Anatoli Tumin
 Professor, AerospaceMechanical Engineering
 Professor, Applied Mathematics  GIDP
 (520) 6214254
 Aerospace & Mechanical Engr., Rm. 000617
 Tucson, AZ 85721
 tumin@email.arizona.edu
Biography
Anatoli Tumin is a Professor of Aerospace and Mechanical Engineering and a Member of the Applied Mathematics program at the University of Arizona. He received his M. S. in physics and applied mathematics from Novosibirsk State University (1972) and his Ph.D. and Dr. Sci. in physics and mathematics from the Moscow Institute of Physics and Technology (MIPT) in 1975 and 1987, respectively. In 1975, he joined the faculty of the Physics Department at MIPT. In 1982, he was employed by the ScientificIndustrial Design Bureau “Molniya,” where he participated (as Lead Designer and Head of Sector) in the design of the Soviet Space Shuttle “Buran.” His responsibilities included prediction of laminarturbulent transition and nonequilibrium heat transfer in reentry flight, specification of tolerances for elements of thermoprotection system, and other aspects of aerothermodynamics relevant to the design of hypersonic vehicles. In 1992, he joined the faculty of TelAviv University. In 2000, he joined the faculty of the University of Arizona. His current research interests include theoretical and computational studies of stability, laminarturbulent transition, and flow control in highspeed flows including real gas effects.
Degrees
 D.S. Physics and Applied Mathematics
 Moscow Institute of Physics and Technology, Russian Federation
 Ph.D. Physics and Applied Mathematics
 Moscow Institute of Physics and Technology, Russian Federation
 M.S. Physics and Applied Mathematics (two degrees)
 Novosibirsk State University, Novosibirsk, Russian Federation
Work Experience
 Professor, University of Arizona, Tucson, Arizona (2000  Ongoing)
 Associate Professor, Tel Aviv University (1993  2002)
 Researcher Fellow, Tel Aviv University (1992  1993)
 Associate Professor, Moscow Institute of Physics and Technology (1990  1992)
 Head of Sector, ScientificIndustrial Design Office (1982  1990)
 Assistant Professor, Moscow Institute of Physics and Technology (1975  1982)
Interests
Teaching
Fluid mechanics, Heat Transfer, Mathematical methods in engineering
Research
Hydrodynamic stability and transition to turbulence in highspeed boundary layers, Aerodynamic heating, Flow control
Courses
201819 Courses

Adv Engineering Analysis
AME 500B (Spring 2019) 
Dissertation
MATH 920 (Spring 2019) 
Adv Engineering Analysis
AME 500A (Fall 2018) 
Compressible Fld Dynamic
AME 536C (Fall 2018) 
Dissertation
MATH 920 (Fall 2018) 
Research
AME 900 (Fall 2018)
201718 Courses

Independent Study
AME 599 (Spring 2018) 
Independent Study
MATH 599 (Spring 2018) 
Research
AME 900 (Spring 2018) 
Research
MATH 900 (Spring 2018) 
Adv Engineering Analysis
AME 500A (Fall 2017) 
Boundary Layers
AME 535 (Fall 2017) 
Independent Study
MATH 599 (Fall 2017) 
Research
AME 900 (Fall 2017)
201617 Courses

Adv Engineering Analysis
AME 500B (Spring 2017) 
Independent Study
AME 599 (Spring 2017) 
Adv Engineering Analysis
AME 500A (Fall 2016) 
Compressible Fld Dynamic
AME 536C (Fall 2016) 
Independent Study
AME 599 (Fall 2016)
Scholarly Contributions
Books
 Neyland, V. Y., & Tumin, A. (1991). Aerothermodynamics of Aerospace Vehicles. Zhukovski: Applied Research, Moscow Institute of Physics and Technology.
 Zhigulev, V. N., & Tumin, A. (1987). Origin of Turbulence. Novosibirsk: Nauka.More infotranslated as NASA TT20340
Journals/Publications
 Fedorov, A., & Tumin, A. (2017). Receptivity of HighSpeed Boundary Layers to Kinetic Fluctuations. AIAA J, 55(7), 23352348.
 Chiquete, C., & Tumin, A. (2012). Stability of detonation in a circular pipe with porous walls. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 370(1960), 668688.More infoPMID: 22213664;Abstract: A stability analysis is carried out taking into account slightly porous walls in an idealized detonation confined to a circular pipe. The analysis is carried out using the normalmode approach and corrections are obtained to the underlying impenetrable wall case results to account for the effect of the slight porosity. The porous walls are modelled by an acoustic boundary condition for the perturbations linking the normal velocity and the pressure components and thus replacing the conventional nopenetration boundary condition at the wall. This new boundary condition necessarily complicates the derivation of the stability problem with respect to the impenetrable wall case. However, exploiting the expressly slight porosity, the modified temporal stability can be determined as a twopoint boundary value problem similar to the case of a nonporous wall. © 2012 The Royal Society.
 Lifshitz, Y., Degani, D., & Tumin, A. (2012). Study of discrete modes branching in highspeed boundary layers. AIAA Journal, 50(10), 22022210.More infoAbstract: The branching of discrete modes in highspeed boundary layers is investigated using parabolized stability equations. The fast and slow discrete modes associated with the fast and slow acoustic modes, respectively, are considered in highspeed boundary layers over adiabatic and cooled walls. Whereas the conventional linear stability theory approach leads to singular behavior in the vicinity of the fastmode synchronization with the entropy and vorticity modes, the parabolized stability equation results do not reveal singular behavior of the solution and are consistent with the available direct numerical simulations of perturbations in highspeed boundary layers. Also, the parabolized stability equation results do not reveal a singular behavior in the vicinity of the point of synchronism of the slow and fast discrete modes. Copyright © 2012 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.
 Fedorov, A., & Tumin, A. (2011). HighSpeed BoundaryLayer instability: Old terminology and a new framework. AIAA Journal, 49(8), 16471657.More infoAbstract: The discrete spectrum of disturbances in highspeed 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 highspeed 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. Multiplemode considerations and direct numerical simulations are needed to clarify this issue. © 2011 by the American Institute of Aeronautics and Astronautics, Inc.
 Tumin, A., Wang, X., & Zhong, X. (2011). Numerical simulation and theoretical analysis of perturbations in hypersonic boundary layers. AIAA Journal, 49(3), 463471.More infoAbstract: Direct numerical simulations on the receptivity of hypersonic boundary layers over a flat plate and a sharp wedge were carried out with twodimensional periodicintime wall blowingsuction introduced into the flow through a slot. The freestream Mach numbers were equal to 5.92 and 8 in the cases of the adiabatic flat plate and sharp wedge, respectively. The perturbation flowfield was decomposed into normal modes with the help of the multimode decomposition technique based on the spatial biorthogonal eigenfunction system. The decomposition allowed for the filtering out of stable and unstable modes hidden behind perturbations of another physical nature. © Copyright 2011 American Institute of Aeronautics and Astronautics, Inc.
 Shalaev, I., & Tumin, A. (2010). Initialvalue problem for perturbations of idealized detonations in circular pipes. Combustion Theory and Modelling, 14(1), 122.More infoAbstract: The initialvalue problem for perturbations of an idealized onereaction detonation in a circular pipe is solved using the Laplace transformin time, Fourier series in the azimuthal angle, and expansion into Bessel's functions of the radial variable. For each radial and azimuthal mode, the inverse Laplace transform can be presented as an expansion of the solution into the normal modes of discrete and continuous spectra. The dispersion relation for the discrete spectrumrequires solving the homogeneous ordinary differential equations for the adjoint system and evaluating an integral through the reaction zone. The solution of the initialvalue problem gives a tool for analysis of the flow receptivity to various types of perturbations in the reaction zone and in the quiescent gas. © 2010 Taylor & Francis.
 Tumin, A. (2009). Comment on "Interaction of small perturbations with shock waves" [Phys. Fluids 16, 4489 (2004)]. Physics of Fluids, 21(7).More infoAbstract: It is pointed out that linearized RankineHugoniot conditions in the noninertial reference frame attached to the instantaneous location of the shock front have the same form as the shock conditions derived in an inertial frame moving at the velocity of the shock at a given time instance. © 2009 American Institute of Physics.
 Lifshitz, Y., Degani, D., & Tumin, A. (2008). On the interaction of turbulent shear layers with harmonic perturbations. Flow, Turbulence and Combustion, 80(1), 6180.More infoAbstract: The problem of coherent perturbations in a turbulent shear layer is considered for the purpose of developing a mathematical model based on a triple decomposition that extracts the coherent components of random fluctuations. The governing equations for the mean and the coherent parts of flow are derived, assuming the eddyviscosity equivalence for the random part of flow, and solved by iterations to provide a coupled solution of the problem as a whole. Calculations agree well with experimental data in the upstream part of the layer where the meancoherent flow interaction is the most important. In this region, the interaction changes the mean flow velocity distribution in such a manner that the neutral stability curve is shifted upstream relative to its position in the undisturbed layer and the perturbation intensity decreases further downstream. Experiments show that the coherent waves suppress the turbulent Reynolds stress production downstream of this region, but the model fails to predict the layer spreading correctly probably due to an inadequate turbulence closure of the mean flow. For the case of a turbulent mixing layer, we suggest a new closure relation that takes into account this coherentrandom interaction. © 2007 Springer Science+Business Media B.V.
 Tumin, A. (2008). Comment: "Instability of isolated planar shock waves" [Phys. Fluids 19, 094102 (2007). Physics of Fluids, 20(2).More infoAbstract: It is shown that Erpenbeck's solution of the initialvalue problem for small perturbations in the presence of shocks [J. J. Erpenbeck, Phys. Fluids 5, 604 (1962); 5, 1181 (1962)] leads to a straightforward and simple method for analysis of rippled shocks as well. Particularly, the result for the ripple amplitude of a shock is the same as the result of Bates derived from an integral equation for the shock displacement function [J. W. Bates, Phys. Rev. E 69, 056313 (2004); Phys Fluids 19, 094102 (2007)]. © 2008 American Institute of Physics.
