The stability and transition of a two-dimensional jet

1960 ◽  
Vol 7 (1) ◽  
pp. 53-80 ◽  
Author(s):  
Hiroshi Sato
Keyword(s):  
Laminar Flow ◽  
Numerical Solutions ◽  
Two Dimensional ◽  
Eigenvalues And Eigenfunctions ◽  
Response Characteristics ◽  
Laminar Jets ◽  
The Stability ◽  
Linear Disturbance ◽  
Good Agreement ◽  
Small Disturbances

A study was made of the transition of a two-dimensional jet. In the region where laminar flow becomes unstable, two kinds of sinusoidal velocity fluctuation have been found; one is symmetrical and the other is anti-symmetrical with respect to the centre line of the jet. The fluctuations grow exponentially at first and develop into turbulence without being accompanied by abrupt bursts or turbulent spots.The response characteristics of laminar jets to artificial external excitation were investigated in detail by using sound as an exciting agent. The effect of excitation was seen to be most remarkable when the frequency of excitation coincides with that of self-excited sinusoidal fluctuations.Numerical solutions of equation of small disturbances superposed on laminar flow were obtained assuming the Reynolds number as infinity. Theoretical eigenvalues and eigenfunctions are in good agreement with experimental results, thus verifying the existence of a region of linear disturbance in the two-dimensional jet.

Laminar Flow in a Uniformly Porous Channel

1964 ◽  
Vol 15 (3) ◽  
pp. 299-310 ◽  
Author(s):  
Thein Wah
Keyword(s):  
Differential Equation ◽  
Reynolds Number ◽  
Laminar Flow ◽  
Numerical Solutions ◽  
Large Reynolds Number ◽  
Two Dimensional ◽  
Series Solutions ◽  
Porous Walls ◽  
Linear Differential ◽  
Good Agreement

SummaryThe flow in a two-dimensional channel with porous walls through which fluid is uniformly injected or extracted is considered. Following Berman, a solution is obtained giving a fourth-order non-linear differential equation which depends on a suction Reynolds number R. Numerical solutions of this equation have been obtained. Series solutions of this equation for small and large Reynolds number are given and are shown to give good agreement with the numerical solutions.

Comparison of Numerical and Perturbation Solutions of Two-Dimensional Nonlinear Water-Wave Problems

1976 ◽  
Vol 20 (03) ◽  
pp. 160-170
Author(s):  
Nils Salvesen ◽  
C. von Kerczek
Keyword(s):  
Perturbation Theory ◽  
Numerical Solutions ◽  
Order Perturbation ◽  
Order Perturbation Theory ◽  
Wave System ◽  
Two Dimensional ◽  
Third Order ◽  
Flow Past ◽  
Wave Problems ◽  
Good Agreement

Numerical solutions of the nonlinear problem of the steady two-dimensional potential flow past a submerged line vortex are obtained using the finite-difference iterative technique previously presented by the authors. These solutions are compared in detail with third-order perturbation theory solutions. It is found that very good agreement is obtained for cases of positive circulation of the vortex with strength large enough to produce downstream waves whose steepness is within 15 percent of the maximum possible steepness of irrotational free waves. These computed waves are as steep as the steepest waves obtained in a certain experiment involving the flow past a two-dimensional hydrofoil. For negative circulation, there is substantial difference between the numerical results and third-order perturbation theory. The failure of the perturbation theory is discussed. Details of the far-downstream wave system obtained by the numerical method are compared with other numerical solutions and very high-order perturbation theory solutions of the free-wave problem. Very good agreement is obtained in most cases.

Further contributions on the two-dimensional flow in a sudden expansion

1997 ◽  
Vol 330 ◽  
pp. 169-188 ◽  
Author(s):  
N. ALLEBORN ◽  
K. NANDAKUMAR ◽  
H. RASZILLIER ◽  
F. DURST
Keyword(s):  
Laminar Flow ◽  
Continuation Method ◽  
Sudden Expansion ◽  
Two Dimensional ◽  
Governing Equations ◽  
Bifurcation Structure ◽  
Bifurcation Points ◽  
Flow Parameters ◽  
Two Dimensional Flow ◽  
The Stability

Two-dimensional laminar flow of an incompressible viscous fluid through a channel with a sudden expansion is investigated. A continuation method is applied to study the bifurcation structure of the discretized governing equations. The stability of the different solution branches is determined by an Arnoldi-based iterative method for calculating the most unstable eigenmodes of the linearized equations for the perturbation quantities. The bifurcation picture is extended by computing additional solution branches and bifurcation points. The behaviour of the critical eigenvalues in the neighbourhood of these bifurcation points is studied. Limiting cases for the geometrical and flow parameters are considered and numerical results are compared with analytical solutions for these cases.

