An Arbitrary Lagrangian¨CEulerian Computing Method for All Flow Speeds

8/2/2019 An Arbitrary LagrangianCEulerian Computing Method for All Flow Speeds

1/14

JOURNAL OF COMPUTATIONAL PHYSICS 135, 203216 (1997)ARTICLE NO. CP975702

An Arbitrary LagrangianEulerian Computing Method forAll Flow Speeds*

C. W. Hirt, A. A. Amsden, and J. L. Cook

University of California, Los Alamos Scientific Laboratory, Los Alamos, New Mexico 87544

Received November 10, 1972

(Eulerian), or be moved in any other prescribed wacause of this flexibility the method is referred to A new numerical technique is presented that has many advan-

tages for obtaining solutions to a wide variety of time-dependent Arbitrary LagrangianEulerian (ALE) technique multidimensional fluid dynamics problems. The method uses a fi- scheme of this nature has previously been reportnite difference mesh with vertices that may be moved with the fluid Trulio [3] for compressible flow problems. This new(Lagrangian), be held fixed (Eulerian), or be moved in any other

nique, however, may be applied to flows at any sprescribed manner, as in the Arbitrary LagrangianEulerian (ALE) since it has an implicit formulation similar to that itechnique. In addition, it employs an implicit formulation similar to

Implicit Continuous-fluid Eulerian (ICE) method that of the Implicit Continuous-fluid Eulerian (ICE) technique, mak-ing it applicable to flows at all speeds. particular, in the limit of infinite sound speed, the d

This paper describesthe basic methodology, presents finite difference equations reduce to a generalization of the Maence approximations, and discusses such matters as stability, accu- And-Cell (MAC) equations for the incompressiblracy, and zoning. In addition, illustrations are included from a num-

vierStokes equations [5].ber of representative calculations. 1974 Academic Press

The advantages of the ICED-ALE method incluability to resolve arbitrary confining boundaries, tovariable zoning for purposes of obtaining optimum r1. INTRODUCTIONtion, to be almost Lagrangian for improved accurproblems where fully Lagrangian calculations are noThere have been many finite difference techniques de-

sible, and to operate with time steps many times lvised for the solution of fluid dynamic problems. As cata- than possible with explicit methods.logued in [1], nearly all of these techniques can be classifiedThe basic ICED-ALE method has been separateas falling into one of two basic categories, depending on

three distinct parts called phases. This separation whether they are written primarily in terms of Lagrangianscribed in Section II. Finite difference approximatioor Eulerian coordinates. Within each of these categories itdiscussed in Section III for Cartesian and cylindricalis further possible to distinguish between those techniquesdinates. Also in Sections II and III is an interpretatapplicable to high speed flows and those applicable to lowthe ICE methodology, which leads to an estimate fo

speed. Further subdivisions are mostly matters of individ-number of iterations necessary to solve the implicit d

ual taste, prejudice, or specialization for specific applica-ence equations. In Section IV some discussion is dir

tions.toward matters of stability, accuracy, choice of mes

In this paper a technique is presented for the solutionThis section is illustrated with a number of repres

of the NavierStokes equations that is both Lagrangiantive calculations.and Eulerian, and that is applicable to flows at all speeds. An attempt has been made to concisely summa

The method uses a finite difference mesh with vertices considerable amount of material in this paper. Howthat may move with the fluid (Lagrangian), be held fixed for the reader interested in a more complete descr

of the difference equations, a flow chart, and a comFORTRAN computer listing for a code based o

Reprinted from Vol. 14, Number 3, March 1974, pages 227253.method described in this paper, Ref. [6] is ava* This work was performed under the joint auspices of the Unitedupon request.States Atomic Energy Commission and the Defense Nuclear Agency

(DNA Subtask HC-061, DNA Work Unit No. 15Calculations at LowII. BASIC METHODOLOGYAltitude, and under Contract DNA001-72-C-0106, NWED Subtask Code

HC-061, Work Unit No. 50.)The finite difference mesh used here consists of Now with Science Applications, Incorporated, P.O. Box 2351,

La Jolla, California 92037. work of quadrilateral cells with vertices labeled by in

203


2/14

204 HIRT, AMSDEN, AND COOK

the surface of V by S and the outward normal on Sthese equations are (see, for example, Ref. [7])

d

dtV dV

S(U u) n dS 0

d

dtVu dV

Su(U u) n dS

V

p dV Vg dV 0

d

dtVE dV

SE(U u) n dSFIG. 1. The assignment of the variables about a cell.

S

pu n dS Vg u dV 0.

pairs (i, j), denoting column i and row j. Fluid variablesare assigned to staggered locations in the mesh as shown In these expressions U is the velocity of the surf

When U 0 the equations are Eulerian, and when Uin Fig. 1. Pressures (p), specific internal energies (I), cellthe equations are Lagrangian. The pressure gradientvolumes (V), and densities () or masses (M) are all as-

in Eq. (6b) could be written as a surface integral, bsigned to cell centers. Coordinates (x, y) and velocity com- cylindrical coordinates a simpler finite difference apponents (u, v) are assigned to cell vertices.mation is obtained directly from the volume integraThe differential equations to be solved arethis is advantageous for the implicit formulation difference equations.

t u 0 (1) The finite difference formulae presented in Secti

are written as approximations to these equations in u

t uu p g (2) the integration volumes are the cells of a moving

difference mesh. In particular, the V in Eqs. (6a) anis the volume of a cell in the mesh, and the V in Eq

E

t Eu pu g u, (3)

is a volume surrounding a vertex. A typical cross sefor the latter is indicated by the dashed line in Fig. 2

where E u u Iand Iis the material specific internal difference in integration volumes is dictated by henergy. In Eqs. (2) and (3) g is a body acceleration (usually defined fluid densities and energies at cell centers gravity) and p is the fluid pressure given by the equation velocities are defined at cell vertices. In Section III diof state approximations are described forall the terms in Eqs.

(6c).The calculations necessary to advance a solutiop f(,I). (4)

step in time, t, are separated into three distinct phThe first phase consists of an explicit Lagrangian ca

For problems involving shock waves it is necessary toadd to p an artificial viscous pressure, q. A suitable formfor q that is linear in the velocity divergence is

q u. (5)

Usually q is replaced by zero in expanding cells, that is,where u is positive. In Ref. [6] details are given forincluding a complete viscous stress, but in this paper thesecomplications are omitted in order to simplify the presenta-tion of the essential ideas of the ICED-ALE method.

The conservation statements of mass, momentum, andenergy contained in Eqs. (1)(3) are more convenient for

FIG. 2. The dashed line encloses the momentum integration vour purposes when integrated over a volume V, which may used for vertex 4. The notation is that used for typical vertex a

finite difference equations appearing in the text.be moving with an arbitrarily prescribed velocity. Denoting


3/14

NUMERICAL FLUID DYNAMICS

tion, except mesh vertices are not moved. Second, an itera- Physically, an iteration offers a means by which presignals can traverse across more than one cell in ation phase adjusts the pressure gradient forces to the ad-step. The iteration is, however, more efficient tvanced time level. This phase, which is optional, eliminatesstraight explicit calculation with reduced time step, bethe usual Courant-like numerical stability condition thatpressure variations are propagated only to the point wlimits sound waves to travel no further than one cell perthey are producing effects no longer considered signifitime step. The mesh vertices are moved to their newThis point is discussed in more detail in Sections III anLagrangian positions after this phase. Finally, in the third

phase, which is also optional, the mesh can be moved to a III. DISCRETE APPROXIMATIONSnew configuration. In this (rezone) phase convective fluxesmust be computed to account for the movement of fluid

A. The Finite Difference Equationsbetween cells as the mesh moves. The calculations inthis last phase are automatically iterated if zones try to A set of finite difference approximations is outlinmove too far in any single step, so that gross rezoning this section for two-dimensional Cartesian (x, y) or can be accomplished without introducing numerical insta- drical (r, z) coordinates. When Cartesian coordinatbilities. desired, all radii, r, appearing in the following equa

This separation of a calculational cycle into a Lagrangian should be replaced by unity. When using cylindrical cphase and a convective flux, or rezone, phase originated nates the equations refer to unit azimuthal angle, nin the Particle-in-Cell numerical method [8], and has since Input data to start a calculation consists of mesh vbeen used in many hydrodynamic computer codes. In the coordinates (x, y), velocities (u, v), cell densities (present technique the different phases can be combined internal energies (I).in various ways to suit the requirements of individual prob- The variables adjusted at each stage of a calculalems. For example, in high speed problems, in which the cycle are indicated by the entries in Table I. For exa

vertex velocities are adjusted explicitly in the first pCourant stability condition is not likely to be violated, anphase one, again in the implicit second phase, and fiexplicit calculation is acceptable and the phase two itera-in phase three, if the mesh is rezoned. In the remation may be omitted, and for an explicit Lagrangian calcula-of this subsection the calculations performed at each tion only phase one is used.and listed as entries in Table I, are expressed for a tPhases one and three are variations of familiar Lagran-mesh vertex, labeled 4 in Fig. 2, or for a typical cell, lagian and Eulerian finite difference techniques, althoughA in Fig. 2. All formulae presented here will employthere are some novel features as described in Section III.ber or letter subscripts for vertex or cell quantities, reThe phase two iteration, however, is new and requires

tively, as shown in Fig. 2. In subsection B are descsome preliminary discussion. The purpose of phase two isthe considerations necessary to impose a variety of bto get time-advanced pressure forces in the Lagrangianary conditions.part of a calculation. The reason for this can be appreciated

from the following argument. In an explicit method pres- 1. Initializing Calculations. For convenience andsure forces can be transmitted only one cell each time step, it is desirable to compute and store several auxiliary qthat is, cells exert pressure forces only on neighboring cells. ties used repeatedly in the following equations. When the time step is chosen so large that sound waves quantities include cell volumes, cell total energies, anshould travel more than one cell the one cell limitation is masses assigned to vertices. Once established at theclearly inaccurate and a catastrophic instability develops. of a calculation, the auxiliary quantities are automatThe instability arises because the explicit pressure gradi- updated in the course of a calculation cycle.ents lead to excessive cell compressions or expansions The volume, V, of a cell such as A in Fig. 2 is

when multiplied by too large a time step. This then leadsto larger pressure gradients the next cycle, which try to VA (r1 r2 r3 )[x1 (y2 y3 ) x2 (y3 y1 )reverse the previous excesses, but since the time step is

x3 (y1 y2 )] (r1 r3 r4 )[x1 (y3 y4 )too large the reversal is also too large and the processrepeats itself with a rapidly increasing amplitude. The over- x3 (y4 y1 ) x4 (y1 y3 )],response to pressure gradients in this fashion is eliminatedby using time-advanced pressure gradients, for then cells where ri xi for cylindrical coordinates and ri cannot compress or expand to the point where the gradi- plane coordinates, i 1, 2, 3, or 4.ents are reversed. The mass contained in a cell can be obtained fro

Unfortunately, the time-advanced pressures depend on product of cell volume and density,the accelerations and velocities computed from those pres-sures, so an iterative solution of the equations is necessary. MA AVA .


