R–5375

IMPLICIT METHOD FOR THE 3DEULER EQUATIONS

MARTIN KYNCLJAROSLAV PELANT

VZLU – AEROSPACE RESEARCH AND TEST ESTABLISHMENTBERANOVYCH 130, 199 05 PRAGUE, CZECH REPUBLIC

c© 2012

R–5375

Contents

1 INTRODUCTION 4

2 THE EULER EQUATIONS 4

3 FVM DISCRETIZATION OF THE INVISCID PROBLEM 53.1 Inner Face Values Discretization (Riemann Solver) . . . . . . . . . . . . . . . . . . . . 83.2 Vijayasundaram Numerical Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.3 Boundary Face Values Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4 THE RIEMANN PROBLEM FOR THE SPLIT EULER EQUATIONS 114.1 The Incomplete Initial Value Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5 BOUNDARY CONDITION BY PREFERENCE OF PRESSURE 14

6 BOUNDARY CONDITION BY PREFERENCE OF VELOCITY 156.1 B.C. for the Impermeable Wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

7 BOUNDARY CONDITION BY PREFERENCE OF TOTAL QUANTITIES AND DIREC-TION OF VELOCITY 16

8 SOFTWARE AND EXAMPLES 19

9 Examples 239.1 Laval Nozzle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239.2 A Body in a Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

10 CONCLUSIONS 26

List of Figures

3-1 Coordinate transformation for the inner edges in 2D. . . . . . . . . . . . . . . . . . . . 83-2 Coordinate transformation for the boundary edge in 2D. . . . . . . . . . . . . . . . . . . 114-1 Structure of the solution of the Riemann problem (16),(17),(18) . . . . . . . . . . . . . . 124-2 The incomplete Riemann problem for the 1D Euler equations. General form of the solution. 145-1 Algorithm for the solution of the problem (16),(17),(31),(33),(32),(34),(35) at the half

line SB = (0, t); t > 0. The value of the pressure p⋆ is prescribed. Possible situationsare illustrated by the pictures showing ΩL ∪ ΩHTL ∪ Ω⋆L region. The sought boundarystate located at the time axis, marked red. . . . . . . . . . . . . . . . . . . . . . . . . . 15

6-1 Algorithm for the solution of the problem (16),(17),(36),(33),(37),(34),(35) at the halfline SB = (0, t); t > 0. Possible situations are illustrated by the pictures showingthe region ΩL ∪ ΩHTL ∪ Ω⋆L with the sought boundary state located at the time axis. . . 16

7-1 Coordinate transformation for the boundary edge Γij in 2D. The axis x1 is in the oppositedirection to the prescribed direction of the velocity. . . . . . . . . . . . . . . . . . . . . 17

7-2 Algorithm for the solution of the problem (24)-(29),(40)-(44) at the half line SB =(0, t); t > 0. In the case of failure, the problem doesn’t have a solution, and the valuefor u⋆ is chosen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

8-1 10% cylindrical bump, initial mesh and boundary numbering. . . . . . . . . . . . . . . . 198-2 C3DEEI: allowed elements, vertices numbering. . . . . . . . . . . . . . . . . . . . . . . 208-3 3D inviscid flow computation (coarse mesh). Mach number and pressure isolines. (C3D) 239-1 3D inviscid computation, coarse mesh with 1896 prismatic elements. Pressure and Mach

number isolines, real time of the computation for chosen methods. (C3DEEI) . . . . . . 24

- 2 -

R–5375

9-2 3D inviscid computation, coarse mesh with 8208 hexahedral elements. Pressure andMach number isolines (C3DEEI). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

9-3 3D inviscid computation, mesh with 109490 prismatic elements. Pressure and Machnumber isolines (C3DEEI). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

9-4 3D inviscid computation, mesh with 118835 prismatic elements. Pressure and Machnumber isolines, real time of the computation for chosen methods. (C3DEEI) . . . . . . 26

- 3 -

R–5375

1 INTRODUCTION

The physical theory of the compressible fluid motion is based on the principles of conservation laws ofmass, momentum, and energy. The mathematical equations describing these fundamental conservationlaws form a system of partial differential equations. In this article we will work with the Euler equations,and we focus on the numerical solution of these equations. The correct design of the boundary condi-tions plays also an important role in the numerical modeling of the processes involved. We choose thewell-known finite volume method to discretize the analytical problem, represented by the system of theequations in generalized (integral) form. To apply this method we split the area of the interest into theelements, and we construct a piecewise constant solution in time. The crucial problem of this methodlies in the evaluation of the so-called fluxes through the edges/faces of the particular elements. In orderto compute these fluxes, various methods can be used. One of the most accurate method (and perhapsthe most accurate method) is based on the solution of the so-called Riemann problem for the 2D/3D splitEuler equations. Unfortunately, the exact solution of this problem cannot be expressed in a closed form,and has to be computed by an iterative process (to given accuracy). Therefore this method is also oneof the most demanding. Nevertheless, on account of the accuracy of the Riemann solver, we decidedto use the analysis of the exact solution also for the discretization of the fluxes through the boundaryedges/faces. The right-hand side initial condition, forming the local Riemann problem, is not known atthe boundary faces. In some cases (far field boundary) it is wise to choose the right-hand side initialcondition here as the solution of the local Riemann problem with given far field values, which givesbetter results than the solution of the linearized Riemann problem, see [2]. Another boundary conditionbased on the exact Riemann problem solution, simulating the impermeable wall on move, was shown in[20, pages 221-225], where the right-hand side initial condition is constructed in a special way, in orderto obtain the desired solution. We show, that the right-hand side initial condition for the local problemcan be partially replaced by the suitable complementary condition. This idea was introduced by RNDr.Jaroslav Pelant, CSc. in [17]. We construct own algorithms for the solution of the boundary problem,and we use it in the numerical examples.

2 THE EULER EQUATIONS

Here we show the Euler equations in 3D. The system describes the conservation laws of mass, momentum,and energy. We present the Euler equations in the differential form

∂w

∂t+

3∑

s=1

∂f s(w)

∂xs= 0 in QT = Ω× (0, T ). (1)

Here w = w(x, t) is the state vector, x ∈ Ω, t denotes the time, QT is called a space-time cylinder, f s

are the inviscid (Euler) fluxes

w = (w1, w2, . . . , wN+2)T = (, v1, . . . , vN , E)T ,

f s(w) = (vs, vsv1 + δs1p, . . . , vsvN + δsNp, (E + p) vs)T

with v = (v1, . . . , vN )T denoting the velocity vector, being the density, p the pressure, E the totalenergy. The system (1):

∂

∂t

v1v2v3E

+∂

∂x1

v1v2

1 + p

v1v2v1v3

(E + p) v1

+∂

∂x2

v2v2v1v2

2 + p

v2v3(E + p) v2

+∂

∂xs

v3v3v1v3v2v2

3 + p

(E + p) v3

= 0,

The equations (1) represent a system of hyperbolic partial differential equations. We assume that thepartial derivatives exist, such that the differential formulation (1) makes sense. Within the numericalsolution of this problem we usually consider the equations in the more general integral form, which

- 4 -

R–5375

allows the discontinuities in the solution. It is necessary to specify the closure conditions. We considerthe thermodynamical relations yielding from the caloric equation of state

p = (γ − 1)(E − v2/2). (2)

The system of the Euler equations is rotationally invariant (see e.g. [5, page 108]). We can choose anew Cartesian coordinate system (x1, x2, x3) such that the origin is at the point σ, and the axis x1 is inthe direction of a chosen unit vector n = (n1, n2, n3), |n| = 1.This transformation can be written

x1x2x3

= Q0

x1x2x3

+ σ, with Q0 =

n

o

p

=

n1 n2 n3

o1 o2 o3p1 p2 p3

. (3)

Here Q0 is the rotation matrix, the elements of the rotation matrix are not all independent. The vectorsn,o,p must have the properties

|n| = |o| = 1, n · o = 0, p = n× o.

We can choose the vector o = (o1, o2, o3) such that n · o = 0, |o| = 1. And then the vector p =(p1, p2, p3) is determined by p = n × o. The axis x2 points in the direction of the vector o, and theaxis x3 in the direction of the vector p. The matrix Q0 is orthogonal, and the vectors are orthogonal unitvectors. The inverse of the orthogonal matrix Q0 is Q−1

0 = QT0 . The transformation of the state vector

w yields the state vector q in the new coordinates. It is

q = Q w,

w = Q−1 q,(4)

with

Q =

1 0 0 0 00 n1 n2 n3 00 o1 o2 o3 00 p1 p2 p3 00 0 0 0 1

, Q−1 =

1 0 0 0 00 n1 o1 p1 00 n2 o2 p2 00 n3 o3 p3 00 0 0 0 1

.

Using this transformation of the coordinates, the Euler Equations keep the form

∂q

∂t+

3∑

s=1

∂f s(q)

∂xs= 0.

3 FVM DISCRETIZATION OF THE INVISCID PROBLEM

Here we describe the so-called finite volume discretization of the problem (1) in the domain Ω. By Ωh letus the denote the polyhedral approximation of Ω. The system of the closed polyhedrons with mutuallydisjoint interiors Dh = Dii∈J , where J ⊂ Z+ = 0, 1, . . . is an index set and h > 0, will be called afinite volume mesh. This system Dh approximates the domain Ω

Ωh =⋃

i∈J

Di.

The elements Di ∈ Dh are called finite volumes. For two neighboring elements Di,Dj we set

Γij = ∂Di ∩ ∂Dj = Γji.

