2614:
2187:
2609:{\displaystyle {\begin{array}{rcl}\left\phi &=&{\frac {\partial }{\partial {\boldsymbol {x}}}}\left(G\star \phi \right)-G\star {\frac {\partial \phi }{\partial {\boldsymbol {x}}}}\\&=&{\frac {\partial }{\partial {\boldsymbol {x}}}}\int _{\Omega }G({\boldsymbol {x}}-{\boldsymbol {r}},\Delta ({\boldsymbol {x}},t))\phi ({\boldsymbol {r}},t)d{\boldsymbol {r}}-G\star {\frac {\partial \phi }{\partial {\boldsymbol {x}}}}\\&=&\left({\frac {\partial G}{\partial \Delta }}\star \phi \right){\frac {\partial \Delta }{\partial x}}+\int _{d\Omega }G(x-r,\Delta (x,t))\phi (r,t){\boldsymbol {n}}dS\end{array}},}
2895:
77:
58:
2864:
3253:
36:
3527:
319:
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3141:
27:. Because the principal difficulty in simulating turbulent flows comes from the wide range of length and time scales, this operation makes turbulent flow simulation cheaper by reducing the range of scales that must be resolved. The LES filter operation is low-pass, meaning it filters out the scales associated with high frequencies.
3417:
2074:
143:
1296:
is central to LES subgrid-scale models, which reconstruct the effect of the sub-filter scales (the highest frequencies). One of the challenges in subgrid modeling is to effectively mimic the cascade of kinetic energy from low to high frequencies. This makes the spectral properties of the
997:
3010:
1664:
1480:
3655:
2846:
3771:
3521:
3247:
2160:
Implementations of filtering operations for all but the simplest flows are inhomogeneous filter operations. This means that the flow either has non-periodic boundaries, causing problems with certain types of filters, or has a non-constant filter width
3271:
1962:
314:{\displaystyle {\overline {\phi ({\boldsymbol {x}},t)}}=\displaystyle {\int _{-\infty }^{\infty }}\int _{-\infty }^{\infty }\phi ({\boldsymbol {r}},t^{\prime })G({\boldsymbol {x}}-{\boldsymbol {r}},t-t^{\prime })dt^{\prime }d{\boldsymbol {r}},}
1556:
1876:
3136:{\displaystyle G({\boldsymbol {x}}-{\boldsymbol {r}})={\begin{cases}{\frac {1}{\Delta }},&{\text{if}}\left|{\boldsymbol {x}}-{\boldsymbol {r}}\right|\leq {\frac {\Delta }{2}},\\0,&{\text{otherwise}}.\end{cases}}}
2150:
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753:
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564:
2738:
387:
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135:
3428:
3412:{\displaystyle G({\boldsymbol {x}}-{\boldsymbol {r}})=\left({\frac {6}{\pi \Delta ^{2}}}\right)^{\frac {1}{2}}\exp {\left(-{\frac {6({\boldsymbol {x-r}})^{2}}{\Delta ^{2}}}\right)}.}
3152:
2937:
2069:{\displaystyle {\begin{array}{rcl}{\overline {\overline {\phi }}}&\neq &{\overline {\phi }},\\G\star G\star \phi =G^{2}\star \phi &\neq &G\star \phi ,\end{array}}}
490:
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992:{\displaystyle {\overline {\hat {\phi }}}({\boldsymbol {k}},\omega )={\hat {\phi }}({\boldsymbol {k}},\omega ){\hat {G}}({\boldsymbol {k}},\omega )}
2085:
1659:{\displaystyle {\overline {\frac {\partial \phi }{\partial s}}}={\frac {\partial {\overline {\phi }}}{\partial s}},\qquad s={\boldsymbol {x}},t.}
1475:{\displaystyle \int _{-\infty }^{\infty }\int _{-\infty }^{\infty }G({\boldsymbol {\xi }},t^{\prime })d^{3}{\boldsymbol {\xi }}dt^{\prime }=1.}
3875:
2181:, or both. This prevents the filter from commuting with derivatives, and the commutation operation leads to several additional error terms:
1214:
1008:
3650:{\displaystyle G({\boldsymbol {x}}-{\boldsymbol {r}})={\frac {\sin {(\pi ({\boldsymbol {x-r}})/\Delta )}}{\pi ({\boldsymbol {x-r}})}}.}
3845:
575:
44:
1292:
The spectral interpretation of the filtering operation is essential to the filtering operation in large eddy simulation, as the
709:
638:
The filtering operation removes scales associated with high frequencies, and the operation can accordingly be interpreted in
2942:
839:
2841:{\displaystyle \left=\left({\frac {\partial G}{\partial \Delta }}\star \phi \right){\frac {\partial \Delta }{\partial t}}.}
518:
3782:
350:
137:. The LES filter operation may be spatial, temporal, or both. The filtered field, denoted with a bar, is defined as:
3766:{\displaystyle {\hat {G}}({\boldsymbol {k}})=H\left(k_{c}-\left|k\right|\right),\qquad k_{c}={\frac {\pi }{\Delta }}.}
40:
3921:
3787:
3516:{\displaystyle {\hat {G}}({\boldsymbol {k}})=\exp {\left(-{\frac {{\boldsymbol {k}}^{2}\Delta ^{2}}{24}}\right)}.}
1305:
Homogeneous LES filters must satisfy the following set of properties when applied to the Navier-Stokes equations.
645:
3242:{\displaystyle {\hat {G}}({\boldsymbol {k}})={\frac {\sin {({\frac {1}{2}}k\Delta )}}{{\frac {1}{2}}k\Delta }}.}
2689:
The two terms both appear due to inhomogeneities. The first is due to the spatial variation in the filter size
103:
3892:
2905:
24:
2902:
There are three filters ordinarily used for spatial filtering in large eddy simulation. The definition of
2894:
2732:
with the temporal derivative leads to an error term resulting from temporal variation in the filter size,
465:
344:
is a convolution kernel unique to the filter type used. This can be written as a convolution operation:
100:
The low-pass filtering operation used in LES can be applied to a spatial and temporal field, for example
3807:
1323:
758:
20:
2622:
76:
57:
23:(LES) is a mathematical operation intended to remove a range of small scales from the solution to the
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Several filter operations which eliminate or minimize these error terms have been proposed.
1095:
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while the second is due to the domain boundary. Similarly, the commutation of the filter
2715:
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2192:
1967:
1675:
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512:
may be split up into a filtered and sub-filtered (denoted with a prime) portion, as
1551:{\displaystyle {\overline {\phi +\psi }}={\overline {\phi }}+{\overline {\psi }}.}
35:
3893:"Lecture Notes on DNS and LES for International Masters Program in Turbulence"
3812:
3252:
1871:{\displaystyle \phi =f\circ g(\phi )-g\circ f(\phi )=f(g(\phi ))-g(f(\phi )),}
1293:
2145:{\displaystyle {\overline {\phi ^{\prime }}}=G\star (1-G)\star \phi \neq 0.}
3526:
2996:
1282:{\displaystyle {\hat {\phi ^{\prime }}}=(1-{\hat {G}}){\hat {\phi }}.}
1062:{\displaystyle {\overline {\hat {\phi }}}={\hat {G}}{\hat {\phi }}.}
1297:
implemented LES filter very important to subgrid modeling efforts.
75:
56:
34:
1314:
The value of a filtered constant must be equal to the constant,
2858:
2898:
Turbulent energy spectrum and effect of filtering operations
462:
respectively. Scales smaller than these are eliminated from
2096:
1461:
1430:
1226:
623:{\displaystyle \phi ^{\prime }=\left(1-G\right)\star \phi .}
584:
548:
294:
278:
237:
3129:
748:{\displaystyle {\hat {\phi }}({\boldsymbol {k}},\omega ),}
2875:
2981:{\displaystyle {\hat {G}}({\boldsymbol {k}},\omega ),}
2874: with: Proper alignment of plots. You can help by
1952:
Filters satisfying these properties are generally not
878:{\displaystyle {\hat {G}}({\boldsymbol {k}},\omega ):}
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This can also be written as a convolution operation,
559:{\displaystyle \phi ={\bar {\phi }}+\phi ^{\prime }.}
521:
498:
468:
438:
418:
398:
353:
330:
179:
146:
106:
3530:
Sharp spectral filter in physical and spectral space
2988:
and a discussion of important properties, is given.