4/14


TABLE I

Variables Updated

Cycle Steps x y u v p E M I

(1) Initializing Calculations p E M

(2) Phase IFirst Part u v E(3) Phase IIImplicit uL vL pL

Lagrangian

(4) Phase ISecond Part xL yL EL L

Rezone

(5) Phase IIIRezone xn1 yn1 u n1 vn1 E n1 Mn1 n1

(6) Auxiliary pn1 In1

For advancing velocities it is necessary to assign a mass The difference equations used to advance the ve

components of vertex 4 in Fig. 2 areto each vertex. In this technique it is assumed that themass in each cell is equally shared between its four cornervertices, so that vertex 4 in Fig. 2, for example, is given

u4 u4 t

2M4r4 [pA(y1 y3 ) pB (y3 y6 )the mass

pc(y6 y8 ) pD(y8 y1 )] tgx ,M4 (MA MB MC MD). (9)

where the tilde over u4 on the left side signifies the terary new value of u4 , and similarly

To ensure energy conservation the total specific energy,E, is directly advanced in time rather than the internal

energy I. The relation between them for the typical cell A v4 v4 t

4M4 pA(r1 r3 )(x3 x1 ) pB(r3 r6 )(x6 in Fig. 2 is

pC(r6 r8 )(x8 x6 ) pD(r8 r1 )(x1 x8 )]

EA IA (u21 u

22 u

23 u

24 v

21 v

22 v

23 v

24 ). (10)

These expressions were obtained from the integraof the equations of motion, Eqs. (6). A mass of 2MAt the beginning of a calculation, E is computed from thebeen assumed to lie within the integration volume, beinput values ofI, u, and v. Thereafter, this relation is usedit includes approximately 1/2 the mass in each celto recoverIfrom E for use in the equation of state pressure,rounding vertex 4, while M4 contains only 1/4 ofEq. (4).surrounding cell mass. It would be possible to us

actual mass contained in the integration volume, bu2. Phase One, First Part. In this step velocities are ad-vanced explicitly in time using pressure gradients and body not necessarily true that this would produce more acc

results. For example, in an incompressible flow the inforces computed from the currently available pressuresand mesh coordinates. If viscous, elastic, or other stresses tion volume may not be constant even though the in

ual volumes of the cells are constant, and thereforare desired, they may be included at this stage as well (seeRef. [6]). The total energy of each cell is also advanced in vertex mass computed before coordinates are m

would not be the same as that computed afterward. Itime to account for the work done by the body forces andother stresses, except those of pressure. Pressure work case, the prescription used here has been successfully

by many other investigators; see, for example, Ref. terms are included only after the implicit pressure calcula-tion in phase two. This delay permits time-advanced pres- For the initial time advancement of energy E, an

iary quantity, Q, is computed for each vertex. This qusures to be used in computing the work and ensures consis-tency with the velocities coming out of phase two. represents the work done on fluid in the integration v


5/14


by all stresses, except pressure. In the present case, again where VA is the current volume of cell A and V* volume the cell would have if its vertices were mreferring to Fig. 2,according to, e.g.,

Q4 t

8M4qA [(u1 u3 )(y1 y3 ) x*4 xn4 u4t, y*4 y4 v4 t, etc.

(v1 v3 )(x3 x1 )](r1 r3 ) It is important that V* be computed in terms of co

nates shifted with velocities accelerated with the new qB [(u3 u6 )(y3 y6 )sures through formulae like Eqs. (11), in which

(v3 v6 )(x6 x3 )](r3 r6 ) replaced by pL.A solution forpLA can be obtained by applying a New qC[(u6 u8 )(y6 y8 ) (12)

Raphson iteration to Eq. (14) considered as an im (v6 v8 )(x8 x6 )](r6 r8 ) equation for pLA through Eqs. (11), (15), and (16)

velocities (u, v) obtained in the previous step are us qD [(u8 u1 )(y8 y1 )initial guesses for the iteration. The iteration procee

(v8 v1 )(x1 x8 )](r8 r1 ) sweeping through the mesh and applying the folladjustments to each cell, once each sweep: t[gxu4 gyv4 ].

(a) Compute V* using the most updated valu

After all the vertex Q values have been computed, the (u, v).total specific energy for each cell is adjusted to(b) Compute new guesses for LA andI

LA from Eqs

(c) Compute a pressure change, pA , accordingEA EA (Q1 Q2 Q3 Q4 ). (13)

pA pLA f(

LA ,I

LA)

SAIn other words, the total energy change at each vertex isassigned to each neighboring cell in proportion to the massthat each cell contributed to the vertex. Total energy is

where the most updated value is used for pLA on theconserved in this process.

side and SA is a relaxation factor to be described.

3. Phase Two, Implicit. When an explicit calculation (d) Adjust the current guess for cell pressure, pis wanted, this step can be omitted. The object of phase

adding

pA to it, and adjust the velocities at the cotwo is to obtain new velocities that have been accelerated of the cell to reflect this pressure change:with time-advanced pressure gradients. Since the time-ad-vanced pressures depend on the densities and energies

u1 u1 t

2M1r1 (y2 y4 ) pAobtained when vertices are moved with their new veloci-

ties, which in turn are functions of the new pressures, thesepressures are defined implicitly and must be determined

v1 v1 t

4M1(r2 r4 )(x4 x2 ) pA

by iteration. The implicit problem can be formulated asfollows: Let a superscript L denote time-advanced values

u2 u2 t

2M2r2 (y3 y1 ) pAand a superscript n denote values at the beginning of a

cycle. The desired pressure, pLA , of cell A will be the solu-tion of the equation

v2 v

2

t

4M2 (r1

r3 )(x1

x3 )pA

pLA f(LA ,I

LA) 0, (14)

u3 u3 t

2M3r3 (y4 y2 ) pA

where the new cell density and energy can be approximatedin terms of their initial values as

