Distributed Control of Two-Dimensional Navier–Stokes Equations in Fourier Spectral Simulations

2017 ◽  
Vol 139 (8) ◽  
Author(s):  
Behrooz Rahmani ◽  
Amin Moosaie
Keyword(s):  
Differential Equations ◽  
Distributed Control ◽  
Nonlinear Equations ◽  
Stokes Equations ◽  
Control Method ◽  
Grid Point ◽  
Nonlinear Partial Differential Equations ◽  
Nonlinear Flow ◽  
Mode Interaction ◽  
Flow Equations

A method for distributed control of nonlinear flow equations is proposed. In this method, first, Takagi–Sugeno (T–S) fuzzy model is used to substitute the nonlinear partial differential equations (PDEs) governing the system by a set of linear PDEs, such that their fuzzy composition exactly recovers the original nonlinear equations. This is done to alleviate the mode-interaction phenomenon occurring in spectral treatment of nonlinear equations. Then, each of the so-obtained linear equations is converted to a set of ordinary differential equations (ODEs) using the fast Fourier transform (FFT) technique. Thus, the combination of T–S method and FFT technique leads to a number of ODEs for each grid point. For the stabilization of the dynamics of each grid point, the use is made of the parallel distributed compensation (PDC) method. The stability of the proposed control method is proved using the second Lyapunov theorem for fuzzy systems. In order to solve the nonlinear flow equation, a combination of FFT and Runge–Kutta methodologies is implemented. Simulation studies show the performance of the proposed method, for example, the smaller settling time and overshoot and also its relatively robustness with respect to the measurement noises.

Coriolis effects on MHD newtonian flow over a rotating non-uniform surface

2020 ◽  
pp. 095440622096973
Author(s):  
Abayomi S Oke ◽  
Winifred N Mutuku ◽  
Mark Kimathi ◽  
Isaac Lare Animasaun
Keyword(s):  
Differential Equations ◽  
Lorentz Force ◽  
Coriolis Force ◽  
Stokes Equations ◽  
Physical Phenomenon ◽  
Nonlinear Partial Differential Equations ◽  
Mhd Flow ◽  
Simultaneous Increase ◽  
Convection Flow ◽  
The Impact

The roles of the simultaneous effect of Coriolis force and Lorentz force (resulting from MHD flow) in Sunspots, solar wind, and many other natural and physical phenomenon is undoubtedly significant. The impact of fluids heated by the Sun is influenced by the rotation of the earth’s surface and this necessitates the study of fluid flow over such surface as the Earth. For this reason, the significance of Coriolis force on MHD free-convection flow of Newtonian fluid over the rotating upper horizontal surface of paraboloid of revolution is explored. The relevant body forces are derived and included in the Navier-Stokes equations to obtain appropriate equations governing the flow. By nondimensionalizing the governing equations using similarity variables, the system of nonlinear partial differential equations is reduced to a system of nonlinear ordinary differential equations which is solved using Runge-Kutta-Gills method along with Shooting technique and the results are depicted graphically. It is observed that simultaneous increase in both Coriolis force and Lorentz force causes an increase in the temperature profile of the flow. It is also observed that the effect of increasing Coriolis force on the Skin Friction and heat transfer rate is counter-balanced by increasing Lorentz force.

scholarly journals Application of the Poor Man's Navier-Stokes Equations to Real-Time Control of Fluid Flow

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
James B. Polly ◽  
J. M. McDonough
Keyword(s):  
Fluid Flow ◽  
Differential Equations ◽  
Real Time ◽  
Stokes Equations ◽  
Discrete Dynamical System ◽  
Nonlinear Partial Differential Equations ◽  
Navier Stokes ◽  
Control Force ◽  
Machining Processes ◽  
The Poor

Control of fluid flow is an important, underutilized process possessing potential benefits ranging from avoidance of separation and stall on aircraft wings to reduction of friction in oil and gas pipelines to mitigation of noise from wind turbines. But the Navier-Stokes (N.-S.) equations, whose solutions describe such flows, consist of a system of time-dependent, multidimensional, nonlinear partial differential equations (PDEs) which cannot be solved in real time using current computing hardware. The poor man's Navier-Stokes (PMNS) equations comprise a discrete dynamical system that is algebraic—hence, easily (and rapidly) solved—and yet which retains many (possibly all) of the temporal behaviors of the PDE N.-S. system at specific spatial locations. Herein, we outline derivation of these equations and discuss their basic properties. We consider application of these equations to the control problem by adding a control force. We examine the range of behaviors that can be achieved by changing this control force and, in particular, consider controllability of this (nonlinear) systemvianumerical experiments. Moreover, we observe that the derivation leading to the PMNS equations is very general and may be applied to a wide variety of problems governed by PDEs and (possibly) time-delay ordinary differential equations such as, for example, models of machining processes.

