“International Multidimensional Engine Modeling User’s Group Meeting at the SAE Congress”

The Detroit Downtown Courtyard by Marriott Hotel, Detroit, MI (USA), April 23rd, 2012

Boundary Conditions and Subgrid Scale Models for LES

Simulation of Internal Combustion Engines

F. Piscaglia, A. Montorfano, A. Onorati

Politecnico di Milano, Italy

The implementation and the combination of advanced boundary conditions and subgrid scale models for Large EddySimulations are presented. The goal is to perform reliable cold flow LES simulations in complex geometries, such ascylinder engines. In the paper, an inlet boundary condition for synthetic turbulence generation is combined with a fullynon reflecting Navier Stokes Characteristic Boundary Condition (NSCBC) for the outlet and with the local DynamicSmagorinsky subgrid scale model, that is not included in the official distribution of OpenFOAM. Validation of the modelshas been performed separately on two steady state flow benches: a backward facing step geometry [1] and a simpleIC engine geometry with one axed central valve [2]. The code developed has been included into a C++ library formulti-dimensional engine modeling based in the OpenFOAM Technology.

Key words: turbulence, Large Eddy Simulation, wall bounded flow, unstructured mesh, OpenFOAM

INTRODUCTION

LES applied to ICE is potentially a reliable tool to sim-ulate turbulent motion during gas exchange phase (swirl,tumble), fuel mixing and combustion, cyclic combustionvariability. However, there are some known issues thatcannot be neglected: boundary conditions in the LES codemust be able both to properly handle acoustic waves andturbulence properties, solution of Navier Stokes (NS) equa-tions in the unstructured meshes typically used for real-world cylinder heads may lead to have a high numericaldissipation that is comparable to νsgs. Also, when simula-tions of complex geometries such as in internal combustionengines are performed, the behavior of the eddy-viscositymodels near a wall represents a second difficulty [3]. Thispaper is divided into two halves, one theoretical and onecomputational. In the first part, two boundary conditions,for synthetic turbulence generation at the inlet and for anacoustically non-reflecting outlet, are presented. The sec-ond part contains results from simulations, which display awide range of interesting features. The anechoic boundarycondition has been used and tested in a three-dimensionaltime-domain CFD approach, that have been employed topredict and analyze the acoustic attenuation performanceof complex perforated muffler geometries, where strong 3Deffects limit the validity of the use of one-dimensional mod-els. Results from LES simulations using the inlet bound-ary condition and the SGS models implemented have beencompared with experimental data on a backward facingstep geometry [1]. Flow conditions were incompressibleand fully turbulent and a separation region with recircula-tion was present; the influence of the inlet boundary condi-tions and of the subgrid models on the recirculation regioncharacteristics and the overall flow quantities have beenstudied. Finally, to explore the suitability of this method-ology for large-eddy simulation (LES) in reciprocating in-ternal combustion engines, simulations on a simple cylinder

configuration, also studied in [2], were carried out and LESvelocities profiles were compared to LDA measurements.

DEVELOPMENT OF NSCBC IN LOCAL COOR-DINATES

In order to have a fully non-reflecting outlet bound-ary condition, that permits waves to pass through theboundaries without spurious reflections, the formulationof the characteristic treatment for Navier-Stokes equations(NSCBC) is needed. The NSCBC theory by Poinsot andLele [4], originally suggested in Cartesian coordinates, hasbeen extended to local coordinates. The resulting systemof equations has been solved by a multistage time steppingscheme to improve the numerical performance. To trans-form the conservative system from a physical domain (ξ,η, ζ) to a computational domain (x, y, z), a change of vari-ables is required because, as it will be discussed later, thecharacteristic theory to include in the compressible sub-sonic Navier Stokes equations refers to a wave element thatis perpendicular to the outlet domain. Being ui (where

Figure 1: For each cell face at the boundary end, a local ref-erence frame (ξ, η, ζ) has been defined. Each reference framehas its origin in the cell face center and the vector ζ is set asperpendicular to the cell face.

