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Aerodynamic characteristics of horizontal tail surfaces
From Summary: "Collected data are presented on the aerodynamic characteristics of 17 horizontal tail surfaces including several with balanced elevators and two with end plates. Curves are given for coefficients of normal force, drag, and elevator hinge moment. A limited analysis of the results has been made. The normal-force coefficients are in better agreement with the lifting-surface theory of Prandtl and Blenk for airfoils of low aspect ratio than with the usual lifting-line theory. Only partial agreement exists between the elevator hinge-moment coefficients and those predicted by Glauert's thin-airfoil theory."
Annual report of the National Advisory Committee for Aeronautics (25th).administrative report including Technical Report nos. 645 to 680
Report includes the National Advisory Committee for Aeronautics letter of submittal to the President, summaries of the committee's activities and research accomplished, bibliographies, and financial report.
Boundary-Layer Transition on the N.A.C.A. 0012 and 23012 Airfoils in the 8-Foot High-Speed Wind Tunnel
No Description Available.
Boundary-Layer Transition on the N.A.C.A. 0012 and 23012 Airfoils in the 8-Foot High-Speed Wind Tunnel, Special Report
Determinations of boundary-layer transition on the NACA 0012 and 2301 airfoils were made in the 8-foot high-speed wind tunnel over a range of Reynolds Numbers from 1,600,000 to 16,800,000. The results are of particular significance as compared with flight tests and tests in wind tunnels of appreciable turbulence because of the extremely low turbulence in the high-speed tunnel. A comparison of the results obtained on NACA 0012 airfoils of 2-foot and 5-foot chord at the same Reynolds Number permitted an evaluation of the effect of compressibility on transition. The local skin friction along the surface of the NACA 0012 airfoil was measured at a Reynolds Number of 10,000,000. For all the lift coefficient at which tests were made, transition occurred in the region of estimated laminar separation at the low Reynolds Numbers and approach the point of minimum static pressure as a forward limit at the high Reynolds Numbers. The effect of compressibility on transition was slight. None of the usual parameters describing the local conditions in the boundary layer near the transition point served as an index for locating the transition point. As a consequence of the lower turbulence in the 8-foot high-speed tunnel, the transition points occurred consistently farther back along the chord than those measured in the NACA full-scale tunnel. An empirical relation for estimating the location of the transition point for conventional airfoils on the basis of static-pressure distribution and Reynolds Number is presented.
The calculated effect of various hydrodynamic and aerodynamic factors on the take-off of a large flying boat
Report presents the results of an investigation made to determine the influence of various factors on the take-off performance of a hypothetical large flying boat by means of take-off calculations. The factors varied in the calculations were size of hull (load coefficient), wing setting, trim, deflection of flap, wing loading, aspect ratio, and parasite drag. The take-off times and distances were calculated to the stalling speeds and the performance above these speeds was separately studied to determine piloting technique for optimum take-off.
Constant-pressure blowers
The conventional axial blowers operate on the high-pressure principle. One drawback of this type of blower is the relatively low pressure head, which one attempts to overcome with axial blowers producing very high pressure at a given circumferential speed. The Schicht constant-pressure blower affords pressure ratios considerably higher than those of axial blowers of conventional design with approximately the same efficiency.
Correlation of cooling data from an air-cooled cylinder and several multicylinder engines
The theory of engine-cylinder cooling developed in a previous report was further substantiated by data obtained on a cylinder from a Wright r-1820-g engine. Equations are presented for the average head and barrel temperatures of this cylinder as functions of the engine and the cooling conditions. These equations are utilized to calculate the variation in cylinder temperature with altitude for level flight and climb. A method is presented for correlating average head and barrel temperatures and temperatures at individual points on the head and the barrel obtained on the test stand and in flight. The method is applied to the correlation and the comparison of data obtained on a number of service engines. Data are presented showing the variation of cylinder temperature with time when the power and the cooling pressure drop are suddenly changed.
Design charts relating to the stalling of tapered wings
An aid in airplane design, charts have been prepared to show the effects of wing taper, thickness ratio, and Reynolds number on the spanwise location of the initial stalling point. Means of improving poor stalling characteristics resulting from certain combinations of the variables have also been considered; additional figures illustrate the influence of camber increase to the wing tips, washout, central sharp leading edges, and wing-tip slots on the stalling characteristics. Data are included from which the drag increases resulting from the use of these means can be computed. The application of the data to a specific problem is illustrated by an example.
