scholarly journals A new multistage Parker-Sochacki method for solving the Troesch’s problem

2020 ◽  
Vol 9 (2) ◽  
pp. 592
Author(s):  
Saheed Akindeinde ◽  
Samuel Adesanya ◽  
Ramosheuw S. Lebelo ◽  
Kholeka C. Moloi
Keyword(s):  
Series Solution ◽  
Numerical Solutions ◽  
Polynomial System ◽  
Step Length ◽  
Integration Interval ◽  
Finite Term ◽  
Comparable Accuracy ◽  
Straightforward Computation ◽  
Very High ◽  
Troesch’S Problem

In this article, we introduce a new method to obtain an approximate analytical solution of the highly unstable Troesch’s problem. In the proposed method, without recourse to any hyperbolic tangent transformation or finite term approximation of the hyperbolic sine function, the problem is recast as a system of projectively polynomials which allows straightforward computation of the series solution of the problem. The radius of convergence  of the series solution to the problem is derived a-priorly in terms of the parameters of the polynomial system. Using a step length ; the problem domain is divided into subintervals, where corresponding subproblems are defined and solved with Parker-Sochacki method with very high accuracy. Highly accurate piecewise continuous approximate solution is thus obtained on the entire integration interval. The obtained solution, which is valid for every choice of the Troesch parameter , showed comparable accuracy to known numerical solutions in the literature. In particular, new results are presented for large values of  in the range [20;500].

scholarly journals An Examination of a Second Order Numerical Method for Solving Initial Value Problems

2020 ◽  
pp. 120-127
Author(s):  
S. E. Fadugba ◽  
S. N. Ogunyebi ◽  
B. O. Falodun
Keyword(s):  
Numerical Method ◽  
Initial Value Problems ◽  
Numerical Solutions ◽  
Nodal Point ◽  
Step Length ◽  
Second Order ◽  
Initial Value ◽  
Good Tool ◽  
Integration Interval ◽  
Relative Errors

This paper presents an examination of a Second Order Convergence Numerical Method (SOCNM) for solving Initial Value Problems (IVPs) in Ordinary Differential Equations (ODEs). The SOCNM has been derived via the interpolating function comprises of polynomial and exponential forms. The analysis and the properties of SOCNM were discussed. Three numerical examples have been solved successfully to examine the performance of SOCNM in terms of accuracy and stability. The comparative study of SOCNM, Improved Modified Euler Method (IMEM), Fadugba and Olaosebikan Scheme (FOS) and the Exact Solution (ES) is presented. By varying the step length, the absolute relative errors at the final nodal point of the associated integration interval are computed. Furthermore, the analysis of the properties of SOCNM shows that the method is consistent, stable, convergent and has second order accuracy. Moreover, the numerical results show that SOCNM is more accurate than IMEM and FOS and also compared favourably with the ES. By varying the step length, there are two-order decrease in the values of the final absolute relative errors generated via SOCNM. Hence, SOCNM is found to be accurate, stable and a good tool for the numerical solutions of IVPs of different characteristics in ODEs.

Flow Field on Helodeck of a Frigate: A Review

2019 ◽  
Vol 161 (A4) ◽  
Keyword(s):  
Flow Field ◽  
Numerical Solutions ◽  
Offshore Structures ◽  
High Speeds ◽  
High Turbulence ◽  
The Subject ◽  
Offshore Industry ◽  
The Military ◽  
Very High ◽  
Made In

