach number 3: distribution of (a) thecomputed Mach number; (b) the computed pressure coefficient on thecenter line and on the cylinder surface

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α o= 6

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M 8 = 15.00

(b) (c)

Figure 13. Hypersonic shock interaction on a circular cylindrical leading edge ata Mach number of 15. (a) problem definition; (b) final adapted mesh;(c) computed Mach number contours

four stages of adaptive mesh enrichment produce the mesh of 14,693 elements shown inFigure 13(b). Figure 13(c) presents the corresponding computed Mach number contours.Mesh enrichment proves to be extremely important in enhancing the resolution of the bowshock and also in allowing the capture of the shock interaction on the leading edge of the


cylinder [79, 95]. The convergence rate remains quite satisfactory during this analysis, andthe mesh adaptivity was performed, using the density and velocity field for the compu-tation of the error indicator, after the L2 norm of the density residual dropped 5 ordersof magnitude on each mesh. Limiter freezing was not employed. The computed variationof pressure over the cylinder on the final mesh when the G/LED scheme, with symmetricminmod limiter and the upwind MUSCL limiter are used, are plotted together with the Roefirst order upwind solution in Figure 14(a). Figure 14(b) shows the corresponding resultsobtained using the MUSCL scheme with the Venkatakrishnan/Thomas limiter and differentRiemann solvers. The solutions produced using the other Riemann solvers are not plotted,as they are basically the same as the results shown. It should be stressed that the surfacepressure at the stagnation point is at least twice as large on the refined mesh as it is on theinitial mesh. This shows the requirement for resolving all flow features and the importanceof mesh adaptivity. A comparison of the Mach number distribution, on the initial and finalmesh, along the line illustrated in Figure 15(a) is presented in Figure 15(b) for the casewhen the limited MUSCL scheme is employed, with the Venkatakrishnan/Thomas limiterand the Roe approximate Riemann solver. A good shock capture for both initial and finalmesh can be noted, and the enhanced resolution when the adapted mesh is used is evidentfrom this figure.

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Figure 14. Computed distribution of pressure over the surface of the cylinder, using thefinal adapted mesh, for the problem of shock interaction on a circular cylindricalleading edge at a Mach number of 15 using different high resolution schemes.(a) Roe first order upwind scheme, Galerkin LED scheme with the symmetricminmod limiter and the upwind MUSCL limiter; (b) limited MUSCL schemeusing Venkatakrishnan/Thomas limiter and Roe and AUSM flux splitting

6.2 Viscous Flows6.2.1 Subsonic flow over a NACA0012 airfoil

The first viscous test case consists of flow over NACA0012 airfoil at 00 incidence at a Machnumber of 0.5 and a Reynolds number of 5000, based upon the airfoil chord. The Prandtlnumber is assumed to be constant at 0.72 and the free stream temperature is T∞ = 5280R.The no slip adiabatic wall condition is imposed at the airfoil surface. A detail of the finestmesh adopted for this laminar viscous flow calculation is shown in Figure 16(a). Thismesh is obtained using the advancing layers approach to build the mesh in the vicinity ofthe solid wall and the standard advancing front technique to fill in the remainder of thecomputational domain. The corresponding predicted Mach number contour distribution,


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Figure 15. Shock interaction on a circular cylindrical leading edge at a Machnumber of 15 using limited MUSCL scheme with the Venkatakrish-nan/Thomas limiter and the Roe flux splitting. (a) definition of theline used for plot; (b) Mach number distribution along this line for theinitial and final meshes

(a)

(b)

Figure 16. Subsonic viscous flow over a NACA0012 airfoil. (a) detail of the mesh;(b) Mach number contour distribution

obtained using a Roe/MUSCL approach, is shown in Figure 16(b). Figures 17(a) and 17(b)present a comparison of the different skin friction distributions produced by using a differentnumber of layers in the structured like mesh region immediately adjacent to the airfoil. In


each case, the normal mesh spacing at the wall was 0.0005 chords. Figure 18 presents acomparison of the values obtained for the pressure coefficient using three different meshes.Here, a fixed number of 18 structured layers of elements were placed around the airfoil,with each mesh displaying a different distribution of spacing in the normal direction. Thespacing distributions were defined be setting values of 0.0005, 0.0010 and 0.0050 chordsfor the thickness of the first layer and a geometric progression for the following layers.Figures 19(a) and 19(b) show the corresponding skin friction distributions. As expected,the pressure coefficient is not very sensitive to the spacing of the viscous layers. However,the skin friction distribution is correct only on the mesh which has the value of 0.0005 chordsfor the first layer of elements. These results show that for the correct prediction of the thinboundary layer and recirculation zone, at least 18 structured layers of triangles, with thevalue of 0.0005 chords for the thickness of the first layer, is required for this implementation.The separation point is at ≈ 81.8% chord for the finest mesh, which is in good agreementwith the results of other computations for this problem [102].