Determination of ground effect from tests of a glider in towed flight
Report presents the results of an investigation made to find the effect of ground on the aerodynamic characteristics of a Franklin PS-2 glider. The lift, the drag, and the angle of attack of the glider in towed flight were determined at several heights from 0.14 to 1.19 span lengths and at various speeds for each height. Two wing arrangements were tested: the plain wing, and the wing with a nearly full-span 30-percent-chord split flap deflected 45 degrees. The experimental results for the plain wing were in good agreement with theoretical values calculated by the method of Wieselsberger for both the angle of attack and the drag coefficient at a height of 0.21 span length; Tani's refinements of the theory had a practically negligible effect on the computed values in this case.
The development of electrical strain gages
The design, construction, and properties of an electrical-resistance strain gage consisting of fine wires molded in a laminated plastic are described. The properties of such gages are discussed and also the problems of molding of wires in plastic materials, temperature compensation, and cementing and removal of the gages. Further work to be carried out on the strain gage, together with instrument problems, is discussed.
DFS dive-control brakes for gliders and airplanes ; and, Analytical study of the drag of the DFS dive-control brake
These two reports are surveys on the progress and present state of development of dive-control flaps for gliders and airplanes. The second article describes how on the basis of wind tunnel and free-flight tests, the drag increase on brake flaps of the type DFS, can be predicted. Pressure records confirm a two-dimensional load distribution along the brake-flap surface Aerodynamically, the location of the brake flaps along the span is of importance for reasons of avoidance of vibration and oscillation phenomena on control and tail surfaces; statically, because of the magnitude of the frontal drag in diving with respect to the bending moments, which may become decisive for the dimensions of the wing attachment and for the wing covering.
Effect of exit-slot position and opening on the available cooling pressure for NACA nose-slot cowlings
Report presents the results of an investigation of full-scale nose-slot cowlings conducted in the NACA 20-foot wind tunnel to furnish information on the pressure drop available for cooling. Engine conductances from 0 to 0.12 and exit-slot conductances from 0 to 0.30 were covered. Two basic nose shapes were tested to determine the effect of the radius of curvature of the nose contour; the nose shape with the smaller radius of curvature gave the higher pressure drop across the engine. The best axial location of the slot for low-speed operation was found to be in the region of maximum negative pressure for the basic shape for the particular operating condition. The effect of the pressure operating condition on the available cooling pressure is shown.
Effect of fuel-air ratio, inlet temperature, and exhaust pressure on detonation
An accurate determination of the end-gas condition was attempted by applying a refined method of analysis to experimental results. The results are compared with those obtained in Technical Report no. 655. The experimental technique employed afforded excellent control over the engine variables and unusual cyclic reproducibility. This, in conjunction with the new analysis, made possible the determination of the state of the end-gas at any instant to a fair degree of precision. Results showed that for any given maximum pressure the maximum permissible end-gas temperature increased as the fuel-air ratio was increased. The tendency to detonate was slightly reduced by an increase in residual gas content resulting from an increase in exhaust backpressure with inlet pressure constant.
Effect of wing loading, aspect ratio, and span loading of flight performances
An investigation is made of the possible improvements in maximum, cruising, and climbing speeds attainable through increase in the wing loading. The decrease in wing area was considered for the two cases of constant aspect ratio and constant span loading. For a definite flight condition, an investigation is made to determine what loss in flight performance must be sustained if, for given reasons, certain wing loadings are not to be exceeded. With the aid of these general investigations, the trend with respect to wing loading is indicated and the requirements to be imposed on the landing aids are discussed.
Effective gust structure at low altitudes as determined from the reactions of an airplane
Measurements of gust structure and gust intensity were made in the lower levels of the atmosphere (0 to 3,500 ft.). An Aeronca C-2 airplane was used as the measuring instrument, the gust structure being derived from the recorded motions of the airplane. Data were also obtained on wind velocities and temperatures as functions of altitude for use in attempting to correlate the gust-structure data with various meteorological quantities. The results indicated little or no correlation between the gust velocity and the gradient distance. The data, however, did indicate that an airplane the size of the Aeronca will respond most frequently to gusts having gradient distance of the order of 30 feet. The maximum true gust velocity measured during the investigation was 25 feet per second.