Numerical Simulation of the Water Entry of Two-Dimensional Body with Flow Separation

2011 ◽  
Vol 138-139 ◽  
pp. 376-381 ◽  
Author(s):  
Yun Bo Li ◽  
Ya Jun Li ◽  
Yan Wang
Keyword(s):  
Free Surface ◽  
Flow Separation ◽  
Water Entry ◽  
Two Dimensional ◽  
Numerical Result ◽  
Runge Kutta Method ◽  
Theory Result ◽  
The Stability ◽  
Stability And Accuracy ◽  
Good Agreement

The water entry of two-dimensional body with flow separation is simulated based on potential theory and boundary element method. The double point model and four-order Runge-Kutta method and jet-cut model and free surface smooth technique and regrinding technique are used to assure the stability and accuracy of the numerical result. A flow separation model is introduced to simulate the water entry of two-dimensional body with knuckle. The free surface elevation and pressure distribution of wedge with knuckle are predicted and compared with other theory result. Good agreement between numerical result and other theory result is indicated that the numerical method is stability and effective.

scholarly journals Analytical Solution of Two-Dimensional Sine-Gordon Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Alemayehu Tamirie Deresse ◽  
Yesuf Obsie Mussa ◽  
Ademe Kebede Gizaw
Keyword(s):  
Numerical Solutions ◽  
Initial Conditions ◽  
Nonlinear Partial Differential Equations ◽  
Test Problems ◽  
Two Dimensional ◽  
Sine Gordon Equation ◽  
Science And Engineering ◽  
Transform Method ◽  
Differential Transform ◽  
Good Agreement

In this paper, the reduced differential transform method (RDTM) is successfully implemented for solving two-dimensional nonlinear sine-Gordon equations subject to appropriate initial conditions. Some lemmas which help us to solve the governing problem using the proposed method are proved. This scheme has the advantage of generating an analytical approximate solution or exact solution in a convergent power series form with conveniently determinable components. The method considers the use of the appropriate initial conditions and finds the solution without any discretization, transformation, or restrictive assumptions. The accuracy and efficiency of the proposed method are demonstrated by four of our test problems, and solution behavior of the test problems is presented using tables and graphs. Further, the numerical results are found to be in a good agreement with the exact solutions and the numerical solutions that are available in literature. We have showed the convergence of the proposed method. Also, the obtained results reveal that the introduced method is promising for solving other types of nonlinear partial differential equations (NLPDEs) in the fields of science and engineering.

On the behaviour of small disturbances in plane Couette flow with a temperature gradient

1965 ◽  
Vol 286 (1404) ◽  
pp. 117-128 ◽  
Keyword(s):  
Rayleigh Number ◽  
Couette Flow ◽  
Reynolds Numbers ◽  
Neutral Stability ◽  
Two Dimensional ◽  
Plane Couette Flow ◽  
Infinitesimal Disturbance ◽  
Prandtl Numbers ◽  
The Stability ◽  
Small Disturbances

The stability of plane Couette flow with a heated lower plate is considered with respect to a two-dimensional infinitesimal disturbance. The eigenvalues are found with the aid of a digital computer as the latent roots of a matrix. Neutral stability curves for various Prandtl numbers at Reynolds numbers up to 150 are obtained by a second method. It is found that the principle of the exchange of stabilities does not hold for this problem. With the aid of Squire’s transformation the conclusion is drawn that all fluids will become unstable at the same value of the Rayleigh number irrespective of whether shear is present or not.

