Numerical Computation of the Heat Transfer and Fluid Mechanics in the Laminar Wall Jet and Comparison to the Self-Similar Solutions

2004 ◽  
Author(s):  
Johnny Issa ◽  
Alfonso Ortega
Keyword(s):  
Boundary Conditions ◽  
Analytical Solutions ◽  
Numerical Solutions ◽  
Fluid Dynamic ◽  
Wall Jet ◽  
Temperature Profiles ◽  
Computational Domain ◽  
Two Dimensional ◽  
Self Similar ◽  
Similar Solutions

Despite its importance as a canonical two-dimensional flow, the laminar wall jet has not been extensively studied using modern computational fluid dynamic methods. As in the laminar boundary layer, existence of analytical self-similar solutions make the problem particularly attractive for validating CFD code, yet we have found little archival work in which it has been used for this purpose. In the present study, we present a numerical investigation of the steady, laminar, and two-dimensional plane wall jet with constant properties. A finite-volume approach is used to solve the governing equations using self-similar inlet boundary conditions for the velocity and temperature profiles. The thermal solution is investigated for isothermal boundary condition at the wall. Velocity and temperature profiles are reported at various locations downstream and show an excellent agreement with the similarity solution obtained by Glauert [1] and Schwarz, et al. [2] respectively. In addition, the skin friction coefficient and the Nusselt number are investigated and compared with the analytical solutions presented by Glauert [1] and Mitachi, et al. [3] respectively, and very good agreement is observed. Despite its simplicity, it is shown that proper convergence of the numerical solutions of the wall jet to the expected analytical solutions requires care in specification of the jet inlet conditions, and the boundary conditions on the computational domain boundaries.

Transient Two-Dimensional Heat Conduction Problem With Partial Heating Near Corners

2016 ◽  
Author(s):  
Robert L. McMasters ◽  
Filippo de Monte ◽  
James V. Beck ◽  
Donald E. Amos
Keyword(s):  
Heat Conduction ◽  
Numerical Integration ◽  
Analytical Solutions ◽  
Numerical Solutions ◽  
Thermal Protection ◽  
Heat Conduction Problem ◽  
Separation Of Variables ◽  
Two Dimensional ◽  
Rectangular Region ◽  
Long Distance

This paper provides a solution for two-dimensional heating over a rectangular region on a homogeneous plate. It has application to verification of numerical conduction codes as well as direct application for heating and cooling of electronic equipment. Additionally, it can be applied as a direct solution for the inverse heat conduction problem, most notably used in thermal protection systems for re-entry vehicles. The solutions used in this work are generated using Green’s functions. Two approaches are used which provide solutions for either semi-infinite plates or finite plates with isothermal conditions which are located a long distance from the heating. The methods are both efficient numerically and have extreme accuracy, which can be used to provide additional solution verification. The solutions have components that are shown to have physical significance. The extremely precise nature of analytical solutions allows them to be used as prime standards for their respective transient conduction cases. This extreme precision also allows an accurate calculation of heat flux by finite differences between two points of very close proximity which would not be possible with numerical solutions. This is particularly useful near heated surfaces and near corners. Similarly, sensitivity coefficients for parameter estimation problems can be calculated with extreme precision using this same technique. Another contribution of these solutions is the insight that they can bring. Important dimensionless groups are identified and their influence can be more readily seen than with numerical results. For linear problems, basic heating elements on plates, for example, can be solved to aid in understanding more complex cases. Furthermore these basic solutions can be superimposed both in time and space to obtain solutions for numerous other problems. This paper provides an analytical two-dimensional, transient solution for heating over a rectangular region on a homogeneous square plate. Several methods are available for the solution of such problems. One of the most common is the separation of variables (SOV) method. In the standard implementation of the SOV method, convergence can be slow and accuracy lacking. Another method of generating a solution to this problem makes use of time-partitioning which can produce accurate results. However, numerical integration may be required in these cases, which, in some ways, negates the advantages offered by the analytical solutions. The method given herein requires no numerical integration; it also exhibits exponential series convergence and can provide excellent accuracy. The procedure involves the derivation of previously-unknown simpler forms for the summations, in some cases by virtue of the use of algebraic components. Also, a mathematical identity given in this paper can be used for a variety of related problems.

Similar Solutions for Laminar Film Condensation with Adverse Pressure Gradients

1973 ◽  
Vol 95 (2) ◽  
pp. 268-270 ◽  
Author(s):  
P. M. Beckett
Keyword(s):  
Numerical Solutions ◽  
Saturated Vapor ◽  
Film Condensation ◽  
Pressure Gradients ◽  
Two Dimensional ◽  
Laminar Film ◽  
Approximate Models ◽  
Laminar Film Condensation ◽  
Similar Solutions

Steady two-dimensional laminar film condensation is investigated when the saturated vapor has the Falkner–Skan mainstream. Numerical solutions and approximate models are discussed with reference to other published work.