 Tumin, A. (2008). Nonparallel flow effects on roughnessinduced perturbations in boundary layers. Journal of Spacecraft and Rockets, 45(6), 11761184.More infoAbstract: The nonparallel flow effects on roughnessinduced perturbations in subsonic boundary layers over a flat plate are explored within the scope of the linearized NavierStokes equations. The receptivity problem is treated under the parallel flow approximation, and the nonparallel flow effects on the perturbations development are taken into account. The analysis is based on the method of multiple scales when intermodal exchange due to the nonparallel character of the flow is neglected. The nonparallel flow effects lead to lower velocity and higher temperature in the wake downstream from the hump. Copyright © 2008 by the American. Institute of Aeronautics and Astronautics, Inc. All rights reserved.
 Tumin, A. (2007). Initialvalue problem for small disturbances in an idealized onedimensional detonation. Physics of Fluids, 19(10).More infoAbstract: The solution of the initialvalue problem for linearized reactive Euler equations is presented as an expansion into modes of discrete and continuous spectra in the case of onedimensional perturbations. The result provides a convenient tool to predict initial amplitudes of the unstable modes depending on the initial perturbation. Examples of initial perturbations introduced into the quiescent gas and into the reaction zone are discussed. It is shown that the discrete spectrum stemming from the solution of the initialvalue problem for an idealized onereaction detonation is equivalent to the spectrum of the conventional normalmode approach. © 2007 American Institute of Physics.
 Tumin, A. (2007). Multidomain spectral collocation method for stability analysis of detonations. AIAA Journal, 45(9), 23562359.More infoAbstract: A study was conducted to demonstrate the effectiveness of the multidomain spectral collocation method used for analyzing the stability of detonations. The process of problem formulation also led to the development of equations, with constant steadystate variables within the induction zone. Researchers also applied a spectral collocation method, to analyze onedimensional detonations at finite activation energy. Researchers also considered onedimensional perturbations in a onedimensional flow of a gas that participated in a firstorder, irreversible reaction without mole and specific heat change, to simplify the spectral collocation method. The steady dimensionless detonation velocity was scaled, with the speed of sound in the gas. Researchers also used the multidomain spectral collocation method, to solve the eigenvalue problem, using two and three domains.
 Tumin, A. (2007). Threedimensional spatial normal modes in compressible boundary layers. Journal of Fluid Mechanics, 586, 295322.More infoAbstract: Threedimensional spatially growing perturbations in a twodimensional compressible 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 used in a decomposition of flow fields derived from computational studies when pressure, temperature, and all the velocity components, together with some of their derivatives, are available. The method can also be used if partial data are available when a priori information may be utilized in the decomposition algorithm. © 2007 Cambridge University Press.
 Tumin, A., Xiaolin, X., & Zhong, X. (2007). Direct numerical simulation and the theory of receptivity in a hypersonic boundary layer. Physics of Fluids, 19(1).More infoAbstract: Direct numerical simulation of receptivity in a boundary layer over a sharp wedge of halfangle 5.3 degrees is carried out with twodimensional perturbations introduced into the flow by periodicintime blowingsuction through a slot. The freestream Mach number is equal to 8. The perturbation flow field downstream from the slot is decomposed into normal modes with the help of the biorthogonal eigenfunction system. Filteredout amplitudes of two discrete normal modes and of the fast acoustic modes are compared with the linear receptivity problem solution. The examples illustrate how the multimode decomposition technique may serve as a tool for gaining insight into computational results. © 2007 American Institute of Physics.
 Zuccher, S., Shalaev, I., Tumin, A., & Reshotko, E. (2007). Optimal disturbances in the supersonic boundary layer past a sharp cone. AIAA Journal, 45(2), 366373.More infoAbstract: Optimal disturbances for the supersonic flow past a sharp cone are computed to assess the effects due to flow divergence. This geometry is chosen because previously published studies on compressible optimal perturbations for flat plate and sphere could not isolate the influence of divergence alone, as many factors characterized the growth of disturbances on the sphere (flow divergence, pressure gradient, centrifugal forces, and dependence of the edge parameters on the local Mach number). Flowdivergence effects result in the presence of an optimal distance from the cone tip for which the optimal gain is the largest possible, showing that divergence effects are stronger in the proximity of the cone tip. By properly reseating the gain, wave number, and streamwise coordinate due to the fact that the boundarylayer thickness on the sharp cone is √3 thinner than the one over the flat plate, it is found that both the gain and the wave number compare fairly well. Moreover, results for the sharp cone collapse into those for the flat plate when the initial location for the computation tends to the final one and when the azimuthal wave number is very large. Results show also that a cold wall enhances transient growth. Copyright © 2006 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.
 Forgoston, E., & Tumin, A. (2006). Threedimensional wave packets in a compressible boundary layer. Physics of Fluids, 18(10).More infoAbstract: A threedimensional wave packet generated by a local disturbance in a twodimensional hypersonic boundary layer flow is studied with the aid of the previously solved initialvalue problem. The solution to this problem can be expanded in a biorthogonal eigenfunction system as a sum of modes consisting of continuous and discrete spectra of temporal stability theory. A specific disturbance consisting of an initial temperature spot is considered, and the receptivity to this initial temperature spot is computed for both the twodimensional and threedimensional cases. Using previous analysis of the discrete and continuous spectrum, the inverse Fourier transform is computed numerically. The twodimensional inverse Fourier transform is calculated for two discrete modes: Mode F and Mode S. The Mode S result is compared with an asymptotic approximation of the Fourier integral, which is obtained using the Gaussian model as well as the method of steepest descent. It is shown that the method of steepest descent provides an excellent approximation to the more computationally intensive numerical evaluation of the inverse Fourier transform. Additionally, the threedimensional inverse Fourier transform is found using an asymptotic approximation of the Fourier integral. A main feature of the resulting threedimensional wave packet is its twodimensional nature, which arises from an association of Mode S with Mack's second mode. © 2006 American Institute of Physics.
 Tumin, A. (2006). Biorthogonal eigenfunction system in the tripledeck limit. Studies in Applied Mathematics, 117(2), 165190.More infoAbstract: The solutions of receptivity problems for a periodicintime actuator placed on the wall in a twodimensional boundary layer and for a twodimensional hump are discussed within the scope of the biorthogonal eigenfunction expansion technique in the limit of high Reynolds number when the tripledeck scaling is imposed. It is shown that the solutions obtained with the help of the biorthogonal eigenfunction system are equivalent to the solutions derived within the scope of the tripledeck theory. © 2006 by the Massachusetts Institute of Technology.
 Zuccher, S., Tumin, A., & Reshotko, E. (2006). Parabolic approach to optimal perturbations in compressible boundary layers. Journal of Fluid Mechanics, 556, 189216.More infoAbstract: Optimal perturbations in compressible, nonparallel boundary layers are considered here. The flows past a flat plate and past a sphere are analysed. The governing equations are derived from the linearized NavierStokes equations by employing a scaling that relies on the presence of streamwise vortices, which are wellknown for being responsible for the 'liftup' effect. Consequently, the energy norm of the inlet perturbation encompasses the wallnormal and spanwise velocity components only. The effect of different choices of the energy norm at the outlet is studied, testing full (all velocity components and temperature) and partial (streamwise velocity and temperature only) norms. Optimal perturbations are computed via an iterative algorithm completely derived in the discrete framework. The latter simplifies the derivation of the adjoint equations and the coupling conditions at the inlet and outlet. Results for the flat plate show that when the Reynolds number is of the order of 103, a significant difference in the energy growth is found between the cases of full and partial energy norms at the outlet. The effect of the wall temperature is in agreement with previous parallelflow results, with cooling being a destabilizing factor for both flat plate and sphere. Flow divergence, which characterizes the boundary layer past the sphere, has significant effects on the transient growth phenomenon. In particular, an increase of the sphere radius leads to a larger transient growth, with stronger effects in the vicinity of the stagnation point. In the range of interesting values of the Reynolds number that are typical of wind tunnel tests and flight conditions for a sphere, no significant role is played by the wallnormal and streamwise velocity components at the outlet. © 2006 Cambridge University Press.
 Forgoston, E., & Tumin, A. (2005). Initialvalue problem for threedimensional disturbances in a compressible boundary layer. Physics of Fluids, 17(8), 114.More infoAbstract: An initialvalue problem is formulated for a threedimensional wave packet in a compressible boundary layer flow. The problem is solved using a Laplace transform with respect to time and Fourier transforms with respect to the streamwise and spanwise coordinates. The solution can be presented as a sum of modes consisting of continuous and discrete spectra of temporal stability theory. Two discrete modes, known as mode S and mode F, are of interest in highspeed flows since they may be involved in a laminarturbulent transition scenario. The continuous and discrete spectrum are analyzed numerically for a hypersonic flow with Mach number M=5.6. The following features are revealed: (1) The synchronism of mode S with acoustic waves at a streamwise wave number α1;→0 is primarily twodimensional; (2) at high angles of disturbance propagation, mode F is no longer synchronized with entropy and vorticity waves; (3) at high angles of disturbance propagation, the synchronism between mode S and mode F is not accompanied by a mode S instability, and at even higher angles of disturbance propagation, mode S and mode F are not synchronized. © 2005 American Institute of Physics.
 Tumin, A., & Reshotko, E. (2005). Receptivity of a boundarylayer flow to a threedimensional hump at finite Reynolds numbers. Physics of Fluids, 17(9), 18.More infoAbstract: The receptivity of boundarylayer flow to a threedimensional hump (an array of humps) at a finite Reynolds number is solved with the help of an expansion of the linearized solution of NavierStokes equations into the biorthogonal eigenfunction system. There are two counterrotating vortices behind the roughness element that bring the highspeed fluid down into the wake region. Depending on the geometry, two relatively highspeed streaks could be observed in the wake downstream from the hump. The results for the flowfield structure are in qualitative agreement with available computational data. The quantitative discrepancy is attributed to the nonlinear character of the receptivity mechanism at the parameters considered in the computational studies. © 2005 American Institute of Physics.
 Zuccher, S., Tumin, A., & Reshotko, E. (2005). Optimal disturbances in compressible boundary layers  Complete energy norm analysis. 4th AIAA Theoretical Fluid Mechanics Meeting.More infoAbstract: In the present work we revise results of transient growth in compressible boundary layers (flat plate and sphere) to consider the complete Mack energy norm at the outlet, without the assumption that the outflow perturbation is comprised solely of streaky structures. Optimal perturbations are still in the form of counterrotating streamwise vortices and this justifies the choice of the scaling in the governing equations. A strong effect of the complete (full) energy norm at the outlet is found for the flat plate in supersonic regimes. No significant effects of the choice of the outlet norm can be appreciated for the sphere, in the range of parameters that are relevant to wind tunnel testing or flight conditions. © 2005 by the authors.
