• Title/Summary/Keyword: Similarity Solution

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Non-Similarity Solution for Two-Dimensional Laminar Jet (이차원 층류제트를 위한 비 상사해)

  • 이상환
    • Transactions of the Korean Society of Mechanical Engineers
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    • v.18 no.1
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    • pp.150-155
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    • 1994
  • An Approximate solution for plane two-dimensional incompressible laminar jet issuing from a finite opening with arbitrary initial profile into the same ambient fluid is proposed. For an arbitrary initial velocity profile, the problem is generated from the well known similarity solution for the jet of infinitesimal opening and provides good approximations in the region where the similarity solution cannot be used as an approximation. The asymptotic behavior of this solution is investigated and it is shown that, as goes downstream, the present solution approachs the similarity solution.

A similarity solution for undrained expansion of a cylindrical cavity in K0-consolidated anisotropic soils

  • Wang, You;Lin, Lin;Li, Jingpei
    • Geomechanics and Engineering
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    • v.25 no.4
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    • pp.303-315
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    • 2021
  • A rigorous and generic similarity solution is developed for assessment of the undrained expansion responses of a cylindrical cavity expansion in K0-consolidated anisotropic soils. A K0-consolidated anisotropic modified Cam-clay (K0-AMCC) model that can represent the initial stress anisotropy and the effects of stress-induced anisotropy is used to model the soil behaviors during cavity expansion. All the seven basic unknowns, the three stress components, the pore water pressure, the particle velocity, the specific volume and the hardening parameter, are reduced to the functions of a dimensionless radial coordinate and are taken as coupled variables to formulate the problem. The governing equations are formulated by making use of the equilibrium equation, the constitutive equation, the consistency condition, the continuity condition and the undrained condition, which are then solved as an initial value problem. The proposed rigorous similarity solution is compared with some well-documented rigorous solutions to validate the solution and to highlight the special expansion responses in anisotropic soils. The results reveal that the present solution can yield more predictions for cavity expansion problems in soils with initial anisotropic stresses.

An Extended Similarity Solution for One-Dimensional Multicomponent Alloy Solidification in the Presence of Shrinkage-Induced Flow (체적수축유동이 있는 일차원 다원합금 응고에 대한 확장된 해석해)

  • Chung, Jae-Dong;Yoo, Ho-Seon;Choi, Man-Soo;Lee, Joon-Sik
    • Transactions of the Korean Society of Mechanical Engineers B
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    • v.24 no.3
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    • pp.426-434
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    • 2000
  • This paper deals with a generalized similarity solution for the one-dimensional solidification of ternary or higher-order multicomponent alloys. The present approach not only retains the existing features of binary systems such as temperature- solute coupling, shrinkage-induced flow, solid-liquid property differences, and finite back diffusion, but also is capable of handling a multicomponent alloy without restrictions on the partition coefficient and microsegregation parameter. For an alloy of N-solute species, governing equations in the mushy region reduce to (N+2) nonlinear ordinary differential equations via similarity transformation, which are to be solved along with the closed-form solutions for the solid and liquid regions. A linearized correction scheme adopted in the solution procedure facilitates to determine the solidus and liquidus positions stably. The result for a sample ternary alloy agrees excellently with the numerical prediction as well as the reported similarity solution. Additional calculations are also presented to show the utility of this study. Finally, it is concluded that the present analysis includes the previous analytical approaches as subsets.

A Similarity Solution of the Characteristics of Micropolar Fluid Flow in the Vicinity of a Wedge (상사해법을 이용한 쐐기형 물체 주위의 미세 극성유체 유동 특성에 관한 연구)

  • Kim, Youn J.
    • Transactions of the Korean Society of Mechanical Engineers B
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    • v.23 no.8
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    • pp.969-977
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    • 1999
  • A similarity solution of a steady laminar flow of micropolar fluids past wedges has been studied. The similarity variables found by Falkner and Skan are employed to reduce the streamwise-dependence in the coupled nonlinear boundary layer equations. Numerical solutions of the equations are then obtained using the fourth-order Runge-Kutta method and the distribution of velocity, micro-rotation, shear and couple stress across the boundary layer are obtained. These results are compared with the corresponding flow problems for Newtonian fluid past wedges with various wedge angles. Numerical results show that, keeping ${\beta}$ constant, the skin friction coefficient is lower for a micropolar fluid, as compared to a Newtonian fluid. For the case of constant material parameter K, however, the velocity distribution for a micropolar fluid is higher than that of a Newtonian fluid.

A Similarity Solution for the Directional Casting of Peritectic Alloys in the Presence of Shrinkage-Induced Flow (체적수축유동이 있는 포정합금의 방향성주조에 대한 상사해)

  • Yu, Ho-Seon;Jeong, Jae-Dong;Lee, Jun-Sik
    • Transactions of the Korean Society of Mechanical Engineers B
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    • v.25 no.4
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    • pp.485-495
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    • 2001
  • This paper presents a similarity solution for the directional casting of binary peritectic alloys in the presence of shrinkage-induced flow. The present model retains essential ingredients of alloy solidification, such as temperature-solute coupling, macrosegregation, solid-liquid property differences, and finite back diffusion in the primary phase. An algorithm for simultaneously determining the peritectic and liquidus positions is newly developed, which proves to be more efficient and stable than the existing scheme. Sample calculations are performed for both hypo- and hyper-peritectic compositions. The results show that the present analysis is capable of properly resolving the solidification characteristics of peritectic alloys so that it can be used for validating numerical models as a test solution.

