The IMMERSOL model of Radiative Heat Transfer, PHOENICS, CHAM Ltd.,. 2012, http://www.cham.co.uk/phoenics/d_polis/d_enc/enc_rad3.htm. 6. S.V.Zhubrin ...
1 COMPUTATIONS OF WALL DISTANCES FOR DISTRIBUTED RESISTANCE ANALOGY by S.V.Zhubrin May, 2012
1. Introduction The calculation of the wall distances is one of the main ingredients in the practical evaluation of near-wall transfer effects [1]. The computations of both the distances to the nearest wall and distances between the walls play a definitive role in the grid generations [2] including those for time-dependent geometries [3], and are employed in a simple algebraic turbulence model which does not require the solution of any partial differential equations [4]. The distance between the bounding walls may also have a clear influence on the radiative transfer of heat within the high temperature industrial units [5]. The wall distances are still one of the key parameters in a number of industrial applications incorporating multi-physics approach [6]. The DRA, Distributed Resistance Analogy, invented for representation of a multitude of inter-connected small-scale elements by a set of distributed empirical resistances to local flow, heat and mass transfer often requires wall distances as an input of quantitative correlations for the latter [7,8,9]. However, the sub-grid nature, say, of a bundle of numerous small inserts found inside the computational control volume makes the task of distance calculations for DRA, both to and between the walls of in-cell inserts, a challenging one. The present contribution introduces a way of computing the average sub-grid wall distances in the framework of computations based on extended differential equation for distance function. The method is described, validated by comparison with a cases which have known solutions, and is already in use for industrial CFD applications. 2. The distance-calculation procedure The present technique is motivated by the wall distance methods developed for flows with multiple internal boundaries of grid-cell scale as described in [2-4]. They calculate the wall distances by solving differential equation for the distance function, from solution of which the values of distances are recovered. The latter process usually involves auxiliary analytically derived equations. Because a simple, approximate, Poisson-equation-based distance approach [4] was found effective in allowing the calculation of distances to be implemented in industrial CFD codes, this approach is taken here as a precursor of current technique. Unlike the original, the present method allows for the distances in sub-cell environment to be computed as detailed below.
2 For the problem of DRA's interpenetrating spaces the method of [4] is generalized so that to handle the calculations in both larger cell-scale sub-domains and in the regions cluttered by small-scale solid inserts of sub-cell size. The generalized transport 2 equation for distance function, , in m , aimed at multi-scale wall distance computations is written as follows:
a v Aw''' 2 0 2
where
a is area porosity, v 2
3
(1)
Aw''' is volumetric surface boundary conditions of 0 at the cell-
is volume porosity, and
area of sub-cell inserts, m /m , with the scale walls.
From the solution of (1) the values of distances are reconstructed as
D 2 2
(2)
L 2
(3)
2
2
where D stands for the distance between walls, and wall, both in m. In (1)-(3)
denotes the magnitude of
xj
is the distance to the nearest
- vector, i.e.
where
L
j 1,3 x j
2
(4)
are the distances along Cartesian coordinate directions.
Equation (1) is valid in a total space which can be partially filled by both scattered solid objects of cell-scale size and by arrays of other smaller-scale inserts so numerous that many of them are contained within a single cell of the computational grid. The thoughts behind -equation generalization may deserve some explanations. The first two terms of (1) hardly require any in-depth discussion, for they are the result of direct application of DRA's interpenetrating-space averaging technique to original Poisson equation given in [4]. The third term quantifies the effect of near-solid within- cell boundary sources, where
2 0.5D 2
(5)
3 represents a local one-dimensional analytical solution for wall diffusion flux of . Discussion As the functional tests show the model produces precisely the correct results for geometrically-simple circumstances with known solutions, e.g. for the space between two parallel walls of infinite extent, and for the dense array of passages between two parallel walls. For all other situations D and L have been found to give plausible results, and the model is already in use for industrial CFD applications [6]. Concluding remarks The present work has shown how DRA based models of technological thermohydraulics can be supplied with a method which allows the calculation of the sub-grid average wall distances which in turn are used as closure of DRA formulations. The method has been successfully tested by comparison with a simple cases which have known solutions. The method proposed has a potential for further extension, for instance to incorporate unsteady change of distances, and further models of subcell-distance evolution can also be included. References 1. H. K. Versteeg, W. Malalasekera, An introduction to computational fluid dynamics: The finite volume method, Longman Group Ltd, London, 1995, pp. 257. 2. P. G. Tucker, C. L. Rumsey, P. R. Spalart, R. E. Bartels, R.T. Biedron, Computations of Wall Distances Based on Differential Equations, AIAA Journal, v. 43, No. 2, February 2005 3. P. G. Tucker, C. L. Rumsey, R. E. Bartels, R.T. Biedron, Transport Equation Based Wall Distance Computations Aimed at Flows With Time-Dependent Geometry, NASA/TM-2003-212680, 2003, , pp. 58 4. D. Agonafer, Liao Gan-Li ,D. B. Spalding, The LVEL Turbulence Model for Conjugate Heat Transfer at Low Reynolds Numbers, Application of CAE/CAD Electronic Systems, EEP ASME v. 18, 1996. 5. The IMMERSOL model of Radiative Heat Transfer, PHOENICS, CHAM Ltd., 2012, http://www.cham.co.uk/phoenics/d_polis/d_enc/enc_rad3.htm 6. S.V.Zhubrin, Computational Model for Performance Predictions in Bayonet-Tube Methane Steam Reformer, 10/2011, doi: 10.13140/2.1.4137.2965 7. S.V. Patankar, D.B.Spalding, Computer Analysis of the three-dimensional flow and heat transfer in a steam generator, Forsch. Ing. Wes., v. 44, 1972, N2, pp.47-52 8. S. B. Beale, S.V.Zhubrin, A Distributed Resistance Analogy for Solid Oxide Fuel Cells, Numerical Heat Transfer, Fundamentals, 06/2005; 47(6), doi:10.1080/10407790590907930 9. V. Agranat, S. Zhubrin, A. Maria, J. Hinatsu, M. Stemp, M. Kawaji, An Integrated Gas-Liquid flow Analyzer and its Application to Performance Predictions in Water Electrolysis Systems, 09/2006, https://www.researchgate.net/publication/237297220
