Orr-Sommerfeld equation

This is the well-known Orr-Sommerfeld equation (OSE) that forms the major tool for the investigation of flow instability. If one considers two-dimensional disturbance field in a two-dimensional mean flow then the above equation transforms to the simpler form,... 

Thus, the Orr-Sommerfeld equation is a fourth order ODE and has same form whether the mean flow is three- or two-dimensional. Same observation can be made with respect to the disturbance held as well for special cases. 

For the stability analysis of fluid dynamical system whose equilibrium solution is given by a parallel or quasi-parallel flow, one has to solve the Orr-Sommerfeld equation depending on whether the mean flow is two- or three-dimensional. For the two-dimensional problem equivalent boundary conditions are given by... 

Q = [a +iaRe —c)Y/ . For boundary layer instability problems, i e —> 00 and then Q >> laj. This is the source of stiffness that makes obtaining the numerical solution of (2.3.21) a daunting task. This causes the fundamental solutions of the Orr-Sommerfeld equation to vary by different orders of magnitude near and far away from the wall. This type of behaviour makes the governing equation a stiff differential equation that suffers from the growth of parasitic error, while numerically solving it. 

In general, Orr-Sommerfeld equation is a fourth order ODE and thus, will have four fundamental solutions whose asymptotic variation for y —> 00, is given by the characteristic exponents of (2.4.3) i.e. 

This is the characteristic equation for the eigenvalues posed by the Orr-Sommerfeld equation that also can be viewed as the dispersion relation of the problem. So the task at hand is to obtain a combination of a and u> for a given Re, such that the solution of OSE satisfies (2.4.8). The stiffness of OSE causes the numerical solution to lose the linear independence of different solution components corresponding to the different fundamental solutions. This is the source of parasitic error growth of any stiff differential equation. To remove this problem in a straight forward manner, one can use the Compound Matrix Method (CMM). 

In this method, instead of working with 4> one works with a new set of variables that are combinations of the fundamental solutions 4> and 3. These new variables all vary with y at the identical rate, removing the stiffness problem. For the Orr-Sommerfeld equation the new variables are (for details about these new variables see Drazin Reid (1981), Sengupta (1992)),... 

Instability Analysis from Solntion of Orr-Sommerfeld Equation... 

The Orr- Sommerfeld equation can be solved either as a temporal or as a spatial instability problem. For disturbance field created as a consequence of a localized excitation inside boundary layers, the temporal growth of the disturbance field is not realistic. It has been observed phenomenologically that for attached flows, instability is usually of a convective t3rpe and obtaining solution by spatial analysis is the appropriate one. In chapter 4, we will note that even for such a problem there can be spatio-temporally growing wave-fronts that dominate in attached boundary layers that are noted to be spatially stable. Such a problem is not evident for flows those are spatially unstable. The monograph by Betchov Criminale (1967) specifically talks about temporal growth of disturbances in shear layers and the readers are referred there for detailed expositions. 

There is another reason for our preference in calculating system response by integrating (2.6.10) directly and not use contour integral (2.6.13) and (2.6.17). This is due to the restrictive condition (2.6.12) needed to hold for Jordan s Lemma to be used. We will show here, that the condition of Jordan s Lemma does not hold for the Orr-Sommerfeld equation - a result that has not been used in stability studies of fluid dynamics. 

Thus, the only possible distinguished limit is (5 = ei, impl3dng that the inner layer of the Orr-Sommerfeld equation is of thickness, S = In terms of the physical variables, the asymptotic value of is then ... 

The summation is over all the spanwise modes. One can use the above ansatz in three-dimensional Navier-Stokes equation and linearize the resultant equations after making a parallel flow approximation to get the following Orr-Sommerfeld equation for the Fourier- Laplace transform f of v ... 

The solution of Orr- Sommerfeld equation has four fundamental modes as... 

