Development and Analysis of Wall Models for Internal Combustion Engine Simulations Using High-speed Micro-PIV Measurements

Abstract

The performance, efficiency and emissions of internal combustion (IC) engines are affected by the thermo-viscous boundary layer region and heat transfer. Computational models for the prediction of engine performance typically rely on equilibrium wall-function models to overcome the need for resolving the viscous boundary layer structure. The wall shear stress and heat flux are obtained as boundary conditions for the outer flow calculation. However, these equilibrium wall-function models are typically derived by considering canonical flow configurations, introducing substantial modeling assumptions that are not necessarily justified for in-cylinder flows. The objective of this work is to assess the validity of several model approximations that are commonly introduced in the development of wall-function models for IC-engine applications. This examination is performed by considering crank-angle resolved high-resolution micro-particle image velocimetry (µ-PIV) measurements in a spark-ignition direct-injection single cylinder engine. Using these measurements, the performance of an algebraic equilibrium wall-function model commonly used in RANS and LES IC-engine simulations is evaluated. By identifying shortcomings of this model, a non-equilibrium differential wall model is developed and two different closures are considered for the determination of the turbulent viscosity. It is shown that both wall models provide adequate predictions if applied inside the viscous sublayer. However, the equilibrium wall-function model consistently underpredicts the shear stress if applied in the log-layer. In contrast, the non-equilibrium wall model provides improved predictions of the near-wall region and shear stress irrespective of the wall distance and the piston location. By utilizing the experimental data, significant adverse pressure gradients due to the large vortical motion inside the cylinder (induced by tumble, swirl and turbulence) are observed and included in the non-equilibrium wall model to further improve the model performance. These investigations are complemented by developing a consistent wall heat transfer model, and simulation results are compared against the equilibrium wall-function model and Woschni’s empirical correlation.

Appendix A: Estimate of Viscous Sublayer Thickness

According to [14], the thickness of the viscous sublayer can be related to the boundary layer thickness using the following expression:

$$ \frac{\delta}{\delta_{\nu}} \approx c_{w} \left( \frac{|U_{p}|\delta}{\nu_{w}}\right)^{\frac{7}{8}} \,, $$

(17)

where c _w = 0.15. An estimate for δ _v as a function of Reynolds number is obtained by inserting $\delta /D \approx \zeta \text {Re}_{p}^{-1/5}$ with ζ = 0.193 [51] into Eq. 17:

$$ \frac{\delta_{\nu}}{D} \approx \frac{\zeta^{1/8}}{c_{w}}\text{Re}_{p}^{-\frac{9}{10}} \,, $$

(18)

indicating that the minimum of δ _ν is obtained at the maximum of Re _p = |U _p |D/ν _w . The piston speed is calculated using the following expression [3]:

$$ \frac{U_{p}}{{\mathcal{U}}_{p}} = -\frac{\pi}{2}\sin\theta\left( 1 +\frac{\cos\theta}{\sqrt{(l/a)^{2} - \sin^{2}\theta}} \right) \, \text{,} $$

(19)

where ${\mathcal {U}}_{p}=2a\omega /\pi$ is the mean piston speed, with ω being the angular velocity, and a the half-stroke or crank radius; 𝜃 is the crank angle, and l is the length of the connecting rod. By neglecting higher order terms, U _p can be approximated as:

$$ \frac{U_{p}}{{\mathcal{U}}_{p}} \approx - \frac{\pi}{2}\sin\theta\,. $$

(20)

The kinematic viscosity is evaluated as ν _w = μ _w /ρ _w , in which the density at the wall, ρ _w , is related to the wall temperature T _w and engine pressure via the ideal gas law. The engine pressure is obtained from the isentropic state relation:

$$ p = p_{\text{BDC}} \left( \frac{V_{\text{BDC}}}{V} \right)^{\gamma} \, \text{,} $$

(21)

where, according to [3],

$$ V = \frac{\pi D^{2} a}{4} \left( \frac{l}{a} + \frac{r+1}{r-1} - \sqrt{\frac{l^{2}}{a^{2}} - \sin^{2}\theta} - \text{cos}\,\theta \right) \approx \frac{\pi D^{2} a}{4} \left( \frac{r+1}{r-1} - \text{cos}\,\theta \right) \,, $$

(22)

where V is the engine volume, p _BDC is the pressure, V _BDC is the cylinder volume at BDC, and r is the compression ratio. With this, the kinematic viscosity at the wall, ν _w , can be evaluated as:

$$ \frac{\nu_{w}}{\nu_{w, \text{BDC}}}\approx \left( \frac{r-1}{2r}\right)^{\gamma} \left( \frac{r+1}{r-1} - \cos\,\theta\right)^{\gamma}\,, $$

(23)

and hence

$$ \frac{\text{Re}_{p}}{\overline{\text{Re}}_{p}} \approx \frac{\frac{\pi}{2}|\sin\theta|}{\left( \frac{r-1}{2r}\right)^{\gamma} \left( \frac{r+1}{r-1} - \cos\theta\right)^{\gamma}}\,, $$

(24)

where $\overline {\text {Re}}_{p}={\mathcal {U}}_{p}D/\nu _{w,\text {BDC}}$ is the mean piston Reynolds number, defined in analogy to the mean piston speed.

The maximum value for Re _p is obtained for 𝜃 between 330 and 340 CAD for 8 ≤ r ≤ 20 and the values for sin𝜃 and cos𝜃 within this range exhibit only small variations. For the current engine configuration (r = 11), the optimal 𝜃 is found to be 334 CAD and with this value, δ _ν,min can be estimated as:

$$ \frac{\delta_{v,\min}}{D} \approx 7.6 \left[\frac{1}{\overline{\text{Re}}_{p}}\left( \frac{r-1}{2r}\right)^{\gamma} \left( \frac{r+1}{r-1} - 0.90\right)^{\gamma}\right]^{\frac{9}{10}}\;, $$

(25)

and the estimate for δ _b,min follows directly through the relation δ _b = 11δ _ν .