2 Aerodynamics
137
2. Supersonic flow
When the supersonic flow is an inviscid potential flow, the governing
equation is nonlinear second-order hyperbolic partial differential equation. Similarly, based on the theory of small perturbation linearization,
the second-order hyperbolic partial differential equation is established
and solved by the characteristic method. In supersonic flow, the effects
of compression wave, expansion wave, and shock wave on the flow are
studied. The shock wave of an ideal gas has no thickness and is a discontinuity in mathematical sense. Rankine in 1870 (a British scientist) and
Hugoniot in 1887(a French scientist) independently derived the relationship between flow parameters before and after shock wave (called
the Rankine Hugoniot relationship) using the continuous equation,
momentum equation, and energy equation. Later, Prandtl established
the relationship between velocity coefficients before and after normal
shock wave. For the small disturbance problem of thin wing, Ackeret put
forward the linear theory of two-dimensional airfoil in 1925, and then
correspondingly appeared the linear theory of three-dimensional wing.
For two-dimensional and three-dimensional steady supersonic flows, the
boundary between disturbed and undisturbed areas is the Mach wave. If
the supersonic flow is accelerated by a series of Mach waves, it is called
expansion wave. Prandtl and his student T. Meyer (1907–1908) established the expansion wave relation. Figure 2.72 shows the oblique shock
wave and the positive shock wave around the head of the body. Figure 2.73
shows the shock system and the Mach disk of supersonic aircraft.
Prandtl Glauert condensation clouds (as shown in Fig. 2.74) are formed
by the condensation of water vapor in the air into clouds when the air
stream passes through the shock wave and compresses the surrounding
air. After the condensation of water vapor into tiny water droplets, it is
like a cloud. But it is not always accompanied by acoustic explosion, and
it is not necessarily the shock wave when the sound barrier is broken.
3. Transonic flow
For the transonic inviscid potential flow, part of the supersonic flow area
(as shown in Fig. 2.75 with the emergence of shock wave) will be around
the flow field. The flow change is complex, and the flow control equation is a second-order nonlinear mixed partial differential equation, which
is difficult to solve theoretically. Especially when the speed of flight or
flow is close to the speed of sound, the aerodynamic performance of the
aircraft changes rapidly, the resistance increases abruptly, and the lift drops
abruptly. The maneuverability and stability of the aircraft are extremely
deteriorated, which is the famous sound barrier in the aviation history.
137
2. Supersonic flow
When the supersonic flow is an inviscid potential flow, the governing
equation is nonlinear second-order hyperbolic partial differential equation. Similarly, based on the theory of small perturbation linearization,
the second-order hyperbolic partial differential equation is established
and solved by the characteristic method. In supersonic flow, the effects
of compression wave, expansion wave, and shock wave on the flow are
studied. The shock wave of an ideal gas has no thickness and is a discontinuity in mathematical sense. Rankine in 1870 (a British scientist) and
Hugoniot in 1887(a French scientist) independently derived the relationship between flow parameters before and after shock wave (called
the Rankine Hugoniot relationship) using the continuous equation,
momentum equation, and energy equation. Later, Prandtl established
the relationship between velocity coefficients before and after normal
shock wave. For the small disturbance problem of thin wing, Ackeret put
forward the linear theory of two-dimensional airfoil in 1925, and then
correspondingly appeared the linear theory of three-dimensional wing.
For two-dimensional and three-dimensional steady supersonic flows, the
boundary between disturbed and undisturbed areas is the Mach wave. If
the supersonic flow is accelerated by a series of Mach waves, it is called
expansion wave. Prandtl and his student T. Meyer (1907–1908) established the expansion wave relation. Figure 2.72 shows the oblique shock
wave and the positive shock wave around the head of the body. Figure 2.73
shows the shock system and the Mach disk of supersonic aircraft.
Prandtl Glauert condensation clouds (as shown in Fig. 2.74) are formed
by the condensation of water vapor in the air into clouds when the air
stream passes through the shock wave and compresses the surrounding
air. After the condensation of water vapor into tiny water droplets, it is
like a cloud. But it is not always accompanied by acoustic explosion, and
it is not necessarily the shock wave when the sound barrier is broken.
3. Transonic flow
For the transonic inviscid potential flow, part of the supersonic flow area
(as shown in Fig. 2.75 with the emergence of shock wave) will be around
the flow field. The flow change is complex, and the flow control equation is a second-order nonlinear mixed partial differential equation, which
is difficult to solve theoretically. Especially when the speed of flight or
flow is close to the speed of sound, the aerodynamic performance of the
aircraft changes rapidly, the resistance increases abruptly, and the lift drops
abruptly. The maneuverability and stability of the aircraft are extremely
deteriorated, which is the famous sound barrier in the aviation history.