LES in complex geometries using boundary body forces · LES in complex geometries using boundary...

Center for Turbulence ResearchProceedings of the Summer Program 1998

171

LES in complex geometriesusing boundary body forces

By R. Verzicco1, J. Mohd-Yusof, P. Orlandi1 AND D. Haworth2

A numerical method is presented which can simulate flows in complex geometrieswith moving boundaries while still retaining all the advantages and the efficiency ofsolving the Navier-Stokes equations on cylindrical grids. The boundary conditionsare applied independently of the grid by assigning body forces over surfaces thatneed not coincide with coordinate lines.

The method has been validated by a large-eddy simulation of the flow in a motoredaxisymmetric piston-cylinder assembly for which experimental measurements areavailable. The comparison of the results has shown a very good agreement for meanand rms velocity profiles, thus confirming the accuracy of the present approach. Thisnumerical method, in addition, runs on a small PC-like workstation ten times fasterthan corresponding simulations on supercomputers.

In large-eddy simulations the dynamic subgrid-scale model is very efficient incombination with the body force procedure because it automatically accounts forthe walls without requiring ad hoc damping functions. This feature is very usefulin the present approach since the body surface in general does not coincide with acoordinate line and the computation of the wall distance is time consuming.

1. IntroductionNumerical simulations are extensively used by industries as a fundamental tool

in the design and test of prototypes. Computer simulations allow for the rapidinvestigation of wide parameter ranges or the testing of novel technological solutionswithout resorting to expensive experimental setups and measurements.

The most detailed numerical simulations up to now have been performed in ex-tremely simple geometries since in these conditions the efficiency of the solutionprocedures is maximized and the storage requirements of the computer codes arethe least. For example, homogeneous isotropic turbulence in a tri-periodic box hasbeen simulated using 5123 nodes or more, and comparable high resolutions havebeen achieved in the plane channel flow or in mixing layers.

Unfortunately, in the majority of industrial applications, the flow is stronglydominated by the domain geometry, and even the simplest flow features can notbe reproduced if the shape of the boundary is not properly accounted for. In such

1 Universita di Roma “La Sapienza” Dipartimento di Meccanica e Aeronautica, via Eudossiana

18 00184 Roma, Italy.

2 Engine Research Department, GM R&D Center, 30500 Mound Road 106, Warren, MI 48090-

9055

172 R. Verzicco, J. Mohd-Yusof, P. Orlandi & D. Haworth

circumstances curvilinear coordinates fitted on the body are currently used even ifmost of the efficiency of the solution procedure is lost. In addition, the complexityof the computer code as well as the number of operations per node and storagerequirements are increased. Therefore, given the finite amount of memory availableon each computer, the maximum number of nodes over which the flow can becomputed is reduced.

Another factor increasing the complexity of the numerical simulations is the pres-ence of moving boundaries. In fact, when boundary fitted meshes are employed, thegrid must be recomputed as the geometry changes, thus considerably increasing thecomputational time. Unfortunately in industrial flows the presence of boundariesin relative motion is the rule rather than the exception, making the computationsvery time consuming.

The last point to be considered is the Reynolds number, which in real applicationsis usually far too high to make the flow affordable by direct numerical simulation(DNS). Industrial flows, in fact, are usually turbulent, and their resolution mustrely on turbulence modeling such as the large-eddy simulation (LES) approach orthe Reynolds averaged Navier-Stokes equations (RANS).

From this scenario it appears that flows of practical interest, owing to the com-bination of complex geometries, moving boundaries, and high Reynolds numbers,make their computation extremely expensive, and the use of an alternative approachis very much desirable.