 Fedorov, A., & Tumin, A. (2004). Evolution of disturbances in entropy layer on blunted plate in supersonic flow. AIAA Journal, 42(1), 8994.More infoAbstract: Linear and nonlinear stability analyses of the entropy layer over a blunted plate are discussed. Results of the linear stability theory are compared with the direct numerical solution of the Euler equations when a disturbance of prescribed frequency is imposed on the mean flow. A solver of the Euler equations based on the spacetime conservation element/solution element method predicts linear and nonlinear dynamics of unstable disturbances with high accuracy. The nonlinear effect demonstrates a trend to saturation of the entropylayer disturbances.
 Gaydos, P., & Tumin, A. (2004). Multimode decomposition in compressible boundary layers. AIAA Journal, 42(6), 11151121.More infoAbstract: Twodimensional spatially growing perturbations in a twodimensional compressible boundary layer are considered within the scope of linearized NavierStokes equations in a quasiparallel flow approximation. Because a spatially growing solution can be expanded into a biorthogonal eigenfunction system, the latter can be utilized for decomposition of flowfields derived from computational studies when pressure, temperature, 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.
 Reshotko, E., & Tumin, A. (2004). Role of Transient Growth in RoughnessInduced Transition. AIAA Journal, 42(4), 766770.More infoAbstract: Surface roughness can have a profound effect on boundarylayer transition. However, the mechanisms responsible for transition with threedimensional distributed roughness have been elusive. Various TollmienSchlichtingbased mechanisms have been investigated in the past but have been shown not to be applicable. More recently, the applicability of transient growth theory to roughnessinduced transition has been studied. A model for roughnessinduced 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.
 Tumin, A., & Reshotko, E. (2004). The problem of boundarylayer flow encountering a threedimensional hump revisited. AIAA Paper, 38063822.More infoAbstract: The receptivity of boundarylayer flow to a threedimensional hump (an array of humps) at a finite Reynolds number is solved with the help of an expansion of the solution of linearized NavierStokes equations into the biorthogonal eigenfunction system. Results of the tripledeck theory are revisited, and it is shown that there is qualitative agreement of the flowfield structure with the finite Reynolds number case. There are a pair of counterrotating vortices behind the roughness element that bring the highspeed fluid down into the wake region. To analyze the flowfield far downstream of the roughness elements, the receptivity problem solution is used as inflow into a marching procedure of solving the linearized boundarylayer equations. The results reveal that there is a reversal of the streamwise velocity perturbation. Downstream from the point of the reversal, the perturbations possess transient growth. Comparison with the optimal disturbances originating at the point of reversal demonstrates that the velocity field in the wake is close to the optimal one. This means that the optimal perturbations are realizable in experiment with an array of humps placed on the wall.
 Fedorov, A., & Tumin, A. (2003). Initialvalue problem for hypersonic boundarylayer flows. AIAA Journal, 41(3), 379389.More infoAbstract: An initialvalue problem is analyzed for a twodimensional wave packet induced by a local twodimensional disturbance in a hypersonic boundary layer. The problem is solved using Fourier transform with respect to the streamwise coordinate and Laplace transform with respect to time. The temporal continuous spectrum is revisited, and the uncertainty associated with the overlapping of continuousspectrum branches is resolved. It is shown that the discrete spectrum's dispersion relationship is nonanalytic because of the synchronization of the first 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 verification of this receptivity mechanism is discussed.
 Tumin, A. (2003). Multimode decomposition of spatially growing pertubations in a twodimensional boundary layer. Physics of Fluids, 15(9), 25252540.More infoAbstract: Threedimensional spatially growing perturbations in a twodimensional incompressible boundary layer are considered within the scope of linearized NavierStokes 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 illposed, but the method might be applied under additional assumptions about the perturbations. © 2003 American Institute of Physics.
 Tumin, A. (2003). Optimal streamwise vortices intended for supersonic mixing enhancement. AIAA Journal, 41(8), 15421546.More infoAbstract: The theory of optimal, spatially growing disturbances is applied to a compressible turbulent mixing layer. It is suggested that the weak, steady disturbances generated by tiny vortex generators be considered as in the tripledecomposition method. The linearized equations for the disturbances are closed via the eddy viscosity model with a turbulent Prandtl number equal to one. The optimization procedure is formulated in terms of Mack's energy norm. As an example, a parallel flow with a hyperbolic tangent velocity profile and a temperature profile following from the CroccoBusemann relation is analyzed. Theoretical results indicate that the optimization procedure leads to streamwise vortices and that it is consistent with experimental findings. The theory provides the possibility of estimating the spacing of the tiny vortex generators placed on the splitter plate. Results show that an increase in the convective Mach number is accompanied by an increase in the transient growth effect.
 Tumin, A. (2003). The spatial stability of natural convection flow on inclined plates. Journal of Fluids Engineering, Transactions of the ASME, 125(3), 428437.More infoAbstract: The spatial stability of a natural convection flow on upwardfacing, heated, inclined plates is revisited. The eigenvalue problem is solved numerically employing two methods: the collocation method with Chebyshev polynomials and the fourthorder RungeKutta method. Two modes, traveling waves and stationary longitudinal vortices, are considered. Previous theoretical models indicated that nonparallel effects of the mean flow are significant for the vortex instability mode, but most of them ignored the fact that the eigenfunctions are dependent on the streamwise coordinate as well, in the present work, the method of multiple scales is applied to take the nonparallel flow effects into consideration. The results demonstrate the stabilizing character of the nonparallel flow effects. The vortex instability mode is also considered within the scope of partial differential equations. The results demonstrate dependence of the neutral point on the initial conditions but, farther downstream, the results collapse onto one curve. The marching method is compared with the quasiparallel normal mode analysis and with theoretical results including correction to nonparallel flow effects. The marching method provides better agreement of theoretical and experimental growth rates.
 Tumin, A., & Ashpis, D. E. (2003). Optimal disturbances in boundary layers subject to streamwise pressure gradient. AIAA Journal, 41(11), 22972300.More infoAbstract: The optimal disturbances/streamwise vortices associated with the transient growth mechanism was performed for boundary layers in the presence of a streamwise pressure gradient. The theory provided the optimal spacing of the control elements in the spanwise direction and their placement in the streamwise direction.
 Tumin, A., & Reshotko, E. (2003). Optimal Disturbances in Compressible Boundary Layers. AIAA Journal, 41(12), 23572363.More infoAbstract: The problem of transient growth in compressible boundary layers is considered within the scope of partial differential equations including the nonparallel flow effects. As follows from previous investigations, the optimal disturbances correspond to steady counterrotating streamwise vortices. The corresponding scaling of the perturbations leads to the governing equations for a Görtlertype of instability, with the Görtler number equal to zero. The iteration procedure employs back and forth marching solutions of the adjoint and original systems of equations. At low Mach numbers, the results agree with those for Blasius boundarylayer flows. In the case of parallel flows, the method leads to the same results obtained for compressible flows within the scope of linearized NavierStokes equations. It is shown that there is an optimal spacing of the streamwise vortices and an optimal location for their excitation.
 Reau, N., & Tumin, A. (2002). Harmonic perturbations in turbulent wakes. AIAA Journal, 40(3), 526530.More infoAbstract: A theoretical model of harmonic perturbations in far turbulent wakes is considered. The proposed model is based on the triple decomposition method. It is assumed that the instantaneous velocities and pressures consist of three distinctive components: the mean (timeaveraged), the coherent (phaseaveraged), and the random (turbulent) motion. The interaction between incoherent turbulent fluctuations and largescale coherent disturbances is incorporated by means of a Newtonian eddy viscosity model. For highamplitude perturbations, the nonlinear feedback to the mean flow is taken into account by means of the coherent Reynolds stresses. The equations for the mean flow are coupled with the linearized equations for the disturbances, taking into account the mean flow nonparallel effects. The model resolves uncertainties noted in previous theories and provides a correct comparison with available experimental data. The effect of the harmonic perturbations on the turbulent wake growth at high amplitudes is investigated as well.
 Reau, N., & Tumin, A. (2002). On harmonic perturbations in a turbulent mixing layer. European Journal of Mechanics, B/Fluids, 21(2), 143155.More infoAbstract: A theoretical model of harmonic perturbations in a turbulent mixing layer is proposed. The model based on the triple decomposition method. It is assumed that the instantaneous velocities and pressure consist of three distinctive components: the mean (time average), the coherent (phase average), and the random (turbulent) motion. The interaction between incoherent turbulent fluctuations and largescale coherent disturbances is incorporated by the Newtonian eddy viscosity model. A slight divergence of the flow is also taken into account, and the results are compared with experimental data. For a high amplitude of the perturbations, the nonlinear feedback to the mean flow is taken into account by means of the coherent Reynolds stresses. The results reveal the possibility of a negative spreading rate of the mixing layer. A simultaneous consideration of the mean flow divergence and nonlinear selfinteraction results in Landaulike amplitude equations. It is observed that the nonlinear term in the amplitude equation is not significant at the levels of amplitude considered. The velocity disturbance profiles of the second harmonic are also presented and, at lowlevel amplitude, they are in good agreement with experiments.
 Reshotko, E., & Tumin, A. (2001). Spatial theory of optimal disturbances in a circular pipe flow. Physics of Fluids, 13(4), 991996.More infoAbstract: 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. © 2001 American Institute of Physics.
 Tumin, A. (2001). A model of spatial algebraic growth in a boundary layer subjected to a streamwise pressure gradient. Physics of Fluids, 13(5), 15211523.More infoAbstract: Steady threedimensional disturbances are considered within the scope of the linearized boundary layer equations. The disturbances are assumed to be periodic in the spanwise direction, and their spanwise scale is assumed to be much larger than the boundary layer thickness. The model is an extension of Luchini's theory [J. Fluid Mech. 327, 101 (1996) to the FalknerSkan boundary layer. © 2001 American Institute of Physics.
