scholarly journals Entrainment of short-wavelength free-stream vortical disturbances in compressible and incompressible boundary layers

2016 ◽  
Vol 797 ◽  
pp. 683-728 ◽  
Author(s):  
Xuesong Wu ◽  
Ming Dong
Keyword(s):  
Boundary Layer ◽  
Boundary Layers ◽  
Free Stream ◽  
Base Flow ◽  
Bypass Transition ◽  
Leading Order ◽  
Compressible Boundary Layers ◽  
Characteristic Wavelength ◽  
Incompressible Boundary ◽  
Edge Layer

The fundamental difference between continuous modes of the Orr–Sommerfeld/Squire equations and the entrainment of free-stream vortical disturbances (FSVD) into the boundary layer has been investigated in a recent paper (Dong & Wu, J. Fluid Mech., vol. 732, 2013, pp. 616–659). It was shown there that the non-parallel-flow effect plays a leading-order role in the entrainment, and neglecting it at the outset, as is done in the continuous-mode formulation, leads to non-physical features of ‘Fourier entanglement’ and abnormal anisotropy. The analysis, which was for incompressible boundary layers and for FSVD with a characteristic wavelength of the order of the local boundary-layer thickness, is extended in this paper to compressible boundary layers and FSVD with even shorter wavelengths, which are comparable with the width of the so-called edge layer. Non-parallelism remains a leading-order effect in the present scaling, which turns out to be more general in that the equations and solutions in the previous paper are recovered in the appropriate limit. Appropriate asymptotic solutions in the main and edge layers are obtained to characterize the entrainment. It is found that when the Prandtl number $\mathit{Pr}<1$, free-stream vortical disturbances of relatively low frequency generate very strong temperature fluctuations within the edge layer, leading to formation of thermal streaks. A composite solution, uniformly valid across the entire boundary layer, is constructed, and it can be used in receptivity studies and as inlet conditions for direct numerical simulations of bypass transition. For compressible boundary layers, continuous spectra of the disturbance equations linearized about a parallel base flow exhibit entanglement between vortical and entropy modes, namely, a vortical mode necessarily induces an entropy disturbance in the free stream and vice versa, and this amounts to a further non-physical behaviour. High Reynolds number asymptotic analysis yields the relations between the amplitudes of entangled modes.

Streak instabilities in boundary layers beneath free-stream turbulence

2014 ◽  
Vol 741 ◽  
pp. 280-315 ◽  
Author(s):  
M. J. P. Hack ◽  
T. A. Zaki
Keyword(s):  
Boundary Layer ◽  
Boundary Layers ◽  
High Speed ◽  
Free Stream ◽  
Adverse Pressure Gradient ◽  
Base Flow ◽  
Bypass Transition ◽  
Low Speed ◽  
Free Stream Turbulence ◽  
Low Speed Streaks

AbstractThe secondary instability of boundary layer streaks is investigated by means of direct stability analysis. The base flow is computed in direct simulations of bypass transition. The random nature of the free-stream perturbations causes the formation of a spectrum of streaks inside the boundary layer, with breakdown to turbulence initiated by the amplification of localized instabilities of individual streaks. The capability of the instability analysis to predict the instabilities which are observed in the direct numerical simulation is established. Furthermore, the analysis is shown to identify the particular streaks that break down to turbulence farther downstream. Two particular configurations of streaks regularly induce the growth of these localized instabilities: low-speed streaks that are lifted towards the edge of the boundary layer, and the local overlap between high-speed and low-speed streaks inside the boundary layer. It is established that the underlying modes can be ascribed to the general classification of inner and outer modes which was introduced by Vaughan & Zaki (J. Fluid Mech., vol. 681, 2011, pp. 116–153). Statistical evaluations show that Blasius boundary layers favour the amplification of outer instabilities. Adverse pressure gradient promotes breakdown to turbulence via the inner mode.

Correlated incompressible and compressible boundary layers

1949 ◽  
Vol 200 (1060) ◽  
pp. 84-100 ◽  
Keyword(s):  
Boundary Layer ◽  
Exact Solution ◽  
Boundary Layers ◽  
Absolute Temperature ◽  
Stream Velocity ◽  
Main Stream ◽  
Boundary Layer Equations ◽  
Compressible Boundary Layers ◽  
Incompressible Boundary ◽  
Similar Solutions

The boundary-layer equations for a compressible fluid are transformed into those for an incompressible fluid, assuming that the boundary is thermally insulating, that the viscosity is proportional to the absolute temperature, and that the Prandtl number is unity. Various results in the theory of incompressible boundary layers are then taken over into the compressible theory. In particular, the existence of ‘similar’ solutions is proved, and Howarth’s method for retarded flows is applied to determine the point of separation for a uniformly retarded main stream velocity. A comparison with an exact solution is used to show that this method gives a closer approximation than does Pohlhausen’s.

