In this Chapter
- Importance of the Subject
- Dynamic Stall
- High Angle of Attack
- Unsteady Wakes behind Bluff Bodies
- Unsteady Boundary Layers
- Computed Examples
- Selected References
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Aircraft agility depends on the control of the unsteady flow field |
Although most aerodynamic flows are treated as steady ones, many others are non
stationary. The variety of non stationary flows is large, and includes transient
regimes, impulsive starts, maneuvering, periodic flows, and flows that are
intrinsically unsteady because of the mechanism of vortex shedding from bluff bodies.
The ability to control general three-dimensional unsteady flows could open new
possibilities in the performance of many aerodynamic systems, including aircraft,
helicopter and wind energy conversion systems.
Other aspects of unsteady flows include: hydrodynamic propulsion (propeller-hull
interaction), flapping wing propulsion, a number of cavity flows, and many heat
transfer problems.
The flows that will be considered are:
The unsteady problems of oscillatory type have been widely studied for airfoils and wings since the 1930s, when the first theories
have been formulated (Theodorsen, 1932).
Dynamic stall affects helicopter rotor blades in forward flight, maneuvering and
descent (because of the asymmetric loads created by the flight dynamics); wind
turbine rotors (because of the unsteady nature of the wind, along with the
atmospheric boundary layer, the presence of the tower, the topography of the terrain,
etc.).
Dynamic stall on airfoils is a particular case of the above. At the highest speed of
a helicopter rotor another peculiar aspect appears: the unsteady shock wave on the
blade.
Other unsteady flows of practical importance include flows past circular cylinders
(von Karman, 1930s), and past spheres. These phenomena are oscillatory only at very
low Reynolds numbers and become fully turbulent and aperiodic at higher speeds. The
largest Reynolds number at which the von Karman vortex street is observed is Re=400,
that corresponds to a Strouhal number St=0.21.
The non dimensional parameter defining the similitude of periodic flows is the
Strouhal number St=fL/V (f=frequency; L=characteristic length; V =
characteristic speed), or the reduced frequency
k=2
Periodic flows on airfoils and wings (plunging, pitching and a combination of the
two) lead to a peculiar effect called dynamic stall. The main reason why
dynamic stall appears is the finite response time of the flow to an incoming
disturbance (for ex. change in angle of attack, free stream turbulence effects,
etc.).
The response time (sometimes called time-lag) is dependent on the viscous
effects, which ultimately lead to energy dissipation. The latter is proportional to
the integral of the hysteresis loop.
The first to provide a mathematical description of airfoils in flutter was Theodorsen
(1932). His theory was based on linearized small perturbation equations.
Besides the cylinder and the sphere at relatively high Reynolds numbers, there is a
number of flows that are unsteady and aperiodic, although they can be treated as
steady. These include airfoils and wings at high angle of attack.
Delta wings, pointed cylinders and prolate spheroids are some examples of
technological interest (fighter aircraft, missile aerodynamics, etc.) Their behavior
is related to the dynamics of the vortices released from the body surface.
Unsteady flows developing in the wakes of bluff bodies, particularly on road
vehicles and aircraft after bodies, are of interest from the point of view of the base
drag that is produced at the rear end. Similar interest in reported for some cavity
flows.
Fundamental studies are available for the flat plate at all speeds (up to hypersonic)
and all angle of attacks (up to 90 degrees). This is by itself a sign of the
importance of this simple device to the understanding of basic fluid dynamic
problems, besides airfoils (Fig. below).
Other systems at angle of attack include blunt and pointed bodies (prolate
spheroids, pointed cylinders), delta wings and low aspect-ratio wings. At the other
end of the technology there is the full aircraft (Lamar, 1992).
Flow separation on these systems is quite complex. On low aspect-ratio wings and
delta wings the flow separation produced a substantial augmentation of lift (besides
drag).
Wakes behind bluff bodies are unsteady (and sometimes periodic) at any realistic
Reynolds number. Simple geometries like the circular cylinder and the sphere have
been investigated for a long time, in order to understand the physics of flow
separation and vortex formation in 3-D. These bodies also present a technical
interest from the point of view of base drag reduction on cars, trucks, aircraft
after bodies and other vehicles.
Flow analysis around bluff bodies such as suspension bridges, tall buildings, and
towers is nowdays an essential element in the engineering process.
The circular cylinder, along with the flat plate, is the body most widely
studied in fluid dynamics and aerodynamics. Drag data for the cylinder are
known from very low Reynolds numbers all the way to hypersonic speeds.
Vortex Shedding. Systematic vortex shedding analysis was first due to von
Karman, who analyzed the breakdown of the symmetric flow. The von Karman vortex
street has become one of the most well known unsteady problem. Impulsive start
was already known to Prandtl (1904), and the rotating cylinder was known to Tollmien
(1931).
Drag data are tabulated for all Reynolds numbers, flow visualizations are
available up to Mach numbers M=12.1 (to the author’s knowledge). Although the
unsteady wake behind the cylinder has been considered for a long time as purely two
dimensional, there are spanwise vortex structures that appear at some Reynolds
numbers. These structures are a function of the cylinder aspect ratio L/D.
References on the circular cylinder can be found in any text of fluid dynamics.
Wakes behind spheres are observed to be steady for Reynolds numbers below
300-400. Above this limit (which also depends on the surface finish) vortices break
off and are periodically released to form vortex loops that are connected like in a
chain.
At Re above 6000 the vortex shedding is very periodic, with Strouhal number
ranging from 0.125 to 0.20, the largest figure being a limit at high Reynolds numbers
(Achenbach, 1974). Similar wakes can be observed behind particles falling in
water. Effects of the surface geometry have been studied for the evaluation of the
aerodynamic performances of sports balls (Metha, 1985).
Bluff bodies other than cylinder and sphere include a wide variety of
configurations. Squared cylinders, elliptic cylinders and parallelepipeds of various
aspect-ratios are used to simulate more complex real-life objects. We will limit
this discussion to road vehicles.
Passenger cars, buses and trucks have blunt trailing edges, and various
cavities. Streamlining has been applied (successfully) to both cars and trucks (on a
lesser degree to buses). However, streamlining on fore bodies has little influence on
flow separation and drag. The after body is instead critical. Among the extensive
studies performed by the car industry it is interesting to report the effect of the
slant angle on both lifting and non lifting bluff bodies.
The amount of research available for unsteady boundary layer is a tiny thing compared
with the body of work carried out on steady, incompressible boundary layers. Typical
cases available in the literature include flows past flat plates and circular
cylinders, flows started from rest, colliding shear layers, and periodic
oscillations.
Oscillating boundary layers are the basis of dynamic stall behavior,
and therefore are studied from the point of view of unsteady separation. Boundary
layer separation is related to the amplitude of the oscillations. For a review of
recent work see Cousteix, 1986.
Importance of the Subject
Oscillatory Flows
Dimensionless Parameters
St. The two
non-dimensional groups are equivalent within a constant. The reduced frequency is
used more often for dynamic stall problems.
Dynamic Stall
Non Oscillatory Flows
High Angle of Attack Aerodynamics
Unsteady Wakes behind Bluff Bodies
Flow Past a Cylinder
Related Material
Flow Past a Sphere
Flow Past other Bluff Bodies
Unsteady Boundary Layers
Related Material
Selected References
