UNIT�E DE RECHERCHEINRIA-SOPHIAANTIPOLIS

Institut Nationalde Rechercheen Informatiqueet en Automatique2004 route des LuciolesB.P. 9306902 Sophia-AntipolisFrance

Rapports de RechercheN�1883Programme 6Calcul scienti�que, Mod�elisationet Logiciel num�eriqueGODUNOV TYPE METHOD ONNON-STRUCTURED MESHESFOR THREE-DIMENSIONALMOVING BOUNDARYPROBLEMS

Boniface NkongaHerv�e GuillardAvril 1993

GODUNOV TYPE METHOD ON NON-STRUCTURED MESHES FORTHREE-DIMENSIONAL MOVING BOUNDARY PROBLEMSBoniface Nkonga, Herv�e Guillard.INRIA, centre de Sophia-Antipolis, B.P. 93 , 06902 Sophia-Antipolis Cedex (France)Tel : 93.65.77.95. e{mail : nkonga@cosinus.inria.frTel : 93.65.77.96. e{mail : guillard@mingus.inria.frKeywords : Hyperbolic problems, Fluid Dynamics, Moving boundaries, Non-structuredmeshes, Godunov method, Three-dimensional ows, Internal combustion engine.Abstract: This paper presents a numerical method for the computation of compress-ible ows in domains whose boundaries move in a well de�ned predictable manner. Themethod uses the space-time formulation by Godunov while the discretization is conductedon non-structured tetrahedral meshes, using Roe's approximate Riemann solver, an im-plicit time stepping and a MUSCL-type interpolation. The computation of the geometricalparameters required to take into account the movement of the boundaries is described.Examples including the calculation of the ow in the cylinder of an internal combustionengine illustrates the possibilities of the method.

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METHODE DE TYPE GODUNOV EN NON STRUCTURE POURDES PROBLEMES TRIDIMENSIONNELS A FRONTIERE MOBILEBoniface Nkonga, Herv�e Guillard.INRIA, centre de Sophia-Antipolis, B.P. 93 , 06902 Sophia-Antipolis Cedex (France)Tel : 93.65.77.95. e{mail : nkonga@cosinus.inria.frTel : 93.65.77.96. e{mail : guillard@mingus.inria.frMots cl�es : Probl�emes Hyperboliques, Dynamique des Fluides, Domaines D�eformables,Maillages non Structur�es, M�ethodes type Godunov, Ecoulements Tridimensionnels, Mo-teur �a Piston.R�esum�e: Nous pr�esentons dans cet article une m�ethode num�erique pour la r�esolutiondes �ecoulements compressibles dans un domaine �a fronti�ere mobile. Dans le cadre d'uneformulation mixte Volume �nis/El�ements �nis sur des maillages non structur�es, on utilisela discr�etisation dans l'espace spatio-temporel introduite par Godunov. Ceci nous permetde g�en�eraliser les solveurs de Riemann et en particulier le solveur approch�e de Roe, deconstruire des m�ethodes implicites par lin�earisation des ux et d'utiliser une interpolationde type MUSCL pour obtenir des sch�emas pr�ecis �a l'ordre deux. Nous d�ecrivons le calculdes param�etres g�eom�etriques n�ecessaire pour prendre en compte le mouvement des fron-ti�eres. Pour illustrer les possibilit�es de la m�ethode, cette derni�ere est appliqu�ee �a plusieurstype d'�ecoulements parmi lesquels les �ecoulements dans le cylindre d'un moteur �a piston.

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1 INTRODUCTIONThis work deals with the computation of the ows of compressible uids in moving geome-tries such as those occurring for instance in reciprocating engines and more generally inmany industrial devices. As the geometry of such industrial devices is inherently complex,the versatility and exibility of non-structured meshes is very appealing. Accordingly, thepresent method is based on the use of tetrahedral meshes. Because compressible owsoften exhibits strong compressions or discontinuities, the spatial approximation uses therobust second-order Euler solver developed in [1, 3, 4]. This solver is also attractive be-cause of its nice matricial properties when coupled with an implicit time stepping. Thispaper explains how the movement of the boundaries is included into this framework ina very simple and natural way. It is organized as follows: Section 2 discusses the ap-proximation of conservation laws on moving meshes. As stressed out by many authors(see e.g. [13] for a review), the key point here is the respect of the so-called geometricalconservation law that is the form taken by the consistency relation on a moving grid. Thetime-space formulation given by Godunov's method provides a very simple and elegantway to automatically satisfy this constraint. Then in Section 3 we provide the computa-tion of the geometrical parameters when the intersection of two spatial control-volumes isa triangular face whose three nodes move with di�erent velocities. As any planar polygo-nal surface can be split into triangular facets, the general case can be treated by repeatedapplication of formula obtained on triangular facets. Actually in Section 4, we special-ized these results to the �nite volume method of [11] where the boundaries of the controlvolumes are composed of an union of plane quadrangular faces. The �fth section dealswith modi�cations that are necessary to implement Roe's approximate Riemann solveron a moving grid. Finally, in the last section we conclude by presenting several test casesand examples illustrating the possibilities of the method.2 CONSERVATION LAWS INMOVINGDOMAINLet D(t) � IRm (m � 3) be the domain �lled by the uid at time t. The displacementof the domain from its position at time t = 0 to its position at time t is given by a map~x = '(~a) from D(0) onto D(t). The space time domain E occupied by the uid betweent = 0 and t = T is de�ned by E = T[t=0D(t):Except when the domain boundaries are �xed, the space-time domain E is not a simpleproduct of a spatial domain by a time interval. This introduces an additional di�cultywith respect to �xed boundary problems.We will denote by J(t) = D~xD~a the jacobian matrix of the map ' and by �(~x; t) thevelocity displacement of the domain D(t) at ~x 2 D(t) :~�(~x; t) = @'@t '�1(~x; t)!3

