Mass 1sfer.

Vol. 18, pp. 1333-1336.

Pergamon Press 1975

Pnnted in Great Britain

LUDWIGPRANDTL,1875-1953

TRIBUTETO LUDWIG

PRANDTL

On 4 February of this year was the 100th anniversary of the birth of Ludwig Prandtl. The theory ofconvective heat transfer could not have been developed without the foundation of modern fluid mechanic8
created by him. We asked therefore Professor Hermann Schlichting, his student and close associate, toreview briefly the work of Ludwig Prandtl.
ERNSTR. G. ECKERT
1. INTRODUCTIONLUDWIG PRANDTL (1875-1953) is generally accepted
to be the father of modern fluid dynamics.
Since about the beginning of this century modern
research in the field of fluid dynamics has achieved
great success. It has been able to provide a theoreticalclarification of observed phenomena which the scienceof classical hydrodynamics
of the preceding century
failed to do. Ludwig Prandtl has a major part in thisachievement. There are essentially three branches of
fluid dynamics which have become particularly
welldeveloped during the last seventy years and Prandtl
has created the basis of all of them; they include
Vol.18.No.12-A

boundarylayer theory,
aerofoil theory and gas
dynamics.
On the occasion of the hundredthanniversary of
Prandtl’s birth a very condensed survey on his scientificlife is given here.
Above is a photograph of Prandtl when he was about
sixty years old. Prandtl was born on 4 February 1875
in the town of Freising in Bavaria; he died on 15 August1953 in Gottingen at the age of nearly 79 years. In theyear 1900, when Prandtl was 25 years old, he wasawarded his doctorate at the University of Munich
under August Foeppl. In the year 1901, after he had
worked in industry at M.A.N. in Niirnberg for one year,1333

1334

Tribute to Ludwig Prandtl

that is at the age of 26 years, he was invited to join
the Faculty of the Technical University at Hanover as
Professor of mechanics.
At M.A.N. he was given the task of improving thedesign of an air duct for the removal of chips. In amanner typical for his approach to problems, he firstmeasured the energy losses which occurred in theindividual parts of the duct. This allowed him torecognize that the largest losses were connected with aconical portion which failed to produce the pressure
recovery expected on the basis of Bernoullits equation,
This problem led him to the phenomenon of flow
separation, which could be explained by his boundary
layer theory. The corresponding paper, read in 1904
at the International Congress of Mathematics, madePrandtl famous at one stroke. The young Prandtl’sachievements were soon recognized by Felix Klein, thegreat organizer of the mathematical and appliedsciences at the University of Gottingen. In 1904 at
Klein’s initiative, Ludwig Prandtl took over the newly
founded Institute of Applied Mechanics at Gottingen
University. Based on this Institute and on the chair of
the same name connected with it, Prandtl developeda most fruitful activity in research and teaching.
In the following I shall restrict myself exclusively toa consideration of Prandtl’s research work in fluid
mechanics, regardless of the fact that he made fundamental contributions
to the other branches ofmechanics (elasticity theory, plasticity).
2. SOME HIGHLIGHTS OF PRANDTL’S SCIENTIFIC WORK
2.1. Boundary layer theory
At the beginning of this century the view prevailed
that it was hopeless to solve the Navier-Stokes equations of a viscous fluid, especially, as far as the problem
of external flow about a body was concerned. For thisreason Prandtl searched for approximate solutions for
low-viscosity fluids, such as air and water, which areimportant in engineering applications. He succeeded
to reduce the mathematical difficulties of the NavierStokes equations to such an extent that the simplified
system of equations could be solved with the aid of thethen existing methods. From a physical point of view,
the simplification consists in the following:Prandtl divided the complete flow field into two
regions: the thin boundary layer which develops very
close to the solid wall, in which the frictional forces
are as important as the inertia forces; and the externalregion, in which the flow is practically frictionless.
What has just been said constitutes the foundationof Prandtl’s boundary layer theory of 1904. The mostimportant result of this theory is the calculation of thepoint of separation of the flow from the wall and theskin friction.
As an example of this theory we give a result for thelaminar boundary layer on a flat plate. In this casethere occurs no separation, because the pressure
gradient is zero along the plate. The coefficient of thelocal skin friction is obtained asequation (1) is compared with experiments both forlaminar and turbulent flow. The agreement between
theory and experiment is excellent.
The concept of the boundary layer has also beenvery successfully applied to problems of heat transfer
between a heated body in an external flow. In thiscase, in addition to the velocity boundary layer, there
exists a temperature boundary layer. The calculationof the temperature boundary layer yields the heattransfer from the heated body to the external flow.
An example of this kind is presented in Fig. 2 for a
hot vertical flat plate in natural convection in air(Prandtl number Pr = 0.73).
The upper diagram gives the temperature distribution in the boundary layer and the lower diagramthe velocity distribution due to convection. Here,besides the Prandtl number, the Grashof number
appears as parameter. It is given by the ratio ofIOLI

I I l/llll
Theory;Theory:BlasiusI

IIIII-Prandtl

1921*5(1908)

011,042Reynolds357number,,05Re,- -37I06Y

FIG. 1. Boundary layer of flat plate; local coefficient of skinfriction (laminar, equation (l), Blasius, 1907).

