Motion of fluids with very little viscosity By L. Prandtl (794397)
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,...—-1,/7L. .--, .,,. ‘f-. .{’ ‘, ‘-~P/*c/-J.//.;,’”’‘/f-J.. ...-,,.;~,.‘.+ i%,.~j’,,:,<.’, ‘:...J,TECHN””ICALMEMORANDUMSNATIONAL ADVISORY COMMITTEE FOR AEROi!AUTICSNo. 452MOTION OF FLUIDS WITH VERY LITTLE VISCOSITYBy L. PrandtlFr’om ‘fVierAbhandlungen zur Hydrodynamtk und AerodynamiklfG~ttingen, 1927,.,; ....’. ..-:.WashingtonMarch, 1928L-,,,,,..!:,, ,,.. —. —....-..— -L:.—INATIONAL ADVISO13X CQMMITT3Z FOR AERONAUT ICS .TECHNICAL MEMORANDUM NO. 452.-.MOTION OF FLUIDS WITH VERY LITTLE VISCOSITY. *By L. Prandtl.In classic hydrodynamicschiefly discussed.the motion of nonviscous fluids isFor the motion of viscous fluids, we havethe differential equation whose evaluation has been well confirindby physical observations.As for solutions of this differentialequation, we have, aside from unidimensional problems like thosegiven by Lord Rayleigh (Proceedings of the London MathematicalSociety, 11page 57 = Papers I“page 474 ff.), only the ones inwhich the inertia of the fluid is disregarded or plays no imporThe bidimensional and tridimensional problems, takingtant role.viscosity and ii~ertia into account, still await solution.Thisis probably due to the troublesome properties of the differentialIn the IiVectorSymbolicslf of Gibbs,** this readsequation.P@+voin whichvof the power;Av~+A(V+p)=kis the velocity;p,pressure;p,k,A2Vthe density;viscosity(1)a functionV,constant.There isalso the continuity equation........—,.
,..,.,div v ,,= ...O..,* lllJeberFlussigkeitsbev~ebmng bei sehr kleiner Reibung. l’This paper was read before the Third International Congress ofMathematicians at Heidelberg in 1904. From ‘fVierAbhandlungenzur Hydrodynmik und Aerodynarllik,llpp. 1-8, G~ttingen, 1927.** aob scalar product,differentiator(A =, i ~ +ajx# +v~c~.product, A Hamilton..Ir2N.A. C.A. Technical Memorandum No. 452.for incompressible fluids, which alone will be here considered...From the differential equation, it is easy to infer that, forsufficiently slow and also slowly changing motions, the factorp,in contrast with the other terms, can be as small as desired, sothat the effect of the inertia can here be disregarded with sufficient approximation.
” Conversely, with sufficiently rapid motion,the quadratic termlocation)(change of velocity due toofChangeis large enough to let the viscosity effect appear quit?sub ordinate.motionv @ A vThe latter almost always happens in cases of fluidoccurring in technology.It is therefore logical simply touse here the equation for non-viscous fluids.It is known, how–ever, that the solutions of this equation generally agree verypoorly with experience.I will recall only the Dir,&’chletsphere,which, according to the theory, should move without friction.I have now set myself the task to investigate systematicallythe laws of motion of a fluid whose viscosity is assumed to bevery small.The viscosity is supposed to be so small that it canbe disregarded wherever there are no great velocity differencesnor accumulative effects.This plan has proved to be very fruit-ful , in that, on the one hand, it produces mathanatical formulas,which enable a solution of the problans and, on the other hand,the agreement with observations promises to be very satisfactory.To mention one instance now: when, for e~mple,in the steadyitio-tion around a sphere, there is a transition from the motion withviscosity to the limit of nonviscosity, then something quite dif-.,,.,3N*A.
C.A. Technical Memorandum No. 4’52ferent from the Dirichlet motion is produced.The latter is thenonly an i-nitialcondition, which is soon disturbed by the effectof an ever-so-small viscosity.I will now take up the individual problems.The force on theunit area, due to the viscosity, isK = kA2If the vortex is represented by(2)Vw = ~ rot v,thenK = -2 k rotw,according to a well-known vector analytical transformation, takinginto consideration thatthat, forw = O,alsodiv v = O.K = O,From this it follows directlythat is, that however great theviscosity, a vortexless flow is possible.If, however, this isnot obtained in certain cases, it is due to the fact that turbu-”lent fluid from the boundary is injected into the vortexless flow.With a periodic or cyclic motion, the effect of viscosity,For perma.that is, the line integraleven when it is very small, can accumulate with time.nence, therefore, the work off K o dsK,along every streamline with cyclic motions, must bezero for a full cycle.‘E~KOds=(V,+p, )-(V,+pl).A general ’formula for the distribution of the vortex can bederived from thisbidimensionalwith -the aid of the Helmholtz vortex laws forrnotionswhich have a flow function Y(cf.