For all (~x; t) 2 E, we consider a �rst-order conservation law:@W@t +r � ~F (W ) = 0 (1)where W is a vector of IRp and ~F a smooth function from IRp into IRp. Let Ci(t)i=1;::::nsbe a splitting of the domain D(t) into no intersecting cells such that:ns[i=1 Ci(t) = D(t) and _Ci \ _Cj = � if i 6= jwe will denote by @Ci(t) the boundary of the cell Ci(t) and by @Cij(t) the intersection ofthe boundaries of the cells Ci(t) and Cj(t) such that :@Ci(t) = [j2T (i) @Cij(t)where T (i) is the set of the neighbors of i. We denote by Qi the space time hyper volumede�ned by the displacement of Ci under the action of the map ' between times t = tnand t = tn+1 and by Sij the intersection of the two hyper volumes Qi and Qj during thesame time interval. Obviously the union of Qi �lls E and we have:Sij = tn+1[t=tn @Cij(t) and @Qi = Ci(tn) [ Ci(tn+1) [� [j2T (i) Sij�Figure 1 provides a graphical representation of the space time and spatial cells Qi and Ciin the two-dimensional case.The discrete equations are obtained by integrating equation (1) over Qi resulting in:Z tn+1tn Z Z ZCi(t) @@tW d~x dt+ Z tn+1tn Z Z@Ci(t) ~F (W ) � ~n(t) dS dt = 0 (2)Using the following identity:ddt (Z Z ZCi(t)Wd~x) = Z Z ZCi(t) @@tWd~x+ Z Z@Ci(t)W ~� � ~n(t) dSequation (2) becomes:V ol(Cn+1i )W n+1i � V ol(Cni )W ni + Xj2T (i) Z tn+1tn Z Z@Ci(t)�~F (W )�W ~�� � ~n(t) dS dt = 0 (3)where W n+1i and W ni are de�ned as the mean value of W over the cell Ci(tn+1) and Ci(tn).Similarly, if we de�ne Wij and ~Fij as the mean value of W and ~F over Sij , we obtain:V ol(Cn+1i )W n+1i � V ol(Cni )W ni +�t Xj2T (i)0@ ~Fij � ~�ij � �ijjj~�ijjjWij1A = 0 (4)4

where the mean value ~�ij of the integral of the normal ~n is de�ned by:~�ij = 1�t Z Z ZSij ~n(t) dS dt = 1�t Z tn+1tn Z Z@Cij(t) ~n(t) dS dt (5)and where we have introduced �ij the mean value of the normal velocity de�ned by:�ij = 1�t jj~�ijjj Z Z ZSij ~� � ~n(t) dS dt = 1�t jj~�ijjj Z tn+1tn Z Z@Cij(t) ~� � ~n(t) dS dt (6)The correct evaluation of the above quantities is the key point in the design of a numericalscheme on a moving grid. This can be understood as follows : let us consider that attime t = 0 the variable W is equal to a constant state W0. If the domain boundaries@D are �xed, the solution of any conservation law of the form (1) will be W (~x; t) =W0 8t. Consequently the integration of (1) over any volume C(t) � D and for any map' respecting the boundaries @D will degenerate into the identity:ddt ZC(t) d~x = Z@C(t) ~� � ~n dS (7)According to (5)-(6) the discrete counterpart of this identity takes the form:V ol(Cn+1i ) � V ol(Cni ) = �t Xj2T (i)0@ jj~�ijjj �ij1A (8)and states that the increase of the volume of a cell must equal the summation of thevolumes swept out by each face of the cell between tn and tn+1. Therefore the respect of(8) is a necessary condition for a numerical scheme to keep constant an uniform initialstate and thus the evaluation of the geometrical parameters (5) and (6) must be carefullydone. This fact was �rst pointed out in [12]. Given an approximation of ~�ij and �ij, asometimes used manner to enforce (8) is to update the volume of the cells by explicitlyde�ning V ol(Cn+1i ) by equation (8). However, in this case, the actual geometrical vol-ume of the cells may di�er from the value given by (8) and it is preferable to devise adirect geometrical way to respect (8). In the one and two-dimensional cases, the correctevaluation of (5) and (6) is described in Godunov's book [6]. Actually the original 1959Godunov's method [7] was designed for a moving as well as for a �xed grid. We now givethe computation of the geometrical parameters (5) and (6) for the case of a triangularfacet moving in a three-dimensional space. As any plane polygonal surface can be splitinto a �nite number of triangles a repeated application of these results will give the generalsolution (see Section 4 for an example).3 GEOMETRICAL PARAMETERSLet T (tn) be a triangular surface in IR3 whose nodes are I(tn), J(tn) and K(tn). Ouraim is to compute the parameters (5) and (6) under the action of the map '. It mayhappen that ' is explicitely known for every (~x; t) for instance by an analytical expression,however when the motion of the domain is complex, the map ' is generally computed5