 Tumin, A., & Reshotko, E. (2001). Spatial theory of optimal disturbances in boundary layers. Physics of Fluids, 13(7), 20972104.More infoAbstract: 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 TollmienSchlichting (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. © 2001 American Institute of Physics.
 Shapiro, I., Shtilman, L., & Tumin, A. (1999). On stability of flow in an annular channel. Physics of Fluids, 11(10), 29842992.More infoAbstract: Linear and nonlinear stability of a flow between the walls of two coaxial cylinders has been investigated. The linear analysis has been carried out within the framework of the temporal linear stability theory. The eigenvalue map has been obtained using the collocation method based on Chebyshev polynomials. The nonlinear analysis is based on a novel parallel code for DNS (direct numerical simulations) of a coaxial pipe flow with periodic boundary conditions in the streamwise direction. It is shown that the major source of finite amplitude instability is associated with the interaction of threedimensional disturbances and streamwise rolls. © 1999 American Institute of Physics.
 Tumin, A. (1998). Subharmonic resonance in a laminar wall jet. Physics of Fluids, 10(7), 17691771.More infoAbstract: Receptivity and stability of a twodimensional laminar wall jet is considered. The wall jet and the disturbance source examined in a direct numerical simulation by S. Wernz and H. F. Fasel (AIAA Paper 960079) are chosen for the analysis. The disturbances are introduced by blowing and suction through a slot in the wall. The disturbances of two frequencies (28 and 56 Hz) are considered. Because two eigenmodes may be unstable in the walljet flow, both of them are taken into account for each frequency. Therefore, the linear receptivity problem is solved for two pairs of eigenmodes, and their development and interaction downstream from the source are analyzed. The analysis allows one to explain the behavior of the fundamental and subharmonic disturbances observed in the numerical simulation. © 1998 American Institute of Physics.
 Tumin, A., & Aizatulin, L. (1997). Instability and receptivity of laminar wall jets. Theoretical and Computational Fluid Dynamics, 9(1), 3345.More infoAbstract: Results of eigenvalue analysis based on global and local eigenvalue considerations are presented. A collocation method with the Chebyshev polynomial approximation has been used for the global eigenvalue analysis. The results explain the appearance of a second unstable mode. In the case of real frequencies with Reynolds number R < 381 there is only one unstable mode. This mode coalesces at R ≈ 381 with a stable mode. At R > 381 they become separated by interchannging their branches, then the second unstable mode occurs. The receptivity problem has been considered with respect to perturbations emanating from a wall. The results illustrate that highfrequency modes have a stronger response than lowfrequency modes. It is shown that the method of expansion in a biorthogonal eigenfunction system and the method used by Ashpis and Reshotko are equivalent with regard to the receptivity problem solution.
 Likhachev, O., & Tumin, A. (1996). Stability of a compressible laminar walljet with heat transfer. Journal of Fluids Engineering, Transactions of the ASME, 118(4), 824828.More infoAbstract: The flow of a plane, laminar, subsonic perfect gas wall jet with heat transfer through the wall was investigated theoretically. For the case under consideration the entire surface was maintained at a constant temperature which differed from the temperature of the ambient gas. The velocity and temperature distribution across the flow were calculated for a variety of temperature differences between the ambient gas and the surface. The boundary layer equations representing these flows were solved by using the IllingworthStewartson transformation, thus extending the classical Glauert's solution to a thermally nonuniform flow. The effects of heat transfer on the linear stability characteristics of the wall jet were assessed by making the local parallel flow approximation. Two kinds of unstable eigenmodes coexisting at moderate Reynolds numbers are significantly affected by the heat transfer. The influence of cooling or heating on the stability of the flow was expected in view of the experience accumulated in incompressible boundary layers, i.e. heating destabilizes and cooling stabilizes the flows. Cooling of the wall affects the small scale disturbances more profoundly, contrary to the results obtained for the large scale disturbances.
 Tumin, A. (1996). Nonlinear interaction of wave trains in a supersonic boundary layer. Physics of Fluids, 8(9), 25522554.More infoAbstract: Nonlinear interaction of disturbances caused by a harmonic point source in supersonic boundary layer over a flat plate is considered. It is assumed that wave lengths are much less than a characteristic scale of nonlinear interaction and the averaging technique on the intermediate scales may be used. The wave trains are considered as sums of narrow wave packets. The parameters of flow and harmonic point source are chosen in accordance with the experiments carried out by Kosinov et al. [Nonlinear Instability of Nonparallel Flows (Springer, New York, 1994)]. © 1996 American Institute of Physics.
 Tumin, A. (1996). Receptivity of pipe Poiseuille flow. Journal of Fluid Mechanics, 315, 119137.More infoAbstract: The receptivity problem is considered for pipe flow with periodic blowsuction through a narrow gap in the pipe wall. Axisymmetric and nonaxisymmetric modes (1, 2, and 3) are analysed. The method of solution is based on global eigenvalue analysis for spatially growing disturbances in circular pipe Poiseuille flow. The numerical procedure is formulated in terms of the collocation method with the Chebyshev polynomials application. The receptivity problem is solved with an expansion of the solution in a biorthogonal eigenfunction system, and it was found that there is an excitation of many eigenmodes, which should be taken into account. The result explains the nonsimilar character of the amplitude distribution in the downstream direction that was observed in experiments.
 Tumin, A., Amitay, M., Cohen, J., & Zhou, M. D. (1996). A normal multimode decomposition method for stability experiments. Physics of Fluids, 8(10), 27772779.More infoAbstract: A method for decomposition of an experimental signal that contains several normal modes with a prescribed frequency, is proposed The method is based on an expansion of solutions for smallamplitude spatially growing disturbances in a biorthogonal eigenfunction system. It is shown that data of one velocity component at a given crossstream section of the flow are in some cases, sufficient for the decomposition. The method is applied successfully to analyze smallamplitude periodic disturbances in a transitional walljet flow. © 1996 American Institute of Physics.
 Likhachev, O., & Tumin, A. (1995). Stability of a compressible laminar walljet with heat transfer. American Society of Mechanical Engineers, Fluids Engineering Division (Publication) FED, 224, 2530.More infoAbstract: The flow of a plane, laminar, subsonic perfect gas wall jet with heat transfer through the wall was investigated theoretically. For the case under consideration the entire surface was maintained at a constant temperature which differed from the temperature of the ambient gas. The velocity and temperature distribution across the flow were calculated for a variety of temperature differences between the ambient gas and the surface. The boundary layer equations representing these flows were solved by using the IllingworthStewartson transformation, thus extending the classical Glauert's solution to a thermally nonuniform flow. The effects of heat transfer on the linear stability characteristics of the wall jet were assessed by making the local parallel flow approximation. Two kinds of unstable eigenmodes coexisting at moderate Reynolds numbers are significantly affected by the heat transfer. The influence of cooling or heating on the stability of the flow was expected in view of the experience accumulated in incompressible boundary layers, i.e. heating destabilizes and cooling stabilizes the flows. Cooling of the wall affects the small scale disturbances more profoundly, contrary to the results obtained for the large scale disturbances.
 Tumin, A. (1995). Threewave nonlinear interaction in a threedimensional compressible boundary layer. International Journal of NonLinear Mechanics, 30(5), 661671.More infoAbstract: A perturbation method for analysis of nonlinear wave interaction in threedimensional compressible boundary layer is developed. The method is based on a biorthogonal eigenfunction system for threedimensional compressible boundary layers. It is assumed that characteristic space and time scales of the disturbances are much less than space and time scales of nonlinear development and an averaging technique in intermediate scales may be applied. The method is a generalization of Zelman's results for twodimensional incompressible boundary layer. As an example, a threewave interaction is considered. A numerical example for socalled Tollmien  Schlichting wave interaction shows the possibility of amplification for rather broad packets without an exact resonance synchronizm for threewave interaction. © 1995.
 Tumin, A. M. (1988). THREEDIMENSIONAL PACKET OF INSTABILITY WAVES IN A SUPERSONIC BOUNDARY LAYER.. Izvestia Sibirskogo otdelenia Akademii nauk SSSR. Seria tehniceskih nauk, 1416.More infoAbstract: Results of numerical calculations of a threedimensional packet of instability waves in a supersonic bombary layer on a flat plate are prescuted. These results are compared with experimental data obtained in a supersonic wind tunnel.
 Tumin, A. M., & Chernov, Y. (1988). Asymptotic analysis of flow instability in a compressible boundary layer on a curved surface. Journal of Applied Mechanics and Technical Physics, 29(3), 390395.
 Tumin, A. M., & Fedorov, A. V. (1984). Instability wave excitation by a localized vibrator in the boundary layer. Journal of Applied Mechanics and Technical Physics, 25(6), 867873.
 Tumin, A. M. (1983). Excitation of TollmienSchlichting waves in the boundary layer by the vibrating surface of an infinite span delta wing. Journal of Applied Mechanics and Technical Physics, 24(5), 670674.
 Tumin, A. M., & Fedorov, A. V. (1983). Excitation of Instability Waves in the Boundary Layer on a Vibrating Surface.. PMTF, Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, 139.More infoAbstract: An analysis of instability waves excitation in a two dimensional weakly inhomogeneous boundary layer on a vibrating surface is presented. Asymptotic expression is obtained for the amplitude of a TollusinSchlichting instability wave in the resonance case when the frequency and length of the wave that characterize the vibrating surface coincide with the corresponding parameters of the unstable excitation at the point of stability loss. Numerical results are presented for a vibrating heated film the flow of a compressible gas.
 Tumin, A. M., & Fedorov, A. V. (1983). Excitation of unstable waves in boundary layer on a vibrating surface. Journal of Applied Mechanics and Technical Physics, 24(3), 348354.
 Tumin, A. M., & Fedorov, A. V. (1983). Spatial growth of disturbances in a compressible boundary layer. Journal of Applied Mechanics and Technical Physics, 24(4), 548554.
 Zhigulev, V. N., Kirkinskiy, A. I., Sidorenko, N. V., & Tumin, A. M. (1980). MECHANISM OF SECONDARY INSTABILITY AND ITS ROLE IN THE GENERATION OF TURBULENCE.. Fluid mechanics. Soviet research, 9(2), 96113.More infoAbstract: The development of disturbances in the boundary layer after loss of stability is analyzed, concentrating on the threedimensional nonlinear perturbing wave. A mechanism of secondary instability of the perturbing wave in the boundary layer is suggested and the role of this mechanism in generation of turbulence is examined. The calculations made are compared with experimental data and a statistical approach to the problem of hydrodynamic instability to reveal the role of secondary instability in the development of flow turbulence.