In this report we illustrate an alternative procedure for simulating complex flowswhile retaining the advantages of orthogonal fixed grids. Specifically, we will testand validate a method in which boundary conditions are assigned independentlyof the grid by prescribing suitable body forces. These forces are such as to yielda desired velocity value v on a given surface, which in general will not coincidewith the coordinate lines and will move in time. Indeed this idea is not new sincePeskin (1972) already used a similar approach. More recently Goldstein, Handler &Sirovich (1993) and Saiki & Biringen (1996), have shown further examples in whichcomplex boundaries were mimicked using body forces on a Cartesian mesh. Themain drawback of all these procedures was the largely reduced stability limit of thetime integration scheme. Goldstein Handler & Sirovich (1993), for example, pointout that when computing the flow around a circular cylinder their time step couldnot be bigger than 5·10−5-5·10−4, which yielded a CFL between 4·10−3-4·10−2. Itis clear that with this restriction this technique is mainly limited to two-dimensionalflows and its validity is largely compromised. A different expression for the forcingwas suggested by Mohd-Yusof (1996,1997) and Mohd-Yusof & Lumley (1994,1996),who, by computing the laminar flow over a riblet and the flow around cylinders,showed the stability limit of the integration scheme to be essentially unaffected bythe forcing. This new forcing was used by Fadlun, Verzicco & Orlandi (1998), whoshowed that, though the results are essentially the same as those obtained by the oldmethod, the time step remains large enough to make three-dimensional simulationsaffordable. In that work the results were validated by an ad hoc experiment.

Although the method is in principle very general, to our knowledge it was never

‘Immersed boundary’ simulations 173

applied before to a real flow of practical interest in which complex geometries,moving boundaries, and turbulence models are required. The main aim of thisreport is to show some results obtained for the computation of a motored piston-cylinder assembly whose Reynolds number is high enough to require the use of anLES approach. The results are compared with an experiment performed in exactlythe same conditions (Morse, Whitelaw & Yianneskis, 1978), indicating that thepresent method is a good candidate to simulate industrial flows in an inexpensiveway.

2. The problem

The configuration chosen to validate the present numerical procedure is a sim-plified axisymmetric piston-cylinder assembly with a fixed central valve. Both,cylinder head and piston are flat (pancake chamber), and a sketch of the geometryis given in Fig. 1.

s

s

c

hd

g

r

rrr

rr

0

r

av

w

s

e

z = z (t)p

α

valve rad.

seat angle

b = 2r

s

r

75 mm

60 mm

16.8 mm

20.8 mm

60 deg.

(s+c):c 3:1

annul. rad.

comp. rat.

stroke

bore

r

h

d

g

r

r

r

s0

e

w

r

v

a

s

6.5 mm

16.5 mm

33 mm

11 mm

3 mm

α

40 mm

13 mm( )

piston motion

z (t) = s + s cos(Ω t)2

Ω 200 rpm = 21 rad/s=

νRe= = 2000Ωs r

s = s + s2

1

1 0

( )

( c ) 30 mm

r

-

-

---

p

Figure 1. Sketch of the problem.

In the experiment the piston was externally motored so that the fluid flowed intothe cylinder from outside during the downward piston motion and vice-versa whenthe piston was moved up. Since the valve was fixed and a tiny annular gap was leftopen between the valve and the cylinder head, no compression phase was includedin the flow dynamics. The piston was driven by a simple harmonic motion at a


speed of 200rpm ' 21rad/s, which for the present geometry yields a mean pistonspeed of V p = 0.4m/s (when averaged over half cycle).

Experimental measurements were available for the validation of the numericalresults. In particular, Morse et al. (1978) used Laser-Doppler anemometry to mea-sure phase-averaged mean and rms radial profiles of axial velocity. The profiles wereavailable at 10mm increments starting from the cylinder head for crank angles 36o

and 144o after top dead center.

3. Equations, solution procedure, and simulation set-up

The governing equations for the LES of the piston-cylinder assembly are thefiltered Navier-Stokes equations with the boundary body force f :

DuDt

= −∇P +∇ · ν[∇u + (∇u)T ]+ f , (1a)

∇ · u = 0. (1b)

Here u is the filtered velocity, P is the sum of the filtered pressure and the trace ofthe sub-grid scale stress tensor while its anisotropic part qij has been parametrizedby the dynamic sub-grid scale model through qij = −2νtSij with νt the turbulentviscosity and Sij the filtered rate of strain tensor. With this notation ν is definedas ν = νt + 1/Re, and the Reynolds number is Re = Ωsrr/ν (see figure 1) withν the kinematic viscosity of the fluid. The turbulent viscosity νt was determinedby a dynamic procedure (Germano et al.1991 and Lilly 1992) that does not requireexternal constants. Since the flow is unsteady and inhomogeneous in time, theaveraging in the computation of the constant C in the expression of the turbulentviscosity (νt = C∆2(2SijSij)1/2, with ∆ the local grid spacing) was performed onlyin the azimuthal direction and among the closest neighbors in the radial and axialdirections. All the points with total viscosity ν smaller than zero were then clipped,and they never exceeded 3% of the total points.