Flame speeds and energy considerations for explosions in a spherical bomb
Simultaneous measurements were made of the speed of flame and the rise in pressure during explosions of mixtures of carbon monoxide, normal heptane, iso-octane, and benzene in a 10-inch spherical bomb with central ignition. From these records, fundamental properties of the explosive mixtures, which are independent of the apparatus, were computed. The transformation velocity, or speed at which flame advances into and transforms the explosive mixture, increases with both the temperature and the pressure of the unburned gas. The rise in pressure was correlated with the mass of charge inflamed to show the course of the energy developed.
Free-spinning wind-tunnel tests of a low-wing monoplane with systematic changes in wings and tails V : effect of airplane relative density
The reported tests are a continuation of an NACA investigation being made in the free-spinning wind tunnel to determine the effects of independent variations in load distribution, wing and tail arrangement, and control disposition on the spin characteristics of airplanes. The standard series of tests was repeated to determine the effect of airplane relative density. Tests were made at values of the relative-density parameter of 6.8, 8.4 (basic), and 12.0; and the results were analyzed. The tested variations in the relative-density parameter may be considered either as variations in the wing loading of an airplane spun at a given altitude, with the radii of gyration kept constant, or as a variation of the altitude at which the spin takes place for a given airplane. The lower values of the relative-density parameter correspond to the lower wing loadings or to the lower altitudes of the spin.
The frequencies of cantilever wings in beam and torsional vibrations
Methods are described for calculating the period and frequency of vibration of cantilever wings and similar structures in which the weight and moment of inertia vary along the span. Both the beam and torsional frequencies may be calculated by these methods. The procedure is illustrated by examples. It is shown that a surprisingly close approximation to the beam frequency may be obtained by a very brief calculation in which the curvature of the wing in vibration is assumed to be constant. A somewhat longer computation permits taking account of the true curvature of the beam by a series of successive approximations which are shown to be strongly convergent. Analogous methods are applied to calculations of the torsional frequency. For the first approximation it is assumed that the angle of twist varies linearly alone the semispan. True variation of the twist is computed by successive approximations which are strongly convergent, as in the case of beam vibrations.
Investigations on the incompletely developed plane diagonal-tension field
This report presents the results of an investigation on the incompletely developed diagonal-tension field. Actual diagonal-tension beams work in an intermediate stage between pure shear and pure diagonal tension; the theory developed by wagner for diagonal tension is not directly applicable. The first part of the paper reviews the most essential items of the theory of pure diagonal tension as well as previous attempts to formulate a theory of incomplete diagonal tension. The second part of the paper describes strain measurement made by the N. A. C. A. to obtain the necessary coefficients for the proposed theory. The third part of the paper discusses the stress analysis of diagonal-tension beams by means of the proposed theory.
Knocking in an internal-combustion engine
The question remains open of the relation between the phenomena of knocking in the engine and the explosion wave. The solution of this problem is the object of this paper. The tests were conducted on an aircraft engine with a pyrex glass window in the cylinder head. Photographs were then taken of various combinations of fuels and conditions.
Longitudinal stability and control with special reference to slipstream effects
Report presents the results of a study of data obtained in the NACA full-scale wind tunnel concerning the effects of interference and of propeller operation on longitudinal stability and control. The data include pitching moments for various power conditions for airplanes with tails removed and with tails set at various stabilizer and elevator angles. A number of surveys of the dynamic pressure and the flow direction in the region of the horizontal tail surface are also included. Results are given for eight airplanes, including a model of a four-engine airplane tested both as a tractor and as a pushed and a model of two-engine pusher. The effects are shown of propeller operation on the downwash angles and the dynamic pressures at the tail and on the pitching-moment contribution of the propeller and the wing.
Mechanical Properties of Flush-Riveted Joints
The strength of representative types of flush-riveted joints has been determined by testing 865 single-shearing, double-shearing, and tensile specimens representing 7 types of rivet and 18 types of joint. The results, presented in graphic form, show the stress at failure, type of failure, and d/t ratio. In general, dimpled joints were appreciably stronger than countersunk or protruding-head joints, but their strength was greatly influenced by constructional details. The optimum d/t ratios have been determined for the several kinds of joints. Photomicrographs of each type show constructional details and, in several instances, cracks in the sheet.
