Solution Of Boundary Layer Equations Research Articles

Stochastic modeling of finite-rate chemistry effects in hydrogen-air turbulent jet diffusion flames with helium dilution

Stochastic simulations of turbulent hydrogen-air jet diffusion flames at three different dilution rates with helium are implemented using the ‘one-dimensional turbulence’ (ODT) model. The approach is based on one-dimensional unsteady solution of boundary layer equations to represent molecular processes and a stochastic implementation of turbulent advection. The 1D scalar and streamwise momentum profiles represent radial profiles within the flames; while, the unsteady evolution of the solution is interpreted as a downstream evolution of the radial scalar and streamwise momentum profiles. Multiple realizations of jet simulations are used to compute conditional statistics of major species, NO, and temperature. The ODT computations are implemented with a five-step reduced mechanism for hydrogen combustion and an optically-thin radiation model. Computed conditional statistics of temperature, major and minor species are compared to the experimental data from a set of documented flames at Sandia National Labs. Reasonable qualitative and quantitative agreement between computed and measured statistics is found, including very good predictions of NO mean and RMS profiles. Both computation and experiment exhibit the role of dilution in enhancing finite-rate chemistry effects, which vary as a function of downstream distance and fuel dilution.

Eigenfunction Expansion Solution for Boundary-Layer Equations in Cylindrical Coordinates: Simultaneously Developing Flow in Circular Tubes

The Generalized Integral Transform Technique (GITT) is employed, via a novel eigenfunction expansion, in the solution of the steady-state continuity, momentum, and energy equations under the boundary-layer formulation and cylindrical coordinates, and applied to the solution of simultaneously developing laminar flow inside circular ducts. The streamfunction formulation is adopted to automatically satisfy the continuity equation and to eliminate the pressure field. A fourth-order eigenvalue problem is thus considered for the velocity field, eliminating the difficulties associated with the singularity at the channel centerline through this recently introduced expansion basis. A thorough analysis of convergence behavior is undertaken for both the velocity and temperature proposed eigenfunction representations, and here illustrated for representative values of governing parameters and positions along the channel. Results for quantities associated with applications, such as the product of the friction factor–Reynolds number and Nusselt numbers, are also computed along the entrance region for different values of the governing parameters, and tabulated for reference purposes. Critical comparisons with previous results in the literature are also performed, in order to validate the numerical code developed and to inspect the adequacy of previously proposed approximate solutions.

Free Convection from a Vertical Permeable Circular Cone with Pressure Work and Non-Uniform Surface Temperature

Laminar free convection from a vertical permeable circular cone maintained at non-uniform surface temperature with pressure work term is considered. Non-similarity solutions for boundary layer equations are found to exist when the surface temperature follows the power law variations with the distance measured from the leading edge. The numerical solutions of the transformed non-similar boundary layer equations are obtained by using two methodologies, namely, (i) a finite difference method and (ii) a series solution or perturbation method for small values of ξ, the dimensionless suction parameter. Solutions obtained in terms of skin-friction coefficient, local rate of heat transfer, velocity and temperature profiles for the values of Prandtl number, pressure work parameter and temperature gradient are displayed in both graphical and tabular forms. Finite difference solutions are compared with the solutions obtained by perturbation technique and found to be in excellent agreement near the leading edge.

Reference Enthalpy Method Developed from Solutions of the Boundary-Layer Equations

A simple average of local enthalpy over the velocity profile is proposed as the proper definition of reference enthalpy for the purpose of quasi-one-dimensional treatment of compressible boundary layers. Use of Van Driest's nearly exact solutions of the laminar boundary-layer equations shows that this definition produces Eckert's reference enthalpy formulation for the special case of an adiabatic wall. For surfaces other than adiabatic, either Eckert's form should be replaced by that of Young and Janssen, or the coefficient in Eckert's viscous heating term should be slightly modified. A similar analysis was conducted for turbulent flows using Whitfield and High's simplified solutions of the turbulent boundary-layer equations. Dorrance's derivation of reference quantities is also addressed. This work provides a theoretical basis for the empirical reference enthalpy formulas of Eckert and others and supplies practical expressions for the reference enthalpy of both laminar and turbulent compressible boundary layers.

