Abstract

The assumption of isotropic turbulence is commonly incorporated into models of the internal combustion engine (ICE) in-cylinder flows. While preliminary analysis with two-dimensional (2D) velocity data indicates that the turbulence may tend to isotropy as the piston approaches top-dead-center (TDC), the validity of this assumption has not been fully investigated, partially due to the lack of three-component velocity data in ICEs. In this work, the velocity was measured using two-dimensional, three-component (2D-3C) particle image velocimetry in a single-cylinder, motored, research engine to investigate the evolution of turbulence anisotropy throughout the compression stroke. Invariants of the Reynolds stress anisotropy tensor were calculated and visualized, through the Lumley triangle, to investigate turbulence states. Results showed the turbulence to be mostly anisotropic, with preferential tendency toward 2D axisymmetry at the beginning of the compression stroke and approaching-isotropy near top-dead-center. Findings provide new insights into turbulence in dynamic, bounded flows to assist with the development of physics-based, quantitative models.

Introduction

Turbulence is a critical parameter in the operation of internal combustion engines (ICEs), as it influences fuel–air mixing, burning rates, and heat transfer [1]. Turbulence analysis has focused both on core [2] and near-wall regions [3] using a combination of experimental [4] and numerical methodologies [5]. Typically, the core flow region inside the engine cylinder is minimally affected by the cylinder walls. This region contains the largest turbulent structures, which develop during the intake stroke and may persist through (at least) part of the compression stroke [6]. These large-scale structures are unsteady, three-dimensional, and often tumble- or swirl-dominated [1]. Treatment of ICE near-wall turbulence has been traditionally based on canonical boundary layer theory, although this approach fails to capture its full complexity [7]. Experimental studies of the interaction between core- and near-wall flows in ICEs suggest that, compared to canonical flows, the core-flow region affects the outer portion of the boundary layer (the log-law region) to a greater extent than wall-generated turbulence [3].

Many models of in-cylinder engine flow rely on the isotropic eddy viscosity hypothesis, in which the turbulence is assumed to possess no directional preference in the smallest turbulence structures [8]. Turbulence anisotropy is quantified through the Reynolds stress, as shown in Eq. (1). A full description of the anisotropy tensor requires knowledge of all three components of the velocity vector at multiple spatial locations, which are experimentally challenging to measure.
aij=uiuj¯23kδij
(1)

Numerical modeling has been used to quantify turbulence anisotropy of in-cylinder engine flows [5,9]. Hamlington and Ihme assumed homogenous, isotropic turbulence at the beginning of the compression stroke to calculate a single component of the anisotropy tensor, a11 from Eq. (1). The a11 term was found to increase throughout the compression stroke until near top-dead-center (TDC) [5]. Miles et al. examined the normal Reynolds stress components that contribute to the normalized anisotropy in the k–ε model. In the rθ coordinate system, it was observed that the r-component increased near TDC, while the θ-component decreased [9]. In a separate investigation, Miles et al. compared numeric modeling results with experimental data. They concluded, based on examining the deviatoric normal stresses used in the isotropic eddy viscosity hypothesis and the turbulent kinetic energy, that the late-cycle turbulence (after TDC) was anisotropic, though the anisotropy was not directly quantified [8].

The evolution of turbulence anisotropy may be visualized using a method proposed by Lumley [10], and commonly referred to as the Lumley triangle. In this analysis, the invariants of the normalized anisotropy tensor (Eqs. (3) and (4)) are calculated and displayed in invariant space. In the definition of the normalized anisotropy tensor (bij) presented in Eq. (2), the ensemble-average is used in place of the time-average commonly employed in canonical flows. This follows the standard Reynolds decomposition for in-cylinder engine flows, in which the instantaneous flow field is decoupled into its (ensemble) average and fluctuating component. The latter contains contributions from both the turbulence and from the cyclic variabilities in the ensemble mean.
bij=aij2k=uiujukuk13δij
(2)
IIb=12bii2=12bijbji
(3)
IIIb=13bii3=13bijbjkbki
(4)

Figure 1 shows a reference Lumley triangle. Turbulence states setting the triangle bounds are noted, along with the isotropic turbulence state at (0,0).