- 5 -

R–5375

Here we will work with the so-called regular meshes, i.e. the intersection of two arbitrary (different)elements is either empty or it consists of a common vertex or a common edge or a common face (in 3D).The boundary ∂Di of each element Di is

∂Di =⋃

Γij∈ΓDi

Γij. (5)

Here the set ΓDi= Γij ; Γij ⊂ ∂Di forms the boundary ∂Di. By nij let us denote the unit outer normal

to ∂Di on Γij. Let us construct a partition 0 = t0 < t1 < . . . of the time interval [0, T ] and denote thetime steps τk = tk+1 − tk. We integrate the system (1) over the set Di × (tk, tk+1). With the integralform of the equations we can study a flow with discontinuities, such as shock waves, too.

∫

Di

∫ tk+1

tk

∂w

∂tdx dt+

∫ tk+1

tk

∫

Di

3∑

s=1

∂f s(w)

∂xsdx dt = 0 (6)

Using the Green’s theorem on Di it is

∫

Di

3∑

s=1

∂f s(w)

∂xsdx =

∫

∂Di

3∑

s=1

f s(w)ns dS. (7)

Here n = (n1, . . . , n3) is the unit outer normal to ∂Di. Further we use (5), and we rewrite (6)

∫

Di

(w(x, tk+1)−w(x, tk)) dx+

∫ tk+1

tk

∑

Γij∈ΓDi

∫

Γij

3∑

s=1

fs(w)(nij)s dS dt = 0 (8)

We define a finite volume approximate solution of the system studied (1) as a piecewise constant vector-valued functions wk

h, k = 0, 1, . . . , where wkh is constant on each element Di, and tk is the time instant.

By wki we denote the value of the approximate solution on Di at time tk. We approximate the integral

over the element Di∫

Di

w(x, tk) dx ≈ |Di|wki . (9)

Further we proceed with the approximation of the fluxes. Usually the flux∑3

s=1 f s(w)(nij)s|Γijis

being approximated by a numerical flux at suitable time instant tl

3∑

s=1

f s(w)(nij)s|Γij≈ H(wl

i,wlj,nij),

with wli, wl

j denoting the approximate solution on the elements adjacent to the edge Γij at the timeinstant tl. In the case of a boundary face the vector wl

j has to be specified. Here we show the numericalflux based on the solution of the Riemann problem for the split Euler equations (shown later in Section4). By wl

Γijlet us denote the state vector w at the center of the edge Γij at the time instant tl, and let

us suppose wlΓij

is known. Evaluation of these values will be a question of the further analysis, here weuse them to approximate the integrals with the one-point rule

∫

Γij

3∑

s=1

f s(w(x, tl))(nij)s dS ≈ |Γij|3∑

s=1

f s(wlΓij

)(nij)s. (10)

Now it is possible to approximate the system (8) by the following explicit finite volume scheme

|Di|(wk+1i −wk

i ) + τk∑

Γij∈ΓDi

|Γij |

3∑

s=1

f s(wkΓij

)(nij)s = 0. (11)

- 6 -

R–5375

With this finite volume formula one computes the values of the approximate solution at the time instanttk+1, using the values from the time instant tk, and by evaluating the values wk

Γijat the faces Γij . In

order to achieve the stability of the used method, the time step τk must be restricted by the so-called CFLcondition, see [5]. The crucial problem of this discretization lies with the evaluation of the edge valueswk

Γij. Or one deals with the problem of finding the face fluxes H(wk

i ,wkj ,nij).

Here we will work with the implicit scheme

|Di|(wk+1i −wk

i ) + τk∑

Γij∈ΓDi

|Γij|H(wk+1i ,wk+1

j ,nij) = 0. (12)

The crucial problem of this discretization lies with the evaluation of the face fluxes H(wk+1i ,wk+1

j ,nij).One possibility is to use the linearization via the Taylor expansion of the vector function H(u,v,n).

We assume that the mapping H(u,v,n) is differentiable. Let DH(u,v,n)Du , DH(u,v,n)

Dv be the Jacobimatrix of the mapping H(u, ., .), H(.,v, .)

DH

Du=

(

∂Hi

∂uk

)5

i,k=1

=

∂H1∂u1

∂H1∂u2

∂H1∂u3

∂H1∂u4

∂H1∂u5

∂H2∂u1

∂H2∂u2

∂H2∂u3

∂H2∂u4

∂H2∂u5

∂H3∂u1

∂H3∂u2

∂H3∂u3

∂H3∂u4

∂H3∂u5

∂H4∂u1

∂H4∂u2

∂H4∂u3

∂H4∂u4

∂H4∂u5

∂H5∂u1

∂H5∂u2

∂H5∂u3

∂H5∂u4

∂H5∂u5

,DH

Dv=

(

∂Hi

∂vk

)5

i,k=1

We will use shorter notation H |kij = H(wki ,w

kj ,nij),

DHDu

∣

∣

∣

k

ij= DH

Du (wki ,w

kj ,nij),

DHDv

∣

∣

∣

k

ij=

DHDv (wk

i ,wkj ,nij). We approximate the flux H |k+1

ij as

H|k+1

ij ≈ H|kij +DH

Du

∣

∣

∣

∣

k

ij

(wk+1

i −wki ) +

DH

Dv

∣

∣

∣

∣

k

ij

(wk+1

j −wkj )

Then the scheme (12) can be writen as

I+τk|Di|

∑

Γij∈ΓDi

|Γij |DH

Du

∣

∣

∣

∣

k

ij

∆wi +τk|Di|

∑

Γij∈ΓDi

|Γij |DH

Dv

∣

∣

∣

∣

k

ij

∆wj = −τk|Di|

∑

Γij∈ΓDi

|Γij |H |kij (13)

Here we used the notation ∆wi = wk+1i − wk

i , i = 1, ..., n with n denoting the number of finitevolumes. The system in a matrix form

A∆w = b. (14)

with∆w = (∆w1,∆w2, ...,∆wn)T , b = (− τk

|D1|

∑

Γ1l∈ΓD1|Γ1l| H |k1l , ...,−

τk|Dn|

∑

Γnl∈ΓDn|Γnl| H |knl)

T ,

and A is the block matrix with blocks

aij =

I+ τk|Di|

∑

l∈ΓDi|Γil|

DHDu

∣

∣

∣

k

il, i = j,

τk|Di|

|Γij |DHDv

∣

∣

∣

k

ij, i 6= j, j ∈ S(i),

0, i 6= j, j /∈ S(i).

The contributions of the face Γij (between the elements Di,Dj ) into the system A∆w = b

. .

I +τk

|Di|∑

Γil∈ΓDi|Γil| DH

Du

∣

∣

∣

k

il

τk|Di|

|Γij | DHDv

∣

∣

∣

k

ij. .

. .

τk|Dj | |Γji| DH

Dv

∣

∣

∣

k

ji. . I +

τk|Dj |

∑

Γjl∈ΓDj|Γjl| DH

Du

∣

∣

∣

k

jl

. . .

.

∆wi.

.

∆wj.

=

.τk

|Di|∑

Γil∈ΓDi|Γil| H|kil

.

.τk

|Dj |∑

Γjl∈ΓDj|Γjl| H|kjl

.

- 7 -

R–5375

Further we assume that the flux H is conservative: H(u,v,n) = −H(u,v,−n), hence

H |kji = − H |kij ,DH

Du

∣

∣

∣

∣

k

ji

= −DH

Dv

∣

∣

∣

∣

k

ij

,DH

Dv

∣

∣

∣

∣

k

ji

= −DH

Du

∣

∣

∣

∣

k

ij

.

The construction of the matrix A and the right side b can be made as follows. We set A = I, b = 0. Then

the flux H |kij and its Jacobians DHDu

∣

∣

∣

k

ij, DH

Dv

∣

∣

∣

k

ijare computed for each face Γij , updating the values in

A, b :

bi = bi +τk|Di|

|Γij | H |kij , bj = bj −τk|Dj |

|Γij | H |kij ,

aii = aii +τk|Di|

|Γij |DHDu

∣

∣

∣

k

ij, aij =

τk|Di|

|Γij |DHDv

∣

∣

∣

k

ij,

ajj = ajj −τk|Dj |

|Γij|DHDv

∣

∣

∣

k

ij, aji = − τk

|Dj ||Γij|

DHDu

∣

∣

∣

k

ij.

The main problem of this discretization lies with the evaluation of the face fluxes H |kij and its Jacobian

DHDu

∣

∣

∣

k

ij, DH

Dv

∣

∣

∣

k

ij.

3.1 Inner Face Values Discretization (Riemann Solver)

One possibility of the face flux evaluation is to approximate the face values wkΓij

and then compute thenumerical flux

H|kij =

3∑

s=1

f s(wkΓij

)(nij)s. (15)

To approximate the face values wkΓij

at time instant tk we solve the simplified system (16) in the vicinity

of the face Γij in time with the initial condition formed by the state vectors wki and wk

j . Using therotational invariance of the Euler equations, the system (1) is expressed in a new Cartesian coordinatesystem x1, x2, x3 with the origin at the center of the gravity of Γij and with the new axis x1 in thedirection of n = (n1, n2, n3), given by the face normal n = nij .

x1

x2

Γij

Di Dj

x1nij

Fig. 3-1: Coordinate transformation for the inner edges in 2D.

