History effects and near equilibrium in adverse-pressure-gradient turbulent boundary layers

Turbulent boundary layers under adverse pressure gradients are studied using well-resolved large-eddy simulations (LES) with the goal of assessing the influence of the streamwise pressure-gradient development. Near-equilibrium boundary layers were characterized through the Clauser pressure-gradient parameter $\unicode[STIX]{x1D6FD}$. In order to fulfil the near-equilibrium conditions, the free stream velocity was prescribed such that it followed a power-law distribution. The turbulence statistics pertaining to cases with a constant value of $\unicode[STIX]{x1D6FD}$ (extending up to approximately 40 boundary-layer thicknesses) were compared with cases with non-constant $\unicode[STIX]{x1D6FD}$ distributions at matched values of $\unicode[STIX]{x1D6FD}$ and friction Reynolds number $Re_{\unicode[STIX]{x1D70F}}$. An additional case at matched Reynolds number based on displacement thickness $Re_{\unicode[STIX]{x1D6FF}^{\ast }}$ was also considered. It was noticed that non-constant $\unicode[STIX]{x1D6FD}$ cases appear to approach the conditions of equivalent constant $\unicode[STIX]{x1D6FD}$ cases after long streamwise distances (approximately 7 boundary-layer thicknesses). The relevance of the constant $\unicode[STIX]{x1D6FD}$ cases lies in the fact that they define a ‘canonical’ state of the boundary layer, uniquely characterized by $\unicode[STIX]{x1D6FD}$ and $Re$. The investigations on the flat plate were extended to the flow around a wing section overlapping in terms of $\unicode[STIX]{x1D6FD}$ and $Re$. Comparisons with the flat-plate cases at matched values of $\unicode[STIX]{x1D6FD}$ and $Re$ revealed that the different development history of the turbulent boundary layer on the wing section leads to a less pronounced wake in the mean velocity as well as a weaker second peak in the Reynolds stresses. This is due to the weaker accumulated effect of the $\unicode[STIX]{x1D6FD}$ history. Furthermore, a scaling law suggested by Kitsios et al. (Intl J. Heat Fluid Flow, vol. 61, 2016, pp. 129–136), proposing the edge velocity and the displacement thickness as scaling parameters, was tested on two constant-pressure-gradient parameter cases. The mean velocity and Reynolds-stress profiles were found to be dependent on the downstream development. The present work is the first step towards assessing history effects in adverse-pressure-gradient turbulent boundary layers and highlights the fact that the values of the Clauser pressure-gradient parameter and the Reynolds number are not sufficient to characterize the state of the boundary layer.

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Governing Parameters of Adverse Pressure Gradient Turbulent Boundary Layers

Volume 1: Flow Manipulation and Active Control; Bio-Inspired Fluid Mechanics; Boundary Layer and High-Speed Flows; Fluids Engineering Education; Transport Phenomena in Energy Conversion and Mixing; Turbulent Flows; Vortex Dynamics; DNS/LES and Hybrid RANS/LES Methods; Fluid Structure Interaction; Fluid Dynamics of Wind Energy; Bubble, Droplet, and Aerosol Dynamics ◽

10.1115/fedsm2018-83110 ◽

2018 ◽

Author(s):

Yvan Maciel ◽

Tie Wei ◽

Ayse G. Gungor ◽

Mark P. Simens

Keyword(s):

Boundary Layer ◽

Reynolds Number ◽

Pressure Gradient ◽

Boundary Layers ◽

Outer Region ◽

Adverse Pressure Gradient ◽

Turbulent Boundary Layers ◽

Inertial Parameter ◽

Velocity Scale ◽

Gradient Parameter

We perform a careful nondimensional analysis of the turbulent boundary layer equations in order to bring out, without assuming any self-similar behaviour, a consistent set of nondimensional parameters characterizing the outer region of turbulent boundary layers with arbitrary pressure gradients. These nondimensional parameters are a pressure gradient parameter, a Reynolds number (different from commonly used ones) and an inertial parameter. They are obtained without assuming a priori the outer length and velocity scales. They represent the ratio of the magnitudes of two types of forces in the outer region, using the Reynolds shear stress gradient (apparent turbulent force) as the reference force: inertia to apparent turbulent forces for the inertial parameter, pressure to apparent turbulent forces for the pressure gradient parameter and apparent turbulent to viscous forces for the Reynolds number. We determine under what conditions they retain their meaning, depending on the outer velocity scale that is considered, with the help of seven boundary layer databases. We find the impressive result that if the Zagarola-Smits velocity is used as the outer velocity scale, the streamwise evolution of the three ratios of forces in the outer region can be accurately followed with these non-dimensional parameters in all these flows — not just the order of magnitude of these ratios. This cannot be achieved with three other outer velocity scales commonly used for pressure gradient turbulent boundary layers. Consequently, the three new nondimensional parameters, when expressed with the Zagarola-Smits velocity, can be used to follow — in a global sense — the streamwise evolution of the stream-wise mean momentum balance in the outer region. This study provides a clear and consistent framework for the analysis of the outer region of adverse-pressure-gradient turbulent boundary layers.

