Modal Stability Theory

2014 ◽  
Vol 66 (2) ◽  
Author(s):  
Matthew P. Juniper ◽  
Ardeshir Hanifi ◽  
Vassilios Theofilis
Keyword(s):  
Stability Theory ◽  
Eigenvalue Problems ◽  
Stokes Equations ◽  
Temporal Stability ◽  
Collocation Methods ◽  
Navier Stokes ◽  
Parallel Flows ◽  
Numerical Solution Methods ◽  
Solution Methods ◽  
Spatio Temporal

This article contains a review of modal stability theory. It covers local stability analysis of parallel flows including temporal stability, spatial stability, phase velocity, group velocity, spatio-temporal stability, the linearized Navier–Stokes equations, the Orr–Sommerfeld equation, the Rayleigh equation, the Briggs–Bers criterion, Poiseuille flow, free shear flows, and secondary modal instability. It also covers the parabolized stability equation (PSE), temporal and spatial biglobal theory, 2D eigenvalue problems, 3D eigenvalue problems, spectral collocation methods, and other numerical solution methods. Computer codes are provided for tutorials described in the article. These tutorials cover the main topics of the article and can be adapted to form the basis of research codes.

scholarly journals Linear and nonlinear spatio-temporal instability in laminar two-layer flows

2010 ◽  
Vol 656 ◽  
pp. 458-480 ◽  
Author(s):  
P. VALLURI ◽  
L. Ó NÁRAIGH ◽  
H. DING ◽  
P. D. M. SPELT
Keyword(s):  
Linear Theory ◽  
Stokes Equations ◽  
Random Noise ◽  
Temporal Stability ◽  
Reynolds Numbers ◽  
Diffuse Interface ◽  
Navier Stokes ◽  
Sharp Change ◽  
Linear And Nonlinear ◽  
Spatio Temporal

The linear and nonlinear spatio-temporal stability of an interface separating two Newtonian fluids in pressure-driven channel flow at moderate Reynolds numbers is analysed both theoretically and numerically. A linear, Orr–Sommerfeld-type analysis shows that most of such systems are unstable. The transition to an absolutely unstable regime is investigated, and is shown to occur in an intermediate range of Reynolds numbers and ratios of the thicknesses of the two layers, for near-density matched fluids with a viscosity contrast. A critical Reynolds number is found for transition from convective to absolute instability of relatively thin films. Results obtained from direct numerical simulations (DNSs) of the Navier–Stokes equations for long channels using a diffuse-interface method elucidate that waves generated by random noise at the inlet show that, near the inlet, waves are formed and amplified strongly, leading to ligament formation. Successive waves coalesce with each other further downstream, resulting in longer larger-amplitude waves further downstream. In the linearly absolute regime, the characteristics of the spatially growing wave near the inlet agree with that of the saddle point as predicted by the linear theory. The transition point from a convective to an absolute regime predicted by linear theory is also in agreement with a sharp change in the value of a healing length obtained from the DNSs.

Investigation of the stability of boundary layers by a finite-difference model of the Navier—Stokes equations

1976 ◽  
Vol 78 (2) ◽  
pp. 355-383 ◽  
Author(s):  
H. Fasel
Keyword(s):  
Finite Difference ◽  
Stability Theory ◽  
Stokes Equations ◽  
Time Dependent ◽  
Linear Stability Theory ◽  
Navier Stokes ◽  
Experimental Measurements ◽  
Navier Stokes Equations ◽  
The Stability ◽  
Small Periodic

The stability of incompressible boundary-layer flows on a semi-infinite flat plate and the growth of disturbances in such flows are investigated by numerical integration of the complete Navier–;Stokes equations for laminar two-dimensional flows. Forced time-dependent disturbances are introduced into the flow field and the reaction of the flow to such disturbances is studied by directly solving the Navier–Stokes equations using a finite-difference method. An implicit finitedifference scheme was developed for the calculation of the extremely unsteady flow fields which arose from the forced time-dependent disturbances. The problem of the numerical stability of the method called for special attention in order to avoid possible distortions of the results caused by the interaction of unstable numerical oscillations with physically meaningful perturbations. A demonstration of the suitability of the numerical method for the investigation of stability and the initial growth of disturbances is presented for small periodic perturbations. For this particular case the numerical results can be compared with linear stability theory and experimental measurements. In this paper a number of numerical calculations for small periodic disturbances are discussed in detail. The results are generally in fairly close agreement with linear stability theory or experimental measurements.

