Large Eddy Simulation of the Piston Boundary Layer Evolution During the Compression Stroke in a Motored Internal Combustion Engine

Near-wall flow is a fundamental problem in fluid mechanics, with extensive research aimed at understanding boundary layer evolution, its role in turbulence generation, and its implications for numerical simulations and modeling.

The energy cascade mechanism is well known and is driven by the breakdown of large eddies through successive instabilities (Pope 2000). In wall-bounded flows, like the ones characterizing ICEs, the physical presence of a wall constrains the development of instabilities, limiting the scales of the vortices. Consequently, the energetic eddies near the wall exhibit smaller dimensions, and the turbulent kinetic energy is added at high wave numbers (De Villiers 2006).

Wall-bounded flows have a second independent spatial energy cascade, called the streak cycle, responsible for generating turbulence within the turbulent boundary layer that is independent of the outer flow (Jiménez 1999; Jiménez and Pinelli 1999; Schoppa and Hussain 2000). This independence prompted significant scientific focus on the effects of the pressure gradients on the boundary layer. To illustrate its impact on the mean flow field, Fig. 1 shows the velocity profiles of three canonical turbulent boundary layers developing on a flat plate under the influence of different streamwise pressure gradients. While the Zero Pressure Gradient (ZPG) profile strictly follows the logarithmic law, the velocity profiles of the Favorable Pressure Gradient (FPG) and Adverse Pressure Gradient (APG) boundary layers show a typical upward and downward shift compared to the logarithmic law, respectively (Fig. 1a). Another difference can be seen in the outer region of the boundary layers, where the APG-related profile shows a distinct gradient, readily visible in the linear scaling of the velocity profile (Fig. 1b). This leads to a pronounced boundary layer wake, i.e., the difference between the maximum velocity at the boundary layer edge and the velocity predicted by the law of the wall. The streamlines are displaced from the wall, resulting in a thickening boundary layer and a more significant velocity deficit. The pronounced wake correlates with the decrease in wall shear stress, increasing the turbulence within the outer layer with the Reynolds stress components approaching each other, implying the weakening of turbulence anisotropy. In contrast, an FPG leads to a global increase in the velocity gradients at the wall and the wall shear stress, causing a decreasing boundary layer thickness and a small boundary layer wake. It is complemented by the suppression of turbulence, which causes a higher level of anisotropy of the near-wall turbulence.

This representation provides only a first impression of the impact of the global pressure gradient on the flow field and will serve as a basic reference when discussing the boundary layer development at the piston surface. As will be demonstrated, the flow structure at a moving piston inside an ICE is characterized by a pressure field exhibiting complex three-dimensional variations in both magnitude and direction. This raises the question of how these variations affect the velocity profile compared to that of flat plate boundary layers with typically unidirectional pressure gradients.

De Villiers (2006) argued that these mechanisms have significant implications for turbulence modeling in the context of CFD. The introduction of turbulence at small scales near the wall and the lack of correlation between these small dissipating and large energy-carrying scales imply significant modeling challenges. The strong anisotropic near-wall turbulence leads to a complex, nonlinear relationship between the Reynolds stresses (\({R_{ij}}\)) and the strain rate tensor (\({S_{ij}}\)), which presents a particular modeling challenge. A Wall-Resolved Large Eddy Simulation (WR-LES) can overcome this challenge, but a true WR-LES necessitates a computational grid with very small and regularly spaced cells near the walls, substantially increasing computational cost, restricting the widespread deployment of WR-LES techniques (Rutland 2011).

Therefore, CFD investigations of ICEs generally employ wall functions in Large Eddy Simulation (LES) (Rutland 2011). However, these models are frequently developed in a Reynolds-Averaged Navier-Stokes (RANS) context and are usually based on canonical flow setups such as the ones presented in Fig. 1, where the outer flow is unidirectional and irrotational. The highly dynamic nature of engine flow does not fulfill these criteria. Therefore, wall models may not account for non-equilibrium effects (Keskinen et al. 2018; Ma et al. 2017), leading to modeling errors in CFD simulation of engines.

Several studies have experimentally investigated these complex boundary layers in ICEs. Attempting to replicate engine-like conditions, investigations within a Model Compression Machine (MCM) revealed intriguing phenomena: In the presence of tumble motion, an APG is induced by the interaction of the large-scale tumble vortex with the piston, leading to the separation of the boundary layer at the piston edge (Borée et al. 2002).

In a pioneering multi-cycle study employing laser Doppler velocimetry in a real engine, Hall and Bracco (1986) demonstrated the presence of a thin boundary layer at the cylinder wall in an engine with a swirling flow configuration under both motored and fired operation. However, the achieved maximum resolution of 500 \(\mu m\) limited the scope of the conclusions. Subsequent investigations by Foster and Witze (1987) and Pierce et al. (1992) in which resolutions of 60 \(\mu m\) were achieved concluded that boundary layers are significantly thinner than predicted by flat plate boundary layer theory.