v3 v3 t

4M3(r2 r4 )(x2 x4 ) pA

LA nA

VA

V* (15)u4 u4

t

2M4r4 (y1 y3 ) pA

v4 v4 t

4M4(r3 r1 )(x3 x1 ) pA .I

LA I

nA

pnA

nA1 V*

VA ,


6/14


The mesh is repeatedly swept and calculations (ad) are ity, this suggests the iteration will be stable onlyeffective Courant number is less than unity,performed once for each cell each sweep, until no cell

exhibits a pressure change violating the inequality

Ceff

xt

N 1, p

pmax , (19)

where x is a typical cell dimension. If it is assumewherepmax is the actual or an estimated maximum pressurethe iteration has been designed to proceed at max

in the mesh and is a suitably chosen small number. Typi-speed, corresponding to an equality in the above ex

cally, is of order 103.sion, then the number of iterations necessary for co

The relaxation number SA , used in Eq. (17) for pA , gence will be of ordermust be chosen to keep the pressure changes in boundand progressing in the right direction, but its exact valueis not crucial. In the ordinary NewtonRaphson procedure

N u tx 1

1/ 2

. SA is the derivative of the function whose root is soughtwith respect to the iteration variable. That is, SA is the rateat which the quantitypA f(A ,IA) changes as the variablepA changes. This rate must be computed using the implicit According to this, the iteration number increases as

relations given by Eqs. (11), (15), and (16). The difference tincreases or decreases, but is independent of the equations can be manipulated [6] to yield an algebraic material sound speed. Thus, once a tolerable level of

sure error, , has been chosen, the implicit schemeexpression for SA , but it is easier to compute the rateverges in a finite number of iterations, regardless oof change numerically. For this purpose a small pressureactual material sound speed. It is this feature of thechange, pT, is chosen, which is usually of order pmax .method [4] that makes it superior to an ordinary exThen the velocity changes induced by this pressure changemethod whose time step must be continually reducin a cell are used to compute the volume change and corre-the sound speed is increased.sponding density and energy changes according to Eqs.

(15). Finally, SA for the cell is set equal to the quotient of4. Phase-Two, Second Part. The final values ofuthe difference between p f(, I), evaluated after and

and pL from the iteration in the previous step are thbefore the change in pressure, and pT. These SA valuesLagrangian values for the cycle. To complete the La

are computed and stored for each cell before the phase gian portion of a cycle, the cell energies must notwo iteration is started. It is unnecessary to update themadjusted for the pressure work terms omitted in sduring an iteration.and vertices must be moved with the fluid to their newFor a given , the number of iterations necessary totions.obtain convergence can be roughly estimated through the

The energy for cell A in Fig. 2 is changed accordfollowing argument. Assume Niterations are necessary forconvergence. Since the iteration acts something like anexplicit calculation, its effective time step must be t/N.

ELA EA t

4MAp12 (r1 r2 )[(u1 u2 )(y1 y2In a flow with Mach number M, pressure variations satisfy

the approximate inequality (v1 v2 )(x2 x1 )]

p23 (r2

r3 )[(u2

u3 )(y2

y3 )

pp

M2. (20) (v2 v3 )(x3 x2 )]

p34 (r3 r4 )[(u3 u4 )(y3 y4 )Therefore, the iteration has an effective Mach number,M2 , and hence an effective sound speed, Ceff, such that (v3 v4 )(x4 x3 )]

p41 (r4 r1 )[(u4 u1 )(y4 y1 )

C2eff u2

, (21) (v4 v1 )(x1 x4 )] ,

where the velocities and pressures are the final valuwhere u is a typical fluid speed. Since the Courant numberfor an explicit calculation must be less than unity for stabil- uL, vL, and pL. Cell-edge pressures are required; p


7/14


example, is the pressure along the left edge of cell A be- vertex can move in any one shift and forces as many reof the rezone calculations as are necessary for those tween vertices 3 and 4. For this boundary pressure a mass

weighting scheme is used, ces t hat exceed t he limit. I n this way the rezone c alculare always stable, even when a gross change in theconfiguration is called for.

p34 MBp

LA MAp

LB

MA MB, (25) Both schemes have been used in connection wi

ICED-ALE formulation, but only the latter will bscribed here, while the former is detailed in Ref. [6]

and similarly for the other edges of cell A. The mass eral prescriptions for choosing vertex rezone velocitiweighted average (25) was recommended by Fromm [10]

described in Section IV-D. For purposes of this seand has led to good results in a variety of test cases.

they are assumed given.Mesh vertices are moved with the fluid to their new loca-

Before the rezone calculations are started all verttions,

locities are converted to momenta and all cell specificgies are converted to total energies so that the re

xL4 xn4 t u

L4 (26) calculations will be rigorously conservative of mass

mentum, and energy.yL4 yn4 tv

L4 .

The adjustments associated with a shift in the poof a typical vertex, say 4 in Fig. 2, proceed as follows

The new velocities are used to move vertices since thisthe vertex is moved to its new location,

makes the explicit Lagrangian portion of a cycle second-order accurate in time. The implicit calculation, however,

xn14 xL4 t U4is only first-order accurate.