The boundary body force f is prescribed at each time step on a given surface sothat the desired velocity v is obtained on that boundary. As shown by Mohd-Yusof(1996), considering the time-discretized version of Eq. (1a), one can write

un+1 − un = ∆t(RHS + f), (2)

where RHS contains the nonlinear, pressure, and viscous terms. If we want un+1 =v then f has to be

f = −RHS +v− un

∆t(3)

in the flow region where we want to mimic the body and zero elsewhere. Of coursein general the surface of the region where we want un+1 = v will not coincide witha coordinate line (Fig. 2); therefore, the value of f on the cell node closest to thesurface but outside the body is linearly interpolated between the actual value insidethe body and the zero value in the flow. The interpolation procedure is consistent


x

r

Figure 2. Vertical section through the meridional planes θ = 0 and θ = π ofthe computational grid (only 1 every 4 grid points are shown) and the body forcedistribution to mimic the geometry of Fig. 1.

with the central finite-difference approximation since both of them are second-orderaccurate. Further details will be given in a forthcoming paper where this procedureis shown to be consistent with the second order accuracy of the scheme.

Equations (1) have been discretized in a cylindrical coordinate system by centralsecond-order accurate finite-difference approximations on a staggered grid. Detailsof the numerical method are given in Verzicco & Orlandi (1996); therefore, onlythe main features are summarized here. In the three-dimensional case, in the limitof ν → 0, the energy is conserved and this holds in the discretized equations.The system is solved by a fractional-step method with the viscous terms computedimplicitly and the convective terms explicitly; the large sparse matrix resulting fromthe implicit terms is inverted by an approximate factorization technique. At eachtime step the momentum equations are provisionally advanced using the pressure atthe previous time step, giving an intermediate non-solenoidal velocity field. A scalarquantity Φ is then introduced to project the non-solenoidal field onto a solenoidalone. The large band matrix associated with the elliptic equation for Φ is reduced toa penta-tridiagonal matrix using trigonometric expansions (FFTs) in the azimuthaldirection; the matrix is then inverted using the FISHPACK package. A hybridlow-storage third-order Runge-Kutta scheme is used to advance the equations in


time. Finally, in cylindrical coordinates the equations are singular at r = 0. Theadvantage of using staggered quantities is that only the radial component of themomentum equation needs to be evolved at the centerline (r = 0), and for thiscomponent we calculate the evolution of qr = rur instead of ur since the formerquantity clearly vanishes on the centerline.

The simulation set-up for the problem described in Section 1 is shown in Fig. 2.The lower boundary of the domain is inflow or outflow with a prescribed velocityprofile depending on the phase of the piston motion. The upper boundary is alsoinflow or outflow, but a convective boundary condition for the velocities has beenused. Thus if ui is the i-th velocity component, the equation

∂ui∂t

+Ci∂ui∂x

= 0 (3)

is solved at the boundary, and Ci is explicitly determined from the previous timestep. The lateral wall of the computational domain is free-slip, and all of the no-slipboundary conditions at the body surfaces are enforced by f .

The axisymmetric simulations have been started from rest, and the transient(typically 2 cycles) has been discarded. In order to save computational time, thethree-dimensional DNS and LES were restarted from the corresponding axisym-metric field, replicated in the azimuthal direction with a random azimuthal velocityfield (|uθ|max = 0.25V p) superimposed. The initial transient was thus reduced toone cycle.

