Singularitätenverfahren zur Ermittlung der Kräfte und Momente auf Körper in Potentialströmungen
| Scientific Paper | |
|---|---|
| Title | Singularitätenverfahren zur Ermittlung der Kräffte und Momente auf Körper in Potentialströmungen |
| Read in full | Link to paper |
| Author(s) | Anton Betz |
| Keywords | density, Vortex, forces, Ether |
| Published | 1932 |
| Volume | 111 |
| Number | 3 |
| No. of pages | 11 |
| Pages | 454-462 |
Read the full paper here
Abstract
Ingenieur-Archiv 111, Band 3, pp. 454-462 (Eingegangen am 14. Juni 1932.) Verlag von Julius Springer - Berlin. This fundamental paper breaks down the different types of fluid dynamic flow configurations and gives the equations of the forces and moments of interaction between the most popular of these fluid dynamic singularities. IMHO the importance of having done that lies in the analytically derived stability of two singularities; the spherical vortex by Professor Hill and the gyration stabilized vortex of Professor Hicks and the earmarking of these singularities as the building blocks of matter formed by the flow of the ether by Carl Friedrick Kraft, Ott C. Hilgenberg, Gustav Le Bon, and others. Crucial to the understanding of this importance is the basic model of the ether as a fluidic medium to which compressible fluid dynamic equations can be applied, including coordinate transformations with the factor (1-v2/c2)1/2. This ether model has also primordial ambient scalar energy density, comparable to the atmospheric pressure of stagnant air. Note that energy density has the same dimensionality as hydrostatic pressure (Joules/cubic meter ~ in lb/cubic inch = lb/square inch). However; when flow singularities arise in the medium the local scalar pressure is reduced by the vectorial energy density of the directional flow field's velocity momentum. This is well expressed by the Bernouilli equation, where the potential (scalar) energy density (pressure) is reduced (minus sign) by the kinetic energy density of the local flowing medium, to give the lower static pressure as measured by an observer in the flow.
The meaning of the negative sign should not be underestimated. Its physical meaning is missing in Relativity. In fluid dynamics it explains the manifestation of forces. Fluid dynamics does not need the postulation of attraction forces at a distance due to gravitons, etc. Singularity flow by itself causes the local scalar pressure reductions and thus the pressure differentials, that create the pushing forces by the ambient scalar pressure, as they act, for example, on finite areas of stable singularities, that exhibit kinetic mass. To say that "forces exist first and therefore we have energy as a secondary" is wrong. Forces are due to field gradients, which are small pressure differentials, negatively superimposed on the high scalar pressure of a mediumand acting on areas.
Overview
This is a paper in classical applied fluid mechanics, not in dissident physics, and it should be read as such. It is by A. Betz of the Kaiser-Wilhelm-Institut für Strömungsforschung in Göttingen — the aerodynamicist of the Betz limit and Prandtl's successor at the institute — and appeared in Ingenieur-Archiv Band III, Heft 5 (1932), pp. 454–462, received 14 June 1932. Its subject is the calculation of the forces and moments acting on a body immersed in a potential flow, when the body is replaced by an equivalent arrangement of flow singularities: sources, sinks, doublets (dipoles) and vortices. Nothing in the paper concerns the aether, relativity, the structure of matter or the origin of force; the aether interpretation recorded in the abstract above is a wiki contributor's commentary on the possible significance of the results, not Betz's own thesis.
The engineering motivation is stated at the outset. Aerodynamic and hydrodynamic bodies — aerofoils, struts, ship hulls, propeller blades — are routinely modelled by distributing singularities inside them so that the superposed flow reproduces the body's shape as a streamline. The pressure distribution then follows from Bernoulli's equation and the forces from integrating the pressures over the surface. Betz's point is that this last, laborious step can be avoided altogether. Because the total force and moment on a body can be obtained from a momentum balance over an arbitrary control surface, and because a control surface can be contracted onto the singularities themselves, the forces and moments can be read directly off the singularity strengths and the undisturbed flow at the singularity locations. The result is a compact catalogue of formulae — the two-page table of Abb. 14–21 with which the paper ends — usable, as he puts it, "für den praktischen Gebrauch rasch und bequem".
The method
Momentum balance on a contracted control surface
Betz begins from the standard result that the resultant force on everything inside a control surface equals the surface integral of the pressures plus the flux of momentum through it. Since the singularity-plus-flow field is known analytically everywhere outside the singularities, the control surface may be deformed at will. Contracting it into small circles (in plane flow) or spheres (in three dimensions) around each singularity leaves an integral in which only two things appear: the quantity characterising the singularity — source strength E or "Ergiebigkeit", circulation Γ or "Wirbelstärke", dipole moment μ — and the undisturbed flow that would exist at that place if the singularity were absent. How the undisturbed flow behaves anywhere else is irrelevant. In general only the velocity, and at most its derivatives, are needed at the singularity points.
For several singularities together the control surface is drawn to enclose each one (S1, S2, … in Abb. 6) with narrow connecting tubes R1, R2, … whose contributions cancel in pairs on integration. The mutual influence of the singularities upon one another then has to be counted, but only at the singularity locations. Betz's finding is that this mutual influence produces no resultant force, and in two-dimensional flow at most a moment.
The elementary cases
The catalogue is built up from three elementary results, each derived rather than quoted.
A source of strength E in a parallel stream of velocity v experiences a force against the flow direction,
- Px = −ρEv
with no transverse force and no moment. The physical reading is that the fluid issuing from the source must be accelerated up to the stream velocity, and the reaction of that acceleration is a drag on the source.
A vortex of circulation Γ in a parallel stream experiences the Kutta–Joukowski force perpendicular to the stream,
- Py = −ρvΓ
with no streamwise force and no moment — the classical lift formula, here obtained as one entry in a general scheme.
