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A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenFri, 08 Dec 2023 21:53:07 +0000A Dewpoint Meter Using Cooling by Expansion of CO2
https://resolver.caltech.edu/CaltechAUTHORS:LIErsi45
Authors: Liepmann, K.; Liepmann, H. W.
Year: 1945
DOI: 10.1063/1.1770319
For use in certain aerodynamical problems a dewpoint meter using the Joule-Thompson effect, with CO2 as cooling agent, has been developed. The instrument described here has some advantages over the common instrument which depends oupon the evaporation of ether. Two slightly different devices have been used successfully.https://authors.library.caltech.edu/records/fxwkt-4fp96The Interaction Between Boundary Layer and Shock Waves in Transonic Flow
https://resolver.caltech.edu/CaltechAUTHORS:20230309-962584000.1
Authors: Liepmann, Hans Wolfgang
Year: 1946
DOI: 10.2514/8.11473
Experiments of transonic flow past a circular arc profile show that the shock-wave pattern and the pressure distribution are strongly dependent upon the state of the boundary layer. A change from laminar to turbulent boundary layer at a given Mach Number changes the flow pattern considerably.
Shock waves can interact with the boundary layer in a manner similar to a reflection from a free jet boundary. These shock waves are not distinctly discernible from pressure distribution measurements.https://authors.library.caltech.edu/records/pzsxd-srp07Investigations of Effects of Surface Temperature and Single Roughness Elements on Boundary-Layer Transition
https://resolver.caltech.edu/CaltechAUTHORS:LIEnacarpt890
Authors: Liepmann, Hans W.; Fila, Gertrude H.
Year: 1947
The laminar boundary layer and the position of the transition point are investigated on a heated flat plate. It was found that the Reynolds number of transition decreases as the temperature of the plate is increased. It is shown from simple qualitative analytical considerations that the effect of variable viscosity in the boundary layer due to the temperature diference produces a velocity profile with an inflection point if the wall temperature is higher than the free-stream temperature. This profile is confirmed by measurements. Furthermore, it is confirmed that, even with large deviation from the Blasius condition, the velocity and temperature profiles are very nearly identical, as predictable theoretically for a Prandtl number [sigma] of the order of 1.0 (for air, [sigma]=0.76). The instability of
injection-point profiles is discussed.
Studies of the flow in the wake of large, two-dimensional
roughness elements are presented. It is shown that a boundary laysr can separate and reattach itself to the wall without having transition take place.https://authors.library.caltech.edu/records/awb3z-zw847Investigation of Effects of Surface Temperature and Single Roughness Elements on Boundary-Layer Transition
https://resolver.caltech.edu/CaltechAUTHORS:LIEnacatn1196
Authors: Liepmann, Hans W.; Fila, Gertrude H.
Year: 1947
The laminar boundaxy layer and the position of the transition point were investigated on a heated flat plate. It was found that the Reynolds number of transition decreases as the temperature of the plate is increased. It is shown from simple qualitative analytical considerations that the effect of variable viscosity in the boundary layer due to the temperature difference produces a velocity profile with an inflection point if the wall temperature is higher than the free-stream temperature. This profile is confirmed by measurements. Furthermore, it is confirmed that even with large deviation from the Blasius condition, the velocity and temperature profiles are very nearly identical, as predictable theoretically for a Prandtl number [sigma] of the order of 1.0 (for air, [sigma] = 0.76). The instability of inflection-point profiles is discussed.
Studies of the flow in the wake of large, two-dimensional roughness elements are presented. It is shown that a boundary layer can separate and reattach itself to the wall without having transition take place.https://authors.library.caltech.edu/records/3x3eq-aj709Investigations of Free Turbulent Mixing
https://resolver.caltech.edu/CaltechAUTHORS:LIEnacatn1257
Authors: Liepmann, Hans Wolfgang; Laufer, John
Year: 1947
A discussion of the integral relations for the flow of the boundary-layer type is presented. It is shown that the characteristic laws of spread of jets, wakes, and so forth, can be obtained directly for the laminar case and, with the help of dimensional reasoning, for the turbulent case as well.
Measurements of the mean velocity, the intensity and scale of the turbulent fluctuations, and of the turbulent shear in a two-dimensional mixing zone are presented. The results of these measurements are compared with the mixing-length theories. It is shown that both mixing length and exchange coefficient vary across the mixing zone. The theories based on the assumption of constant mixing length or exchange coefficient are thus in error.
