edge theory

EdgeTheoretical Formulation

Defence and Security, Systems and TechnologyFOI dnr 03-2870

March 2007Issue 4.1.0

EdgeTheoretical Formulation

FOI dnr 03-2870ISSN-1650-1942

Defence and Security, Systems and TechnologyFOI dnr 03-2870

March 2007Issue 4.1.0

FOI dnr 03-2870 Edge – Theoretical Formulation Issue 4.1.0

Contents

1 Introduction 7

1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Theoretical Formulation 9

2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Geometrical Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3.1 Governing Equations for System Rotation . . . . . . . . . . . . . . 14

2.3.2 Modification for Moving and Deforming Grids . . . . . . . . . . . 14

2.4 Fluid Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.4.1 Calorically Perfect Gas . . . . . . . . . . . . . . . . . . . . . . . . 15

2.4.2 Thermally Perfect Gas, Singel Gas . . . . . . . . . . . . . . . . . . 16

2.4.3 Thermally Perfect Gas, Multiple Gases . . . . . . . . . . . . . . . 16

2.5 Turbulence Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.5.1 Spalart & Allmaras One-equation Model . . . . . . . . . . . . . . 17

2.5.2 Eddy Viscosity Two-Equation Models . . . . . . . . . . . . . . . . 17

2.5.3 Explicit Algebraic Reynolds Stress Models (EARSM) . . . . . . . 18

2.5.4 Differential Reynolds Stress Models . . . . . . . . . . . . . . . . . 20

2.5.5 DES and Hybrid RANS-LES Models . . . . . . . . . . . . . . . . 21

2.5.6 Numerical Treatment . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.6 Spatial Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.6.1 Inviscid Fluxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.6.2 Viscous Fluxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.7 Steady State Time Integration and Multigrid . . . . . . . . . . . . . . . . . 30

2.7.1 Multigrid Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.7.2 Multistage Runge–Kutta . . . . . . . . . . . . . . . . . . . . . . . 32

2.7.3 Implicit Residual Smoothing . . . . . . . . . . . . . . . . . . . . . 33

2.8 Local Low-Speed Preconditioning . . . . . . . . . . . . . . . . . . . . . . 34

2.8.1 Preconditioning Matrix . . . . . . . . . . . . . . . . . . . . . . . . 34

2.9 Time Accurate Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . 35

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2.9.1 Explicit Time Accurate . . . . . . . . . . . . . . . . . . . . . . . . 35

2.9.2 Implicit Time Accurate . . . . . . . . . . . . . . . . . . . . . . . . 36

2.10 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.10.1 Connectivity Boundary Conditions . . . . . . . . . . . . . . . . . . 38

2.10.2 Wall Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . 38

2.10.3 External Boundary Conditions . . . . . . . . . . . . . . . . . . . . 39

2.11 Aeroelastic Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.11.1 Aeroelastic Analysis using Natural Mode Shapes . . . . . . . . . . 43

2.11.2 Structural Deformation and Moving Grids . . . . . . . . . . . . . . 45

2.11.3 Prescribed Motion Solutions . . . . . . . . . . . . . . . . . . . . . 46

2.11.4 Generalized Forces . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.11.5 Coupled Solution of the Modal Equations of Motion . . . . . . . . 47

2.12 Vortex Generator Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

A Appendix 51

A.1 3D Inviscid Jacobians and Eigenvectors . . . . . . . . . . . . . . . . . . . 51

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1 Introduction

Edge is a parallelized CFD flow solver system for solving 2D/3D viscous/inviscid, com-pressible flow problems on unstructured grids with arbitrary elements. Edge can be usedfor both steady state and time accurate calculations including manoeuvres and aeroelasticsimulations. There is also an inviscid adjoint flow solver that can be used for gradient basedshape optimization and mesh adaption techniques. There is also functionality for meshdeformation, postprocessing, file format conversion and other utility tasks.

The Edge system comprises five major components:

• Preprocessor – constructs the dual mesh and edge based datasets.

• Edge flow solver – performs the main flow computation.

• Helper programs – a general “toolbox” for manipulating input and output data.

• Application programs – programs for shape optimization mesh deformation and aeroe-lasticity.

• Xedge – a graphical user interface (GUI) for Edge.

The flow solver employs an edge-based formulation which uses a node-centered finite-volume technique to solve the governing equations. The control volumes are non-overlappingand are formed by a dual grid, which is computed from the control surfaces for each edgeof the primary input mesh. The relationship between the dual and primary input meshes isillustrated in Figure 1. In this example, a set of hexagonal control volumes are constructedfrom a simple triangular input mesh. In any Edge mesh, all the mesh elements are con-nected through matching faces. Edge meshes therefore may not contain hanging nodes. Inthe flow solver, the governing equations are integrated explicitly towards steady state withRunge–Kutta time integration. Convergence is accelerated using agglomeration multigridand implicit residual smoothing. Time accurate computations can be performed using a semiimplicit, dual time stepping scheme which exploits convergence acceleration technique viaa steady state from inner iteration procedure. This document contains the theoretical formu-lation of Edge. Details on the governing equations as well as the discretization can be foundhere. For more practical information on how to install and run Edge, see the Edge UserGuide. Implementation details for software developers are not included in this document,such information is included in the Edge Coding Guidelines and, of course, in the sourcecode itself.

7


1.1 Notation

V Volume of cellV l Volume of cell at mesh level lVi Volume of cell iU Conservative variablesV Primitive variablesW Characteristic variables

ndim Space dimensionn = (nx, ny, nz) Normal vector

n0i Normal vector to the surface between node 0 and node iS0i Surface area between node 0 and node iνi Node ixi Coordinates for node imi Number of neighboring nodes to node iFI Inviscid fluxFV Viscous flux

8


2 Theoretical Formulation

2.1 Overview

Edge is a flow solver for unstructured grids of arbitrary elements. Edge solves the Reynolds–Averaged Navier–Stokes compressible equations in either a steady frame of reference or ina frame with system rotation. Turbulence can be modeled with differential eddy viscositymodels or explicit algebraic Reynolds stress models. The solver is based on an edge-basedformulation and uses a node-centered finite-volume technique to solve the governing equa-tions. The control volumes are non-overlapping and are formed by a dual grid obtained fromthe control surfaces for each edge. All elements are connected through matching faces. Thegoverning equations are integrated explicitly towards steady state with Runge–Kutta timeintegration. The convergence is accelerated with agglomeration multigrid and implicit resid-ual smoothing. For time accurate analysis, this convergence procedure is used as a driverof an implicit time stepping, “dual time steps”. Edge contains different spatial discretiza-tions for the mean flow as well as the turbulence, different gas models, steady state andtime accurate time integration, low speed preconditioning etc. Applications include shapeoptimization and aeroelasticity.

2.2 Geometrical Considerations

The finite-volume technique requires control volumes surrounding the unknowns in thenodes. The control volumes are non-overlapping and are formed by the dual grid ob-tained from the control surfaces at the edges. The dual grid is supplied by the preprocessor,Berglind (2000), as an input to the flow solver. The grid with its dual grid is depicted in theexample in two dimensions in Figure 1.

ν0

ν1

ν2

ν3

n0S0

V0

Figure 1. The input grid (solid) also denoted triangular grid and its dual grid (dashed) forming the control volumes.

9


The coordinates of the input grid are provided for each node and the connectivity is suppliedin an edge based manner where an edge connects two nodes. In addition to the node numbersa control surface nS is supplied for each edge where n is the normalized normal vector andS the area. The control surfaces of all edges emerging from a node enclose the controlvolume of a node. The surface vector for each edge is oriented from the node with thefirst index to the node with the second index. The sum of the surface vectors over a closedvolume is equal to the null vector, ∑

i

niSi = 0. (1)

This check is performed in the preprocessor for all control volumes of the dual grid. If eachedge surface is considered to lie in a plane, the volume of the cone formed by the edgesurface and one of the nodes 0 and k is

12ndim

n0kS0k · (xk − x0), (2)

where ndim is the space dimension. Quantities with two sub indices denote edge quantitieswhere the indices denote the two nodes connecting the edge.

All edge volumes of a node sum up to the control volume, i.e.

V0 =1

2ndim

m0∑k=1

n0kS0k · (xk − x0), (3)

where m0 is the number of neighbor nodes to node ν0 or, equivalently, the number of edgesconnected to node ν0.

To close the control volumes at boundaries, control surfaces are supplied at the boundary.In Figure 2 the control surfaces to node ν3 are given at all edges connected to ν3, whereasthe edges to the boundary node ν1 do not close the control volume. To close it, a controlsurface is given separately for the boundary node ν1.

ν1 ν2

ν3ν4

ν5

Figure 2. Control volumes at an inner node and a boundary node.

At a corner point where two or more boundaries meet the boundary control surface is splitinto control surfaces for each boundary condition separately. In that case, the boundarynode may occur in several boundary conditions. In addition to the control surface suppliedfor each boundary node, an inner point is also supplied at all boundaries. The inner nodesare used in some of the boundary conditions described in Section 2.10 on page 37 and theinner node is chosen as the end node of the contiguous edge closest to the boundary surface.In the example above, node ν3 is the inner node to node ν1.

In 3D a similar discretization is done. The centroid dual consists of triangular facets be-tween the centroids of the cells, the faces and the edges. The control volume of a grid point

10


ν1

ν2

ν3

ν4

E1

E2

E4 E6

F1

F2

F3

F4

C

Figure 3. Facets for the edges E1, E2, E4 and E6 of a tetrahedron.

consists of faces intersecting the midpoint of each edge. The faces for all edges contigu-ous to an internal grid point form a closed surface. For a boundary grid point additionalboundary faces have to be added to the control volume.

Each face associated with an edge consists of several triangular facets, see Figure 3. Thefacets are defined by the points: the center of the grid cell C, the center of a face F and themidpoint of the edge E. In Figure 3, the facets associated to the edge E1 are E1 − C − F3

and E1 − C − F4. If the surface vectors for two adjacent triangles with one common edgeare added, the resulting surface vector will be one half of the cross product of the diagonals.The surface vector associated with an edge is computed by adding the surface vectors of thecorresponding facets.

At the boundaries, the surfaces that close the boundary control volumes have to be com-puted. In a loop over the boundary surface elements the surfaces for each node is accumu-lated, see Figure 4. The contribution to node i from the triangle 1− 2− i in Figure 4 is thesurface vectors for the triangles C − E2 − i and C − i− E3.

11


12

i

C

E2

E3

1 2

3

4

5

6

7i

Figure 4. Node surface for a boundary control volume.

2.3 Governing Equations

The Reynolds–Averaged Navier–Stokes (RANS) equations written in a Cartesian frame ofreference rotating with an angular velocity Ω, can be expressed as

∂U

∂t+∇ · FI +∇ · FV = Q, (4)

where FI and FV are respectively the inviscid and viscous flux matrices and Q is the vectorof source terms. On integrated form the same equations become∫

Ω

∂U

∂tdΩ +

∑faces

(FIn)S +∑faces

(FV n)S =∫

ΩQdΩ. (5)

The RANS equations are obtained by time averaging the Navier–Stokes system, with first-order closure, based on Boussinesq’s assumption:

− ρw′′i w

′′j = µt

[∂wi

∂xj+∂wj

∂xi− 2

3(∇w)δij

]− 2

3ρkδij , (6)

with wi the relative velocity component in the xi-direction.

The flux matrices resolve into Cartesian components

FI = fI1eTx + fI2e

Ty + fI3e

Tz ,

FV = fV1eTx + fV2e

Ty + fV3e

Tz ,

(7)

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where

U =

ρρw1

ρw2

ρw3

ρE

, fIi =

ρwi

p∗δ1i + ρwiw1

p∗δ2i + ρwiw2

p∗δ3i + ρwiw3

(ρE + p∗)wi

, fVi =

0τi1τi2τi3

qi + wjτij +(µ+

µt

σk

)ki

(8)

and where the shorthand notation, i, is used to denote derivatives with respect to xi. Thedensity and the pressure in (8) are time averaged values related to the instantaneous valuethrough:

q = q + q′, (9)

where q is the time averaged value and q′ the fluctuating part and

q′ = 0. (10)

The energy, temperature and velocity components are density weighted averages defined as

q =ρq

ρ, (11)

where k is the turbulent kinetic energy and is defined as

k =12wiwi. (12)

Below, all superscripts that denote an averaging are removed for clarity. All variables aresupposed to be averaged. In contrast to the laminar case, both the static pressure and thetotal energy contain contributions from the turbulence kinetic energy k and are defined as

p∗ = p+23ρk, (13)

andE = e+

23wiwi + k. (14)

respectively. The stresses and the heat fluxes are given by

τij = (µ+ µt)[∂wi

∂xj+∂wj

∂xi− 2

3(∇w)δij

], (15)

qi = (κ+ κt)∂T

∂xi. (16)

The viscosity may either be constant (input variable ICOVIC = 1) or may be derived fromSutherland’s law (ICOVIC = 0).

