Momentum Thickness Reynolds Number Research Articles

Correlation-Based Transition Modeling for Unstructured Parallelized Computational Fluid Dynamics Codes

A new correlation-based transition model has been developed, which is built strictly on local variables. As a result, the transition model is compatible with modern computational fluid dynamics techniques such as unstructured grids and massively parallel execution. The model is based on two transport equations, one for intermittency and one for a transition onset criterion in terms of momentum-thickness Reynolds number. A number of validation papers have been published on the basic formulation of the model. However, until now the full model correlations have not been published. The main goal of the present paper is to publish the full model and release it to the research community so that it can continue to be further validated and possibly extended or improved. Included in this paper are a number of test cases that can be used to validate the implementation of the model in a given computational fluid dynamics code. The authors believe that the current formulation is a significant step forward in engineering transition modeling, as it allows the combination of transition correlations with general-purpose computational fluid dynamics codes. There is a strong potential that the model will allow the first-order effects of transition to be included in everyday industrial computational fluid dynamics simulations.

Effects of large-scale free stream turbulence on a turbulent boundary layer

Results of a wind tunnel experiment in which there are systematic variations of free stream turbulence above a flat-plate boundary layer are presented. Upstream of the plate, an active grid generates free stream turbulence varying in intensity from 0.25% to 10.5%. The momentum thickness Reynolds number of the boundary layer varies from 550 to nearly 3000. In all cases, the ratio of the free stream turbulence length scale to the boundary layer depth is greater than unity. Hotwire measurements show that, at high turbulence intensities, the effects of the free stream turbulence extend deep into the boundary layer, affecting the wall stress as well as the small-scale (derivative) statistics. Premultiplied energy spectra show a double peak. At very low free stream turbulence intensities these peaks are associated with the inner and outer scales of the turbulent boundary layer, but at high turbulence intensities the free stream energy peak dominates over the boundary layer’s outer scale. The implications of the effect of the large free stream turbulence scales on the small, near-wall scales is discussed with reference to recent high Reynolds number experiments in a turbulent boundary layer without free stream turbulence [Hutchins and Marusic, Philos. Trans. R. Soc. London, Ser. A 365, 647 (2007)].

An Integral Solution for Heat Transfer in Accelerating Turbulent Boundary Layers

An equilibrium thermal wake strength parameter is developed for a two-dimensional turbulent boundary layer flow and is then used in the combined thermal law of the wall and the wake to give an approximate temperature profile to insert into the integral form of the thermal energy equation. After the solution of the integral x momentum equation, the integral thermal energy equation is solved for the local Stanton number as a function of position x for accelerating turbulent boundary layers. A simple temperature distribution in the thermal “superlayer” is part of the present modeling. The analysis includes a dependence of the hydrodynamic and thermal wake strengths on the momentum thickness and enthalpy thickness Reynolds numbers, respectively. An approximate dependence of the turbulent Prandtl number, in the “log” region, on the strength of the favorable pressure gradient is proposed and incorporated into the solution. The resultant solution for the Stanton number distribution in accelerated turbulent flows is compared with experimental data in the literature. A comparison of the present predictions is also made to a finite difference solution, which uses the turbulent kinetic energy—turbulent dissipation model of turbulence, for a few cases of accelerating flows.

Rotta skin friction law and Schoenherr formula

This paper examines the Rotta skin friction law (1950) and the Schoenherr empirical formula (1932) for flat plate turbulent boundary layers in the light of direct skin friction measurements conducted in the range of the momentum thickness Reynolds number up to Rθ = 2.13 × 105. First, we demonstrate that the skin friction data almost perfectly follow the Rotta skin friction law derived by assuming the existence of the logarithmic overlap region and by using the velocity profile constants determined by Nagib et al (2007 Phil. Trans. R. Soc. A 365 755–70). Next, we clarify the high Reynolds number asymptotic form of the Rotta law and derive the Schoenherr empirical formula from the asymptotic relation. The Schoenherr formula is shown to be a good approximation to the Rotta skin friction law for Rθ greater than 1.1×104. Above all, the present effort yields a new formula for the local skin friction, which is almost identical with the Rotta skin friction law in accuracy, with the relative deviation being less than 0.05% in the range of Cf less than 0.0037.

