2012 - Linear global instability of non-orthogonal incompressibleswept attachment-line boundary-layer flow (original) (raw)
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A simple extension of the classic Görtler-Hämmerlin (1955) (GH) model, essential for three-dimensional linear instability analysis, is presented. The extended Görtler-Hämmerlin model classifies all three-dimensional disturbances in this flow by means of symmetric and antisymmetric polynomials of the chordwise coordinate. It results in one-dimensional linear eigenvalue problems, a temporal or spatial solution of which, presented herein, is demonstrated to recover results otherwise only accessible to the temporal or spatial partial-derivative eigenvalue problem (the former also solved here) or to spatial direct numerical simulation (DNS). From a numerical point of view, the significance of the extended GH model is that it delivers the three-dimensional linear instability characteristics of this flow, discovered by solution of the partial-derivative eigenvalue problem by , at a negligible fraction of the computing effort required by either of the aforementioned alternative numerical methodologies. More significant, however, is the physical insight which the model offers into the stability of this technologically interesting flow. On the one hand, the dependence of three-dimensional linear disturbances on the chordwise spatial direction is unravelled analytically. On the other hand, numerical results obtained demonstrate that all linear three-dimensional instability modes possess the same (scaled) dependence on the wallnormal coordinate, that of the well-known GH mode. The latter result may explain why the three-dimensional linear modes have not been detected in past experiments; criteria for experimental identification of three-dimensional disturbances are discussed. Asymptotic analysis based on a multiple-scales method confirms the results of the extended GH model and provides an alternative algorithm for the recovery of three-dimensional linear instability characteristics, also based on solution of onedimensional eigenvalue problems. Finally, the polynomial structure of individual three-dimensional extended GH eigenmodes is demonstrated using three-dimensional DNS, performed here under linear conditions. † Present address: Rockwell Scientific,
Journal of Fluid Mechanics, 2003
A simple extensión of the classic Gortler-Hámmerlin (1955) (GH) model, essential for three-dimensional linear instability analysis, is presented. The extended Gortler-Hámmerlin model classifies all three-dimensional disturbances in this flow by means of symmetric and antisymmetric polynomials of the chordwise coordínate. It results in one-dimensional linear eigenvalue problems, a temporal or spatial solution of which, presented herein, is demonstrated to recover results otherwise only accessible to the temporal or spatial partial-derivative eigenvalue problem (the former also solved here) or to spatial direct numerical simulation (DNS). From a numerical point of view, the significance of the extended GH model is that it delivers the three-dimensional linear instability characteristics of this flow, discovered by solution of the partial-derivative eigenvalue problem by , at a negligible fraction of the computing effort required by either of the aforementioned alternative numerical methodologies. More significant, however, is the physical insight which the model offers into the stability of this technologically interesting flow. On the one hand, the dependence of three-dimensional linear disturbances on the chordwise spatial direction is unravelled analytically. On the other hand, numerical results obtained demónstrate that all linear three-dimensional instability modes possess the same (scaled) dependence on the wallnormal coordínate, that of the well-known GH mode. The latter result may explain why the three-dimensional linear modes have not been detected in past experiments; criteria for experimental identification of three-dimensional disturbances are discussed. Asymptotic analysis based on a multiple-scales method confirms the results of the extended GH model and provides an alternative algorithm for the recovery of three-dimensional linear instability characteristics, also based on solution of onedimensional eigenvalue problems. Finally, the polynomial structure of individual three-dimensional extended GH eigenmodes is demonstrated using three-dimensional DNS, performed here under linear conditions. f Present address: Rockwell Scientific, P.O. Box 1085, Thousand Oaks, CA 91358, USA.
Linear Instability of Plane Couette and Poiseuille Flows
—It is shown that linear instability of plane Couette flow can take place even at finite Reynolds numbers Re > Re th ≈ 139, which agrees with the experimental value of Re th ≈ 150 ± 5 [16, 17]. This new result of the linear theory of hydrodynamic stability is obtained by abandoning traditional assumption of the longitudinal periodicity of disturbances in the flow direction. It is established that previous notions about linear stability of this flow at arbitrarily large Reynolds numbers relied directly upon the assumed separation of spatial variables of the field of disturbances and their longitudinal periodicity in the linear theory. By also abandoning these assumptions for plane Poiseuille flow, a new threshold Reynolds number Re th ≈ 1035 is obtained, which agrees to within 4% with experiment—in contrast to 500% discrepancy for the previous estimate of Re th ≈ 5772 obtained in the framework of the linear theory under assumption of the " normal " shape of disturbances [2].