The laminar thermal conductivity is found from the viscosity and a user specified Prandtlnumber Pr (variable PRREF of the input file)

κ =µCp

Pr. (17)

The turbulent viscosity follows from the turbulence model. Turbulence models are dis-cussed in Section 2.4.2 on page 16. The turbulent conductivity is obtained from the turbu-lent viscosity through a turbulent Prandtl number

κt =µtCp

Prt. (18)

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2.3.1 Governing Equations for System Rotation

By setting the input option IROT = 1 configurations with a constant system rotation can becalculated in a steady state manner. The system rotation is entered through the input optionOMGRO which is the rotation vector Ω giving the magnitude and direction of the rotation.The relation between the relative velocity and the absolute velocity then becomes

w = u− Ω× r. (19)

In order to achieve high accuracy also in the farfield of the computational domain the equa-tions are not solved in the form displayed in (8). Rather equation (19) is substituted intoequation (8). After some algebraic manipulations the fluxes and source terms take on thefollowing form

U =

ρρu1

ρu2

ρu3

ρEr

, fIi =

ρwi

p∗δ1i + ρwiu1

p∗δ2i + ρwiu2

p∗δ3i + ρwiu3

(ρEr + p∗)wi

, fVi =

0τi1τi2τi3

qi + wjτij +(µ+

µt

σk

)ki

,

(20)and the source term is defined as

Q =

0−ρΩ× u

0

, (21)

whereEr = E − u · (Ω× r) (22)

which has a contribution from the centrifugal force which is regarded as a potential. Notethat the viscous flux has not changed and that the source term has become simpler. Themain advantage of this formulation is that when computing the fluxes the contribution fromthe system rotation is always obtained at the cell face and never averaged from cell centers.Since the stored variables are the absolute velocities several of the numerical routines mustbe adapted such that they add the missing part of the relative flow.

2.3.2 Modification for Moving and Deforming Grids

A number of parameters in the input file control the use of moving grids in Edge. In theAEROELASTICS set of input parameters, IAEOPT controls how the grid motion is han-dled. Details about the the aeroelastic module can be found in Smith (2005).

Normal Edge simulation in fixed grids have IAEOPT = 100. However, with IAEOPT =101, the present version of Edge enables excitation of a set of grid perturbation fields (howthis is done will be discussed in greater detail below). With IAEOPT = 102, a coupledfluid/structures analysis is performed. The moving grids concept modifies flux terms in theequations. Instead of the inviscid flux vectors being

fIi =

ρwi

p∗δ1i + ρwiw1

p∗δ2i + ρwiw2

p∗δ3i + ρwiw3

(ρE + p∗)wi

=

0δ1i

δ2i

δ3i

wi

p∗ +

ρρw1

ρw2

ρw3

ρE

wi,

14


in a fixed coordinates system, they now become

fIi =

0δ1i

δ2i

δ3i

wi

p∗ +

ρρw1

ρw2

ρw3

ρEr

(wi −

dxi

dt

).

Not only the flux vectors but also surface normals and volume for the control cells have tobe computed repeatedly, and thus part of the preprocessor analysis has been brought intothe solver.

2.4 Fluid Modeling

Different gas models can be chosen (input variable ITYGAS), as explained below.

2.4.1 Calorically Perfect Gas

The perfect gas law is used as constitutive equation,

p = ρrT, (23)

where r is the gas constant for the perfect gas under consideration defined as

r =R

M, (24)

where R is the universal gas constant and M the molecular weight of the perfect gas. Theratio of specific heats γ at constant pressure cp and at constant volume cv is defined as

γ =cpcv. (25)

When the gas temperature is low enough so that the vibrational and electronic modes arenot excited the internal energy of the gas will be proportional to the temperature and thusthe specific heats and γ are constants. The gas is called calorically perfect. This is the usualassumption of moderate speed aerodynamics.

The relation between r and cp is given by

r =γ − 1γ

cp. (26)

The values of γ and cp are user input, respectively in the variables GAMMA and CP. Thestatic pressure is obtained from the conservative variables through the following relation

p = (γ − 1)[ρE − (ρw)2

2ρ

]. (27)

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2.4.2 Thermally Perfect Gas, Singel Gas

At higher temperatures, the variation of the internal energy with temperature is given bythe excitation of the translational, rotational, vibrational and electronic modes of the gasmolecules.

Under conditions where no chemical reactions or ionization occur the internal energy of thegas is now a function of temperature only which makes the gas thermally perfect.

The perfect gas law is still used as constitutive equation, but the specific heats are nowfunctions of the temperature. These functions are user input, cf. input variables IGTAB andIGTTAB.

The static pressure is now obtained as

p = ρrT (e), (28)

with T obtained implicitly from the relation of the internal energy to temperature.

2.4.3 Thermally Perfect Gas, Multiple Gases

The thermally perfect gas assumption may also be used for calculation of the mixing ofseveral thermally perfect gases where an equation is solved for each gas and its fraction ofthe total density, its mass fraction specie ψj , 0 ≤ ψj ≤ 1. Any arbitrary number of species(NSPEC ≥ 1) may be specified. For each specie a transport equation is solved. Theseequations, formulated in the form of equation (5), look like:∫

Ω

∂ρψj

∂tdΩ +

∑faces

(ρψjw · n)S +∑faces

(µσ + µtσt)∂ψj

∂nS = 0 (29)

where σ and σt are the Schmidt and turbulent Schmidt numbers respectively. The totaldensity will satisfy

nspec∑j=1

ψjρj = ρ, (30)

where nspec is the number of species.

2.5 Turbulence Models

For calculations of turbulent flows, a variety of turbulence models are available, whichare categorized into three different groups, namely, RANS (Reynolds–Averaged Navier–Stokes), DES (Detached Eddy Simulation) and hybrid RANS-LES, as well as LES (LargeEddy Simulation). The modelling approach is specified by the parameter ITURB, that is,setting ITURB = 2 for RANS modelling, ITURB = 3 for DES and hybrid RANS-LESmodelling, ITURB = 4 for LES modelling. The choice of different models under theselected modelling approach is made with the variable TURB MOD NAME. The availableturbulence models are shortly discussed below.

16


2.5.1 Spalart & Allmaras One-equation Model

The Spalart & Allmaras (1994) model is chosen by setting TURB MOD NAME =’Spalart-Allmaras one-eq model’. This model solves one transport equation for a quantity ν which isequivalent to the eddy viscosity. Since turbulence is characterised by two scales, e.g. veloc-ity and length scales, and the model only solves for one propoerty additional information isneeded. The Spalart & Allmaras model uses the wall distance, that would be active throughthe complete boundary layer, not only in the viscous sub layer. The detailed definition ofthe model is given in the reference, and is not repeted here. The model is integrated all theway to the wall which requires a good resolution normal to the wall (y+ ∼ 1).

2.5.2 Eddy Viscosity Two-Equation Models

In these turbulence models two additional transport equations for the turbulent kinetic en-ergy k and some auxiliary quantity (ε, ω or τ ) is solved. The models below are integratedall the way to the wall which requires a good resolution normal to the wall (y+ ∼ 1).

2.5.2.1 The Wilcox Standard k − ω Turbulence Model

The standard k−ω turbulence model by Wilcox (1988) is used if TURB MOD NAME =’stdWilcox k-omega’. The additional equations can be put in the form of (4) with

U =(ρkρω

), fIi =

(ρkwi

ρωwi

), −fVi =

((µ+ µtσ

∗)k,i

(µ+ µtσ)ω,i

), (31)

and

Q =

(Pk − β∗ρkω

γω

kPk − βρω2

), (32)

where Pk is the production of turbulent kinetic energy given by

Pk =∂wi

∂xj

(µ

[∂wi

∂xj+∂wj

∂xi− 2

3(∇w)δij

]− 2

3ρkδij

). (33)

The eddy viscosity is given by

µT = ρk

ω. (34)

Finally, the closure coefficients are as follow

γ = 0.556, β = 0.075, β∗ = 0.09, σ = 0.5, and σ∗ = 0.5. (35)

One of the major problems with the Wilcox k−ω model is that the model has an unphysicalfree stream dependency.

2.5.2.2 Low Reynolds Number (LRN) k − ω Models

Low Reynolds number (LRN) models are developed to account for the viscous and wall-damping effects. For linear two-equation models, this has usually been achieved by meansof some empirical damping functions, which also help in many modelling variants to at-tain correct asymptotic properties when integrated to the wall surface. The LRN k − ω

17


model by Wilcox (1994) has been derived on the basis of the Wilcox standard k−ω model.This suggests that the LRN Wilcox model returns to the standard version, when all thedamping functions are set to a constant value of unity. The model is chosen by settingTURB MOD NAME =’Wilcox LRN k-omega’.

The LRN model by Peng et al. (1997) is a modified version of the k−ω model. It is cast ina transformed form from k − ε type model by some approximations and, consequently, hasa cross diffusion term in the ω equation, which helps to suppress the freestream sensitivity.The cross diffusion term takes the following form

Cωµt

k

∂k

∂xk

∂ω

∂xk. (36)

Unlike the following Kok TNT k − ω model, this LRN model employs the cross diffusionterm over the whole boundary layer. The model constants and damping functions are cal-ibrated for internal flows characterized by recirculation with separation and reattachment.The model is chosen by setting TURB MOD NAME =’PDH LRN k-omega’.

2.5.2.3 The Kok TNT k − ω Turbulence Model

The k − ω model by Kok (2000) is derived for overcoming the unphysical free streamdependency. The equations are on the same form as the k − ω model in (31) but with anadded cross diffusion term. Also the model coefficients are modified. The cross diffusionterm is written as

σdρ

ωmax

(0,∂k

∂xk

∂ω

∂xk

). (37)

The model is chosen by setting TURB MOD NAME =’Kok TNT k-omega’.

2.5.2.4 The Menter BSL and SST k − ω Turbulence Models

The k−ω model by Menter (1994) is also derived for overcoming the unphysical free streamdependency. Also here the equations are on the same form as the k − ω model in (31) butwith an added cross diffusion term as in (37). The model is a blending of the standardk− ε model in the outer part of the boundary layer and the Wilcox k− ω model in the nearwall part of the boundary layer. The k − ε model is transformed into k − ω resulting in across diffusion term and modified coefficients which are blended according to a blendingfunction.

In the SST model an additional modification is made, compared to the BSL model. A limiteron the eddy viscosity is added so that uv ≤ a1k, with a1 = 0.31. This is in accordance withthe Bradshaw assumption, and will reduce the turbulence level around stagnation regionsand in adverse pressure gradient boundary layers compared to a standard eddy-viscositymodel and the prediciton of separated flows are improved.

The models are chosen by setting TURB MOD NAME =’Menter BSL k-omega’ or ’MenterSST k-omega’.

18


2.5.3 Explicit Algebraic Reynolds Stress Models (EARSM)

The EARSM by Wallin & Johansson (2000) is a rational approximation of a full Reynoldsstress transport model in the weak equilibrium limit where the Reynolds stress anisotropymay be considered constant in time and space. The Reynolds stress tensor is explicitlyexpressed in terms of the velocity gradient and the turbulence scales.

2.5.3.1 The Wallin & Johansson EARSM

The EARSM may be written in a common way where the anisotropy tensor a is written interms of the strain- and rotation rate tensors S and Ω as

a =10∑

λ=1

βλT(λ), (38)

where the β coefficients are functions of the five invariants of S and Ω. The T’s are

T(1) = S,

T(2) = S2 − 13

IISI,

T(3) = Ω2 − 13

IIΩI,

T(4) = SΩ−ΩS,

T(5) = S2Ω−ΩS2,

T(6) = SΩ2 + Ω2S− 23

IVI,

T(7) = S2Ω2 + Ω2S2 − 23

VI,

T(8) = SΩS2 − S2ΩS2,

T(9) = ΩSΩ2 −Ω2SΩ,

T(10) = ΩS2Ω2 −Ω2S2Ω.

