Alain J. Kassab

An Efficient Localized Radial Basis Function Meshless Method for Fluid Flow and Conjugate Heat Transfer

A localized radial basis function (RBF) meshless method is developed for coupled viscous fluid flow and convective heat transfer problems. The method is based on new localized radial-basis function (RBF) expansions using Hardy Multiquadrics for the sought-after unknowns. An efficient set of formulae are derived to compute the RBF interpolation in terms of vector products thus providing a substantial computational savings over traditional meshless methods. Moreover, the approach developed in this paper is applicable to explicit or implicit time marching schemes as well as steady-state iterative methods. We apply the method to viscous fluid flow and conjugate heat transfer (CHT) modeling. The incompressible Navier‐Stokes are time marched using a Helmholtz potential decomposition for the velocity field. When CHT is considered, the same RBF expansion is used to solve the heat conduction problem in the solid regions enforcing temperature and heat flux continuity of the solid/fluid interfaces. The computation is accelerated by distributing the load over several processors via a domain decomposition along with an interface interpolation tailored to pass information through each of the domain interfaces to ensure conservation of field variables and derivatives. Numerical results are presented for several cases including channel flow, flow in a channel with a square step obstruction, and a jet flow into a square cavity. Results are compared with commercial computational fluid dynamics code predictions. The proposed localized meshless method approach is shown to produce accurate results while requiring a much-reduced effort in problem preparation in comparison to other traditional numerical methods. DOI: 10.1115/1.2402181

Conjugate Heat Transfer Effects on a Realistic Film-Cooled Turbine Vane

A conjugate heat transfer solver has been developed and applied to a realistic film-cooled turbine vane for a variety of blade materials. The solver used for the fluid convection part of the problem is the Glenn-HT general multiblock heat transfer code. The solid conduction module is based on the Boundary Element Method (BEM), and is coupled directly to the flow solver. A chief advantage of the BEM method is that no volumetric grid is required inside the solid – only the surface grid is needed. Since a surface grid is readily available from the fluid side of the problem, no additional gridding is required. This eliminates one of the most time consuming elements of the computation for complex geometries. Two conjugate solution examples are presented - a high thermal conductivity Inconel nickel-based alloy vane case and a low thermal conductivity silicon nitride ceramic vane case. The solutions from the conjugate analyses are compared with an adiabatic wall convection solution. It is found that the conjugate heat transfer cases generally have a lower outer wall temperature due to thermal conduction from the outer wall to the plenum. However, some locations of increased temperature are seen in the higher thermal conductivity Inconel vane case. This is a result of the fact that film cooling is a two-temperature problem, which causes the direction of heat flux at the wall to change over the outer surface. Three-dimensional heat conduction in the solid allows for conduction heat transfer along the vane wall in addition to conduction from outer to inner wall. These effects indicate that the conjugate heat transfer in a complicated geometry such as a film-cooled vane is not governed by simple one-dimensional conduction from the vane surface to the plenum surface, especially when the effects of coolant injection are included.

A generalized boundary integral equation for isotropic heat conduction with spatially varying thermal conductivity

Abstract In this paper we derive a generalized fundamental solution for the BEM solution of problems of steady state heat conduction with arbitrarily spatially varying thermal conductivity. This is accomplished with the aid of a singular nonsymmetric generalized forcing function, D , with special sampling properties. Generalized fundamental solutions, E , are derived as locally radially symmetric responses to this nonsymmetric singular forcing function, D , at a source point ξ. Both E and D are defined in terms of the thermal conductivity of the medium. Although locally radially symmetric, E varies within the domain as the source point, ξ changes position. A boundary integral equation is formulated. Examples of generalized fundamental solutions are provided for various thermal conductivities along with the corresponding forcing function, D . Here, four numerical examples are provided. Excellent results are obtained with our formulation for variations of thermal conductivity ranging from quadratic and cubic in one dimension to exponential in two dimensions. Problems are solved in regular and irregular regions. Current work is under way investigating extensions of this general approach to further applications where nonhomogeneous property variations are an important consideration.