4. Results

Since the flow inside the piston engine is turbulent, unsteady, and fully three-dimensional, the numerical simulation must cope with all of these aspects. Nev-ertheless, in order to test the procedure, before considering the whole problem wehave simulated the flow at a lower Reynolds number with the hypothesis of axisym-metry. This simulation served also as a guideline for the analysis of the large-scaledynamics, which in the fully turbulent case become less clear. The results are givenin Fig. 3 where azimuthal vorticity maps at six instants within a cycle are shown.At t = 0 the piston starts its descent and the fluid is sucked from outside into thecylinder through the annular gap between the central valve and the cylinder head.This generates a high speed jet which impinges on the side wall and eventuallyseparates owing to viscous effects. At t = π/2 the velocity of the jet is the highest,and the flow is completely dominated by the toroidal vortex, which grows in sizeas the piston moves down. It should be noted that if Vp(t) is the piston speed, onaccount of mass conservation, the jet velocity should be about 10Vp(t); in reality,the jet velocity is even higher since the boundary layers developing on the walls ofthe annular gap reduce the effective area.

The interaction of the jet with the horizontal and vertical walls generates a varietyof smaller structures already visible at t = π/2. Later on, when the piston decreasesits speed, the jet does not have enough momentum to penetrate the flow down tothe piston, but rather generates structures which remain localized close to the valve.


t = 0 t = π/4 t = π/2

t = π t = 3π/2 t = 7π/4

Figure 3. Contour plots of azimuthal vorticity at Re = 625, (129 × 385 grid).positive negative values (∆ω = ±2.5).

The complex flow structure emerging from the interaction of all the recirculatingvortices is shown at t = π when the piston is at its lowest position. As the pistonstarts moving up, the flow structures are pushed out of the cylinder through thetiny channel again, generating a high speed jet but now in the exhaust region.At t = 7π/4 the piston is slowly reaching the top-dead-center position, and only aresidual motion is left inside the cylinder; a new cycle will start with similar featuresat t = 2π. Note that below the piston a secondary flow is generated by the piston


t = π/4 t = π/2

Figure 4. Contour plots of azimuthal vorticity at Re = 315; three-dimensionalsimulation without azimuthal perturbation (97 × 85 × 193 grid). positive

negative values (∆ω = ±2.5).

motion and by the mass flow imposed at the lower boundary. This flow, however,does not interact with the primary flow since the piston moves as a solid body asconfirmed by the absence of vorticity inside the body.

Very similar dynamics were observed by Eaton & Reynolds (1987, 1989), who bysmoke visualization and high-speed photography were able to capture the flow dy-namics in an axisymmetric motored piston-cylinder assembly. In their description,however, it is clearly stated that when the piston reaches the bottom-dead-center(BDC) position, the flow is complex and very three-dimensional in nature. Ha-worth & Jansen (1997) simulated the same flow described in this report and theyalso found the flow to be fully three-dimensional and turbulent. These observationsmotivated us to set as the ultimate goal the three-dimensional simulation at thesame Reynolds number as the experiment of Morse et al. (1978). However, beforeproceeding to the full simulation a few intermediate steps are necessary.

The first is to verify that the boundary body forces do not introduce any unphys-ical perturbation making the flow artificially three-dimensional. For this reason wehave performed a three-dimensional simulation without initial azimuthal perturba-tion and we have verified that indeed the flow remains axisymmetric at least duringthe first two cycles. In Fig. 4 two vertical sections are shown of the flow at Re = 315,confirming that the symmetry about the central axis is preserved. Note that the


t = π t = 3π/2 t = 7π/4

t = 0 t = π/4 t = π/2

Figure 5. Contour plots of azimuthal vorticity at Re = 315; three-dimensionalsimulation with azimuthal perturbation (97×85×193 grid). positive neg-ative values (∆ω = ±2.5).

