2777:, and the flow resistance is calculated to approach infinity. In reality, aerodynamic and thermodynamic perturbations get amplified strongly near the sonic speed, but a singularity does not occur. An explanation for this is that the linearized small-disturbance potential equation above is not valid, since it assumes that there are only small variations in Mach number within the flow and absence of compression shocks and thus is missing certain nonlinear terms. However, these become relevant as soon as any part of the flow field accelerates above the speed of sound, and become essential near
105:
2714:. The introduction of this relation allowed the design of aircraft which were able to operate in higher subsonic speed areas. Originally all these results were developed for 2D flow. Göthert eventually realized in 1946 that the geometric distortion induced by the PG transformation renders the simple 2D Prandtl Rule invalid for 3D, and properly stated the full 3D problem as described above.
35:
1142:
1732:
2083:
1549:
2340:
1386:
682:
977:
249:
1560:
352:
1926:
1918:
2721:
to supersonic-freestream flows in 1925. Like for the subsonic case, the supersonic case is valid only if there are no transonic effect, which requires that the body be slender and the freestream Mach is sufficiently far above unity.
1397:
2201:
1226:
530:
2624:
884:
2206:
1231:
982:
2523:
1788:
1198:
734:
1137:{\displaystyle {\begin{aligned}{\bar {x}}&=x\\{\bar {y}}&=\beta y\\{\bar {z}}&=\beta z\\{\bar {\alpha }}&=\beta \alpha \\{\bar {\phi }}&=\beta ^{2}\phi \end{aligned}}}
833:
1790:
geometry. It can be solved by incompressible methods, such as thin airfoil theory, vortex lattice methods, panel methods, etc. The result is the transformed perturbation potential
2808:
2680:
2405:
2767:
1727:{\displaystyle V_{\infty }{\bar {n}}_{\bar {x}}+{\bar {\phi }}_{\bar {y}}{\bar {n}}_{\bar {y}}+{\bar {\phi }}_{\bar {z}}{\bar {n}}_{\bar {z}}=0\quad {\mbox{(on body surface)}}}
434:
153:
1817:
53:
475:
2165:
945:
502:
381:
2470:
2193:
2078:{\displaystyle C_{p}=-2{\frac {\phi _{x}}{V_{\infty }}}=-{\frac {2}{\beta ^{2}}}{\frac {{\bar {\phi }}_{\bar {x}}}{V_{\infty }}}={\frac {1}{\beta ^{2}}}{\bar {C}}_{p}}
137:
915:
260:
1218:
969:
2550:
2125:
1822:
2435:
1544:{\displaystyle {\bar {\phi }}_{{\bar {x}}{\bar {x}}}+{\bar {\phi }}_{{\bar {y}}{\bar {y}}}+{\bar {\phi }}_{{\bar {z}}{\bar {z}}}=0\quad {\mbox{(in flow field)}}}
522:
2335:{\displaystyle {\begin{aligned}C_{p}&={\frac {C_{p0}}{\beta }}\\c_{l}&={\frac {c_{l0}}{\beta }}\\c_{m}&={\frac {c_{m0}}{\beta }}\end{aligned}}}
1381:{\displaystyle {\begin{aligned}{\bar {n}}_{\bar {x}}&=\beta n_{x}\\{\bar {n}}_{\bar {y}}&=n_{y}\\{\bar {n}}_{\bar {z}}&=n_{z}\end{aligned}}}
3006:
677:{\displaystyle {\vec {V}}=\nabla \phi +V_{\infty }{\hat {x}}=(V_{\infty }+\phi _{x}){\hat {x}}+\phi _{y}{\hat {y}}+\phi _{z}{\hat {z}}}
739:
and in addition that there is no transonic flow, approximately stated by the requirement that the local Mach number not exceed unity.
3044:
71:
2774:
2731:
2562:
841:
147:
Inviscid compressible flow over slender bodies is governed by linearized compressible small-disturbance potential equation:
2694:
The PG transformation works well for all freestream Mach numbers up to 0.7 or so, or once transonic flow starts to appear.