After the vertices have been moved, new densities, L, yn14 yL4 t V4 ,

are computed for each cell as the quotient of cell massdivided by the new cell volume. where U4 , V4 are the rezone velocities specified fo

The phase one and phase two calculations contained in vertex.the previous steps comprise an implicit Lagrangian method When vertex 4 is moved, the lines connecting itthat is stable for any Courant number, C t/x, where C neighbors 1, 3, 6, and 8 sweep out volumes contais the fluid speed of sound. When the sound speed becomes mass and total energy that must be exchanged betvery much larger than the fluid speed, this method ap- the adjacent cells. For example, if vertex 4 moves tproaches a variant of the incompressible Lagrangian right, the grid line connecting 4 to 3 sweeps out of

method described in Ref. [11]. and adds to cell B a volume equal to

5. Phase Three, Rezone. As is well known, Lagrangiancell methods are not adequate for describing flows under- V

t

3(2r4 r3 )[U4 (y3 y4 ) V4 (x4 x3 )].

going large distortions. In the present method, the devasta-ting effects of large distortions are eliminated by moving

Associated with this volume exchange there will athe mesh vertices with respect to the fluid so as to maintaina mass and total energy exchange between the cellsa reasonable mesh structure. Whenever a vertex is movedmass or energy per unit volume assigned to this vorelative to the fluid, however, there must be an exchangecan be computed in various ways. It is well knownof material among the cells surrounding the vertex. Thisuse of a simple average of the quantities on either sexchange, which can be interpreted as a convective flux,the line leads to a computational instability, but tis expressed by the second terms in Eqs. (6).

stable calculation can be obtained by weighting the avEither the convective flux adjustments can be performed in favor of the value in the cell from which the quafor the entire mesh at one time using only values of theis subtracted. This is the upstream or donor cell convfluid variables coming out of the Lagrangian portions offlux approximation. Thus, the mass subtracted from the calculation to compute the new values after rezoning,and added to cell B isor each vertex can be separately adjusted, with the values

arising from each adjustment used in subsequent calcula-tions for other vertices. The former method requires extra

M1

2(V V)

MA

VA

1

2(V V)

MB

VB,

storage for quantities needed both before and after ad-justing. The latter method requires no extra storage andhas the additional advantage that individual vertices can where is the donor cell weighting factor. When

the flux is centered and when 1 the flux is full be rezoned repeatedly if necessary. In fact, a simple schemehas been devised that automatically limits the distance a cell. The best choice for is discussed in Section


8/14


The corresponding total energy subtracted from cell A and rigid boundaries may be classified as free-slip or nand may be given prescribed motions. Combinatioadded to cell B isthese conditions can be used to simulate a great variproblem situations.

(ME ) 1

2(V V)

MAEA

VA

1

2(V V)

MBEB

VB. In nearly all cases, the setting of boundary conditi

accomplished by making adjustments to the velocit(30)

the boundary vertices. These adjustments must beformed before and after the phase-one calculation

Similar formulae are used for the exchanges of mass and after each iteration in phase two.energy between the other pairs of cells surrounding the

Consider a vertex located on the top or bottom bouvertex.

of the mesh, with coordinates (xc , yc ). Coordinates A shift in vertex 4 is also accompanied by a momentumvertex to the left will be denoted by (xL , yL) and thexchange between vertices 1, 3, 6, and 8, because 4 is athe right by (xR , yR). For a vertex on the left or rightcorner of the control volumes for these vertices. For exam-of the mesh, the following discussion will apply pro

ple, when vertex 4 is moved, the surface connecting verticesbelow is read for left and above is read for r

4 and 2 sweeps out a volume,The simplest boundary condition to impose is for a

no-slip wall on which the fluid velocity is set equal prescribed wall velocity.V

t

3(2r4 r2 )[U4 (y2 y4 ) V4 (x4 x2 )]. (31)

A rigid free-slip boundary is more difficult to ha

since it is only the fluid velocity normal to the bouThe mass in this volume is (MA/VA) V and the u- that is constrained. If the normal direction to the boumomentum it contains is approximated as at vertex (xc , yc) is defined as the direction normal

line connecting (xL , yL) with (xR , yR), then the coboundary condition is achieved by replacing the ve

(Mu) 1

2

MA

VA[(V V) u3 (V V) u1 ].

at the vertex by

(32)

uc un sin uc cos2 vc sin cos

This momentum change must be subtracted from vertex1 and added to vertex 3. Similar exchanges are computed

vc un cos uc cos sin vc sin2 ,

for the vertex pairs (3, 6), (6, 8), and (8, 1). The v-momen-

tum is handled in the same way, withv

replacing u in the where un is the prescribed boundary velocity in the nabove formula. The scalar is again chosen as zero for adirection positive when directed to the right of the v

centered momentum flux and as unity for an upstream orpointing from (xL , yL) to (xR , yR). The angle is donor cell flux.mined from

When these exchanges among the cells and vertices sur-rounding the moved vertex have been completed, newvolumes are computed for the cells (A, B, C, and D) and cos

xR xL

[(xR xL)2

(yR yL)2 ]1/ 2

.the mesh is ready to have any other, or even the same,vertex moved to a new location.

This transformation leaves the tangential fluid velociAfter all vertices have been moved to the positions de-changed while replacing the normal fluid velocity bsired, the total cell masses and energies are converted backIf the boundary vertices are to move, as in a Lagrato densities and specific energies, and vertex momenta are

calculation, then further refinement of this boundarconverted back to velocities.dition will be needed to keep the vertices on the bou

6. Auxiliary Calculations. New specific internal ener- when it is curved, since Eq. (33) only keeps a vertgies, I, are computed from E by subtracting the average the local tangent to the boundary.cell kinetic energy according to Eq. (10). Finally, new cell Prescribed inflow and outflow boundaries are impressures may be computed from the defining equation of by setting fluid velocities at the boundary vertices tstate in terms of the new values of and I. desired values.

At a free surface the tangential and normal stressB. Boundary Conditions

zero, and no special conditions are required in thisHowever, since free surface vertices receive accelerMany kinds of boundary conditions are possible. In this

section are given the prescriptions for rigid boundaries, from only one side, some caution must be exercisedthe tangential accelerations vary significantly in a direinflow and outflow boundaries, and free boundaries. Also,


9/14


with a 0.5 and 0.8. The tube has unit radiuthe segment computed is initially 5 units long. Thestep was 0.01. The top of the mesh is a continuative ouboundary, while the left edge is an axis of cylindricalmetry (i.e., a rigid free-slip wall). The right edge omesh is a free surface, except that an additional foimposed on these vertices to represent the stress that wbe generated in an elastic confining membrane. Theis assumed to be incompressible with density 1.0.

