# Measurements of fully developed turbulent flow in a trapezoidal duct

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## Abstract

The turbulence characteristics of fully developed isothermal air flows through a symmetric trapezoidal duct were examined experimentally using Pitot tube and hot-wire anemometry over a Reynolds number range of 3.7–11.6×10^{4}. The measurements included local wall shear stress and the cross-sectional distributions of mean axial velocity, secondary velocities and Reynolds stresses. Four secondary flow cells were detected in a symmetric half of the duct. Although secondary velocity components were typically less than about 1% of the bulk axial velocity, their effect was especially pronounced on the distributions of turbulent kinetic energy and local wall shear stress.

## Keywords

Reynolds Number Turbulent Kinetic Energy Axial Velocity Reynolds Stress Flow Cell## List of symbols

*a, b, c, d*trapezoidal duct dimensions (Fig. 1)

*A, B*coefficients in log law (Table 1)

*D*_{h}equivalent hydraulic diameter

*f*Darcy friction factor, (

*2D*_{ h }/*ϱU*_{ b }^{2}) (*dP/dx*)*k*turbulent kinetic energy per unit mass, \(\tfrac{1}{2}(\overline {u^2 } + \overline {v^2 } + \overline {w^2 } )\)

*k*^{+}dimensionless turbulent kinetic energy,

*k/(ū*^{*})^{2}*P*static pressure

*Re*Reynolds number,

*ϱU*_{ b }*D*_{ h }/*μ**s*distance along inclined wall, measured from top corner (Fig. 1)

*u, v, w*fluctuating components of the velocities in the

*x, y, z*directions*u*^{+},*v*^{+},*w*^{+}dimensionless turbulence intensities; √

*u*^{2}/*ū*^{*},*√v*^{2}/*ū*^{*},*√w*^{2}/*ū*^{*}*u*^{*}local friction velocity, (

*τ*_{ w }/*ϱ*)^{1/2}*ū*^{*}average friction velocity, (

*¯gt/ϱ*)^{1/2}*Ū*axial mean velocity (time-average)

*U*_{b}average mean axial velocity

*U*_{sec}resultant of

*¯V*and*¯W*, (*¯V*^{2}+*¯*^{2})^{1/2}*U*^{+}dimensionless velocity,

*Ū/u*^{*}*¯V, ¯W*mean velocities in the

*y, z*directions (secondary velocities)*x*axial direction

*y, 2*horizontal and vertical directions (Fig. 1)

*z*^{+}dimensionless distance from (and normal to) a wall,

*zu*^{*}/*v*- \(\hat z\)
distance from wall (at

*y*=0) to location of maximum axial velocity*μ*laminar dynamic viscosity

*v*kinematic viscosity

*ϱ*air density

*τ*_{w}local wall shear stress

*¯τ*_{w}average of local wall shear stresses over all walls

*¯τ*average wall shear stress, (

*dP/dx*) (*D*_{ h }/4)*φ*corner angle of trapezoidal duct (Fig. 1)

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