i=ξ, η, ζ), the flow velocity in the local reference frameand

mi = ρ · ui (1)

the momentum density in the i-th direction (still writtenfor the local reference frame), under the assumption of in-viscid flow the system of governing equations in local co-ordinates for a generic boundary end takes the form:

∂ρ

∂t+

1

c2[L2 +

1

2(L5 + L1)] +

∂m2

∂η+∂m3

∂ζ= 0

∂p

∂t+

1

2(L5 + L1) +

∂[(ρE + p)u2]

∂η

∂[(ρE + p)u3]

∂ζ= −∇ · q

∂u1∂t

+1

2ρc(L5 − L1) +

∂m1u2∂η

+∂m1u3∂ζ

= 0

∂u2∂t

+ L3 +∂m2u2∂η

+∂m2u3∂ζ

= −∂p∂η

∂u3∂t

+ L4 +∂m3u2∂η

+∂m3u3∂ζ

= −∂p∂ζ

where Li represents the amplitude variation of the ithcharacteristic wave crossing the boundary [5]:

L =

L1

L2

L3

L4

L5

=

λ1

(∂p∂ξ − ρc

∂u1

∂ξ

)λ2

(c2 ∂ρ∂ξ −

∂p∂ξ

)λ3

∂u2

∂ξ

λ4∂u3

∂ξ

λ5

(∂p∂ξ + ρc∂u1

∂ξ

)

(2)

Li are associated with each characteristic velocity λi:λ1 = u1 − cλ2 = λ3 = λ4 = u1

λ5 = u1 + c

(3)

In particular, in (3), c is the local speed of sound, λ1 andλ5 are the velocity of sound waves moving along the posi-tive and negative stream-wise direction, λ2 is the velocityfor entropy advection, while λ3 and λ4 are the velocity atwhich u2 and u3 are advected in the ξ direction.

Application of the LODI relations into the NSequations (anechoic b.c.)

The implementation of the non reflecting boundary condi-tion to model an anechoic termination requires some cau-tion. The NSCBC approach sets values for the wave am-plitude variations in the viscous multi-dimensional case byexamining a Local One Dimensional Inviscid (LODI) prob-lem [4], where transverse, viscous and reaction terms areneglected:

∂ρ

∂t+

1

c2[L2 +

1

2(L5 + L1)] = 0

∂p

∂t+

1

2(L5 + L1) = 0

∂u1∂t

+1

2ρc(L5 − L1) = 0

∂u2∂t

+ L3 = 0

∂u3∂t

+ L4 = 0

(4)

For subsonic flows at the exit, the eigenvalue λ1 = u1−cin (3) associated to the wave amplitude L1 is negative andthe disturbance travels inward the boundary at the speedof sound relative to the moving fluid. The application ofthe LODI relations into the Navier Stokes equations, thatdoes not necessarily imply a hyperbolic nature of the sys-tem, is questionable. In particular, if the ingoing charac-teristic is set to zero (perfect anechoic condition), the prob-lem results to be ill-posed and the information about thestatic pressure in the outer domain never feeds back intothe computational domain [6, 7, 8]. For a well-posed prob-lem [6], corrections must be added to the treatment of thenon-reflecting NSCBC, that becomes therefore partially re-flective. It is important to note that partial reflection alsoincludes the contributions of spurious reflections causedby the numerical approximation of hyperbolic equationsat the boundary [9]; this effect looks stronger if inviscidboundary conditions for the primitive variables are speci-fied and if transverse terms in the equations are neglected.The acoustic theory proposed to characterize the actualreflection coefficient of numerical ”non-reflecting” bound-ary condition using LRM (Linear Relaxation Method) pro-posed by Rudy and Strikwerda [6] has been applied toderive a formulation of the reflected characteristic in gen-eralized coordinates [10]:

L1 = K · (p− p∞) = σ · |1−M2|√2Jρl

(p− p∞) (5)

that has been included in Eq. (4). In Eq. (5), M isthe local Mach number of the boundary cell, l is the char-acteristic length of the domain. The preferred range forconstant σ is in the interval [0.1; π]. The difference L5-L1 in Eq. (4) can be still calculated by one side upwinddifferencing.