Then the derivatives with respect to x2, x3 are neglected and we get the so-called split 3D Euler equations,see [5, page 138]:

∂q

∂t+

∂f1(q)

∂x1= 0. (16)

The values wki and wk

j adjacent to the face Γij are known, forming the initial conditions

q(x1, 0) = qL = Q wki , x1 < 0, (17)

q(x1, 0) = qR = Q wkj , x1 > 0. (18)

- 8 -

R–5375

The transformation matrix Q is defined in (4). In this work we will refer to these initial conditions asto the left-hand side initial condition (17) and the right-hand side initial condition (18). The problem(16), (17), (18) has a unique “solution” in (−∞,∞) × (0,∞), the analysis can be found in [5, page138], we will show the analysis of this problem later in Section 4. Let qRS(qL, qR, x1, t) denote thesolution of this problem at the point (x1, t). We are interested in the solution of this local problem at thetime axis, which is the sought solution in the local coordinates qΓij

= qRS(qL, qR, 0, t). The backwardtransformation (4) of the state vector qΓij

into the global coordinates is

wkΓij

= Q−1qΓij= Q−1 qRS(Q wk

i ,Q wkj , 0, t). (19)

The constructed numerical flux can be written as

H(u,v,n) :=

3∑

s=1

f s

(

Q−1 qRS(Q u,Q v, 0, t))

ns. (20)

The described process of finding the face values wkΓij

is independent on the choice of the vectors o,p,determining the transformation matrix Q defined in (4), see [7].

3.2 Vijayasundaram Numerical Flux

Another possibility of the face flux discretization is to use some of the properties of the solved system(1). The system is a first order quasilinear system of partial differential equations, see [4, page 385], andthe flux can be written

3∑

s=1

f s(w)ns =3∑

s=1

nsAs(w)w, with As =Df s

Dw.

Using the rotational invariance of the equations of the inviscid flow and the transformation of thecoordinates defined in (4) it is

3∑

s=1

nsAs(w)w = Q−1A1(Qw)Qw = Q−1TΛT−1Qw = TΛT−1w.

Here A1(Qw) = TΛT−1 is diagonizable matrix (due to the hyperbolicity of the problem), shown in [5].Matrices Λ, T = Q−1T, T−1 = T−1Q depend on w as

Λ = diag(u − a, u, u, u, u + a),

T =

1 1 1 1 1U − n1a U U − o1a U − p1a U + n1aV − n2a V V − o2a V − p2a V + n2aW − n3a W W − o3a W − p3a W + n3a

h− au |v|2

2|v|2

2 − av |v|2

2 − aw h+ au

,

˜T−1 =1

a2

1

2

(

γ1

2|v|2 + ua

)

− 1

2(n1a+ γ1U) − 1

2(n2a+ γ1V ) − 1

2(n3a+ γ1W ) γ1

2

a2 − a(v + w)− γ1

2|v|2 γ1U + (o1 + p1)a γ1V + (o2 + p2)a γ1W + (o3 + p3)a −γ1

va −o1a −o2a −o3a 0wa −p1a −p2a −p3a 0

1

2

(

γ1

2|v|2 − ua

)

1

2(n1a− γ1U) 1

2(n2a− γ1V ) 1

2(n3a− γ1W ) γ1

2

.

Here U = w2/w1, V = w3/w1,W = w4/w1 are the components of the velocity in global coordinates,and u = n1U + n2V + n3W, v = o1U + o2V + o3W, w = p1U + p2V + p3W are the components ofthe velocity in local coordinates x1, x2, x3 defined in (3). Further a =

√

γp/ is the speed of the sound,

h = a2/γ1 +|v|2

2 is the total specific enthalpy, and γ1 = γ − 1.

- 9 -

R–5375

Now we approximate the flux∑3

s=1 f s(wkΓij

)(nij)s = TΛT−1(wkΓij

)wkΓij

≈ H |kij

H |kij = P+

(

wki +wk

j

2,n

)

wki + P−

(

wki +wk

j

2,n

)

wkj , (21)

withP+ = TΛ+T−1, P− = TΛ−T−1,

Λ± = diag(λ±1 , λ

±2 , λ

±3 , λ

±4 , λ

±5 ), λ+

i = max(λi, 0), λ−i = min(λi, 0).

The Vijayasundaram numerical flux can be written as (see [4, page 456] )

H(u,v,n) = P+

(

u+ v

2,n

)

u+ P−

(

u+ v

2,n

)

v.

The Jacobi mapping of such a numerical flux at Γij and time instant tk can be approximated byDHDu

∣

∣

∣

k

ij, DH

Dv

∣

∣

∣

k

ijwith

DH

Du

∣

∣

∣

∣

k

ij

= P+

(

wki +wk

j

2,n

)

,DH

Dv

∣

∣

∣

∣

k

ij

= P−

(

wki +wk

j

2,n

)

. (22)

In practical applications we can choose the directiono =

(−n2/√

1− n23, n1/

√

1− n23, 0), n3 ≤ n2,

(−n3/√

1− n22, 0, n1/

√

1− n22), n3 > n2,

and then compute p = n× o.

3.3 Boundary Face Values Discretization

For the boundary faces we choose the discretization (15), where wkΓij

must be given or computed

H|kij =3∑

s=1

f s(wkΓij

)(nij)s.

LetΓij be the face of the element Di laying at the boundary of the computational area. To approximatethe face values wk

Γijat time instant tk we solve the incomplete simplified system (16) in the vicinity of

the face Γij in time with the initial condition determined by the state vector wki . Similarily to Section 3.1

we use the rotational invariance of the Euler equations, and we express the system (1) in a new Cartesiancoordinate system x1, x2, x3 with the origin at the center of gravity of Γij and with the new axis x1 in thedirection of vector n = (n1, n2, n3). The vector n corresponds to the face normal vector nij pointingout of the volume Di, however for certain boundary conditions it is convenient to define n in a differentway (still pointing out of the volume Di). Then the derivatives with respect to x2, x3 are neglected andwe get the so-called split 3D Euler equations (16):

∂q

∂t+

∂f1(q)

∂x1= 0.

The state vector wki on the element Di adjacent to the face Γij is known, forming the initial condition

for the local problem (16),(17)

q(x1, 0) = qL = Q wki , x1 < 0.

The transformation matrix Q is defined in (4). We seek the boundary state qΓij= q(0, t). Once the local

state vector qΓijis known, we use transformation (4) to evaluate the face state vector wk

Γijin global

coordinates, it is wkΓij

= Q−1qΓij.

- 10 -

R–5375

x1

x2

Γij

Di

x1nij

Fig. 3-2: Coordinate transformation for the boundary edge in 2D.

We suppose that the solution of the problem (16),(17),(23) in ΩB is a solution of the problem(16),(17),(18) in ΩB, described later in Section 4, for some (unknown) qR satisfying the pressurepositivity condition (30). By adding properly chosen equations into the system (16),(17) it is possibleto reconstruct the boundary state qB such that the system (16),(17),(23) has a unique solution in ΩB ,see Section 4. We will refer to these added equations as to complementary conditions. The suitablecomplementary conditions may give the unique boundary state qB, as the solution of the incompletesystem (16),(17) at the half line SB = (0, t); t > 0. Several choices of the complementary conditionswill be discussed further. Here we introduce the initial-boundary value problem for the 3D split Eulerequations. We consider the system (16) in the set ΩB = (−∞, 0) × (0,∞), equipped with the initialcondition (17), and the boundary condition

q(0, t) = qB , t > 0. (23)

The problem of the boundary condition is to choose the boundary state qB such a way that the system(16),(17),(23) is well-posed, i.e. it has a unique solution (entropy weak) in the considered set ΩB. Wesuppose that the solution of the problem (16),(17),(23) in ΩB is a solution of the problem (16),(17),(18) inΩB, described later in Section 4, for some (unknown) qR satisfying the pressure positivity condition (30).By adding properly chosen equations into the system (16),(17) it is possible to reconstruct the boundarystate qB such that the system (16),(17),(23) has a unique solution in ΩB , see Section 4. We will refer tothese added equations as to complementary conditions. The suitable complementary conditions maygive the unique boundary state qB , as the solution of the incomplete system (16),(17) at the half lineSB = (0, t); t > 0. Several choices of the complementary conditions will be discussed further in theSection 5 - 7.

4 THE RIEMANN PROBLEM FOR THE SPLIT EULER EQUATIONS

For many numerical methods dealing with the two or three dimensional equations, describing the com-pressible flow, it is useful to solve the split Riemann problem. Here we will follow in 3D, analogouslyit is possible to proceed in 2D. It is the system of the equations (16), equipped with the initial conditions(17),(18)

∂q

∂t+

∂f1(q)

∂x1= 0, q(x1, 0) =

qL, x1 < 0,qR, x1 > 0.

‘Split’ means here that we still have 5 equations in 3D, but the dependence on x2, x3 is neglected,and we deal with the system for one space variable x1. The system (16) is considered in the setQ∞ = (−∞,∞) × (0,+∞). The solution of this simplified system is useful when dealing with theinviscid fluxes within various numerical methods.