Distortion and necking of a viscous inclusion in Stokes flow

1989 ◽  
Vol 422 (1862) ◽  
pp. 173-192 ◽  
Keyword(s):  
Viscous Fluid ◽  
Stokes Flow ◽  
Numerical Solutions ◽  
Finite Amplitude ◽  
Elliptic Solution ◽  
Small Perturbations ◽  
All Solutions ◽  
Elliptic Profile ◽  
The Stability ◽  
Small Disturbances

Spence & Wilmott (1988) considered the deformation of a slender inclusion of highly-viscous fluid in a Stokes flow of less viscous fluid, and derived a coupled pair of equations to describe its evolution. The equations possess self-preserving solutions for elliptical inclusions, previously known from the work of Bilby et al . (1975, 1976) and Bilby & Kolbuszewski (1977). In the present paper numerical solutions are presented for non-elliptic shapes. These have been obtained on a CRAY XMP-22 by use of an eigenfunction expansion suggested by the elliptic solution, leading to an initial value problem for the coefficients. The solutions show that large deformations can develop in finite time from small perturbations of the ellipse, particularly at large viscosity ratios. Special attention is directed to the phenomenon of boudinage, whereby alternate swelling and necking develops as the inclusion is stretched by the Stokes flow. This appears to characterize all solutions that depart significantly from pure elliptic shape. In the Appendix the stability of the elliptic profile to small disturbances is examined. It is found that all subharmonics of a given disturbance are excited to a finite amplitude that increases with the viscosity ratio, but that higher harmonics are not excited in linear theory, although nonlinear coupling leads to the eventual excitation of all modes.

Acoustic destablilization of vortices

1979 ◽  
Vol 290 (1372) ◽  
pp. 353-371 ◽  
Keyword(s):  
Boundary Condition ◽  
Growth Rate ◽  
Mach Number ◽  
Acoustic Waves ◽  
Polar Angle ◽  
Inviscid Flow ◽  
Two Dimensional ◽  
Low Mach Number ◽  
The Stability ◽  
Good Agreement

A line vortex which has uniform vorticity 2Ω 0 in its core is subjected to a small two-dimensional disturbance whose dependence on polar angle is e imθ . The stability is examined according to the equations of compressible, inviscid flow in a homentropic medium. The boundary condition at infinity is that of outgoing acoustic waves, and it is found that this capacity to radiate leads to a slow instability by comparison with the corresponding incompressible vortex which is stable. Numerical eigenvalues are computed as functions of the mode number m and the Mach number M based on the circumferential speed of the vortex. These are compared with an asymptotic analysis for the m = 2 mode at low Mach number in which it is found that the growth rate is (π/ 32) M 4 Ω 0 in good agreement with the numerical results.

The stability of an incompressible two-dimensional wake

1972 ◽  
Vol 51 (2) ◽  
pp. 233-272 ◽  
Author(s):  
G. E. Mattingly ◽  
W. O. Criminale
Keyword(s):  
Reference Frames ◽  
Two Dimensional ◽  
Mean Velocity ◽  
Spatial Reference ◽  
Spatial Reference Frames ◽  
The Mean ◽  
Temporal And Spatial ◽  
The Stability ◽  
Transverse Oscillations ◽  
Small Disturbances

The growth of small disturbances in a two-dimensional incompressible wake has been investigated theoretically and experimentally. The theoretical analysis is based upon inviscid stability theory wherein small disturbances are considered from both temporal and spatial reference frames. Through a combined stability analysis, in which small disturbances are permitted to amplify in both time and space, the relationship between the disturbance characteristics for the temporal and spatial reference frames is shown. In these analyses a quasi-uniform assumption is adopted to account for the continuously varying mean-velocity profiles that occur behind flat plates and thin airfoils. It is found that the most unstable disturbances in the wake produce transverse oscillations in the mean-velocity profile and correspond to growing waves that have a minimum group velocity.Experimentally, the downstream development of the wake of a thin airfoil and the wave characteristics of naturally amplifying small disturbances are investigated in a water tank. The disturbances that develop are found to produce transverse oscillations of the mean-velocity profile in agreement with the theoretical prediction. From the comparison of the experimental results with the predictions for the characteristics of the most unstable waves via the temporal and spatial analyses, it is concluded that the stability analysis for the wake is to be considered solely from the more realistic spatial viewpoint. Undoubtedly, this conclusion is also applicable to other highly unstable flows such as jets and free shear layers.In accordance with the disturbance vorticity distribution as determined from the spatial model, a description of the initial development of a vortex street is put forth that contrasts with the description given by Sato & Kuriki (1961).

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