scholarly journals A Bypass Transition Model for Boundary Layers

1993 ◽  
Author(s):  
Mark W. Johnson
Keyword(s):  
Boundary Layer ◽  
Boundary Layers ◽  
Free Stream ◽  
Transition Model ◽  
Wall Region ◽  
Bypass Transition ◽  
Pressure Gradients ◽  
Turbulence Level ◽  
Velocity Fluctuations ◽  
Laminar Boundary Layers

Experimental data for laminar boundary layers developing below a turbulent free stream shows that the fluctuation velocities within the boundary layer increase in amplitude until some critical level is reached which initiates transition. In the near wall region, a simple model, containing a single empirical parameter which depends only on the turbulence level and length scale, is derived to predict the development of the velocity fluctuations in laminar boundary layers with favourable, zero or adverse pressure gradients. A simple bypass transition model which considers the streamline distortion in the near wall region brought about by the velocity fluctuations suggests that transition will commence when the local turbulence level reaches approximately 23%. This value is consistent with experimental findings. This critical local turbulence level is used to derive a bypass transition prediction formula which compares reasonably with start of transition experimental data for a range of pressure gradients (λθ = −0.01 to 0.01) and turbulence levels (Tu = 0.2% to 5%). Further improvement to the model is proposed through prediction of the boundary layer distortion, which occurs due to Reynolds stresses generated within the boundary layer at high free stream turbulence levels and also through inclusion of the effect of turbulent length scale as well as turbulence level.

A Bypass Transition Model for Boundary Layers

1994 ◽  
Vol 116 (4) ◽  
pp. 759-764 ◽  
Author(s):  
M. W. Johnson
Keyword(s):  
Boundary Layer ◽  
Boundary Layers ◽  
Free Stream ◽  
Transition Model ◽  
Wall Region ◽  
Bypass Transition ◽  
Pressure Gradients ◽  
Turbulence Level ◽  
Velocity Fluctuations ◽  
Laminar Boundary Layers

Experimental data for laminar boundary layers developing below a turbulent free stream show that the fluctuation velocities within the boundary layer increase in amplitude until some critical level is reached, which initiates transition. In the near-wall region, a simple model, containing a single empirical parameter, which depends only on the turbulence level and length scale, is derived to predict the development of the velocity fluctuations in laminar boundary layers with favorable, zero, or adverse pressure gradients. A simple bypass transition model, which considers the streamline distortion in the near-wall region brought about by the velocity fluctuations, suggests that transition will commence when the local turbulence level reaches approximately 23 percent. This value is consistent with experimental findings. This critical local turbulence level is used to derive a bypass transition prediction formula, which compares reasonably with start of transition experimental data for a range of pressure gradients (λ θ = −0.01 to 0.01) and turbulence levels (Tu = 0.2 to 5 percent). Further improvement to the model is proposed through prediction of the boundary layer distortion, which occurs due to Reynolds stresses generated within the boundary layer at high free-stream turbulence levels and also through inclusion of the effect of turbulent length scale as well as turbulence level.

Spatial optimal growth in three-dimensional compressible boundary layers

2012 ◽  
Vol 704 ◽  
pp. 251-279 ◽  
Author(s):  
David Tempelmann ◽  
Ardeshir Hanifi ◽  
Dan S. Henningson
Keyword(s):  
Boundary Layer ◽  
Boundary Layers ◽  
Three Dimensional ◽  
Plate Boundary ◽  
Optimal Growth ◽  
Flat Plates ◽  
Curvature Effects ◽  
Physical Mechanisms ◽  
Compressible Boundary Layers ◽  
Incompressible Boundary

AbstractThis paper represents a continuation of the work by Tempelmann et al. (J. Fluid Mech., vol. 646, 2010b, pp. 5–37) on spatial optimal growth in incompressible boundary layers over swept flat plates. We present an extension of the methodology to compressible flow. Also, we account for curvature effects. Spatial optimal growth is studied for boundary layers over both flat and curved swept plates with adiabatic and cooled walls. We find that optimal growth increases for higher Mach numbers. In general, extensive non-modal growth is observed for all boundary layer cases even in subcritical regions, i.e. where the flow is stable with respect to modal crossflow disturbances. Wall cooling, despite stabilizing crossflow modes, destabilizes disturbances of non-modal nature. Curvature acts similarly on modal as well as non-modal disturbances. Convex walls have a stabilizing effect on the boundary layer whereas concave walls have a destabilizing effect. The physical mechanisms of optimal growth in all studied boundary layers are found to be similar to those identified for incompressible flat-plate boundary layers.

