Special Issue Article

International J of Engine Research2017, Vol. 18(1-2) 15–25� IMechE 2017Reprints and permissions:sagepub.co.uk/journalsPermissions.navDOI: 10.1177/1468087416686699journals.sagepub.com/home/jer

Non-equilibrium wall-modeling forinternal combustion enginesimulations with wall heat transfer

Peter C Ma1, Mark Greene2, Volker Sick2 and Matthias Ihme1

AbstractHeat transfer affects the performance and phasing of internal combustion engines. Correlations and equilibrium wall-function models are typically employed in engine simulations to predict heat transfer. However, many studies have shownthat significant errors are expected, owing to the failure of fundamental assumptions in deriving equilibrium wall-functionmodels. Non-equilibrium wall models provide a more accurate way of describing the near-wall region of in-cylinder flows.In this study, simultaneous high-speed high-resolution particle image velocimetry and heat-flux measurements are con-ducted in an optically accessible engine. The experiments are performed under both motored and fired conditions at twodifferent engine speeds. The experimental data are utilized to assess the performance of different models in predicting thethermoviscous boundary layer. These models include commonly used heat transfer correlations, equilibrium and modifiedwall-function models, and a recently developed non-equilibrium wall model. It is found that the equilibrium wall-functionmodel significantly underpredicts the heat flux under both motored and fired conditions. By considering heat releaseeffects in the boundary layer, the non-equilibrium wall model is shown to be able to adequately capture the structure anddynamics of both momentum and thermal boundary layers in comparison with experimental measurements, demonstrat-ing its improved performance over previously employed correlation functions and the equilibrium model.

KeywordsInternal combustion engines, non-equilibrium wall model, particle image velocimetry, heat transfer, boundary layer

Date received: 6 October 2016; accepted: 23 October 2016

Introduction

Convective heat transfer plays an important role in theperformance of internal combustion (IC) engines.1 Itwas also observed that heat transfer can influence com-bustion phasing in homogeneous charge compressionignition (HCCI) engines.2 Therefore, significant prog-ress has been made to enable the quantitative predic-tion of heat transfer, as summarized by Borman andNishiwaki3 and Heywood.4 Most of the proposed cor-relations that have been developed over the past eightyyears5–9 provide spatially averaged heat transfer coeffi-cients and, as such, lack a detailed description of localconvective heat transfer. The spatially averaged corre-lations are often experimentally based on heat-flux andwall-temperature measurements at a single point in thecombustion chamber. However, experiments utilizingmore than one heat-flux probe at different locations inthe combustion chamber show that wall temperature,and therefore heat flux, varies spatially at any instantduring the cycle.2

Multidimensional numerical simulations offeropportunities for a quantitative prediction of fluid flowand heat transfer processes in IC engines. However, theunsteady and turbulent nature of in-cylinder flowsmakes modeling of the turbulent closures in the trans-port equations a challenging task. Borman andNishiwaki3 concluded that the fundamental problem inmodeling is the lack of detailed data regarding the gas-side velocity and temperature distribution, and vitalquestions concerning boundary-layer models to beexplored experimentally. A more detailed analysis andtreatment of heat transfer properties in experimentsand simulations is therefore needed.

1Department of Mechanical Engineering, Stanford University, USA2Department of Mechanical Engineering, University of Michigan, USA

Corresponding author:

Matthias Ihme, 488 Escondido Mall, Building 500 Room 500A, Stanford,

CA 94305, USA.

Email: mihme@stanford.edu

Lucht et al.10 utilized single-point coherent anti-Stokes Raman scattering (CARS) in an engine to char-acterize the temperature profile in the vicinity of thetoroidal cylinder head. The curved head allowed mea-surements as close as 25 mm from the surface and datawere presented for different operating conditions andcrank angles. The use of thermocouples to measureheat flux on cylinder walls is widely established. Theinstantaneous heat flux at the surface of a cylinder headin a motored diesel engine was measured by Lawton8

using a fast-response surface thermocouple. Heat fluxfrom the cylinder wall to the flow was observed evenwhen the bulk temperature of the flow was higher thanthe wall temperature. Nijeweme et al.11 conductedinstantaneous heat-flux measurements using 12 fast-response thermocouples that were installed in the com-bustion chamber of a pent-roof, spark-ignition (SI)engine, and measurements were obtained for motoredand fired conditions at different engine speeds, throttlesettings, and ignition timings. Phosphor thermometryusing a high-speed laser for in-cylinder surface tem-perature measurements was conducted by Fuhrmann etal.12 at the spark-plug dummy inside an optically acces-sible IC engine. Later, Fuhrmann et al.13 obtained two-dimensional surface-temperature measurements undermotored and fired conditions in a full-metal engineusing thermographic phosphors.