Similarity solutions of boundary layer equations for a special non-Newtonian fluid in a special coordinate systme

Two dimensional equations of steady motion for third order fluids are expressed in a special coordinate system generated by the potential flow corresponding to an inviscid fluid. For the inviscid flow around an arbitrary object, the streamlines are the phicoordinates and velocity potential lines are psi-coordinates which form an orthogonal curvilinear set of coordinates. The outcome, boundary layer equations, is then shown to be independent of the body shape immersed into the flow. As a first approximation, assumption that second grade terms are negligible compared to viscous and third grade terms. Second grade terms spoil scaling transformation which is only transformation leading to similarity solutions for third grade fluid. By using Lie group methods, infinitesimal generators of boundary layer equations are calculated. The equations are transformed into an ordinary differential system. Numerical solutions of outcoming nonlinear differential equations are found by using combination of a Runge-Kutta algorithm and shooting technique.

Laminar boundary layer flow over a horizontal permeable flat plate

An analysis is performed to study the laminar boundary layer flow over a porous flat plate with injection or suction imposed at the wall. Two different analyses techniques are used for the solution of boundary layer equations: similarity solution and numerical solution. The effect of uniform suction/injection on the heat transfer is discussed. The constant surface temperature thermal boundary condition is used for the horizontal flat plate. The effect of Prandtl number on heat transfer is also investigated. A scale analysis is performed to get more insight into the Prandtl effect. Friction coefficients and Nusselt numbers are calculated for constant fluid injection/suction along the plate. The results indicate that the suction enhances the heat transfer coefficient while injection causes a decrease in heat transfer.

An Efficient Way of Specifying Profile Inflow Boundary Conditions

An efficient way to systematically reproduce inflow profiles across a compressible, turbulent boundary layer for the mean flow variables, as well as the turbulent quantities in one-equation and two-equation turbulence-models, from given external flow conditions and one boundary layer parameter, is described. The reproduced profiles are checked against both experimental results and numerical solution of boundary layer equations. Theoretical analysis shows that a new form of density-weighted velocity, rather the Van Driest density-weighted velocity, obeys the linear-law at the viscous sublayer in a compressible turbulent boundary layer. Especially for non-adiabatic wall at hypersonic Mach numbers, where there are large density gradients, these two kinds of density-weighted velocity could differ considerably. Therefore, the new form of density-weighted velocity proposed in this paper should be employed for the viscous sublayer. It is also shown that power-law fitting for the streamwise velocity gives an unacceptable profile in the viscous sublayer. An efficient way to specify the normal velocity profile is also proposed and tested. The reproduced normal velocity at the boundary layer edge is found to agree remarkably well with the numerical solution of boundary layer equations.

Numerical Simulation of Continuous Stirred Ultrafiltration Process: An Approach Based on Moving Boundary Layer Concept

A mass transfer model based on unsteady state mass balance over concentration boundary layer coupled with gradual development of so called "gel" or "cake" layer has been formulated in this study. The model considers the boundary layer problem simultaneously with the film theory of mass transfer and resistance in series model. The resulting equations are solved first by reducing the set of partial differential equations to nonlinear equations utilizing orthogonal collocation technique and then applying multidimensional Newton–Raphson method with suitable seeding scheme to solve the above-mentioned nonlinear equations together with some additional constraints. The problem of determination of membrane surface concentration is eliminated by this simultaneous solution of boundary layer equation together with gel layer. The model is found to be capable of predicting permeate flux and rejection under different experimental conditions. The main feature of this model is that it takes into account realistically the effect of gel layer formation within concentration boundary layer whereas in most of the previous studies this effect has not been considered. The dependency of concentration profile, gel thickness, and permeate concentration with different operating variables are also studied and the variations are found to be quite reasonable.

Exact solutions of boundary layer equations of a special non-Newtonian fluid over a stretching sheet

Exact solutions of boundary layer equations of a special non-Newtonian fluid over a stretching sheet

Integral transform solution of boundary layer equations in stream function-only formulation

The boundary layer equations for steady incompressible laminar channel flow are solved through the integral transform method, by adopting the stream function-only formulation of the governing equations, instead of the more commonly used primitive variables formulation. This hybrid numerical–analytical approach provides benchmark results under user-prescribed accuracy targets and is recognized in the validation of purely numerical schemes. The relative merits of the stream function formulation are illustrated through numerical results for the convergence behavior in the case of a plane Poiseuille flow.

Numerical solution of boundary layer equations in compressible cross-flow to a cylinder

Two-dimensional, steady-state, compressible boundary layer and energy equations are solved numerically using finite difference method in cylindrical coordinates. The heat transfer from and to an isolated cylinder for cross-flow is calculated up to the separation point iteratively. The velocity and temperature profiles, and local Nusselt numbers around the cylinder are obtained for the constant wall temperature boundary condition at different oncoming velocities. The results are obtained for compressible and incompressible flow and given in diagrams.