On continuous spectra of the Orr–Sommerfeld/Squire equations and entrainment of free-stream vortical disturbances

2013 ◽  
Vol 732 ◽  
pp. 616-659 ◽  
Author(s):  
Ming Dong ◽  
Xuesong Wu
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Free Stream ◽  
Stokes Equations ◽  
Low Frequency ◽  
Streamwise Velocity ◽  
Bypass Transition ◽  
Leading Order ◽  
Local Boundary ◽  
Asymptotic Suction

AbstractSmall-amplitude perturbations are governed by the linearized Navier–Stokes equations, which are, for a parallel or nearly parallel shear flow, customarily reduced to the Orr–Sommerfeld (O-S) and Squire equations. In this paper, we consider continuous spectra (CS) of the O-S and Squire operators for the Blasius and asymptotic suction boundary layers, and address the issue of whether and when continuous modes can represent free-stream vortical disturbances and their entrainment into the shear layer. For the Blasius boundary layer, we highlight two particular properties of the CS: (i) the eigenfunction of a continuous mode simultaneously consists of two components with wall-normal wavenumbers $\pm {k}_{2} $, a phenomenon which we refer to as ‘entanglement of Fourier components’; and (ii) for low-frequency disturbances the presence of the boundary layer forces the streamwise velocity in the free stream to take a much larger amplitude than those of the transverse velocities. Both features appear to be non-physical, and cast some doubt about the appropriateness of using CS to characterize free-stream vortical disturbances and their entrainment into the boundary layer, a practice that has been adopted in some recent studies of bypass transition. A high-Reynolds-number asymptotic description of continuous modes and entrainment is present, and it shows that the entanglement is a result of neglecting non-parallelism, which has a leading-order effect on the entrainment. When this effect is included, entanglement disappears, and moreover the streamwise velocity is significantly amplified in the edge layer when ${R}^{- 1} \ll \omega \ll 1$, where $R$ is the Reynolds number based on the local boundary-layer thickness. For the asymptotic suction boundary layer, which is an exactly parallel flow, both temporal and spatial CS may be defined mathematically. However, at a finite $R$ neither of them represents the physical process of free-stream vortical disturbances penetrating into the boundary layer. The latter must instead be characterized by a peculiar type of continuous modes whose eigenfunctions increase exponentially with the distance from the wall. In the limit $R\gg 1$, all three types of CS are identical at leading order, and hence can be used to represent free-stream vortical disturbances and their entrainment. Low-frequency disturbances are found to generate a large-amplitude streamwise velocity in the boundary layer, which is reminiscent of longitudinal streaks.

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.

Heat Transfer Committee and Turbomachinery Committee Best Paper of 1996 Award: The Path to Predicting Bypass Transition

1997 ◽  
Vol 119 (3) ◽  
pp. 405-411 ◽  
Author(s):  
R. E. Mayle ◽  
A. Schulz
Keyword(s):  
Boundary Layer ◽  
Kinetic Energy ◽  
Energy Equation ◽  
Free Stream ◽  
Bypass Transition ◽  
Turbulence Level ◽  
Free Stream Turbulence ◽  
Energy Profiles ◽  
Computer Codes ◽  
Good Agreement

A theory is presented for calculating the fluctuations in a laminar boundary layer when the free stream is turbulent. The kinetic energy equation for these fluctuations is derived and a new mechanism is revealed for their production. A methodology is presented for solving the equation using standard boundary layer computer codes. Solutions of the equation show that the fluctuations grow at first almost linearly with distance and then more slowly as viscous dissipation becomes important. Comparisons of calculated growth rates and kinetic energy profiles with data show good agreement. In addition, a hypothesis is advanced for the effective forcing frequency and free-stream turbulence level that produce these fluctuations. Finally, a method to calculate the onset of transition is examined and the results compared to data.

Recent Developments in Calculation Methods for Turbulent Boundary Layers With Pressure Gradients and Heat Transfer

1966 ◽  
Vol 33 (2) ◽  
pp. 429-437 ◽  
Author(s):  
J. C. Rotta
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Temperature Distribution ◽  
Velocity Profile ◽  
Boundary Layers ◽  
Turbulent Boundary Layers ◽  
Pressure Gradients ◽  
Calculation Methods ◽  
Recent Developments ◽  
Incompressible Boundary

A review is given of the recent development in turbulent boundary layers. At first, for the case of incompressible flow, the variation of the shape of velocity profile with the pressure gradient is discussed; also the temperature distribution and heat transfer in incompressible boundary layers are treated. Finally, problems of the turbulent boundary layer in compressible flow are considered.

The Path to Predicting Bypass Transition

1996 ◽  
Author(s):  
R. E. Mayle ◽  
A. Schulz
Keyword(s):  
Boundary Layer ◽  
Kinetic Energy ◽  
Energy Equation ◽  
Free Stream ◽  
Bypass Transition ◽  
Turbulence Level ◽  
Free Stream Turbulence ◽  
Energy Profiles ◽  
Computer Codes ◽  
Good Agreement

A theory is presented for calculating the fluctuations in a laminar boundary layer when the free stream is turbulent. The kinetic energy equation for these fluctuations is derived and a new mechanism is revealed for their production. A methodology is presented for solving the equation using standard boundary layer computer codes. Solutions of the equation show that the fluctuations grow at first almost linearly with distance and then more slowly as viscous dissipation becomes important. Comparisons of calculated growth rates and kinetic energy profiles with data show good agreement. In addition, a hypothesis is advanced for the effective forcing frequency and free-stream turbulence level which produce these fluctuations. Finally, a method to calculate the onset of transition is examined and the results compared to data.

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