The solution of this problem is fundamentally the same as the solution of the Riemann problem forthe 1D Euler equations, see [5, page 138]. In fact, the solution for the pressure, the first component of

- 11 -

R–5375

the velocity, and the density is exactly the same as in one-dimensional case. It is a characteristic featureof the hyperbolic equations, that there is a possible raise of discontinuities in solutions, even in the casewhen the initial conditions are smooth, see [3, page 390]. Here the concept of the classical solution is toorestrictive, and therefore we seek a weak solution of this problem. To distinguish physically admissiblesolutions from nonphysical ones, entropy condition must be introduced, see [3, page 396]. By the solutionof the problem (16),(17),(18) we mean the weak entropy solution of this problem in Q∞. The analysisto the solution of this problem can be found in many books, i.e. [5], [3], [20]. The general theoremon the solvability of the Riemann problem can be found in [5, page 88]. Here we summarize, that theproblem has a unique solution for certain choice of the initial conditions. This solution can be writtenfor t > 0 in the similarity form q(x1, t) = q(x1/t), where q(x1/t) : IR → IR3 ([5, page 82]). Nowwe will show the form of the possible solution q = q(x1, t) of the Riemann problem (16),(17),(18). Thesolution is piecewise smooth and its general form can be seen in Fig. 4-1, where the system of half linesis drawn. These half lines define regions, where q is either constant or given by a smooth function. Let

t

x10

x1t =sHL sTL=

x1t

x1t =u⋆

x1t =sTR sHR=

x1t

ΩHTL

ΩL

Ω⋆L Ω⋆R

ΩR

ΩHTR

Fig. 4-1: Structure of the solution of the Riemann problem (16),(17),(18)

us define the open sets called wedges (see Fig. 4-1):

ΩL = (x1, t);x1t < sHL; t > 0,

ΩHTL = (x1, t); sHL <x1t < sTL; t > 0,

Ω⋆L = (x1, t); sTL < x1t < u⋆; t > 0,

Ω⋆R = (x1, t); u⋆ <x1t < sTR; t > 0,

ΩHTR = (x1, t); sTR < x1t < sHR; t > 0,

ΩR = (x1, t); sHR < x1t ; t > 0.

We will refer to the set ΩHTL as to the left wave, and the set ΩHTR will be called the right wave. Thesolution in ΩHTL and in ΩHTR is continuous. Let us denote

q|ΩHTL= qHTL, q|ΩHTR

= qHTR.

The solution in ΩL, Ω⋆L, Ω⋆R, ΩR is constant (see e.g. [5, page 128]), we write

q|ΩL= qL,

q|Ω⋆L= q⋆L,

q|Ω⋆R= q⋆R,

q|ΩR= qR.

It is more convenient to use the vector of primitive variables (, u, v, w, p) rather then the vector ofconservative variables (, u, v, w,E) in solving the Riemann problem. The exact solution of theRiemann problem has three waves in general, illustrated in Fig. 4-1. The wedges ΩL and Ω⋆L areseparated by the left wave (either 1-shock wave, or 1-rarefaction wave). There is a contact discontinuitybetween the regions Ω⋆L and Ω⋆R. Wedges Ω⋆R and ΩR are separated by the right wave (either 3-shockwave, or 3-rarefaction wave).

For a certain choice of the initial condition the solution is the same on either side of the particularwave, and the wave diminishes. This situation is covered in the general example with three waves

- 12 -

R–5375

in the solution. For example, given the initial condition with qL = qR, there is no discontinuity inthe solution. General analogy to this example is the solution with three waves separating the regionsΩL, Ω⋆L, Ω⋆R, ΩR with the same constant solution. The solution for the primitive variables can bedescribed as follows:

(, u, v, w, p)|ΩL= (L, uL, vL, wL, pL),

(, u, v, w, p)|Ω⋆L= (⋆L, u⋆, vL, wL, p⋆),

(, u, v, w, p)|Ω⋆R= (⋆R, u⋆, vR, wR, p⋆),

(, u, v, w, p)|ΩR= (R, uR, vR, wR, pR).

The folowing relations for these variables hold:

u⋆ = uL+

−(p⋆ − pL)

(

2(γ+1)L

p⋆+γ−1γ+1

pL

) 12

, p⋆ > pL

2γ−1

aL

[

1 −(

p⋆pL

)(γ−1)/2γ]

, p⋆ ≤ pL

(24)

⋆L =

L

γ−1γ+1

pLp⋆

+1

pLp⋆

+γ−1γ+1

, p⋆ > pL

L

(

p⋆pL

) 1γ , p⋆ ≤ pL

(25)

s1TL =

uL − aL

√

γ+12γ

p⋆pL

+γ−12γ

, p⋆ > pL

u⋆ − aL

(

p⋆pL

)(γ−1)/2γ, p⋆ ≤ pL

(26)

u⋆ = uR +

(p⋆ − pR)

(

2(γ+1)R

p⋆+γ−1γ+1

pR

) 12

, p⋆ > pR,

− 2γ−1

aR

[

1 −(

p⋆pR

)(γ−1)/2γ]

, p⋆ ≤ pR

(27)

⋆R =

R

p⋆pR

+γ−1γ+1

γ−1γ+1

p⋆pR

+1, p⋆ > pR

R

(

p⋆pR

) 1γ , p⋆ ≤ pR

(28)

s3TR =

uR + aR

√

γ+12γ

p⋆pR

+γ−12γ

, p⋆ > pR

u⋆ + aR

(

p⋆pR

)(γ−1)/2γ, p⋆ ≤ pR

(29)

Here aL =√

γpL/L, aR =√

γpR/R, and γ denotes the adiabatic constant. Further s1TL denotes”unknown left wave speed”, s3TR ”unknown right wave speed”. Note, that the system (24) - (29) is thesystem of 6 equations for 6 unknowns p⋆,u⋆,⋆L ⋆R,s1TL,s3TR. The solution of this system leads to anonlinear algebraic equation, and one cannot express the analytical solution of this problem in a closedform. The problem has a solution only if the pressure positivity condition is satisfied

uR − uL <2

γ − 1(aL + aR). (30)

We will use some of these relations to construct and solve the initial-boundary value problem which willbe the original result of this work.

RemarkOnce the pressure p⋆ is known, the solution on the left-hand side of the contact discontinuity dependsonly on the left-hand side initial condition (17). And similarily, with p⋆ known, only the right-hand sideinitial condition (18) is used to compute the solution on the right-hand side of the contact discontinuity.

4.1 The Incomplete Initial Value Problem

In this part we deal with the incomplete Riemann problem for the Euler equations described by the system(16) equipped with the left-hand side initial condition (17). This problem is not uniquely solvable in(−∞,∞)× (0,∞). The short analysis of this problem will help us to understand the construction of theboundary conditions performed in Sections 5 - 7.We have shown the unique entropy weak solution of the initial value problem (16),(17), equippedwith the initial condition (18) formed by an arbitrary state qR satisfying the pressure positivity condition(30). Therefore we may seek the solution of the incomplete initial problem (16),(17) in the same generalform, consisting of 4 constant states qL, q⋆L, q⋆R, qR separated by 3 waves (see Fig. 4-1). Now theprimitive variables in the right region ΩR are unknown.

Here we will point out the number of unknowns and the number of missing equations in particularregions in order to compose a uniquely solvable system.

• The solution in ΩL (across 0 wave)Here the solution L, uL, vL, wL, pL is known from the initial condition (17).

- 13 -

R–5375

x10

L, uL, vL, wL, pL

s1TL

(17)R, uR, vR, wR, pR

s3TR

u⋆, p⋆

⋆LvLwL

⋆RvRwR

Fig. 4-2: The incomplete Riemann problem for the 1D Euler equations. General form of the solution.

• The solution in ΩL ∪ΩHTL ∪ Ω⋆L (across 1 wave)There are three unknowns in the region Ω⋆L. It is the density ⋆L, the pressure p⋆, and the velocityu⋆. Also the speed s1TL of the left wave determining the position of the region ΩHTL is notknown. The solution components in ΩL∪ΩHTL∪Ω⋆L region must satisfy the system of equations(24)-(26). It is a system of three equations for four unknowns. We have to add another equation inorder to get the uniquely solvable system in ΩL ∪ ΩHTL ∪ Ω⋆L.

• The solution in ΩL ∪ΩHTL ∪ Ω⋆L ∪ Ω⋆R (accross 2 waves)There are four unknowns determining the solution in ΩL ∪ ΩHTL ∪ Ω⋆L region: the density ⋆L,the pressure p⋆, the velocity u⋆, and the speed of the left wave s1TL. Further the density ⋆R andthe velocity components vR, wR in the Ω⋆R region are not known. We deal with seven unknowns,whereas the system of three equations (24),(25),(26) must be satisfied. We have to add fourproperly chosen equations into this system of equations in order to get uniquely solvable systemfor 7 unknowns sought in ΩL ∪ ΩHTL ∪ Ω⋆L ∪ Ω⋆R region.

• The solution in ΩL ∪ΩHTL ∪ Ω⋆L ∪ Ω⋆R ∪ ΩHTR ∪ ΩR (accross 3 waves)We are interested in 11 unknowns ⋆R, ⋆L, p⋆, u⋆, R, pR, uR, vR, wR, s

1TL, s

3TR. There are 6

equations (24)-(29), that must be satisfied. We have to add 5 equations in order to get the uniquelysolvable system.