BEM/FVM conjugate heat transfer analysis of a three‐dimensional film cooled turbine blade

We report on the progress in the development and application of a coupled boundary element/finite volume method temperature‐forward/flux‐back algorithm developed to solve conjugate heat transfer arising in 3D film‐cooled turbine blades. We adopt a loosely coupled strategy where each set of field equations is solved to provide boundary conditions for the other. Iteration is carried out until interfacial continuity of temperature and heat flux is enforced. The NASA‐Glenn explicit finite volume Navier‐Stokes code Glenn‐HT is coupled to a 3D BEM steady‐state heat conduction solver. Results from a CHT simulation of a 3D film‐cooled blade section are compared with those obtained from the standard two temperature model, revealing that a significant difference in the level and distribution of metal temperatures is found between the two. Finally, current developments of an iterative strategy accommodating large numbers of unknowns by a domain decomposition approach is presented. An iterative scheme is developed along with a physically‐based initial guess and a coarse grid solution to provide a good starting point for the iteration. Results from a 3D simulation show the process that converges efficiently and offers substantial computational and storage savings.

Solving inverse heat conduction problems using trained POD-RBF network inverse method

The article presents advances in the approach aiming to solve inverse problems of steady state and transient heat conduction. The regularization of ill-posed problem comes from the proper orthogonal decomposition (POD). The idea is to expand the direct problem solution into a sequence of orthonormal basis vectors, describing the most significant features of spatial and time variation of the temperature field. Due to the optimality of proposed expansion, the majority of the basis vectors can be discarded practically without accuracy loss. The amplitudes of this low-order expansion are expressed as a linear combination of radial basis functions (RBF) depending on both retrieved parameters and time. This approximation, further referred as trained POD-RBF network is then used to retrieve the sought-for parameters. This is done by resorting to least square fit of the network and measurements. Numerical examples show the robustness and numerical stability of the scheme.

Solving Transient Nonlinear Heat Conduction Problems by Proper Orthogonal Decomposition and the Finite-Element Method

ABSTRACT A method of reducing the number of degrees of freedom and the overall computing time by combining proper orthogonal decomposition (POD) with the finite-element method (FEM) has been devised. The POD-FEM technique can be applied both to linear and nonlinear problems. At the first stage of the method a standard FEM time-stepping procedure is invoked. The temperature fields obtained for the first few time steps undergo statistical analysis, yielding an optimal set of globally defined trial and weighting functions for the Galerkin solution of the problem at hand. The resulting set of ordinary differential equations (ODEs) is of greatly reduced dimensionality when compared with the original FEM formulation. For linear problems, the set can be solved either analytically, resorting to the modal analysis technique, or by time stepping. In the case of nonlinear problems, only time stepping can be applied. The focus of this article is on the time-stepping approach, in which the generation of the FEM-POD matrices, requiring some additional matrix manipulations, can be embedded in the assembly of standard FEM matrices. The gain in execution times comes from the significantly shorter time of solution of the set of algebraic equations at each time step. Numerical results are presented for both linear and nonlinear problems. In the case of linear problems, the derived time-stepping technique is compared with the standard FEM and the modal analysis. For nonlinear problems the proposed POD-FEM approach is compared with the standard FEM. Good accuracy of the POD-FEM solver has been observed. Controlling the error introduced by the reduction of the degrees of freedom in POD is also discussed.

Localized Meshless Modeling of Natural-Convective Viscous Flows

A localized radial-basis function (RBF) collocation meshless method is developed for natural-convection heat transfer problems in fully viscous fluid flows. The expansion method is based on the localized collocation of polynomial-augmented Hardy multiquadrics RBF, and it is efficiently formulated to generate derivative fields through simple inner products of small-order vectors. The solution of the Navier-Stokes equations is formulated using a third-order-accurate explicit fractional time-stepping method and a velocity-correction scheme. Several cases are studied and confirmed to attain accurate results when compared to classical benchmark solutions as well as numerical predictions provided by the commercial computational fluid dynamics code Fluent.

