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Research Papers: Fundamental Issues and Canonical Flows

# Turbulent Boundary Layer With Negligible Wall Stress

[+] Author and Article Information
Noor Afzal

Faculty of Engineering & Technology, Aligarh Muslim University, Aligarh 202002, India

J. Fluids Eng 130(5), 051205 (May 07, 2008) (15 pages) doi:10.1115/1.2903754 History: Received May 05, 2006; Revised August 18, 2007; Published May 07, 2008

## Abstract

The turbulent boundary layer subjected to strong adverse pressure gradient near the separation region has been analyzed at large Reynolds numbers by the method of matched asymptotic expansions. The two regions consisting of outer nonlinear wake layer and inner wall layer are analyzed in terms of pressure scaling velocities $Up=(νp′∕ρ)1∕3$ in the wall region and $Uδ=(δp′∕ρ)1∕2$ in the outer wake region, where $p′$ is the streamwise pressure gradient and $ρ$ is the fluid density. In this work, the variables $δ$, the outer boundary layer thickness, and $Uδ$, the outer velocity scale, are independent of $ν$, the molecular kinematic viscosity, which is a better model of fully developed mean turbulent flow. The asymptotic expansions have been matched by Izakson–Millikan–Kolmogorov hypothesis leading to open functional equations. The solution for the velocity distribution gives new composite log-half-power laws, based on the pressure scales, providing a better model of the flow, where the outer composite log-half-power law does not depend on the molecular kinematic viscosity. These new composite laws are better and one may be benefited from their limiting relations that for weak pressure gradient yield the traditional logarithmic laws and for strong adverse pressure gradient yield the half-power laws. During matching of the nonlinear outer layer two cases arise: One where $Uδ∕Ue$ is small and second where $Uδ∕Ue$ of order unity (where $Ue$ is the velocity at the edge of the boundary layer). In the first case, the lowest order nonlinear outer flow under certain conditions shows equilibrium. The outer flow subjected to the constant eddy viscosity closure model is governed by the Falkner–Skan equation subjected to the matching condition of finite slip velocity on the surface. The jet- and wakelike solutions are presented, where the zero velocity slip implying the point of separation, which compares well with Coles traditional wake function. In the second case, higher order terms in the asymptotic solutions for nearly separating flow have been estimated. The proposed composite log-half-power law solution and the limiting half-power law have been well supported by extensive experimental and direct numerical simulation data. For moderate values of the pressure gradient the data show that the proposed composite log-half-power laws are a better model of the flow.

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## Figures

Figure 1

Velocity profile data of Skare and Krogstad (14) for adverse pressure gradient in terms of pressure velocity scaling and comparison with half-power law: (a) inner wall layer variables and (b) outer layer variables

Figure 2

Velocity profile data of Dengel and Fernholz (9) for adverse pressure gradient in terms of pressure velocity scaling and comparison with half-power law: (a) inner wall layer variables and (b) outer layer variables

Figure 3

Velocity profile data of Angele and Klingmann (11) for adverse pressure gradient in terms of pressure velocity scaling and comparison with half-power law: (a) inner wall layer variables and (b) outer layer variables

Figure 4

Comparison of the velocity profile from the DNS data of Na and Moin (33) under strong adverse pressure gradient at X=150, 155, 157, 158; --- half-power law (26); — DNS data (after Skote and Henningson (13)); here, X=x∕δ0*

Figure 5

Comparison of the velocity profile in the flow reattachment region, X=412 and X=500, from the DNS data of Skote and Henningson (32) under adverse pressure gradient with the half-power law 26

Figure 6

Comparison of the velocity profile at X=300 from the APG3 DNS data of Skote and Henningson (30) under adverse pressure gradient with the present composite laws based on pressure the velocity scaling with axial coordinate yp in terms of (a) logarithmic scale and (b) half-power scale

Figure 7

Slope of half-power law under adverse pressure gradient data: KY: Kader and Yaglom (4), Af: Afzal (3), and T: A=2∕k, k=0.41

Figure 8

Wall layer intercept of half-power law under adverse pressure gradient; — present proposal: kC=ΛRp[ln(ΛRp)3∕2−C1] for C1=0

Figure 9

Outer wake layer intercept of half-power law under adverse pressure gradient; — present proposal: kE=Λ[−lnΛ−E1] for E1=1.5

Figure 10

Wake- and jetlike velocity profile solutions from the lowest order outer nonlinear wake layer with Clauser eddy viscosity model from Falkner–Skan equations 47,48,48,48

Figure 11

Comparison of Coles wake function with the solution of the velocity profile from the lowest order outer nonlinear wake layer equations with constant eddy viscosity model for zero velocity the slip Cf=bS=0, predicting m=−0.165858

Figure 12

The skin friction parameter Cf∕αC and wall slip velocity parameter bS=US∕Ue from the solution of the lowest order outer nonlinear wake layer equations from the constant eddy viscosity model

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