The Mixing of Two Parallel Streams of Dissimilar Fluids—Part I: Analytical Development

1971 ◽  
Vol 38 (2) ◽  
pp. 301-309 ◽  
Author(s):  
R. L. Baker ◽  
L. N. Tao ◽  
H. Weinstein
Keyword(s):  
Differential Equation ◽  
Boundary Condition ◽  
Ordinary Differential Equation ◽  
Schmidt Number ◽  
Similarity Transformation ◽  
Velocity Ratio ◽  
Density Ratio ◽  
Large Density ◽  
Parallel Streams ◽  
Analytical Development

The mixing region between dissimilar fluids is investigated in the region where the similarity transformation is valid. The treatment is complete in that laminar and turbulent cases both with and without large density differences are considered. The Schmidt number is an arbitrary input to the problem and may be varied. The ordinary differential equation resulting from the similarity transformation is integrated numerically and some solutions are presented. The three boundary conditions are proper; the so-called arbitrary third boundary condition is treated as originally suggested by von Karman and extended to the case of large density difference. Illustrations of the effects of varying velocity ratio, density ratio, and Schmidt number are presented.

Film Cooling With Large Density Differences Between the Mainstream and the Secondary Fluid Measured by the Heat-Mass Transfer Analogy

1977 ◽  
Vol 99 (4) ◽  
pp. 620-627 ◽  
Author(s):  
D. R. Pedersen ◽  
E. R. G. Eckert ◽  
R. J. Goldstein
Keyword(s):  
Mass Transfer ◽  
Film Cooling ◽  
Heat Mass Transfer ◽  
Velocity Ratio ◽  
Density Ratio ◽  
Main Stream ◽  
Constant Property ◽  
Film Cooling Effectiveness ◽  
Large Density ◽  
Porous Strip

The effect of large density differences on film cooling effectiveness was investigated through the heat-mass transfer analogy. Experiments were performed in a wind tunnel where one of the plane walls was provided with a porous strip or a row of holes with three-diameter lateral spacing and inclined 35 deg into the main stream. Helium, CO2, or refrigerant F-12, was mixed with air either in small concentrations to approach a constant property situation or in larger concentration to produce a large density difference and injected through the porous strip or the row of holes into the mainstream. The resulting local gas concentrations were measured along the wall. The density ratio of secondary to mainstream fluid was varied between 0.75 and 4.17 for both injection systems. Local film effectiveness values were obtained at a number of positions downstream of injection and at different lateral positions. From these lateral average values could also be calculated. The following results were obtained. The heat mass-transfer analogy was verified for injection through the porous strip or through holes at conditions approaching a constant property situation. Neither the Schmidt number, nor the density ratio affects the film effectiveness for injection through a porous strip. The density ratio has a strong effect on the film effectiveness for injection through holes. The film effectiveness for injection through holes has a maximum value for a velocity ratio (injection to free stream) between 0.4 and 0.6. The center-line effectiveness increases somewhat with a decreasing ratio of boundary layer thickness to injection tube diameter.

scholarly journals A Novel Method for Solving the Bagley-Torvik Equation as Ordinary Differential Equation

2019 ◽  
Vol 14 (8) ◽  
Author(s):  
Yong Xu ◽  
Qixian Liu ◽  
Jike Liu ◽  
Yanmao Chen
Keyword(s):  
Differential Equation ◽  
Boundary Condition ◽  
Ordinary Differential Equation ◽  
Diffusion Equation ◽  
Fractional Derivative ◽  
Separation Of Variables ◽  
Computational Cost ◽  
Diffusion Equations ◽  
Novel Method ◽  
High Computational Efficiency

We present a novel method to solve the Bagley-Torvik equation by transforming it into ordinary differential equations (ODEs). This method is based on the equivalence between the Caputo-type fractional derivative (FD) of order 3/2 and the solution of a diffusion equation subjected to certain initial and boundary conditions. The key procedure is to approximate the infinite boundary condition by a finite one, so that the diffusion equation can be solved by separation of variables. By this procedure, the Bagley-Torvik and the diffusion equations together are transformed to be a set of ODEs, which can be integrated numerically by the Runge-Kutta scheme. The presented method is tested by various numerical cases including linear, nonlinear, nonsmooth, or multidimensional equations, respectively. Importantly, high computational efficiency is achieved as this method is at the expense of linearly increasing computational cost with the solution domain being enlarged.

scholarly journals Approximate analytical solutions of MHD viscous flow

2016 ◽  
Vol 13 (1) ◽  
pp. 79-87
Author(s):  
Vishwanath Basavaraj Awati
Keyword(s):  
Differential Equation ◽  
Boundary Layer ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Viscous Flow ◽  
Similarity Transformation ◽  
Nonlinear Differential Equations ◽  
Shrinking Sheet ◽  
Incompressible Viscous Flow ◽  
Approximate Analytical Solutions

The paper presents the semi-numerical solution for the magnetohydrodynamic (MHD) viscous flow due to a shrinking sheet caused by boundary layer of an incompressible viscous flow. The governing three partial differential equations of momentum equations are reduced into ordinary differential equation (ODE) by using a classical similarity transformation along with appropriate boundary conditions. Both nonlinearity and infinite interval demand novel mathematical tools for their analysis. We use fast converging Dirichlet series and Method of stretching of variables for the solution of these nonlinear differential equations. These methods have the advantages over pure numerical methods for obtaining the derived quantities accurately for various values of the parameters involved at a stretch and also they are valid in much larger parameter domain as compared with  HAM, HPM, ADM and the classical numerical schemes.