by a numerical procedure and is only known on a discrete set of points. Therefore wemake the assumption that the velocities of the three nodes I, J and K are constant (butdi�erent) during a time step �t and that the velocity of any point P belonging at timetn over T (tn) is a linear interpolation of the velocities of the nodes de�ning the facet. Let(�; �) denote the barycentric coordinate of P 2 T (tn) :~Xn = ~XP (tn) = (1� � � �) ~XI(tn) + � ~XJ (tn) + � ~XK(tn)then the velocity of P is :~�(P ) = (1� �� �)~�I + �~�J + � ~�Kand thus ~�(P ) is a constant and for any t 2 [tn; tn+1] we have :~XP (t) = ~Xn + (t� tn)~�(P )= (1 � �� �) ~XI(t) + � ~XJ (t) + � ~XK(t) (9)From this equation, we deduce that P (t) belongs to the triangle I(t), J(t), K(t) for anyt 2 [tn; tn+1] (the facet remains plane), that the barycentric coordinates (�; �) of Premain constant during the time step and that ~XP (t) varies linearly in time :~XP (t) = (1 � s) ~Xn + s ~Xn+1; with s = t� tn�tWith these relations it is now an easy task to compute ~�ij, we have �rst:~�(t) = Z ZT (t) ~n(t) dS = 12( ~XJ (t)� ~XI(t)) ^ ( ~XK(t)� ~XI(t))and it is readily found that~�T = 1�t Z tn+1tn Z ZT (t) ~n(t) dS dt = 1�t Z tn+1tn ~�(t) dt = 13(~�n + ~�n+1 + ~��) (10)where ~�n+1 and ~�n are respectively the integrals of the normal unit on T (tn) and T (tn+1),and the vector ~�� is de�ned by:2~�� = ( ~XnJ � ~XnI ) ^ ( ~Xn+1K � ~Xn+1I )2 + ( ~Xn+1J � ~Xn+1I ) ^ ( ~XnK � ~XnI )2The computation of �T proceeds in the same way. We �rst compute the volume �V sweptout by the facet T (Figure 2 ) :�V = Z tn+1tn Z ZT (t)~� � ~n(t) dS dt = Z tn+1tn ~n(t) � "Z ZT(t) ~� dS# dt= ~�G � Z tn+1tn ~n(t) areafT (t)g dt! = �t~�G � ~�Twhere ~�G is the velocity of the gravity center of the facet. We then obtain :�T = �V�t jj~�T jj = ~�Tjj~�T jj! � ~�I + ~�J + ~�K3 ! (11)This ends the computation of geometrical parameters for a triangular facet.6

4 MUSCL-FEM METHODIn this section we apply the previous results to the MUSCL-FEM method by [1, 4, 3].The three-dimensional version of this method is developped in ([11]). This method canbe understood either as a classical Galerkin �nite element method stabilized by upwindterms [2] or as a �nite volume method on non-structured simplicial meshes. For ease ofpresentation, we adopt here the second point of view. Given a non-structured tetrahedralmesh, to each node i of the mesh is associated a cell de�ned by the midpoints of theedges of the tetrahedra having i as a node, the gravity center of these tetrahedra and thegravity center of the faces of these tetrahedra. Figure 3 displays the intersection of sucha control volume with a tetrahedra. Let us split the boundary of the i{cell by an unionof partial boundaries associated to each edge:@Ci = [j2T (i) @CijFrom Figure 3, it is seen that @Cij is itself an union on Tij, the set of tetrahedra having[i; j] as an edge, of quadrangular surfaces :@Cij = [�2Tij(@Cij \ � )Let i; j; k; l be the four nodes de�ning such a tetrahedra � , @Cij \ � is de�ned by the fourpoints fM1; G1; G; G3g (see Figure 3 ) whose coordinates are :~XM1 = 12( ~Xi + ~Xj); ~XG1 = 13( ~Xi + ~Xj + ~Xl);~XG3 = 13( ~Xi + ~Xj + ~Xk); ~XG = 14( ~Xi + ~Xj + ~Xl + ~Xk): (12)The following relations are easily obtained from the de�nition of fM1; G1; G; G3g :���!M1G1 = 16(� ~Xi � ~Xj + 2 ~Xl); ��!GG1 = 112( ~Xi + ~Xj + ~Xl � 3 ~Xk);���!M1G3 = 16(� ~Xi � ~Xj + 2 ~Xk) and ��!GG3 = 112( ~Xi + ~Xj + ~Xk � 3 ~Xl):and it is readily checked that these relations imply :���!G1M1 ^ ���!G3M1 = 2 (��!G3G ^ ��!G1G) (13)which states that @Cij \ � (t) is a planar quadrangle. A quite remarkable consequenceof our assumption on the piecewise linearity of the velocity displacement of the mesh isthat the relations (12) holds for all t 2 [tn; tn+1]. Therefore, the planar quadrangularsurface fM1; G1; G; G3g remains plane during the time interval [tn; tn+1]. We note thatthis feature is not so common for moving grid algorithms. In particular, the deforma-tion of hexahedral meshes usually results in the formation of non-planar boundaries ofthe cells which introduces a certain arbitrariness in the de�nition of the volume of the cells.7