The tumbling and swirling flows in the cylinder havea significant influence on the engine performance,14 andthe momentum boundary layer controls the near-wallbehavior of the flow and turbulence characteristics. Thefirst velocity measurements recorded in the boundarylayer of an engine were conducted by Hall and Bracco15

using laser Doppler velocimetry (LDV). Measurementswere made at a location as close as 500 mm from thecylinder wall in a non-fired and a skip-fired SI engineoperating at 1200 revolutions per minute (RPM).Foster and Witze16 carried out LDV measurementsnear a toroidally contoured engine head at motoredand skip-fired engine conditions. The apex at half thecylinder radius enabled LDV measurement as close as60 mm from the wall. Boundary-layer thicknesses werefound to be less than 200 mm for high swirl flow and700 mm–1000 mm for low swirl. The boundary-layerthickness increased for both swirl levels when the enginewas fired. Pierce et al.17 investigated the near-wall velo-city for five different engine geometries. Both LDV andmulti-exposure particle image visualizations were usedin two- and four-stroke engines. Measurements up towithin 50 mm of the cylinder-head wall were achievedfor custom geometries. For more typical flush-cylinderwalls, a wall resolution of 400 mm was achieved. Ahigh-speed particle image velocimetry (PIV) techniquefor in-cylinder boundary-layer flow measurements wasdemonstrated by Alharbi and Sick.18 The evolution ofthe boundary-layer flow at the cylinder head of amotored four-valve engine during compression andexpansion phases was examined. By extending this tech-nique, Jainski et al.19 made PIV measurements at three

different engine speeds, and a wall resolution of lessthan 150 mm was achieved.

Reliable numerical simulations of IC engines rely onthe accurate characterization of the boundary-layerstructure, as this directly affects turbulence generationand wall heat transfer. Since the thickness of thisboundary layer is on the sub-millimeter scale, substan-tial computational resources are required to resolve theboundary-layer structure computationally. To addressthis issue, different modeling approaches have beenproposed. Most common are one-dimensional20,21

and multidimensional22–24 models. Multidimensionalapproaches, such as Reynolds-averaged Navier–Stokes(RANS) and large-eddy simulation (LES) techniques,solve the governing equations for mass, momentum,energy, and species conservation in the in-cylinder coreregion, together with an appropriate turbulence closuremodel. For the characterization of the near-wall region,so-called wall-function models are widely employed.These models are of practical interest, owing to theirsimplicity and computational efficiency for engineeringcomputations. In this approach, the near-wall region isnot resolved and the effect on heat transfer and flowstructure is modeled using a self-similar boundary-layerprofile. However, several modeling assumptions areinvoked to derive analytical expressions, and it wasshown in previous studies that the equilibrium assump-tions are not adequate to model the dynamic evolutionsof the viscous boundary layer in IC engines.25 Wall-function models were found to underpredict the heatflux substantially, compared with experimental mea-surements, and efforts were taken to improve theperformance.24,26,27

Zonal approaches,28 such as detached eddy simula-tion (DES) and wall-modeled LES (WMLES) methodsprovide alternatives to wall-function models withhigher predictive fidelity. In the WMLES approach, amesh with high resolution is embedded between thematching location and the wall. The RANS equationsare solved in the embedded mesh with no-slip boundaryconditions applied at the wall, and the wall shear stressand heat flux are fed back to the outer LES as wallboundary conditions. For DES, different turbulencemodels from RANS and LES are applied for near-walland engine-core regions separated by an appropriatematching location. Hasse et al.29 applied a DES modelto study cycle-to-cycle variations, and compared theensemble-averaged and instantaneous velocity fieldswith optical measurements. A DES approach was alsoused for simulations of a motored diesel engine byBottone et al.30 and it was found that, for most cases, agood agreement of results for pressure, swirl number,and tumble number was observed from the DES andthe LES solutions with the mean RANS profiles inmulticycle simulations. Ma et al.25 utilized high-resolution PIV measurements to evaluate the perfor-mance of non-equilibrium wall models in a pent-roofSI engine under motored conditions, and concludedthat non-equilibrium effects due to pressure gradients

16 International J of Engine Research 18(1-2)

during compression require consideration of the accu-rate representation of in-cylinder boundary layers.

The objective of this study is to conduct simulta-neous high-speed high-resolution PIV and wall heat-flux measurements at the cylinder head and study thenear-wall region of IC engines. The performance of dif-ferent wall models and heat-flux correlations for theprediction of the viscous and thermal boundary layersin IC engines is examined through comparisons withexperimental measurements. The non-equilibrium wallmodel formulated by Ma et al.25 is extended to includethe heat-release term for the modeling of chemical reac-tions under fired conditions. The focus of this study isto evaluate the capabilities of different models, includ-ing heat transfer correlations, equilibrium wall-functionmodels, and the non-equilibrium wall model, in pre-dicting the wall-heat transfer at the cylinder head forboth motored and fired cases.

