scholarly journals A Unified View of the Law of the Wall Using Mixing-Length Theory

1973 ◽  
Vol 24 (1) ◽  
pp. 55-70 ◽  
Author(s):  
V C Patel
Keyword(s):  
Friction Velocity ◽  
Critical Value ◽  
Wall Region ◽  
Pressure Gradients ◽  
Mixing Length ◽  
Stress Distributions ◽  
General Validity ◽  
Law Of The Wall ◽  
Unified View ◽  
Mixing Length Theory

SummaryIt is shown that, if the well-known mixing-length formula is regarded simply as a relationship between the velocity and the stress distributions in the wall region of a turbulent flow, then a truly universal distribution of mixing length is sufficient to describe the experimentally observed departures of the velocity distribution from the usual law of the wall as a result of severe pressure gradients and transverse surface curvature. Comparisons have been made with a wide variety of experimental data to demonstrate the general validity of the mixing-length model in describing the flow close to a smooth wall.An extension of the re-laminarisation criterion of Patel and Head, and some experimental evidence, suggest that the thick axisymmetric boundary layer on a slender cylinder placed axially in a uniform stream cannot be maintained in a fully turbulent state for values of the Reynolds number, based on friction velocity and cylinder radius, below a certain critical value.

Characteristics of turbulent boundary layers at low Reynolds numbers with and without transpiration

1970 ◽  
Vol 42 (4) ◽  
pp. 769-802 ◽  
Author(s):  
Roger L. Simpson
Keyword(s):  
Boundary Layers ◽  
Reasonable Agreement ◽  
Eddy Viscosity ◽  
Outer Region ◽  
Reynolds Numbers ◽  
Wall Region ◽  
Mixing Length ◽  
Low Reynolds Numbers ◽  
Velocity Defect ◽  
Law Of The Wall

An extension to Coles's (1956) ‘law of the wall–law of the wake’ formulation for incompressible unblown boundary layers with momentum thickness Reynolds numberReθ> 6000. is made forReθ< 6000. It is found that κ = 0·40, the von Kármán constant forReθ> 6000, is replaced by Ω = 0·40 (Reθ/6000)⅛forReθ< 6000. Based upon the data of Simpson (1967) this formulation is extended to injection and undersucked (dθ/dx> 0) flows in ‘law of the wall’ and ‘velocity-defect’ representations. This law of the wall for the logarithmic turbulent region and Reichardt's sublayer variation of εM/ν are used to obtain a continuous expression for εM/ν as a function ofU+,V+w, andReθfor the wall region. This expression is in reasonable agreement with the generated εM/ν blowing results and in less agreement with the unblown and suction results. Eddy viscosity and mixing length results confirm that εM/δ*U∞∝ Ω2andl/δ ∝ Ω for the outer region and that εM/δ*U∞andl/δ are substantially independent of blowing and moderate suction, as also reflected by the velocity defect representation for injection and suction.

Analysis of Thrust Bearings Operating in Turbulent Regime

1988 ◽  
Vol 110 (3) ◽  
pp. 555-560 ◽  
Author(s):  
M. Harada ◽  
H. Aoki
Keyword(s):  
Steady State ◽  
Thrust Bearing ◽  
Three Dimensional ◽  
Cross Coupling ◽  
Fluid Film ◽  
Turbulent Regime ◽  
Pressure Gradients ◽  
Mixing Length ◽  
Thrust Bearings ◽  
Mixing Length Theory

This paper relates to the turbulent motion in the lubricant fluid film with centrifugal effects and the lubrication theory for thrust bearings operating in turbulent regime. Using Prandtl’s mixing-length theory, three-dimensional turbulent velocity distributions, including pressure gradients and centrifugal effects, are calculated, and the cross-coupling of nonplanar flow of the lubricant fluid film is discussed. From these results, turbulent lubrication equations with centrifugal effects are derived. Applying these lubrication equations to a sectorial inclined thrust bearing, the steady-state characteristics and the dynamic ones are calculated.