2641:
is the vector normal to the surface of the boundary
1672:If notation is introduced for operator commutation
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3660:The filter kernel in spectral space is given by:
3539:The filter kernel in physical space is given by:
3422:The filter kernel in spectral space is given by:
3265:The filter kernel in physical space is given by:
3146:The filter kernel in spectral space is given by:
3004:The filter kernel in physical space is given by:
382:{\displaystyle {\overline {\phi }}=G\star \phi .}
3838:Large Eddy Simulation for Incompressible Flows
3256:Gaussian filter in physical and spectral space
3831:
3829:
3827:
80:The same DNS velocity field filtered using a
61:The same DNS velocity field filtered using a
8:
1884:then this third property can be expressed as
412:uses cutoff length and time scales, denoted
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675:{\displaystyle \phi ({\boldsymbol {x}},t),}
130:{\displaystyle \phi ({\boldsymbol {x}},t)}
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3000:Box filter in physical and spectral space
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1149:also has an associated cutoff frequency
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114:
2932:{\displaystyle G({\boldsymbol {x}},t)}
2855:Classic large eddy simulation filters
1092:has an associated cutoff wave number
832:can be filtered by the corresponding
7:
485:{\displaystyle {\overline {\phi }}.}
3755:
3612:
3489:
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1608:
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1383:
1349:{\displaystyle {\overline {a}}=a,}
1079:
773:{\displaystyle {\boldsymbol {k}},}
419:
213:
208:
194:
189:
14:
2634:{\displaystyle {\boldsymbol {n}}}
492:Using this definition, any field
41:direct numerical simulation (DNS)
3631:
3597:
3370:
2862:
3736:
1635:
1564:3. Commutation with derivatives
81:
62:
45:homogeneous decaying turbulence
39:A velocity field produced by a
3870:. Cambridge University Press.
3690:
3682:
3676:
3638:
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1787:
1769:
1757:
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1679:
1435:
1414:
1270:
1261:
1255:
1240:
1231:
1201:{\displaystyle {\hat {\phi }}}
1192:
1122:and the temporal filter width
1050:
1038:
1019:
986:
972:
966:
957:
943:
937:
925:
911:
901:
869:
855:
849:
836:of the filter kernel, denoted
825:{\displaystyle {\hat {\phi }}}
816:
739:
725:
719:
666:
652:
534:
283:
248:
242:
221:
167:
153:
124:
110:
1:
1301:Homogeneous filter properties
780:the spatial wave number, and
3840:(Third ed.). Springer.
3783:Computational fluid dynamics
2101:
1997:
1980:
1976:
1704:for two arbitrary operators
1616:
1597:
1540:
1527:
1514:
1332:
1309:1. Conservation of constants
1172:{\displaystyle \omega _{c}.}
1024:
906:
634:Definition in spectral space
474:
359:
171:
96:Definition in physical space
3938:
3788:Filter (signal processing)
1294:spectra of turbulent flows
1179:The unfiltered portion of
455:{\displaystyle \tau _{c},}
2680:{\displaystyle d\Omega .}
1142:{\displaystyle \tau _{c}}
803:the temporal frequency.
2705:{\displaystyle \Delta ,}
1940:{\displaystyle \left=0.}
796:{\displaystyle \omega ,}
3836:Sagaut, Pierre (2006).
2654:{\displaystyle \Omega }
2174:{\displaystyle \Delta }
1085:{\displaystyle \Delta }
425:{\displaystyle \Delta }
25:Navier-Stokes equations
3891:Laval, Jean-Philippe.
3864:Pope, Stephen (2000).