Alharbi and Sick (2010) advanced near-wall flow studies using combined Particle Image Velocimetry (PIV) and Particle Tracking Velocimetry (PTV) with a 45 \(\mu m\) spatial resolution, observing sub-millimeter eddies at the cylinder head in a pent-roof engine. Jainski et al. (2013) achieved a similar resolution with PIV in the cylinder head region at higher engine speeds and showed that boundary layer thickness decreases as engine speed increases. It was also highlighted that significant deviations from the log-law in the range of \(30 \leq {y^ + } \leq 50\) exist.

In a study using highspeed PIV data, MacDonald et al. (2017) found a similar discrepancy and proposed that both the mean shear stress and dissipation of the large-scale core flow turbulence contribute to turbulence generation in engines, due to higher Reynolds stress intensities near the wall compared to classical boundary layer theory.

Shimura et al. (2019) conducted an experimental study on the piston top boundary layer, finding no adherence to the log-law and suggesting possible flow relaminarization due to an FPG during compression. A detailed discussion of the experimental results of the engine numerically investigated in this study is given at the end of the introduction. In summary, newer experimental configurations demonstrated resolving the relevant ICE boundary layer resolutions below 50 \(\mu m\) is necessary.

Concerning 3D-CFD studies, extensive efforts have been spent developing suitable wall functions (Rutland 2011; Keskinen et al. 2018; Ma et al. 2017; Li et al. 2021; Nuutinen et al. 2014). Wu et al. (2019) compared RANS and LES simulations using wall modeling to experimental data and concluded that classical log-law-based wall functions failed to predict experimental measurements. In a follow-up study (Wu et al. 2020), the authors observed a poor experimental match using LES with conjugate heat transfer, possibly due to the inadequate wall models used. Delayed Detached Eddy Simulations using the k-\(\omega\)-Shear Stress Transport model (DDES—k-\(\omega\)-SST) showed deviations from the log-law, with boundary layer separation occurring at Intake Valve Closing (IVC) at the cylinder head and anisotropic near-wall turbulence (Fan et al. 2018).

A combined numerical and experimental work by Buhl et al. (2017) analyzed the tumble motion and boundary layer at the piston top in the intake stroke. Experimentally, a “flying PIV” setup was used (see also Koehler et al. (2015)), which enabled the evaluation of the flow in a radial plane 1.5 mm above the moving piston. Scale adaptive simulations (SAS-SST), with a high resolution of the piston boundary layer, were carried out, and it was found that the boundary layer thickness varied along the piston surface significantly, possibly due to an interaction between the near-wall flow with the tumble flow structure. Departures to the classical boundary layers profile were also observed. All the above studies highlight the need for a better understanding of the near-wall structures and that current wall functions cannot be readily applied in ICEs.

More recent advances in the availability of computational resources allowed for the first fully wall-resolved LES simulations of the TCIII pancake engine by Giannakopoulos et al. (2020) as well as first Direct Numerical Simulation (DNS) of engine-like configurations (Schmitt et al. 2015, 2016a, b; Mandanis et al. 2018).

The Darmstadt optically accessible research engine has been widely deployed to investigate near-wall flow. Recent experimental works on this engine have made considerable scientific progress on this topic: Renaud et al. (2018) studied the boundary layer development on the piston during the compression phase at four operating points with varying engine speeds and intake manifold pressures (0.95/0.4 bar intake pressure and 800/1500 rotations per minute (rpm)). This data serves as experimental reference data for the present work. A resolution down to 20 \(\mu m\) in the wall normal direction was achieved. The authors conclude that although the flow conditions over the flat piston appear at first glance to be similar to that of classical canonical setups, the observed boundary layer profile showed profound differences from those observed on flat plates. The authors discuss that the boundary layer exhibits turbulent properties despite the low momentum thickness based Reynolds number \(R{e_\theta }\), probably due to the ingress of turbulent structures of the core flow into the boundary layer. This data set was later extended to 2500 rpm by Schmidt et al. (2023). Even at higher engine speeds, no log-law region was observed, and an inverse scaling of the boundary layer-thickness with the bulk Reynolds number was found. The authors also evaluated conditioned statistics by binning the data for fast and slow engine cycles. They noted that the degree of adherence to the law of the wall differs in terms of position and time evolution due to cycle-to-cycle variations.

A first DNS study of a single compression stroke of the engine was carried out by Giannakopoulos et al. (2023) at 800 rpm and 0.4 bar intake pressure. More recently, the methodology was extended to higher engine speeds and higher intake pressure (0.95 bar) over multiple cycles by Danciu et al. (2024). However, as only a single or low number of cycles were achieved, spatial averaging was used to evaluate near-wall quantities. Similar results were observed as with the experimental counterpart: No log-law region was found, and the boundary layer evolution was highly dynamic with strong local and temporal changes.

The literature revealed the current scientific barriers in the understanding and modeling of turbulent wall boundary layers in ICEs:

This work aims to advance the understanding of ICE boundary layer flows by conducting and analyzing high-resolution (wall-resolved) LES across the entire piston top over multiple engine cycles. This enables a detailed examination of the boundary layer with temporally ensemble averaged data akin to experimental studies. Consequently, the flow evolution of the boundary layer during the compression phase of the Darmstadt optical engine under motored conditions has been numerically investigated.

The remainder of this paper is structured as follows: First, the numerical setup is presented in Sect. 2. After the results are analyzed in Sect. 3: The global flow is compared to experimental data and analyzed. This is followed by an evaluation of the piston boundary layer along the piston symmetry line and the piston valve line. It should be noted that the assessment of velocity transformations is not within the scope of this study. Finally, a global view of the entire piston top is presented, and averaged data is compared to single cycle LES realizations.

The CFD software used is based on the open source code OpenFOAM-2.4.x^® and TFMotion (Pati 2022), an in-house library for ICE simulations. Capturing the motion of the valves and piston in engine simulations over a complete engine cycle is achieved using a key-grid approach. Here, body-fitted computational grids are deformed according to the engine’s prescribed motion by solving Laplacian-based mesh motion equations. Because the computational grids can only be deformed for a limited crank angle range, within which mesh quality criteria are met, a collection of meshes is needed to capture a complete engine cycle. These are generated in a pre-processing step. Therefore, the solution must be mapped from mesh to mesh during the simulation. This approach ensures a high mesh quality during the simulation and has been validated thoroughly (Pati et al. 2020; Pati 2022; Paredi et al. 2017; Lucchini et al. 2014).

In the context of LES, minimizing the amount of mesh-to-mesh solution mapping is crucial to negate the destruction of turbulent structures due to interpolation errors. Furthermore, reducing the cells’ deformation, and thus their aspect ratio (AR), is important. To circumvent these shortcomings, a cell layer addition and removal routine (Montorfano et al. 2014) has been deployed to capture the piston motion. This strategy has been combined with the Laplacian cell motion to capture the valve motion (Pati 2022). This cell layer A/R (addition/removal) is used in the intake, compression, and expansion phases.

In this work, LES is performed with implicit filtering, where the numerical grid itself acts as the filter. The resolved scales are directly determined by the mesh size, while a sub-grid scale turbulence model explicitly accounts for the effects of unresolved turbulence. Therefore, an appropriate grid resolution and aspect ratio must be achieved, whilst maintaining reasonable computational cost. To this end two meshing strategies are employed: During the gas exchange and expansion phase a base mesh with a maximum cell size of 0.5 mm inside the cylinder and a refinement of 0.25 mm towards all combustion chamber walls was used. This mesh contains approximately 5.7 million cells at IVC, excluding intake and exhaust ports.

To study the piston boundary layer during compression, a piston wall-resolved mesh was employed from IVC until \({20^ \circ }{\text{CA bTDC}}\). Details of this mesh can be seen in Fig. 2. The piston refined region is a block-structured hexahedral O-Grid with a wall-normal cell size of down to 25 \(\mu m\) of the first wall cell, to achieve a \({y^ + } < 1\) resolution. The cells grow in the wall-normal direction by an expansion factor of 1.04 for the first 35 cells and are then kept static to reach a total of 50 cells within the refinement region, resulting in a total height of 4 mm. The radial resolution is 80 \(\mu m\) to avoid high aspect ratios in the cells closest to the wall in order to minimize errors due to turbulence filtering. The cells in this mesh region are not deformed. Cell layers are removed to capture the piston motion in the area labeled “Layer AR” in Fig. 2. Above this area, there is another static mesh region with a resolution of 0.5 mm cells. Towards the other combustion chamber walls the mesh is again only refined to 0.25 mm. The two mesh regions are connected using an Arbitrary Mesh Interface (AMI). The total cell count at IVC of this grid is 24.4 million.

This multi-domain meshing strategy provides a high mesh resolution within the piston boundary layer while maintaining reasonable computational performance by reducing mesh resolution in the bulk flow without resorting to cells with high expansion ratios. At IVC, the base mesh is mapped to the wall-resolved mesh. In order to verify that 80% of the turbulent kinetic energy is resolved by the LES, as required by Pope (2000), the Kolmogorov length-based quality index LES_QI\(_\eta\) is recommended to be above 0.75 (Celik et al. 2005). Figure 3 shows the LES_QI\(_\eta\) in the tumble plane for a single LES realization at \({60^ \circ }{\text{CA bTDC}}\) of the piston-resolved mesh. LES_QI\(_\eta\) is always above \(0.8\) within the bulk flow region and is close to one within the highly refined boundary layer region.