Reynolds number of this simulation has been halved with respect to the axisym-metric case, and the reason is that the same radial and axial grid spacing could notbe maintained with 97 grid points in the azimuthal direction. Given the reducedradial and axial resolution, we had to reduce the Reynolds number to fully resolveall the flow scales without resorting to a turbulence model. It could be argued that


t = π/2

t = π

Figure 6. (Left) Contour plots of azimuthal vorticity at Re = 2000 (65×65×151grid). positive negative values (∆ω = ±2.5). (Center) meridionalvelocity vectors. (Right) horizontal velocity vectors in a section 15mm below thecylinder head.

the reduced Reynolds number is the reason for the symmetry conservation; how-ever, the same simulation with an initial azimuthal perturbation shows that indeedthis is not the case. In Fig. 5 six vorticity snapshots taken across two consecutivecycles are reported showing that strong three-dimensionalities develop and that,differently from the axisymmetric case, the flow does not repeat itself after each


Legend

ωmax

ωmin

Piston Cycles

-2000

-1500

-1000

-500

0

500

1000

1500

2000

0 1 2

Figure 7. Time evolution of the azimuthal vorticity extrema for the large-eddy simulation at Re = 2000. The dotted line shows the time law of the pistondisplacement (in arbitrary units).

cycle. This implies that phase averages are needed to compute mean profiles andhigher-order moments, thus requiring the simulation of the flow over several cycles.Additional information obtained by this simulation is that with this grid the DNSat the present Re is already at the resolution limit; therefore, a turbulence modelis needed to simulate higher Re cases.

As stated in Section 2, the experiment performed by Morse et al. (1978) wasat Re = 2000, which is definitely too high to be affordable with DNS. For thisreason a large-eddy simulation with the dynamic subgrid-scale model was carriedout, and some representative results are given in Fig. 6. A computer animation ofthe flow provided considerable information on the in-cylinder dynamics, which isimpossible to summarize with a limited number of snapshots. From Fig. 6, however,it is already evident that the flow degenerates into small scale structures even if theunderlying large-scale flow, already evidenced in the axisymmetric simulations, isstill discernible from the meridional velocity vectors. The motion in the meridionalplanes is coupled with intense azimuthal velocity fluctuations whose magnitude iscomparable with the vertical velocity. As already indicated by the DNS at low Re,this flow is time periodic only in a statistical sense; therefore, the calculation of thestatistics requires phase averages in order to converge. This is confirmed by thetime evolution of the azimuthal vorticity extrema over few cycles given in Fig. 7where the piston motion is shown for reference. It can be noted that, while the lowfrequency dynamics follow the piston motion, the fluctuations are related to thesmall scales, thus reflecting the turbulent character of the flow.

In Fig. 8 is shown the comparison between the numerical and experimental meanand rms radial profiles of axial velocity at the crank angle 36o after TDC with theprofiles 10mm axially spaced starting from the cylinder head. The numerical results


Legend

10mm

−UxV p

<U′2>12

V p

-2

0

2

4

6

8

10

0 5 10 15 20 25 30 35

0

1

2

3

4

0 5 10 15 20 25 30 35

Legend

20mm

−UxV p

<U′2>12

V p

-2

0

2

4

6

8

10

0 5 10 15 20 25 30 35

0

1

2

3

4

0 5 10 15 20 25 30 35

Legend

30mm

−UxV p

<U′2>12

V p

r r

-2

0

2

4

6

8

10

0 5 10 15 20 25 30 35

0

1

2

3

4

0 5 10 15 20 25 30 35

Figure 8. Mean (left) and rms (right) radial profiles of axial velocity at 36o

after TDC for the flow at Re = 2000. numerical simulation; symbols for theexperimental results.

are azimuthally and phase averaged over three piston cycles. While it is clear thatthe rms profiles would certainly benefit from some additional averaging, the meanprofiles already show a good agreement with the experiment. Figure 9 shows thesame profiles as Fig. 8 but for a crank angle of 144o after TDC, and again theresults are very satisfactory. The same quality of agreement is obtained for theother profiles at 144o respectively at 40, 50, 60, and 70mm below the cylinder head,but they are not shown for the sake of conciseness.

