|
Organizers |
Turbulence model for boundary layers in strong pressure gradients
by
Alexandru Dumitrache
Institute of Statistics and Applied Mathematics, Romanian Academy, ROMANIA
Coauthors: Horia Dumitrescu (Institute of Statistics and Applied Mathematics, Romanian Academy, Romania)
The interest in accurately calculating two-dimensional boundary layer flows is old but it remains very important due to large number of fluid flow devices that works under these conditions.
A number of differential and integral formulations have been used over the years in an attempt to predict those flows. The simplest of those methods employ the mixing length or eddy viscosity concepts, which have been proved very useful in equilibrium boundary layers. This approach assumes that the shear stress profile at same distance downstream from the origin of the boundary layers is uniquely related to the mean - flow conditions at this station. In other words, the flow exhibits similarity from station to station and only a local velocity and length scale is required for a full description. The experimental observations indicate that in the case of strong adverse pressure gradient and/or separation the shear stress profile depends not only on the local velocity profile and the distance from the wall but also on the upstream development of the turbulence (i.e. the "history" of the turbulence). This observation has led some researches to develop models that attempt to account for this lag of the large eddy structure to the local conditions of the shear flow. Bradshaw, Ferris and Atwell were among the first to employ the turbulence kinetic energy (T.K.E.) equation as the governing equation for the development of Reynolds shear stress along the boundary layer. Assuming that the ratio of Reynolds shear stress and the turbulence kinetic energy is constant along a streamline and introducing empirical functions for the dissipation and diffusion terms, they transformed the T.K.E. to an equation for the development of the shear stress.
Johnson and King, Dumitrescu and Dumitrache, proposed a hybrid Reynolds stress/eddy viscosity model for the separating flow. Starting from the experimental observation that the maximum shear stress is the proper scaling parameter for strong adverse pressure gradient and separating flows, they proposed an ordinary differential equation for -[`uv]max that was heuristically deduced from the differential turbulence kinetic energy equation. The constant in the eddy viscosity model for the outer region is adjusted so that the maximum shear stress predicted from the model coincides with the value predicted from the ordinary differential equation.
McDonald and Camarata have proposed the use of the integral form of the T.K.E. equation, which avoids the specification of the diffusion function. The integral T.K.E. equation in used for specifying a modified mixing length value for the outer region, while the dissipation is given as a function of the local stress and a dissipation length, the value of which is kept constant throughout the calculation.
The present paper attempts to describe a new, general approach to the prediction of boundary layer flows in the presence of strong adverse pressure gradients and/or separation.
The basic features of the turbulence model include:
1) the use of the integrated T.K.E. equation as a way of taking into account "history" effects on the mixing length,
2) the use of \taumax as the correct scaling factor,
3) the specification of a rigorous model for the distribution of the shear stress profile in the near wall region, where the assumption \tau = \taumax is not valid for strong, adverse pressure gradients,
4) the contribution of the normal stresses as separation is approached, and
5) the use of an empirical function that attempts to take into account the change of the turbulence structure near and at separation. All of this results in an auxiliary equation for the constant in the outer region mixing length model.
Comparisons of predictions and measurements for steady, two dimensional cases with strong, adverse pressure gradients are presented.
Date received: January 29, 2004
Copyright © 2004 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cakt-41.