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1284:
551:
142:
1227:
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the flow and a diverging duct (dA > 0) decreases velocity of the flow. For supersonic flow, the opposite occurs due to the change of sign of (1 â M). A converging duct (dA < 0) now decreases the velocity of the flow and a diverging duct (dA > 0) increases the velocity of the flow. At Mach = 1, a special case occurs in which the duct area must be either a maximum or minimum. For practical purposes, only a minimum area can accelerate flows to Mach 1 and beyond. See table of sub-supersonic diffusers and nozzles.
748:
796:
841:
1168:
394:
566:, who invented it. As subsonic flow enters the converging duct and the area decreases, the flow accelerates. Upon reaching the minimum area of the duct, also known as the throat of the nozzle, the flow can reach Mach 1. If the speed of the flow is to continue to increase, its density must decrease in order to obey conservation of mass. To achieve this decrease in density, the flow must expand, and to do so, the flow must pass through a diverging duct. See image of de Laval Nozzle.
1294:
241:
is irrelevant. Once the speed of the flow approaches the speed of sound, however, the Mach number becomes all-important, and shock waves begin to appear. Thus the transonic regime is described by a different (and much more complex) mathematical treatment. In the supersonic regime the flow is dominated by wave motion at oblique angles similar to the Mach angle. Above about Mach 5, these wave angles grow so small that a different mathematical approach is required, defining the
233:
1140:
570:
881:
864:
113:." In truth, the barrier to supersonic flight was merely a technological one, although it was a stubborn barrier to overcome. Amongst other factors, conventional aerofoils saw a dramatic increase in drag coefficient when the flow approached the speed of sound. Overcoming the larger drag proved difficult with contemporary designs, thus the perception of a sound barrier. However, aircraft design progressed sufficiently to produce the
1217:
817:. Further, the name "normal" is with respect to geometry rather than frequency of occurrence. Oblique shocks are much more common in applications such as: aircraft inlet design, objects in supersonic flight, and (at a more fundamental level) supersonic nozzles and diffusers. Depending on the flow conditions, an oblique shock can either be attached to the flow or detached from the flow in the form of a
201:
properties change mainly in the flow direction rather than perpendicular to the flow. However, an important class of compressible flows, including the external flow over bodies traveling at high speed, requires at least a 2-dimensional treatment. When all 3 spatial dimensions and perhaps the time dimension as well are important, we often resort to computerized solutions of the governing equations.
197:). The Lagrangian approach follows a fluid mass of fixed identity as it moves through a flowfield. The Eulerian reference frame, in contrast, does not move with the fluid. Rather it is a fixed frame or control volume that fluid flows through. The Eulerian frame is most useful in a majority of compressible flow problems, but requires that the equations of motion be written in a compatible format.
1260:
air slows down to subsonic before it enters the turbojet engine. This is accomplished with one or more oblique shocks followed by a very weak normal shock, with an upstream Mach number usually less than 1.4. The airflow through the intake has to be managed correctly over a wide speed range from zero to its maximum supersonic speed. This is done by varying the position of the intake surfaces.
254:
these waves are simply concentric spheres. As the sound-generating point begins to accelerate, the sound waves "bunch up" in the direction of motion and "stretch out" in the opposite direction. When the point reaches sonic speed (M = 1), it travels at the same speed as the sound waves it creates. Therefore, an infinite number of these sound waves "pile up" ahead of the point, forming a
46:(the ratio of the speed of the flow to the speed of sound) is smaller than 0.3 (since the density change due to velocity is about 5% in that case). The study of compressible flow is relevant to high-speed aircraft, jet engines, rocket motors, high-speed entry into a planetary atmosphere, gas pipelines, commercial applications such as abrasive blasting, and many other fields.
1155:
angle. Flow can expand around either a sharp or rounded corner equally, as the increase in Mach number is proportional to only the convex angle of the passage (ÎŽ). The expansion corner that produces the
PrandtlâMeyer fan can be sharp (as illustrated in the figure) or rounded. If the total turning angle is the same, then the P-M flow solution is also the same.