Validation

The performance of non-reflecting boundary conditions candepend on a number of different factors [11]: the cut-offratio, the ratio between frequency and wavelength (gassound velocity), the angle of the incident waves relative tothe boundary normal, the mean flow velocity, the damp-ing constant σ of the LRM. The effect of these parameterson the performance of the boundary condition was inves-tigated, for a fixed value of the damping constant σ in (5),that was set to 0.2. Tests were performed on the shocktube geometry shown Fig. 2: the gas modeled was airat ambient temperature (293 K), the pressure differencep1 − p2 between air at high and low pressure was 650 Pa,a non-reflecting boundary conditions was set at the openend of the duct. The simulation time was set accuratelyto allow the compression wave front to completely crossthe boundary and to avoid any interference between wavesreflected by the other boundaries.

For a fixed value of K in Eq. (5) (which is the case inany computation), different frequencies of the signal arenot reflected in the same way: usually, waves at high fre-quency will easily leave the computational domain (R→ 0)whereas low frequency waves will be strongly reflected.

2

Figure 2: Schematic representation on the XY plane of the 3Dshock-tube geometry. Initial pressure conditions: p1=100650Pa, p2=100000 Pa.

0 200 400 600 800 1000 1200 1400 1600 1800 2000

10−4

10−3

10−2

10−1

100

f [Hz]

|R|

α = 0 deg

α = 15 deg

α = 30 deg

α = 45 deg

α = 60 deg

Figure 3: Magnitude R of the numerical reflection coefficientwith varying wave angles.

The frequency that splits the different behavior of theboundary condition is the cut-off frequency (fc), that sep-arates reflected waves (f < fc) by waves that will leave thedomain (f > fc). In terms of energy content, half of theacoustic energy is fed back into the computational domainat the frequency fc. A reliable parameter to evaluate theperformance of a non-reflecting boundary condition is thereflection coefficient R, that represents the permeability ofthe boundary towards low frequency waves and that canbe calculated according to the characteristic theory as:

R(f, α) =L1(f, α)

L5(f, α)(6)

The behavior of the reflection coefficient has been stud-ied in a range of the incident pressure wave angle α inbetween 0 and 60 degrees. In Fig. 3, the amplitude of thewave reflection coefficient |R| of Eq. (6) as a function ofthe frequency is analyzed. As expected, R is higher at lowfrequency and constantly decreases in the low frequencyrange [0 600] Hz. Differences in magnitude for the reflec-tion coefficient R do not seem appreciable with differentwave angles α for frequencies lower than 600 Hz. Beyondthis value, the reflection coefficient generally increases withthe wave angle. An exception has been found with a waveangle α = 60◦. In this case, results look very similar tothe case with a wave having normal incidence; the expla-nation of this unexpected behavior is still under investi-gation. From the frequency of about 600 Hz, the slopeof R becomes positive. The boundary condition has beenapplied to model an anechoic termination, when simula-tions of acoustic silencers for internal combustion enginesapplications were carried out. The Transmission Loss fordifferent configurations of silencers have been calculatedand some results are depicted in Fig. 4. A more detailed

Figure 4: Transmission loss of different flow reverse chamberswith zero mean flow [12]. Experimental data have been pro-vided by AVL GmbH.

description and an extensive validation of the approach isreported in [12].

SYNTHETIC TURBULENCE INLET B.C.

One of the possible way to have realistic turbulence atthe inflow boundary of a LES domain is to superpose ran-dom fluctuations onto a mean velocity profile. However,care must be taken to meet some physical requirements,that would ensure turbulence is not destroyed by theNavier-Stokes solver and that it possess some properties of“real” turbulence. Therefore, synthetic fluctuations mustbe divergence-free, and they must have a specified spec-tral energy content together with time- and space- corre-lation. This can be achieved by generating the fluctuatingcomponent as a sum of Fourier modes with wavenumbersκ1 . . . κN . Following the procedure developed by Davidson[13], the fluctuating component of velocity is calculated as:

u′′i (xj) = 2

N∑n=1

un cos(κnj xj + ψn

)σni (7)

Orientation in space of wavenumber vector κj (as well asthe phase angle ψn) is chosen randomly according to apredefined statistical distribution.