A discussion of the energy balance of the fluctuating motion is given and the triple point correlation is estimated.https://authors.library.caltech.edu/records/ebd4r-t6m96On the Spectrum of Isotropic Turbulence
https://resolver.caltech.edu/CaltechAUTHORS:LIEnacatn2473
Authors: Liepmann, H. W.; Laufer, J.; Liepmann, Kate
Year: 1951
Measurements of the spectrum and correlation functions at large Reynolds number (RN ~ 10^5 based on the grid mesh) have been made, as well as a series of accurate spectrum measurements at lower Reynolds number (RN ~ 10^4).
The results are compared with the theoretical laws proposed in recent years. It is found that the measurements at large Reynolds numbers exhibit a range of frequencies where the spectrum is nearly of the form n^- 5/3.
The largest part of the spectrum in the initial stage of decay at the lower Reynolds number was found to follow closely the simple spectrum A/[B + n^2] , where A and B are constants and n is the frequency of fluctuation. At x/M = 1000 (where x is the distance behind the grid and M is the mesh size) the spectrum approaches a Gaussian distribution.
The second, fourth, and sixth moments of the spectrum have been computed from the measurements and are discussed In relation to theoretical results.
The significance of the number of zeros of the fluctuating velocity u(t) is discussed and examples of measurements for the determination of the microscale of turbulence [lambda] from zero counts are given.https://authors.library.caltech.edu/records/58rp5-bsf15On Reflection of Shock Waves from Boundary Layers
https://resolver.caltech.edu/CaltechAUTHORS:LIEnacarpt1100
Authors: Liepmann, H. W.; Roshko, A.; Dhawan, S.
Year: 1952
Measurements of the reflection characteristics of shock waves from a flat surface with a laminar and turbulent boundary layer are presented. The investigations were carried out at Mach numbers from about 1.3 to 1.5 and a Reynolds number of 0.9 x 10^4.
THe difference in the shock-wave interaction with laminar and turbulent boundary layers, first found in transonic flow is confirmed and ,investigated in detail for supersonic flow. The relative upstream influence of a shock wave impinging on a given boundary layer has been measured for both laminar and turbulent layers. The upstream influence of a shock wave in the laminar layer is found to be of the order of 50 bounday-layer thicknesses as compared with about 5 in the turbulent case. Separation almost always occurs in the laminar boundary layer. The separation is restricted to a region of finite extent upstream of the the shock wave. In the turbulent case no separation was found. A model of the flow near the point of impingement of the shock wave on the boundary layer is given for both cases. The difference between impulse-type and step-type shock waves is discussed and their interactions with the boundary layer are compared.
Some general considerations on the experimental production of shock waves from wedges and cones are presented, as well as a discussion of boundary layer in supersonic flow. A few exampies of reflection of shock waves from supersonic shear layers are also presented.https://authors.library.caltech.edu/records/2haej-rzc24Counting Methods and Equipment for Mean-Value Measurements in Turbulence Research
https://resolver.caltech.edu/CaltechAUTHORS:LIEnacatn3037
Authors: Liepmann, H. W.; Robinson, M. S.
Year: 1953
This report deals with methods of measuring the probability distributions and mean values of random functions as encountered in turbulence research. Applications to the measurement of probability distributions of the axial velocity fluctuation u(t) and its derivative du/dt in isotropic turbulence are shown. The assumption of independent probabilities of u(t) and du/dt, which has been used as an approximation in the application of zero counts to the measurement of the microscale of turbulence [lambda], is investigated. The results indicate that the assmuption is satisfied within a few percent and that there is, so far, no evidence that the systematic difference between [lambda] measured from zero counts and [lambda] measured independently can be traced entirely to the statistical dependence of u and du/dt.
The chronological development of apparatus is described, concluding with the present 10-channel statistical analyzer based upon a system of pulse amplitude modulation followed by an amplitude discriminator and a counter. A discussion of the relative merits of various systems is included to indicate the reasons for this choice.https://authors.library.caltech.edu/records/dgqbv-ydk93Shearing-Stress Measurements by use of a Heated Element
https://resolver.caltech.edu/CaltechAUTHORS:LIEnacatn3268
Authors: Liepmann, H. W.; Skinner, G. T.
