Stability of a laminar boundary layer of a power-law no n-newtonian liquid

The stability of a laminar boundary layer of a power-law non-Newtonian fluid is studied. The validity of the Squire theorem on the possibility of reducing the flow stability problem for a power-law fluid relative to three-dimensional disturbances to a problem with two-dimensional disturbances is demonstrated. A numerical method of integrating the generalized Orr-Sommerfeld equation is constructed on the basis of previously proposed [1] transformations. Stability characteristics of the boundary layer on a longitudinally streamlined semiinfinite plate are considered.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 101–106, January–February, 1976.     

Acoustic Field Effect on Laminar Turbulent Transition on a Swept Wing in the Favorable Pressure Gradient Region

The development of disturbances in a three-dimensional boundary layer on a swept wing model is studied both under natural conditions and for artificial excitation of traveling waves by an acoustic field. It is found that steady-state streamwise structures are formed in the three-dimensional boundary layer; under natural conditions a wave packet leading to turbulence is detected. When the flow is exposed to the action of an acoustic field at a frequency from the wave packet, disturbances whose velocity along the streamwise structures is equal to 0.55 of the oncoming flow velocity are formed, while the laminar-turbulent transition is displaced upstream.     

Experimental investigation of the development of harmonic disturbances in the boundary layer on a flat plate at mach number M=4

The development of three-dimensional wave packets artificially introduced into a boundary layer has been experimentally investigated. The measurements were made by the hot-wire anemometer method in the boundary layer on a flat plate at a Mach number M = 4. The artificial disturbances were introduced into the boundary layer by means of an electric discharge. A Fourier analysis of the data made it possible to obtain the wave characteristics of the plane waves. The composition of the disturbances was analyzed and those most dangerous from the instability standpoint were identified. The data obtained are compared with the results of experiments carried out at M = 2. The differences in the data are discussed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 54–58, November–December, 1990.     

Three-dimensional waves in a boundary layer

As is known [1], two-dimensional waves develop in the boundary layer and then become three-dimensional waves with increase of the Reynolds number R. Since Squire [2] has shown that the linear growth of three-dimensional waves is more intense than that of the two-dimensional, it is natural that the behavior of three-dimensional waves in the boundary layer is explained by nonlinear intersection [3], However, Gaster [4] has noted that although disturbances which increase with time are usually considered, experimentally we observe disturbances which grow in space. (Squire's proof does not extend to this case.) It has been shown that the spatially growing disturbances cannot explain the occurrence of the three-dimensional waves (in the linear formulation).The author wishes to thank his scientific advisor G. I. Petrov and also A. A. Zaitsev for valuable discussions of the study.     

Development of three-dimensional disturbances in a boundary layer

In the region of transition from a two-dimensional laminar boundary layer to a turbulent one, three-dimensional flow occurs [1–3]. It has been proposed that this flow is formed as the result of nonlinear interaction of two-dimensional and three-dimensional disturbances predicted by linear hydrodynamic stability theory. Using many simplifications, [4, 5] performed a calculation of this interaction for a free boundary layer and a boundary layer on a wall with a very coarse approximation of the velocity profile. The results showed some argreement with experiment. On the other hand, it is known that disturbances of the Tollmin—Schlichting wave type can be observed at sufficiently high amplitude. This present study will use the method of successive linearization to calculate the primary two- and three-dimensional disturbances, and also the average secondary flow occurring because of nonlinear interaction of the primary disturbances. The method of calculation used is close to that of [4, 5], the disturbance parameters being calculated on the basis of a Blazius velocity profile. A detailed comparison of results with experimental data [1] is made. It developed that at large disturbance amplitude the amplitude growth rate differs from that of linear theory, while the spatial distribution of disturbances agree s well with the distribution given by the natural functions and their nonlinear interaction. In calculating the secondary flow an experimental correction was made to the amplitude growth rate.     

