Tony W. H. Sheu

A numerical revisit of backward-facing step flow problem

In the present study we take a fresh look at a laminar flow evolving into a larger channel through a step configured in a backward-facing format. We conduct steady three-dimensional Navier–Stokes flow analysis in the channel using the step geometry and flow conditions reported by Armaly et al. This allows a direct comparison with the results of physical experiments, thus serving to validate the numerical results computed in the range of 100⩽Re⩽1000. Results show that there is generally excellent agreement between the present results and the experimental data for Re=100 and 389. Fair agreement for Re=1000 is also achieved, except in the streamwise range of 15⩽x⩽25. The main difference stems from the fact that the roof eddy is not extended toward the midspan in the channel with a span width 35 times of the height of the upstream channel. In the present study we also reveal that the flow at the plane of symmetry develops into a two-dimensional-like profile only when the channel width is increased up to 100 t...

Side wall effects on the structure of laminar flow over a plane-symmetric sudden expansion

Abstract Computational investigations have been performed in order to study the side-wall effect on a fluid downstream of a channel expansion which is plane. The expansion ratio under investigation is 3 and the aspect ratios are 3, 3.5, 3.75, 4, 5, 6, 7, 8, 9, 10, 12, 18, 24, 48, in the three-dimensional analyses. For the flow with a value of Re=60 , results show symmetric nature of the flow when the channel aspect ratio has a value less than 3.5. Beyond this critical aspect ratio, flow symmetry can no longer be sustained due to the Coanda effect. This confirms the experimental observation that a decrease in aspect ratio has a stabilizing effect. Unless the aspect ratio is increased further to a value above 12, flow in the third dimension plays an essential role to characterize the inherent nature of the flow. In this study, we also confine ourselves to studying flow separation, reattachment, and recirculation by employing a theoretically rigorous theory of topology. Much insight into the vortical flow structure can be revealed from limiting streamlines, on which critical points, such as spiral focal points and saddles, are plotted.

Eddy structures in a transitional backward-facing step flow

In the present study, flow simulation has been carried out in a backward-facing step channel defined by an expansion ratio of 2.02 and a spanwise aspect ratio of 8 to provide the physical insight into the longitudinal and spanwise flow motions and to identify the presence of Taylor-Gvortices. The Reynolds numbers have been taken as 1000 and 2000, which fall in the category of transitional flow. The present simulated results were validated against the experimental and numerical data and the comparison was found to be satisfactory. The simulated results show that the flow becomes unsteady and exhibits a three-dimensional nature with the Kelvin-Helmholtz instability oscillations and Taylor-Glongitudinal vortices. The simulated data were analysed to give an in-depth knowledge of the complex interactions among the floor and roof eddies, and the spiralling spanwise flow motion. Destabilization of the present incompressible flow system, with the amplified Reynolds number due to the Kelvin-Helmholtz and Taylor-G¨ ortler instabilities, is also highlighted. A movie is available with the online version of the paper.

Flow topology in a steady three-dimensional lid-driven cavity

Abstract We present in this paper a thorough investigation of three-dimensional flow in a cubical cavity, subject to a constant velocity lid on its roof. In this steady-state analysis, we adopt the mixed formulation on tri-quadratic elements to preserve mass conservation. To resolve difficulties in the asymmetric and indefinite large-size matrix equations, we apply the BiCGSTAB solution solver. To achieve stability, weighting functions are designed in favor of variables on the upstream side. To achieve accuracy, the weighting functions are properly chosen so that false diffusion errors can be largely suppressed by the equipped streamline operator. Our aim is to gain some physical insight into the vortical flow using a theoretically rigorous topological theory. To broaden our understanding of the vortex dynamics in the cavity, we also study in detail the longitudinal spiralling motion in the flow interior.

An element-by-element BICGSTAB iterative method for three-dimensional steady Navier-Stokes equations

Construction of a stabilized Galerkin upwind finite element model for steady and incompressible Navier-Stokes equations in three dimensions is the main theme of this study. In the time-independent context, the weighted residuals statement is kept biased in favor of the upstream flow direction by adding an artificial damping term of physical plausibility to the Galerkin framework. This upwind approach has significant advantage of seeking solutions free from cross-stream diffusion error. Finite element solutions have been found by mixed formulation, implemented in quadratic cubic elements which are characterized as possessing the so-called LBB (Ladyzhenskaya-Babuska-Brezzi) condition. An element-by-element BICGSTAB solution solver is intended to alleviate difficulties regarding the asymmetry and indefiniteness arising from the use of a mixed formulation for incompressible fluid flows. The developed three-dimensional finite element code is first rectified by solving a problem amenable to analytic solution. A well-known lid-driven cavity flow problem in a cubical cavity is also studied.

Some physical insights into a two-row finned-tube heat transfer

A three-dimensional numerical study was conducted to broaden our knowledge of the conjugate heat transfer in a finned-tube heat exchanger element. A finite volume discretization method and a SIMPLE-based solution algorithm were applied to working differential equations and their discrete counterparts for computation of gas velocities and temperatures. Since the heat transfer between the gaseous and solid phases is determined by the complex flow structure, calculations for three-dimensional thermally and hydrodynamically developing laminar flows are performed by iteratively solving the heat conduction equation for the plate fin and conservation equations for the gas phase via the coupling boundary condition. The emphasis of this study is directed toward numerical exploration of the flow structure. To this end, the underlying topological theory shows the promise of being a powerful tool for the study of flow details. In dry conditions, some insight in the heat-transfer capability of the two-row finned-tube heat exchanger can be gained by examining the span-averaged Nusselt number and span-averaged pressure drop in the flow passage.

