scholarly journals An asymptotic investigation of the stationary modes of instability of the boundary layer on a rotating disc

1986 ◽  
Vol 406 (1830) ◽  
pp. 93-106 ◽  
Keyword(s):  
Boundary Layer ◽  
Nonlinear Problems ◽  
Wave Number ◽  
Viscous Effects ◽  
Rotating Disc ◽  
Expansion Procedure ◽  
Asymptotic Investigation ◽  
Starting Point ◽  
Consistent Manner ◽  
High Reynolds

The paper investigates high-Reynolds-number stationary instabilities in the boundary layer on a rotating disc. The investigation demonstrates that, in addition to the inviscid mode found by Gregory, Stuart & Walker ( Phil. Trans. R. Soc. Lond. A 248, 155 (1955)) at high Reynolds numbers, there is a stationary short-wavelength mode. This mode has its structure fixed by a balance between viscous and Coriolis forces and cannot be described by an inviscid theory. The asymptotic structure of the wave-number and orientation of this mode is obtained, and a similar analysis is given for the inviscid mode. The expansion procedure provides the capacity of taking non-parallel effects into account in a self-consistent manner. The inviscid solution of Gregory et al . is modified to take account of viscous effects. The expansion procedure used is again capable of taking non-parallel effects into account. The results obtained suggest why the inviscid approach of Gregory et al . should give a good approximation to the experimentally measured orientation of the vortices. The results also explain partly why the inviscid analysis should not give such a good approximation to the wavenumber of the vortices. The asymptotic analysis of both modes provides a starting point for the corresponding nonlinear problems.

A nonlinear, asymptotic investigation of the stationary modes of instability of the three-dimensional boundary layer on a rotating disc

1987 ◽  
Vol 413 (1845) ◽  
pp. 497-513 ◽  
Keyword(s):  
Boundary Layer ◽  
Three Dimensional ◽  
Nonlinear Effects ◽  
Finite Amplitude ◽  
Neutral Stability ◽  
Lower Branch ◽  
Rotating Disc ◽  
Dimensional Boundary ◽  
High Reynolds ◽  
Set Up

There exist two types of stationary instability of the flow over a rotating disc corresponding to the upper, inviscid mode and the lower-branch mode, which has a triple-deck structure, of the neutral stability curve. The linear problem has been investigated by P. Hall ( Proc. R. Soc. Lond. A 406, 93-106 (1986)) and the asymptotic structure of the wavenumber and orientation of these modes has been obtained. Here, a nonlinear investigation of high Reynolds number, stationary instabilities in the three-dimensional boundary layer on a rotating disc is given for the lower branch mode. By considering nonlinear effects and following the framework set up by Hall, asymptotic solutions are obtained that enable the finite amplitude growth of a disturbance close to the neutral location to be described.

‘Inactive’ motion and pressure fluctuations in turbulent boundary layers

1967 ◽  
Vol 30 (2) ◽  
pp. 241-258 ◽  
Author(s):  
P. Bradshaw
Keyword(s):  
Boundary Layer ◽  
Outer Layer ◽  
Pressure Fluctuations ◽  
Turbulent Energy ◽  
Viscous Sublayer ◽  
Wave Number ◽  
Noticeable Effect ◽  
Frequency Spectra ◽  
Active Motion ◽  
Order Of Magnitude

Townsend's (1961) hypothesis that the turbulent motion in the inner region of a boundary layer consists of (i) an ‘active’ part which produces the shear stress τ and whose statistical properties are universal functions of τ and y, and (ii) an ‘inactive’ and effectively irrotational part determined by the turbulence in the outer layer, is supported in the present paper by measurements of frequency spectra in a strongly retarded boundary layer, in which the ‘inactive’ motion is particularly intense. The only noticeable effect of the inactive motion is an increased dissipation of kinetic energy into heat in the viscous sublayer, supplied by turbulent energy diffusion from the outer layer towards the surface. The required diffusion is of the right order of magnitude to explain the non-universal values of the triple products measured near the surface, which can therefore be reconciled with universality of the ‘active’ motion.Dimensional analysis shows that the contribution of the ‘active’ inner layer motion to the one-dimensional wave-number spectrum of the surface pressure fluctuations varies as τ2w/k1 up to a wave-number inversely proportional to the thickness of the viscous sublayer. This result is strongly supported by the recent measurements of Hodgson (1967), made with a much smaller ratio of microphone diameter to boundary-layer thickness than has been achieved previously. The disagreement of the result with most other measurements is attributed to inadequate transducer resolution in the other experiments.