Stability of a compressible boundary layer

The problem of stability in a compressible boundary layer, as opposed to an incompressible layer, involves many parameters and requires consideration of three-dimensional perturbations. The transverse component of the velocity, the thermal regime at the wall, etc., take on great significance. Investigation of all aspects of this problem requires systematic calculations performed by electronic computers. There do exist a few calculations of stability of a compressible boundary layer with respect to three-dimensional disturbances for particular cases. It follows from those studies (see, for example, [1]) that consideration of three-dimensional perturbations and of the transverse component of the basic flow velocity is important. Many aspects of this problem remain uninvestigated. Aside from the sheer cumbersomeness of the problem, there exist purely mathematical difficulties connected with the presence of a small parameter with higher derivatives in the differential equations for the perturbations, which causes losses in accuracy of calculation. In this present study an algorithm will be developed for solution of the problem of stability of a compressible boundary layer relative to three-dimensional disturbances with consideration of the transverse component of the basic velocity. Calculations are performed for a boundary layer on a plane thermally insulating plate, and the effects of the transverse velocity component and the three-dimensionality of the perturbations on stability at various Mach numbers are demonstrated.     

Linear stability of three-dimensional boundary layers

Stability of compressible three-dimensional boundary layers on a swept wing model is studied within the framework of the linear theory. The analysis based on the approximation of local self-similarity of the mean flow was performed within the Falkner-Skan-Cooke solution extended to compressible flows. The calculated characteristics of stability for a subsonic boundary layer are found to agree well with the measured results. In the case of a supersonic boundary layer, the results calculated for a Mach number M = 2 are also in good agreement with the measured spanwise scales of nonstationary vortices of the secondary flow. The calculated growth rates of disturbances, however, are substantially different from the measured values. This difference can be attributed to a high initial amplitude of disturbances generated in the experiment, which does not allow the linear stability theory to be applied. The evolution of natural disturbances with moderate amplitudes is fairly well predicted by the theory. The effect of compressibility on crossflow instability modes is demonstrated to be insignificant. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 2, pp. 3–14, March–April, 2008.     

Extension of e~N method to general three-dimensional boundary layers

In order to extend the e~N method to general three-dimensional boundary layers, the conservation law of the imaginary parts for the wave parameters with a fixed wave vector is deduced. The compatibility relationship(CR) and the general theory of ray tracing(RT), which have been extensively used in conservative systems, are applied to a general three-dimensional boundary layer belonging to non-conservative systems. Two kinds of e~N methods, i.e., the e~N -CR method and the e~N -RT method, are established.Both the two kinds of methods can be used to predict the evolutions of the spanwise wavenumber and the amplitude of the disturbances in general three-dimensional boundary layers. The reliability of the proposed methods is verified and validated by performing a direct numerical simulation(DNS) in a hypersonic general three-dimensional boundary layer over an aircraft model. The results are also compared with those obtained by other e~N methods, indicating that the proposed methods have great potential applications in improving the transition prediction accuracy in general three-dimensional boundary layers.     

Secondary instability modes generated by a Tollmien-Schlichting wave scattering from a bump

The plane finite-amplitude Tollmien-Schlichting wave interaction with a three-dimensional bump on a wall is considered for plane channel flow. The scattering of this wave leads to the production of unsteady three-dimensional disturbances which transform into growing secondary instability modes. The generation of such modes is studied assuming the three-dimensional disturbances to be small in comparison with the primary plane instability wave. The solution predicts that secondary disturbance amplification takes place only within a narrow wedge downstream of the bump. The qualitative comparison of results with experiments on turbulent wedge origination at an isolated roughness in a boundary layer is presented.     

