Body-fitted Grid System Research Articles

Development of a hybrid method for improved accuracy of mould filling computational models for castings

ABSTRACT Computational modelling of mould filling is a component of paramount importance for the design of gating systems of castings. The main methods are the body-fitted grid system and the Cartesian grid system. While the former comes with high accuracy but high calculation times, the latter is fast but less accurate. A hybrid method combining a three-dimensional Cartesian cut-cell method and a porous media approach has been further developed. The calculated filling patterns compared favourably with high-speed video images of sand mould filling during pouring of ductile iron, and with visual images recorded through an acrylic glass-water model device simulating the gravity die casting process. In addition, numerical experiments were conducted to demonstrate the improved filling accuracy of the developed hybrid method. To this goal, three different symmetry tests with and without sub-mesh were designed and evaluated. All tests convincingly demonstrated the computational time and accuracy advantages of the developed method.

Comparison of the blocked-off and embedded boundary methods in radiative heat transfer problems in 2D complex enclosures at radiative equilibrium

Radiative heat transfer in participating media at radiative equilibrium in two-dimensional complex geometries will be investigated using two Cartesian boundary treatments, i.e. the blocked-off method and the embedded boundary method. The main advantages of Cartesian formulation are to simplify grid generation and using efficient Cartesian solvers for complicated problems. Angular and spatial discretization of the radiative transfer equation are performed using the discrete ordinates method and the finite volume method, respectively. The accuracy of both methods in solution of radiative heat transfer problems in irregular geometries are verified by comparison with benchmark solutions from literatures. Then, in order to investigate all features of the blocked-off and embedded boundary methods, two radiative problems with complex enclosures that contain gray absorbing and emitting medium at radiative equilibrium are examined. The results obtained from the two methods are compared with each other as well as with results obtained by body-fitted grid system. It has been shown that for a medium at radiative equilibrium with Cartesian formulation, the embedded boundary method is the method of choice, especially for calculations near the complex boundaries. It should be noted that for optically thick media, both blocked-off and embedded boundary methods show poor performance.

Flow Simulation Around Cambered Airfoil by Using Conformal Mapping and Intermediate Domain in Lattice Boltzmann Method

In this paper the developed interpolation lattice Boltzmann method is used for simulation of unsteady fluid flow. It combines the desirable features of the lattice Boltzmann and the Joukowski transformation methods. This approach has capability to simulate flow around curved boundary geometries such as airfoils in a body fitted grid system. Simulation of unsteady flow around a cambered airfoil in a non-uniform grid for the first time is considered to show the capability of this method for modeling of fluid flow around complex geometries and complicated long-term periodic flow phenomena. The developed solver is also coupled with a fast adaptive grid generator. In addition, the new approach retains all the advantages of the standard lattice Boltzmann method. The Strouhal number, the pressure, the drag and the lift coefficients obtained from the simulations agree well with classical computational fluid dynamics simulations. Numerical studies for various test cases illustrate the strength of this new approach.

Mixing of electrokinetically-driven power-law fluids in zigzag microchannels

Purpose – The purpose of this paper is to study the mixing performance of the electrokinetically-driven power-law fluids in a zigzag microchannel. Design/methodology/approach – The Poisson-Boltzmann equation, the Laplace equation, the modified Cauchy momentum equation, and the convection-diffusion equation are solved to describe the flow characteristics and mixing performance of power-law fluids in the zigzag microchannel. A body-fitted grid system and a generalized coordinate transformation method are used to model the grid system and transform the governing equations, respectively. The transformed governing equations are solved numerically using the finite-volume method. Findings – The mixing efficiency of dilatant fluids is higher than that of pseudoplastic fluids. In addition, the mixing efficiency can be improved by increasing the width of the zigzag blocks or extending the total length of the zigzag block region. Originality/value – The results presented in this study provide a useful insight into potential strategies for enhancing the mixing performance of the power-law fluids in a zigzag microchannel. The results of this study also provide a useful source of reference for the development of efficient and accurate microfluidic systems.

