Calculation of the steady flow past a sphere at low and moderate Reynolds numbers

1971 ◽  
Vol 48 (4) ◽  
pp. 771-789 ◽  
Author(s):  
S. C. R. Dennis ◽  
J. D. A. Walker
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Equations Of Motion ◽  
Reynolds Numbers ◽  
Legendre Functions ◽  
Sphere Diameter ◽  
Axially Symmetric ◽  
Legendre Series ◽  
Flow Past ◽  
Flow Past A Sphere

The steady axially symmetric incompressible flow past a sphere is investigated for Reynolds numbers, based on the sphere diameter, in the range 0·1 to 40. The formulation is a semi-analytical one whereby the flow variables are expanded as series of Legendre functions, hence reducing the equations of motion to ordinary differential equations. The ordinary differential equations are solved by numerical methods. Only a finite number of these equations can be solved, corresponding to an approximation obtained by truncating the Legendre series at some stage. More terms of the series are required asRincreases and the present calculations were terminated atR= 40. The calculated drag coefficient is compared with the results of previous investigations and with experimental data. The Reynolds number at which separation first occurs is estimated as 20·5.

Calculation of incompressible flow past a circular cylinder at moderate Reynolds numbers

1969 ◽  
Vol 37 (1) ◽  
pp. 95-114 ◽  
Author(s):  
Robert Leigh Underwood
Keyword(s):  
Reynolds Number ◽  
Circular Cylinder ◽  
Differential Equations ◽  
Incompressible Flow ◽  
Equations Of Motion ◽  
Pressure Coefficient ◽  
Reynolds Numbers ◽  
Number Range ◽  
Flow Parameters ◽  
Flow Past

The steady, two-dimensional, incompressible flow past a circular cylinder is calculated for Reynolds numbers up to ten. An accurate description of the flow field is found by employing the semi-analytical method of series truncation to reduce the governing partial differential equations of motion to a system of ordinary differential equations which can be integrated numerically. Results are given for Reynolds numbers between 0.4 and 10.0 (based on diameter). The Reynolds number at which separation first occurs behind the cylinder is found to be 5.75. Over the entire Reynolds number range investigated, characteristic flow parameters such as the drag coefficient, pressure coefficient, standing eddy length, and streamline pattern compare favourably with available experimental data and numerical solution results.

Multibody Dynamics on Differentiable Manifolds

2021 ◽  
Vol 16 (4) ◽  
Author(s):  
Edward J. Haug
Keyword(s):  
Euclidean Space ◽  
Differential Equations ◽  
Vector Space ◽  
Ordinary Differential Equations ◽  
Multibody Dynamics ◽  
Equations Of Motion ◽  
Differentiable Manifold ◽  
Kinematics And Dynamics ◽  
Variational Equations ◽  
Differentiable Manifolds

Abstract Topological and vector space attributes of Euclidean space are consolidated from the mathematical literature and employed to create a differentiable manifold structure for holonomic multibody kinematics and dynamics. Using vector space properties of Euclidean space and multivariable calculus, a local kinematic parameterization is presented that establishes the regular configuration space of a multibody system as a differentiable manifold. Topological properties of Euclidean space show that this manifold is naturally partitioned into disjoint, maximal, path connected, singularity free domains of kinematic and dynamic functionality. Using the manifold parameterization, the d'Alembert variational equations of multibody dynamics yield well-posed ordinary differential equations of motion on these domains, without introducing Lagrange multipliers. Solutions of the differential equations satisfy configuration, velocity, and acceleration constraint equations and the variational equations of dynamics, i.e., multibody kinematics and dynamics are embedded in these ordinary differential equations. Two examples, one planar and one spatial, are treated using the formulation presented. Solutions obtained are shown to satisfy all three forms of kinematic constraint to within specified error tolerances, using fourth-order Runge–Kutta numerical integration methods.

scholarly journals A Simple Approximation for Axially Symmetric Diffraction of Plane Shocks by Cones

1984 ◽  
Vol 39 (4) ◽  
pp. 320-324 ◽  
Author(s):  
Shanbing Yu ◽  
Hans Grönig
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Simple Approximation ◽  
Axially Symmetric ◽  
Circular Arcs ◽  
Shock Dynamics

By Whitham’s shock dynamics, axially symmetric diffraction problems of plane shocks by cones are reduced to the integration of a system of ordinary differential equations. This paper gives a simpler approach. Approximating the Mach stems by circular arcs, the problems are reduced to the solution of algebraical equations, and the calculating is much simplified. In the cases of small or large cone angles the estimation is even easier.

