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Analytical investigation of boundary layer growth and swirl intensity decay rate in a pipe

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Abstract

In this research, the developing turbulent swirling flow in the entrance region of a pipe is investigated analytically by using the boundary layer integral method. The governing equations are integrated through the boundary layer and obtained differential equations are solved with forth-order Adams predictor-corrector method. The general tangential velocity is applied at the inlet region to consider both free and forced vortex velocity profiles. The comparison between present model and available experimental data demonstrates the capability of the model in predicting boundary layer parameters (e.g. boundary layer growth, shear rate and swirl intensity decay rate). Analytical results showed that the free vortex velocity profile can better predict the boundary layer parameters in the entrance region than in the forced one. Also, effects of pressure gradient inside the boundary layer is investigated and showed that if pressure gradient is ignored inside the boundary layer, results deviate greatly from the experimental data.

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Abbreviations

C :

Constant of tangential velocity definition

C 0 :

Constant of tangential velocity definition at the pipe inlet

D :

Pipe diameter, m

E = (δ z /δ θ )1/7:

One-seventh power of the axial/tangential boundary layer thicknesses

F :

Force that act on the boundary layer in axial direction, N

\({\dot{m}}\) :

Mass flow rate, kg s−1

p :

Static pressure, N m−2

p δ :

Static pressure outside the boundary layer, N m−2

\({\dot {P}}\) :

Momentum rate, m Kg s−2

R :

Radius of pipe, m

Re :

Reynolds Number based on pipe diameter

r, θ, z:

Coordinates

S :

Swirl intensity

S 0 :

Swirl intensity at the pipe inlet

U z0 :

Axial velocity at the pipe inlet, m s−1

U z :

Axial velocity at the edge of boundary layer, m s−1

U θ :

Tangential velocity at the edge of boundary layer, m s−1

u z :

Axial velocity in the boundary layer, m s−1

u θ :

Tangential velocity in the boundary layer, m s−1

δ z :

Axial boundary layer thickness, m

δ θ :

Tangential boundary layer thickness, m

δ*:

Displacement thickness, m

υ :

Kinematic viscosity, m2s−1

ρ :

Density, Kg m−3

θ*:

Momentum thickness, m

τ z :

Axial wall shear stress, N m−2

τ θ :

Tangential wall shear stress, N m−2

References

  1. Algifri A.H., Bhardwaj R.K., Rao Y.V.N.: Eddy viscosity in decaying swirl flow in a pipe. Appl. Sci. Res. 45, 287–302 (1988)

    Article  Google Scholar 

  2. Lucca-Negro O., O’Dohery T.: Vortex breakdown: a review. Prog. Energy Combust. Sci. 27, 431–481 (2001)

    Article  Google Scholar 

  3. Alekseenko S.V., Kuibin P.A., Okulov V.L., Shtork S.I.: Helical vortices in swirl flow. J. Fluid Mech. 382, 195–243 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  4. Concha F.: Flow pattern in Hydrocyclones. KONA 25, 97–132 (2007)

    Google Scholar 

  5. Taylor G.I.: The boundary layer in the converging nozzle of swirl atomizer. Quart. J. Mech. Appl. Math. 3, 129–139 (1950)

    Article  MathSciNet  MATH  Google Scholar 

  6. Talbot L.: Laminar swirling pipe flow. J. Appl. Mech. 21, 1–7 (1954)

    MATH  Google Scholar 

  7. Weber H.E.: The boundary layer inside a conical surface due to swirl. J. Appl. Mech. 23, 587–592 (1956)

    MathSciNet  MATH  Google Scholar 

  8. Murray J.D.: The boundary layer on a flat plate in a stream with uniform shear. J. Fluid Mech. 11, 309–316 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  9. Kreith F., Sonju K.: The decay of a turbulent swirl in a pipe. J. Fluid Mech. 22, 257–271 (1965)

    Article  MATH  Google Scholar 

  10. Rochino, A., Lavan, Z.: Analytical Investigations of Incompressible Turbulent Swirling Flow in Stationary Ducts. ASME Transactions. J. Appl. Mech., pp. 151–158 (1969)

  11. Najafi A.F., Saidi M.H., Sadeghipour M.S., Souhar M.: Boundary layer solution for the turbulent swirling decay flow through a fixed pipe: SBR at the inlet. Int. J. Eng. Sci. 43, 107–120 (2005)