1920:
in the transformed space. The physical linearized pressure coefficient is then obtained by the inverse transformation
2552:
and subsequently the forces and moments. No simple results are possible, except in special cases. For example, using
2710:
had taught the transformation in his lectures about 1922, however the first rigorous proof was published in 1928 by
2475:
1740:
1150:
693:
2770:
745:
2703:
109:
2936:"Ebene und räumliche Strömung bei hohen Unterschallgeschwindigkeiten: Erweiterung der Prandtl'schen Regel"
2780:
2640:
2348:
2739:
3077:
244:{\displaystyle \phi _{xx}+\phi _{yy}+\phi _{zz}=M_{\infty }^{2}\phi _{xx}\quad {\mbox{(in flow field)}}}
96:-flow calculation methods. It also allows applying incompressible-flow data to compressible-flow cases.
3017:
386:
2963:
1793:
2938:[Plane and Three-Dimensional Flow at High Subsonic Speeds: Extension of the Prandtl Rule],
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93:
439:
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920:
480:
359:
347:{\displaystyle V_{\infty }n_{x}+\phi _{y}n_{y}+\phi _{z}n_{z}=0\quad {\mbox{(on body surface)}}}
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2170:
114:
89:
17:
1913:{\displaystyle {\bar {\phi }}_{\bar {x}},{\bar {\phi }}_{\bar {y}},{\bar {\phi }}_{\bar {z}}}
897:
436:
are the surface-normal vector components. The unknown variable is the perturbation potential
2971:
2637:→ ∞ this reduces to the 2D case, since in incompressible 2D flow for a flat airfoil we have
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104:
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687:
The above formulation is valid only if the small-disturbance approximation applies,
2956:
Proceedings of the Royal
Society A: Mathematical, Physical and Engineering Sciences
1391:
The small-disturbance potential equation then transforms to the
Laplace equation,
3034:
2935:
140:
477:, and the total velocity is given by its gradient plus the freestream velocity
2985:
2976:
2951:
2942:(in German) (127), Berlin: Zentrale fuer Wissenschaftliches Berichtswesen
2702:
The interest in compressibility research emerged after the WWI, when the
1200:
geometry will then have normal vectors whose x components are reduced by
2472:
scalings do NOT apply. Instead, it is necessary to work with the scaled
1737:
This is the incompressible potential-flow problem about the transformed
838:
The
Prandtl–Glauert (PG) transformation uses the Prandtl–Glauert factor
3026:
Ludwig
Prandtl memorial lecture, GAMM 2005, March 28th - April 1st 2005
2683:
2810:
The more correct nonlinear equation does not exhibit the singularity.
254:
together with the small-disturbance flow-tangency boundary condition.
3018:"Die Entwicklung des Pfeilflügels, eine technische Herausforderung"
2525:
geometry as given above, and use the Göthert's Rule to compute the
103:
3020:[The evolution of the swept-wing, a technical challenge]
1554:
and the flow-tangency boundary condition retains the same form.
3062:] (in German). Vol. 2 (4th ed.). Springer Verlag.
28:
2437:
geometry. This 2D-only result is known as the
Prandtl Rule.
2633:
is the wing's aspect ratio. Note that in the 2D case where
3036:
The dynamics and thermodynamics of compressible fluid flow
2952:"The Effect of Compressibility on the Lift of an Aerofoil"
2619:{\displaystyle C_{L}={\frac {2\pi \alpha }{\beta +2/AR}}}
88:
is a mathematical technique which allows solving certain
2997:
Foundations of aerodynamics: bases of aerodynamic design
879:{\displaystyle \beta \equiv {\sqrt {1-M_{\infty }^{2}}}}
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for a flat elliptical wing, the lift coefficient is
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may be too technical for most readers to understand
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894:dimensions and angle of attack by the factor of
2993:Kuethe, Arnold Martin; Chow, Chuen-Yen (1976).