To compute the geometrical parameters (5,6), let us consider T1(t) = fM1(t) ; G1(t) ; G3(t)gand T2(t) = fG1(t) ; G(t) ; G3(t)g; the two triangles that form the quadrangle @Cij \ � (t)and let us use the results of Section 3. We note a�ij(t) the area of @Cij \ � (t), a1(t) anda2(t) respectively areas of T1 and T2. Equation (13) implies that:3 a1(t) = 6 a2(t) = 2 a�ij(t) 8 t 2 [tn; tn+1] (14)We note ~��ij(t) the value of the integral of the normal on @Cij \ � (t). We have:~��ij(t) = Z Z@Cij(t)\�(t)~n(t) dS = a�ij(t)~n� (t)= 32 Z ZT1(t) ~n(t) dS = 32a1(t)~n� (t)= 3 Z ZT2(t) ~n(t) dS = 3 a2(t)~n�(t)The application of the formula established for a triangular facet in Section 3 gives thecontribution ~��ij of the element � to the mean normal associated to the segment [i; j]:~��ij = 1�t Z tn+1tn Z Z@Cij(t)\�(t)~n(t) dS dt = 1�t Z tn+1tn ~��ij(t) dt = 13� ~��;n+1ij + ~��;�ij + ~��;nij �(15)With~��;n+1ij = 32( ~Xn+1G1 � ~Xn+1G ) ^ ( ~Xn+1G3 � ~Xn+1G ); ~��;nij = 32( ~XnG1 � ~XnG) ^ ( ~XnG3 � ~XnG)and 2 ~��;�ij = 32( ~Xn+1G1 � ~Xn+1G ) ^ ( ~XnG3 � ~XnG) + 32( ~XnG1 � ~XnG) ^ ( ~Xn+1G3 � ~Xn+1G ):and �nally we obtain the mean value of the normal by summing the contribution of alltetrahedra having [i; j] as an edge : ~�ij = X�2Tij ~��ij (16)The contribution ��ij of the tetrahedra � to the normal speed �ij associated with theedge [i; j] is computed in the same way. For the two triangles T1 and T2, the assumptionmade on the mesh displacement implies that:Z ZT1;2(t)~� dS = a1;2(t) ~�(GT1;2)where ~�(GT1); ~�(GT2) are the (constant) speeds of the gravity center of T1; T2. Then��ij = 1�tjj~�ijjj Z tn+1tn Z Z@Ci(t)\�(t)~� � ~n� (t) dS dt= 1�tjj~�ijjj Z tn+1tn ( a1(t)~n�(t) � ~�(GT1) + a2(t)~n�(t) � ~�(GT2)) dt8

Combining this relation with equation (15) we obtain:��ij = 13jj~�ijjj( 2~�(GT1) + ~�(GT2)) � ~��ij = ~��ij � ~��ijjj~�ijjj (17)where the mean value velocity ~��ij is given by:~��ij = 2~�(GT1) + ~�(GT23 = 136�13~�i + 13~�j + 5~�k + 5~�l�and �nally the mean value of the normal velocity is obtained by summing the contributionof all tetrahedra having [i; j] as an edge:�ij = X�2Tij ��ij = 1jj~�ijjj X�2Tij ~��ij � ~��ij (18)5 THE ROE SCHEME ON MOVING MESHThe previous scheme is now applied to the solution of the three-dimensional Euler's equa-tions. In this case, the vector W and the ux ~F take the form:W = 0BBBBB@ ��u�v�wE 1CCCCCA ~F = 0B@ F (W )G(W )H(W )1CAwith F (W ) = 0BBBBB@ �u�u2 + p�uv�uwu(E + p)1CCCCCA ; G(W ) = 0BBBBB@ �v�uv�v2 + p�vwv(E + p)1CCCCCA ; H(W ) = 0BBBBB@ �w�uw�vw�w2 + pw(E + p)1CCCCCAThe pressure p is related to the other variables by the perfect gases state law following:p = ( � 1)�E � 12�(u2 + v2 + w2)� is the speci�c heat ratio.In accordance with Godunov's method, the valuesWij and ~Fij used to approximateW andthe ux ~F on the hyper-surface Sij will be obtained from the solution of a 1-D Riemannproblem along the normal ~�ij and between the two states Wj and Wi:8>>><>>>: @W@t + @F@n = 0W (n; tn) = ( Wi for n < 0Wj for n > 0 (19)9

where n is the co-ordinate along ~n = ~�ij=jj~�ij jj and F = ~F � ~n = nxF + nyG + nzH.Using the notation # = ~u � ~n the ux F is given by:F = (�#; �u # + p nx; �v # + p ny; �w # + p nz ; # (E + p)) (20)The solution Wexact(n; t) of the Riemann problem (19) is constant on any curve de�nedby: nt = �. The approximate values of W and ~F:~n will then be chosen as the values ofW and F on the straight line de�ned by the equation : nt = �ij where �ij is the average"slope" of the (hyper)-surface Sij separating cells Qi and Qj.Of course the exact solution Wexact(�) of the Riemann problem can be used to calculatethe hyperbolic ux. However to reduce the computational cost, we use Roe's approximateRiemann solver [10] which is based on a linearization of the uxes. In 1-D, the modi�cationof Roe's scheme necessary to take into account the movement of the grid is detailed in[8]. The extension to the system (19)(20) is straightforward and we obtain:�ij = ~Fij � ~�ij � �ijjj~�ijjjWij' jj~�ijjj2 n F(Wi) + F(Wj) � �ij(Wj +Wi ) + j ~A� �ij Idj � (Wi �Wj) o (21)where ~A = A( ~W ) is the Roe's matrix, ~T and ~� respectively the matrix of the eigenvectorsand eigenvalues of ~A, j ~A� �ij Idj = ~T�1j~� � �ijIdj ~T . The components of the vector ~Wknown as Roe's average are de�ned by:~u = uip�i + ujp�jp�i +p�j ; ~v = vip�i + vjp�jp�i +p�j ;~w = wip�i + wjp�jp�i +p�j ; H = E + p� ; and ~H = Hip�i +Hjp�jp�i +p�j :The previous scheme is �rst order accurate. It can easily be extended to second orderaccuracy using the MUSCL technique introduced by Van Leer [9] and extended to nonstructured mesh in [3]. In this case, the two states W+i and W�j de�ning the Riemannproblem are given by : 8>>>><>>>>: W+i = W ni + ~Pi � �!ij2W�j = W nj � ~Pj � �!ij2 (22)where ~Pi is an approximation of rW on the spatial cell Ci(tn) de�ned as an averageon the set of tetrahedra having i as a node of the constant values of rW given by P1interpolation on each simplex.To complete the description of the numerical method, we note that it is sometimesuseful to dispose of an implicit scheme for advancing in time the solution. We used here10

