Finite Analytic Numerical Solution Axisymmetric Navier-Stokes and Energy Equations

1983 ◽  
Vol 105 (3) ◽  
pp. 639-645 ◽  
Author(s):  
Ching-Jen Chen ◽  
Young Hwan Yoon
Keyword(s):  
Heat Transfer ◽  
Numerical Solution ◽  
Numerical Method ◽  
Stream Function ◽  
Analytic Solution ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Inlet Temperature ◽  
Local Analytic Solution

Connective heat transfer for steady-state laminar flow in axisymmetric coordinates is considered. Numerical solutions for flow pattern and temperature distribution are obtained by the finite analytic numerical method applied to the Navier-Stokes equations expressed in terms of vorticity and stream function, and the energy equation. The finite analytic numerical method differs from other numerical methods in that it utilizes a local analytic solution in an element of the problem to construct the total numerical solution. Finite analytic solutions of vorticity, stream function, temperature, and heat transfer coefficients for flow with Reynolds numbers of 5, 100, 1000, and 2000, and Prandtl numbers of 0.1, 1.0, and 10.0 with uniform grid sizes, are reported for an axisymmetric pipe with a sudden expansion and contraction. The wall temperature is considered to be isothermal and differs from the inlet temperature. It is shown that the finite analytic is stable, converges rapidly, and simulates the convection of fluid flow accurately, since the local analytic solution is capable of simulating automatically the influence of skewed convection through the element boundary on the interior nodal values, thereby minimizing the false numerical diffusion.

Validation of a Numerical Method for Unsteady Flow Calculations

1993 ◽  
Vol 115 (1) ◽  
pp. 110-117 ◽  
Author(s):  
M. Giles ◽  
R. Haimes
Keyword(s):  
Heat Transfer ◽  
Flat Plate ◽  
Numerical Method ◽  
Stokes Equations ◽  
Companion Paper ◽  
Ideal Gas ◽  
Euler Method ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Loss Prediction

This paper describes and validates a numerical method for the calculation of unsteady inviscid and viscous flows. A companion paper compares experimental measurements of unsteady heat transfer on a transonic rotor with the corresponding computational results. The mathematical model is the Reynolds-averaged unsteady Navier–Stokes equations for a compressible ideal gas. Quasi-three-dimensionality is included through the use of a variable streamtube thickness. The numerical algorithm is unusual in two respects: (a) For reasons of efficiency and flexibility, it uses a hybrid Navier–Stokes/Euler method, and (b) to allow for the computation of stator/rotor combinations with arbitrary pitch ratio, a novel space–time coordinate transformation is used. Several test cases are presented to validate the performance of the computer program, UNSFLO. These include: (a) unsteady, inviscid flat plate cascade flows (b) steady and unsteady, viscous flat plate cascade flows, (c) steady turbine heat transfer and loss prediction. In the first two sets of cases comparisons are made with theory, and in the third the comparison is with experimental data.

Numerical Solutions of a Viscous Uniform Approach Flow Past Square and Diamond Cylinders

2002 ◽  
Author(s):  
Charles Dalton ◽  
Wu Zheng
Keyword(s):  
Stream Function ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Elliptic Partial Differential Equation ◽  
Navier Stokes ◽  
Central Difference ◽  
Uniform Approach ◽  
Flow Past ◽  
Line Force ◽  
Vorticity Transport

Numerical results are presented for a uniform approach flow past square and diamond cylinders, with and without rounded corners, at Reynolds numbers of 250 and 1000. This unsteady viscous flow problem is formulated by the 2-D Navier-Stokes equations in vorticity and stream-function form on body-fitted coordinates and solved by a finite-difference method. Second-order Adams-Bashforth and central-difference schemes are used to discretize the vorticity transport equation while a third-order upwinding scheme is incorporated to represent the nonlinear convective terms. A grid generation technique is applied to provide an efficient mesh system for the flow. The elliptic partial differential equation for stream-function and vorticity in the transformed plane is solved by the multigrid iteration method. The Strouhal number and the average in-line force coefficients agree very well with the experimental and previous numerical values. The vortex structures and the evolution of vortex shedding are illustrated by vorticity contours. Rounding the corners of the square and diamond cylinders produced a noticeable decrease on the calculated drag and lift coefficients.

scholarly journals Natural Convection of Non-Newtonian Power-Law Fluid in a Square Cavity with a Heat-Generating Element