In this work, an engine speed of 800 rpm with an average intake flow pressure of 0.95 bar under motored operation (not fired condition) is investigated. In Fig. 4, the whole numerical domain, including the boundary conditions, is presented. At the inlet and outlet, total pressure boundary conditions are imposed according to the experimental measurements. The length of the inlet and outlet corresponds to the locations of experimental pressure and temperature sensors.

At all walls, a no-slip velocity boundary condition is imposed. The wall temperature of the piston, cylinder head, valves, and liner are set to a constant value of 343.15 K. This is slightly above the coolant temperature of 333.15 K to account for the engine heating up even under motored conditions. The intake and exhaust ports are set to the coolant temperature (333.15 K). The intake and exhaust pipes are adiabatic walls.

To expedite the time-to-solution, we employed the following parallelization approach: First, the solution of a converged multi-cycle Reynolds-Averaged Navier-Stokes (RANS) simulation is mapped onto the Large Eddy Simulation (LES) grid at Exhaust Valve Closing (EVC). The initial conditions were then perturbed using a digital filter approach (Klein et al. 2003), which introduces random fluctuations to the velocity fields to ensure unique initial conditions for each simulation. This perturbation technique is applied to four separate setups, generating four unique initial conditions, resulting in four parallel simulation strands. To further mitigate the influence of the initial conditions and ensure the statistical independence of the four simulation strands, we discarded the first two full cycles from each strand. This strategy ensures that each simulation strand evolves independently. Using this approach, a total of 33 cycles were simulated and analyzed. Previous studies of this engine have shown that, under the given operating conditions, 25 to 50 cycles are typically required for reasonable statistical convergence (Baum et al. 2014; Barbato et al. 2023). The authors in Barbato et al. (2023) also note that beyond 25 cycles, only minor improvements in convergence were observed.

The simulation employs the Pressure-Implicit with Splitting of Operators (PISO) algorithm for solving the Favre-filtered Navier-Stokes and energy equations, combined with an implicit Euler time-stepping. During the gas exchange, the Courant-Friedrichs-Lewy-Number (CFL) number is kept below one (CFL\(< 1\)) using a variable time step. During the compression, a constant time step of 0.005 \(^ \circ\)CA (1.041 \(\mu\)s) was used, resulting in a time step well below the Kolmogorov time scale, that for this type of engine at this rpm is in the order of \(\approx 10\,\, \mu\)s (Heywood 2018; Heim et al. 2014). Given the small timestep used, the Euler time integration method is appropriate for LES (Park 2006), while offering high stability. The 2nd order accuracy central differencing scheme was used in the piston boundary layer.

The density \(\rho\) and viscosity \(\nu\) of the gas are calculated based on the ideal gas and Sutherland’s laws (Sutherland 1893).

Turbulence closure is achieved through the eddy viscosity hypothesis, using the \(\sigma\)-model approach according to Nicoud et al. (2011). Hereby the sub-grid eddy viscosity \({\nu _{SGS}}\) is modelled as:

with the grid size \(\Delta\) and the Favre-filtered velocity \(\vec u\). The coefficients \({\sigma _i}\) are based singularly on the velocity gradient tensor of the resolved scales \({g_{ij}} = \partial {u_i}/\partial {x_j}\) where the coefficient \({C_m}\) is set to a constant value equal to \({C_m} = 1.5\). At all walls without high mesh refinement wall functions according to Spalding (1961) were used. At the refined piston wall, no wall functions were set.

Unless specified otherwise, quantities are ensemble averaged at the indicated crank angle degree in post-processing and near-wall flow quantities are then calculated on the averaged data.

Under motored conditions (without combustion), heat fluxes play a minor role, and no chemical reactions are present. For this reason, the global flow field is arguably the most critical metric for validating engine CFD simulations under this operating condition.

First, a comparison between the simulation and experimental ensemble averaged in-plane velocity magnitude in the central tumble plane, is presented in Fig. 5. In all plots shown, the exhaust side is positioned on the right and the intake side on the left (see also Fig. 6).

The flow is characterized by a large-scale tumble motion, which is induced during the intake stroke and persists throughout most of the compression phase, only breaking down close to Top Dead Center (TDC). The tumble center, located right of the spark plug on the exhaust side of the combustion chamber, moves towards the cylinder head during the compression. The simulation captures the evolution of the tumble flow well. The tumble center is matched at \({60^ \circ }{\text{CA bTDC}}\) and \({45^ \circ }{\text{CA bTDC}}\). The tumble center is slightly lower in the simulation at \({30^ \circ }{\text{CA bTDC}}\) compared to the experiment.

As a result of the tumble flow, a flow impingement is observed on the exhaust side of the piston. The flow then continues parallel along the flat piston, detaching as it is forced upwards by the cylinder wall at the intake side.

