Copyright © A. Filippone (1996-2001). All Rights Reserved.

In this Chapter

Flow phenomena at low Reynolds numbers are more complicated than those occurring at high Reynolds numbers (flow regimes typical of flight), and to some extent poorly understood. It is particularly interesting to report that it is not quite clear how a low Reynolds number airfoil section should look like (Lissaman, 1983).

The aerodynamics of bluff bodies, instead, seems more advanced, and the technical literature reports cases at Reynolds number as low as a small fraction of unity. Reynolds numbers for lifting bodies are in the range 50,000 to 500,000. For bluff bodies they are much lower. Both streamlined and bluff bodies are reported.

Low Reynolds numbers
Figure 1: Low Reynolds Number Aerodynamics

Aerodynamics of Streamlined Bodies

The low Reynolds number regime leads to some peculiar features, namely:

  • Low resistance of a laminar boundary layer to adverse pressure gradients;
  • Appearance of limited areas of flow separation (bubbles)
  • Turbulence transition triggered by boundary layer instability
  • Effects of free stream disturbances and surface conditions
  • 3-D effects in otherwise 2-D flows
  • non linear lift/drag characteristics
  • lift and drag hysteresis at static conditions
  • bifurcations of boundary layer states

Bifurcation of boundary layer states yields non unique and un-symmetric solutions even for symmetric conditions.

Theories of slender body aerodynamics at low Reynolds number have been developed by several authors, including Burgers (1938), Taylor (1969), Lighthill (1975). At very low Reynolds number the inertial forces are negligible.

Separation Bubble

A separation bubble is a region of locally separated flow on the airfoil. The extent of this region depends on the operational parameters (Reynolds number, angle of attack, free stream turbulence), and airfoil geometry (thickness, camber, surface quality). Depending on a complicated combination among the above quantities the bubble can be short or long, it can contract or extend with the increasing angle of attack.

Figure 2: Laminar Separation Bubble

A long separation bubble usually starts far behind the leading edge, causes a collapse of the leading edge pressure peak and modifies the total pressure distribution on the upper side of the airfoil. This type of bubble is associated with a large loss in lift.

A short bubble is just behind the leading edge, does not alter macroscopically the surface pressure distribution, and changes only slightly the lift coefficient.

bubble lift

Figure 3: Cp characteristics

Turbulent Transition

Bubble reattachment and airfoil characteristics are strongly dependent on turbulent transition. A bubble reattaches as turbulent, transition occurring at some location within the bubble.

At very low Reynolds number a delayed transition may prevent bubble reattachment, and thus cause a premature stall and a consistent loss of lift. For this reason accurate knowledge of transition is necessary.

Turbulent Transition

Figure 4: Laminar bubble and turbulent transition on low speed airfoil, angle of attack 10 degs. (courtesy of Luke Brown)

Factors affecting Transition

The most general physical causes that trigger turbulent flow transition on a solid wall are the following:

  • External pressure gradients
  • Temperature
  • Surface roughness
  • External disturbances and acoustic waves

Forced Transition

If transition does not occur by natural means, it can be forced by operating of the surface roughness or adding transition trips of appropriate size and shape. One simple criterion sometimes applied to predict bubble reattachment is the Owen-Klanfer criterion, that consists in evaluating the Reynolds number based on boundary layer thickness.

Predicting Transition

There are several theoretical methods for predicting turbulent transition. Some methods commonly used in aerodynamics include the Michel method, the eN method. The accuracy of these methods (or any other methods currently known) is not always sufficient for computing airfoil characteristics as those reported below.

Lift/Drag Characteristics

Lift and drag characteristics are affected by the Reynolds number in a way that is unknown at the speeds proper of commercial flight. The extent of the viscous flow and the separated region (e.g. the size and behavior of the separation bubble).

Fig. 4 and Fig. 5 show two different, albeit typical, lift curves at Reynolds numbers below 100,000.

bubble lift

Figure 4: Airfoil Lift characteristics

In the figure above the lift curve is dominated by the laminar separation bubble (B). When the bubble contracts with the increasing incidence, the bubble lift decreases slightly, then increases again and the airfoil finally stalls with a trailing edge separation.

bubble lift

Figure 5: Airfoil Lift characteristics

The case above shows a a hysteresis loop (I), that occurs when the airfoil flow at increasing angle of attack features different characteristics of those at decreasing angle of attack. This result

bubble drag

Figure 6: Drag characteristics

Aerodynamics of Bluff Bodies

The bluff bodies considered at low Reynolds number are almost exclusively of spheric shape, since they include the motion of liquid drops bubbles and particulate in air, that are of general technical interest (for example, cavitation problems, combustion, fluidized beds, magneto- hydrodynamics; diffusion problems, etc.).

At the limit of zero Reynolds number the Navier-Stokes equations are reduced to a condition of equilibrium for the pressure, whose fundamental solution is a point force called stokeslet (Hancock, 1953). In practice the Reynolds number cannot be zero, so small inertia forces are present.

The drag coefficient of solid particles and bubbles has been widely investigated, and often some correlations are given: Oseen’s equation for Re < 1; For reference, the drag coefficient of a bubble is CD = 10 at Re = 1; CD = 2.6 at Re = 42. Data are also available for slender cylinders.

Current Research Topics

For streamlined bodies (airfoils and wings) there is ongoing research in the fields of wind tunnel testing, airfoil design, turbulent transition studies for gliding, human-powered flight, solar-powered flight, wind turbine blades, micro air vehicles.

For bluff bodies the field of research includes a wide array of industrial multi-phase flows, wherein at least one phase is a solid particulate (exhaust gases, fluidized beds, combustion problems), or a bubble (cavitation), dynamics of sprays and jets.

Transition prediction is an interesting topic for the most advanced CFD methods. The traditional Reynolds averaged Navier-Stokes equations deal with ensemble averaged equations, and therefore is unable to resolve the large scale eddies. The large eddy simulation (LES) methods model the turbulence in the sub-grid scale.

As an alternative, direct Navier-Stokes solution (DNS) may be used. This approach requires very fine grids and small steps to resolve the Kolmogoroff scale. Both LES and DNS are still at a development stage to predict how well they can predict turbulent transition at small Reynolds numbers.

Related Material

On the Web

These sites are not part of the domain. There is no control over their content or availability.

  • Low Speed Airfoils at the University of Illinois

Selected References

  • Mueller, TJ (editor). Fixed and Flapping Wing Aerodynamics for Micro Air Vehicles, AIAA Progress in Aeronautics and Astronautics, Vol 195, Reston, VA, 2001.

  • Proceedings of the Conference on Low Reynolds Number Airfoil Aerodynamics, UNDAS-CP-77B123, Notre Dame, Indiana, June 1985.

  • Proceedings of the Aerodynamics at Low Reynolds Number 10^4 < Re < International Conference, The Royal Aeronautical Society, London, Oct. 1986.

  • AGARD, Low Reynolds Number Vehicles , AGARDograph AG-288, 1985.

  • Selig M, Lyon C, Giguere P, Ninham C, and Guglielmo J., Summary of Low-Speed Airfoil Data, Vol. 2, SoarTech Publ., Virginia Beach, VA, 1996.

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Copyright © A. Filippone (1996-2001). All Rights Reserved.