an implicit linearized scheme where the approximate linearization of the uxes writes as:�ij(W n;W n+1) = 12f A(W ni ) �W n+1i +A(W nj ) �W n+1j��ij W n+1i � �ij W n+1j+ j ~A(W ni ;W nj )� �ijIdj � (W n+1i �W n+1j ) g (23)Let �W = W n+1 �W n, the previous equation can be re{cast in �-scheme form:�ij(W n;W n+1) = �ij(W n; �W ) + �ij(W n;W n) (24)In this case, the explicit part of the ux �ij(W n;W n) will be calculated with a secondorder accurate method while the implicit part will be evaluated by �rst-order formula.Although this scheme is formally �rst order accurate, in the sequel the resulting schemewill be denoted by \implicit second order accurate" to distinguish it with a pure �rst-order implicit scheme. Indeed many experiments reveal that this scheme is much moreaccurate than a �rst order one.Let us end with some indications on the numerical boundary conditions. We supposethat the whole boundary � of the spatial domain represent a solid wall. In the �nitevolume method used here, the boundary � is composed of an union of plane quadrilaterallike ( i; M3; ; G3 ;M1 ) of Figure 3. Let �i be equal to � \ @Ci, where i is a nodebelonging to the boundary. The geometrical parameters ~n�i and ��i are obtain in theway that is described in section 3, taking into account the fact that the area of �i is 1=3the area of the triangle (i; j; k) (see �gure 3). The slip condition on �i then reads asfollow: ~vi � ~n�i = ��i (25)This condition is imposed in a weak form by taking it into account in the computation ofthe ux crossing the boundary �i. After appropriate linearization, the resulting boundary ux is given by: �IL�i = ��i(W n) + �tjj~n�ijjA�i �Wi (26)where the explicit part ��i is given by:�T�i(W n) = �tjj~n�ijj �0; nx�ipni ; ny�ipni ; nz�ipni ; ��ipni � (27)and the implicit matrix A�i(W n) reads as:0BBBBBBBBBBBBBBB@ 0 0 0 0 0� jj~vni jj22 nx�i �uni �nx�i �vni �nx�i �wni �nx�i �nx�i� jj~vni jj22 ny�i �uni �ny�i �vni �ny�i �wni �ny�i �ny�i� jj~vni jj22 nz�i �uni �nz�i �vni �nz�i �wni �nz�i �nz�ijj~vni jj22 ��i �uni ���i �vni ���i �wni ���i ���i1CCCCCCCCCCCCCCCA (28)11

with � = ( � 1).6 NUMERICAL RESULTSSolutions of half-Riemann problemsLet us consider a rectangular tube containing a uid at rest, closed at the right end bya piston. We note Wl = (�l; ul; pl) the initial state of the uid. At time t = 0, we movesuddenly the piston with a constant velocity up. The position of the right extremity ofthe tube is then given by: xp(t) = xp(0)+ t up. The exact solution of this (half)-Riemannproblem consists of two states Wl and Wf connected by a rarefaction wave if up > 0 andby a shock wave if up < 0. For up > 0 (rarefaction wave) the state Wf is given by:8>>>>>>>><>>>>>>>>: �f = �l 1 � ( � 1)up2 s �l pl! 2 �1uf = uppf = pl 1 + ( � 1)up2 s �l pl! 2 �1while if up < 0 (shock wave) and if we call � the shock speed, the expression of Wf isgiven by:8>>><>>>: �f = �l �� � upuf = uppf = pl + ��lup with � = (1 + )up4 0@1 +vuut1 + 16 pl�l(1 + )2 u2p 1ANote that in the two cases, the piston plays the role of the contact discontinuity in thecomplete Riemann problem.We begin to investigate the capabilities of the method by considering the previous twoproblems. The numerical code used here is a two dimensional one and in order to simulatea 1-D experiment, the mesh is 101 points in the x-direction by 3 points in the y-direction.A �rst-order scheme with a CFL number of 0.6 has been used. Initially, the pressureand density have constant values equal to unity while the velocity is zero everywhere. Attime t=0, the piston (localized on the right in Figure 4 and Figure 5) is suddenly movedwith a velocity up = 1 resulting in the development of a rarefaction wave in the cylinderor with a velocity up = �1 resulting in the creation of a shock wave. Figure 4 showsthe velocity and density obtained for the rarefaction wave case while Figure 5 displaysthe results for the moving shock wave. For reference we have plotted in Figure 6 thesame quantities in the case of the classical Sod shock tube problem where the contactdiscontinuity plays the role of the piston. It can be seen that the results obtained withthe moving boundary algorithm is of the same quality than the ones obtained in the caseof a �xed mesh. No oscillation are present in the shock and rarefaction waves pro�les.Near the moving boundary (right boundary), a smearing of the contact discontinuitycan be noticed in the results. A small overshot due to the boundary conditions can beseen on the density pro�les at this location, this overshot is absent from the pressure andvelocity (not displayed here) pro�les, indicating that this problem is related to the contactdiscontinuity that remains attached to the piston in its displacement.12