Convective Transport Around a Rotating Square Cylinder at Moderate Reynolds Numbers

Two-dimensional numerical simulation is performed to analyze the thermofluidic transport around a rotating square cylinder in an unconfined medium. The convective transport originates as a consequence of the interaction between a uniform free-stream flow and the flow evolving due to the rotation of the sharp-edged body. A finite volume-based method and a body-fitted grid system along with the moving boundaries are used to obtain the numerical solution of the incompressible Navier–Stokes and energy equations. The Reynolds number based on the free-stream flow is considered in the range 10 ≤ Re ≤ 200, and the dimensionless rotational speed of the cylinder is kept 0 ≤ Ω ≤ 5. Depending on the Reynolds number and the rotational speed of the cylinder, the transport characteristics change. For the range 10 ≤ Re < 50, the flow remains steady irrespective of the rotational speed. In the range 50 ≤ Re ≤ 200, regular low-frequency Kármán vortex shedding (VS) is observed up to a critical rate of rotation (Ωcr ). Beyond Ωcr , the global convective transport shows a steady nature. The rotating circular cylinder also shows likewise degeneration of Kármán VS at some critical rotational speed. However, the heat transfer behavior varies significantly with a rotating circular cylinder. Such thermofluidic transport around a spinning square in an unconfined free-stream flow is reported for the first time.

Numerical Study of the Laminar Flow Past a Rotating Square Cylinder at Low Spinning Rates

The fluid dynamic interaction between a uniform free stream flow and the rotation induced flow from a sharp edged body is numerically investigated. A two-dimensional (2D) finite volume based computation is performed in this regard to simulate the laminar fluid flow around a rotating square cylinder in an unconfined medium. Body fitted grid system along with moving boundaries is used to obtain the numerical solution of the incompressible Navier–Stokes equations. The Reynolds number based on the free stream flow is kept in the range 10≤Re≤200 with a dimensionless rotational speed of the cylinder in the range 0≤Ω≤5. At low Re=10, the flow field remains steady irrespective of the rotational speed. For 50≤Re≤200, regular low frequency Kármán vortex shedding (VS) is observed up to a critical rate of rotation (Ωcr). Beyond Ωcr, the global flow shows steady nature, although high frequency oscillations in the aerodynamic coefficients are present. The rotating circular cylinder also shows likewise degeneration of Kármán VS at some critical rotational speed. However, significant differences can be seen at higher rotation. Such fluid dynamic transport around a spinning square in an unconfined free stream flow is reported for the first time.

Simulation of fluid flow in a body-fitted grid system using the lattice Boltzmann method

The ability of the interpolation supplemented lattice Boltzmann method (ISLBM) diminishes in simulation of fluid flow around complex geometries and it is nearly impossible to use this method in body-filled grid systems. In this paper, a developed version of the interpolation supplemented lattice Boltzmann method is proposed to remove the limitations of the original ISLBM. Combination of the ISLBM and the Joukowski transformation is the basis of the method. In fact, using the Joukowski transformation, the physical domain with a body-fitted grid system is mapped to the computational domain with a uniform Cartesian grid system such that the conventional ISLBM can be easily applied. The results are compared with those of a Navier-Stokes solver and there is good agreement between these two results.

Prediction and experimental verification of vortex flow in draft tube of Francis turbine based on CFD

The three-dimensional unsteady turbulent flow in the draft tube of a Francis turbine was calculated by using the SIMPLE algorithm with the body-fitted coordinate and tetrahedral grid system with the help of FLUENT software. This calculation was based on the Navier-Stokes equations and LES (large-eddy simulation) model. The flowing situations in the draft tube under three operating conditions of this turbine were successfully calculated. The PIV (particle image velocimetry) was used to measure the distribution of flow field at the draft tube cone section at 4 horizontal sections from the outlet of turbine runner. Specialized image analysis software was used to process the experimental results. Through comparison, the results of numerical simulations were in good coincidence with the experimental results.

Numerical approach to hole shape effect on film cooling effectiveness over flat plate including internal impingement cooling chamber