Local RBFs Based Collocation Methods for Unsteady Navier-Stokes Equations

2015 ◽  
Vol 7 (4) ◽  
pp. 430-440 ◽  
Author(s):  
Xueying Zhang ◽  
Xin An ◽  
C. S. Chen
Keyword(s):  
Linear Systems ◽  
Numerical Experiments ◽  
Stokes Equations ◽  
Sparse Matrix ◽  
High Accuracy ◽  
Collocation Methods ◽  
Navier Stokes ◽  
Sparse Linear Systems ◽  
Navier Stokes Equations ◽  
Weight Coefficients

AbstractThe local RBFs based collocation methods (LRBFCM) is presented to solve two-dimensional incompressible Navier-Stokes equations. In avoiding the ill-conditioned problem, the weight coefficients of linear combination with respect to the function values and its derivatives can be obtained by solving low-order linear systems within local supporting domain. Then, we reformulate local matrix in the global and sparse matrix. The obtained large sparse linear systems can be directly solved instead of using more complicated iterative method. The numerical experiments have shown that the developed LRBFCM is suitable for solving the incompressible Navier-Stokes equations with high accuracy and efficiency.

Onset of absolutely unstable behaviour in the Stokes layer: a Floquet approach to the Briggs method

2021 ◽  
Vol 928 ◽  
Author(s):  
Alexander Pretty ◽  
Christopher Davies ◽  
Christian Thomas
Keyword(s):  
Growth Rate ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Steady Flows ◽  
Navier Stokes Equations ◽  
Electron Stream ◽  
Stokes Layer ◽  
Symmetry Constraints ◽  
Spatio Temporal ◽  
Spatial Locations

For steady flows, the Briggs (Electron-Stream Interaction with Plasmas. MIT Press, 1964) method is a well-established approach for classifying disturbances as either convectively or absolutely unstable. Here, the framework of the Briggs method is adapted to temporally periodic flows, with Floquet theory utilised to account for the time periodicity of the Stokes layer. As a consequence of the antiperiodicity of the flow, symmetry constraints are established that are used to describe the pointwise evolution of the disturbance, with the behaviour governed by harmonic and subharmonics modes. On coupling the symmetry constraints with a cusp-map analysis, multiple harmonic and subharmonic cusps are found for each Reynolds number of the flow. Therefore, linear disturbances experience subharmonic growth about fixed spatial locations. Moreover, the growth rate associated with the pointwise development of the disturbance matches the growth rate of the disturbance maximum. Thus, the onset of the Floquet instability (Blennerhassett & Bassom, J. Fluid Mech., vol. 464, 2002, pp. 393–410) coincides with the onset of absolutely unstable behaviour. Stability characteristics are consistent with the spatio-temporal disturbance development of the family-tree structure that has hitherto only been observed numerically via simulations of the linearised Navier–Stokes equations (Thomas et al., J. Fluid Mech., vol. 752, 2014, pp. 543–571; Ramage et al., Phys. Rev. Fluids, vol. 5, 2020, 103901).

