Near critical phenomena in laminar boundary layers

Asymptotic analysis of boundary layer separation in the limit of large Reynolds number Re→∞ has shown that in a number of cases which are of importance from a practical point of view solutions of the resulting interaction equations describing two-dimensional (2-D) steady flows exist up to a limiting value Γ_c of the relevant controlling parameter Γ only while two branches of solutions exist in a regime Γ<Γ_c. The present study aims at a better understanding of near critical flows |Γ-Γ_c|→0 and in particular the changes of the flow behaviour associated with the passage of Γ through Γ_c.     

Similar laminar boundary layers in incompressible micropolar fluids

Summary A search for similar solutions reveals as only possible similar boundary layer flow in micropolar fluids the flow near a stagnation point. The corresponding equations have been solved numerically by means of a shooting method. Consideration is given not only to the coupling parameterC ₁ and the microdiffusivity parameterC ₂ but also to the microinertia parameterC ₃. It is shown that macroscopic properties of steady boundary layer flows are not very much affected by these parameters, while the microrotation and therefore the inner structure of the layer is very sensitive to all three parameters. These properties of the microstructure can become important in certain unsteady flow problems; then also the macroscopic behaviour may be different to the behaviour of Newtonian fluids.

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Zusammenfassung In der vorliegenden Untersuchung wird gezeigt, daß ähnliche Grenzschichten in mikropolaren Flüssigkeiten nur in der Nähe eines Staupunkts existieren. Die zugehörigen gewöhnlichen Differentialgleichungen werden mit einem Einschießverfahren numerisch gelöst. Neben dem KopplungsparameterC ₁ und dem MikrorotationsparameterC ₂ wird dabei auch der Einfluß der Mikroträgheit im ParameterC ₃ berücksichtigt. Es zeigt sich, daß diese Parameter die makroskopischen Eigenschaften stationärer Grenzschichtströmungen relativ wenig beeinflussen, während sich die Mikrorotation und damit die innere Struktur der Grenzschicht mit diesen Parametern sehr stark ändern kann. Man kann vermuten, daß diese Eigenschaften mikropolarer Flüssigkeiten bei instationären Vorgängen durchaus auch im makroskopischen Verhalten zu größeren Abweichungen gegenüber newtonschen Flüssigkeiten führen können.

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With 6 figures and 1 table     

Second-order effects in unsteady laminar compressible three-dimensional stagnation-point boundary layers

The flow, heat and mass transfer at the stagnation point of a three-dimensional body in unsteady laminar compressible fluid with variable properties have been studied using a second-order boundary-layer theory when the basic potential flow admits selfsimilarity. Both nodal and saddle point regions have been considered. The equations governing the flow have been solved numerically using an implicit finite-difference scheme. It is observed that the enhancement or reduction in the skin friction and heat transfer due to the second-order boundary layers depends upon the values of the parameter characterizing the unsteadiness in the free-stream velocity, the nature of the stagnation point, the variation of the density-viscosity product across the boundary layer, mass transfer and the wall temperature. The suction increases the skin friction and heat transfer whereas injection does the opposite.     

Vortex structure in laminar boundary layers on a concave wall

Naturally occurring Görtler vortices have been investigated, using laser anemometry, in the laminar concave-surface boundary layers of water channels with 90° bends. Distributions of streamwise and spanwise velocities are presented, supported by flow visualization. Amplification was found to cease at Görtler numbers in the region of 9, followed by increasing distortion and spanwise wandering of the vortices. Development of a vortex system from an imposed disturbance has also been predicted by numerical computations.     

Stability characteristics of laminar boundary layers over compliant walls

The stability characteristics of laminar boundary layers over compliant walls was studied by the linear theory. Unlike the previous authors, the coupled motion of the fluid and solid was required to satisfy the continuity conditions of both the velocity and stress at the interface. Results of calculations show that as the speed ratio or density ratio exceeds a certain threshold value, the two types of unstable waves will no longer be distinguishable, and the tangential component of the disturbance stress is no longer negligible. So the neglect of it, as the previous authors did, is unjustified. The project is supported by the National Natural Science Fundation of China.     

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.     

The role of the fluid relaxation time in laminar elastico-viscous boundary layers

Summary The stress relaxation effect is investigated in the flow of elastico-viscous fluids in laminar boundary layers. Examples of the exact similar solutions ofFalkner andSkan class have been extended to include the relaxation effect. In order to separate the effects of the relaxation and the first normal-stress difference, a rheological equation with a new material time derivative is used. This derivative is referred to a co-ordinate system attached to the principal directions of the strain rate tensor.The results show a reduction in the skin friction owing to the elasticity of the fluid in boundary layers having increasing shear rate downstream. Conversely skin friction increases in flows with decreasing shear rates downstream. Comparison with a former work yields the role of the first normal-stress difference. The results can explain the observed downstream shift of the separation point in elastico-viscous flow past a submerged cylinder.