Optimal disturbances above and upstream of a flat plate with an elliptic-type leading edge

Adjoint-based iterative methods are employed to compute linear optimal disturbances in a spatially growing boundary layer around an elliptic leading edge. The Lagrangian approach is used where an objective function is chosen and constraints are assigned. The optimisation problem is solved using power iterations combined with a matrix-free formulation, where the state is marched forward in time with a standard direct numerical simulation solver and backward with the adjoint solver until a chosen convergence criterion is fulfilled. We consider the global and, more relevant to receptivity studies, the upstream localised optimal initial condition leading to the largest possible energy amplification at time T. We find that the two-dimensional initial condition with the largest potential for growth is a Tollmien–Schlichting-like wave packet that includes the Orr mechanism and is located inside the boundary layer downstream of the leading edge. Three-dimensional optimal disturbances induce streaks by the lift-up mechanism. Requiring the optimal initial condition to be localised upstream of the plate enables us to better study the effects of the leading edge on the boundary layer receptivity mechanisms. Two-dimensional upstream disturbances are inefficient at triggering unstable eigenmodes, whereas three-dimensional disturbances induce streamwise streaks with significant growth.     

Nonlinear Instability in the Region of Laminar-Turbulent Transition in Supersonic Three-Dimensional Flow over a Flat Plate

Direct numerical simulation is applied to obtain laminar-turbulent transition in supersonic flow over a flat plate. It is shown that, due to the nonlinear instability, Tollmien–Schlichting waves generated in the boundary layer lead to the formation of oblique disturbances in the flow. These represent a combination of compression and expansion waves, whose intensities can be two orders higher than that of external harmonic disturbances. The patterns of the three-dimensional flow over the plate are presented and the structures of the turbulent flat-plate boundary layers are described for the freestream Mach numbers M = 2 and 4.     

Sensitivity Analysis Using Adjoint Parabolized Stability Equations for Compressible Flows

An input/output framework is used to analyze the sensitivity of two- and three-dimensional disturbances in a compressible boundary layer for changes in wall and momentum forcing. The sensitivity is defined as the gradient of the kinetic disturbance energy at a given downstream position with respect to the forcing. The gradients are derived using the parabolized stability equations (PSE) and their adjoint (APSE). The adjoint equations are derived in a consistent way for a quasi-two-dimensional compressible flow in an orthogonal curvilinear coordinate system. The input/output framework provides a basis for optimal control studies. Analysis of two-dimensional boundary layers for Mach numbers between 0 and 1.2 show that wall and momentum forcing close to branch I of the neutral stability curve give the maximum magnitude of the gradient. Forcing at the wall gives the largest magnitude using the wall normal velocity component. In case of incompressible flow, the two-dimensional disturbances are the most sensitive ones to wall inhomogeneity. For compressible flow, the three-dimensional disturbances are the most sensitive ones. Further, it is shown that momentum forcing is most effectively done in the vicinity of the critical layer. This revised version was published online in July 2006 with corrections to the Cover Date.     

Growth of artificially induced disturbances in a supersonic boundary layer

This work proposes a method of inducing artificial disturbances of adjustable amplitude in a supersonic boundary layer. Using the proposed method, an experimental study is made of the development of a three-dimensional wave packet of low intensity at a frequency of 20 kHz in the boundary layer of a flat plate at Mach number M = 2.0. The Fourier components of the wave packet are determined. The data obtained are compared with the results of calculating the linear stability of the supersonic boundary layer in a plane-parallel flow approximation.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 37–43, September–October, 1984.     

Development of three-dimensional disturbances in a boundary layer with pressure gradient

The results of an experimental investigation of the three-dimensional stability of a boundary layer with a pressure gradient are presented. A low-turbulence subsonic wind tunnel was employed. The development of a three-dimensional wave packet of oscillations harmonic in time in the boundary layer on a model wing is studied. The amplitudephase distributions of the pulsations in the wave packet are subjected to a Fourier analysis. Spectral (with respect to the wave numbers) decomposition of the oscillations enables the flow stability with respect to plane waves with different directions of propagation to be examined. The results are compared with the corresponding data obtained in flat plate experiments. The effect of the pressure gradient on the development of the three-dimensional spectral components of the disturbances and the dispersion properties of the flow is analyzed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 85–91, May–June, 1988.     