Fig. 1
Lumley triangle depicting specific turbulence states
Fig. 1
Lumley triangle depicting specific turbulence states
Close modal

Turbulence anisotropy analysis in a reciprocating internal combustion engine using the Lumley triangle has been conducted experimentally by Zentgraf et al. [11]. Velocity data were measured on a tumble plane centered between the intake and exhaust valves at two crank angles: one during the intake stroke ((270 before top-dead-center (BTDC)) and the second during the compression stroke (90 BTDC). The authors found a higher degree of isotropy at 90 BTDC compared to 270 BTDC, and more variability in turbulence states as the engine speed increased. The experimental data from Zentgraf et al. provided a comparative basis for large-eddy simulations conducted by He et al. [12], which revealed significant influence from intake-flow structures (e.g., the intake jet) on the anisotropy of the flow. Tendency toward isotropy away from the solid boundaries was also observed, consistent with findings by MacDonald et al. [3].

Using anisotropy tensor analysis, Soni et al. [13] demonstrated the application of a barycentric triangle technique to identify turbulence states in a single-cycle simulation of the gas exchange processes in a reciprocating ICE.

This paper details the experimental application of the anisotropy tensor invariant analysis to a reciprocating ICE during the compression stroke. Anisotropy tensor components were calculated from two-dimensional, three-component (2D-3C) velocity data, measured using stereoscopic particle image velocimetry at high temporal resolution (i.e., one velocity field every four crank angles). The enhanced temporal resolution and measurement of all three velocity components facilitate a novel assessment of the turbulence anisotropy evolution throughout the compression stroke (i.e., from 176 CA BTDC to 4 CA BTDC).

The main objectives of this research are to provide a rigorous, experimental assessment of the core-flow turbulence evolution throughout the compression stroke and assess the applicability of the isotropic turbulence assumption.2

Experimental Setup

To obtain 2D-3C velocity data, stereo particle image velocimetry experiments were performed in a motored, single-cylinder, reciprocating internal combustion engine. In stereo particle image velocimetry, the out-of-plane velocity component is extracted by reconstructing planar images simultaneously acquired from two angled cameras [15,16].

Optical access is enabled by a flat quartz top and side-mounted quartz windows (Fig. 2). The engine, with bore and stroke dimensions of 75 mm and 82 mm, respectively, was motored at a constant 620 rpm by an electric motor.

Fig. 2
Top engine view showing the field-of-view (red rectangle) and light sheet entrance point. The cameras are positioned above the quartz top.
Fig. 2
Top engine view showing the field-of-view (red rectangle) and light sheet entrance point. The cameras are positioned above the quartz top.
Close modal

The light sheet was created using a dual-cavity, diode-pumped, Nd:YLF (λ = 527 nm), with peak pulse energy of 10 mJ, per cavity, at 1 kHz. The beam was shaped into a sheet using the lens arrangement listed in Table 1. The focal lengths and positions were determined a priori through ray-tracing software, to ensure that the focal length laid at the center of the field-of-view (FoV). The resulting, one millimeter-thick light-sheet entered the engine through the side-mounted windows, parallel to the piston top surface, as shown in Fig. 3.

Fig. 3
Laser sheet position relative to the side optical window and the piston
Fig. 3
Laser sheet position relative to the side optical window and the piston
Close modal
Table 1

Light-sheet optics

LensLens typeNominal focal length (mm)
1Plano-concave−74.3
2Plano-convex150
3Rectangular cylindrical plano-concave−25.0
LensLens typeNominal focal length (mm)
1Plano-concave−74.3
2Plano-convex150
3Rectangular cylindrical plano-concave−25.0

Two high-speed, CMOS cameras (Phantom v9.1) were mounted in forward-scatter configuration to acquire the images. The camera repetition rate was set at 2000 frames-per-second (fps) (twice the laser frequency) to capture images in frame-straddling mode. At this rate, the cameras operated at 960 pixel × 720 pixel resolution.

