# zbMATH — the first resource for mathematics

On the logarithmic mean profile. (English) Zbl 1183.76766
Summary: Elements of the first-principles-based theory of T. Wei et al. [J. Fluid Mech. 522, 303–327 (2005; Zbl 1065.76106)], P. Fife et al. [Multiscale Model. Simul. 4, No. 3, 936–959 (2005; Zbl 1108.76030); J. Fluid Mech. 532, 165–189 (2005; Zbl 1086.76022); J. Fluid Mech. 573, 371–398 (2007; Zbl 1133.76317); Discrete Contin. Dyn. Syst. 24, No. 3, 781–807 (2009; Zbl 1165.76020)] are clarified and their veracity tested relative to the properties of the logarithmic mean velocity profile. While the approach employed broadly reveals the mathematical structure admitted by the time averaged Navier-Stokes equations, results are primarily provided for fully developed pressure driven flow in a two-dimensional channel. The theory demonstrates that the appropriately simplified mean differential statement of Newton’s second law formally admits a hierarchy of scaling layers, each having a distinct characteristic length. The theory also specifies that these characteristic lengths asymptotically scale with distance from the wall over a well-defined range of wall-normal positions, $$y$$. Numerical simulation data are shown to support these analytical findings in every measure explored. The mean velocity profile is shown to exhibit logarithmic dependence (exact or approximate) when the solution to the mean equation of motion exhibits (exact or approximate) self-similarity from layer to layer within the hierarchy. The condition of pure self-similarity corresponds to a constant leading coefficient in the logarithmic mean velocity equation. The theory predicts and clarifies why logarithmic behaviour is better approximated as the Reynolds number gets large. An exact equation for the leading coefficient (von Kármán coefficient $$\kappa$$) is tested against direct numerical simulation (DNS) data. Two methods for precisely estimating the leading coefficient over any selected range of y are presented. These methods reveal that the differences between the theory and simulation are essentially within the uncertainty level of the simulation. The von Kármán coefficient physically exists owing to an approximate self-similarity in the flux of turbulent force across an internal layer hierarchy. Mathematically, this self-similarity relates to the slope and curvature of the Reynolds stress profile, or equivalently the slope and curvature of the mean vorticity profile. The theory addresses how, why and under what conditions logarithmic dependence is approximated relative to the specific mechanisms contained within the mean statement of dynamics.

##### MSC:
 76F40 Turbulent boundary layers
Full Text:
##### References:
 [1] DOI: 10.2514/1.15617 · doi:10.2514/1.15617 [2] DOI: 10.1017/S0022112009006624 · Zbl 1181.76084 · doi:10.1017/S0022112009006624 [3] DOI: 10.1017/S0022112005003988 · Zbl 1086.76022 · doi:10.1017/S0022112005003988 [4] DOI: 10.1017/S0022112004001958 · Zbl 1065.76106 · doi:10.1017/S0022112004001958 [5] DOI: 10.3934/dcds.2009.24.781 · Zbl 1165.76020 · doi:10.3934/dcds.2009.24.781 [6] DOI: 10.1137/040611173 · Zbl 1108.76030 · doi:10.1137/040611173 [7] DOI: 10.1017/S0022112006003958 · Zbl 1133.76317 · doi:10.1017/S0022112006003958 [8] DOI: 10.1063/1.3013635 · Zbl 1182.76237 · doi:10.1063/1.3013635 [9] Townsend, The Structure of Turbulent Shear Flow (1976) · Zbl 0325.76063 [10] Cantwell, Introduction to Symmetry Analysis (2002) [11] Tennekes, First Course in Turbulence (1972) [12] DOI: 10.1016/j.paerosci.2007.01.001 · doi:10.1016/j.paerosci.2007.01.001 [13] DOI: 10.1063/1.3005858 · Zbl 1182.76714 · doi:10.1063/1.3005858 [14] Barenblatt, Scaling, Self-Similarity and Intermediate Asymptotics (1996) · Zbl 0907.76002 · doi:10.1017/CBO9781107050242 [15] Schlichting, Boundary Layer Theory (2000) · Zbl 0940.76003 · doi:10.1007/978-3-642-85829-1 [16] DOI: 10.1007/BF00536208 · Zbl 0509.76060 · doi:10.1007/BF00536208 [17] Pope, Turbulent Flows (2000) · Zbl 0966.76002 · doi:10.1017/CBO9780511840531 [18] DOI: 10.1017/S0022112000001580 · Zbl 0959.76503 · doi:10.1017/S0022112000001580 [19] DOI: 10.1017/S0022112095003351 · Zbl 0849.76030 · doi:10.1017/S0022112095003351 [20] DOI: 10.1017/S0022112082001311 · Zbl 0517.76057 · doi:10.1017/S0022112082001311 [21] DOI: 10.1115/1.1840903 · doi:10.1115/1.1840903 [22] DOI: 10.1063/1.870250 · Zbl 1149.76503 · doi:10.1063/1.870250 [23] DOI: 10.1017/S0022112000002408 · Zbl 1007.76067 · doi:10.1017/S0022112000002408 [24] DOI: 10.1063/1.3006423 · Zbl 1182.76550 · doi:10.1063/1.3006423 [25] DOI: 10.1063/1.869966 · Zbl 1147.76463 · doi:10.1063/1.869966 [26] DOI: 10.1063/1.2972935 · Zbl 1182.76530 · doi:10.1063/1.2972935 [27] Millikan, Proceedings of Fifth International Congress of Applied Mechanics pp 386– (1939) [28] DOI: 10.1017/S0022112008003637 · Zbl 1175.76020 · doi:10.1017/S0022112008003637 [29] DOI: 10.1063/1.1343480 · Zbl 1184.76351 · doi:10.1063/1.1343480 [30] DOI: 10.1098/rsta.2006.1944 · Zbl 1152.76407 · doi:10.1098/rsta.2006.1944 [31] DOI: 10.2514/1.18911 · doi:10.2514/1.18911 [32] Kawamura, Turbulence Heat and Mass Transfer 3 (Proceedings of the Third Intl Symp. on Turbulence Heat and Mass Transfer pp 15– (2000) [33] Izakson, Tech. Phys. USSR IV 2 pp 155– (1937) [34] DOI: 10.1063/1.2162185 · doi:10.1063/1.2162185 [35] Hansen, Similarity Analyses of Boundary Value Problems in Engineering (1964) · Zbl 0137.22603 [36] DOI: 10.1017/S0022112008002760 · Zbl 1147.76015 · doi:10.1017/S0022112008002760 [37] DOI: 10.1115/1.3101858 · doi:10.1115/1.3101858 [38] DOI: 10.1017/S0022112002003270 · Zbl 1032.76500 · doi:10.1017/S0022112002003270 [39] DOI: 10.1017/S0022112098002419 · Zbl 0941.76510 · doi:10.1017/S0022112098002419 [40] DOI: 10.1016/j.ijheatmasstransfer.2005.07.035 · Zbl 1188.76215 · doi:10.1016/j.ijheatmasstransfer.2005.07.035
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.