Efficient methods to account for variable property effects in numerical momentum and heat transfer solutions Y. Jin, H. Herwig ⇑ Institute for Thermo-Fluid Dynamics, TU Hamburg-Harburg, D-21073 Hamburg, Germany article info Article history: Received 12 July 2010 Received in revised form 29 October 2010 Accepted 19 November 2010 Available online 13 January 2011 Keywords: Variable properties Asymptotic method Heat transfer abstract Numerical solutions of complex turbulent flows that have been achieved under the assumption of constant properties can be corrected with respect to the influence of variable properties for arbitrary (small) heat transfer rates and arbitrary (Newtonian) fluids with an asymptotic approach. It is based on the Taylor series expansion of all properties with respect to temperature. Two methods, one with a special manipulation to circumvent higher order equations are introduced and discussed in detail. Finally it is shown, how DNS and LES solutions, which need extremely large CPU-times, can be treated with respect to the influence of variable properties, at least approximately. Ó 2010 Elsevier Ltd. All rights reserved. 1. Introduction ‘‘Variable property effects’’ is a theoretical construct since a real fluid always is subject to variable properties when changes in tem- perature or pressure occur. Their influence compared to a corre- sponding situation but with artificially constant properties may be small and thus neglected in a first approximation. Those artifi- cial ‘‘constant property results’’ may then be corrected with respect to the initially neglected effects due to the variability of the fluid properties. This concept assumes small variable property effects and therefore is not applicable when the flow itself is basically gen- erated by a variable property (like natural convection, generated by density variations) or strongly affected by it (like compressible flow, determined by density variations). The variable property correction of a constant property solution can be accomplished in different ways. Basically there are three methods which are widely used in this context as discussed below. 1.1. Property ratio method Results in terms of the nondimensional friction factor f and Nusselt number Nu are gained by multiplying the constant prop- erty results f cp and Nu cp with a property ratio correction factor, i.e. f ¼ f cp Y a  1 a  2  ma ; Nu ¼ Nu cp Y a  1 a  2  na ð1Þ Here a  1 , a  2 are properties (q / , l / , k / , c  p ) at two different tempera- tures and m a , n a are empirical exponents. Studies with this ap- proach are Li et al. [1] and Mahmood et al. [2], for example. 1.2. Reference temperature method The constant property results in terms of f and Nu are evaluated at a certain temperature for the properties that appear in f and Nu. This so-called reference temperature, T  R ¼ T  1 þ jðT  2  T  1 Þ ð2Þ between two characteristic temperatures T  1 and T  2 of the problem must be chosen such that f and Nu determined under the assump- tion of constant properties (and with T  1 or T  2 as reference temper- ature) give the results for variable properties. For that purpose the factor j in (2) must be determined properly. Studies using this method are Jayari et al. [3] and Debrestian and Anderson [4], for example. 1.3. Asymptotic method Since the effects of variable properties are assumed to be small the variable property solution can be taken as a perturbation of the constant property solution. In a systematic approach the problem is treated as a regular perturbation problem with a (small) pertur- bation parameter e linked to the heat transfer rate. The constant property solution is that for e = 0 and variable property effects are described as 1st, 2nd, ... , nth order effects of an asymptotic expansion with respect to the perturbation parameter e. This approach in an early paper has been suggested by Carey and 0017-9310/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2010.12.004 ⇑ Corresponding author. Tel.: +49 40 42878 3044; fax: +49 40 42878 4169. E-mail address: h.herwig@tu-harburg.de (H. Herwig). International Journal of Heat and Mass Transfer 54 (2011) 2180–2187 Contents lists available at ScienceDirect International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt