1. Ch9 Linearized Flow 9.1 Introduction
up to the middle 1950s, before CFD comes
A uniform flow is changed, or perturbed, only slightly
Small-perturbation theories
Frequently (but not always) linear theory, e.g. acoustic theory in Sec. 7.5.
Highlighting some important physical aspects of the flow, explicitly identifying trends and governing parameters, providing practical formulas for the rapid estimation of aerodynamic forces and pressure distributions.

2. -A slender body immersed in a uniform flow

3. 9.2 Linearized Velocity Potential Equation
where , , denotes velocity perturbations from the uniform flow
total velocity potential
Define a new velocity potential － perturbation velocity potential ，
then
where

4. Also ，
substitute into
－ perturbation-velocity potential equation

5. or －
(*)
throughout the flow
(**)
Substitute (**) into (*), and algebraically rearranging

6. － an exact equation for irrotational, isentropic flow
linear
nonlinear
－ an exact equation for irrotational, isentropic flow
Now specialize to the case of small perturbation, i.e. , , , are small compared to ，

7. ， 0≦M∞≦0.8 and M∞≧1.2 （ transonic flow (0.8≦M∞≦1.2) is excluded ）
the magnitude of
2. M∞≦5 (approximately) （ hypersonic flow (M∞≧5 ) is excluded ） ， or
approximate equations are valid for subsonic ＆ supersonic flow only

8. Note： The real physical problems associated with
transonic flow：mixed subsonic –supersonic regions with possible shocks, and extreme sensitivity to geometry changes at sonic conditions.
hypersonic flow：strong shock waves closed to the geometric boundaries, i.e., thin shock layers, as well as high enthalpy, and hence high-temperature conditions in the flow.

9. 9.3 Linearized Pressure Coefficient

10. －exact

11. Consider small perturbations：，
－linearized pressure coefficient
valid for small perturbations
depends only on x-component of the perturbation velocity

12. 9.4 Linearized Subsonic Flow
-Take incompressible results (theory or experiment) and modify them to take compressibility into account.
-Applied for any 2-D shape, including
which satisfies the assumptions of small perturbations.
2-D flow over an airfoil
Flow over a bumpy or wavy body

13. Consider the compressible subsonic flow over a thin airfoil at small angle of attack (i.e. small perturbations)
－inviscid flow boundary condition holds at the surface , i.e at surface // to the surface if

14. for subsonic compressible flow over a 2-D airfoil；
transformed to a familiar incompressible form by considering a transformed coordinate system ，
－transformed perturbation velocity potential

15. ，
Laplace’s equation for incompressible flow
represents an incompressible flow in space, which is related to compressible flow in (x, y) space

16. The shape of airfoil is given by y=f(x) and in (x, y) and space, respectively.
The shape of the airfoil in (x, y) ＆ space is the same.
The compressible flow over an airfoil in (x, y) space transforms to the incompressible flow over the same airfoil in space.

17. Denoting the incompressible perturbation velocity in the direction by , where
Prandtl-Glauert rule
－incompressible pressure coefficient in space
L：Lift force is perpendicular to the V∞
S：a reference area, for a wing, usually the platform area of the wing
：a reference length, for an airfoil, usually the chord length

18. ，
－Prandtl-Glauert rule
valid up to M∞≒0.7
An important effect of compressibility on subsonic flowfields
－Compressibility strengthens the disturbance to the flow introduced by a solid body！

19. c.f.
incompressible flow
－A perturbation of given strength reaches further away from the surface in compressible flow.
－The spatial extent of the disturbed flow region is increased by compressibility.
－The disturbance reaches out in all directions, both upstream and downstream.
In classical inviscid incompressible flow theory
d’Alembert’s paradox：a 2-D closed body experiences no aerodynamic drag.
∵ No friction and its associated separated flow.
∴the pressure distributions over the forward and rearward portions of the body exactly cancel in the flow direction.

