[Paper Review] Mean velocity scaling in plane turbulent wall jets

Gupta, Abhishek, Choudhary, Harish+3 more|arXiv (Cornell University)|May 1, 2019

Aerodynamics and Acoustics in Jet Flows21 citations

TL;DR

This paper proposes a local, self-similar scaling framework for plane turbulent wall jets based on streamwise variations of kinematic momentum rate M and kinematic viscosity ν, demonstrating that velocity and length scales scale universally with local conditions rather than initial or far-downstream conditions. It identifies two universal scaling layers—inner (wall) and outer (full-free jet)—whose overlap yields a Reynolds-number-dependent power-law profile, which becomes universal when transformed via an intermediate variable η√Reτ, with strong agreement between the mean-velocity overlap layer and the momentum-balance mesolayer.

ABSTRACT

Studies in the literature on plane turbulent wall jets on flat surfaces, have invariably considered either the nozzle initial conditions or the asymptotic conditions far downstream, as scaling parameters for the streamwise variations of length and velocity scales. These choices, however, do not square with the notion of self similarity which is essentially a "local" concept. We first demonstrate that the streamwise variations of velocity and length scales in wall jets show remarkable scaling with local parameters i.e. there appear to be no imposed length and velocity scales. Next, it is shown that the mean velocity profile data suggest existence of two distinct layers - the wall (inner) layer and the full-free jet (outer) layer. Each of these layers scales on the appropriate length and velocity scales and this scaling is observed to be universal i.e. independent of the local friction Reynolds number. Analysis shows that the overlap of these universal scalings leads to a Reynolds-number-dependent power-law velocity variation in the overlap layer. It is observed that the mean-velocity overlap layer corresponds well to the momentum-balance mesolayer and there appears to be no evidence for an inertial overlap; only the meso-overlap is observed. Introduction of an intermediate variable absorbs the Reynolds-number dependence of the length scale in the overlap layer and this leads to a universal power-law overlap profile for mean velocity in terms of the intermediate variable.

Motivation & Objective

• To resolve the long-standing challenge of scaling mean velocity in plane turbulent wall jets.
• To determine whether streamwise development of velocity and length scales depends on initial conditions (ICs) or far-downstream conditions (FCs), or on local parameters.
• To test the hypothesis of self-similarity in wall jets by identifying universal scaling layers independent of Reynolds number and ICs/FCs.
• To investigate the existence and nature of an inertial overlap layer and to reconcile the mean-velocity overlap with physical mesolayers such as the momentum-balance mesolayer.
• To develop a universal power-law profile for the mean velocity in the overlap region by introducing an intermediate variable that absorbs Reynolds-number dependence.

Proposed method

• Analyzes experimental and DNS data from multiple sources to assess streamwise scaling of Umax, zT, and Uτ using local kinematic momentum rate M and ν as scaling parameters.
• Identifies two distinct scaling layers: an inner (wall) layer scaled on Uτ and ν, and an outer (full-free jet) layer scaled on Umax and zT, both universal across Reynolds numbers.
• Applies overlap analysis between the inner and outer universal scaling laws to derive a Reynolds-number-dependent power-law profile for the mean velocity in the overlap region.
• Introduces an intermediate variable η√Reτ (or z+/√Reτ) to collapse data across Reynolds numbers, absorbing Reτ dependence and yielding a universal power-law profile.
• Compares the mean-velocity overlap layer with the momentum-balance mesolayer using DNS data to validate the physical consistency of the intermediate variable.
• Performs systematic data collapse and plotting across low- and high-Reynolds-number regimes to test universality and layer behavior.

Experimental results

Research questions

• RQ1 Do streamwise variations of velocity and length scales in wall jets depend on initial conditions (ICs) or far-downstream conditions (FCs), or on local kinematic momentum rate M and viscosity ν?
• RQ2 Do the inner (wall) and outer (full-free jet) layers in wall jets scale universally, independent of Reynolds number and ICs/FCs?
• RQ3 Does the overlap region between the inner and outer scaling laws exhibit a universal power-law profile when transformed via an intermediate variable?
• RQ4 Is the mean-velocity overlap layer physically consistent with the momentum-balance mesolayer in wall jets?
• RQ5 Is there evidence for an inertial overlap layer between the inner and outer scaling regions, or is only a meso-overlap observed?

Key findings

• Streamwise variations of Umax, zT, and Uτ scale universally with local kinematic momentum rate M and viscosity ν, indicating self-similar development independent of ICs and FCs.
• The wall jet exhibits two universal scaling layers: an inner layer scaled on Uτ and ν, and an outer layer scaled on Umax and zT, both independent of Reτ and ICs/FCs.
• The overlap of these universal layers produces a Reynolds-number-dependent power-law mean velocity profile in the overlap region.
• The mean-velocity overlap layer closely coincides with the momentum-balance mesolayer, validated by DNS data.
• Introducing the intermediate variable η√Reτ (or z+/√Reτ) collapses all experimental data onto a single universal power-law curve over 42% of a decade in the intermediate variable (0.7 < η√Reτ < 3).
• No inertial overlap is observed; only a meso-overlap exists, and the outer scaling extent increases with Reτ, indicating stronger outer flow influence at higher Reynolds numbers.