The various functions desired from a frontline warship such as a frigate, corvette or a destroyer, coupled with the requirement of very high speeds and economic viability restricting the size, necessitates a very dense arrangement of weapons and sensors on the top deck and superstructure. Accordingly, Navies across the world have faced several problems with respect to functions for which a good aerodynamic design for these structures is essential. Major issues include smoke nuisance created due to impinging of the ship's exhaust gases on to the top deck leading to possible suction by engine intakes and high turbulence in the ship's air-wake leading to ship aircraft interface concerns. The flow field on the helodeck is extremely complex due to its geometry and interaction with the wake of the ship’s superstructure. A knowledge of this complexity is essential for ensuring safe helo operations on the helodeck. The problem of ship helicopter interaction has hogged the lime light in recent times, due to rising demand for design of warships for increased stealth, especially in the past two decades. Consequently, several researchers in countries with advanced Navies have invested considerable resources towards evolving both experimental and numerical solutions for the problem. However, given the military nature of the operations, open literature on the subject containing details of such research, which can be used as reference material for present work, are limited. Considering the complexities involved in the problem, an attempt has been made in this paper to holistically review the widely scattered and limited literature in this field. A good amount of literature on marine helo applications emerge from the offshore industry. Keeping in mind that the fields of warship design and offshore structures are dissimilar and have their peculiar problems, informed conclusions have been made in drawing lessons from available literature.

scholarly journals Fourier series solution for an anisotropic and layered configuration of the dispersive Henry Problem

2018 ◽  
Vol 54 ◽  
pp. 00014
Author(s):  
B. Koohbor ◽  
M. Fahs ◽  
B. Belfort ◽  
B. Ataie-Ashtiani ◽  
C. T. Simmons
Keyword(s):  
Fourier Series ◽  
Diffusion Coefficients ◽  
Series Solution ◽  
Numerical Solutions ◽  
Numerical Models ◽  
Salt Transport ◽  
Spectral Space ◽  
Finite Element Code ◽  
Henry Problem ◽  
Lower Diffusion

Henry Problem (HP) still plays an important role in benchmarking numerical models of seawater intrusion (SWI) as well as being applied to practical and managerial purposes. The popularity of this problem is due to having a closed-form semi-analytical (SA) solution. The early SA solutions obtained for HP were limited to extensive assumptions that restrict its application in practical works. Several further studies expended the generality of the solution by assuming lower diffusion coefficients or including velocity-dependent dispersion in the results. However, all these studies are limited to homogeneous and isotropic domains. The present work made an attempt to improve the reality of the SA solution obtained for dispersive HP by considering anisotropic and stratified heterogeneous coastal aquifers. The solution is obtained by defining Fourier series for both stream function and salt concentration, applying a Galerkin treatment using the Fourier modes as trial functions and solving the flow and the salt transport equations simultaneously in the spectral space. In order to include stratified heterogeneity, a special depth-hydraulic conductivity model is applied that can be solved analytically without significant mathematical complexity. Several examples are proposed and studied. The results show excellent agreement between the SA and numerical solutions obtained with an in-house advanced finite element code.

Fan Similarity Model for the Fan–Intake Interaction Problem

2017 ◽  
Vol 140 (5) ◽  
Author(s):  
Mauro Carnevale ◽  
Feng Wang ◽  
Anthony B. Parry ◽  
Jeffrey S. Green ◽  
Luca di Mare
Keyword(s):  
Numerical Solutions ◽  
Mutual Influence ◽  
Three Dimensional ◽  
Test Case ◽  
Present Contribution ◽  
Shock System ◽  
Aero Engines ◽  
Linearized Model ◽  
Bypass Ratio ◽  
Very High

Very high bypass ratio turbofans with large fan tip diameter are an effective way of improving the propulsive efficiency of civil aero-engines. Such engines, however, require larger and heavier nacelles, which partially offset any gains in specific fuel consumptions. This drawback can be mitigated by adopting thinner walls for the nacelle and by shortening the intake section. This binds the success of very high bypass ratio technologies to the problem of designing an intake with thin lips and short diffuser section, which is well matched to a low speed fan. Consequently, the prediction of the mutual influence between the fan and the intake flow represents a crucial step in the design process. Considerable effort has been devoted in recent years to the study of models for the effects of the fan on the lip stall characteristics and the operability of the whole installation. The study of such models is motivated by the wish to avoid the costs incurred by full, three-dimensional (3D) computational fluid dynamics (CFD) computations. The present contribution documents a fan model for fan–intake computations based on the solution of the double linearization problem for unsteady, transonic flow past a cascade of aerofoils with finite mean load. The computation of the flow in the intake is reduced to a steady problem, whereas the computation of the flow in the fan is reduced to one steady problem and a set of solutions of the linearized model in the frequency domain. The nature of the approximations introduced in the fan representation is such that numerical solutions can be computed inexpensively, while the main feature of the flow in the fan passage, namely the shock system and an approximation of the unsteady flow encountered by the fan are retained. The model is applied to a well-documented test case and compares favorably with much more expensive 3D, time-domain computations.

