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3604:{\displaystyle {\frac {\partial {\overline {u_{i}^{\prime }u_{j}^{\prime }}}}{\partial t}}+{\bar {u}}_{k}{\frac {\partial {\overline {u_{i}^{\prime }u_{j}^{\prime }}}}{\partial x_{k}}}=-{\overline {u_{i}^{\prime }u_{k}^{\prime }}}{\frac {\partial {\bar {u}}_{j}}{\partial x_{k}}}-{\overline {u_{j}^{\prime }u_{k}^{\prime }}}{\frac {\partial {\bar {u}}_{i}}{\partial x_{k}}}+{\overline {{\frac {p^{\prime }}{\rho }}\left({\frac {\partial u_{i}^{\prime }}{\partial x_{j}}}+{\frac {\partial u_{j}^{\prime }}{\partial x_{i}}}\right)}}-{\frac {\partial }{\partial x_{k}}}\left({\overline {u_{i}^{\prime }u_{j}^{\prime }u_{k}^{\prime }}}+{\frac {\overline {p^{\prime }u_{i}^{\prime }}}{\rho }}\delta _{jk}+{\frac {\overline {p^{\prime }u_{j}^{\prime }}}{\rho }}\delta _{ik}-\nu {\frac {\partial {\overline {u_{i}^{\prime }u_{j}^{\prime }}}}{\partial x_{k}}}\right)-2\nu {\overline {{\frac {\partial u_{i}^{\prime }}{\partial x_{k}}}{\frac {\partial u_{j}^{\prime }}{\partial x_{k}}}}}}
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5192:{\displaystyle {\overline {\frac {\partial \left({\bar {u}}_{i}+u_{i}^{\prime }\right)}{\partial t}}}+{\overline {\left({\bar {u}}_{j}+u_{j}^{\prime }\right){\frac {\partial \left({\bar {u_{i}}}+u_{i}^{\prime }\right)}{\partial x_{j}}}}}={\overline {\left({\bar {f}}_{i}+f_{i}^{\prime }\right)}}-{\frac {1}{\rho }}{\overline {\frac {\partial \left({\bar {p}}+p^{\prime }\right)}{\partial x_{i}}}}+\nu {\overline {\frac {\partial ^{2}\left({\bar {u}}_{i}+u_{i}^{\prime }\right)}{\partial x_{i}\partial x_{j}}}}.}
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4639:{\displaystyle {\frac {\partial \left({\bar {u_{i}}}+u_{i}^{\prime }\right)}{\partial t}}+\left({\bar {u}}_{j}+u_{j}^{\prime }\right){\frac {\partial \left({\bar {u}}_{i}+u_{i}^{\prime }\right)}{\partial x_{j}}}=\left({\bar {f}}_{i}+f_{i}^{\prime }\right)-{\frac {1}{\rho }}{\frac {\partial \left({\bar {p}}+p^{\prime }\right)}{\partial x_{i}}}+\nu {\frac {\partial ^{2}\left({\bar {u}}_{i}+u_{i}^{\prime }\right)}{\partial x_{i}\partial x_{j}}}.}
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2085:{\displaystyle {\frac {\partial {\bar {u}}_{i}}{\partial t}}+{\bar {u}}_{j}{\frac {\partial {\bar {u}}_{i}}{\partial x_{j}}}={\bar {f}}_{i}-{\frac {1}{\rho }}{\frac {\partial {\bar {p}}}{\partial x_{i}}}+\nu {\frac {\partial ^{2}{\bar {u}}_{i}}{\partial x_{j}\partial x_{j}}}-{\frac {\partial {\overline {u_{i}^{\prime }u_{j}^{\prime }}}}{\partial x_{j}}}.}
1754:{\displaystyle {\frac {\partial {\bar {u}}_{i}}{\partial t}}+{\bar {u}}_{j}{\frac {\partial {\bar {u}}_{i}}{\partial x_{j}}}+{\overline {u_{j}^{\prime }{\frac {\partial u_{i}^{\prime }}{\partial x_{j}}}}}={\bar {f}}_{i}-{\frac {1}{\rho }}{\frac {\partial {\bar {p}}}{\partial x_{i}}}+\nu {\frac {\partial ^{2}{\bar {u}}_{i}}{\partial x_{j}\partial x_{j}}}.}
5630:{\displaystyle {\overline {u_{i}u_{j}}}={\overline {\left({\bar {u}}_{i}+u_{i}^{\prime }\right)\left({\bar {u_{j}}}+u_{j}^{\prime }\right)}}={\overline {{\bar {u_{i}}}{\bar {u}}_{j}+{\bar {u}}_{i}u_{j}^{\prime }+u_{i}^{\prime }{\bar {u}}_{j}+u_{i}^{\prime }u_{j}^{\prime }}}={\bar {u}}_{i}{\bar {u}}_{j}+{\overline {u_{i}^{\prime }u_{j}^{\prime }}}}
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The RANS equations have been widely utilized as a model for determining flow characteristics and assessing wind comfort in urban environments. This computational approach can be executed through direct calculations involving the solution of the RANS equations, or through an indirect method involving
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A model for testing performance was determined that, when combined with the vortex lattice (VLM) or boundary element method (BEM), RANS was found useful for modelling the flow of water between two contrary rotation propellers, where VLM or BEM are applied to the propellers and RANS is used for the
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is arbitrarily close to the average. This means that given transient data over a sufficiently large time, the average can be numerically computed within some small error. However, there is no analytical way to obtain an upper bound on
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1330:{\displaystyle {\frac {\partial u_{i}}{\partial t}}+u_{j}{\frac {\partial u_{i}}{\partial x_{j}}}=f_{i}-{\frac {1}{\rho }}{\frac {\partial p}{\partial x_{i}}}+\nu {\frac {\partial ^{2}u_{i}}{\partial x_{j}\partial x_{j}}}}
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the training of machine learning algorithms using the RANS equations as a basis. The direct approach is more accurate than the indirect approach but it requires expertise in numerical methods and
2373:{\displaystyle \rho {\frac {\partial {\bar {u}}_{i}}{\partial t}}+\rho {\bar {u}}_{j}{\frac {\partial {\bar {u}}_{i}}{\partial x_{j}}}=\rho {\bar {f}}_{i}+{\frac {\partial }{\partial x_{j}}}\left}
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are useful in the derivation of the RANS equations. Using these properties, the Navier–Stokes equations of motion, expressed in tensor notation, are (for an incompressible
Newtonian fluid):
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and the convection by the mean flow. This change is balanced by the mean body force, the isotropic stress owing to the mean pressure field, the viscous stresses, and apparent stress
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5783:{\displaystyle {\frac {\partial u_{i}}{\partial x_{i}}}={\frac {\partial {\bar {u}}_{i}}{\partial x_{i}}}+{\frac {\partial u_{i}^{\prime }}{\partial x_{i}}}=0}
2526:{\displaystyle {\bar {S}}_{ij}={\frac {1}{2}}\left({\frac {\partial {\bar {u}}_{i}}{\partial x_{j}}}+{\frac {\partial {\bar {u}}_{j}}{\partial x_{i}}}\right)}
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585:. This nonlinear Reynolds stress term requires additional modeling to close the RANS equation for solving, and has led to the creation of many different
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2778:{\displaystyle \rho {\bar {u}}_{j}{\frac {\partial {\bar {u}}_{i}}{\partial x_{j}}}=\rho {\bar {f_{i}}}+{\frac {\partial }{\partial x_{j}}}\left.}