20. ∴d’Alembert’s paradox can be generalized to include subsonic compressible flow as well as incompressible flow.
Similar results are obtained from nonlinear subsonic calculations (thick bodies at large angle of attack)

21. Ex 9.1
An uniform upstream M∞ flow over a wavy wall
Using the small perturbation theory, derive an equation for ＆
sol：
Assume
f(x) only
f(y) only

22. A1, A2, B1, B2 are determined by BCs.
y→∞ , V ( ) remains finite. →A2=0 → 2.

23. ＃

24. ＃
－the same cosine variation as the shape of the wall, but is 180˚ out of phase.
－symmetrical distribution
∴ no net force in x-direction.
no drag.

25. as
if incompressible M∞≈0

26. 9.5 Improved Compressibility Corrections
－linearized solutions are influenced predominantly by free-stream conditions; they do not fully recognize change in local regions of the flow
nonlinear phenomena
Improved compressibility correction
－Laitone
local pressure coefficient
local Mach No., can be related to M∞

27. Cp0<<1
－Karman and Tsien：hodograph solution of the nonlinear equations of motion along with a simplified “tangent gas” equation of state.
－Karman-Tsien rule

28. 9.6 Linearized Supersonic Flow
Linearized perturbation-velocity potential equation for 2-D flow
for subsonic flow
for supersonic flow
－elliptic P.D.E.
－hyperbolic P.D.E.
Consider the supersonic flow over a body or surface which introduces small changes in the flowfield, e.g. flow over a thin airfoil,
over a mildly wavy wall,
over a small hump in a surface.

29. －wave equation if g=0 , if f=0,
－lines of =const. correspond to x-λy=const. (left-running Mach lines)
if f=0,
－lines of =const. correspond to x+λy=const. (right-running Mach lines)
－disturbances propagate along Mach lines. ∴the flowfield upstream of a disturbance does not feel the presence of the disturbance.
c.f.
M∞<1, disturbances propagate everywhere in the flowfield.( upstream ＆ downstream )

30. Letting g=0 , ， B.C. on the surface
(local surface inclination w.r.t. V∞)

32. ，
consistent with
a net pressure imbalance
a drag (wave drag)
For the upper surface,

33. Note：
Although shock waves do not appear explicitly within the framework of linearized theory, their consequence in terms of wave drag are reflected in the linearized results.
d’Alembert’s paradox does not apply to supersonic flows！！

35. For M∞=2, linearized theory yields reasonably accurate results for Cp whenθ<4°
Note：CL ＆ CD are more accurate at large angle of attack then one would initially expect.

36. Ex. 9.2
An uniform supersonic flow over the same wavy wall in ex. 9.1
？ Cp？
sol：
Let g=0
B.C.

37. ＃

38. The perturbations do not disappear at y→∞ ↔ subsonic
Note：
The perturbations do not disappear at y→∞ ↔ subsonic ,
← characteristic lines
unsymmetrical streamlines
Cpw is a sine variation, whereas the wall is a cosine shape →a net force in the x-direction.(wave drag)
c.f.

39. 9.7 Critical Mach Number MA=1 M∞=Mcr
Mc ≡M∞ at which sonic flow is first encountered in the airfoil
Assuming isentropic flow,
MA=1
M∞=Mcr

40. －derived from fundamental gas dynamics, independent of the size or shape of the airfoil.
+ Prandtl-Glauert rule
(or Laitone rule
or Karman-Tsien rule)
Mcr for a given airfoil
Cp0 (measured or calculated) different for different airfoil

41. Thin airfoil
mild expansion over the top surface
Cp0 small ( curve B is low )
Mcr large
c.f.
Thick airfoil
stronger expansion
Cp0 larger ( curve B is higher )
Mcr lower
∴An airfoil designed for a high Mcr must have a thin airfoil.

42. 1 > Drag-divergence Mach Number, MDD > Mcr
At MDD, the drag (CD) is massively increased.
“sonic barrier”
before 1947
when M∞ > Mcr

43. Mcr ↑ MDD ↑ ( MDD ↑ more by “supercritical” airfoil)
The total pressure loss associated with the weak λ-shock will be small, however, the adverse pressure gradient induced by the shock tends to separate the boundary layer on the top surface, causing a large pressure drag CD ↑ dramatically
Mcr ↑ MDD ↑ ( MDD ↑ more by “supercritical” airfoil)
ch 14
Two ways to increase Mcr：
(1) ↓ ( thinner airfoil ) ﹟

44. (2) sweep the wing

45. The flow behaves as if the airfoil section is thinner.