5 BOUNDARY CONDITION BY PREFERENCE OF PRESSURE

In this section we construct the boundary condition, i.e. the state qB in (23), preferring the given valuefor the pressure pGIVEN > 0. This boundary condition corresponds to the real-world problem, whenwe deal with the experimentally obtained pressure distribution at the boundary. We add the followingcomplementary conditions into the system (16),(17)

p⋆ := pGIVEN, (31)

pR = p⋆, (32)

⋆R := GIVEN, (33)

vR := vGIVEN, (34)

wR := wGIVEN. (35)

Here p⋆ denotes the pressure in Ω⋆L ∪ Ω⋆R part of the solution of the Riemann problem for the splitEuler equations, shown in Section 4. The conditions (31), (32) prescribe the given pressure wherever it

- 14 -

R–5375

is possible, solution in ΩL is governed by the condition (17). We seek the boundary state qB as theunique solution of the problem (16),(17),(31)-(35) at the half line SB = (0, t); t > 0. The system(24)-(29), (31)-(35) is uniquely solvable. The algorithm for the construction of the primitive variablesB , uB , vB , wB , pB at the half-line SB is shown in Fig. 5-1. The complete analysis of this problem isshown in [7]

p⋆ > pL p⋆≤ pL

s1 = uL −√

γpLL

√

γ+12γ

p⋆pL

+γ−12γ

u⋆ = uL − (p⋆ − pL)

(

2(γ+1)ρL

p⋆+γ−1γ+1

pL

) 12

⋆L = L

γ−1γ+1

pLp⋆

+1

pLp⋆

+γ−1γ+1

aL =√

γpLL

, sHL = uL − aL

u⋆ = uL + 2γ−1

aL

(

1 −(

p⋆pL

)(γ−1)/2γ)

sTL = u⋆ − aL

(

p⋆pL

)(γ−1)/2γ

⋆L = L

(

p⋆pL

)1/γ

s1 > 0 s1 ≤ 0 sHL ≥ 0 sHL < 0u⋆ ≥ 0 u⋆ < 0 sTL ≥ 0 sTL < 0

u⋆ ≥ 0 u⋆ < 0

OUTLET OUTLET INLET OUTLET OUTLET OUTLET INLETpB = pLuB = uLvB = vLwB = wLB = L

pB = p⋆uB = u⋆vB = vLwB = wLB = ⋆L

pB = p⋆uB = u⋆vB = vRwB = wRB = ⋆R

pB = pLuB = uLvB = vLwB = wLB = L

uB

=2

γ+

1

[

aL

+γ−

12

uL

]

vB

=vL

wB

=w

L

pB

=pL

[

2γ+

1+

γ−

1(γ+

1)aL

uL

]2γ

γ−

1

B

=L

[

2γ+

1+

γ−

1(γ+

1)aL

uL

]2

γ−

1

pB = p⋆uB = u⋆vB = vLwB = wLB = ⋆L

pB = p⋆uB = u⋆vB = vRwB = wRB = ⋆R

Fig. 5-1: Algorithm for the solution of the problem (16),(17),(31),(33),(32),(34),(35) at the half lineSB = (0, t); t > 0. The value of the pressure p⋆ is prescribed. Possible situations are illustratedby the pictures showing ΩL ∪ ΩHTL ∪ Ω⋆L region. The sought boundary state located at the time axis,marked red.

6 BOUNDARY CONDITION BY PREFERENCE OF VELOCITY

In this section we construct the boundary condition, i.e. the state qB in (23), preferring the given valuefor the velocity uGIVEN. We add the following complementary conditions into the system (16),(17)

u⋆ := uGIVEN, (36)

uR = u⋆, (37)

⋆R := GIVEN,

vR := vGIVEN,

wR := wGIVEN.

Here u⋆ denotes the first component of the velocity in the region Ω⋆L ∪ Ω⋆R.We seek the boundary state qB as the unique solution of the problem (16),(17),(36), (33)-(35) at

the half line SB = (0, t); t > 0. The system (24)-(29), (36), (33)-(35) is uniquely solvable. Thealgorithm for the construction of the primitive variables B , uB , vB , wB , pB at the half-line SB is shownin Fig. 6-1. The complete analysis of this problem is shown in [7].

- 15 -

R–5375

u⋆ < uL u⋆≥ uL

D = 4LγpL + 2L(γ+12

)2(uL − u⋆)2

p⋆ = 12

(

2pL + γ−12

L(uL − u⋆)2 + (uL − u⋆)

√D

)

s1 = uL −√

γpLL

√

γ+12γ

p⋆pL

+γ−12γ

⋆L = L

γ−1γ+1

pLp⋆

+1

pLp⋆

+γ−1γ+1

aL =√

γpLL

, sHL = uL − aL

p⋆ = pL

(

−u⋆+uL+ 2γ−1

aL

2γ−1

aL

)

2γγ−1

sTL = u⋆ − aL

(

p⋆pL

)(γ−1)/2γ

⋆L = L

(

p⋆pL

)1/γ

s1 ≥ 0 s1 < 0 sHL ≥ 0 sHL < 0u⋆ ≥ 0 u⋆ < 0 sTL ≥ 0 sTL < 0

u⋆ ≥ 0 u⋆ < 0

OUTLET OUTLET INLET OUTLET OUTLET OUTLET INLETpB = pLuB = uLvB = vLwB = wL := L

pB = p⋆uB = u⋆vB = vLwB = wLB = ⋆L

pB = p⋆uB = u⋆vB = vRwB = wRB = ⋆R

pB = pLuB = uLvB = vLwB = wLB = L

pB

=pL

[

2γ+

1+

γ−

1(γ+

1)aL

uL

]2γ

γ−

1

uB

=2

γ+

1

[

aL

+γ−

12

uL

]

vB

=vL

wB

=w

L

B

=L

[

2γ+

1+

γ−

1(γ+

1)aL

uL

]2

γ−

1

pB = p⋆uB = u⋆vB = vLwB = wLB = ⋆L

pB = p⋆uB = u⋆vB = vRwB = wRB = ⋆R

Fig. 6-1: Algorithm for the solution of the problem (16),(17),(36),(33),(37),(34),(35) at the half line SB =(0, t); t > 0. Possible situations are illustrated by the pictures showing the region ΩL ∪ΩHTL∪Ω⋆L

with the sought boundary state located at the time axis.

6.1 B.C. for the Impermeable Wall

As a special case of the complementary condition (36) we prescribe zero normal velocity, i.e. we choose

u⋆ := 0. (38)

Further we use the analysis of the system (16),(17),(36), shown above in 6. We seek the boundary stateqB as the unique solution of the problem (16),(17),(38) at the half line SB = (0, t); t > 0. Using theanalysis in 6 we construct the solution of the system (16),(17),(38) in primitive variables. It is

BuBvBwBpB

=

L

γ−1γ+1

pLp⋆

+1

pLp⋆

+γ−1γ+1

0vLwLp⋆

, with p⋆ =2pL+

γ+12

Lu2L

+uL

√

4LγpL+2L

(γ+12

)2u2L

2, uL > 0,

L

(

1 + γ−12

uLaL

) 2γ−1

0vLwL

pL

(

1 +γ−12

uLaL

)2γ

γ−1

, uL ≤ 0.

(39)

Here L, uL, vL, wL, pL are prescribed, and aL =√

γpL/L. The solution for the first component ofthe velocity uB is equal to the prescribed velocity u⋆ = 0 in this special case.

7 BOUNDARY CONDITION BY PREFERENCE OF TOTAL QUANTI-TIES AND DIRECTION OF VELOCITY

Here we will construct the boundary state wkΓij

, following the procedure and notation described above

in 3.3. The state vector wki on element Di at time instant tk is known. Our aim is to prescribe the inlet

- 16 -

R–5375

boundary values for the total pressure, total temperature, and the direction of the velocity, if possible.Let us denote the prescribed values by po, θo,d. We use the transformation of this problem into alocal coordinate system x1, x2, x3, shown in 3.3, with the axis x1 being in the opposite direction of theprescribed velocity direction, given by n = −d, |d| = 1. The situation is depicted in Fig. 7-1.

x1

x2

Γij

Di

x1

x2

n

v

Fig. 7-1: Coordinate transformation for the boundary edge Γij in 2D. The axis x1 is in the oppositedirection to the prescribed direction of the velocity.

Then the derivatives with respect to x2, x3 are neglected. Our aim is to evaluate the boundary stateqB = q(0, t), such that the system (16),(17),(23) is uniquely solvable.

We add the complementary conditions

p⋆ = po

(

1−γ − 1

2a20u2⋆

)γ/(γ−1)

, (40)

θ⋆R = θo

(

1−γ − 1

2a2ou2⋆

)

, (41)

u⋆ < 0, (42)

p⋆ = pR,

vR = 0, (43)

wR = 0. (44)

Here θo > 0, po > 0 are given constants and a2o = γRθo. Further R denotes the characteristic gasconstant, and γ is the adiabatic constant. The equations (40),(41) are considered for the velocity

u⋆ ∈ (−√

2a2o(γ−1) ,

√

2a2o(γ−1)). The equations (41),(40) come from the idea, that the total pressure po and

the total temperature θo are known (prescribed) in the region Ω⋆R.The sought solution for qB of the system (24)-(29),(40)-(44) was described in [7]. The algorithm for

the construction of the primitive variables B , uB , vB , wB , pB at the half-line SB is shown in Fig. 7-2.The complete analysis of this problem is shown in [7]. Here we state that the system may not have asolution for the certain choice of the initial data. In that case the prescribed quantities cannot be preferredat the boundary. In practical applications we use the boundary condition preffering the pressure (Section5) with the prescribed pressure pGIVEN = po in this unfortunate case. At last, the sought state vector isevaluated as wk

Γij= Q−1qB . We use this boundary condition in the case, when the total quantities are

known at the inlet.