 Zhigulev, V. N., Sidorenko, N. V., & Tumin, A. M. (1980). Generation of instability waves in a boundary layer by external turbulence. Journal of Applied Mechanics and Technical Physics, 21(6), 774778.
Proceedings Publications
 Luna, K., & Tumin, A. (2019, January). Receptivity of HighSpeed Boundary Layers in Binary Mixture of Gases to Kinetic Fluctuations,. In AIAA Paper 20191382.
 Wu, L., & Tumin, A. (2019, January). Receptivity of Turbulent Compressible Mixing Layers to Localized Energy Deposition,. In AIAA Paper 20191652.
 Edwards, L., & Tumin, A. (2018, January). Receptivity to Kinetic Fluctuations: A Multiple Scales Approach. In AIAA Paper 20181075.
 Edwards, L., & Tumin, A. (2017, 1). Real gas effects on receptivity to kinetic fluctuations. In AIAA Paper 20170070.
 Edwards, L., & Tumin, A. (2017, 6). Analysis of Receptivity to Kinetic Fluctuations in the ReentryF Flight Experiment. In AIAA Paper 20173633.
 Fedorov, A., & Tumin, A. (2016, June). Receptivity of HighSpeed Boundary Layers to Kinetic Fluctuations. In AIAA Paper No. 20163191.
 Sivasubramanian, J., Tumin, A., & Fasel, H. F. (2016, June). The Reynolds Number Effect on Receptivity to a Localized Disturbance in a Hypersonic Boundary Layer. In AIAA Paper No. 20164246.
 Klentzman, J., & Tumin, A. (2014, September 812). The Second Mode in Highenthalpy Boundary Layers in Chemical nonequilibrium. In IUTAMABCM 8th Symposium on Laminar Turbulent Transition.
 Klentzman, J., & Tumin, A. (2013, January). Stability and receptivity of high speed boundary layers in oxygen. In AIAA Paper 20132882.More infoAbstract: The stability and receptivity of high speed boundary layers in binary mixtures of oxygen are investigated including chemical nonequilibrium effects. The analysis is conducted for twodimensional perturbations for both an inviscid and a viscous model and the results are compared to examine the impact of viscous effects. It is found that the viscosity effects stabilize the flow, and the main impact on temperature and mass fraction perturbations occurs in the viscous sublayer and in the critical layer. The biorthogonal eigenfunction system is used to study the receptivity of the boundary layers including real gas effects. The results demonstrate that the receptivity coeffcients have two maxima associated with the branching points in the discrete spectrum. The maxima become stronger when the wall temperature is low.
 Klentzman, J., Ulker, E., & Tumin, A. (2012, January). Projection of the solution of the linearized navierstokes equations in reacting high speed boundary layers onto discrete modes. In AIAA Paper 20123149.More infoAbstract: The problem of multimode decomposition of small perturbations in highspeed boundary layers in chemical nonequilibrium is addressed using the discretized adjoint approach. The solution of the linearized NavierStokes equations is considered in the quasiparallel flow approximation using the normal mode analysis, and the ordinary differential equations for the amplitude functions are discretized using fourthorder finite differences. The discretization leads to a system of linear algebraic equations in the form of the generalized eigenvalue problem. It is straightforward to define the left eigenvectors (eigenvectors of the adjoint problem) and to formulate the biorthogonality condition for the discrete modes. Assuming that there is a complete system of eigenfunctions of the discrete and continuous spectra, the biorthogonality condition allows for the finding of the projection of a solution onto the discrete modes. The biorthogonality condition is also utilized for solving the receptivity problem with perturbations introduced at the wall. © 2012 by Jill Klentzman.
 Lifshitz, Y., Degani, D., & Tumin, A. (2012, January). Study of discrete modes branching in highspeed boundary layers. In AIAA Paper 20120919.More infoAbstract: The branching of discrete modes in highspeed boundary layers is investigated using the Parabolized Stability Equations (PSE). The fast and slow discrete modes associated with the fast and slow acoustic modes, respectively, are considered in highspeed boundary layers over adiabatic and cooled walls. Whereas the conventional Linear Stability Theory (LST) approach leads to singular behavior in the vicinity of the fast mode synchronization with the entropy and vorticity modes, the PSE results do not reveal singular behavior of the solution and are consistent with the available Direct Numerical Simulations (DNS) of perturbations in highspeed boundary layers. Also, the PSE results do not reveal a singular behavior in the vicinity of the point of synchronism of the slow and fast discrete modes. Copyright © 2012 by the authors.
 Rodríguez, D., Tumin, A., & Theofilis, V. (2011, January). Towards the foundation of a global modes concept. In AIAA Paper 20113603.More infoAbstract: A contribution is presented, intended to provide theoretical foundations for the ongoing efforts to employ global instability theory for the analysis of the classic boundarylayer flow, and address the associated issue of appropriate inflow/outflow boundary conditions to close the PDEbased global eigenvalue problem in open flows. Starting from a theoretically clean and numerically simple application, in which results are also known analytically and thus serve as a guidance for the assessment of the performance of the numerical methods employed herein, a sequence of issues is systematically built into the target application, until we arrive at one representative of open systems whose instability is presently addressed by global linear theory applied to open flows, the latter application being neither tractable theoretically nor straightforward to solve by numerical means. Experience gained along the way is documented. It regards quantification of the departure of the numerical solution from the analytical one in the simple problem, the generation of numerical boundary layers at artificially truncated boundaries, no matter how far the latter are placed from the region of highest flow gradients and, ultimately the impractically large number of (direct and adjoint) modes necessary to project an arbitrary initial perturbation and follow its temporal evolution by a global analysis approach, a finding which may question the purported robustness reported in the literature of the recovery of optimal perturbations as part of global analyses yielding underresolved eigenspectra. Copyright © 2011 by Daniel Rodriguez.
 Tumin, A. (2011, June). The biorthogonal eigenfunction system of linear stability equations: A survey of applications to receptivity problems and to analysis of experimental and computational results. In AIAA Paper 20113244.More infoAbstract: The concept of a biorthogonal eigenfunction system (BES) of linear stability equations has been utilized for receptivity problems in boundary layers, wall jets, pipe flow, and detonations. Other applications of the BSE include the analysis of experimental and computational studies of perturbations. In the present survey, applications of the BSE to receptivity problems and to the analysis of experimental and computational data illustrate the main features of the method. © 2011 by the author.
 Fedorov, A., & Tumin, A. (2010, January). Branching of discrete modes in highspeed boundary layers and terminology issues. In AIAA Paper 20105003.More infoAbstract: Discrete spectrum of disturbances in highspeed boundary layers is discussed with emphasis on singularities caused by synchronization of the normal modes. Numerical examples illustrate different spectrum structures and jumps from one structure to another with small variations of basicflow parameters. It is shown that this singular behavior is due to branching of the dispersion curves in the synchronization region. Depending on locations of the branch points, the spectrum contains one or two unstable modes. In this connection, the terminology used for instability of highspeed boundary layers is clarified. It is emphasized that the spectrum branching can cause significant difficulties in stability analyses based on traditional LST and PSE methods, which ignore the coupling between different modes. Multiplemode considerations are needed to resolve these difficulties and obtain uniformly valid solutions. © 2010 by thr authors.
 Tumin, A., Wang, X., & Zhong, X. (2010, January). Direct numerical simulation and theoretical analysis of perturbations in hypersonic boundary layers. In IUTAM Bookseries, 18, 427432.More infoAbstract: Direct numerical simulations of receptivity in a boundary layer over a flat plate and a sharp wedge were carried out with twodimensional perturbations introduced into the flow by periodicintime blowingsuction through a slot. The free stream Mach numbers are equal to 5.92 and 8 in the cases of adiabatic flat plate and sharp wedge, respectively. The perturbation flow field was decomposed into normal modes with the help of the multimode decomposition technique based on the spatial biorthogonal eigenfunction system. The decomposition allows filtering out the stable and unstable modes hidden behind perturbations having another physical nature. © 2010 Springer Science+Business Media B.V.
 Tumin, A., Wang, X., & Zhong, X. (2010, January). Numerical simulation and theoretical analysis on hypersonic boundarylayer receptivity to wall blowingsuction. In AIAA Paper 20100534.More infoAbstract: Direct numerical simulations on the receptivity of hypersonic boundary layers over a flat plate and a sharp wedge were carried out with twodimensional periodicintime wall blowingsuction introduced into the flow through a slot. The freestream Mach numbers are equal to 5.92 and 8 in the cases of adiabatic flat plate and sharp wedge, respectively. The perturbation flow field was decomposed into normal modes with the help of the multimode decomposition technique based on the spatial biorthogonal eigenfunction system. The decomposition allows for the filtering out of the stable and unstable modes hidden behind perturbations of another physical nature. © 2010 by the authours.
 Shalaev, I., & Tumin, A. (2009, January). Initialvalue problem for perturbations of idealized detonations in circular pipes. In AIAA Paper 20090438.More infoAbstract: The initialvalue problem for perturbations of an idealized onereaction detonation in a circular pipe is solved using the Laplace transform in time, Fourier series in the azimuthal angle, and expansion into Bessel's functions of the radial variable. For each radial and azimuthal mode, the inverse Laplace transform can be presented as an expansion of the solution into the normal modes of discrete and continuous spectra. The dispersion relation for the discrete spectrum requires solving the homogeneous ordinary differential equations for the adjoint system and evaluating an integral through the reaction zone. The solution of the initialvalue problem gives a convenient tool for analysis of the flow receptivity to various types of perturbations in the reaction zone and in the quiescent gas. Copyright © 2009 by A. Tumin.