Mechanical properties of flush-riveted joints
Report discusses the strength of several types of flush-riveted joints, including single-shearing, double-shearing, and tensile specimens. The stress at failure, type of failure, and d/t ratio are provided. Dimpled joints were found to be the strongest, but their strength was heavily influenced by the details of their construction.
Mechanism of flutter.a theoretical and experimental investigation of the flutter problem
The results of the basic flutter theory originally devised in 1934 and published as NACA Technical Report no. 496 are presented in a simpler and more complete form convenient for further studies. The paper attempts to facilitate the judgement of flutter problems by a systematic survey of the theoretical effects of the various parameters. A large number of experiments were conducted on cantilever wings, with and without ailerons, in the NACA high-speed wind tunnel for the purpose of verifying the theory and to study its adaptability to three-dimensional problems. The experiments included studies on wing taper ratios, nacelles, attached floats, and external bracings. The essential effects in the transition to the three-dimensional problem have been established. Of particular interest is the existence of specific flutter modes as distinguished from ordinary vibration modes. It is shown that there exists a remarkable agreement between theoretical and experimental results.
Piston temperatures in an air-cooled engine for various operating conditions
As part of a program for the study of piston cooling, this report presents the results of tests conducted on a single-cylinder, air-cooled, carburetor engine to determine the effect of engine operating conditions on the temperatures at five locations on the piston.
Preliminary investigation of the flying qualities of airplanes
The National Advisory Committee for Aeronautics is undertaking an investigation of the flying qualities of airplanes. The work consists in the determination of the significant qualities susceptible of measurement, the development of the instruments required to make the measurements, and the accumulation of data on the flying qualities of existing airplanes, which data are to serve as a basis for quantitative specifications for the flying qualities of future designs. A tentative schedule of measurable flying qualities has been prepared and the instruments needed for their measurements have been assembled. A trial of the schedule and the instruments has been made using the Stinson SR-8e airplane. The results showed that, although the original schedule and instruments are basically satisfactory some further development is required to eliminate nonessential items and to expedite flight testing. The report describes and discusses the work done with this airplane.
Preliminary wind-tunnel investigation of an NACA 23012 airfoil with various arrangements of venetian-blind flaps
Report presents the results of an investigation made in the NACA 7 by 10-foot wind tunnel of a large-chord NACA 23012 airfoil with several arrangements of venetian-blind flaps to determine the aerodynamic section characteristics as affected by the over-all flap chord, the chords of the slats used to form the flap, the slat spacing, the number of slats and the position of the flap with respect to the wing. Complete section data are given in the form of graphs for all the combinations tested.
Principles, practices, and progress of noise reduction in airplanes
No Description Available.
Propeller rotation noise due to torque and thrust
Sound pressure of the first four harmonics of rotation from a full-scale two-blade propeller were measured and are compared with values calculated from theory. The comparison is made (1) for the space distribution with constant tip speed and (2) for fixed space angles with variable tip speed. A relation for rotation noise from an element of radius developed by Gutin is given showing the effect of number of blades on the rotation noise.
Stability of Castering Wheels for Aircraft Landing Gears
A theoretical study was made of the shimmy of castering wheels. The theory is based on the discovery of a phenomenon called kinematic shimmy. Experimental checks, use being made of a model having low-pressure tires, are reported and the applicability of the results to full scale is discussed. Theoretical methods of estimating the spindle viscous damping and the spindle solid friction necessary to avoid shimmy are given. A new method of avoiding shimmy -- lateral freedom -- is introduced.
Static thrust and power characteristics of six full-scale propellers
Static thrust and power measurements were made of six full-scale propellers. The propellers were mounted in front of a liquid-cooled-engine nacelle and were tested at 15 different blade angles in the range from -7 1/2 degrees to 35 degrees at 0.75r. The test rig was located outdoors and the tests were made under conditions of approximately zero wind velocity.
Stress distribution in and equivalent width of flanges of wide, thin-wall steel beams
The use of different forms of wide-flange, thin-wall steel beams is becoming increasingly widespread. Part of the information necessary for a national design of such members is the knowledge of the stress distribution in and the equivalent width of the flanges of such beams. This problem is analyzed in this paper on the basis of the theory of plane stress. As a result, tables and curves are given from which the equivalent width of any given beam can be read directly for use in practical design. An investigation is given of the limitations of this analysis due to the fact that extremely wide and thin flanges tend to curve out of their plane toward the neutral axis. A summary of test data confirms very satisfactorily the analytical results.