- 17 -

R–5375

1. compute u⋆

yesuL ≥ 0

no

F (0) ≥ 0 yes u⋆ = root of (*S)

no

Failed, choose u⋆ = min 0, uL + 2γ−1aL

uL + 2γ−1aL > −

√

2a2o(γ−1) yes F (min 0, uL + 2

γ−1aL) < 0

yes

no

no

Failed, choose

u⋆ = −√

2a2o(γ−1) + ǫ

−√

2a2o(γ−1) ≥ uL yes u⋆, use (*R)

no

F (uL) < 0 yes u⋆, use (*R)

no

u⋆ = root of (*S)

(*S) F (u) = po

(

1 − γ−1

2a2o

u2

)γ/(γ−1)

− pL − γ+14

L(uL − u)2 − (uL−u)

2

√

4LγpL + 2L(γ+12

)2(uL − u)2,

(*R) u⋆ =2

(

uL+ 2γ−1

aL

)

−√

DIS

2

[

1+2a2

L(γ−1)a2

o

(

popL

)(γ−1)/γ] , DIS = 4

(

popL

)(γ−1)/γ 2a2L

(γ−1)a2o

[

2a2o

(γ−1)+

4a2L

(γ−1)2

(

popL

)(γ−1)/γ−(

uL +2aLγ−1

)2]

.

2. compute uB , pB, B

uB = u⋆, pB = po

(

1− γ−12a20

u2⋆

)γ/(γ−1), B = po

Rθo

(

1− γ−12a2o

u2⋆

)1/(γ−1).

Fig. 7-2: Algorithm for the solution of the problem (24)-(29),(40)-(44) at the half line SB =(0, t); t > 0. In the case of failure, the problem doesn’t have a solution, and the value for u⋆ ischosen.

- 18 -

R–5375

8 SOFTWARE AND EXAMPLES

The described methods and original boundary conditions were coded in C programming language, andused in own programs for the solution of the compressible gas flow on unstructured meshes in 3D.Application of the presented boundary conditions on moving meshes was tested in [12, 6, 13, 18]. In thiswork we further used the program ANGENER [1] for the refinement of the triangular meshes, TETGEN(http://tetgen.berlios.de) for the tetrahedral mesh generation, and TECPLOT (http://www.tecplot.com)for the visualization of the computed results. Here we will shortly describe the developed program, usedto compute the shown examples.

C3DEEIAs a part of this work we attach (attachment [C3DEEI]) the latest version of the developed code for the

solution of the 3D compressible flow, written in C programming language. It is a further developmentof the program shown in [7], with the implicit time marching implemented, as described in Section 3. Itcan be used for the numerical solution of the 3D Euler Equations on tetrahedral grids. It is based on thefinite volume method. An explicit and implicit time discretization is used. We used the GMRES methodwith ILUK preconditioning in order to solve the system (14) in each time step, SPARSEKIT [19] wasused. The software is written for the parallel use, OPENMP directive is used. To approximate the facevalues at the inner faces, the program solves local initial-value Riemann problem for the 3D split Eulerequations, shown in Section 4 or the Vijayasundaram numerical flux is used. At the boundary faces, theinitial-boundary modification to this problem is solved, as shown in Section 5 - 7. We will show the useof this program on the example of channel flow. We chose the channel with 10% cylindrical bump at thelower wall, as a natural extension of the so-called GAMM channel to 3D. The geometry of the GAMMchannel can be found in [5, page 277]. Here we extend this 2D geometry into a 3D channel with thethickness 0.2. The initial mesh is shown in Fig. 8-1.

1 2

3

4

5

Fig. 8-1: 10% cylindrical bump, initial mesh and boundary numbering.

The direction of the flow is from left to right. The flow regime in test case is set to the inlet Machnumber M = 0.67. In this regime a shock is generated near the lower wall. The following data wereused in our tests:

• initial condition - constant state in the whole domain,o = 1.5, vo1 = 205.709277, vo2 = 0, vo3 = 0, po = 101000.

• inlet boundary condition - total quantities and velocity directionThe boundary condition from Section 7 was used, withθo = 255.639, po = 136461,d = (1, 0, 0). These values were obtained using the equations (40)and (41) for the data in initial condition.

• outlet boundary condition - preference of the pressureThe boundary condition from Section 5. with described averaging technique was used, withp⋆ = 101000.

- 19 -

R–5375

• impermeable wall boundary conditionThe boundary condition in Section 6 was used in the case of the inviscid flow.

The computational results are shown in Fig. 8-3. Anisotropic refinement method, shown and coded in[21], was used with minor changes in the code (refinement programs are also attached).

Input and Output Data Files FormatThe 3D solution is computed on tetrahedral, prismatic, and hexahedral elements, and all the solver pa-rameters are given in one input ini file. All comments in the input files start with the sign * and end withthe end of the line.

mesh file format:

The mesh file with tetrahedral, prism, and hexahedral elements is written in ASCII and uses the followingformat

npoin number of pointsnvol number of volume elementsnbfaces number of boundary facesx1 y1 z1 coordinates of the 1st point (=vertex 1)· · ·xnpoin

ynpoinznpoin

coordinates of the npointh pointP 11 P 2

1 · · · PN11 element 1 : index of vertices

· · ·P 1nvol

P 2nvol

· · · PNnvolnvol

element nvol : index of verticesB1

1 B21 · · · B

M11 I1 boundary face 1: index of vertices, followed

· · · by the index of the boundary

B1nbfaces

B2nbfaces

· · · BMnbfacesnbfaces

Inbfacesboundary face nbfaces: vertices, index

npp number of periodical pairs (points)Q1

1 Q21 periodical pair 1 (index of points)

· · ·Q1

nppQ2

nppperiodical pair npp

The point coordinates are followed by the element vertices and boundary face vertices. The coordinatesof the vertex i are xi, yi, zi. Each volume element Dj is given by the index of its vertices P 1

j P 2j · · ·

PNjj , j = 1...nvol , volume element has Nnvol vertices, Nnvol ∈ 4, 6, 8. The geometric ordering of

the vertices is shown in Fig. 8-2. Each boundary face i = 1...nbfaces is given by the indexes of itsvertices B1

i , B2i , ...B

Mii , Mi ∈ 3, 4, and by the index of the boundary condition Ii ∈ 1, ..., NBC.

The periodical point pairs are given by the vertices indexes Q1i Q

2i , i = 1...npp.

1

2

3

4

1

2

3

4

5

6

1

2 3

4

5

6 7

8

Fig. 8-2: C3DEEI: allowed elements, vertices numbering.

ini file format:The format of the ini file, with the initial solver data, will be shown on the example. There is always

- 20 -

R–5375

a keyword followed by the input value. Input and output physical values are given in the InternationalSystem of Units (SI). The following keywords are recognized:

• EQTYPEEquations to solve: euler, NS, NS-FVMFEM-split, NS-FVMFEM-nosplit.

• TIMEINTEGRATE_PROCTime integration to use: euler, ec, rk4, implicit_euler, implicit_eulerN.(Euler, Euler-Cauchy, 4-points Runge-Kutta, Implicit Euler, Implicit Euler - N iterations of the Newton method)

• MATRIXSOLVER SOLVERMatrix solver to use (implicit method)SPARSEKIT_ILUKGMRES Sparskit GMRES + ILUK preconditioner (default).SPARSEKIT_ILUKBICGSTAB Sparskit BiCGSTAB + ILUK preconditioner.CUSP_PBiCGSTAB_CPU CUSP package, BiCGSTAB (TESTING, so far bad preconditioner).CUSP_PGMRES_CPU CUSP package, GMRES (TESTING, so far bad preconditioner).

• MATRIXSOLVER MAXITER NMatrix solver, maximal number of iterations (50 default)

• MATRIXSOLVER RESTART NMatrix solver, restart after N iterations (applied with GMRES iterative solver, 25 default)

• MATRIXSOLVER EPSrelative GStopping criterium for the matrix solver, default G=1e-8

• EFLUXInviscid flux to use: Riemann, Vijayasundaram.

• EFLUX_DHDWInviscid flux Jacobi matrix estimate: N, VS.N compute derivatives numerically.VS via Vijayasundaram flux with fixed matrixes.

• SPACERECONSTRUCTION_PROCSpace reconstruction used to achieve the higher order method:none No reconstruction, constant approximations are used (FVM).FVM0 No reconstruction, constant approximations are used (FVM).BarthJespersen_P Barth-Jespersen limiter is applied on primitive variables.BarthJespersen_C Barth-Jespersen limiter is applied on conservative variables.VanAlbada_P Van Albada limiter is applied on primitive variables (hexaxedral meshes).VanAlbada_C Van Albada limiter is applied on conservative variables (hexaxedral meshes).

• SPACERECONSTRUCTION_START 10000Computes 10000 iterations with the 1st order method and then it continues with the higher order reconstruction.

• SPACERECONSTRUCTION_GRADIENT_RECOMPUTE_STEP 60Gradients for the reconstruction are recomputed after every 60 iterations.

• GRIDNAME X.gridThe name of the base file with the computational mesh. The given string determines also the file-names of the other output and input files. These files are X.grid.sol andX.grid.sol.bin for the input/output solution file in ASCII and binary format, X.grid.dat for the output TECPLOT file, X.grid.stat for the output computationalstatistics, X.grid.conv for the output convergence statistics.In the case of the MPI run, program works with the base file-name, altered by prefix with a number. This number denotes the id of the splitted domain part (0 .. nparts-1) for example0_X.grid, 1_X.grid are grid files. Further, the files with volume aliases and send and receive indexes are used with MPI (input files): 0_X.grid.vid,1_X.grid.vid,0_X.grid.recv,1_X.grid.recv,0_X.grid.send and 1_X.grid.send.