Year: 1954
The rate of local heat transfer from a solid surface to a moving fluid is related to the local skin frinction. Measurements of the heat transmission from small elements embedded in the surface of a solid can thus be used to botain local skin-friction coefficients. This method was applied by Fage and Falkner for laminar boundary layers and by Ludwieg for turbulent boundary layers. The present report discussed the possible range of application of such an instrument in low- and high-speed flow and presents experimental data to show that a very simple instrument can be used to obtain laminar and turbulent skin-friction coefficients with a single calibration. The instrument consists of an ordinary hot-wire cemented into a groove in the surface. The heat loss from the wire is proportional to the cube root of the wall shearing stress, and the constant of proportionality may be found by one calibration, for example, in laminar flow.https://authors.library.caltech.edu/records/z1sfr-x9a98On the Contribution of Turbulent Boundary Layers to the Noise inside a Fuselage
https://resolver.caltech.edu/CaltechAUTHORS:CORnacatm1420
Authors: Corcos, G. M.; Liepmann, H. W.
Year: 1956
The following report deals i preliminary fashion with the transmission through a fuselage of random noise generated on the fuselage skin by a turbulent boundary layer. The concept of attenuation is abandoned and instead the problem is formulated as a sequence of two linear couplings: the turbulent boundary layer fluctuations excite the fuselage skin in lateral vibrations and the skin vibrations induce sound inside the fuselage. The techniques used are those required to determine the response of linear systems to random forcing functions of several variables. A certain degree of idealization has been resorted to. Thus the boundary layer is assumed locally homogeneous, the fuselage skin is assumed flat, unlined and free from axial loads and the "cabin" air is bounded only by the vibrating plate so that only outgoing waves are considered. Some of the details of the statistical description have been simplified in order to reveal the basic features of the problem.
The results, strictly applicable only to the limiting case of thin boundary layers, show that the sound pressure intensity is proportional to the square of the free stream density, the square of cabin air density and inversely proportional to the first power of the damping constant and to the second power of the plate density. The dependence on free stream velocity and boundary layer thickness cannot be given in general without a detailed knowledge of the characteristics of the pressure fluctuations in the boundary layer (in particular the frequency spectrum). For a flat spectrum the noise intensity depends on the fifth power of the velocity and the first power of the boundary layer thickness. This suggests that boundary layer removal is probably not an economical means of decreasing cabin noise.
In general, the analysis presented here only reduces the determination of cabin noise intensity to the measurement of the effect of any one of four variables (free stream velocity, boundary layer thiclkness, plate thickness or the characteristic velocity of propagation in the plate).
The plate generates noise by vibrating in resonance over a wide range of frequencies and increasing the damping constant is consequently an effective method of decreasing noise generation.
One of the main features of the results is that the relevent quantities upon which noise intensity depends are non-dimensional numbers in which boundary layer and plate properties enter as ratios. This is taken as an indication that in testing models of structures for boundary layer noise it is not sufficient to duplicate in the model the structural characteristics of the fuselage. One must match properly the characteristics of the exciting pressure fluctuations to that of the structure.https://authors.library.caltech.edu/records/3q4c9-e8r53Magnetically driven cylindrical shock waves
https://resolver.caltech.edu/CaltechAUTHORS:LIEpof61
Authors: Liepmann, H. W.; Vlases, G.
Year: 1961
DOI: 10.1063/1.1706428
Nearly every experiment on shock-wave propagation uses plane waves. Cylindrical and spherical waves are more difficult to produce, usually decay fast and do not offer any particular advantages. In magneto-fluid dynamics, however, it is possible to produce cylindrical waves easily and the axisymmetric geometry is a natural choice to study motion across magnetic field lines.https://authors.library.caltech.edu/records/0t38w-km896A 17-inch Diameter Shock Tube for Studies in Rarefied Gasdynamics
https://resolver.caltech.edu/CaltechAUTHORS:20130509-141109412
Authors: Liepmann, H. W.; Roshko, Anatol; Coles, Donald; Sturtevant, Bradford
Year: 1962
DOI: 10.1063/1.1746625
A shock tube for studying problems in rarefied gasdynamics is described. The motivation for operating at low density (to increase the length and time scales of certain interesting flows) and the effect of low density on the performance and design of the shock tube are discussed. In order to guarantee uniform and reproducible shock waves of moderate strength, the configuration of the tube is conventional. However, innovations are introduced (for example in the suspension, the pumping system, and the diaphragm loading and rupturing mechanism) to simplify the operation of the large facility. Care in the design of the tube as a vacuum system has resulted in a leak rate of less than 0.01 μ Hg per hour. A series of shakedown runs at relatively high pressures has shown, for example, that the reproducibility of a given shock Mach number is ±0.6%.https://authors.library.caltech.edu/records/z8ejr-5kj55Structure of a Plane Shock Layer
https://resolver.caltech.edu/CaltechAUTHORS:20120921-134953075
Authors: Liepmann, H. W.; Narashima, R.; Chahine, M. T.