Direct numerical simulation of turbulence in a nominally zero-pressure-gradient flat-plate boundary layer

A nominally-zero-pressure-gradient incompressible boundary layer over a smooth flat plate was simulated for a continuous momentum thickness Reynolds number range of 80 ≤ Reθ ≤ 940. Transition which is completed at approximately Reθ = 750 was triggered by intermittent localized disturbances arising from patches of isotropic turbulence introduced periodically from the free stream at Reθ = 80. Streamwise pressure gradient is quantified with several measures and is demonstrated to be weak. Blasius boundary layer is maintained in the early transitional region of 80 < Reθ < 180 within which the maximum deviation of skin friction from the theoretical solution is less than 1%. Mean and second-order turbulence statistics are compared with classic experimental data, and they constitute a rare DNS dataset for the spatially developing zero-pressure-gradient turbulent flat-plate boundary layer. Our calculations indicate that in the present spatially developing low-Reynolds-number turbulent flat-plate boundary layer, total shear stress mildly overshoots the wall shear stress in the near-wall region of 2–20 wall units with vanishing normal gradient at the wall. Overshoots as high as 10% across a wider percentage of the boundary layer thickness exist in the late transitional region. The former is a residual effect of the latter. The instantaneous flow fields are vividly populated by hairpin vortices. This is the first time that direct evidence (in the form of a solution of the Navier–Stokes equations, obeying the statistical measurements, as opposed to synthetic superposition of the structures) shows such dominance of these structures. Hairpin packets arising from upstream fragmented Λ structures are found to be instrumental in the breakdown of the present boundary layer bypass transition.

Near-surface turbulence in the atmospheric boundary layer

Measurements of the near-surface turbulence in the atmospheric boundary layer have been made using hot-wire probes above the salt flats of northwestern Utah, where the momentum thickness Reynolds number, R θ , is O ( 10 6 ) , and the surface is smooth and nearly devoid of flow obstructions. The measurements were made with arrays of up to 24 parallel straight sensors and with a modular 12-sensor probe capable of measuring all of the components of the instantaneous velocity vector and velocity gradient tensor. Measurements were also made in a laboratory wind tunnel at R θ = 1730 using 22 straight sensors. The data analysis focuses on the effects of the Reynolds number on turbulence properties and on the physics of the dissipation rate of turbulent kinetic energy. Some properties are found to be dependent on the Reynolds number when normalized with inner variables, while others are not. Among those that show the significant Reynolds number dependence are the rms and the skewness factor of the streamwise velocity fluctuations. Significant differences in flow structure, particularly those related to high rates of dissipation, are implied by the data. The joint PDF and covariance integrand of streamwise and wall normal vorticity fluctuations show less preferred orientation of the vorticity vector in the buffer layer at R θ of O ( 10 6 ) than at R θ = 1070 . The largest contribution to the dissipation rate, at O ( 10 6 ) is by the ∂ w / ∂ z velocity gradient, while this term makes a quite small contribution to the dissipation rate at low R θ . Here w and z are the spanwise velocity fluctuations and direction, respectively. Conditional analysis in the streamwise-wall normal ( x − y ) plane based on high instantaneous dissipation rate shows that the typical high dissipation rate events are generally similar at high and low Reynolds numbers, but display some significant differences.

