As an object moves through the atmosphere, the gas molecules of the atmosphere near the object are disturbed and move around the object. Aerodynamic forces are generated between the gas and the object. The magnitude of these forces depend on the shape of the object, the speed of the object, the mass of the gas going by the object and on two other important properties of the gas; the viscosity, or stickiness, of the gas and the compressibility, or springiness, of the gas. To properly model these effects, aerodynamicists use similarity parameters which are ratios of these effects to other forces present in the problem. If two experiments have the same values for the similarity parameters, then the relative importance of the forces are being correctly modelled.
Aerodynamic forces depend in a complex way on the viscosity of the gas. As an object moves through a gas, the gas molecules stick to the surface. This creates a layer of air near the surface, called a boundary layer, which, in effect, changes the shape of the object. The flow of gas reacts to the edge of the boundary layer as if it was the physical surface of the object. To make things more confusing, the boundary layer may separate from the body and create an effective shape much different from the physical shape. And to make it even more confusing, the flow conditions in and near the boundary layer are often unsteady (changing in time). The boundary layer is very important in determining the drag of an object. To determine and predict these conditions, aerodynamicists rely on wind tunnel testing and very sophisticated computer analysis.
The important similarity parameter for viscosity is the Reynolds number. The Reynolds number expresses the ratio of inertial (resistant to change or motion) forces to viscous (heavy and gluey) forces. From a detailed analysis of the momentum conservation equation, the inertial forces are characterized by the product of the density rho times the velocity V times the gradient of the velocity dV/dx. The viscous forces are characterized by the dynamic viscosity coefficient mu times the second gradient of the velocity d^2V/dx^2. The Reynolds number Re then becomes:
Re = (rho * V * dV/dx) / (mu * d^2V/dx^2)
The gradient of the velocity is proportional to the velocity divided by a length scale L. Similarly, the second derivative of the velocity is proportional to the velocity divided by the square of the length scale. Then:
Re = (rho * V * V/L) / (mu * V / L^2)
Re = (rho * V * L) / mu
The Reynolds number is a dimensionless number.
The Reynolds number can be further simplified if we use the kinematic viscosity nu that is equal to the dynamic viscosity divided by the density:
nu = mu / rho
Re = V * L / nu
Reynolds Number is used to determine whether a flow will be laminar or turbulent. If Re is high (>2100), inertial forces dominate viscous forces and the flow is turbulent; if Re is low (<1100), viscous forces dominate and the flow is laminar.