Year: 1962
DOI: 10.1063/1.1706527
The structure of a plane shock wave is discussed and the expected range of applicability of the Navier‐Stokes equations within the shock layer is outlined. The shock profiles are computed using the Bhatnagar‐Gross‐Krook model of the Boltzmann equation and a uniformly converging iteration scheme starting from the Navier‐Stokes solution. It is shown that the Navier‐Stokes solution remains a good approximation in the high‐pressure region of the shock layer up to approximately the point of maximum stress for all shock strengths. In the low‐pressure region, the correct profiles deviate with increasing shock strength from the Navier‐Stokes solution. The physical significance of the kinetic model used and the relation of the present study to previous theoretical and experimental work is discussed.https://authors.library.caltech.edu/records/aq5fx-3dc35Shock Tubes in Rarefied Gas Flow Research
https://resolver.caltech.edu/CaltechAUTHORS:20170801-130148469
Authors: Coles, D.; Liepmann, H. W.; Roshko, A.; Sturtevant, B.
Year: 1968
In any real fluid motion there exists regions in space-time
in which the fluid is far from thermodynamic equilibrium.
The relative extent of these non-equilibrium regions is
determined by the ratio of the molecular relaxation times and the corresponding length scales to the macroscopic time and space scales appropriate to the flow. Gas flow within such non-equilibrium regions is properly called "rarefied". In recent years the shock tube has become a rather efficient tool in the investigations of rarefied gas flows and I intend to illustrate progress in this use through a discussion of some very recent and typical investigations of the GALCIT group carried out under NASA sponsorship.https://authors.library.caltech.edu/records/vr9ed-ex370Shock Tubes in Rarefied Gas Flow Research
https://resolver.caltech.edu/CaltechAUTHORS:20130510-081002532
Authors: Coles, D.; Liepmann, H. W.; Roshko, A.; Sturtevant, B.
Year: 1968
The flow
within a shock wave is governed by the relaxation times of
the molecular degrees of freedom. Advances in shock tube
design and instrumentation have made it possible in recent
years to resolve all the relaxation times including the shortest,
corresponding to the translational degree of freedom.
The shock tube thus becomes an important tool for critical
experiments in the study of the range of applicability of the
Navier-Stokes equations and similar approximations and of
the character of solutions of the Boltzmann equation. Significant
progress has been made recently in the understanding
of the most obvious such problem, the flow within a shock in
a monatomic gas. Theory and experiment are now in substantial
agreement and the over-all process of energy exchange
is understood. Problems connected with shock wave
reflection from real walls have made progress but a host of
problems remain to be studied including surface interaction
effects. The extension of this type of shock tube research to
more complicated systems, reacting gases, gas mixtures,
and the like has begun and some progress can be reported.
Recent experimental progress is illustrated by a number of
measurements made at GALCIT in the 17- and 6-in. shock
tubes. Sophistications in shock tube design and instrumentation
will be discussed.https://authors.library.caltech.edu/records/yxtj9-f8y97Shock Tubes in Rarefied Gas Flow Research
https://resolver.caltech.edu/CaltechAUTHORS:20130509-142125916
Authors: Coles, D.; Liepmann, H. W.; Roshko, A.; Sturtevant, B.
Year: 1969
DOI: 10.1063/1.1692612
The flow within a shock wave is governed by the relaxation times of the molecular degrees of freedom.
