Translation of Prandtl's original paper on the boundary layer

1. On Fluid Motion with Very Small Friction¹²

By L. Prandtl (Hannover) (With one plate of figures.)

In classical hydrodynamics, the motion of a frictionless fluid is treated predominantly. For a viscous fluid we possess the differential equation of motion, whose formulation is well confirmed by physical observations. Of solutions of this differential equation—apart from one-dimensional problems such as those given, among others, by Lord Rayleigh³—one has only those in which the inertia of the fluid is neglected, or at least plays no role. The two- and three-dimensional problem that takes account of both friction and inertia still awaits solution. The reason for this lies, no doubt, in the unpleasant properties of the differential equation. In Gibbs’s vector notation it reads⁴:

( velocity, density, force potential, pressure, viscosity constant); in addition comes the continuity equation: for incompressible fluids, which alone are to be treated here, simply

From the differential equation one can easily see that for sufficiently slow and also slowly varying motions the factor becomes arbitrarily small compared with the other terms, so that here, with sufficient approximation, the influence of inertia may be neglected. Conversely, for sufficiently rapid motion the term quadratic in , (change of velocity due to change of position), becomes large/compiler large enough that the viscous action appears entirely negligible. In the cases of fluid motion that arise in engineering, the latter is almost always true. It is therefore natural simply to use the equation for an inviscid fluid. One knows, however, that the familiar solutions of this equation usually agree very poorly with experience; I need only recall the Dirichlet sphere, which according to theory should move without resistance.

I have therefore set myself the task of systematically investigating the laws of motion of a fluid whose friction is assumed to be very small. The friction is to be so small that it may be neglected everywhere except where large differences in velocity occur, or where an accumulating effect of friction takes place. This plan has proved very fruitful, since on the one hand it leads to mathematical formulations that make it possible to master the problems, and on the other hand it promises a very satisfactory agreement with observation. To mention one point at once: if, for example, in steady flow around a sphere one passes from the viscous motion to the limit of vanishing viscosity, one obtains something quite different from the Dirichlet motion. The Dirichlet motion is only an initial state, which is soon disturbed by the action of even the smallest viscosity.

I now turn to the individual questions. The force on the unit cube that arises from viscosity is

If one denotes by the vorticity, then by a well-known vector-analytic transformation, taking into account that , one obtains: From this it follows immediately that for also ; that is, even with arbitrarily strong viscosity, an irrotational motion is a possible motion. If nevertheless in certain cases it does not persist, this is because vortical fluid from the boundary pushes into the irrotational region.

In any periodic or cyclic motion, over a long duration the effect of viscosity, even if very small, can accumulate.

One must therefore require for the steady (persistent) state that the work of , i.e. the line integral along each streamline, be equal to zero for a complete cycle in cyclic motions; for flows periodic in space one has

For two-dimensional motions, for which a stream function ⁵ exists, one can derive from this, with the aid of Helmholtz’s vortex theorems, a general statement about the distribution of vorticity. For plane motion one obtains⁶:

For closed streamlines this becomes zero; hence the simple result follows that within a region of closed streamlines the vorticity takes on a constant value.

For axisymmetric motions with flow in meridian planes, for closed streamlines the vorticity is proportional to the radius: ; this yields a force in the direction of the axis.

By far the most important question of the problem is the behavior of the fluid at the walls of solid bodies. The physical processes in the boundary layer between fluid and solid body are represented sufficiently well if one assumes that the fluid adheres to the walls, i.e. that there the velocity is everywhere zero, or equal to the body velocity. If now the viscosity is very small and the path of the fluid along the wall is not too long, then already in the immediate vicinity of the wall the velocity will have its normal value. In the thin transition layer the abrupt differences in velocity then give rise, despite the small viscosity constant, to noticeable effects.

This problem is best treated by making systematic neglects in the general differential equation. If one takes as small of second order, then the thickness of the transition layer is small of first order, likewise the normal component of the velocity. Transverse differences of pressure can be neglected, as can any curvature of the streamlines. The pressure distribution is imposed on our transition layer by the outer (free) fluid.

For the plane problem, which alone I have treated so far, one obtains in the steady state (-direction tangential, -direction normal, and the corresponding velocity components) the differential equation

and in addition

If, as usual, is completely given, and furthermore the profile of is given at the initial cross-section, then every such problem can be handled numerically: by quadratures one can obtain from each the corresponding ; with this one can, with the aid of one of the known approximation methods⁷, repeatedly advance one step in the X-direction. A difficulty consists, however, in various singularities that occur at the solid boundary.

The simplest case of the states of motion treated here is that water flows along a plane thin plate. Here a reduction of variables is possible; one can set

By numerical solution of the resulting differential equation one arrives at a formula for the drag:

( width, length of the plate, speed of the undisturbed water relative to the plate). The course of is given in Fig 1.

But the result most important for applications of these investigations is that in certain cases, at a location completely determined by the external conditions, the fluid stream detaches from the wall (cf. Fig 2). Thus a layer of fluid set into rotation by friction at the wall pushes out into the free fluid and there, effecting a complete transformation of the motion, plays the same role as Helmholtz’s separation sheets. When the viscosity constant is changed, only the thickness of the vortex layer changes (it is proportional to ); everything else remains unchanged; one may therefore, if one wishes, pass to the limit and still retain the same flow pattern.

As a closer discussion shows, the necessary condition for the detachment of the jet is that along the wall, in the direction of the flow, a pressure increase is present. What magnitude this pressure increase must have in particular cases can only be learned from the numerical evaluation of the problem still to be carried out. As a plausible reason for separation one may state that in the presence of a pressure increase the free fluid converts part of its kinetic energy into potential energy. The transition layers, however, have already lost a large part of their kinetic energy; they no longer possess enough to penetrate into the region of higher pressure, and therefore yield sideways to it.