(39)

The Reynolds stress tensor is then related to the anisotropy as

ρuiuj = ρk

(aij +

23δij

), (40)

which can be rewritten as an effective eddy viscosity, µt, and an extra contribution to theexpression for the stress tensor (15), which reads

ρkaextraij . (41)

2.5.3.2 The length-scale determining (ω) models

This constitutive relation for the Reynolds stress tensor can, in principle, be coupled toany two-equation model platform. In Edge the EARSM is available with the standard k−ωmodel by Wilcox (1988) (TURB MOD NAME =’W&J EARSM + std k-omega’), the k−ωmodel by Kok (2000) (TURB MOD NAME =’W&J EARSM + Kok TNT k-omega’) and

19


the k − ω model by Menter (1994) (TURB MOD NAME =’W&J EARSM + Menter BSLk-omega’).

These k − ω models are basically derived for a standard eddy-viscosity relation but the re-quirements from an EARSM are different. A new k − ω model derived with the Wallin &Johansson (2000) EARSM and, thus, completely consistent with it is the model by Hell-sten (2005) (TURB MOD NAME =’W&J EARSM + Hellsten k-omega’). This model isextensively tested in different flows and is the recommended default model.

2.5.3.3 Curvature Correction

In strong rotation and curvature the EARSM approximation is not perfecty valid and a fullReynolds stress model is to prefere. However, an approximation of the missing terms isgiven by Wallin & Johansson (2002). The approximation is given in terms of gradientsof the strain- and rotation rate tensors. The model has some stability problems, mostlybecause of the dependency on second spatial derivatives, and should be used with caution.However, in cases with strong swirl the model is expected to give improvements. Thecurvature correction is available together with the standard k − ω model by Wilcox (1988)(TURB MOD NAME =’W&J CC-EARSM + std k-omega’), the k − ω model by Menter(1994) (TURB MOD NAME =’W&J CC-EARSM + Menter BSL k-omega’), and the k−ωmodel by Hellsten (2005)(TURB MOD NAME =’W&J CC-EARSM + Hellsten k-omega’).

2.5.4 Differential Reynolds Stress Models

The transport equation for the Reynolds stress tensor may be derived from the Navier–Stokes equations

Duiuj

Dt= Pij − εij + Πij +Dij . (42)

The terms represent production, dissipation, pressure-strain rate, and diffusion (molecularand turbulent), respectively. One should particularly note that the production term is explicitin the Reynolds stresses,

Pij = −uiuk∂Uj

∂xk− ujuk

∂Ui

∂xk(43)

whereas the other terms need to be modelled.

In differential Reynolds stress models (DRSM), or Reynolds stress transport (RST) models,all different terms in (42) are kept or modelled which results in a transport equation forevery individual Reynolds stress component. In general three-dimensional mean-flows thisimplies six equations due to symmetry in the Reynolds stress tensor.

The dissipation rate tensor εij is usually decomposed into an isotropic part and a deviationfrom that, εij = ε(eij + 2δij/3). First, the total dissipation rate ε is modelled through atransport equation, similar to the ε equation in the K–ε models. Also other alternatives toε, such as ω or τ exist. The dissipation rate anisotropy eij is typically explicitly modelled interms of the Reynolds stress anisotropy or included into the modelling of the pressure strainrate.

The standard Wilcox ω model, used together with the DRSM model reads

DωDt

= αω

KP − βω2 +

∂

∂xl

[(ν +

ν(eff)T

σω

)∂ω

∂xl

], (44)

20


C01 C1

1 C2 C∗2 C3 C4 C5

W&J 4.6 1.24 0.47 0 2 0.56 0SSG 3.4 1.8 0.8 1.30 1.25 0.40 4.2

Table 1. The values of the C coefficients for different pressure-strain models.

from which the dissipation ε = β∗ωK.

The diffusion is modelled using a simple gradient diffusion model

Dij =∂

∂xl

[(ν +

ν(eff)T

σK

)∂uiuj

∂xl

], (45)

with the effective viscosity defined as ν(eff)T = K/ω.

The modelling of the pressure strain rate is given in the following form that includes manyof the linear or quasi-linear models in litterature, such as e.g. the SSG model Spezialeet al. (1991). A general form for the pressure-strain rate and dissipation rate anisotropy eijlumped together reads

Πij

ε− eij = −1

2

(C0

1 + C11

Pε

)aij +

(C2 −

C∗2

2√aklalk

)τSij

+C3

2τ

(aikSkj + Sikakj −

23aklSlkδij

)− C4

2τ (aikΩkj − Ωikakj) (46)

+C5

4

(aikakj −

13aklalkδij

)where τ = K/ε is the turbulent timescale. The C coefficients for the models available inEdge are given in Table 1.

The two differential Reynolds stress models are available together with the standard k − ωmodel by Wilcox (1988) (TURB MOD NAME =’W&J DRSM + std k-omega’),(TURB MOD NAME =’SSG DRSM + std k-omega’) and thek − ω model by Hellsten (2005) (TURB MOD NAME =’W&J DRSM + Hellsten k-omega’),(TURB MOD NAME =’SSG DRSM + Hellsten k-omega’).

2.5.5 DES and Hybrid RANS-LES Models

DES (Detached Eddy Simulation) and other hybrid methods, combining RANS (Reynolds–Averaged Navier–Stokes) and LES (Large Eddy Simulation) in the turbulence model, areemerging turbulence modelling approaches. Note that DES is one type of “hybrid RANS-LES models”. The development of such methods has been originally motivated due toaeronautical applications, where the turbulent flow is characterized by unsteadiness, mas-sive separation and vortical motions. In order to avoid using huge near-wall grid resolutionas required in well-resolved LES for high-Reynolds number flows, these models often useRANS mode in the wall boundary layer being coupled with LES mode in the off-wall re-gion and in regions where the flow is “detached” from wall surfaces (e.g. afterbody flows).The DES and other hybrid RANS-LES models are often viewed as a promising compro-mise between RANS modelling (in terms of computational efficiency) and LES (in terms ofcomputational accuracy).

21


The use of LES mode in hybrid RANS-LES methods implies that instantaneous, large-scaleflow structures are resolved in the LES region. When DES or other RANS-LES models areselected, the simulation should be carried out being time-dependent (by setting ITIMAQ >0) and usually in a three-dimensional computational domain.

2.5.5.1 The Spalart-Allmaras DES Model

The DES modelling approach was first proposed in 1997 by the pioneering work of Spalartet al. (1997). This approach employs the Spalart–Allmaras (S-A) one-equation model inthe wall boundary layer in combination with its SGS (SubGrid Scale) modelling variant inthe LES region away from the wall. The same turbulence transport equation is thus invokedin both RANS and LES regions through a switch of turbulence length scales.

The S-A model solves a transport equation for a working eddy viscosity, νt. For details,the readers should refer to Spalart & Allmaras (1994) for the S-A RANS model and Spalartet al. (1997) for its DES version. Nevertheless, without repeating the details about the modelconstants and empirical functions, the equation for νt is defined as

∂νt

∂t= Cb1Sνt +

1σ

∂

∂xj

[(ν + νt)

∂νt

∂xj

]+ Cb2

∂νt

∂xj

∂νt

∂xj

− Cw1fw

[νt

l

]2

(47)

with

S = Ω +νt

κ2l2fν2 (48)

where Ω is the magnitude of vorticity, l is the local turbulent length scale, κ is the vonKarman constant and σ is a model constant. The turbulence eddy viscosity is computed byµt = fν1ρνt. The C’s and f ’s with different subscripts are model constants and empiricalmodel functions.

When the turbulent length scale, l, is taken as being the local wall distance, d, namely, l = d,the model takes the form of the original S-A RANS model. The SGS model in the off-wallLES region is adjusted from the RANS model by switching the length scale l from the localwall distance to a SGS turbulence length scale in association to the local cell size. This isdone by the formulation of l = min(d,Cdes∆), where ∆ is the maximum size of each cellin the three directions for structured grid and Cdes is a model constant. With unstructuredgrid, we have estimated ∆ by taking the maximum edge of each cell. The model constant,Cdes, has been calibrated in LES for decaying, homogeneous, isotropic turbulence, whichgives Cdes = 0.65 for best simulation of turbulence energy decay. Note that, when the νt-equation functions as a transport equation for SGS turbulence, under the assumption of localequilibrium the model is similar to the well-known SGS model by Smagorinsky (1963) (butwith a linear alignment between νt and Ω). The Spalart-Allmaras DES model is chosen bysetting ITURB = 3 and TURB MOD NAME =’Spalart-Allmaras DES model’.

2.5.5.2 The Peng Hybrid RANS-LES Model

The hybrid RANS-LES model by Peng (2005) is based on an algebraic formulation forthe turbulent eddy viscosity for both the RANS and LES modes. Simple algebraic (zero-equation) RANS models have been proved very robust in modelling attached boundarylayers. The use of an algebraic RANS mode in hybrid RANS-LES modelling is thus an

22


effective choice to pave the near-wall attached boundary layer. On the LES side, a presum-able argumentation is that the unresolved SGS turbulence resulting from a filtering process(with a sufficiently small filter width) is more isotropic and thus its modelling may also beaccomplished with a relatively simple and amenable approximation.

The Peng hybrid RANS-LES model combines an algebraic mixing-length type RANS modein the wall layer with the SGS model by Smagorinsky (1963). The RANS eddy viscosity isformulated by

µt = ρl2µ|S| (49)

where the length scale lµ = fµκd and fµ is damping function. In the off-wall LES region,the SGS eddy viscosity with the Smagorinsky model reads

µsgs = ρ(Cs∆)2|S| (50)

with Cs = 0.12 and ∆ is the filter width. The matching between the RANS and LESmodes is accomplished by modifying the RANS turbulent length scale over the RANS-LESinterface into lµ = lµfs so that µt = ρl2µ|S| in the RANS region, where fs is a matchingfunction. The eddy viscosity, µh, in the hybrid RANS-LES model is computed by

µh =µt if lµ < ∆µsgs if lµ ≥ ∆

(51)

Apart from its use for external aerodynamic flows, the use of the model for internal flowswith moderate separation has also been demonstrated by Peng (2006). This hybrid modeldoes not solve for any additional trubulence transport equation. It is thus more computa-tionally efficient than the S-A DES model with a reduction of CPU time by about 20%. Thishybrid model functions by setting ITURB = 3 and TURB MOD NAME =’Peng hybridHYB0 model’.

2.5.6 Numerical Treatment

2.5.6.1 Restriction of the Time Step

Depending on the input variable IUPWKZ either a central scheme (IUPWKZ = 0) or anupwind scheme (IUPWKZ = 1) is used to discretize the convective terms of the turbu-lent equations, cf. also Section 2.6 on page 25. The order of the upwind method may bespecified. The parameter IORDKZ = 1 for first order and IORDKZ = 2 for second orderaccuracy.

To control the stability of the calculation, especially in the beginning of the computation,the local time step is restricted by the residual divided by the solution

∆t∗ =∆t

1− ∆tCFL

min(

2R(ρk)ρk

, 2R(ρε)ρε

, ρ−) , (52)

where R is the residual and ρ− is the spectral radius of the negative source terms cor-responding to a point implicit treatment of these source terms. This restriction preventsnegative values of the turbulent quantities. A similar restriction is applied in the multigridcycles when updating the solution from coarse grid corrections, for more information seeEliasson & Wallin (2000).

23


2.5.6.2 Stability for DRSM

The hyperbolic and viscous stability limits were investigated by Eisfeld (2003). The spectralradius for the hyperbolic part reads

λ(h) = Uknk +√

(2K2 + a) (53)

and for the viscous part

λ(p) = max[c1µ

ρ, c2

µT

ρ

](54)

where c1 and c2 depend on the different diffusion and model coefficients, and

µT =ρK

ω(55)

The remaining completely local source term reads

Soij = ρε

[−1

2C0

1aij −23δij +

14C5

(aikakj −

13IIaδij

)](56)

where ε = β∗Kω and with the ω equation

So(ω) = −βρω2 (57)

The equation to be analysed is, thus,

∂ [ρuiuj , ρω]∂t

= J

ρuiuj

ρω

(58)

where J is the Jacobian. Since uiuj can be diagonalized, also the above equation canbe, and the evolution of the Reynolds stresses may be replaced by the evolution of theeigenvalues of the Reynolds stresses, ri

∂ [ρλi, ρω]∂t

= J

ρriρω

(59)

The Jacobian becomes a 4× 4 matrix determined by

Jαβ =∂Soα

∂(ρrβ)(60)

where Soα = [So11, So22, So33, So(ω)] and rβ = [r1, r2, r3, ω].