The remainder of this paper has the following struc-ture. The next section describes the optical engine and theexperimental setup for the wall heat-flux and PIV mea-surements. Models with different fidelities for the predic-tion of the near-wall region in IC engines are thenpresented. Results for the viscous and thermal boundarylayers follow. The paper finishes with conclusions.

Experimental setup

Measurements of surface temperature, heat-flux, andvelocity fields were conducted at the engine head in thetransparent combustion chamber III (TCC-III) engine,which is described in detail by Schiffmann et al.31 TheTCC-III engine was designed with a simplified architec-ture to facilitate analysis and modeling, while maximiz-ing optical access for simultaneous PIV measurements.The TCC-III engine contains a flat piston surface, a flathead, two symmetrical valves, a centrally located sparkplug, simplified port and runner geometries, and fea-tures a 92 mm bore full quartz cylinder liner and a pis-ton window to allow optical access to the pancake-shaped combustion chamber. A schematic of the TCC-III engine is provided in Figure 1, where yellow compo-nents represent quartz windows. The engine contains aBowditch-style piston extension that provides for opti-cal access into the engine cylinder through a window inthe center of the piston. Intake air properties are accu-rately controlled with sonic orifices and intake systemheaters. The engine cooling water and intake air tem-perature as measured at the intake port were main-tained at 80�C. Boundary conditions in the TCC-IIIengine are documented by five pressure transducerslocated in the intake and exhaust ports, the entrance tothe intake plenum, the exit of the exhaust plenum, andwithin the cylinder. The phase notation used through-out this paper is crank angle degrees after top dead cen-ter exhaust (CAD aTDCe) with top dead centercompression at 360 CAD. The engine specifications aregiven in Table 1. Surface temperature and heat-flux

measurements were conducted under motored and firedengine operation at 500 RPM and 1300 RPM. For thefired experiments, the engine was operated with homo-geneous stoichiometric propane–air mixtures with aspark timing of 342 CAD.

Surface temperature and heat flux were measuredwith a microsecond response Medtherm TCS-244-JU(JU-.156)-72-11340 heat transfer probe, positioned35.5 mm from the cylinder axis, as shown in Figure 2.One J-type thermocouple was set at the probe surface,with a second J-type thermocouple set 3.96 mm into thedepth of the probe material. Thermocouple voltageswere sampled using RC Electronics DTX-520CJ cold-junction module assemblies including electronic cold-junction temperature compensation. Thermocouplesignals were amplified and converted to a 0 V–10 V sig-nal linearized to temperature by an RC Electronics

Figure 1. Transparent combustion chamber III (TCC-III) engineand cylinder geometry. Yellow components represent quartzwindows.

Figure 2. Location of heat transfer probe on cylinder head.Yellow is quartz.

Ma et al. 17

DTX-5120 thermocouple conditioner mounted in aDTX-5017 rackmount chassis. The 0 V–10 V linear ther-mocouple signals were recorded by the crank-anglebased A&D Phoenix AM system, which also recordedthe high-speed engine pressure measurements. Surfaceand in-depth temperatures were recorded every 0.5CAD throughout the cycle. The Medtherm heat-fluxprobe was installed vertically in a spare spark-plug holeand mounted flush with the TCC-III engine head. Thelocation of the probe within the cylinder is shown inFigure 2.

The instantaneous heat flux was calculated from thesurface and in-depth temperatures following the proce-dure set forth by Nijeweme et al.11

qw = lmTs � Tid

l

+ lm

XNn=1

ffiffiffiffiffiffinvp

2am(An +Bn)cos(nvt)

+ lm

XNn=1

ffiffiffiffiffiffinvp

2am(Bn � An)sin(nvt)

ð1Þ

where qw is the instantaneous heat flux calculated atthe surface, l is the depth of the reference temperaturemeasurement. lm and am are the material thermal con-ductivity and thermal diffusivity of the probe material,Ts and Tid are the cycle-averaged surface and in-depthtemperatures, v is the angular velocity, and An and Bn

are the coefficients of the Fourier series expansion of thesurface temperature trace. The uncertainties of the heat-flux measurements, evaluated from the standard error,are about 5% and less than 1% near top dead center(TDC) for motored and fired conditions, respectively.

The PIV measurements were conducted in a43 6 mm2 field of view adjacent to the cylinder head atthe location depicted in Figure 2. The beams of twoQuantronix Hawk II lasers were combined with aBrewster plate before being formed into a sheet by abeam homogenizer and cylindrical lens. The light sheetthickness measured about 0.5 mm at the PIV interroga-tion volume. Seed was generated by a TSI 9306 six-jetatomizer using Dow Corning 510 50 cSt silicone oil asthe seed oil, with a nominal particle diameter of 1 mm.The seed–air mixture from the atomizer was combinedwith the remaining engine intake air upstream of theintake plenum.