70= 0664(laminar, Blasius)(1)
P%/2f&W
with p as density, U, as velocity of the external flow,r. as shearing stress at the wall, and Re, = U,xjv as
the local Reynolds number with x as the distance fromthe leading edge of the plate.
In Fig. 1 the coefficient of local skin friction of-=c’FIG. 2. Temperature and velocity distribution in the laminarboundary layer on a hot vertical flat plate in natural convection. Theory after E. Pohlhausen, 1921.Experiments after
E. Schmidt and W. Beckmann, 1930.

Tribute to Ludwig Prandtl
buoyancy forces to inertial forces. This figure demonstrates that also for the temperature boundary layer
there is very good agreement between theory andexperiment.
2.2. Turbulent flow
The concept of the boundary layer which was applied
by Prandtl and his collaborators at first to laminar flow
proved to be very successful also for turbulent flow.
A milestone in this respect was the resistance law for
spheres and cylinders of circular cross-section. In thediagram of the drag coefficient against Reynolds number there is a sudden decrease of the drag coefficientwith increasing Reynolds number at a critical Reynolds
number. With his famous experiment using a tripping
wire on a sphere Prandtl was able to demonstrate thatthe steep decrease of the drag coefficient at the critical
Reynolds number is a consequence of the transition
from laminar to turbulent flow in the boundary layer.
Since the turbulent boundary layer is less sensitive toseparation than the laminar boundary layer, the transition from laminar to turbulent flow shifts the separation point on the surface backwards and thus gives asmaller wake and a smaller drag.
For fully developed turbulent flow Prandtl introduced the concept of the mixing length, and thus wasable to establish a semiempirical theory starting from
Reynolds’ apparent stresses due to the turbulent
velocity fluctuations.
This mixing length concept, which was introduced
by Prandtl as early as 1925, proved very useful bothto flow problems in pipes and also for free turbulence
as for instance in jets and wakes.
In Fig. 1 the skin friction for the turbulent boundary
layer on the flat plate as obtained by theory andexperiment is also given in complete agreement.
2.3. Stability theory
In addition to his numerous theoretical and experimental studies on the problems of developed turbulence, Prandtl occupied himself uninterruptedly in thetwenties with the problem of the origin of turbulence.
This problem had earlier, that is in the eighties of thepreceding century, troubled Reynolds, after he had
obtained his fundamental experimental results. Thehypothesis that the appearance of turbulence is theresult of an instability developed in the laminar motion
can be traced to him (Reynolds’ hypothesis). Nevertheless, almost half a century was needed to demonstrate the truth of this hypothesis, and a further twentyyears elapsed before an experimental verification forthis theory was produced. The mathematicians activein this field in the first decade of this century derived
the Orr-Sommerfeld equation in the frame-work of alinear stability theory. They tried to demonstrate theexistence of unstable laminar flow by seeking solutionsof this equation with amplified superimposed disturbances. But the efforts of the mathematicians wereunsuccessful for a long time. Prandtl and his collaborators started to work on this problem about 1920.
However, a theory developed by Tietjens proved to beinadequate for the calculation of a critical Reynolds
number for the boundary layer on a flat plate.
It was left to Prandtl’s associate W. Tollmien to
discover the decisive breakthrough that led to thesolution of this problem in 1929. Tollmien was able,for the first time, to calculate theoretically a critical

