Rectangular Computational Domain Research Articles

An efficient shock capturing scheme for two-dimensional, open channel, unsteady flows in a generalised coordinate system

An efficient algorithm based on flux difference splitting is presented for the solution of two-dimensional, open channel flows. A transformation maps a nonrectangular, physical domain into a rectangular one. The governing equations are then the shallow water equations, including terms of slope and friction, in a generalised coordinate system. A regular mesh on a rectangular computational domain can then be employed. The resulting scheme is efficient, has good jump capturing properties and the advantage of using boundary/body-fitted meshes.

Thermohydrodynamic Lubrication Analysis Incorporating Thermal Expansion Across the Film

The study reported in this paper deals with the development of a thermohydrodynamic computational procedure for evaluating the pressure, temperature and velocity distributions in fluid films with fixed geometry between the stationary and moving bearing surfaces. The velocity variations and the heat generation are assumed to occur in a central zone with the same length and width as the bearing but with a significantly smaller thickness than the fluid film thickness. The thickness of the heat generation (shear) zone is developed empirically for the best fit with experimentally determined peak pressures for a journal bearing with a fixed film geometry operating in the laminar regime. A transient thermohydrodynamic computational model with a transformed rectangular computational domain is utilized. The analysis can be readily applied to any given film geometry. The computed distribution of the pressure in the film is in excellent agreement with the experimental findings for different oils and speeds. The developed procedure gives an analytical basis for explaining the “Fogy effect” where significant pressures can be generated in slider bearings with parallel surfaces as a result of the thermal expansion of the film in the direction of the thickness. The procedure confirms the experimentally determined square root relationship between the pressure and the sliding velocity reported in references [1–4]. The normalized pressure profiles computed for the different conditions of the journal bearings are identical to those obtained by isoviscous theory.

Thermal modelling and stress analysis in the continuous casting of arbitrary sections

A three‐dimensional parabolic heat conduction model has been developed to analyze heat transfer and solidification in variously shaped continuous casters. A fixed grid source based enthalpy method was used to model solidification during a casting process. In order to correctly model arbitrary cross‐sectional shaped castings, a body‐fitted coordinate transformation (BFT) technique was employed to transform the physical space on to a rectangular computational domain. The transformed governing equations and boundary conditions were solved in the transformed plane. Two arbitrarily shaped casters, having dog‐bone, and hexagonal cross‐sections, were tested in this study. The governing equations were discretized using the control‐volume finite difference approach, wherein an implicit solution technique was employed to solve the set of discretized algebraic equations.From the temperature field predicted by heat flow model, a quasi‐non‐linear stress analysis was carried out within the solidifying shell of the beam blank (dog bone) section. The effect of roller arrangement on the distortion of solidified shell was investigated.

Numerical Simulation of Nonisothermal Capillary Interfaces

An effective algorithm for analysis of nonisothermal capillary interfaces has been developed. The algorithm is based on a coordinate transformation method. The unknown physical domain is mapped onto a rectangular computational domain, with the explicit form of the mapping function not being known. The field variables and the mapping function are determined simultaneously using Picard-type iteration. The algorithm is based on the second-order finite difference discretization and delivers the second-order accuracy, even for very large interfacial distortions.

A numerical scheme for two-dimensional, open channel flows with non-rectangular geometries

An algorithm based on flux difference splitting is presented for the solution of two-dimensional, open channel flows. A transformation maps a non-rectangular, physical domain into a rectangular one. The governing equations are then the shallow water equations, including terms of slope and friction, in a generalized coordinate system. A regular mesh on a rectangular computational domain can then be employed. The resulting scheme has good jump capturing properties and the advantage of using boundary/body-fitted meshes. The scheme is applied to a problem of flow in a river whose geometry induces a region of supercritical flow.

