A trigonometric series expansion method and two similar modified methods for the Orr-Sommerfeld equation are presented. These methods use the trigonometric series expansion with an auxiliary function added to the high...A trigonometric series expansion method and two similar modified methods for the Orr-Sommerfeld equation are presented. These methods use the trigonometric series expansion with an auxiliary function added to the highest order derivative of the unknown function and generate the lower order derivatives through successive integra- tions. The proposed methods are easy to implement because of the simplicity of the chosen basis functions. By solving the plane Poiseuille flow (PPF), plane Couette flow (PCF), and Blasius boundary layer flow with several homogeneous boundary conditions, it is shown that these methods yield results with the same accuracy as that given by the conventional Chebyshev collocation method but with better robustness, and that ob- tained by the finite difference method but with fewer modal number.展开更多
This paper analyses the effects of small injection/suction Reynolds number, Hartmann parameter, permeability parameter and wave number on a viscous incompressible electrically conducting fluid flow in a parallel porou...This paper analyses the effects of small injection/suction Reynolds number, Hartmann parameter, permeability parameter and wave number on a viscous incompressible electrically conducting fluid flow in a parallel porous plates forming a channel. The plates of the channel are parallel with the same constant temperature and subjected to a small injection/suction. The upper plate is allowed to move in flow direction and the lower plate is kept at rest. A uniform magnetic field is applied perpendicularly to the plates. The main objective of the paper is to study the effect of the above parameters on temporal linear stability analysis of the flow with a new approach based on modified Orr-Sommerfeld equation. It is obtained that the permeability parameter, the Hartmann parameter and the wave number contribute to the linear temporal stability while the small injection/suction Reynolds number has a negligible effect on the stability.展开更多
The stability of fluid flow in a horizontal layer of Brinkman porous medium with fluid viscosity different from effective viscosity is investigated. A modified Orr-Sommerfeld equation is derived and solved numerically...The stability of fluid flow in a horizontal layer of Brinkman porous medium with fluid viscosity different from effective viscosity is investigated. A modified Orr-Sommerfeld equation is derived and solved numerically using the Chebyshev collocation method. The critical Reynolds number Re, the critical wave number ac and the critical wave speed cc are computed for various values of porous parameter and ratio of viscosities. Based on these parameters, the stability characteristics of the system are discussed in detail. Streamlines are presented for selected values of parameters at their critical state.展开更多
In this work,we present a theoretical study on the stability of a two-dimensional plane Poiseuille flow of magnetic fluids in the presence of externally applied magnetic fields.The fluids are assumed to be incompressi...In this work,we present a theoretical study on the stability of a two-dimensional plane Poiseuille flow of magnetic fluids in the presence of externally applied magnetic fields.The fluids are assumed to be incompressible,and their magnetization is coupled to the flow through a simple phenomenological equation.Dimensionless parameters are defined,and the equations are perturbed around the base state.The eigenvalues of the linearized system are computed using a finite difference scheme and studied with respect to the dimensionless parameters of the problem.We examine the cases of both the horizontal and vertical magnetic fields.The obtained results indicate that the flow is destabilized in the horizontally applied magnetic field,but stabilized in the vertically applied field.We characterize the stability of the flow by computing the stability diagrams in terms of the dimensionless parameters and determine the variation in the critical Reynolds number in terms of the magnetic parameters.Furthermore,we show that the superparamagnetic limit,in which the magnetization of the fluids decouples from hydrodynamics,recovers the same purely hydrodynamic critical Reynolds number,regardless of the applied field direction and of the values of the other dimensionless magnetic parameters.展开更多
In this work the Laguerre basis for the biharmonic equation introduced by Jie Shen is employed in the spectral solution of self-similar problems of the boundary layer theory.An original Petrov-Galerkin formulation of ...In this work the Laguerre basis for the biharmonic equation introduced by Jie Shen is employed in the spectral solution of self-similar problems of the boundary layer theory.An original Petrov-Galerkin formulation of the Falkner-Skan equation is presented which is based on a judiciously chosen special basis function to capture the asymptotic behaviour of the unknown.A spectral method of remarkable simplicity is obtained for computing Falkner-Skan-Cooke boundary layer flows.The accuracy and efficiency of the Laguerre spectral approximation is illustrated by determining the linear stability of nonseparated and separated flows according to the Orr-Sommerfeld equation.The pentadiagonal matrices representing the derivative operators are explicitly provided in an Appendix to aid an immediate implementation of the spectral solution algorithms.展开更多