213:(M) is defined as the ratio of the speed of an object (or of a flow) to the speed of sound. For instance, in air at room temperature, the speed of sound is about 340 m/s (1,100 ft/s). M can range from 0 to â, but this broad range falls naturally into several flow regimes. These regimes are subsonic,
1208:
Wind tunnels can be divided into two categories: continuous-operating and intermittent-operating wind tunnels. Continuous operating supersonic wind tunnels require an independent electrical power source that drastically increases with the size of the test section. Intermittent supersonic wind tunnels
1154:
As opposed to the flow encountering an inclined obstruction and forming an oblique shock, the flow expands around a convex corner and forms an expansion fan through a series of isentropic Mach waves. The expansion "fan" is composed of Mach waves that span from the initial Mach angle to the final Mach
888:
Based on the level of flow deflection (ÎŽ), oblique shocks are characterized as either strong or weak. Strong shocks are characterized by larger deflection and more entropy loss across the shock, with weak shocks as the opposite. In order to gain cursory insight into the differences in these shocks, a
546:
where dP is the differential change in pressure, M is the Mach number, Ï is the density of the gas, V is the velocity of the flow, A is the area of the duct, and dA is the change in area of the duct. This equation states that, for subsonic flow, a converging duct (dA < 0) increases the velocity of
54:
The study of gas dynamics is often associated with the flight of modern high-speed aircraft and atmospheric reentry of space-exploration vehicles; however, its origins lie with simpler machines. At the beginning of the 19th century, investigation into the behaviour of fired bullets led to improvement
1175:
A PrandtlâMeyer compression is the opposite phenomenon to a
PrandtlâMeyer expansion. If the flow is gradually turned through an angle of ÎŽ, a compression fan can be formed. This fan is a series of Mach waves that eventually coalesce into an oblique shock. Because the flow is defined by an isentropic
240:
These flow regimes are not chosen arbitrarily, but rather arise naturally from the strong mathematical background that underlies compressible flow (see the cited reference textbooks). At very slow flow speeds the speed of sound is so much faster that it is mathematically ignored, and the Mach number
1259:
Perhaps the most common requirement for oblique shocks is in supersonic aircraft inlets for speeds greater than about Mach 2 (the F-16 has a maximum speed of Mach 2 but doesn't need an oblique shock intake). One purpose of the inlet is to minimize losses across the shocks as the incoming supersonic
1130:
PrandtlâMeyer fans can be expressed as both compression and expansion fans. PrandtlâMeyer fans also cross a boundary layer (i.e. flowing and solid) which reacts in different changes as well. When a shock wave hits a solid surface the resulting fan returns as one from the opposite family while when
871:
Oblique shock waves are similar to normal shock waves, but they occur at angles less than 90° with the direction of flow. When a disturbance is introduced to the flow at a nonzero angle (Ύ), the flow must respond to the changing boundary conditions. Thus an oblique shock is formed, resulting in a
253:
As an object accelerates from subsonic toward supersonic speed in a gas, different types of wave phenomena occur. To illustrate these changes, the next figure shows a stationary point (M = 0) that emits symmetric sound waves. The speed of sound is the same in all directions in a uniform fluid, so
200:
Finally, although space is known to have 3 dimensions, an important simplification can be had in describing gas dynamics mathematically if only one spatial dimension is of primary importance, hence 1-dimensional flow is assumed. This works well in duct, nozzle, and diffuser flows where the flow
169:
involve only two unknowns: pressure and velocity, which are typically found by solving the two equations that describe conservation of mass and of linear momentum, with the fluid density presumed constant. In compressible flow, however, the gas density and temperature also become variables. This
149:
There are several important assumptions involved in the underlying theory of compressible flow. All fluids are composed of molecules, but tracking a huge number of individual molecules in a flow (for example at atmospheric pressure) is unnecessary. Instead, the continuum assumption allows us to
803:
Normal shock waves can be easily analysed in either of two reference frames: the standing normal shock and the moving shock. The flow before a normal shock wave must be supersonic, and the flow after a normal shock must be subsonic. The
Rankine-Hugoniot equations are used to solve for the flow
150:
consider a flowing gas as a continuous substance except at low densities. This assumption provides a huge simplification which is accurate for most gas-dynamic problems. Only in the low-density realm of rarefied gas dynamics does the motion of individual molecules become important.
