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HIGHSPEED AERODYNAMICS

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UNIT-3

3. DIFFERENTIAL EQUATIONS OF MOTION FOR STEADY COMPRESSIBLEFLOWS 9Small perturbation potential theory, Solutions for supersonic flows. Mach

waves and Mach angles, Prandti-Glauert affine transformation relations for

subsonic flows, Linearised two dimensional supersonic flow theory, Lift, drag

pitching moment and centre of pressure of supersonic profiles.

Small perturbation potential theory

LINEARIZED FLOW

Transport yourself back in time to the year 1940, and imagine that you are an

aerodynamicist responsible for calculating the lift on the wing of a high-

performance fighter plane. You recognize that the airspeed is high enough so

that the well-established incompressible flow techniques of the day will give

inaccurate results. Compressibility must be taken into account. However, you

also recognize that the governing equations for compressible flow are nonlinear,

and that no general solution exists for these equations. Numerical solutions are

out of the question! So, what do you do?The only practical recourse is to seek

assumptions regarding the physics of the flow which will allow the governing

equations to become linear, but which at the same time do not totally

compromise the accuracy of the real problem. In turn, these linear equations can

be attacked by conventional mathematical techniques.

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Comparison between uniform and

perturbed flows

There are a number of practical aerodynamic problems where, on a physicalbasis, a uniform flow is changed, or perturbed, only slightly. One such example

is the flow over a thin airfoil illustrated in the above figure. The flow is

characterized by only a small deviation of the flow from its original uniform

state. The analyses of such flows are usually called small-perturbation theories.

Small-perturbation theory is frequently (but not always) linear theory, an

example is the acoustic theory, where the assumption of small perturbations

allowed a linearized solution. Linearized solutions in compressible flow always

contain the assumption of small perturbations, but small perturbations do not

always guarantee that the governing equations can be linearized.

LINEARIZED VELOCITY POTENTIAL EQUATION

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LINEARIZED PRESSURE COEFFICIENTThe pressure coefficient Cp is defined as

where p is the local pressure, and p, , and V are the pressure, density, and

velocity, respectively, in the uniform free stream. The pressure coefficient is simply

a non-dimensional pressure difference; it is extremely useful in fluid dynamics.

An alternative form of the pressure coefficient, convenient for compressible flow,

can be obtained as follows

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Substitute it in the above equation, we get

The above equation is an alternative form of Cp expressed in terms of and M

rather than , and V. It is still an exact representation of Cp.

We now proceed to obtain an approximate expression for Cp which is consistent

with linearized theory. Since the total enthalpy is constant,

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The above equation becomes,

and the above equation gives,

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The above equation is still an exact expression. However considering small

perturbations:

Hence the above equation is of the form

Thus, the previous equation can be expressed in the form of the above equation as

follows, neglecting higher-order terms:

Substituting the above equation in the below equation,

We get,

The above equation becomes,

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The above equation gives the linearized pressure coefficient, valid for smallperturbations. Note its particularly simple form; the linearized pressure coefficient

depends only on the x component of the perturbation velocity.

Prandtl-Glauert rule

It is a similarity rule, which relates incompressible flow over a given two-

dimensional profile to subsonic compressible flow over the same profile.

where Cp0is the incompressible pressure coefficient.

The above equation is called the Prandtl-Glauert rule.

Consider the compressible subsonic flow over a thin airfoil at small angle of attack

(hence small perturbations), as sketched in the Fig 9.2 (pp.259). The usual inviscid

flow boundary condition must hold at the surface, i e., the flow velocity must be

tangent to the surface. Referring to Fig. 9 2, at the surface this boundary condition

is

We have the linearized perturbation-velocity potential equation.

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Note that this is an approximate equation and no longer represent the exact physicsof the flow.

1. The perturbations must be small.

2. Transonic flow 0.8 < M< 1.2) is excluded.

3. Hypersonic flow (M> 5) is excluded.

This equation is valid for subsonic and supersonic flow only. However, this equation

has the striking advantage that it is linear.

In summary, we have demonstrated that subsonic and supersonic flows lend

themselves to approximate, linearized theory for the case of irrotational, isentropicflow with small perturbations. In contrast, transonic and hypersonic flows cannot be

linearized, even with small perturbations. This is another example of the consistency

of nature.Note some of the physical problems associated with transonic flow (mixed

subsonic-supersonic regions with possible shocks, and extreme sensitivity to

geometry changes at sonic conditions) and with hypersonic flow (strong shock waves

close to the geometric boundaries, i e., thin shock layers, as well as high enthalpy,

and hence high-temperature conditions in the flow). Just on an intuitive basis, we

would expect such physically complicated flows to be inherently nonlinear. For the

remainder of this chapter, we will consider linear flows only; thus, we will deal with

subsonic and supersonic flows.

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xxx

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An airfoil(in American English) or aerofoil(in British English) is the shape of a wing or

blade (of a propeller, rotor or turbine) or sail as seen in cross-section.

An airfoil-shaped body moved through a fluid produces an aerodynamic force. The

component of this force perpendicular to the direction of motion is called lift. The

component parallel to the direction of motion is called drag. Subsonic flight airfoils have

a characteristic shape with a rounded leading edge, followed by a sharp trailing edge,often with asymmetric camber. Foils of similar function designed with water as the

working fluid are called hydrofoils.

