Supersonic flow past cones of general cross-section

1962 ◽  
Vol 13 (3) ◽  
pp. 383-399 ◽  
Author(s):  
P. M. Stocker ◽  
F. E. Mauger
Keyword(s):  
Supersonic Flow ◽  
Differential Equations ◽  
Stream Function ◽  
Cross Section ◽  
Bluff Body ◽  
Numerical Solutions ◽  
Independent Variables ◽  
Flow Past ◽  
Vortical Layer

The differential equations representing the supersonic flow of a gas past a cone of any cross-section are integrated numerically, using a method similar to those used for bluff-body problems. A stream function is used as one of the independent variables and this is particularly suitable for determining the singular ‘vortical layer’. The method is here applied to the cases of elliptic cones at zero yaw and circular cones at incidence. The results are compared with experiment and with other numerical solutions.

Numerical Solutions of a Viscous Uniform Approach Flow Past Square and Diamond Cylinders

2002 ◽  
Author(s):  
Charles Dalton ◽  
Wu Zheng
Keyword(s):  
Stream Function ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Elliptic Partial Differential Equation ◽  
Navier Stokes ◽  
Central Difference ◽  
Uniform Approach ◽  
Flow Past ◽  
Line Force ◽  
Vorticity Transport

Numerical results are presented for a uniform approach flow past square and diamond cylinders, with and without rounded corners, at Reynolds numbers of 250 and 1000. This unsteady viscous flow problem is formulated by the 2-D Navier-Stokes equations in vorticity and stream-function form on body-fitted coordinates and solved by a finite-difference method. Second-order Adams-Bashforth and central-difference schemes are used to discretize the vorticity transport equation while a third-order upwinding scheme is incorporated to represent the nonlinear convective terms. A grid generation technique is applied to provide an efficient mesh system for the flow. The elliptic partial differential equation for stream-function and vorticity in the transformed plane is solved by the multigrid iteration method. The Strouhal number and the average in-line force coefficients agree very well with the experimental and previous numerical values. The vortex structures and the evolution of vortex shedding are illustrated by vorticity contours. Rounding the corners of the square and diamond cylinders produced a noticeable decrease on the calculated drag and lift coefficients.

scholarly journals Dual Stratified Nanofluid Flow Past a Permeable Shrinking/Stretching Sheet Using a Non-Fourier Energy Model

2019 ◽  
Vol 9 (10) ◽  
pp. 2124 ◽  
Author(s):  
Najiyah Safwa Khashi’ie ◽  
Norihan Md Arifin ◽  
Ezad Hafidz Hafidzuddin ◽  
Nadihah Wahi
Keyword(s):  
Mass Transfer ◽  
Differential Equations ◽  
Transfer Rate ◽  
Energy Equation ◽  
Numerical Solutions ◽  
Thermal Relaxation ◽  
Physical Parameters ◽  
Nanofluid Flow ◽  
Local Skin ◽  
Flow Past

The present study emphasizes the combined effects of double stratification and buoyancy forces on nanofluid flow past a shrinking/stretching surface. A permeable sheet is used to give way for possible wall fluid suction while the magnetic field is imposed normal to the sheet. The governing boundary layer with non-Fourier energy equations (partial differential equations (PDEs)) are converted into a set of nonlinear ordinary differential equations (ODEs) using similarity transformations. The approximate relative error between present results (using the boundary value problem with fourth order accuracy (bvp4c) function) and previous studies in few limiting cases is sufficiently small (0% to 0.3694%). Numerical solutions are graphically displayed for several physical parameters namely suction, magnetic, thermal relaxation, thermal and solutal stratifications on the velocity, temperature and nanoparticles volume fraction profiles. The non-Fourier energy equation gives a different estimation of heat and mass transfer rates as compared to the classical energy equation. The heat transfer rate approximately elevates 5.83% to 12.13% when the thermal relaxation parameter is added for both shrinking and stretching cases. Adversely, the mass transfer rate declines within the range of 1.02% to 2.42%. It is also evident in the present work that the augmentation of suitable wall mass suction will generate dual solutions. The existence of two solutions (first and second) are noticed in all the profiles as well as the local skin friction, Nusselt number and Sherwood number graphs within the considerable range of parameters. The implementation of stability analysis asserts that the first solution is the real solution.

Analysis of Turbulent Flow Past a Class of Semi-Infinite Bodies

1986 ◽  
Vol 108 (2) ◽  
pp. 157-165
Author(s):  
A. M. Abdelhalim ◽  
U. Ghia ◽  
K. N. Ghia
Keyword(s):  
Flat Plate ◽  
Stream Function ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Analytical Data ◽  
Flow Region ◽  
Two Dimensional ◽  
Infinite Body ◽  
Similarity Type ◽  
Flow Past

This study was undertaken with the primary purpose of developing an analysis for flow past a class of two-dimensional and axisymmetric semi-infinite bodies. The time-averaged Navier-Stokes equations for these flows are derived in surface-oriented conformal coordinates (ξ, η) in terms of similarity-type vorticity and stream-function variables. Turbulence closure is achieved by means of a two-equation turbulence model utilizing the kinetic energy k and its dissipation rate ε which enable determination of the isotropic eddy viscosity. The coupled vorticity and stream-function equations are solved simultaneously using an incremental formulation of the factored alternating-direction implicit scheme. The turbulence equations for k and ε are solved by the standard ADI method. Numerical solutions are obtained for the thin flat plate and compared with available experimental and analytical data. Also, results are obtained for flow over a parabola and compared with the flat-plate results in order to assess the effects of longitudinal curvature on the flow results. Finally, solutions are obtained for flow past a two-dimensional semi-infinite body with a shoulder, at Red = 24,000. The computed results have the same general trend as the experimental data; possible causes for the differences within the separated-flow region are cited.

