Eli Reshotko

Transient growth: A factor in bypass transition

Transient growth arises through the coupling between slightly damped, highly oblique (nearly streamwise) T–S and Squire modes leading to algebraic growth followed by exponential decay in a region that is subcritical with respect to the T–S neutral curve. A weak transient growth can also occur for two dimensional or axisymmetric modes since the Orr–Sommerfeld operator and its compressible counterpart are not self-adjoint, therefore their eigenfunctions are not strictly orthogonal. So transient growth is a candidate mechanism for many examples of bypass transition. The original transient growth theories were all temporal. However spatial growth formulations are now emerging including their extension to compressible flow with pressure gradient and heat transfer. The relevance to bypass transition is examined through several examples including Poiseuille pipe flow, the hypersonic blunt body paradox and distributed roughness effects.

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

The vibrating ribbon problem revisited

A revised formal solution of the vibrating ribbon problem of hydrodynamic stability is presented. The initial formulation of Gaster (1965) is modified by application of the Briggs method and a careful treatment of the complex double Fourier transform inversions. Expressions are obtained in a natural way for the discrete spectrum as well as for the four branches of the continuous spectra. These correspond to discrete and branch-cut singularities in the complex wavenumber plane. The solutions from the continuous spectra decay both upstream and downstream of the ribbon, with the decay in the upstream direction being much more rapid than that in the downstream direction. Comments and clarification of related prior work are made.

Transition Issues for Atmospheric Entry

*Much has been learned about the physics underlying the transition process at supersonic and hypersonic speeds through years of analysis, experiment and computation. The application has been principally to simple shapes like plates, cones and spherical nosetips. But the shapes of the new entry vehicles are not simple. They will invariably be at angle of attack so three dimensional effects will be very important, as will roughness effects due to ablation. This paper will review the physics basis of our present understanding of the transition process. Further, because of the complex geometries, it will address the need for careful experimental work as per the guidelines enunciated years ago by the U.S. Transition Study Group. Following these guidelines is essential to obtaining reliable, usable data for use in refining transition estimation techniques . I. Introduction Entry vehicles descend rapidly through the atmosphere and decelerate as the drag forces increase with the increasing atmospheric density. The vehicle starts out fully laminar. Since Reynolds numbers increase rapidly through the des cent, transition tends to move very rapidly over the whole vehicle over a narrow range of altitudes. O ne tries to minimize the aerodynamic heating loads in entry so as to minimize the thermal protection needs of the vehicle. This means delaying transition to as low an altitude as possible . The history of high -performance entry vehicles begins with the development of the ICBM in the 1950’s. These v ehicles were essentially cone -cylinders with large nose bluntness to minimize stagnation point heating. The nose materials were often subliming ablators to take advantage of the further reduction in stagnation region heating due to the surface blowing from the subliming surface. Transition would move forward from the cylinder to the cone as the vehicle descende d through the atmosphere. If transition occurred asymmetrically on the cylinder or cone (leading to drag asymmetry), it was essential that the asymmetric effects be small enough to be corrected by the control system of the missile so as to minimize the cir cle of error about the target. In some cases, the transition unexpectedly moved forward onto the spherical nose , a phenomenon named the “blunt -body -paradox.” A study in that time period by Allen and Eggers 1 showed that for orbital and sub -orbital entry ( < 8 km/sec) , the convective aerodynamic heating rate was directly related to the vehicle’s ballistic coefficient, (W/C DA). The lower the ballistic coefficient, the lower the heating rates in entry. Hence the tendency to low weight and high drag coefficient. The highly blunted capsul e shapes of the Mercury, Gemini, Apollo and Soyuz vehicles with a blat ing heat shields are a consequence of this argument . At entry speeds above about 10 km/sec , the shock layers ahead of the entering blunt shapes become lumino us and radiate. These radiative heat transfer rates are highly density dependent. Thus the blunt body may not be the best entry shape for supercircular entry speeds. Studies by Allen et al 2,3 show that to minimize total heat transfer or total ablated mass , the optimum shape gets progressively slenderer as the entry speed is increased. Not enough was done with these vehicles to identify the major transition issues.