 Chiquete, C., Shalaev, I., & Tumin, A. (2008, January). Receptivity of plane idealized onereaction detonation to threedimensional perturbations. In 46th AIAA Aerospace Sciences Meeting and Exhibit.More infoAbstract: Solution of the initialvalue problem for small threedimensional perturbations in a plane idealized onedimensional detonation is presented as an expansion into modes of discrete and continuous spectra in the case of threedimensional perturbations. The result provides a tool to predict initial amplitudes of the unstable modes depending on the initial perturbation. An example of initial perturbation introduced into the reaction zone is discussed. It is shown that the discrete spectrum stemming from the solution of the initialvalue problem for an idealized onereaction detonation is equivalent to the spectrum of the conventional normalmode approach. Copyright © 2008 by the authors.
 Tumin, A. (2008, January). Nonparallel flow effects on roughnessinduced perturbations in boundary layers. In 46th AIAA Aerospace Sciences Meeting and Exhibit.More infoAbstract: The nonparallel flow effects on roughnessinduced perturbations in subsonic boundary layers over a flat plate are explored within the scope of the linearized NavierStokes equations. The receptivity problem is treated under the parallel flow approximation, and the nonparallel flow effects on the perturbations development are taken into account. The analysis is based on the method of multiple scales when intermodal exchange due to the nonparallel character of the flow is neglected. The nonparallel flow effects lead to lower velocity and higher temperature in the wake downstream from the hump. Copyright © 2008 by the author.
 Chiquete, C., & Tumin, A. (2007, January). Biorthogonal eigenfunction system for supersonic inviscid flow past a flat plate. In Collection of Technical Papers  37th AIAA Fluid Dynamics Conference, 1, 452469.More infoAbstract: In order to illustrate the application of the biorthogonal eigenfunction system to receptivity problems and to multimode decomposition, a study case is chosen so that all steps of the method are accompanied by analytical solutions. The receptivity of an inviscid supersonic flow past a flat plate to localized periodicintime perturbations emanating from the wall is revisited within the scope of the method of the biorthogonal eigenfunction system for linearized Euler's equations. In addition, application of the biorthogonal eigenfunction system to projection of computational results onto modes of continuous spectra is shown.
 Tumin, A. (2007, January). Outlook for theoretical modeling of isolated roughnessinduced perturbations in turbulent boundary layers (invited). In Collection of Technical Papers  37th AIAA Fluid Dynamics Conference, 1, 602613.More infoAbstract: Theoretical modeling of perturbations introduced into turbulent boundary layers by an isolated hump can be carried out within the scope of the Reynoldsaveraged NavierStokes equations. The main obstacle is the closure problem. Having chosen a closure model that is suitable for the boundary layer, one may expect that it will not be adequate for the perturbations. As is known from asymptotic analysis of laminar boundary layers with roughness elements, the physics and the governing equations depend on the roughness length, width, and height. Even at small heights of roughness element, one should expect that the governing equations are nonlinear. These and other questions relevant to roughnessinduced perturbations are discussed in the lecture.
 Tumin, A. (2007, January). Stability of idealized onereaction detonations revisited. In Collection of Technical Papers  45th AIAA Aerospace Sciences Meeting, 17, 1196911996.More infoAbstract: It is shown that the discrete spectrum stemming from the solution of the initialvalue problem for an idealized onereaction detonation is equivalent to the spectrum of the conventional normalmode approach. Solution of the initialvalue problem also leads to a method for expanding the perturbation field into modes of discrete and continuous spectra. In addition, it gives a convenient tool to predict initial amplitudes of the unstable modes depending on the initial perturbation. A multidomain spectral collocation method is utilized for analysis of the discrete spectrum. The method provides an efficient tool for computation of the eigenvalue map in studies of the stability of detonations.
 Forgoston, E., Viergutz, M., & Tumin, A. (2006, January). Numerical and asymptotical study of threedimensional wave packets in a compressible boundary layer. In Collection of Technical Papers  36th AIAA Fluid Dynamics Conference, 2, 970985.More infoAbstract: A threedimensional wave packet generated by a local disturbance in a twodimensional hypersonic boundary layer flow is studied with the aid of the previously solved initialvalue problem. The solution can be presented as a sum of modes consisting of continuous and discrete spectra of temporal stability theory. Two discrete modes, known as Mode S and Mode F, are of interest in highspeed flows since they may be involved in a laminarturbulent transition scenario. The continuous and discrete spectra are analyzed numerically for a hypersonic flow. A comprehensive study of the spectrum is performed, including Reynolds number, Mach number and temperature factor effects. A specific disturbance consisting of an initial temperature spot is considered, and the receptivity to this initial temperature spot is computed for both the twodimensional and threedimensional cases. Using the analysis of the discrete and continuous spectrum, the inverse Fourier transform is computed numerically. The twodimensional inverse Fourier transform is calculated for Mode F and Mode S. The Mode S result is compared with an asymptotic approximation of the Fourier integral, which is obtained using a Gaussian model as well as the method of steepest descent. Additionally, the threedimensional inverse Fourier transform is found using an asymptotic approximation. Using the inverse Fourier transform computations, the development of the wave packet is studied, including effects due to Reynolds number, Mach number and temperature factor.
 Reshotko, E., & Tumin, A. (2006, January). Application of transient growth theory to bypass transition. In Solid Mechanics and its Applications, 129, 8393.More infoAbstract: Transient growth arises through the coupling between slightly damped, highly oblique (nearly streamwise) TS and Squire modes leading to algebraic growth followed by exponential decay in a region that is subcritical with respect to the TS neutral curve. A weak transient growth can also occur for two dimensional or axisymmetric modes since the OrrSommerfeld operator and its compressible counterpart are not selfadjoint, therefore their eigenfunctions are not strictly orthogonal. So transient growth is a candidate mechanism for many examples of bypass transition. The relevance to bypass transition is examined through the example of the hypersonic blunt body paradox. © 2006 Springer, Printed in the Netherlands.
 Tumin, A. (2006, January). Receptivity of compressible boundary layers to threedimensional wall perturbations. In Collection of Technical Papers  44th AIAA Aerospace Sciences Meeting, 18, 1344513463.More infoAbstract: Receptivity of compressible boundary layers to threedimensional perturbations at the wall is solved with the help of the biorthogonal eigenfunction system. The method allows computation of normal mode amplitudes of the discrete and continuous spectra. Considered examples of boundary layers over a flat plate include periodicintime blowing and suction through the wall at freestream Mach numbers M = 2 and 4.5, and an array of roughness elements at M = 0.5 and 2. Results with the periodicintime actuator are compared with earlier results that were obtained by direct numerical integrations in the complex plane of the streamwise wavenumber. The main input into perturbation outside the boundary layer is associated with the fast acoustic waves. Perturbations associated with the entropy and vorticity modes have their maxima at the edge of the boundary layer, and they decay far from the edge (toward the Mach wave). In the case of roughness elements placed on the wall, there are counterrotating streamwise vortices, a wake region downstream from the hump, and highspeed streaks at both sides of the hump. Temperature perturbation is positive in the wake region and negative on the sides. In the case of a cold wall, there is a lowtemperature streak above the wake region that is attributed to displacement of the cold gas by the hump. In the supersonic boundary layer, in addition to the perturbations inside the boundary layer, the perturbations also have relatively large amplitudes in the vicinity of the Mach waves generated by the roughness elements.
 Tumin, A. (2006, January). Threedimensional spatial normal modes in compressible boundary layers. In AIAA Paper 20061109, 18, 1341613444.More infoAbstract: Threedimensional spatially growing perturbations in a twodimensional compressible boundary layer are considered within the scope of linearized NavierStokes 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, 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 twodimensional. At high angles of disturbance propagation, the fast discrete mode is no longer synchronized with entropy and vorticity modes.
 Tumin, A., Degani, D., & Lifshitz, Y. (2006, January). Theoretical studies of harmonic perturbations in turbulent shear flows for purpose of flow control. In Collection of Technical Papers  3rd AIAA Flow Control Conference, 2, 10841100.More infoAbstract: A survey of progress in theoretical studies of coherent perturbations in turbulent shear flows (mixing layer, far wake, and boundary layer with adverse pressure gradient) is presented.
 Tumin, A., Wang, X., & Zhong, X. (2006, January). Direct numerical simulation and the theory of receptivity in a hypersonic boundary layer. In Collection of Technical Papers  44th AIAA Aerospace Sciences Meeting, 18, 1339713415.More infoAbstract: Direct numerical simulation of receptivity in a boundary layer over a sharp wedge of halfangle 5.3 degrees was carried out with twodimensional perturbations introduced into the flow by periodicintime blowingsuction through a slot. The free stream Mach number was equal to 8. The perturbation flow field downstream from the slot was decomposed into normal modes with the help of the biorthogonal eigenfunction system. Filteredout 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.
 Zuccher, S., Shalaev, I., Tumin, A., & Reshotko, E. (2006, January). Optimal disturbances in the supersonic boundary layer past a sharp cone. In Collection of Technical Papers  44th AIAA Aerospace Sciences Meeting, 18, 1347813493.More infoAbstract: Optimal disturbances for the supersonic flow past a sharp cone are computed in order to assess the effects due to flow divergence. This geometry is chosen because previously published studies on compressible optimal perturbations for flat plate and sphere did not allow to discriminate the influence of divergence alone, as many factors characterized the growth of disturbances on the sphere (flow divergence, centrifugal forces and dependence of the edge parameters on the local Mach number). Flowdivergence effects result in the presence of an optimal distance from the cone tip for which the optimal gain is the largest possible, showing that divergence effects are stronger in the proximity of the cone tip. By properly rescaling the gain, wavenumber and streamwise coordinate due to the fact that the boundary layer on the sharp cone is √3 thinner than the one over the flat plate, it is found that both the gain and the wavenumber compare fairly well. Moreover, results for the sharp cone collapse into those for the flat plate when the initial location for the computation tends to the final one and when the azimuthal wavenumber is very large. Results show also that a cold wall enhances transient growth.