Large-eddy simulation of separated flow over a three-dimensional axisymmetric hill

The paper examines, by means of highly resolved large-eddy simulations, the fluid mechanic behaviour of an incompressible, turbulent flow separating from an axisymmetric, hill-shaped obstacle. The hill is subjected to a turbulent flat-plate boundary layer of thickness half of the hill height, developing within a large duct in which the hill is placed. The Reynolds number based on the hill height and the free-stream velocity is 130000, and the momentum-thickness Reynolds number of the boundary layer is 7300. Extensive and detailed experimental data are available for these same conditions, and these are compared with the simulations. Two entirely independent simulations are reported, undertaken by different groups with different codes and different grids. They are shown to agree closely in most respects, and any differences are carefully identified and discussed. Mean-flow properties, pressure distributions and turbulence characteristics are reported and compared with the experimental data, with particular emphasis being placed on the separation region that is formed in the leeward of the hill and on the near wake. The connection between the wall topology of the mean flow and the secondary motions in the wake is discussed in detail. Finally, some aspects of the unsteady flow are analysed, including the unsteadiness of the separation process and structural features and coherent motions in the wake.

Effect of rheological properties on drag reduction in turbulent boundary layer flow

Direct numerical simulation of a zero-pressure gradient drag-reducing turbulent boundary layer of viscoelastic fluids was systematically performed at the momentum-thickness Reynolds number Reθ0=500 and Weissenberg number We=25 using constitutive equation models such as the Oldroyd-B, the finitely extensible nonlinear elastic Peterlin model at the maximum chain extensibility parameters L2=100, 1000, and 10 000, and the Giesekus model at the mobility factors α=0.01, 0.001, and 0.0001, where the ratios of solvent viscosity to zero shear rate solution viscosity, β, were 0.9, 0.99, and 0.999. For the case that the elongational viscosity for the steady elongational flow was identical, the streamwise variation in the drag reduction (DR) was thoroughly investigated, and then the effects of rheological properties such as the elongational and shear viscosities and the first and the second normal stress differences on DR were clarified. It is found that the streamwise profile of DR shifts downstream with the decrease in the first normal stress difference. The shear-thinning property and the first normal stress difference slightly affect the maximum DR, while the decrease in the magnitude of the second normal stress difference results in the decrease in the maximum DR.

Large-Eddy simulations of circular synthetic jets in quiescent surroundings and in turbulent cross-flow

Large-Eddy simulation (LES) is used to investigate the physical processes involved in the injection of a synthetic (zero-net-mass-flux) jet in quiescent surroundings and into a zero-pressure-gradient, turbulent boundary layer over a flat plate at conditions corresponding to experimental data obtained by others. The former case is studied because it constitutes a key ingredient of the latter. Specifically, it allows the computational framework to be verified, basic properties of synthetic jets to be studied, and the sensitivity of the jet structure to the details of the computational representation of the jet orifice to be investigated. In particular, the validity of representing the circular orifice, within a multi-block H-topology mesh, alternatively, by an equivalent square orifice, or by means of a solid-cell-blocking technique or via local mesh distortion, is examined. The boundary layer ahead of the jet is generated by a separate precursor simulation at a momentum-thickness Reynolds number Re θ = 2380 , providing the main simulation with a full and accurate description of the unsteady conditions at the computational inflow boundary. Time- and phase-averaged results are presented and compared against experimental data.

Influence of adverse pressure gradient on rough-wall turbulent flows

This paper presents an experimental investigation of adverse pressure gradient turbulent flow over two rough surfaces and a reference smooth surface. The adverse pressure gradient was produced in an asymmetric diffuser whose opening angle was 3°. The rough surfaces comprised sand grains and gravels of nominal mean diameters of 1.55 mm and 4.22 mm, respectively. The tests were conducted at an approach flow velocity of 0.5 m/s and the momentum thickness Reynolds number varied from 900 to 3000. A particle image velocimetry technique was used for the velocity measurements. Profiles of the mean velocity, turbulent intensities, Reynolds stress ratios, mixing length, eddy viscosity and the production terms were then obtained to document the effects of adverse pressure gradient (APG) on low Reynolds number rough-wall turbulent boundary layers. The results indicate that APG thickens the boundary layer and roughness sublayer. The APG and surface roughness also enhanced the production of turbulence as well as the turbulence level when compared with the smooth-wall data.