According to the foregoing, the treatment of a definite flow phenomenon breaks into two parts that interact with each other: on the one hand the free fluid, which can be treated as inviscid according to Helmholtz’s vortex laws; on the other hand the transition layers at the solid boundaries, whose motion is regulated by the free fluid but which in turn, by emitting vortex sheets, give the characteristic stamp to the free motion.

I have attempted in a few cases to follow the process more closely by drawing the streamlines; however, the results make no claim to quantitative correctness. Insofar as the motion is irrotational, one makes good use in drawing of the fact that the streamlines and the lines of constant velocity potential form an orthogonal (square-meshed) curvilinear net.

Fig 3 and Fig 5 show the beginning of the motion around a wall projecting into the stream in two stages. The irrotational initial motion is rapidly transformed by a separation sheet (dashed) issuing from the edge of the obstacle and winding up spirally; the vortex moves farther and farther away and leaves behind, at the finally stationary separation sheet, water at rest.

How the analogous process takes place for a circular cylinder can be seen from Fig 4 and Fig 6; the layers of fluid set into rotation by friction are again indicated by dashes. The separation surfaces extend here too, in the steady state, to infinity.

All these separation surfaces are known to be unstable; if a small sinusoidal disturbance is present, then motions arise such as are shown in Fig 7 and Fig 8. One sees how, through the interlocking of the streams, distinctly separated vortices form.

The vortex sheet is rolled up in the interior of these vortices, as indicated in Fig 9. The lines of this figure are not streamlines, but rather such as would be obtained by adding dyed fluid.

I shall now briefly report on experiments I undertook for comparison with the theory. The experimental apparatus (shown in Fig 10 in elevation and plan) consists of a 1½ m long trough with an intermediate floor. The water is set into circulation by a paddle wheel and enters the upper channel, ordered and calmed by a guide apparatus and four screens , fairly free of vorticity; at the object to be investigated is installed. Suspended in the water is a mineral consisting of fine, shiny flakes (iron mica); thereby all regions of the water that are appreciably deformed, especially all vortices, become visible through a peculiar sheen produced by the orientation of the flakes located there.

The photograms assembled on the plate were obtained in this way. In all of them the flow goes from left to right. Nos. 1–4 treat the motion at a wall projecting into the flow. One recognizes the separation surface issuing from the edge; in 1 it is still very small, in 2 it is already covered with strong disturbances; in 3 the vortex extends over the entire picture; 4 shows the “steady state.” One also notices a disturbance above the wall; since in the corner, owing to the stagnation of the water stream, a higher pressure prevails, the fluid stream also detaches here with time (cf. p. 488). The various streaks visible in the “irrotational” part of the flow (especially in Nos. 1 and 2) arise from the fact that at the beginning of the motion the fluid was not completely at rest.

Nos. 5 and 6 give the flow around a circularly curved obstacle, or, if one prefers, through a gradually constricted and then widened channel. No. 5 shows a stage shortly after the beginning of the motion. The upper separation surface is wound into a spiral, the lower is stretched out and breaks up into very regular vortices. On the convex side near the right end one notices the beginning of a separating flow; No. 6 shows the steady state in which the flow separates approximately at the narrowest cross-section.

Nos. 7–10 show the flow around a circular cylindrical obstacle (a post). No. 7 shows the beginning of separation, 8 and 9 further stages; between the two vortices a streak is visible—this consists of water that, before the beginning of separation, had belonged to the transition layer. No. 10 shows the steady state. The wake of vortical water behind the cylinder swings to and fro, hence the asymmetric instantaneous shape.

The cylinder contains a slit running along a generator; if one sets this as in Nos. 11 and 12 and sucks water out of the cylinder interior with a hose, one can intercept the transition layer of one side. If it is absent, its effect— separation—must also fail. In No. 11, which corresponds in time to No. 9, one sees only one vortex and the streak. In No. 12 (steady state) the flow, although—as one sees—only a vanishingly small part of the water enters the interior of the cylinder, nevertheless adheres closely to the cylinder wall up to the slit; but now a separation surface has formed at the plane outer wall of the trough (a first indication of this phenomenon is already visible in 11). Since in the widening flow opening the velocity must decrease and hence the pressure rises⁸, the conditions for separation of the flow from the wall are given; thus even this striking phenomenon finds its justification in the theory presented.

1. ¹I wrote this as a standard typst document compiling to a PDF and let my blog setup compile it to HTML. See source for details.
2. ²English translation aided by GPT5.2. Figures were extracted with AI assisted tool use and manual finetuning. Original paper “Über Flüssigkeitsbewegung bei sehr kleiner Reibung.” available at https://www.damtp.cam.ac.uk/user/tong/fluids/prandtl.pdf. In the course of looking for the missing plate of figures, I found that there is an english translation already, at https://ntrs.nasa.gov/citations/19930090813. Unfortunately the quality of this scan is not very good so I have not bothered to include the plate.
3. ³Proceedings Lond. Math. Soc. 11, p. 57 = Papers I, p. 474 ff.
4. ⁴ scalar product, vector product, Hamiltonian differential operator
5. ⁵Cf. Encyclopedia of the Mathematical Sciences IV 14, 7.

6. ⁶After Helmholtz, the vorticity of a fluid element remains proportional to its length in the direction of the vorticity axis; therefore in steady plane motion is constant on each streamline , i.e. ; with this,

7. ⁷Cf., e.g., Zeitschr. f. Math. u. Physik, vol. 46, p. 435 (Kutta).
8. ⁸It holds that along each streamline.