Tensile elastic properties of typical stainless steels and nonferrous metals as affected by plastic deformation and by heat treatment
A general discussion is given of the relationships between stress, strain, and permanent set. From stress-set curves are derived proof stresses based on five different percentages of permanent set. The influence of prior plastic extension on these values is illustrated and discussed. A discussion is given of the influence of work-hardening, rest interval, and internal stress on the form of the proof stress-extension curve.
Tests of a gust-alleviating flap in the gust tunnel
Tests were made to determine the effectiveness of a long-period dynamically overbalanced flap in reducing airplane accelerations due to atmospheric gusts. For two gust shapes, one gust velocity, one forward velocity, and one wing loading, a series of flights was made with the flap locked and was then repeated with the flap free to operate. The records were evaluated by routine methods. The results indicate that the flap reduced the maximum acceleration increment 39 percent for a severe gust but with a representative gust shape (a sharp-edge gust), the reduction was only 3 percent. The results also indicate that the flap tended to reduce the longitudinal stability of the airplane. Computations made of the effectiveness and the action of the flap were in good agreement with the experimental results.
A theoretical study of lateral stability with an automatic pilot
The influence of automatic operation of the aileron and rudder controls on the lateral stability of an airplane is discussed. The control deflections are assumed to be proportional to the deviations and to the rates of deviation of the airplane from steady-flight conditions. The effects of changes in the types of deviation governing control application are considered. For one simple method of control in which the aileron deflection is proportional to the angle of bank and the rudder deflection is proportional to the angle of yaw, the effect of lag in control application is studied and regions of stability with and without lag are given. For the simple control with lag, curves are included that show the variations in the roots of the stability equation with changes in the amount of control applied. It is concluded that, although the simple control provides a satisfactory means of varying most of the lateral stability characteristics, the stability in azimuth will always be poor for such a control.
The unsteady lift of a wing of finite aspect ratio
Unsteady-lift functions for wings of finite aspect ratio have been calculated by correcting the aerodynamic inertia and the angle of attack of the infinite wing. The calculations are based on the operational method.
Stresses in single-spar wing constructions with incompletely built-up ribs
It is shown that the force distribution resulting from incomplete ribs in single spar wing structures may be determined with the aid of the shear field method by a statistically indeterminate computation. A numerical computation is given of the force distribution of a wing structure whose two neighboring incomplete ribs with web missing in half the section are torsionally loaded.
Aerodynamics of rotating-wing aircraft with blade-pitch control
From Introduction: "In the present report, with the aid of the usual computation methods, a rotor is investigated the pitch of whose blades is capable of being controlled in such a manner that it varies linearly with the flapping angle. To test the effect of this linkage on the aircraft performance, the theory is applied to an illustrative example."
Damping formulas and experimental values of damping in flutter models
The problem of determining values of structural damping for use in flutter calculations is discussed. The concept of equivalent viscous damping is reviewed and its relation to the structural damping coefficient g introduced in NACA Technical Report No. 685 is shown. The theory of normal modes is reviewed and a number of methods are described for separating the motions associated with different modes. Equations are developed for use in evaluating the damping parameters from experimental data. Experimental results of measurements of damping in several flutter models are presented.
Experimental contribution to the study of combustion in compression-ignition engines
The purpose of this paper is to explain the differences in thermodynamic qualities of two oils and to try to make this differentiation clear enough so that it can be applied to two oils extremely alike, as, for instance a given oil and the same oil to which a small amount of another substance has been added.
Experimental Determination of Exhaust Gas Thrust, Special Report
This investigation presents the results of tests made on a radial engine to determine the thrust that can be obtained from the exhaust gas when discharged from separate stacks and when discharged from the collector ring with various discharge nozzles. The engine was provided with a propeller to absorb the power and was mounted on a test stand equipped with scales for measuring the thrust and engine torque. The results indicate that at full open throttle at sea level, for the engine tested, a gain in thrust horsepower of 18 percent using separate stacks, and 9.5 percent using a collector ring and discharge nozzle, can be expected at an air speed of 550 miles per hour.
