numerical simulation of flows encountered during mold-filling

Numerical simulation of flows encountered duringmold-filling

K. S. Chan, K. Pericleous, and M. Cross

Centre for Numerical Modelling and Process Analysis, Thames Polytechnic, London, UK

This paper describes a model to simulate flows and interface activity during the filling of threedimensional metal-casting molds. A technique developed by Liu Jun at Imperial College, based onthe use of a conserved scalar variable to represent the liquid with an adaptation of the van Leerscheme to define the instantaneous position of the interface, has been adopted. Computations havebeen performed to simulate two distinct filling problems, one dealing with the slow filling of a largesand-casting mold and the other dealing with the much more rapid situation encountered in pressuredie casting. The computations were performed by using the PHOENICS CFD code, and the resultswere compared against a water model experiment of the die-casting case.

Keywords: 3-D mold filling, free surface flows, van Leer scheme, scalar equation method

Introduction

The simulation of aspects of the casting of metals hasoccupied a good deal of effort over the past few years(see Refs. 1-4 for reviews of current activities). Oneof the key components in the costing of some components and in continuous casting is the filling of themold. The filling process has to proceed in such a waythat excessive liquid metal surface wave action and themore complicated breaking or atomization phenomenaare avoided. If wave action becomes excessive, thensurface impurities, flux cover layers, or air bubblesmay be entrained into the metal. In the latter case, asthe metal component solidifies, flawed castings willresult. In the casting of thin sections, for example,there are complex interactions between the filling, heattransfer, and solidification phenomena that must bequantified if their design and manufacturing route is tobe optimized. Hence there is considerable interest indeveloping a practical means of simulating the fillingof molds by liquid metals to enable detailed analysisof the phenomena involved in the optimization of acasting process route.

Research on the development of algorithms and codesto simulate the filling process has been underway fora decade or more. Most of the early work was published by Stoehr and Hwang.' Their approach is basedupon the volume-of-fluid (VOP) method originally developed by Hirt and Nichols" and encapsulated in various regular meshlike codes. Most of the original studies were done in two dimensions because thecomputational time for a simulation is prodigious.Comparisons with experiment have proved to be en-

Address reprint requests to Dr. Chan at Thames Polytechnic, Centrefor Numerical Modelling and Process Analysis, Wellington St., London SE18 6PF, UK.Received 15 February 1991; accepted 19August 1991

624 Appl. Math. Modelling, 1991,Vol. 15, Nov.lDec.

couraging, and recently, the VOP-based algorithms havebeen extended to three-dimensional problems and evenembedded within commercial products.

The VOP algorithm uses the solution of a conservedscalar variable to track the liquid metal and then attempts to preserve the shape of the metal-air interfacethrough the donor-acceptor scheme of Ramshaw andTrapp." It is the preservation of the metal-air interfacethat makes the computational effort so large. Certainly,one of the reasons why the VOF scheme requires somuch effort is its explicit formulation. Recently, LiuJun" and Liu Jun and Spalding? have described a procedure that is similar in approach to the VOP algorithmbut uses the van Leer!" scheme to model and preservethe shape of the liquid metal surface. The basic technique has been validated in two dimensions for a collapsing liquid column and "sloshing" in a half-rotatedvessel. In a recent investigation it has been shown thatthe van Leer scheme is arguably the best in modellingshock-capturing problems both accurately and econornically.!' The objective of this paper is to exploitand assess the VOF-van Leer scheme in the modellingof filling of molds by liquid metals in three dimensions.Two problems have been selected for demonstrationpurposes, which are typical of those encountered incasting. One deals with the relatively slow filling of alarge sand-casting mold, and the other with a muchfaster pressure die-casting process. In the latter case,the simulation results are favorably compared with waterexperiments carried out at BNP-Pulmer. 12

Mathematical model

Governing equationsThe filling problem is described by the equations offluid motion, which for constant properties become theNavier-Stokes equations:

e 1991 Butterworth-Heinemann

(3)

(2)

Numerical simulation of flows: K. S. Chan et al.