For a more in-depth quantitative analysis, the velocity field is compared along specific sampling lines shown in Fig. 6 alongside the global coordinate system. The red circle indicates the origin of the global coordinate system. Figure 7 presents the in-plane velocity components (\({{\text{u}}_{\text{x}}}\) and \({{\text{u}}_{\text{y}}}\)) sampled along the lines described before. Velocity is sampled at IVC for both the simulation and experimental data. The ensemble averaged LES data aligns well with the experimental data, remaining within the range of one standard deviation of the experimental measurement.

For the horizontal line samples (y = − 15 mm and y = − 25 mm, Fig. 7a and b respectively), both flow components are positive on the intake side (x \(<\) 0 mm). Close to the exhaust side (x \(>\) 0 mm), both velocity components decrease and the \({{\text{u}}_{\text{y}}}\) component changes its sign due to the tumble motion. The effect of the tumble motion is also evident on the vertical line sample at the exhaust side (x = 20 mm, Fig. 7d). Here, the \({u_y}\) component is negative for large parts of the combustion chamber, indicating the downward flow towards the piston where it impinges. Here the \({{\text{u}}_{\text{y}}}\) component becomes positive due to the piston motion. The \({{\text{u}}_{\text{x}}}\) component is negative towards the piston wall, indicating that the flow moves along the piston from the exhaust to the intake side. The vertical line sample between the intake valves (x = − 20 mm, Fig. 7 (c)) indicates a consistent flow towards the cylinder head (\({{\text{u}}_{\text{y}}}{{ > 0}}\) for the entire line sample). Overall, a good agreement between the simulation and experiment was achieved.

To better illustrate the boundary layer on the piston, it is useful to present the velocity field in the piston’s reference frame. In Fig. 8, the piston velocity \({\vec u_{{\text{piston}}}}\) has been subtracted from the velocity field. This reveals that the boundary layer on the piston predominantly flows parallel from the exhaust to the intake side, particularly around the piston center, with minimal normal velocity components. On the exhaust side, distinct regions of flow attachment can be observed, similar to an impinging jet flow, while on the intake side, flow reversal and detachment are evident.

This analysis reveals fundamental differences from the bulk flow of a canonical boundary layer flow, where the bulk flow is generally unidirectional and irrotational. The outer flow of a piston boundary layer is the result of the global tumble induced during the intake and its compression by the piston’s upward motion, and it changes spatially and temporally throughout the engine cycle.

For near-wall assessments of the data, a new local coordinate system is deployed. The origin of this new local coordinating system is anchored at the piston center and moves according to its motion. Velocities are considered positive if directed from the exhaust to the intake side.

Figure 9 shows the wall-parallel velocity (\({{\text{u}}_{\text{x}}}\)) versus the distance from the wall \(({{\text{y}}_{{\text{wall}}}})\) at the piston center at \({60^ \circ }{\text{CA bTDC}}\), \({45^ \circ }{\text{CA bTDC}}\) and \({30^ \circ }{\text{CA bTDC}}\). All three velocity profiles show a very high-velocity gradient near the wall, followed by a velocity peak, and then a constant decrease towards the bulk flow. This decrease in the velocity profile is caused by the fact that the velocity decreases towards the tumble center.

For all crank angles, the velocity peak is approximately 1 mm away from the wall, and the velocity gradient remains relatively constant throughout the compression. The maximum velocity decreases as the piston moves closer to TDC due to the dissipation of the tumble motion.

A good agreement between simulation and experiment can be observed, except for a slight over-prediction at \({30^ \circ }{\text{CA bTDC}}\) and \({45^ \circ }{\text{CA bTDC}}\). This can be attributed to the slight difference in the tumble center location in the simulation.

Regarding the velocity gradient at the wall (noted in the legend), there is a good agreement between the LES and the experiment, indicating that the simulation correctly captures the wall shear stress. For both data sets least square regression within the viscous sub-layer was used to determine the velocity gradient.

In order to evaluate the boundary layer using the dimensionless wall distance \({y^ + } = y{u_\tau }/\nu\) and velocity \({u^ + } = {u_x}/{u_\tau }\) with the friction velocity \({u_\tau } = \sqrt {{\tau _w}/\rho }\) and the wall-shear stress \({\tau _w} = \rho \nu {\left( {\partial {u_x}/\partial y} \right)_{{y_{wall}}}}\), the (temperature dependent) viscosity \(\nu\) and density \(\rho\) must be known. Here the viscosity \(\nu\) and density \(\rho\) are however singular reference values, that are assumed constant throughout the boundary layer as these definitions stem from incompressible flows without a temperature boundary layer. In engines, of course, they are not constant throughout the boundary layer, nor during the engine cycle, due to the compression and expansion. In addition, in most experimental studies of engines, only the in-cylinder pressure is measured directly. Hence, these thermo-physical quantities can only indirectly be calculated by the in-cylinder pressure, through an in-cylinder temperature estimation. Nevertheless, given the presence of a temperature boundary layer in the engine, a decision must be made regarding the reference temperature for which these thermo-physical reference values are calculated.