Energies ◽  
2019 ◽  
Vol 12 (11) ◽  
pp. 2149 ◽  
Author(s):  
Darya S. Loenko ◽  
Aroon Shenoy ◽  
Mikhail A. Sheremet
Keyword(s):  
Heat Transfer ◽  
Stream Function ◽  
Power Law ◽  
Stokes Equations ◽  
Heat Removal ◽  
Modern Technology ◽  
Navier Stokes ◽  
Local Source ◽  
Thermal Conductivity Ratio ◽  
Square Cavity

Development of modern technology in microelectronics and power engineering necessitates the creation of effective cooling systems. This is made possible by the use of the special fins technology within the cavity or special heat transfer liquids in order to intensify the heat removal from the heat-generating elements. The present work is devoted to the mathematical modeling of thermogravitational convection of a non-Newtonian fluid in a closed square cavity with a local source of internal volumetric heat generation. The behavior of the fluid is described by the Ostwald-de Waele power law model. The defining Navier–Stokes equations written using the dimensionless stream function, vorticity and temperature are solved using the finite difference method. The effects of the Rayleigh number, power-law index, and thermal conductivity ratio on heat transfer and the flow structure are studied. The obtained results are presented in the form of isolines of the stream function and temperature, as well as the dependences of the average Nusselt number and average temperature on the governing parameters.

Optimized Accelerating Parameters of Convergence for the Navier-Stokes Equations

1992 ◽  
Author(s):  
Bashar S. AbdulNour
Keyword(s):  
Stream Function ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Relaxation Method ◽  
Reynolds Numbers ◽  
Computer Time ◽  
Navier Stokes ◽  
Computational Time ◽  
Navier Stokes Equations ◽  
Successive Over Relaxation

Abstract An over-relaxation procedure, that includes weighing factors, is applied to the steady, two-dimensional Navier-Stokes equations in order to reduce the computational time. The benefits obtained from this strategy are illustrated by the problem of viscous flow in the entrance region of an unconstricted and a constricted channel. The describing equations are expressed in terms of the stream function and vorticity. The convergence domain for the Successive Over-Relaxation method and the optimum values of the accelerating parameters, which consist of the over-relaxation and weighting factors for both the stream function and vorticity, are discussed. Numerical solutions are obtained for Reynolds numbers ranging from 20 to 2000. The computer time is reduced by as much as a factor of six using the optimum values of the accelerating parameters.

Validation of a Numerical Method for Unsteady Flow Calculations

1991 ◽  
Author(s):  
Michael Giles ◽  
Robert Haimes
Keyword(s):  
Heat Transfer ◽  
Flat Plate ◽  
Numerical Method ◽  
Stokes Equations ◽  
Companion Paper ◽  
Ideal Gas ◽  
Euler Method ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Loss Prediction

This paper describes and validates a numerical method for the calculation of unsteady inviscid and viscous flows. A companion paper compares experimental measurements of unsteady heat transfer on a transonic rotor with the corresponding computational results. The mathematical model is the Reynolds-averaged unsteady Navier-Stokes equations for a compressible ideal gas. Quasi-three-dimensionality is included through the use of a variable streamtube thickness. The numerical algorithm is unusual in two respects: a) for reasons of efficiency and flexibility it uses a hybrid Navier-Stokes/Euler method, and b) to allow for the computation of stator/rotor combinations with arbitrary pitch ratio a novel space-time coordinate transformation is used. Several test cases are presented to validate the performance of the computer program, UNSFLO. These include: a) unsteady, inviscid flat plate cascade flows, b) steady and unsteady, viscous flat plate cascade flows, c) steady turbine heat transfer and loss prediction. In the first two sets of cases comparisons are made with theory, and in the third the comparison is with experimental data.

COMPARATIVE STUDY OF THERMAL FLOWS WITH DIFFERENT FINITE VOLUME AND LATTICE BOLTZMANN SCHEMES

2004 ◽  
Vol 15 (02) ◽  
pp. 307-319 ◽  
Author(s):  
AHMAD AL-ZOUBI ◽  
GUNTHER BRENNER
Keyword(s):  
Heat Transfer ◽  
Comparative Study ◽  
Finite Volume ◽  
Lattice Boltzmann ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Hybrid Approach ◽  
Navier Stokes ◽  
Steady Flows ◽  
Parallel Plates

In the present paper, a comparative study of numerical solutions for steady flows with heat transfer based on the finite volume method (FVM) and the relatively new lattice Boltzmann method (LBM) is presented. In the last years, the LB methods have challenged the classical FV methods to solve the Navier–Stokes equations and have proven to be superior in accuracy and efficiency for certain applications. Most of these studies were related to the transport of mass and momentum. In the meantime, significant effort has been invested in the application of the LBM to simulate flows including heat transfer. The studies in the present paper are the analysis of performance and accuracy aspects of LBM applied to the prediction of these flows. For a fully developed laminar flow between parallel plates, analytical solutions for the heat transfer in fully developed thermal boundary layers are available and may be compared with the respective numerical results. Finally, a hybrid approach is proposed to circumvent numerical problems of the thermal LB methods.

