Anatoli Tumin

Spatial theory of optimal disturbances in boundary layers

A spatial theory is proposed for the linear transient growth of disturbances in a parallel boundary layer. Following from the consideration of a signaling problem, the spatial development of disturbances downstream of a source may be presented as a sum of decaying eigenmodes and Tollmien–Schlichting (TS) like instability modes. Therefore, the problem of optimal disturbances may be considered as an initial value problem on the subset of the decaying eigenmodes and a TS wave, and a standard optimization procedure may be applied for evaluation of the optimal transient growth. The results indicate that the most significant transient growth is associated with stationary streamwise vortices. Numerical examples illustrate that favorable pressure gradient decreases the overall amplification. Effects of compressibility and the wall cooling are investigated as well.

Role of Transient Growth in Roughness-Induced Transition

Surface roughness can have a profound effect on boundary-layer transition. However, the mechanisms responsible for transition with three-dimensional distributed roughness have been elusive. Various Tollmien-Schlichting-based mechanisms have been investigated in the past but have been shown not to be applicable. More recently, the applicability of transient growth theory to roughness-induced transition has been studied. A model for roughness-induced transition is developed that makes use of computational results based on the spatial transient growth theory pioneered by the present authors. For nosetip transition, the resulting transition relations reproduce the trends of the Reda and passive nosetip technology (PANT) data and account for the separate roles of roughness and surface temperature level on the transition behavior

Multimode decomposition of spatially growing perturbations in a two-dimensional boundary layer

Three-dimensional spatially growing perturbations in a two-dimensional incompressible boundary layer are considered within the scope of linearized Navier–Stokes equations. The Cauchy problem is solved under the assumption of a finite growth rate of the disturbances. It is shown that the solution can be presented as an expansion into a biorthogonal eigenfunction system. The result can be utilized for decomposition of flow fields derived from computational studies when pressure and all velocity components, together with their derivatives, are available. The method can be used also in a case where partial data are available when a priori information leads to consideration of a finite number of modes. In the case of a continuous spectrum, the problem of decomposition based on partial information is ill-posed, but the method might be applied under additional assumptions about the perturbations.

Three-dimensional spatial normal modes in compressible boundary layers

Three-dimensional spatially growing perturbations in a two-dimensional compressible boundary layer are considered within the scope of linearized Navier{Stokes equations. The Cauchy problem is solved under the assumption of a flnite growth rate of the disturbances. It is shown that the solution can be presented as an expansion into a biorthogonal eigenfunction system. The result can be utilized for decomposition of ∞ow flelds derived from computational studies when pressure, temperature, and all the velocity components, together with some of their derivatives, are available. The method can be used also if partial data are available when a priori information may be utilized in the decomposition alogorithm. Properties of the discrete spectrum for a boundary layer over a cone with an adiabatic wall at the edge Mach number 5.6 is explored. It is shown that the synchronism of the slow discrete mode with acoustic waves at a low frequency or a low Reynolds number is primarily two-dimensional. At high angles of disturbance propagation, the fast discrete mode is no longer synchronized with entropy and vorticity modes.

Laminar–turbulent transition in Poiseuille pipe flow subjected to periodic perturbation emanating from the wall. Part 2. Late stage of transition

Transition in a fully developed circular pipe flow was investigated experimentally by introducing periodic perturbations. The simultaneous excitation of helical modes having indices m = ±1,±2 and ±3 was chosen. The experiments revealed that the late stage of transition is accompanied by the formation of streaky structures that are associated with peaks and valleys in the azimuthal distribution of the streamwise velocity disturbance. The breakdown to turbulence starts with the appearance of spikes in the temporal traces of the velocity. Spectral characteristics of these spikes and the direction of their propagation relative to the wall are similar to those in boundary layers. Analysis of the data suggests the existence of a high-shear layer in the instantaneous velocity profile. Additional experiments in which a very weak, steady flow was added locally to the periodic axisymmetric perturbation were also carried out. These experiments resulted in the generation of a single peak in the azimuthal distribution of the disturbance amplitude. The characteristics of the transition process (spikes, vortical patterns etc.) within this peak were similar to ones observed in the helical excitation experiments

Direct Numerical Simulation and the Theory of Receptivity in a Hypersonic Boundary Layer