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Figure 17. Subsonic viscous flow over a NACA0012 airfoil showing the effect of using differentnumbers of structured like layers of elements immediately adjacent to the airfoil.(a) distribution of Cf , the skin friction coefficient; (b) detail of the Cf distributionin the vicinity of the trailing edge

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Figure 18. Subsonic viscous flow over a NACA0012 airfoil showing the effect on the pres-sure coefficient over the airfoil of using different mesh spacing distributionsin the normal direction in the structured like layers of elements immediatelyadjacent to the airfoil


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Figure 19. Subsonic viscous flow over a NACA0012 airfoil showing the effect of usingdifferent mesh spacing distributions in the normal direction in the structuredlike layers of elements immediately adjacent to the airfoil. (a) distribution ofCf , the skin friction coefficient; (b) detail of the Cf distribution in the vicinityof the trailing edge

6.2.2 Supersonic flow past a flat plate

A basic validation test for any solution algorithm for viscous flow is the comparison of thepredicted laminar boundary layer development on a flat plate with the exact solution dueto van Driest [140]. The comparison will indicate if there is excessive artificial dissipation inthe numerical scheme. The example chosen consists of a free stream flow at a Mach numberof 4.0 and a temperature of 392.40R. The Reynolds number, based upon the length of theplate, is 4×106. The local Prandtl number is assumed to be constant and equal to 0.75. Thecomputational mesh employed consists of a grid with 20,000 elements and 10,201 nodes.This is a triangulation of a structured 101× 101 quadrilateral grid with stretched elementsin the boundary layer region [79, 93]. The initial conditions simulate a flat plate suddenlyexposed to the free stream. The no slip adiabatic wall condition is imposed at the plate.The far field boundary condition follows that described in section 3.5. Figure 20(a) showsthe comparison between the similarity solution of van Driest [140] and the velocity profilescomputed using the G/LED algorithm. The velocity distribution values, normalized by thefree stream velocity, are displayed at different sections downstream of the leading edge. Thevertical scale on the figure is the dimensionless length

Y =x2

x1

√Rx (69)

where Rx denotes the Reynolds number computed using the distance, x1, from the leadingedge.

The computed velocity profiles are seen to be in good agreement with the analyticalsolution. An indication of the higher resolution achieved with these schemes can be ob-tained from Figure 20(b), which shows the results computed by the first order upwind,the MUSCL and the G/LED schemes. It was observed that the adoption of the minmod,MUSCL or van Albada limiters, with the G/LED scheme, leads to almost identical results.In Figure 21(a), the solution obtained using the MUSCL scheme with the AUSM splittingand the Venkatakrishnan/Thomas limiter is shown for different sections downstream of thenose of the plate. A good prediction of the boundary layer development is obtained, whencompared with the similarity solution, and the performance is similar to the MUSCL/ROE


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Figure 20. Supersonic viscous flow over a flat plate: comparison of velocity profilescomputed (a) at different distances downstream from the nose of theplate using the G/LED scheme; (b) at x1 = 0.5 computed with differentschemes

results of Figure 20(b). With the G/LED formulation, only small differences were ob-served when different supports, symmetric or upwind, were used for the computation ofthe limiters, as illustrated in Figure 21(b), where the minmod limiter is adopted. Forthis application, the influence of the use of different supports for the limiter calculation isshown to be insignificant. As a result, we can say that the artificial dissipation induced bythese limiters is basically the same, in this simple test case, and both MUSCL and G/LEDschemes succeed in predicting the laminar boundary layer, presenting good agreement withthe theoretical solution.

6.2.3 Hypersonic flow over a compression corner

The final example consists of hypersonic flow over a 240 compression corner. The freestream Mach number is 14.1 and the Reynolds number is 103, 680, based upon a flat platelength of 1.44 ft. The temperature of the fluid in the free stream is 1600R and the localPrandtl number is constant at the value 0.72. The temperature of the wall is held fixed at5350R. The Reynolds number is low enough to ensure that the flow remains completelylaminar and the free stream temperature is low enough so that there are no significantreal gas effects [121]. A schematic description of the expected flow behavior is shown inFigure 22.

