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Fundamental Issues and Canonical Flows

Spatial Resolution of the Axisymmetric Turbulent Statistics Along Thin Circular Cylinders at High Transverse Curvatures and Low-Re

[+] Author and Article Information
Stephen A. Jordan

 Naval Undersea Warfare Center, Newport, RI 02841stephen.jordan@navy.mil

J. Fluids Eng 134(9), 091206 (Aug 22, 2012) (9 pages) doi:10.1115/1.4007269 History: Received April 13, 2012; Revised July 18, 2012; Published August 22, 2012; Online August 22, 2012

After three decades of accumulated experimental and numerical results, a comprehensive understanding of the spatial evolution of axisymmetric turbulent boundary layers (ATBL) along long thin cylinders still eludes both scientists and engineers. While experimentalists dealt with axial alignment complexities, computationalists lacked proper inflow boundary conditions. Herein, we correct this latter deficiency and initiate an investigation of the thin cylinder turbulence under low Reynolds numbers and high transverse curvatures (boundary layer thicknesses to radius). Using the large-eddy simulation (LES) methodology, we are particularly interested in the radial propagation of the transverse curvature on the ATBL statistics. A ten-simulation matrix was constructed to examine these effects with validation against the experimental evidence. These tests investigated the ATBL maturity up to transverse curvatures approaching 2 orders of magnitude. A recently developed turbulent inflow procedure for the thin cylinder was implemented that couples a dynamic form of Spalding’s expression for rescaling the mean streamwise velocity with recycling of all superimposed turbulent fluctuations. The technique specifically circumvents intensive computations from the cylinder leading edge, and the rescaling-recycling combination minimizes the inflow turbulent regeneration length under very high transverse curvatures. After the initial transition phase in each LES computation, the respective numerical uncertainty was quantified to ensure sufficient spatial resolution within the discretized domain for resolving the energy-bearing scales of the turbulent motion. For the present low-Re conditions, the strength of the log layer steadily diminishes under continuous rise in the transverse curvature whereas the scaled fluctuating intensities elevate (except for the dominate shear stresses) with no sign towards full maturity. Each simulation reveals a boundary layer thickness that grows downstream by a factor of 7 relative to the momentum thickness with a linear influence of the transverse curvature on the wall-shear stress coefficient.

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Copyright © 2012 by American Society of Mechanical Engineers
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Figures

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Figure 1

Sketch of inflow production for turbulent boundary layer growth along long thin cylinders; L = xmax (cylinder length)

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Figure 2

Downstream distribution of inflow parameters κ, κβ, α for first test case (Table 1) Rea  = 586, Reθ  = 778, a+  = 37, and δi /a = 11

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Figure 3

Energy spectra and uncertainty estimates of LES test case Rea  = 586, δi /a = 11, and Reθ  = 778; (a) averages along azimuthal direction at Y+ ≈ 56 and (b) error estimates using cell Jacobians (in wall units) at adjacent cell to cylinder periphery (J1+) and interior cell at Y+ ≈ 56 (J56+); (a) streamwise energy spectra, (b) numerical uncertainty

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Figure 4

Resolved axial mean velocity profiles (in wall units) and scaled streamwise normal Reynolds stress for test case Rea  = 586, Reθ  = 778, a+  = 37, and δi /a = 11; (a) axial mean velocity, (b) streamwise normal Reynolds stress

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Figure 5

Axial mean velocity profiles and scaled streamwise normal Reynolds stress for test cases Rea  = 586, 778 ≤ Reθ  ≤ 5528, 37 ≤ a+  ≤ 39, and 11 ≤ δ/a ≤ 73; (a) axial mean velocity, (b) streamwise normal Reynolds stress

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Figure 10

Transverse curvature effects on the tangential wall-friction coefficient including comparisons to the experimental measurements [(5),6,11] and DNS/LES computations using periodic streamwise boundary conditions [17-19]; the power curve (solid line shown in (b)) was fitted to previously published experiment data (5 ≤ δ/a ≤ 8) and DNS/LES computations (10 ≤ δ/a ≤ 49)

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Figure 9

Spatial evolution of the axisymmetric turbulent boundary layer along the thin cylinder relative to the flat plate and the temporal results

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Figure 8

Velocity defect profiles (inner scaling, uτ ) of the present LES computations (lines) and the experimental results of Willmarth [5] (symbols)

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Figure 7

Composites of the resolved shear (−vx vr ) and normal azimuthal (vφ ) Reynolds stresses for 586 ≤ Rea  ≤ 620, 788 ≤ Reθ  ≤ 5528, 37 ≤ a+  ≤ 39, and 11 ≤ δi /a ≤ 73; (a) normal azimuthal Reynolds stress, (b) shear Reynolds stress

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Figure 6

Comparisons between selected experimental and resolved axial mean velocities and streamwise normal Reynolds stresses for 620 ≤ Rea  ≤ 785, 2903 ≤ Reθ  ≤ 4350, 38 ≤ a+  ≤ 144, and 7.2 ≤ δi /a ≤ 38; open symbols denote laboratory measurements and lined profiles depict present LES computations; (a) axial mean velocity, (b) normal Reynolds stress

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