A numerical investigation of the rolling-up of vortex sheets

1980 ◽  
Vol 373 (1752) ◽  
pp. 67-91 ◽  
Keyword(s):  
Numerical Integration ◽  
Numerical Investigation ◽  
Equations Of Motion ◽  
Additional Term ◽  
Vortex Sheet ◽  
Time Step ◽  
Vortex Sheets ◽  
Numerical Instabilities ◽  
Sinusoidal Disturbance ◽  
Induced Velocity

In this paper the development of a vortex sheet due to an initially sinusoidal disturbance is calculated. When determining the induced velocity in points of the vortex sheet, it can be represented by concentrated vortices but it is shown that it is analytically more correct to add an additional term that represents the effect of the immediate neighbourhood of the point considered. The equations of motion were integrated by a Runge-Kutta technique to exclude numerical instabilities. The time step was determined by the requirement that a quantity (Hamiltonian) that remains invariant as a result of the equations of motion, should not change more than a certain amount in the numerical integration of the equations of motion. One difficulty is that if a greater number of concentrated vortices are introduced to represent the vortex sheet, the effect of round-off errors becomes more important. The number of figures retained in the computations limits the number of concentrated vortices. Where the round-off errors have been kept sufficiently small, a process of rolling-up of vorticity clearly occurs. There is no point in pursuing the calculations much beyond this point, first because the representation of the vortex sheet by concentrated vortices becomes more and more inaccurate and secondly because viscosity will have the effect of transforming the rolled-up vortex sheet into a region of vorticity.

Application of the Modified Rosenbrock Algorithm for Transient Rotordynamic Analysis

2012 ◽  
Author(s):  
Anand Srinivasan ◽  
Dhruv Kumar
Keyword(s):  
Numerical Integration ◽  
Equations Of Motion ◽  
Mathematical Formulation ◽  
Single Step ◽  
Degree Of Freedom ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Forcing Function ◽  
Transient Simulations

It is well known that transient rotordynamic analyses involve numerical integration of the equations of motion in order to study the response of the system under an applied forcing function. A common problem that arises in such simulations is the choice of step-size that needs to be used to obtain numerically stable results. Traditional numerical integration techniques such as the Runge-Kutta algorithms not only require splitting up second order differential equations as two first order equations, but also necessitate multiple integrations at each time-step, thus increasing the solution time. The Newmark-beta and Wilson-theta algorithms are some of the prevalent methods that have been used for transient simulations in rotordynamics. However, those single-step methods are only conditionally stable, and require iterations to converge to a solution at each time step, thus making it pseudosingle-step. In the more recent years, a modified form of the Rosenbrock algorithm has been proposed as a numerically stable and true single-step mathematical formulation for the integration of structural dynamics problems. In this paper, the modified Rosenbrock algorithm has been applied to a transient start-up multi-degree-of-freedom rotordynamics problem. A constant time step-size algorithm has been used for the simulations, and results of the transient analysis have been presented. The fact that a multi-degree-of-freedom system can be solved without condensation of the higher order modes makes the superior numerical damping characteristics of the algorithm become evident.

scholarly journals Geometry and dynamics of vortex sheets in 3 dimension

2002 ◽  
pp. 55-76 ◽  
Author(s):  
D.A. Burton ◽  
R.W. Tucker
Keyword(s):  
Fluid Velocity ◽  
Normal Component ◽  
Vortex Sheet ◽  
The Self ◽  
Vortex Filament ◽  
Vortex Sheets ◽  
Filament Dynamics ◽  
De Rham ◽  
Induced Velocity

We consider the properties and dynamics of vortex sheets from a geometrical, coordinate-free, perspective. Distribution-valued forms (de Rham currents) are used to represent the fluid velocity and vorticity due to the vortex sheets. The smooth velocities on either side of the sheets are solved in terms of the sheet strengths using the language of double forms. The classical results regarding the continuity of the sheet normal component of the velocity and the conservation of vorticity are exposed in this setting. The formalism is then applied to the case of the self-induced velocity of an isolated vortex sheet. We develop a simplified expression for the sheet velocity in terms of representative curves. Its relevance to the classical Localized Induction Approximation (LIA) to vortex filament dynamics is discussed. .

scholarly journals Numerical Integration Scheme Using Singular Perturbation Method

2013 ◽  
Author(s):  
Gibin Gil ◽  
Ricardo G. Sanfelice ◽  
Parviz E. Nikravesh
Keyword(s):  
Singular Perturbation ◽  
Perturbation Method ◽  
Numerical Integration ◽  
Equations Of Motion ◽  
Computation Time ◽  
Integration Scheme ◽  
Singular Perturbation Method ◽  
Time Step ◽  
Numerical Integration Scheme ◽  
Deformable Bodies

Some multi degree-of-freedom dynamical systems exhibit a response that contain fast and slow variables. An example of such systems is a multibody system with rigid and deformable bodies. Standard numerical integration of the resultant equations of motion must adjust the time step according to the frequency of the fastest variable. As a result, the computation time is sacrificed. The singular perturbation method is an analysis technique to deal with the interaction of slow and fast variables. In this study, a numerical integration scheme using the singular perturbation method is discussed, its absolute stability condition is derived, and its order of accuracy is investigated.