Numerical approach have been conducted on a flat, three-dimensional discrete-hole film cooling geometries that included the mainflow, injection tubes, impingement chamber, and supply plenum regions. The effects of blowing ratio and hole’s shape on the distributions of flow field and adiabatic film cooling effectiveness over a flat plate collocated with two rows of injection holes in staggered-hole arrangement were studied. The blowing ratio was varied from 0.3 to 1.5, while the density ratio of the coolant to mainstream is kept at 1.14. The geometrical shapes of the vent of the cooling holes are cylindrical round, simple angle (CYSA), forward-diffused, simple angle (FDSA) and laterally diffused, simple angle (LDSA). Diameter of different shape of cooling holes in entrance surface are 5.0 mm and the injection angle with the main stream in streamwise and spanwise are 35° and 0° respectively. Ratio of the length of the cooling holes and the diameter in the entrance surface is 3.5. The distance between the holes in the same row as well as to the next row is three times the diameter of hole in the entrance surface. The governing equation is the fully elliptic, three-dimensional Reynolds-averaged Navier–Stokes equations. The mesh used in the finite-volume numerical computation is the multi-block and body-fitted grid system. The simulated streamwise distribution of spanwise-averaged film cooling effectiveness exhibited that low Reynolds number k– ε model can give close fit to the experimental data of the previous investigators. Present study reveals that (1) the geometrical shape of the cooling holes has great effect on the adiabatic film cooling efficiency especially in the area near to the cooling holes. (2) The thermal-flow field over the surface of the film-cooled tested plate dominated by strength of the counter-rotating vortex pairs (CRVP) that generated by the interaction of individual cooling jet and the mainstream. For LDSA shape of hole, the CRVP are almost disappeared. The LDSA shape has shown a highest value in distribution of spanwise-averaged film cooling effectiveness when the blowing ratio increased to 1.5. It is due to the structure of the LDSA is capable of reducing the momentum of the cooling flow at the vent of the cooling holes, thus reduced the penetration of the main stream. (3) The structure of the LDSA can also increase the lateral spread of the cooling flow, thus improves the spanwise-averaged film cooled efficiency.

The effect of an insoluble surfactant on the skin friction of a bubble

In this paper, we develop a novel moving mesh method suitable for solving axisymmetric free-boundary problems, including the Marangoni effect induced by surfactant or temperature variation. This method employs a body-fitted grid system where the gas–liquid interface is one line of the grid system. We model the surfactant equation of state with a non-linear Langmuir law, and, for simplicity, we limit ourselves to the situation of an insoluble surfactant. We solve complicated dynamic boundary conditions accurately on the gas–liquid interface in the framework of finite-volume methods. Our method is used to study the effect of a surfactant on the skin friction of a bubble in a uniaxial flow. For the limiting case where the surface diffusivity is zero, the effect of a tangential stress generated by the surface tension gradient, allows us to explain a new phenomenon in high concentration regimes: larger surface tension, but also larger deformation. Furthermore, this condition leads to the formation of boundary layers and flow separation at high Reynolds numbers. The influence of these complex flow patterns is examined.

An arbitrary Lagrangian Eulerian method for moving-boundary problems and its application to jumping over water

We develop an ALE (Arbitrary Lagrangian Eulerian) moving mesh method suitable for solving two-dimensional and axisymmetric moving-boundary problems, including the interaction between a free-surface and a solid structure. This method employs a body-fitted grid system where the gas–liquid interface and solid–liquid interface are lines of the grid system, and complicated dynamic boundary conditions are incorporated naturally and accurately in a Finite-Volume formulation. The resulting nonlinear system of mass and momentum conservation is then solved by a fractional step (projection) method. The method is validated on the uniform flow passing a cylinder (a two-dimensional flow with a solid structure) and several problems of bubble dynamics (axi-symmetrical flows with a free surface) for both steady and unsteady flows. Good agreement with other theoretical, numerical and experimental results is obtained. A further application is the investigation of a two-dimensional mechanical strider (a mass-spring system) interacting with a water surface, demonstrating the ability of the method in handling the interaction between a solid structure and a free surface. We find that the critical compression required to jump off the water surface varies linearly with spring constant for stiff springs and algebraically with exponent 0.7 for weak springs.

A Numerical and Experimental Investigation of the Slot Film-Cooling Jet With Various Angles

Numerical simulations coupled with laser Doppler velocimetry (LDV) experiments were carried out to investigate a slot jet issued into a cross flow, which is relevant in the film cooling of gas turbine combustors. The film-cooling fluid injection from slots or holes into a cross flow produces highly complicated flow fields. In this paper, the time-averaged Navier-Stokes equations were solved on a collocated body-fitted grid system with the shear stress transport k−ω, V2F k−ϵ, and stress-ω turbulence models. The fluid flow and turbulent Reynolds stress fields were compared to the LDV experiments for three jet angles, namely, 30, 60, and 90 deg, and the jet blowing ratio is ranging from 2 to 9. Good agreement was obtained. Therefore, the present solution procedure was also adopted to calculations of 15 and 40 deg jets. In addition, the temperature fields were computed with a simple eddy diffusivity model to obtain the film-cooling effectiveness, which, in turn, was used for evaluation of the various jet cross-flow arrangements. The results show that a recirculation bubble downstream of the jet exists for jet angles larger than 40 deg, but it vanishes when the angle is &lt;30deg, which is in good accordance with the experiments. The blowing ratio has a large effect on the size of the recirculation bubble and, consequently, on the film cooling effectiveness. In addition, the influence of boundary conditions for the jet and cross flow are also addressed in the paper.

