UC Irvine

The enhancement of skin friction drag and surface heat flux by the transition to turbulence is a crucial physical phenomenon in wall-bounded flows. An interpretable mapping of how various flow phenomena such as turbulence and pressure gradient influence these key surface quantities is desirable for advancing our understanding of fundamental flow physics, as well as informing engineering design analysis and developing efficient flow control techniques. To accomplish such a mapping, in this study, integral forms based on the first-moment of conservation equations are developed. The angular momentum integral (AMI) equation, obtained from the first moment of the momentum equation, yields an identity for the skin friction coefficient (friction drag). Furthermore, the moment of (total) enthalpy integral (MTEI) equation, derived from the first moment of the en- ergy equation, provides a mapping for the Stanton number (surface heat flux). This first-moment approach uniquely isolates the skin friction (or surface heat flux) of a laminar BL in a single term that depends only on the Reynolds number (or Peclet number) most relevant to the flow’s engineer- ing context, hence other terms are interpreted as augmentations or reductions relative to the laminar case having the same Reynolds number (or Peclet number).

In the case of zero-pressure-gradient incompressible transitional BLs, the AMI and MTEI equations examine the peak friction drag and surface heat flux during the transition. These tools demonstrate and quantify how the streamwise growth of the BL and the mean wall-normal flux resist the extreme growth of turbulent enhancement via Reynolds shear stress. This rapid growth of turbulence during transition imposes near-wall streamwise acceleration that results in a negative wall-normal velocity very close to the wall. For a fully turbulent regime, the explicit turbulent enhancement is the primary process of near-wall momentum (or heat) flux, not molecular transport. Consequently, the other flow phenomena weakly impact the skin friction (or surface heat flux).

The AMI analysis of turbulent flows subjected to strong favorable pressure gradients presents a substantial reduction of the turbulent enhancement due to a phenomenon referred to as reversion. The AMI analysis captures this complex phenomenon caused by flow acceleration and exhibits re-laminarization which deactivates the turbulence. Conversely, adverse pressure gradients cause marginal alteration in turbulent enhancement downstream, suggesting a weak correlation between the total Reynolds shear stress and the strength of the pressure gradient. Additionally, the AMI equation introduces a pressure gradient parameter, which compared to the classic Clauser parame- ter, offers a more robust similarity in turbulent statistics between two flows with a distinct upstream history.

In high-speed boundary layers, e.g., supersonic vehicles, due to severe heating, surface heat flux is more critical than friction drag. The analyze the surface heat flux and friction drag the first-moment integral approach is extended to compressible flows, considering the variation of density, viscosity, and thermal conductivity across the boundary layer. The AMI results quantify how the variation of mean density inside a high-speed turbulent boundary layer impacts the momentum transport and reduces the friction drag. From an alternative viewpoint, the results demonstrate how the effect of compressibility on laminar boundary layers can be utilized to develop a mapping between the skin friction coefficient of the incompressible and compressible turbulent flows. The MTEI analysis similarly demonstrates how the mean density alters the impact of turbulence on the transport of total enthalpy and Stanton number. In doing so, the MTEI results highlight the relative role of turbulent fluxes of enthalpy and mean kinetic energy on the Stanton number.

These results suggest that the first-moment integral equations could be a valuable tool for evaluating flow control schemes. In the case of turbulent boundary layers over an airfoil with surface suction or blowing, only minor variations occurred in the turbulent enhancement. The AMI equa- tion is also applicable to evaluate more complex control methods such as using porous mediums. Beyond its role as an analysis tool, the concept of first-moment integral equations holds promise for the development of computationally efficient turbulent models. The AMI analysis quantifies approximately 80% of the turbulent enhancement in wall-bounded flows originates from the outer layer of the flow. Therefore, solving the integral form of the Navier-Stokes equation, focusing on resolving the outer layer, could provide a promising platform for turbulent modeling.