Solenoidality is enforced by generating velocity vectorsσi that lie on a plane orthogonal to κj . Spectral mode

3

Figure 5: Instantaneous velocity field at T = 2Tc. (a) fixedprofile; (b) plane mapping; (c) synthetic turbulence [14].

amplitude un is calculated according to a prescribed spec-trum shape. In this work, a modified version of the VonKarman energy spectrum is used:

E(κ) = Au2rmsκe

(κ/κe)4

[1 + (κ/κe)2]17/6

exp[−2(κ/κη)2

](8)

where:

- κη = ε1/4ν−3/4 is the maximum wavenumber, corre-sponding to Kolmogorov lengthscale

- A = 1.456 is a model’s constant

- κe = 9π/55 A/L is a function of the integral length-scale (L).

Time correlation is obtained by applying Billson tempo-ral filter [15] to the ‘raw’ fluctuations given by (7):

(u′)m

= a (u′)m−1

+ b (u′′)m

(9)

where (T is the integral timescale):

a = exp (−∆t/T ), b =√

1− a2

The algorithm for synthetic turbulence generation is ap-plied at each temporal integration step. The additionalcomputational cost implied by the b.c. for synthetic tur-bulence generation has been estimated about 10% of thetotal simulation time.

RESULTS

To explore the suitability of this methodology for large-eddy simulation (LES) in reciprocating internal com-bustion engines, the local dynamic Smagorinsky model[16] subgrid-scale model has been implemented in theOpenFOAM® technology. Results coming from LES sim-ulations performed by using the implemented SGS modeland implicit LES have been compared to experiments fortwo steady state flow benches: a backward facing step ge-ometry [1] and a simple IC engine geometry with one axedcentral valve [2].

Backward facing step

The configuration with a sudden expansion, the backwardfacing step (BFS), has been simulated to reproduce

the first difficulty occurring in IC engines: the strongdetachment due to the increase of the diameter betweenthe intake ports and the cylinder. Figure 6 shows thedomain used by Eaton et. al [1] in the experiments. Theturbulent boundary layer was generated by positioningtwo trips: the former was located on the step wall, thelatter was located on the wall opposite to the step. Bothtrips were placed at the same distance from the stepbut their heights were different, so the boundary layerthickness was different on the step wall and on the wallopposite to the step. The results obtained with LES formean velocity profiles, as well as for velocity fluctuations,have been quantitatively compared to the experimentalLDA measurements [1] in terms of both mean and RMSvelocities.

Figure 6: Schematic of the backward facing step geometry [1]simulated; L1=0.4953 m, L2= , W1=0.0762 m, W2=0.127 m,W3=0.015 m, H=0.0508 m.

In the computations object of this paper, only a smallportion of the domain of the BFS was modeled; the inletchannel length L1 was chosen to have fully developed flownear the step, while the outlet channel length L2 was sup-posed to be long enough to not be affected by the smallnumerical reflections coming by the outlet boundary condi-tion. The fluid-dynamic information recorded by a RANSsimulation on a precursor domain has been used to recon-struct the turbulent fluctuations at the boundary inlet bythe synthetic turbulence inlet b.c. described above. Anunsteady convective boundary condition has been used forthe outlet. Periodic boundary conditions were set on the z-normal surfaces, while all the other surfaces of the domainwere set as adiabatic walls.

Simulations were carried out on a fully structured meshhaving about 800000 hexahedral cells both by implicit SGSmodels (where the dissipation is given by discretization er-rors of monotone convection algorithms) and by an explicitalgorithm, the local dynamic Smagorinsky SGS model.Mean inlet velocity of the flow was 11.6 m/s (Re' 50000)and the values of grid resolution were ∆x+=200, ∆y+=0.8,∆z+=20, where x, y, z denote the streamwise, wall-normaland spanwise directions. No wall functions were used, theboundary layer was directly resolved at the walls. Despitegrid resolution looked consistent with the guidelines fromthe literature on wall bounded flows [17], by Fig. 7 it is ap-parent how the mean velocity and the turbulent intensitywere not properly captured, in particular in the vicinity ofthe walls; this has been found also on test performed onfiner grids (up to 4M cells). Preliminary results shown hereon the BFS geometry evidenced the need to implement asubgrid viscosity model that could improve the quality ofthe predictions on the cases shown in the paper.