Images were captured using a Vision Research v7.3Phantom camera fitted with a 200 mm Nikon NikkorMacro lens and extension bellows. The camera randouble frame images at 3 kHz, capturing PIV imagepairs as frequently as 1 CAD at 500 RPM. A –4 m focallength cylindrical meniscus lens was placed between theengine cylinder and camera lens to optically correct thedistortions generated by viewing through the quartzcylinder in the direction shown in Figure 1. Timing ofthe lasers and the camera acquisition was controlled bya LaVision High-Speed Controller with laser pulsedelays from 5 ms–50 ms dependent upon engine phase.

The image processing for PIV was conducted usingLaVision’s DaVis 8.2.3.3902 software to achieve vectorgrids with a spatial resolution of 200 mm. On average, thestatistical uncertainty of the ensemble-averaged mean velo-city, evaluated from the standard error, is less than 10%.

Mathematical models

Models studied for predicting the thermoviscousboundary layers in IC engines with different levels offidelity are presented in this section. For all models, thechamber properties are evaluated based on isentropiccompression and expansion processes under motoredconditions, and a GT-POWER simulation computed inGTise v7.4.0 is used for the evaluation of chamberpressure and temperature under fired conditions. Wall-related quantities are evaluated based on the wall tem-perature, which is assumed to be equal to the coolingwater temperature.

Heat transfer correlations

Empirical correlations for the prediction of instanta-neous convective heat transfer in IC engines have beenreported in the literature, and are usually derived basedon steady-state boundary-layer flow assumptions. Areview of different types of heat transfer correlations isgiven by Borman and Nishiwaki.3 Two commonly usedempirical correlations for heat transfer will be consid-ered in this study.

One of the most widely used correlations isAnnand’s model5, which can be expressed as

h= c1lB�1Re0:7 + c2

T4 � T4w

T� Twð2Þ

where h is the convective heat transfer coefficient, l isthe heat conductivity, B is the bore diameter, Re is theReynolds number, T is the volume-averaged in-cylindergas temperature, and Tw is the wall temperature.Standard SI units are used. The Reynolds number canbe evaluated from the mean piston speed and the

Table 1. Specifications of the transparent combustion chamberIII (TCC-III) engine.

Parameter Value

Bore 92 mmStroke 86 mmConnecting rod 231 mmDisplacement 570 cm3

Clearance volume 64 cm3

Compression ratio 10:1Intake valve open 712 CADIntake valve close 240 CADExhaust valve open 484 CADExhaust valve close 12 CADIntake pressure 40 kPaIntake temperature 808C

CAD: crank angle degrees.

18 International J of Engine Research 18(1-2)

cylinder bore, and the coefficients c1 =0:35 andc2 =4:33 10�9W/(m2K4) are model constants for con-ductive and radiative heat transfer, respectively.

The commonly used Woschni model6 can beexpressed as

h=3:26B�0:2p0:8T�0:53U0:8 ð3Þ

in which standard SI units are used, except kilopascals(kPa) for pressure. U is the characteristic gas speed,which is defined as6

U= j1Sp + j2VTr

prVr(p� pm) ð4Þ

for compression and combustion strokes, where Sp isthe mean piston speed, subscript ‘‘r’’ refers to a refer-ence state, such as the end of the intake stroke, and pmis the pressure under motored conditions. Note thatpr, pm, and p are required to take the same units toensure dimensional consistency of equation (4). Themodel coefficients are taken as j1 =6:18 andj2 =3:243 10�3m/(s K).

For these empirical heat transfer correlations, thewall heat flux, qw, can then be evaluated from the defi-nition of the heat transfer coefficient h= qw=(T� Tw).

Wall-function models

The equilibrium wall-function model, commonlyemployed to describe the near-wall boundary layer in IC-engine simulations, was derived by invoking several assump-tions to obtain a closed-form analytic expression.22,23,25

Detailed derivations of the near-wall region for flat-plateboundary layers and related steady-state quasi-one-dimensional flows can be found in standard textbooks.32

The boundary-layer structure can be normalized bya viscous scale, which is defined through the shear velo-city ut =

ffiffiffiffiffiffiffiffiffiffiffiffiffitw=rw

pand the wall shear stress

tw =mw

∂�u

∂y

����w

ð5Þ

where �u is the wall-parallel velocity component, theoverbar denotes ensemble-averaged quantities, y is thewall distance, and rw and mw are the density anddynamic viscosity at the wall, respectively. The velocityand wall distance can be expressed in terms of an innervelocity u+ = �u=ut and inner wall coordinate,y+ = y=dn, respectively, where dn = nw=ut is the vis-cous length scale and nw =mw=rw is the kinematic visc-osity at the wall.

