In 1904, 29-year old Ludwig Prandtl introduced an approach to understanding and analyzing flows around bodies that produced a massive breakthrough.
In a paper presented during the Third Congress of Mathematicians at Heidelberg, Germany, Prandtl introduced the boundary layer. This concept would eventually lay down the foundations of modern aeronautical engineering, and today is at the base of most modern calculations of skin friction, heat transfer, and flow separation.
Prandtl was born on February 4, 1875, in Freising, Bavaria. In 1894 he started his scientific studies at the Technische Hochschule in Munich, from where he graduated six years later with a Ph.D. in solid mechanics. In 1901, Prandtl became Professor of Mechanics in the Mathematical Engineering Department at the Technische Hochschule in Hanover (today’s University of Hannover). There he developed his brand-new interest in fluid mechanics (branch of physics concerned with the motions of fluids and specifically with the relationships among force and matter). The effect of his 1904 paper was so significant that he became Director of the Institute of Applied Mechanics later that year at the renowned University of Göttingen.
Prandtl’s contribution to the advancement of aerodynamics did not end up with the boundary layer. His name is also associated with many other significant advances in the field, such as his famous lifting-line theory for finite wings that he developed when working on low-speed aerodynamics and airfoils.
After leading the world of aerodynamics for most of the first part of the twentieth century, Prandtl died in 1953. Still, his contribution makes him be considered today by many as the father of modern aerodynamics. But, his findings apply not only to air but also to any fluid flowing around a body: in sailing, we find Prandtl associated with aerodynamics (e.g., airflow around a sail) and hydrodynamics (e.g., water flow around a keel).
Two regions of flow
Prandtl was the first to realize that the effects of the fluid’s viscosity, i.e., the effects of friction, were only relevant in places where two adjacent streamlines had substantially different velocities. However, he noticed that for the vast majority of the flow away from the body, the differences in velocities among streamlines were relatively small, while only in a thin layer of flow just over the body surface were these velocity differences high.
This finding led him to conceptualize the flow around bodies separated in two different areas: an area away from the body where the fluid can be treated as being inviscid (no viscosity, i.e., friction plays no role), and a thin region immediately adjacent to the body surface where the flow can be considered as viscous. He named this thin layer the boundary layer.
As we discussed in “Flows in sailing,” this approach to flows around bodies by separating them into two different regions has produced substantial results:
- the application of inviscid fluid theory to the region outside the boundary layer adequately predicts the pressure distribution, flow separation, lift, and induced drag on the body;
- the application of viscous fluid theory to the boundary layer adequately predicts skin friction, heat transfer (important topic for high-speed flows, particularly supersonic), and flow separation;
But how can inviscid theory predict pressure distribution over the surface of a body when the body itself is surrounded by a region of flow where viscosity plays a fundamental role? While inviscid calculations provide correct pressure values at the outer edge of the boundary layer, it is a fact that this pressure does not change within the boundary layer: the pressure value remains unaltered from the outer edge to the body surface in a direction perpendicularly to it. This phenomenon is true only for attached boundary layers (see section “Separation” below).
Inside the boundary layer
Real-life flows are viscous. Due to viscosity, the molecules in the flow in contact with the surface of a body do not move. They remain stuck to the surface, which means that their velocity is always zero. This fact, known as the no-slip condition, is always true regardless of the smoothness of the body surface.
Above the body surface, the flow velocity increases until it reaches the outer edge of the boundary layer, where it moves with the speed of the external flow. The boundary layer thickness, which is comparatively very thin regarding the outer flow, is defined as the distance from the body surface to the point where the velocity is 99% of the freestream speed.
The reason behind this velocity variation (velocity gradient) within the boundary layer is viscosity: because of attraction forces among fluid molecules in proximity, the first layer of flow, which remains fixed to the surface, will slow down the second layer causing it to move at a relatively low speed regarding the first one. As we move up further away from the body surface through millions of layers, this process continues until the flow velocity eventually reaches the speed of the freestream.
Based on the type of viscous flow, we can distinguish two different regions within the boundary layer:
- the laminar boundary layer and;
- the turbulent boundary layer.
The flow in the laminar boundary layer is laminar. In this region, the streamlines are smooth and regular.
On the other hand, in the turbulent boundary layer, the flow is turbulent. In it, the streamlines break up, the fluid elements move irregularly, and the boundary layer thickness grows bigger than that of the laminar one.
If the freestream flow is laminar, the boundary layer over a body will also start as laminar. Then, at some distance from the leading edge, because of the development of unsteadiness and disturbances introduced by surface roughnesses, the laminar flow will become unstable within a region known as the transition region (often considered a third boundary layer region) until eventually developing into entirely turbulent. The critical value is the distance from the leading edge to the point where the transition from laminar to turbulent flow occurs.
The velocity profile perpendicular to the surface is different in the laminar and turbulent boundary layers. In the laminar one, the flow velocity increases smoothly from zero to the freestream velocity as one moves away from the surface. However, in the turbulent one, the velocity through the layers remains reasonably close to the freestream velocity before rapidly decreasing to zero at the body surface. The viscous effects (e.g., friction) within the turbulent flow will be more significant than within the laminar flow because of these higher velocities.
Based on what we have just discussed, keeping the laminar boundary layer as long as possible will be necessary to reduce the friction drag as much as possible. However, in certain applications, the disadvantage of a higher friction drag is often overcome by the higher flow velocities and momentum developed in the turbulent layer, which will help delay or even completely avoid flow separation. This is the reason why, for example, golf balls display dimples.
A flow moving around a body will either speed up or slow down to follow the body geometry. Thanks to Bernoulli, we know that, within the inviscid part of the flow, the pressure decreases when velocity increases, and the pressure increases when the velocity increases.
What happens inside the boundary layer? Here Bernoulli is not applicable. But, as we have seen, the flow pressure at the edge of the boundary layer is transmitted down unchanged until the body surface. When a fluid element inside the boundary layer flows downstream to a lower pressure region, the lower pressure drags the element along, helping the flow to keep moving downstream following the body geometry. But, when a fluid element moves downstream from lower to higher pressures, the higher pressure works against the flow, slowing it further down.
The velocity of the flow located outside the boundary layer may help overcome the effects of higher pressures downstream. However, for the layers of flow within the boundary layer, especially those closer to the body surface, which are already moving at a very slow speed, the higher pressure can make the flow stop completely and even reverse its direction.
We call recirculating flow the flow that reverses due to the effect of adverse pressure gradients. When recirculating flow occurs, the “normal” flow moves downstream on top of the recirculating flow region, increasing the boundary layer’s thickness.
When the pressure increases in the flow direction, we say there is an adverse pressure gradient. If the adverse pressure gradient is moderate (the pressure increases moderately in the flow direction), the flow may remain attached to the body. However, if the flow encounters high adverse pressure gradients (i.e., the pressure increases too rapidly), recirculating flow will happen, and the boundary layer will separate from the body surface.
Separation is, therefore, a viscous effect, and it occurs when the boundary layer detaches from the body surface. It can happen on both laminar and turbulent boundary layers, although due to higher speeds and higher energy, turbulent boundary layers are less prone to it. Detached flows can later reattach to the body surface if the pressure gradient is favorable.
Separation increases what is known as pressure drag (resistance component due to pressure imbalance between the from and rear parts of a body). Furthermore, when the flow separates without reattachment, we experience what is defined as a stalled condition (the lift force will be reduced drastically).
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