Comparison of the numerical and self-similar solutions of Sedov’s problem on a point explosion in gas

2020 ◽  
Vol 15 (3-4) ◽  
pp. 212-216
Author(s):  
R.Kh. Bolotnova ◽  
V.A. Korobchinskaya
Keyword(s):  
Gas Dynamics ◽  
Numerical Solutions ◽  
Similarity Theory ◽  
Compressible Medium ◽  
Initial Moment ◽  
Point Explosion ◽  
Perfect Gas ◽  
Gas Dynamics Equations ◽  
Self Similar ◽  
Similar Solutions

Comparative analysis of solutions of Sedov’s problem of a point explosion in gas for the plane case, obtained by the analytical method and using the open software package of computational fluid dynamics OpenFOAM, is carried out. A brief analysis of methods of dimensionality and similarity theory used for the analytical self-similar solution of point explosion problem in a perfect gas (nitrogen) which determined by the density of uncompressed gas, magnitude of released energy, ratio of specific heat capacities and by the index of geometry of the explosion is given. The system of one-dimensional gas dynamics equations for a perfect gas includes the laws of conservation of mass, momentum, and energy is used. It is assumed that at the initial moment of time there is a point explosion with instantaneous release of energy. Analytical self-similar solutions for the Euler and Lagrangian coordinates, mass velocity, pressure, temperature, and density in the case of plane geometry are given. The numerical simulation of considered process in sonicFoam solver of OpenFOAM package built on the PISO algorithm was performed. For numerical modeling the system of differential equations of gas dynamics is used, including the equations of continuity, Navier-Stokes motion for a compressible medium and conservation of internal energy. Initial and boundary conditions were selected in accordance with the obtained analytical solution using the setFieldsDict, blockMeshDict, and uniformFixedValue utilities. The obtained analytical and numerical solutions have a satisfactory agreement.

Similarity solutions for viscous vortex cores

1992 ◽  
Vol 238 ◽  
pp. 487-507 ◽  
Author(s):  
Ernst W. Mayer ◽  
Kenneth G. Powell
Keyword(s):  
Numerical Solutions ◽  
Stokes Equations ◽  
Axial Flow ◽  
Similarity Solutions ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
The Core ◽  
Axial Pressure Gradient ◽  
Self Similar ◽  
Similar Solutions

Results are presented for a class of self-similar solutions of the steady, axisymmetric Navier–Stokes equations, representing the flows in slender (quasi-cylindrical) vortices. Effects of vortex strength, axial gradients and compressibility are studied. The presence of viscosity is shown to couple the parameters describing the core growth rate and the external flow field, and numerical solutions show that the presence of an axial pressure gradient has a strong effect on the axial flow in the core. For the viscous compressible vortex, near-zero densities and pressures and low temperatures are seen on the vortex axis as the strength of the vortex increases. Compressibility is also shown to have a significant influence upon the distribution of vorticity in the vortex core.

Computational Fluid Dynamic Prediction of Noise From a Cold Turbulent Propane Jet

2008 ◽  
Author(s):  
Tanya S. Stanko ◽  
Derek B. Ingham ◽  
Michael Fairweather ◽  
Mohamed Pourkashanian
Keyword(s):  
Boundary Conditions ◽  
Numerical Solutions ◽  
Fluid Dynamic ◽  
Broadband Noise ◽  
Navier Stokes ◽  
Mean Flow ◽  
Dynamic Prediction ◽  
Design Changes ◽  
Turbulent Jet Flow ◽  
Velocity Information

Numerical solutions of a turbulent jet flow are used to provide velocity information throughout a simple cold turbulent propane jet at a Reynolds number of 68,000. Predictions provided by the Reynolds-averaged Navier-Stokes simulations, based on a Reynolds stress turbulence model, are compared with experimental data available in the literature. The effect of the modelled inlet boundary conditions on the predicted flow field is described, and the discrepancy between the simulation results and experiment measurements is found to be less than the corresponding variations due to uncertainness in the experimental boundary conditions. In addition, these solutions are used as the basis for noise predictions for the jet based on Lighthill’s theory using the Goldstein broadband noise source formalization that postulates axisymmetric turbulence superposed on the mean flow. The latter model provides an aeroacoustic tool that is reasonable in identifying components or surfaces that generate significant amounts of noise, thereby providing opportunities for early design changes to aircraft and gas turbine components.