(au au2 uu uw) ap (a2u a2u a2u)p -+-+a-+a- = -~+}L -+-+- +S (1)at ax ay ez ax ax2 ay2 az2 x

(av uv av2 VW) ap (a2v a2v a2v)

p -+a-+-+a- = -~+}L -+-+- +Sat ax ay az ay ax2 ay2 az2 y

(aw uw vw aW

2) ap (a2w a2w a2w)p -+a-+a-+- = -~+}L -+-+- +Sat ax ay az az ax2 ay2 at2 z

(7)

(4)

(9)

(8)

where subscripts a and I refer to air and liquid, respectively.

Where the Reynolds number is high, the flow canbe assumed to be turbulent. The Navier-Stokes equations can then be expressed in time-averaged form withinteractions between fluctuating and mean fields represented by the Reynolds stress tensor. The Reynoldsstress tensor is evaluated in terms of mean velocitygradients with the aid of a turbulent viscosity }Lt, whichis in turn evaluated in each fluid by using the twoequation k, IE model.'? Hence

k2}Lt=C!J.·P

IE

and

where k and IE represent the turbulent kinetic energyper unit mass and its dissipation rate, respectively,each being the subject of an additional transport equation. The model is a "switch-on" option in thePHOENICS code" and has been described in detail inmany publications, including reviews by Markatos'?and Nallasamy." For this reason it will not be described here.

The k, E turbulence model is compatible with thepresent method, and it was used for some of the moldfilling calculations performed; however, it increasedthe computational cost considerably, since it sloweddown convergence. In most calculations it was foundadequate to use a mean effective viscosity to representturbulent diffusion within each fluid. The linear interpolation formula of equation (8) was again used at theinterface.

Future work will include modifications to the standard k, IE model to account for the damping effect ofthe liquid surface on turbulent fluctuations.

The van Leer scheme for reducing numericaldiffusionAlthough equations (7) and (8) can be modified to ensure that properties retain their original values in allbut the interface cells? to reduce the ill effects of numerical diffusion, interface smearing was still found tobe a problem, especially when first-order differencingschemes such as "upwind" are used to solve equation(6). Hirt and Nichols" in their VOF method adopt thewell known "donor-acceptor" scheme of Ramshawand Trapp? to combat numerical diffusion smearing. Inthe present work, the van Leer'? scheme is adopted as

(5)

(6)

where u, D, and ware the velocity components in cartesian directions x, y, and z; p is the pressure in the fluid;and p and }L are the fluid density and viscosity, respectively. The source terms Sx, S", and S, representexternal influences and body forces (gravity, centrifugal, Coriolis, Lorenz, etc.).

The above equations are supplemented by the continuity equation, which can be written in the form

D(ln p) au au aw--+.,-+-+-=0

Dt ax ay az

au + av + aw = 0ax ay az

Equation (4) is expressed in terms of volumetric conservation to emphasize the use of the so-called GALA 13

algorithm in the numerical procedure, which eliminatesdifficult numerical problems arising from large densityvariations across the interface.

The two fluids are distinguished by the conservedscalar quantity ct>, which has a value of 1 in the liquidand 0 in air and obeys the equation

or, for incompressible flow,

Fluid propertiesAt the free surface the fluid density p and viscosity }Lvary with time as the interface position changes relative to a fixed grid. In the present method the instantaneous values of p and }L are deduced from the valuesof ct> by using the following relations:

act> + u act> + v act> + w act> = 0at ax ay az

Equation (6) has no diffusive terms and hence impliesthat ct> is influenced only by convection. Since the twofluids are immiscible, exact solutions of ct> should exhibit values of 1 in the liquid and 0 in air, the two fieldsbeing separated by the interface, where ct> will acquireintermediate values.

The above equations with appropriate boundaryconditions need to be solved in both the liquid and airspaces inside the mold within a "single-phase" framework, in contrast, say, to the variable blockage technique (e.g., Ref. 14), which neglects the air space, orthe two-fluid algorithm IPSA, 15 which solves two setsofNavier-Stokes equations, one for each phase as usedby Aldham et al. 16

Appl. Math. Modelling, 1991, Vol. 15, Nov./Dec. 625


N

InIII ue- - .. - - -- - - +p- - - - .. - - - .~W w eIIIIS

S

Figure 1. A typical grid cell

being the simplest to implement and most economicalof the TVD schemes available (e.g., see Ref. 11). Consider the cell of Figure 1, in which p represents thecell node and E, W, N, and S represent the East, West,North, and South neighbors, respectively. To calculatethe convected flux of ep into the East neighbor, weneed to determine the value of epe at the east cell interface. The velocity there is u, stored at the cell facein the staggered grid convention. Then, according tothe fully implicit upwind formula,

4>e = tPP for Lie > 0

and

epe = epE for u, < 0

epp and cPE being the values of ep at the grid nodes atthe end of a time step.