As already mentioned the flow has very strong variation from cycle to cycle that


Legend

10mm

−UxV p

<U′2>12

V p

-10

-5

0

5

10

0 5 10 15 20 25 30 35

0

1

2

3

4

5

0 5 10 15 20 25 30 35

Legend

20mm

−UxV p

<U′2>12

V p

-10

-5

0

5

10

0 5 10 15 20 25 30 35

0

1

2

3

4

5

0 5 10 15 20 25 30 35

Legend

30mm

−UxV p

<U′2>12

V p

r r

-10

-5

0

5

10

0 5 10 15 20 25 30 35

0

1

2

3

4

5

0 5 10 15 20 25 30 35

Figure 9. The same as Fig. 8 at 144o after TDC.

could be of some interest to engineers in determining how much the flow deviatesfrom its average properties. In order to show these fluctuations we report in Fig. 10some profiles at 36o ATDC and 10mm below the cylinder head for three differ-ent cycles. We can see that the mean profiles are quite smooth (note that theseprofiles are already azimuthally averaged, the instantaneous azimuthal section isless smooth); the radial shift of the peak is quite small and its intensity constantwithin 20%. The rms profiles, on the other hand, fluctuate more with oscillationsup to 50%; these large fluctuations are likely to be the reason for the discrepanciesobserved in Figs. 8 and 9 between numerical and experimental rms profiles.

Before concluding this section we wish to briefly summarize the results of some


Legend

−UxV p

<U′2>12

V p

r r

-2

0

2

4

6

8

10

0 5 10 15 20 25 30 350

0.51

1.52

2.53

3.54

4.55

0 5 10 15 20 25 30 35

Figure 10. Mean (left) and rms (right) radial profiles of axial velocity at 36o

after TDC for the flow at Re = 2000, 10mm below the cylinder head. cycle1, cycle 2, and cycle 3. The symbols are the experimental results.

Legend

−UxV p

<U′2>12

V p

r r

-4

-2

0

2

4

6

8

10

12

0 5 10 15 20 25 30 350

0.5

1

1.5

2

2.5

3

3.5

4

0 5 10 15 20 25 30 35

Figure 11. mean (left) and rms (right) radial profiles of axial velocity at 36o afterTDC for the flow at Re = 2000, 10mm below the cylinder head. dynamicmodel and fine grid, dynamic model and coarse grid, Smagorinskymodel and coarse grid. The symbols are the experimental results.

further comparisons. In particular, LES of this piston flow has also been carried outon a coarser grid using both the dynamic and the Smagorinsky models. The resultsfor the same quantities as Fig. 10 are shown in Fig. 11. The profiles obtained by thecoarser grid show fairly good agreement with the experiments and the fine-grid case,and the dynamic model behaves consistently with the expectations, i.e. it is moreactive as the grid becomes coarser. The Smagorinsky model (with the constantset equal to 0.2) shows bigger discrepancies with the experiments, yielding a morepeaked jet and reduced velocity fluctuations. This behavior is also consistent withthe more dissipative nature of this turbulence model, which tends to smooth allvelocity gradients independently of their laminar or turbulent character. It shouldbe noted that this result was obtained for one particular value of the constant C,and a tuning of this constant could improve the results. However, Fig. 5 shows that


the flow is very inhomogeneous in space, thus a single value of the constant mightnot be sufficient for the description of the flow dynamics.

5. ConclusionsIn this report we have described an alternative procedure for simulating complex

flow of industrial interest in a very inexpensive way. This technique is based on theuse of body forces which allow the assignment of boundary conditions independentlyof the grid. The main advantage of this procedure is that the above forces can beprescribed on a simple cylindrical mesh so that all the advantages and the efficiencyof solving the Navier-Stokes equations in simple constant-metric coordinates areretained when dealing with complex geometries.

In order to validate this procedure the flow in a motored axisymmetric piston-cylinder assembly has been simulated and the results have shown a very good agree-ment with the experiments. The selected flow was particularly suitable for testingthe numerical procedure since it has complex geometries with moving boundariesand its Reynolds number is high enough to require a turbulence model. These arestandard requirements for industrial flows and a robust numerical procedure mustefficiently cope with all of them.