4

0 0.6 1.20

0.025

0.05

0.075

U/Uin

[−]

y [

m]

x/H = −9.75

0 0.2 0.40

0.025

0.05

0.075

u’/Uin

[−]

y [

m]

0 0.6 1.20

0.025

0.05

0.075

U/Uin

[−]

x/H = −3

0 0.2 0.40

0.025

0.05

0.075

u’/Uin

[−]

0 0.6 1.20

0.025

0.05

0.075

U/Uin

[−]

x/H = −0.75

0 0.2 0.40

0.025

0.05

0.075

u’/Uin

[−]

−0.4 0 1.2−0.05

−0.025

0

0.025

0.05

0.075

U/Uin

[−]

x/H = 4

0 0.4−0.05

−0.025

0

0.025

0.05

0.075

u’/Uin

[−]

−0.4 0 1.2−0.05

−0.025

0

0.025

0.05

0.075

U/Uin

[−]

x/H = 6

0 0.4−0.05

−0.025

0

0.025

0.05

0.075

u’/Uin

[−]

−0.4 0 1.2−0.05

−0.025

0

0.025

0.05

0.075

U/Uin

[−]

x/H = 8

0 0.4−0.05

−0.025

0

0.025

0.05

0.075

u’/Uin

[−]

0 1.2−0.05

−0.025

0

0.025

0.05

0.075

U/Uin

[−]

x/H = 12

0 0.4−0.05

−0.025

0

0.025

0.05

0.075

u’/Uin

[−]

0 1.2−0.05

−0.025

0

0.025

0.05

0.075

U/Uin

[−]

x/H = 16

0 0.4−0.05

−0.025

0

0.025

0.05

0.075

u’/Uin

[−]

Experim. Implicit LES local dynSmag

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(l)

(m)

(n)

(o)

(p)

(q)

(r)

Figure 7: Backward facing step geometry: comparison of the velocity profiles between the LES simulation and PIV measurements[1] at different locations along the streamwise direction; a-h) mean axial velocity i-r) mean axial velocity fluctuations.

Cold flow IC simulation

A simple IC engine geometry (consisting of a sudden ex-pansion with ratio 3.5 and one axis-centered valve), re-ported in Fig. 8, has been also tested. Measurementsof mean flow velocity and of velocity fluctuations (alongthe radial and tangential directions) were available on twoplanes located at a distance of 20 mm and 70 mm from thecylinder top, respectively. Similar studies on this configu-ration have already been done in [2].

20 mm70 mm

plane 1 plane 2 Ld = 300 mm

Lu = 104 mm

Di = 16 mm

De = 34 mm

D = 120 mm

h = 10 mm

Ds = 27.6 mm

Ls = 4.24 mm

Ld

D

De

Di

Ds

Ls

h

Figure 8: Two-dimensional schematization for the axial sym-metric piston-cylinder assembly of [2], including coordinate sys-tem.

In this work, two computational grids were used for thesame geometry: the former was made of 1.4 million ofcells, the latter had about 8 million of cells. Mean veloc-ity at the inlet was 65 m/s and no explicit subgrid modelhas been applied (implicit LES); simulations with explicitSGS modeling are running at the time this paper is written.Comparison of the current LES with LDA measurementsare shown in Fig. 9. Quantities of graphs a-c are referredto the upper plane (20 mm below the cylinder top), whileplots d-f are referred to the lower plane (70 mm below thecylinder top). Results have been averaged both in time,after a steady-state has been reached, and in space, alongthe circumferential direction. Mean velocities are captured