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Zusammenfassung Die Wirkung der Spannungsrelaxation in der laminaren Grenzschichtströmung einer viskoelastischen Flüssigkeit wird untersucht. Beispiele exakter Ähnlichkeitslösungen vomFalkner-Skan-Typ werden so verallgemeinert, daß sie Relaxationseffekte mit einschließen. Zum Zweck der Trennung der Wirkung von Relaxation und erster Normalspannungs-Differenz wird eine rheologische Stoffgleichung mit einer neuen materiellen Zeitableitung verwendet. Diese ist auf ein mit den Hauptachsen des Deformationsgeschwindigkeits-Tensors verbundenes Koordinatensystem bezogen.Die Ergebnisse zeigen eine Verminderung der Wandreibung bei Grenzschichten mit stromabwärts zunehmender Schergeschwindigkeit infolge der Flüssigkeitselastizität, während im umgekehrten Fall die Wandreibung zunimmt. Ein Vergleich mit früheren Untersuchungen macht die Rolle der ersten Normalspannungs-Differenz deutlich. Die Ergebnisse vermögen die beobachtete Stromabwärts-Verschiebung der Grenzschichtablösung bei einem eingetauchten Zylinder zu deuten.

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Presented at the British Society of Rheology Conference on General Rheology and Stretching Flows, Edinburgh, Sept. 7–9, 1977.

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With 2 figures     

An experimental study of receptivity of acoustic waves in laminar boundary layers

The receptivity of a laminar boundary layer developing on a flat plate was studied with two- and three-dimensional roughness elements. The layer was subjected to acoustic waves from speakers orientated perpendicular to the surface of the plate. Visualization of the transition patterns were obtained by heating temperature sensitive liquid crystals on the plate and observing the cooling patterns associated with the different flow regimes. Hot-wire data showed that the most amplified Tollmien-Schlichting waves dominated the downstream growth when the roughnesses were placed within the linearly unstable regime. The receptivity depended upon the size and aspect ratio of the three-dimensional roughness as predicted by Choudhari and Kerschen 1990. This research was partially funded by the Office of Naval Research under Contract N00014-89-J-1400. Their support is gratefully acknowledged. We also thank one of the reviewers for his helpful comments.     

Direct numerical simulation of laminar breakdown in high-speed,axisymmetric boundary layers

The laminar breakdown of high-speed, axisymmetric boundary-layer flow is simulated numerically by solving the compressible Navier-Stokes equations using spectral collocation and high-order compact-difference techniques. Numerical test cases include Mach 4.5 flow along a hollow cylinder and Mach 6.8 flow along a sharp cone. From initial states perturbed by second-mode primary and subharmonic (H-type) secondary disturbances, the well-resolved (temporal) calculations proceed well into the laminar breakdown stages, characterized by saturation of the primary and secondary instability waves, explosive growth of higher harmonics, and rapid increase in the wall shear stress. The numerical results qualitatively replicate two previously unexplained phenomena which have been observed in high-speed transition experiments: the appearance of so-called rope-like waves and the precursor transition effect, in which transitional flow appears to originate near the critical layer well upstream of the transition location at the wall. The numerical results further reveal that neither of these effects can be explained, even qualitatively, by linear stability theory alone. Structures of rope-like appearance are shown to arise from secondary instability. Whereas certain features of the precursor transition effect also emerge from secondary instability theory, its nature is revealed to be fundamentally nonlinear.This research was accomplished largely while the first author was in residence as a National Research Council Associate at NASA Langley Research Center, Hampton, VA 23665, USA.     

Similarity solution of thermal boundary layers for laminar narrow axisymmetric jets

In this work, the similarity equation describing the thermal boundary layers of laminar narrow axisymmetric jets is derived based on boundary layer assumptions. The equation is solved exactly. Some properties of the thermal jet are discussed. By introducing new-defined non-dimensional coordinates, the similarity solution results in a “universal” format. The results can also be applied in the boundary layer problem of species diffusion.     

Active flow control of laminar boundary layers for variable flow conditions

This work investigates the stability of a fxLMS controller for active wave cancelation of broad-band Tollmien–Schlichting disturbances in a flat plate boundary-layer with a single DBD plasma actuator. In particular the influence of a changing free stream velocity and the resulting off-design operation of the control algorithm is analyzed up to an unstable behavior. As the main reason for unstable controller operation in the off-design case the difference between actual and predicted phase angle of the disturbances at the position of the error sensor is identified. A method for an online adjustment of the secondary-path model to different free-stream velocities is presented. Finally a wall-bounded method based on the disturbances phase speed is developed that can cope with changes of the physical secondary path not only due to changes of the free-stream velocity but also due to changes of the pressure distribution. This method enables the extension of the stable operation range of the control system significantly.     

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     

An integral procedure for a two-phase boundary layer in laminar film condensation

A general integral procedure has been developed for the analysis of laminar film condensation heat transfer. A solution of the gravity force condensation problem with constant properties is presented, including the effect of the drag due to an initially stationary body of pure saturated vapor. The present integral method, based on the two-phase boundary layer theory, assumes a finite density-viscosity ratio between the liquid and vapor phases, and can, in fact, be employed under almost all possible two dimensional and axisymmetric geometrical configurations. An extensive comparison of the calculation results with available exact solutions reveals an excellent performance of the present integral procedure.     

Dispersed phase motion in laminar boundary layer flow over a wedge

The motion of a dispersed phase in the laminar boundary layer on a wedge is considered with allowance for the effect of not only the Stokes force, which coincides in direction with the flow velocity, but also the transverse force (Saffman force) resulting from the transverse nonuniforrnity of the flow over the individual particle [1–3].Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.6, pp. 34–42, November–December, 1993.In conclusion, the authors wishes to thank S. V. Manuilovich for assisting with the numerical calculations.