A Two-Dimensional Cylindrical Transient Conduction Solution Using Green's Functions

2014 ◽  
Vol 136 (10) ◽  
Author(s):  
Robert L. McMasters ◽  
James V. Beck
Keyword(s):  
Parameter Estimation ◽  
Boundary Conditions ◽  
Green's Functions ◽  
Bessel Functions ◽  
Numerical Solutions ◽  
Green’S Functions ◽  
Two Dimensional ◽  
Trigonometric Functions ◽  
Convective Boundary Conditions ◽  
Convective Boundary

There are many applications for problems involving thermal conduction in two-dimensional (2D) cylindrical objects. Experiments involving thermal parameter estimation are a prime example, including cylindrical objects suddenly placed in hot or cold environments. In a parameter estimation application, the direct solution must be run iteratively in order to obtain convergence with the measured temperature history by changing the thermal parameters. For this reason, commercial conduction codes are often inconvenient to use. It is often practical to generate numerical solutions for such a test, but verification of custom-made numerical solutions is important in order to assure accuracy. The present work involves the generation of an exact solution using Green's functions where the principle of superposition is employed in combining a one-dimensional (1D) cylindrical case with a 1D Cartesian case to provide a temperature solution for a 2D cylindrical. Green's functions are employed in this solution in order to simplify the process, taking advantage of the modular nature of these superimposed components. The exact solutions involve infinite series of Bessel functions and trigonometric functions but these series sometimes converge using only a few terms. Eigenvalues must be determined using Bessel functions and trigonometric functions. The accuracy of the solutions generated using these series is extremely high, being verifiable to eight or ten significant digits. Two examples of the solutions are shown as part of this work for a family of thermal parameters. The first case involves a uniform initial condition and homogeneous convective boundary conditions on all of the surfaces of the cylinder. The second case involves a nonhomogeneous convective boundary condition on a part of one of the planar faces of the cylinder and homogeneous convective boundary conditions elsewhere with zero initial conditions.

Gas Turbine Temperature Prediction Using Unsteady CFD and Realistic Non-Uniform 2D Combustor Exit Properties

2008 ◽  
Author(s):  
Fre´de´ric N. Felten ◽  
Semir Kapetanovic ◽  
D. Graham Holmes ◽  
Michael Ostrowski
Keyword(s):  
Boundary Conditions ◽  
Gas Turbine ◽  
Turbine Blades ◽  
Computational Cost ◽  
Computational Domain ◽  
Two Dimensional ◽  
Gas Temperature ◽  
Sliding Mesh ◽  
Flow Gradients ◽  
The Impact

Typical Computational Fluid Dynamics (CFD) studies performed on High Pressure Turbines (HPT) do not include the combustor domain in their analyses. Boundary conditions from the combustor exit have to be prescribed at the inlet of the computational domain for the first HPT nozzle. It is desirable to include the effect of combustor non-uniformities and flow gradients in order to enhance the accuracy of the aerodynamics and heat transfer predictions on the nozzle guide vanes and downstream turbine blades. The present work is the continuation of steady and quasi-unsteady studies performed previously by the authors. A fully unsteady nonlinear approach, also referred to as sliding mesh, is now used to investigate a first HPT stage and the impact of realistic non-uniformities and flow gradients found along the exit plane of a gas turbine combustor. Two Turbine Inlet Boundary Conditions (TIBC) are investigated. Simulations using a two-dimensional TIBC dependant on both the radial and circumferential directions are performed and compared to baseline analyses, where the previous two-dimensional TIBC is circumferentially averaged in order to generate inlet boundary conditions dependant only on the radial direction. The two elements included in the present work, combustor pitchwise non-uniformities and full unsteady blade row interactions are shown to: (1) alter the gas temperature profile predictions up to ±5%; (2) modify the surface temperature predictions by ±8% near the trailing edge of the vane suction side; (3) increase the overall pressure losses by roughly 1%, and (4) modified the ingestion behavior of the purge cavity flow. In addition, keeping in mind the tradeoff between improved predictions and computational cost, the use of an unsteady sliding mesh formulation, instead of a quasiunsteady frozen gust, reveals the importance of the two-way unsteady coupling between adjacent blade rows for temperature and pressure predictions.

An Approximate Solution of the Laminar Boundary Layer on a Flat Plate with Uniform Suction

1955 ◽  
Vol 59 (538) ◽  
pp. 697-698
Author(s):  
S. J. Peerless ◽  
D. B. Spalding
Keyword(s):  
Boundary Layer ◽  
Boundary Conditions ◽  
Present Method ◽  
Integral Method ◽  
Boundary Layer Thickness ◽  
Numerical Solutions ◽  
Approximate Methods ◽  
Boundary Layer Problems ◽  
Momentum Integral ◽  
Similar Solutions

Boundary layer problems may be divided into two classes: (a) those for which similar solutions can be found, i.e. where the boundary conditions are such that similar profiles differing only in scale factor exist at different sections; and (b) those where the boundary conditions do not effect similarity, so that the development of the boundary layer must be calculated in stages. The latter class are known as “continuation problems,” and very few numerical solutions have been obtained because of the labour involved.Approximate methods of solving continuation problems are known, using the Karman momentum integral method (e.g. Ref. 1) or variants. Some of these methods make use of velocity profiles calculated for “similar” boundary layers. This note presents a new approximate method which uses “similar” profiles but avoids using the momentum integral. Instead of characterising the boundary layer thickness by the “momentum thickness,” which needs to be calculated yet is of less direct interest, the wall shear stress is used; this stress usually has to be calculated in any case and the present method is therefore comparatively simple.

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