Computation of compressible quasi-axisymmetric slender vortex flow and breakdown

Analysis and computation of steady, compressible, quasi-axisymmetric flow of an isolated, slender vortex are considered. The compressible Navier-Stokes equations are reduced to a simpler set by using the slenderness and quasi-axisymmetry assumptions. The resulting set along with a compatibility equation are transformed from the diverging physical domain to a rectangular computational domain. Solving for a compatible set of initial profiles and specifying a compatible set of boundary conditions, the equations are solved using a type-differencing scheme. Vortex breakdown locations are detected by the failure of the scheme to converge. Computational examples include isolated vortex flows at different Mach numbers, external axial-pressure gradients and swirl ratios. Excellent agreement is shown for a bench-mark case between the computed results using the slender vortex equations and those of a full Navier-Stokes solver.

Study on the Flow Fields of Irregular-Shaped Domains by an Algebraic Grid-Generation Technique

A finite-difference solution algorithm is developed to solve the Navier-Stokes equations in a nonorthogonal curvilinear coordinate system. The governing equations are written in the strong-conservative-law form. A rectangular computational domain is yielded by Morettis transformation. The matrix of the transformation can be determined by direct analytic differentiation. The discretized conservation equations are derived on a control volume basis and solved by the extended SIMPLE calculation procedure. Numerical results obtained by employing the present methodology wil be compared with results from analytical solutions. The relative merits of three numerical representations for approximating the convection terms in the momentum equations are compared

On the Numerical Solution of the Dynamically Loaded Hydrodynamic Lubrication of the Point Contact

The transient analysis of hydrodynamic lubrication of a point-contact is presented. A body-fitted coordinate system is introduced to transform the physical domain to a rectangular computational domain, enabling the use of the Newton-Raphson method for determining pressures and locating the cavitation boundary, where the Reynolds boundary condition is specified. In order to obtain the transient solution, an explicit Euler method is used to effect a time march. The transient dynamic load is a sinusoidal function of time with frequency, fractional loading, and mean load as parameters. Results include the variation of the minimum film thickness and phase-lag with time as functions of excitation frequency. The results are compared with the analytic solution to the transient step bearing problem with the same dynamic loading function. The similarities of the results suggest an approximate model of the point contact minimum film thickness solution.

EXTENSION OF THE SIMPLE ALGORITHM TO HEAT TRANSFER IN TIME-PERIODIC FLOWS WITH MOVING BOUNDARIES

The Semi-Implicit Method for Pressure-Linked Equations (SIMPLE)algorithm for heat transfer and fluid flow problems is extended to time-periodic situations. A vectorized line group method for solving the system of associated algebraic equations in a rectangular two-dimensional computational domain is developed to speed up the computations. A multiblock procedure with the line group method is used to solve a piston-driven oscillating heat transfer problem. The numerical results obtained show some interesting new phenomena and agree with analytical results where such comparisons are possible.

Numerical study of the flowfields in a combustion chamber with a nonorthogonal curvilinear coordinate system

Abstract A finite‐difference solution is developed to study the two‐dimensional flowfields of a combustion chamber. The methodology is also applied to the steady in‐cylinder flows, where the boundary of the cylinder head is a curved wall. The curved wall is described by a cubic spline interpolation function. The governing equations are expressed in axisymmetric nonorthogonal curvilinear coordinates. A rectangular computational domain is yielded by Moretti's transformation and the task of numerically generating the body‐fitted coordinates is avoided. Hence, the matrix of the coordinate transformation can be determined by direct analytic differentiation. The discretized conservation equations are derived on a control volume basis and solved by the extended SIMPLE calculation procedure. The program is run on an IBM PC/AT. Numerical results obtained by employing the present methodology are compared with results from an analytical solution. The relative merits of three numerical representations, i.e., power‐la...