Mixing Length in the Wall Region of Turbulent Boundary Layers

1977 ◽  
Vol 28 (2) ◽  
pp. 97-110 ◽  
Author(s):  
R A McD Galbraith ◽  
S Sjolander ◽  
M R Head
Keyword(s):  
Boundary Layers ◽  
Universal Constant ◽  
Turbulent Boundary Layers ◽  
Wall Region ◽  
Mixing Length ◽  
Law Of The Wall ◽  
Universal Law

SummaryEvidence is presented to show that the universal law of the wall has a wider range of validity than the assumptionl= ky, with k a universal constant. If an effective value of k is defined for the wall region its value is shown to vary between wide limits, and keffcan be correlated with other parameters describing the flow in the wall region.

scholarly journals The resilience of the logarithmic law to pressure gradients: evidence from direct numerical simulation

2009 ◽  
Vol 643 ◽  
pp. 163-175 ◽  
Author(s):  
RODERICK JOHNSTONE ◽  
GARY N. COLEMAN ◽  
PHILIPPE R. SPALART
Keyword(s):  
Numerical Simulation ◽  
Shear Stress ◽  
Direct Numerical Simulation ◽  
Poiseuille Flow ◽  
Scaling Laws ◽  
Scaling Law ◽  
Local Stress ◽  
Wall Region ◽  
Pressure Gradients ◽  
Law Of The Wall

Wall-bounded turbulence in pressure gradients is studied using direct numerical simulation (DNS) of a Couette–Poiseuille flow. The motivation is to include adverse pressure gradients, to complement the favourable ones present in the well-studied Poiseuille flow, and the central question is how the scaling laws react to a gradient in the total shear stress or equivalently to a pressure gradient. In the case considered here, the ratio of local stress to wall stress, namely τ+, ranges from roughly 2/3 to 3/2 in the ‘wall region’. By this we mean the layer believed not to be influenced by the opposite wall and therefore open to simple, universal behaviour. The normalized pressure gradients p+ ≡ dτ+/dy+ at the two walls are −0.00057 and +0.0037. The outcome is in broad agreement with the findings of Galbraith, Sjolander & Head (Aeronaut. Quart. vol. 27, 1977, pp. 229–242) relating to boundary layers (based on measured profiles): the logarithmic velocity profile is much more resilient than two other, equally plausible assumptions, namely universality of the mixing length ℓ = κy and that of the eddy viscosity νt = uτκy. In pressure gradients, with τ+ ≠ 1, these three come into conflict, and our primary purpose is to compare them. We consider that the Kármán constant κ is unique but allow a range from 0.38 to 0.41, consistent with the current debates. It makes a minor difference in the interpretation. This finding of resilience appears new as a DNS result and is free of the experimental uncertainty over skin friction. It is not as distinct in the (rather strong) adverse gradient as it is in the favourable one; for instance the velocity U+ at y+ = 50 is lower by 3% on the adverse gradient side. A plausible cause is that the wall shear stress is small and somewhat overwhelmed by the stress and kinetic energy in the bulk of the flow. The potential of a correction to the ‘law of the wall’ based purely on p+ is examined, with mixed results. We view the preference for the log law as somewhat counter-intuitive in that the scaling law is non-local but also as becoming established and as highly relevant to turbulence modelling.

A law of the wall for compressible turbulent boundary layers with air injection

1969 ◽  
Vol 37 (3) ◽  
pp. 449-456 ◽  
Author(s):  
L. C. Squire
Keyword(s):  
Experimental Data ◽  
Boundary Layers ◽  
Present Report ◽  
Turbulent Boundary Layers ◽  
Air Injection ◽  
Mixing Length ◽  
Law Of The Wall ◽  
Friction Measurements ◽  
Mixing Length Theory ◽  
Considerable Body

A considerable body of experimental data now exists concerning turbulent boundary layers with air injection at the wall, both at subsonic and at supersonic speeds. In the present report these data for Mach numbers up to 6·5 have been analyzed to find the parameters which occur in the law of the wall as deduced from mixing-length theory. Although the absolute values of the parameters are subject to error because of the lack of accurate skin-friction measurements, the trends of these parameters with Mach number and injection mass flow are clearly defined.

