Closure to “Discussions of ‘Application of a General Boundary Layer Analysis to Turbulent Boundary Layers Subjected to Strong Favorable Pressure Gradients’” (1976, ASME J. Fluids Eng., 98, pp. 129–130)

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.

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Self-similar compressible turbulent boundary layers with pressure gradients. Part 1. Direct numerical simulation and assessment of Morkovin’s hypothesis

Journal of Fluid Mechanics ◽

10.1017/jfm.2019.670 ◽

2019 ◽

Vol 880 ◽

pp. 239-283 ◽

Cited By ~ 3

Author(s):

Christoph Wenzel ◽

Tobias Gibis ◽

Markus Kloker ◽

Ulrich Rist

Keyword(s):

Numerical Simulation ◽

Boundary Layer ◽

Mach Number ◽

Direct Numerical Simulation ◽

Boundary Layers ◽

Turbulent Boundary Layers ◽

Shear Stresses ◽

Pressure Gradients ◽

Mean Flow ◽

Self Similar

A direct numerical simulation study of self-similar compressible flat-plate turbulent boundary layers (TBLs) with pressure gradients (PGs) has been performed for inflow Mach numbers of 0.5 and 2.0. All cases are computed with smooth PGs for both favourable and adverse PG distributions (FPG, APG) and thus are akin to experiments using a reflected-wave set-up. The equilibrium character allows for a systematic comparison between sub- and supersonic cases, enabling the isolation of pure PG effects from Mach-number effects and thus an investigation of the validity of common compressibility transformations for compressible PG TBLs. It turned out that the kinematic Rotta–Clauser parameter $\unicode[STIX]{x1D6FD}_{K}$ calculated using the incompressible form of the boundary-layer displacement thickness as length scale is the appropriate similarity parameter to compare both sub- and supersonic cases. Whereas the subsonic APG cases show trends known from incompressible flow, the interpretation of the supersonic PG cases is intricate. Both sub- and supersonic regions exist in the boundary layer, which counteract in their spatial evolution. The boundary-layer thickness $\unicode[STIX]{x1D6FF}_{99}$ and the skin-friction coefficient $c_{f}$, for instance, are therefore in a comparable range for all compressible APG cases. The evaluation of local non-dimensionalized total and turbulent shear stresses shows an almost identical behaviour for both sub- and supersonic cases characterized by similar $\unicode[STIX]{x1D6FD}_{K}$, which indicates the (approximate) validity of Morkovin’s scaling/hypothesis also for compressible PG TBLs. Likewise, the local non-dimensionalized distributions of the mean-flow pressure and the pressure fluctuations are virtually invariant to the local Mach number for same $\unicode[STIX]{x1D6FD}_{K}$-cases. In the inner layer, the van Driest transformation collapses compressible mean-flow data of the streamwise velocity component well into their nearly incompressible counterparts with the same $\unicode[STIX]{x1D6FD}_{K}$. However, noticeable differences can be observed in the wake region of the velocity profiles, depending on the strength of the PG. For both sub- and supersonic cases the recovery factor was found to be significantly decreased by APGs and increased by FPGs, but also to remain virtually constant in regions of approximated equilibrium.

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A Simplified Form of the Auxiliary Equation for Use in the Calculation of Turbulent Boundary Layers

Journal of the Royal Aeronautical Society ◽

10.1017/s0368393100068383 ◽

1958 ◽

Vol 62 (567) ◽

pp. 215-219

Author(s):

T. J. Black

Keyword(s):

Boundary Layer ◽

Turbulent Boundary Layer ◽

Boundary Layers ◽

Turbulent Boundary Layers ◽

Auxiliary Equation ◽

Pressure Gradients ◽

Momentum Thickness ◽

Form Parameter ◽

New Type ◽

Simple Quadrature

A New type of auxiliary equation is given for calculating the development of the form-parameter H in turbulent boundary layers with adverse pressure gradients. The chief advantage of this new method lies in the rapidity and ease of calculation which has been achieved, without apparent sacrifice of accuracy.Whereas the growth of momentum thickness in the turbulent boundary layer can now be rapidly calculated by methods involving only simple quadrature, the prediction of the form parameter development remains a laborious task, while the results obtained do not always appear to justify the complexity of the calculations.

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Experimental Studies on Wake-Induced Transition of Turbulent Boundary Layers

Volume 7: Fluids and Heat Transfer, Parts A, B, C, and D ◽

10.1115/imece2012-86486 ◽

2012 ◽

Author(s):

Keiji Takeuchi ◽

Susumu Fujimoto ◽

Eitaro Koyabu ◽

Tetsuhiro Tsukiji

Keyword(s):

Boundary Layer ◽

Flat Plate ◽

Boundary Layers ◽

Turbulence Intensity ◽

Intensity Distribution ◽

Experimental Studies ◽

Turbulent Boundary Layers ◽

Bypass Transition ◽

Pressure Gradients ◽

Hot Wire

Wake-induced bypass transition of boundary layers on a flat plate subjected to favorable and adverse pressure gradients was investigated. Detailed boundary layer measurements were conducted using two hot-wire probes. A spoked-wheel-type wake generator was used to create periodic wakes in front of the flat plate. The main focus of this study was to reveal the effect of the Strouhal number, which changed by using different numbers of wake-generating bars, on the turbulence intensity distribution and the transition onset position of the boundary layer on the flat plate using two hot-wire probes.