Isentropic compressionIn this example, the experimental conditions are the same as previously but the piston ismoved in a smooth manner in order to avoid the generation of shock waves and to keepthe ow isentropic. The law giving the piston position is representative of the motion ofa piston in an internal combustion engine and reads:xp(t) = �(l + � + �2 ) + �2 cos(�) +s�2 � �24 sin2(�) (29)where �(t) (�(t) is the Crank angle) equals �(0) + 2� !t, and the parameters l, �, �,�(0) and ! take the values:l = 0:132 cm; � = 8:9 cm; � = 15:5 cm; �(0) = ��; ! = 2000 tr=mnAn analytical solution of this problem can be found using a low Mach number approxi-mation. Let us note _xp(t), �xp(t) the velocity and the acceleration of the piston, then upto terms of order Ma the solution of this problem writes:�(~x; t) = �l xp(0)xp(t)! ; u(~x; t) = x _xp(t)xp(t)! ;p(~x; t) = pl xp(0)xp(t)! + �l xp(0) �xp(t)2 x2p(t) x2p(t)3 � x2! :For this low Mach number experiment, the numerical results are obtained using the im-plicit second order scheme described in section 4. A (21 � 5) triangular mesh is usedand the time step is constant corresponding in formula (29) to a variation �� = �=360.This corresponds to a CFL number in the interval [7,100]. Figure 7 displays the resultat � = ��=9. It is seen that the agreement with the analytical solution is excellent:the velocity pro�le is linear and except some small oscillations near the boundaries, thequadratic behaviour of the dynamic pressure is well reproduced. These numerical resultswere obtained using a MUSCL approximation without the gradient limitation usuallyassociated to TVD scheme for the capture of discontinuity waves. For this very smoothsolution, gradient limitation have not been found necessary. Figure 8 show that �rst orderimplicit scheme failed to compute the quadratic perturbation of the pressure. Indeed itseems that for this type of ow where the spatial variation of pressure and density arevery small, second order accuracy are necessary.2-D model of a pitching NACA 0012 airfoilWe consider now an airfoil experiencing a periodic pitching between �1 and +1 degree.The angle of attack at time t is de�ned by �(t) = sin(�t2 ). The initial conditions arethe converged solution of the steady problem with a null angle of attack. The numericalscheme is explicit and uses the MUSCL technique with gradient limitation to obtainsecond-order accuracy. A maximum CFL number of 0.5 and a mesh of 2280 vertices hasbeen used. Figure 9 displays the results obtained after a half cycle calculation. We observethat the inertia of the movement induces several changes with respect to the steady state13

solutions. In particular, even for a null angle of attack, the solution is not symmetric ascan be seen in Figure 9.b where the shock position on the extrados is shift with respectto the shock position on the intrados. Figure 9.d displays the concurent evolution of theangle of attack and lift de�ned by R(airfoil) p ny dl. It can be seen that the two cuves areout of phase and that a negative lift can correspond to a positive angle of attack when thislatter is diminishing while on the contrary a positive lift can be obtained for a negativeangle of attack when this one is increasing.Flow in a three-dimensional piston engineWe end this Section by some results obtained on a three-dimensional computation of the ow inside the cylinder of a piston engine. This type of application is characterized by alow or very low Mach number, complex geometries, intrinsically three-dimensional e�ectsand strong deformation of the meshes due to the movement of the piston. The geometryunder study is an industrial combustion chamber used in some Renault vehicles. Figures10 and 11 displays two views of the chamber at di�erents time during the cycle. Onlyone half of the chamber is displayed as it is symmetric. The radius of the cylinder is 4 cmand at time t=0 (bottom dead center ; Figure 10) its height is 8:45 cm while at top deadcenter the height is reduced to 0:10 cm. The roof closing the cylinder head is 1:6 cm high.Figures 10 and 11 also show the mesh used in the computation. It consists of a 4032nodes, 20166 elements non-structured tetraedral mesh built by the 3-D Delaunay-Voronoialgorithm by [5]. The motion of the piston is given by the law de�ned by (29) where nowthe parameters l; �; �; �(0); ! have the value:l = 0:100 cm; � = 8:35 cm; � = 14:0 cm; �(0) = ��; ! = 1200 tr=mnIn order to simulate the motion of the ow at the end of the induction stroke, at timet = 0 a recirculating motion of horizontal axis (called a tumble in engineer's litterature)is arti�cially created. The initial conditions are then described by the following relations:�l = 1:19 10�3 g=cm3; vl = 0 cm=s; pl = 106 (cgs);ul(~x) = �K(r)z � 3:6r ; wl(~x) = K(r)xrr = qx2 + (z � 3:6)2 K(r) = C � sin (0:14�r) sin (0:16�r)(r + 10�3 )0:8The corresponding velocity �eld on the symmetry plane is plotted in �gure 12.The computation uses the second order linearized implicit scheme described in Section 4without gradient limitation. The time step is constant and equivalent to a variation of�� = �180. The computation presented here extends from � = �� (bottom dead center) to� = �. As can be seen in Figure 13, at � = ��3 , the piston motion have compress the initialvortex and the rotation axis has moved toward the right of the chamber. An explanationmay be that the piston moving upward accelerates the left side of the initial recirculatingzone while it decellerates the right side. Figure 14 displaying stream{surfaces (surfacestangential to the velocity �eld) shows that the compressed tumble is accompanied by14

an helicoidal motion from the side of the cylinder toward the symmetry plane. At topdead center ( � = 0:) we notice an important change into the structure of the ow.The roof and the spherical part of the head chamber perturb signi�cantly the tumblestructure and change its rotation axis that becomes gradually more and more inclined aswe move from the symmetry plane toward the spherical side of the chamber head (Figure15). During the expansion phase after top dead center, the piston moves downward andthus accelerates now the right part of the ow. This destroys largely the tumble motionand no recirculating region can now be seen in the symmetry plane (Figures 17 and 18). Actually, a closer examination of the velocity �eld reveals that the ow has now animportant horizontal component with a small recirculating motion of near vertical axisthat moves slowly to the right of the chamber (Figure 19, 20 and 21 ). Finally we note thatthe ow intensity, measured by the maximum velocity norm, remains important duringthe computation : at � = �=3 the maximum velocity is still of the order of 14 m/s (notethat at this time, the piston velocity is 5m/s).Concluding remarks.A numerical algorithm for the computation of three-dimensional ows on moving grids hasbeen developped and tested. Due to the space-time formulation of Godunov's method,this technique incorporates the movement of the grid in a very simple way that auto-matically satisfy the geometrical conservation law. Beside that, one of the advantagesof this technique is that its results from straightforward modi�cations of existing Eule-rian codes. This has been exampli�ed by the implementation of the technique into athree dimensional method designed to use non-structured tetrahedral meshes. Second-order and implicit versions of the method have also been built. Numerical examples haveshown that when combined with unstructured meshes, this technique can be e�cientlyused for a broad range of applications from very subsonic ow to transonic ows. Worksare presently undertaken to extend the method to true second-order (in time and space)accuracy and to add to it reggridding capabilities to deal with the appearance of large orsmall cells induced by the displacement of the boundaries.AcknowledgementsThis work has been supported in part by RENAULT S.A. We address special thanks toOlivier Bailly of "Direction de la Recherche" of RENAULT for his help in setting up thethree dimensional problem of Section 5.15