1335

Reynolds number for the boundary layer on a flat plateat zero incidence in good agreement with experiment.
In the years to follow, H. Schlichting extended thistheory and proved that the critical Reynolds number
depends strongly, among others, on the pressure
gradient along the wall.
It is strange to report that these theoretical results
for a long time were not completely accepted by thespecialists working outside Germany. Fourteen years
were to elapse until the careful measurements performed by H. L. Dryden and his colleagues fully
confirmed this theory in the year 1943.
2.4. Experimental technique (wind tunnels)
A short time after his arrival in Gottingen, namely
towards the end of the year 1906, Prandtl’s thoughts
turned in a completely new direction. This development
was to prove as important for his scientific attainments
as his efforts on behalf of flows with separation andturbulence. This was the dawn of the new era ofaeronautics. Otto Lilienthal executed in 1896 the firstgliding flight. In the United States, the brothers Wrightachieved the first powered flight in the year 1903.
At the beginning of the twentieth century, Germanengineers were working on the development of theairship. This task was supported by the Motor
Luftschiff Studiengesellschaft (Society for the Study ofAirships). Prandtl participated in this developmentagain with Felix Klein’s encouragement-inthat hedesigned an installation for the testing of models in awind tunnel. This wind tunnel came into operation in1908. It operated in competition with another type ofwind tunnel constructed simultaneously by G. Eiffel
in Paris and became so successful that other windtunnels were modelled after it in many countries.
The essential characteristic of the first wind tunnelof 1908 is that the air circulates in a closed duct. The
important parameters were:Working section: 2 x 2 = 4mZMaximum speed : 10 m/s = 36 km/h
Firslt$r’
Power installed:
30 hp.
>
During World War I Prandtl built his second windtunnel, together with his collaborator A. Betz, whichcame into operation in 1917. This tunnel was equippedwith a nozzle of large contraction ratio, and its maximum speed was considerably higher than that of thefirst tunnel. The important characteristics of the second
Giittingen tunnel were :Working section: 4m2(D = 2.25 m)
Maximum speed: 5Om/s = 180 km/h
Secotlr~lt~lPower installed : 400 hp.
1This type of wind-tunnel (the so-called Goettingentype), conceived by Prandtl, soon became one of themost important experimental aids for the performanceof all manner of experiments on bodies in flow.
This tunnel exerted in all countries a large influenceon the development, on the science of aerodynamics,
and the art of aeronautical engineering.
2.5. wing theory
The wind tunnels just described were not only forinvestigations on airships, but also for airfoils. As aresult of his work in the field of the aerodynamics ofwings undertaken during the first decade of this century,Prandtl found himself in competition with F. W.

1336

Tribute to Ludwig Prandtl

Lanchester of England. Both endeavored, at first independently of each other, to determine the effect offinite span of a wing on its aerodynamic properties.
In close relation with this problem, they attempted toexplain the so-called induced drag around the edgesof the wing. Carl Runge, the other important mathematician of Giittingen, deserves to be given the credit
for having brought together in GGttingen the two
aerodynamicists-Prandtland
Lanchester-for a scientific discussion. It was also on Runge’s initiative that
Lanchester’s book concerning the aerodynamics offlight was translated from the English into the German
language.
The continuous efforts expanded by Prandtl tounderstand the observed characteristics of flows, todescribe them, as far as possible, theoretically andnumerically, and to explore them by the scientificmethod, characterized all publications which appeared
during these pioneering times.
A=Ito7

Converted

to A= 5

result on the lift and drag of an airfoil as a functionof the aspect ratio A is given. For the drag coefficient,cg, in terms of the lift coefficient, c,,. he could provethat(2)

subject to the condition that A is large. In this figure
the measured curves of lift coefficient against dragcoefficient are given for five different aspect ratios A as
five separate curves. However, when they are reducedwith the aid of equation (2) to an aspect ratio A = 5.
they trace a single curve. Thus, experiment brilliantly
confirmed Prandtl’s wing theory.
Prandtl has also contributed numerous papers tocompressible flow (gas dynamics) and meteorologicalproblems; but space does not allow to deal withthese here.
His own publications, including his books and contributions to hand-books, sum up to the considerable
number of 168; the number of doctorates reached 83.
3. PRANDTL

FIG. 3. Wing of finite span; polar curves for different aspectratios (Prandtl-Betz, 1921)..4 = 6/l = aspect ratio.

In the years 1918 and 1919 Prandtl submitted twopapers to the Giittingen Academy of Science on his
discoveries concerning wing theory. This publication
contained Prandtl’s theory of the lifting line for a wingof finite span, the theory of the multiplane, the theoryof the wing of minimum induced drag, as well as manyother contributions. This is an achievement of the samerank as his boundary layer theory of 1904. I shall report
here one result, which is most important from thepractical point of view. In Fig. 3 Prandtl’s classical

AS A UNIVERSITYPROFESSOR

Prandtl’s career as an active university professor
extended over a period of more than 45 years, because
he was appointed at the young age of 26 years to an
ordinary professorship of mechanics at the Technical
University of Hanover in 1901. At Felix Klein’s behest,Prandtl had accepted in 1904 the invitation extended
to him by the University of GGttingen to occupy thenewly created chair of applied mechanics. In 1947 heretired from the professorship at Gattingen at the ageof 72. During the whole of this long period of time,
Prandtl applied himself to his lectures with great
devotion. He remained faithful to Giittingen to the endof his life. Prandtl repeatedly stressed the fact that thelink between the pure and the applied science was forhim important as well as indispensable and that thecultivation of this spirit was a unique feature of life at
Giittingen University. A large number of students wasintroduced by him to the science of mechanics, and,in particular, of fluid mechanics. Many of them wereinduced to continue as doctoral candidates and to work
under Prandtl in one of the institutes directed by him.
I think, that this account clearly proves that thescience of mechanics, particularly the science of fluid
mechanics, progressed enormously through Ludwig
Prandtl; he placed, so to say, this science in a newepoch.
H. SCHLICHTING
Giittingen