A Numerical Study of Forced Convection Casson Nanofluid Flow Past a Wedge With Melting Process

2020 ◽  
Author(s):  
Mehari Fentahun Endalew ◽  
Subharthi Sarkar
Keyword(s):  
Differential Equations ◽  
Skin Friction ◽  
Numerical Solutions ◽  
Numerical Study ◽  
Wedge Angle ◽  
Nanofluid Flow ◽  
Local Skin ◽  
Casson Nanofluid ◽  
Flow Past ◽  
Highly Nonlinear

Abstract A numerical investigation is carried out to analyze steady two dimensional Casson nanofluid flow past a wedge with melting. The partial differential equations that govern the nanofluid flow are transformed into highly nonlinear coupled ordinary differential equations by employing similarity transformation. Thereafter, numerical solutions of these governing equations are obtained by MATLAB routine bvp4c. A special case of the present study is compared with an existing solution in literature and is found to be in good agreement. The effects of pertinent physical entities on the nanofluid velocity, nanofluid temperature, and nanoparticle concentration are represented graphically, while skin friction, Nusselt number, and Sherwood number are recorded in tabular form. We observed that, with an increase of wedge angle parameter, nanofluid velocity and local skin friction increase. However, when the melting parameter increases, nanofluid temperature and heat transfer rate decrease. This study would be useful in unfurling novel applications of Casson nanofluids in cooling devices and heat sinks.

Self-induced interactions

1982 ◽  
Vol 379 (1776) ◽  
pp. 217-230 ◽  
Keyword(s):  
Boundary Layer ◽  
Partial Differential Equations ◽  
Supersonic Flow ◽  
Differential Equations ◽  
Pressure Gradient ◽  
Skin Friction ◽  
Laminar Boundary Layer ◽  
Numerical Solutions ◽  
Adverse Pressure Gradient ◽  
Series Expansions

A laminar boundary layer in supersonic flow can evolve spontaneously from an undisturbed form to a separated state, where an adverse pressure gradient thickens the boundary layer, thus displacing the external streamlines, which leads to the original pressure gradient. A linearized study by Lighthill was later generalized by Stewartson in a triple-deck analysis, in which the equations for the main deck are ∂ u /∂ x + ∂ v /∂ y ═ 0, u ∂ u /∂ x + v ∂ u /∂ y ═ – d p /d x + ∂ 2 u /∂ y 2 , subject to u ═ v ═ 0 at y ═ 0, u → y as x → – ∞, and u → y – ∫ x -∞ p(t) d t as y → ∞. The problem is here studied by means of the series expansions u ═ y – ∑ ∞ n ═ 1 a n e nkx f' n (y) , v ═ ∑ ∞ n ═ 1 nka n e nkx f n (y) , p ═ ∑ ∞ n ═ 1 a n e nkx . This gives a sequence of equations for the f n (y) , of which the first 15 have been solved. Appropriate series for the pressure p and the skin-friction т have been derived and analysed, and previous numerical solutions of the partial differential equations by Williams have been well confirmed, in some instances to greater accuracy. Among the conclusions reached are the following, (i) The value of p at separation is calculated to be 1.02594744. (ii) As x → ∞, p tends to a constant value p 0 ═ 1.7903, compared with the value 1.800 given by Williams. (iii) In the separated region, the most negative value taken by т is –0.1494081.

Normal form method in search for periodic solutions of ordinary differential equations. Case of the fourth order

2018 ◽  
pp. 24-34
Author(s):  
V. F. Edneral ◽  
O. D. Timofeevskaya
Keyword(s):  
Normal Form ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Periodic Solutions ◽  
Fourth Order ◽  
Numerical Solutions ◽  
Numerical Calculations ◽  
Computational Time ◽  
Resonant Normal Form ◽  
Normal Form Method

Introduction:The method of resonant normal form is based on reducing a system of nonlinear ordinary differential equations to a simpler form, easier to explore. Moreover, for a number of autonomous nonlinear problems, it is possible to obtain explicit formulas which approximate numerical calculations of families of their periodic solutions. Replacing numerical calculations with their precalculated formulas leads to significant savings in computational time. Similar calculations were made earlier, but their accuracy was insufficient, and their complexity was very high.Purpose:Application of the resonant normal form method and a software package developed for these purposes to fourth-order systems in order to increase the calculation speed.Results:It has been shown that with the help of a single algorithm it is possible to study equations of high orders (4th and higher). Comparing the tabulation of the obtained formulas with the numerical solutions of the corresponding equations shows good quantitative agreement. Moreover, the speed of calculation by prepared approximating formulas is orders of magnitude greater than the numerical calculation speed. The obtained approximations can also be successfully applied to unstable solutions. For example, in the Henon — Heyles system, periodic solutions are surrounded by chaotic solutions and, when numerically integrated, the algorithms are often unstable on them.Practical relevance:The developed approach can be used in the simulation of physical and biological systems.

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