Effects of Reynolds Number and Free-Stream Turbulence on Boundary Layer Transition in a Compressor Cascade

2000 ◽  
Author(s):  
Heinz-Adolf Schreiber ◽  
Wolfgang Steinert ◽  
Bernhard Küsters
Keyword(s):  
Boundary Layer ◽  
Free Stream ◽  
Reynolds Numbers ◽  
Boundary Layer Transition ◽  
Bypass Transition ◽  
Layer Transition ◽  
Free Stream Turbulence ◽  
High Reynolds Numbers ◽  
High Turbulence ◽  
High Reynolds

An experimental and analytical study has been performed on the effect of Reynolds number and free-stream turbulence on boundary layer transition location on the suction surface of a controlled diffusion airfoil (CDA). The experiments were conducted in a rectilinear cascade facility at Reynolds numbers between 0.7 and 3.0×106 and turbulence intensities from about 0.7 to 4%. An oil streak technique and liquid crystal coatings were used to visualize the boundary layer state. For small turbulence levels and all Reynolds numbers tested the accelerated front portion of the blade is laminar and transition occurs within a laminar separation bubble shortly after the maximum velocity near 35–40% of chord. For high turbulence levels (Tu > 3%) and high Reynolds numbers transition propagates upstream into the accelerated front portion of the CDA blade. For those conditions, the sensitivity to surface roughness increases considerably and at Tu = 4% bypass transition is observed near 7–10% of chord. Experimental results are compared to theoretical predictions using the transition model which is implemented in the MISES code of Youngren and Drela. Overall the results indicate that early bypass transition at high turbulence levels must alter the profile velocity distribution for compressor blades that are designed and optimized for high Reynolds numbers.

Circulation control in magnetohydrodynamic rotating flows

2017 ◽  
Vol 829 ◽  
pp. 328-344 ◽  
Author(s):  
V. D. Borisevich ◽  
E. P. Potanin ◽  
J. Whichello
Keyword(s):  
Magnetic Field ◽  
Boundary Layer ◽  
Exact Analytical Solution ◽  
External Rotation ◽  
Three Dimensional ◽  
Flow Characteristics ◽  
Rotating Cylinder ◽  
Rotating Disc ◽  
Angular Velocities ◽  
Energy Equations

A model of a laminar viscous conducting flow, near a dielectric disc in a uniform magnetic field and in the presence of external rotation, is considered, where there is a uniform suction and an axial temperature gradient between the flow and the disc’s surface. It is assumed that the parameters of the suction or the magnetohydrodynamic (MHD) interaction are such that the nonlinear inertial terms, related to the circulation flow, are negligible in the differential equations of the MHD boundary layer on a rotating disc. Analysis of the motion and energy equations, taking the dependence of density on temperature into account, is carried out using the Dorodnitsyn transformation. The exact analytical solution for the boundary layer and heat transfer equations is obtained and analysed, neglecting the viscous and Joule dissipation. The dependence of the flow characteristics in the boundary layer on the rate of suction and the magnetic field induction is studied. It is shown that the direction of the radial flow in the boundary layer on a disc can be changed, not only by variation of the ratio between the angular velocities in the external flow and the boundary layer, but also by changing the ratio of the temperatures in these two flows, as well as by varying the hydrodynamic Prandtl number. The approximate calculation of a three-dimensional flow in a rotating cylinder with a braking disc (or lid) is carried out, demonstrating that a magnetic field slows the circulation velocity in a rotating cylinder.

Comparison of Inviscid Computations With Theory and Experiment in Vibrating Transonic Compressor Cascades

1990 ◽  
Author(s):  
G. A. Gerolymos ◽  
E. Blin ◽  
H. Quiniou
Keyword(s):  
Boundary Layer ◽  
Plate Theory ◽  
Strong Shock ◽  
Acoustic Resonance ◽  
Boundary Layer Separation ◽  
Computational Method ◽  
Viscous Effects ◽  
Transonic Compressor ◽  
Boundary Layer Interaction ◽  
Interaction Regions

The prediction of unsteady flow in vibrating transonic cascades is essential in assessing the aeroelastic stability of fans and compressors. In the present work an existing computational code, based on the numerical integration of the unsteady Euler equations, in blade-to-blade surface formulation, is validated by comparison with available theoretical and experimental results. Comparison with the flat plate theory of Verdon is, globally, satisfactory. Nevertheless, the computational results do not exhibit any particular behaviour at acoustic resonance. The use of a 1-D nonreflecting boundary condition does not significantly alter the results. Comparison of the computational method with experimental data from started and unstarted supersonic flows, with strong shock waves, reveals that, notwithstanding the globally satisfactory performance of the method, viscous effects are prominent at the shock wave/boundary layer interaction regions, where boundary layer separation introduces a pressure harmonic phase shift, which is not presicted by inviscid methods.