The impact of this choice of the reference temperature is highlighted in Fig. 10b. Here, wall distance and velocity at the piston center at \({45^ \circ }{\text{CA bTDC}}\) of the CFD are non-dimensionalized (or scaled) according to three different temperature reference samples within the boundary layer: wall (\({T_{{\text{wall}}}}\)), bulk (\({T_{{\text{bulk}}}}\)), and mean (\({T_{{\text{mean}}}}\)), as shown schematically in Fig. 10a. To clarify: the temperatures are sampled from the ensemble averaged temperature field at the indicated crank angle and xz-location normal to the piston. \({T_{{\text{bulk}}}}\) is sampled 4 mm from the piston wall, \({T_{{\text{wall}}}}\) at 0 mm from the piston wall. The mean temperature is then calculated as \({T_{{\text{mean}}}}({\text{x,z}}{{\text{,}}^ \circ }{\text{CA}}) = 0.5 \times ({T_{{\text{bulk}}}}({\text{x,z}}{{\text{,}}^ \circ }{\text{CA}}) + {T_{{\text{wall}}}}({\text{x,z}}{{\text{,}}^ \circ }{\text{CA}}))\). Therefore, mean temperature here does not refer to an average temperature over the entire engine stroke or across the entire piston surface.

The profile outside of the viscous sublayer (\({y^ + } > 5\)) is pushed downwards as the sampling position of the reference temperature is moved from the wall towards the bulk, showing a strong sensitivity, especially in the log-law region and its intersection. Literature discrepancies in this choice are evident. For example, Giannakopoulos et al. (2023) used wall values, Renaud et al. (2018) used the mean temperature for some near wall quantities, and in the study by Shimura et al. (2019) the (estimated) in-cylinder bulk temperature is used. This effect can also introduce uncertainties if the wall temperature is unknown and must be estimated. However, different scaling assumptions do not invalidate findings about the presence of a log-law region itself, merely the relative position of the scaled velocity profile to the log-law region with ZPG.

From this point on, the temperature to calculate the thermo-physical reference values used for non-dimensionalization of the boundary layer profile is \({{\text{T}}_{{\text{mean}}}}\) at the sampled location and crank angle. Since temperature values are readily available in CFD, both the experimental and simulation data are normalized using thermo-physical reference values extracted from CFD results, ensuring consistent normalization.

It is also noteworthy, that, when sampling along the piston symmetry line (see also Fig. 11a), the bulk and mean temperature are almost constant, as can be inferred from Fig. 11b. This is because the temperature field inside the combustion chamber at the investigated crank angles does not show high stratification. This agrees with the findings of Danciu et al. (2024). The wall temperature is also, of course, constant as it is set as the boundary condition. Due to this low temperature stratification the reference density shown in Fig. 11b also shows very low variability across the piston. Thus, changes in the normalized profiles at different piston locations at the same crank angle are unrelated to variations in the thermo-physical reference values.

We would also like to note that a semi-local scaling (not shown) approach where local values of the density and viscosity are used for normalization, as suggested by Huang et al. (1995) and reported by other studies to collapse the velocity profiles (Schmitt et al. 2015; Giannakopoulos et al. 2023; Schmidt et al. 2023) in engines, only leads to a collapse of the profiles when sampling at the piston center at different crank angles, but not when sampling at other piston locations. Also no linear log-law region is recovered by this semi-local scaling. It should also be noted that the flow in the combustion chamber during compression is at low Mach numbers. Therefore, compressibility effects associated with supersonic flows are negligible.

Figure 12 compares the nondimensionalized velocity profiles \({u^ + }\) and wall-distance \({y^ + }\) of the simulation and the experiment at the piston center. A robust agreement between simulation and experiment is evident. The pressure gradient \(\nabla p\) from the CFD simulation is also indicated in the legend of Fig. 12. A positive pressure gradient indicates an APG and a negative pressure gradient indicates a FPG. An FPG is observed at the piston center only at \({60^ \circ }{\text{CA bTDC}}\) and \({45^ \circ }{\text{CA bTDC}}\). At \({30^ \circ }{\text{CA bTDC}}\) the CFD predicts a APG despite the shape of the velocity profile being preserved. This confirms the hypothesis put forth by Renaud et al. (2018) regarding the acceleration of the boundary layer by an FPG, with the caveat that at later stages of the compression, the pressure gradient inverts before the piston center. This does not imply a fundamentally different flow pattern during the later compression stage. As will be discussed in more detail later, both regions of FPG and APG are generally observed across the piston, with the inversion occurring around the piston center. Due to the continued squish of the tumble flow structure caused by the piston’s upward motion and the shift in the tumble center, the position of this inversion zone tends to migrate towards the exhaust side during the compression stroke.