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Pressure Gradient Effects on Smooth- and Rough-Surface Turbulent Boundary Layers—Part II: Adverse Pressure Gradient

Journal of Fluids Engineering ◽

10.1115/1.4027475 ◽

2014 ◽

Vol 137 (1) ◽

Cited By ~ 12

Author(s):

Ju Hyun Shin ◽

Seung Jin Song

Keyword(s):

Boundary Layer ◽

Reynolds Number ◽

Pressure Gradient ◽

Boundary Layers ◽

Boundary Layer Thickness ◽

Adverse Pressure Gradient ◽

Turbulent Boundary Layers ◽

Roughness Effects ◽

Mean Velocity ◽

Velocity Defect

An experimental investigation has been conducted to identify the effects of pressure gradient and surface roughness on turbulent boundary layers. In Part II, smooth- and rough-surface turbulent boundary layers with and without adverse pressure gradient (APG) are presented at a fixed Reynolds number (based on the length of flat plate) of 900,000. Flat-plate boundary layer measurements have been conducted using a single-sensor, hot-wire probe. For smooth surfaces, compared to the zero pressure gradient (ZPG) boundary layer, the APG boundary layer has a higher mean velocity defect throughout the boundary layer and lower friction coefficient. APG decreases the streamwise normal Reynolds stress for y less than 0.4 times the boundary layer thickness and increases it slightly in the outer region. For rough surfaces, APG reduces the roughness effects of increasing the mean velocity defect and normal Reynolds stress for y less than 23 and 28 times the average roughness height, respectively. Consistently, for the same roughness, APG decreases the integrated streamwise turbulent kinetic energy. APG also decreases the roughness effect on the friction coefficient, roughness Reynolds number, and roughness shift. Compared to the ZPG boundary layers, the roughness effects on integral boundary layer parameters—boundary layer thickness and momentum thickness—are weaker under APG. Thus, contrary to the favorable pressure gradient (FPG) in part I, APG reduces the roughness effects on turbulent boundary layers.

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Pressure gradient effects on the large-scale structure of turbulent boundary layers

Journal of Fluid Mechanics ◽

10.1017/jfm.2012.531 ◽

2013 ◽

Vol 715 ◽

pp. 477-498 ◽

Cited By ~ 86

Author(s):

Zambri Harun ◽

Jason P. Monty ◽

Romain Mathis ◽

Ivan Marusic

Keyword(s):

Boundary Layer ◽

Reynolds Number ◽

Pressure Gradient ◽

Boundary Layers ◽

Amplitude Modulation ◽

Large Scale ◽

Outer Region ◽

Adverse Pressure Gradient ◽

Turbulent Boundary Layers ◽

Small Scale

AbstractResearch into high-Reynolds-number turbulent boundary layers in recent years has brought about a renewed interest in the larger-scale structures. It is now known that these structures emerge more prominently in the outer region not only due to increased Reynolds number (Metzger & Klewicki, Phys. Fluids, vol. 13(3), 2001, pp. 692–701; Hutchins & Marusic, J. Fluid Mech., vol. 579, 2007, pp. 1–28), but also when a boundary layer is exposed to an adverse pressure gradient (Bradshaw, J. Fluid Mech., vol. 29, 1967, pp. 625–645; Lee & Sung, J. Fluid Mech., vol. 639, 2009, pp. 101–131). The latter case has not received as much attention in the literature. As such, this work investigates the modification of the large-scale features of boundary layers subjected to zero, adverse and favourable pressure gradients. It is first shown that the mean velocities, turbulence intensities and turbulence production are significantly different in the outer region across the three cases. Spectral and scale decomposition analyses confirm that the large scales are more energized throughout the entire adverse pressure gradient boundary layer, especially in the outer region. Although more energetic, there is a similar spectral distribution of energy in the wake region, implying the geometrical structure of the outer layer remains universal in all cases. Comparisons are also made of the amplitude modulation of small scales by the large-scale motions for the three pressure gradient cases. The wall-normal location of the zero-crossing of small-scale amplitude modulation is found to increase with increasing pressure gradient, yet this location continues to coincide with the large-scale energetic peak wall-normal location (as has been observed in zero pressure gradient boundary layers). The amplitude modulation effect is found to increase as pressure gradient is increased from favourable to adverse.