2518:{\displaystyle {\bar {x}}{\bar {y}}{\bar {z}}}
1783:{\displaystyle {\bar {x}}{\bar {y}}{\bar {z}}}
1193:{\displaystyle {\bar {x}}{\bar {y}}{\bar {z}}}
729:{\displaystyle |\nabla \phi |\ll V_{\infty }}
8:
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2407:are the incompressible-flow values for the
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2127:and also the lift and moment coefficients
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828:{\displaystyle \leftM_{\infty }^{2}<1}
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72:Learn how and when to remove this message
56:, without removing the technical details.
2904:
2880:
2868:
2846:
2717:The PG transformation was extended by
143:. Notice the infinite limit at Mach 1.
2916:
2773:. The singularity is also called the
54:make it understandable to non-experts
7:
2803:{\displaystyle M_{\infty }\simeq 1.}
2675:{\displaystyle c_{l0}=2\pi \alpha ,}
2400:{\displaystyle C_{p0},c_{l0},c_{m0}}
2762:{\displaystyle M_{\infty }\simeq 1}
951:component of the normal vectors by
383:is the freestream Mach number, and
3028:(in German), Universität Luxemburg
2789:
2748:
2029:
1966:
1569:
886:. It consists of scaling down all
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809:
792:
721:
702:
591:
563:
549:
504:which is assumed here to be along
489:
368:
269:
210:
25:
2769:the PG transformation features a
2088:which is known as Göthert's rule
429:{\displaystyle n_{x},n_{y},n_{z}}
33:
1716:
1533:
336:
233:
2509:
2497:
2485:
2063:
2017:
2005:
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1833:
1812:{\displaystyle {\bar {\phi }}}
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446:
86:Prandtl–Glauert transformation
18:Prandtl-Glauert transformation
1:
2706:tips started to reach M=0.8.
3054:Truckenbrodt, Erich (1996).
2167:are increased by the factor
470:{\displaystyle \phi (x,y,z)}
139:as a function of freestream
3033:Shapiro, Ascher H. (1953).
2775:Prandtl–Glauert singularity
2732:Prandtl–Glauert singularity
2160:{\displaystyle c_{l},c_{m}}
1819:or its gradient components
940:{\displaystyle \beta ^{2},}
497:{\displaystyle V_{\infty }}
376:{\displaystyle M_{\infty }}
3094:
2729:
2100:, the net result is that
2465:{\displaystyle 1/\beta }
2188:{\displaystyle 1/\beta }
1220:from the original ones:
132:{\displaystyle 1/\beta }
100:Mathematical formulation
2940:Lilienthal Gesellschaft
2442:three-dimensional flows
910:{\displaystyle \beta ,}
3039:. Vol. 1. Wiley.
2977:10.1098/rspa.1928.0039
2934:Göthert, B.H. (1940),
2854:Kuethe & Chow 1976
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1214:
1213:{\displaystyle \beta }
1194:
1138:
965:
964:{\displaystyle \beta }
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911:
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678:
518:
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430:
377:
348:
245:
144:
133:
110:Prandtl–Glauert factor
3016:Meier, H.-U. (2005),
2805:
2764:
2736:Near the sonic speed
2677:
2621:
2547:
2545:{\displaystyle C_{p}}
2520:
2467:
2432:
2402:
2337:
2190:
2162:
2122:
2120:{\displaystyle C_{p}}
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2098:two-dimensional flow
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108:Plot of the inverse
2968:1928RSPSA.118..113G
2684:Thin airfoil theory
2554:Lifting-Line Theory
2430:{\displaystyle xyz}
873:
818:
219:
2907:, p. 113–119.
2800:
2759:
2704:aircraft propeller
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3008:978-0-471-50953-0
2895:, pp. 178–9.
2893:Truckenbrodt 1996
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517:{\displaystyle x}
341:
340:(on body surface)
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92:flow problems by
82:
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16:(Redirected from
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2962:(779): 113–119.
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2856:, pp. 248-.