Higher-order schemes with CIP method and adaptive Soroban grid towards mesh-free scheme

A new class of body-fitted grid system that can keep the third-order accuracy in time and space is proposed with the help of the CIP (constrained interpolation profile/cubic interpolated propagation) method. The grid system consists of the straight lines and grid points moving along these lines like abacus – Soroban in Japanese. The length of each line and the number of grid points in each line can be different. The CIP scheme is suitable to this mesh system and the calculation of large CFL (>10) at locally refined mesh is easily performed. Mesh generation and searching of upstream departure point are very simple and almost mesh-free treatment is possible. Adaptive grid movement and local mesh refinement are demonstrated.

INVESTIGATION OF RADIATIVE HEAT TRANSFER IN COMPLEX GEOMETRIES USING BLOCKED-OFF, MULTIBLOCK, AND EMBEDDED BOUNDARY TREATMENTS

A prediction of radiative heat transfer in a complex geometry is performed using different boundary treatments such as blocked-off, spatial-multiblock, and embedded boundary methods. The formulation of embedded boundary treatment for finite volume is derived. The finite-volume method (FVM) is used to model the radiative transfer in an absorbing and emitting medium which is maintained at an isothermal condition and enclosed by cold and black walls. While the body-fitted grid system is used for the spatial multiblock treatment, the Cartesian grid system is chosen for the others. Their results are compared and discussed for three different cases, including a trapezoidal enclosure, a semicircular enclosure with internal block, and an incinerator-shaped enclosure. The accuracy obtained by application of each treatment is shown to be highly satisfactory. Consequently, each treatment is suitable for modeling the radiative heat transfer in the complex geometry. However, the solution obtained by the blocked-off treatment yields some errors compared with the others, since the Cartesian grid used in the blocked-off treatment cannot configure exactly the complex boundaries. Especially, the radiative heat flux on the nonorthogonal wall is largely underestimated due to its stepwise description of the wall.

레벨셋법을 이용한 이동 집중격자 생성법에 대한 연구

Eulerian개념을 사용한 격자계 내 임의의 경계면 주위 점성유동 해석에서, 운동하며 변형하는 경계면 근방 해의 정도를 향상시키기 위해서 격자생성시 경계면으로 격자점들을 집중시켜주는 레벨셋법에 바탕을 둔 격자변형법을 도입하였다. 본 연구에서는 격자점들을 경계면 근방으로 집중되는 정도를 용이하게 조절할 수 있도록 새로운 형태의 모니터함수를 제시하였다. 집중격자계를 사용함으로 얻어지는 향상된 해의 정도의 검증을 위하여 바닥에 고정된 반원 실린더 주위 정상유동에 대하여 가상경계법을 함께 사용하여 해석하였다. 수치계산결과는 물체적합 격자계를 사용해서 얻은 결과와 매우 잘 일치하였으며, 집중격자법을 사용하지 않은 해석결과보다 향상된 결과를 보여주었다. 수치계산의 또 다른 예제로서 다수의 고정된 물체주위 유동해석으로 확장 적용하여 공학적 유용성을 검증하였다. 마지막으로 이동 집중격자계의 생성법의 적용을 위해서 움직이면서 변형을 일으키는 2차원 기포상승문제를 해석하였다. 수치해석결과에서 격자점들은 매시간 기포의 변형에 맞추어 적합하게 집중된 형태를 잘 보여주었으며, 고정된 격자계를 사용한 결과와 잘 일치하였다. In order to improve the accuracy of the solution near the boundary in an analysis of viscous flow around an arbitrary boundary which move and be deformed using an Eulerian concept, a level-set based grid deformation method is introduced to concentrate grid points near the boundary. This paper presents a new monitor function which can easily control the level of the concentration of grid points along the boundary. Computations for steady flow around a semi-circular cylinder mounted on the bottom of the flow domain were carried out to check the improvement of the solution using the adaptive grid system with an immersed boundary method. The present numerical results show a good agreement with the solutions obtained by a body fitted grid system and more accurate solutions than those computed with non-adaptive grid system. For the validation of mechanical usefulness of the present method, an expanded analysis of flow around multi-body fixed in the flow domain was carried out. Finally, the present moving adaptive grid method was applied to a two-dimensional bubble rise problem. The computed results show well adapted grid points around the boundary of the bubble at every time and a good agreement with the result calculated by fixed grid system.