Three kinds of data are included in Fig. 3. In Figshown the computing grid after 275 calculational cBulges of successive pressure pulses are evident alonouter tube boundary. In Figs. 3b and 3c are the corresing velocity vector and pressure contour plots. A rof high pressure is located under each radial bulge low pressure under each depression.

In this example the mesh was continuously rezonkeep the radial grid lines fixed, while the axial gridwere adjusted to be equally spaced along each radia

No detailed comparisons with theoretical or experFIG. 3. Calculation simulating the pulsating flow of fluid pumped tal data have been attempted with this problem, si

through an elastic tube. Plots shown are the computing grid, velocity is only presented here as a qualitative example. Somvectors, and isobars after 275 calculational cycles, or at t 2.641.

tailed calculations illustrating the accuracy of the ICALE technique are presented in the next section.

normal to the free surface. The one-sided calculations canthen be a poor approximation, and it may be necessary to IV. GENERAL REMARKSextrapolate the acceleration from within the fluid out to

To make the ICED-ALE technique presented ithe free surface.paper a useful tool, it is necessary to give consideratContinuative outflow boundaries are always trouble-such matters as computational stability, accuracy, presome for low speed flows, since influences from these

tions for rezoning, automatic timestep control, markeboundaries can be felt upstream. The goal of a continuative ticle techniques, etc. These topics are discussed in thboundary is to permit outflow with a minimum of upstreamlowing paragraphs.disturbance. A prescription that has worked in ALE is to

set the velocities of the boundary vertices equal to theA. Test Calculationsvelocities located at vertices immediately inside the bound-

ary. This replacement should be made before and afterA useful problem to test a compressible flow code

phase one, but notafter each iteration in phase two. Duringshock tube, in which a long straight cylinder is divide

phase two the continuative boundary velocities are permit-two compartments by a central diaphragm. On on

ted to adjust to whatever pressure changes occur duringof the diaphragm there is a gas, say, of density 0

the iteration. The use of this prescription in a low speedinternal energy I 0.18, while on the other side th

application, however, must be carefully checked in eachhas density 0.1 at the same energy. A calculation b

case to be sure it is not causing unwanted upstream influ-

by removing the diaphragm with the gases at resences.assume -law gases with 5/3. The pressure diffeFor phase-three rezoning of and E, values of thesein the two gases drives a shock into the less densquantities must be specified in cells outside the boundarywhile a rarefaction moves into the denser gas. A coif flow is to take place across the boundary.surface trails behind the shock.A calculational example that illustrates the use of several

Both a Lagrangian and an Eulerian calculationkinds of boundary conditions is shown in Fig. 3. This figurebeen made for this problem, using 60 zones of sizeillustrates the result of a calculation of fluid pumped peri-0.333 and time step t 0.1 (nondimensional uniodically through an elastic tube. The bottom edge of theused throughout). The artificial viscosity coefficientmesh is an inflow boundary with an assigned periodic in-in both cases was 0.04. In Figs. 4a and 4b, the dflow velocity of the formand velocity profiles are shown at t 10.0 for each cation in comparison with the theoretical predictions. v a sin2 t,


10/14


FIG. 4b. Density and velocity profiles for a Eulerian calculatioFIG. 4a. Density and velocity profiles for a Lagrangian calculationof the shock tube problem at t 10.0. The solid line is the theoretical pre- shock tube problem at t 10.0. The solid line is the theoretical prediction.

cases the calculations are stable. For the largest timonly three cycles are needed to reach t 10.0, anresults are typical for standard finite difference calculationsshock has moved approximately 15 zones in this timof similar problems.

Some accuracy has been lost at the largest t, buA more significant test of the ICED-ALE method is

in exceptional cases would one choose a time stepillustrated in Fig. 5 where the velocity profiles are shownallowed the shock to move five cells each cycle. In ge

for three calculations of the same problem in which thethe time step should be chosen to give reasonable re

time step was successively t 0.1, 1.333, and 3.333. In alltion for the time scales of interest. If the shock is of printerest, a time step would be used in which the traverses no more than one cell each step. By contran incompressible or very low speed flow the timeshould be chosen to have fluid particles moving onevery few cycles, but in this limit compression waves stravel across many cells each time step.

A simple example of an incompressible fluid calcucan be obtained by repeating the shock tube calcu

described above using gases with a very large sound and with gravity added and directed along the concylinder axis. A hydrostatic pressure should be estabin the tube when its ends are closed, to balance the grtional acceleration, and the fluid should remain aThis is indeed the case as can be seen in Fig. 6 wthe pressure obtained after 145 iterations is plottefunction of height, x. The equation of state was chothis case to be p a2( 0(x)) where the sound FIG. 5. Comparison of shock tube velocity profiles obtained usingis a 105 and 0(x) is the initial density distributionthree different time increments. The profiles obtained with t 0.1 andtime step was t 0.01, the gravity acceleration wat 3.333 are shown at t 10.0, the profile obtained with t 1.333 is

shown at t 10.66. and the convergence criterion was 104.


11/14


The most important stability considerations are aated with phase three. It is well known that forwardand centered space differencing ( 0) for the convein phase three is unstable. Stability can be achievincreasing the magnitude of. However, to prevent uessary numerical smoothing the magnitude of shoukept as small as possible. An optimal choice migdeveloped along the lines of an idea by Boris [13], buhas not yet been done. As a rule of thumb, shoube less than V/V where V is either given by Eqand V is the average volume of cells on either side mass flux boundary, or V is given by Eq. (31) andthe volume of the cell containing the momentumboundary.

