Algebraic Grid Generation Research Articles

CFD investigation of supersonic two parallel expansions nozzles

This paper aims to validate the design contours and result parameters of the newly developed Dual Expansion Nozzle (DEN) using the Fastran software when it enters the non-adaptation regime with the increase in altitude and variation of the Nozzle Pressure Ratio (NPR). DEN design and Computational Fluid Dynamic (CFD) application utilize the High Temperature (HT) model defined by a calorically imperfect and thermally perfect gas, providing good accuracy compared to PG model. The computational domain is decomposed into subdivisions of structured grids using the algebraic grid generator software CFD-GEOM. The mesh convergence was analyzed using three mesh resolutions: coarse, medium, and fine. A comparison was made between the two conventional nozzles, MLN and the newly developed BPN, as both are currently used in aerospace propulsion to enhance aerodynamic performance. If a rocket engine operates under strongly over-expanded conditions with ambient pressure considerably higher than the nozzle exit pressure, and by increasing NPR value, the flow separates from the wall and can lead to high side loads. These loads are reduced in DEN. The flow separation point is observed close to the exit section for the DEN in comparison with Minimum Length Nozzle (MLN) and Best Performance Nozzle (BPN) for the same NPR, resulting in low flow separation and low side load effects for DEN. This result demonstrates a small loss in the exit Mach number, leading to less loss in the thrust coefficient and a small effect of boundary layer friction for DEN. Enhanced over-expanded aerodynamics is observed in DEN compared to MLN and BPN. The application is made for air with an exit Mach number equal to 3.00.

Application of numerical grid generation for improved CFD analysis of multiphase screw machines

Algebraic grid generation is widely used for discretization of the working domain of twin screw machines. Algebraic grid generation is fast and has good control over the placement of grid nodes. However, the desired qualities of grid which should be able to handle multiphase flows such as oil injection, may be difficult to achieve at times. In order to obtain fast solution of multiphase screw machines, it is important to further improve the quality and robustness of the computational grid. In this paper, a deforming grid of a twin screw machine is generated using algebraic transfinite interpolation to produce initial mesh upon which an elliptic partial differential equations (PDE) of the Poisson’s form is solved numerically to produce smooth final computational mesh. The quality of numerical cells and their distribution obtained by the differential method is greatly improved. In addition, a similar procedure was introduced to fully smoothen the transition of the partitioning rack curve between the rotors thus improving continuous movement of grid nodes and in turn improve robustness and speed of the Computational Fluid Dynamic (CFD) solver. Analysis of an oil injected twin screw compressor is presented to compare the improvements in grid quality factors in the regions of importance such as interlobe space, radial tip and the core of the rotor. The proposed method that combines algebraic and differential grid generation offer significant improvement in grid quality and robustness of numerical solution.

Algebraic generation of single domain computational grid for twin screw machines. Part I. Implementation

Special attention is required for generation of computational grids in highly deforming working chambers of twin screw machines for 3D CFD calculations. Two approaches for customised grid generation are practically available. The first is an algebraic grid generation and the second is a differential decomposition method. This paper reports on new developments in the algebraic approach that has the advantages associated with both algebraic and differential methods. Two control functions are introduced for regularisation of the initial algebraic distribution. One is based on an analytical control function in transformed coordinate system while the other uses background blocking structure in order to guide the initial algebraic distribution towards a single computational mesh. This paper presents implementation and grid characteristics of these new functions. Developed grids have been tested and results from flow calculations on a dry air compressor have been validated in part II of the paper [29].It was possible to achieve two distinct characteristics desirable in a twin screw rotor domain mesh. Firstly, it is possible to independently control grid refinement in the interlobe region thereby providing better accuracy in representation of the leakage gaps. Secondly and most importantly, it is possible now to eliminate the non-conformal interface between the two rotor domains thereby producing a single domain structured grid for the rotors, while still maintaining the fully hexahedral cell topology. An improvement in the global orthogonality of the cells was achieved. Despite of a decrement in the Face warp quality, aspect ratio of cells retained similar scale.