Heat transfer in a conjugate heat exchanger with a wavy fin surface

Abstract A three-dimensional computational study on conjugate heat exchangers was conducted. Attention was specifically directed towards studying extended surfaces used to increase heat transfer. The strategy adopted in the present investigation of forced convection in a flow passage was to use the finite volume method. Our implementation incorporated a SIMPLE-based semi-implicit solution algorithm which was applied to working equations formulated within the single-phase catalog. The analysis allowed for marked changes in thermodynamic and flow properties. To justify using the proposed numerical model to simulate this conjugate heat transfer problem, we considered first a heat exchanger with a plane fin simply because experimental data are available for comparison. This validation study was followed by a study of how a newly designed fin pattern can provide increased heat transfer. The efficiency has been judged by considering several aspects, namely the span-averaged pressure drop, Nusselt number and heat flux. To better illuminate the flow and heat transfer characteristics in a flow passage bounded by two fins having wavy geometries, we have plotted solutions in a three-dimensional format.

Simulation of nonlinear Westervelt equation for the investigation of acoustic streaming and nonlinear propagation effects

This study investigates the influence of blood flow on temperature distribution during high-intensity focused ultrasound (HIFU) ablation of liver tumors. A three-dimensional acoustic-thermal-hydrodynamic coupling model is developed to compute the temperature field in the hepatic cancerous region. The model is based on the nonlinear Westervelt equation, bioheat equations for the perfused tissue and blood flow domains. The nonlinear Navier-Stokes equations are employed to describe the flow in large blood vessels. The effect of acoustic streaming is also taken into account in the present HIFU simulation study. A simulation of the Westervelt equation requires a prohibitively large amount of computer resources. Therefore a sixth-order accurate acoustic scheme in three-point stencil was developed for effectively solving the nonlinear wave equation. Results show that focused ultrasound beam with the peak intensity 2470 W/cm(2) can induce acoustic streaming velocities up to 75 cm/s in the vessel with a diameter of 3 mm. The predicted temperature difference for the cases considered with and without acoustic streaming effect is 13.5 °C or 81% on the blood vessel wall for the vein. Tumor necrosis was studied in a region close to major vessels. The theoretical feasibility to safely necrotize the tumors close to major hepatic arteries and veins was shown.

AN INCOMPRESSIBLE NAVIER-STOKES MODEL IMPLEMENTED ON NONSTAGGERED GRIDS

The present study aims to develop an effective finite-difference model for solving incompressible Navier-Stokes equations. For the sake of programming simplicity, discretization of equations is made on nonstaggered grids without oscillatory solutions arising from the decoupling of the velocity and pressure fields. For the sake of computational efficiency, both segregated and alternating direction implicit (ADI) solution algorithms are employed to reduce the matrix size and, in turn, the CPU time. For the sake of numerical accuracy, a convection-diffusion-reaction finite-difference scheme is employed to provide nodally exact solutions in each ADI solution step. The convective instability problem is thus eliminated, since each convective term is modeled analytically even in multidimensional cases. The validity of the proposed numerical model is rigorously justified by solving one- and two-dimensional problems, which are amenable to analytical solutions. The simulated solutions for the scalar prototype equation agree well with the exact solutions and provide a very high spatial rate of convergence. The same is true for the simulated results of the Navier-Stokes equations.

Topological flow structures in backward-facing step channels

Abstract The present paper is intended to solve the steady-state Navier-Stokes equations for different Reynolds numbers. Through out this paper, the incompressible fluid will be considered in three-dimensional channels with different spans. The flow field under investigation was characterized as having a backward-facing step across which a fully-developed three-dimensional channel flow expanded into the channel with an expansion ratio of 1.9432. Numerical solutions for this backward-facing step problem were obtained on the basis of the step height, 0.9423, various spans, taking on values up to 10, and Reynolds numbers as high as 800. Of the different flow conditions that were considered, we elaborate on the flow topology under the conditions of an intermediate Reynolds number, Re = 389, and the largest width of the channel, 10. Following Lighthill [Lighthill, M., Attachment and separation in three-dimensional flow. In Laminar Boundary Layers , Vol. 2(6), ed. L. Rosenhead, II. Oxford University Press, 1963, pp. 72–82.] [1], we apply topology theory, which provides a rigorous mathematical foundation for studying kinematically possible flows. The present computational results, together with the inferred flow topology, reveal details of the flow structure which suggest a mechanism for the development of strongly three-dimensional flow with increasing Reynolds numbers. The computation of ‘oil-flow’ streamlines improves the visualization of the flow field and helps sketch the complicated flow patterns by clarifying the three-dimensional flow separation just behind the step. The scope of this enhancement to improved visualization of flow structure is also extended to the flow reattachment on the floor as well as the roof recirculatory flow pattern, manifested itself by the upstream separation and downstream reattachment surfaces. Notably addressed is the separation-reattachment phenomenon emanating only from the roof near the two side walls.