On absolute linear instability analysis of plane Poiseuille flow by a semi-analytical treatment
Journal of the Brazilian Society of Mechanical Sciences and Engineering, 37 (2): 495–505 (2015), DOI: 10.1007/s40430-014-0187-2, 2015
"The absolute linear hydrodynamic instability of the plane Poiseuille flow is investigated by solving the Orr– Sommerfeld equation using the semi-analytical treatment of the Adomian decomposition method (ADM). In order to use the ADM, a new zero-order ADM approximation is defined. The results for the spectrum of eigenvalues are obtained using various orders of the ADM approximations and discussed. A comparative study of the results for the first, second and third eigenvalues with the ones from a previously published work is also presented. A monotonic trend of approach of decreasing relative error with the increase of the orders of ADM approximation is indicated. The results for the first, second and third eigenvalues show that they are in good agreement within 1.5 % error with the ones obtained by a previously published work using the Chebyshev spectral method. The results also show that the first eigenvalue is positioned in the unstable zone of the spectrum, while the second and third eigenvalues are located in the stable zone."
On the stability of attachment-line boundary layers. Part 1. The incompressible swept Hiemenz flow
Journal of Fluid Mechanics, 1996
The stability of the incompressible attachment-line boundary layer is studied by solving a partial-differential eigenvalue problem. The basic flow near the leading edge is taken to be the swept Hiemenz flow which represents an exact solution of the Navier-Stokes (N-S) equations. Previous theoretical investigations considered a special class of two-dimensional disturbances in which the chordwise variation of disturbance velocities mimics the basic flow and renders a system of ordinarydifferential equations of the Orr-Sommerfeld type. The solution of this sixth-order system by showed that the two-dimensional disturbance is stable provided that the Reynolds number R < 583.1. In the present study, the restrictive assumptions on the disturbance field are relaxed to allow for more general solutions. Results of the present analysis indicate that unstable perturbations other than the special symmetric two-dimensional mode referred to above do exist in the attachment-line boundary layer provided R > 646. Both symmetric and antisymmetric two-and three-dimensional eigenmodes can be amplified. These unstable modes with the same spanwise wavenumber travel with almost identical phase speeds, but the eigenfunctions show very distinct features. Nevertheless, the symmetric twodimensional mode always has the highest growth rate and dictates the instability. As far as the special two-dimensional mode is concerned, the present results are in complete agreement with previous investigations. One of the major advantages of the present approach is that it can be extended to study the stability of compressible attachment-line flows where no satisfactory simplified approaches are known to exist.
Linear global instability of non-orthogonal incompressible swept attachment-line boundary-layer flow
Journal of Fluid Mechanics, 2012
Flow instability in the non-orthogonal swept attachment-line boundary layer is addressed in a linear analysis framework via solution of the pertinent global (BiGlobal) partial differential equation (PDE)-based eigenvalue problem. Subsequently, a simple extension of the extended Görtler–Hämmerlin ordinary differential equation (ODE)-based polynomial model proposed by Theofilis et al. (2003) for orthogonal flow, which includes previous models as special cases and recovers global instability analysis results, is presented for non-orthogonal flow. Direct numerical simulations have been used to verify the analysis results and unravel the limits of validity of the basic flow model analysed. The effect of the angle of attack, mathitAoA\mathit{AoA}mathitAoA, on the critical conditions of the non-orthogonal problem has been documented; an increase of the angle of attack, from mathitAoA=0\mathit{AoA}= 0mathitAoA=0 (orthogonal flow) up to values close to lrmpi/2\lrm{\pi} / 2lrmpi/2 which make the assumptions under which the basic flow is de...
The instability of oscillatory plane Poiseuille flow
Journal of Fluid Mechanics, 1982
The instability of oscillatory plane Poiseuille flow, in which the pressure gradient is time-periodically modulated, is investigated by a perturbation technique. The Floquet exponents (i.e. the complex growth rates of the disturbances to the oscillatory flow) are computed by series expansions, in powers of the oscillatory to steady flow velocity amplitude ratio, about the values of the growth rates of the disturbances of the steady flow. It is shown that the oscillatory flow is more stable than the steady flow for values of Reynolds number and disturbance wave number in the vicinity of the steady flow critical point and for values of frequencies of imposed oscillation greater than about one tenth of the frequency of the steady flow neutral disturbance. At very high and low values of imposed oscillation frequency, the unsteady flow is slightly less stable than the steady flow. These results hold for the values of the velocity amplitude ratio at least up to 0·25.
The stability of two-dimensional linear flows of an Oldroyd-type fluid
Journal of Non-Newtonian Fluid Mechanics, 1985
A linear stability analysis is made for an Oldroyd-type fluid undergoing steady two-dimensional flows in which the velocity field is a linear function of position throughout an unbounded region. This class of basic flows is characterized by a parameter X which ranges from X = 0 for simple shear flow to X = 1 for pure extensional flow. The time derivatives in the constitutive equation can be varied continuously from co-rotational to co-deformational as a parameter /I varies from 0 to 1. The linearized disturbance equations are analyzed to determine the asymptotic behavior as time t --) co of a spatially periodic initial disturbance. It is found that unbounded flows in the range 0 < X Q 1 are unconditionally unstable with respect to periodic initial disturbances which have lines of constant phase parallel to the inlet streamline in the plane of the basic flow. When the Weissenberg number is sufficiently small, only disturbances with sufficiently small wavenumber a3 in the direction normal to the basic flow plane are unstable. However, for certain values of 8, critical Weissenberg numbers are found above which flows are unstable for all values of the wavenumber (Ye.