To enable comparison with the available experimental and numerical data, the scalingpresented by Rudy et al [121] is adopted. This means that the plotted wall coefficientsare log(50Cp), log(1000Ch) and 50Cf , where the pressure, heat transfer and skin frictioncoefficients, Cp, Ch and Cf respectively, are defined by

Cp = pw/(∞U∞U∞

2) Ch = qw/[∞U∞(H∞ − Hw)] Cf = τw/(

∞U2∞2

) (70)

and the subscripts w and ∞ refer to wall and free stream values. The wall values are definedin section 5.4. The computed velocity vector distribution is presented in Figure 23, wherequalitative aspects of the flow field, such as the presence of a separation zone and the formof the shock boundary layer interaction, are well represented.

The resolution of the leading edge shock, the substantially thin boundary layer on theramp, the separation zone and the shock shock and shock boundary layer interactions arevery demanding, making this problem a challenging application for the validation of any


(a)

0

2

4

6

8

10

12

14

16

0 0.2 0.4 0.6 0.8 1 1.2

Y*

U1/Uoo

X=0.5 X=0.7 X=0.9

van Driest

(b)

0

2

4

6

8

10

12

14

16

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Y*

U1/Uoo

LED-Upw LED-Sym van Driest

Figure 21. Supersonic viscous flow over a flat plate: comparison of velocity profilescomputed (a) at different distances downstream from the nose of theplate using the MUSCL/AUSM formulation; (b) using G/LED schemewith different supports for the limiter calculation

Flow

Leading Edge

Shock

Compression

Fan

Induced

ResultantShock

Slip Surface

Expansion Fan

RecirculationPoint

Separation

ReattachmentPoint

BoundaryLayer

Shock

Figure 22. Schematic representation of hypersonic flow over a 240 compressioncorner

numerical algorithm for the solution of the Navier Stokes equations. This problem hasbeen extensively studied in the literature and both numerical [122] and experimental [47]data are available for comparison. Computations were made using a triangulation of a 111by 101 structured grid [79, 93], to allow for comparison with other numerical results onthe same mesh. The domain of computation was extended ahead of the leading edge ofthe flat plate. This extension was found to be necessary to remove non physical behaviorfrom the solution in the region near the leading edge of the plate. The no slip isothermalcondition was imposed at the wall. The implementation of the far field boundary conditionfollows the process described in section 3.5. For this problem, both the G/LED and theMUSCL scheme were tested with different limiters and stencils and with different sets oflimited variables [79, 93]. The adoption of symmetric limiters with the G/LED schemeproduces more diffusive shock representations and, overall, the results are not as good asthose obtained using upwind biased limiters. Furthermore, it was found that the resultsproduced were closest to those obtained by other methods when the van Albada limiterwas used. Evidence from other researchers shows that accurate prediction of this complex


Figure 23. Hypersonic flow over a 240 compression corner: computed velocity vec-tor distribution

flow field depends strongly on a good resolution of the shock wave which emanates fromthe leading edge of the plate. This demanding strong shock at the leading edge, with theMach number dropping from 14.1 to 0.0, requires the addition of the correct amount ofartificial dissipation, in order to damp oscillations without compromising the capture ofthe discontinuity, and a very refined mesh in the vertical direction, to resolve properly theshock in the neighbourhood of the leading edge. Rudy et al [121, 122] have presented wallplots for this problem which were computed using the structured mesh flow solver CFL3D.Their results are presented in Figure 24, together with some representative results obtainedusing the schemes surveyed here.

The wall coefficients are computed according to the definitions given in equation (70).In Figure 24(a), the predicted variation of the pressure coefficient, for both the G/LED andthe MUSCL schemes, using the Roe Riemann solver, is compared with the results of theCFL3D code. Apart from the region very close to the separation point and at the pressurepeak, the three solutions are very similar. The results of the MUSCL scheme appearto be the most smeared and the results of CFL3D are the sharpest near the separationpoint. The comparison of the predicted variation of the heat transfer coefficient for thethree schemes is given in Figure 24(b). Again, CFL3D gives sharper resolution near theseparation point and slightly different values in the separation zone, where the MUSCL andG/LED predictions are very close. Grid convergence studies [41] indicate similar differencesthrough the separation zone. Figure 24(c) shows the predicted variation of the skin frictioncoefficient. In this case, the main discrepancies occur in the region downstream of thereattachment point. The difference is small but is present throughout the ramp and thepredicted peak is higher here than with the CFL3D algorithm. Skin friction predictionswhich are closer to the CFL3D results are produced if the G/LED method and the upwindbiased MUSCL limiter is adopted. These results are not so close to the CFL3D results overthe remainder of the wall and are not shown here. The results of experimental measurementshave also been included in these three figures. It is apparent that the comparison betweennumerical prediction and experiment is not good and the present computations tend tosupport the claim [122] that significant 3D effects were present in the original experiment.


