moving meshes

ELSEVIER Comput. Methods Appl. Mech. Engrg. 113 (1994) 183-204

Computer methods in applied

mechanics and engineering

Godunov type method on non-structured meshes for three-dimensional moving boundary problems

B. Nkonga, H. Guillard INRIA, centre de Sophia-Antipolis, B.P. 93, 06902 Sophia-Antipolis Cedex, France

Received 30 March 1993

Abstract

This paper presents a numerical method for the computation of compressible flows in domains whose boundaries move in a well defined predictable manner. The method uses the space-time formulation by Godunov while the discretization is conducted on non-structured tetrahedral meshes, using Roe's approximate Riemann solver, an implicit time stepping and a MUSCL-type interpolation. The computation of the geometrical parameters required to take into account the movement of the boundaries is described. Examples including the calculation of the flow in the cylinder of an internal combustion engine illustrates the possibilities of the method.

1. Introduction

This work deals with the computation of the flows of compressible fluids in moving geometries such as those occurring for instance in reciprocating engines and more generally in many industrial devices. As the geometry of such industrial devices is inherently complex, the versatility and flexibility of non-structured meshes is very appealing. Accordingly, the present method is based on the use of tetrahedral meshes. Because compressible flow often exhibits strong compressions or discontinuities, the spatial approximation uses the robust second-order Euler solver developed in [1-3]. This solver is also attractive because of its nice matrix properties when coupled with an implicit time stepping. This paper explains how the movement of the boundaries is included into this framework in a very simple and natural way. It is organized as follows: Section 2 discusses the approximation of conservation laws on moving meshes. As stressed by many authors (see e.g. [4] for a review), the key point here is the respect of the so-called geometrical conservation law, that is the form taken by the consistency relation on a moving grid. The time-space formulation given by Godunov's method provides a very simple and elegant way to automatically satisfy this constraint. Then in Section 3 we provide the computation of the geometrical parameters when the intersection of two spatial control-volumes is a triangular face whose three nodes move with different velocities. As any planar polygonal surface can be split into triangular facets, the general case can be treated by repeated application of formula obtained on triangular facets. In Section 4, we specialize these results to the finite volume method of [5] where the boundaries of the control volumes are composed of a union of plane quadrangular faces. The fifth section deals with modifications that are necessary to implement Roe's approximate Riemann solver on a moving grid. Finally, in the last section we conclude by presenting several test cases and examples illustrating the possibilities of the method.

184 B. Nkonga, H. Guillard I Comput. Methods Appl. Mech. Engrg. 113 (1994) 183-204

2. Conservation laws in a moving domain

Let ~(t) C R m (m ~< 3) be the domain filled by the fluid at time t. The displacement of the domain from its position at time t = 0 to its position at time t is given by a map x = ~p(a) from ~(0) onto ~(t). The space-time domain ~ occupied by the fluid between t = 0 and t = T is defined by

T = U ~(t) .

t=0

Except when the domain boundaries are fixed, the space-time domain ~ is not a simple product of a spatial domain by a time interval. This introduces an additional difficulty with respect to fixed boundary problems.

We will denote by J(t)= Dx/Da the Jacobian matrix of the map ~o and by x(x, t) the velocity displacement of the domain ~(t) at x E ~(t):

a~p -1 It(x, t) =-~- (~o (x, t)).

For all (x, t) E ~, we consider a first-order conservation law,

aW O----i- + V. F(W) = 0, (1)

where W is a vector of R p and F a smooth function from R v into R p. Let qgi(t) (i = 1 , . . . , ns) be a splitting of the domain ~(t) into non-intersecting cells such that

O~i( t )=~(t ) and ~i fq~j=ql i f i# j ; i=l

we will denote by O~i(t ) the boundary of the cell ~i(t) and by (9~q(t) the intersection of the boundaries of the cells ~;(t) and ~j(t) such that

tgc~i(l) = U a~q(t), jEff( i )

where ~(i) is the set of neighbors of i. We denote by ~ the space-time hyper volume defined by the displacement of c~ under the action of the map ~p between times t = t" and t = t "+~ and by ~j the intersection of the two hyper volumes ~/and ~j. during the same time interval. Obviously the union of ~ fills ~ and we have

tn+l "~iJ = t=t"U (9%(t) and O,~i --- c~i(tn) U C~i(tn+l) u (lE~J(i) ~i]) "

Figure 1 provides a graphical representation of the space-time and spatial cells ~; and ~ in the two-dimensional case.