Numerical Investigation of Hydromagnetic Effect on the Natural Convection Heat Transfer from Circular Cylinder in an Enclosed Enclosure

The natural convection heat transfer from horizontal circular cylinder placed in a square enclosure is investigated numerically. The study deals with the effect of magnetic field on the flow and heat transfer characteristics. The investigation employs different Hartman numbers (0, 50, 100, 200), different Rayleigh numbers (103, 104, and 105) with constant enclosure width to cylinder diameter ratios W/D = 2.5. The study included the solving of the governing equations in the form of the vorticity-stream function to be fitted with coordinate system. The algebraic grid generation is used to generate initial transformation. The elliptic grid generation is used to fit with physical domain between the heated horizontal cylinder and the enclosure into a computational domain. The resulting equations are solved using finite difference method that based on the finite volume. The research studied the influence of the variation the Hartman number on the local and average Nusselt numbers, flow patterns and temperature distributions with different Rayleigh numbers. The effect of Hartman numbers on the flow patterns and temperature distributions will be displayed using streamlines and isotherms. The results show that the conduction heat transfer is the dominant mode for low Rayleigh numbers. The convection heat transfer is the dominant mode of the heat transfer for high Rayleigh number in absence of the magnetic effect. The convection heat transfer convert to conduction mode at high Rayleigh numbers due to the effect of the magnetic field in the fluid. Also, the results show that the behavior of the local Nusselt numbers for Ha = 0 are unique and differ from those of other higher Hartman numbers for all Rayleigh numbers.

The use of shock-detecting sensor to improve the stability of Lattice Boltzmann Model for high Mach number flows

Attempts to simulate compressible flows with moderate Mach number to relatively high ones using Lattice Boltzmann Method (LBM) have been made by numerous researchers in the recent decade. The stability of the LBM is a challenging problem in the simulation of compressible flows with different types of embedded discontinuities. The present study proposes an approach for simulation of inviscid flows by a compressible LB model in order to enhance the robustness using a combination of Essentially NonOscillatory (ENO) scheme and Shock-Detecting Sensor (SDS) procedure. A sensor is introduced with adjustable parameters which is active near the discontinuities and affects less on smooth regions. The validity of the improved model to capture shocks and to resolve contact discontinuity and rarefaction waves in the well-known benchmarks such as, Riemann problem, and shock reflection is investigated. In addition, the problem of supersonic flow in a channel with ramp is simulated using a skewed rectangular grid generated by an algebraic grid generation method. The numerical results are compared with analytical ones and those obtained by solving the original model. The numerical results show that the presented scheme is capable of generating more robust solutions in the simulation of compressible flows and is almost free of oscillations for high Mach numbers. Good agreements are obtained for all problems.

Numerical Investigation of Prandtl Number Effects on The Natural Convection Heat Transfer From Circular Cylinder in An Enclosed Enclosure

In the present work, the natural convection heat transfer from horizontal circular cylinder situated in a square enclosure is investigated numerically. The work investigates the effect of Prandtl numbers on the flow and heat transfer characteristics. The study uses different Prandtl numbers (0.03, 0.7, 7, and 50), different Raylieh numbers (104, 105, and 106) and different enclosure width to cylinder diameter ratios W/D (1.667, 2.5 and 5). The work included the solution of the governing equations in the vorticity-stream function formulation which were transformed into body fitted coordinate system. The transformations are based initially on algebraic grid generation and elliptic grid generation to map the physical domain between the heated horizontal cylinder and the enclosure into a computational domain. The disecritization equation system are solved by using finite difference method. The code build using Fortran 90 to execute the numerical algorithm.The results were compared with previous numerical results, which showed good agreement. The effect of Prandtl number variation on the average Nusselt numbers, flow patterns and isotherms with different Raylieh numbers and enclosure width ratios were investigated. The flow patterns and temperature distributions are presented by means of streamlines and isotherms, respectively. The results show that the streamlines and isotherms for Pr=0.03 are unique and differ from those of other higher Prandtl numbers for all enclosure widths and Ra≥105. The streamlines and isotherms for Pr≥0.7 are nearly similar and independent of Prandtl number. The same behaviors as streamlines and isotherms occur with Nusselt number for lower and higher values of Prandtl numbers with all ratios of enclosure width to cylinder diameter.