(a)

0

0.5

1

1.5

2

0 0.5 1 1.5 2 2.5

Cp

X1

Exper. G-LED

MUSCL-ROECFL3D

(b)

-0.5

0

0.5

1

1.5

2

0 0.5 1 1.5 2 2.5

Ch

X1

Exper. G-LED

MUSCL-ROECFL3D

(c)

-0.5

0

0.5

1

1.5

2

0 0.5 1 1.5 2 2.5

Cf

X1

Exper. G-LED

MUSCL-ROECFL3D

Figure 24. Hypersonic flow over a 240 compression corner showing a comparisonof computed surface distributions. (a) pressure coefficient; (b) heattransfer coefficient; (c) skin friction coefficient

In general, a good correlation was found between the predictions of the different highresolution schemes considered. The results demonstrate that all the analysed schemescan accurately predict complex flow configurations which incorporate viscous inviscid in-teractions. However, for viscous computations, it has been found that the split velocityoption for the AUSM scheme produces oscillations in the solution in the vicinity of viscouswalls [79, 93]. For these problems, therefore, the use of the split Mach version is essentialfor a meaningful result. It was also found that the higher order MUSCL/AUSM schemegives the sharpest shock resolution, with the best results when compared to the predic-tions obtained by CFL3D. Although the MUSCL/AUSM presents some oscillations for thepressure distribution in the vicinity of the leading edge [79, 93], the overall numerical re-sults obtained for the applications analysed, the efficiency in terms of CPU time and alsothe favorable properties of the AUSM formulation make this scheme particularly promisingfor viscous hypersonic complex flow computations, and in particular for blunt body flowsimulations. The main reason for the appearance of the pressure wiggles is believed to bethat suggested by Wada and Liou [142] to explain the behavior observed in 1D simulationof the hypersonic colliding flow [79, 91], i.e. that they are due to the fact that the massflux in the AUSM scheme does not directly take into account the density behind the shockwave. The results obtained in the first attempt to use the modified AUSMV and AUSMDVformulations of Wada and Liou [142] eliminate the oscillations but were not altogether suc-


cessful, with the solutions resembling first order predictions. The close connection betweenthe AUSMV and the van Leer flux vector splitting justify the bad results obtained in theAUSMV computation. The free parameter [91, 142], which regulates the FD or FV natureof the AUSMDV scheme, has an important role for viscous simulations. An incorrect choiceof the value for this parameter might be the reason for the bad results obtained using theAUSMDV approach. Further investigation is required before a definite conclusion can bereached on this issue.

7 CONCLUDING REMARKS

The development of high resolution algorithms for the solution of the compressible Eulerand laminar Navier Stokes equations, on general unstructured triangular meshes, has beendescribed. Apart from the storage advantages associated with the use of the edge baseddata structure, especially for the 3D extension of the formulation, the computational imple-mentation is found to require less CPU time than the implementation of the same algorithmusing the conventional finite element data structure [115]. Furthermore, the implementa-tion of these high resolution schemes on a generic triangular discretization relies on theavailability of the edge based data structure and is directly extendible for 3D problems ontetrahedral meshes. The combination of a Galerkin finite element procedure with a rationalway of supplying additional numerical dissipation by means of a limiting procedure of LEDtype proves to be successful in producing accurate and robust algorithms. The flexibility ofadapting the mesh and the inclusion of a mechanism to prevent the appearance of negativevalues for the thermodynamic variables allows enhancement of the computed solution andprevents numerical instabilities in regions where the solution has very low values of pres-sure and density. The adaptive remeshing procedure [117] was not exploited by the authorsin the applications presented here and represents an alternative to the mesh enrichmenttechnique employed.