rather well both by the coarse and the fine mesh in bothplanes. On the other hand, RMS of the fluctuating compo-nent of the velocity exhibits a significant deviation from themeasured values, especially near the cylinder walls. Thepoor predictions in the wall region are most likely due tothe insufficient resolution of the grid to solve the near-wallturbulence. Although it is difficult to study in detail thedynamics of the near-wall turbulence for such a complexcase (no reference test-cases exist), it might be useful toevaluate the scaled mesh resolution at the wall as an in-dex of spatial resolution. In the simulation using the 1.4M grid, the scaled mesh sizes are: ∆x+tg ≈ 200, n+ ≈ 10and ∆x+ax ≈ 500, while in the 8 M grid the resolution is∆x+tg ≈ 100, n+ ≈ 5 and ∆x+ax ≈ 500, being n the wall-normal (radial) direction, xtg the tangential coordinate andxax the axial coordinate. These high values indicate thatthe dissipation at the wall is not correctly evaluated, andthis has a strong influence on the predictions of the fluc-tuating velocity components.

CONCLUSIONS

In this paper, LES simulations of a simple internalcombustion engine geometry have been performed. First,boundary conditions for LES have been implemented andvalidated. Quantitative comparisons have been made be-tween the experimental and numerical velocity profilesfor different mesh sizes of the geometries tested. Turbu-lent fluctuations are not properly captured near the walls,where the turbulent viscosity looks underestimated. Thiserror, that is more apparent with implicit LES, is stillpresent with a refined mesh, because the grid resolutionused was not high enough to directly resolve the bound-ary layer at the walls. On the other hand, proper grid

5

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

r/R [−]

U/U

in [

−]

−1 −0.5 0 0.5 10

0.1

0.2

0.3

0.4

0.5

r/R [−]

uP

rim

e/U

in [

−]

−1 −0.5 0 0.5 10

0.1

0.2

0.3

0.4

0.5

r/R [−]

wP

rim

e/U

in [

−]

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

r/R [−]

U/U

in [

−]

−1 −0.5 0 0.5 10

0.1

0.2

0.3

0.4

0.5

r/R [−]

uP

rim

e/U

in [

−]

−1 −0.5 0 0.5 10

0.1

0.2

0.3

0.4

0.5

r/R [−]

wP

rim

e/U

in [

−]

Experim.

1.4M cells

8M cells

(a) (b) (c)

(d) (e) (f)

Figure 9: Comparison of the velocity profiles between the LES simulation and the LDA measurements. Plane x=20 mm: a) meanaxial velocity b) mean axial velocity fluctuations c) mean tangential velocity fluctuations. Plane x=70 mm: c) mean axial velocityd) mean axial velocity fluctuations e) mean tangential velocity fluctuations.

Figure 10: Iso-surface of an instantaneous snapshot of the ve-locity flow field. Calculation by implicit LES of the axial sym-metric piston-cylinder assembly [2]. Mesh size: 1.4 million cells.

resolution would lead to grids having a number of cellsthat would be too high for the available computational re-sources. To have a better prediction of the scaling at thewall, the dynamic Smagorinsky subgrid model has beenimplemented into the OpenFOAM Technology. Results onthe BFS geometry show an improved agreement when thesubgrid model is used, despite several tasks are still open;in particular, it is not really clear what is the minimummesh refinement needed to conclude which kind of meshis more appropriate for LES computation of internal com-

bustion engine geometries. This topic is currently underinvestigation.

Recently, a Wall-Adapting Local Eddy-viscosity(WALE) subgrid scale model has been implemented in theOpenFOAM Technology. The WALE model is based onthe square of the velocity gradient tensor and it accountsfor the effects of both the strain and the rotation rate ofthe smallest resolved turbulent fluctuations and it recoversthe proper y3 near-wall scaling for the eddy viscositywithout requiring dynamic pressure; hence, it is supposedto be a very reliable model for ICE simulation. Currentresearch activity is involving the testing and the validationof this model on engine test cases.

ACKNOWLEDGMENTS

Simulations results shown in the paper were carriedout on the computational resources of CINECA (Bologna,Italy) through the Italian SuperComputing Resource Allo-cation (ISCRA) framework, project LES4ICE.

CONTACT

Prof. Federico Piscaglia, Ph.D.

Politecnico di Milano

ph. (+39) 02 2399 8620

e-mail: federico.piscaglia@polimi.it

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