scholarly journals A Numerical Study of Unsteady Natural Convection in a Rectangular Enclosure: The Effect of Variable Thermodynamic and Transport Properties

2009 ◽  
Author(s):  
K. M. Akyuzlu ◽  
M. Chidurala
Keyword(s):  
Mathematical Model ◽  
Transport Properties ◽  
Temperature Difference ◽  
Wall Temperature ◽  
Numerical Experiments ◽  
Working Fluid ◽  
Two Dimensional ◽  
Constant Property ◽  
Variable Property ◽  
Circulation Patterns

A two-dimensional, mathematical model is adopted to investigate the development of buoyancy driven circulation patterns and temperature contours inside a rectangular enclosure (with aspect ratio of one) filled with a compressible fluid (Pr = 0.72). One of the vertical walls of the enclosure is kept at a higher temperature than the opposing vertical wall. The top and the bottom of the enclosure are assumed insulated. The physics based mathematical model for this problem consists of conservation of mass, momentum (two-dimensional, unsteady Navier-Stokes equations for compressible flows) and energy equations for the enclosed fluid subjected to appropriate boundary conditions. The compressibility of the working fluid is represented by an ideal gas relation. Thermodynamic and transport properties of the fluid are assumed to be function of temperature. The governing equations are discretized using second order accurate central differencing for spatial derivatives and second order finite differencing based on Taylor expansion for time derivatives. The resulting nonlinear equations are then linearized using Newton’s linearization method. The set of algebraic equations that result from this process are then put into a matrix form and solved using a Coupled Modified Strongly Implicit Procedure (CMSIP) for the unknowns of the problem. Grid independence and time convergence studies were carried out on different mesh sizes and also on a stretched orthogonal mesh to determine the accuracy of the square mesh adopted for the present study. Numerical experiments were carried out for a benchmark case (driven cavity flows) to verify the accuracy of the CMSIP, the proposed solution procedure. Numerical experiments were then carried out to simulate the development of the buoyancy driven circulation patterns for Rayleigh (Ra) numbers between 103 and 106. Also a parametric study was carried out (where Ra number was kept constant) to determine the effect of variations in wall temperature difference and reference length on the velocity and temperature fields. The effects of variable fluid properties on circulation patterns, temperature distributions, vertical and horizontal velocity profiles, and heat transfer from the walls of the enclosure were determined in a separate set of numerical experiments. Finally, unsteady thermal and hydrodynamic behavior of the working fluid was studied by imposing a sudden wall temperature change in the square enclosure. It is concluded that there is notable difference between the results of the variable property and the constant property models. Also, the variable property model predicts lower values for wall heat fluxes and Nu number than the constant property one. This seems to be more true when the temperature difference between the hot and cold walls of the enclosure is larger.

A Numerical Study of Unsteady Natural Convection in a Rectangular Enclosure: The Effect of Compressibility

2004 ◽  
Author(s):  
K. M. Akyuzlu ◽  
Y. Pavri ◽  
A. Antoniou
Keyword(s):  
Mathematical Model ◽  
Stokes Equations ◽  
Numerical Study ◽  
Vertical Wall ◽  
Ideal Gas ◽  
Working Fluid ◽  
Two Dimensional ◽  
Algebraic Equations ◽  
Rectangular Enclosure ◽  
Circulation Patterns

A two-dimensional, mathematical model is adopted to investigate the development of buoyancy driven circulation patterns and temperature contours inside a rectangular enclosure filled with a compressible fluid (Pr=1.0). One of the vertical walls of the enclosure is kept at a higher temperature then the opposing vertical wall. The top and the bottom of the enclosure are assumed insulated. The physics based mathematical model for this problem consists of conservation of mass, momentum (two-dimensional Navier-Stokes equations) and energy equations for the enclosed fluid subjected to appropriate boundary conditions. The working fluid is assumed to be compressible through a simple ideal gas relation. The governing equations are discretized using second order accurate central differencing for spatial derivatives and first order forward finite differencing for time derivatives where the computation domain is represented by a uniform orthogonal mesh. The resulting nonlinear equations are then linearized using Newton’s linearization method. The set of algebraic equations that result from this process are then put into a matrix form and solved using a Coupled Modified Strongly Implicit Procedure (CMSIP) for the unknowns (primitive variables) of the problem. A numerical experiment is carried out for a benchmark case (driven cavity flow) to verify the accuracy of the proposed solution procedure. Numerical experiments are then carried out using the proposed compressible flow model to simulate the development of the buoyancy driven circulation patterns for Rayleigh numbers between 103 and 105. Finally, an attempt is made to determine the effect of compressibility of the working fluid by comparing the results of the proposed model to that of models that use incompressible flow assumptions together with Boussinesq approximation.

A Numerical Study of Unsteady Natural Convection in a Rectangular Enclosure: Transition From Laminar to Turbulent Flow

2010 ◽  
Author(s):  
K. M. Akyuzlu ◽  
M. Chidurala
Keyword(s):  
Mathematical Model ◽  
Finite Difference ◽  
Turbulent Flow ◽  
Numerical Study ◽  
Second Order ◽  
Working Fluid ◽  
Two Dimensional ◽  
Rectangular Enclosure ◽  
Circulation Patterns ◽  
Vertical Walls

A two-dimensional, mathematical model is adopted to investigate the development of buoyancy driven circulation patterns and temperature stratification inside a rectangular enclosure. One of the vertical walls of the enclosure is kept at a higher temperature then the opposing vertical wall. The top and the bottom of the enclosure are assumed insulated. The physics based mathematical model for this problem consists of conservation of mass, momentum (two-dimensional, unsteady Navier-Stokes equations for turbulent compressible flows), and energy equations for the enclosed fluid subjected to appropriate boundary conditions. A standard two equation turbulence model is used to model the turbulent flow in the enclosure. The compressibility of the working fluid is represented by an ideal gas relation. The conservation equations are discretized using an implicit finite difference technique which employs second order accurate central differencing for spatial derivatives and second order (based on Taylor expansion) finite differencing for time derivatives. The linearized finite difference equations are solved using a Coupled Modified Strongly Implicit Procedure (CMSIP) for the unknowns of the problem. Numerical experiments were then carried out to simulate the development of the buoyancy driven circulation patterns inside rectangular enclosures (with aspects ratios 0.5, 1 and 1.5) filled with a compressible fluid (Pr = 0.72). Experiments were repeated for various wall temperature differences which corresponded to Rayleigh numbers between 104 and 106. Changes in unsteady circulation patterns, temperature contours, and vertical and horizontal velocity profiles were predicted while the flow inside the enclosure transferred from laminar to turbulent flow due to the sudden temperature change imposed on the vertical walls of the enclosure. Only the results of the enclosure with aspect ratio one is presented in this paper. These results indicate that this transition is characterized by unicellular circulation patterns breaking up in to multicellular formations and increase in the values of the predicted wall heat fluxes and Nusselt number as flow becomes turbulent.

A Study of Unsteady Mixed Convection in a Lid Driven Flow Inside a Rectangular Cavity

2010 ◽  
Author(s):  
K. M. Akyuzlu ◽  
K. Hallenbeck
Keyword(s):  
Heat Transfer ◽  
Mathematical Model ◽  
Second Order ◽  
Rectangular Cavity ◽  
Constant Speed ◽  
Working Fluid ◽  
Two Dimensional ◽  
Driven Cavity ◽  
Circulation Patterns ◽  
Vertical Walls

A numerical study is conducted to identify the unsteady characteristics of momentum and heat transfer in lid-driven cavity flows. The cavity under study is filled with a compressible fluid and is of rectangular shape. The bottom of the cavity is insulated and stationary where as the top of the cavity (the lid) is pulled at constant speed. The vertical walls of the cavity are kept at constant but unequal temperatures. A two-dimensional, mathematical model is adopted to investigate the shear and buoyancy driven circulation patterns inside this rectangular cavity. This physics based mathematical model consists of conservation of mass, momentum (two-dimensional, unsteady Navier-Stokes equations for compressible flows) and energy equations for the enclosed fluid subjected to appropriate boundary and initial conditions. The compressibility of the working fluid is represented by an ideal gas relation and its thermodynamic and transport properties are assumed to be function of temperature. The governing equations are discretized using second order accurate central differencing for spatial derivatives and second order finite differencing (based on Taylor expansion) for the time derivatives. The resulting nonlinear equations are then linearized using Newton’s linearization method. The set of algebraic equations that result from this process are then put into a matrix form and solved using a Coupled Modified Strongly Implicit Procedure (CMSIP) for the unknowns of the problem. Grid independence and time convergence studies were carried out to determine the accuracy of the square mesh adopted for the present study. Two benchmark cases (driven cavity and rectangular channel flows) were studied to verify the accuracy of the CMSIP. Numerical experiments were then carried out to simulate the unsteady development of the shear and buoyancy driven circulation patterns for different Richardson numbers in the range of 0.036<Ri<100 where the Re number is kept less than 2000 to assure laminar flow conditions inside the cavity. Simulations start with a stagnant fluid subjected to a sudden increase in one of the walls temperature. At the same time the upper lid of the cavity is accelerated, instantaneously, to a constant speed. The circulation patterns, temperature contours, vertical and horizontal velocity profiles were generated at different times of the simulation, and wall heat fluxes and Nusselt numbers were calculated for the steady state conditions. Only the results for a square cavity are presented in this paper. These results indicate that the heat transfer rates at the vertical walls of the cavity are enhanced with the decrease in Richardson number.

A Numerical Study of Thermoacoustic Oscillations in a Rectangular Channel Using CMSIP Method

2009 ◽  
Author(s):  
K. M. Akyuzlu ◽  
K. Albayrak ◽  
C. Karaeren
Keyword(s):  
Mathematical Model ◽  
Velocity Field ◽  
Finite Difference ◽  
Second Order ◽  
Component Model ◽  
Working Fluid ◽  
Two Dimensional ◽  
Throttle Valve ◽  
Heated Channel ◽  
Thermoacoustic Oscillations

This paper presents a mathematical model that was developed to study instabilities (primarily thermoacoustic oscillations) experienced inside a channel (with a rectangular cross section) heated symmetrically (from its top and bottom.) The heated channel is configured to simulate a combustion chamber of a rocket hybrid rocket motor and is connected to a converging–diverging nozzle in the downstream and to a plenum with a flow straightener in the upstream side. The working fluid is supplied from a pressurized storage tank to the upstream plenum through a throttle valve. A multi-component approach is used to model this test apparatus. In this integrated component model, the unsteady flow through the throttle valve and the nozzle is assumed to be one-dimensional and isentropic where as the flow in the forward plenum and the heated channel is assumed to be a two-dimensional, unsteady, compressible, turbulent, and subsonic. The physics based mathematical model of the flow in the channel consists of conservation of mass, momentum (two-dimensional Navier-Stokes) and energy equations subject to appropriate boundary conditions as defined by the physical problem stated above. The working fluid is assumed to be compressible where the density of the fluid is related to the pressure and temperature of the fluid through a simple ideal gas relation. The governing equations are discretized using second order accurate central differencing for spatial derivatives and second order accurate (based on Taylor expansion) finite difference approximations for temporal derivatives. The resulting nonlinear equations are then linearized using Newton’s linearization method. The set of algebraic equations that result from this process are then put into a matrix form and solved using a Coupled Modified Strongly Implicit Procedure (CMSIP) for the unknowns (primitive variables, i.e., pressure, temperature, and the velocity field) of the problem. The turbulence model equations and the unsteady flow equation for the throttle valve are solved using a second order accurate explicit finite difference technique. Convergence and grid independence studies were done to determine the optimum mesh size and computational time increment. Furthermore, two benchmark cases (unsteady driven cavity and laminar channel flows) were simulated using the developed numerical model to verify the accuracy of the proposed solution procedure. Numerical experiments were then carried out to simulate the thermoacoustic oscillations inside rectangular channels with various aspect ratios ranging from 5 to 20 for various operating conditions (i.e., for Re numbers between 102 and 106) and to determine the flow regions where these oscillations are sustained. The numerical simulation results indicate that the mathematical model for the gas flow in the heated channel predicts the expected unsteady temperature and pressure distributions, and the velocity field, successfully. Furthermore, it is concluded that the proposed integrated component model is successful in generating the characteristics of the instabilities associated with thermal, hydrodynamic, and thermoacoustic oscillations in heated channels.

