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Prediction of Boundary Layer Flow Transition under Non-Zero Pressure Gradient Conditions using Boundary-Fitted Technique
Abstract
A numerical technique to investigate the transitional boundary layer flow over a flat plate subjected to freestream turbulence and non-zero pressure gradient is presented. The technique solves the full equations of fluid motion rather than the boundary-layer equations which are neglect the y-momentum equation and preserve the pressure variation only along the streamwise direction. In the present technique, the y-momentum equation remains employed to maintain the full-step computation and furthermore the pressure is treated as the flow variable and the effect of the pressure variation along the streamwise direction is deliberately converted to the variation of domain thickness along the same direction according to the test section of Coupland (1993), for T3C1, T3C2 and T3C4 cases. The boundary-fitted technique to transform the nonuniform grids to the uniform rectangular grids is used. The performance of two well-known turbulence models: the k-ε model of Launder and Sharma (1974) and the SST turbulence model of Menter (1994), in predicting the transitional boundary layer is assessed against the experimental data.
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An approach to numerical convection is presented that exclusively yields upstream-centered schemes. It starts from a meshwise approximation of the initial-value distribution by simple basic functions, e.g., Legendre polynomials. In every mesh the integral of the distribution is conserved. The overall approximation need not be continuous. The approximate distribution is convected explicitly and then remapped meshwise in terms of the basic functions. The weights of the basic functions that approximate the initial values in a mesh may be determined by finite differencing, but the most accurate schemes are obtained by least-squares fitting. In the latter schemes, the weights of the basic functions must be regarded as independent state quantities and must be stored separately. Examples of second-order and third-order schemes are given, and the accuracy of these schemes is discussed. Several monotonicity algorithms, designed to prevent numerical oscillations, are indicated. Numerical examples are given of linear and nonlinear wave propagation, also regarding monotonicity.
The generality of the energy-dissipation model of turbulent heat and
momentum transport, developed by Jones and Launder (1972, 1973), is
tested by applying the model to predict the swirling flows generated by
a rotating disk in a quiescent atmosphere. It is found that this
turbulence model can predict accurately the flow, heat transfer, and
mass transfer near a rotating disk. This is significant for the problem
of predicting convective heat transfer rates in turbine disks.
This paper describes a new nonlinear eddy-viscosity model of turbulence designed with a view to predicting flow far from equilibrium, including transition. The scheme follows earlier UMIST practice in adopting a cubic relation between the stress and the strain/ vorticity tensors but broadens the range of flows to which the model applies by including a third transport equation for an anisotropy parameter of the stress field. Applications are shown for transition on a flat plate at different levels of free-stream turbulence, for the normal impingement of a turbulent jet on a flat plate, and for the flow around a turbine blade. The model is shown to generate much more realistic predictions than what is said to be the best of the linear eddy-viscosity schemes.
A new correlation-based transition model has been developed, which is built strictly on local variables. As a result, the transition model is compatible with modern CFD techniques such as unstructured grids and massively parallel execution. The model is based on two transport equations, one for intermittency and one for a transition onset criterion in terms of momentum thickness Reynolds number. The proposed transport equations do not attempt to model the physics of the transition process (unlike e.g. turbulence models), but form a framework for the implementation of transition correlations into general-purpose CFD methods. The transition model was initially developed for turbomachinery flows. The main goal of the present paper is to validate the model for predicting transition in aeronautical flows. An incremental approach was used to validate the model, first on 2D airfoils and then on progressively more complicated test cases such as a 3-element flap, a 3D transonic wing and a full helicopter configuration. In all cases good agreement with the available experimental data was observed. The authors believe that the current formulation is a significant step forward in engineering transition modeling, as it allows the combination of transition correlations with general purpose CFD codes. There is a strong potential that the model will allow the 1 st order effects of transition to be included in everyday industrial CFD simulations.
A new transport equation for the intermittency factor was proposed to predict separated and transitional boundary layers under low-pressure turbine airfoil conditions. The intermittent behavior of the transitional flows is taken into account and incorporated into computations by modifying the eddy viscosity, ut, with the intermittency factory. Turbulent quantities are predicted by using Menter's two-equation turbulence model (SST). The intermittency factor is obtained from a transport equation model, which not only can reproduce the experimentally observed streamwise variation of the intermittency in the transition zone, but also can provide a realistic cross-stream variation of the intermittency profile. In this paper, the intermittency model is used to predict a recent separated and transitional boundary layer experiment under low pressure turbine airfoil conditions. The experiment provides detailed measurements of velocity, turbulent kinetic energy and intermittency profiles for a number of Reynolds numbers and freestream turbulent intensity conditions and is suitable for validation purposes. Detailed comparisons of computational results with experimental data are presented and good agreements between the experiments and predictions are obtained.
The predictive performance of three low-Reynolds-number turbulence models, one based on the linear (Boussinesq) stress-strain relationship and two adopting cubic forms, is examined by reference to transitional flat-plate boundary layers subjected to streamwise pressure gradient and moderate free-stream turbulence, the latter causing bypass transition. One of the cubic models involves the solution of two transport equations, while the other involves three equations, one of which relates to the second invariant of turbulence anisotropy. The investigation demonstrates that, for the conditions examined, the non-linear models return a more credible representation of transition than the linear variant, although none of the models may be said to be entirely satisfactory. Thus, while non-linear modelling shows promise and offers inherent advantages through the use of a more general stress-strain linkage and elaborate calibration, further refinement is needed in the present cubic variants for transitional flows.
Two new two-equation eddy-viscosity turbulence models will be presented. They combine different elements of existing models that are considered superior to their alternatives. The first model, referred to as the baseline (BSL) model, utilizes the original k-omega model of Wilcox In the inner region of the boundary layer and switches to the standard k -epsilon model in the outer region and in free shear flows. It has a performance similar to the Wilcox model, but avoids that model's strong freestream sensitivity. The second model results from a modification to the definition of the eddy-viscosity in the BSL model, which accounts for the effect of the transport of the principal turbulent shear stress. The new model is called the shear-stress transport-model and leads to major improvements in the prediction of adverse pressure gradient flows.
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