scholarly journals MHD Slip Flow Past an Extending Surface with Third Type Boundary Condition and Thermal Radiation Effects

2019 ◽  
Vol 24 (3) ◽  
pp. 577-590
Author(s):  
A.D.M. Gururaj ◽  
S. Dhanasekar ◽  
V. Parthiban
Keyword(s):  
Boundary Condition ◽  
Differential Equations ◽  
Thermal Radiation ◽  
Ordinary Differential Equations ◽  
Radiation Effects ◽  
Slip Flow ◽  
Nonlinear Ordinary Differential Equations ◽  
Convective Boundary ◽  
Flow Past ◽  
Interaction Factor

Abstract MHD slip flow past an extending surface with third type (convective) boundary condition and thermal radiation is analysed. The governing momentum and energy equations are converted into set of nonlinear ordinary differential equations using appropriate similarity transformations. The Fourth-Order Runge-Kutta shooting method is applied for obtaining the numerical solution of the resulting nonlinear ordinary differential equations. The numerical results for velocity and temperature distribution are found for different values of the vital parameters, namely: the magnetic interaction factor, slip factor, convective factor, Prandtl number and radiation factor and are presented graphically, and discussed.

Expansions at small Reynolds numbers for the flow past a sphere and a circular cylinder

1957 ◽  
Vol 2 (3) ◽  
pp. 237-262 ◽  
Author(s):  
Ian Proudman ◽  
J. R. A. Pearson
Keyword(s):  
Circular Cylinder ◽  
Boundary Conditions ◽  
Reynolds Numbers ◽  
Polar Coordinates ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Flow Past ◽  
Small Reynolds Numbers ◽  
Cylindrical Polar Coordinates ◽  
Flow Past A Sphere

This paper is concerned with the problem of obtaining higher approximations to the flow past a sphere and a circular cylinder than those represented by the well-known solutions of Stokes and Oseen. Since the perturbation theory arising from the consideration of small non-zero Reynolds numbers is a singular one, the problem is largely that of devising suitable techniques for taking this singularity into account when expanding the solution for small Reynolds numbers.The technique adopted is as follows. Separate, locally valid (in general), expansions of the stream function are developed for the regions close to, and far from, the obstacle. Reasons are presented for believing that these ‘Stokes’ and ‘Oseen’ expansions are, respectively, of the forms $\Sigma \;f_n(R) \psi_n(r, \theta)$ and $\Sigma \; F_n(R) \Psi_n(R_r, \theta)$ where (r, θ) are spherical or cylindrical polar coordinates made dimensionless with the radius of the obstacle, R is the Reynolds number, and $f_{(n+1)}|f_n$ and $F_{n+1}|F_n$ vanish with R. Substitution of these expansions in the Navier-Stokes equation then yields a set of differential equations for the coefficients ψn and Ψn, but only one set of physical boundary conditions is applicable to each expansion (the no-slip conditions for the Stokes expansion, and the uniform-stream condition for the Oseen expansion) so that unique solutions cannot be derived immediately. However, the fact that the two expansions are (in principle) both derived from the same exact solution leads to a ‘matching’ procedure which yields further boundary conditions for each expansion. It is thus possible to determine alternately successive terms in each expansion.The leading terms of the expansions are shown to be closely related to the original solutions of Stokes and Oseen, and detailed results for some further terms are obtained.

Water Hammer Peak Pressures and Decay Rates of Transients in Smooth Lines With Turbulent Flow

2018 ◽  
Vol 140 (6) ◽  
Author(s):  
David A. Hullender
Keyword(s):  
Transfer Function ◽  
Differential Equations ◽  
Turbulent Flow ◽  
Ordinary Differential Equations ◽  
Damping Ratio ◽  
Reynolds Numbers ◽  
Water Hammer ◽  
Decay Rates ◽  
Pressure Transients ◽  
Peak Pressures

Transient pressure peak values and decay rates associated with water hammer surges in fluid lines are investigated using an analytical method that has been formulated, in a previous publication, to simulate pressure transients in turbulent flow. The method agrees quite well with method of characteristics (MOC) simulations of unsteady friction models and has been verified with experimental data available for Reynolds numbers out to 15,800. The method is based on the formulation of ordinary differential equations from the frequency response of a pressure transfer function using an inverse frequency algorithm. The model is formulated by dividing the line into n-sections to distribute the turbulence resistance along the line at higher Reynolds numbers. In this paper, it will be demonstrated that convergence of the analytical solution is achieved with as few as 5–10 line sections for Reynolds numbers up to 200,000. The method not only provides for the use of conventional time domain solution algorithms for ordinary differential equations but also provides empirical equations for estimating peak surge pressures and transient decay rates as defined by eigenvalues. For typical sets of line and fluid properties, the trend of the damping ratio of the first or dominate mode of the pressure transients transfer function is found to be an approximate linear function of a dimensionless parameter that is a function of the Reynolds number. In addition, a reasonably accurate dimensionless trend formula for estimates of the normalized peak pressures is formulated and presented.