Linear Stability Analysis(LSA)of parallel shear flows,via local and global approaches,is presented.The local analysis is carried out by solving the Orr-Sommerfeld(OS)equation using a spectral-collocation method based ...Linear Stability Analysis(LSA)of parallel shear flows,via local and global approaches,is presented.The local analysis is carried out by solving the Orr-Sommerfeld(OS)equation using a spectral-collocation method based on Chebyshev polynomials.A stabilized finite element formulation is employed to carry out the global analysis using the linearized disturbance equations in primitive variables.The local and global analysis are compared.As per the Squires theorem,the two-dimensional disturbance has the largest growth rate.Therefore,only two-dimensional disturbances are considered.By its very nature,the local analysis assumes the disturbance field to be spatially periodic in the streamwise direction.The global analysis permits a more general disturbance.However,to enable a comparison with the local analysis,periodic boundary conditions,at the inlet and exit of the domain,are imposed on the disturbance.Computations are carried out for the LSA of the Plane Poiseuille Flow(PPF).The relationship between the wavenumber,a,of the disturbance and the streamwise extent of the domain,L,in the global analysis is explored for Re=7000.It is found that a and L are related by L=2pn/a,where n is the number of cells of the instability along the streamwise direction within the domain length,L.The procedure to interpret the results from the global analysis,for comparison with local analysis,is described.展开更多
An important aspect of the Orr Sommerfeld problem, which governs the linear stability of parallel shear flows, is concerned with the study of the temporal and spatial spectra for large but finite values of the Reynold...An important aspect of the Orr Sommerfeld problem, which governs the linear stability of parallel shear flows, is concerned with the study of the temporal and spatial spectra for large but finite values of the Reynolds number R . By using only outer (WKB) approximations which are valid in the "complete" sense, we are able to derive approximations to the eigenvalue relation for channel flows, pipe flow, and boundary layer flows which are all remarkably simple and which have a relative error of order ( αR) -1/2 . In this paper, we discuss briefly the basic ideas involved in the derivation of these approximations for boundary layer flows. We then present some results to illustrate the effectiveness of these new approximations. For example, we are even able to compute eigenvalues which lie arbitrarily close to the continuous spectra where all previous numerical treatments have failed.展开更多
基金supported by the National Natural Science Foundation of China(Nos.11221062,11521091,and 91752203)
文摘A trigonometric series expansion method and two similar modified methods for the Orr-Sommerfeld equation are presented. These methods use the trigonometric series expansion with an auxiliary function added to the highest order derivative of the unknown function and generate the lower order derivatives through successive integra- tions. The proposed methods are easy to implement because of the simplicity of the chosen basis functions. By solving the plane Poiseuille flow (PPF), plane Couette flow (PCF), and Blasius boundary layer flow with several homogeneous boundary conditions, it is shown that these methods yield results with the same accuracy as that given by the conventional Chebyshev collocation method but with better robustness, and that ob- tained by the finite difference method but with fewer modal number.
文摘This paper analyses the effects of small injection/suction Reynolds number, Hartmann parameter, permeability parameter and wave number on a viscous incompressible electrically conducting fluid flow in a parallel porous plates forming a channel. The plates of the channel are parallel with the same constant temperature and subjected to a small injection/suction. The upper plate is allowed to move in flow direction and the lower plate is kept at rest. A uniform magnetic field is applied perpendicularly to the plates. The main objective of the paper is to study the effect of the above parameters on temporal linear stability analysis of the flow with a new approach based on modified Orr-Sommerfeld equation. It is obtained that the permeability parameter, the Hartmann parameter and the wave number contribute to the linear temporal stability while the small injection/suction Reynolds number has a negligible effect on the stability.
基金supported by the Research Grants Council of the Hong Kong Special Administrative Region,China(Grant No.HKU 715510E)
文摘The stability of fluid flow in a horizontal layer of Brinkman porous medium with fluid viscosity different from effective viscosity is investigated. A modified Orr-Sommerfeld equation is derived and solved numerically using the Chebyshev collocation method. The critical Reynolds number Re, the critical wave number ac and the critical wave speed cc are computed for various values of porous parameter and ratio of viscosities. Based on these parameters, the stability characteristics of the system are discussed in detail. Streamlines are presented for selected values of parameters at their critical state.