Vibration analysis of multiple degrees of freedom mechanical system with dry friction

2012 ◽  
Vol 227 (7) ◽  
pp. 1505-1514 ◽  
Author(s):  
SD Yu ◽  
BC Wen
Keyword(s):  
Mechanical System ◽  
Dry Friction ◽  
Degrees Of Freedom ◽  
Simple Procedure ◽  
Mathematical Problem ◽  
Equations Of Motion ◽  
Inequality Constraints ◽  
Integration Scheme ◽  
Time Step ◽  
Multiple Degrees Of Freedom

This article presents a simple procedure for predicting time-domain vibrational behaviors of a multiple degrees of freedom mechanical system with dry friction. The system equations of motion are discretized by means of the implicit Bozzak–Newmark integration scheme. At each time step, the discontinuous frictional force problem involving both the equality and inequality constraints is successfully reduced to a quadratic mathematical problem or the linear complementary problem with the introduction of non-negative and complementary variable pairs (supremum velocities and slack forces). The so-obtained complementary equations in the complementary pairs can be solved efficiently using the Lemke algorithm. Results for several single degree of freedom and multiple degrees of freedom problems with one-dimensional frictional constraints and the classical Coulomb frictional model are obtained using the proposed procedure and compared with those obtained using other approaches. The proposed procedure is found to be accurate, efficient, and robust in solving non-smooth vibration problems of multiple degrees of freedom systems with dry friction. The proposed procedure can also be applied to systems with two-dimensional frictional constraints and more sophisticated frictional models.

scholarly journals Remarks on Stationary and Uniformly-rotating Vortex Sheets: Rigidity Results

2021 ◽  
Author(s):  
Javier Gómez-Serrano ◽  
Jaemin Park ◽  
Jia Shi ◽  
Yao Yao
Keyword(s):  
Angular Velocity ◽  
Disjoint Union ◽  
Vortex Sheet ◽  
Vortex Sheets ◽  
Smooth Curves ◽  
Rigidity Results ◽  
Positive Vorticity ◽  
Constant Vorticity ◽  
Concentric Circles ◽  
Finite Disjoint Union

AbstractIn this paper, we show that the only solution of the vortex sheet equation, either stationary or uniformly rotating with negative angular velocity $$\Omega $$ Ω , such that it has positive vorticity and is concentrated in a finite disjoint union of smooth curves with finite length is the trivial one: constant vorticity amplitude supported on a union of nested, concentric circles. The proof follows a desingularization argument and a calculus of variations flavor.

scholarly journals Collapse of n Point Vortices, Formation of the Vortex Sheets and Transport of Passive Markers

Energies ◽  
2021 ◽  
Vol 14 (4) ◽  
pp. 943
Author(s):  
Henryk Kudela
Keyword(s):  
Differential Equations ◽  
Point Vortex ◽  
Optimization Procedure ◽  
Vortex Sheet ◽  
Point Vortices ◽  
Vortex Sheets ◽  
Impermeable Barrier ◽  
Initial Location ◽  
Passive Tracers ◽  
Initial Iterate

In this paper, the motion of the n-vortex system as it collapses to a point in finite time is studied. The motion of vortices is described by the set of ordinary differential equations that we are able to solve analytically. The explicit formula for the solution demands the initial location of collapsing vortices. To find the collapsing locations of vortices, the algebraic, nonlinear system of equations was built. The solution of that algebraic system was obtained using Newton’s procedure. A good initial iterate needs to be provided to succeed in the application of Newton’s procedure. An unconstrained Leverber–Marquart optimization procedure was used to find such a good initial iterate. The numerical studies were conducted, and numerical evidence was presented that if in a collapsing system n=50 point vortices include a few vortices with much greater intensities than the others in the set, the vortices with weaker intensities organize themselves onto the vortex sheet. The collapsing locations depend on the value of the Hamiltonian. By changing the Hamiltonian values in a specific interval, the collapsing curves can be obtained. All points on the collapse curves with the same Hamiltonian value represent one collapsing system of vortices. To show the properties of vortex sheets created by vortices, the passive tracers were used. Advection of tracers by the velocity induced by vortices was calculated by solving the proper differential equations. The vortex sheets are an impermeable barrier to inward and outward fluxes of tracers. Arising vortex structures are able to transport the passive tracers. In this paper, several examples showing the diversity of collapsing structures with the vortex sheet are presented. The collapsing phenomenon of many vortices, their ability to self organize and the transportation of the passive tracers are novelties in the context of point vortex dynamics.

