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John O | April 2019

Researchers add new insights to one of the foundational texts in fluid mechanics

By Josh Perry, Editor


Engineers at the University of Pittsburgh (Pa.) used linear stability theory and direct numerical simulations to discover the fluid instabilities in the model for katabatic slope flows originally proposed by German scientist Ludwig Prandtl, who is considered the father of modern aerodynamics.


Pitt researchers have discovered new insights into the foundational text for modern aerodynamics. (University of Pittsburgh/YouTube)


Prandtl’s book, “Essentials of Fluid Dynamics,” was translated from the original German in 1952 and has been adapted and revised numerous times by engineering students and researchers. It is currently available under the title, “Prandtl – Essentials of Fluid Mechanics.”


According to a report from the university, the last three pages of Prandtl’s original work have been largely ignored by the fluid mechanics community and those mathematical solutions had disappeared from the current version. That was until Pitt researchers, using supercomputers, explored those equations to find new insights.


“Katabatic slope flows are gravity-driven winds common over large ice sheets or during nighttime on mountain slopes, where cool air flows downhill,” the article explained. “Understanding those winds are vital for reliable weather predictions, which are important for air quality, aviation and agriculture. But the complexity of the terrain, the stratification of the atmosphere and fluid turbulence make computer modeling of winds around mountains difficult. Since Prandtl’s model does not set the conditions for when a slope flow would become turbulent, that deficiency makes it difficult, for example, to predict weather for the area around Salt Lake City in Utah, where the area’s prolonged inversions create a challenging environment for air quality.”


In the age of supercomputers, it is easier for engineers to calculate the impact of complex terrains. Researchers discovered that the original formulation is “prone to unique fluid instabilities” because of the slope angle and a new dimensionless number.


One of the key findings is that a single dimensionless number cannot determine the dynamic stability of katabatic slope, which is the current model for the meteorology and fluid dynamics community. While the Richardson number is part of the equations, researchers have discovered there are other factors at play.


The research was recently published in the Journal of Fluid Mechanics. The abstract stated:


“We investigate the stability of the Prandtl model for katabatic slope flows using both linear stability theory and direct numerical simulations. Starting from Prandtl’s analytical solution for uniformly cooled laminar slope flows, we use linear stability theory to identify the onset of instability and features of the most unstable modes.


“Our results show that the Prandtl model for parallel katabatic slope flows is prone to transverse and longitudinal modes of instability. The transverse mode of instability manifests itself as stationary vortical flow structures aligned in the along-slope direction, whereas the longitudinal mode of instability emerges as waves propagating in the base-flow direction. Beyond the stability limits, these two modes of instability coexist and form a complex flow structure crisscrossing the plane of flow.


“The emergence of a particular form of these instabilities depends strongly on three dimensionless parameters, which are the slope angle, the Prandtl number and a newly introduced stratification perturbation parameter, which is proportional to the relative importance of the disturbance to the background stratification due to the imposed surface buoyancy flux.


“We demonstrate that when this parameter is sufficiently large, then the stabilizing effect of the background stratification can be overcome. For shallow slopes, the transverse mode of instability emerges despite meeting the Miles–Howard stability criterion of Ri>0.25.


“At steep slope angles, slope flow can remain linearly stable despite attaining Richardson numbers as low as 3×10−3.”


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