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Measurements in adverse-pressure-gradient turbulent boundary layers with a step change in surface roughness

Journal of Fluid Mechanics ◽

10.1017/s0022112075002200 ◽

1975 ◽

Vol 70 (3) ◽

pp. 573-593 ◽

Cited By ~ 19

Author(s):

W. H. Schofield

Keyword(s):

Surface Roughness ◽

Pressure Gradient ◽

Boundary Layers ◽

Leading Edge ◽

Adverse Pressure Gradient ◽

Turbulent Boundary Layers ◽

Step Change ◽

Internal Layer ◽

Mean Velocity ◽

Law Of The Wall

The response of turbulent boundary layers to sudden changes in surface roughness under adverse-pressure-gradient conditions has been studied experimentally. The roughness used was in the ‘d’ type array of Perry, Schofield & Joubert (1969). Two cases of a rough-to-smooth change in surface roughness were considered in the same arbitrary adverse pressure gradient. The two cases differed in the distance of the surface discontinuity from the leading edge and gave two sets of flow conditions for the establishment and growth of the internal layer which develops downstream from a change in surface roughness. These conditions were in turn different from those in the zero-pressure-gradient experiments of Antonia & Luxton. The results suggest that the growth of the new internal layer depends solely on the new conditions at the wall and scales with the local roughness length of that wall. Mean velocity profiles in the region after the step change in roughness were accurately described by Coles’ law of the wall-law of the wake combination, which contrasts with the zero-pressure-gradient results of Antonia & Luxton. The skin-friction coefficient after the step change in roughness did not overshoot the equilibrium distribution but made a slow adjustment downstream of the step. Comparisons of mean profiles indicate that similar defect profile shapes are produced in layers with arbitrary adverse pressure gradients at positions where the values of Clauser's equilibrium parameter β (= δ*τ−10dp/dx) are similar, provided that the pressure-gradient history and local values of the pressure gradient are also similar.

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Scaling of Favorable Pressure Gradient Turbulent Boundary Layers With Eventual Relaminarization

Volume 2: Fora ◽

10.1115/fedsm2005-77486 ◽

2005 ◽

Author(s):

Rau´l Bayoa´n Cal ◽

Xia Wang ◽

Luciano Castillo

Keyword(s):

Boundary Layer ◽

Pressure Gradient ◽

Boundary Layers ◽

Reynolds Shear Stress ◽

Velocity Profiles ◽

Turbulent Boundary Layers ◽

Reynolds Stresses ◽

Mean Velocity ◽

Favorable Pressure Gradient ◽

The Mean

Applying similarity analysis to the RANS equations of motion for a pressure gradient turbulent boundary layer, Castillo and George [1] obtained the scalings for the mean deficit velocity and the Reynolds stresses. Following this analysis, Castillo and George studied favorable pressure gradient (FPG) turbulent boundary layers. They were able to obtain a single curve for FPG flows when scaling the mean deficit velocity profiles. In this study, FPG turbulent boundary layers are analyzed as well as relaminarized boundary layers subjected to an even stronger FPG. It is found that the mean deficit velocity profiles diminish when scaled using the Castillo and George [1] scaling, U∞, and the Zagarola and Smits [2] scaling, U∞δ*/δ. In addition, Reynolds stress data has been analyzed and it is found that the relaminarized boundary layer data decreases drastically in all components of the Reynolds stresses. Furthermore, it will be shown that the shape of the profile for the wall-normal and Reynolds shear stress components change drastically given the relaminarized state. Therefore, the mean velocity deficit profiles as well as Reynolds stresses are found to be necessary in order to understand not only FPG flows, but also relaminarized boundary layers.