3767:
3651:
3531:
3517:
3413:
3257:
3243:
3137:
3001:
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2933:
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2726:
2706:
2681:
2655:
2635:
2610:
2175:
2146:
2070:
1941:
1872:
1738:
1718:
1698:
1660:
1552:
1476:
1350:
1283:
1202:
1173:
1143:
1116:
1115:{\displaystyle k_{c},}
1086:
1063:
993:
879:
826:
797:
774:
749:
700:
676:
642:. For a scalar field
624:
560:
506:
486:
456:
426:
406:
383:
338:
315:
131:
92:
73:
54:
47:. The domain size is
3808:Large eddy simulation
3768:
3652:
3535:Sharp spectral filter
3529:
3518:
3414:
3255:
3244:
3138:
2999:
2983:
2934:
2897:
2843:
2727:
2707:
2682:
2656:
2636:
2611:
2176:
2156:Inhomogeneous filters
2147:
2071:
1942:
1873:
1739:
1719:
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1203:
1174:
1144:
1117:
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1064:
994:
880:
827:
798:
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750:
701:
699:{\displaystyle \phi }
677:
625:
561:
507:
505:{\displaystyle \phi }
487:
457:
427:
407:
384:
339:
316:
132:
79:
60:
38:
21:large eddy simulation
3667:
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3011:
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2716:
2693:
2665:
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2623:
2188:
2165:
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1963:
1894:
1754:
1728:
1708:
1676:
1574:
1498:
1372:
1324:
1215:
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1153:
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1076:
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891:
840:
807:
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759:
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576:
519:
496:
466:
436:
416:
396:
351:
328:
144:
104:
1410:
1392:
217:
198:
31:Homogeneous filters
3763:
3647:
3532:
3513:
3409:
3258:
3239:
3133:
3128:
3002:
2978:
2929:
2900:
2838:
2722:
2702:
2677:
2651:
2631:
2606:
2601:
2171:
2142:
2066:
2064:
1956:, meaning, first:
1954:Reynolds operators
1937:
1868:
1734:
1714:
1694:
1656:
1548:
1472:
1393:
1375:
1346:
1279:
1198:
1169:
1139:
1112:
1082:
1059:
989:
875:
822:
793:
770:
745:
696:
672:
620:
556:
502:
482:
452:
422:
402:
392:The filter kernel
379:
334:
311:
310:
200:
181:
127:
93:
74:
55:
19:in the context of
3922:Turbulence models
3877:978-0-521-59886-6
3798:Fourier transform
3758:
3679:
3642:
3502:
3441:
3398:
3339:
3325:
3234:
3225:
3203:
3165:
3121:
3101:
3065:
3055:
2955:
2892:
2891:
2833:
2802:
2760:
2725:{\displaystyle G}
2513:
2482:
2447:
2333:
2306:
2256:
2215:
2104:
2000:
1983:
1979:
1924:
1737:{\displaystyle g}
1717:{\displaystyle f}
1630:
1619:
1600:
1596:
1543:
1530:
1517:
1335:
1273:
1258:
1234:
1195:
1072:The filter width
1053:
1041:
1027:
1022:
969:
940:
909:
904:
852:
834:Fourier transform
819:
722:
684:Fourier transform
537:
477:
405:{\displaystyle G}