The four eigenvalues of the Jacobian become

λ1 = −2βω λ2 = −β∗ω λ3,4 = −β∗ω

(12C0

1 ±16C5

√32IIa

)(61)

and the spectral radius is the magnitude of the smallest one (largest negative)

λmax = max

[2β, β∗, β∗

(12C0

1 +16C5

√32IIa

)]ω (62)

24


2.5.6.3 Realizable Time Stepping for DRSM

Realizability for DRSM is related to positivity for the K-ω equations. The first requirementis that the model itself sustain realisable solutions. The second requirement is that the spacediscretization is monotonic, which is fulfilled by TVD schemes. These aspects will not bediscussed here.

Unrealizable solutions may occure during the iteration to steady state even if the model andspatial discretization is “realizable”. This is because of the iteration procedure and mayoccure both for implicit and explicit methods. Explicit methods will be discussed in thefollowing.

The updating of the Reynolds stresses τij ≡ uiuj is not that obvious since the individualcomponents could be negative without violating realisability. The criteria is that τ shold bepositive definite, i.e. all of the eigenvalues should be positive, but the eigenvalues are ratherexpensive to compute. A better measure is the determinant of τ , which is positive if alleigenvalues are positive. However, the determinant is positive also if two of the eigenvaluesare negative, but this state may only be reached by passing through det (τ) = 0. A specialtechnique is applied to restrict the time step in order to preserve a positive definite uiuj .

2.5.6.4 Free Stream and Initial Conditions

Initially, free stream values for k and ω are imposed. A good estimation of these free streamvalues can be derived from the turbulence level TUFREE (typically 1% or less) and the ratioof free stream turbulent and laminar viscosity VRFREE

k∞ =32(TUFREE u∞)2, µT∞ = VRFREE µ∞. (63)

2.6 Spatial Discretization

The finite volume discretization is obtained by applying the integral formulation of thegoverning equations in (4) to the control volume surrounding the unknown at node

∂

∂t(U0V0) +

m0∑k=1

FI0kn0kS0k +

m0∑k=1

FV0kn0kS0k = Q0V0, (64)

where m0 is the number of neighbors to node ν0. The surfaces S0k enclose the controlvolume for node ν0 and form the dual grid illustrated in Figure 1 in 2D for a given triangu-lation. The flux vectors FI0k

and FV0kare computed on the edge consisting of nodes ν0 and

νk where S0k is given, the source term Q0 is computed directly at the node.

2.6.1 Inviscid Fluxes

The schemes for the inviscid flux FI0kconsidered here are based on a central discretization

with dissipation terms of either artificial dissipation type or upwind flux difference splittingtype.

25


2.6.1.1 Central Scheme with Artificial Dissipation

The inviscid flux across the cell face between nodes ν0 and ν1 is computed as

FI01 = FI

(U0 + U1

2

)− d01, (65)

where d01 denotes the artificial dissipation. A blend of second and fourth order differencesare chosen as artificial dissipation, this corresponds to a blend of first and third order differ-ences for the fluxes. The following form has been shown to be suitable

d01 = (ε(2)01 (U1 − U0)− ε(4)01 (∇2U1 −∇2U0))ϕ01λ01, (66)

where ∇2 denotes the undivided Laplacian operator and where

∇2U0 =m0∑k=1

(Uk − U0) = −m0U0 +m0∑k=1

Uk. (67)

The local spectral radius λ01 is defined as

λ01 = (|u01 · n01|+ c01)S01, (68)

where u01 = (u0 + u1)/2 and c01 = (c0 + c1)/2 denote the cell face speed and cell facespeed of sound respectively. The normal direction of the control surface to the edge betweennodes ν0 and ν1 is denoted by n01 and S01 denotes its size.

The factor ϕ01 is introduced to account for the stretching in the grid. It is defined as

ϕ01 = 4ϕ0ϕ1

ϕ0 + ϕ1, (69)

where ϕ0 is defined as the ratio between the integrated spectral radius,

ϕ0 =

λ0

4λ01

p

, (70)

where

λ0 =m0∑k=0

(u0k · nk + c0k)Sk =m0∑k=1

λ0k, (71)

and where λ0k is given in (68). The factor p = 0.3 was chosen to have a close resemblancewith the Martinelli eigenvalue scaling for structured grids, Martinelli (1987). This gives adissipation proportional to the local spectral radius λ01 in the direction of the stretching. Inthe other directions a value slightly larger than the local spectral radius is obtained.

The function ε(2)01 is chosen to be active in the neighborhood of shocks and small in smoothregions of the flow

ε(2)01 = κ(2)

(∣∣∣∣∣m0∑k=1

(pk − p0)

∣∣∣∣∣ /m0∑k=1

(pk + p0)

)s2 (72)

where κ(2) is a constant coefficient provided as input (VIS2) and s2 is a scaling factor toreduce the dependency on the number of neighbors. The size of s2 is chosen such that for

26


six neighbors the dissipation equals the dissipation in a structured scheme and a unit valueis obtained,

s2 =3(m0 +m1)m0m1

. (73)

The fourth order difference dissipation is switched off in the vicinity of shocks

ε(4)01 = max(0, κ(4) − ε

(2)01 )s4, (74)

where κ(4) is another user defined constant (VIS4) and s4 is a scaling factor chosen inaccordance with s2

s4 =s224. (75)

The turbulent equations have VIS2KZ and VIS4KZ corresponding to VIS2 and VIS4.

On coarser grids a simplified form of the artificial dissipation operator based on secondorder differences only is used to save computational time but also to increase the amount ofdissipation. The coarse grid operator looks like

d01 = ε(0)01 (U1 − U0)ϕ01λ01, (76)

whereε(0)01 = κ(0)s2

ndim

3, (77)

and where κ(0) is another user defined constant (VIS0).

At a boundary the artificial dissipation should not contribute, i.e. the flux on the boundaryis set to zero. In addition, a boundary condition on the second derivatives in the undividedLaplacian (67) is required. Following Mavriplis the conservative variables are extrapolatedlinearly which corresponds to a normal second derivative,

∂2U

∂n2= 0. (78)

Requiring no normal derivative of the variables in the computation of the Laplacian is equiv-alent to only account for the contributions along the boundary. With the notation in Figure2 the undivided Laplacian becomes

∇2U1 =∑

k=2,...,5

(Uk − U1) = U2 − 2U1 + U5. (79)

The pressure sensor in equation (72) is not modified at a boundary, nor are the scalingfactors s2 and s4. In Figure 2 there are four flux contributions to the residual in node ν1,d12, d13, d14 and d15, the number of legs to node ν1 is m1 = 4.

2.6.1.2 Upwind Schemes

In addition to the central scheme, upwind schemes of second order accuracy are available.The upwind scheme is of Roe flux difference splitting type as opposed to the more com-monly used MUSCL type upwind schemes. The main reason for this is to have a scheme assimilar as possible to what is available in Euranus, the structured counterpart.

27


As for the central scheme, the inviscid term is computed as a central part with additionaldissipation. The central part is computed as an average of the fluxes through

FI01 =12

(FI(U0) + FI(U1))− d01, (80)

compare with relation (65).

The upwind dissipation d01 is here computed as

d01 =12RΛR−1 (U1 − U0) =

12RΛL−1 (V1 − V0) =

12RΛ dW01, (81)

where U and V denote the conservative and the primitive variables respectively, the prim-itive variables being the ones used in the computer code. The characteristic variables aredenoted by dW01 = L−1(V1− V0) = R−1(U1−U0). The tensor R is the right eigenvectormatrix to the flux Jacobian,

∂FI

∂q= RΛR−1, (82)

where the diagonal tensor Λ contains the eigenvalues. A similar expression can be obtainedfor the tensor L belonging to the primitive variables.

The diagonal matrix Λ is obtained as

Λ = |Λ∗|(I − Φ), (83)

where Φ is a diagonal matrix with limiters for second order accuracy. Note that Φ = 0 fora first order scheme. For a Roe flux difference splitting scheme the components of R, L, Λmust be computed from the Roe averaged variables (IROEAV = 1)

ρ01 =√ρ0√ρ1,

u01 =u0√ρ0 + u1

√ρ1√

ρ0 +√ρ1

,

H01 =H0√ρ0 +H1

√ρ1√

ρ0 +√ρ1

,

c201 = (γ − 1)[H01 − |u01|2

].

(84)

However, arithmetic averages usually provide good solutions and are computationally lessexpensive and may be used as an option (IROEAV = 0).

The diagonal matrix Λ∗ in equation (83) contains the eigenvalues adjusted with an entropyfix to prevent the eigenvalues to become zero and produce unphysical solutions. The fol-lowing entropy fix is used for each of the eigenvalues

|λi|∗ =

λ2

i + δ2

2δ, |λi| ≤ δ,

|λi|, |λi| > δ,

(85)

where δ is a small fraction of the spectral radius, usually around 5%. The variable ENTRFXdetermines the size of the entropy fix. Two values for ENTRFX are to be specified. Thefirst one applies to the linear field (the characteristic variables that propagate with speedu · n), the second value to the non-linear field (the characteristic variables that propagatewith speed u · n ± c). For the turbulent equations the corresponding variable is TURFIX,see the Appendix in Edge User Guide.

28


To achieve second order accuracy (IORDER = 2) the limiter Φ in (83) has to be computedfrom divided differences of the solution. Here we chose differences of the characteristics toavoid oscillations in the pressure, although it is computationally more expensive.

Gradients of all primitive variables are needed to compute the characteristics in the nodes.The gradient in a node is computed by evaluating the surface integral of the gradient theorem

∇v0 =1V0

∮∂V0

vn dS ≡ 1V0

m0∑k=1

12(vk + v0)n0kS0k, (86)

where v denotes a component of the primitive variables. The node valued characteristicsdW0 and dW1 may be obtained as

dW0 = L0(∇V0 · (x1 − x0)),dW1 = L1(∇V1 · (x1 − x0)),

(87)

in addition to the face value dW01 in (81).

However, on a structured Cartesian grid these characteristics correspond to a broad stencil,dWi = (Wi+1 + Wi−1)/2 using the expression in (81). To have a scheme that providesidentical results compared to a structured scheme on a regular grid the following expressionis used to compute the node valued characteristics

dW0 = 2L0(∇V0 · (x1 − x0))− dW01,

dW1 = 2L1(∇V1 · (x1 − x0))− dW01,(88)

which, on a regular Cartesian structured grid corresponds to

dW01 = dWi+1/2 = Wi+1 −Wi,

dW0 = Wi −Wi−1,

dW1 = Wi+2 −Wi+1,

(89)

i.e. dW0 and dW1 correspond to the left and right compact differences for the cell facei+ 1/2 flux.

There are three different limiters that can be used, the minmod limiter, the van Leer limiterand the superbee limiter. The variable LIMITE controls the type of limiter, the minmodlimiter is the default and recommended limiter. The minmod function chooses the argumentwith the smallest amplitude provided all arguments have the same sign. If the signs aredifferent the limiter is zero and the scheme reduces locally to a first order accurate one.

Two different ways of limiting are available. In the first way the minmod limiter is computedas

φdw01 = minmod(dw0, dw01, dw1) (90)

where φ and dw denote a component of Φ and dW respectively. This way of limiting issimilar to the new family of symmetric TVD schemes used in Euranus, the parameter forthis is IARDIS = 2.

The second way of limiting is more consistent with the flux difference splitting technique,IARDIS = 1. The limiter is then computed depending on the sign of the eigenvalues of theflux Jacobian,

φdw01 =

minmod(dw0, dw01), λi ≥ 0,minmod(dw01, dw1), λi < 0.

(91)

At boundaries no particular modification to the scheme is made. The numerical flux due tothe upwind dissipation is zero.

29


2.6.2 Viscous Fluxes

A compact discretization of the thin-layer approximation (IFULNS = 0) or the fully vis-cous terms (IFULNS = 1) is used. A thin-layer approximation only contains the normalderivatives of the viscous terms.