• TECPLOT_OUT 1Output TECPLOT file is created if given 1, no file if 0 is given.

• CONV_OUT 0Output convergence file is created if set to 1, no file if set to 0.

• SPLIT_REDMAP_OUT 1Output temporary file redmap for the refinement of the selected elements, with the program meshrefine.

• ANIZO_RPF_OUT 0Output temporary file anizoRPF for the anisotropic refinement point function, which can be used with the program meshrefanizo.

• IC ruvwp rho v1 v2 v3 pInitial condition, here rho v1 v2 v3 p are the given initial values for density, velocity components and the pressure. This initial condition is not used if the solution fileX.grid.sol or X.grid.sol.bin is presented in the execution directory. Example: IC ruvwp 1.29233 0 0 0 101325

• CONST KAPA 1.4Adiabatic gas constant, given as γ = 1.4.

• CONST R 287.04Ideal gas constant, given as R = 287.04.

• CONST RMI_REF 1.716e-05Viscosity at the θref temperature.CONST T_REF 273.15The reference temperature θref .CONST S_T 110.4The Sutherland temperature S.The Sutherland’s law constants, used for the correction of the dynamic viscosity at each face

µ = µref

(

θ

θref

)3/2 θref + S

θ + S. (45)

- 21 -

R–5375

• CONST RKA 0.0211Ideal gas constant, given as k = 0.0211.

• OUT_PHYSICAL_TIMESTEP 10.0003Periodical output, time-step between each output.

• STOP LINFTY 1e-9Stop the computation when residual drops to the given minimum.

• STOP PHYTIME 999Stop the computation when the given physical time is reached.

• STOP ITERATIONS 100000Stop the computation when the given iteration 100000 is reached. If started from the iteration with the higher number, computations are stopped after 100000 iterations.

• USE_NORMED_VALUES 1All values within the solver are normed to the critical ones. This may enhance the overall accuracy. Input and output files are still written in SI. If USE_NORMED_VALUES 0 isgiven, normed values are not used.

• CONST ACCURACY 1e-7Accuracy used within the solution of the edge values problem. It is used by the Riemann solver and for the boundary conditions evaluation.

• CONST CFL 20000Stability time-step restriction, first step. (CFL ¡ 1 for explicit method)

• CONST CFLMIN 0.85Stability time-step restriction, minimal CFL within implicit method.

• CONST CFLMAX 100000Stability time-step restriction, maximal CFL within implicit method.

• PRINT_ITERATION 100Controls the standard output stream, prints values each 100 iterations.

• BCBoundary conditions. Keyword is followed by the number of the boundaries. Then the data for each boundary follows. The id of the boundary is followed by the used boundarycondition with necessary data.Example (3D GAMM channel):BC61 inlet fixed_to_T_direction 1.5 205.709277 0 0 1010002 outlet pavg 1010003 wallT 273.154 wallT 273.155 periodic 66 periodic 53 4 5 6 1 2This example shows 6 boundaries. The indexes 1-6 correspond to the indexes of the boundaries in the mesh file. At the boundary 1 we prescribe inlet boundary condition conservingthe total pressure, the total temperature, and the direction of the velocity given by the vector, with values computed from the given fixed state. Boundaries 3 and 4 represent thewalls with the given temperature T = 273.15. Boundary 2 is an outlet boundary with the pressure p⋆ = 101000 in average given. Boundaries 5 and 6 are periodical. Thesequence 3 4 5 6 1 2prescribes the boundary priority for the points laying on multiple boundaries (boundary 3 has the highest priority).

ini file example (3D GAMM channel):

EQTYPE eulerTIMEINTEGRATE_PROC implicit_eulerCONST CFL 20000CONST CFLMIN 0.85CONST CFLMAX 100000MATRIXSOLVER SOLVER SPARSEKIT_ILUKGMRESMATRIXSOLVER MAXITER 50MATRIXSOLVER RESTART 35MATRIXSOLVER EPSrelative 1e-6SPACERECONSTRUCTION_PROC FVM0SPACERECONSTRUCTION_START 5000SPACERECONSTRUCTION_GRADIENT_RECOMPUTE_STEP 60GRIDNAME X.gridTECPLOT_OUT 1CONV_OUT 0CONST KAPA 1.4CONST R 287.04OUT_PHYSICAL_TIMESTEP 100STOP LINFTY 1e-8STOP PHYTIME 50.25STOP ITERATIONS 50USE_NORMED_VALUES 1CONST ACCURACY 1e-6PRINT_ITERATION 1IC ruvwp * 1 rho v1 v2 v3 p kappa1.5 205.709277 0 0 101000BC* no boundaries + each boundary condition + boundaries priority for dual mesh61 inlet fixed_to_T_direction 1.5 205.709277 0 0 1010002 outlet p 1010003 wall4 wall5 periodic 66 periodic 5EOF

- 22 -

R–5375

Fig. 8-3: 3D inviscid flow computation (coarse mesh). Mach number and pressure isolines. (C3D)

solution file format:The solution file stores the finite volume solution in the primitive variables for each element. Solutionfile is written in ASCII. Example:

nvol 5(1) v1(1) v2(1) v3(1) p(1). . .(nvol) v1(nvol) v2(nvol) v3(nvol) p(nvol)iterationphysical time

Here nvol is the number of volume elements, 5 is the count of the solution components, γ is adiabaticconstant, symbol (i) denotes the density, v1(i), v2(i), v3(i) are the components of the velocity, andp(i) is the pressure on element Di (i is the element index determined by the the mesh file). Solutionbinary file stores the values in the same order as the solution file. It is preferred in the case of thecomputation restart.

Program runThe program c3deei is coded with OPENMP directive, and it uses all the systems processors for thecomputation by default. We tested the program on the Linux system, the Intel C/C++ compiler was usedfor the compilation.

9 Examples

Here we show the computational results of the compressible gas flow, obtained with the own-developedcodes. For other computational results see also [8], [11], [9], [10], [14], [15], [16]. The showncomputations ran on the personal computer with two Intel(R) Xeon(R) CPU E5520 @ 2.27GHzprocessors.

9.1 Laval Nozzle

A de Laval nozzle is a convergent-divergent tube used to accelerate subsonic flow to a supersonic speed.Its purpose is to achieve a homogeneous parallel flow in the test section, without shock waves and otherdisturbances. This can be done with careful shaping the cambered sides.In the converging part, the speed of the flow is increased due to the reducing cross-section area of thetube. The velocity reaches the speed of sound (M = 1) in the throat of the nozzle and cannot beincreased beyond M = 1. We say that the flow is choked. Then the flow is expanded into a supersonicspeed in the divergent part of the tube.The specific nozzle we show here is a part of the supersonic wind tunnel at the department of high-speedaerodynamics in VZLU. It is a rectangular tube with cambered upper and lower sides. The design of the

- 23 -

R–5375

nozzle was chosen to yield the Mach number M = 2.135, the length of the nozzle is 2 meters. At theinlet, the nozzle is connected to the standard atmosphere, and through the outlet it is connected to thelow-pressure reservoir. To simulate real conditions, we use the inlet boundary condition conserving thetotal pressure po = 101325, total temperature To = 273.15 and the uniform direction of the velocity,perpendicular to the inlet boundary, see Section 7. At the outlet we use the boundary conditionpreferring the pressure (Section 5.), with the prescribed value p∗ = 10490.3808184132. This value is anadiabatic estimate, obtained with the use of the equation (40) for given po and M. Results were shownin [16], [7], computed on structured and unstructured meshes, adaptively refined according to thesolution. Here we add more results, computed with the use of the described implicit method on thethree-dimensional unstructured mesh and we show the comparison of the real computational time forthe chosen methods, see Fig. 9-1 - 9-3. Here (VS+VS) means the Vijayasundaram numerical flux +Jacobian computed using (22), (RS+VS) means the exact Riemann Solver + Jacobian computed using(22), (RS+N) means the Riemann solver numerical flux + Jacobian evaluated numerically. The speedadvantage of the implicit method is apparent even for the computation on the coarse meshes.

method used CFL iterations real time speedup1 explicit 0.85 1725 0m28.213s 1.00 ×

2 implicit (RS+N) 8-308 48 0m13.322s 2.12 ×

3 implicit (VS+VS) 8-339 55 0m11.702s 2.41 ×

4 implicit (RS+VS) 8-662 54 0m10.498s 2.69 ×

Fig. 9-1: 3D inviscid computation, coarse mesh with 1896 prismatic elements. Pressure and Machnumber isolines, real time of the computation for chosen methods. (C3DEEI)

- 24 -

R–5375

method used nproc CFL iterations real time speedup1 explicit 1 0.85 30627 33m21.782s 1.00 ×

2 explicit 4 (OMP) 0.85 30627 12m51.273s 2.60 ×

3 explicit 8 (OMP) 0.85 30627 7m42.946s 4.32 ×

4 implicit (RS+N) 1 132-4406 57 2m35.913s 12.84 ×

4 implicit (VS+VS) 1 132-2985 66 2m3.313s 16.23 ×

5 implicit (RS+VS) 1 132-3918 66 1m57.246s 17.11 ×

Fig. 9-2: 3D inviscid computation, coarse mesh with 8208 hexahedral elements. Pressure and Machnumber isolines (C3DEEI).

method used nproc CFL iterations real time speedup1 explicit 1 0.85 46262 699m57.692s 1.00 ×

2 implicit 1 50 - 6674.01 134 46m47.029s 14.03 ×

Fig. 9-3: 3D inviscid computation, mesh with 109490 prismatic elements. Pressure and Mach numberisolines (C3DEEI).