Natural convection in an enclosure having a vertical sidewall with time-varying temperature

1996 ◽  
Vol 329 ◽  
pp. 65-88 ◽  
Author(s):  
Ho Sang Kwak ◽  
Jae Min Hyun
Keyword(s):  
Heat Transfer ◽  
Natural Convection ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Numerical Study ◽  
Temperature Fields ◽  
Navier Stokes ◽  
Time Varying ◽  
Vertical Sidewall ◽  
Boussinesq Fluid

A numerical study is performed for time-varying natural convection of an incompressible Boussinesq fluid in a sidewall-heated square cavity. The temperature at the cold sidewall Tc is constant, but at the hot sidewall a time-varying temperature condition is prescribed, $ T_H = \overline{T_H} + \Delta T^{\prime} \sin ft $. Comprehensive numerical solutions are found for the time-dependent Navier–Stokes equations. The numerical results are analysed in detail to show the existence of resonance, which is characterized by maximal amplification of the fluctuations of heat transfer in the interior. Plots of the dependence of the amplification of heat transfer fluctuations on the non-dimensional forcing frequency ω are presented. The failure of Kazmierczak & Chinoda (1992) to identify resonance is shown to be attributable to the limitations of the parameter values they used. The present results illustrate that resonance becomes more distinctive for large Ra and Pr ∼ 0(1). The physical mechanism of resonance is delineated by examining the evolution of oscillating components of flow and temperature fields. Specific comparisons are conducted for the resonance frequency ωr between the present results and several other previous predictions based on the scaling arguments.

A Simultaneous Variable Solution Procedure for Laminar and Turbulent Flows in Curved Channels and Bends

2000 ◽  
Vol 122 (3) ◽  
pp. 552-559 ◽  
Author(s):  
Jianrong Wang ◽  
Siamack A. Shirazi
Keyword(s):  
Experimental Data ◽  
Numerical Solution ◽  
Numerical Method ◽  
Solution Method ◽  
Stokes Equations ◽  
Turbulence Models ◽  
Solution Procedure ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Curved Channels

Direct Numerical Simulation of turbulent flow requires accurate numerical techniques for solving the Navier-Stokes equations. Therefore, the Navier-Stokes equations in general orthogonal and nonorthogonal coordinates were employed and a simultaneous variable solution method was extended to solve these general governing equations. The present numerical method can be used to accurately predict both laminar and turbulent flow in various curved channels and bends. To demonstrate the capability of this numerical method and to verify the method, the time-averaged Navier-Stokes equations were employed and several turbulence models were also implemented into the numerical solution procedure to predict flows with strong streamline curvature effects. The results from the present numerical solution procedure were compared with available experimental data for a 90 deg bend. All of the turbulence models implemented resulted in predicted velocity profiles which were in agreement with the trends of experimental data. This indicates that the solution method is a viable numerical method for calculating complex flows. [S0098-2202(00)01803-4]

A Numerical Study of the Influence of Disc Geometry on the Flow and Heat Transfer in a Rotating Cavity

1990 ◽  
Author(s):  
B. L. Lapworth ◽  
J. W. Chew
Keyword(s):  
Heat Transfer ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Numerical Study ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Model Calculations ◽  
Flow And Heat Transfer ◽  
Rotating Cavity ◽  
Radial Outflow

Numerical solutions of the Reynolds-averaged Navier-Stokes equations have been used to model the influence of cobs and a bolt cover on the flow and heat transfer in a rotating cavity with an imposed radial outflow of air. Axisymmetric turbulent flow is assumed using a mixing length turbulence model. Calculations for the non-plane discs are compared with plane disc calculations and also with the available experimental data. The calculated flow structures show good agreement with the experimentally observed trends. For the cobbed and plane discs, Nusselt numbers are calculated for a combination of flow rates and rotational speeds; these show some discrepancies with the experiments, although the calculations exhibit the more consistent trend. Further calculations indicate that differences in thermal boundary conditions have a greater influence on Nusselt number than differences in disc geometry. The influence of the bolt cover on the heat transfer has also been modelled, although comparative measurements are not available.

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