On the continuum theory for the large Reynolds number spherical expansion into a near vacuum

1975 ◽  
Vol 68 (4) ◽  
pp. 625-638 ◽  
Author(s):  
N. C. Freeman ◽  
R. S. Johnson ◽  
S. Kumar ◽  
W. B. Bush
Keyword(s):  
Reynolds Number ◽  
Numerical Solutions ◽  
Continuum Theory ◽  
Large Reynolds Number ◽  
Heat Conducting ◽  
Sonic Point ◽  
Spherical Expansion ◽  
High Reynolds ◽  
Background Gas ◽  
Very High

The steady, spherically symmetric flow of a compressible gas is considered. The gas is both viscous and heat-conducting. In the limit of very high Reynolds number (= α−1, α → 0) and correspondingly low pressure at infinity, the structure of the whole flow field is discussed. The five regions that arise by virtue of the limit α → 0 are briefly considered. Special care is given to the matching across the overlap domains and the first region (close to, but outside, the sonic point) and the fifth (where the pressure adjusts to its ambient value) are carefully examined. It is argued that the application of appropriate matching principles, together with judicious use of numerical solutions, allows an arbitrary pressure and temperature to be assigned to the background gas.

Theoretical and Numerical Solutions of Maglev Train Induced Vibration

2013 ◽  
Vol 459 ◽  
pp. 271-277
Author(s):  
S.H. Ju ◽  
C.C. Leong ◽  
Y.S. Ho
Keyword(s):  
Control System ◽  
Numerical Scheme ◽  
Large Time ◽  
Numerical Solutions ◽  
Step Length ◽  
Dynamic Interaction ◽  
Maglev Train ◽  
Time Step ◽  
Large Time Step ◽  
Interaction Problem

This paper proposed an efficient method based on theoretical equations to solve the dynamic interaction problem between the Timoshenko beam and maglev vehicles. A systematic PI numerical scheme is developed for the control system of the maglev train. The major advantage is that only one simple equation required in the control calculation, although the original control system is fairly complicated. Numerical simulations indicate that a large time step length can be used in the proposed method to obtain stable and accurate results.

A Series Solution for the Input Admittance of a Circular Loop Antenna over a Lossy Earth

2019 ◽  
Vol 12 (2) ◽  
pp. 113-117
Author(s):  
Osman M. Alsemaid ◽  
Sameir M. Ali Hamed ◽  
Saif A. Mahmood
Keyword(s):  
Current Distribution ◽  
Series Solution ◽  
Numerical Solutions ◽  
Loop Antenna ◽  
Fourier Series Expansion ◽  
Image Method ◽  
Input Admittance ◽  
Circular Loop ◽  
Numerical Integrations ◽  
Good Agreement

Background: This study deals with the derivation of expressions for the current distribution and the input admittance of a circular loop antenna (CLA) over a lossy Earth in a series form. Methods: The analysis is based on the image method, Fourier series expansion and the direct integration of the vector potential. Result: The results for the current distribution of the study are in very good agreement with those corresponding results available in the literature, which checks the correctness of the formulations of the study. Conclusion: The provided solution in this paper is simple, straightforward and can be applied without performing any numerical integrations, which makes it preferred compared to the numerical solutions that are available in the literature.