Direct numerical simulation of receptivity in a boundary layer over a sharp wedge of half-angle 5:3 degrees was carried out with two-dimensional perturbations introduced into the ∞ow by periodic-in-time blowing-suction through a slot. The free stream Mach number was equal to 8. The perturbation ∞ow fleld downstream from the slot was decomposed into normal modes with the help of the biorthogonal eigenfunction system. Filtered-out amplitudes of two discrete normal modes and of the fast acoustic modes are compared with the linear receptivity problem solution. The examples ilustrate how the multimode decomposition technique may serve as a tool for gaining insight into computational results. I. Introduction The progress being made in computational ∞uid dynamics provides an opportunity for reliable simulation of such complex phenomena as laminar-turbulent transition. The dynamics of ∞ow transition depends on the instability of small perturbations excited by external sources. Computational results provide complete information about the ∞ow fleld, which would be impossible to measure in real experiments. However, validation of the results might be a challenging problem. Sometimes, numerical simulations of small perturbations in boundary layers are accompanied by comparisons with results obtained within the scope of the linear stability theory. In principle, this is possible in the case of a ∞ow possessing an unstable mode. Far downstream from the actuator, the perturbations might be dominated by the unstable mode, and one may compare the computational results for the velocity and temperature perturbation proflles and their growth rates with the linear stability theory. This analysis does not work when the amplitude of the unstable mode is comparable to that of other modes, or when one needs to evaluate the amplitude of a decaying mode. Recently, a method of normal mode decomposition was developed for two- and three- dimensional perturbations in compressible and incompressible boundary layers. 1{3 The method is based on the expansion of solutions of linearized Navier{Stokes equations for perturbations of prescribed frequency into the normal modes of discrete and continuous spectra. The instability modes belong to the discrete spectrum, whereas the continuous spectrum is associated with vorticity, entropy, and acoustic modes. Because the problem of perturbations within the scope of the linearized Navier{Stokes equations is not self-adjoint, the eigenfunctions representing the normal modes are not orthogonal. Therefore, the eigenfunctions of the adjoint problem are involved in the computation of the normal modes’ weights. Originally, the method based on the expansion into the normal modes was used for analysis of discrete modes (Tollmien{Schlichting{like modes) only. After clariflcation of uncertainties associated with the continuous spectra in Ref. 1, the method was also applied to the analysis of roughness-induced perturbations. 4{6 In order to flnd the amplitude of a normal mode, one needs proflles of the velocity, temperature, and pressure perturbations, together with some of their streamwise derivatives given at only one station downstream from the disturbance source. Because computational results can provide all the necessary information about the perturbation fleld, the application of the multimode decomposition is straightforward. However, the flrst

Spatial theory of optimal disturbances in a circular pipe flow

A spatial theory of linear transient growth for disturbances in a circular pipe is presented. Following from the consideration of a signaling problem, the spatial development of disturbances downstream of a source may be presented as a sum of decaying eigenmodes. Therefore, the problem of optimal disturbances in the pipe flow may be considered as an initial value problem on the subset of the downstream decaying eigenmodes, and a standard optimization procedure may be applied for evaluation of the optimal transient growth. Examples are presented for spatial transient growth of axisymmetric and nonaxisymmetric disturbances. It is shown that stationary disturbances may achieve more significant transient growth than nonstationary ones. The maximum of the transient growth exists at azimuthal index m=1 for stationary perturbations, whereas nonstationary perturbations may achieve their maxima at higher azimuthal indices.

High-Speed Boundary-Layer Instability: Old Terminology and a New Framework

The discrete spectrum of disturbances in high-speed boundary layers is discussed with emphasis on singularities caused by synchronization of the normal modes. Numerical examples illustrate different spectral structures and jumps from one structure to another with small variations of basic flow parameters. It is shown that this singular behavior is due to branching of the dispersion curves in the synchronization region. Depending on the locations of the branch points, the spectrum contains an unstable mode or two. In connection with this, the terminology used for instability of high-speed boundary layers is clarified. It is emphasized that the spectrum branching may cause difficulties in stability analyses based on traditional linear stability theory and parabolized stability equations methods. Multiple-mode considerations and direct numerical simulations are needed to clarify this issue.

Initial-Value Problem for Hypersonic Boundary-Layer Flows

An initial-value problem is analyzed for a two-dimensional wave packet induced by a local two-dimensional disturbance in a hypersonic boundary layer. The problem is solved using Fourier transform with respect to the streamwisecoordinateandLaplacetransform with respectto time. The temporal continuous spectrum isrevisited, and the uncertainty associated with the overlapping of continuous-spectrum branches is resolved. It is shown that thediscretespectrum’ s dispersion relationship isnonanalyticbecause of thesynchronization of thee rst mode with the vorticity/entropy waves of the continuous spectrum. However, the inverse Laplace transform is regular at the synchronism point. Characteristics of the wave packet generated by an initial temperature spot are numerically calculated. It is shown that the hypersonic boundary layer is highly receptive to vorticity/entropy disturbances in the synchronism region. The feasibility of experimental verie cation of this receptivity mechanism is discussed.

The Blunt Body Paradox — A Case for Transient Growth

The ‘blunt body paradox’ refers to the early transition on spherical forebodies (even those that are highly polished) observed at supersonic and hypersonic freestream speeds both in flight and in wind tunnels. This transition occurs usually in the subsonic portion of the flow behind the bow shock wave (see for example Buglia 1961), a region of highly favorable pressure gradient that is stable to T-S waves. Surface cooling leads to even earlier transition. This phenomenon, identified in the mid-1950’s, has defied clear explanation. It has always been prominent on Morkovin’s list of unsolved problems. The tentative suggestions are generally roughness related since stagnation point boundary layers are very thin. But no connection has been made between the microscopic roughness on the surface and the features of the observed early transition such as location, sensitivity to surface temperature level, etc. This has led the present authors to seek an explanation through transient growth.