1201:
406:
One-dimensional (1-D) flow refers to flow of gas through a duct or channel in which the flow parameters are assumed to change significantly along only one spatial dimension, namely, the duct length. In analysing the 1-D channel flow, a number of assumptions are made:
1073:
128:
and ballistic ranges with the use of optical techniques to document the findings. Theoretical gas dynamics considers the equations of motion applied to a variable-density gas, and their solutions. Much of basic gas dynamics is analytical, but in the modern era
1106:
Incoming flow is first turned by angle ÎŽ with respect to the flow. This shockwave is reflected off the solid boundary, and the flow is turned by â ÎŽ to again be parallel with the boundary. Each progressive shock wave is weaker and the wave angle is increased.
1238:
Blowdown type supersonic wind tunnels offer high
Reynolds number, a small storage tank, and readily available dry air. However, they cause a high pressure hazard, result in difficulty holding a constant stagnation pressure, and are noisy during operation.
812:
Although one-dimensional flow can be directly analysed, it is merely a specialized case of two-dimensional flow. It follows that one of the defining phenomena of one-dimensional flow, a normal shock, is likewise only a special case of a larger class of
1242:
Indraft supersonic wind tunnels are not associated with a pressure hazard, allow a constant stagnation pressure, and are relatively quiet. Unfortunately, they have a limited range for the
Reynolds number of the flow and require a large vacuum tank.
1196:
are used for testing and research in supersonic flows, approximately over the Mach number range of 1.2 to 5. The operating principle behind the wind tunnel is that a large pressure difference is maintained upstream to downstream, driving the flow.
1246:
There is no dispute that knowledge is gained through research and testing in supersonic wind tunnels; however, the facilities often require vast amounts of power to maintain the large pressure ratios needed for testing conditions. For example,
1209:
are less expensive in that they store electrical energy over an extended period of time, then discharge the energy over a series of brief tests. The difference between these two is analogous to the comparison between a battery and a capacitor.
1115:
An irregular reflection is much like the case described above, with the caveat that ÎŽ is larger than the maximum allowable turning angle. Thus a detached shock is formed and a more complicated reflection known as Mach reflection occurs.
791:
increases across the shock. When analysing a normal shock wave, one-dimensional, steady, and adiabatic flow of a perfect gas is assumed. Stagnation temperature and stagnation enthalpy are the same upstream and downstream of the shock.
337:
157:
where the flow velocity at a solid surface is presumed equal to the velocity of the surface itself, which is a direct consequence of assuming continuum flow. The no-slip condition implies that the flow is viscous, and as a result a
739:
782:
Normal shock waves are shock waves that are perpendicular to the local flow direction. These shock waves occur when pressure waves build up and coalesce into an extremely thin shockwave that converts kinetic energy into
540:
245:
regime. Finally, at speeds comparable to that of planetary atmospheric entry from orbit, in the range of several km/s, the speed of sound is now comparatively so slow that it is once again mathematically ignored in the
258:. Upon achieving supersonic flow, the particle is moving so fast that it continuously leaves its sound waves behind. When this occurs, the locus of these waves trailing behind the point creates an angle known as the
108:
Accompanying the improved conceptual understanding of gas dynamics in the early 20th century was a public misconception that there existed a barrier to the attainable speed of aircraft, commonly referred to as the
1090:, the flow transitions from a strong to weak oblique shock. With Ύ = 0°, a normal shock is produced at the limit of the strong oblique shock and the Mach wave is produced at the limit of the weak shock wave.
1263:
Although variable geometry is required to achieve acceptable performance from take-off to speeds exceeding Mach 2 there is no one method to achieve it. For example, for a maximum speed of about Mach 3, the
1158:
The
PrandtlâMeyer expansion can be seen as the physical explanation of the operation of the Laval nozzle. The contour of the nozzle creates a smooth and continuous series of PrandtlâMeyer expansion waves.