Analytical solutions for the unsteady MHD rotating flow over a rotating sphere near the equator

Open Physics ◽  
2011 ◽  
Vol 9 (1) ◽  
Author(s):  
Erik Sweet ◽  
Kuppalapalle Vajravelu ◽  
Robert Gorder
Keyword(s):  
Steady State ◽  
Differential Equations ◽  
Analytical Solutions ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Nonlinear Partial Differential Equations ◽  
Coupled System ◽  
Polar Coordinates ◽  
Rotating Flow ◽  
Rotating Sphere

AbstractIn this paper we investigate the three-dimensional magnetohydrodynamic (MHD) rotating flow of a viscous fluid over a rotating sphere near the equator. The Navier-Stokes equations in spherical polar coordinates are reduced to a coupled system of nonlinear partial differential equations. Self-similar solutions are obtained for the steady state system, resulting from a coupled system of nonlinear ordinary differential equations. Analytical solutions are obtained and are used to study the effects of the magnetic field and the suction/injection parameter on the flow characteristics. The analytical solutions agree well with the numerical solutions of Chamkha et al. [31]. Moreover, the obtained analytical solutions for the steady state are used to obtain the unsteady state results. Furthermore, for various values of the temporal variable, we obtain analytical solutions for the flow field and present through figures.

The Behaviour of Fluid-Conveying Pipes, Supported at Both Ends, by the Complete Extensible Nonlinear Equations of Motion

2006 ◽  
Author(s):  
Yahya Modarres-Sadeghi ◽  
Michael P. Pai¨doussis ◽  
Alexandra Camargo
Keyword(s):  
Differential Equations ◽  
Flow Velocity ◽  
Nonlinear Equations ◽  
Equations Of Motion ◽  
Nonlinear Partial Differential Equations ◽  
Pitchfork Bifurcation ◽  
Critical Flow Velocity ◽  
Order Of Magnitude ◽  
Nonlinear Equations Of Motion ◽  
Fluid Conveying Pipes

In this paper, the post-divergence behaviour of fluid-conveying pipes supported at both ends is studied using the complete extensible nonlinear equations of motion. The two coupled nonlinear partial differential equations are discretized via Galerkin’s method and the resulting set of ordinary differential equations is solved by Houbolt’s finite difference method and also using AUTO. Typically, the pipe is stable and retains its original static equilibrium position up to where it loses stability by a supercritical pitchfork bifurcation. By increasing the flow velocity, the amplitude of the buckled position increases, but no secondary instability can be observed thereafter, in agreement with Holmes’ results for his simplified model. The effect of different parameters on the behaviour of the pipe has been studied. By increasing the externally applied tension, or by increasing the gravity parameter, the critical flow velocity for the pitchfork bifurcation increases. The pitchfork bifurcation is subcritical if the nondimensional externally imposed tension, is greater than the nondimensional axial rigidity. The solution in the vicinity of the critical point for this case is confirmed to be subcritical, although the fold and the stable non-trivial solution thereafter could not be seen — perhaps because the model is correct to only third-order of magnitude. Dynamic instabilities may be possible for a pipe hinged at both ends but free to slide axially at the downstream end, according to preliminary results.

A Bulk-Flow Analysis of Static and Dynamic Characteristics of Eccentric Circumferentially-Grooved Liquid Annular Seals

2004 ◽  
Vol 126 (2) ◽  
pp. 316-325 ◽  
Author(s):  
Mihai Arghir ◽  
Jean Frene
Keyword(s):  
Differential Equations ◽  
Dynamic Characteristics ◽  
Pressure Field ◽  
Stokes Equations ◽  
Numerical Procedure ◽  
Bulk Flow ◽  
Elliptic Type ◽  
Flow Equations ◽  
Numerical Predictions ◽  
Annular Seals

The bulk-flow equations used for inertia dominated thin-film flows is an attractive model for the analysis of circumferentially grooved annular seals because the solutions based on the numerical integration of the complete Navier-Stokes equations can be very time-consuming. By using three types of control volumes and some user-tuned constants, the bulk-flow model can be used for calculating the static and the dynamic characteristics. Until now, this has been carried out for centered seals where the flow is governed by ordinary differential equations but no solutions have yet been given for eccentric working conditions. In this latter case, the model is governed by partial differential equations of an elliptic type. The main problem is that for describing the groove effects, the pressure field must incorporate the concentrated drop or recovery effects that occur at the interface between the groove and the land zone. This means that the numerical procedure used for solving the elliptic equations should be able to handle a pressure field having discontinuous values and discontinuous first order derivatives. In the present work, the method used for integrating the system of bulk-flow equations is the SIMPLE algorithm. The algorithm is extended for handling pressure jumps by adding two pressure values on each side of the discontinuity. These values are then expressed in terms of cell centered pressures by imposing the mass conservation and the generalized Bernoulli equation at the discontinuity. This numerical solution is original and has never previously been presented in the finite volume related literature. Comparisons between the numerical predictions (leakage flow rate and rotordynamic coefficients) and experimental data taken from the literature Marquette and Childs (1997) are subsequently presented for an eccentric ten-groove annular seal.