In addition to the choice of there is a more fundaFIG. 6. Hydrostatic pressure profile after 145 iterations in the firsttal stability and accuracy requirement inherent in cycle (dots) compared with the theoretical profile (solid line). After three

cycles the calculated results lie on the solid line. three. Material cannot be fluxed through more thacell in one time step, because the flux approximationsbeen based on the implicit assumption of exchange

between neighboring cells or vertices. Thus, the fluThese examples show that the ICED-ALE method is ume to cell volume ratio, V/V, must never be allowstable for arbitrary Courant numbers and is accurate for exceed unity. Since the V in Eqs. (28) or (31) is prboth high speed and low speed problems, although some tional to t, this limitation is really a limitation oaccuracy is lost in processes with time scales not well re- time step.solved by the chosen time step. In practice only one of the forms, Eqs. (28) or (3

needed for limiting t, and coupled with Eq. (34) B. Computational Stability restrictions can be used to automatically control the

step in a program. In Ref. [6], the automatic controA rigorous stability analysis cannot be performed for

is coupled with the option of an automatic determinthe ICED-ALE technique presented here, but good esti-

of the viscosity coefficients, and , to optimize stamates can be made based on analogies with simpler

and efficiency.schemes for linear equations with constant cell sizes [12].

When phase one is followed by phase two there is nostability restriction on the distance a sound wave may prop- C. Coupling Alternate Mesh Verticesagate in a time step. However, when viscous effects are

Accelerations computed at a vertex (i, j) with the dincluded in the phase one calculations, as in Ref. [6], for

ence equations presented in Section III are indepeexample, there is a stability condition that limits the dis-

of the position of the vertex within the integration rtance over which momentum can diffuse in one time step

outlined in Fig. 2. Intuitively it is expected that theto be less than one cell width. Violation of this condition

accurate results will be obtained when the vertex is loresults in a rapidly growing and oscillating instability. A

at the center of the integration region, a conditiongood estimate for the restriction on the time step, t, needed

can often be arranged with proper rezoning of the in this case is

However, this insensitivity to the location of the ce

vertex is symptomatic of a common problem in finite dence methods in which the shortest resolvable wave le

t 2(2 )

1x 2

1

y 21, (34) (2x) are not sufficiently damped.

An example of the problems that may arise is contin the velocity vector plot shown in Fig. 7a. The pro

where and are the first and second coefficients of consists of an incompressible fluid flow directed fromviscosity and where x and y are the effective cell sizes tom to top with unit speed entering the bottom odefined as mesh and passing around a rectangular block. The co

ing mesh was treated as Eulerian with square cells throut. Vectors are drawn from each mesh vertex. Rigid

x (x1 x2 x3 x4 ), slip conditions were imposed on the boundaries oblock. The region of difficulty extends off the trailingy (y2 y3 y4 y1 ).


12/14


previously described for the calculation of fluid sloin a rectangular tank [2]. In Ref. [16] a simple techis presented for the automatic construction of gridfollow curved boundaries, have increased resolutionlected regions, etc.

When considering a problem involving more thamaterial, no special techniques are needed if interbetween the different materials coincide with meshOf course these interface lines must be moved witfluid in Lagrangian fashion, which means that severe dtions of interfaces cannot be allowed. Unfortunatelimposes a limitation on some multimaterial applica

The kind of difficulties that may be encountereillustrated in Fig. 8. In this example, a heavy, incompible fluid of nondimensional density 2 was above a lfluid of unit density. The calculational mesh initiallysisted of rectangular cells having equal masses. Agravitational acceleration was directed downward, pring an unstable situation. A half cosine velocity per

tion was applied to the interface at the start of the caFIG. 7. Velocity vectors for flow around a rectangular block. Theseplots, taken at the same time in two different calculations, show the tion. The fluid configuration (mesh) is shown at timesappearance before and after coupling of the alternate mesh vertices. in Fig. 8a and 0.123 in Fig. 8b. Arrows along the s

the mesh mark the interface intersections. The meeither side of the interface was continuously rezon

(top) of the block where the velocity vectors are seen tobe approximately orthogonal, but no rezoning prescr

alternate direction on adjacent vertices.is likely to be found that will permit the calculati

An effective means of eliminating this undesirable fea-proceed significantly further than shown in Fig. 8b. P

ture was developed for an early version of the ALE methodting slip tangentially along the interface would hel

[14]. The idea is to introduce a small restoring force on eacheven with this the rolling up of the interface bec

vertex to keep it more in line with neighboring vertices. Forprogressively harder to define with the initial rectan

vertex 4 in Fig. 2 the restoring acceleration is

1

anc t1

4(u1 u3 u6 u8 ) u4 . (35)

This acceleration is treated as arising from a body forceand is added to g for purposes of computing the work donein Eq. (12). The coefficient anc implies that this relaxationin the velocity field has a characteristic time of anc timesteps. Note, however, that ifanc 1, the technique becomesidentical to a procedure introduced by Lax many yearsago [15]. To avoid the difficulty of that procedure as t

0, it would be better to define anc

anc/

t, in which a

nc isthe actual relaxation time, rather than the number of cycles

for relaxation.The effect of using expression (35) can be seen in Fig.

7b, where the alternating velocities have been removed.This calculation, with the vertex coupler, agrees very wellwith a calculation of the same problem using the Marker-and-Cell method [5].

D. Zoning and RezoningFIG. 8. Calculation of an interface instability flow. Arrows i

Many choices are available for the construction of suit- the Lagrangian interface. Even with continuous rezoning of the ivertices, severe grid distortion will soon terminate the calculatioable meshes. For example, a variety of options have been


13/14


fluid motions. Figure 9, for example, shows a sequenmarker particle configurations obtained in the courscalculation of an intense explosion in the atmosphereleft edge of the computation region is an axis of cylinsymmetry. Initially particles were densely, but unifodistributed in a semicircular region about the centhe explosion. Additional particles with a much smparticle density were placed in the remainder of theputing region. As the problem proceeded, the mescontinuously enlarged to approximately double its orsize, leaving a region without particles around the edges of the mesh. The particle configurations shosubsequent collapse of the initial hot bubble and the ftion of a buoyant vortex ring. Although it is not evin Fig. 9, the mesh was continuously rezoned to tranupward with the hot material. Many calculations otype have been performed and have been shown tomore accurate results than are obtainable with staEulerian techniques [17].