With density and transport quantities inside theboundary layer assumed to be constant, the momentumand temperature equations can be decoupled into twoseparate ordinary differential equations. The momen-tum equation can then be integrated analytically usingthe mixing-length closure in its asymptotic limits of theviscous sublayer and the log layer; along with the buf-fer layer, they are referred to as the law-of-the-wall.32

The equilibrium wall-function model is described bythe analytic form22,23

u+ =y+ if y+ \ 111kln(y+)+B if y+511

�ð6Þ

where k=0:41 is the von-Karman constant andB=5:2 is the log-law constant. In analogy to themomentum boundary layer, a thermal law-of-the-wallcan be written as33

T+ =Pry+ if y+ \ 11Prt

1klny+ +B

� �+P if y+511

�ð7Þ

where T+ =(T� Tw)=Tt and the shear temperature is

Tt =qw

rwcput

ð8Þ

where cp is the specific heat at constant pressure, Pr isthe laminar Prandtl number, Pr t is the turbulentPrandtl number, and the term P takes the form33

P=Prtp=4

sin(p=4)

A+

k

� �12 Pr

Prt� 1

� �PrtPr

� �14

ð9Þ

where A+ =26 is the van Driest constant.Several variations of thermal wall-function models

exist in the literature.24,26,27,34 Rakopoulos et al.26 devel-oped a modified wall-function model, which includesthe pressure work term in the energy equation and var-iations in density and transport quantities. Keum et al.27

considered the effect of variable density and variableviscosity in the wall-function model formulation andapplied this model to an HCCI engine. In the following,the modified wall-function model of Rakopoulos et al.will be used in comparison with the equilibrium wall-function model. The model is formulated as

T+ =1

0:4767ln y+ +

1

0:4767 Pr

� �

� 1

0:4767ln 40+

1

0:4767 Pr

� �+10:2384

+P+ y+ � 40+117:31(0:4767+1=Pr)

0:4767+1=Pr

� �

ð10Þ

where

T+ =rwutcpT

qwln

Tw

T

� �ð11Þ

and

P+ =nw

qwut

∂�p

∂tð12Þ

Equation (10) can be used regardless of the walldistance.

For wall-function models, the wall heat flux, qw, canbe evaluated directly from the model formulations, giventhe chamber temperature and thermoviscous propertiesin the boundary layer. Note that for wall-function

Ma et al. 19

models, the effects of combustion processes are neglectedand the boundary layer is assumed to be inert; thisassumption may be valid only for the viscous sublayer.

Non-equilibrium wall model

Non-equilibrium wall models are formulated fromensemble-averaged variable-density, low-Mach Navier–Stokes equations32 by retaining all contributions fromthe transient, convection, pressure gradient, and pres-sure work terms in the momentum and temperatureequations. In addition, we also retain temperature-dependent variations in density and transport proper-ties, which are not included in the algebraic form of theequilibrium wall-function model. An attempt is madeto model the combustion process near the wall byincluding the heat-release term in the temperatureequation. The set of governing equations, describingconservation of mass, momentum, and energy, can bewritten as

∂�r

∂t+

∂

∂x(�r �u)+

∂

∂y(�r�v)=0 ð13aÞ

�r∂�u

∂t+ �u

∂�u

∂x+ �v

∂�u

∂y

� �= � ∂�p

∂x

+∂

∂y(�m+mt)

∂�u

∂y

� � ð13bÞ

�rcp∂T

∂t+ �u

∂T

∂x+ �v

∂T

∂y

� �=

∂�p0∂t

+∂

∂y(�l+ lt)

∂T

∂y

� �+ _Q

ð13cÞ

which is augmented by the ideal-gas state relation

�p0 = �rRT ð13dÞ

In equations (13a) to (13b), �v is the wall-normal velo-city, R is the gas constant, the ensemble-averagedhydrodynamic pressure gradient ∂�p=∂x is assumed tobe independent of the wall-normal direction, �p0 is thehomogeneous thermodynamic pressure, mt is the turbu-lent viscosity, lt is the turbulent conductivity, and _Q isthe heat release rate. A detailed derivation and theassumptions made can be found in Ma et al.25

Turbulence is closed by a two-equation k–v model.35

The pressure gradient is evaluated from PIV measure-ments by solving a pressure Poisson equation.25 Theheat-release term _Q is taken from simulation results ofthe GT-POWER model. The streamwise convectionterms were omitted, resulting in a one-dimensional wallmodel.25 To account for spatial variations along thewall-parallel direction, these terms can be retained.These terms, however, only become relevant if the spa-tial variations evolve on scales that are comparable tothe boundary-layer thickness.