9.2 A Body in a Channel

Here we present a computational result of the 3D non-stationary inviscid channel flow at Mach numberM = 0.67, the flow is subsonic in this example. A body immersed in the flowing fluid establishes acertain wave pattern which evolves in time and eventually exits the channel. The inlet is located left,outlet right, other boundaries are considered as wall. The simple geometry of 8% double circular arcDCA08 was chosen as a body.Initial condition:

• = 1.5,v = (205.709277, 0, 0), p = 101000.

Boundary conditions:

• inlet (left): BC. Section 7., with po = 136461, θo = 255.639, direction d = (1, 0, 0),

• outlet (right): BC. Section 5., p = 101000,

• walls: BC. Section 6., u = 0,

- 25 -

R–5375

• periodical boundary condition (front and rear boundary).

The computational results and comparison of the real computational time for the chosen methods areshown in Fig. 9-4. The significant speedup (236 ×) was achieved with the use of the implicit method.The notation (A25) means that 25 iterations (at most) were allowed in solving (14) with the iterativeGMRES method. The correct choice of the boundary conditions is also important in this case (testedwith the explicit method). It can be shown that the fixed boundary conditions give incorrect results nearboundaries, the linearized boundary condition reflects the waves into the domain, leading to theoscillations in the solution. The suggested boundary conditions do not suffer from these drawbacks.

method used CFL iterations real time speedup1 explicit 0.85 27370 626m44.229s 1.00 ×

2 implicit (A50) 80000-100000 8 4m13.803s 148.16 ×

3 implicit (A50) 80000-200000 7 4m4.848s 153.58 ×

3 implicit (A25) 80000-244141 7 2m38.985s 236.53 ×

Fig. 9-4: 3D inviscid computation, mesh with 118835 prismatic elements. Pressure and Mach numberisolines, real time of the computation for chosen methods. (C3DEEI)

10 CONCLUSIONS

In this paper we worked with the system of equations describing the compressible fluid flow in 3D. Weapplied the finite volume method for the discretization of these equations, and we have shown theimplicit method, which is considerably faster than the explicit method. The so-called Riemann problemfor the 3D split Euler equations was presented, as the initial-value problem equipped with left-hand sideand right-hand side initial-value condition. The analysis of this problem can be found in many books,i.e. see [5], [20]. The solution of the Riemann problem led us to the system of nonlinear algebraicequations, which must be solved with the use of an iterative process. In order to discretize the values atthe boundary, the original initial-value problem was modified, as shown in [17]. The right-hand sideinitial-value condition is not known at the boundary faces, and has to be either prescribed or replaced bysuitable conditions. In order to prescribe some variable (for example pressure) by chosen value at theboundary, the local modified Riemann problem must have a solution, otherwise the value cannot beused at the boundary. This was the main idea behind the construction of the shown boundary conditionsby preference of certain variable. On the contrary to the solution of the initial-value Riemann problem,the solution of the modified boundary problems can be written in a closed form. Therefore it is notcomputationally expensive to use the constructed boundary conditions in the code.The algorithms for the solution of the boundary problems were coded and implemented into theown-developed software for the solution of the inviscid compressible gas flow (the Euler equations).Numerical examples show superior behaviour of the implicit method.AcknowledgmentThis result originated with the support of Ministry of Industry and Trade of Czech Republic for thelong-term strategic development of the research organisation.

- 26 -

R–5375

REFERENCES

[1] V. Dolejsı. Angener user’s guide. Department of Numerical Mathematics,Faculty of Mathematicsand Physics,Charles University, Prague, http://www.karlin.mff.cuni.cz/˜dolejsi/angen/angen.htm,2005.

[2] V. Dolejsı. Discontinuous Galerkin method for the numerical simulation of unsteady compressibleflow. WSEAS Transactions on Systems,5(5):1083-1090, 2006.

[3] M. Feistauer. Mathematical Methods in Fluid Dynamics. Longman Scientific & Technical,Harlow, 1993.

[4] M. Feistauer. Numerical methods for compressible flow. In J. Neustupa and P. Penel, editors,Mathematical Fluid Mechanics. Recent Results and Open Questions. Birkhauser, Basel, 2001.

[5] M. Feistauer, J. Felcman, and I. Straskraba. Mathematical and Computational Methods forCompressible Flow. Oxford University Press, Oxford, 2003.

[6] M. Kyncl J. Pelant. Applications of the Navier-Stokes equations for viscous turbulent flow onsteady or moving grids with the (k,w) turbulent model. Czech Aerospace Proceedings / Leteckyzpravodaj, 1:2–6, 2008.

[7] M. Kyncl. Numerical solution of the three-dimensional compressible flow. Master’s thesis,Prague, 2011. Doctoral Thesis.

[8] M. Kyncl and J. Pelant. Applications of the Navier-Stokes equations for 3d viscous laminar flowfor symmetric inlet and outlet parts of turbine engines with the use of various boundary conditions.Technical report R3998, VZLU, Beranovych 130, Prague, 2006.

[9] M. Kyncl and J. Pelant. Applications of the Navier-Stokes equations for 2d viscous, compressibleturbulent flow on steady grids with the EARSM turbulent model. Technical report R4300, VZLU,Beranovych 130, Prague, 2007.

[10] M. Kyncl and J. Pelant. Applications of the Navier-Stokes equations for 3d viscous, compressiblelaminar and turbulent flow in the axis–symmetrical channels as parts of turbine engines withsteady or moving walls. Technical report R4301, VZLU, Beranovych 130, Prague, 2007.

[11] M. Kyncl and J. Pelant. Applications of the Navier-Stokes equations for viscous turbulent flow onsteady or moving grids with the (k,omega) turbulent model. Technical report R4153, VZLU,Beranovych 130, Prague, 2007.

[12] M. Kyncl and J. Pelant. Applications of the Navier-Stokes equations for 3d viscous, compressiblelaminar and turbulent flow in the channels with steady or moving walls. Technical report R4415,VZLU, Beranovych 130, Prague, 2008.

[13] M. Kyncl and J. Pelant. Applications of the Navier-Stokes equations for 3d viscous compressibleturbulent flow with the (k, ω) or EARSM turbulent model. Technical report R4607, VZLU,Beranovych 130, Prague, 2009.

[14] M. Kyncl and J. Pelant. Numerical simulation of the forced airfoil oscillation with the (k, ω) orEARSM turbulent model. Technical report R4606, VZLU, Beranovych 130, Prague, 2009.

[15] M. Kyncl and J. Pelant. Applications of the Navier-Stokes equations for 3d viscous compressibleturbulent channel flow with the use of various boundary conditions. Technical report R4916,VZLU, Beranovych 130, Prague, 2010.

[16] M. Kyncl, J. Pelant, and J. Simak. Numerical solution of the transonic compressible gas flowthrough Laval nozzles. Technical report R4917, VZLU, Beranovych 130, Prague, 2010.

- 27 -

R–5375

[17] J. Pelant. Arti reports VZLU, z-65, z-67 to z-73. Prague, 1996-2000.

[18] A. Prachar, M. Kyncl, and J. Pelant. Numerical simulations of the oscillating wing/airfoilproblem. Technical report R4915, VZLU, Beranovych 130, Prague, 2010.

[19] Youcef Saad. Sparsekit: a basic tool kit for sparse matrix computations. University of Minnesota,1994.

[20] E. F. Toro. Riemann Solvers and Numerical Methods for Fluid Dynamics. Springer, Berlin, 1997.

[21] M. Zoubek. Adaptive methods for the solution of three dimensional flow. Master’s thesis, CharlesUniversity, Prague, 2001. (in Czech).

- 28 -

R–5375

VZLU– Aerospace Research and Test Establishment, Beranovych 130, 199 05 PragueCzech Republic

REPORT DOCUMENTATION PAGE

1. Report No. 2. Report Date 3. Account No. 4. No. of Pages 5. Publication Code

6. Title and Subtitle

7. Author’s department (Abbr. – No. – Name)

8. Autor(s) (Name, Signature) Head of Department (Name, Signature)

9. Responsible Person (Name, Signature) 10. Head of Department (Name, Signature) 11. Technical Director (Name, Signature)

12. Abstract

13. Key Words

R–5375 May 2012 IP4401 29 + CD FIR

IMPLICIT METHOD FOR THE 3D EULER EQUATIONS

MOT – 4400 – Engines

Martin Kyncl Jaroslav Pelant Jirı Fiala

Martin Kyncl Jaromır Lamka Viktor Kucera

We work with the system of equations describing non-stationary inviscid compressible fluid flow in 3D, i.e. theEuler equations, and we focus on the numerical solution of these equations and on the boundary conditions. Inparticular, we choose the implicit finite volume method. Many boundary conditions (i.e. fixed, linearized) bringnon-physical errors into the solution, slowing down the convergent process, or even ruining the solution in thewhole domain. We use the analysis of the Riemann problem for the construction of the boundary conditions inorder to match the experimental data. We show, that the unknown one-side initial condition for the local Riemannproblem can be partially replaced by the suitable complementary condition. We suggest such complementaryconditions (by preference of pressure, velocity, total quantities,...) giving physically relevant data. Our methodfor the construction of the boundary conditions is robust and its use accelerates the solver convergence to thedesired solution. As a part of this work we present own-developed software for the solution of the 3D Eulerequations, using the implicit method.

compressible gas flow, boundary conditions, finite volume method, implicit method

- 29 -