References[1] A. DERVIEUX. Steady euler simulations using unstructured meshes. Partial dif-ferential equations of hyperbolic type and applications, Geymonat ed., pages 33{111,1987. World Scienti�c, Singapore.[2] A. DERVIEUX, L. FEZOUI, and F. LORIOT. On high resolution extensions oflagrange-galerkin �nite-element schemes. Rapport de Recherche INRIA, 1703, juin1992.[3] L. FEZOUI. R�esolution des �equation d'euler par un sch�ema de van leer en �el�ements�nis. Rapport INRIA, 358, 1988.[4] L. FEZOUI and B. STOUFFLET. A class of implicit upwind schemes for eulersimulation with un-structured meshes. Journal of Computational Physics, 84, 1989.[5] P. L. GEORGE, F. HECHT, and E. SALTEL. Maillage automatique de domainestridimensionnels quelconques. Rapport de recherche INRIA, 1021, 1989.[6] S. GODUNOV. R�esolution num�erique des probl�emes multidimensionnels de la dy-namique des gaz. MIR Moscou, 1979.[7] S.K. GODUNOV. A di�erence scheme for numerical computation of discontinoussolutions of equations of uids dynamics. Math. Sb., 47:271{290, 1959.[8] A. HARTEN and J.M. HYMAN. A self-adjusting grid for the computations of weaksolutions of hyperbolic conservation laws. Journal of Comp. Physics, 50:235{269,1983.[9] B. VAN LEER. Towards the ultimate conservative di�erence scheme: 2) monotonic-ity and conservation combined in a second-order scheme. J. Comp. Phys., 14:361{370,1974.[10] P.L. ROE. Approximate riemann solver, parameter vectors and di�erence schemes.Journal of Computational Physics, 43:357{372, 1981.[11] H. STEVE. Sch�emas Implicites Lin�earis�es D�ecentr�e Pour La R�esolution des Equa-tions d'Euler en Plusieurs Dimensions. Th�ese de l'Universit�e de Provence Aix-Marseille 1, Juillet 1988.[12] P.D. THOMAS and C.K. LOMBARD. . AIAA Journal, 17:1030, 1979.[13] M. VINOKUR. An analysis of �nite-di�erence and �nite-volume formulation ofconservation laws. Journal of Computational Physics, 81:1{52, 1989.16

x

y

t

o

GI

ijS

iC

t=t

(n)

(n+1)

t=t

(n+1)

(n)

(n+1)

(n+1)

Ci

I

G

(n)

(n)

iFigure 1: Hyper control-volume Qi and spatial cell Ci for tow-dimensional case

17

n(t)

n(t)

I

JK

Figure 2: Volume �V sweep out by a triangular facet (I,J.K).18

i

j

3G

M

G 1

G 2

G 3 G

M 1

M 2

M 3

kkl

G 3

M 1 G

1

GFigure 3: Intersection of @Ci with a neighboring element.19

1.50 0.00

DENSITY

Max = 0.99999

Min = 0.36739

X

0.1

0000

E+

01 0

.300

00E

+00

****************************************************************************************************

*

1.50 0.00

PRESSURE

Max = 1.0000

Min = 0.27359

X

0.1

0000

E+

01 0

.200

00E

+00

****************************************************************************************************

*

Figure 4: Rarefaction wave: First-order scheme solution (stars) compared to analyticalresult (solid line). 0.75 0.00

DENSITY

Max = 2.3550

Min = 1.0000

X

0.2

4000

E+

01 0

.850

00E

+00 ******************************************************************

*

*

*

*

*****************************

*

*

0.75 0.00

PRESSURE

Max = 2.9266

Min = 1.0000

X

0.3

0000

E+

01 0

.700

00E

+00

******************************************************************

*

*

*

*

******************************

*Figure 5: Shock wave: First-order scheme solution (stars) compared to analytical result(solid line). 20

1.00 0.00

PRESSURE

Max = 1.0000

Min = 0.10000

X

0.1

0000

E+

01 0

.100

00E

+00

****************************************************************************************************

****************************************************************************************************

*

1.00 0.00

DENSITY

Max = 1.0000

Min = 0.12500

X

0.1

0000

E+

01 0

.125

00E

+00

****************************************************************************************************

****************************************************************************************************

*

Figure 6: First-order numerical solution of the classical Sod shock tube problem. 0.47 0.00

REDUCED PRESSURE

Max = 0.19414E-04

Min = 0.23396E-07

X

0.2

2000

E-0

4 0

.000

00E

+00

* ** * * *

* **

**

**

**

*

*

*

*

**

0.47 0.00

VELOCITY

Max = -0.80659E-01

Min = -407.12

X

0.0

0000

E+

00-0

.410

00E

+03

**

**

**

**

**

**

**

**

**

**

*Figure 7: Isentropic compression � = ��=9: Second order implicit scheme solution (with-out gradient limitation) (stars) compared to analytical result (solid line). The reducedpressure is de�ned by (�(x) = p(x)�pminpmax ). 21

0.47 0.00

REDUCED PRESSURE

Max = 0.21866E-03

Min = 0.

