Method For Boundary-layer Equations Research Articles

The Lie-group shooting method for boundary layer equations in fluid mechanics

In this paper, we propose a Lie-group shooting method to tackle two famous boundary layer equations in fluid mechanics, namely, the Falkner-Skan and the Blasius equations. We can employ this method to find unknown initial conditions. The pivotal point is based on the erection of a one-step Lie group element G(T) and the formation of a generalized mid-point Lie group element G(r). Then, by imposing G(T) = G(r) we can seek the missing initial conditions through a minimum discrepancy of the target in terms of the weighting factor r ɛ (0,1). It is the first time that we can apply the Lie-group shooting method to solve the boundary layer equations. Numerical examples are worked out to persuade that this novel approach has good efficiency and accuracy with a fast convergence speed by searching r with the minimum norm to fit two targets.

The Lie-group shooting method for boundary layer equations in fluid mechanics

The Lie-group shooting method for boundary layer equations in fluid mechanics

Effect on noncondensable gas on laminar film condensation along a vertical plate fin

Abstract Laminar film condensation along a vertical plate fin in the presence of a noncondensable gas is numerically investigated. In the present analysis, the gas mass fraction and temperature at the liquid–vapor interface is regarded as the spatial variation along the fin surface. Governing boundary layer equations together with their corresponding boundary conditions for the condensation system and the one-dimensional fin heat conduction equation with its negligible tip leakage are cast into dimensionless forms. The resulting system of equations is solved by using the central finite-difference approximation for the fin and local nonsimilarity method for boundary layer equations. Results show that the fin efficiency decreases with increasing the thermal resistance ratio of fin-to-liquid Nc, and relatively small amounts of the noncondensable gas in the bulk of the vapor have a significant effect on the fin efficiency for small values of Nc. In addition, the interface temperature is not equal to the fin temperature along the vertical fin, and the local heat transfer coefficient is not uniform, either.

Integral transform method for boundary layer equations in simultaneous heat and fluid flow problems

The two‐dimensional steady boundary layer equations, for simultaneous heat and fluid flow within ducts, are handled through the generalized integral transform technique. The momentum and energy equations are integral transformed by eliminating the transversal coordinate and reducing the PDE’s into an infinite system of coupled non‐linear ordinary differential equations for the transformed potentials. An adaptively truncated version of this ODE system is numerically handled through well known initial value problem solvers, with automatic precision control procedures. The explicit inversion formulae are then recalled to provide analytic expressions for velocity and temperature fields and related quantities of practical interest. Typical examples are presented in order to illustrate the hybrid numerical analytical approach and its convergence behaviour.

Local relaxation method for boundary layer equations with separation

In the context of the laminar steady two-dimensional flow of an incompressible Newtonian fluid in a hilly channel for large values of the Reynolds number, we propose a numerical model of the boundary layer equations, when separation appears. Therefore, when the separation of the flow occurs, we consider the mathematical problem, justified by a formal asymptotic analysis, as elliptic.

An integral spline method for boundary layer equations

Abstract : An integral procedure using spline polynomials is described for the two dimensional boundary layer equations. This is a modified finite-element (MFE) formulation, wherein each term in the equations, rather than each independent variable, is approximated with a spline curve fit. Therefore, this is not a true finite-element or Galerkin method and the conventional spline relationships between functional and derivative values still apply. The only difference between the present integral formulation and our earlier differential collocation procedures is in the treatment of the governing differential equations. The differential methods are more suited to non-conservation equations; the present integral formulation is more desirable for conservation or divergence form of the equations. Boundary layer solutions using conventional second-order finite difference collocation, the second and fourth order Keller Box Scheme, and fourth order spline collocation or MFE methods are compared. Conservation and non-conservation forms are considered. Finally, the extension of the MFE formulation to three-coordinate parabolic systems and for the transonic small disturbance equations is briefly described. (Author)