The system of partial differential equations is solvednumerically using a second-order central difference spa-tial discretization and a third-order Runge–Kutta timeadvancement. A temperature-dependent viscosity is

considered using Sutherland’s law. Thermal and turbu-lent conductivities are evaluated through laminar andturbulent Prandtl numbers using standard values ofPr = 0.71 and Prt = 0.9, respectively.

Boundary conditions for velocity at the matchinglocation, yp, are prescribed from the PIV data, and no-slip boundary conditions are applied at the wall. Inaddition, the temperature equation is constrained byprescribing a constant wall temperature and the tem-perature at the matching point is prescribed from theengine-core temperature as a boundary condition. Wallboundary conditions for the k–v model equations canbe found in Wilcox35 and consist of a homogeneousDirichlet condition for k and an analytic expression forv. The upper boundary condition for k is evaluatedfrom experimental data, and a homogeneous Neumannboundary condition is applied for v at the matchinglocation. Initial conditions for the velocity profile arefrom experimental data. Initial conditions for the tem-perature are prescribed by an error function, which isobtained from the steady-state solution.

Since the temperature profile is explicitly solved dur-ing the simulation, the instantaneous wall heat flux inthe non-equilibrium wall model can be directly evalu-ated from its definition.

Results and discussion

The wall models introduced in the previous section,including the non-equilibrium wall model, the equili-brium wall-function model, and the modified wall-function model of Rakopoulos et al.,26 as well as theheat transfer correlations of Annand5 and Woschni6,are applied to the TCC-III engine under both motoredand fired conditions at 500 RPM. The matching loca-tion is set to yp =2.0 mm, unless otherwise stated.

Figure 3 shows the results of velocity profiles at fourselective crank angles predicted by the non-equilibriumwall model and the equilibrium wall-function model, incomparison with the PIV measurements. It can be seenthat, owing to the non-equilibrium effects, the velocityprofile in the boundary layer behaves distinctly differentcompared with the equilibrium profile from the wall-function model. At 300 CAD, although the wall-functionmodel is in good agreement with experiments in thenear-wall region, the underprediction further away fromthe wall is apparent. Note that the non-monotonic beha-vior of the near-wall structure cannot be captured withan equilibrium wall-function model. Similar results areseen for 330 CAD. After TDC, the equilibrium wall-function model predicts thinner inner layers. In contrast,by taking into account the variable thermotransportproperties, as well as convection and pressure gradients,the non-equilibrium wall model predicts velocity profilesin good agreement with the measurements.

The shear velocity, ut, can be evaluated from thevelocity profiles; a comparison with experimentallydetermined results is shown in Figure 4. For the

20 International J of Engine Research 18(1-2)

experimental data, a linear regression routine was usedto evaluate the velocity gradient from the PIV measure-ments;18,19 the shear velocity was then computed fromequation 5, and thermodynamic quantities are evalu-ated from isentropic assumptions and Sutherland’s law.Details can be found in our previous works.19,25 Wallmodels are applied with different matching locations toassess the sensitivity. In Figure 4, the relative errorbetween the model predictions and measurements isplotted as a function of matching locations and crankangles. It can be seen that the behavior of the non-equilibrium wall model is insensitive to the matchinglocation and that the relative error is mostly below10%, except for a small region around 340 CAD,

where the wall-parallel velocity changes sign and theshear velocity from the experiment is close to zero. Thewall-function model shows significant difficulties inpredicting the shear velocity before and after TDC ifthe matching location is outside the viscous sublayer,resulting in an error that can exceed 80% of the mea-surements. The performance improves when the match-ing location is moved closer to the wall.

The buffer layer thickness is defined as db =11dn,corresponding to the location of the buffer layer basedon the law-of-the-wall. The locations of db atdifferent crank angles are displayed by the white curvesin Figure 4. Results under fired conditions (not shown)are similar to those under motored conditions. The buf-fer layer thickness decreases slowly before 340 CAD,because the monotonic decrease in the kinematic visc-osity at the wall is at a similar speed to the increase inthe bulk gas velocity from the production in turbulencedue to compression. As can be seen in Figure 4, for theengine at a relatively low speed, the buffer layer thick-ness remains below 1 mm before TDC and increasessignificantly during the expansion stroke. Note that,owing to the low turbulence level during the expansionstroke, the flow near the wall may relaminarize orintermittency of the turbulence may occur, and hencethe canonical boundary-layer structures might not beapplicable anymore. A typical mesh resolution at thewall is of the order of 1–2 mm for typical RANS andLES simulations with wall functions applied.25 The firstgrid point is typically claimed to be put in the log layerso that the law-of-the-wall can be used to estimate thewall shear stress. However, combustion processes mayextend into the boundary layer1 below the locationwhere wall functions are applied and, without takinginto account the chemical reactions in the boundarylayer, significant underprediction in the heat flux canbe expected.