The viscous stresses for the momentum equations can be divided as

τijnj = (τijnj)tl + (τijnj)tan, (92)

where (τijnj)tl contains only normal derivatives and leads to a thin-layer discretization ifonly this term is considered. A fully viscous approximation is obtained if also the remainingpart of the viscous terms (τijnj)tan are added. The components of the normal are heredenoted n = (nx, ny, nz) = (n1, n2, n3). The thin-layer part can then be formulated as,see Gnoffo (1990),

(τijnj)tl = µ

(∂ui

∂n+

13

(∂uj

∂nnj

)ni

). (93)

The normal derivatives in equation (93) can be approximated on the edges as in Haasel-bacher et al. (1999) where

∂ϕ01

∂n=

ϕ1 − ϕ0

|x1 − x0|(94)

with the notation from Figure 1 and where the normal is directed from node ν0 to node ν1.With this formulation only two points are involved in computing the normal gradients at theedges and hence automatically leads to a compact second derivative.

By recalling the identity of the Laplace’s equation∫Ω

∆φdV =∮

∂Ω

∂φ

∂ndS, (95)

the following approximation of the Laplace’s equation at node is ν0 obtained as

∆ϕ0 ≈1V0

m0∑k=1

S0kϕk − ϕ0

|xk − x0|. (96)

The remaining parts of the viscous terms contain gradients which may be added using theGreen–Gauss formulation in (86) and thus a fully viscous approach can be obtained.

2.7 Steady State Time Integration and Multigrid

2.7.1 Multigrid Strategy

Multigrid is used to speed up the rate of convergence. The preprocessor agglomerates thecoarser grids that are read in by the flow solver. The input variable NGRID denotes thenumber of grid levels to be used in the flow solver.

The multigrid method is based on the FAS approach. The V (input parameter MGRSTR =1), W (MGRSTR = 2) and F -cycle (MGRSTR = 3) can be chosen by the user, or ifpreferred their saw tooth variant (MGRSTR = 4, 5, 6 for respectively V , W , F saw tooth).

30


In the saw tooth variant no smoothing (i.e. Runge–Kutta solver) will be applied in thecoarse-to-fine part of the cycle. As a result, the solution is only prolongated when passingfrom coarse to fine.

Consider a set of meshes denoted with an index l = 1, . . . , L with L the finest level. TheNavier–Stokes problem on the finest level can be written as

∂UL

∂t+NL(UL) = 0, (97)

whereNL(UL) is the spatial discretization of the Navier–Stokes operator on the finest meshL. The problem is then approximated on coarser levels l as

∂U l

∂t+Nl(U l) = Fl, (98)

with Fl the forcing function, defined recursively as

Fl = Nl(I ll+1U

l+1) + I ll+1[Fl+1 −Nl+1(U l+1)], (99)

where I ll+1 and I l

l+1 represent restriction operators of respectively the unknowns and theresiduals from finer mesh l to coarser mesh l + 1. They are defined as

I ll+1R

l+1 =∑

Rl+1, (100)

I ll+1U

l+1 =∑V l+1U l+1∑V l+1

, (101)

where Rl+1 is defined asRl+1 = Fl+1 −Nl+1(U l+1) (102)

and V represents the cell volume. The summation in equation (100) and (101) is over thefine cells contained within a coarse cell.

After temporal discretization equation (98) becomes

S∆U l +Nl(U l(0)) = Fl, (103)

where U l(0) is the current solution on mesh l which has to be smoothed. One has

U l(0) = I ll+1U

l+1, (104)

where ∆U l is an update of U l(0) and is to be calculated.

The operator S is the smoother. It is the operator that corresponds to the chosen timeintegration method, in this case an explicit Runge–Kutta time stepping.

Note that the number of smoothing sweeps (i.e. the number of times the Runge–Kuttaoperator) can be chosen by the user for each grid level (NSWPFC, cf. also below). Oncethe solution on the coarsest mesh is smoothed, the coarse-to-fine sweep of the multigridcycle is initiated. The current solutions on finer grids are updated with the solution on thenext coarser level

U l = U l + I ll−1(U

l−1 − I l−1l U l). (105)

The operator I ll−1 is a prolongation operator, currently of injection type, from coarser mesh

l−1 to finder mesh l. A similar procedure as in equation (52) is used to guarantee positivityfor concerned quantities.

31


In the basic V , W , F cycle (MGRSTR = 1, 2, 3 respectively) the new solution on thefiner mesh is smoothed before proceeding to the next finer level, by solving (100), withU l(0) = U l. The number of sweeps of the smoothing operator is user input, NSWPCF.

The user has also the possibility to smooth the corrections after prolongation (second term inthe right-hand-side of equation (105)) before adding them to the current finer grid solution,by applying the residual smoothing operator to the corrections. This is governed by theinput parameter SMCOR which has the same definition as RSMPAR, cf. Section 2.7.3 onpage 33.

The computing cost of a multigrid cycle is reduced by using simplifying assumptions oncoarser (i.e. all levels but the finest) grids (input parameter MGSIMP). If MGSIMP = 1, afirst order accurate upwind scheme is used on coarser levels (if IUPWIN = 1), else a morediffusive central scheme is used on coarser levels (IUPWIN = 0).

In order to create a good initial solution full multigrid is also available; it is used if theinput parameter IFULMG = 1 which is default. The calculations start then on the coarsestmesh. After a residual drop of RESFMG orders of magnitude, the solution is prolongatedto the next grid level. Two-grid cycles are now applied until the residual dropped once moreRESFMG orders, after which the third grid level is included and three-grid cycles are used.The procedure is repeated until the finest grid is reached. The variable RESFMG is userinput. The input variable NFMGCY gives the maximum number of cycles spent in eachstage of the full multigrid.

2.7.2 Multistage Runge–Kutta

An explicit q-stage Runge–Kutta scheme for the equation

dU

dt= F (U), (106)

can be writtenu1 = un + α1∆tF (un),

u2 = un + α2∆tF (u1),...

uq = un + ∆tF (uq−1),

un+1 = uq.

(107)

The coefficients αi determine the stability area and the order of accuracy of the Runge–Kutta scheme. They can be chosen in such a way that they suit the problem to be solved.

The local time step is computed for each node ν0 according to

∆t0 = min(

CFLV0

λ0,CFLVIS

V0

λv0

), (108)

where V0 is the volume and λ0 is the integrated convective spectral radius according toequation (71). The corresponding viscous spectral radius λv0 is defined as

λv0 =m0∑k=1

µ0k2ndimS2

k

ρ0k, (109)

32


where µ0k is the sum of the dynamic and turbulent viscosity in a turbulent calculation on theedge between nodes ν0 and νk, i.e. µ0k = (µ0 + µk)/2. The parameters CFL and CFLVISare user input and chosen according to the stability region of the Runge–Kutta scheme used.

If the input variable CFLVIS < 0 the viscous CFL number, CFLVIS, is replaced by theinviscid CFL number, CFL. The actual local time step is then computed by weighting theinviscid and viscous time step according to a harmonic mean formula

∆t0 = CFLV0

1λ0

1λv0

1λ0

+1λv0

(110)

For explicit time accurate calculations (ITIMAQ = 2), global time stepping is used, i.e. inall cells the same time step is used, which is, for stability reasons, the minimum of the localtime steps.

For steady state calculations a 3-stage, first order accurate scheme which provides goodsmoothing for both central and upwind schemes is recommended and is default,

α1 = 0.66667, α2 = 0.66667, and α3 = 1.0. (111)

This scheme allows good smoothing properties at with one computation of the artificialviscosity in the first stage. The parameters for the Runge–Kutta scheme are the number ofstages (NSTAGE) and the coefficients (IRKCO). The first element of IRKCO correspondsto the first coefficient, the second element to the second, and so forth. A maximum of tencoefficients (i.e. a 10-stage scheme) is foreseen.

The default setting is such that the numerical dissipation is calculated only once insteadof in all Runge–Kutta stages. It is computed in the first stage. This approach reduces thecomputational cost substantially. The disadvantage is that the dissipative terms need to bestored separately.

The recalculation of the dissipative residuals is governed by the variables ISWV and IBOTH.For an m-stage Runge–Kutta scheme the m first elements of this vector are used, the firstelement corresponding to the first stage, the second element to the second stage and so on.If the element of ISWV is zero it means that the dissipative residual should not be calculatedat the corresponding stage, and the latest available dissipative residuals will be used. Notethat for consistency the first value of ISWV should always be one.

2.7.3 Implicit Residual Smoothing

To increase the maximum time step, implicit residual smoothing is employed. The smooth-ing is employed for each residual to all equations independently. The smoothing for nodeν0 can for the residual of each unknown be written

R0 = R0 + ε∇2R0, (112)

where ε is a constant, ∇2 is the undivided Laplacian defined in equation (67). Here R0

indicates a new smoothed residual. This may be written

(1 +m0ε)R0 − ε

m0∑k=1

Rk = R0. (113)

33


For a structured solver a tridiagonal system of equations is obtained. For the unstructuredsolver a sparse, diagonally dominant matrix is obtained. The sparse matrix is not invertedexactly, instead a few Jacobi iterations of (113) are done. This results in the followingiterative scheme for the residual smoothing

Rn+10 =

R00 + ε

m0∑k=1

Rnk

1 + εm0, ε ≥ 0. (114)

Usually two iterations give sufficient smoothing, the parameter for the number of iterationsis NSMCYC.

Theoretically for structured grids where implicit residual smoothing is combined with thedirect solution of tridiagonal systems, the time step can be increased according to the valueof ε,

α =CFL∗

CFL≤√

4ε+ 1, (115)

where CFL and CFL∗ are the CFL numbers of the unsmoothed and smoothed scheme.Typical values using a structured grid is α = 2 corresponding to a doubling of the CFLnumber and to ε = 0.75.

Here, where a few Jacobi iterations are carried out, usually

α =CFL∗

CFL≤ 1.3, (116)

for good efficiency. This corresponds to a value of approximately ε ≤ 0.2. The inputparameter RSMPAR = α and the default is RSMPAR = 1.3.

On stretched grids only smoothing in the direction of the stretching should be used. Thesmoothing in the other directions must be reduced or removed. With structured grids thisis achieved by letting the coefficient ε be a function of geometric quantities. A similarprocedure is used here.

The stretching is accounted for as follows

Rn+10 =

R00 + εm0

m0∑k=1

RnkΨk/

m0∑k=1

Ψk

1 + εm0, ε ≥ 0, (117)

where Ψk contains the geometric information

Ψk =S2

0k

|xk − x0|. (118)

This type of smoothing increases slightly the smoothing in the direction of the stretchingand it is removed in the other directions. On a regular grid it reduces to equation (114). Thissmoothing is also applied to the corrections in the multigrid procedure. To account for thestretching the parameter IRESMO = 1 should be used.

34


2.8 Local Low-Speed Preconditioning

2.8.1 Preconditioning Matrix

In order to overcome problems with stiffness of Euler and Navier–Stokes equations in lowspeed flows the low speed preconditioning has been added to the code. It is based on theTurkel’s type of preconditioner and Navier–Stokes equations are modified as

MΓ−1M−1

∫Ω

∂U

∂tdΩ +

∑faces

(FIn)S +∑faces

(FV n)S =∫

ΩdΩ, (119)

where Γ−1 is the preconditioning matrix for the primitive variables V = (ρ, u, v, w, p)defined as

Γ−1 =

−δ + 1 0 0 0c2 + δβ2 − β2

c2β2

0 1 0 0αu

ρβ2

0 0 1 0αv

ρβ2

0 0 0 1αw

ρβ2

−δc2 0 0 0c2 + δβ2

β2

, (120)

and M = ∂U/∂V is the transformation matrix between conservative and primitive vari-ables. The coefficient α varies between −1 and 1, δ is in subsonic flow field equal to oneand in supersonic flow field equal to zero. The coefficient β is calculated using

β2 = min(max(K1u2loc,K2u

2inf), c

2), (121)

where uloc is the local velocity in the mesh cell. There are, however, problems with stabilityespecially in the regions of rapid changes of velocity – typical examples could be stagnationpoint or shock wave – boundary layer interaction. Stability problems in the stagnation pointcould be removed by increasing constant resulting in covering this problematic region withconstant β.