787:. The waves thus overtake and reinforce one another, forming a finite shock wave from an infinite series of infinitesimal sound waves. Because the change of state across the shock is highly irreversible,
558:
Therefore, to accelerate a flow to Mach 1, a nozzle must be designed to converge to a minimum cross-sectional area and then expand. This type of nozzle â the converging-diverging nozzle â is called a
1251:
has the largest supersonic wind tunnel in the world and requires the power required to light a small city for operation. For this reason, large wind tunnels are becoming less common at universities.
1147:
To this point, the only flow phenomena that have been discussed are shock waves, which slow the flow and increase its entropy. It is possible to accelerate supersonic flow in what has been termed a
451:
and diffuser flows is altered. Using the conservation laws of fluid dynamics and thermodynamics, the following relationship for channel flow is developed (combined mass and momentum conservation):
647:
956:
951:
124:
Historically, two parallel paths of research have been followed in order to further gas dynamics knowledge. Experimental gas dynamics undertakes wind tunnel model experiments and experiments in
940:
774:. Because changes downstream can only move upstream at sonic speed, the mass flow through the nozzle cannot be affected by changes in downstream conditions after the flow is choked.
760:
As previously mentioned, in order for a flow to become supersonic, it must pass through a duct with a minimum area, or sonic throat. Additionally, an overall pressure ratio, P
133:
applies computing power to solve the otherwise-intractable nonlinear partial differential equations of compressible flow for specific geometries and flow characteristics.
1098:
Due to the inclination of the shock, after an oblique shock is created, it can interact with a boundary in three different manners, two of which are explained below.
268:
1346:
380:
360:
1592:
105:) also contributed significantly to the principles considered fundamental to the study of modern gas dynamics. Many others also contributed to this field.
744:
can be obtained, where M is the Mach number and Îł is the ratio of specific heats (1.4 for air). See table of isentropic flow Mach number relationships.
678:
1615:
582:
Ultimately, because of the energy conservation law, a gas is limited to a certain maximum velocity based on its energy content. The maximum velocity,
788:
457:
1086:
as the x and y-components of the fluid velocity V. With the Mach number before the shock given, a locus of conditions can be specified. At some ÎŽ
91:
889:
shock polar diagram can be used. With the static temperature after the shock, T*, known the speed of sound after the shock is defined as,
66:
At the beginning of the 20th century, the focus of gas dynamics research shifted to what would eventually become the aerospace industry.
1675:
1755:
1639:
1248:
1600:
1362:
183:
1430:
1068:{\displaystyle {\begin{aligned}M_{2x}^{*}&={\frac {V_{x}}{a^{*}}}\\M_{2y}^{*}&={\frac {V_{y}}{a^{*}}}\end{aligned}}}
595:
1554:
945:
with R as the gas constant and Îł as the specific heat ratio. The Mach number can be broken into
Cartesian coordinates
1623:
1531:
1508:
1485:
1743:
895:
182:
is the appropriate state equation. Otherwise, more complex equations of state must be considered and the so-called
1644:
768:, of approximately 2 is needed to attain Mach 1. Once it has reached Mach 1, the flow at the throat is said to be
1148:
1577:
1898:
1778:
1668:
1723:
130:
86:, a student of Prandtl, continued to improve the understanding of supersonic flow. Other notable figures (
1733:
1151:, after Ludwig Prandtl and Theodore Meyer. The mechanism for the expansion is shown in the figure below.
1125:
189:
Fluid dynamics problems have two overall types of references frames, called
Lagrangian and Eulerian (see
1180:
region (flow that travels through the oblique shock), a slip line results between the two flow regions.
1351:
55:
in the accuracy and capabilities of guns and artillery. As the century progressed, inventors such as
828:
2088:
1975:
1958:
1705:
1661:
818:
83:
1953:
751:
Isentropic flow relationship table. Equations to relate the field properties in isentropic flow.
1853:
1341:
1193:
661:
175:
79:
447:
As the speed of a flow accelerates from the subsonic to the supersonic regime, the physics of
2093:
554:
Table showing the reversal in the physics of nozzles and diffusers with changing Mach numbers
190:
162:
forms on bodies traveling through the air at high speeds, much as it does in low-speed flow.