A Strategy to Accelerate the Numerical Integration in the Dynamic Simulation of Multibody Systems

1997 ◽  
Author(s):  
Jesús Cardenal ◽  
Javier Cuadrado ◽  
Eduardo Bayo
Keyword(s):  
Multibody Systems ◽  
Equations Of Motion ◽  
Stiff Systems ◽  
Step Size ◽  
Step Method ◽  
Integral Of Motion ◽  
Time Step ◽  
Time Step Size ◽  
Variable Time ◽  
Numerical Integrator

Abstract This paper presents a multi-index variable time step method for the integration of the equations of motion of constrained multibody systems in descriptor form. The basis of the method is the augmented Lagrangian formulation with projections in index-3 and index-1. The method takes advantage of the better performance of the index-3 formulation for large time steps and of the stability of the index-1 for low time steps, and automatically switches from one method to the other depending on the required accuracy and values of the time step. The variable time stepping is accomplished through the use of an integral of motion, which in the case of conservative systems becomes the total energy. The error introduced by the numerical integrator in the integral of motion during consecutive time steps provides a good measure of the local integration error, and permits a simple and reliable strategy for varying the time step. Overall, the method is efficient and powerful; it is suitable for stiff and non-stiff systems, robust for all time step sizes, and it works for singular configurations, redundant constraints and topology changes. Also, the constraints in positions, velocities and accelerations are satisfied during the simulation process. The method is robust in the sense that becomes more accurate as the time step size decreases.

Numerical Investigation of Cowl Lip Adjustments for a Rocket-Based Combined-Cycle Inlet in Takeoff Regime

2016 ◽  
Vol 33 (3) ◽  
Author(s):  
Lei Shi ◽  
Xiaowei Liu ◽  
Guoqiang He ◽  
Fei Qin ◽  
Xianggeng Wei ◽  
...  
Keyword(s):  
Numerical Integration ◽  
Numerical Investigation ◽  
Flow Separation ◽  
Static Condition ◽  
Combined Cycle ◽  
Variable Geometry ◽  
Overall Performance ◽  
Negative Impacts

AbstractNumerical integration simulations were performed on a ready-made central strut-based rocket-based combined-cycle (RBCC) engine operating in the ejector mode during the takeoff regime. The effective principles of various cowl lip positions and shapes on the inlet operation and the overall performance of the entire engine were investigated in detail. Under the static condition, reverse cowl lip rotation in a certain range was found to contribute comprehensive improvement to the RBCC inlet and the entire engine. However, the reverse rotation of the cowl lip contributed very little enhancement of the RBCC inlet under the low subsonic flight regime and induced extremely negative impacts in the high subsonic flight regime, especially in terms of a significant increase in the drag of the inlet. Changes to the cowl lip shape provided little improvement to the overall performance of the RBCC engine, merely shifting the location of the leeward area inside the RBCC inlet, as well as the flow separation and eddy, but not relieving or eliminating those phenomena. The results of this study indicate that proper cowl lip rotation offers an efficient variable geometry scheme for a RBCC inlet in the takeoff regime.

scholarly journals NONCOMMUTATIVE METAFLUID DYNAMICS

2006 ◽  
Vol 21 (03) ◽  
pp. 505-516 ◽  
Author(s):  
A. C. R. MENDES ◽  
C. NEVES ◽  
W. OLIVEIRA ◽  
F. I. TAKAKURA
Keyword(s):  
Phase Space ◽  
Nonlinear Equations ◽  
Equations Of Motion ◽  
Additional Term ◽  
Dissipative Force ◽  
Covariant Form ◽  
Covariant Quantization ◽  
Noncommutative Spaces ◽  
Classical Phase Space

In this paper we define a noncommutative (NC) metafluid dynamics.1,2 We applied the Dirac's quantization to the metafluid dynamics on NC spaces. First class constraints were found which are the same obtained in Ref. 4. The gauge covariant quantization of the nonlinear equations of fields on noncommutative spaces were studied. We have found the extended Hamiltonian which leads to equations of motion in the gauge covariant form. In addition, we show that a particular transformation3 on the usual classical phase space (CPS) leads to the same results as of the ⋆-deformation with ν = 0. Besides, we have shown that an additional term is introduced into the dissipative force due to the NC geometry. This is an interesting feature due to the NC nature induced into model.

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