    Article  Google Scholar 

  12. Hsieh K.T., Rajamani R.K.: Mathematical model of the hydrocyclone based on physics of fluid flow. AIChE J. 37, 735–746 (1991)

    Article  Google Scholar 

  13. Vyas, A.B., Majdalani, J.: The bidirectional vortex. Part 1: an exact inviscid solution. In: 39th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit, Huntsville, 5052 (2003)

  14. Vyas, A.B., Majdalani, J.: The bidirectional vortex. Part 2: viscous core corrections. In: 39th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit, Huntsville, 5053 (2003)

  15. Majdalani J., Rienstra S.W.: On the bidirectional vortex and other similarity solution in spherical coordinates. Z. Angew. Math. Phys. 58, 289–308 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Vyas A.B., Majdalani J.: Exact solution of the bidirectional vortex. AIAA J. 44, 2208–2216 (2006)

    Article  Google Scholar 

  17. Yajnik K.S., Subbaiah M.V.: Experiments on swirling turbulent flows: part 1. Similarity in swirling flows. J. Fluid Mech. 60, 665–687 (1973)

    Article  Google Scholar 

  18. Kitoh O.: Experimental study of turbulent swirling flow in a straight pipe. J. Fluid Mech. 225, 445–479 (1991)

    Article  Google Scholar 

  19. Algifri A.H., Bhardwaj R.K., Rao Y.V.N.: Turbulence measurement in decaying swirl flow in a pipe. Appl. Sci. Res. 45, 233–250 (1988)

    Article  Google Scholar 

  20. Chang F., Dhir V.K.: Mechanisms of heat transfer enhancement and slow decay of swirl in tubes using tangential injection. Int. J. Heat Fluid Flow 16, 78–87 (1995)

    Article  Google Scholar 

  21. Kobayashi T., Yoda M.: Modified k−ε model for turbulent swirling flow in a straight pipe. JSME Int. J. 30, 66–71 (1987)

    Google Scholar 

  22. He, P., Salcudean, M., Gartshore, I.S.: A numerical simulation of hydrocyclones. Trans. IChemE 77, Part A 429–441 (1999)

    Google Scholar 

  23. Ma L., Ingham D.B., Wen X.: Numerical modeling of the fluid and particle penetration through small sampling cyclones. J. Aerosol Sci. 31, 1097–1119 (2000)

    Article  Google Scholar 

  24. Yoshida H.: Three-dimensional simulation of air cyclone and particle separation by a revised-type cyclone. Colloids Surf. 109, 1–12 (1996)

    Article  Google Scholar 

  25. Hansen, K.G., Madsen, J.: A computational and experimental study of the internal flow in a scaled pressure-swirl atomizer. M.Sc. Thesis, Aalborg Universitet Esbjerg (2001)

  26. Evans W.K., Suksangpanomrung A., Nowakowski A.F.: The simulation of the flow within a hydrocyclone operating with an air core and with an inserted metal rod. Chem. Eng. J. 143, 51–61 (2008)

    Article  Google Scholar 

  27. Narasimha M., Brennan M., Holtham P.N.: Large eddy simulation of hydrocyclone-prediction of air-core diameter and shape. Int. J. Miner. Process. 80, 1–14 (2006)

    Article  Google Scholar 

  28. Schlichting H.: Boundary Layer Theory. McGraw-Hill, New York (1973)

    Google Scholar 

  29. Burden, R.L., Faires, J.D.: Numerical Analysis, 7th edn. Brooks/Cole, Belmont, ISBN: 0534 382169 (2000)

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Authors and Affiliations

  1. School of Mechanical Engineering, Sharif University of Technology, Tehran, Iran

    Reza Maddahian, Azadeh Kebriaee, Bijan Farhanieh & Bahar Firoozabadi

Authors
  1. Reza Maddahian
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  2. Azadeh Kebriaee
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  3. Bijan Farhanieh
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  4. Bahar Firoozabadi
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Correspondence to Reza Maddahian.

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Maddahian, R., Kebriaee, A., Farhanieh, B. et al. Analytical investigation of boundary layer growth and swirl intensity decay rate in a pipe. Arch Appl Mech 81, 489–501 (2011). https://doi.org/10.1007/s00419-010-0424-9

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  • DOI: https://doi.org/10.1007/s00419-010-0424-9

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