‘fEncy-klopadie der mathematischen Wissenschaftenjlt Vol. IV, 14, 7).N.A. C.A. Technical Memorandum No. 452With steady flow we obtain*_gf=w2+r?+(L+P1)2k\vods”With closed streamlines this becomes zero.Hence we obtain thesimple result that, within a region of closed streamlines, thevortex assumes a constant value.withthe flow in meridian planes, the vortex for closed stream-linesK=For axially symmetrical motionsis proportional to the rad.i.usw = cr.4 kcThis gives a forcein the direction of the axis.The most important aspect of the problem is the behavior ofthe fluid on the surface of the solid body.Sufficient accountcan be taken of the physical phenomena in the boundary layer between the fluid and the solid body by assuming that the fluid adheres to the surface and that, therefore, the velocity is eitherzero or equal to the velocity of the body.viscosityIf, however, theis very slight and the path of the flow along the suwface is not too long, then the velocity will have its normal valuein immediate proximity to the surface.In the thin transitionlayer, the great velocity differences will then produce noticeableeffects in spite of the small viscosity constants.This problem can be handled best by systanatic omissions inthe general differential equation.Ifkis taken as small in*According to Helmholtz, the vortex of a particle is permanentlyproportional to ‘itslength in the direction of the vortex axis.Hence we have, with steady even flow on each streamline(v = const.), w constant, consequently w = fv .
Herewith“ /Kods=2kjrotwods=2kff(~)jrotvods==2kf~($)/vods.1!5N.A. C.A. Technical Memorandum No. 452the second order, then the thickness of the transition layer willbe small in the,. first order, like the normal components of thevelocity.The lateral pressure differenc~ can be disregarded, aslikewise any curvature of the streamlines.The pressure distri-bution will be impressed on the transition layer by the free fluid.For the problem which has thus fa,rbeen discussed, we obtainin the steady condition (X-direction tangential, Y-direction nor–.‘real, * and v the corresponding velocity components) the diffe~ential equation”If, as usual,udp/dxis given throughout, as also the course offor the initial cross section, then every numerical problem ofthis kind can be numerically solved, by obtaining the corresponding~u/3xby squaring everyu.Thus we can always make prog-ress in the X–direction wit”n the aid of one of the well-knownapproximationmethods (Cf.
Kutta, ‘iZeitschrift f!u?Math. undPhysik, llVol. 46, p.435).One difficulty, however, consists inthe various singularities developed on the solid surface.Thesimplest case of the conditions h.mrc considered is when the waterflows along a flat thin plate.Here a reduction of the variablesu = f(-l=).By the numerical so\Jx/lution of the resulting differential equation, we obtain for theis possible and we can write1)(I6N.A. C.A. Technical MemorandumNo= 452,.. ;.,,..:drag the formulaR=l.1(bwidth,... b~’”k P ZZ length of plate,opposite plate).U.Uo3velocity of undisturbed waterFigure 1 shows the course ofu.The most important practical result of these investigationsis that, in certain cases, the flow separates from the surfaceat a point entirely determined by external conditions (Fig.
2).A fluid layer, which is set in rotation by the friction on thewall, is thus forced into ”the free fluid and, in accomplishing acomplete transformation of the flew, plays the same role as the...Helmholz separation layers. A change in the viscosity constantsksimply changes the thickness of the turbulent layer (propor–.——~)everything else remaining untional to the quantity1 pu /’It is therefore possible to pass to the limit k = Ochanged.and still retain the same flow figure.As shown by closer consideration, the necessary conditionfor the separation of the flow is that there should be a pressureincrease along the surface in the uirection of the flow.Thenecessary magnitude of this pressure increase in definite casescan be determined only by the numerical evaluation of the problemwhichis yet to be undertaken.AS a plausible reason for theseparation of the, flo,w, it may be stated that, with a pressure,,increase, the free fluid, its kinetic energy is partially converted into potential energy.The transition layers, however, havelost a large part of their kinetic energy and no longer possess7,N.A.
C.A. Technical Memorandum No. 452enough energy to penetrate the region of higher pressure.They arethereforelaterally.‘.!,,.deflected–.,Ac@rdingprocessanother.to the preceding, the treatment of a given flowis resolved into two components mutually related to oneOn the one hand, we have the free fluid, which can betreated as nonviscous according to the Helmholtz vortex laws,while, on the other hand, we ‘have the transition layers on thesolid boundaries, whose motion is determined by the free fluid,but which, in their-turn, impart their characteristic impress to“the free flow by the emission of turbulent layers.I have attempted,morein a few cases,” to illustrate the processclearly by diagrams of the streamlines, though no claim ismade to quantitative accuracy.In so far as the flow is vortex-free, one can, in drawing, take advantage of the circumstance,that the streamlines forfia quadratic system of curves with thelines of constant potential.Figures 3-4 show, in two stages, the beginning of the flowaround a wall projecting into the current.The vortex-freeini-tial flow is rapidly transformed by a spiral separating layer.The vortex continually advances, leaving still water behind thefinallystationary separating layer.Figures 5–6 illustrate the analogous process with a cylinder.The fluid layers set in rotation by the friction are plainly indicated.Here also the separating layers extend into infinity.All these separating layers are labile.If a slight sinoidal8N.A.
C.A. Technical Memorandum No. 452is present, motions develop as shown in Figures 7-8.disturbanceIt is clearly seen how separate vortices are developed by the mu–tual interference of the flows.The vortex layer is rolled up in–side these vortices, as shown in Figure 9.The lines of thisfigure are not streamlines, but such as were obtained by using acolored liquid.I will now briefly describe experiments which I undertookfor comparison with the theory.The experimental apparatus (Fig.10) consists of a tank 1.5 m (nearly 5 feet) long with an intermediate bottom.The water is set in motion by a paddle wheeland, after passing through the deflecting apparatussievesaand fourenters the upper channel comparatively free from vor–b,tices, the object to be tested being introduced atscales of micaceousc.iron ore are suspended in the water.FineThesescales indicate the nature of the flow, especially as regprds thevortices, by the peculiarities of their reflection due to theirorientation.The accompanying photographs were obtained in this manner,the flow being from left to right.a wall projecting into the current.Nos.
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