Advances in shock-tube design and instrumentation in recent years have made it possible to resolve all the
relaxation times including the shortest, corresponding to the translational degrees of freedom. The shock
tube thus becomes an important tool for critical experiments in the study of the range of applicability of
the Navier-Stokes equations and similar approximations and of the character of solutions of the Boltzmann
equation. Significant progress has recently been made in the understanding of the most obvious such problem,
the flow within a shock in a monatomic gas. Theory and experiment are now in substantial agreement and
the over-all process of energy exchange is understood. Progress has been made in problems connected with
shock wave reflection from real walls, but a host of others remain to be studied including surface interaction
effects. The extension of this type of shock-tube research to more complicated systems, reacting gases, gas
mixtures, and the like has begun and some progress can be reported. Recent experimental progress is illustrated
by a number of measurements made in the 6- and 17-in. shock tubes at the California Institute of
Technology.https://authors.library.caltech.edu/records/fd0ez-w3x85Cryogenic shock tube
https://resolver.caltech.edu/CaltechAUTHORS:LIEpof73
Authors: Liepmann, H. W.; Cummings, J. C.; Rupert, Viviane C.
Year: 1973
DOI: 10.1063/1.1694339
Two shock tubes have been developed which allow for partial or full immersion of the test sections within a cryogenic bath. One tube is used for the study of shock-wave interactions with strong density gradients, and the other to obtain very large shock Mach numbers with ideal gas conditions in all flow regions.https://authors.library.caltech.edu/records/1gjmj-c2d02Fluid Dynamics of Liquid Helium
https://resolver.caltech.edu/CaltechAUTHORS:20120813-092543887
Authors: Liepmann, H. W.
Year: 1975
DOI: 10.1137/0128058
Liquid helium at low temperatures owes its existence to h through the zero point energy classically it should be solid. ^(4)He the common isotope, owes its peculiar behavior as a fluid to its spin and hence again to h; classically the difference between ^(3)He and ^(4)He should be trivial.
In liquid helium flow we deal with a system which still shows all the usual behavior of a liquid plus
some additional strange properties which reflect directly macroscopic quantum effects. The governing
equations of motion due largely to Landau and London are, except in their linearized form, not as well
founded and most certainly less well confirmed than one would like. Consequently, the experimental
fluid dynamicist working with helium should have a field day exploring flow problems in an atmosphere
more adventureous than with any ordinary fluid. This indeed is often the case. One does, however,
ruefully discover that some of the more interesting and significant flow configurations which one likes
to study in this strange field are by no means sufficiently well explored in the corresponding classical
cases. One therefore likes to design simple fluid flow experiments which bring out the essentially new
properties of He II and permit an experimental contribution to, or decision among, the theories of
He II flow. In this spirit, experiments associated with the propagation of shock waves in liquid helium
have been initiated at GALCIT. The design and construction of a cryogenic shock tube and its application
to liquid helium are discussed in this paper.https://authors.library.caltech.edu/records/w0051-b6b50Control of laminar-instability waves using a new technique
https://resolver.caltech.edu/CaltechAUTHORS:20120627-155913693
Authors: Liepmann, H. W.; Brown, G. L.; Nosenchuck, D. M.
Year: 1982
DOI: 10.1017/S0022112082001025
A new technique using surface-film activators has been developed to induce and control laminar-instability waves by periodic heating. A flat plate was instrumented
and installed in the GALCIT High-speed Water Tunnel with flush-mounted surface heaters and probes. Extremely two-dimensional naturally occurring Tolmien-Schlichting (TS) waves were observed along with the subsequent formation of turbulent spots. Laminar-instability waves were then excited in a controlled fashion using the surface-mounted heaters. A preliminary experiment on cancellation of
excited laminar-instability waves was carried out. Finally, turbulent spots were produced using amplitude-modulated bursts to form Gaussian TS wave packets.
Flow visualization, along with wall shear measurements, was used to infer the velocity and vorticity field near the wall.https://authors.library.caltech.edu/records/n5ref-p7d21Active control of laminar-turbulent transition
https://resolver.caltech.edu/CaltechAUTHORS:20120717-085352812
Authors: Liepmann, H. W.; Nosenchuck, D. M.
Year: 1982
DOI: 10.1017/S0022112082001037
Instability waves, commonly called T-S waves, can be introduced in a laminar boundary layer by periodic heating of flush-mounted heating elements. Experiments have demonstrated that nearly complete cancellation of a T-S wave excited in this way can be achieved by using a second downstream heating element with a suitable phase shift. As one application of the technique, a single element together with a feedback loop activated by measured wall shear stress has been used to reduce the amplitude of naturally occurring laminar instability waves. A significant increase in the transition Reynolds number has been achieved.https://authors.library.caltech.edu/records/mmvts-41k88Nonlinear Interactions in the Fluid Mechanics of Helium II
https://resolver.caltech.edu/CaltechAUTHORS:20161018-152858115
Authors: Liepmann, H. W.; Laguna, G. A.