For the algebraic approaches, the use of a symmetric stencil for the computation of thelimiters leads, in general, to the addition of more numerical dissipation as the support forthe calculation of the limiter function is enlarged. On the other hand, the improvementin resolution normally achieved with the use of upwind biased limiters for the steady statecomputations performed here, and the fact that no improved convergence behavior wasobserved, leads to the conclusion that the upwind biased limiters should be preferred forsteady state simulations. This is further supported by the good results obtained with theuse of the one sided MUSCL geometric approach, which uses an upwind biased support forthe limiters. The mixing of limiters, when characteristic variables are limited and when thephysics of the application so justify, can represent a good option. Also, when a transientsimulation is performed, the use of certain limiters, when computed using upwind biasedstencil, leads to non physical solutions, as discussed in Part II [92], and then the conclusionin favour of upwind limiters is not shared. Although the influence of limiters on accuracy,stability and convergence rate has been shown to be important for the solution of the Eulerequations, most limiters tend to produce reasonable results for inviscid flows. However,when the Navier Stokes equations are considered, especially at high Reynolds number andin the hypersonic regime, the choice of limiter was found to be crucial to ensure that thenumerical dissipation induced does not exceed the physical dissipation. Upwind biasedlimiters, which are computed using a more physical and smaller support, tend in general togive more accurate results.

It was observed that the inclusion of a mechanism to freeze the limiters, or the inclusionof an element of background diffusion, may be necessary if good convergence behavior is tobe achieved, independent of the support used to compute the limiters. Further investigation,in terms of accuracy, stability and convergence performance, is required in order to analyse


the effects of the inclusion of a background diffusion and in order to examine the combinedeffects of the limiter choice and of the entropy correction, as was done for 1D problems inPart II [92]. A preliminary observation showed that the convergence rate is also influencedby the type of grid used for the computation. Shock aligned grids and coarser discretizationsnormally led to faster convergence than highly skewed grids and finer discretizations.

A CFL number much smaller than the total variation stability limit was always foundto be necessary when the AUSM or WPS schemes were adopted, which is in accordancewith the stability limits presented by Liou et al [68, 69, 70]. Instability of the computationsoccurred more frequently with the use of the Osher Riemann solver, normally at the startof the transient. Restarting from another solution was used when this was encountered .

The implementation of the different schemes is performed in the same program frame-work. This allows for the comparison of the relative performance of different schemes, butalso in some way is a constraint on the performance of a specific scheme. Furthermore, inthese implementations, little effort was made to achieve specific improvements in memoryand CPU time requirements. With this in mind, the switched artificial viscosity approachwas found to be by far the most efficient approach, in terms of CPU time necessary toperform a computation. The algebraic approach requires at least twice as much CPU timeas the switched artificial viscosity approach. The MUSCL approach requires at least threetimes the amount of CPU time used with the switched artificial viscosity approach. Withinthe switched artificial viscosity schemes, the scalar scaling alternative is approximately 30%cheaper than the matrix option. For the algebraic schemes, the use of the Osher Riemannsolver requires roughly twice as much CPU time as the use of the Roe Riemann solver. Interms of the geometric schemes, the hybrid FD/FV splitting schemes, AUSM and WPS,are the most efficient, with the Roe splitting requiring approximately 50% more CPU timeand the Osher splitting requiring approximately 90% more CPU time. These numbers wereobtained for small scale computations and using sequential computers, giving solely a roughidea of the performance in terms of CPU time for the schemes analysed.

Several other issues related to the accurate and robust simulation of hypersonic flowshave been addressed and the numerical results obtained demonstrate the potential of thisclass of schemes to deal with a wide range of compressible flow applications. Bad per-formance of shock capturing schemes is often experienced when they are directly appliedfor nearly incompressible or very low Mach number flows [146], as the acoustic wave speedgoes to infinity, resulting in an ill conditioned problem. The methods suitable for compress-ible equations can be adapted, normally using some variant of the pseudo compressibilitymethod introduced by Chorin [17, 46, 108]. Another option, is to use specific methodsspecially developed for incompressible applications [18, 49, 135, 151]. A systematic andcareful comparative study of all discussed schemes for the present applications and othermore challenging applications is necessary for definite conclusions. However, we hope thatthe assessment of the performance of the various procedures, during the validation study,furnishes elements which are helpful to reduce uncertainties and to give an insight intoelements and parameters existent in each procedure.