A dipole of moment μ in a parallel stream, formed by letting a source and sink of separation a coalesce with μ = Ea held fixed, experiences no net force at all in a uniform stream, but does experience a moment about its own centre,
- M = ρμv sin φ
where φ is the angle between the dipole axis and the flow direction. The moment vanishes when the axis lies along the stream and is greatest when it lies across it — the familiar tendency of an elongated body in a potential flow to turn broadside on.
Non-uniform flows
The interesting cases arise when the ambient flow is not uniform, so that the source and the sink of a dipole sit at points where the velocity differs. If they lie one behind the other along the stream with velocities v1 and v2, the resultant is Px = ρE(v1 − v2), and letting them coalesce with a linear velocity gradient gives
- Px = −ρμ ∂v/∂x
so that a dipole is pushed against its own axis in an accelerating flow. If instead the pair lies across the stream, the two velocities are equal in magnitude but differ in direction by an angle δ, giving a transverse force Py = ρEv·2 sin(δ/2) acting through the intersection point of the two velocity vectors, at a distance l = (a/2) cot(δ/2) from the line joining them; in the dipole limit
- Py = ρμv/l
directed opposite to the dipole axis in a diverging flow and along it in a converging one — the reverse of the streamwise case — plus the moment M = Pyl = ρμv, identical with the parallel-flow moment.
For a dipole at an arbitrary angle φ the three results combine into
- Px = −ρμ(∂v/∂x) cos φ, Py = ρμ(v/l) sin φ, M = ρμv sin φ
and the transverse force vanishes as l → ∞, that is, as the flow becomes parallel. Betz then specialises the ambient non-uniform flow to the two cases of practical interest. For the field of a source of strength E0 at distance s he uses v = E0/(2πs) with ∂v/∂x = −v/s for a line source, and v = E0/(4πs2) with ∂v/∂x = −2v/s for a point source. For a curved flow of radius of curvature r he uses ∂v/∂r = −v/r, obtaining a streamwise force Px = −ρμ(v/r) sin φ acting at the centre of curvature, a moment M = Pxr = −ρμv sin φ, and a radially outward force Pr = ρμ(v/r) cos φ arising because the forces on source and sink are no longer parallel but inclined by δ = (a cos φ)/r. The curved flow is then identified with the field of a straight vortex of circulation Γ0, for which v = Γ0/(2πr).
The table
The paper closes with a two-page tabulation, Abb. 14–21, of the force components and moment for the most important combinations, with a small pictorial diagram of each arrangement. Sign conventions are fixed explicitly: the flow always runs left to right, +Px is the component along the stream, +Py the component perpendicular to it reckoned to the left, +M the anticlockwise moment, and the tabulated moment always refers to the point of the singularity. The formulae hold for both point and line singularities, with the forces and moments per unit length in the latter case. The entries run: source in parallel flow; vortex in parallel flow; dipole in parallel flow; source in the field of a point or line source (Px = −ρEE0/4πs2 and −ρEE0/2πs respectively, M = 0); vortex in the field of a point or line source (Py = −ρE0Γ/4πs2 and −ρE0Γ/2πs, M = 0); dipole in the field of a point or line source; dipole in the field of a vortex; and finally the dipole in a general flow,
- Px = −ρμ[(∂v/∂x) cos φ − (v/r) sin φ], Py = ρμ[(v/l) sin φ − (v/r) cos φ], M = ρμv sin φ.
Assessment
Judged as what it is, this is a clean and useful piece of classical applied mathematics. The method is sound, the derivations are elementary and complete, the sign conventions are stated, and the end product is exactly what an engineer of 1932 would have wanted: a lookup table replacing a surface integration. The insight that the mutual interaction of singularities contributes no resultant force, and in plane flow only a moment, is the kind of structural simplification that makes such a table possible at all. The recovery of the Kutta–Joukowski lift Py = −ρvΓ as a single line of a general scheme, alongside the source drag −ρEv and the dipole moment ρμv sin φ, shows the economy of the approach. Betz's results are standard textbook material today and are not in dispute.
Its limitations are those of potential-flow theory generally, and Betz does not pretend otherwise: the fluid is inviscid and irrotational outside the singularities, so there is no boundary layer, no separation and no viscous drag. In this framework the drag of a closed body in a uniform stream is necessarily zero (d'Alembert's paradox), and the source drag −ρEv is not a real drag on a solid body but the reaction to injecting fluid — a distinction that matters if the formulae are read too physically. The three-dimensional cases are given only for the arrangements listed; nothing is said about stability, time dependence or compressibility.
That last point bears on the wiki abstract prefixed to this record, which is a contributor's editorial note rather than the author's summary, and which makes claims the paper does not support. Betz derives no stability result for any singularity; Hill's spherical vortex and Hicks's gyrating vortex are nowhere mentioned, nor are Krafft, Hilgenberg or Le Bon; the paper contains no aether, no compressible medium, no factor √(1 − v2/c2) and no discussion of the nature of force or of gravitation. Its fluid is incompressible and its context is aeronautical engineering at Göttingen. Readers coming to this record for the aether-vortex model of matter should be clear that the physics they are looking for is in the abstract's commentary and in the work of the authors it names, not in Betz's nine pages.
Two bibliographic points should also be noted. The page title carries a typographical error, "Kräffte" for Kräfte — the correct spelling appears in the running head of every page of the original. And the infobox author "Anton Betz" is almost certainly wrong: the byline reads "Von A. Betz, Göttingen" and the paper is signed from the Kaiser-Wilhelm-Institut für Strömungsforschung, which was directed by Albert Betz (1885–1968).