Computational fluid dynamics analysis of surgical adjustment of left ventricular assist device implantation to minimise stroke risk

Background. Currently, mechanical support is the most promising alternative to cardiac transplantation. Ventricular assist devices (VADs) were originally used to provide mechanical circulatory support in patients awaiting planned heart transplantation (‘bridge-to-transplantation’ therapy). The success of short-term bridge devices led to clinical trials evaluating the clinical suitability of long-term support (‘destination’ therapy) with left ventricular assist devices (LVADs). The first larger scale, randomised trial that tested long-term support with an LVAD reported a 44% reduction in the risk of stroke or death in patients with an LVAD. In spite of the success of LVADs as bridge-to-transplantation and long-term support, patients managed by these devices are still at risk of several adverse events. The most devastating complication is caused by embolisation of thrombi formed within the LVAD or inside the heart into the brain. Prevention of thrombi formation is attempted through anticoagulation management and by improving LVADs design; however, there is still significant occurrence of thromboembolic events in patients. Investigators have reported that the incidence of thromboembolic cerebral events ranges from 14% to 47% over a period of 6–12 months. Methods and approach. An alternative method to reduce the incidence of cerebral embolisation is proposed by the co-authors, and the hypothesis is that it is possible to minimise the number of thrombi flowing into the carotid and vertebral arteries by an optimal placement of the LVAD outflow conduit, with or without the addition of aortic bypass connecting the ascending aorta and the innominate artery (IA), or left carotid artery. This paper presents the computational fluid dynamics (CFD) analysis of the aortic arch haemodynamics using a representative geometry of the human aortic arch with or without an alternative aortic bypass. In order to study the trajectory of the thrombi within the aortic arch bed, the CFD code, Fluent 6.3, is utilised to resolve the flow field and to solve the Lagrangian particle tracking of thrombi released randomly at the inlet of the LVAD cannula. Results. Results are presented for simulations of thrombi in the range of 2–5 mm. The percentage of individual diameter as well as aggregate diameter thrombi flowing to the carotid and vertebral arteries as a function of LVAD conduit placement and aortic bypass implantation is reported. The influence of the LVAD conduit implantation and bypass reveals a nearly 50% variation in predicted cerebral embolism rates. Conclusions. The adjustment of the location of the anastomosis of the LVAD outflow cannula as well as its angle of incidence plays a significant role in the level of thromboembolisms. By proper adjustment in this CFD study of a synthetic model of an aortic arch bed, we found that nearly a 50% reduction in cerebral embolism could be achieved for a configuration consisting of a shallow angle of implantation over a baseline normal incidence of the LVAD cannula. Within the limitations of our model, we have established that the LVAD implantation geometry is an important factor and should be taken into consideration when implanting an LVAD. It is possible that other parameters such as distance of the LVAD outflow cannula to the root of the IA could affect the thrombi embolisation probabilities. However, the results of this study suggest that the risk of stroke may be significantly reduced by as much as 50% by tailoring the VAD implantation by a simple surgical manoeuvre. The results of this line of research may ultimately lead to techniques that can be used to estimate the optimal LVAD configuration in a patient-specific manner by pre-operative imaging.

An Efficient Singular Superposition Technique for Cavity Detection and Shape Optimization

A method is presented for the efficient solution of the inverse geometric problem of detection of subsurface cavities and flaws using thermographic techniques. Here, a superposition of clusters of sources/sinks with a boundary-element solution of the forward problem offers a numerical scheme that does not require remeshing of the interior geometry as the inverse problem is solved iteratively to detect the flaw or cavity. The approach offers tremendous advantage in reducing the computational burden involved in remeshing and presents a promising technique for three-dimensional applications.

Film Cooling Effectiveness From a Single Scaled-Up Fan-Shaped Hole: A CFD Simulation of Adiabatic and Conjugate Heat Transfer Models

This paper documents a computational investigation of the film cooling effectiveness of a 3-D gas turbine endwall with one fan-shaped cooling hole. The simulations were performed for adiabatic and conjugate heat transfer models. Turbulence closure was investigated using three different turbulence models; the realizable k-e model, the SST k-ω model, as well as the v2 –f turbulence model. Results were obtained for a blowing ratio of one, and a coolant-to-mainflow temperature ratio of 0.54. The simulations used a dense, high quality, O-type, hexahedral grid with three different schemes of meshing for the cooling hole: hexahedral-, hybrid-, and tetrahedral-topology grid. The computed flow/temperature fields are presented, in addition to local, two-dimensional distribution of film cooling effectiveness for the adiabatic and conjugate cases. Results are compared to experimental data in terms of centerline film cooling effectiveness downstream cooling-hole, the predictions with realizable k-e turbulence model exhibited the best agreement especially in the region for (2 ≤ x/D ≤ 6). Also, the results show the effect of the conjugate heat transfer on the temperature (effectiveness) field in the film cooling hole region and, thus, the additional heating up of the cooling jet itself.Copyright