The cameras were equipped with two 105 mm (Nikon Nikkor) lenses at the minimum f-stop setting (f/4) to maximize collection efficiency. In the arrangement, shown schematically in Fig. 4, the angles between cameras (45°) were maximized within experimental constraints to reduce the error in the out-of-plane velocity component [16]. The lenses were fitted with Scheimpflug adapters to ensure uniform focus from each camera perspective throughout the FoV.

Fig. 4
Camera and lens angles selected for the stereoscopic particle image velocimetry experiments
Fig. 4
Camera and lens angles selected for the stereoscopic particle image velocimetry experiments
Close modal

The flow was seeded with olive oil droplets. generated with a six-jet atomizer (TSI model 9306). The atomizer settings were adjusted to achieve approximately 20–30 particles per 32 pixel × 32 pixel interrogation window [16].

The droplet size distribution, while encompassing nominal droplet diameters (dp) up to 4 μm, is significantly skewed toward dp1.5μm. Based on the estimated Kolmogorov timescale (∼30 μs) for this engine at 600 rpm, a 1.5 μm nominal droplet diameter yields approximately 95% response to turbulence fluctuations [17].

The lasers, cameras, and piston positions were synchronized using an in-house developed LABVIEW VI and a pulse generator (BNC model 575). A crankshaft-mounted rotary encoder supplied the main input synchronization signal.

Images of the seeded flow were captured every four crank angles during the compression stroke, between 176 CA before top-dead-center (BTDC) and 4 CA BTDC. At each crank angle, the ensemble consisted of 114 cycles. This sample size was limited by optical window fouling. Engine and triggering parameters are summarized in Table 2, while the stereoscopic PIV parameters are listed in Table 3.

Table 2

Engine and triggering parameters

Engine speed (rpm)620
Trigger signal frequency (Hz)3720
Laser frequency (Hz)930
Camera frequency (Hz)1860
Crank angles between image pairs (CA)4
Time between laser pulses (μs)20
Number of cycles114
Intake valve open (CA)47–188
Exhaust valve open (CA)540–680
Engine speed (rpm)620
Trigger signal frequency (Hz)3720
Laser frequency (Hz)930
Camera frequency (Hz)1860
Crank angles between image pairs (CA)4
Time between laser pulses (μs)20
Number of cycles114
Intake valve open (CA)47–188
Exhaust valve open (CA)540–680
Table 3

Stereo-PIV parameters

Camera angle (deg)45
Field of view in xy plane (mm × mm)12.7 × 18.5
Magnification0.45
Spatial resolution (μm/pixel)25
In-plane spatial resolution based on a 32 × 32 pixel2 interrogation window (μm × μm)800 × 800
Camera angle (deg)45
Field of view in xy plane (mm × mm)12.7 × 18.5
Magnification0.45
Spatial resolution (μm/pixel)25
In-plane spatial resolution based on a 32 × 32 pixel2 interrogation window (μm × μm)800 × 800

Data Processing

Image preprocessing algorithms were developed (MATLAB 2019a) to subtract the background noise induced by internal light reflections off the piston surface and cylinder walls. An ensemble average of a background image set (i.e., without seeding) was calculated at each crank angle and subtracted from the instantaneous images. Image intensity disparities caused by laser-to-laser output energy fluctuations were mitigated via an intensity normalization filter.

After preprocessing, the data sets were imported into a particle image velocimetry software (LaVision's davis 8.4) to calculate the velocity fields. Image calibration was accomplished using a dual-plane calibration target (LaVision 025-3.3) and a third-order polynomial fit. A built-in self-calibration routine [18] was then applied to reduce disparities between the calibration image and light-sheet locations.