An Experimental Study of Natural Convection in a Rectangular Enclosure Using PIV Measurement Techniques

2011 ◽  
Author(s):  
K. M. Akyuzlu ◽  
J. Farkas
Keyword(s):  
Experimental Study ◽  
Natural Convection ◽  
Measurement Techniques ◽  
Flow Patterns ◽  
Working Fluid ◽  
Two Dimensional ◽  
Rectangular Enclosure ◽  
Aspect Ratios ◽  
Circulation Patterns ◽  
Piv Measurement

An experimental study is conducted to determine the circulation patterns inside a rectangular enclosure due to natural convection using a Particle Image Velocimeter (PIV). Experiments were conducted using two different fluids (air and water) and for rectangular enclosures with aspect ratios 0.5 and 1.0. Natural convection in enclosures has been experimentally studied in the past. Many of these studies cited in the literature use some kind of an optical method like interferograms, shadowgraphs, streak photographs, or multi-exposure photographs to visualize the flow patterns in the enclosure. The present study employs a commercial two-dimensional PIV to capture, instantaneously, the circulation patterns inside the test section. The test cavity in the present setup is of rectangular shape, which is 5 inches (127 mm) wide, where the height of the enclosure can be changed to obtain aspect ratios of 0.5 and 1.0. The depth of the rectangular enclosure measures 12 inches (305 mm) to minimize the effect of walls normal to the two dimensional flow patterns that are expected in this type of arrangement. The walls of the cavity are made of Aluminum plates. These plates are kept at constant but different temperatures during the experiments. In the present study, hollow glass sphere particles with 10 microns in diameter were used as seeding for water experiments and fine particles/flakes of ash generated from burned incense were used as seeding in the air experiments. For each working fluid, the experiments were repeated for different aspect ratios and for different wall temperature differences which corresponded to Rayleigh numbers in the range of 106 and 107. Velocity fields were captured at steady state for each experiment using the two-dimensional PIV system. Numerical studies were also carried out using a commercial CFD software. Comparisons of the numerical and experimental results indicate a good match in terms of circulation patterns and velocity magnitudes in the core of the buoyancy driven flow. Discrepancies in measured and predicted values of velocities are more pronounced near to the boundaries of the enclosure. Separate measurements with finer interrogation areas and different PIV setting were required to improve the accuracy of the measurements near the corners (top and bottom) of the enclosure. The results of these measurements are also presented.

A Parametric Study of Laminar Mixed Convection in a Square Cavity Using Numerical Simulation Techniques

2012 ◽  
Author(s):  
K. Hallenbeck ◽  
K. M. Akyuzlu
Keyword(s):  
Heat Transfer ◽  
Mixed Convection ◽  
Parametric Study ◽  
Numerical Experiments ◽  
Working Fluid ◽  
Two Dimensional ◽  
Square Cavity ◽  
Laminar Mixed Convection ◽  
Prandtl Numbers ◽  
Nusselt Number Correlation

A parametric study is conducted using numerical experimentation to construct an empirical Nusselt number correlation in terms of Richardson and Prandtl numbers for laminar mixed convection in a square cavity. The square cavity under study is assumed to be filled with a compressible fluid. The bottom of the cavity is insulated and stationary where as the top of the cavity (the lid) is pulled at constant speed. The vertical walls of the cavity are kept at constant but unequal temperatures. A two-dimensional, mathematical model is adopted to predict the momentum and heat transfer inside this rectangular cavity. This physics based mathematical model consists of conservation of mass, momentum (two-dimensional, unsteady Navier-Stokes equations for compressible flows) and energy equations for the enclosed fluid subjected to appropriate boundary and initial conditions. The compressibility of the working fluid is represented by an ideal gas relation. The thermodynamic and transport properties of the working fluid are assumed to be constant. The governing equations are discretized using second order accurate central differencing for spatial derivatives and second order finite differencing (based on Taylor expansion) for the time derivatives. The resulting nonlinear equations are then linearized using Newton’s linearization method. The set of algebraic equations that result from this process are then put into a matrix form and solved using a Coupled Modified Strongly Implicit Procedure (CMSIP) for the unknowns of the problem. Grid independence and time convergence studies were carried to determine the accuracy of the square mesh adopted for the present study. Two benchmark cases (driven cavity and rectangular channel flows) were studied to verify the accuracy of the CMSIP. Numerical experiments were then carried out to simulate the heat transfer characteristics of mixed convection flow for different Richardson numbers in the range of 0.036<Ri<1.00 where the Reynolds number is kept less than 2000 to ensure laminar flow conditions inside the cavity. The velocity vector field maps (circulation patterns) and temperature contours, and temperature profiles along the horizontal axes were generated for different Prandtl numbers ranging from 0.3 to 1. Wall heat fluxes and Nusselt numbers were determined for each parametric study. The collected data from the numerical experiments were then used to construct an empirical Nusselt number correlation in terms of Richardson and Prandtl numbers.

