Begell House Inc.
Hybrid Methods in Engineering
HME
1099-2391
1
3
1999
ON KAPLUN LIMITS AND THE MULTILAYERED ASYMPTOTIC STRUCTURE OF THE TURBULENT BOUNDARY LAYER
31
10.1615/HybMethEng.v1.i3.10
Atila P. Silva
Freire
Mechanical Engineering Department, Federal University of Rio de Janeiro, Av. Moniz de Aragao 420, 21945-972 Rio de Janeiro, Brazil
In the present work, some formal properties of singular perturbation equations are studied through the concept of "equivalent in the limit" of Kaplun, so that a proposition for the principal equations is derived. The proposition shows that if there is a principal equation at a point (η, 1) of the (Ξ × Σ) product space, Ξ space of all positive continuous functions in (0, 1], Σ = (0, 1], then there is also a principal equation at a point (η, ε) of (Ξ × Σ), ε = first critical order. The converse is also true. The proposition is of great implication because it ensures that the asymptotic structure of a singular perturbation problem can be determined by a first-order analysis of the formal domains of validity. The turbulent boundary layer asymptotic structure is then studied by application of Kaplun limits to three test cases: the zero-pressure boundary layer, the separating boundary layer and the shock-wave interacting boundary layer. As it turns out, different asymptotic structures are found, depending on the test cases considered. However, before we consider the real turbulent boundary layer problem, the basics of the theory are illustrated by the study of a model equation that mimics turbulent flow passed over a flat surface. The model equation was chosen for being relatively simple while retaining most of the features of the real problem. This allows one to easily grasp the main concepts and ideas wthout being hampered by unnecessary details. Results show that a two-layered structure is derived, which, however, is different from the classic structure commonly found in the literature, and hence is capable of explaining the flow separation phenomenon. A skin-friction equation resulting from a matching process, and universal laws resulting from local approximated equations are carefully interpreted and evaluated.
ALTERNATE APPLICATION OF REGULAR AND MODIFIED CONJUGATE GRADIENT METHOD TO TWO-DIMENSIONAL INVERSE HEAT CONDUCTION
18
10.1615/HybMethEng.v1.i3.20
Woo-Seung
Kim
Department of Mechanical Engineering, Hanyang University, 1271 Sa-1 dong, Sangnok-gu, Ansan, Kyeonggi-do, 425-791, Korea
M. N.
Ozisik
Mechanical and Aerospace Engineering Department North Carolina State University, Raleigh, N. C. 27607
Eui-Rak
Choi
Department of Mechanical Engineering, Hanyang University, Seoul 133-791, Korea
A two-dimensional transient inverse heat conduction problem involving the estimation of the unknown location, (X*, Y), and timewise varying unknown strength, G(τ), of a line heat source embedded inside a rectangular bar with insulated boundaries has been solved simultaneously. The regular conjugate gradient method (RCGM) and the modified conjugate gradient method (MCGM), with adjoint equation, are used alternately to estimate the unknown strength G(τ) of the source term, while the parameter estimation approach is used to estimate the unknown location (X*, Y*) of the line heat source. The alternate use of the regular and the modified conjugate gradient methods alleviates the convergence difficulties encountered at the initial and final times (i.e., τ = 0 and τ = τf) and hence stabilizes the computation and fastens the convergence of the solution. In order to examine the effectiveness of this approach under severe test conditions, the unknown strength G(τ) is chosen in the form of rectangular, triangular, and sinusoidal functions.
SELF-SIMILAR COMPRESSIBLE LAMINAR BOUNDARY LAYERS BEHIND A NORMAL SHOCK WAVE ADVANCING INTO A STATIONARY FLUID
14
10.1615/HybMethEng.v1.i3.30
P. G. P.
Toro
Institute de Aeronaútica e Espaço - IAE - Centre Técnico Aeroespacial - CTA São José dos Campos - SP 12228-904 - Brazil
Z.
Rusak
Department of Mechanical Engineering, Aeronautical Engineering, and Mechanics, Rensselaer Polytechnic Institute, Troy, NY 12180-3590 - USA
H. T.
Nagamatsu
Department of Mechanical Engineering, Aeronautical Engineering, and Mechanics, Rensselaer Polytechnic Institute, Troy, NY 12180-3590 - USA
L. N.
Myrabo
Department of Mechanical Engineering, Aeronautical Engineering, and Mechanics, Rensselaer Polytechnic Institute, Troy, NY 12180-3590 - USA
A new methodology of deriving a self-similar solution of the compressible laminar boundary layer equations is applied to investigate the airflow behind a normal shock wave advancing into a stationary fluid bounded by a solid wall. Modified Levy-Mangier and Dorodnitsyn-Howarth transformations are used to solve the boundary layer flow on a smooth flat plate with no external pressure gradient. These transformations describe the similarity variables in terms of a power of the density that appears in the viscosity-temperature power law relation. This results in an explicit relation between the stream function and the temperature fields described by a closed coupled system of nonlinear ordinary differential equations. Solutions of the velocity and temperature profiles and the influence of the wall to external flow temperature ratio are presented for various shock wave strengths.
