Conceit o

Dr. Nikos J. Mourtos AE 264

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SUBSONIC FLOW THEORY

1.0 LINEARIZED 2 D SUBSONIC FLOW

Compressible, subsonic flow over a thin airfoil @ small aoa.

y = f x( ) = function that defines the shape of the airfoil in the

x,y( ) space Tangency Condition (TC) for inviscid flow

dfdx =

V + u

= tan For small perturbations

u V and tan TC

dfdx

V

Now

=y

Tangency Condition for Linearized Theory

=y =V

dfdx

Write the LPVPE for 2-D flow

xx +yy = 0 1M2

We can transform this eq. to the familiar Laplace eq. for 2-D incompressible flow via a new coordinate system:

= x = y In this transformed space, we define a transformed perturbation velocity

potential as

,( ) = x,y( ) to convert the 2D-LPVPE in terms of the transformed variables:

x =1

y = 0

x = 0

y = the derivatives of

in the

x,y( ) space are related to the derivatives of

in

,( ) space according to:


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x =x =

1 x =

1

x +

x

=1

=

xx =

y =y =

1 y =

1

y +

y

=

=

yy =

substitute into the 2D-LPVPE

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+ = 0 + = 0 This is Laplaces eq., which also governs incompressible flow. Hence,

represents an incompressible flow in

,( ) space, which is related to a compressible flow

in the

x,y( ) space. Shape of the airfoil:

y = f x( ) in the

x,y( ) space

= g ( ) in the

,( ) space Transform the TC

Vdfdx =

y =

1 y =

Applying the TC in the

,( ) space

Vdgd =

The RHS in the last 2 eqs. are equal; equating the LHS:

dfdx =

dgd This eq. says that the slope of the airfoil in the

x,y( ) space and the

,( ) space is the same. This confirms that the transformations we have been using relates the compressible flow over an airfoil in the

x,y( ) space to the incompressible flow over the same airfoil in the

,( ) space. 1.1 The Linearized Subsonic Pressure Coefficient Following up with the pressure coefficient:

Cp = 2 u V

= 2

Vx =

2V1 x =

2V1

Define the perturbation velocity component in the

- direction by

u = / Then

Cp =1

2u V


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Since the

,( ) space corresponds to incompressible flow from

Cp = 2 u V

we get the incompressible pressure coefficient

Cp0 = 2u V

Combine last 2 eqs:

Cp =Cp01M2

=Cp0

Prandtl Glauert rule Remember that the L, D, and M on an airfoil can be found by

p, distributionsurface . For inviscid flow

= 0 and D = 0 d'Alembert's paradox( ) , hence the L and M on an airfoil can be found by

p distributionsurface . Since the p distribution for subsonic compressible flow is related to the p distribution for incompressible flow through the Prandtl Glauert rule it can be shown that:

Cl =Cl 01M2

=Cl0

Cm =Cm01M2

=Cm0

Effect of compressibility is to increase the magnitude of

Cl and Cm Note that

M 1limCp,Cl ,Cm = thats because linearized theory breaks down near

M =1 Prandtl Glauert rule is accurate up to

M 0.7 Compressibility Effects

u = x =1 x =

1

=u

=u

1 M2

M u i.e., compressibility strengthens the disturbance to the flow by a solid body

In comparison to incompressible flow, a perturbation of a given strength reaches farther away from the surface in compressible flow.

The disturbance reaches out in ALL directions, both upstream and downstream. DAlemberts paradox is valid also for compressible subsonic flow (check Prandtl Glauert rule for cp)


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2.0 IMPROVED COMPRESSIBILITY CORRECTIONS

Attempt to take into consideration some of the nonlinear aspects of the flow. Came about during WWII. Karman Tsien rule

Cp =Cp0

+ M2

1+

Cp02

Laitones rule

Cp =Cp0

+M2 1+

12 M

2

2

Cp0

2.1 Compressibility Corrections to Lift Slope Summary of Related Incompressible Flow Eqs: Lift slope: High AR (AR > 4) straight (unswept) wing with an elliptical lift distribution

a = a01+ a0

A Prandtl

High AR (AR > 4) straight (unswept) wing with a non-elliptical lift distribution

a = a01+ a0

A 1+( )

=

= f (AR,) induced factor for drag

induced factor for lift slope Low AR (AR < 4) straight (unswept) wing with an elliptical lift distribution

a = a0

1+ a0A

2

+a0A

H.B. Helmbold


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Low AR swept wing

a = a0 cos

1+ a0 cosA

2

+a0 cos

A

Kuchemann Compressibility Corrections airfoil lift slope

a0,comp =a0

1M2=a0

Span efficiency factor for lift slope

e1 = (1+ )1 Compressible lift slope for a high-AR straight (unswept) wing

acomp =a0,comp

1+ a0,compe1A

=a0

+a0,compe1A

Modified Helmbolds eq. for a low-AR straight wing (replace

a0 by

a0 /)

acomp =a0

1M2 +a0

e1A

2

+a0A

Modified Kuchemann eq. for a low-AR straight wing (replace

a0 by

a01M,n2

=a0

1M2 cos2 , where

M,n is the Mach number normal to the half-chord line of the wing, which is swept by and angle

