A New Approach for the Derivation of the Similarity Equation of the 2-D Unsteady Boundary Layer Equations

2012 ◽  
Author(s):  
Md. Abdus Sattar
Keyword(s):  
Boundary Layer ◽  
Numerical Solutions ◽  
Similarity Solutions ◽  
Length Scale ◽  
Local Similarity ◽  
Unsteady Boundary Layer ◽  
Similarity Equation ◽  
Boundary Layer Equations ◽  
Dimensional Boundary ◽  
The One

A local similarity equation for the hydrodynamic 2-D unsteady boundary layer equations has been derived based on a time dependent length scale initially introduced by the author in solving several unsteady one-dimensional boundary layer problems. Similarity conditions for the potential flow velocity distribution are also derived. This derivation shows that local similarity solutions exist only when the potential velocity is inversely proportional to a power of the length scale mentioned above and is directly proportional to a power of the length measured along the boundary. For a particular case of a flat plate the derived similarity equation exactly corresponds to the one obtained by Ma and Hui[1]. Numerical solutions to the above similarity equation are also obtained and displayed graphically.

Singularities in solutions of the three-dimensional laminar-boundary-layer equations

1985 ◽  
Vol 160 ◽  
pp. 257-279 ◽  
Author(s):  
James C. Williams
Keyword(s):  
Boundary Layer ◽  
Laminar Boundary Layer ◽  
Numerical Solutions ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Unsteady Boundary Layer ◽  
Boundary Layer Flows ◽  
Boundary Layer Equations ◽  
New Type ◽  
Steady Laminar

The three-dimensional steady laminar-boundary-layer equations have been cast in the appropriate form for semisimilar solutions, and it is shown that in this form they have the same structure as the semisimilar form of the two-dimensional unsteady laminar-boundary-layer equations. This similarity suggests that there may be a new type of singularity in solutions to the three-dimensional equations: a singularity that is the counterpart of the Stewartson singularity in certain solutions to the unsteady boundary-layer equations.A family of simple three-dimensional laminar boundary-layer flows has been devised and numerical solutions for the development of these flows have been obtained in an effort to discover and investigate the new singularity. The numerical results do indeed indicate the existence of such a singularity. A study of the flow approaching the singularity indicates that the singularity is associated with the domain of influence of the flow for given initial (upstream) conditions as is prescribed by the Raetz influence principle.

Keller-Box Scheme to Mixed Convection Flow Over a Solid Sphere with the Effect of MHD

2021 ◽  
Vol 6 (1) ◽  
pp. 97
Author(s):  
Mohammad Ghani ◽  
Wayan Rumite
Keyword(s):  
Boundary Layer ◽  
Free Convection ◽  
Mixed Convection ◽  
Numerical Solutions ◽  
Solid Sphere ◽  
Convection Flow ◽  
Boundary Layer Equations ◽  
Buoyancy Forces ◽  
Dimensional Boundary ◽  
External Forces

Mixed convection is the combination of a free convection caused by the buoyancy forces due to the different density and a forced convection due to external forces that increase the heat exchange rate. This means that, in free convection, the effect of external forces is significant besides buoyancy forces. In this study the fluid type with viscoelastic effect is non-Newtonian. The viscoelastic fluids that pass over a surface of a sphere form a thin layer, which due to their dominant viscosity is called by the border layer. The obtained limiting layer is analyzed with the thickness of the boundary layer-  near the lower stagnating point, then obtained dimensional boundary layer equations, continuity, momentum, and energy equations. These dimensional boundary layer equations are then transformed into non-dimensional boundary layer equations by using non-dimensional variables. Further, the non-dimensional boundary layer equations are transformed into ordinary differential equations by using stream function, so that obtained the non-similar boundary layer equations. These non-similar boundary layer equations are solved numerically by using finite difference method of Keller-Box. The discretization results are non-linear and it should be linearized using newton linearization technique. The numerical solutions are analyzed the effect of Prandtl number, viscoelastic, mixed convection, and MHD parameters towards velocity profile, temperature profile, and wall temperature.

Non-similarity solutions to the corner boundary-layer equations (and the effects of wall transpiration)

1999 ◽  
Vol 400 ◽  
pp. 125-162 ◽  
Author(s):  
PETER W. DUCK ◽  
SIMON R. STOW ◽  
MANHAR R. DHANAK
Keyword(s):  
Boundary Layer ◽  
Numerical Solutions ◽  
A Priori ◽  
Leading Edge ◽  
Flow Region ◽  
Similarity Solutions ◽  
Leading Order ◽  
Governing Equations ◽  
Boundary Layer Equations ◽  
Pressure Term