Figure 5 shows the heat-flux measurements underboth motored and fired conditions for two enginespeeds at 500 RPM and 1300 RPM. For motored con-ditions, the peak heat flux for the higher engine speedis more than twice that of the lower engine speed. Thisis expected, owing to the stronger turbulence inducedby the higher engine speed. The heat flux under fired

Figure 3. Velocity profiles predicted by non-equilibrium and equilibrium models in comparison with PIV measurements undermotored conditions for engine speed at 500 RPM. Vertical dashed lines show the matching location.

Figure 4. Relative error (in percentage) in shear velocitybetween (a) equilibrium and (b) non-equilibrium models and PIVmeasurements under motored conditions for engine speed at500 RPM. The solid curve shows the buffer layer location,where y+ = 11.

Ma et al. 21

conditions is an order of magnitude larger than thatunder motored conditions, owing to the combustion.The reversal of the heat flux is present for all cases.Owing to the higher pressure rise caused by combustionat fired conditions, a greater chance for negative heatflux is expected under fired conditions.11 The reasonfor the negative heat flux during the expansion processis explained in detail by Lawton.8 Specifically, the pres-sure work term is homogeneous in the boundary layer;however, the temperature close to the wall is lower thanthat in the engine-core region, and therefore the signifi-cant pressure work term will result in a non-monotonictemperature profile with a negative heat flux and a bulktemperature value higher than the wall temperature.

Temperature profiles predicted from the non-equilibrium wall model and the equilibrium wall-function model under motored and fired conditions areshown in Figures 6 and 7, respectively. It can seen thatthe difference in temperature profiles between the twomodels is significant. For the results in Figure 6 undermotored conditions, the equilibrium wall-functionmodel predicts a thicker thermal boundary layer beforeTDC. During expansion, the non-equilibrium modelshows a reversal of the temperature profile while theequilibrium model does not show this behavior. Asshown in the budget analysis by Rakopoulos et al.,26

the pressure work term dominates the contribution inthe energy equation in the region close to TDC, andthe evolution of the pressure work term also explainsthe occurrence of the negative heat transfer duringexpansion.8 For the case under fired conditions, shownin Figure 7, the wall-function model predicts a thickerthermal boundary layer during both compression and

expansion. At CAD 390, owing to the inclusion of theheat-release term, the non-equilibrium model has a sig-nificantly higher temperature near the wall, whereasthe wall-function model yields a monotonic equilibriumtemperature profile.

Results for heat-flux predictions under motored con-ditions are displayed in Figure 8 in comparison with

Crank Angle [CAD]0 120 240 360 480 600 720

Hea

t Flu

x [k

W/m

2]

–200

0

200

400

600

800

1000

1200

500 RPM, motored500 RPM, fired1300 RPM, motored1300, RPM fired

Figure 5. Heat-flux measurements under motored and firedconditions for engine speeds at 500 RPM and 1300 RPM.

Figure 6. Temperature profiles predicted by non-equilibriumand equilibrium models under motored conditions for enginespeed at 500 RPM.

Figure 7. Temperature profiles predicted by non-equilibriumand equilibrium models under fired conditions for engine speedat 500 RPM.

22 International J of Engine Research 18(1-2)

experimental data. The non-equilibrium wall modelshows good agreement with the measurements, and thenegative heat transfer after TDC is well captured. Theequilibrium wall-function model underpredicts the heatflux over all CADs shown in Figure 8, and negativevalues of heat flux are not captured. This is in agree-ment with findings from previous studies, where it wasfound that equilibrium wall-function models are notor-ious in underpredicting the heat flux in the cylinder.11

The modified wall-function model of Rakopoulos etal.26 shows similar behavior to the non-equilibriumwall model, but overpredicts heat flux before TDC.One of the inputs for the model of Rakopoulos et al.26

is the shear velocity, ut, which, for the present analysis,can be taken from the PIV measurements. However, itwas found that by using the instantaneous shear velo-city from experiments, severe overprediction beforeTDC and underprediction after TDC was observed.Therefore, the results shown in Figure 8 were obtainedusing a constant shear velocity evaluated at 270 CAD.The two heat transfer correlation models yield similarresults with the negative heat transfer after TDC notcaptured.

Figure 9 shows the cumulative heat flux undermotored conditions from different wall models in com-parison with measurements. The wall heat flux is inte-grated from 270 CAD in time. As can be seen fromFigure 9, owing to the overprediction before TDC forthe non-equilibrium wall model, the cumulative heatflux shows the corresponding overprediction. Similarresults are obtained from the modified wall-functionmodel of Rakopoulos et al.26 The equilibrium wall-function model underpredicts the cumulative heat fluxas expected and the two correlations show discrepanciescompared with the non-equilibrium wall model, owing

to the misprediction of the negative heat flux afterTDC. Overall, the non-equilibrium wall model pro-duces good agreement with experiments for motoredconditions and shows improved performance in pre-dicting the wall-heat flux compared with the equili-brium wall-function model and the correlation-typemodels.