A Novel Numerical Scheme for Outlet Boundary Conditions in Large Density Ratio Lattice Boltzmann Model

2016 ◽  
Author(s):  
Long Li ◽  
Yongwen Liu
Keyword(s):  
Boundary Condition ◽  
Lattice Boltzmann ◽  
Numerical Scheme ◽  
Density Ratio ◽  
Two Phase ◽  
Droplet Shape ◽  
Large Density ◽  
Large Velocity ◽  
Large Density Ratio ◽  
Outlet Boundary Condition

In the past decades, the Lattice Boltzmann method has gained much success in variety fields especially in multiphase flow, porous media flow, and other complex flow, and become a promising method for computational fluid dynamic (CFD). The outlet boundary condition (OBC) and its numerical scheme are critical issues in CFD, which may influence the accuracy and stability of the calculation. The common OBCs i.e. Neumann boundary condition (NBC), extrapolation boundary condition (EBC), and convection boundary condition (CBC), which have been widely investigated in single-phase LB model, have rarely been investigated in multiphase LB model. The previous research on the OBCs for two-phase LB model only aims at small density ratio. While in most industrial applications, the density ratio often ranges from a hundred to a thousand, and a large density ratio would bring some problems such as parasitic current and bad stability in LB method. Lee and Fischer have proposed an improved LB model which is suitable for large density ratio two-phase flow. In order to assess the OBCs for large density ratio LB model, the OBCs are investigated. And it is found that the existing OBC numerical scheme cannot be directly applied to the large density ratio LB model. In present study, a novel numerical scheme for the OBCs is proposed assuming that the outlet velocity is gained by the outlet boundary condition instead of the momentum equation which is an improvement of previous scheme, and it can be used in large density ratio LB model. The performance of the proposed OBC scheme is examined for different density ratios. The results show that the proposed OBC scheme could converge in a stable manner. Comparing with the reference flow condition, the CBC scheme shows a better performance than the NBC scheme and the EBC scheme. The NBC scheme would lead a large droplet deformation, large velocity peaks at the outlet, and large errors for both small and large density ratio. And the EBC scheme keeps a good droplet shape, but it would lead large velocity peaks at the outlet and large error when large density ratio is considered. The CBC scheme always shows superior performance including a good droplet shape, smooth outlet velocity profile, and small errors no matter whether the density ratio is small or large. Hence the CBC scheme could be applied in large density ratio LB model for the outlet boundary condition, which has a good accuracy and stability in the calculation.

Problem for an ordinary differential equation with a general boundary condition

2021 ◽  
Vol 21 (2) ◽  
pp. 9-14
Author(s):  
L.Kh. Gadzova ◽  
Keyword(s):  
Differential Equation ◽  
Boundary Condition ◽  
Ordinary Differential Equation ◽  
Fractional Order ◽  
Unique Solvability ◽  
Explicit Representation ◽  
General Boundary ◽  
General Boundary Condition ◽  
General Representation

For an ordinary differential equation of fractional order with a general boundary condition, a general representation of the solution of the equation is found, a condition of unique solvability is found, and an explicit representation of the solution is constructed.

scholarly journals Traveling wave solutions for (3 + 1) dimensional equations arising in fluid mechanics

2014 ◽  
Vol 3 (4) ◽  
Author(s):  
Najeeb Alam Khan ◽  
Hassam Khan
Keyword(s):  
Differential Equation ◽  
Boundary Condition ◽  
Ordinary Differential Equation ◽  
Traveling Wave ◽  
Couple Stress ◽  
Traveling Wave Solutions ◽  
Wave Parameter ◽  
Fluid Models ◽  
Restrictive Assumption ◽  
Wave Solutions

AbstractIn this note, traveling wave solutions for (3 + 1) dimensional fluid models of incompressible flow are considered. The governing partial differential equations of two models are reduced to ordinary differential equation by employing wave parameter and exact solutions are obtained. It is shown that these fluid models allow 3 + 1 dimensional solutions amongst each other. The methodology used in this work is independent of symmetric consideration and other restrictive assumption. Finally, a set of example of boundary condition is discussed for the couple stress fluid. It is observed that velocity profile strongly depends upon couple stress parameter.

Modified Outlet Boundary Condition Schemes for Large Density Ratio Lattice Boltzmann Models

2017 ◽  
Vol 139 (5) ◽  
Author(s):  
Long Li ◽  
Xiaodong Jia ◽  
Yongwen Liu
Keyword(s):  
Boundary Condition ◽  
Distribution Function ◽  
Lattice Boltzmann ◽  
Density Ratio ◽  
Neumann Boundary ◽  
Momentum Balance ◽  
Two Phase ◽  
Large Density ◽  
Large Density Ratio ◽  
Density Ratios

Outlet boundary conditions (OBCs) and their numerical descriptions are critical to computational fluid dynamics (CFD) since they have significant influence on the numerical accuracy and stability. They present significant challenges to the two-phase lattice Boltzmann (LB) method, especially in the limit of large density ratio. In this study, three commonly used OBCs: convection boundary condition (CBC), Neumann boundary condition (NBC), and extrapolation boundary condition (EBC), are investigated and improved on basis of two LB models for large density ratios (single and double distribution function models). The existing numerical schemes for the OBCs are not directly applicable to the LB models because of the deviation of the momentum balance at the outlet boundary. The deviation becomes substantial at a large density ratio. Thus, in this work, modified OBC schemes are proposed to make the OBCs suitable for the two-phase LB models by adding an independent equation to obtain the outlet velocity. Numerical tests on droplet flowing in a channel are performed to evaluate the performance of the modified OBC schemes. Results indicate that the modified OBC schemes may be extended to tackle large density ratio situations. The modified NBC and EBC schemes are only suitable for the LB model with single distribution function. Three modified CBC schemes exhibit optimum performance for both single and double distribution function LB models which can be implemented for large density ratios.

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