The near-wall analysis is continued in Fig. 13 by sampling along the symmetry line of the piston (z = 0 mm, varying x, see Fig. 11). Only data at \({45^ \circ }{\text{CA bTDC}}\) is presented for brevity, but the flow behavior is analogous at \({60^ \circ }{\text{CA bTDC}}\) and \({30^ \circ }{\text{CA bTDC}}\). Figure 13a shows the absolute velocity, and Fig. 13b the nondimensionalized \({u^ + }{\kern 1pt} - {\kern 1pt} {y^ + }\) velocity profile. In Fig. 13a, the color map corresponds to the position along the symmetry line, with yellow indicating the exhaust side and purple the intake side. In Fig. 13b the color map corresponds to the pressure gradient, with APG regions colored blue and FPG regions colored in red (the pressure gradient value is included in the legend).

Generally, regions of FPG (towards the exhaust side), which accelerate the flow, and APG (towards the intake side), which decelerate the flow, are observed. The flow accelerates from the exhaust side towards the piston center, where the velocities reach their peak. On the intake side, the flow slows and detaches, as evidenced by the velocity profile at x = − 29 mm in Fig. 13a, which shows a reverse flow near the wall.

The effects of the boundary layer thickening and the typical increase of the velocity deficit associated with boundary layers influenced by APG as discussed in Sect. 1 (see also Fig. 1b) can readily be observed in Fig. 13a. It is, however, clear that the flow structure at the piston is characterized by a much more complex pressure field that varies in both sign and intensity, leading to very different velocity profile shapes than those found in the generic flat-plate boundary layers with unidirectional pressure gradients.

It is also evident from the non-dimensional velocity profiles in Fig. 13b that the boundary layer along the piston symmetry line does not adhere to the classical log-law region for all sampled locations. In areas with FPG, the profiles remain below the log-law region, and in regions of APG, they rise above the log-law region. This is the opposite of what is expected from canonical flow (see Fig. 1a). The highest deviation below and above the log-law region is observed close to the attachment and detachment points of the flow on the piston, respectively. We wish to emphasize that this comparison with canonical flows is not intended to imply that the boundary layer behavior in engines is expected to follow canonical boundary layers precisely. Given that the boundary layer flow first accelerates and then decelerates due to variations in the pressure gradient and the outer flow, the deviation from the log-law region is unsurprising. Nonetheless, the development of wall functions, even for models specifically designed for engine applications, employ or calibrate model constants based on precisely such canonical boundary layers or the universal law of the wall (Li et al. 2021; Nuutinen et al. 2014), due to the lack of comprehensive near wall engine data. Therefore, identifying these differences and generating such reference data, to in the future test these models is essential to understanding error sources and improving models. Given the unsteady nature of the flow and pressure gradient shown here, wall models should consider these non-equilibrium effects (Ma et al. 2017; Li et al. 2021).

Figure 14 shows a plot of selected boundary layer metrics along the piston symmetry line, namely the nominal boundary layer thickens \(\delta\), the displacement thickness \({\delta ^*}\), the momentum thickness \(\theta\), the momentum-based Reynolds number \(R{e_\theta }\), as well as the shape factor \(H\). These are defined according the equations 3-7, where the star subscript indicates the location at the peak velocity.

The grey area in Fig. 14b indicates flow separation. \(R{e_\theta }\) rapidly grows from the impingement point to the regions of high FPG. Then, it continues to grow only moderately in the center region. It increases stronger again in the area of higher APG and even more strongly at the detachment point. However, it does not increase monotonically.

The \(H\)-factor evolution presents an interesting finding. According to Schlichting and Gersten (2006), in a typical laminar to turbulent transition of a boundary layer over a flat plate, the \(H\)-factor will decrease from \(H = 2.6\) in laminar regions to \(H = 1.4\) in the turbulent region. As noted by Renaud et al. (2018) and confirmed by the simulation data, the shape factor at the piston center is close to \(H = 1.4\), which would point towards a turbulent boundary layer, despite a low Reynolds number of around \(R{e_\theta } = 75\), for which laminar flow would be expected. In fact, at the stagnation point, where the flow attaches on the exhaust side, the shape factor immediately shows a value of approximately \(H = 1.5\). Moving along the piston (increasing \(R{e_\theta }\)), \(H\) remains nearly constant. In the flow detachment zone, \(H\) increases rapidly, moving towards values associated with laminar flow.

This lends credence to the interpretations made by Buhl et al. (2017) and Renaud et al. (2018). They suggest that the strongly turbulent bulk flow in engines leads to an ingress of turbulence into the boundary layer. Thus, the boundary layer shows characteristics associated with turbulent boundary layers at remarkably low \(R{e_\theta }\) values. On the other hand, \(R{e_\theta }\) strongly increases in the presence of a strong pressure gradient, whether adverse or favorable.