The incompressible boundary layer in the corner formed by two intersecting, semi-infinite planes is investigated, when the free-stream flow, aligned with the corner, is taken to be of the form U∞F(x), x representing the non-dimensional streamwise distance from the leading edge. In Dhanak & Duck (1997) similarity solutions for F(x) = xn were considered, and it was found that solutions exist for only a range of values of n, whilst for ∞ > n > −0.018, approximately, two solutions exist. In this paper, we extend the work of Dhanak & Duck to the case of non-90° corner angles and allow for streamwise development of solutions. In addition, the effect of transpiration at the walls of the corner is investigated. The governing equations are of boundary-layer type and as such are parabolic in nature. Crucially, although the leading-order pressure term is known a priori, the third-order pressure term is not, but this is nonetheless present in the leading-order governing equations, together with the transverse and crossflow viscous terms.Particular attention is paid to flows which develop spatially from similarity solutions. It turns out that two scenarios are possible. In some cases the problem may be treated in the usual parabolic sense, with standard numerical marching procedures being entirely appropriate. In other cases standard marching procedures lead to numerically inconsistent solutions. The source of this difficulty is linked to the existence of eigensolutions emanating from the leading edge (which are not present in flows appropriate to the first scenario), analogous to those found in the computation of some two-dimensional hypersonic boundary layers (Neiland 1970; Mikhailov et al. 1971; Brown & Stewartson 1975). In order to circumvent this difficulty, a different numerical solution strategy is adopted, based on a global Newton iteration procedure.A number of numerical solutions for the entire corner flow region are presented.

Three-Dimensional Boundary Layers on Cones at Small Angles of Attack

1968 ◽  
Vol 35 (4) ◽  
pp. 634-640 ◽  
Author(s):  
O. Pinkus ◽  
S. B. Cousin
Keyword(s):  
Boundary Layer ◽  
Numerical Solutions ◽  
Three Dimensional ◽  
Equivalent Radius ◽  
Wind Tunnel Model ◽  
Boundary Layer Equations ◽  
Tunnel Model ◽  
Dimensional Boundary ◽  
Heat Transfer Effects ◽  
Incipient Separation

Based on an expression given by Cooke, an equation is derived for a three-dimensional “equivalent radius” for cones at a small angle of attack. This function when used in any of the available axisymmetric boundary-layer equations yields corresponding solutions for yawed cones. Expressions for the streamlines along which the above equations are to be integrated are also derived. The method yields a line of possible incipient separation or wake formation in addition to the boundary-layer properties in both the longitudinal and circumferential direction. Numerical solutions including heat transfer effects are presented for a wind tunnel model and compared with experimental results.

scholarly journals The onset of instability in unsteady boundary-layer separation

1996 ◽  
Vol 315 ◽  
pp. 223-256 ◽  
Author(s):  
K. W. Cassel ◽  
F. T. Smith ◽  
J. D. A. Walker
Keyword(s):  
Boundary Layer ◽  
Boundary Layer Separation ◽  
External Pressure ◽  
Theoretical Description ◽  
Unsteady Boundary Layer ◽  
Layer Separation ◽  
Boundary Layer Equations ◽  
Dimensional Boundary ◽  
High Reynolds ◽  
Inviscid Instability

The process of unsteady two-dimensional boundary-layer separation at high Reynolds number is considered. Solutions of the unsteady non-interactive boundary-layer equations are known to develop a generic separation singularity in regions where the pressure gradient is prescribed and adverse. As the boundary layer starts to separate from the surface, however, the external pressure distribution is altered through viscous—inviscid interaction just prior to the formation of the separation singularity; hitherto this has been referred to as the first interactive stage. A numerical solution of this stage is obtained here in Lagrangian coordinates. The solution is shown to exhibit a high-frequency inviscid instability resulting in an immediate finite-time breakdown of this stage. The presence of the instability is confirmed through a linear stability analysis. The implications for the theoretical description of unsteady boundary-layer separation are discussed, and it is suggested that the onset of interaction may occur much sooner than previously thought.

Similarity Solutions of Laminar, Incompressible Boundary Layer Equations of Non-Newtonian Fluids

1968 ◽  
Vol 90 (1) ◽  
pp. 71-74 ◽  
Author(s):  
A. G. Hansen ◽  
T. Y. Na
Keyword(s):  
Boundary Layer ◽  
Numerical Solutions ◽  
Engineering Application ◽  
Similarity Solutions ◽  
Newtonian Fluids ◽  
Important Conclusion ◽  
Arbitrary Continuous Function ◽  
Boundary Layer Equations ◽  
Eyring Model ◽  
Layer Flow

A similarity analysis is made of the steady, two-dimensional, laminar boundary layer flow of non-Newtonian fluids. The important conclusion drawn from this analysis is that for non-Newtonian fluids of any model, similarity solutions exist only for the case of a wedge flow of 90 deg. The only limitation is that the stress and rate-of-strain can be related by an arbitrary continuous function. The result in this paper further extends a recent work by Lee and Ames [11]. Numerical solutions for the Powell-Eyring model are presented in dimensionless form as an engineering application of the theory.

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