Some effort have been pursued by the authors in order to improve the computationalperformance of the of schemes by exploiting: anisotropic mesh generation/adaptation [80,81, 84, 105, 111], vector/parallel implementation [96, 104, 106], alternative data struc-tures [145, 96] and multigrid acceleration [67, 115, 116]. These topics will be reviewedin a future paper. Other important related topics that could be investigated include theuse of higher order reconstruction and multidimensional upwind discretization. Finally,extensions should be undertaken in order to deal with a more comprehensive class of appli-cations, e.g. incompressible flows, unsteady flows, three dimensional applications, turbulentflows, real gas effects, coupled problems, and some of these extensions have already beenreviewed [104].


ACKNOWLEDGEMENTS

P. R. M. Lyra would like to acknowledge the financial support provided by the BrazilianResearch Council (CNPq), under grant numbers 204506–90.5 and 305263–88.9(RE).

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APPENDIX A: DETERMINATION OF THE EDGE–BASED DISCRETEEQUATION

In this appendix we deduce the form of equation (18), presented in section 3.2 for theGalerkin finite element formulation when an edge based data structure is adopted. Theexpressions for the computation of the edge weighting coefficients Cj

IISand boundary face

coefficients Df , given in equation (19), and the properties of these coefficients stated inequation (20), are also demonstrated. Consider the patch of elements shown in Figure 25,where the sides IIS and I3I lie on the boundary. The right hand side of the discretestatement given in equation (12), after introducing the results of equations (15) and (16),can be rewritten as

RHSI =mI∑S=1

∑E∈IIS

2∑j=1

[ΩE

3∂NI

∂xj

]E

(F Ij + F IS

j)

−

∑E∈I

2∑j=1

[ΩE

3∂NI

∂xj

]E

(F Ij) −

∑B∈I

[ΓB

6(2F

nI + F

nJ)

] (71)


I

Ω2

Ω3

Ω1

I1

I3

I2

Ι S

Figure 25. Typical patch of triangular elements sharing a boundary node I

using the notation defined in section 3.2. To make things simple, we consider only thedirection x1 and we omit the subscript and superscript 1 in what follows. Using Green’slemma, and the fact that the shape function NI is zero on the sides ISI1, I1I2 and I2I3 dueto the compact nature in finite element formulation, we have that∫

Ω

∂NI

∂xdΩ =

∫Γ

NIndΓ or∑E∈I

[ΩE

∂NI

∂x

]E

=ΓIIS

2nIIS

+ΓI3I

2nI3I (72)

To simplify the development further, consider only the contributions of the boundary faceIIS in equation (71). Introducing equation (72) into equation (71), we can write

RHS(IIS)I =

[ΩE1

3∂NI

∂x

∣∣∣∣E1

](F I +F IS

)−[nIIS

ΓIIS

6

](F I) −

[nIIS

ΓIIS

6

](2F I +F IS

) (73)

After adding and subtracting the term[nIIS

ΓIIS

12

](F I + F IS

) (74)

equation (73) may be easily rearranged as

RHS(IIS)I = −

[nIIS

ΓIIS

12− ΩE1

3∂NI

∂x

∣∣∣∣E1

](F I + F IS

)

+[−ΓIIS

12

]nIIS

(4F I + 2F IS+ F I − F IS

)

(75)

where the terms inside the square brackets represent the weighting coefficients C and D foredge IIS in direction x1. Exactly the same arguments can be used for the direction x2 andthis enables us to write the general forms given in equations (18) and (19). Referring againto Figure 25, but now considering a generic direction j, the property of the weights givenin equation (20)(a) can be verified if we rewrite it as

RHS(IIS)I = −1

3

(2∑E∈I

ΩE

[∂NI

∂xj

]E

)

+13

ΓjIIS

4nj

IIS+

ΓjI3I

4nj

I3I

−−Γj

IIS

4nj

IIS− Γj

I3I

4nj

I3I

= 0

(76)


where the definition of the coefficient given in equation (19) has been used. This is equivalentto

−23

∫Ω

∂NI

∂xdΩ +

23

∫Γ

NIndΓ = 0 (77)

which is true from Green’s lemma of equation (72). The asymmetric property of the edgeweights, represented in equation (20)(b), can be demonstrated using geometrical arguments.Observe that the derivatives ∂NI/∂xj |E and the outward normal vector nIIS

have the sameabsolute value as ∂NIS

/∂xj |E and nISI respectively, but opposite sign. It follows that thecoefficients Cj

IISand Cj

ISI , defined according to equation (19), have the same absolute valueand opposite sign.

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Phone: 34-93-4106035; Fax: 34-93-4016517

E-mail: [email protected]

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