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Next, each instantaneous quantity can be split into time-averaged and fluctuating components, and the resulting equation time-averaged, to yield:
496:{\displaystyle \rho {\bar {u}}_{j}{\frac {\partial {\bar {u}}_{i}}{\partial x_{j}}}=\rho {\bar {f}}_{i}+{\frac {\partial }{\partial x_{j}}}\left.}
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Finally, since integration in time removes the time dependence of the resultant terms, the time derivative must be eliminated, leaving:
3771:{\displaystyle \nu {\overline {{\frac {\partial u_{i}^{\prime }}{\partial x_{k}}}{\frac {\partial u_{j}^{\prime }}{\partial x_{k}}}}}}
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The left hand side of this equation represents the change in mean momentum of a fluid element owing to the unsteadiness in the
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for the mean term (since an overbar is sometimes used to represent a vector). In this case, the fluctuating term
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4738:{\displaystyle {\overline {\frac {\partial \left({\bar {u}}_{i}+u_{i}^{\prime }\right)}{\partial x_{i}}}}=0}
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5948:. (Su: Ocean Engineering Group, Department of Civil, Architectural and Environmental Engineering The
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5824:"On the Dynamical Theory of Incompressible Viscous Fluids and the Determination of the Criterion"
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4216:{\displaystyle {\frac {\partial \left({\bar {u_{i}}}+u_{i}^{\prime }\right)}{\partial x_{i}}}=0}
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3796:(CFD), as well as substantial computational resources to handle the complexity of the equations.
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847:{\displaystyle u({\boldsymbol {x}},t)={\bar {u}}({\boldsymbol {x}})+u'({\boldsymbol {x}},t),}
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Splitting each instantaneous quantity into its averaged and fluctuating components yields,
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3961:{\displaystyle {\bar {X}}=\lim _{T\to \infty }{\frac {1}{T}}\int _{t_{0}}^{t_{0}+T}x\,dt.}
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will be used to represent the instantaneous, mean, and fluctuating terms, respectively.
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The basic tool required for the derivation of the RANS equations from the instantaneous
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5898:"On velocity correlations and the solutions of the equations of turbulent fluctuation"
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636:. Reynolds decomposition refers to separation of the flow variable (like velocity
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is turbulent dissipation rate. All RANS models are based on the above equation.
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574:{\displaystyle \left(-\rho {\overline {u_{i}^{\prime }u_{j}^{\prime }}}\right)}
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5931:"Application of a BEM/RANS Interactive Method to Contra-Rotating Propellers"
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owing to the fluctuating velocity field, generally referred to as the
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BenMoshe, Nir; Fattal, Eyal; Leitl, Bernd; Arav, Yehuda (June 2023).
5857:
1428:{\displaystyle {\frac {\partial {\bar {u}}_{i}}{\partial x_{i}}}=0}
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This follows from the mass conservation equation which gives,
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5995:"Using Machine Learning to Predict Wind Flow in Urban Areas"
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3660:{\displaystyle {\overline {u_{i}^{\prime }u_{j}^{\prime }}}}
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1127:{\displaystyle {\frac {\partial u_{i}}{\partial x_{i}}}=0}
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Su, Yiran; Kinnas, Spyros A.; Jukola, Hannu (June 2017).
5875:(14. print. ed.). Cambridge, Mass. : MIT Press.
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5952:; Jukola: Steerprop Ltd. P.O. Box 217, FI-26101
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3968:For this to be a well-defined term, the limit (
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5946:International Symposium on Marine Propulsion
140:) are time-averaged equations of motion for
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69:"Reynolds-averaged Navier–Stokes equations"
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134:Reynolds-averaged Navier–Stokes equations
120:Learn how and when to remove this message
18:Reynolds-averaged Navier-Stokes equations
5238:{\displaystyle {\overline {u_{i}u_{j}}}}
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4646:Time-averaging these equations yields,
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3611:This equation is very complicated. If
2092:On further manipulations this yields,
7:
5871:Tennekes, H.; Lumley, J. L. (1992).
5199:Note that the nonlinear terms (like
172:, these equations can be written in
58:adding citations to reliable sources
2533:is the mean rate of strain tensor.
144:. The idea behind the equations is
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2792:The time evolution equation of
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676:{\displaystyle {\overline {u}}}
609:{\displaystyle {\overline {.}}}
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5978:: CS1 maint: date and year (
5950:University of Texas at Austin
3782:Applications (RANS modelling)
6048:Computational fluid dynamics
5873:A first course in turbulence
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3794:computational fluid dynamics
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3652:
3596:
3477:
3409:
3354:
3316:
3232:
3069:
2984:
2921:
2843:
2788:Equations of Reynolds stress
2762:
2360:
2056:
1603:
668:
624:Derivation of RANS equations
601:
589:. The time-average operator
561:
480:
27:Turbulence modeling approach
4052:such that integration from
3671:is obtained. The last term
969:{\displaystyle u^{\prime }}
703:{\displaystyle u^{\prime }}
6064:
5822:Reynolds, Osborne (1895).