The discrete equations are obtained by integrating Eq. (1) over ~ resulting in

f,'o"+' f f f,,,,, -gi W':U dt f,. f f,,,,,, V(W,',(O dS dt=O . (2) Using the identity,

WI . n(t) dS ,

Eq. (2) becomes

Vl(C~7+l'bW"+l-Vl(~7)W, ' + , - i ~ I"+' fro jEJ'(i) n q~i(t) (F(W) - W)" n(t) dS dt = O, (3)

B. Nkonga, H. Guillard I Comput. Methods Appl. Mech. Engrg. 113 (1994) 183-204 185

t

:I n+3

:t (n)

0

~I1 J- ] C

n+ 1 )

i

Fig. I. Hyper control-volume .~, and spatial cell ~ for the two-dimensional case.

n n + 1 where WI '' and W i are defined as the mean value of W over the cell ~i(t ) and ~i(t"). Similarly, if we define Wii and Fii as the mean value of W and F over ~/j, we obtain

Vol (~7+')w7 +' -Vo l (~7)w; ' + ,at 5". (F , j . , , ) , j - o~,/ll.,jllw,.j) = 0, (4) jEff(i)

where the mean value ~/o of the integral of the normal n is defined by

/ f /~ , 1 ff,/fo = n(t) dS dt (5) 1 n(t) dS dt -~ %(o ~o = -~- , ,, and where we have introduced the mean value of the normal velocity o-~j defined by

' ff , ' f'""f/o - - K" n(t) dS dt . (6) ~' J - a/ l l . , j l l ;, K .n ( t )dSdt - mll . , / l l " %,(,) The correct evaluation of the above quantities is the key point in the design of a numerical scheme on a moving grid. This can be understood as follows: let us consider that at time t = 0, the variable W is equal to a constant state W 0. If the domain boundaries 09 are fixed, the solution of any conservation law of the form (1) will be W(x, t )= W o Vt. Consequently the integration of (1) over any volume C(t) C ~ and for any map ~p respecting the boundaries 09 will degenerate into the identity:

(t) C(t)

According to Eqs. (5), (6), the discrete counterpart of this identity takes the form

Vol(~,' . '+')-Vol(Cg~')=A/ ~ (llrl,jllo-,j) (8) icer(i)

and states that the increase in the volume of a cell must equal the summat ion of the volumes swept out by each face of the cell between t" and t "1. Therefore the respect of (8) is a necessary condition for a numerical scheme to keep constant a uniform initial state and thus the evaluation of the geometrical

186 B. Nkonga. H. Guillard I Comput. Methods Appl. Mech. Engrg. 113 (1994) 183-204

parameters (5) and (6) must be carefully done. This fact was first pointed out in [6]. Given an approximation of 7/~; and o-~i, a method sometimes used to enforce (8) is to update the volume of the cells by explicitly defining Vol(~7 1) by Eq. (8). However, in this case, the actual geometrical volume of the cells may differ from the value given by (8) and it is preferable to devise a direct geometrical method to respect (8). In the one- and two-dimensional cases, the correct evaluation of (5) and (6) is described in [7]. Actually Godunov's original 1959 method [8] was designed for a moving as well as for a fixed grid. We now give the computation of the geometrical parameters (5) and (6) for the case of a triangular facet moving in a three dimensional space. As any plane polygonal surface can be split into a finite number of triangles, a repeated application of these results will give the general solution (see Section 4 for an example).

3. Geometrical parameters

Let T(t") be a triangular surface in lt~ 3 whose nodes are l(t"), J(t") and K(t'). Our aim is to compute the parameters (5) and (6) under the action of the map ~0. It may happen that ~0 is explicitly known for every (x,t) for instance by an analytical expression, however, when the motion of the domain is complex, the map ~o is generally computed by a numerical procedure and is only known on a discrete set of points. Therefore we make the assumption that the velocities of the three nodes I, J and K are constant (but different) during a time step At and that the velocity of any point P belonging at time t" over T(t") is a linear interpolation of the velocities of the nodes defining the facet. Let (a,/3) denote the barycentric coordinate of P E T(t'):

X" = Xp(t" ) = (1 - a -/3)Xt(t" ) + aXj(t") +/3Xr(t" ) ;

then the velocity of P is

K(P) = (1 - ct - /3)K t + aKj +/3K K

and thus K(P) is a constant and for any t@ [t', t"+l], we have

xp( t ) = x" + (t - t ' )K (P )

= (1 - ct - /3)Xt(t) + aXj(t) +/3XK(t). (9)