Low speed preconditioning is turned on by the parameter IPREPA = 1. The parametersK1 and K2 are set with the parameters RKBET1 and RKBET2 respectively. There is also aglobal Mach number RM0 above which no preconditioning is used. Usually RM0 = 1.

Recommended values of the preconditioning constants are α = ALPHA = 0, K1 =RKBET1 = 1, K2 = RKBET2 = 4, RM0 = 1, and K3 = RKBET3 = 0.33.

2.9 Time Accurate Calculations

There are two ways to perform time accurate calculations, either explicit Runge–Kutta timemarching with a global time step or implicit time marching with explicit subiterations.

35


2.9.1 Explicit Time Accurate

Explicit time accurate calculation is available by setting the parameter ITIMAQ = 2. Inthat case a global time step is automatically computed from the minimum local time step inall domains. The calculation of the local time step can be found in equations (108) – (110).A global time step is computed provided the parameter ISGLTS = 0 which is default.

Optionally, the parameter ISGLTS = 1 and the time step can be set by the user using theparameter DELTAT.

A Runge–Kutta scheme of at least second order accuracy in time is recommended, a goodchoice is the fourth order accurate, four stage Runge–Kutta with coefficients

α1 = 0.25, α2 =13, α3 = 0.5, and α4 = 1. (122)

The explicit time accurate option is not compatible with multigrid (NGRID = 1) or implicitresidual smoothing (RSMPAR ≤ 0).

2.9.2 Implicit Time Accurate

The implicit solver is written as

β1(UV)n+1 + β0(UV)n + β−1(UV)n−1

∆t+ γ1R(Un+1) + γ0R(Un) = 0, (123)

where UV denotes the unknowns times the volume. The coefficients β1, β0, β−1 and γ1,γ0 can be chosen to yield desired accuracy and stability. The implicit solver is availablethrough ITIMAQ = 1, the parameters β1, β0, β−1 are available with names BETNP1,BETN, BETNM1 and γ1, γ0 with names GAMNP1, GAMN respectively.

Introduce the pseudo time τ , denote the dependent variables Un+1 by U∗(τ) and considerthe steady state problem

V n+1dU∗

dτ+R∗(U∗) = 0, (124)

where

R∗(U∗) =β1V

n+1

∆tU∗ + γ1R(U∗) +Q, (125)

and

Q =β0(UV)n + β−1(UV)n−1

∆tU∗ + γ0R(Un), (126)

is a constant source term. As steady state in pseudo time is approached

dU∗

dτ→ 0 ⇒ U∗ → Un+1. (127)

Within each real time step, the set of ordinary differential equations (124) is solved usingan explicit Runge–Kutta method. To accelerate the convergence, we can adopt the samemultigrid strategy combined with local time stepping and implicit residual smoothing as fora steady state calculation.

The local time step will be influenced by the scaling of the residual and the additional termof the dependent variables in equation (124). The local time step is obtained as

∆τ0 = min(

∆t0γ1

,CFLVIS∆tβ1

), (128)

36


where ∆t0 is the original steady state local time step in equation (108) or (110) and ∆t isthe implicit time step (parameter DELTA) specified by the user.

The user can specify any three-level implicit scheme, the best scheme (which is default) isthe second order accurate backward difference scheme

β1 =32, β0 = −2, β−1 =

12, γ1 = 1, and γ0 = 0, (129)

which is A-stable and provides good smoothing also for large implicit time steps.

The user must supply a convergence criteria using the coefficient RESTAQ where the inneriterations are interrupted and the solution is advanced to the next time level when

log(max(|R(ρ)|)) < RESTAQ. (130)

Other residuals than the density can be specified with the variable NRRES. Note that con-vergence must be obtained in each time step before the solution can be advanced to the nexttime level. The user is recommended to put a value of RESTAQ such that the integratedforces don’t change when the convergence criteria is satisfied. If the convergence fails thetime step (DELTA) is most likely too large.

The user can also specify the minimum and maximum number of inner iterations (parame-ters ITMNAQ and ITMXAQ respectively.)

The different numerical and physical models combine as for a steady state solution. Moredetails of the implicit time accurate method is found in Eliasson & Nordstrom (1996).

2.10 Boundary Conditions

Implemented boundary conditions in the Edge code are given in Table 2.

Most boundary conditions use a so called weak formulation, i.e. the boundary conditionsare imposed through the flux and all unknowns on these boundaries are updated like anyinterior unknown.

A few exceptions exist though, e.g. on viscous walls the velocity is imposed stronglythrough a no-slip condition. A strong formulation implies that values of a strongly im-posed variable on the boundary are explicitly fixed, i.e. they remain at their imposed valuesand are not considered as unknowns.

Some of the boundary conditions are described theoretically below. The notations fromFigure 2 are used in the following. Note that only the central, inviscid terms and the viscousterms that contribute to the fluxes on the boundaries, the dissipative fluxes are set to zero.

2.10.1 Connectivity Boundary Conditions

At all connecivity boundaries an inviscid flux is added to these boundaries by using theaverage between the value on the boundary and the value on the connected boundary. Inthe case of rotation symmetry the velocity on the connected boundary is roatated beforeaveraged. On a propeller disk boundary a volume force is added to simulate the presence ofa propeller. Only the inviscid fluxes are added, the viscous fluxes have so far been neglected.

37


Notation MeaningConnectivity boundary conditions

Periodic A connectivity through a translational vector.Rotation A connectivity through a rotational matrix.Propeller Propeller disk model connectivity.

Wall boundary conditionsWeak euler Zero normal velocity.Weak adiabatic No slip. Assumes zero normal derivative of the temperature.Weak isothermal No slip. Assumes constant wall temperature.

Symmetry boundary conditionsWeak euler Zero normal velocity.

External boundary conditionsweak characteristic Characteristic boundary conditions.weak charact. vortcor Characteristic boundary conditions with vortex correction in 2D.weak total states Subsonic inflow boundary condition.strong total states pressure Subsonic inflow boundary condition.weak inlet Subsonic inflow boundary condition.weak static pressure Subsonic outflow boundary condition.weak extrapolation Extrapolation of all variables.weak free stream All variables specified to free stream.propulsion inlet Propulsion inlet.propulsion outlet Propulsion outlet.nacelle inlet Nacelle inlet.nacelle exhaust Nacelle exhaust.mass flow inlet Subsonic inflow boundary condition.mass flow outlet Subsonic outflow boundary condition.mass flow outlet JWS Subsonic outflow boundary condition.Mach outlet Subsonic outflow boundary condition.

Table 2. Available boundary conditions.

38


2.10.2 Wall Boundary Conditions

2.10.2.1 Euler Wall and Symmetry Plane(Weak euler)

At an Euler wall the normal component of the velocity is zero

u1 · n = 0, (131)

and hence the inviscid wall flux becomes

f1 =

0

p1nxSp1nySp1nzS

0

. (132)

The fluxes are added to the residuals for node ν1. The same boundary condition is used fora symmetry plane. Note that condition (131) is only implied through the flux, the unknownvelocity itself will not necessarily satisfy this condition exactly. This is a consequence ofusing a weak boundary condition.

2.10.2.2 Viscous Wall (Weak Adiabatic, Weak Isothermal)

On a viscous wall the velocity is imposed strongly through a no-slip boundary condition

u1 = 0. (133)

Both a weak and a strong formulation have been tested and evaluated. The strong formula-tion has shown better rates of convergence and has therefore been the chosen one. With astrong formulation, the residual of the velocity on the boundary does not need to be solvedfor since the velocity will be kept constant according to equation (133). This also impliesthat the fluxes for the velocity need not to be computed. In addition to the velocity, for aturbulent calculation, also the turbulent quantities are imposed strongly through

k1 = 0, (134)

ω1 =6Cwallµ1

βρ1|x1 − x3|2, (135)

where the value ω1 on the boundary is obtained as for structured grids recommended byHellsten (1998). The constants β = 0.075, Cwall = 1.5 usually and |x1−x3| is the distancefrom the wall node ν1 to the closest internal node ν3 for |x1 − xk|, see Figure 2.

At an iso-thermal wall there is a contribution from the viscous terms to the energy equation.The remaining boundary flux becomes

fE1 = −(κ+ κT )∂T

∂n1, (136)

where the gradient on the boundary node is computed as the difference of the temperaturein the interior node and the wall node

∂T

∂n1=

T3 − T1

|x1 − x3|− Tw − T1

|x1 − x3|. (137)

39


where Tw is the specified wall temperature.

At an adiabatic wall there is no contribution from the viscous terms to the energy equationat a wall since the temperature gradient is zero,

∂T

∂n1= 0, (138)

and hence the boundary flux is zero for both the density and energy equation.

2.10.3 External Boundary Conditions

2.10.3.1 Farfield (Weak Characteristic)

Characteristic boundary conditions are used at the farfield. These boundary conditions canbe used for both subsonic and supersonic in- and outflow where the characteristics are eitherset from free stream quantities for ingoing characteristics or extrapolated from the interiorfor outgoing characteristics.

Primitive variables are used and stored in the program since they lead to sparse and compu-tationally less expensive expressions.

Given a set of local primitive variables V1 = (ρ, u, v, w, p)T at the boundary a new setof primitive variables V ′

1 have to be computed to be used in the flux evaluation for nodeν1. The characteristic variables are denoted W and the relation between the primitive andcharacteristic variables is

V = LW, (139)

where L is given in Appendix A.1 with its corresponding inverse.

By computing characteristics based on both local and free stream primitive variables,

WL = L−1VL, W∞ = L−1V∞ (140)

either local or the free stream characteristics are used depending on the sign of the eigen-value. The local variables VL are computed by local extrapolation,

VL = V1, (141)

and the components of the transformation matrices L and L−1 are computed using freestream values and the local surface normal vector,

L = L(V∞, n), L−1 = L−1(V∞, n). (142)

Depending on the sign of the eigenvalue the components of characteristics W1 can be ob-tained

w1i =

WLi, λi > 0,W∞i, λi ≤ 0,

(143)

where λi denotes the ith eigenvalue belonging to the ith characteristic. The variables V1 canthen be obtained from (139) from which the flux on the boundary can be computed.

40


2.10.3.2 Farfield with Vortex Correction (Weak Charact. Vort)

Exactly the same boundary condition as the one above with the difference that the freestream values are no longer constant but depend on the integrated lift using a so calledvortex correction method. This boundary condition is intended for 2D subsonic externalproblems where the external boundary is located relatively close to the computed object.It gives in general an increased accuracy of the solution. Local boundary data cannot besupplied.

2.10.3.3 Total States Inlet (Weak Total States)

Total pressure, total temperature and a flow direction are specified to constant or varyingconditions. The Mach number is extrapolated upstream. The boundary condition is intendedas an subsonic inflow condition for internal aerodynamics.

2.10.3.4 Total States Inlet (Strong Total States Pressure)

This boundary condition is similar to the condition above. The difference is that the staticpressure is extrapolated upstream, the other variables are extrapolated. An additional dif-fernece is the strong condition implied on all variables.

2.10.3.5 Incompressible Inlet (Weak Inlet)

Density and velocity specified to constant or varying free stream conditions. Pressure isextrapolated if the flow is locally subsonic, it is specified if it is supersonic. The boundarycondition is intended for an internal inflow boundary.

2.10.3.6 Pressure Outlet (Weak Static Pressure)

A boundary condition intended for subsonic outflows where the static pressure is specified.It may also be used at local supersonic outflow conditions, in that case the pressure is notspecified (all variables extrapolated).

2.10.3.7 Extrapolation (Weak Extrapolation)

All variables are extrapolated with this condition. Note that theoretically this boundarycondition is only stable for supersonic outflow.

2.10.3.8 Free stream (Weak Free Stream)

All variables specified weakly to free stream values. Local varying free stream data may besupplied at the boundary, any variable may be spefied locally. Note that theoretically thisboundary condition is only stable for supersonic inflow.

41


2.10.3.9 Propulsion Inlet

Boundary condition for coupling to a ram jet module. This boundary condition is not avail-able in the official Edge distribution.

2.10.3.10 Propulsion Outlet

Boundary condition for coupling to a ram jet module. This boundary condition is not avail-able in the official Edge distribution.

2.10.3.11 Nacelle Inlet

This boundary condition is intended for an engine nacelle infow to which the capture area εis specified. Details about this boundary condition is given in Conway (2004). The bound-ary may be connected to the nacelle exhaust boundary.