3990:{\displaystyle {\bar {X}}}
3835:{\displaystyle {\bar {X}}}
1038:{\displaystyle {\bar {u}}}
976:is represented instead by
942:{\displaystyle {\bar {u}}}
168:flow of an incompressible
3669:turbulence kinetic energy
5938:www.marinepropulsors.com
4026:chaotic dynamical system
5245:) can be simplified to
3813:The true time average (
630:Navier–Stokes equations
162:Navier–Stokes equations
5849:10.1098/rsta.1895.0004
5784:
5631:
5239:
5193:
4739:
4640:
4217:
4114:
4093:
4073:
4046:
4018:
3991:
3962:
3856:
3836:
3772:
3661:
3605:
2779:
2527:
2374:
2086:
1755:
1429:
1358:
1331:
1128:
1064:
1039:
1010:
990:
970:
943:
914:
894:
848:
754:
704:
677:
650:
634:Reynolds decomposition
610:
575:
497:
146:Reynolds decomposition
6012:10.3390/atmos14060990
5785:
5632:
5240:
5194:
4740:
4641:
4218:
4115:
4094:
4074:
4072:{\displaystyle t_{0}}
4047:
4019:
4017:{\displaystyle t_{0}}
3992:
3963:
3857:
3837:
3773:
3662:
3606:
2780:
2528:
2375:
2087:
1756:
1430:
1359:
1357:{\displaystyle f_{i}}
1332:
1129:
1065:
1040:
1011:
991:
971:
944:
915:
895:
849:
755:
705:
678:
651:
611:
576:
498:
178:Cartesian coordinates
5649:
5249:
5203:
4748:
4650:
4226:
4133:
4104:
4083:
4056:
4036:
4001:
3972:
3866:
3846:
3817:
3675:
3615:
2800:
2540:
2386:
2096:
1769:
1438:
1373:
1341:
1137:
1081:
1049:
1020:
1000:
980:
953:
924:
904:
858:
764:
718:
687:
660:
640:
593:
513:
184:
54:improve this article
5896:P. Y. Chou (1945).
5840:1895RSPTA.186..123R
5755:
5620:
5605:
5537:
5522:
5485:
5467:
5375:
5328:
5145:
4996:
4920:
4867:
4799:
4701:
4596:
4458:
4390:
4337:
4276:
4183:
4024:. In the case of a
3944:
3743:
3704:
3650:
3635:
3576:
3537:
3475:
3460:
3407:
3352:
3314:
3299:
3284:
3207:
3165:
3067:
3052:
2982:
2967:
2919:
2904:
2841:
2826:
2760:
2745:
2358:
2343:
2054:
2039:
1583:
1562:
559:
544:
478:
463:
5780:
5741:
5627:
5606:
5591:
5523:
5508:
5471:
5453:
5361:
5314:
5235:
5189:
5131:
4982:
4906:
4853:
4785:
4735:
4687:
4636:
4582:
4444:
4376:
4323:
4262:
4213:
4169:
4110:
4089:
4069:
4042:
4014:
3987:
3958:
3910:
3899:
3852:
3832:
3768:
3729:
3690:
3657:
3636:
3621:
3601:
3562:
3523:
3461:
3446:
3393:
3338:
3300:
3285:
3270:
3193:
3151:
3053:
3038:
2968:
2953:
2905:
2890:
2827:
2812:
2775:
2746:
2731:
2523:
2370:
2344:
2329:
2082:
2040:
2025:
1751:
1569:
1548:
1425:
1354:
1327:
1124:
1075:Reynolds operators
1073:The properties of
1063:{\displaystyle u'}
1060:
1035:
1006:
986:
966:
939:
910:
890:
844:
750:
700:
673:
646:
606:
571:
545:
530:
493:
464:
449:
6043:Turbulence models
5915:10.1090/qam/11999
5902:Quart. Appl. Math
5882:978-0-262-20019-6
5772:
5730:
5705:
5684:
5625:
5576:
5557:
5542:
5496:
5444:
5422:
5409:
5385:
5355:
5302:
5279:
5233:
5184:
5180:
5119:
5082:
5078:
5041:
5018:
5005:
4970:
4948:
4942:
4900:
4841:
4818:
4814:
4773:
4727:
4723:
4675:
4631:
4570:
4534:
4497:
4475:
4432:
4412:
4364:
4311:
4291:
4256:
4205:
4163:
4113:{\displaystyle T}
4092:{\displaystyle T}
4045:{\displaystyle T}
4030:strange attractor
3984:
3908:
3884:
3878:
3855:{\displaystyle x}
3842:) of a variable (
3829:
3766:
3760:
3721:
3655:
3599:
3593:
3554:
3498:
3480:
3416:
3412:
3361:
3357:
3319:
3260:
3235:
3224:
3182:
3138:
3115:
3090:
3072:
3030:
3005:
2987:
2942:
2924:
2872:
2857:
2846:
2765:
2710:
2675:
2656:
2630:
2606:
2581:
2556:
2516:
2491:
2470:
2445:
2422:
2399:
2363:
2308:
2273:
2254:
2222:
2204:
2179:
2154:
2136:
2118:
2077:
2059:
2011:
1973:
1942:
1923:
1906:
1886:
1871:
1846:
1821:
1806:
1788:
1746:
1708:
1677:
1658:
1641:
1621:
1606:
1600:
1540:
1515:
1490:
1475:
1457:
1417:
1392:
1325:
1265:
1238:
1212:
1165:
1116:
1032:
1009:{\displaystyle u}
989:{\displaystyle u}
936:
913:{\displaystyle U}
799:
738:
712:Reynolds operator
671:
649:{\displaystyle u}
618:Reynolds operator
604:
587:turbulence models
564:
483:
433:
408:
387:
362:
319:
300:
268:
250:
225:
200:
174:Einstein notation
130:
129:
122:
104:
16:(Redirected from
6055:
6017:
6016:
6014:
5990:
5984:
5983:
5977:
5969:
5963:
5961:
5935:
5926:
5920:
5919:
5917:
5893:
5887:
5886:
5868:
5862:
5861:
5851:
5819:
5790:
5789:
5787:
5786:
5781:
5773:
5771:
5770:
5769:
5756:
5754:
5749:
5736:
5731:
5729:
5728:
5727:
5714:
5713:
5712:
5707:
5706:
5698:
5690:
5685:
5683:
5682:
5681:
5668:
5667:
5666:
5653:
5643:
5637:
5636:
5634:
5633:
5628:
5626:
5621:
5619:
5614:
5604:
5599:
5589:
5584:
5583:
5578:
5577:
5569:
5565:
5564:
5559:
5558:
5550:
5543:
5538:
5536:
5531:
5521:
5516:
5504:
5503:
5498:
5497:
5489:
5484:
5479:
5466:
5461:
5452:
5451:
5446:
5445:
5437:
5430:
5429:
5424:
5423:
5415:
5411:
5410:
5405:
5404:
5395:
5391:
5386:
5381:
5380:
5376:
5374:
5369:
5357:
5356:
5351:
5350:
5341:
5333:
5329:
5327:
5322:
5310:
5309:
5304:
5303:
5295:
5285:
5280:
5275:
5274:
5273:
5264:
5263:
5253:
5244:
5242:
5241:
5236:
5234:
5229:
5228:
5227:
5218:
5217:
5207:
5198:
5196:
5195:
5190:
5185:
5179:
5178:
5177:
5165:
5164:
5151:
5150:
5146:
5144:
5139:
5127:
5126:
5121:
5120:
5112:
5103:
5102:
5092:
5091:
5083:
5077:
5076:
5075:
5062:
5061:
5057:
5056:
5055:
5043:
5042:
5034:
5022:
5021:
5019:
5011:
5006:
5001:
4997:
4995:
4990:
4978:
4977:
4972:
4971:
4963:
4954:
4949:
4944:
4943:
4941:
4940:
4939:
4926:
4925:
4921:
4919:
4914:
4902:
4901:
4896:
4895:
4886:
4874:
4872:
4868:
4866:
4861:
4849:
4848:
4843:
4842:
4834:
4824:
4819:
4813:
4805:
4804:
4800:
4798:
4793:
4781:
4780:
4775:
4774:
4766:
4753:
4752:
4744:
4742:
4741:
4736:
4728:
4722:
4721:
4720:
4707:
4706:
4702:
4700:
4695:
4683:
4682:
4677:
4676:
4668:
4655:
4654:
4645:
4643:
4642:
4637:
4632:
4630:
4629:
4628:
4616:
4615:
4602:
4601:
4597:
4595:
4590:
4578:
4577:
4572:
4571:
4563:
4554:
4553:
4543:
4535:
4533:
4532:
4531:
4518:
4517:
4513:
4512:
4511:
4499:
4498:
4490:
4478:
4476:
4468:
4463:
4459:
4457:
4452:
4440:
4439:
4434:
4433:
4425:
4413:
4411:
4410:
4409:
4396:
4395:
4391:
4389:
4384:
4372:
4371:
4366:
4365:
4357:
4344:
4342:
4338:
4336:
4331:
4319:
4318:
4313:
4312:
4304:
4292:
4290:
4282:
4281:
4277:
4275:
4270:
4258:
4257:
4252:
4251:
4242:
4230:
4222:
4220:
4219:
4214:
4206:
4204:
4203:
4202:
4189:
4188:
4184:
4182:
4177:
4165:
4164:
4159:
4158:
4149:
4137:
4127:
4121:
4119:
4117:
4116:
4111:
4098:
4096:
4095:
4090:
4078:
4076:
4075:
4070:
4068:
4067:
4051:
4049:
4048:
4043:
4023:
4021:
4020:
4015:
4013:
4012:
3996:
3994:
3993:
3988:
3986:
3985:
3977:
3967:
3965:
3964:
3959:
3943:
3936:
3935:
3925:
3924:
3923:
3909:
3901:
3898:
3880:
3879:
3871:
3862:) is defined by
3861:
3859:
3858:
3853:
3841:
3839:
3838:
3833:
3831:
3830:
3822:
3811:
3777:
3775:
3774:
3769:
3767:
3762:
3761:
3759:
3758:
3757:
3744:
3742:
3737:
3724:
3722:
3720:
3719:
3718:
3705:
3703:
3698:
3685:
3682:
3666:
3664:
3663:
3658:
3656:
3651:
3649:
3644:
3634:
3629:
3619:
3610:
3608:
3607:
3602:
3600:
3595:
3594:
3592:
3591:
3590:
3577:
3575:
3570:
3557:
3555:
3553:
3552:
3551:
3538:
3536:
3531:
3518:
3515:
3504:
3500:
3499:
3497:
3496:
3495:
3482:
3481:
3476:
3474:
3469:
3459:
3454:
3444:
3438:
3430:
3429:
3417:
3408:
3406:
3401:
3392:
3391:
3381:
3380:
3375:
3374:
3362:
3353:
3351:
3346:
3337:
3336:
3326:
3325:
3320:
3315:
3313:
3308:
3298:
3293:
3283:
3278:
3268:
3261:
3259:
3258:
3257:
3241:
3236:
3231:
3230:
3226:
3225:
3223:
3222:
3221:
3208:
3206:
3201:
3188:
3183:
3181:
3180:
3179:
3166:
3164:
3159:
3146:
3139:
3134:
3133:
3124:
3121:
3116:
3114:
3113:
3112:
3099:
3098:
3097:
3092:
3091:
3083:
3075:
3073:
3068:
3066:
3061:
3051:
3046:
3036:
3031:
3029:
3028:
3027:
3014:
3013:
3012:
3007:
3006:
2998:
2990:
2988:
2983:
2981:
2976:
2966:
2961:
2951:
2943:
2941:
2940:
2939:
2926:
2925:
2920:
2918:
2913:
2903:
2898:
2888:
2882:
2880:
2879:
2874:
2873:
2865:
2858:
2856:
2848:
2847:
2842:
2840:
2835:
2825:
2820:
2810:
2804:
2784:
2782:
2781:
2776:
2771:
2767:
2766:
2761:
2759:
2754:
2744:
2739:
2729:
2721:
2720:
2712:
2711:
2703:
2690:
2689:
2677:
2676:
2668:
2657:
2655:
2654:
2653:
2637:
2632:
2631:
2626:
2625:
2616:
2607:
2605:
2604:
2603:
2590:
2589:
2588:
2583:
2582:
2574:
2566:
2564:
2563:
2558:
2557:
2549:
2532:
2530:
2529:
2524:
2522:
2518:
2517:
2515:
2514:
2513:
2500:
2499:
2498:
2493:
2492:
2484:
2476:
2471:
2469:
2468:
2467:
2454:
2453:
2452:
2447:
2446:
2438:
2430:
2423:
2415:
2410:
2409:
2401:
2400:
2392:
2379:
2377:
2376:
2371:
2369:
2365:
2364:
2359:
2357:
2352:
2342:
2337:
2327:
2319:
2318:
2310:
2309:
2301:
2288:
2287:
2275:
2274:
2266:
2255:
2253:
2252:
2251:
2235:
2230:
2229:
2224:
2223:
2215:
2205:
2203:
2202:
2201:
2188:
2187:
2186:
2181:
2180:
2172:
2164:
2162:
2161:
2156:
2155:
2147:
2137:
2135:
2127:
2126:
2125:
2120:
2119:
2111:
2103:
2091:
2089:
2088:
2083:
2078:
2076:
2075:
2074:
2061:
2060:
2055:
2053:
2048:
2038:
2033:
2023:
2017:
2012:
2010:
2009:
2008:
1996:
1995:
1982:
1981:
1980:
1975:
1974:
1966:
1962:
1961:
1951:
1943:
1941:
1940:
1939:
1926:
1925:
1924:
1916:
1909:
1907:
1899:
1894:
1893:
1888:
1887:
1879:
1872:
1870:
1869:
1868:
1855:
1854:
1853:
1848:
1847:
1839:
1831:
1829:
1828:
1823:
1822:
1814:
1807:
1805:
1797:
1796:
1795:
1790:
1789:
1781:
1773:
1760:
1758:
1757:
1752:
1747:
1745:
1744:
1743:
1731:
1730:
1717:
1716:
1715:
1710:
1709:
1701:
1697:
1696:
1686:
1678:
1676:
1675:
1674:
1661:
1660:
1659:
1651:
1644:
1642:
1634:
1629:
1628:
1623:
1622:
1614:
1607:
1602:
1601:
1599:
1598:
1597:
1584:
1582:
1577:
1564:
1561:
1556:
1546:
1541:
1539:
1538:
1537:
1524:
1523:
1522:
1517:
1516:
1508:
1500:
1498:
1497:
1492:
1491:
1483:
1476:
1474:
1466:
1465:
1464:
1459:
1458:
1450:
1442:
1434:
1432:
1431:
1426:
1418:
1416:
1415:
1414:
1401:
1400:
1399:
1394:
1393:
1385:
1377:
1363:
1361:
1360:
1355:
1353:
1352:
1336:
1334:
1333:
1328:
1326:
1324:
1323:
1322:
1310:
1309:
1296:
1295:
1294:
1285:
1284:
1274:
1266:
1264:
1263:
1262:
1249:
1241:
1239:
1231:
1226:
1225:
1213:
1211:
1210:
1209:
1196:
1195:
1194:
1181:
1179:
1178:
1166:
1164:
1156:
1155:
1154:
1141:
1133:
1131:
1130:
1125:
1117:
1115:
1114:
1113:
1100:
1099:
1098:
1085:
1069:
1067:
1066:
1061:
1059:
1044:
1042:
1041:
1036:
1034:
1033:
1025:
1015:
1013:
1012:
1007:
995:
993:
992:
987:
975:
973:
972:
967:
965:
964:
948:
946:
945:
940:
938:
937:
929:
919:
917:
916:
911:
899:
897:
896:
891:
865:
853:
851:
850:
845:
831:
823:
809:
801:
800:
792:
777:
759:
757:
756:
751:
740:
739:
734:
726:
709:
707:
706:
701:
699:
698:
682:
680:
679:
674:
672:
664:
655:
653:
652:
647:
615:
613:
612:
607:
605:
597:
580:
578:
577:
572:
570:
566:
565:
560:
558:
553:
543:
538:
528:
502:
500:
499:
494:
489:
485:
484:
479:
477:
472:
462:
457:
447:
439:
435:
434:
432:
431:
430:
417:
416:
415:
410:
409:
401:
393:
388:
386:
385:
384:
371:
370:
369:
364:
363:
355:
347:
334:
333:
321:
320:
312:
301:
299:
298:
297:
281:
276:
275:
270:
269:
261:
251:
249:
248:
247:
234:
233:
232:
227:
226:
218:
210:
208:
207:
202:
201:
193:
150:Osborne Reynolds
125:
118:
114:
111:
105:
103:
62:
38:
30:
21:
6063:
6062:
6058:
6057:
6056:
6054:
6053:
6052:
6023:
6022:
6021:
6020:
5992:
5991:
5987:
5970:
5959:
5957:
5933:
5928:
5927:
5923:
5895:
5894:
5890:
5883:
5870:
5869:
5865:
5821:
5820:
5816:
5811:
5803:Favre averaging
5799:
5794:
5793:
5761:
5757:
5737:
5719:
5715:
5695:
5691:
5673:
5669:
5658:
5654:
5647:
5646:
5644:
5640:
5590:
5566:
5547:
5486:
5434:
5412:
5396:
5392:
5342:
5338:
5334:
5292:
5291:
5287:
5286:
5265:
5255:
5254:
5247:
5246:
5219:
5209:
5208:
5201:
5200:
5169:
5156:
5152:
5109:
5108:
5104:
5094:
5093:
5067:
5063:
5047:
5031:
5027:
5023:
4960:
4959:
4955:
4931:
4927:
4887:
4883:
4879:
4875:
4831:
4830:
4826:
4825:
4806:
4763:
4762:
4758:
4754:
4746:
4745:
4712:
4708:
4665:
4664:
4660:
4656:
4648:
4647:
4620:
4607:
4603:
4560:
4559:
4555:
4545:
4544:
4523:
4519:
4503:
4487:
4483:
4479:
4422:
4421:
4417:
4401:
4397:
4354:
4353:
4349:
4345:
4301:
4300:
4296:
4283:
4243:
4239:
4235:
4231:
4224:
4223:
4194:
4190:
4150:
4146:
4142:
4138:
4131:
4130:
4128:
4124:
4102:
4101:
4081:
4080:
4059:
4054:
4053:
4034:
4033:
4004:
3999:
3998:
3970:
3969:
3927:
3915:
3864:
3863:
3844:
3843:
3815:
3814:
3812:
3808:
3803:
3784:
3749:
3745:
3725:
3710:
3706:
3686:
3683:
3673:
3672:
3620:
3613:
3612:
3582:
3578:
3558:
3543:
3539:
3519:
3516:
3487:
3483:
3445:
3439:
3418:
3383:
3382:
3363:
3328:
3327:
3269:
3266:
3262:
3249:
3245:
3213:
3209:
3189:
3171:
3167:
3147:
3144:
3140:
3125:
3122:
3104:
3100:
3080:
3076:
3037:
3019:
3015:
2995:
2991:
2952:
2931:
2927:
2889:
2883:
2862:
2849:
2811:
2805:
2798:
2797:
2794:Reynolds stress
2790:
2730:
2700:
2678:
2662:
2658:
2645:
2641:
2617:
2595:
2591:
2571:
2567:
2546:
2538:
2537:
2505:
2501:
2481:
2477:
2459:
2455:
2435:
2431:
2428:
2424:
2389:
2384:
2383:
2328:
2298:
2276:
2260:
2256:
2243:
2239:
2212:
2193:
2189:
2169:
2165:
2144:
2128:
2108:
2104:
2094:
2093:
2066:
2062:
2024:
2018:
2000:
1987:
1983:
1963:
1953:
1952:
1931:
1927:
1910:
1876:
1860:
1856:
1836:
1832:
1811:
1798:
1778:
1774:
1767:
1766:
1735:
1722:
1718:
1698:
1688:
1687:
1666:
1662:
1645:
1611:
1589:
1585:
1565:
1547:
1529:
1525:
1505:
1501:
1480:
1467:
1447:
1443:
1436:
1435:
1406:
1402:
1382:
1378:
1371:
1370:
1344:
1339:
1338:
1314:
1301:
1297:
1286:
1276:
1275:
1254:
1250:
1242:
1217:
1201:
1197:
1186:
1182:
1170:
1157:
1146:
1142:
1135:
1134:
1105:
1101:
1090:
1086:
1079:
1078:
1052:
1047:
1046:
1018:
1017:
998:
997:
978:
977:
956:
951:
950:
922:
921:
902:
901:
856:
855:
816:
762:
761:
727:
716:
715:
690:
685:
684:
658:
657:
638:
637:
626:
591:
590:
583:Reynolds stress
529:
520:
516:
511:
510:
448:
422:
418:
398:
394:
376:
372:
352:
348:
345:
341:
322:
306:
302:
289:
285:
258:
239:
235:
215:
211:
190:
182:
181:
170:Newtonian fluid
154:turbulent flows
126:
115:
109:
106:
63:
61:
51:
39:
28:
23:
22:
15:
12:
11:
5:
6061:
6059:
6051:
6050:
6045:
6040:
6035:
6033:Fluid dynamics
6025:
6024:
6019:
6018:
5985:
5966:Google Scholar
5954:Rauma, Finland
5921:
5888:
5881:
5863:
5813:
5812:
5810:
5807:
5806:
5805:
5798:
5795:
5792:
5791:
5779:
5776:
5768:
5764:
5760:
5753:
5748:
5744:
5740:
5734:
5726:
5722:
5718:
5711:
5704:
5701:
5694:
5688:
5680:
5676:
5672:
5665:
5661:
5657:
5638:
5624:
5618:
5613:
5609:
5603:
5598:
5594:
5587:
5582:
5575:
5572:
5563:
5556:
5553:
5546:
5541:
5535:
5530:
5526:
5520:
5515:
5511:
5507:
5502:
5495:
5492:
5483:
5478:
5474:
5470:
5465:
5460:
5456:
5450:
5443:
5440:
5433:
5428:
5421:
5418:
5408:
5403:
5399:
5389:
5384:
5379:
5373:
5368:
5364:
5360:
5354:
5349:
5345:
5337:
5332:
5326:
5321:
5317:
5313:
5308:
5301:
5298:
5290:
5283:
5278:
5272:
5268:
5262:
5258:
5232:
5226:
5222:
5216:
5212:
5188:
5183:
5176:
5172:
5168:
5163:
5159:
5155:
5149:
5143:
5138:
5134:
5130:
5125:
5118:
5115:
5107:
5101:
5097:
5089:
5086:
5081:
5074:
5070:
5066:
5060:
5054:
5050:
5046:
5040:
5037:
5030:
5026:
5017:
5014:
5009:
5004:
5000:
4994:
4989:
4985:
4981:
4976:
4969:
4966:
4958:
4952:
4947:
4938:
4934:
4930:
4924:
4918:
4913:
4909:
4905:
4899:
4894:
4890:
4882:
4878:
4871:
4865:
4860:
4856:
4852:
4847:
4840:
4837:
4829:
4822:
4817:
4812:
4809:
4803:
4797:
4792:
4788:
4784:
4779:
4772:
4769:
4761:
4757:
4734:
4731:
4726:
4719:
4715:
4711:
4705:
4699:
4694:
4690:
4686:
4681:
4674:
4671:
4663:
4659:
4635:
4627:
4623:
4619:
4614:
4610:
4606:
4600:
4594:
4589:
4585:
4581:
4576:
4569:
4566:
4558:
4552:
4548:
4541:
4538:
4530:
4526:
4522:
4516:
4510:
4506:
4502:
4496:
4493:
4486:
4482:
4474:
4471:
4466:
4462:
4456:
4451:
4447:
4443:
4438:
4431:
4428:
4420:
4416:
4408:
4404:
4400:
4394:
4388:
4383:
4379:
4375:
4370:
4363:
4360:
4352:
4348:
4341:
4335:
4330:
4326:
4322:
4317:
4310:
4307:
4299:
4295:
4289:
4286:
4280:
4274:
4269:
4265:
4261:
4255:
4250:
4246:
4238:
4234:
4212:
4209:
4201:
4197:
4193:
4187:
4181:
4176:
4172:
4168:
4162:
4157:
4153:
4145:
4141:
4122:
4109:
4088:
4066:
4062:
4041:
4011:
4007:
3983:
3980:
3957:
3954:
3951:
3947:
3942:
3939:
3934:
3930:
3922:
3918:
3913:
3907:
3904:
3897:
3894:
3891:
3887:
3883:
3877:
3874:
3851:
3828:
3825:
3805:
3804:
3802:
3799:
3798:
3797:
3789:
3783:
3780:
3765:
3756:
3752:
3748:
3741:
3736:
3732:
3728:
3717:
3713:
3709:
3702:
3697:
3693:
3689:
3680:
3654:
3648:
3643:
3639:
3633:
3628:
3624:
3598:
3589:
3585:
3581:
3574:
3569:
3565:
3561:
3550:
3546:
3542:
3535:
3530:
3526:
3522:
3513:
3510:
3507:
3503:
3494:
3490:
3486:
3479:
3473:
3468:
3464:
3458:
3453:
3449:
3442:
3436:
3433:
3428:
3425:
3421:
3415:
3411:
3405:
3400:
3396:
3390:
3386:
3378:
3373:
3370:
3366:
3360:
3356:
3350:
3345:
3341:
3335:
3331:
3323:
3318:
3312:
3307:
3303:
3297:
3292:
3288:
3282:
3277:
3273:
3265:
3256:
3252:
3248:
3244:
3239:
3234:
3229:
3220:
3216:
3212:
3205:
3200:
3196:
3192:
3186:
3178:
3174:
3170:
3163:
3158:
3154:
3150:
3143:
3137:
3132:
3128:
3119:
3111:
3107:
3103:
3096:
3089:
3086:
3079:
3071:
3065:
3060:
3056:
3050:
3045:
3041:
3034:
3026:
3022:
3018:
3011:
3004:
3001:
2994:
2986:
2980:
2975:
2971:
2965:
2960:
2956:
2949:
2946:
2938:
2934:
2930:
2923:
2917:
2912:
2908:
2902:
2897:
2893:
2886:
2878:
2871:
2868:
2861:
2855:
2852:
2845:
2839:
2834:
2830:
2824:
2819:
2815:
2808:
2789:
2786:
2774:
2770:
2764:
2758:
2753:
2749:
2743:
2738:
2734:
2727:
2724:
2719:
2716:
2709:
2706:
2699:
2696:
2693:
2688:
2685:
2681:
2674:
2671:
2665:
2661:
2652:
2648:
2644:
2640:
2635:
2629:
2624:
2620:
2613:
2610:
2602:
2598:
2594:
2587:
2580:
2577:
2570:
2562:
2555:
2552:
2545:
2521:
2512:
2508:
2504:
2497:
2490:
2487:
2480:
2474:
2466:
2462:
2458:
2451:
2444:
2441:
2434:
2427:
2421:
2418:
2413:
2408:
2405:
2398:
2395:
2368:
2362:
2356:
2351:
2347:
2341:
2336:
2332:
2325:
2322:
2317:
2314:
2307:
2304:
2297:
2294:
2291:
2286:
2283:
2279:
2272:
2269:
2263:
2259:
2250:
2246:
2242:
2238:
2233:
2228:
2221:
2218:
2211:
2208:
2200:
2196:
2192:
2185:
2178:
2175:
2168:
2160:
2153:
2150:
2143:
2140:
2134:
2131:
2124:
2117:
2114:
2107:
2101:
2081:
2073:
2069:
2065:
2058:
2052:
2047:
2043:
2037:
2032:
2028:
2021:
2015:
2007:
2003:
1999:
1994:
1990:
1986:
1979:
1972:
1969:
1960:
1956:
1949:
1946:
1938:
1934:
1930:
1922:
1919:
1913:
1905:
1902:
1897:
1892:
1885:
1882:
1875:
1867:
1863:
1859:
1852:
1845:
1842:
1835:
1827:
1820:
1817:
1810:
1804:
1801:
1794:
1787:
1784:
1777:
1750:
1742:
1738:
1734:
1729:
1725:
1721:
1714:
1707:
1704:
1695:
1691:
1684:
1681:
1673:
1669:
1665:
1657:
1654:
1648:
1640:
1637:
1632:
1627:
1620:
1617:
1610:
1605:
1596:
1592:
1588:
1581:
1576:
1572:
1568:
1560:
1555:
1551:
1544:
1536:
1532:
1528:
1521:
1514:
1511:
1504:
1496:
1489:
1486:
1479:
1473:
1470:
1463:
1456:
1453:
1446:
1424:
1421:
1413:
1409:
1405:
1398:
1391:
1388:
1381:
1351:
1347:
1321:
1317:
1313:
1308:
1304:
1300:
1293:
1289:
1283:
1279:
1272:
1269:
1261:
1257:
1253:
1248:
1245:
1237:
1234:
1229:
1224:
1220:
1216:
1208:
1204:
1200:
1193:
1189:
1185:
1177:
1173:
1169:
1163:
1160:
1153:
1149:
1145:
1123:
1120:
1112:
1108:
1104:
1097:
1093:
1089:
1058:
1055:
1031:
1028:
1005:
985:
963:
959:
935:
932:
909:
889:
886:
883:
880:
877:
874:
871:
868:
864:
843:
840:
837:
834:
830:
826:
822:
819:
815:
812:
808:
804:
798:
795:
789:
786:
783:
780:
776:
772:
769:
749:
746:
743:
737:
733:
730:
723:
697:
693:
670:
667:
645:
625:
622:
603:
600:
569:
563:
557:
552:
548:
542:
537:
533:
526:
523:
519:
492:
488:
482:
476:
471:
467:
461:
456:
452:
445:
442:
438:
429:
425:
421:
414:
407:
404:
397:
391:
383:
379:
375:
368:
361:
358:
351:
344:
340:
337:
332:
329:
325:
318:
315:
309:
305:
296:
292:
288:
284:
279:
274:
267:
264:
257:
254:
246:
242:
238:
231:
224:
221:
214:
206:
199:
196:
189:
138:RANS equations
128:
127:
42:
40:
33:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
6060:
6049:
6046:
6044:
6041:
6039:
6036:
6034:
6031:
6030:
6028:
6013:
6008:
6004:
6000:
5996:
5989:
5986:
5981:
5975:
5967:
5955:
5951:
5947:
5943:
5939:
5932:
5925:
5922:
5916:
5911:
5907:
5903:
5899:
5892:
5889:
5884:
5878:
5874:
5867:
5864:
5859:
5855:
5850:
5845:
5841:
5837:
5833:
5829:
5825:
5818:
5815:
5808:
5804:
5801:
5800:
5796:
5777:
5774:
5766:
5762:
5746:
5742:
5732:
5724:
5720:
5709:
5699:
5686:
5678:
5674:
5663:
5659:
5642:
5639:
5611:
5607:
5596:
5592:
5585:
5580:
5570:
5561:
5551:
5544:
5528:
5524:
5513:
5509:
5505:
5500:
5490:
5476:
5472:
5468:
5458:
5454:
5448:
5438:
5431:
5426:
5416:
5401:
5397:
5387:
5377:
5366:
5362:
5358:
5347:
5343:
5335:
5330:
5319:
5315:
5311:
5306:
5296:
5288:
5281:
5270:
5266:
5260:
5256:
5224:
5220:
5214:
5210:
5186:
5174:
5170:
5161:
5157:
5147:
5136:
5132:
5128:
5123:
5113:
5105:
5099:
5087:
5084:
5072:
5068:
5058:
5048:
5044:
5035:
5028:
5015:
5012:
5007:
4998:
4987:
4983:
4979:
4974:
4964:
4956:
4950:
4936:
4932:
4922:
4911:
4907:
4903:
4892:
4888:
4880:
4869:
4858:
4854:
4850:
4845:
4835:
4827:
4820:
4810:
4801:
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2795:
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623:
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521:
517:
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469:
465:
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450:
443:
440:
436:
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423:
412:
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389:
381:
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338:
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327:
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294:
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272:
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204:
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187:
179:
175:
171:
167:
163:
159:
155:
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143:
139:
135:
124:
121:
113:
110:November 2010
102:
99:
95:
92:
88:
85:
81:
78:
74:
71: –
70:
66:
65:Find sources:
59:
55:
49:
48:
43:This article
41:
37:
32:
31:
19:
6002:
5998:
5988:
5964:– via
5958:. Retrieved
5956:). p. 1
5937:
5924:
5905:
5901:
5891:
5872:
5866:
5831:
5827:
5817:
5641:
4125:
3809:
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1366:
1072:
627:
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137:
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116:
107:
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90:
83:
76:
64:
52:Please help
47:verification
44:
5944:, Finland:
5834:: 123–164.
3667:is traced,
920:instead of
6038:Turbulence
6027:Categories
6005:(6): 990.
5999:Atmosphere
5809:References
166:stationary
158:turbulence
142:fluid flow
80:newspapers
5908:: 38–54.
5759:∂
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5974:cite web
5797:See also
1057:′
821:′
760:. Thus,
732:′
164:. For a
5836:Bibcode
2382:where,
632:is the
94:scholar
5960:2 July
5879:
5856:
1337:where
1045:, and
854:where
96:
89:
82:
75:
67:
5942:Espoo
5934:(PDF)
5858:90643
5854:JSTOR
3801:Notes
616:is a
101:JSTOR
87:books
5980:link
5962:2021
5877:ISBN
180:as:
132:The
73:news
6007:doi
5910:doi
5844:doi
5832:186
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56:by
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1472:t
1462:i
1452:u
1423:0
1420:=
1412:i
1408:x
1397:i
1387:u
1350:i
1346:f
1320:j
1316:x
1307:j
1303:x
1292:i
1288:u
1282:2
1268:+
1260:i
1256:x
1247:p
1233:1
1223:i
1219:f
1215:=
1207:j
1203:x
1192:i
1188:u
1176:j
1172:u
1168:+
1162:t
1152:i
1148:u
1122:0
1119:=
1111:i
1107:x
1096:i
1092:u
1054:u
1027:u
1004:u
984:u
958:u
931:u
908:U
888:)
885:z
882:,
879:y
876:,
873:x
870:(
867:=
863:x
842:,
839:)
836:t
833:,
829:x
825:(
818:u
814:+
811:)
807:x
803:(
794:u
788:=
785:)
782:t
779:,
775:x
771:(
768:u
748:)
745:0
742:=
729:u
722:(
692:u
666:u
644:u
599:.
568:)
551:j
547:u
536:i
532:u
518:(
491:.
487:]
470:j
466:u
455:i
451:u
437:)
428:i
424:x
413:j
403:u
390:+
382:j
378:x
367:i
357:u
343:(
336:+
331:j
328:i
314:p
304:[
295:j
291:x
278:+
273:i
263:f
253:=
245:j
241:x
230:i
220:u
205:j
195:u
136:(
123:)
117:(
112:)
108:(
98:·
91:·
84:·
77:·
50:.
20:)
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