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944:
943:
938:
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916:
914:
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908:
885:
883:
882:
877:
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872:
867:
852:
834:
832:
831:
826:
817:
812:
803:
799:
798:
796:
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776:
735:
733:
732:
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712:
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683:
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664:
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648:
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623:
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614:
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579:
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566:
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523:
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476:
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468:
435:
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382:
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353:
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329:
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319:
318:
306:
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296:
295:
283:
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273:
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250:
248:
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240:
236:
232:
231:
218:
213:
201:
200:
185:
184:
169:
168:
138:
136:
135:
130:
125:
77:
70:
66:
63:
57:
37:
36:
29:
21:
3093:
3092:
3088:
3087:
3086:
3084:
3083:
3082:
3068:
3067:
3066:
3060:Fluid Mechanics
3053:
3047:
3032:
3021:
3015:
3009:
2992:
2946:
2933:
2929:
2924:
2923:
2915:
2911:
2903:
2899:
2891:
2887:
2879:
2875:
2867:
2860:
2852:
2848:
2843:
2838:
2825:Hermann Glauert
2816:
2784:
2779:
2778:
2743:
2738:
2737:
2734:
2728:
2712:Hermann Glauert
2700:
2692:
2644:
2639:
2638:
2592:
2581:
2566:
2561:
2560:
2532:
2527:
2526:
2474:
2473:
2446:
2445:
2444:, these simple
2413:
2412:
2384:
2368:
2352:
2347:
2346:
2329:
2328:
2310:
2301:
2291:
2288:
2287:
2269:
2260:
2250:
2247:
2246:
2228:
2219:
2209:
2200:
2199:
2169:
2168:
2147:
2134:
2129:
2128:
2107:
2102:
2101:
2094:
2056:
2044:
2024:
1998:
1984:
1961:
1951:
1930:
1925:
1924:
1884:
1855:
1826:
1821:
1820:
1792:
1791:
1739:
1738:
1684:
1658:
1629:
1603:
1574:
1564:
1559:
1558:
1537:(in flow field)
1487:
1444:
1401:
1396:
1395:
1375:
1374:
1364:
1357:
1331:
1328:
1327:
1317:
1310:
1284:
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1280:
1270:
1260:
1234:
1225:
1224:
1202:
1201:
1149:
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1131:
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1117:
1110:
1095:
1094:
1081:
1066:
1065:
1052:
1037:
1036:
1023:
1008:
1007:
997:
976:
975:
953:
952:
924:
919:
918:
896:
895:
840:
839:
787:
777:
753:
749:
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743:
716:
692:
691:
652:
627:
599:
586:
558:
529:
528:
506:
505:
484:
479:
478:
438:
437:
416:
403:
390:
385:
384:
363:
358:
357:
320:
310:
297:
287:
274:
264:
259:
258:
237:(in flow field)
220:
189:
173:
157:
152:
151:
113:
112:
102:
78:
67:
61:
58:
50:help improve it
47:
38:
34:
23:
22:
15:
12:
11:
5:
3091:
3089:
3081:
3080:
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3013:
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2858:
2845:
2844:
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2837:
2834:
2833:
2832:
2827:
2822:
2820:Ludwig Prandtl
2815:
2812:
2799:
2796:
2791:
2787:
2758:
2755:
2750:
2746:
2730:Main article:
2727:
2724:
2708:Ludwig Prandtl
2699:
2696:
2691:
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2222:
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2216:
2212:
2208:
2207:
2184:
2180:
2176:
2154:
2150:
2146:
2141:
2137:
2114:
2110:
2093:
2090:
2086:
2085:
2072:
2065:
2062:
2051:
2047:
2043:
2038:
2031:
2027:
2019:
2016:
2007:
2004:
1991:
1987:
1983:
1978:
1975:
1968:
1964:
1958:
1954:
1948:
1945:
1942:
1937:
1933:
1905:
1902:
1893:
1890:
1883:
1876:
1873:
1864:
1861:
1854:
1847:
1844:
1835:
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1805:
1802:
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1764:
1761:
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1735:
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1715:
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1693:
1690:
1679:
1676:
1667:
1664:
1657:
1650:
1647:
1638:
1635:
1624:
1621:
1612:
1609:
1602:
1595:
1592:
1583:
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1552:
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1532:
1529:
1521:
1518:
1509:
1506:
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1478:
1475:
1466:
1463:
1453:
1450:
1443:
1435:
1432:
1423:
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1410:
1407:
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1371:
1367:
1363:
1360:
1358:
1352:
1349:
1340:
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1330:
1329:
1324:
1320:
1316:
1313:
1311:
1305:
1302:
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1269:
1266:
1263:
1261:
1255:
1252:
1243:
1240:
1233:
1232:
1209:
1186:
1183:
1174:
1171:
1162:
1159:
1145:
1144:
1129:
1124:
1120:
1116:
1113:
1111:
1106:
1103:
1097:
1096:
1093:
1090:
1087:
1084:
1082:
1077:
1074:
1068:
1067:
1064:
1061:
1058:
1055:
1053:
1048:
1045:
1039:
1038:
1035:
1032:
1029:
1026:
1024:
1019:
1016:
1010:
1009:
1006:
1003:
1000:
998:
993:
990:
984:
983:
960:
936:
931:
927:
906:
903:
871:
866:
862:
858:
855:
850:
847:
836:
835:
824:
821:
816:
811:
807:
802:
794:
790:
784:
780:
774:
771:
768:
765:
762:
759:
756:
752:
737:
736:
723:
719:
715:
711:
707:
704:
700:
685:
684:
670:
667:
659:
655:
651:
645:
642:
634:
630:
626:
620:
617:
611:
606:
602:
598:
593:
589:
585:
582:
576:
573:
565:
561:
557:
554:
551:
548:
542:
539:
513:
491:
487:
466:
463:
460:
457:
454:
451:
448:
445:
423:
419:
415:
410:
406:
402:
397:
393:
370:
366:
355:
354:
335:
332:
327:
323:
317:
313:
309:
304:
300:
294:
290:
286:
281:
277:
271:
267:
252:
251:
230:
227:
223:
217:
212:
208:
204:
199:
196:
192:
188:
183:
180:
176:
172:
167:
164:
160:
128:
124:
120:
101:
98:
94:incompressible
80:
79:
41:
39:
32:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3090:
3079:
3076:
3075:
3073:
3061:
3057:
3056:Fluidmechanik
3052:
3048:
3046:9780471066910
3042:
3038:
3037:
3031:
3027:
3019:
3014:
3010:
3004:
2999:
2998:
2991:
2987:
2983:
2978:
2973:
2969:
2965:
2961:
2957:
2953:
2949:
2945:
2941:
2937:
2932:
2931:
2926:
2918:
2913:
2910:
2906:
2901:
2898:
2894:
2889:
2886:
2882:
2877:
2874:
2870:
2865:
2863:
2859:
2855:
2850:
2847:
2840:
2835:
2831:
2830:Jakob Ackeret
2828:
2826:
2823:
2821:
2818:
2817:
2813:
2811:
2797:
2794:
2785:
2776:
2772:
2756:
2753:
2744:
2733:
2725:
2723:
2720:
2719:Jakob Ackeret
2715:
2713:
2709:
2705:
2697:
2695:
2689:
2687:
2685:
2669:
2666:
2663:
2660:
2657:
2652:
2649:
2645:
2636:
2632:
2610:
2607:
2603:
2599:
2596:
2593:
2588:
2585:
2582:
2576:
2571:
2567:
2559:
2558:
2557:
2555:
2537:
2533:
2506:
2494:
2482:
2459:
2455:
2451:
2443:
2438:
2424:
2421:
2418:
2410:
2392:
2389:
2385:
2381:
2376:
2373:
2369:
2365:
2360:
2357:
2353:
2323:
2318:
2315:
2311:
2305:
2303:
2296:
2292:
2282:
2277:
2274:
2270:
2264:
2262:
2255:
2251:
2241:
2236:
2233:
2229:
2223:
2221:
2214:
2210:
2198:
2197:
2196:
2182:
2178:
2174:
2152:
2148:
2144:
2139:
2135:
2112:
2108:
2099:
2091:
2089:
2070:
2060:
2049:
2045:
2041:
2036:
2025:
2014:
2002:
1989:
1985:
1981:
1976:
1973:
1962:
1956:
1952:
1946:
1943:
1940:
1935:
1931:
1923:
1922:
1921:
1900:
1888:
1881:
1871:
1859:
1852:
1842:
1830:
1800:
1771:
1759:
1747:
1713:
1710:
1700:
1688:
1674:
1662:
1655:
1645:
1633:
1619:
1607:
1600:
1590:
1578:
1565:
1557:
1556:
1555:
1530:
1527:
1516:
1504:
1491:
1484:
1473:
1461:
1448:
1441:
1430:
1418:
1405:
1394:
1393:
1392:
1369:
1365:
1361:
1359:
1347:
1335:
1322:
1318:
1314:
1312:
1300:
1288:
1275:
1271:
1267:
1264:
1262:
1250:
1238:
1223:
1222:
1221:
1207:
1181:
1169:
1157:
1127:
1122:
1118:
1114:
1112:
1101:
1091:
1088:
1085:
1083:
1072:
1062:
1059:
1056:
1054:
1043:
1033:
1030:
1027:
1025:
1014:
1004:
1001:
999:
988:
974:
973:
972:
958:
950:
934:
929:
925:
904:
901:
893:
889:
869:
860:
856:
853:
848:
845:
822:
819:
814:
805:
800:
788:
782:
778:
769:
766:
763:
757:
754:
750:
742:
741:
740:
717:
713:
705:
690:
689:
688:
665:
657:
653:
649:
640:
632:
628:
624:
615:
604:
600:
596:
587:
580:
571:
559:
555:
552:
546:
537:
527:
526:
525:
511:
485:
461:
458:
455:
452:
449:
443:
421:
417:
413:
408:
404:
400:
395:
391:
364:
333:
330:
325:
321:
315:
311:
307:
302:
298:
292:
288:
284:
279:
275:
265:
257:
256:
255:
228:
225:
221:
215:
206:
202:
197:
194:
190:
186:
181:
178:
174:
170:
165:
162:
158:
150:
149:
148:
142:
126:
122:
118:
111:
106:
99:
97:
95:
91:
87:
76:
73:
65:
62:November 2011
55:
51:
45:
42:This article
40:
31:
30:
27:
19:
3078:Aerodynamics
3059:
3055:
3035:
3025:
2996:
2959:
2955:
2939:
2912:
2905:Glauert 1928
2900:
2888:
2881:Göthert 1940
2876:
2869:Shapiro 1953
2849:
2735:
2716:
2701:
2693:
2682:as given by
2634:
2630:
2628:
2441:
2439:
2408:
2344:
2097:
2095:
2087:
1736:
1553:
1390:
1146:
948:
891:
887:
837:
738:
686:
356:
253:
146:
90:compressible
85:
83:
68:
59:
43:
26:
2948:Glauert, H.
2771:singularity
2726:Singularity
2690:Limitations
2411:(unscaled)
141:Mach number
2917:Meier 2005
2836:References
3001:. Wiley.
2986:1364-5021
2841:Citations
2795:≃
2790:∞
2754:≃
2749:∞
2667:α
2664:π
2594:β
2589:α
2586:π
2510:¯
2498:¯
2486:¯
2460:β
2324:β
2283:β
2242:β
2183:β
2064:¯
2046:β
2030:∞
2018:¯
2006:¯
2003:ϕ
1986:β
1977:−
1967:∞
1953:ϕ
1944:−
1904:¯
1892:¯
1889:ϕ
1875:¯
1863:¯
1860:ϕ
1846:¯
1834:¯
1831:ϕ
1804:¯
1801:ϕ
1775:¯
1763:¯
1751:¯
1704:¯
1692:¯
1678:¯
1666:¯
1663:ϕ
1649:¯
1637:¯
1623:¯
1611:¯
1608:ϕ
1594:¯
1582:¯
1570:∞
1520:¯
1508:¯
1495:¯
1492:ϕ
1477:¯
1465:¯
1452:¯
1449:ϕ
1434:¯
1422:¯
1409:¯
1406:ϕ
1351:¯
1339:¯
1304:¯
1292:¯
1268:β
1254:¯
1242:¯
1208:β
1185:¯
1173:¯
1161:¯
1128:ϕ
1119:β
1105:¯
1102:ϕ
1092:α
1089:β
1076:¯
1073:α
1060:β
1047:¯
1031:β
1018:¯
992:¯
959:β
926:β
902:β
865:∞
857:−
849:≡
846:β
810:∞
793:∞
779:ϕ
764:γ
722:∞
714:≪
706:ϕ
703:∇
669:^
654:ϕ
644:^
629:ϕ
619:^
601:ϕ
592:∞
575:^
564:∞
553:ϕ
550:∇
541:→
490:∞
444:ϕ
369:∞
312:ϕ
289:ϕ
270:∞
222:ϕ
211:∞
191:ϕ
175:ϕ
159:ϕ
127:β
3072:Category
2950:(1928).