Streamwise evolution of longitudinal vortices in a turbulent boundary layer

2009 ◽  
Vol 623 ◽  
pp. 27-58 ◽  
Author(s):  
OLA LÖGDBERG ◽  
JENS H. M. FRANSSON ◽  
P. HENRIK ALFREDSSON
Keyword(s):  
Boundary Layer ◽  
Turbulent Boundary Layer ◽  
Vortex Core ◽  
Cross Flow ◽  
General Point ◽  
Natural Setting ◽  
Viscous Effects ◽  
Array Configuration ◽  
The Cross ◽  
Flow Plane

In this experimental study both smoke visualization and three-component hot-wire measurements have been performed in order to characterize the streamwise evolution of longitudinal counter-rotating vortices in a turbulent boundary layer. The vortices were generated by means of vortex generators (VGs) in different configurations. Both single pairs and arrays in a natural setting as well as in yaw have been considered. Moreover three different vortex blade heights h, with the spacing d and the distance to the neighbouring vortex pair D for the array configuration, were studied keeping the same d/h and D/h ratios. It is shown that the vortex core paths scale with h in the streamwise direction and with D and h in the spanwise and wall-normal directions, respectively. A new peculiar ‘hooklike’ vortex core motion, seen in the cross-flow plane, has been identified in the far region, starting around 200h and 50h for the pair and the array configuration, respectively. This behaviour is explained in the paper. Furthermore the experimental data indicate that the vortex paths asymptote to a prescribed location in the cross-flow plane, which first was stated as a hypothesis and later verified. This observation goes against previously reported numerical results based on inviscid theory. An account for the important viscous effects is taken in a pseudo-viscous vortex model which is able to capture the streamwise core evolution throughout the measurement region down to 450h. Finally, the effect of yawing is reported, and it is shown that spanwise-averaged quantities such as the shape factor and the circulation are hardly perceptible. However, the evolution of the vortex cores are different both between the pair and the array configuration and in the natural setting versus the case with yaw. From a general point of view the present paper reports on fundamental results concerning the vortex evolution in a fully developed turbulent boundary layer.

The Estimation of Laminar Skin Friction, including the Effects of Distributed Suction

1960 ◽  
Vol 11 (1) ◽  
pp. 1-21 ◽  
Author(s):  
N. Curle
Keyword(s):  
Boundary Layer ◽  
Skin Friction ◽  
Laminar Boundary Layer ◽  
Simple Formula ◽  
Outer Part ◽  
Order Correction ◽  
Viscous Effects ◽  
Viscous Forces ◽  
Exact Theory ◽  
Total Head

SummaryStratford's analysis of the laminar boundary layer near separation uses two physical ideas. In the outer part of the boundary layer, where viscous effects are small, the development is given by the condition that the total head is constant along streamlines, apart from a second-order correction for viscosity. Near the wall, however, viscous forces must balance the pressure forces, and the profile adjusts itself accordingly. Quantitatively these ideas yield a simple formula for predicting separation, which has been found to be particularly accurate.In this paper it is indicated how the same approach may be used to yield the full distribution of skin friction along the wall. Further, the effects of suction may be incorporated into the method. Physically, suction affects the outer part of the boundary layer in that the streamlines are drawn towards the wall when suction is applied. At the wall, the balance between viscous and pressure forces is influenced by the momentum of the fluid which is sucked away. When these effects are accounted for quantitatively, the resulting formula for the skin friction is still very simple.Several examples are considered, and comparison is made with exact theory and with approximate results by other methods. It is indicated that the method has a useful range of validity.

Free-stream coherent structures in parallel boundary-layer flows

2014 ◽  
Vol 752 ◽  
pp. 602-625 ◽  
Author(s):  
Kengo Deguchi ◽  
Philip Hall
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Coherent Structures ◽  
Free Stream ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Homotopy Continuation ◽  
Boundary Layer Flows ◽  
Navier Stokes Equations ◽  
High Reynolds

AbstractOur concern in this paper is with high-Reynolds-number nonlinear equilibrium solutions of the Navier–Stokes equations for boundary-layer flows. Here we consider the asymptotic suction boundary layer (ASBL) which we take as a prototype parallel boundary layer. Solutions of the equations of motion are obtained using a homotopy continuation from two known types of solutions for plane Couette flow. At high Reynolds numbers, it is shown that the first type of solution takes the form of a vortex–wave interaction (VWI) state, see Hall & Smith (J. Fluid Mech., vol. 227, 1991, pp. 641–666), and is located in the main part of the boundary layer. On the other hand, here the second type is found to support an equilibrium solution of the unit-Reynolds-number Navier–Stokes equations in a layer located a distance of $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}O(\ln \mathit{Re})$ from the wall. Here $\mathit{Re}$ is the Reynolds number based on the free-stream speed and the unperturbed boundary-layer thickness. The streaky field produced by the interaction grows exponentially below the layer and takes its maximum size within the unperturbed boundary layer. The results suggest the possibility of two distinct types of streaky coherent structures existing, possibly simultaneously, in disturbed boundary layers.

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