The prediction of shock structure in noncircular jets using conformal mapping with a pseudospectral hybrid scheme

Broadband shock associated noise, in supersonic jets operating off-design, is generated by the interaction of large-scale structures in the jet mixing layer and the quasiperiodic shock cell structure of the jet. In this work, the shock cell structure of supersonic jets with noncircular exit geometry is modeled using a linear analysis. The analysis requires the solution of the equations of hydrodynamic stability for zero frequency. The model takes into account the finite thickness of the jet shear layer using realistic velocity and density profiles. The effects of the shear layer turbulence are included by incorporating eddy-viscosity terms. The numerical method involves the transformation of the cross section of a given jet to a rectangular computational domain using a series of conformal mappings. A pseudospectral hybrid scheme is used in the computational domain to set up the finite-differencing scheme. The variation of the pressure fluctuation with downstream distance is given for an elliptic jet of aspect ratio 2. Comparisons with experimental data are made, where possible. This work is an extension of earlier work [P. J. Morris, T. R. S. Bhat, and G. Chen, J. Sound Vib. 132(2), 199–211 (1989)], where the shear layer was approximated by a vortex sheet. [Work supported by NASA Langley Research Center under NASA Grant NAG-1-657.]

Computation of incompressible potential flow using von Mises coordinates

Incompressible potential flow over symmetric airfoils has been formulated in terms of von Mises coordinates. This formulation provides a rectangular computational domain with Dirichlet boundary conditions for the single unknown y( x, ψ). The governing second order nonlinear pde is solved by SLOR on uniform and clustered grids. The method proves to be accurate, efficient and easy to code.

Laminar flow and heat transfer in a periodically converging‐diverging channel

AbstractA finite difference solution for laminar viscous flow through a sinusoidally curved converging‐diverging channel is presented. The physical wavy domain is transformed into a rectangular computational domain in order to simplify the application of boundary conditions on the channel walls. The discretized conservation equations for mass, momentum and energy are derived on a control volume basis. The pseudo‐diffusive terms that arise from the co‐ordinate transformation are treated as source terms, and the resulting system of equations is solved by a semi‐implicit procedure based on line relaxation. Results are obtained for both the developing and the fully developed flow for a Prandtl number of 0.72, channel maximum width‐to‐pitch ratio of 1.0, Reynolds number ranging from 100 to 500 and wall amplitude‐to‐pitch ratio varying from 0.1 to 0.25. Results are presented here for constant fluid properties and for a prescribed wall enthalpy only.

Laminar fluid flow in a twisted elliptic tube

Twisted tubes are important in generating secondary flow and fluid mixing in a cross-stream direction. In the present paper, the fully developed laminar fluid flow through twisted elliptic tubes is numerically analyzed up to a large twist ratio by using the finite difference method. The elliptic solution domain is transformed to a rectangular computational domain by using a new, simple, and analytic transformation containing a removable singularity. The effect of the two parameters, the tube twist ratio and the aspect ratio of the ellipse, is investigated with respect to their role in determining the axial and circumferential velocities, the streamline patterns, and the resistance coefficients.

FLOW THROUGH A CONVERGING-DIVERGING TUBE WITH CONSTANT WALL ENTHALPY

A control volume finite-difference approach for the solution of axisymmetric laminar viscous flow through a converging-diverging pipe is presented. The pipe wall has a sinusoidal shape and constant enthalpy. The physical wavy domain is transformed into a rectangular computational domain to simplify the application of boundary conditions on the solid boundary. The governing equations in primitive variables are discretized on a control volume basis that ensures global as well as local conservation of mass, momentum, and energy. The pseudodiffusive terms that arise from the coordinate transformation are treated as source terms, and the resulting system of equations is solved by a semi-implicit procedure based on line relaxation. Results are obtained for both the developing and the fully developed flow for a Prandtl number of 0.72, ratio of pipe maximum diameter to pitch of 1.0, Reynolds number ranging from 100 to 500, and ratio of wall amplitude to pitch varying from 0.1 to 0.25.

A study of the motion in rotating containers using a boundary fitted co‐ordinate system

AbstractNumerical solutions are often inaccurate because conventional co‐ordinate systems do not represent the complex physical boundaries accurately. In the present work, the numerical solution of linear shallow water wave equations has been obtained by transforming the physical domain into a rectangular computational domain using elliptic differential operators. This work is part of a programme to develop three‐dimensional body‐fit grid systems for environmental flows. Solutions have been obtained for a cylindrical container and also a parabolic container. The initial conditions chosen are the ones for which analytical solutions exist. The numerical solutions compare well with analytical solutions.

