8
1 The Experimental Approach in Aerodynamic Design
Table 1.3 Mean molecular
free path in the atmosphere
Altitude (km)
λ (m)
20
10 −6
70
10 −3
110
1
150
10
At “aeronautical” altitudes, that is to say, less than about 20 km, the mean molecular free path is less than one micron, so much smaller than the size of the vehicle,
which is of the order of several meters (see Table 1.3), this justifies the assumption
of a fluid as a continuum and the validity of the Navier-Stokes equations. At 70 km,
the mean free path becomes of the order of a millimetre, which is not very small
vis-à-vis the re-entry body, but still small enough to consider the air as a continuous
medium, except perhaps in some areas where the flow expands strongly. Beyond
100 km, the average mean free path becomes comparable to the length of the vehicle
and the assumption of continuous medium for the air is again invalid.
From dimensional analysis, the physical quantities of the continuous model are
defined by four quantities, being a unit of length adapted, the density, ρ, temperature,
T, and the velocity, V, for a given reference state. To measure the relative weights
of the different terms of the equations, it is usual to introduce additional reference
quantities. For example, if we are interested in the equation of motion that reflects a
balance of inertial forces, pressure and viscosity, it is convenient to introduce two new
reference quantities: one for the pressure ρV
2 or density, another for the viscosity
term ρVL. By introducing these two additional terms into the momentum equation
two dimensionless numbers are obtained and are very important in aerodynamics.
These are:
– the Mach number, M = V /a, ratio of the velocity V to the local speed of sound, a
and,
– the Reynolds number, Re = ρVL/μ, ratio of the inertial forces to the viscosity
forces where μ is the molecular or dynamic viscosity of the fluid.
These two numbers are critical dynamic similarity parameters for the extrapolation
of the actual wind tunnel conditions to flight conditions or vice versa (see Sect. 2.4).
As shown in Fig. 1.8, the aerodynamics of aircraft and land vehicles is characterised by very high values of Reynolds number (several million) reflecting the low
viscosity of the air.
Thus, for an Airbus A380, the Reynolds number at cruise, based on the average
chord of the wing, is approximately 70 million. Since this is a large value, some terms
of the Navier-Stokes equations become negligible and we can consider that the flow
in the entirety is almost a “perfect” fluid. By neglecting viscosity and thermal conductivity; the flow can be modelled by a simplified set of equations called the Euler equations. The viscous effects (friction, thermal conduction) are then modelled in confined
regions (boundary layers and/or mixing layers, wakes) which have very large variations in velocity and/or temperature, compensating for the very low values of
1 The Experimental Approach in Aerodynamic Design
Table 1.3 Mean molecular
free path in the atmosphere
Altitude (km)
λ (m)
20
10 −6
70
10 −3
110
1
150
10
At “aeronautical” altitudes, that is to say, less than about 20 km, the mean molecular free path is less than one micron, so much smaller than the size of the vehicle,
which is of the order of several meters (see Table 1.3), this justifies the assumption
of a fluid as a continuum and the validity of the Navier-Stokes equations. At 70 km,
the mean free path becomes of the order of a millimetre, which is not very small
vis-à-vis the re-entry body, but still small enough to consider the air as a continuous
medium, except perhaps in some areas where the flow expands strongly. Beyond
100 km, the average mean free path becomes comparable to the length of the vehicle
and the assumption of continuous medium for the air is again invalid.
From dimensional analysis, the physical quantities of the continuous model are
defined by four quantities, being a unit of length adapted, the density, ρ, temperature,
T, and the velocity, V, for a given reference state. To measure the relative weights
of the different terms of the equations, it is usual to introduce additional reference
quantities. For example, if we are interested in the equation of motion that reflects a
balance of inertial forces, pressure and viscosity, it is convenient to introduce two new
reference quantities: one for the pressure ρV
2 or density, another for the viscosity
term ρVL. By introducing these two additional terms into the momentum equation
two dimensionless numbers are obtained and are very important in aerodynamics.
These are:
– the Mach number, M = V /a, ratio of the velocity V to the local speed of sound, a
and,
– the Reynolds number, Re = ρVL/μ, ratio of the inertial forces to the viscosity
forces where μ is the molecular or dynamic viscosity of the fluid.
These two numbers are critical dynamic similarity parameters for the extrapolation
of the actual wind tunnel conditions to flight conditions or vice versa (see Sect. 2.4).
As shown in Fig. 1.8, the aerodynamics of aircraft and land vehicles is characterised by very high values of Reynolds number (several million) reflecting the low
viscosity of the air.
Thus, for an Airbus A380, the Reynolds number at cruise, based on the average
chord of the wing, is approximately 70 million. Since this is a large value, some terms
of the Navier-Stokes equations become negligible and we can consider that the flow
in the entirety is almost a “perfect” fluid. By neglecting viscosity and thermal conductivity; the flow can be modelled by a simplified set of equations called the Euler equations. The viscous effects (friction, thermal conduction) are then modelled in confined
regions (boundary layers and/or mixing layers, wakes) which have very large variations in velocity and/or temperature, compensating for the very low values of