Deforming grid generation and CFD analysis of variable geometry screw compressors

The most common type of twin screw machines are twin screw compressors. These normally contain rotors of uniform pitch and profile along the rotor length. However, in some cases such as in twin screw vacuum pumps with very large pressure ratios, the variable pitch rotors are often used to improve efficiency. The limited use of rotors with variable pitch and/or section profile is mainly due to manufacturing constraints. In order to analyse the performance of such machines by means of Computational Fluid Dynamics (CFD), it is necessary to produce a numerical mesh capable of calculating 3D transient fluid flows within their working domains.An algebraic grid generation algorithm applicable to unstructured grid, Finite Volume Method (FVM) for variable pitch and variable profile screw machines is described in this paper. The grid generation technique has been evaluated for an oil free air compressor with “N” profile rotors of 3/5 lobe configuration. The performance was obtained by calculations with commercial CFD code. The grid generation procedure provides mesh of the required quality and results from CFD calculations are presented to compare performance of constant pitch rotors, variable pitch rotors and variable profile rotors. The variable pitch and variable profile rotors achieve steeper internal pressure rise and a larger discharge area for the same pressure ratio. Variable pitch rotors achieve reduced sealing line length in high pressure domains.

Numerical Analysis of the Internal Flow in an Annular Flow Channel Type Oil Damper

The purpose of the present study is to clarify the fluid flow of an oil damper through numerical analysis in order to obtain an exact value of the damping coefficient of an oil damper. The finite difference method (FDM) was used to solve the governing equation of the fluid flow generated by a moving piston. Time steps evolved according to the fractional step method, and the arbitrary Lagrangian–Eulerian (ALE) method was adopted for the moving boundary. In order to stabilize the computation in the moving boundary problem, a masking method with a single block grid system was used. In other words, algebraic grid generation using a stretching function was used for the moving piston in the cylinder of the oil damper. The time-dependent coordinate system in the physical domain, which coincides with the contour of the moving boundary, is transformed into a fixed rectangular coordinate system in the computational domain. The computational results were compared with experimentally obtained results and the approximate analytical solution. The results of the present analysis exhibit good agreement with the experimental results over various widths of the annular flow channel between the piston and cylinder.

Free surface flow around a ship model using an interface-capturing method

A free surface, turbulent flow code around ship hulls using the Slightly Compressible Flow formulation is used for simulating the flow around a Wigley hull. The code uses finite differences and the numerical scheme was marched in time using the 2nd order explicit Runge–Kutta method. Turbulent flows were simulated using the Baldwin–Lomax turbulence model. The grid is generated using an algebraic grid generator. The free surface treatment method is an interface-capturing one, on a fixed grid on which the kinematic free surface equation is solved. Several boundary conditions were implemented and their behavior was assessed. The flow conditions are the ones for which experimental and numerical results exist. The results show very good agreement with the experimental results.

An investigation of natural convection in a three dimensional tapered chimney without Boussinesq assumption

Natural convection in a three dimensional tapered chimney is investigated numerically. In order to investigate the natural convection under a high temperature difference situation, Boussinesq assumption is not adopted instead of consideration of the compressibility of fluid. Methods of Roe scheme, preconditioning and dual time stepping are used to solve governing equations for a low speed compressible flow. Coordinates transformation of algebraic grid generation and non-reflecting boundary condition are used to facilitate computation processes. The results reveal that in the expanding duct local Nusselt numbers distributed around the corner region are larger than those distributed around the central region. An available correlation equation is proposed and is well consistent with the numerical results.

Numerical Investigation of Natural Convection Heat Transfer from Square Cylinder in a Vented Enclosure