Unfortunately, the first-order upwind approach ishighly diffusive. Van Leer's method introduces second-order terms in time and space, and in addition itpreserves monotonicity, thus preventing the spuriousoscillations that often accompany higher-order methods. The method has been proven particularly successful in preserving sharp discontinuities in onedimensional tests (e.g., see Ref. II).

Consider pure advection of a scalar eP(x, t) by avelocity utx, t):

aep + u aep = 0 (10)at ax

Equation (10) can be solved on a regular grid for anyadvection method with a forward time step. Referringto Figure 1, we get

LI' Ii.tep~+l=ep~_ ~x(epe-4>w) (II)

for positive u.: Hence the updated value of epp can bededuced once the cell face values of cP are known. Thevan Leer approach determines the face values in termsof local gradients of ep. Hence

ep" = epp + ox(aeplax)p(1 - Ueotlox)/2 for u, > 0

epe = cPE - ox (a4>/aX)E(l + u, ot/ox)12 for u, < 0(12)

626 Appl. Math. Modelling, 1991, Vol. 15, Nov.lDec.

The gradient (a¢/ax)p is best computed'? as

(a¢/ax)p = 2 sgn (8e) min {18el,o·5(18el + 18wl),18w l}/ox(13)

with

sgn (8,,) = + 1 if s,» 0

= -1 if 8"< 0

Further, to ensure monotonicity, if 8" . 8", < 0 then(aep/ax)p = 0, thus reverting to upwind conditions.

Similar formulas can be written for cPw, ¢,Il <P." and<PI in a three-dimensional node. The convective flux ofep is then split into three parts, f... fy, and t,.., one foreach direction. The order of flux calculation alternatesbetween time steps to ensure that there is no directionalbias. It is also ensured that conservation of ep is maintained at each adjustment. Hence the net flux in thex-direction is given by

i, = 8t (Ae V"eP" - A w UWcPll') (14)

Similar formulations apply for t, and fz.It should be noted that the derivation of the gra

dients requires values of ep at the beginning of the timestep that render the approach explicit. To avoid instability, the interface must not jump a full cell during atime step, that is,

8t < min {18x/ul, ISy/vl, ISz!wl} (15)

in accordance with the Courant stability criterion.

Solution procedure

In the applications that follow, the partial differentialequations (1)-(4) and (9) have been solved subjectto appropriate boundary conditions by means of theSIMPLEST algorithm" embodied in the PHOENICScode. Equation (6) and relationships (7)-(13) werecoded in FORTRAN in a user attachment section ofPHOENICS. Although PHOENICS was used as thevehicle for the simulations shown, the van Leer attachment is not code specific. It can therefore be incorporated into any single-phase CFD code, providedthat adequate access to the code exists via user interface routines, as shown schematically in Figure 2.

The "staggered" grid arrangement is adopted forthe allocation of flow variables, with velocities placedat the cell faces. Convection terms in all equationsexcept (6) follow the "upwind-differencing" convention.

The solution procedure is composed of the followingsteps:

1. Solve momentum equations using a guessed pressure field.

2. Compute continuity errors and set up the pressurecorrection equation by differentiating the continuityequations with respect to pressure.

Module

General Purpose

CFn

1. User/ Rout ines

(IO,P,P.}

Van Leer

Marker Fluid

Attachment

Numerical simulation of flows: K. S. Chan et et.

Function

Problem Specification:• Grid and Geometry• Boundary Conditions• Solution Control• Output Control

Problem Solution:• Solve k, e equations• Update properties• Volume continuity• Momentum equations• Compute residuals

User Interface:• Connexions to main

solver and attachment

Van Leer Solver:• Compute face values of II'• Compute advection fluxes• Update II' cell-by-cell

ensuring conservation• Compute "mixture"

properties

Figure 2. Mold-filling attachment to CFD code

Applications

= 1· 177 m3/s

= 7650 kg/rn?= 1·5 X 1O- sm2/s

= 1· 5 X 1O-2m2/s

3. Apply corrections to pressure and velocities to ensure that volume conservation prevails.

4. Repeat until errors are substantially reduced.5. Update the scalar field using the latest velocity val

ues.6. Update corresponding fluid properties.