Next, the performance of different models is exam-ined for fired conditions. Figure 10 shows the wallheat-flux predictions of different wall models in com-parison with measurements. It can be seen that the

Crank Angle [CAD]270 300 330 360 390 420 450

Hea

t Flu

x [k

W/m

2]

–25

0

25

50

75

100

125

ExperimentNon-equilibriumEquilibriumRakopoulosAnnandWoschni

Figure 8. Heat flux predicted by different models incomparison with measurements under motored conditions forengine speed at 500 RPM.

Figure 9. Cumulative heat flux predicted by different models incomparison with measurements under motored conditions forengine speed at 500 RPM.

Figure 10. Heat flux predicted by different models incomparison with measurements under fired conditions forengine speed at 500 RPM.

Ma et al. 23

equilibrium wall-function model significantly underpre-dicts the heat flux under fired conditions by almost anorder of magnitude. This deficiency can be attributedto the fact that the heat-release and pressure-workterms are neglected in the wall-function model. Themodified wall-function model of Rakopoulos et al.26

originally formulated for motored conditions is shownto provide a better performance, compared with theequilibrium wall-function model. The two correlation-type models give better behavior immediately afterTDC, with larger heat-flux predictions compared withthe wall-function models. The non-equilibrium wallmodel provides the best performance and accuratelycaptures the combustion phase, though still with under-prediction of the heat flux. The improved performanceof the non-equilibrium wall model over the other mod-els results from the inclusion of the heat-release term,which takes into account the heat release from combus-tion. The underprediction from the non-equilibriumwall model may come from the uncertainties of thetemperature boundary conditions and the heat-releaserate, as well as the cylinder pressure; these are all evalu-ated from the GT-POWER model.

Figure 11 shows results for the cumulative heat fluxunder fired conditions for different models in com-parison with experimental data. The results are con-sistent with the results shown in Figure 10. Wall-function models and correlations yield underpredic-tions of the heat flux at different levels. The model ofAnnand5 gives a similar amount of heat flux at theend, owing to the significant overprediction of theheat flux after the combustion processes, as can beseen from Figure 10. The non-equilibrium wall modelgives the best results.

Conclusions

Simultaneous high-speed high-resolution PIV and wallheat-flux measurements were conducted at the cylinderhead in an optically accessible IC engine under bothmotored and fired conditions. The experimental datawere utilized to assess the performance of a series ofwall models to predict the wall heat flux at the cylinderhead. These models include two spatially averaged heattransfer correlations, the equilibrium wall-functionmodel, its modification by Rakopoulos et al.,26 and anewly developed non-equilibrium wall model.25 Thefollowing conclusions can be drawn from this study:

1. The non-equilibrium wall model was found to pro-vide improved performance in predicting velocityprofiles and shear velocities, compared with theequilibrium wall-function model. The relative errorin the shear velocity with respect to measurementsis below 10% for most of the engine cycle.

2. The peak heat flux under motored conditions iswell predicted by the non-equilibrium wall modelwhereas the equilibrium wall-function modelunderpredicts the heat flux by a factor of two andmore.

3. The non-equilibrium wall model is able to predictthe reversal of the heat flux during the expansion,owing to the pressure gradient term, whereas thisfeature is completely missed by correlation-typemodels and the equilibrium wall-function model.

4. The equilibrium wall-function model shows sub-stantial underpredictions of heat flux under firedconditions by almost an order of magnitude, com-pared with measurements.

5. By taking into account combustion processes, pres-sure work, and variable thermotransport propertiesin the boundary layer, the non-equilibrium wallmodel is able to capture the peak heat flux underfired conditions.

The current non-equilibrium wall model utilizes theGT-POWER model to evaluate the heat-release rate.The performance of the non-equilibrium wall modelcan be further improved by considering combustionmodels with higher fidelity; this is the subject of futurework.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interestwith respect to the research, authorship, and/or publi-cation of this article.

Funding

The author(s) disclosed receipt of the following finan-cial support for the research, authorship, and/or publi-cation of this article: This work was supported by theNSF/DOE Advanced Combustion Engine Project(grant number CBET-1258609). Peter C Ma and

Figure 11. Cumulative heat flux predicted by different modelsin comparison with measurements under fired conditions forengine speed at 500 RPM.

24 International J of Engine Research 18(1-2)

Matthias Ihme acknowledge support through the Ford-Stanford Alliance program (grant number C2015-0590).

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