基金P.Z.S.PAZ is grateful for the financial support provided by Coordination for the Improvement of Higher Education Personnel-Brazil(CAPES)(Finance Code 001)National Council for Scientific and Technological Development-Brazil(CNPq)during the course of this research.F.R.CUNHA acknowledges the financial support of CNPq(No.305764/2015-2)Y.D.SOBRAL acknowledges the financial support of University of Brasilia(Call DPI/DPG No.02/2021).
文摘In this work,we present a theoretical study on the stability of a two-dimensional plane Poiseuille flow of magnetic fluids in the presence of externally applied magnetic fields.The fluids are assumed to be incompressible,and their magnetization is coupled to the flow through a simple phenomenological equation.Dimensionless parameters are defined,and the equations are perturbed around the base state.The eigenvalues of the linearized system are computed using a finite difference scheme and studied with respect to the dimensionless parameters of the problem.We examine the cases of both the horizontal and vertical magnetic fields.The obtained results indicate that the flow is destabilized in the horizontally applied magnetic field,but stabilized in the vertically applied field.We characterize the stability of the flow by computing the stability diagrams in terms of the dimensionless parameters and determine the variation in the critical Reynolds number in terms of the magnetic parameters.Furthermore,we show that the superparamagnetic limit,in which the magnetization of the fluids decouples from hydrodynamics,recovers the same purely hydrodynamic critical Reynolds number,regardless of the applied field direction and of the values of the other dimensionless magnetic parameters.
文摘In this work the Laguerre basis for the biharmonic equation introduced by Jie Shen is employed in the spectral solution of self-similar problems of the boundary layer theory.An original Petrov-Galerkin formulation of the Falkner-Skan equation is presented which is based on a judiciously chosen special basis function to capture the asymptotic behaviour of the unknown.A spectral method of remarkable simplicity is obtained for computing Falkner-Skan-Cooke boundary layer flows.The accuracy and efficiency of the Laguerre spectral approximation is illustrated by determining the linear stability of nonseparated and separated flows according to the Orr-Sommerfeld equation.The pentadiagonal matrices representing the derivative operators are explicitly provided in an Appendix to aid an immediate implementation of the spectral solution algorithms.
文摘Linear Stability Analysis(LSA)of parallel shear flows,via local and global approaches,is presented.The local analysis is carried out by solving the Orr-Sommerfeld(OS)equation using a spectral-collocation method based on Chebyshev polynomials.A stabilized finite element formulation is employed to carry out the global analysis using the linearized disturbance equations in primitive variables.The local and global analysis are compared.As per the Squires theorem,the two-dimensional disturbance has the largest growth rate.Therefore,only two-dimensional disturbances are considered.By its very nature,the local analysis assumes the disturbance field to be spatially periodic in the streamwise direction.The global analysis permits a more general disturbance.However,to enable a comparison with the local analysis,periodic boundary conditions,at the inlet and exit of the domain,are imposed on the disturbance.Computations are carried out for the LSA of the Plane Poiseuille Flow(PPF).The relationship between the wavenumber,a,of the disturbance and the streamwise extent of the domain,L,in the global analysis is explored for Re=7000.It is found that a and L are related by L=2pn/a,where n is the number of cells of the instability along the streamwise direction within the domain length,L.The procedure to interpret the results from the global analysis,for comparison with local analysis,is described.
文摘An important aspect of the Orr Sommerfeld problem, which governs the linear stability of parallel shear flows, is concerned with the study of the temporal and spatial spectra for large but finite values of the Reynolds number R . By using only outer (WKB) approximations which are valid in the "complete" sense, we are able to derive approximations to the eigenvalue relation for channel flows, pipe flow, and boundary layer flows which are all remarkably simple and which have a relative error of order ( αR) -1/2 . In this paper, we discuss briefly the basic ideas involved in the derivation of these approximations for boundary layer flows. We then present some results to illustrate the effectiveness of these new approximations. For example, we are even able to compute eigenvalues which lie arbitrarily close to the continuous spectra where all previous numerical treatments have failed.