From this equation, we deduce that P(t) belongs to the triangle l(t), J(t), K(t) for any t ~ [t', t "+1] (the facet remains plane), that the barycentric coordinates (a,/3) of P remain constant during the time step and that Xe(t ) varies linearly in time:

t - t" Xp(t) = (1 -s )X" +sX "+1 , with s - At

With these relations, it is now an easy task to compute r/i;; we have first

f fr n(t) dS = (Xj(t) - Xt(t)) ^ (Xr(t) - Xi(t)) t~(t) = ~,>

and it is readily found that

1 f [ '+ ' f f r 1 '"+' nr=---~ , mn(t) dS dt=--~ , t~( / )d /=( /~ '+~"+l+~*) , (10)

where p,+l and p" are the integrals of the normal unit on T(t') and T(t'+~), respectively, and the vector p* is defined by

n+l 1 (yn+l n+l 2t~*=(x;-x'/)^(x"/'-x, )+3,--, -x, )^(x~.-xT). The computation of ~r proceeds in the same way. We first compute the volume 8V swept by the facet T (Fig. 2):

B. Nkonga, H. Guillard / Comput. Methods Appl. Mech. Engrg. 113 (1994) 183-204 187

I Fig. 2. Volume 8V swept out by a triangular facet (1, J, K).

8V= K " n ( t ) dS dt = n( t ) " K dS dt " ( t ) " ( t )

(fS' ) = Kc" n n ( t ) area{T(t)} dt = At K G r/r where K c is the velocity of the gravity center of the facet. We then obtain

~T- Atl I ' ITl l - ~ " 3 "

This ends the computation of geometrical parameters for a triangular facet.

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4. MUSCL-FEM method

In this section we apply the previous results to the MUSCL-FEM method by [1-3]. The three- dimensional version of this method is developed in [5]. This method can be understood either as a classical Galerkin finite element method stabilized by upwind terms [9] or as a finite volume method on non-structured simple meshes. For ease of presentation, we adopt here the second point of view. Given a non-structured tetrahedral mesh, to each node i of the mesh is associated a cell defined by the midpoints of the edges of the tetrahedra having i as a node, the gravity center of these tetrahedra and the gravity center of the faces of these tetrahedra. Fig. 3 displays the intersection of such a control volume with a tetrahedra. Let us split the boundary of the /-cell by a union of partial boundaries associated with each edge:


k

J M

G 3 G

Fig. 3. Intersection of 0q~, with a neighboring element.

U a%. jE~(1)

From Fig. 3, it is seen that aCij is itself a union on T~j, the set of tetrahedra having [i, ]] as an edge, of quadrangular surfaces:

o%= U rE Ti i

Let i, j, k, l be the four nodes defining such a tetrahedra z, Ocgii n r is defined by the four points {Ml; Gl; G; G3) (see Fig. 3) whose coordinates are

XM 1 .~. I (X i .~_ X j ) , Xc I = 1 (X i q_ Xj -~ X I ) ,

Xo3=(X, + Xj + X,), Xc=(X, + Xj + X, + Xk). (12)

The followings relations are easily obtained from the definition of {M~; G~; G; G3}:

M~G, =~(-X, -X j+2Xt) , GG, =~2(X,+Xj+Xt--3Xk), MIG 3 = l ( - -X i - -X j "~ 2Xk) , GG 3 =-~2(Xi -~ Xj dr X k - 3Xt) .

and it is readily checked that these relations imply

G~M~ ^ G3M ~ = 2(G3G ^ G~G) , (13)

which states that a~ij n ~-(t) is a planar quadrangle. A quite remarkable consequence of our assumption on the piecewise linearity of the velocity displacement of the mesh is that the relations (12) holds for all


t E [t", t "-+1]. Therefore, the planar quadrangular surface {M1; G~; G; Ga} remains plane during the time interval [t", t'+~]. We note that this feature is not so common for moving grid algorithms. In particular, the deformation of hexahedral meshes usually results, in the formation of non-planar boundaries of the cells which introduces a certain arbitrariness in the definition of the volume of the cells. To compute the geometrical parameters (5, 6), let us consider Tl(t ) = {Ml(t); Gl(t); Ga(t)} and T2(t ) = {G~(t); G(t); G3(t)}, the two triangles that form the quadrangle 0% Ml-(t) and let us use the results of Section 3. We denote by ai~.(t ) the area of O~i/N ~-(t), al(t ) and a2(t ) areas of T 1 and T 2, respectively. Eq. (13) implies that

3al(t ) -- 6a2(t ) -- 2a~.(t) VtE [t ~, t '+l] . (14)

We denote by /.t~(t) the value of the integral of the normal on 0% M ~-(t). We have