Figure 14b shows the evolution of different boundary layer thickness values definitions. It can be observed that the nominal boundary layer thickness \(\delta\) shows much higher values compared to displacement \({\delta ^*}\) and momentum \(\theta\) thickness. Generally, all boundary layer thickness values increase sharply after the flow attachment on the exhaust side. As the flow progresses along the piston center line, the displacement \({\delta ^*}\) and momentum \(\theta\) thickness increase at a slower rate and even briefly decrease at approximately x = 17 mm. This coincides with a drop in the \(R{e_\theta }\) values, similar to the behavior observed in generic turbulent boundary layer flows under FPG, where the acceleration of the flow leads to decreasing \(R{e_\theta }\) (Jakirlić et al. 2006). However, the boundary layer thickness subsequently begins to slowly increase again around the piston center. This indicates a non-stationary boundary layer. The value of the nominal boundary layer thickness \(\delta\) fluctuates as the outer bulk velocity varies due to the tumble flow. The boundary layer thickness undergoes a rapid growth in the flow detachment zone. The causal factors of this observed behavior such as pressure gradient anisotropy, transient vorticity interaction and momentum transport mechanisms, which are relevant to modeling, should be investigated in future studies

Moving away from the tumble plane Fig. 15 shows the near-wall velocity profile and non-dimensionalized velocity on the line bisected by the valve plane and the piston (hereafter referred to as valve line, see Fig. 11). A very similar behavior, as was observed along the piston symmetry line, is found. There is an initial acceleration due to regions of FPG, followed by a deceleration due to regions of APG at the opposing liner wall with flow detachment at x = − 29 mm. Again, no adherence to the log-law region is found for all points sampled.

Figure 16 shows some boundary layer metrics on the piston top at 45\(^ \circ\)CA bTDC. The metrics investigated are: the wall shear stress, the boundary layer thickness \({\delta ^*}\), the pressure gradient oriented along the flow, and the alignment of the pressure gradient. The latter is defined as \([\nabla p \cdot \vec u]/[|\nabla p| \cdot |\vec u|]\) and it varies from − 1 (perfectly aligned FPG), to 0 (pressure gradient orthogonal to the flow), to 1 (perfectly aligned APG).

The ensemble averaged fields in the left column give insight into the general flow behavior. The wall shear stress is low where the flow attaches to the piston top on the exhaust side. It increases steadily due to the acceleration of the flow towards the piston center. As the flow decelerates, the wall shear stress reduces. As the flow detaches on the intake side, the wall shear stress undergoes a significant decrease.

The displacement thickness increases significantly over a considerable portion of the piston, extending from the exhaust to the intake side and reaching a notable peak in the vicinity of the flow detachment zone. Within the detachment area, the displacement thickness becomes close to zero. The detachment zone is also clearly visible in the visualization of pressure gradient in flow direction (\(\nabla p \cdot \vec u/|\vec u|\)) as well as alignment between the pressure gradient and velocity (\([\nabla p \cdot \vec u]/[|\nabla p| \cdot |\vec u|]\)).

The former shows two broad regions of FPG and APG, separated by a sharp pressure inversion zone, left of the piston center. It is crescent-shaped towards the spanwise liner walls (i.e. in the z direction). The latter shows that the pressure gradient is generally well aligned with the flow, especially along the piston symmetry line. Again, the pressure inversion region is very sharply defined. It should be noted that in the detachment zone, there is not an inversion of the pressure gradient but rather an inversion of the flow (typical of detached flows).

Some areas of misalignment between the pressure gradient and the velocity vector exist in the spanwise outer regions. This indicates yet another difference between engine and canonical flow setups that has to be accounted for: Lateral pressure acting on 3D boundary layers requires modification of wall models to account for this effect (Lozano-Durán et al. 2020).

Figure 16’s center and right columns show instantaneous data of two individual cycles. For all quantities presented, strong cycle-to-cycle fluctuations exist. It is imperative to point out that flow features in the ensemble averaged fields are not readily visible in the instantaneous snapshots. The pressure inversion zone is not sharply defined, and local flow structures with APG exist in regions of, for example, average FPG. The same can be said about all other quantities shown in Fig. 16, reflecting the boundary layer’s dynamic behavior in a temporal and spatial sense, driven by the attachment and detachment of local turbulent vortical structures.

The complexity of the flow can be summarized as follows: A turbulent bulk flow impinges on the horizontal piston surfacea parallel surface, leading to a non-stationary, potentially already turbulent, boundary layer, which is accelerated by a FPG, that is then subjected to a sudden reversal (APG). This leads to deceleration and detachment. In parts, this flow has some similarities to wall-impinging jets. Studies concerning different types of impinging jets have also found strong deviations (Krumbein et al. 2017; Lav et al. 2022) from the classical log-law. Some similarities may also be drawn to the by-pass transition, which is very relevant in turbomachinery (Ghasemi et al. 2014). Moreover, the pressure gradient acting laterally on the boundary layer is present in some areas.