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Research Papers: Flows in Complex Systems

# A Reduced-Order Model of the Mean Properties of a Turbulent Wall Boundary Layer at a Zero Pressure Gradient

[+] Author and Article Information
Lei Xu, Zvi Rusak

Department of Mechanical,
Aerospace and Nuclear Engineering,
Rensselaer Polytechnic Institute,
Troy, NY 12180

Luciano Castillo

Department of Mechanical Engineering,
National Wind Resource Center,
Texas Tech University,
Lubbock, TX 79409

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received October 12, 2013; final manuscript received December 13, 2013; published online January 27, 2014. Assoc. Editor: Mark F. Tachie.

J. Fluids Eng 136(3), 031103 (Jan 27, 2014) (16 pages) Paper No: FE-13-1609; doi: 10.1115/1.4026418 History: Received October 12, 2013; Revised December 13, 2013

## Abstract

A novel two-equations model for computing the flow properties of a spatially-developing, incompressible, zero-pressure-gradient, turbulent boundary layer over a smooth, flat wall is developed. The mean streamwise velocity component inside the boundary layer is described by the Reynolds-averaged Navier–Stokes equation where the Reynolds shear stress is given by an extended mixing-length model. The nondimensional form of the mixing length is described by a polynomial function in terms of the nondimensional wall normal coordinate. Moreover, a stream function approach is applied with a leading-order term described by a similarity function. Two ordinary differential equations are derived for the solution of the similarity function along the wall normal coordinate and for its streamwise location. A numerical integration scheme of the model equations is developed and enables the solution of flow properties. The coefficients of the mixing-length polynomial function are modified at each streamwise location as part of solution iterations to satisfy the wall and far-field boundary conditions and adjust the local boundary layer thickness, $δ99.4$, to a location where streamwise speed is 99.4% of the far-field streamwise velocity. The elegance of the present approach is established through the successful solution of the various flow properties across the boundary layer (i.e., mean streamwise velocity, viscous stress, Reynolds shear stress, skin friction coefficient, and growth rate of boundary layer among others) from the laminar regime all the way to the fully turbulent regime. It is found that results agree with much available experimental data and direct numerical simulations for a wide range of $Reθ$ based on the momentum thickness ($Reθ$) from $15$ up to $106$, except for the transition region from laminar to turbulent flow. Furthermore, results shed light on the von Kármán constant as a function of $Reθ$, the possible four-layer nature of the mean streamwise velocity profile, the universal profiles of the streamwise velocity and the Reynolds shear stress at high $Reθ$, and the scaling laws at the outer region.

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## Figures

Fig. 1

Mean velocity profile, Reθ = 1928

Fig. 2

Scaled viscous stress and scaled Reynolds shear stress versus y+, Reθ = 1928

Fig. 15

Accumulated skin friction coefficient: computation (solid line), empirical line according to the formula in Eq. (3) (dashed line), and Blasius analytical laminar boundary layer line (dashed-dotted line)

Fig. 3

Mean velocity profile, Reθ≃2900

Fig. 4

Scaled viscous stress and scaled Reynolds shear stress versus y+, Reθ≃2900

Fig. 5

Mean velocity profile, Reθ = 5024

Fig. 6

Scaled viscous stress and scaled Reynolds shear stress versus y+, Reθ = 5024

Fig. 7

Mean velocity profile, Reθ = 6662

Fig. 8

Scaled viscous stress and scaled Reynolds shear stress versus y+, Reθ = 6662

Fig. 9

Mean velocity profile, Reθ = 23,119

Fig. 10

Scaled viscous stress and scaled Reynolds shear stress versus y+, Reθ = 23,119

Fig. 11

Mean velocity profile, Reθ = 31,000

Fig. 12

Scaled viscous stress and scaled Reynolds shear stress versus y+, Reθ = 31,000

Fig. 13

Reθ versus Rex

Fig. 14

Local skin friction coefficient: computation (solid line), Eq. (44) from George [38] (dashed line), experimental measurements of Österlund [7] (diamonds), De Graaff and Eaton [25] (circles), and Blasius analytical laminar boundary layer line (dashed-dotted line)

Fig. 16

Reynolds number based on boundary layer thickness, present computation (solid line), Prandtl's empirical relationship in Eq. (46) (dashed-dotted line), Blasius laminar flow solution in Eq. (45) (dashed line)

Fig. 17

The computed value of von Kármán constant as a function of Reθ

Fig. 21

Four-layer-region of a flat plate boundary layer at Reθ = 7.13 × 104

Fig. 18

Ratio of the viscous stress gradient and the Reynolds stress gradient with various Reynolds numbers

Fig. 19

Mean velocity profiles in wall coordinates for Reθ = 53  to 5.79 ×105, and universal profile according to solution of Eq. (51)

Fig. 20

Reynolds shear stress profiles in wall coordinates, Reθ = 2,391 - 5.79 × 105, and universal profile according to the formula in Eq. (52)

Fig. 23

Zagarola and Smits scaling law, computed (solid line), Österlund's experiment at Reθ = 2532 (plus), Österlund's experiment at Reθ = 5156 (circle), Österlund's experiment at Reθ = 10,161 (triangle), Österlund's experiment at Reθ = 25,779 (diamond)

Fig. 24

George and Castillo scaling law, computed (solid line), Österlund's experiment at Reθ = 2532 (plus), Österlund's experiment at Reθ = 5156 (circle), Österlund's experiment at Reθ = 10,161 (triangle), Österlund's experiment at Reθ = 25,779 (diamond)

Fig. 22

von Kármán scaling law, computed (solid line), Österlund's experiment at Reθ = 2532 (plus), Österlund's experiment at Reθ = 5156 (circle), Österlund's experiment at Reθ = 10,161 (triangle), Österlund's experiment at Reθ = 25,779 (diamond)

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