~ f f n( t )dS =a~(t)n'(t) I~q(t) = a%(,)n,(,)

ffr n(t) dS -3 " - -~al(t)n (t) " 1(o

= 3 f f n(t) dS = 3a2(t)n'(t ) . JJT 2(0 The application of the formula established for a triangular facet in Section 3 gives the contribution ~/i~ of the element z to the mean normal associated to the segment [i, j]:

lf,,""ffo lftt'*t " n ( t ) dS dt = -~ . I&q(l) dt = . . . . +l ,.* ~., " tPq + P,/ +/Zq }, (15) ~i/= ~- , %(t)n,(0 with

(x~ .+1 "."=_~r" -x~)^ " -x" ~l.t//,n + 1 = 3 I __ ..yn+IG ) ^ k" [yn+l - -X GG3 ), I~ij 2 \ ' " GI n (XG3 G)

2/.ti~.* _ 3/Vn+l _XG+I X" 3 . . . . . . +1 n+l - ~, . .~, )^(~-x~) -x~)^ ) + _vl.,~c~ (Xc 3 - Xc ,

and finally we obtain the main value of the normal by summing the contribution of all tetrahedra having [i, j] as an edge:

lf~i]= E 71~i1" (16) ~.~ jrq

The contribution o-q of the tetrahedra ~- to the normal speed o-q associated with the edge [i, j] is computed in the same way. For the two triangles T 1 and T2, the assumption made on the mesh displacement implies that

f fr K dS = aL2(t)lc(Gr, 2), 1.2(0

where K(Gr,) , K(Gr2 ) are the (constant) speeds of the gravity center of T 1, T 2. Then

1 - to . n~( t ) dS dt O'iJ A/ll~qll o ,~,(,)n.(,)

1 f'"+' - Afll~qll " {a1(t)n~(t)'K(Gr') + a,(t)n'(t).K(a~)} dt. Combining this relation with Eq. (15), we obtain

,r .r ,. 1 " gi /" lqii (17)

o- . = 311.1.11 {2g(Gr ' ) + ec(Gr2)} " ~q - I1.1,;11 '

190 B. Nkonga, H. Guillard / Comput. Methods Appl. Mech. Engrg. 113 (1994) 183-204

where the mean value velocity ~~ is given by

2K(GT,) + K(GT 2) 1 K~ -- 3 -- 36 (13Ki + 13Kj + 5K k + 5Kt)

and finally the mean value of the normal velocity is obtained by summing the contribution of all tetrahedra having [i, j] as an edge:

1 E K,~ rL~ (18) "rE.~q

5. The Roe scheme on a moving mesh

The previous scheme is now applied to the solution of three-dimensional Euler's equations. In this case, the vector W and the flux F take the form

with

W= LH(W)J

/ puw / / pow / Lu(E + p)J Lv(E + p)J

F ] puw I . |pw-+p

Lw(E + p)

The pressure p is related to the other variables by the perfect gases state law,

p = (3 ' - 1)(E-P(U 2 + v 2 + w2)),

where 3' is the specific heat ratio. In accordance with Oodunov's method, the values W~j and F# used to approximate W and the flux F

on the hyper-surface ~j will be obtained from the solution of a 1-D Riemann problem along the normal ~ij and between the two states Wj and W~:

OW O,~ at + -~--n =0 '

W(n t") , ={W~. frn

B. Nkonga. H. Guillard / Comput. Methods Appl. Mech. Engrg. 113 (1994) 183-204 191

to take into account the movement of the grid is detailed in [11]. The extension to the system (19), (20) is straightforward and we obtain

% -- F , . ,1,j - '~,11 "1,11 W,,

II,l,l___JI {~(w,) + ~(w,) - ,~,j(wj + w,) + I,~ - ~rqlal (W~ IV;) } (21) _ _ . - - 2

where A = A(I,~') is the Roe's matrix, T and ,~ the matrices of the eigenvectors and eigenvalues of * , ~ - I ~ ~

I - o-ql.I = T IA - o-ql,t IT, respectively. The components of the vector W known as Roe s average are defined by

7= tT- v~, + v~ v~, + v~

= w,v~, + wHg, H : e + p ~ _ H,V~, + H, Vg, v~,. + V~ p v~, + v~fi~

The previous scheme is first order accurate. It can easily be extended to second order accuracy using the MUSCL technique introduced by Van Leer [12] and extended to a non-structured mesh in [2]. In this case, the two states W, + and W[ defining the Riemann problem are given by

+ ,, ij W/ ,, ij W, = W, + P , . - ] , = Wj - P j .~ , (22)

where P~ is an approximation of ~TW on the spatial cell ~(t") defined as an average on the set of tetrahedra having i as a node of the constant values of VW given by P1 interpolation on each simplex.