In the present work, the natural convection heat transfer from horizontal cylinder with square cross section situated in a square enclosure, vented symmetrically from the top and the bottom was investigated numerically. The work investigate the effect of the Ra, enclosure width and opening size of the enclosure on the streamlines, isotherms and heat transfer results. The numerical work included the solution of the governing equations in the vorticity-stream function formulation which were transformed into body fitted coordinate system. The transformations are based initially on algebraic grid generation and elliptic grid generation to map the physical domain between the heated horizontal cylinder and the vented enclosure into a computational domain. A hybrid scheme finite volume based finite difference method was used. The study included the following ranges of the studied variables:- 0 < Ra ≤ 6.5× 105 1.5 ≤ W/H ≤ 4 0.375 < O/H ≤ 4 The numerical results were compared with experimental results, which showed good agreement. The effect of cylinder cross section, Ra, enclosure width, and opening size on the Nu, mass flowrate, flow patterns and isotherms were investigated. The results show that the cylinder cross section has a large influence on the results especially the Nu. The Nu is proportional with Ra and inversely proportional with enclosure width and opening size. The flow patterns and isotherms display the flow and temperature behaviors with changing studied variables. The results show that the starting of natural convection heat transfer depended on the cylinder cross-section, enclosure width and opening size in addition with Ra. In addition, the results display that the hydrodynamic and thermal boundary layer thickness decreases with increasing Ra. Nomenclature

Natural convection in a cavity with a sloping upper surface filled with an anisotropic porous material

Steady natural convection in a viscous incompressible fluid flowing in a vertical permeable non-rectangular cavity is considered. The permeability of the porous medium and the thermal diffusivity of the fluid are assumed to be anisotropic in nature. Using the Boussinesq approximation and the Darcy law to model the porous medium, the governing equations—recast in terms of a stream function—are solved numerically. An algebraic grid generation technique is employed to transform the non-rectangular domain to a square domain. The influence of a number of parameters, arising due to geometrical and physical aspects, is discussed in relation to the flow and heat transfer.

An investigation of compressible forced convection in a three dimensional tapered chimney by CUDA computation platform

Compressible forced convection in a three dimensional tapered chimney is investigated numerically. Schemes of Roe and preconditioning are adopted to resolve compressible flow problem under a low speed velocity situation. Coordinates transformation of algebraic grid generation and non-reflecting boundary condition is used to execute the coordinated transformation of the tapered duct and decrease the computation domain, respectively. Computing procedures are performed on the CUDA computation platform which is developed recently and is a very effective tool. Due to the variation of the geometry of three dimensional tapered chimney, the flow velocity increases in the central region and decreases near the corner region. Therefore, heat transfer rates are enhanced remarkably in the central region and become inferior abruptly near the corner region. The efficiency of CUDA computation platform is marvelous, and the possibility of capability of super computation of individualization may be realized by using the CUDA computation platform.

A Fast Algebraic Grid Generation Algorithm Base on an Improved Kamaugh Map

A fast algebraic grid generation algorithm is proposed and a improved Kamaugh map data structure for the new algorithm is designed in the paper. The new created vertex inside of the region is based on algebraic operation of the corner vertices and the middle vertices of boundary broken lines on the region and the program data structure is a improved Kamaugh map data structure, so the algorithm has the merit of simple algebraic formula, quickly executing speed and powerful technology for the four-sided region enclosed by four broken lines, the algorithm is realized on the four-sided region and B-spline surface on the complex surface. The algorithm is extensively applicable for shape modeling in computer aided geometric design, industrial prototype design and reverse engineering.

An assessment of the boundary conditions for free surface simulations using an interface-capturing method

A free surface, turbulent flow code around ship hulls using the Slightly Compressible Flow formulation is used for simulating the flow around a Wigley hull. The code uses finite differences and the numerical scheme was marched in time using the 2nd order explicit Runge-Kutta method. Turbulent flows were simulated using the Baldwin-Lomax turbulence model. The grid is generated using an algebraic grid generator. The free surface treatment method is an interface-capturing one, on a fixed grid on which the kinematic free surface equation is solved. Several boundary conditions were implemented and their behavior was assessed. The flow conditions are the ones for which experimental and numerical results exist. The results show good agreement with the experimental results.

Turbulent free-surface flow around a Wigley hull using the slightly compressible flow formulation

The slightly compressible flow formulation is applied to the free-surface, three-dimensional turbulent flow around a Wigley hull. Two turbulence models (large eddy simulation and Baldwin–Lomax) are used and compared. The simulation conditions are the ones for which experimental and numerical results exist. The computational grid is built using an algebraic grid generator with the model fixed in space. The codes use the interface-capturing technique for computing the free-surface displacements and the Beam and Warming scheme for marching in time the numerical model. The results compare well with the experimental data available.