Effect of Variable Properties Within a Boundary Layer With Large Freestream-to-Wall Temperature Differences

2014 ◽  
Vol 136 (5) ◽  
Author(s):  
Nathan J. Greiner ◽  
Marc D. Polanka ◽  
James L. Rutledge ◽  
Jacob R. Robertson
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Turbulent Flows ◽  
Wall Temperature ◽  
Ratio Method ◽  
Variable Properties ◽  
Temperature Ratio ◽  
Adiabatic Wall ◽  
Constant Property ◽  
Variable Property

Modern gas-turbine engines are characterized by high core-flow temperatures and significantly lower turbine-surface temperatures. This can lead to large property variations within the boundary layers on the turbine surfaces. However, cooling of turbines is generally studied near room temperature, where property variation within the boundary layer is negligible. The present study first employs computational fluid dynamics to validate two methods for quantifying the effect of variable properties in a boundary layer: the reference temperature method and the temperature ratio method. The computational results are then used to expand the generality of the temperature ratio method by proposing a slight modification. Next, these methods are used to quantify the effect of variable properties within a boundary layer on measurement techniques, which assume constant properties. Both low-temperature flows near ambient and high-temperature flows with a freestream temperature of 1600 K are considered under both laminar and turbulent conditions. The results show that variable properties have little effect on laminar flows at any temperature or turbulent flows at low temperatures such that constant property methods can be validly employed. However, variable properties are seen to have a profound effect on turbulent flows at high temperatures. For the high-temperature turbulent flow considered, the constant property methods are found to overpredict the convective heat transfer coefficient by up to 54.7% and underpredict the adiabatic wall temperature by up to 209 K. Utilizing the variable property techniques, a new method for measuring the adiabatic wall temperature and variable property heat-transfer coefficient is proposed for variable property flows.

Two Dimensional Unsteady State Mathematical Model for dispersion of Chlorine in Water in a Channel

2011 ◽  
Vol 3 (8) ◽  
pp. 503-505
Author(s):  
Jaipal Jaipal ◽  
◽  
Rakesh Chandra Bhadula ◽  
V. N Kala V. N Kala
Keyword(s):  
Mathematical Model ◽  
Two Dimensional ◽  
Unsteady State

scholarly journals Computer Modeling of Aerosol Emissions Spread in the Atmosphere

2019 ◽  
Vol 97 ◽  
pp. 05023 ◽  
Author(s):  
Daler Sharipov ◽  
Sharofiddin Aynakulov ◽  
Otabek Khafizov
Keyword(s):  
Boundary Layer ◽  
Mathematical Model ◽  
Atmospheric Boundary Layer ◽  
Computer Modeling ◽  
Numerical Experiments ◽  
Climatic Factors ◽  
Numerical Algorithms ◽  
Aerosol Emissions ◽  
The Given ◽  
And Diffusion

The paper deals with the development of mathematical model and numerical algorithms for solving the problem of transfer and diffusion of aerosol emissions in the atmospheric boundary layer. The model takes into account several significant parameters such as terrain relief, characteristics of underlying surface and weather-climatic factors. A series of numerical experiments were conducted based on the given model. The obtained results presented here show how these factors affect aerosol emissions spread in the atmosphere.

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