The velocity vector fields were calculated using four passes: two passes using a 64 × 64 pixel2 interrogation window with 75% overlap, and two passes using a 32 × 32 pixel2 window with 75% overlap. No smoothing or interpolation was applied to the resulting vector fields. They were visually inspected and outlier vectors throughout the field-of-view were removed using the built-in universal outlier detection filter.

Uncertainty Analysis

Errors in particle image velocimetry (PIV) arise from many sources, including experimental setup, image acquisition (e.g., discrepancies in laser light intensities), camera calibration, and during pre and postdata processing. The uncertainty calculation method built-in davis 8.4 was used in this work to quantify the uncertainty in the velocity magnitude and investigate the dependence (if any) of this uncertainty on piston position (i.e., crank angle) and spatial location. An in-depth description of the method can be found in Ref. [19]. As an overview, two images and their PIV displacement field are used as inputs. The images are dewarped, based on the displacement field, and divided into smaller windows (similar to those used during PIV processing). The correlation function is then examined on a pixel basis around the displacement in each direction to estimate the velocity uncertainty. When tested on synthetic data with artificially added noise, this method showed adequate agreement with the expected error [19].

Using this algorithm, uncertainty values for the ensemble-averaged flow fields were extracted, according to Eq. (5), where σ represents the uncertainty, θ is the crank angle, Ncycle represents the total number of cycles, and n indicates the individual cycle number.
σV(x,y,θ)=nNcycleσV(x,y,θ,n)Ncycle
(5)

Histograms were then created to examine the range of uncertainties in the velocity magnitude. Results, shown in Fig. 5, indicate that uncertainties in the velocity magnitude are below 2.5 m/s.

Fig. 5
Uncertainty in the ensemble averaged velocity for all crank angles and cycles
Fig. 5
Uncertainty in the ensemble averaged velocity for all crank angles and cycles
Close modal

Next, the variation in the uncertainty of the velocity magnitude was investigated as a function of piston position to identify any potential experimental challenges throughout the compression stroke. In Fig. 6, uncertainties in the ensemble average velocities are normalized by their corresponding velocity magnitudes. Overall, the uncertainty remained below 20% throughout the compression stroke, with some variations in the distribution as a function of piston position.

Fig. 6
Normalized uncertainty histograms of the ensemble velocity magnitude throughout the compression stroke
Fig. 6
Normalized uncertainty histograms of the ensemble velocity magnitude throughout the compression stroke
Close modal

Finally, uncertainty histograms were created to examine variations as a function of spatial location. This was accomplished by first dividing the field-of-view into five regions, as shown in Fig. 7. Uncertainty histograms for each region were created for the entire FoV and each of the five quadrants shown in Fig. 7. Figure 8 shows results at 268 CA. The uncertainty histograms for each quadrant are similar to the uncertainty histogram of the entire field-of-view, indicating negligible dependence of the uncertainty in the measured velocities on spatial location. This also suggests that spatial variations in experimental setup parameters (e.g., light-sheet) minimally influenced the results. Based on this uncertainty analysis, there is also confidence that any potential spatial variations in anisotropy results are not attributable to differences in the velocity uncertainty among quadrants.

Fig. 7
Field-of-view regions used in conditional data sampling. Valve location and laser path shown for reference.
Fig. 7
Field-of-view regions used in conditional data sampling. Valve location and laser path shown for reference.
Close modal
Fig. 8
Uncertainty histograms of the ensemble velocity magnitude, calculated for the center region and four adjacent quadrants for 268 CA (92 BTDC)
Fig. 8
Uncertainty histograms of the ensemble velocity magnitude, calculated for the center region and four adjacent quadrants for 268 CA (92 BTDC)
Close modal

Results

Statistical Convergence.