)

a = a0 cos

1M2 cos2 +a0 cosA

2

+a0 cosA


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3.0 CRITICAL MACH NUMBER

Critical Mach Number: the free-stream Mach number @ which the flow on the wing first becomes sonic @ the min pressure (max velocity) point on the airfoil. NB: The min pressure point is NOT @ the max (t/c) location even for a symmetrical airfoil @

= 0 ! This is because when the flow adjusts its velocity as it goes around the airfoil, it takes into account the geometry of the entire airfoil, not simply the local highest point. Very important to know, because @ Mcruise slightly higher than Mcr the airfoil experiences a dramatic rise in drag If A is a point on the airfoil, from isentropic flow eqs:

pAp

=pA / p0p / p0

=1+ 12 M

2

1+ 12 MA2

1


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Using the pressure coefficient for compressible flow:

CpA =2

M2pAp

1

Combine the 2 eqs:

CpA =2

M21+ 12 M

2

1+ 12 MA2

1

1

Now if the local MA = 1, by definition

M = Mcr and Cp,A = Cp,cr so

Cp,cr =2

Mcr21+ 12 Mcr

2

1+ 12

1

1

This eq. gives us the Cp @ any point in the flow where the local Mach = 1. This eq. is a universal aerodynamic eq. from isentropic flow; it has no connection with the shape of any given airfoil. How to find the Mcr of a given airfoil

Plot

Cp,cr = f (M) from the universal eq. above. Note that according to this eq.

Cp,cr as

M = Mcr Obtain (experimentally or theoretically) the low speed (incompressible)

Cp0 @ the min pressure point of the airfoil. Use any of the compressibility corrections to plot the variation of

Cpwith

M . Note that according to any of these eqs.

Cp as

M The point @ which these two curves intersect represents the point @ which the local flow velocity @ the min pressure point of the airfoil is sonic. By definition, the free-stream Mach number @ this intersection point is the

Mcrof the airfoil.


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4.0 DRAG DIVERGENCE MACH NUMBER

Divergence Mach Number: the free-stream Mach number @ which the drag on a body begins to increase rapidly as the Mach number increases. This rapid increase can cause the drag coefficient to rise to more than ten times its low speed value. The large increase in drag is associated with strong shock waves that cause BL separation on the airfoil surface. A thinner airfoil will have a higher Mcr and a higher Mdiv.

Sound Barrier: a myth started by the Prandtl-Glauert formula, which predicts

CD as M 1. Wind tunnel tests @ the time were only available for Mach close to 1 but not for M = 1. Convair F-102: could not achieve M > 1 in level flight; had to dive, then level off:


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Convair F 102


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What is striking about the design of the X 1? The tail airfoil was thinner than the wing airfoil. Why? All moving tail. Why?


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To cope with the high D near M = 1, the designers of high speed ac in WWII used 2 features: Thin airfoils

o Bell X-1: 2 sets of wings 10% thick (NACA 65-110) for M < 1, 8% thick (NACA 65-108) for M > 1 operations. Typical ac at the time had wings ~ 15% thick or thicker. Thinner airfoils on the horizontal stabilizer to ensure that when the wing encountered compressibility effects (ex. buffeting) the tail and elevator were still free of such problems to be fully functional for stability & control

o Bell X-1: tail is 6% thick (NACA 65-006) All-moving tails to maintain control in case elevator effectiveness was lost


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4.1 Swept Wings Adolf Busemann (1935) & R.T. Jones (1945) Consider a straight wing with a t/c = 0.15 airfoil. Now the same wing swept back through

= 450 A streamline going over the wing now sees an airfoil as thick as before (t2 = t1) but with a longer chord (c2 = c1 / cos450) This makes the effective t/c = 0.106


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North American F-86 (Korean war)


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4.2 Area Rule

Kuchemann (early 50s) Richard Witcomb

o Zero-lift drag rise near M = 1 is due primarily to shock waves o Shock wave formations about complex swept-wing / body combinations @ zero-lift near M = 1 are similar to those that occur for a body of revolution with the same axial development of cross-sectional area normal to the airstream A(x). o Idea: no abrupt changes in A(x) should occur; indent the body where the wing is mounted, so that the combination has nearly the same A(x) as the original body alone: coke bottle fuselage shape.

Area ruling reduces peak D by a factor of 2 near M = 1


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