Abstract
In general, flow in an annulus will be strongly asymmetric because of unequal curvatures and, possibly, dissimilar texture of the two boundaries. As a result, the turbulent flow structure in such geometries possesses a number of features which can not satisfactorily be accounted for with a simple model of turbulence. The present paper provides predictions based on a more elaborate model, developed by the author, in which differential transport equations are solved for the turbulent shear stress, the turbulent kinetic energy and the rate of the turbulence energy dissipation, simultaneously with the mean momentum equation. The complete model thus consists of a closed set of four coupled non-linear partial differential equations. Predictions of flow in annuli with both smooth and rough cores with various radius ratios display in all cases very good agreement with experimental results of various authors.
Zusammenfassung
Die Strömung im Ringkanal ist im allgemeinen stark unsymmetrisch wegen unterschiedlicher Wandkrümmungen und u. U. auch Wandtexturen. Die dadurch hervorgerufene Struktur der turbulenten Strömung kann mit einfachen Turbulenzmodellen nicht befriedigend beschrieben werden. In dieser Arbeit wird ein vom Autor entwickeltes verfeinertes Modell verwendet, in dem Transport-Differentialgleichungen gelöst werden für die turbulente Schubspannung, die turbulente kinetische Energie und die Energiedissipation der Turbulenz, zusammen mit der Gleichung des mittleren Impulses. Das vollständige Modell besteht also aus einem geschlossenen Satz von vier gekoppelten nichtlinearen partiellen Differentialgleichungen. Voraussagen für die Ringkanalströmung mit glatter und rauher Innenwand und verschiedenen Durchmesserverhältnissen stimmen sehr gut mit von verschiedenen Autoren gemessenen Werten überein.
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Abbreviations
- Br:
-
free constant in universal velocity expression for rough wall
- Bs:
-
free constant in universal velocity expression for smooth wall
- Bs1 :
-
free constant in universal velocity expression for inner wall of a smooth annulus
- Ce :
-
constant in turbulence energy equation
- Cs, Cs1, Cs2 :
-
constants in shear stress equation
- Cε, Csε1, Csε2 :
-
constants in dissipation equation
- e:
-
turbulent kinetic energy
- f:
-
friction factor
- L:
-
turbulence dissipation length scale
- P:
-
mean pressure
- Re:
-
mean flow Reynolds number
- r:
-
radius
- r1, r2 :
-
radius of inner and outer wall of an annulus respectively
- re :
-
radius at which turbulent kinetic energy is minimum
- rM :
-
radius at which mean velocity is maximum
- r0 :
-
radius at which turbulent shear stress is zero
- U:
-
mean velocity
- Ui :
-
mean velocity component in i-direction
- UM :
-
maximum mean velocity
- Uτ1, Uτ2 :
-
friction velocity at inner and outer wall respectively
- U+ :
-
dimensionless velocity: U/Uτ
- Ui :
-
fluctuating velocity component in i-direction
- ũi :
-
r.m.s. of the fluctuating velocity component
- xi :
-
Cartesian coordinates (i=1 mean flow direction, i=2 direction normal to the wall)
- x +2 :
-
dimensionless coordinate: x2 Uτ/v
- ɛ :
-
rate of dissipation of turbulent kinetic energy:\(v\overline {(\vartheta _{U_i } /\vartheta _{X_j } )^2 }\)
- κ :
-
von Karman constant
- ν :
-
kinematic viscosity
- τ :
-
total shear stress
- τ 1,τ 2 :
-
shear stress at inner and outer wall of an annulus respectively
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Hanjalić, K. Prediction of turbulent flow in annular ducts with differential transport model of turbulence. Wäarme- und Stoffübertragung 7, 71–78 (1974). https://doi.org/10.1007/BF01369514
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DOI: https://doi.org/10.1007/BF01369514