DISTURBANCES IN A BOUNDARY LAYER INTRODUCED BY A LOW INTENSITY ARRAY OF VORTICES

To acquire insight into the role of free-stream turbulence on laminar-turbulent transition, the interaction between a boundary layer and an array of single-wavenumber vortices convected at the mean free-stream velocity is studied analytically and numerically. For small amplitudes, the effect of a spectrum can be obtained by superposition. The flow field is taken to be the sum of the steady laminar field (Blasius) plus a flow field ascribable to the effects of the vortex array. This latter flow field is further subdivided into the portion that exists in the absence of the plate (the vortex array itself) plus a flow field representing the alteration to that array due to the shearing mean flow and no-slip and impermeability conditions at the plate surface. This last portion of the flow field is described by a nonhomogeneous Orr–Sommerfeld equation with phase speed unity and real wavenumber. The forcing function depends on the mean flow and on the free-stream disturbance array. The problem is not an eigenvalu...

Hybrid Reynolds-Averaged Navier-Stokes/Large-Eddy Simulations of Supersonic Turbulent Mixing

Aturbulentmixinglayerformedfromtwosupersonicstreamsinitially separatedbyasplitterplateisinvestigated withahybridcomputationalmethodusingaReynolds-averagedNavier ‐Stokes(RANS)approachforwall-bounded regions and a large-eddy simulation (LES) approach for the turbulent mixing region. Simulations of the mixing layer predict a vortex shedding originating from the splitter base and then a rapid transition to turbulence, which is in agreement with experimental observations. As a result, the potential of the hybrid method is demonstrated for e ows in which a geometric feature, such as the blunt splitter plate considered here, provides the dominant unsteady feature leading to turbulence, without imposing additional ine ow disturbances. Parametric studies of LES subgrid modeling settings, variations in the spanwise computational domain, and wall temperature settings in the RANS region were conducted. The subgrid model cases, using a baseline computational grid with small spanwise computational domain, overpredicted the mixing layer turbulencelevels but only showed small variation in the mixing layer predictions with large changes in the model parameters. The cases examining wider spanwise domains enabled more turbulent energy to be released in the spanwise direction, which in turn reduced axial and vertical turbulence levels. Finally, prescribing the wall temperatures in the RANS regions instead of using the more traditional adiabatic wall boundary conditions further reduced turbulence levels and enabled reasonable agreement with experimental data.

Is Retheta/Me a Meaningful Transition Criterion?

usually Reθ/Me = const. Also since Reθ varies as the square root of a length Reynolds number, any scatter in Reθ is greatly amplified when inverted to get a length transition Reynolds number or a length to transition. There is no apparent physics basis for this form of transition correlation. In this paper, the roles of modal (T-S, crossflow, Gortler, etc) and non-modal (transient growth) mechanisms on the transition process will be described. For modal disturbances, the physics based transition estimations are done by e methods. On the other hand, transient growth methods have been successfully applied in cases where the transition is due to surface roughness. These methods take into account many of the factors that can affect transition such as pressure gradient, surface temperature level, suction or blowing, surface roughness, etc. They also can suggest design changes that can lead to transition delay. Successful

Microfabricated Shear Stress Sensors, Part 1: Design and Fabrication

The design and fabrication of shear stress sensors based on the floating-element method and polysilicon-surface-micromachining technology is reported. Three designs have been developed for microfabrication, two including monolithic integration of mechanical sensor elements with on-chip circuitry. The first design is four-mask standard polysilicon-surface-micromachining process to develop passive floating-element sensors with optically determined deflection sensitivity. The second-generation devices are fabricated in a six-mask modified N-channel metal-oxide-semiconductor process, where sensor elements and signal conditioning circuitry have been integrated on the sensor die for amplified voltage output. The third design modifies the commercially available micromachined by replacing the accelerometer element with a shear-stress-sensitive floating element, enabling active sensing for linear response and self-test features

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.