Copyright © A. Filippone (1999-2003). All Rights Reserved.

Unsteady Aerodynamics

In this Chapter

Aerobatics 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.

Importance of the Subject

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:

  • Oscillatory Flows
  • Non Oscillatory Flows
    • Vortex dynamics at high angle of attack
    • Unsteady wakes behind bluff bodies

Oscillatory Flows

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.

Dimensionless Parameters

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 St. The two non-dimensional groups are equivalent within a constant. The reduced frequency is used more often for dynamic stall problems.

Dynamic Stall

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.


Non Oscillatory Flows

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.

High Angle of Attack Aerodynamics

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).


Unsteady Wakes behind Bluff Bodies

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.

Flow Past a Cylinder

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.

Related Material

Flow Past a Sphere

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).

Flow Past other Bluff Bodies

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.


Unsteady Boundary Layers

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.

Related Material

Selected References

  • Theodorsen T. General Theory of Aerodynamic Instability and the Mechanism of Flutter, NACA TR 496, 1935.

  • AGARD. Unsteady Aerodynamics – Fundamentals and Applications to Aircraft Dynamics, AGARD CPP-386, May 1985.

  • Smith FT. Steady and Unsteady Boundary Layer Separation, in Ann. Rev. Fluid Mech, Vol 18, pages 197-220, 1986.

  • Leishman J Gordon. Principles of Helicopter Aerodynamics., Cambridge Univ Press, 2000 (Chapter 8).

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Copyright © A. Filippone (1999-2003). All Rights Reserved.