Simple solutions of an opening in elastic-brittle plastic rock mass by total strain and incremental approaches

  • Park, Kyungho
    • Geomechanics and Engineering
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    • v.13 no.4
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    • pp.585-600
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    • 2017
  • This study deals with simple solutions for a spherical or circular opening excavated in elastic-brittle plastic rock mass compatible with a linear Mohr-Coulomb (M-C) or a nonlinear Hoek-Brown (H-B) yield criterion. Based on total strain approach, the closed-form solutions of stresses and displacement are derived simultaneously for circular and spherical openings using original H-B and M-C yield criteria. Two simple numerical procedures are proposed for the solution of generalized H-B and M-C yield criteria. Based on incremental approach, the similarity solution is derived for circular and spherical openings using generalized H-B and M-C yield criteria. The classical Runge-Kutta method is used to integrate the first-order ordinary differential equations. Using three data sets for M-C and H-B models, the results of the radial displacements, the spreading of the plastic radius with decreasing pressure, and the radial and circumferential stresses in the plastic region are compared. Excellent agreement among the solutions is obtained for all cases of spherical and circular openings. The importance of the use of proper initial values in the similarity solution is discussed.

TWO-DIMENSIONAL STAGNATION FLOW TOWARD A PLANE WALL COATED WITH MAGNETIC FLUID OF UNIFORM THICKNESS (균일 두께의 자성유체 피막이 있는 평면 벽을 향하는 2차원 정체 유동)

  • Ko, Hyung-Jong;Kim, Kyoung-Hoon;Kim, Se-Woong
    • Journal of computational fluids engineering
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    • v.12 no.4
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    • pp.20-27
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    • 2007
  • Two-dimensional stagnation flow toward a plane wall coated with magnetic fluid of uniform thickness is investigated. The flow field is represented as a similarity solution of the Navier-Stokes equation for this incompressible laminar flow. The resulting third order ordinary differential equation is solved numerically by using the shooting method and by determining two shooting parameters so as to satisfy the boundary and interface conditions. Features of the flow including streamline patterns are investigated for the varying values of density ratio, viscosity ratio, and Reynolds number. An adverse flow with double eddy pair in magnetic fluid region is found to emerge as the Reynolds number becomes higher than a threshold value. The results for the interface velocity, interface and wall shear stress, and boundary layer and displacement thickness are also presented.

Conjugate Heat Transfer of Laminar Film Condensation Along a Horizontal Plate (강제대류 층류 막응축에서 복합열전달)

  • Lee Euk-Soo
    • Transactions of the Korean Society of Mechanical Engineers B
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    • v.30 no.3
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    • pp.238-245
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    • 2006
  • This paper proposes appropriate conjugate parameters and dimensionless temperatures to analysis the conjugate problem of heat conduction in solid wall coupled with laminar film condensation flow adjacent to horizontal flat plate. An efficient methods for some fluids are proposed for its solution. The momentum and energy balance equations are reduced to a nonlinear system of ordinary differential equations with four parameters: the Prandtl number, Pr, Modified Jacob number, $Ja^*/Pr$, defined by an overall temperature difference, a property ratio $\sqrt{\rho_l{\mu}_l/{\rho_v{\mu}_v}$ and the conjugate parameter $\zeta$. The obtained similarity solution reveals the effect of the conjugate parameter, and the results are compared with the simplified solution. The variations of the heat transfer rates as well as the interface temperature and frictions along the plate are shown explicitly.

EXISTENCE AND NON-UNIQUENESS OF SOLUTION FOR A MIXED CONVECTION FLOW THROUGH A POROUS MEDIUM

  • Hammouch, Zakia;Guedda, Mohamed
    • Journal of applied mathematics & informatics
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    • v.31 no.5_6
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    • pp.631-642
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    • 2013
  • In this paper we reconsider the problem of steady mixed convection boundary-layer flow over a vertical flat plate studied in [6],[7] and [13]. Under favorable assumptions, we prove existence of multiple similarity solutions, we study also their asymptotic behavior. Numerical solutions are carried out using a shooting integration scheme.

A study on effect of heat transfer of condensation including noncondensable gas over a flat plate (불응축가스가 평판위 응축열전달에 미치는 영향에 관한 연구)

  • 양대일;정형호
    • Journal of Advanced Marine Engineering and Technology
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    • v.24 no.1
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    • pp.25-30
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    • 2000
  • In present paper, mass transfer over a flat plate with film condensation including noncondesable gas is analyzed with the help of similarity methods. Couette flow was assumed in liquid film and boundary-layer approximation was used in the ambient flow. Governing equations were transformed into the ordinary differential equtions by the similarity methods. Runge-Kutta and shooting method were used in order to fine the effect of mass transfer on the velocity and concentrations at the liquid-vapor interface.

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