To complete the description of the numerical method, we note that it is sometimes useful to dispose of an implicit scheme for advancing the solution in time. Here we used an implicit linearized scheme where the approximate linearization of the fluxes is written as

~q(W,, W, ,+, )=4{A(W, , ) .W, ,+ I ,,+, ,,+, ,,+t i + A(W; ' ) .Wj - o'/;W~ - crqW; ~ n n (Wn+l , ,+ l +]A(Wi ,W i ) -oq l , , l ' , ~ -W; )}. (23)

Let cSW= W "+t - W". The previous equation can then be re-cast in 3-scheme form as

~v(W", W ''+') = qsq(W", 8W) + q)q(W", W"). (24)

In this case, the explicit part of the flux q)q(W", W") will be calculated with a second order accurate method while the implicit part will be evaluated by a first-order formula. Although this scheme is formally first order accurate, in the sequel the resulting scheme will be denoted by 'implicit second order accurate' to distinguish it from a pure first order implicit scheme. Indeed many experiments reveal that this scheme is much more accurate than a first order one.

Let us end with some indications of the numerical boundary conditions. We suppose that the whole boundary F of the spatial domain represents a solid wall. In the finite volume method used here, the boundary P is composed of a union of plane quadrilateral like (i, M3, G 3, M, ) of Fig. 3. Let F/be equal to / " fl 3~, where i is a node belonging to the boundary. The geometrical parameters nt; and o-1;. are obtained by the method described in Section 3, taking into account the fact that the area of F, is 1 / 3 the area of the triangle (i, j, k) (see Fig. 3). The slip condition on F i then reads as follows:

v~.nr, = crt~. (25)

This condition is imposed in a weak form by taking it into account in the computation of the flux crossing the boundary F~. After appropriate linearization, the resulting boundary flux is given by

4,~, a = 4%(W") + AtIIn,:IIA,;~W , , (26)

where the explicit part ~bj; is given by


~ t W n X n y n Z n tl )=Atlln ,ll[0 n ,p, n p, n ,p, or p,], and the implicit matrix Ar~(W") reads as

0 0 0 0 0

X /1 X n X

/3 nr ~ _ui /3nr ~ _vi /3nl ~ _wi,/3nr /3nr

/3 ~ n y . y ,, r " Y /3n~ r~ -ui/3nr~ -or/3nr, -wi/3nr~ .

11o711"- n z /3 - - f -

IIvTll 2 or,: -ui' /3orr, -v;'/3% -wT /3or , /3or ,

with/3 = (y - 1).

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6. Numerical results

6.1. Solutions of half-Riemann problems

Let us consider a rectangular tube containing a fluid at rest, closed at the right end by a piston. We denote by W~ = (p~, ut, Pt) the initial state of the fluid. At time t = 0, we move the piston suddenly with a constant velocity up. The position of the right extremity of the tube is then given by xp(t) = xp(O) + tUp. The exact solution of this (half)-Riemann problem consists of two states W t and W I connected by a rarefaction wave if up > 0 and by a shock wave if Up < 0. For up > 0 (rarefaction wave) the state W I is given by

Pl = Pt( 1

Uf = t/p

pl = pt(1 +

while if up

Y ~/Pt/

Y 3~Pt/

< 0 (shock wave) and if we call o- the shock speed, the expression of W r is given by or

Pf = Pt or _ up

( l+y)up( v _16yp, .'~ u i=u p with o" = 4 + 1 + pl(1 + r)2u2 ] "

Pf = Pt + orPtUp

Note that in the two cases, the piston plays the role of the contact discontinuity in the complete Riemann problem.

We begin to investigate the capabilities of the method by considering the previous two problems. The numerical code used here is a two-dimensional one an in order to simulate a 1-D experiment, the mesh is 101 points in the x-direction by 3 points in the y-direction. A first order scheme with a CFL number of 0.6 has been used. Initially, the pressure and density have constant values equal to unity while the velocity is zero everywhere. At time t = 0, the piston (localized on the right in Fig. 4 and Fig. 5) is suddenly moved with a velocity up = 1 resulting in the development of a rarefaction wave in the cylinder or with a velocity up =-1 resulting in the creation of a shock wave. Fig. 4 shows the velocity and density obtained for the rarefaction wave case while Fig. 5 displays the results for the moving shock

DENSITY

o

0.00 1.50 0.00 ci Max = 0.99999 d Max = 1.(3000

Min = 0.36739 Min = 0.27359

PRESSURE

!

1.50

Fig. 4. Rarefaction wave: first order scheme solution (stars) compared to analytical result (solid line).


!

DENSITY 1 PRESSURE

00 Max= 2.3550 0.75

Min -- 1.0000

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ./

00 Max = 2.9266

Sin = 1.0~0

!