The lift on an airfoil is primarily the result of its angle of attack and shape. When oriented

at a suitable angle, the airfoil deflects the oncoming air, resulting in a force on the airfoil

in the direction opposite to the deflection. This force is known as aerodynamic force and

can be resolved into two components: Lift and drag. Most foil shapes require a positive

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Any object with an angle of attack in a moving fluid, such as a flat plate, a building, or

the deck of a bridge, will generate an aerodynamic force (called lift) perpendicular to the

flow. Airfoils are more efficient lifting shapes, able to generate more lift (up to a point),

and to generate lift with less drag.

A lift and drag curve obtained in wind tunnel testing is shown on the right. The curverepresents an airfoil with a positive camber so some lift is produced at zero angle of

attack. With increased angle of attack, lift increases in a roughly linear relation, called the

slope of the lift curve. At about 18 degrees this airfoil stalls, and lift falls off quickly

beyond that. The drop in lift can be explained by the action of the upper-surface boundary

layer, which separates and greatly thickens over the upper surface at and past the stall

angle. The thickened boundary layer's displacement thickness changes the airfoil's

effective shape, in particular it reduces its effective camber, which modifies the overall

flow field so as to reduce the circulation and the lift. The thicker boundary layer also

causes a large increase in pressure drag, so that the overall drag increases sharply near

and past the stall point.

Airfoil design is a major facet of aerodynamics. Various airfoils serve different flight

regimes. Asymmetric airfoils can generate lift at zero angle of attack, while a symmetric

airfoil may better suit frequent inverted flight as in an aerobatic airplane. In the region of

the ailerons and near a wingtip a symmetric airfoil can be used to increase the range of

angles of attack to avoid spin-stall. Thus a large range of angles can be used without

boundary layer separation. Subsonic airfoils have a round leading edge, which is

naturally insensitive to the angle of attack. The cross section is not strictly circular,

however: the radius of curvature is increased before the wing achieves maximum

thickness to minimize the chance of boundary layer separation. This elongates the wing

and moves the point of maximum thickness back from the leading edge.

Supersonic airfoils are much more angular in shape and can have a very sharp leading

edge, which is very sensitive to angle of attack. A supercritical airfoil has its maximum

thickness close to the leading edge to have a lot of length to slowly shock the supersonic

flow back to subsonic speeds. Generally such transonic airfoils and also the supersonic

airfoils have a low camber to reduce drag divergence. Modern aircraft wings may have

different airfoil sections along the wing span, each one optimized for the conditions in

each section of the wing.

Movable high-lift devices, flaps and sometimes slats, are fitted to airfoils on almost every

aircraft. A trailing edge flap acts similarly to an aileron; however, it, as opposed to an

aileron, can be retracted partially into the wing if not used.

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As an object moves through a fluid, the velocity of the fluid varies around the surface of

the object. The variation of velocity produces a variation of pressure on the surface of the

object as shown by the the thin red lines on the figure. Integrating the pressure times thesurface area around the body determines the aerodynamic force on the object. We can

consider this single force to act through the average location of the pressure on the

surface of the object. We call the average location of the pressure variation the center of

pressurein the same way that we call the average location of the weight of an object the

center of gravity. The aerodynamic force can then be resolved into two components, lift

and drag, which act through the center of pressure in flight.

Determining the center of pressure is very important for any flying object. To trim an

airplane, or to provide stability for a model rocket or a kite, it is necessary to know the

location of the center of pressure of the entire aircraft. How do engineers determine the

location of the center of pressure for an aircraft which they are designing?

In general, determining the center of pressure (cp) is a very complicated procedure

because the pressure changes around the object. Determining the center of pressure

requires the use of calculus and a knowledge of the pressure distribution around the body.

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Pitching moment

In aerodynamics, the pitching momenton an airfoil is the moment (or torque) produced

by the aerodynamic force on the airfoil if that aerodynamic force is considered to be

applied, not at the center of pressure, but at the aerodynamic center of the airfoil. The

pitching moment on the wing of an airplane is part of the total moment that must bebalanced using the lift on the horizontal stabilizer.

The lift on an airfoil is a distributed force that can be said to act at a point called the

center of pressure. However, as angle of attack changes on a cambered airfoil, there is

movement of the center of pressure forward and aft. This makes analysis difficult when

attempting to use the concept of the center of pressure. One of the remarkable properties

of a cambered airfoil is that, even though the center of pressure moves forward and aft, if

the lift is imagined to act at a point called the aerodynamic center the moment of the lift

force changes in proportion to the square of the airspeed. If the moment is divided by the

dynamic pressure, the area and chord of the airfoil, to compute a pitching moment

coefficient, this coefficient changes only a little over the operating range of angle ofattack of the airfoil. The combination of the two concepts of aerodynamic center and

pitching moment coefficient make it relatively simple to analyse some of the flight

characteristics of an aircraft.

A graph showing coefficient of pitching moment with respect to angle of attack. Thenegative slope for positive indicates stability in pitching.