Sand-casting moldThe simple mold shown in Figure 3 is first considered.Fluid enters through a top inlet (A) and leaves througha lateral outlet (B). The outlet is placed level with thetop of the main mold cavity to ensure that no fluidleaves until the cavity is full. At the far end of the moldaway from the inlet the mold section deepens to forma kind of reservoir. The outlet is placed halfway between the inlet and the far wall. The following fluidproperties and flow data are used:

Liquid volume flow rateLiquid densityAir kinematic viscosity (20CC)Air effective kinematic viscosity

Liquid kinematic viscosity = 10 -6 m2/s

Liquid effective kinematic viscosity = 10 -3 m2/sCavity volume = 1 . 147 x 10-4 m3

Hence expected fill time = 0 . 975 s

The calculation is assumed to be turbulent and isothermal. Wall friction is applied to all solid surfaces,and the outlet is assumed to be at a zero referencepressure.

Figure 4 shows the cartesian grid used, which hasNx = 12, Ny = 11, and Nz = 17 cells, that is, a totalof 2244 cells. This grid was chosen as a compromisebetween computational time, accuracy, and surfacedetail resolution in preference to an initial coarser gridof 1200 cells. For this grid a time step of 0 . 001 s waschosen to ensure stability under equation (15). At least25 iterative sweeps were needed to ensure convergenceat each time step.

A complete run required 760 time steps to almostfillthe chamber (see below). Total run time on a NORSK·DATA ND5900 minicomputer was 41 hours. TheND5900 performs as a 0.5-Mflop machine on CFD codes.

The results of the calculation are shown graphicallyin Figures 5-15 at selected time steps. The develop-



Figure 3. Sand mold geometry

Figure 4. Computational grid

Figure 5. Liquid sUrface at t = 0.03 S

Figure 6. Liquid surface at t = 0.09 s

628 Appl. Math . Modelling, 1991, Vol. 15, Nov.lDec.


I ' 1/1t-" ..-ur

_ e-,

~:: :: '\ i.'~t:---,., - - - - ~

,I ' • " '1.. - - .:1,

~

~- -:-1t I _ -,

\ \ - ,~ - -

_____ : 3.~2ITt." . 1• •

Figure 8. Velocity vectors and liqu id surface in section t = 0.15 s


Figure 10. Liqu id surface at t = 0.21 s




I -~~~1...,- -

t\! ~"':--.:. ~ ' \ po",- .

~'"

~ \" l 1 I L p-IX """

Figure 14. Velocity vectors and liquid surface in section at t =0.33 s



ment of the airlliquid interface is represented as theisosurface at which ¢ = 0 . 5. In addition, selectedvector maps indicate the flow behavior of both the airand liquid masses .

Figure 5 at the first time step shows in perspectivethe initial liquid column as it enters the mold. Figure6 shows the position of the liquid surface at the pointat which it reaches the lip of the end reservoir precipice. Figure 7 shows the subsequent jet of liquid impacting on the end wall and spreading. Figure 8 showsa section of the surface in side view, togethe r with theflow field vectors. Thejet impingement and subsequentspreading can be seen, and a recirculation cell thatdevelops below the jet as air is trapped in the endchamber is also apparent. Above the liquid surface, airinitially follows the liquid movement until it reachesthe end wall. There it reverses and flows along the topof the mold toward the exit.

Figure 9 follows Figure 7 in time and shows theprogression of the liquid layer in the end chamber asit first flows along the base wall, forming a third interface.

Figures 10 and 11 show the lower interface closingupon itself in a complex manner, trapping a quantityof air within it, while on the top interface a wave develops that moves away from the end wall toward theoutlet. As it approaches the outlet, it forms a crest(Figure 11), which is amplified by fast-flowing air thatis trying to squeeze past it as it escapes through theoutlet. The wave results in liquid escaping the moldbefore the end section is full. Subsequent time stepsshow a gradual filling of the end chamber (Figures12-14) and diminution of the trapped air volume. Figure 15 shows that the mold is nearly full; however, thefilling process takes longer than expected, since theliquid escapes prematurely, owing to wave action.

To alleviate the wave problem and reduce the possibility of air being trapped in the casting, the outletwas placed above the end chamber. Figure 16 showsthe new geometry and the jet of fluid a moment priorto impact. Figure 17 indicates that after impact, theflow behaves as before ; however, later time steps indicate that less air is trapped (Figure 18), the postim pact wave is absent, and the cavity is almost full beforeany liquid leaves through the outlet.



T I HGR 4DU4llREOlICII(ino 4RE A

Figure 19. Die-casting water experiment simulated

Figure 17. Case 2: Liquid surface at t = 0.18 s

Figure 16. Case 2: Liquid surface at t = 0.12 s

Figure 18. Case 2: Liquid surface at t = 0.36 sFigure20. Sketch showing runner and cavity

Die-casting simulationThe water experiment of Ref. 12 was then simulatedto compare calculated air-water interfaces against photographic evidence.

The experiment being simulated is shown in Figure19. Following impact at the end of the plunger, wateris forced at fairly high velocity (10-50 m/s) into a vertical cavity through a runner. The runner-cavity arrangement is shown schematically in Figure 20. Forthis simulation a grid of Nx = 6, Ny = 19, Nz = 17,that is, 1938 cells, was used. However , to ensure numerical stability, time steps between 10- 4 sand 10-5

s were necessary, depending on inlet velocity . This ledto long computer times (up to 150 hours) per simulation, highlighting one of the deficiencies of the technique that being explicit in time needs to satisfy theCourant criterion.

Calculations are still in progress, with further re-

finements being introduced to the model to account (a)for air compressibility and (b) for the correct hydrauliclosses in the runner/cavity slot. Figures 21 and 22 showthe interface progress compared to the experiment andalso the resulting flow field. The main characteristicsof the flow are well represented, although differencesexist close to the overflow vents.

Conclusions

A new three-dimensional technique for handling airliquid interfaces has been demonstrated, as applied toliquid metal-casting processes . The technique showsrealistic results and is in the process of being validatedby experiments. Interesting features of surface behavior such as waves or air voids can be easily analyzed ,and remedies to casting geometry can be readily introduced to effect improvements. This was demonstrated in the sand-casting mold, in which wave motionwas suppressed simply by changing the outlet position.

630 Appl. Math. Modelling, 1991, Vol. 15, Nov.lDec.

- : 3.1967E+<J 1 rn/s

Figure 22. Case 3: Experiment and simulation of air/water interface at t = 0.005 s

Figure 21. Case 3: Experiment and simulation of air/water interface at t = 0.0018 s

ReferencesBrody, H. D. and Apelian, D., eds, Modelling ofCosting andWelding Processes. TMS-AIME, Warrendale, PA, U.S.A., 1981

2 Dantzig, J. A. and Berry, J. T., eds, Modelling ofCasting andWelding Processes, Vol. 2. TMS-AIME, Warrendale, PA,U.S.A., 1984

3 Kow, S. and Mehrabian, R., eds. Modelling of Casting andWelding Processes, Vol. 3. TMS-AIME, Warrendale, PA,U.S.A., 1987

4 Giamei, E. and Abbasclian, G. J., eds, Vol. 4. Modelling ofCasting and Welding Processes, Vol. 4. TMS-AIME, Warrendale, PA, U.S.A., 1989

5 Stoehr, R. and Hwang, W. S. Modelling the flow of moltenmetal having a free surface during entry into molds. Modellingof Casting and Welding Processes, Vol. 2, ed. J. A. Dantzigand J. T. Berry. TMS-AIME. U.S.A., 1984, p. 47

6 Hirt, C. W. and Nichols, B.D. Volume of fluid (YoF) methodfor the dynamics of free boundaries. J. Comput, Phys. 1982,39,201

7 Ramshaw, J. D. and Trapp, J. A. A numerical technique forlow speed homogeneous two-phase flow with sharp interfaces.J. Comput, Phys, 1976,21,438