Dynamics of a Basketball Rolling Around the Rim

2005 ◽  
Vol 128 (2) ◽  
pp. 359-364
Author(s):  
C. Q. Liu ◽  
Fang Li ◽  
R. L. Huston
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
Dynamical Equations ◽  
First Order ◽  
Coaching Strategies ◽  
Ball Motion

Governing dynamical equations of motion for a basketball rolling on the rim of a basket are developed and presented. These equations form a system of five first-order, ordinary differential equations. Given suitable initial conditions, these equations are readily integrated numerically. The results of these integrations predict the success (into the basket) or failure (off the outside of the rim) of the basketball shot. A series of examples are presented. The examples show that minor changes in the initial conditions can produce major changes in the subsequent ball motion. Shooting and coaching strategies are recommended.

scholarly journals Nonlinear Response of Cantilever Beams to Combination and Subcombination Resonances

1998 ◽  
Vol 5 (5-6) ◽  
pp. 277-288 ◽  
Author(s):  
Ali H. Nayfeh ◽  
Haider N. Arafat
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Virtual Work ◽  
Equations Of Motion ◽  
Nonlinear Ordinary Differential Equations ◽  
Response Curves ◽  
First Order ◽  
Aspect Ratios

The nonlinear planar response of cantilever metallic beams to combination parametric and external subcombination resonances is investigated, taking into account the effects of cubic geometric and inertia nonlinearities. The beams considered here are assumed to have large length-to-width aspect ratios and thin rectangular cross sections. Hence, the effects of shear deformations and rotatory inertia are neglected. For the case of combination parametric resonance, a two-mode Galerkin discretization along with Hamilton’s extended principle is used to obtain two second-order nonlinear ordinary-differential equations of motion and associated boundary conditions. Then, the method of multiple scales is applied to obtain a set of four first-order nonlinear ordinary-differential equations governing the modulation of the amplitudes and phases of the two excited modes. For the case of subcombination resonance, the method of multiple scales is applied directly to the Lagrangian and virtual-work term. Then using Hamilton’s extended principle, we obtain a set of four first-order nonlinear ordinary-differential equations governing the amplitudes and phases of the two excited modes. In both cases, the modulation equations are used to generate frequency- and force-response curves. We found that the trivial solution exhibits a jump as it undergoes a subcritical pitchfork bifurcation. Similarly, the nontrivial solutions also exhibit jumps as they undergo saddle-node bifurcations.

Control of Vibration Amplitude, Frequency and Damping of an Electrostatically Actuated Microbeam Using Capacitive, Inductive and Resistive Elements

2010 ◽  
Author(s):  
Abdolreza Pasharavesh ◽  
Y. Alizadeh Vaghasloo ◽  
A. Fallah ◽  
M. T. Ahmadian
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Vibration Amplitude ◽  
Differential Quadrature Method ◽  
Equations Of Motion ◽  
Nonlinear Partial Differential Equations ◽  
Electrical Current ◽  
Oscillatory System ◽  
Electrostatically Actuated ◽  
In Series

In this study vibration amplitude, frequency and damping of a microbeam is controlled using a RLC block containing a capacitor, resistor and inductor in series with the microbeam. Applying this method all of the considerable characteristics of the oscillatory system can be determined and controlled with no change in the geometrical and physical characteristics of the microbeam. Euler-Bernoulli assumptions are made for the microbeam and the electrical current through the microbeam is computed by considering the microbeam deflection and its voltage. Considering the RLC block, the voltage difference between the microbeam and the substrate is calculated. Two coupled nonlinear partial differential equations are obtained for the deflection and the voltage. The one parameter Galerkin method is employed to transform the equations of motion to a set of nonlinear coupled ordinary differential equations. Differential quadrature method (DQM) is implemented to solve the governing nonlinear ordinary differential equations. The effect of the controller parameters such as capacitance, resistance and inductance on the amplitude, frequency and damping is studied. Also the internal resonance between the electrical and mechanical parts of the system is studied. Results indicate using these elements, amplitude, frequency and damping can be controlled as desired by the user.

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