On the criteria for reverse transition in a two-dimensional boundary layer flow

1969 ◽  
Vol 35 (2) ◽  
pp. 225-241 ◽  
Author(s):  
M. A. Badri Narayanan ◽  
V. Ramjee
Keyword(s):  
Longitudinal Velocity ◽  
Outer Region ◽  
Transition Process ◽  
Velocity Profiles ◽  
Wall Region ◽  
Two Dimensional ◽  
Mean Velocity ◽  
Law Of The Wall ◽  
Reverse Transition ◽  
Turbulence Decay

Experiments on reverse transition were conducted in two-dimensional accelerated incompressible turbulent boundary layers. Mean velocity profiles, longitudinal velocity fluctuations $\tilde{u}^{\prime}(=(\overline{u^{\prime 2}})^{\frac{1}{2}})$ and the wall-shearing stress (TW) were measured. The mean velocity profiles show that the wall region adjusts itself to laminar conditions earlier than the outer region. During the reverse transition process, increases in the shape parameter (H) are accompanied by a decrease in the skin friction coefficient (Cf). Profiles of turbulent intensity (u’2) exhibit near similarity in the turbulence decay region. The breakdown of the law of the wall is characterized by the parameter \[ \Delta_p (=\nu[dP/dx]/\rho U^{*3}) = - 0.02, \] where U* is the friction velocity. Downstream of this region the decay of $\tilde{u}^{\prime}$ fluctuations occurred when the momentum thickness Reynolds number (R) decreased roughly below 400.

The Effects of Adverse Pressure Gradients on Momentum and Thermal Structures in Transitional Boundary Layers: Part 2—Fluctuation Quantities

1996 ◽  
Vol 118 (4) ◽  
pp. 728-736 ◽  
Author(s):  
S. P. Mislevy ◽  
T. Wang
Keyword(s):  
Boundary Layer ◽  
Shear Stress ◽  
Transition Region ◽  
Boundary Layers ◽  
Reynolds Shear Stress ◽  
Turbulent Shear ◽  
Wall Region ◽  
Pressure Gradients ◽  
Baseline Case ◽  
Near Wall

The effects of adverse pressure gradients on the thermal and momentum characteristics of a heated transitional boundary layer were investigated with free-stream turbulence ranging from 0.3 to 0.6 percent. Boundary layer measurements were conducted for two constant-K cases, K1 = −0.51 × 10−6 and K2 = −1.05 × 10−6. The fluctuation quantities, u′, ν′, t′, the Reynolds shear stress (uν), and the Reynolds heat fluxes (νt and ut) were measured. In general, u′/U∞, ν′/U∞, and νt have higher values across the boundary layer for the adverse pressure-gradient cases than they do for the baseline case (K = 0). The development of ν′ for the adverse pressure gradients was more actively involved than that of the baseline. In the early transition region, the Reynolds shear stress distribution for the K2 case showed a near-wall region of high-turbulent shear generated at Y+ = 7. At stations farther downstream, this near-wall shear reduced in magnitude, while a second region of high-turbulent shear developed at Y+ = 70. For the baseline case, however, the maximum turbulent shear in the transition region was generated at Y+ = 70, and no near-wall high-shear region was seen. Stronger adverse pressure gradients appear to produce more uniform and higher t′ in the near-wall region (Y+ < 20) in both transitional and turbulent boundary layers. The instantaneous velocity signals did not show any clear turbulent/nonturbulent demarcations in the transition region. Increasingly stronger adverse pressure gradients seemed to produce large non turbulent unsteadiness (or instability waves) at a similar magnitude as the turbulent fluctuations such that the production of turbulent spots was obscured. The turbulent spots could not be identified visually or through conventional conditional-sampling schemes. In addition, the streamwise evolution of eddy viscosity, turbulent thermal diffusivity, and Prt, are also presented.

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