This equation can be used to describe the onset of instability, when a suitable mean flow is defined. We note that this equation is very generic for all incompressible flows (steady or unsteady flows), as it is based on full Navier-Stokes equation without making any assumptions. In Sengupta et al. (2006a) this equation has been used to explain the classical linear instability theory for parallel flows showing exactly identical TS waves obtained from Orr-Sommerfeld equation. In section 4.3, this is fully explained with the development of the actual equations and results. For the computational data, a mean flow was taken at t = 20 as representative undisturbed flow and the right hand side of (3.5.2) was calculated and plotted as shown in Fig. 3.9- at some representative times. 

The governing equation for the Fourier-Laplace transform is given by the following Orr-Sommerfeld equation,... 

The Bromwich contour for point A was chosen in the a -plane on a line extending from -20 to 4-20 that is below and parallel to the areal axis at a distance of 0.009 and in the w-plane it extended from -1 to - -1, above and parallel to the u>reai axis at a distance of 0.02. For the other points, the Bromwich contour in the a- plane is located at a distance of 0.001 below the Ureal axis. The choice of the Bromwich contour in the a- plane was such that all the downstream propagating eigenvalues lie above it. Orr-Sommerfeld equation was solved along these contours with 8192 equidistant points in the a- plane and 512 points in the w-plane. Orr-Sommerfeld equation was solved taking equidistant 2400 points across the shear layer in the range 0 < 2/ < 6.97. Spatial stability analysis produced waves for the four points of Fig. 4.2 with the properties shown in Table 4.1. 

These are the well-known Orr-Sommerfeld equations for mixed convection flows, that show the disturbance normal velocity and the temperature fields to be coupled, constituting a sixth order differential system. Equations (6.4.10) and (6.4.11) are to be solved subject to the six boundary conditions ... 

Sengupta, T.K. (1992). Solution of the Orr-Sommerfeld equation for high wave numbers. Comput. and Fluids, 21(2), 302-304. 

Instability Analysis from Solution of Orr-Sommerfeld Equation 43... 

The investigations being very similar for these two flows, we only report here on what was done for the two-layer Poiseuille flow. In [79], the Orr-Sommerfeld equations are rigorously derived, for the second type of perturbations via Laplace and Fourier... 

The numerical study of the Orr-Sommerfeld equations requires to discretize the dy operators in equation (25). As in [84], the spectral tau-Chebychev approximations are often used, though pseudo-spectral [85] or finite element techniques [86] may be chosen too. 

Under the long wavelength and quasistationary approximations and with the use of the linearized forms of the hydrodynamic and thermodynamic boundary conditions, first, we solve the Orr-Sommerfeld equation for the amplitude of perturbed part of the stream function from the Navier-Stokes equations. Second, we solve the equation for the amplitude of perturbed part of the temperature in the liquid film. The dispersion relation for the fluctuation of the solid-liquid interface is determined by the use of these solutions. From the real and imaginary part of this dispersion relation, we obtain the amplification rate cr and the phase velocity =-(7jk as follows ... 

Using the solution to the Orr-Sommerfeld equation, we can derive the relation between the amplitude of the liquid-air surface and the amplitude of the solid-liquid interface 4 = Since the value of a is small in the lower... 

The last section in this chapter is a brief introduction to stability of parallel shear flows. We consider three basic issues (i) Rayleigh s equation for inviscid flows, (ii) Rayleigh s necessary condition on an inflection point for inviscid instability, and (iii) a derivation of the Orr-Sommerfeld equation and Squire s theorem. 

To derive the Orr-Sommerfeld equation, we substitute (12-319) into (12-318) and linearize by retaining terms that are linear in e l. If we consider a typical disturbance mode in the form... 

This condition would, in principle, allow a specific estimate for the critical Reynolds number for stability as a function of ax, assuming that the eigenfunctions of the Orr-Sommerfeld equation had not been calculated. In the absence of explicit solutions for right-hand side could be estimated with trial functions to provide an upper bound on the critical value of Re. For present purposes, we simply note that the argument of Synge proves that there is a critical Reynolds number for stability of a steady, 2D unidirectional flow. 

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