0.75

Fig. 5. Shock wave: first order scheme solution (stars) compared to analytical result (solid line).

wave. For reference, we have plotted in Fig. 6 the same quantit ies in the case of the classical Sod shock tube prob lem where the contact discontinuity plays the role of the piston. It can be seen that the results obta ined with the moving boundary algorithm is of the same quality as those obta ined in the case of a fixed mesh. No osci l lat ions are present in the shock and rarefact ion waves profi les. Near the moving boundary (right boundary) , a smear ing of the contact discontinuity can be noticed in the results. A

194 B. Nkonga. H. Guillard I Comput. Methods Appl. Mech. Engrg. 113 (1994) 183-204

PRESSURE

" t I ,i i II

i DENSITY

i000 ,00 0 ,00 Max = 1.0000 o Max = 1.0000 /din= 0.10000 Mill= 0.12500

Fig. 6. First order numerical solution of the classical Sod shock tube problem.

small overshot due to the boundary conditions can be seen on the density profiles at this location, this overshot is absent from the pressure and velocity (not displayed here) profiles, indicating that this problem is related to the contact discontinuity that remains attached to the piston in its displacement.

6.2. Isentropic compression

In this example, the experimental conditions are the same as previously but the piston is moved in a smooth manner in order to avoid the generation of shock waves and to keep the flow isentropic. The law giving the piston position is representative of the motion of a piston in an internal combustion engine and reads

xp( t )=- I+13+~- +~cos(O)+ 13-~ a- ~ - ~- sin-(O ), (29)

where O(t) (the Crank angle) equals 0(0)+ 2~-oJt, and the parameters 1, a, /3, 0(0) and w take the va lues :

/ = 0.132 cm, a = 8.9cm, /3 = 15.5 cm, 0(0)= - iv , w = 2000 tr /min

An analytical solution of this problem can be found using a low Mach number approximation. Let us denote by At(t), .i:r(t) the velocity and the acceleration of the piston, then up to terms of order Ma the solution of this problem is written as

_

. .

p(x , t )=p, \~} + k 3 -x .

For this low Mach number experiment, the numerical results are obtained using the implicit second order scheme described in Section 4. A (21 5) triangular mesh is used and the time step is constant corresponding in formula (29) to a variation 60 =-~/360. This corresponds to a CFL number in the

B. Nkonga. H. Guillard / Comput. Methods Appl. Mech. Engrg. 113 (1994) 183-204 195

REDUCED PRESSURE

- " . . - . .

~0.1~ '~'*~47 d Max = 0.194141:I-4N. "

Min = 0.23396E-07

~ .~ VELOCITY

5i Max = -0.80659E-01 Nin = -4117.12

Fig. 7. lsentropic compression 0 = -~r/9. Second order implicit scheme solution (without gradient limitation) (stars) compared to analytical result (solid line) The reduced pressure is defined by ll(x)= (p(x)- p,,,)/Pm,x.

interval [7, 100]. Fig. 7 displays the result at 0 = - ' r r /9 . It is seen that the agreement with the analytical solution is excellent: the velocity profile is linear and except for some oscillations near the boundaries, the quadratic behaviour of the dynamic pressure is well reproduced. These numerical results were obtained using a MUSCL approximation without the gradient limitation usually associated with a TVD scheme for the capture of discontinuity waves. For this very smooth solution, gradient limitation has not been found necessary Fig. 8 shows that the first order implicit scheme failed to compute the quadratic perturbation of the pressure. Indeed it seems that for this type of flow where the spatial variation of pressure and density are very small, second order accuracy is necessary.

REDU(~D PRESSURE

! .-"

---.S~.:. 0 ~ 0~47 _0.00 0.47

Max-- 0.21866E-03 ~ Max = -10.456 /din-- 0. /din = -396.95

Fig. 8. Isentropic compression 0 = -lr/9. First order implicit scheme solution (stars) compared to analytical result (solid line). The reduced pressure is defined by ll(x)= (p(x)- P.,,.)/P,,,,,x.


6.3. 2-D model of a pitching NACA 0012 airfoil

We now consider an airfoil experiencing a periodic pitching between -1 and + 1 degrees. The angle of attack at time t is defined by O(t) -- sin(~t/2). The initial conditions are the converged solution of the steady problem with a null angle of attack. The numerical scheme is explicit and uses the MUSCL technique with gradient limitation to obtain second order accuracy. A maximum CFL number of 0.5 and a mesh of 2280 vertices has been used. Fig. 9 displays the results obtained after a half cycle calculation. We observe that the inertia of the movement induces several changes with respect to the steady state solutions. In particular, even for a null angle of attack, the solution is not symmetric as can be seen in Fig. 9(b) where the shock position on the extrados is shifted with respect to the shock position on the intrados. Fig. 9(d) displays the concurrent evolution of the angle of attack and lift defined by f(air foil) pny dl. It can be seen that the two curves are out of phase and that a negative lift can correspond to a positive angle of attack when the latter is diminishing while on the contrary a positive lift can be obtained for a negative angle of attack when the latter is increasing.