界面活性剤水溶液の抵抗低減乱流境界層における乱れ構造 : 抵抗低減率の異なる流れ場の比較(流体工学,流体機械)

In this study, turbulence structures of the drag-reducing turbulent boundary layer in the surfactant solution at the momentum-thickness Reynolds number Re_θ≃800 were compared between at the large and small drag reduction ratios. Turbulence statistics and structures were obtained by the two-component LDV system and the PIV system for the streamwise and wall-normal (x-y) section and the streamwise and spanwise (x-z) section. The surfactant solution used here was a mixture of cetyltrimethyl ammonium chloride (CTAC) with sodium salicylate as counterion, which was dissolved in tap water. The concentration of CTAC was 65ppm, and the molar ratio of counterion to CTAC was two. The drag reduction ratios DR=78% and 34% were obtained at the solution temperature T=30℃ and 35℃, respectively. The quadrant analysis of streamwise and wall-normal velocity fluctuations revealed that at DR=78%, the positive contribution of the sweep and ejection events to the Reynolds shear stress was comparable with the negative contribution of the inward-interaction and outward-interaction events, so that the Reynolds shear stress was almost zero. At DR=78%, both streamwise and spanwise velocity fluctuations were attenuated largely across the turbulent boundary layer. On the other hand, at DR=34%, turbulence structures were similar to those of water as a whole, although the scale was larger compared to the water.

Lagrangian measurements of inertial particle accelerations in a turbulent boundary layer

Two-dimensional Lagrangian acceleration statistics of inertial particles in a turbulent boundary layer with free-stream turbulence are determined by means of a particle tracking technique using a high-speed camera moving along the side of the wind tunnel at the mean flow speed. The boundary layer is formed above a flat plate placed horizontally in the tunnel, and water droplets are fed into the flow using two different methods: sprays placed downstream from an active grid, and tubes fed into the boundary layer from humidifiers. For the flow conditions studied, the sprays produce Stokes numbers varying from 0.47 to 1.2, and the humidifiers produce Stokes numbers varying from 0.035 to 0.25, where the low and high values refer to the outer boundary layer edge and the near-wall region, respectively. The Froude number is approximately 1.0 for the sprays and 0.25 for the humidifiers, with a small variation within the boundary layer. The free-stream turbulence is varied by operating the grid in the active mode as well as a passive mode (the latter behaves as a conventional grid). The boundary layer momentum-thickness Reynolds numbers are 840 and 725 for the active and passive grid respectively. At the outer edge of the boundary layer, where the shear is weak, the acceleration probability density functions are similar to those previously observed in isotropic turbulence for inertial particles. As the boundary layer plate is approached, the tails of the probability density functions narrow, become negatively skewed, and their peak occurs at negative accelerations (decelerations in the streamwise direction). The mean deceleration and its root mean square (r.m.s.) increase to large values close to the plate. These effects are more pronounced at higher Stokes number. In the vertical direction, there is a slight downward mean deceleration and its r.m.s., which is lower in magnitude than that of the streamwise component, peaks in the buffer region. Although there are free-stream turbulence effects, and the complex boundary layer structure plays an important role, a simple model suggests that the acceleration behaviour is dominated by shear, gravity and inertia. The results are contrasted with inertial particles in isotropic turbulence and with fluid particle acceleration statistics in a boundary layer. The background velocity field is documented by means of hot-wire anemometry and laser Doppler velocimetry measurements. These appear to be the first Lagrangian acceleration measurements of inertial particles in a shear flow.