Boundary layer flow of Reiner-Philippoff fluids

An analysis is made of the boundary layer flow of Reiner-Philippoff fluids. This work is an extension of a previous analysis by Hansen and Na [A.G. Hansen and T.Y. Na, Similarity solutions of laminar, incompressible boundary layer equations of non-Newtonian fluids. ASME 67-WA/FE-2, presented at the ASME Winter Annual Meeting, November (1967)], where the existence of similar solutions of the boundary layer equations of a class of general non-Newtonian fluids were investigated. It was found that similarity solutions exist only for the case of flow over a 90° wedge and, being similar, the solution of the non-linear boundary layer equations can be reduced to the solution of non-linear ordinary differential equations. In this paper, the more general case of the boundary layer flow of Reiner-Philippoff fluids over other body shapes will be considered. A general formulation is given which makes it possible to solve the boundary layer equations for any body shape by a finite-difference technique. As an example, the classical solution of the boundary layer flow over a flat plate, known as the Blasius solution, will be considered. Numerical results are generated for a series of values of the parameters in the Reiner-Philippoff model.

Similarity solutions of boundary layer equations for second order fluids

In a previous paper [ Int. J. Engng Sci. 30, 523 (1992)] boundary layer equations for second order fluids are derived. The general isovector fields for these equations are found by using exterior calculus. From the four independent isovectors only one provides useful solutions which is a scaling transformation. Using this isovector, the ordinary differential equations leading to the similarity solutions are found. The numerical solution of the equations are presented and the profile corresponding to the solution is discussed. The shear stress of the similarity solution is also calculated.

Viscoelastic flow past an infinite plate with suction and constant heat flux

The solution of the title problem, discussed by various researchers in the literature, is re-examined. The solution of boundary layer equations is obtained in a closed form.

Discontinuous solutions of boundary layer equations with a positive pressure gradient

Discontinuous solutions of boundary layer equations with a positive pressure gradient

Heat transfer and shear stress at the end wall of a vortex chamber

The results of experimental and theoretical investigations of turbulent shear stress and heat transfer in boundary layers formed on the end walls of vortex chambers are reported. Laser-Doppler anemometry is employed for diagnostics of the flow. Heat transfer with the end wall is studied under the boundary condition q w = const.. The theoretical analysis is based on an integral approach to the solution of boundary layer equations. It has been found that the mechanisms of development of dynamic and thermal boundary layers can be described using the dependences for one-dimensional plate streamlining if the spatial character of the flow is taken into account by a special function. The analytical formulas for determining the basic parameters characterizing aerodynamics and heat transfer in spatial end boundary layers have been obtained.

Calculation of the free surface of a flow on a concrete overflow dam or on a chute

1. The method of calculating flow on an overflow dam based on numerical solution of boundary-layer equations is presently preferable in comparison with the published methods based on integral relations of the boundary layer. 2. To determine the value of the velocity coefficient φ in any section on the spillway, it is recommended to use the graph obtained on the basis of the proposed method.

Accurate solutions of boundary-layer equations for moving permeable surfaces

New accurate solutions of the equations of a laminar boundary layer are obtained for steady flow induced by the motion of a continuous solid surface at constant velocity.

An asymptotic theory of turbulent separation

The problem addressed in this paper is the turbulent counterpart of the classical Goldstein analysis of laminar separation points. In the present paper we present a new asymptotic theory of turbulent boundary layers and we use it to determine the singular behavior of solutions of the boundary layer equations at the point of zero skin friction in a prescribed pressure distribution. In the first part of the paper we describe a new asymptotic theory of turbulent boundary layers which is based on formal expansions in terms of two parameters: Reynolds number Re → ∞ and α → 0 , where α is a turbulence model constant appearing in the zero equation turbulence model employed in the present study. In order to establish the accuracy of the theory, solutions obtained with the new theory for equilibrium turbulent flow are compared with “exact” solutions of Mellor and Gibson. In the second part of the paper the local behavior of solutions of the turbulent boundary layer equations with a zero equation turbulent model is established using the new asymptotic theory. It is demonstrated that the solution of the turbulent boundary layer equations are singular at separation points if the pressure distribution is prescribed just as in the laminar case. However, the nature of the singularity is different in laminar and turbulent flow. We show that the approach to zero skin friction is linear in turbulent flow as compared to the square root behavior of the laminar “Goldstein” singularity.

Singular solution of boundary-layer equations on a moving surface

We consider a plane steady flow of incompressible fluid in the boundary layer which develops on a surface moving downstream. We obtain a singular solution of Prandtl's equations, continuously extendable through the simultaneous vanishing point of the friction and the longitudinal velocity vector component. Such a solution is realized, in particular, in a flow past a rotating circular cylinder.