2.10.3.12 Nacelle Exhaust

This boundary condition is intended for an engine nacelle exhaust boundary. It is requiredto specify pressure and temperature ratios. Details about this boundary condition is givenin Conway (2004).

2.10.3.13 Mass Flow Inlet

This boundary condition is can be prescribed at a subsonic inflow boundary which requiresgiven value of mass flow. This boundary condition is in principal identical to the total statesinflow boundary condition where the value of total pressure is a function of mass flowJirasek (2006). The procedure is as follows. First value of velocity normal to the boundaryis found from the value of desired mass flow mr and density ρ

un =mr

ρ. (144)

Then, the Laval number is defined as

λn =wn

c∗=

√u2

n + v2n

c∗. (145)

Finally, total pressure is calculated as

p0 = p

(1− γ − 1

γ + 1λ2

n

)− γγ−1

. (146)

2.10.3.14 Mass Flow Outlet

This boundary condition can be prescribed at a subsonic outflow boundary which requiresgiven value of mass flow. This boundary condition is in principal identical to the staticpressure inflow boundary condition where the value of static pressure is a function of massflow Jirasek (2006). This procedure is shown graphically in figure 5. For each value of total

42


Figure 5. Mass flow and static pressure as function of Laval number.

state and given value of mass flow there is a corresponding value of static pressure. Themathematical way is to solve the the implicit equation for velocity un

mr = ρ0un

(1− γ − 1

2c20u2

n

) 1γ−1

S (147)

where total states are known from solution and mr is desired value of mass flow. Velocityun is then used to calculate the value of Laval number

λn =un

co√

2γ+1

(148)

where c0 is known value of the total speed of sound. The value of λn is then used todetermine required value of static pressure on the outflow boundary

p = p0

(1− γ − 1

γ + 1λ2

n

) γγ−1

. (149)

2.10.3.15 Mass Flow Outlet JWS

This boundary condition relaxes the value of the static pressure at the outflow boundary toits desired value using equation Slater (2001)

pk+1 = pk

(1 + Φ(Mr − Mk)/M

), (150)

where Φ is relaxation parameter and Mr is the desired mass flow through the outflow bound-ary. The value of relaxation parameter is supplied in boundary conditions file. Recom-mended value is Φ = −0.02.

2.10.3.16 Mach Outlet

This condition is similar to the mass flow outflow boundary condition Jirasek (2006). It isbasically the static pressure outflow boundary condition, where the value of static pressureis expressed as as function of desired Mach number M and known value of total pressurep0

p = p0

(1 +

γ − 12

M2

)− γγ−1

. (151)

43


2.11 Aeroelastic Applications

Aeroelasticity is a multi-physics discipline, involving the interaction of aerodynamics andstructural mechanics and is of major importance in the aircraft industry. An excellent reviewof the literature on Aeroelasticity is provided in Battoo (1999). The general process foraeroelastic computations comprises the following cycle of operations:

• Compute aerodynamic loads on all moving surfaces, given the geometry and its rateof change.

• Transform the aerodynamic loads to a load distribution on the structure.

• Integrate the structural equations to update the structural coordinates and velocities.

• Transfer the structural motion to the moving surface.

• Update the flow solution for the new surface position and velocity.

The aeroelastic functionality in the present version of Edge is restricted to a modal repre-sentation of the structural model. In such a formulation, the time varying coordinates x(t)of the structural grid are represented as a linear combination of a set of constant modal basisvectors, φk, of the form

x(t) = x0 +Nm∑k=1

qk(t)φk. (152)

Here, x(t) is a long column vector containing the coordinates (degrees of freedom) of nodesin the structural grid, and x0 is the same vector at time zero x(0). The scalar variables qk(t),are Lagrangian generalized coordinates also called “modal” coordinates. The modal basisvectors, φk, also known as “modeshapes”, are vectors of the same class as x(0).

2.11.1 Aeroelastic Analysis using Natural Mode Shapes

For a very wide class of mechanical systems, the dynamics for small displacements can beaccurately represented by a linear differential equation of the form

Mx+ Cx+Kx = f, (153)

where, as in equation (152), x is the vector of structural coordinates, and f(t) is the cor-responding vector of structural forces. The mass, damping and stiffness matrices, M , C,and K respectively, are real and symmetric. The three structural matrices are in most casessparsely populated and in practice usually hidden in the variables of a finite element struc-tural model, for example a Nastran model.

For the special case when the structural damping matrix, C, and the vector of externalforces, f , are both zero, equation (153) reduces to a generalized eigenvalue problem,

Kφk = ω2kMφk. (154)

The solution eigenvectors φk comprise a set of “normal modes” with modal angular fre-quencies, ωk, where k ∈ [1, N ] and N is the dimension of x (number of structural degrees

44


of freedom). It can be shown that the eigenvectors, φk, satisfy the orthogonality conditions

φTj Mφk = akδjk,

φTj Kφk = akωkδjk,

(155)

in which the normalization constant, ak, is known as the “generalized mass” for mode k.The eigenvalue problem is usually solved using an algorithm which extracts the eigenvectorsin order of increasing ωk and the modal basis is truncated at order Nm N , such thatωk < ωNm for k < Nm.

Using the truncated modal basis, the structural equation of motion (153) can be reduced toa set ofNm scalar equations, coupled only through the external force term. Using (152) andpremultiplying (153) by the transpose of φk and using the orthogonality conditions (155)the equation of motions can be reduced to the form,

akqk + 2ζkakωkqk + akω2kqk = Qk, k ∈ [1, Nm], (156)

where ζk is the damping ratio for mode k and

Qk = φTk f, (157)

is the corresponding generalized force, or “modal” force. It is assumed that the structuraldamping matrix, C, is a linear combination of the mass and stiffness matrices M and K.This is a standard approximation and is known as “proportional” or “Rayleigh” damping.

The modal equation of motion (156) is a simple diagonal system and, unlike equation (153)can be integrated without any matrix operations. However, this requires that the modalgeneralized forcesQk must be calculated at each time step and this presents a problem. Theforces and modeshapes in equation (157) are defined on the structural grid but the forcescomputed by the flow solver are defined on the wetted surface boundary grid and these areusually completely separate point sets. This is known as the spatial coupling problem andarises in all partitioned aeroelastic methods. For small displacements and modal systems,however, the solution is relatively simple.

We assume that there exists a constant linear transformation matrix, H , which maps dis-placements from the structural grid to the wetted surface grid. This transformation can beapplied to the modal basis vectors to generate a corresponding set of modeshapes definedon the wetted surface

φWk = HφS

k , (158)

where the labels W and S refer respectively to the wetted surface and structural grids.

We now impose an energy conservation condition: that the work done by an arbitrary in-finitesimal displacement is invariant under the inter-grid transformation H . This may bewritten in the form

δW = (δxS)T fS = (δxW )T fW , (159)

where the labeling is the same as in equation (158) and δW is a scalar work integral. Fromthis, it follows that for any linear displacement transformation, H , there is a unique, match-ing transformation which maps forces from the wetted surface to the structural grid

HT fW = fS . (160)

From equation (157) it can be seen that the modal forceQk is dimensionally a work integraland is also invariant under a conservative transformation of the form given in equation (158)and (160).

45


For any modal system it is therefore necessary only to find a suitable linear transformationand to compute a set of modeshapes transformed onto the wetted surface grid. The forcetransformation is hidden in the integral Qk so the explicit transformation matrix, H , is notrequired. This transformation of modeshapes is performed using the program aedbelast inthe Edge system.

2.11.2 Structural Deformation and Moving Grids

In any aeroelastic simulation, the change in shape of the elastic body is determined by thesolution of the structural mechanics or structural dynamics equations. The volume meshwhich forms the computational domain for the flow solver must in turn be deformed suchthat it remains conformed to and connected with the moving boundary surface. The meshdeformation must also be such that the cell volumes remain positive and mesh quality prop-erties are not compromised.

It must be emphasized that in Edge, and similar unstructured codes, mesh deformationoperations are applied to the primary grid nodes. In an aeroelastic computation, the dualgrid metrics, surfaces and volumes, are re-computed after every incremental deformationof the primary grid. However, the connectivity of the mesh, including the agglomerationmultigrid structure, remains constant.

Mesh deformation is in general a rather complicated problem and there are several differentmethods available. One approach, which is used in the Tritet mesh generator, is to treat thevolume mesh as a quasi-elastic system and solve a field problem. The relevant componentsof Tritet are supplied in the Edge mesh adaption system and are accessed via the commandline script aetribend.

In the present version of Edge, elastic moving grids are implemented using a modal “per-turbation field” method in which the main deformation computation is performed “off line”in a preparatory operation. The programs used for this are aexbset, adbelast, aetribend andaeputpert and the procedure is described in the Edge User Manual.

The perturbation field method is based on an extension of the “modeshape” into the volumemesh. For each wetted surface modeshape, a matching deformed mesh is calculated using amesh deformation tool. This mesh has the same connectivity as the base mesh so the differ-ence can be represented simply by a “perturbation field” comprising the node deflections.The modal perturbation fields are stored and the time-varying coordinates of the primarymesh are rapidly computed using an expression similar to (152),

X(t) = X0 +Nm∑k=1

qk(t)∆Xk, (161)

where X0 is the coordinates field of the basic mesh read from Edge.bmsh and ∆Xk is theperturbation field for mode k computed in preprocessing, using the ae programs of the Edgesystem.

The perturbation field method is inefficient in two respects: a full volume vector field isstored for each mode and the deflection fields are computed for deflections of arbitrarymagnitude rather than for the actual deflection of the moving surface. It is also limitedexclusively to modal systems. It has been implemented in Edge mainly for continuity withthe older structured mesh code Euranus. In future versions of Edge it will be superseded bya system using “on line” mesh deformation.

46


2.11.3 Prescribed Motion Solutions

For prescribed motion solutions, the Edge.amot input file is used to set the exact behavior ofthe vector of modal coordinates, q(t), as a function of time. The vector, q(t), is thus a knowninput signal and the computed vector of aerodynamic modal forces, Q(t), is an output. Thetime history of both these modal vectors is recorded in the residuals file Edge.bres.

The main application for prescribed solutions is the computation of aerodynamic transferfunctions, which is a standard practice in the Aircraft industry. Typically, the motion wouldbe prescribed as single mode input, with a selected pulse profile and generalized forceswould be computed for the whole mode set. Using digital signal processing techniques,a linear, frequency domain SIMO (Single Input Multiple Output) analysis would be per-formed to extract one row of a transfer function matrix, H(ω), as defined in the expression

Q(ω) = H(ω)q(ω). (162)

Linear transfer functions of this form can be used for flutter prediction and dynamic loadscalculations. More advanced, system identification, procedures are also available whichcan capture the non-linear dynamics of the modal system. However, no such functions areprovided in the present code.

Another application for modal prescribed motion solutions is the simulation of large dis-placement motions whilst still using the “perturbation field” method. For example, largeangle, rigid body rotations can be modeled by scheduling a sequence of perturbation fieldsfor incremental rotations.

2.11.4 Generalized Forces

The generalized aerodynamic forces of (156) and (157) are surface integrals of pressure andviscous forces over the moving wall boundary. The integrals may be written

Qaerodynamick =

∮wall-boundary

∆XTk · (pn+ τn)dS, k = 1, . . . , Nm, (163)

and are computed at each time step. Together with the modal coordinates these forces arestored in the residual time history file Edge.bres. The displacements needed for this integralare only on the boundary. Hence they have the same form regardless if the perturbation fieldis given in the whole of the flow domain or only on the boundary. The generalized forcesQk can be used in post processing to obtain aerodynamic transfer functions or directly inthe integration of the structural equations of motion (156).

2.11.5 Coupled Solution of the Modal Equations of Motion

The equations (156) are now written in the form

Aq +Dq + Eq = Q. (164)

where the matrix A is the diagonal modal generalized mass matrix with elements ak on theprincipal diagonal, D is a diagonal generalized damping matrix with elements 2ζkakωk, Eis the diagonal generalized stiffness matrix with elements akω

2k on the principal diagonal

and, Q, is the vector of generalized forces. The generalized force vector, Q, includes the

47


aerodynamic surface loads, (163), and any forces on the structural grid, such as gravitation,centrifugal forces and direct mechanical inputs like weapon release loads.