Analysis of aerothermal loads on spherical dome protuberances

Hypersonic flow over spherical dome protuberances was investigated to determine increased pressure and heating loads to the surface. The configuration was mathematically modeled in a time-dependent three-dimensional analysis of the conservation of mass, momentum (Navier-Stokes), and energy equations. A boundary mapping technique was used to obtain a rectangular parallelepiped computational domain, a MacCormack explicit time-split predictor-corrector finite difference algorithm was used to obtain solutions. Results show local pressures and heating rates for domes one-half, one, and two boundary layer thicknesses high were increased by factors on the order of 1.4, 2, and 6, respectively. Flow over the lower dome was everywhere attached while flow over the intermediate dome had small windward and leeside separations. The higher dome had an unsteady windward separation region and a large leeside separation region. Trailing vortices form on all domes with intensity increasing with dome height. Discussion of applying the results to a thermally bowed thermal protection system are presented.

Numerical simulation of reconnection in an emerging magnetic flux region

The resistive MHD equations are numerically solved in two dimensions for an initial-boundary-value problem which simulates reconnection between an emerging magnetic flux region and an overlying coronal magnetic field. The emerging region is modelled by a cylindrical flux tube with a poloidal magnetic field lying in the same plane as the external, coronal field. The plasma betas of the emerging and coronal regions are 1.0 and 0.1, respectively, and the magnetic Reynolds number for the system is 2 × 103. At the beginning of the simulation the tube starts to emerge through the base of the rectangular computational domain, and, when the tube is halfway into the computational domain, its position is held fixed so that no more flux of plasma enters through the base. Because the time-scale of the emergence is slower than the Alfven time-scale, but faster than the reconnection time-scale, a region of closed loops forms at the base. These loops are gradually opened and reconnected with the overlying, external magnetic field as time proceeds. The evolution of the plasma can be divided into four phases as follows: First, an initial, quasi-steady phase during which most of the emergence is completed. During this phase, reconnection initially occurs at the slow rate predicted by the Sweet model of diffusive reconnection, but increases steadily until the fast rate predicted by the Petschek model of slow-shock reconnection is approached. Second, an impulsive phase with large-scale, super-magnetosonic flows. This phase appears to be triggered when the internal mechanical equilibrium inside the emerging flux tube is upset by reconnection acting on the outer layers of the flux tube. During the impulsive phase most of the flux tube pinches off from the base to form a cylindrical magnetic island, and temporarily the reconnection rate exceeds the steady-state Petschek rate. (At the time of the peak reconnection rate, the diffusion region at the X-line is not fully resolved, and so this may be a numerical artifact.) Third, a second quasi-steady phase during which the magnetic island created in the impulsive phase is slowly dissipated by continuing, but low-level, reconnection. And fourth, a static, non-evolving phase containing a potential, current-free field and virtually no flow. During the short time in the impulsive phase when the reconnection rate exceeds the steady-state Petschek rate, a pile-up of magnetic flux at the neutral line occurs. At the same time the existing Petschek-slow-mode shocks are shed and replaced by new ones; and, for a while, both new and old sets of slow shocks coexist.

Algebraic grid generation

Three methods are described for transforming grids in bounded two- and three- dimensional physical domains into a uniform grid in a rectangular computational domain. The methods are based on mathematical interpolation functions and do not require the solution of differential equations or the use of complex variables. They are simply referred to as algebraic methods and are called transfinite interpolation, the multisurface method, and the two-boundary technique. The primary advantage of the methods is that they provide explicit control of physical grid shape and physical grid spacing. Secondly, they require relatively few computations. Consequently, the application of inter-active computer graphics in conjunction with the methods is advocated for rapid generation of grids. The basic mathematical structure of each method is described, and particular attention is given to surface representation, parameter variable generation, and control function generation. Physical boundary topology and grid derivative requirements are presented.