Applications of parabolized stability equation for predicting transition position in boundary layers

The phenomenon of laminar-turbulent transition exists universally in nature and various engineering practice.The prediction of transition position is one of crucial theories and practical problems in fluid mechanics due to the different characteristics of laminar flow and turbulent flow.Two types of disturbances are imposed at the entrance,i.e.,identical amplitude and wavepacket disturbances,along the spanwise direction in the incompressible boundary layers.The disturbances of identical amplitude are consisted of one two-dimensional(2D) wave and two three-dimensional(3D) waves.The parabolized stability equation(PSE) is used to research the evolution of disturbances and to predict the transition position.The results are compared with those obtained by the numerical simulation.The results show that the PSE method can investigate the evolution of disturbances and predict the transition position.At the same time,the calculation speed is much faster than that of the numerical simulation.     

Roughness-induced generation of crossflow vortices in three-dimensional boundary layers

The receptivity theory of Goldstein and Ruban is extended within the nonasymptotic (quasi-parallel) framework of Zavol'skii et al. to predict the roughness-induced generation of stationary and nonstationary instability waves in three-dimensional, incompressible boundary layers. The influence of acoustic-wave orientation, as well as that of different types of roughness geometries, including isolated roughness elements, periodic arrays, and two-dimensional lattices of compact roughness shapes, as well as random, but spatially homogeneous roughness distributions, is examined. The parametric study for the Falkner-Skan-Cooke family of boundary layers supports our earlier conjecture that the initial amplitudes of roughness-induced stationary vortices are likely to be significantly larger than the amplitudes of similarly induced nonstationary vortices in the presence of acoustic disturbances in the free stream. Maximum unsteady receptivity occurs when the acoustic velocity fluctuation is aligned with the wave-number vector of the unsteady vortex mode. On the other hand, roughness arrays that are oriented somewhere close to the group velocity direction are likely to produce higher instability amplitudes. Limitations of the nonasymptotic theory are discussed, and future work is suggested.Financial support for this work was provided by the Theoretical Flow Physics Branch at the NASA Langley Research Center, Hampton, VA 23681, under Contract No. NAS1-19299.     

One method for describing boundary layers

Flow models taking into account pipe roughness are presented. Beta functions are described for some types of roughness. The modeling results are compared with data of classical experiments.     

Heat addition effect on the instability of the crossflow in a three-dimensional boundary layer

The heat addition effect on the stability characteristics of the crossflow in the swept-wing boundary layer is investigated. The physical mechanism of the effect is established. The calculations performed illustrate the variation in the instability mode growth rates under the heating of the surface in a flow and at bulk energy addition.     

An approximate method for magnetic boundary layers

An approximate method is derived for obtaining the flow in magnetic boundary layers with aligned fields which were discussed by Stewartson (1965). On comparing with exact, computed solutions the method is found to be accurate when applied to forward stagnation point flow for the whole, allowable range of the magnetic field strength parameter. It is then applied to the layer that develops from the forward stagnation point on a circular cylinder placed in a uniform stream and values obtained for the point at which it separates from the surface.     

Three-dimensional instability analysis of boundary layers perturbed by streamwise vortices

A parametric study is presented for the incompressible, zero-pressure-gradient flat-plate boundary layer perturbed by streamwise vortices. The vortices are placed near the leading edge and model the vortices induced by miniature vortex generators (MVGs), which consist in a spanwise-periodic array of small winglet pairs. The introduction of MVGs has been experimentally proved to be a successful passive flow control strategy for delaying laminar-turbulent transition caused by Tollmien–Schlichting (TS) waves. The counter-rotating vortex pairs induce non-modal, transient growth that leads to a streaky boundary layer flow. The initial intensity of the vortices and their wall-normal distances to the plate wall are varied with the aim of finding the most effective location for streak generation and the effect on the instability characteristics of the perturbed flow. The study includes the solution of the three-dimensional, stationary, streaky boundary layer flows by using the boundary region equations, and the three-dimensional instability analysis of the resulting basic flows by using the plane-marching parabolized stability equations. Depending on the initial circulation and positioning of the vortices, planar TS waves are stabilized by the presence of the streaks, resulting in a reduction in the region of instability and shrink of the neutral stability curve. For a fixed maximum streak amplitude below the threshold for secondary instability (SI), the most effective wall-normal distance for the formation of the streaks is found to also offer the most stabilization of TS waves. By setting a maximum streak amplitude above the threshold for SI, sinuous shear layer modes become unstable, as well as another instability mode that is amplified in a narrow region near the vortex inlet position.     