FIG. 9. An ICED-ALE calculation of an intense explosion in the

Marker particles are moved with the local fluid veatmosphere, showing the marker particle configuration at 0.5, 10.0, 20.0,each time step. In previous particle techniques [5], theand 30.0 s. The expansion of the mesh is indicated by the growing frame,

and the rise of the hot bubble through the ambient atmosphere is evident velocity at each particle is computed as a linear intein the relative particle motion. tion in both x and y directions among the neares

vertex velocities. In the present method, however, theinterpolation technique is difficult to apply directl

array of cells. Nevertheless, this calculation agrees remark-cause the mesh consists of arbitrary quadrilateral

ably well, up to the time shown in the last figure, with aWith arbitrary cells it is even difficult to determine w

calculation performed by Daly using a two-fluid Marker-cell a particle is located in. To overcome these prob

and-Cell method [5].an auxiliary rectangular mesh of uniform zones is supposed over the general ALE mesh. Each cycle, the auxE. Incompressible Flowmesh is assigned a velocity field linearly interpolated

For an incompressible flow it is not desirable to compute the general mesh. Particles can then be moved in thethe pressure for phase one from the equation of state Eq. way [5] with respect to the auxiliary mesh. To interp(4). If a calculation is attempted with the sound speed from the ALE to the rectangular mesh, a sweep is significantly larger than u/1/ 2, where u is a typical fluid through the vertices of the ALE mesh. For each vspeed, then small density variations possibly remaining its location in the rectangular mesh is determined andafter the phase-three rezoning may be magnified into large its momentum and mass are distributed linearly to thpressure variations when used in the equation of state for nearest rectangular mesh vertices. When all ALE vethe start of the next cycle. These pressures can then gener- have been swept, the total momentum accumulated aate velocity fluctuations that may be impossible to elimi- rectangular vertex is divided by the total mass accumunate in the following phase-two iteration. The problem may there, resulting in a velocity field that can be used to be avoided by omitting the equation of state calculation in the particles. Boundary conditions must be set in the

phase one and using instead the pressures remaining from iary mesh as appropriate.the previous phase-two iteration as a first guess for thenext cycle. In practice, of course, the iteration can be ACKNOWLEDGMENTSstarted with any reasonable guess for the initial pressure,

The authors are pleased to acknowledge contributions from but the better the first guess the sooner convergence ismembers of Group T-3 at the Los Alamos Scientific Laboratory

obtained. Thus, the equation of state calculation is omittedparticular thank J. U. Brackbill, T. D. Butler, R. A. Gentry, F. H. H

before phase one when Mach numbers are less than ap- and H. M. Ruppel.proximately 1/ 2, but retained otherwise.

REFERENCESF. Marker Particles

1. F. H. Harlow, Numerical Methods for Fluid Dynamics, an AnIn some problems it is convenient to use Lagrangian Bibliography, Report LA-4281, Los Alamos Scientific Labo

Los Alamos, NM, 1969.marker particles to aid in the visualization of complicated


14/14


2. C. W. Hirt, Proceedings of the Second International Conference on 1955; A. A. Amsden, Report LA-3466, Los Alamos Scientific Ltory, 1966.Numerical Methods in Fluid Dynamics, Berkeley, 1970.

9. M. L. Wilkins, Report UCRL-7322, Rev. 1, Lawrence Ra3. J. G. Trulio, Report AFWL-TR-66-19, Air Force Weapons Labora-Laboratory, Livermore, CA, 1969.tory, Kirtland Air Force Base, 1966.

10. J. E. Fromm, Report LA-2535, Los Alamos Scientific Lab4. F. H. Harlow and A. A. Amsden, J. Comput. Phys. 8, 197 (1971).(1961).

5. F. H. Harlow and J. E. Welch, Phys. Fluids 8, 2182 (1965); J. E.11. C. W.Hirt, J.L. Cook,and T.D. Butler,J.Comput. Phys. 5, 103Welch, F. H. Harlow, J. P. Shannon, and B. J. Daly, Report LA-12. C. W. Hirt, J. Comput. Phys. 2, 339 (1968).3425, Los Alamos Scientific Laboratory, 1966; A. J. Chorin, Math.

Comput. 22, 745 (1968). 13. J. P. Boris and D. L. Book, J. Comput. Phys. 11, 38 (1973).6. A. A. Amsden and C. W. Hirt, Yaqui: An Arbitrary Lagrangian 14. T. D. Butler, Proceedings of the Second International Confer

Eulerian Computer Program for Fluid Flows at All Speeds, Report Numerical Methods in Fluid Dynamics, Berkeley, 1970.LA-5100, Los Alamos Scientific Laboratory, 1973. 15. P. D. Lax, Comm. Pure Appl. Math. 7, 159 (1954).

7. G. K. Batchelor, An Introduction to Fluid Dynamics, Cambridge 16. A. A. Amsden and C. W. Hirt, J. Comput. Phys. 11, 348 (19Univ. Press, Cambridge, 1967. 17. C. W. Hirt and J. L. Cook, Proc. Atomic Effects Symposium

1973.8. F. H. Harlow, Report LAMS-1956, LosAlamos Scientific Laboratory,

Documents

An Arbitrary Lagrangian¨CEulerian Computing Method for All Flow Speeds