 Ergin, F. G., Choudhari, M., Fischer, P., & Tumin, A. (2005, June). Transient growth: Experiments, DNS and theory. In 4th International Symposium on Turbulence and Shear Flow Phenomena, 2, 583588.More infoAbstract: Transient growth of linearly stable disturbances is believed to play an important role in the subcritical transition of laminar boundary layers and the selfsustained nature of boundary layer fluctuations in a fully turbulent flow. Prior work on transient growth has focused on identifying the optimum initial disturbances that result in maximum transient growth. This paper addresses the companion issue of receptivity of those disturbances, the mechanism that determines the actual magnitudes of transient growth that are realized in a given physical situation. A synergistic combination of experimental, computational, and theoretical approaches is used to quantify the flow receptivity to surface roughness in a Blasius boundary layer. Results reveal the nonoptimality of the transient growth factors involved as well as the sensitive dependence of flow perturbations to the geometric characteristics of the roughness distribution. Direct numerical simulations (DNS) are compared in detail with experimental results, results obtained from linear receptivity theory and optimal disturbance calculations. DNS shows good agreement with the experimental results. Differences between the linear theory and DNS are attributed to nonlinear receptivity mechanisms. Results also support the proposal by Fransson et al. (2004) that disagreement between optimal disturbances and experiments/DNS may be attributed to differences involving the wall normal location of the streamwise vortex initiating the transient growth.
 Forgoston, E., & Tumin, A. (2005, January). Threedimensional wave packet in a hypersonic boundary layer. In AIAA Paper 20050099, 1488114892.More infoAbstract: A threedimensional wave packet generated by a local disturbance in a hypersonic boundary layer flow is studied with the aid of the previously solved initialvalue problem. The solution to this problem can be expanded in a biorthogonal eigenfunction system as a sum of discrete and continuous modes. A specific disturbance consisting of an initial temperature spot is considered, and the receptivity to this initial temperature spot is computed for both the twodimensional and threedimensional cases. Using previous analysis of the discrete and continuous spectrum, we numerically compute the inverse Fourier transform. The twodimensional inverse Fourier transform is found for Mode S, and the result is compared with the asymptotic approximation of the Fourier integral. Due to the synchronism between Mode F and entropy/vorticity modes, it is necessary to deform the path of integration around the associated branch cut. Additionally, the inverse Fourier transform for a prescribed spanwise wave number is computed for threedimensional Mode S.
 Forgoston, E., Tumin, A., & Ashpis, D. E. (2005, January). Distributed blowing and suction for the purpose of streak control in a boundary layer subjected to a favorable pressure gradient. In AIAA Paper 20055195.More infoAbstract: An analysis of the optimal control by blowing and suction in order to generate streamwise velocity streaks is presented. The problem is examined using an iterative process that employs the Parabolized Stability Equations for an incompressible fluid along with its adjoint equations. In particular, distributions of blowing and suction are computed for both the normal and tangential velocity perturbations for various choices of parameters.
 Lifshitz, Y., Degani, D., & Tumin, A. (2005, June). Coherent perturbations in a turbulent boundary layer subjected to a highly adverse pressure gradient. In AIAA Paper 20054809.More infoAbstract: Coherent perturbations in a turbulent boundary layer subjected to a highly adverse pressure gradient are considered. The governing equations are the RANS equations with Boussinesqtype closure for the perturbation of background Reynolds stresses. Stability analysis of experimental velocity profiles in a boundary layer over a concave wall demonstrated agreement of theoretical and experimental data. A theoretical model taking into account the interaction of the coherent perturbations and the mean flow via the coherent Reynolds stresses is suggested. It is shown that coherent perturbations affect the mean velocity profile in such a way that the flow becomes more stable. © 2005 by the authors.
 Tumin, A. (2005, January). Biorthogonal eigenfunction system in the tripledeck limit. In AIAA Paper 20050524, 24992514.More infoAbstract: The solutions of receptivity problems for a periodicintime actuator placed on the wall in a twodimensional boundary layer and for a twodimensional hump are discussed within the scope of the biorthogonal eigenfunction expansion technique in the limit of high Reynolds number when the tripledeck scaling is imposed. It is shown that the solutions obtained with the help of the biorthogonal eigenfunction system are equivalent to the solutions derived within the scope of the tripledeck theory.
 Alvarez, J. O., Kerschen, E. J., & Tumin, A. (2004, January). A theoretical model for cavity acoustic resonances in subsonic flow. In Collection of Technical Papers  10th AIAA/CEAS Aeroacoustics Conference, 1, 532543.More infoAbstract: Acoustic resonances leading to high unsteady pressure levels may occur in flow past cavities. The resonance involves a coupling between the downstreampropagating instability wave on the shear layer spanning the open face of the cavity, and acoustic waves propagating within and external to the cavity. These elements of the disturbance field are coupled by the scattering processes that occur at the upstream and downstream ends of the cavity. We develop a theoretical prediction method that combines propagation models in the central region of the cavity with scattering models for the end regions. In our analyses of the scattering processes at the cavity ends, the squarecorner geometry is treated exactly, by a method employing the WienerHopf technique. The shear layer is approximated as a vortex sheet in the edge scattering analyses, but finite shearlayer thickness is accounted for in analyzing the propagation of the waves along the length of the cavity. The global analysis leads to a prediction for the resonant frequencies which has a form similar to the Rossiter formula, but contains no empirical constants. In addition to prediction of the frequency, our theory also determines the temporal growth or decay rate of each mode. Finally, our theory also predicts the influence of secondary feedback loops involving other components of the unsteady field. The theoretical predictions are validated by comparison with experimental data.
 Forgoston, E., & Tumin, A. (2004, June). Initialvalue problem for threedimensional disturbances in a hypersonic boundary layer. In AIAA Paper 20042243.More infoAbstract: An initialvalue problem is formulated for a threedimensional wave packet in a hypersonic boundary layer flow. The problem is solved using a Laplace transform with respect to time and Fourier transforms with respect to the streamwise and spanwise coordinates. The solution can be presented as a sum of modes consisting of continuous and discrete spectra of temporal stability theory. Two discrete modes, known as Mode S and Mode F, are of interest since they may be involved in a laminarturbulent transition scenario. The continuous and discrete spectrum are analyzed numerically, and the following features are revealed: (1) the synchronism of Mode S with acoustic waves at low wave number is primarily twodimensional; (2) at high angles of dis turbance propagation, Mode F is no longer synchronized with entropy and vorticity waves; (3) at high angles of disturbance propagation, the synchronism between Mode S and Mode F no longer leads to a Mode S instability, and at even higher angles of disturbance propagation, Mode S and Mode F are not synchronized. © 2004 by authors. Published by the American Institute of Aeronautics and Astronautics, Inc.
 Joshi, H., & Tumin, A. (2004, January). Centrifugal instability in a turbulent wall jet over a circular cylinder. In AIAA Paper No. 20041108, 69616972.More infoAbstract: The vortex instability mode is considered in a turbulent wall jet over a circular cylinder. The linear and nonlinear stability analyses are carried out within the scope of the parabolized stability equations. The results demonstrate that the neutral point may depend on the initial conditions because of the transient growth phenomenon but, farther downstream, the results collapse onto one curve. The marching method is compared with a quasiparallel normal mode analysis. It is shown that the growth rate is overestimated in the normal mode approach. The results indicate that the zero harmonic (with respect to the spanwise direction) generated due to nonlinear effects may shift the maximum measured velocity profile farther from the surface than it is in a walljet over a flat wall.
 Lifshitz, Y., Degani, D., & Tumin, A. (2004, January). On harmonic perturbations in a turbulent mixing layer. In AIAA Paper 20042653.More infoAbstract: The problem of coherent perturbations in a turbulent mixing layer is revisited. The governing equations for perturbations are derived from the RANS equations with a Boussinesqtype closure model. The proposed theoretical model utilizes the idea of separating the "fast" and the "slow" scales in a manner similar to the PSE concept. However, the equations for the amplitude functions are not simplified (not parabolized). Emulation of an actuator (oscillating flap) is introduced into the computational analysis via inhomogeneous boundary conditions. The actuator size and amplitude, the effect of coherent perturbation on the base flow, and the nonlinear effects in the development of the coherent signal are investigated. © 2004 by authors. Published by the American Institute of Aeronautics and Astronautics, Inc.
 Tumin, A., & Reshotko, E. (2004, June). Optimal disturbances in the boundary layer over a sphere. In AIAA Paper 20042241.More infoAbstract: Optimal steady perturbations in the boundary layer over a sphere are considered within the scope of the parabolized stability equations (PSE). The flow parameters at the edge of the boundary layer correspond to a highspeed freestream flow of a calorically perfect gas, and the boundary layer velocity and temperature profiles are obtained using the localsimilarity approximation. The governing PSE equations are derived from the linearized NavierStokes equations in spherical coordinates within the scope of the concept of optimal perturbations as streamwise vortices. Analysis of the transient growth phenomenon revealed that an increase of the sphere radius leads to an increase of the transient growth, and that the transient growth effect is stronger in the vicinity of the stagnation point. Similarly to previous results for compressible boundary layers, cooling of the wall destabilizes the boundary layer flow. © 2004 by the authors.
 Gaydos, P., & Tumin, A. (2003, January). Multimode decomposition in compressible boundary layers. In AIAA Paper 20033724.More infoAbstract: Twodimensional spatially growing perturbations in a twodimensional compressible boundary layer are considered within the scope of linearized NavierStokes equations. Because a spatially growing solution can be expanded into a biorthogonal eigenfunction system, the latter can be utilized for decomposition of flow fields derived from computational studies when pressure, temperature, 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. © 2003 by the authors. Published by the American Institute of Aeronautics and Astronautics, Inc.
 Kerschen, E. J., & Tumin, A. (2003, January). A theoretical model of cavity acoustic resonances based on edge scattering processes. In 41st Aerospace Sciences Meeting and Exhibit.More infoAbstract: Acoustic resonances leading to high unsteady pressure levels may occur in flow past cavities. The resonance involves a coupling between the downstreampropagating instability wave on the shear layer spanning the open face of the cavity, and acoustic waves propagating within the cavity. These elements of the disturbance field are coupled by the scattering processes that occur at the upstream and downstream ends of the cavity. We develop a theoretical prediction method that combines propagation models in the central region of the cavity with scattering models for the end regions. In our analyses of the scattering processes at the cavity ends, the squarecorner geometry is treated exactly, by a method employing the WienerHopf technique. The shear layer is approximated as a vortex sheet in the edge scattering analyses, but finite shearlayer thickness is accounted for in analyzing the propagation of the waves along the length of the cavity. The global analysis leads to a prediction for the resonant frequencies which has much in common with the Rossiter formula, but contains no empirical constants. The analysis also determines the temporal growth (or decay) rate of each mode, thereby providing the stability boundaries in parameter space. Comparisons are made with existing experimental data. © 2003 by the authors.