Effect of wall-cooling on Mack-mode instability in high speed flat-plate boundary layers

The instability of the Mack mode is destabilized by wall-cooling in a high speed boundary layer. The aim of this paper is to study the mechanism of the wallcooling effect on the Mack mode instability by numerical methods. It is shown that the wall-cooling can destabilize the Mack mode instability, similar to the previous conclusions with the exception that the Mack mode instability can be stabilized by wall-cooling if the wall temperature is extremely low. The reversed wall temperature is related to a freestream condition. If the Mach number increases to a large enough value, e.g., about7, the reversed wall temperature will tend to be zero. It seems that the Mack mode instability is determined by the region between the boundary layer edge and the critical layer. When the wall temperature decreases, this region becomes wider, and the boundary layer becomes more unstable. Additionally, a relative supersonic unstable mode can be observed when the velocity of the critical layer is less than 1- 1/M a or is cancelled by the wall-cooling effect. These results provide a deeper understanding on the wall-cooling effect in high speed boundary layers.     

PSE as applied to problems of secondary instability in supersonic boundary layers

Parabolized stability equations (PSE) approach is used to investigate prob-lems of secondary instability in supersonic boundary layers. The results show that the mechanism of secondary instability does work, whether the 2-D fundamental disturbance is of the first mode or second mode T-S wave. The variation of the growth rates of the 3-D sub-harmonic wave against its span-wise wave number and the amplitude of the 2-D fundamental wave is found to be similar to those found in incompressible boundary layers. But even as the amplitude of the 2-D wave is as large as the order 2%, the maximum growth rate of the 3-D sub-harmonic is still much smaller than the growth rate of the most unstable second mode 2-D T-S wave. Consequently, secondary instability is unlikely the main cause leading to transition in supersonic boundary layers.     

Maximum density effects on thermal instability of horizontal laminar boundary layers

Linear stability theory is used to investigate the onset of longitudinal vortices in laminar boundary layers along horizontal semi-infinite flat plates heated or cooled isothermally from below by considering the density inversion effect for water using a cubic temperature-density relationship. The analysis employs non-parallel flow model incorporating the variation of the basic flow and temperature fields with the streamwise coordinate as well as the transverse velocity component in the disturbance equations. Numerical results for the critical Grashof number Gr _L ^* =Gr _X ^* /Re _X< ^{Emphasis>/3/2} are presented for thermal conditions corresponding to –0.5 ₁–2.0 and –0.8 ₂1.2.Nomenclature a wavenumber, 2/ - D operator, d/d - F (f–f)/2 - f dimensionless stream function - g gravitational acceleration - G eigenvalue, Gr _L/Re_L - Gr _L Grashof number based on L - Gr _X Grashof number based on X - L characteristic length, (X/U)^1/2 - M number of divisions in y direction - P pressure - Pr Prandtl number, / - p dimensionless pressure, P/( ² /Re _L) - Re _L, Re_X Reynolds numbers, (U L/)=Re _X< ^1/2 and (U), respectively - T temperature - U, V, W velocity components in X, Y, Z directions - u, v, w dimensionless perturbation velocities, (U, V, W)/U - X, Y, Z rectangular coordinates - x, y, z dimensionless coordinates, (X, Y, Z)/L - thermal diffusivity - coefficient of thermal expansion - ₁, ₂ temperature coefficients for density-temperature relationship - similarity variable, Y/L=y - dimensionless temperature disturbance, /T - dimensionless wavelength of vortex rolls, 2/a - ₁, ₂ thermal parameters defined by equation (12) - kinematic viscosity - density - dimensionless basic temperature, (T _b T )/T - –1 - T temperature difference, (T _w–T ) - * critical value or dimensionless disturbance amplitude - prime, disturbance quantity or differentiation with respect to - b basic flow quantity - max value at a density maximum - w value at wall - free stream condition     

Experimental investigation of high-frequency secondary disturbances in a swept-wing boundary layer

Institute of Theoretical and Applied Mechanics, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 36, No. 3, pp. 74–83, May–June, 1995.     