Evolution of disturbances in a laminarized supersonic boundary layer on a swept wing

Experimental data on stability of a three-dimensional supersonic boundary layer on a swept wing are presented. The experiments are performed on a swept wing model with a lenticular profile with a 40° sweep angle of the leading edge at a zero angle of attack. The supersonic boundary layer on the swept wing was laminarized with the use of distributed roughness. A pioneering study of interaction of traveling and stationary disturbances is performed. Some specific features of this interaction are identified. The main reason for turbulence emergence in a supersonic boundary layer on a swept wing is demonstrated to be secondary crossflow instability. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 2, pp. 40–46, March–April, 2008.     

Swept-Wing Boundary Layer Receptivity to a Steady Free-Stream Vortex Disturbance

A single trailing vortex developed behind a micro-wing immersed in a free stream was used to study the vortex receptivity of a swept-wing boundary layer. As a result of the interaction, longitudinal-velocity disturbances develop in the boundary layer. On the swept wing, disturbance transformation occurs near the leading edge and is accompanied by the formation of a wave packet consisting of waves typical of cross-flow instability. Disturbances with other characteristics are also detected. These disturbances may be attributable to distributed boundary-layer receptivity to the free-stream vortex disturbance considered.     

Development of wedge-shaped disturbances originating from a point source in a three-dimensional boundary layer

A recent study predicted possibility of existence of a new instability due to the curvature of external streamlines in three-dimensional boundary layers, besides the familiar cross-flow instability, but no reliable evidence of this phenomenon has yet been obtained in experiments. In expectation of dispersive development of the two instabilities, the present study deals with small disturbances induced by continuous forcing from a point source in the boundary layer along a yawed circular cylinder, and attempts to describe their spatial development into wedge-shaped distribution with a linear stability theory, which is applicable to both of the above instabilities. Unlike plane-wave disturbances, the point-source disturbances have an important peculiarity that their propagation is governed by a complex group velocity, and a new method based on the complex property of the group velocity is presented to predict the paths of propagation along which growth rates of disturbances are integrated. Results of this stability calculation clarify important differences in development between the cross-flow disturbances and the streamline-curvature disturbances. These differences will make it possible to observe the new mode of disturbances separately from the other in experiments.     

On secondary instability with respect to three dimensional subharmonic disturbances in boundary layer

The general equations of secondary instability with respect to three-dimensional subharmonic disturbances are derived and applied to Blasius boundary layer in the present paper. The theoretical results of evolution and spatial distribution of subharmonic disturbances are compared with experimental data. The results show the important role of the process of route to transition in low-disturbance environments, and indicate that spatial mode is more rational than temporal mode. Project supported by the National Natural Science Foundation of China Current address: Graduate School, University of Science and Technology of China, Beijing, 100039     

The effect of free stream turbulence on the momentum,heat and mass transfer during flow around a sphere

On the basis of our own experimental investigations of the effect of the fluid stream turbulence on the mean heat transfer, a possibility of significant (up to about 60 per cent) intensification of the process was shown. In the second part of this work, the authors attempted to analyze phenomena which result in heat layer. The significant role of three-dimensional disturbances of the Goertler type in the process of formation of “pseudo-laminar” boundary layer, was emphasized.     

Mechanism of transition in a hypersonic sharp cone boundary layer with zero angle of attack

Firstly, the steady laminar flow field of a hypersonic sharp cone boundary layer with zero angle of attack was computed.Then,two groups of finite amplitude T-S wave disturbances were introduced at the entrance of the computational field,and the spatial mode transition process was studied by direct numerical simulation (DNS) method. The mechanism of the transition process was analyzed.It was found that the change of the stability characteristics of the mean flow profile was the key issue.Furthermore,the characteristics of evolution for the disturbances of different modes in the hypersonic sharp cone boundary layer were discussed.