Max= 1.35

(a)

Min= 0.03

l I

Max= 1.33

(b)

/ ] /N

Min= 0.00

(c)

Min 0 03 Max= 1.35

0.1

0

-0.1

(d)

/ /

i i / /

TUN

Fig. 9. Pitching NACA 0012 airfoil (Ma~ = 0.85). Mach lines for an angle of attack O = 1 (a), O = 0 (b) and 0 -- -1" (c). In (d) evolution of the angle of attack (solid line and scale on the right) and the lift (dashed line and scale on the left) are plotted.


6.4. Flow in a three-dimensional piston engine

We end this section with some results obtained on a three-dimensional computat ion of the flow inside the cylinder of a piston engine. This type of application is characterized by a low or very low Mach number, complex geometries, intrinsically three-dimensional effects and strong deformation of the

Fig. 10. Computational domain at bottom dead center (t = 0 and 0 = -rr).

98 B. Nkonga, H. Guillard / Cornput. Methods Appl. Mech. Engrg. 113 (1994) 183-204

Fig. 11. Computational domain at top dead center (0 = 0).

~ i ~

~' ~ ~. . , , . _ . _ .

" ~' 'P r

Fig. 12. Initial velocity field on the symmetry plane (# = -at , bottom dead center) Vm,,~ = 22.23 m/s .


meshes due to the movement of the piston. The geometry under study is an industrial combustion chamber used in some Renault vehicles. Figs. 10 and 11 displays two views of the chamber at different times during the cycle. Only one half of the chamber is displayed as it is symmetric. The radius of the cylinder is 4 cm and at time t = 0 (bottom dead center; Fig. 10) its height is 8:45 cm while at top dead center the height is reduced to 0.10 cm. The roof closing the cylinder head is 1.6 cm high. Figs. 10 and 11 also show the mesh used in the computation. It consists of a 4032 node, 20 166 element non- structured tetrahedral mesh built by the 3-D Delaunay-Voronoi algorithm [13]. The motion of the piston is given by the law defined by (29) where now the parameters l, a, /3, 0(0), to have the value

1 = 0.100 cm, a = 8.35 cm, /3 = 14.0cm, 0(0)= -~r , to = 1200 t r /min

In order to simulate the motion of the flow at the end of the induction stroke, at time t=0 a recirculating motion of horizontal axis (called a tumble in the engineering literature) is artificially created. The initial conditions are then described by the following relations:

Pt = 1.19 10 -3 g /cm 3 ,

tit(x) = -K ( r ) - - z -3 .6

vl =0cm/s ,

X w,(x) = K(r) r

Pt = 106 (cgs) ,

r = ~/x z + (z - 3.6) 2 , sin(0.14~rr) sin(0.16"rrr) K(r) = C (r + 10-3)8

The corresponding velocity field on the symmetry plane is plotted in Fig. 12. The computation uses the second order linearized implicit scheme described in Section 4 without gradient limitation. The time step is constant and equivalent to a variation of 80 = ~r/180. The computation presented here extends from 0 = -~r (bottom dead center) to 0 = "rr. As can be seen in Fig. 13, at 0 = -a t /3 , the piston motion has compressed the initial vortex and the rotation axis has moved towards the right of the chamber. An explanation may be that the piston moving upwards accelerates the left side of the initial recirculating zone while it decelerates the right side. Fig. 14 displaying stream-surfaces (surfaces tangential to the velocity field) shows that the compressed tumble is accompanied by a helicoidal motion from the side of the cylinder towards the symmetry plane. At top dead center (0 = 0), we notice an important change in the structure of the flow. The roof and the spherical part of the head chamber perturb the tumble structure significantly and change its rotation axis which becomes gradually more and more inclined as

Fig. 13. Velocity field on the symmetry plane (O = -~/3, piston is moving upward). V.,,, = 19.61 cm/s.


Fig. 14. Global flow structure: 0 = -~r /3 and Vm,,~ = 19.61 cm/s .

Fig. 15. Global flow structure at top dead center (0 = 0). Vm~ x = 22.0 cm/s .


Fig. 16. Velocity field on the symmetry plane at top dead center (0 = 0). V,.,x = 22.0 cm/s .

Fig. 17. Velocity field on the symmetry plane (0 = ~r/6, piston is moving downward). Vm, x = 13.8cm/s.

Fig. 18. Velocity field on the symmetry plane (0 = l r /3 , piston is moving downward). Vma ~ = 14.6cm/s.

