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Application of the similarity theory including variable property effects to a complex benchmark problem

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Abstract

An asymptotic method to account for variable property effects, recently described in this journal, is now applied to a complex benchmark geometry. It is a room which is ventilated by forced convection through inlet and outlet slit nozzles at the top and bottom of the side walls. Four heating elements standing on the ground floor add heat with constant heat flux density of varying strength. CFD solutions with the full coverage of all property temperature dependencies of air and SF6 are compared with asymptotic results (ACFD), applied for these fluids. ACFD results are given as systematic expansions with respect to a heat transfer parameter \({\varepsilon}\) which serves as perturbation parameter. First and second order asymptotic results of the Nußelt number at the surface of the heating elements are shown as well as temperature distributions along the adiabatic walls of the room. Special attention is given to the reference Nußelt numbers of zero order \({(\varepsilon=0)}\) which are those for constant properties only for pure forced convection.

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Abbreviations

Symbol :

Name   Unit

ρ*:

Density   (kg/m3)

\({c_p^\ast}\) :

Heat capacity   (J/kg/K)

k*:

Heat conductivity   (W/mK)

μ*:

Dynamic viscosity   (kg/ms)

a*:

Variable fluid property –

T*:

Temperature   (K)

p*:

Pressure   (Pa)

ΔT*:

Temperature difference   (K)

L*:

Characteristic length   (m)

\({\dot {q}^\ast}\) :

Heat flux density   (W/m2)

Symbol:

Name

Nu :

Nußelt number

Re :

Reynolds number

Pr :

Prandtl number

Gr :

Grashof number

\({\varepsilon}\) :

Temperature difference

K a , Ka2, K an :

K-values, a = ρ, μ, k, c p

A a , A aa , A ab , Aa2:

A-values, a = ρ, μ, k, c p

R :

Reference state

c p :

Constant properties

0B :

Bousinnesq approximation

0i:

Ideal gas

References

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Authors and Affiliations

  1. Hamburg University of Technology, Thermo-Fluid Dynamics (M-21), Denickestr. 17 (Building L), 21073, Hamburg, Germany

    Y. Jin & H. Herwig

Authors
  1. Y. Jin
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  2. H. Herwig
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Correspondence to H. Herwig.

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Jin, Y., Herwig, H. Application of the similarity theory including variable property effects to a complex benchmark problem. Z. Angew. Math. Phys. 61, 509–528 (2010). https://doi.org/10.1007/s00033-010-0060-9

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  • DOI: https://doi.org/10.1007/s00033-010-0060-9

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