1283:
332:{\displaystyle \mu =\arcsin \left({\frac {a}{V}}\right)=\arcsin \left({\frac {1}{M}}\right)}
1808:
1569:
1461:
Moving the Stars: Christian
Doppler - His Life, His Works and Principle and the World After
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8:
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166:
39:
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550:
365:
345:
799:
The
Rankine-Hugoniot equations relate conditions before and after a normal shock wave.
229:
flow. The figure below illustrates the Mach number "spectrum" of these flow regimes.
2017:
1982:
1763:
1619:
1596:
1573:
1550:
1527:
1504:
1481:
1359:
especially "Commonly Considered Thermodynamic Processes" and "Laws of Thermodynamics"
1326:
1316:
387:
171:
154:
141:
102:
2067:
1938:
1923:
1888:
1798:
734:{\displaystyle {\frac {{\text{property}}_{1}}{{\text{property}}_{2}}}=f(M,\gamma )}
672:
Using conservations laws and thermodynamics, a number of relationships of the form
242:
1943:
1226:
747:
2032:
2012:
1970:
1965:
1793:
1684:
1438:
795:
563:
559:
96:
56:
35:
27:
1167:
840:
2042:
2022:
2007:
2002:
1948:
1933:
1908:
1903:
1893:
1873:
1863:
1828:
1773:
1738:
1715:
1356:
784:
777:
422:
194:
159:
71:
67:
393:
63:
sought to understand the physical phenomena involved through experimentation.
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2052:
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2027:
1992:
1987:
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1868:
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1838:
1833:
1803:
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1523:
858:
814:
535:{\displaystyle dP\left(1-M^{2}\right)=\rho V^{2}\left({\frac {dA}{A}}\right)}
435:
416:
382:
represents the velocity of the object. Although named for Austrian physicist
247:
226:
179:
170:
requires two more equations in order to solve compressible-flow problems: an
110:
87:
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1997:
1913:
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1331:
1293:
1177:
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118:
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1823:
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1500:
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210:
43:
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1700:
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429:
383:
255:
222:
218:
125:
75:
60:
880:
411:
Ratio of duct length to width (L/D) is †about 5 (in order to neglect
1546:
1431:"Research in Supersonic Flight and the Breaking of the Sound Barrier"
863:
832:
Attached shock wave shown on a X-15 Model in a supersonic wind tunnel
259:
214:
1589:
The Dynamics and Thermodynamics of Compressible Fluid Flow, Volume 1
1131:
one hits a free boundary the fan returns as a fan of opposite type.
412:
114:
1653:
1406:
1321:
121:, the X-1 officially achieved supersonic speed in October 1947.
31:
1216:
178:
equation. For the majority of gas-dynamic problems, the simple
448:
70:
and his students proposed important concepts ranging from the
1269:
1265:
778:
Non-isentropic 1D channel flow of a gas - normal shock waves
667:
204:
1540:
1200:
30:
that deals with flows having significant changes in fluid
1347:
Lagrangian and Eulerian specification of the flow field
577:
1268:
used rectangular inlets with adjustable ramps and the
1463:, Pollauberg, Austria:Living Edition Publishers, 2005
954:
898:
681:
598:
460:
442:
368:
348:
271:
1230:
Langley indraft supersonic wind tunnel vacuum sphere
642:{\displaystyle V_{\text{max}}={\sqrt {2c_{p}T_{t}}}}
1541:Oosthuizen, Patrick H.; Carscallen, W. E. (2013) .
1640:NASA Beginner's Guide to Compressible Aerodynamics
1471:
1176:region (flow that travels through the fan) and an
1067:
934:
733:
641:
534:
374:
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331:
1563:
1375:
1287:XB-70 rectangular inlets with ramps (not visible)
1162:
2080:
1517:
1397:
1134:
386:, these oblique waves were first discovered by
1400:"Fundamentals of Compressible Fluid Mechanics"
59:advanced the field, while researchers such as
1669:
1254:
935:{\displaystyle A^{*}={\sqrt {\gamma RT^{*}}}}
362:represents the speed of sound in the gas and
1612:Hypersonic and High Temperature Gas Dynamics
755:
1645:Virginia Tech Compressible Flow Calculators
1093:
1676:
1662:
1453:
1204:Supersonic wind tunnel classification list
1188:
1220:Blowdown supersonic wind tunnel schematic
668:Isentropic flow Mach number relationships
205:Mach number, wave motion, and sonic speed
1609:
1564:Zucker, Robert D.; Biblarz, O. (2002) .
