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Fundamental Issues and Canonical Flows

An Inflow Method for Axisymmetric Turbulent Boundary Layers Along Very Long Slender Cylinders

[+] Author and Article Information
Stephen A. Jordan

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

J. Fluids Eng 134(5), 051202 (May 07, 2012) (11 pages) doi:10.1115/1.4006512 History: Received November 02, 2011; Revised March 22, 2012; Published May 03, 2012; Online May 07, 2012

Generating acceptable inflow conditions for the turbulent boundary layer (TBL) growth along long thin cylinders is a challenging task. Previous production methods such as rescale/recycling, artificial turbulence, and antecedent databases are difficult to implement because the downstream physics do not conform to consistent scaling laws. An alternate inflow approach that involves only recycling the fluctuating elements coupled with a dynamic form of Spalding’s relationship for assigning the mean quantities shows promise for spatially resolving the axisymmetric turbulence along the thin cylinder. Applying this inflow technique for resolving the turbulent scales along a flat plate at a tested momentum-based Reynolds number of Reθ  = 670 showed excellent agreement with the experimental data as well as the analytical results from the momentum-integral method. A minor adjustment length of approximately two inflow TBL thicknesses was necessary to attain consistent streamwise growth of the boundary layer as well as a simultaneous reduction of the skin friction. Unlike the flat plate, implementing the inflow technique for the thin cylinder required a feedback mechanism during the early transition phase to capture the downstream realistic turbulence. This initial process invoked downstream evaluation of the three parameters that comprise Spalding’s relationship that were periodically fed upstream to the inflow boundary. The validation test case (Reθ  = 620) showed excellent agreement with the experimental measurements in terms of the radial profiles (in cylinder wall units) of the streamwise mean and the normal Reynolds stress. Both the adjustment and turbulence de-correlation axial lengths were under two boundary layer thicknesses from the inlet boundary. Given a useful inflow technique for the thin cylinder permits much needed numerical investigations to complement the present scarcity in the experimental evidence and address numerous unknown characteristics of the TBL spatial growth.

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Copyright © 2012 by American Society of Mechanical Engineers
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Figures

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Figure 1

Inflow production technique for turbulent boundary layer growth along the slender cylinder

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Figure 2

Temporal (transient phase) and final spatial evaluation of the dynamic parameters in Spalding’s modified formula for the flat plate test case; no wake α ≡ 0. (a) temporal, (b) spatial.

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Figure 3

Predicted spatial growth and friction coefficient reduction for the flat plate test case: (a) TBL growth, (b) friction coefficient

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Figure 4

Predicted streamwise mean velocity profiles (in wall units) and associated scaled normal Reynolds stress for the flat plate test case: (a) streamwise mean, (b) normal Reynolds stress

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Figure 5

Energy spectra and uncertainty estimates of LES test case Rea = 586, δi /a = 16 and Reθ = 933; (a) averages along azimuthal direction at Y+ ≈ 50 and (b) error estimates using cell Jacobians (in wall units) at adjacent cell to cylinder periphery (J+ 1 ) and interior cell at Y+ ≈ 50 (J+ 50 ). (a) Streamwise energy spectra, (b) numerical uncertainty.

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Figure 6

Initial inlet boundary profile and downstream radial profiles of streamwise mean velocity (in thin cylinder wall units) for test case Rea = 586 and δi /a = 12. (a) Streamwise inflow, (b) downstream profiles.

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Figure 7

Spatial distribution of inflow parameters with L2 error norm and streamwise mean velocity profiles using wake function for inflow conditions; test case Rea = 586 and δi /a = 12. (a) Inflow parameters, (b) Streamwise mean velocity.

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Figure 8

Superposition of the boundary layer growth of an auxiliary test case (δi /a = 14.3, δf /a = 16.0) onto an extension of the first test case (δi /a = 12.0, δf /a = 15.8)

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Figure 9

Spatial distribution of inflow parameters (second test case) and streamwise mean velocity profiles for test cases δi /a = 12, 16 (Rea = 586). (a) Inflow parameters, (b) streamwise mean velocity.

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Figure 10

Spatial distribution of inflow parameters and streamwise two-point correlations of the axial fluctuations; validation test case Rea = 620, δi /a = 27. (a) Inflow parameters, (b) two-point correlations.

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Figure 11

Streamwise mean velocity and normal Reynolds stress (u′)2 profiles with comparison to the experimental data for test case Rea  = 620. (a) Streamwise mean velocity, (b) Reynolds stress.

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Figure 12

Comparisons of defect velocity (in wall units) between the LES results (test case Rea  = 620) and the experimental data at equivalent transverse ratios (δ/a = 27)

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Figure 13

Turbulent boundary growth of three test cases in comparison to the flat plate, periodic boundary conditions, and relevant experimental data

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