 Tumin, A., & Ashpis, D. E. (2003, January). Optimal disturbances in boundary layers subject to streamwise pressure gradient. In AIAA Paper 2003 4242.More infoAbstract: An analysis of the nonmodal growth of perturbations in a boundary layer in the presence of a streamwise pressure gradient is presented. The analysis is based on PSE equations for an incompressible fluid. Examples with FalknerSkan profiles indicate that a favorable pressure gradient decreases the nonmodal growth while an unfavorable pressure gradient leads to an increase of the amplification. It is suggested that the transient growth mechanism be utilized to choose optimal parameters of tripping elements on a lowpressure turbine (LPT) airfoil. As an example, a boundarylayer flow with a streamwise pressure gradient corresponding to the pressure distribution over a LPT airfoil is considered. It is shown that there is an optimal spacing of the tripping elements and that the transient growth effect depends on the starting point. The amplification is found to be small at the LPT.s very low Reynolds numbers, but there is a possibility to enhance the transient energy growth by means of wall cooling.
 Tumin, A., & Reshotko, E. (2003, January). Optimal disturbances in compressible boundary layers. In AIAA Paper No. 20030792.More infoAbstract: The problem of transient growth in compressible boundary layers is considered within the scope of partial differential equations. As follows from previous investigations, the optimal disturbances correspond to steady counterrotating streamwise vortices. The corresponding scaling of the perturbations leads to the governing equations for a Görtler type of instability, with the Görtler number equal to zero. The iteration procedure employs back and forth marching solutions of the adjoint and original systems of equations. At low Mach numbers, the results agree with those for Blasius boundarylayer flows. In the case of parallel flows, the method leads to the same results obtained for compressible flows within the scope of linearized NavierStokes equations. It is shown that there is an optimal spacing of the streamwise vortices and an optimal location for their excitation. © 2003 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.
 Fedorov, A., & Tumin, A. (2002, January). Evolution of disturbances in entropy layer on a blunted plate in supersonic flow. In AIAA Paper No. 20022847.More infoAbstract: Linear and nonlinear stability analyses of the entropy layer over a blunted plate are discussed. Results of the linear stability theory are compared with the direct numerical solution of the Euler equations when a disturbance of prescribed frequency is imposed on the mean flow. A solver of the Euler equations based on the SpaceTime CE/SE method predicts linear and nonlinear dynamics of unstable disturbances with high accuracy. A relatively coarse grid allows for simulation of nonlinear disturbances with the massflux amplitude up to 9%. The nonlinear effect demonstrates a trend to saturation of the entropy layer disturbances. © 2002 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.
 Reshotko, E., & Tumin, A. (2002, June). Investigation of the role of transient growth in roughnessinduced transition. In AIAA Paper No. 20022805.More infoAbstract: Surface roughness can have a profound effect on boundary layer transition. However, the mechanisms responsible for transition with threedimensional distributed roughness have been elusive. Various TS based mechanisms have been investigated in the past but have been shown not to be applicable. During the past year, the applicability of transient growth theory to roughness induced transition has been studied. A model for roughnessinduced transition is developed that makes use of computational results based on the spatial transient growth theory pioneered by the present authors. For zero pressure gradient flows, the resulting transition relation is reminiscent of the transition correlations found for slender hypersonic vehicles. For nosetip transition, the resulting transition relations reproduce the trends of the Reda and PANT data and carefully account for the separate roles of roughness and surface temperature level on the transition behavior. © 2002 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.
 Tumin, A. (2002, January). Optimal streamwise vortices intended for supersonic mixing enhancement. In AIAA Paper No. 2002 – 0728.More infoAbstract: The theory of optimal, spatially growing disturbances is applied to a compressible turbulent mixing layer. It is suggested that the weak steady disturbances generated by tiny vortex generators be considered as in the triple decomposition method. The linearized equations for the disturbances are closed via the eddy viscosity model with a turbulent Prandtl number equal to one. The optimization procedure is formulated in terms of Mack's energy norm. As an example, a parallel flow with a hyperbolic tangent velocity profile and a temperature profile following from the CroccoBusemann relation is analyzed. Theoretical results indicate that the optimization procedure leads to streamwise vortices, and it is consistent with experimental findings. The theory provides the possibility of estimating the spacing of the tiny vortex generators placed on the splitter plate. Results show that an increase in the convective Mach number is accompanied by an increase in the transient growth effect. © 2002 The American Institute of Aeronautics and Astronautics Inc. All rights reserved.
 Yang, H., & Tumin, A. (2002, January). A model of coherent disturbances in compressible mixing layers. In American Society of Mechanical Engineers, Fluids Engineering Division (Publication) FED, 257, 14371442.More infoAbstract: A theoretical model of harmonic perturbations in a compressible turbulent mixing layer is proposed. The model is based on the triple decomposition method. It is assumed that the instantaneous velocities, temperature, and pressure consist of three distinctive components: mean (timeaveraged), coherent (phaseaveraged), and random (turbulent) motion. The interaction between incoherent turbulent fluctuations and largescale coherent disturbances is incorporated by the Newtonian eddy viscosity model. The governing equations for the coherent disturbances have the same form as in laminar flow with substitution of the Reynolds number and the Prandtl number by their turbulent counterparts. A slight divergence of the flow is also taken into account. Theoretical results and comparison with experimental data reveal the significance of interaction between the coherent and random constituents of the flow.
 Yang, H., & Tumin, A. (2002, January). On harmonic perturbations in compressible mixing layers. In AIAA Paper No. 20022854.More infoAbstract: A theoretical model of harmonic perturbations in a compressible turbulent mixing layer is proposed. The model utilizes the triple decomposition method. It is assumed that the instantaneous velocities, temperature, and pressure consist of three distinctive components: mean (timeaveraged), coherent (phaseaveraged), and random (turbulent) motion. The interaction between incoherent turbulent fluctuations and largescale coherent disturbances is incorporated by the Newtonian eddy viscosity model. The governing equations for the coherent disturbances have the same form as in laminar flow with substitution of the Reynolds number and the Prandtl number by their turbulent counterparts. A slight divergence of the flow is also taken into account. Theoretical results and comparison with experimental data reveal the significance of interaction between the coherent and random constituents of the flow. The model is applied to the case of a mixing layer over a cavity in order to predict the resonance frequencies. © 2002 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.
 Fedorov, A., & Tumin, A. (2001, January). Initialvalue problem for hypersonic boundary layer flows. In AIAA Paper No. 20012781.More infoAbstract: An initialvalue problem is analyzed for a twodimensional wave packet induced by a local twodimensional disturbance in a hypersonic boundary layer. The problem is solved using Fourier transform with respect to the streamwise coordinate and Laplace transform with respect to time. It is shown that the solution can be presented as an expansion in the biorthogonal eigenfunction system. This provides a compact and robust formalism for theoretical and numerical studies of excitation and evolution of wave packets generated by local sources. The temporal continuous spectrum is revisited, and the uncertainty associated with the overlapping of continuousspectrum branches is resolved. It is shown that the behavior of the discrete spectrum's dispersion relationship is nonanalytic due to its relationship with the synchronization of the first or second mode with the vorticity/entropy waves of the continuous spectrum. The characteristics of the wave packet are numerically calculated using an expansion to the biorthogonal eigenfunction system, which comprises modes of discrete and continuous spectra. It is shown that the hypersonic boundary layer is highly receptive to vorticity/ entropy disturbances in the synchronism region. The feasibility of experimental verification of this receptivity mechanism is discussed. © 2001 The American Institute of Aeronautics and Astronautics, Inc. All rights reserved.
 Reau, N., & Tumin, A. (2000, June). On harmonic perturbations in turbulent wakes. In AIAA Paper No. 2000 – 2539.More infoAbstract: A theoretical model of harmonic perturbations in turbulent wakes is considered. The proposed model is based on the triple decomposition method. It is assumed that the instantaneous velocities and pressures consist of three distinctive components: the mean (time average), the coherent (phase average) and the random (turbulent) motion. The interaction between incoherent turbulent fluctuations and large scale coherent disturbances is incorporated by means of a Newtonian eddy viscosity model. A slight divergence of the flow is also taken into account. For highamplitude perturbations, the nonlinear feedback to the mean flow is taken into account by means of the coherent Reynolds stresses. The equations for the mean flow are coupled with the linearized equations for the disturbances taking account of the mean flow nonparallel effects. The model resolves uncertainties noticed in previous theories and provides a correct comparison with available experimental data. The results demonstrate the effect of harmonic perturbations on the turbulent wake growth parameter G=LqUJ6 u0, where L0 is the halfwidth of the wake; uq is the centreline velocity deficit; Q is the momentum thickness and Ux is the free stream velocity. At the initial stage of disturbance amplification, the amplitude parameter G increases. At high levels of the amplitude, downstream of the neutral point of the induced disturbance, G has a plateau or even decreases, and further downstream a growth of the parameter is recovered. © 2000 The American Institute of Aeronautics and Astronautics Inc. All right Reserved.
 Tumin, A. M., & Fedorov, A. V. (1985, January). EXCITATION OF INSTABILITY WAVES BY A VIBRATOR LOCALIZED IN THE BOUNDARY LAYER.. In LaminarTurbulent Transition, 295302.More infoAbstract: A theoretical model of the disturbances generated by the vibrations of the streamlined surface in the boundary layer is discussed. The analysis was performed with the aid of the expansion of the solution in eigenfunctions of linearized NavierStokes equations with nonparallelism of the flow in a real boundary layer taken into account. High efficiency of the excitation of TollmienSchlichting (TS) waves has been found.
Presentations
 Edwards, L., & Tumin, A. (2017, 11). Receptivity to Kinetic Fluctuations: A Multiple Scales Approach. 70th Annual Meeting of the APS Division of Fluid Dynamics. Dener CO: APS.
 Tumin, A., & Edwards, L. D. (2016, November). Real Gas Effects on Receptivity to kinetic fluctuations. the 69th Annual Meeting of the APS Division of Fluid Dynamics..
 Tumin, A., & Klentzman, J. (2014, June 1520). The Second Mode in Highenthalpy Boundary Layers in Chemical Nonequilibrium. 17th US National Congress of Theoretical and Applied Mechanics. Michigan State University, East Lansing, MI.