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Similarity treatment of moving-equilibrium turbulent boundary layers in adverse pressure gradients

Journal of Fluid Mechanics ◽

10.1017/s0022112078002621 ◽

1978 ◽

Vol 89 (2) ◽

pp. 305-342 ◽

Cited By ~ 46

Author(s):

B. A. Kader ◽

A. M. Yaglom

Keyword(s):

Boundary Layer ◽

Pressure Gradient ◽

Boundary Layers ◽

Local Equilibrium ◽

Adverse Pressure Gradient ◽

Velocity Profiles ◽

Experimental Information ◽

Turbulent Boundary Layers ◽

Stream Velocity ◽

Pressure Gradients

Dimensional analysis is applied to the velocity profile U(y) of turbulent boundary layers subjected to adverse pressure gradients. It is assumed that the boundary layer is in moving or local equilibrium in the sense that the free-stream velocity U∞ and kinematic pressure gradient α = ρ−1dP/dx vary only slowly with the co-ordinate x. This assumption implies a rather complicated general equation for the velocity gradient dU/dy which may be considerably simplified for several specific regions of the flow. A general family of velocity profiles is derived from the simplified equations supplemented by some experimental information. This family agrees well with almost all existing data on velocity profiles in adverse-pressure-gradient turbulent boundary layers. It may be used for the derivation of a skin-friction law which predicts satisfactorily the values of the wall shear stress at any non-negative value of the pressure gradient. The variation of the boundary-layer thickness with x is also predicted by dimensional considerations.

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Calculation of some turbulent wall currents

Scientific journal of the Ternopil national technical university ◽

10.33108/visnyk_tntu2021.01.089 ◽

2021 ◽

Vol 1 (101) ◽

pp. 89-93

Author(s):

Vitalii Mamchuk ◽

Leonid Romaniuk

Keyword(s):

Experimental Data ◽

Boundary Layer ◽

Mathematical Model ◽

Turbulent Boundary Layer ◽

Boundary Layers ◽

Turbulent Boundary Layers ◽

Positive Pressure ◽

Pressure Gradients ◽

Turbulent Wall

A mathematical model for the calculation of turbulent boundary layers and wall stream has been developed. The results of calculations are compared with the results of other authors on the compliance of the calculated values with the experimental data. The currents that are formed under the influence of positive pressure gradients and lead to the phenomenon of separation of the turbulent boundary layer are studied.

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Structures in turbulent boundary layers subjected to adverse pressure gradients

Journal of Fluid Mechanics ◽

10.1017/s0022112009990814 ◽

2009 ◽

Vol 639 ◽

pp. 101-131 ◽

Cited By ~ 56

Author(s):

JOUNG-HO LEE ◽

HYUNG JIN SUNG

Keyword(s):

Boundary Layer ◽

Pressure Gradient ◽

Boundary Layers ◽

Adverse Pressure Gradient ◽

Turbulent Boundary Layers ◽

Stochastic Estimation ◽

Pressure Gradients ◽

Linear Stochastic Estimation ◽

Mean Width ◽

The Mean

The effects of adverse pressure gradients on turbulent structures were investigated by carrying out direct numerical simulations of turbulent boundary layers subjected to adverse and zero pressure gradients. The equilibrium adverse pressure gradient flows were established by using a power law free-stream distribution U∞ ~ xm. Two-point correlations of velocity fluctuations were used to show that the spanwise spacing between near-wall streaks is affected significantly by a strong adverse pressure gradient. Low-momentum regions are dominant in the middle of the boundary layer as well as in the log layer. Linear stochastic estimation was used to provide evidence for the presence of low-momentum regions and to determine their statistical properties. The mean width of such large-scale structures is closely associated with the size of the hairpin-like vortices in the outer layer. The conditionally averaged flow fields around events producing Reynolds stress show that hairpin-like vortices are the structures associated with the production of outer turbulence. The shapes of the vortices beyond the log layer were found to be similar when their length scales were normalized according to the boundary layer thickness. Estimates of the conditionally averaged velocity fields associated with the spanwise vortical motion were obtained by using linear stochastic estimation. These results confirm that the outer region of the adverse pressure gradient boundary layer is populated with streamwise-aligned vortex organizations, which are similar to those of the vortex packet model proposed by Adrian, Meinhart & Tomkins (J. Fluid Mech., vol. 422, 2000, pp. 1–54). The adverse pressure gradient augments the inclination angles of the packets and the mean streamwise spacing of the vortex heads in the packets.

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Scaling of adverse-pressure-gradient turbulent boundary layers

Journal of Fluid Mechanics ◽

10.1017/s0022112092001873 ◽

1992 ◽

Vol 238 ◽

pp. 699-722 ◽

Cited By ~ 33

Author(s):

P. A. Durbin ◽

S. E. Belcher

Keyword(s):

Boundary Layer ◽

Pressure Gradient ◽

Boundary Layers ◽

Turbulent Boundary Layers ◽

Wall Layer ◽

Small Range ◽

Surface Shear Stress ◽

Pressure Gradients ◽

Mean Flow ◽

The Mean

An asymptotic analysis is developed for turbulent boundary layers in strong adverse pressure gradients. It is found that the boundary layer divides into three distinguishable regions: these are the wall layer, the wake layer and a transition layer. This structure has two key differences from the zero-pressure-gradient boundary layer: the wall layer is not exponentially thinner than the wake; and the wake has a large velocity deficit, and cannot be linearized. The mean velocity profile has a y½ behaviour in the overlap layer between the wall and transition regions.The analysis is done in the context of eddy viscosity closure modelling. It is found that k-ε-type models are suitable to the wall region, and have a power-law solution in the y½ layer. The outer-region scaling precludes the usual ε-equation. The Clauser, constant-viscosity model is used in that region. An asymptotic expansion of the mean flow and matching between the three regions is carried out in order to determine the relation between skin friction and pressure gradient. Numerical calculations are done for self-similar flow. It is found that the surface shear stress is a double-valued function of the pressure gradient in a small range of pressure gradients.

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