Correlations for modeling transitional boundary layers under influences of freestream turbulence and pressure gradient

This paper presents mathematical expressions for two significant parameters which control the onset location and length of transition in the γ– Re θ transition model of Menter et al. [Menter, F.R., Langtry, R.B., Volker, S., Huang, P.G., 2005. Transition modelling for general purpose CFD codes. In: ERCOFTAC International Symposium on Engineering Turbulence Modelling and Measurements]. The expressions are formulated and calibrated by means of numerical experiments for predicting transitional boundary layers under the influences of freestream turbulence and pressure gradient. It was also found that the correlation for transition momentum thickness Reynolds number needs only to be expressed in terms of local turbulence intensity, so that the more complex form that includes pressure gradient effects is unnecessary. Transitional boundary layers on a flat plate both with and without pressure gradients are employed to assess the performance of these two expressions for predicting the transition. The results show that the proposed expressions can work well with the model of Menter et al. (2005).

The sound from mixing layers simulated with different ranges of turbulence scales

The role of turbulence scales in generating far-field sound in free shear flows is studied via direct numerical simulations of temporally developing, Mach 0.9 mixing layers. Four flows were simulated, starting from the same initial conditions but with Reynolds numbers that varied by a factor of 12. Above momentum thickness Reynolds number Reδm≈300, all the mixing layers radiate over 85% of the acoustic energy of the apparently asymptotically high-Reynolds-number value that we are able to compute. Turbulence energy and pressure wavenumber spectra show the expected Reynolds number dependence; the two highest Reynolds number simulations show evidence of an inertial range and Kolmogorov scaling at the highest wavenumbers. Far-field pressure spectra all decay much more rapidly with wavenumber than the corresponding near-field spectra and show significantly less sensitivity to Reynolds number. Low wavenumbers account for nearly all of the radiated acoustic energy. Far-field streamwise wavenumber pressure spectra scale well with the layer momentum thickness, consistent with the insensitivity to Reynolds number of the largest turbulence structures. At higher wavenumbers the streamwise spectra scale best with the Taylor microscale. Interestingly, none of the spanwise far-field pressure spectra scale well with momentum thickness despite doing so in the near-field turbulence. Instead they scale well at all wavenumbers with the turbulence microscale. Implications of these results for large-eddy simulation of jet noise are discussed.

Turbulent wall pressure fluctuation measurements on a towed model at high Reynolds numbers

Turbulent wall pressure fluctuation measurements were made in water on a towed model of length 129.8 (m) and diameter 3.8 (cm) for steady speeds from 6.2 (m/s) to 15.5 (m/s). The drag on the model was measured with a strut mounted load cell which provided estimates of the momentum thickness and friction velocity. Momentum thickness Reynolds numbers Reθ varied from 4.8 × 105 to 1.1 × 106. The ratio of momentum thickness to viscous length scale is significantly greater than for flat plate cases at comparable Reθ. The effectiveness of inner and outer velocity and length scales for collapsing the pressure spectra are discussed. The wavenumber–frequency spectra show a convective ridge at higher frequencies similar to flat plate boundary layers. At low frequencies, energy broad in wavenumber extends outside the convective ridge and acoustic cone, with no characteristic wave speed. Wall pressure cross-spectral levels scaled with similarity variables are shown to increase with increasing tow speed, and to follow decay constants consistent with flat plate cases. The convection velocities also display features similar to flat plate cases.

Modified amplitude and strouhal number scaling for correction of turbulent wall pressure fluctuations

Pressure fluctuations were measured in an external turbulent boundary layer over a buoyantly propelled axisymmetric body of revolution. Data were measured for three cases, resulting in axial length Reynolds numbers of 6.88x10 , 4.27x10 , and 3.21x10 at the measurement locations. The fresh water measurements were made in a fully developed turbulent boundary layer, following natural transition, with a near zero (very mildly adverse) pressure gradient. The salt water measurements were made in a favorable pressure gradient following a flow trip to force transition. The momentum thickness Reynolds number was greater than 4400 for all measurements, and the data were scaled with outer variables. The turbulence data were corrected using Corcos correction factors and then further scaled in both amplitude and frequency by the square root of (transducer radius/displacement thickness), in an attempt to resolve spatial resolution effects. This resulted in excellent agreement among the spectra to a modified Strouhal number of 1, which was the range of validity for the data. A second data set (constant freestream velocity and increasing displacement thickness with downstream measurement location), scaled somewhat better with the ratio of transducer radius to displacement thickness.