The modal equation of motion is discretized using a centered difference scheme for the timederivatives with a three point time averaging of the modal coordinates and forces defined as

q =14(qn+1 + qn + qn−1), (165)

and

Q =η

2Qn+1 +

1− η

2Qn +

η

2Qn−1, (166)

respectively. The weights on the modal coordinates, qn, are chosen such that, for the un-forced, undamped case, D = 0; Q = 0 and the total energy is conserved exactly. Formally,En+1/2 − En−1/2 = 0,∀n, where,

En+1/2 =1

2(∆t)2(qn − qn−1)TM(qn − qn−1) +

14(qn + qn−1)TM(qn + qn−1). (167)

The same weights are usually chosen for the modal forces, η = 1/2, which gives a robustand accurate scheme. Note that the scheme is implicit in qn+1 except for the special casewhen η = 0.

The update formula for qn+1 is given by

qn+1 = A−11

(−A0q

n −A−1qn−1 +

η

2Qn+1 +

1− η

2Qn +

η

2Qn−1

)(168)

where

A1 = ν2M +12νD +

14, A−1 = ν2M − 1

2νD +

14, A0 = 2ν2M +

12, (169)

and ν = 1/∆t.

Due to the orthogonality relations (155) matrices A1, A0 and A−1 are diagonal, whichmeans that the modal coupled equation of motion can be integrated without solving aninverse problem.

At each time step, n, the update equation (168) is iterated within the inner loop of the dualtime stepping process, until the specified convergence is reached. The update equation issolved very rapidly, however, each time the vector of modal coordinates changes, a moretime-consuming process is initiated: the primary mesh is moved and the metrics of the dualmesh are recomputed. The coupled process can be speeded up by reducing the number ofstructural updates within the inner loop using the parameter NITIN in the main input file.

Modal parameters, are set using the file Edge.amop. The main modal parameters are thenatural frequency, ωk, generalized mass, ak, damping ratio, ζk, and the initialization valuesfor the modal coordinates and velocities. The modal frequency and generalized mass can besupplied together with the structural modeshape data via the MODES DEFINITION datasetin the, ∼.amod, file and are automatically carried through to the Edge.bmos file which isread into Edge. However, these values are overridden by any settings in the Edge.bmos file.

48


2.12 Vortex Generator Model

The vortex generator model enables to calculate the effect of vortex generator without needto embed the actual geometry of the vortex generator into the computational mesh Jirasek(2005). The model is based on Bender, Anderson and Yagle model which adds the sourceterm ~Li to the Navier–Stokes equations

∆Vi∆ρ~ui

∆t=

∑j

~FM∆S + ~Li (170)

∆Vi∆ρE∆t

=∑

j

~FE∆S + ~ui~Li (171)

The source term is calculated as

~Li = CV GSV G∆Vi

Vmαρu2~l (172)

where CV G is the model constant, SV G is the planparallel area of vortex generator, ∆Vi

is the volume of the cell where the force is calculated, Vm is the sum of volumes of cellswhere the force term is applied, α is the angle of local velocity ~u to the vortex generator and~l is the unit vector on which the side force acts. The source term is calculated on the meansurface of the vortex generators and then linearly redistributed to the surrounding points ofthe computational mesh. In the implementation, the model constant CV G is kept constantand has value CV G = 10.

49


References

BATTOO, R. S. 1999 An introductory guide to the literature in Aeroelasticity. AeronauticalJournal 2401, 511–518.

BERGLIND, T. 2000 An Agglomeration Algorithm for Navier–Stokes Grids. AIAA Paper2000–2254.

CONWAY, S. 2004 Implementation and Validation of an Engine Nacelle Boundary Con-dition in Edge 3.2. Scientific Report FOI-R--1320--SE. Computational Physics Depart-ment, Aeronautics Division, FOI.

EISFELD, B. 2003 Die Eigenwerte des Systems aus RANS- und Reynolds-Spannungsgleichungen. Tech. Rep. IB 124-2003/1. DLR.

ELIASSON, P. & NORDSTROM, J. 1996 Computations and Measurements of UnsteadyPressure on a Delta Wing with an Oscillating Flap. In Proceedings to ECCOMAS 1996,pp. 478–484.

ELIASSON, P. & WALLIN, S. 2000 Positive Multigrid Scheme with Two-Equation Turbu-lence Models. In Proceedings to ECCOMAS 2000.

GNOFFO, P. A. 1990 An Upwind-Biased, Point-Implicit Relaxation Algorithm for Viscous,Compressible Perfect-Gas Flows. Technical Report NASA TP-2953. NASA.

HAASELBACHER, A., MCGUIRK, J. & PAGE, G. 1999 Finite Volume Discretization As-pects on Mixed Unstructured Grids. AIAA Journal 37 (2), 177–184.

HELLSTEN, A. 1998 On the solid-wall boundary condition for w in the k – ω - type turbu-lence modes. Scientific Report Report No B-50, Series B. Helsinki University of Tech-nology.

HELLSTEN, A. 2005 New advanced k–ω turbulence model for high-lift aerodynamics.AIAA Journal 43 (9), 1857–1869.

JIRASEK, A. 2005 Vortex-generator model and its application to flow control. Journal ofAircraft 42 (6), 1486–1491.

JIRASEK, A. 2006 Mass flow boundary conditions for subsonic inflow and outflow bound-ary. AIAA Journal, 44 (5), 939–947.

KOK, J. C. 2000 Resolving the dependence on freestream vales of the k − ω turbulencemodel. AIAA Journal 38, 1292–1295.

MARTINELLI, L. 1987 Calculation of Viscous Flows with Multigrid Methods. PhD thesis,MAE Department, Princeton University.

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MENTER, F. R. 1994 Two-equation eddy-viscosity turbulence models for engineering ap-plications. AIAA Journal 32 (8), 1598–1605.

PENG, S.-H. 2005 Hybrid RANS-LES modelling based on zero- and one-equation modelsfor turbulent flow simulation. In Proceedings of 4th Int. Symp. Turb. and Shear FlowPhenomena, , vol. 3, pp. 1159–1164.

PENG, S.-H. 2006 Algebraic hybrid RANS-LES modelling applied to incompressible andcompressible turbulent flows. AIAA Paper 2006–3910.

PENG, S.-H., DAVIDSON, L. & HOLMBERG, S. 1997 A modified low-Reynolds-numberk − ω model for recirculating flows. ASME J. Fluids Engng 119, 867–875.

SLATER, J. W. 2001 Verification assessment of flow boundary conditions for cfd analysisof supersonic inlet flows. AIAA Paper 2001–3882.

SMAGORINSKY, J. 1963 General circulation experiments with the primitive equations.Monthly Weather Review 91 (3), 99–164.

SMITH, J. 2005 Aeroelastic functionality in Edge, Initial Implementation and Validation.Scientific Report FOI-R--1485--SE. Computational Physics Department, Aeronautics Di-vision, FOI.

SPALART, P. R. & ALLMARAS, S. R. 1994 A one-equation turbulence model for aerody-namic flows. La recherche Aerospatiale 1, 5–21.

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SPEZIALE, C. G., SARKAR, S. & GATSKI, T. G. 1991 Modelling the pressure-strain cor-relation of turbulence. J. Fluid Mech. 227, 245–272.

WALLIN, S. & JOHANSSON, A. V. 2000 An explicit algebraic Reynolds stress model forincompressible and compressible turbulent flows. J. Fluid Mech. 403, 89–132.

WALLIN, S. & JOHANSSON, A. V. 2002 Modelling streamline curvature effects in explicitalgebraic Reynolds stress turbulence models. International Journal of Heat and FluidFlows 23 (5), 721–730.

WILCOX, D. C. 1988 Reassessment of the Scale Determining Equation for Advanced Tur-bulence Models. AIAA Journal 26 (11), 1299–1310.

WILCOX, D. C. 1994 Simulation of Transition with a Two-Equation Turbulence Model.AIAA J. 32, 1192–1198.

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52


A Appendix

A.1 3D Inviscid Jacobians and Eigenvectors

The 3D Euler system is written as

∂

∂tq +

∂

∂xF +

∂

∂yG+

∂

∂zH = 0, (173)

where q = (ρ, ρu, ρv, ρw,E)T are the conservative variables. Introducing the Jacobian ofthe conservative variables A, B, and C gives

∂

∂tq +A

∂

∂xq +B

∂

∂yq + C

∂

∂zq = 0, (174)

Alternatively the primitive variables may be used, denoted v = (ρ, u1, u2, u3, p)T

∂

∂tv + A

∂

∂xv + B

∂

∂yv + C

∂

∂zv = 0, (175)

where A, B, and C are the Jacobians of the primitive variables, denoted non-conservativeJacobians. With the transformation matrix

M =∂

∂vq, (176)

between primitive and conservative variables the following relations between the Jacobianshold

A = M−1AM, B = M−1BM, and C = M−1CM. (177)

Introducing the notationAn = nxA+ nyB + nzC, (178)

the conservative Jacobians can be made diagonal as follows

An = RΛR−1, (179)

and the non-conservative Jacobians

An = nxA+ nyB + nzC, (180)

can be made diagonal asAn = LΛL−1. (181)

The diagonal matrix Λ contains the eigenvalues to the Jacobians An and A, the matrices Rand L contain the right eigenvectors as columns to the conservative and non-conservative

53


Jacobians, respectively. The eigenvectors given below are scaled in such a way that thecharacteristic variables have the dimension of a pressure. Note that the scaling of the eigen-vectors is not unique, nor is the choice of the eigenvectors.

The matrices are the following

M =

1 0 0 0 0u1 ρ 0 0 0u2 0 ρ 0 0u3 0 0 ρ 0u2

2 ρu1 ρu2 ρu31

γ−1

(182)

M−1 =

1 0 0 0 0u1ρ

1ρ 0 0 0

u2ρ 0 1

ρ 0 0u3ρ 0 0 1

ρ 0u2

2 −(γ − 1)u1 −(γ − 1)u2 −(γ − 1)u3 γ − 1

(183)

L =

nxc2

ny

c2nzc2

1c2

1c2

0 nzρc

ny

ρcnxρc

nxρc

nzρc 0 nx

ρcny

ρcny

ρcny

ρcnxρc 0 nz

ρcnzρc

0 0 0 1 1

(184)

L−1 =

nxc

2 0 nzρc −nyρc −nx

nyc2 −nzρc 0 nzρc −ny

nzc2 nyρc −nxρc 0 −nz

0 nxρc2

nyρc2

nzρc2

12

0 −nxρc2 −nyρc

2 −nzρc2

12

(185)

Λ =

u · n 0 0 0 0

0 u · n 0 0 00 0 u · n 0 00 0 0 u · n+ c 00 0 0 0 u · n− c

(186)

R = ML =

nxc2

ny

c2nzc2

1c2

1c2

u1nx

c2u1ny−cnz

c2u1nz+cny

c2u1+cnx

c2u1−cnx

c2u2nx+cnz

c2u2ny

c2u2nz−cnx

c2u2+cny

c2u2−cny

c2u3nz−cny

c2u3ny+cnx

c2u3nz

c2u3+cnz

c2u3−cnz

c2

nx|u|22c2

+ a1 ny|u|22c2

+ a2 nz|u|22c2

+ a3H+c(u·n)

c2H−c(u·n)

c2

(187)

wherea1 = (u2nz − u3ny)/c,a2 = (u3nx − u1nz)/c,a3 = (u1ny − u2nx)/c.

(188)

54


R−1 =

nxc

2b2 b1u1nx b1u2nx + nzc b1u3nx − nyc −b1nx − b4nyc

2b2 b1u1ny − nzc b1u2ny b1u3ny + nxc −b1ny − b5nzc

2b2 b1u1nz + nyc b1u2nz − nxc b1u3nz −b1nz − b6c2

2 b3cnx−b1u1

2cny−b1u2

2cnz−b1u3

2γ−1

2c2

2 b3−cnx−b1u1

2−cny−b1u2

2−cnz−b1u3

2γ−1

2

(189)

whereb1 = γ − 1,

b2 = 1− (γ − 1)M2/2,

b3 = (γ − 1)M2/2− (u · n)/c,b4 = c(u2nz − u3ny),b5 = c(u3nx − u1nz),b6 = c(u1ny − u2nx),

(190)

and where H is the total enthalpy

H =E + p

ρ=

c2

γ − 1+

12|u|2 (191)

and c the speed of sound and M = |u|/c the Mach number.

55


56

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