The Boundary Layer

Life in Moving Fluids, by Steven Vogel.

In Chapter 1 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I introduce a concept from fluid dynamics called the boundary layer.

The behavior of a sphere moving through a fluid illustrates how flow behavior depends on Reynolds number… At very high Reynolds number, viscosity is small but still plays a role because of the no-slip boundary condition at the sphere surface. A thin layer of fluid, called the boundary layer, sticks to the solid surface, causing a large velocity gradient and therefore significant viscous drag.

For readers who want to know more about the boundary layer, let me quote the start of Chapter 8 of Steven Vogel’s masterpiece Life in Moving Fluids.

At the interface between a stationary solid and a moving fluid, the velocity of the fluid is zero. This, of course, defines the no-slip condition… The immediate corollary of the no-slip condition is that near every such surface is a gradient in the speed of flow. Entirely within the fluid, speed changes from that of the solid to what we call the “free stream” velocity some distance away. Shearing motion is inescapably associated with a gradient in speed, so in these gradients near surfaces, viscosity, fluids’ antipathy to shear, works its mischief, giving rise to skin friction and consequent power consumption. The gradient region is associated with the term “boundary layer”…

The boundary layer… wasn’t so much discovered as it was invented, in the early part of this century, as a great stroke of genius of Ludwig Prandtl. Recognizing the origin of this notion is crucial. In the basic differential equations for moving fluids, the Navier-Stokes equations, some terms result from the inertia of fluids and some from their viscosity… The Reynolds number gives an indication of the relative importance of inertia and viscosity… At Reynolds numbers below unity, inertia can be ignored and nicely predictive rules nonetheless derived-[such as] Stokes’ law for the drag of a sphere… At high Reynolds number, one might expect to get away with neglecting viscosity… It may sound neat, but it all too commonly gets us in trouble-results diverge from physical reality, drag vanishes, and d’Alembert has his paradox.

Prandtl reconciled practical and theoretical fluid mechanics at high Reynolds numbers by recognizing that viscosity could never be totally ignored. What changes with Reynolds number was where it had to be taken into account; initially it mattered everywhere, but as the Reynolds number increased well above unity, viscosity made a difference only in the gradient regions near surfaces. These regions might be small, and they might get ever smaller… as the Reynolds number increased; but as long as the no-slip condition held, a place had to exist where shear rates were high and viscosity was significant. Prandtl called the place in question… the “boundary layer.” In general, a higher Reynolds number implies a thinner boundary layer but a higher shear rate in that boundary layer.

To learn more about the biological significance of the boundary layer see the Chapter 9 in Life in Moving Fluids, which is all about “Life in Velocity Gradients.”

Boundary Layer Theory, by Schlichting and Gersten.

If you want a more rigorous and mathematical analysis of boundary layers, I recommend Boundary Layer Theory by Hermann Schlichting and his student Klaus Gersten. The eighth edition of this book (2000) is cited in IPMB; a revised and updated ninth edition was published in 2017. Schlichting and Gersten write

At the end of the 19th century, fluid mechanics had split into two different directions which hardly had anything more in common. On one side was the science of theoretical hydrodynamics, emanating from Euler’s equations of motion and which had been developed to great perfection. However this had very little practical importance, since the results of this so-called classical hydrodynamics were in glaring contradiction to everyday experience. This was particularly true in the very important case of pressure loss in tubes and channels, as well as that of the drag experienced by a body moved through a fluid. For this reason, engineers, on the other side, confronted by the practical problems of fluid mechanics, developed their own strongly empirical science, hydraulics. This relied upon a large amount of experimental data and differed greatly from theoretical hydrodynamics in both methods and goals.

It is the great achievement of [German scientist] Ludwig Prandtl [1875–1953] which, at the beginning of this century, set forth the way in which these two diverging directions of fluid mechanics could be unified. He achieved a high degree of correlation between theory and experiment, which, in the first half of this century, has led to unimagined successes in modern fluid mechanics. It was already known then that the great discrepancy between the results in classical hydrodynamics and reality was, in many cases, due to neglecting the viscosity effects in the theory. Now the complete equations of motion of viscous flows (the Navier Stokes equations) had been known for some time. However, due to the great mathematical difficulty of these equations, no approach had been found to the mathematical treatment of viscous flows (except in a few special cases). For technically important fluids such as water and air, the viscosity is very small, and thus the resulting viscous forces are small compared to the remaining forces (gravitational force, pressure force). For this reason it took a long time to see why the viscous forces ignored in the classical theory should have an important effect on the motion of the flow.

In his lecture on “Über Flüssigkeitbewegung bei sehr kleiner Reibung” (On Fluid Motion with Very Small Friction) at the Heidelberg mathematical congress in 1904, L. Prandtl… showed how a theoretical treatment could be used on viscous flows in cases of great practical importance. Using theoretical considerations together with some simple experiments, Prandtl showed that the flow past a body can be divided into two regions: a very thin layer close to the body (boundary layer) where the viscosity is important, and the remaining region outside this layer where the viscosity can be neglected. With the help of this concept, not only was a physically convincing explanation of the importance of the viscosity in the drag problem given, but simultaneously, by hugely reducing the mathematical difficulty, a path was set for the theoretical treatment of viscous flows. Prandtl supported his theoretical work by some very simple experiments in a small, self-built water channel, and in doing this reinitiated the lost connection between theory and practice. The theory of the Prandtl boundary layer or the frictional layer has proved to be exceptionally useful and has given considerable stimulation to research into fluid mechanics since the beginning of this century. Under the influence of a thriving flight technology, the new theory developed quickly and soon became, along with other important advances-airfoil theory and gas dynamics — a keystone of modern fluid mechanics.

Introductory Fluid Mechanics L19 p2 — The Boundary Layer Concept.
E. Bodenschatz — Ludwig Prandtl (1875–1953).

Originally published at http://hobbieroth.blogspot.com.

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Professor of Physics at Oakland University and coauthor of the textbook Intermediate Physics for Medicine and Biology.

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Brad Roth

Professor of Physics at Oakland University and coauthor of the textbook Intermediate Physics for Medicine and Biology.