ON THE LIMITATIONS OF THE SINGLE-DOMAIN APPROACH FOR COMPUTATION OF CONVECTION IN COMPOSITE CHANNELS — COMPARISONS WITH EXACT SOLUTIONS
16
10.1615/HybMethEng.v1.i3.40
Ming
Xiong
Department of Mechanical and Aerospace Engineering, North Carolina State University, Campus Box 7910, Rayleigh, North Carolina 27695-7910, USA
In this work, comparisons between the numerical solutions for forced convective flow in composite channels, partly occupied by a homogeneous (clear) fluid and partly by a fluid saturated porous medium, and the exact solutions for the same problems are carried out. This is done to establish limitations of the single-domain approach utilized to obtain the numerical solutions. These limitations result from the assumptions that are implicitly invoked once the single-domain approach is utilized to model fluid flow in the interface region between the porous medium and a clear fluid. It is shown that the single-domain approach results in correct matching of the shear stress at the porous/fluid interface only if the adjustable coefficient in the representation for the excess stress equals zero and also if the effective viscosity of the porous medium equals to the fluid viscosity.
ANALYTICAL SOLUTION OF ABLATION PROBLEM WITH NONLINEAR COUPLING EQUATION
14
10.1615/HybMethEng.v1.i3.50
A. J.
Diniz
Universidade Estadual Paulista, Faculdade de Engenharia de Ilha Solteira, Departamento de Engenharia Mecanica, Brasil
JoÃ£o Batista
Campos Silva
Universidade Estadual Paulista, Faculdade de Engenharia de Ilha Solteira, Departamento de Engenharia Mecanica, Brasil
E. L.
Zaparoli
Centro Tecnico Aeroespacial, Instituto Tecnologico de Aeronautica, 12228-904 — Sao Jose dos Campos, Sao Paulo, Brasil
The conduction of heat inside solids involving a process of ablation is of interest in the field of aerospace engineering. The solution to this kind of problem, which involves heat transfer and phase changing, shows some special difficulties, mainly due to the moving boundaries. Heat transfer by ablation in different geometries submitted to transient boundary heat flux is studied analytically. In this paper the Integral Transform Technique and the Generalized Integral Transformed Technique are applied for the solution of equations for the preablative and the ablative phases, respectively. The results of interest are the ablation depth and the speed of the moving boundary.
COMPARISON OF ALTERNATIVE SOLUTION TECHNIQUES TO THE ONE-DIMENSIONAL WAVE EQUATION USING MATLAB
14
10.1615/HybMethEng.v1.i3.60
William B.
Mills
Applied Research and Development Division, Tetra Tech, Inc., 3746 Mt. Diablo Blvd., #300, Lafayette, California 94549, USA 94549
The performance of three solution techniques (finite difference, method of lines, and the generalized integral transform technique) applied to the same one-dimensional hyperbolic wave equation with a temporally oscillating velocity is described. Performance of the algorithms is measured in terms of computational speed to attain the same root-mean-square error (Erms) compared with the closed-form solution, over the spatial domain at a specified time. While all three techniques were able to attain the strict Erms, computational times were typically less for the method of lines technique for most alternative conditions simulated. This conclusion is specific to the problem examined and may change for other problems.
Solutions were programmed in MATLAB. Most of the integrals that appeared in the generalized integral transform technique solution were evaluated symbolically by MATLAB to increase accuracy and computational speed.
PERTURBATION METHOD WITH Mathematica
15
10.1615/HybMethEng.v1.i3.70
Mikhail D.
Mikhailov
Applied Mathematics Center, PO Box 384, Sofia, Technical University, Sofia, Bulgaria; and Mechanical Engineering Department—EE/COPPE/UFRJ, Universidade Federal do Rio de Janeiro, Cidade Universitaria, CP 68.503, Rio de Janeiro, RJ, 21945-970, Brasil
A Mathematica package is developed providing rules for perturbation and decomposition method. Using this package examples from the well-known book by Aziz and Na are solved. All solutions are transformed by using Shanks transformation. It is demonstrated that transformed solutions are remarkably accurate for large values of the small parameter.