A spectral collocation method for compressible,non-similar boundary layers

An efficient and highly accurate algorithm based on a spectral collocation method is developed for numerical solution of the compressible, two-dimensional and axisymmetric boundary layer equations. The numerical method incorporates a fifth-order, fully implicit marching scheme in the streamwise (timelike) dimension and a spectral collocation method based on Chebyshev polynomial expansions in the wall-normal (spacelike) dimension. The discrete governing equations are cast in residual form and the residuals are minimized at each marching step by a preconditioned Richardson iteration scheme which fully couples energy, momentum and continuity equations. Preconditioning on the basis of the finite difference analogues of the governing equations results in a computationally efficient iteration with acceptable convergence properties. A practical application of the algorithm arises in the area of compressible linear stability theory, in the investigation of the effects of transverse curvature on the stability of flows over axisymmetric bodies. The spectral collocation algorithm is used to derive the non-similar mean velocity and temperature profiles in the boundary layer of a ‘fuselage’ (cylinder) in a high-speed (Mach 5) flow parallel to its axis. The stability of the flow is shown to be sensitive to the gradual streamwise evolution of the mean flow and it is concluded that the effects of transverse curvature on stability should not be ignored routinely.     

A numerical method for calculating unsteady two-dimensional laminar boundary layers

Summary A method for calculating unsteady two dimensional boundary layers in laminar incompressible flow has been developed and tested. No restrictive assumptions are made regarding the time-dependent terms in the boundary-layer equation. The differential equations are solved with an implicit difference scheme similar to that employed for steady two-dimensional boundary layers. At each step, here, known conditions at three stations are used to calculate the conditions at a new (fourth) station. The entire field is covered by a succession of these steps.

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Übersicht Es wird ein Verfahren zur Berechnung von zeitabhängigen zweidimensionalen Grenzschichten in laminarer inkompressibler Strömung entwickelt und auf seine Genauigkeit geprüft. Dabei werden keine einschränkenden Annahmen über die zeitabhängigen Glieder in der Grenzschichtgleichung gemacht. Die Differentialgleichungen werden durch ein implizites Differenzen-Verfahren gelöst, das dem gewöhnlich für stetige zweidimensionale Grenzschichten verwendeten ähnlich ist. Hier werden für jeden Schritt schon bekannte Werte an drei Stellen benutzt, um die Werte an einer neuen (vierten) Stelle zu berechnen. Das ganze Feld wird auf diese Weise schrittweise überdeckt.

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This work was done while the author was at The John Hopkins University, on leave from the Royal Aircraft Establishment, Farnborough. It was supported in part by a grant from the National Science Foundation (Geophysical Fluid Mechanics, GA-641X). British Crown Copyright, reproduced with the permission of the Controller, Her Britannic Majesty's Stationery Office.     

Linear spatial instability analysis in 3D boundary layers using plane-marching 3D-LPSE

It is widely accepted that a robust and efficient method to compute the linear spatial amplified rate ought to be developed in three-dimensional (3D) boundary layers to predict the transition with the e ^N method, especially when the boundary layer varies significantly in the spanwise direction. The 3D-linear parabolized stability equation (3D-LPSE) approach, a 3D extension of the two-dimensional LPSE (2D-LPSE), is developed with a plane-marching procedure for investigating the instability of a 3D boundary layer with a significant spanwise variation. The method is suitable for a full Mach number region, and is validated by computing the unstable modes in 2D and 3D boundary layers, in both global and local instability problems. The predictions are in better agreement with the ones of the direct numerical simulation (DNS) rather than a 2D-eigenvalue problem (EVP) procedure. These results suggest that the plane-marching 3D-LPSE approach is a robust, efficient, and accurate choice for the local and global instability analysis in 2D and 3D boundary layers for all free-stream Mach numbers.     

Constructing boundary layers for a shallow shell by the energy-asymptotic method

The energy-asymptotic method is used to construct the boundary layers and the internal stress-strain state of a shallow shell. The basic equations are represented in the form of the displacement method. The efficiency of the method is shown. Pridneprovskaya Academy of Building and Architecture, Dnepropetrovsk, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 36, No. 2, pp. 99–104, February, 2000     

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.     