2814:See also
2409:original
947:and the
2964:Bibcode
2927:Sources
2698:History
2092:Results
48:Please
3043:
3005:
2984:
2629:where
2345:where
3058:[
3022:(PDF)
1147:This
3041:ISBN
3003:ISBN
2982:ISSN
2440:For
2096:For
890:and
820:<
84:The
2972:doi
2960:118
52:to
3074::
3024:,
2980:.
2970:.
2958:.
2954:.
2861:^
2798:1.
2686:.
2635:AR
2631:AR
2195::
971::
524:.
3049:.
3011:.
2988:.
2974::
2966::
2919:.
2883:.
2871:.
2786:M
2757:1
2745:M
2670:,
2661:2
2658:=
2653:0
2650:l
2646:c
2611:R
2608:A
2604:/
2600:2
2597:+
2583:2
2577:=
2572:L
2568:C
2538:p
2534:C
2507:z
2495:y
2483:x
2456:/
2452:1
2425:z
2422:y
2419:x
2393:0
2390:m
2386:c
2382:,
2377:0
2374:l
2370:c
2366:,
2361:0
2358:p
2354:C
2319:0
2316:m
2312:c
2306:=
2297:m
2293:c
2278:0
2275:l
2271:c
2265:=
2256:l
2252:c
2237:0
2234:p
2230:C
2224:=
2215:p
2211:C
2179:/
2175:1
2153:m
2149:c
2145:,
2140:l
2136:c
2113:p
2109:C
2071:p
2061:C
2050:2
2042:1
2037:=
2026:V
2015:x
1990:2
1982:2
1974:=
1963:V
1957:x
1947:2
1941:=
1936:p
1932:C
1901:z
1882:,
1872:y
1853:,
1843:x
1772:z
1760:y
1748:x
1714:0
1711:=
1701:z
1689:n
1675:z
1656:+
1646:y
1634:n
1620:y
1601:+
1591:x
1579:n
1566:V
1531:0
1528:=
1517:z
1505:z
1485:+
1474:y
1462:y
1442:+
1431:x
1419:x
1370:z
1366:n
1362:=
1348:z
1336:n
1323:y
1319:n
1315:=
1301:y
1289:n
1276:x
1272:n
1265:=
1251:x
1239:n
1182:z
1170:y
1158:x
1123:2
1115:=
1086:=
1063:z
1057:=
1044:z
1034:y
1028:=
1015:y
1005:x
1002:=
989:x
949:x
935:,
930:2
905:,
892:z
888:y
870:2
861:M
854:1
823:1
815:2
806:M
801:]
789:V
783:x
773:)
770:1
767:+
761:(
758:+
755:1
751:[
718:V
710:|
699:|
666:z
658:z
650:+
641:y
633:y
625:+
616:x
610:)
605:x
597:+
588:V
584:(
581:=
572:x
560:V
556:+
547:=
538:V
512:x
486:V
465:)
462:z
459:,
456:y
453:,
450:x
447:(
422:z
418:n
414:,
409:y
405:n
401:,
396:x
392:n
365:M
334:0
331:=
326:z
322:n
316:z
308:+
303:y
299:n
293:y
285:+
280:x
276:n
266:V
229:x
226:x
216:2
207:M
203:=
198:z
195:z
187:+
182:y
179:y
171:+
166:x
163:x
123:/
119:1
75:)
69:(
64:)
60:(
46:.
20:)
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