Results have shown that the dynamic subgrid-scale model is particularly suitable;in fact this model automatically switches off close to the walls without requiring adhoc damping functions (like the van Driest function for the Smagorinsky model).This feature is very useful in the present numerical method since the body surfacein general is not a coordinate line and the computation of the wall distance becomesdifficult and time consuming. In addition, in the presence of moving boundaries thisdistance should be recomputed every time step with a further increase of the CPUtime.

The efficiency of the proposed method can be illustrated by considering that mostof the results were obtained from simulations run on a PC workstation with 128MBof RAM in about a week using a grid of about 6 · 105 gridpoints (about 24 hoursper engine cycle). Comparative results for the same flow were obtained by Haworth& Jansen (1997), using the fully compressible formulation on an unstructured de-forming mesh of about 1.5 · 105 gridpoints running on a single processor Cray T-90,or 8-processor SGI Origin 2000, occupying 600MB of RAM for a time of 30-40 CPUhours per engine cycle.

We are indebted to M. Fatica for countless suggestions and for providing his helpin the numerical simulations and computer animations. We acknowledge W. H.Cabot for the advice in the implementation of the LES model in the code. R. V.wishes to acknowledge the ERO-US Army (European Research Office US Army) fora partial support under contract n. N68171-98-M-5645.

REFERENCES

Eaton, A. R. & Reynolds, W. C. 1987 High-Speed Photography of Smoke-Marked Flow in a Motored Axisymmetric Engine. ASME Paper 87-FE-10.


Eaton, A. R. & Reynolds, W. C. 1989 Flow Structure and Mixing in a MotoredAxisymmetric Engine. SAE Paper 890321.

El Thary, S. H. & Haworth D. C. 1996 A perspective on the state-of-the-artin IC engine combustion modeling. SIAM Sixth International Conference onCombustion, New Orleans.

Fadlun, E. A., Verzicco, R. & Orlandi, P. 1998 Flussi in geometrie complessecon forze di massa su griglie Cartesiane: simulazioni numeriche e validazionesperimentale. Master Thesis Dept. Meccanica & Aeronautica (1998) (in italian).

Germano, M., Piomelli, U., Moin, P. & Cabot, W. H. 1991 A dynamicsubgrid-scale eddy viscosity model. Phys. of Fluids A. 3, 1760-1765.

Goldstein, D., Handler, R. & Sirovich, L. 1993 Modeling a no-slip flowboundary with an external force field. J. Comp. Phys. 105, 354-366.

Haworth, D. C. & Jansen, K. 1997 Large-Eddy Simulation on UnstructuredDeforming Meshes: Towards Reciprocating IC Engines. Submitted to Computerand Fluids.

Lilly, D. K. 1992 A proposed modification of the Germano subgrid-scale closuremethod. Phys. of Fluids A. 4, 633-635.

Mohd-Yusof, J. 1996 Interaction of Massive Particles with Turbulence. Ph. D.Thesis, Cornell University.

Mohd-Yusof, J. 1997 Combined immerse-boundary/B-spline methods for sim-ulations of flows in complex geometries. CTR Annual Research Briefs 1997.NASA Ames/Stanford Univ., 317-327.

Mohd-Yusof, J. & Lumley, J. L. 1994 Simulation of Flow Around CylindersUsing Boundary Forcing. Bulletin of the Am. Phys. Soc. 39(9).

Mohd-Yusof, J. & Lumley, J. L. 1996 Improved Immersed Boundary Tech-niques for Complex Flows. Bulletin of the Am. Phys. Soc. 41(9).

Morse, A. P., Whitelaw, J. H. & Yianneskis, M. 1978 Turbulent flow mea-surement by Laser Doppler Anemometry in a motored reciprocating engine.Report FS/78/24. Imperial College Dept. Mech Eng.

Peskin, C. S. 1972 Flow patterns around hearth valves: a numerical method . J.Comp. Phys. 10, 252-271.

Saiki, E. M. & Biringen, S. 1996 Numerical simulation of a cylinder in uniformflow: application of a virtual boundary method. J. Comp. Phys. 123, 450-465.

Verzicco, R. & Orlandi, P. 1996 A finite-difference scheme for three-dimensionalincompressible flow in cylindrical coordinates. J. Comp. Phys. 123, 402-413.

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