Given the limited sample size per crank angle (114 cycles), a statistical convergence check of the mean and root-mean-square (rms) of the velocity components (u, v, and w) was conducted. This was accomplished by calculating a running average at several locations within the field-of-view, over several crank angles. A result for the center of the field of view at approximately midstroke (268 CA) is shown for the ensemble average (Fig. 9) and for the root-mean-square of the velocity (Fig. 10). Convergence in the mean and rms of the in-plane velocity components is achieved at approximately 60 cycles. The mean and rms of the out-of-plane velocity component converge at 60 and 100 cycles, respectively. The discontinuity in the running mean and rms of the out-of-plane velocity component at around 70 cycles is a random occurrence at the reported crank angle (268 CA). For most crank angles and spatial locations, the running mean and rms were continuous. These results indicate that the 114-cycle sample size is adequate for the present analysis.

Fig. 9
Running average of the mean velocity components, showing statistical convergence at approximately 60 cycles
Fig. 9
Running average of the mean velocity components, showing statistical convergence at approximately 60 cycles
Close modal
Fig. 10
Running average of the rms of u, v, and w, showing statistical convergence at approximately 60 cycles for u, and v, and 100 cycles for w
Fig. 10
Running average of the rms of u, v, and w, showing statistical convergence at approximately 60 cycles for u, and v, and 100 cycles for w
Close modal

Velocity Vector Fields.

Ensemble-averaged velocity fields are shown in Fig. 11 for a portion of the crank angles. Every fourth vector is shown. Consistent with the location of the intake valve (Fig. 2), a clockwise rotating (swirl-like) structure can be identified in the sequence, with a swirl center (indicated by a black dot) displacing as the piston moves from near bottom-dead-center (CA = 216) to top-dead-center (CA = 356). Due to field-of-view limitations, at some crank angles, the vortex center is estimated to fall outside the image. It is interesting to note that, although this large-scale mean flow structure was likely generated during the intake stroke, it persists through the very end of the compression stroke (during which both valves are closed), leading to a predominantly negative velocity component, u, in the x-direction. As expected, the instantaneous fields (not shown) include a less directed (more random) motion induced by the fluctuating velocity components (u, v, w) albeit with some crank-angle dependence.

Fig. 11
Ensemble averaged vector fields. The black dot indicates the approximate center of the observed vortex with clockwise rotation. Every fourth vector is shown.
Fig. 11
Ensemble averaged vector fields. The black dot indicates the approximate center of the observed vortex with clockwise rotation. Every fourth vector is shown.
Close modal

Anisotropy Tensor Invariant Analysis.

After the preliminary flow field assessment just described, the normalized anisotropy tensor invariants were calculated according to Eqs. (3) and (4) for every crank angle. Results were visualized by locating turbulence states on the Lumley triangle. Figure 12 shows results for all spatial locations within the field-of-view at 268 CA (around midcompression stroke). The figure comprises approximately 10,700 turbulence states (all spatial locations at the given crank angle).

Fig. 12
Lumley Triangle showing all turbulence states at 268 CA
Fig. 12
Lumley Triangle showing all turbulence states at 268 CA
Close modal

From Fig. 12, it becomes readily apparent that the flow is significantly anisotropic, with many turbulence states lying on the 2D axisymmetric turbulence bound (right leg of the triangle).

To investigate the spatial dependence of turbulence states, the anisotropy data were conditionally sampled based on spatial location and piston position. The FoV was divided into five regions: a center area and four quadrants surrounding it, as shown in Fig. 7. Quadrant 1 is nearest the intake valve, while quadrant 4 is the region nearest the exhaust valve. The corresponding Lumley triangles for 268 CA are shown in Fig. 13.

Fig. 13
Midstroke (268 CA) states of turbulence subdivided into each quadrant and the center, as shown on Lumley triangles
Fig. 13
Midstroke (268 CA) states of turbulence subdivided into each quadrant and the center, as shown on Lumley triangles
Close modal