8 Liu Jun. Computer modelling of flows with a free surface.Ph.D. Thesis, CPDU Imperial College, London, 1986

9 Liu Jun and Spalding, D. B. Numerical simulation of flowswith moving interfaces. Physicotlhem. Hydrodyn. 1988, 10(5/6),625-637

10 van Leer, B. Towards the ultimate conservative differencescheme. IV: A new approach to numerical convection. J. Comput. Phys. 1977,23,276

II Leonard, B. P. Universal limiter for transient interpolationmodelling of the advective transport equations: The ULTIMATE conservative difference scheme. Tech. Memo 100916,NASA, Lewis Research Center, Cleveland, OH, U.S.A., 1988

12 Booth, S. E. and Allsop, D. P. Cavity flow studies at BNFMetals Technology Centre, using water modelling techniques.Paper presented at the J2th International Die Casting Congress,Minneapolis, Minn., U.S.A., 1985

13 Spalding, D. B. A method for computing steady and unsteadyflows possessing discontinuities of density. CHAM Report 910/2,Wimbledon, London, UK, 1974

14 Pericleous, K. and Rhodes, N. The modelling of a cyclonicflotation machine. CHAM Report 3251, Wimbledon, London,UK,1984

15 Spalding, D. B. Developments in the IPSA procedure for numerical computation of multi phase flow. Second NationalSymposium on Numerical Properties and Methodologies, ed.T. M. Shih, 1981, pp. 421-436

16 Aldham, C., Rhodes, N. and Tatchell, D. Three-dimensionalcalculations of explosion containment in fast breeder reactors.Pub!. 82-PE-3, ASME, Fluids Eng Division, 1982

17 Launder, B. E. and Spalding, D. B. The numerical computationof turbulent flows. Comput . Methods Appl. Mech. Engrg, 1974,3,269-289

18 Spalding, D. B. PHOENICS: A general purpose computer program for multi-dimensional one- and two-phase flows. Math.Comput, Simulation 1981, 23, 267

19 Markatos, N. C. The mathematical modelling of turbulent flows.Appl. Math. Modelling 1986, 10,3

20 Nallasamy, M. Turbulence models and their applications to theprediction of internal flows: A review. Comput, Fluids 1987,15(2), 151-194

21 Spalding, D. B. Mathematical modelling of fluid-mechanics,heat-transfer and chemical reaction processes. A lecture course,CPDU Rept. HTS/80/1, Imperial College, London, 1980

Numerical simulation of flows: K. S. Chan et et.to BNF for providng the test geometries and experimental results. K. S. Chan acknowledges SERC forreceipt of a research studentship.

r

L ,

Lx

T

- : 2.2326E+01 mls

II r

\ I, - " \ \ I

~~~- ~ : qn~1r;7 , - '\ 11~ - ~ - , I !\ 1 \

'/ ' , , .• I \ \ ' \~ " ' . • • I \~~ '. -, ::' .:.:: z :. ! { ~ ....

, I ill: I I • I '" •

I. I

I ' I II.

I ., ..I SO

I :,1:rH:' f . I

" 'f... I;' ,~ , ,I I! '- I I I I

KV~::: " , .<;: :

'~~-" -' .' ~~

The van Leer scheme adopted for these calculationswas chosen for its discontinuity, retaining characteristics, simplicity of implementation, and suitability forthree-dimensional computations. Although it too introduces some numerical diffusion in the computations, its effects on the interface are small and easilycontrolled.

The method can be readily extended to include heattransfer and solidification. In the present cartesian form,highly irregular shapes can be modelled by "carving"shapes out of a cartesian grid by using cell blockagefactors. However, this process is labor intensive, andfuture extensions may necessitate the adoption of bodyfitted grids or nonstructured FE type elements.

One of the aims of the project is to produce a practical mathematical tool for foundry engineers to use.For this reason, ways of reducing computer run timeswill be investigated. For example, it is thought thatgreat economy can result if the extent of the calculationdomain for ep is restricted at each time step to cellscontaining and adjacent to the interface.

Acknowledgments

The authors are grateful to CHAM Limited for allowingtheir code PHOENICS to be used for this project and

..,

. Ii4IIJ

Appl. Math. Modelling, 1991, Vol. 15, Nov.lDec. 631

numerical simulation of flows encountered during mold-filling

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