Aerodynamic Drag and Lift and d'Alembert's Paradox

Some Misconceptions in Fluid Dynamics

According to the classical theory of an ideal fluid flow, the drag force for objects moving through a medium should be zero. This apparent paradox (d'Alembert's Paradox) is attributed to the assumption of an inviscid fluid in this theory, which, according to this view, would not allow for any dissipation of the kinetic energy of the object in the medium (see Reference). However, this view neglects the fact that even if the gas molecules do not interact with each other (i.e. if the gas is inviscid), they still collide with the object, which must result in a change of momentum and energy.

Consider a plate moving frontally through a gas with velocity +v. A molecule moving vertically towards the plate with velocity -u in the lab frame has a velocity -u-v in the reference frame of the plate and hence bounces back from the front side with the velocity u+v (as the plate is much heavier than the molecule), i.e. in the lab frame the velocity of the molecule is now u+2v. Correspondingly, at the backside of the plate, a molecule with velocity u in the lab frame has a velocity u-v in the reference frame of the plate and thus after reflection a velocity -u+v, i.e. in the lab frame the velocity is now -u+2v. This means in the lab frame the kinetic energy of the molecules being reflected from the front has increased from m/2*u² to m/2*(u+2v)² =m/2*u² +2*m*u*v + 2*m*v², i.e. an increase of +2*m*u*v + 2*m*v².
Molecules reflected from the back side on the other hand have changed their energy from m/2*u² to m/2*(-u+2v)² = m/2*u² -2*m*u*v + 2*m*v² i.e. a change of -2*m*u*v + 2*m*v².
Adding and averaging the two contributions one obtains therefore the average increase of the kinetic energy of the molecules with mass m hitting the plate with velocity v as ΔK = 2*m*v². (Note: this value will actually be smaller by about factor 1/2 as the molecules won't be reflected straight back but will be scattered into the whole half-space due to the roughness of the plate's surface). (One can derive this result also from the energy and momentum conservation equations applied an elastic collision, with the same result if one assumes the plate mass as large compared to the molecular mass).

Let's apply this result to air (although the latter is not strictly inviscid in the above defined sense): if the plate has a cross section of 1m² and a velocity of 10 m/sec it will collide with 3.1026 molecules per second, which, assuming a molecule mass of 4.5.10-26 kg, amounts to a total mass of M=13 kg per second. Replacing m with M in the above result for the energy gain of a molecule, one sees that the kinetic energy gained by the air molecules per second corresponds to the kinetic energy of a plate of about 26 kg (taking the above mentioned correction into account), i.e. a plate of with a mass of 26 kg, a cross section of 1m² and a velocity of 10m/sec would lose its energy within about one second. This result looks actually quite realistic and shows that the assumption of a non-viscous gas can account for observed drag in air, and this obviously suggests also that the same holds for the aerodynamic lift.

Even if one takes the molecular interaction into account, the molecules will lose only some of the velocity 2v that they gained on collision with the plate: if one consider that half of the molecules were actually initially not flying towards the plate and average the velocity gained with these, one still has a velocity gain of +v, i.e. the initially (on average) resting molecules will be imparted a velocity such that they co-move with the plate. (interestingly this is just what is observed in form of the 'stagnation' of the airflow at the rear and front of real airfoils).

The usual aerodynamic lift can therefore in principle well be explained assuming an inviscid medium as the orientation and shape of the wing in combination with the velocity relative to the air leads to more molecules hitting the wing from below than from above. The connection between the drag and lift force is hereby given by the orientation of the surface elements with regard to the incident airflow ( if α is the angle between the normal of the surface element and the airstream, the resultant net force is proportional to cos(α) , which can then be decomposed into the horizontal component i.e. the drag (~cos2(α)) and the vertical component i.e. the lift (~sin(α).cos(α)).
Everything else like the airflow pattern around the wing etc. is only a secondary consequence of this due to the actual viscosity of the air, i.e. hydrodynamics may explain what effect an object moving through air has on the latter, but it does not actually give the causal reason why an airplane flies (see the page Bernoulli's Principle and Aerodynamic Lift for more on this).

In contrast to the usual aerodynamic lift, the well known Magnus effect due to the rotation of objects does however not exist in an inviscid gas:
consider a rotating ball that is moving through an inviscid gas (i.e. molecules interacting with the ball but not with each other): if the surface of the ball would be mathematically smooth, then the rotation would actually be without any effect at all because the air molecules would just bounce off like for a non-rotating sphere, but even for a realistic rough surface (obviously a surface can not be smoother than about 1 atomic radius), the overall effect still cancels to zero: the pressure on the side rotating against the airstream is higher at the front but smaller at the back (and the other way around for the co-rotating side) so overall there is no resultant force on the ball but merely a torque that slows down the rotation.
Hydrodynamics arguments (i.e. Bernoulli's principle) are therefore required to explain the Magnus effect but not for the aerodynamic lift.

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Thomas Smid (M.Sc. Physics, Ph.D. Astronomy)
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