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231:
140:
136:
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1472:Liepmann, Hans W.; Roshko, A. (1957) .
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807:
401:
2081:
1518:John, James E.; Keith, T. G. (2006) .
1393:
1391:
1249:Arnold Engineering Development Complex
875:
852:
432:(i.e. a reversible adiabatic process),
184:non ideal compressible fluids dynamics
1657:
1363:Non-ideal compressible fluid dynamics
1272:used circular inlets with adjustable
1119:
872:change in the direction of the flow.
656:is the specific heat of the gas and T
38:, flows are usually treated as being
1501:McGraw-Hill Science/Engineering/Math
1428:
578:Maximum achievable velocity of a gas
1683:
1388:
1297:SR-71 round inlets with inlet cones
1143:PrandtlâMeyer expansion fan diagram
13:
443:Converging-diverging Laval nozzles
397:Wave motion and the speed of sound
145:Breakdown of fluid mechanics chart
14:
2105:
1633:
1543:Introduction to Compressible Flow
1101:
844:Bowshock example for a blunt body
236:Mach number flow regimes spectrum
1398:Genick BarâMeir (May 21, 2007).
82:, and supersonic nozzle design.
1610:Anderson, John D. Jr. (2000) .
1495:Anderson, John D. Jr. (2003) .
1183:
1422:
1163:PrandtlâMeyer compression fans
728:
716:
1:
1385:, 4th Ed., McGrawâHill, 2007.
1368:
1724:Computational fluid dynamics
1566:Fundamentals of Gas Dynamics
1383:Fundamentals of Aerodynamics
1171:Basic PM compression diagram
1135:PrandtlâMeyer expansion fans
589:, that a gas can attain is:
153:A related assumption is the
131:Computational fluid dynamics
7:
1587:Shapiro, Ascher H. (1953).
1305:
1149:PrandtlâMeyer expansion fan
1126:Prandtl-Meyer expansion fan
10:
2110:
1255:Supersonic aircraft inlets
1123:
856:
49:
1754:
1714:
1691:
756:Achieving supersonic flow
16:Branch of fluid mechanics
1497:Modern Compressible Flow
1094:Oblique shock reflection
423:Steady vs. Unsteady Flow
262:angle or Mach angle, Ό:
1734:NavierâStokes equations
1474:Elements of Gasdynamics
1194:Supersonic wind tunnels
1189:Supersonic wind tunnels
573:Nozzle de Laval diagram
80:supersonic wind tunnels
1429:Anderson, John D. Jr.
1352:PrandtlâMeyer function
1342:Isentropic nozzle flow
1298:
1288:
1231:
1221:
1205:
1172:
1144:
1069:
936:
885:
868:
867:Diagram of obstruction
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833:
800:
752:
735:
662:stagnation temperature
643:
574:
555:
536:
398:
376:
356:
333:
237:
176:conservation of energy
146:
34:. While all flows are
1756:Dimensionless numbers
1706:Archimedes' principle
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1203:
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572:
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537:
396:
377:
357:
334:
235:
191:Joseph-Louis Lagrange
186:(NICFD) establishes.
144:
137:Introductory concepts
1593:Ronald Press Company
1111:Irregular reflection
952:
896:
808:Two-dimensional flow
679:
596:
458:
402:One-dimensional flow
366:
346:
269:
1441:on 25 December 2017
1337:Heat capacity ratio
1312:Incompressible flow
1029:
976:
884:Shock polar diagram
876:Shock polar diagram
853:Oblique shock waves
167:incompressible flow
84:Theodore von KĂĄrmĂĄn
26:) is the branch of
1899:KeuleganâCarpenter
1478:Dover Publications
1299:
1289:
1232:
1222:
1206:
1173:
1145:
1120:PrandtlâMeyer fans
1065:
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834:
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575:
556:
532:
399:
372:
352:
329:
238:
174:for the gas and a
147:
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1602:978-0-471-06691-0
1327:Equation of state
1317:Conservation laws
1303:
1302:
1236:
1235:
1059:
1006:
930:
850:
849:
708:
700:
688:
637:
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388:Christian Doppler
375:{\displaystyle V}
355:{\displaystyle a}
323:
296:
172:equation of state
165:Most problems in
155:no-slip condition
20:Compressible flow
2101:
1678:
1671:
1664:
1655:
1654:
1629:
1606:
1583:
1568:(2nd ed.).