A numerical model for predicting bubble formation in a 3D fluidized bed

Fluidized bed systems have the potential to be widely used in the power generation, mineral processing and chemical industries. One factor limiting their increased use is the lack of adequate design techniques for scaling such systems. A model has been developed for simulating gas–solid fluidized bed plant. The model uses a multiphase Eulerian–Eulerian technique to predict the transient behaviour of fluidized bed systems. The commercial CFD code CFX is used as the computational framework for solving the discretized equations. To overcome the problem of accurate geometrical representation experienced in previous models a body fitted grid system is employed. The model is used to predict isothermal flow in a three-dimensional bubbling fluidized bed. Predictions of the three-dimensional model show bubble formation with gas bubbles or voids preferentially moving along the centre of the bed. Predicted behaviour is qualitatively consistent with experimental observations.

Transient conjugate heat transfer on a naturally cooled body of arbitrary shape

Transient conjugate heat transfer relating the heat conduction inside a solid body of arbitrary shape and the natural convection around the solid is studied in the present investigation. A single set of governing equations covering both solid and fluid is solved by using the weighting function scheme along with the NAPPLE algorithm and the SIS solver. All of the computations are performed on a Cartesian grid system. Numerical results of velocities and isotherms are presented for Pr = 7 and Gr = 10 7 and 10 8 . An interesting observation on formation, growth, merging and decay of several plumes is discussed. The solution shows that there is a dead fluid inside the cavity under the solid body due to an upward temperature gradient. The maximum temperature thus occurs there in a late stage of the heat transfer process. An increase in the Grashof number is found to enhance the heat transfer. Through this study, the present numerical method is shown to have a good performance for conjugate heat transfer on solid body of arbitrary shape without the need of a body-fitted grid system.

An experimental and numerical study on the flow over two-dimensional hills

Abstract An experimental and numerical investigation on the flow over two-dimensional hilly terrain is presented. Experiments for single hills and continous double hills are performed in a boundary-layer wind tunnel, and mean velocity profiles, turbulence characteristics, and surface pressure distributions are measured. The numerical model developed for the present work is based on the finite-volume-method and the SIMPLEC algorithm with a non-orthogonal body-fitted grid system. Several turbulence models are tested for the validation of the prediction accuracy in separated flow cases. Comparisons of the mean velocity profiles and surface pressure distributions between the numerical predictions and the measurements show good agreement. The linear theory provides generally good prediction of speed-up characteristics at the hill top for the hill slope of 0.3, which is defined as the ratio of the hill height to the base length at the upwind mid-height of the hill. Flow separation occurs in the hill slope of 0.5, and the measured reattachment points are compared with the numerical prediction. The low-Reynolds-number model with an orthogonal grid is found to predict the separated flow better than the other turbulence models.

Vortex Shedding from a Rotating Regular Polygonal Prism in a Uniform Flow.

Synchronized vortex shedding from a rotating regular polygonal prism in a uniform flow is investigated numerically and experimentally The Karman vortex in a wake behind the prism rotating in a wind tunnel is investigated by using a hot-wire anemometer. The experimental results show that the Karman vortex shedding is synchronized with the rotation of the prism for a wide range of spill parameters. In numerical analysis, a body-fitted grid system that coincides with a moving boundary of the rotating prism is installed and the 2-dimensional Navier-Stokes equation is solved using the finite difference method. The numerical results show good agreement with the experimental ones. It is found from the experimental and numerical results that the general relation-ship between spin parameter S and Strouhal number St for an N-sided rotating polygonal prism is St=N/(κπ)·S(κ=1, 2, 3···).

Numerical Analysis of Flow between Eccentric Rotating Cylinders.

The flow between eccentric rotating cylinders is studied numerically using a curvilinear coordinate system. The outer cylinder is fixed and the inner cylinder rotates at a constant speed. The three-dimensional Navier-Stokes equations and the equation of continuity are solved. A body-fitted grid system is generated and the basic equations in a physical domain are transformed into a calculational domain and solved by means of a finite difference method. A series of computations is carried out at Re=130 to 500 and the eccentric ratio is 0.1 to 0.5. The flow patterns, velocity vectors and vorticities are obtained. The gap width between two eccentric cylinders changes in a circumferential direction and the maximum Taylor vortex is found in the region where the gap width becomes narrow in the flow direction and the vortex becomes weakest in the region where the gap width is expanding. When the Re number is slightly larger than the critical Re number, the Taylor vortex flow and parallel stable flow coexist in a fluid between two eccentric rotating cylinders.