Year: 1984
DOI: 10.1146/annurev.fl.16.010184.001035
Besides its practical importance in a host of technical applications, fluid
mechanics retains its intrinsic interest as a physical discipline. The
governing equations are nonlinear, and hence the motion of fluids
demonstrates the complexities of solution of a nonlinear field theory, a fact
that has been appreciated more and more in recent times. The most striking
manifestations of this nonlinearity are shock waves and turbulence,
corresponding to nonlinear wave and vortex interactions, respectively.https://authors.library.caltech.edu/records/shc3c-g6327Turbulence Management and Relaminarisation
https://resolver.caltech.edu/CaltechAUTHORS:20170728-155935781
Authors: Liepmann, H. W.; Narasimhan, R.
Year: 1988
DOI: 10.1007/978-3-642-83281-9
The last two decades have witnessed an intensifying
effort in learning how to manage flow turbulence: it has in
fact now become one of the most challenging and prized technological goals in fluid dynamics. The goal itself is of course not new. More than a hundred years ago, Reynolds already listed factors conducive to laminar and to turbulent flow (including among them curvature and acceleration). Furthermore, it is in retrospect clear that there were several early instances of successful turbulence management. Examples are the reduction in drag achieved with a ring-trip placed on the front of a sphere or the insertion of a splitter-plate behind a circular cylinder; by the early 1950s there were numerous exercises at boundary layer control. Although many of these studies were interesting and suggestive, they led to no spectacularly successful practical application, and the effort petered out in the late 1950s.https://authors.library.caltech.edu/records/w6dn2-dqa63On the acoustic radiation from boundary layers and jets
https://resolver.caltech.edu/CaltechAUTHORS:LIEgalcit54
Authors: Liepmann, H. W.
Year: 2008
In the following a general discussion of aerodynamically created sound is given. The study is essentially theoretical in nature, but arrives at a description of the physical phenomena in such a fashion as to yield an immediate access to experiments.
First, the problem of aerodynamic noise is defined and two simple mechanical analogues discussed. Then, the general equations of motion of a viscous, compressible fluid are rearranged in a form suitable for a comparison with Lighthill's approach. However, this approach is not being followed through. Instead, the concept of induced velocities due to displacement effects is put forward and carried through for noise produced by boundary layer flow. The same concept is then extended to describe the sound field created by a jet.https://authors.library.caltech.edu/records/7hmt5-ctr70Chemical Reactions in Turbulent Mixing Flows
https://resolver.caltech.edu/CaltechAUTHORS:20141114-143811731
Authors: Liepmann, H. W.; Broadwell, J. E.; Dimotakis, P. E.
Year: 2014
This is a continuing effort in both gas phase and liquid phase mixing, chemical reactions and combustion, in moderate to high Reynolds number turbulent free shear flows. This is primarily an experimental investigation closely supported by theoretical and modeling efforts, as well as specific diagnostics developments, as dictated by specific needs of the experimental program.https://authors.library.caltech.edu/records/zx7z1-bf968Investigations of the Interaction of Boundary Layer and Shock Waves in Transonic Flow
https://resolver.caltech.edu/CaltechAUTHORS:20151111-142642000
Authors: Liepmann, H. W.
Year: 2015
A discussion is given of the interaction between shock waves and boundary layer and of the formation of shocks in transonic flow, based on measurements of transonic flow past a 12% thick circular arc profile. It is found that:
a. The shock wave pattern at a given Mach number can be completely altered by changing the boundary layer.
b. Shock waves can interact with a boundary layer in a manner similar to the reflection of a wave from a free jet boundary. Shock waves do not necessarily cause boundary layer separation.
c. There exist two types of possible transonic flow past a given symmetrical boundary. One symmetrical about the maximum thickness point and one asymmetrical about this point. The first can be identified with the known symmetrical potential solutions. In this case, recompression begins without a shock wave. The asymmetrical case is characterized by an expansion of the flow up to the shock wave. The shock wave in the symmetrical case is related to the "limiting" line of potential theory, the shock wave in the asymmetrical case to the shock waves occurring in de Laval nozzles.https://authors.library.caltech.edu/records/90t28-1zn74