1560:
1545:(2nd ed.).
1537:
1522:(3rd ed.).
1514:
1499:(3rd ed.).
1491:
1464:
1457:
1451:
1450:
1448:
1446:
1437:. Archived from
1435:history.nasa.gov
1426:
1420:
1418:
1416:
1414:
1404:
1395:
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160:boundary layer
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74:to supersonic
72:boundary layer
68:Ludwig Prandtl
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180:ideal gas law
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117:. Piloted by
116:
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111:sound barrier
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2094:Aerodynamics
1729:Aerodynamics
1611:
1588:
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1542:
1520:Gas Dynamics
1519:
1496:
1473:
1460:
1455:
1443:. Retrieved
1439:the original
1434:
1424:
1411:. Retrieved
1382:
1377:
1332:Gas kinetics
1262:
1258:
1245:
1241:
1237:
1207:
1192:
1184:Applications
1178:anisentropic
1174:
1157:
1153:
1146:
1129:
1114:
1105:
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887:
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804:conditions.
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341:
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199:
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164:
152:
148:
123:
119:Chuck Yeager
107:
92:Luigi Crocco
65:
53:
36:compressible
24:gas dynamics
23:
19:
18:
2063:Weissenberg
1413:January 23,
211:Mach number
126:shock tubes
95: [
76:shock waves
44:Mach number
2083:Categories
1983:Richardson
1764:Archimedes
1701:Hydraulics
1579:0471059676
1369:References
1274:inlet cone
430:isentropic
384:Ernst Mach
256:Shock wave
223:hypersonic
219:supersonic
61:Ernst Mach
2068:Womersley
1959:turbulent
1939:Ohnesorge
1924:Marangoni
1889:Iribarren
1814:Damköhler
1799:Capillary
1547:CRC Press
1055:∗
1026:∗
1002:∗
973:∗
926:∗
915:γ
905:∗
819:bow shock
726:γ
497:ρ
476:−
309:
282:
273:μ
260:Mach wave
215:transonic
42:when the
2043:Suratman
2033:Strouhal
2013:Sherwood
1976:magnetic
1971:Reynolds
1966:Rayleigh
1954:magnetic
1794:Brinkman
1445:14 April
1306:See also
699:property
687:property
428:Flow is
413:friction
250:regime.
115:Bell X-1
2023:Stanton
2018:Shields
2008:Scruton
2003:Schmidt
1949:Prandtl
1934:Nusselt
1909:Laplace
1904:Knudsen
1894:Kapitza
1879:Görtler
1874:Grashof
1864:Galilei
1829:Deborah
1774:Bagnold
1407:ibiblio
1322:Entropy
789:entropy
660:is the
652:where c
50:History
32:density
2053:Ursell
2048:Taylor
2038:Stuart
2028:Stokes
1993:Rossby
1988:Roshko
1944:PĂ©clet
1929:Morton
1869:Graetz
1859:Froude
1849:Eötvös
1839:Eckert
1834:Dukhin
1804:Cauchy
1769:Atwood
1622:
1599:
1576:
1553:
1530:
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1484:
1078:with V
771:choked
562:after
449:nozzle
342:where
306:arcsin
279:arcsin
225:, and
101:, and
2058:Weber
1998:Rouse
1914:Lewis
1884:Hagen
1854:Euler
1844:Ekman
1819:Darcy
1779:Bejan
1570:Wiley
1403:(PDF)
1270:SR-71
1266:XB-70
1082:and V
99:]
88:Meyer
1919:Mach
1824:Dean
1789:Bond
1784:Biot
1620:ISBN
1616:AIAA
1597:ISBN
1574:ISBN
1551:ISBN
1528:ISBN
1505:ISBN
1482:ISBN
1447:2018
1419:>
1415:2020
415:and
209:The
193:and
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1088:max
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