Experimental study of the instability of convective boundary layers under the action of vibration

The wave instability of convective boundary layers in a horizontal cylindrical layer of ethanol under the action of vertical hamonic high-frequency vibration is studied. A strong destabilizing effect of the vibration on the stability of the convective boundary layers is detected. In the plane of the gravity and vibration Rayleigh numbers (Ra and R _V ), the excitation limit of the wave instability is determined. The periods of the temperature oscillations caused by the waves in the boundary layers near the inner and outer cavity boundaries are studied as functions of the Rayleigh numbers. Perm’. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 32–40, May–June, 1998.     

Amplitude method of prediction of laminar-turbulent transition on a swept-wing

The amplitudemethod of prediction of laminar-turbulent transition on a swept-wing initiated by the simultaneous action of free-stream turbulence and surface roughness is developed. Generation of growing intrinsic perturbations in the boundary layer is described by the mechanism of distributed generation, i.e., an external perturbation generates an instability mode with the same wavelength and frequency. Generation occurs in a small neighborhood of the neutral point at which the parameters of an external perturbation and the instability modes coincide. The development of proper perturbations in the boundary layer is described by the nonlinear method of parabolized stability equations (PSE). The criterion of laminar-turbulent transition is the combined amplitude. i.e., transition begins at a point at which the sum of amplitudes of steady and traveling modes reaches a critical value.     

Numerical method for unsteady compressible boundary layers driven by compression or expansion wave

A numerical analysis is presented for the unsteady compressible laminar boundary layer driven by a compression or expansion wave. Approximate or series expansion methods have been used for the problems because of the characteristics of the governing equations, such as non-linearity, coupling with the thermal boundary layer equation and initial conditions. Here a transformation of the governing equations and the numerical linearization technique are introduced to deal with the difficulties. First, the governing equations are transformed for the initial conditions by Howarth and semisimilarity variables. These transformations reduce the number of independent variables from three to two and the governing equations from partial to ordinary differential equations at the initial point. Next, the numerical linearization technique is introduced for the non-linearity and the coupling with the thermal boundary layer equation. Because the non-linear terms are linearized without sacrifice of numerical accuracy, the solutions can be obtained without numerical iterations. Therefore the exact numerical solution, not approximate or series expansion, can be obtained. Compared with the approximate or series expansion method, this method is much improved. Results are compared with the series expansion solutions.     

Spontaneous condensation in boundary layers

It is a well known effect that accumulation of non condensable gas causes a high heat transfer resistance during direct contact condensation in binary steam- nitrogen mixtures. But especially with high pressures and low water temperatures a second effect reduces heat transfer additionally. Fog forms within in the steam-nitrogen boundary layer and the steam condenses at the water droplets of the fog layer and reaches the cooling water interface no longer. The convective mass transfer to the cooling water interface diminishes and no heating up of the water layer takes place. This effect was observed with experiments at stratified two phase flow run with pressures up to 2.0 MPa. The interface temperature has proved to be the most important parameter for fog formation. The paper explains this effect by means of film theory. It offers correlations to quantify the effect and to calculate the minimum interface temperature to avoid spontaneous condensation. Received on 29 May 1998     

Experimental study of nonlinear processes in a swept-wing boundary layer at the mach number M=2

Results of experiments aimed at studying the linear and nonlinear stages of the development of natural disturbances in the boundary layer on a swept wing at supersonic velocities are presented. The experiments are performed on a swept wing model with a lens-shaped airfoil, leading-edge sweep angle of 45°, and relative thickness of 3%. The disturbances in the flow are recorded by a constant-temperature hot-wire anemometer. For determining the nonlinear interaction of disturbances, the kurtosis and skewness are estimated for experimentally obtained distributions of the oscillating signal over the streamwise coordinate or along the normal to the surface. The disturbances are found to increase in the frequency range from 8 to 35 kHz in the region of their linear development, whereas enhancement of high-frequency disturbances is observed in the region of their nonlinear evolution. It is demonstrated that the growth of disturbances in the high-frequency spectral range (f > 35 kHz) is caused by the secondary instability.