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Turbulent boundary layers in decreasing adverse pressure gradients

Journal of Fluid Mechanics ◽

10.1017/s0022112066001344 ◽

1966 ◽

Vol 26 (3) ◽

pp. 481-506 ◽

Cited By ~ 23

Author(s):

A. E. Perry

Keyword(s):

Boundary Layer ◽

Pressure Gradient ◽

Adverse Pressure Gradient ◽

Turbulent Boundary Layers ◽

Pressure Gradients ◽

Streamwise Direction ◽

Mean Velocity ◽

Boundary Layer Development ◽

Regional Similarity ◽

Layer Development

The results of a detailed mean velocity survey of a smooth-wall turbulent boundary layer in an adverse pressure gradient are described. Close to the wall, a variety of profiles shapes were observed. Progressing in the streamwise direction, logarithmic, ½-power, linear and$\frac{3}{2}$-power distributions seemed to form, and generally each predominated at a different stage of the boundary-layer development. It is believed that the phenomenon occurred because of the nature of the pressure gradient imposed (an initially high gradient which fell to low values as the boundary layer developed) and attempts are made to describe the flow by an extension of the regional similarity hypothesis proposed by Perry, Bell & Joubert (1966). Data from other sources is limited but comparisons with the author's results are encouraging.

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Experimental results for the transpired turbulent boundary layer in an adverse pressure gradient

Journal of Fluid Mechanics ◽

10.1017/s0022112075001474 ◽

1975 ◽

Vol 69 (2) ◽

pp. 353-375 ◽

Cited By ~ 45

Author(s):

P. S. Andersen ◽

W. M. Kays ◽

R. J. Moffat

Keyword(s):

Boundary Layer ◽

Shear Stress ◽

Turbulent Boundary Layer ◽

Pressure Gradient ◽

Boundary Layer Equation ◽

Adverse Pressure Gradient ◽

Stream Velocity ◽

Pressure Gradients ◽

Mean Velocity ◽

Displacement Thickness

An experimental investigation of the fluid mechanics of the transpired turbulent boundary layer in zero and adverse pressure gradients was carried out on the Stanford Heat and Mass Transfer Apparatus. Profiles of (a) the mean velocity, (b) the intensities of the three components of the turbulent velocity fluctuations and (c) the Reynolds stress were obtained by hot-wire anemometry. The wall shear stress was measured by using an integrated form of the boundary-layer equation to ‘extrapolate’ the measured shear-stress profiles to the wall.The two experimental adverse pressure gradients corresponded to free-stream velocity distributions of the type u∞ ∞ xm, where m = −0·15 and −0·20, x being the streamwise co-ordinate. Equilibrium boundary layers (i.e. flows with velocity defect profile similarity) were obtained when the transpiration velocity v0 was varied such that the blowing parameter B = pv0u∞/τ0 and the Clauser pressure-gradient parameter $\beta\equiv\delta_1\tau_0^{-1}\,dp/dx $ were held constant. (τ0 is the shear stress at the wall and δ1 is the displacement thickness.)Tabular and graphical results are presented.

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An Approximate Method for the Calculation of Turbulent Boundary Layers in Diffusers

Aeronautical Quarterly ◽

10.1017/s0001925900010404 ◽

1957 ◽

Vol 8 (1) ◽

pp. 58-77 ◽

Cited By ~ 1

Author(s):

J. F. Norbury

Keyword(s):

Boundary Layer ◽

Shear Stress ◽

Pressure Gradient ◽

Approximate Method ◽

Boundary Layers ◽

Stress Gradient ◽

Adverse Pressure Gradient ◽

Turbulent Boundary Layers ◽

Form Parameter ◽

Parameter Equation

SummaryAn approximate method is described for the calculation of turbulent boundary layers in which the turbulence is developed before the commencement of the adverse pressure gradient, as in most diffuser layers. It is based on a method due to Spence which has been modified and also extended to the calculation of three-dimensional diverging layers such as occur in ducts whose breadth is increasing. The velocity profiles occurring in a diverging layer are examined and it is shown that the inner part obeys the universal logarithmic law, as in two-dimensional layers. This result is used to obtain an equation for the form parameter in diverging layers, by substitution in the equation of motion and incorporation of the equations of momentum and continuity for diverging flow. The form parameter equation contains a term involving the gradient of shear stress at y = θ and values of this term are obtained by the analysis of experimental data and the substitution of known values for all the other terms in the form parameter equation. Values of the term involving shear stress gradient are then correlated in terms of known boundary layer quantities, and the resulting correlation allows the formulation of a step-by-step method for the solution of the form parameter equation. This may be used in conjunction with the momentum equation to predict the boundary layer growth. It was not found possible to effect a satisfactory correlation for boundary layers on lifting aerofoils, in which the turbulence develops within the adverse pressure gradient, and the method cannot be used for the prediction of such layers. The results of a number of calculations are given.

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