A generalized vortex theory of the screw propeller and its application
The vortex theory as presented by the author in earlier papers has been extended to permit the solution of the following problems: (1) the investigation of the relation between thrusts and torque distribution and energy loss as given by the induction of helical vortex sheets and by the parasite drag; (2) the checking of the theorem of Betz of the rigidly behaving helical vortex sheet of minimum induced energy loss; (3) the extension of the theory of the screw propeller of minimum energy loss for the inclusion of parasite-drag distribution along the blades. A simple system of diagrams has been developed to systematize the design of airplane propellers for a wide range of parasite-drag distribution along the blades.
An investigation of sheet-stiffener panels subjected to compression loads with particular reference to torsionally weak stiffeners
A total of 183 panel specimens of 24ST aluminum alloy with nominal thickness of 0.020, and 0.040 inch with extruded bulb-angle sections of 12 shapes spaced 4 and 5 inches as stiffeners were tested to obtain the buckling stress and the amplitude of the maximum wave when buckled. Bulb angles from 3 to 27 1/2 inches long were tested as pin-end columns. The experimental data are presented as stress-strain and column curves and in tabular form. Some comparisons with theoretical results are presented. Analytical methods are developed that make it possible for the designer to predict with reasonable accuracy the buckling stress and the maximum-wave amplitude of the sheet in stiffened-panel combinations. The scope of the tests was insufficient to formulate general design criteria but the results are presented as a guide for design and an indication of the type of theoretical and experimental work that is needed.
Measurement of the forces acting on gliders in towed flight
The magnitude, the direction, and the fluctuations of tow forces exerted upon gliders by towing them aloft behind an automobile were measured under a variety of conditions covering a range from gentle to severe types of operation. For these tests, the glider towing force did not exceed 1.6 of the gross weight of the glider. V-G records obtained during the towed-flight period as well as during the subsequent return glide to earth showed accelerations in the range from 3 to -1 g. The results of preliminary airplane tow tests are also presented.
A new method of studying the flow of the water along the bottom of a model of a flying-boat hull
A new method of studying the flow of the water along the bottom of a model of a flying-boat hull is described. In this method, the model is fitted with a transparent bottom and is divided down the center line by a bulkhead. The flow is observed and photographed through one-half of the model by means of the diffused illumination from a battery of lamps contained in the other half of the model. Photographs of the flow, particularly of the changes that occur when the step ventilates, are shown. The results of the present investigation indicate that the method has considerable promise, chiefly in connection with motion-picture studies.
The strength of shell and tubular spar wings
The report is a survey of the strength problems arising on shell and tubular spar wings. The treatment of the shell wing strength is primarily confined to those questions which concern the shell wing only; those pertaining to both shell wing and shell body together have already been treated in TM 838. The discussion of stress condition and compressive strength of shell wings and tubular spar wings is prefaced by several considerations concerning the spar and shell design of metal wings from the point of view of strength.
Testing of high-octane fuels in the single-cylinder airplane engine
One of the most important properties of aviation fuels for spark-ignition engines is their knock rating. The CFR engine tests of fuels of 87 octane and above does not always correspond entirely to the actual behavior of these fuels in the airplane engine. A method is therefore developed which, in contrast to the octane number determination, permits a testing of the fuel under various temperatures and fuel mixture conditions. The following reference fuels were employed: 1) Primary fuels; isooctane and n-heptane; 2) Secondary fuels; pure benzene and synthetic benzine.
Theoretical and experimental investigations of the drag of installed aircraft radiators
The present report proposes the determination of the absolute magnitude of the total radiator drag and, in addition, of the different causes of the radiator drag.
Wind-tunnel tests of the NACA 45-125 airfoil: a thick airfoil for high-speed airplanes
Report discussing an investigation of the pressure distribution, profile drag, and location of transition for a NACA 45-125 airfoil for the purpose of aiding in the development of a thick wing for high-speed airplanes. The critical speed, pressure distribution, and drag of the airfoil are detailed.
Application of the methods of gas dynamics to water flows with free surface I : flows with no energy dissipation
The application is treated in sufficient detail to facilitate as much as possible its application by the engineer who is less familiar with the subject. The present work was undertaken with two objects in view. In the first place, it is considered as a contribution to the water analogy of gas flows, and secondly, a large portion is devoted to the general theory of the two-dimensional supersonic flows.