202 B. Nkonga. H. Guillard / Comput. Methods Appl. Mech. Engrg. 113 (1994) 183-204

Fig. 19. Velocity field on the skin of the head chamber (0 = 7/18). V,,~x = 20.0 cm/s.

we move from the symmetry plane towards the spherical side of the chamber head (Fig. 15). During the expansion phase after top dead center, the piston moves downward and thus now accelerates the right part of the flow. This largely destroys the tumble motion and no recirculating region can now be seen in the symmetry plane (Figs. 17 and 18). A closer examination of the velocity field reveals that the flow now has an important horizontal component with a small recirculating motion of near vertical axis that moves slowly to the right of the chamber (Figs. 19, 20 and 21). Finally, we note that the flow intensity, measured by the maximum velocity norm, remains important during the computation: at 0 = ~r/3 the maximum velocity is still of the order of 14 m/s (note that at this time, the piston velocity is 5 m/s) .

Fig. 20. Velocity field on the skin of the head chamber (0 = ~r/6). Vm,,~ = 13.8cm/s.


Fig. 21. Velocity field on the skin of the head chamber (0 = ~r/3). Vm, ~ = 14.6cm/s.

7. Concluding remarks

A numerical algorithm for the computation of three-dimensional flows on moving grids has been developed and tested. Due to the space-t ime formulation of Godunov's method, this technique incorporates the movement of the grid in a very simple way that automatically satisfies the geometrical conservation law. One of the advantages of this technique is that it results from straightforward modifications of existing Eulerian codes. This has been exemplified by the implementation of the technique into a three-dimensional method designed to use non-structured tetrahedral meshes. Second order and implicit versions of the method have also been built. Numerical examples have shown that when combined with unstructured meshes, this technique can be efficiently used for a broad range of applications from very subsonic flow to transonic flows. Work is presently underway to extend the method to true second-order (in time and space) accuracy and to add to it re-gridding capabilities to deal with the appearance of large or small cells induced by the displacement of the boundaries.

Acknowledgment

This work has been supported in part by Renault S.A. We address special thanks to Olivier Bailly of 'Direction des Recherches' of Renault for his help in setting up the three-dimensional problem of Section 5.

References

[1] A. Dervieux, Steady Euler simulations using unstructured meshes, in: Geymonat, ed., Partial Differential Equations of Hyperbolic Type and Applications (World Scientific, Singapore, 1987) 33-11.

[2] L. Fezoui, R6solution des 6quations d'euler par un sch6ma de van leer en 616ments finis, Rapport INRIA, 358 (1988). [3] L. Fezoui and B. Stoufflet, A class of implicit upwind schemes for Euler simulation with un-structured meshes, .I. Comput.

Phys. 84 (1989). [4] M. Vinokur, An analysis of finite-difference and finite-volume formulation of conservation laws, J. Comput. Phys. 81 (1989)

1-52. [5] H. Steve, Schemas Implicites Lin6aris~s D6centr6 Pour La R~solution des Equations d'Euler en Plusieurs Dimensions,

Th~se de l'Universit6 de Provence Aix-MarseiUe 1, 1988.

204 B. Nkonga. H. Guillard / Comput. Methods Appl. Mech. Engrg. 113 (1994) 183-204

[6] P.D. Thomas and C.K. Lombard, ??? AIAA J. 17 (1979) 1030. 17] S.K. Godunov, R6solution Num6rique des Probl~mes Multidimensionnels de la Dynamique des Gaz (MIR, Moscow, 1979). [8] S.K. Godunov, A difference scheme for numerical computation of discontinuous solutions of equations of fluids dynamics,

Math. Sb. 47 (1959) 271-290. [9] A. Dervieux, L. Fezoui and F. Loriot, On high resolution extensions of lagrange-galerkin finite-element schemes, Rapport

de Recherche INRIA, 1703 (1992). [10] P.L. Roe, Approximate Riemann solver, parameter vectors and difference schemes, J. Comput. Phys. 43 (1981) 357-372. [ 11] A. Harten and J.M. Hyman, A self-adjusting grid for the computations of weak solutions of hyperbolic conservation laws, J.

Comput. Phys. 50 (1983) 235-269. [12] B. Van Leer, Towards the ultimate conservative difference scheme: 2. Monotonicity and conservation combined in a

second-order scheme, J. Comput. Phys. 14 (1974) 361-370. [13] P.L. George, F. Hecht and E. Saltel, Maillage automatique de domaines tridimensionnels quelconques, Rapport de

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FIG. 21. Electron emission and negative spark develop- ng from a sensitive spot of a cathode in nitrogen at 30 atm under im

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