1.2 Review of the Theory
9
Fig. 1.8 Reynolds numbers
based on the length L of the
vehicle for a kinematic
viscosity equal to 15 × 10 −6
m 2 /s
viscosity and/or thermal conductivity. There are two main types of
confined regions depending on whether the velocity varies along
the main direction of the flow or in a direction normal to the
wall. The first case corresponds to the shock region or layer whose
thickness varies inversely with Reynolds number, the second, the boundary
layers at the walls and wakes whose thickness varies as the inverse of the square
roots (laminar case) or to a power (turbulent case) of the Reynolds number. At an
altitude of 20 km if the Mach number of the vehicle is 10, the thickness of the
shock will be of the order of mean molecular free path, therefore very small for
a body whose length is several meters. However, at 150 km the mean free path is
approximately 10 m, for the same Mach number: thence the thickness of the shock
is of the order of the size of the vehicle.
Euler’s equations are first-order equations that model a physics corresponding to
non-linear propagation phenomena without diffusion (transport by velocity and by
acoustic waves). When the flow is subsonic, the perturbations by the flow propagate both downstream and upstream of the source of perturbation in what is usually
referred as the elliptical propagation. For instance an observer on the ground perceives the noise of a plane in subsonic flight when it approaches or when it moves
away. When the flow is supersonic, the perturbations propagate only downstream
in the hyperbolic condition; here the same observer does not perceive the sound of
an airplane coming towards him in supersonic flight. The Euler equations have the
particularity of allowing discontinuous solutions: shock waves and contact discontinuities (slip lines). A fluid particle passing through a shock sees its entropy and
pressure increase. For a stationary shock normal to the direction of flow, the velocity
is supersonic upstream of the shock and subsonic downstream. Over the slip line,
the fluid particles slide on both sides of the discontinuity, the entropy being able to
undergo an arbitrary jump while the pressure remains continuous. Contact discontinuities and shocks are the limiting forms; on one hand we have the wall boundary
9
Fig. 1.8 Reynolds numbers
based on the length L of the
vehicle for a kinematic
viscosity equal to 15 × 10 −6
m 2 /s
viscosity and/or thermal conductivity. There are two main types of
confined regions depending on whether the velocity varies along
the main direction of the flow or in a direction normal to the
wall. The first case corresponds to the shock region or layer whose
thickness varies inversely with Reynolds number, the second, the boundary
layers at the walls and wakes whose thickness varies as the inverse of the square
roots (laminar case) or to a power (turbulent case) of the Reynolds number. At an
altitude of 20 km if the Mach number of the vehicle is 10, the thickness of the
shock will be of the order of mean molecular free path, therefore very small for
a body whose length is several meters. However, at 150 km the mean free path is
approximately 10 m, for the same Mach number: thence the thickness of the shock
is of the order of the size of the vehicle.
Euler’s equations are first-order equations that model a physics corresponding to
non-linear propagation phenomena without diffusion (transport by velocity and by
acoustic waves). When the flow is subsonic, the perturbations by the flow propagate both downstream and upstream of the source of perturbation in what is usually
referred as the elliptical propagation. For instance an observer on the ground perceives the noise of a plane in subsonic flight when it approaches or when it moves
away. When the flow is supersonic, the perturbations propagate only downstream
in the hyperbolic condition; here the same observer does not perceive the sound of
an airplane coming towards him in supersonic flight. The Euler equations have the
particularity of allowing discontinuous solutions: shock waves and contact discontinuities (slip lines). A fluid particle passing through a shock sees its entropy and
pressure increase. For a stationary shock normal to the direction of flow, the velocity
is supersonic upstream of the shock and subsonic downstream. Over the slip line,
the fluid particles slide on both sides of the discontinuity, the entropy being able to
undergo an arbitrary jump while the pressure remains continuous. Contact discontinuities and shocks are the limiting forms; on one hand we have the wall boundary