Turbulence structural measurements using a comprehensive laser–Doppler velocimeter in two- and three-dimensional turbulent boundary layers

The advanced ‘comprehensive’ laser–Doppler velocimeter is used to acquire spatially and temporally resolved turbulence structural measurements in high Reynolds number two- and three-dimensional turbulent boundary layers. The new instrument directly measures three-dimensional particle trajectories at high repetitions. These trajectories are analyzed in post-processing to obtain fluctuating velocity gradient tensor fields, which lead to direct measurements of turbulent viscous dissipation rates. Such data acquired in two- and three-dimensional boundary layers with an approach flow momentum thickness Reynolds number, Re θ = 7500 are presented. Results indicate that anisotropy of the dissipation rate of Reynolds stresses persists to similar heights in viscous wall units as obtained with direct numerical simulations at lower Reynolds numbers. Measurements in a three-dimensional turbulent boundary layer in the vicinity of a wing/body junction also indicate that a reduction in the value of the velocity/pressure-gradient correlations for the Reynolds normal stresses reduces the turbulent energy redistribution and contributes to reduced shear stress magnitudes, as observed previously through DNS (Moin, P., Shih, T.-H., Driver, D., Mansour, N.N., 1990. Direct numerical simulation of a three-dimensional turbulent boundary layer, Phys. Fluids A 2 (10), 1846–1853).

Realization of the Turbulent Boundary Layer over the Rough Wall Satisfied the Conditions of Complete Similarity and Its Mean Flow Quantities

Mean flow quantities have been investigated for the equilibrium boundary layer evolving over the rough wall with a roughness height that is proportional to streamwise distance. The wall shear stress measured by drag balance with a floating element device of a zero displacement mechanism. The local skin friction coefficient is independent of both streamwise distance and momentum thickness Reynolds number and has a value of 0.00826. The boundary layer thickness is proportional to the streamwise distance. The relative roughness height then maintains a constant value. The log-law profile has the same slope of that of the smooth wall turbulent boundary layer in the logarithmic liner part. The value of the roughness function increases in the streamwise direction. The wake parameter approaches to a constant value of approximately 0.70. In addition, the value is consistent with the result determined from the analytical method using the momentum integral equation and Coles's wake law. For the analytical result, the wake parameter decreases with increasing friction parameter.

Analytical model of the time developing turbulent boundary layer

We present an analytical model for the time-developing turbulent boundary layer (TD-TBL) over a flat plate. The model provides explicit formulae for the temporal behavior of the wall-shear stress and both the temporal and spatial distributions of the mean streamwise velocity, the turbulence kinetic energy and Reynolds shear stress. The resulting profiles are in good agreement with the DNS results of spatially-developing turbulent boundary layers at momentum thickness Reynolds number equal to 1430 and 2900. Our analytical model is, to the best of our knowledge, the first of its kind for TD-TBL.

108 粘弾性流体の抵抗低減壁乱流に関する直接数値シミュレーション(2)(OS1-2 流れの抵抗低減,OS1 流れの抵抗抵減,オーガナイズドセッション)

Direct numerical simulation of a zero-pressure gradient drag-reducing turbulent boundary layer of viscoelastic fluids was performed at the momentum-thickness Reynolds number Re_θ_0=500 and Weissenberg number We=25 using constitutive equation models such as the Oldroyd-B, Giesekus (the mobility factor α=0.01, 0.001, 0.0001) and FENE-P models (the extensibility parameter L^2=100, 1000, 10000). It was found that the maximum drag reduction ratio DR(= 45%) for the Giesekus model with α=0.0001 was the largest. In the case of the same elongational viscosity, the drag reduction ratio for the Giesekus model is larger than that for the FENE-P model.