The flow is highly anisotropic and the spatial dependence of turbulence states is apparent. Although shown for 268 CA (92 BTDC midcompression) in Fig. 13, this trend is observed throughout the compression stroke. The turbulence near the center of the engine (black region) shows the strongest tendency to isotropy. Among the outer regions, quadrant 1 contains the most isotropic turbulence states. Recalling that quadrant 1 is closest to the intake valve, this finding may seem counterintuitive due to the strong directionality often imposed by the intake valve on in-cylinder flows. It should be noted, however, that during compression both valves are closed. As previously shown, the flow within the cylinder during the compression stroke is strongly influenced by a large-scale vortical structure, the center of which appears to rotate around the center of the cylinder as the piston moves toward TDC. This vortex center is often found within quadrant 1, which may explain the higher tendency to isotropy in this region compared to the remaining outer quadrants. Most other turbulence states in quadrant 1 fall on the oblate-spheroid (squashed spheroid) leg of the Lumley triangle. The turbulence in quadrants two, three, and four are predominantly axisymmetric (oblate-spheroid).

To investigate the dependence of turbulence states on piston position, Lumley triangles were created for the center region (black) at multiple crank angles throughout the compression stroke. Results, shown in Fig. 14, demonstrate that even in this region, the turbulence is highly anisotropic at the beginning of the compression stroke (e.g., 184 ATDC intake). Turbulence states shift between 2D-axisymmetry and isotropy throughout the compression stroke, tending toward isotropy near top-dead-center. While these results may not be generalized to all reciprocating engines, findings are consistent with observations from previous data sets [3] and more rigorously validated through the anisotropy tensor analysis presented here. These findings also suggest that turbulence isotropy assumptions may be spatially limited and confined to the end of the compression stroke.

Fig. 14
Development of the turbulence in the center region of the flow throughout the compression stroke
Fig. 14
Development of the turbulence in the center region of the flow throughout the compression stroke
Close modal

Conclusions

In this paper, the evolution of turbulence states during the compression stroke in a motored, single-cylinder, optical engine has been quantified through anisotropy tensor invariant analysis and visualized using the Lumley triangle. The analysis was performed on two-dimensional, three-component velocity data measured via stereoscopic particle image velocimetry.

Statistical convergence of the instantaneous and root-mean-square of all velocity components was verified to occur within the 114-sample size. Full uncertainty analysis of the velocity data revealed experimental uncertainty, (normalized by the velocity magnitude) of up to 20% near bottom-dead-center, decreasing to predominantly 10% midstroke, and increasing again to 20% near top-dead-center. The ensemble-averaged velocity vector fields revealed a large-scale vortical structure, which was found to persist throughout the compression stroke, with its center displacing around the cylinder as the piston moved toward top-dead-center. The anisotropy analysis revealed the following:

  • With 2D-3C velocity data, a Lumley triangle analysis provides a rigorous and effective metric for quantifying and visualizing the state and evolution of the turbulence within the engine cylinder and throughout the cycle.

  • The turbulence is mainly anisotropic throughout the compression stroke, exhibiting a strong preference for 2D axisymmetry, mainly encompassed by the oblate (flatten spheroid) category.

  • Within a limited spatial region, the turbulence tends to isotropy late in the compression stroke. This indicates that the isotropic eddy viscosity hypothesis is, at best, valid over only a spatially and temporally limited portion of the engine cycle.

These findings complement previous studies revealing the anisotropic nature of ICE turbulence, uniquely quantifying its temporal evolution throughout the compression stroke, and providing experimental evidence to support the refinement of isotropic turbulence assumptions built into computational models.

Acknowledgment

Any opinion, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

Funding Data

  • National Science Foundation (Grant No: CBET-1258609; Funder ID: 10.13039/100000001).

Nomenclature

     
  • aij =

    anisotropy tensor

  •  
  • bij =

    normalized anisotropy tensor

  •  
  • dp =

    particle diameter

  •  
  • i =

    cycle

  •  
  • II, III =

    invariants of the anisotropy tensor

  •  
  • k =

    turbulent kinetic energy

  •  
  • u' =

    velocity fluctuations

  •  
  • δij =

    Kronecker delta

  •  
  • θ =

    crank angle

Footnotes

2

This paper was presented at the ASME Internal Combustion Engines Fall Technical Conference 2020 [14].

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