Analysis of the dripping–jetting transition in compound capillary jets

2010 ◽  
Vol 649 ◽  
pp. 523-536 ◽  
Author(s):  
M. A. HERRADA ◽  
J. M. MONTANERO ◽  
C. FERRERA ◽  
A. M. GAÑÁN-CALVO
Keyword(s):  
Stability Analysis ◽  
Convective Instability ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
The Core ◽  
Outer Interface ◽  
Spatio Temporal ◽  
Capillary Jets ◽  
The Stability

We examine the behaviour of a compound capillary jet from the spatio-temporal linear stability analysis of the Navier–Stokes equations. We map the jetting–dripping transition in the parameter space by calculating the Weber numbers for which the convective/absolute instability transition occurs. If the remaining dimensionless parameters are set, there are two critical Weber numbers that verify Brigg's pinch criterion. The region of absolute (convective) instability corresponds to Weber numbers smaller (larger) than the highest value of those two Weber numbers. The stability map is affected significantly by the presence of the outer interface, especially for compound jets with highly viscous cores, in which the outer interface may play an important role even though it is located very far from the core. Full numerical simulations of the Navier–Stokes equations confirm the predictions of the stability analysis.

On perturbations of Jeffery-Hamel flow

1988 ◽  
Vol 186 ◽  
pp. 559-581 ◽  
Author(s):  
W. H. H. Banks ◽  
P. G. Drazin ◽  
M. B. Zaturska
Keyword(s):  
Stokes Equations ◽  
Temporal Stability ◽  
Pitchfork Bifurcation ◽  
Critical Value ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Nonlinear Perturbations ◽  
Navier Stokes Equations ◽  
Weakly Nonlinear ◽  
Rigid Plane

We examine various perturbations of Jeffery-Hamel flows, the exact solutions of the Navier-Stokes equations governing the steady two-dimensional motions of an incompressible viscous fluid from a line source at the intersection of two rigid plane walls. First a pitchfork bifurcation of the Jeffery-Hamel flows themselves is described by perturbation theory. This description is then used as a basis to investigate the spatial development of arbitrary small steady two-dimensional perturbations of a Jeffery-Hamel flow; both linear and weakly nonlinear perturbations are treated for plane and nearly plane walls. It is found that there is strong interaction of the disturbances up- and downstream if the angle between the planes exceeds a critical value 2α2, which depends on the value of the Reynolds number. Finally, the problem of linear temporal stability of Jeffery-Hamel flows is broached and again the importance of specifying conditions up- and downstream is revealed. All these results are used to interpret the development of flow along a channel with walls of small curvature. Fraenkel's (1962) approximation of channel flow locally by Jeffery-Hamel flows is supported with the added proviso that the angle between the two walls at each station is less than 2α2.

Dual solutions of magnetohydrodynamic boundary layer flow and a linear temporal stability analysis

2020 ◽  
Vol 234 (6) ◽  
pp. 553-561 ◽  
Author(s):  
Golam Mortuja Sarkar ◽  
Bikash Sahoo
Keyword(s):  
Boundary Layer ◽  
Stability Analysis ◽  
Stokes Equations ◽  
Velocity Ratio ◽  
Temporal Stability ◽  
Dual Solutions ◽  
Navier Stokes ◽  
Temperature Profiles ◽  
Linear Ordinary Differential Equations ◽  
Similarity Transformations

In this paper, we have considered the two-dimensional stagnation point flow and heat transfer of an electrically conducting viscous fluid over an exponentially stretching sheet. The Navier-Stokes equations are reduced into a system of highly non-linear ordinary differential equations by similarity transformations. The resulting systems are then solved numerically by shooting method. The effects of suction/injection parameters on the boundary layer are discussed in detail. Our numerical results reveal that for a particular range of the velocity ratio parameter, dual solutions exist. Interestingly these two solution branches show opposite characters in the velocity and temperature profiles. Thus, it is worthwhile to carry a stability analysis of these two solutions to determine the feasible solution. A linear temporal stability analysis has been carried out, and the stability is tested by the sign of the smallest eigenvalue. The smallest eigenvalues are found by two different numerical schemes, which agree well up to the desired accuracy. The effects of the pertinent flow parameters in the velocity and temperature profiles are discussed in detail and are shown graphically.

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