Approximate Analytical Solution of a Power-Law Pseudoplastic Fluid in a Boundary Layer over a Porous Plate: A New Application of Multi-Stage Parker-Sochacki Method

2021 ◽  
Vol 55 ◽  
pp. 1-14
Author(s):  
Saheed Ojo Akindeinde ◽  
Olusegun Adebayo Adewumi ◽  
Ramoshweu Solomon Lebelo
Keyword(s):  
Boundary Layer ◽  
Nonlinear Boundary ◽  
Polynomial System ◽  
Porous Plate ◽  
Optimal Choice ◽  
Infinite Domain ◽  
Pseudoplastic Fluid ◽  
Boundary Layer Equations ◽  
Multi Stage ◽  
Very High

In this paper, based on Parker-Sochacki method for solving a system of differential equations,a multistage technique is developed for solving the nonlinear boundary layer equations of powerlawfluid on infinite domain. The problem domain is split into subintervals over which the boundaryvalue problem is replaced with a sequence of subproblems. In a shooting-like approach, the boundarycondition at infinity is converted to an equivalent initial condition. By recasting the problem as apolynomial system of first-order autonomous equations, the sub-problems are solved with Parker-Sochacki method with very high accuracy. The interval of convergence of the solution is deriveda-priorly in terms of the parameters of the polynomial system, which guides optimal choice of thediscretization parameter. The technique yielded a convergent piecewise continuous solution over theproblem domain. The results obtained, demonstrated graphically and in tables, compared well withexisting ones in the literature.

Effects of Protocol Step Length on Biomechanical Measures in Swimming

2015 ◽  
Vol 10 (2) ◽  
pp. 211-218 ◽  
Author(s):  
Tiago M. Barbosa ◽  
Kelly de Jesus ◽  
J. Arturo Abraldes ◽  
João Ribeiro ◽  
Pedro Figueiredo ◽  
...  
Keyword(s):  
Sagittal Plane ◽  
Step Length ◽  
Interlimb Coordination ◽  
System Parameters ◽  
Evaluation Procedures ◽  
Video Images ◽  
Coordination Behavior ◽  
Incremental Protocol ◽  
Swimming Kinematics ◽  
Very High

Background:The assessment of energetic and mechanical parameters in swimming often requires the use of an intermittent incremental protocol, whose step lengths are corner stones for the efficiency of the evaluation procedures.Purpose:To analyze changes in swimming kinematics and interlimb coordination behavior in 3 variants, with different step lengths, of an intermittent incremental protocol.Methods:Twenty-two male swimmers performed n × di variants of an intermittent and incremental protocol (n ≤ 7; d1 = 200 m, d2 = 300 m, and d3 = 400 m). Swimmers were videotaped in the sagittal plane for 2-dimensional kinematical analysis using a dualmedia setup. Video images were digitized with a motion-capture system. Parameters that were assessed included the stroke kinematics, the segmental and anatomical landmark kinematics, and interlimb coordination. Movement efficiency was also estimated.Results:There were no significant variations in any of the selected variables according to the step lengths. A high to very high relationship was observed between step lengths. The bias was much reduced and the 95%CI fairly tight.Conclusions:Since there were no meaningful differences between the 3 protocol variants, the 1 with shortest step length (ie, 200 m) should be adopted for logistical reasons.

Numerical Study of the Steady Axisymmetric Flow Through a Disk-Type Prosthetic Heart Valve in a Constant Diameter Chamber

1977 ◽  
Vol 99 (2) ◽  
pp. 91-97 ◽  
Author(s):  
F. N. Underwood ◽  
T. J. Mueller
Keyword(s):  
Shear Stress ◽  
Heart Valve ◽  
Numerical Solutions ◽  
Numerical Study ◽  
Reverse Flow ◽  
Axisymmetric Flow ◽  
Prosthetic Heart Valve ◽  
High Shear ◽  
Flow Through ◽  
Very High

Numerical solutions for the steady axisymmetric flow through a disk-type prosthetic heart valve were obtained for Reynolds numbers from 20 to 1300. Stream function, vorticity, and shear and normal stress plots are presented. Comparison of the length of the separated flow region downstream of the disk with experimental data shows good agreement through Reynolds number 500. The maximum value of the shear stress occurred on the upstream corner of the disk. These detailed results clearly identify regions of very high shear and normal stresses (erythrocyte deformation or damage), regions of very low or very high shear stress at walls (atheromatous lesions), and the extent of separated or reverse flow regions (thrombosis).

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