Numerical Analysis of the Internal Flow in an Oil Damper (Analysis Based on the Two-Dimensional Cylindrical Coordinates to the Damper with Annular Cross-Sectional Flow Channel)

The purpose of this study is to know the fluid flow of an oil damper by a numerical analysis, so as to obtain the exact values of the damping coefficient of an oil damper. The Finite Difference Method (FDM) was used for solving the governing equation of the fluid caused by the moving piston. Each time step proceeded in the Fractional Step method, and the Arbitrary Lagrangian-Eulerian (ALE) method was adopted for the moving boundary. In order to stabilize the computation in the moving boundary problem, we employed the masking method with a single block grid system. That is, the algebraic grid generation using stretching function was used for the moving piston. The time dependent coordinate system in physical domain which coincides with a contour of moving boundary is transformed into a fixed rectangular coordinate system in computational domain. The computational results were compared with the experimental ones and the approximate analytical solution. The results of present analysis show good agreement with the experimental results over a wide range of piston diameter.

Application of Gas-Kinetic BGK Scheme for Solving 2-D Compressible Inviscid Flow around Linear Turbine Cascade

Fluid flows within turbomachinery tend to be extremely complex. Understanding such flows is crucial in the effort to improve current turbomachinery designs. Hence, computational approaches can be used to great advantage in this regard. In this paper, gas-kinetic BGK (Bhatnagar-Gross-Krook) scheme is developed for simulating compressible inviscid flow around a linear turbine cascade. BGK scheme is an approximate Riemann solver that uses the collisional Boltzmann equation as the governing equation for flow evolutions. For efficient computations, particle distribution functions in the general solution of the BGK model are simplified and used for the flow simulations. Second-order accuracy is achieved via the reconstruction of flow variables using the MUSCL (Monotone Upstream-Centered Schemes for Conservation Laws) interpolation technique together with a multistage Runge-Kutta method. A multi-zone H-type mesh for the linear turbine cascades is generated using a structured algebraic grid generation method. Computed results are compared with available experimental data and found to be in agreement with each other. In order to further substantiate the performance of the BGK scheme, another test case, namely a wedge cascade, is used. The numerical solutions obtained via this test are validated against analytical solutions, which showed to be in good agreement.

An algebraic grid optimization algorithm using condition numbers

In this paper we present an algorithm able to provide geometrically optimal algebraic grids by using condition numbers as quality measures. In fact, the solution of partial differential equations (PDEs) to model complex problems needs an efficient algorithm to generate a good quality grid since better geometrical grid quality is gained, faster accuracy of the numerical solution can be kept. Moving from classical approaches, we derive new measures based on the condition numbers of appropriate cell matrices to control grid uniformity and orthogonality. We assume condition numbers in appropriate norms as building blocks of objective functions to be minimized for grid optimization. This optimization procedure improves the mixed algebraic grid generation method first discussed in [C. Conti, R. Morandi, D. Scaramelli, Using discrete uniformity property in a mixed algebraic method, Appl. Numer. Math. 49 (4) (2004) 355–366. [3]]. The whole algorithm is able to cheaply generate optimal algebraic grids providing optimal location of the control points defining a small set of free parameters in the tensor product of the mixed algebraic method.

An algebraic grid optimization algorithm using condition numbers

In this paper we present an algorithm able to provide geometrically optimal algebraic grids by using condition numbers as quality measures. In fact, the solution of partial differential equations (PDEs) to model complex problems needs an efficient algorithm to generate a good quality grid since better geometrical grid quality is gained, faster accuracy of the numerical solution can be kept. Moving from classical approaches, we derive new measures based on the condition numbers of appropriate cell matrices to control grid uniformity and orthogonality. We assume condition numbers in appropriate norms as building blocks of objective functions to be minimized for grid optimization. This optimization procedure improves the mixed algebraic grid generation method first discussed in [C. Conti, R. Morandi, D. Scaramelli, Using discrete uniformity property in a mixed algebraic method, Appl. Numer. Math. 49 (4) (2004) 355-366. [3]]. The whole algorithm is able to cheaply generate optimal algebraic grids providing optimal location of the control points defining a small set of free parameters in the tensor product of the mixed algebraic method.