A Model of a Turbulent Boundary Layer With a Non-Zero Pressure Gradient

TL;DR

This paper presents a model of turbulent boundary layers with pressure gradients, identifying two distinct self-similar regions and how their scaling laws depend on pressure gradient and Reynolds number, supported by experimental data.

Contribution

The authors introduce a two-region model for turbulent boundary layers with pressure gradients, detailing how scaling laws vary with pressure conditions and Reynolds number, supported by experimental validation.

Findings

The power law exponent $eta$ varies with pressure gradient sign.

Adverse pressure gradients increase $eta$, favorable decrease it.

Experimental data aligns with the proposed model.

Abstract

According to a model of the turbulent boundary layer proposed by the authors, in the absence of external turbulence the intermediate region between the viscous sublayer and the external flow consists of two sharply separated self-similar structures. The velocity distribution in these structures is described by two different scaling laws. The mean velocity u in the region adjacent to the viscous sublayer is described by the previously obtained Reynolds-number-dependent scaling law $\phi = u/u_*=A\eta^{\alpha}$, $A=\frac{1}{\sqrt{3}} \ln Re_{\Lambda}+ \frac 52$, $\alpha=\frac{3}{2\ln Re_{\Lambda}}$, $\eta = u_* y/\nu$. (Here $u_*$ is the dynamic or friction velocity, y is the distance from the wall, $\nu$ the kinematic viscosity of the fluid, and the Reynolds number $Re_{\Lambda}$ is well defined by the data) In the region adjacent to the external flow the scaling law is different: $\phi= B\eta^{\beta}$. The power $\beta$ for zero-pressure-gradient boundary layers was found by processing various experimental data and is close (with some scatter) to 0.2. We show here that for non-zero-pressure-gradient boundary layers, the power $\beta$ is larger than 0.2 in the case of adverse pressure gradient and less than 0.2 for favourable pressure gradient. Similarity analysis suggests that both the coefficient B and the power $\beta$ depend on $Re_{\Lambda}$ and on a new dimensionless parameter P proportional to the pressure gradient. Recent experimental data of Perry, Maru\v{s}i\'c and Jones (1)-(4) were analyzed and the results are in agreement with the model we propose.