scholarly journals Quantum Algorithms for Nonlinear Equations in Fluid Mechanics

2020 ◽  
Author(s):  
Rene Steijl
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fluid Mechanics ◽  
Nonlinear Equations ◽  
Quantum Algorithms ◽  
Nonlinear Partial Differential Equations ◽  
Floating Point ◽  
Partial Differential ◽  
Linear Ordinary Differential Equations ◽  
Near Term

In recent years, significant progress has been made in the development of quantum algorithms for linear ordinary differential equations as well as linear partial differential equations. There has not been similar progress in the development of quantum algorithms for nonlinear differential equations. In the present work, the focus is on nonlinear partial differential equations arising as governing equations in fluid mechanics. First, the key challenges related to nonlinear equations in the context of quantum computing are discussed. Then, as the main contribution of this work, quantum circuits are presented that represent the nonlinear convection terms in the Navier–Stokes equations. The quantum algorithms introduced use encoding in the computational basis, and employ arithmetic based on the Quantum Fourier Transform. Furthermore, a floating-point type data representation is used instead of the fixed-point representation typically employed in quantum algorithms. A complexity analysis shows that even with the limited number of qubits available on current and near-term quantum computers (<100), nonlinear product terms can be computed with good accuracy. The importance of including sub-normal numbers in the floating-point quantum arithmetic is demonstrated for a representative example problem. Further development steps required to embed the introduced algorithms into larger-scale algorithms are discussed.

Similarity Solutions of the Force-free Magnetic Field Equations

1994 ◽  
Vol 49 (9) ◽  
pp. 902-912
Author(s):  
E. W. Richter
Keyword(s):  
Magnetic Fields ◽  
Differential Equations ◽  
Nonlinear Equations ◽  
Linear Equations ◽  
Nonlinear Partial Differential Equations ◽  
Similarity Solutions ◽  
Two Dimensional ◽  
Field Equations ◽  
Reduced Equations ◽  
Group Invariant Solutions

Abstract Force-free magnetic fields are described as solutions of special nonlinear partial differential equations which are replaced frequently through linear equations. To record the diversity of the structures of these fields, a discussion of the nonlinear equations is necessary. For this purpose the method of similarity analysis is used. The Lie symmetry groups admitted by the nonlinear equations for force-free magnetic fields are presented. To record and classify the different types of group-invariant solutions, one-and two-dimensional optimal systems of subalgebras are listed. The reduced equations of the two-dimensional optimal system are systems of ordinary differential equations, and their solutions define similarity solutions which are force-free magnetic fields. Only in one case is it necessary to calculate similarity solutions numerically. The corresponding reduced equations are a nonautonomous dynamical system with the similarity variable in the place of time.

scholarly journals Shock Wave Solutions for Some Nonlinear Flow Models Arising in the Study of a Non-Newtonian Third Grade Fluid

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Taha Aziz ◽  
R. J. Moitsheki ◽  
A. Fatima ◽  
F. M. Mahomed
Keyword(s):  
Shock Wave ◽  
Differential Equations ◽  
Fluid Model ◽  
Nonlinear Partial Differential Equations ◽  
Nonlinear Flow ◽  
Third Grade ◽  
Physical Parameters ◽  
Third Grade Fluid ◽  
Wave Solutions ◽  
Grade Fluid

This study is based upon constructing a new class of closed-form shock wave solutions for some nonlinear problems arising in the study of a third grade fluid model. The Lie symmetry reduction technique has been employed to reduce the governing nonlinear partial differential equations into nonlinear ordinary differential equations. The reduced equations are then solved analytically, and the shock wave solutions are constructed. The conditions on the physical parameters of the flow problems also fall out naturally in the process of the derivation of the solutions.

scholarly journals A hybrid boundary integral/Taylor series approach to some nonlinear equations from fluid mechanics

1987 ◽  
Vol 28 (3) ◽  
pp. 279-286 ◽  
Author(s):  
C. J. Coleman
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fluid Mechanics ◽  
Nonlinear Equations ◽  
Taylor Series ◽  
Nonlinear Partial Differential Equations ◽  
Boundary Integral ◽  
Two Dimensional ◽  
Boundary Integral Methods ◽  
Integral Methods

AbstractWe show that a combination of Taylor series and boundary integral methods can lead to an effective scheme for solving a class of nonlinear partial differential equations. The method is illustrated through its application to an equation from two dimensional fluid mechanics.

Exact Solutions of the Unsteady Navier-Stokes Equations

1989 ◽  
Vol 42 (11S) ◽  
pp. S269-S282 ◽  
Author(s):  
C. Y. Wang
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Stokes Equations ◽  
Nonlinear Partial Differential Equations ◽  
Similarity Solutions ◽  
Navier Stokes ◽  
Partial Differential ◽  
Navier Stokes Equations

The unsteady Navier-Stokes equations are a set of nonlinear partial differential equations with very few exact solutions. This paper attempts to classify and review the existing unsteady exact solutions. There are three main categories: parallel, concentric and related solutions, Beltrami and related solutions, and similarity solutions. Physically significant examples are emphasized.

Sign in / Sign up

Export Citation Format

Share Document