X

0.2

0000

E-0

3 0

.000

00E

+00

*

*

* ** * * * * * * * * * * * *

* *

*

*

0.47 0.00

VELOCITY

Max = -10.456

Min = -396.95

X

0.0

0000

E+

00-0

.410

00E

+03

* **

**

**

**

**

**

**

**

**

* *Figure 8: Isentropic compression � = ��=9: First order implicit scheme solution (stars)compared to analytical result (solid line). The reduced pressure is de�ned by (�(x) =p(x)�pminpmax ).22

(a) (b)Max= 1.35 Min= 0.03 Max= 1.33 Min= 0.00(c) (d)

-0.1 -0.85

0 0

0.1 0.85

0 1 2 3 4 5 6 7 8Time

Max= 1.35 Min= 0.03Figure 9: Pitching NACA 0012 airfoil (Ma1 = 0:85): Mach lines for an angle of attack� = 1o (a) , � = 0o (b) and � = �1o (c). On �gure (d) evolution of the angle ofattack (solid line and scale on the right) and the lift (dashed line and scale on the left)are plotted. 23

Figure 10: Computational domain at bottom dead center (t = 0 and � = ��).24

Figure 11: Computational domain at top dead center (� = 0).25

Figure 12: Initial velocity �eld on the symmetry plane (� = �� : Bottom Dead Center)Vmax = 22:23m=s.26

Figure 13: Velocity �eldon the symmetry plane (� = ��=3 : piston is mouving upward).Vmax = 19:61 cm=s.27

Figure 14: Global ow structure: � = ��=3 and Vmax = 19:61 cm=s.28

Figure 15: Global ow structure at Top Dead Center (� = 0). Vmax = 22:0 cm=s.Figure 16: Velocity �eld on the symmetry plane at Top Dead Center (� = 0). Vmax =22:0 cm=s. 29

Figure 17: Velocity �eld on the symmetry plane (� = �=6 : piston is moving downward).Vmax = 13:8 cm=s.Figure 18: Velocity �eld on the symmetry plane (� = �=3 : piston is moving downward).Vmax = 14:6 cm=s.

30

Figure 19: Velocity �eld on the skin of the head chamber (� = �=18). Vmax = 20:0 cm=s.

Figure 20: Velocity �eld on the skin of the head chamber (� = �=6). Vmax = 13:8 cm=s.31

Figure 21: Velocity �eld on the skin of the head chamber (� = �=3). Vmax = 14:6 cm=s.32

List of Figures1 Hyper control-volume Qi and spatial cell Ci for tow-dimensional case : : : 172 Volume �V sweep out by a triangular facet (I,J.K). : : : : : : : : : : : : : 183 Intersection of @Ci with a neighboring element. : : : : : : : : : : : : : : : 194 Rarefaction wave: First-order scheme solution (stars) compared to analyt-ical result (solid line). : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 205 Shock wave: First-order scheme solution (stars) compared to analyticalresult (solid line). : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 206 First-order numerical solution of the classical Sod shock tube problem. : : 217 Isentropic compression � = ��=9: Second order implicit scheme solution(without gradient limitation) (stars) compared to analytical result (solidline). The reduced pressure is de�ned by (�(x) = p(x)�pminpmax ). : : : : : : : : 218 Isentropic compression � = ��=9: First order implicit scheme solution(stars) compared to analytical result (solid line). The reduced pressure isde�ned by (�(x) = p(x)�pminpmax ). : : : : : : : : : : : : : : : : : : : : : : : : : 229 Pitching NACA 0012 airfoil (Ma1 = 0:85): Mach lines for an angle ofattack � = 1o (a) , � = 0o (b) and � = �1o (c). On �gure (d) evolution ofthe angle of attack (solid line and scale on the right) and the lift (dashedline and scale on the left) are plotted. : : : : : : : : : : : : : : : : : : : : : 2310 Computational domain at bottom dead center (t = 0 and � = ��). : : : : 2411 Computational domain at top dead center (� = 0). : : : : : : : : : : : : : 2512 Initial velocity �eld on the symmetry plane (� = �� : Bottom DeadCenter) Vmax = 22:23m=s. : : : : : : : : : : : : : : : : : : : : : : : : : : 2613 Velocity �eldon the symmetry plane (� = ��=3 : piston is mouving up-ward). Vmax = 19:61 cm=s. : : : : : : : : : : : : : : : : : : : : : : : : : : 2714 Global ow structure: � = ��=3 and Vmax = 19:61 cm=s. : : : : : : : : : : 2815 Global ow structure at Top Dead Center (� = 0). Vmax = 22:0 cm=s. : : : 2916 Velocity �eld on the symmetry plane at Top Dead Center (� = 0). Vmax =22:0 cm=s. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2917 Velocity �eld on the symmetry plane (� = �=6 : piston is moving down-ward). Vmax = 13:8 cm=s. : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3018 Velocity �eld on the symmetry plane (� = �=3 : piston is moving down-ward). Vmax = 14:6 cm=s. : : : : : : : : : : : : : : : : : : : : : : : : : : : 3019 Velocity �eld on the skin of the head chamber (� = �=18). Vmax =20:0 cm=s. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3120 Velocity �eld on the skin of the head chamber (� = �=6). Vmax = 13:8 cm=s. 3121 Velocity �eld on the skin of the head chamber (� = �=3). Vmax = 14:6 cm=s. 3233