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Turbulent flow in converging nozzles, part one: boundary layer solution

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Abstract

The boundary layer integral method is used to investigate the development of the turbulent swirling flow at the entrance region of a conical nozzle. The governing equations in the spherical coordinate system are simplified with the boundary layer assumptions and integrated through the boundary layer. The resulting sets of differential equations are then solved by the fourth-order Adams predictor-corrector method. The free vortex and uniform velocity profiles are applied for the tangential and axial velocities at the inlet region, respectively. Due to the lack of experimental data for swirling flows in converging nozzles, the developed model is validated against the numerical simulations. The results of numerical simulations demonstrate the capability of the analytical model in predicting boundary layer parameters such as the boundary layer growth, the shear rate, the boundary layer thickness, and the swirl intensity decay rate for different cone angles. The proposed method introduces a simple and robust procedure to investigate the boundary layer parameters inside the converging geometries.

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Abbreviations

P :

static pressure, Pa

R, θ, φ :

spherical coordinates

R o :

radius of cone at the inlet

S :

swirl intensity

S o :

swirl intensity at the cone inlet

U Φ o , U R o :

inlet rotational and axial velocities at the edge of the boundary layer

U Φ,U R :

rotational and axial velocities at the edge of the boundary layer

u R , u θ , u φ :

velocities in the R, θ, and φ directions inside the boundary layer

u*:

shear velocity

y :

wall distance

y + :

non-dimensional wall distance

α :

half cone angle

δ R :

axial boundary layer thickness

δ φ :

rotational boundary layer thickness

\( \bar \delta _R \) :

non-dimensional axial boundary layer thickness

\( \bar \delta _\phi \) :

non-dimensional rotational boundary layer thickness

υ :

kinematic viscosity

Ω:

constant of the rotational velocity

ρ :

density

τ Rθ :

axial wall shear stress

τ φ θ :

rotational wall shear stress

o :

inlet condition

−:

non-dimensional parameter

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Authors and Affiliations

  1. Center of Excellence in Energy Conversion, School of Mechanical Engineering, Sharif University of Technology, P. O. Box 11365-9567, Tehran, Iran

    R. Maddahian, B. Farhanieh & B. Firoozabadi

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  1. R. Maddahian
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  2. B. Farhanieh
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  3. B. Firoozabadi
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Correspondence to R. Maddahian.

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Maddahian, R., Farhanieh, B. & Firoozabadi, B. Turbulent flow in converging nozzles, part one: boundary layer solution. Appl. Math. Mech.-Engl. Ed. 32, 645–662 (2011). https://doi.org/10.1007/s10483-011-1446-6

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  • DOI: https://doi.org/10.1007/s10483-011-1446-6

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