362:
337:{\displaystyle G}
174:
3929:
3907:
3906:
3904:
3902:
3897:
3888:
3882:
3881:
3861:
3852:
3851:
3833:
3803:Frequency domain
3772:
3770:
3769:
3764:
3759:
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3732:
3728:
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3024:
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2985:
2984:
2979:
2965:
2957:
2956:
2948:
2938:
2936:
2935:
2930:
2919:
2887:
2884:
2866:
2859:
2847:
2845:
2844:
2839:
2834:
2832:
2824:
2816:
2814:
2810:
2803:
2801:
2793:
2785:
2775:
2771:
2761:
2759:
2748:
2731:
2729:
2728:
2723:
2711:
2709:
2708:
2703:
2686:
2684:
2683:
2678:
2660:
2658:
2657:
2652:
2640:
2638:
2637:
2632:
2630:
2615:
2613:
2612:
2607:
2602:
2592:
2530:
2529:
2514:
2512:
2504:
2496:
2494:
2490:
2483:
2481:
2473:
2465:
2452:
2448:
2446:
2445:
2436:
2428:
2417:
2400:
2377:
2363:
2355:
2344:
2343:
2334:
2332:
2331:
2319:
2311:
2307:
2305:
2304:
2295:
2287:
2276:
2272:
2257:
2255:
2254:
2242:
2230:
2226:
2216:
2214:
2213:
2201:
2180:
2178:
2177:
2172:
2151:
2149:
2148:
2143:
2105:
2100:
2099:
2090:
2075:
2073:
2072:
2067:
2065:
2036:
2035:
2001:
1993:
1984:
1972:
1971:
1946:
1944:
1943:
1938:
1930:
1926:
1925:
1923:
1912:
1877:
1875:
1874:
1869:
1743:
1741:
1740:
1735:
1723:
1721:
1720:
1715:
1703:
1701:
1700:
1697:{\displaystyle }
1695:
1665:
1663:
1662:
1657:
1646:
1631:
1629:
1621:
1620:
1612:
1606:
1601:
1595:
1587:
1579:
1578:
1557:
1555:
1554:
1549:
1544:
1536:
1531:
1523:
1518:
1513:
1502:
1481:
1479:
1478:
1473:
1465:
1464:
1452:
1447:
1446:
1434:
1433:
1421:
1409:
1404:
1391:
1386:
1355:
1353:
1352:
1347:
1336:
1328:
1288:
1286:
1285:
1280:
1275:
1274:
1266:
1260:
1259:
1251:
1236:
1235:
1230:
1229:
1220:
1207:
1205:
1204:
1199:
1197:
1196:
1188:
1178:
1176:
1175:
1170:
1165:
1164:
1148:
1146:
1145:
1140:
1138:
1137:
1121:
1119:
1118:
1113:
1108:
1107:
1091:
1089:
1088:
1083:
1068:
1066:
1065:
1060:
1055:
1054:
1046:
1043:
1042:
1034:
1028:
1023:
1015:
1013:
998:
996:
995:
990:
979:
971:
970:
962:
950:
942:
941:
933:
918:
910:
905:
897:
895:
884:
882:
881:
876:
862:
854:
853:
845:
831:
829:
828:
823:
821:
820:
812:
802:
800:
799:
794:
779:
777:
776:
771:
766:
754:
752:
751:
746:
732:
724:
723:
715:
705:
703:
702:
697:
681:
679:
678:
673:
659:
629:
627:
626:
621:
610:
606:
588:
587:
565:
563:
562:
557:
552:
551:
539:
538:
530:
511:
509:
508:
503:
491:
489:
488:
483:
478:
470:
461:
459:
458:
453:
448:
447:
431:
429:
428:
423:
411:
409:
408:
403:
388:
386:
385:
380:
363:
355:
343:
341:
340:
335:
320:
318:
317:
312:
306:
298:
297:
282:
281:
263:
255:
241:
240:
228:
216:
211:
199:
197:
192:
175:
170:
160:
148:
136:
134:
133:
128:
117:
91:
72:
52:
3937:
3936:
3932:
3931:
3930:
3928:
3927:
3926:
3912:
3911:
3910:
3900:
3898:
3895:
3890:
3889:
3885:
3878:
3867:Turbulent Flows
3863:
3862:
3855:
3848:
3835:
3834:
3825:
3821:
3793:Fluid mechanics
3779:
3737:
3717:
3704:
3703:
3699:
3665:
3664:
3620:
3576:
3544:
3543:
3537:
3488:
3476:
3475:
3469:
3465:
3427:
3426:
3388:
3377:
3359:
3353:
3349:
3314:
3310:
3301:
3300:
3270:
3269:
3263:
3261:Gaussian filter
3216:
3184:
3151:
3150:
3127:
3126:
3116:
3107:
3106:
3071:
3067:
3060:
3040:
3009:
3008:
2994:
2941:
2940:
2904:
2903:
2888:
2882:
2879:
2872:needs expansion
2857:
2825:
2817:
2794:
2786:
2783:
2779:
2752:
2746:
2742:
2737:
2736:
2714:
2713:
2691:
2690:
2663:
2662:
2643:
2642:
2621:
2620:
2600:
2599:
2518:
2505:
2497:
2474:
2466:
2463:
2459:
2457:
2450:
2449:
2437:
2429:
2335:
2323:
2316:
2309:
2308:
2296:
2288:
2262:
2258:
2246:
2239:
2234:
2205:
2199:
2195:
2186:
2185:
2163:
2162:
2158:
2091:
2084:
2083:
2063:
2062:
2048:
2043:
2027:
2006:
2005:
1990:
1985:
1961:
1960:
1916:
1901:
1897:
1892:
1891:
1752:
1751:
1726:
1725:
1706:
1705:
1674:
1673:
1622:
1607:
1588:
1580:
1572:
1571:
1503:
1496:
1495:
1456:
1438:
1425:
1370:
1369:
1322:
1321:
1303:
1221:
1213:
1212:
1181:
1180:
1156:
1151:
1150:
1129:
1124:
1123:
1099:
1094:
1093:
1074:
1073:
1007:
1006:
889:
888:
838:
837:
805:
804:
782:
781:
757:
756:
708:
707:
688:
687:
644:
643:
636:
596:
592:
579:
574:
573:
543:
517:
516:
494:
493:
464:
463:
439:
434:
433:
414:
413:
394:
393:
349:
348:
326:
325:
289:
273:
232:
149:
142:
141:
102:
101:
98:
89:
85:
70:
66:
51:
48:
33:
12:
11:
5:
3935:
3933:
3925:
3924:
3914:
3913:
3909:
3908:
3883:
3876:
3853:
3846:
3822:
3820:
3817:
3816:
3815:
3810:
3805:
3800:
3795:
3790:
3785:
3778:
3775:
3774:
3773:
3762:
3757:
3754:
3749:
3744:
3740:
3735:
3731:
3726:
3723:
3720:
3716:
3711:
3707:
3702:
3698:
3695:
3692:
3688:
3684:
3678:
3675:
3658:
3657:
3646:
3640:
3636:
3633:
3630:
3626:
3623:
3617:
3614:
3610:
3606:
3602:
3599:
3596:
3592:
3589:
3586:
3582:
3579:
3573:
3570:
3566:
3562:
3558:
3554:
3551:
3536:
3533:
3524:
3523:
3512:
3507:
3501:
3495:
3491:
3485:
3480:
3472:
3468:
3463:
3460:
3457:
3454:
3450:
3446:
3440:
3437:
3420:
3419:
3408:
3403:
3395:
3391:
3384:
3380:
3375:
3372:
3369:
3365:
3362:
3356:
3352:
3347:
3344:
3338:
3335:
3329:
3321:
3317:
3313:
3309:
3304:
3299:
3296:
3292:
3288:
3284:
3280:
3277:
3262:
3259:
3250:
3249:
3238:
3232:
3229:
3224:
3221:
3213:
3210:
3207:
3202:
3199:
3194:
3190:
3187:
3181:
3178:
3174:
3170:
3164:
3161:
3144:
3143:
3130:
3125:
3117:
3115:
3112:
3109:
3108:
3105:
3100:
3097:
3092:
3088:
3083:
3079:
3075:
3070:
3061:
3059:
3054:
3051:
3046:
3045:
3043:
3038:
3035:
3031:
3027:
3023:
3019:
3016:
2993:
2990:
2977:
2974:
2971:
2968:
2964:
2960:
2954:
2951:
2928:
2925:
2922:
2918:
2914:
2911:
2890:
2889:
2869:
2867:
2856:
2853:
2849:
2848:
2837:
2831:
2828:
2823:
2820:
2813:
2809:
2806:
2800:
2797:
2792:
2789:
2782:
2778:
2774:
2770:
2767:
2764:
2758:
2755:
2751:
2745:
2721:
2701:
2698:
2676:
2673:
2670:
2650:
2629:
2617:
2616:
2605:
2598:
2595:
2591:
2587:
2584:
2581:
2578:
2575:
2572:
2569:
2566:
2563:
2560:
2557:
2554:
2551:
2548:
2545:
2542:
2539:
2536:
2533:
2528:
2525:
2521:
2517:
2511:
2508:
2503:
2500:
2493:
2489:
2486:
2480:
2477:
2472:
2469:
2462:
2458:
2456:
2453:
2451:
2444:
2440:
2435:
2432:
2426:
2423:
2420:
2416:
2412:
2409:
2406:
2403:
2399:
2395:
2392:
2389:
2386:
2383:
2380:
2376:
2372:
2369:
2366:
2362:
2358:
2354:
2350:
2347:
2342:
2338:
2330:
2326:
2322:
2317:
2315:
2312:
2310:
2303:
2299:
2294:
2291:
2285:
2282:
2279:
2275:
2271:
2268:
2265:
2261:
2253:
2249:
2245:
2240:
2238:
2235:
2233:
2229:
2225:
2222:
2219:
2212:
2208:
2204:
2198:
2194:
2193:
2170:
2157:
2154:
2153:
2152:
2141:
2138:
2135:
2132:
2129:
2126:
2123:
2120:
2117:
2114:
2111:
2108:
2103:
2098:
2094:
2077:
2076:
2061:
2058:
2055:
2052:
2049:
2047:
2044:
2042:
2039:
2034:
2030:
2026:
2023:
2020:
2017:
2014:
2011:
2008:
2007:
2004:
1999:
1996:
1991:
1989:
1986:
1982:
1978:
1975:
1969:
1968:
1950:
1949:
1948:
1947:
1936:
1933:
1929:
1922:
1919:
1915:
1910:
1907:
1904:
1900:
1886:
1885:
1881:
1880:
1879:
1878:
1867:
1864:
1861:
1858:
1855:
1852:
1849:
1846:
1843:
1840:
1837:
1834:
1831:
1828:
1825:
1822:
1819:
1816:
1813:
1810:
1807:
1804:
1801:
1798:
1795:
1792:
1789:
1786:
1783:
1780:
1777:
1774:
1771:
1768:
1765:
1762:
1759:
1746:
1745:
1733:
1713:
1693:
1690:
1687:
1684:
1681:
1669:
1668:
1667:
1666:
1655:
1652:
1649:
1645:
1641:
1638:
1634:
1628:
1625:
1618:
1615:
1610:
1604:
1599:
1594:
1591:
1586:
1583:
1566:
1565:
1561:
1560:
1559:
1558:
1547:
1542:
1539:
1534:
1529:
1526:
1521:
1516:
1512:
1509:
1506:
1490:
1489:
1485:
1484:
1483:
1482:
1471:
1468:
1463:
1459:
1455:
1451:
1445:
1441:
1437:
1432:
1428:
1424:
1420:
1416:
1413:
1408:
1403:
1400:
1396:
1390:
1385:
1382:
1378:
1364:
1363:
1362:which implies,
1359:
1358:
1357:
1356:
1345:
1342:
1339:
1334:
1331:
1316:
1315:
1311:
1310:
1302:
1299:
1290:
1289:
1278:
1272:
1269:
1263:
1257:
1254:
1248:
1245:
1242:
1239:
1233:
1228:
1224:
1194:
1191:
1168:
1163:
1159:
1136:
1132:
1111:
1106:
1102:
1081:
1070:
1069:
1058:
1052:
1049:
1040:
1037:
1031:
1026:
1021:
1018:
1000:
999:
988:
985:
982:
978:
974:
968:
965:
959:
956:
953:
949:
945:
939:
936:
930:
927:
924:
921:
917:
913:
908:
903:
900:
874:
871:
868:
865:
861:
857:
851:
848:
818:
815:
792:
789:
769:
765:
755:a function of
744:
741:
738:
735:
731:
727:
721:
718:
695:
671:
668:
665:
662:
658:
654:
651:
635:
632:
631:
630:
619:
616:
613:
609:
605:
602:
599:
595:
591:
586:
582:
567:
566:
555:
550:
546:
542:
536:
533:
527:
524:
501:
481:
476:
473:
451:
446:
442:
421:
401:
390:
389:
378:
375:
372:
369:
366:
361:
358:
333:
322:
321:
309:
305:
301:
296:
292:
288:
285:
280:
276:
272:
269:
266:
262:
258:
254:
250:
247:
244:
239:
235:
231:
227:
223:
220:
215:
210:
207:
203:
196:
191:
188:
184:
178:
173:
169:
166:
163:
159:
155:
152:
126:
123:
120:
116:
112:
109:
97:
94:
87:
68:
49:
32:
29:
13:
10:
9:
6:
4:
3:
2:
3934:
3923:
3920:
3919:
3917:
3894:
3887:
3884:
3879:
3873:
3869:
3868:
3860:
3858:
3854:
3849:
3847:3-540-26344-6
3843:
3839:
3832:
3830:
3828:
3824:
3818:
3814:
3811:
3809:
3806:
3804:
3801:
3799:
3796:
3794:
3791:
3789:
3786:
3784:
3781:
3780:
3776:
3760:
3752:
3747:
3742:
3738:
3733:
3729:
3724:
3721:
3718:
3714:
3709:
3705:
3700:
3696:
3693:
3673:
3663:
3662:
3661:
3644:
3621:
3608:
3587:
3580:
3577:
3571:
3560:
3549:
3542:
3541:
3540:
3534:
3528:
3510:
3505:
3499:
3493:
3483:
3470:
3466:
3461:
3458:
3455:
3435:
3425:
3424:
3423:
3406:
3401:
3393:
3382:
3360:
3354:
3350:
3345:
3342:
3336:
3333:
3327:
3319:
3311:
3307:
3302:
3297:
3286:
3275:
3268:
3267:
3266:
3260:
3254:
3236:
3227:
3222:
3219:
3205:
3200:
3197:
3188:
3185:
3179:
3159:
3149:
3148:
3147:
3123:
3113:
3110:
3103:
3098:
3090:
3086:
3077:
3068:
3057:
3049:
3041:
3036:
3025:
3014:
3007:
3006:
3005:
2998:
2991:
2989:
2975:
2969:
2966:
2949:
2923:
2920:
2909:
2896:
2886:
2877:
2873:
2870:This section
2868:
2865:
2861:
2860:
2854:
2852:
2835:
2829:
2811:
2807:
2804:
2790:
2780:
2776:
2772:
2768:
2765:
2762:
2756:
2743:
2735:
2734:
2733:
2719:
2699:
2687:
2674:
2668:
2603:
2596:
2593:
2582:
2579:
2576:
2570:
2561:
2558:
2555:
2546:
2543:
2540:
2537:
2531:
2523:
2519:
2515:
2509:
2491:
2487:
2484:
2470:
2460:
2454:
2433:
2424:
2421:
2418:
2410:
2404:
2401:
2390:
2381:
2378:
2364:
2356:
2345:
2336:
2313:
2292:
2283:
2280:
2277:
2273:
2269:
2266:
2263:
2259:
2236:
2231:
2227:
2223:
2220:
2217:
2196:
2184:
2183:
2182:
2155:
2139:
2136:
2133:
2130:
2124:
2121:
2118:
2112:
2109:
2106:
2092:
2082:
2081:
2080:
2059:
2056:
2053:
2050:
2045:
2040:
2037:
2032:
2028:
2024:
2021:
2018:
2015:
2012:
2009:
2002:
1994:
1987:
1973:
1959:
1958:
1957:
1955:
1934:
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641:
640:Fourier space
633:
617:
614:
611:
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571:
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3899:. Retrieved
3886:
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2883:January 2020
2880:
2876:adding to it
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2159:
2079:and second,
2078:
1951:
1488:2. Linearity
1304:
1291:
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637:
568:
391:
323:
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16:
15:
3901:27 January
3819:References
3813:Turbulence
2992:Box filter
82:box filter
63:box filter
3756:Δ
3753:π
3715:−
3677:^
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3471:−
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3120:otherwise
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67:Δ =
17:Filtering
3916:Category
3777:See also
1744:, where
3874:
3844:
2619:where
324:where
3896:(PDF)
3903:2020
3872:ISBN
3842:ISBN
2939:and
2661:and
1724:and
1208:is:
1002:or,
682:the
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