scholarly journals A universal velocity profile for smooth wall pipe flow

2019 ◽  
Vol 878 ◽  
pp. 834-874 ◽  
Author(s):  
Brian J. Cantwell
Keyword(s):  
Reynolds Number ◽  
Shear Stress ◽  
Velocity Profile ◽  
Shape Function ◽  
Velocity Profiles ◽  
Momentum Equation ◽  
Additive Constant ◽  
Model Parameters ◽  
Mixing Length ◽  
Model Profile

The most important unanswered questions in turbulence regard the nature of turbulent flow in the limit of infinite Reynolds number. The Princeton superpipe (PSP) data comprise 26 velocity profiles that cover three orders of magnitude in the Reynolds number from $Re=19\,639$, to $Re=20\,088\,000$ based on pipe radius and pipe centreline velocity. In this paper classical mixing length theory is combined with a new mixing length model of the turbulent shear stress to solve the streamwise momentum equation and the solution is used to approximate the PSP velocity profiles. The model velocity profile is uniformly valid from the wall to the pipe centreline and comprises five free parameters that are selected through a minimization process to provide an accurate approximation to each of the 26 profiles. The model profile is grounded in the momentum equation and allows the velocity derivative, Reynolds shear stress and turbulent kinetic energy production to be studied. The results support the conclusion that logarithmic velocity behaviour near the wall is not present in the data below a pipe Reynolds number somewhere between $Re=59\,872$, and $Re=87\,150$. Above $Re=87\,150$, the data show a very clear, nearly logarithmic, region. But even at the highest Reynolds numbers there is still a weak algebraic dependence of the intermediate portion of the velocity profile on both the near-wall and outer flow length scales. One of the five parameters in the model profile is equivalent to the well-known Kármán constant, $k$. The parameter $k$ increases almost monotonically from $k=0.4034$ at $Re=87\,150$ to $k=0.4190$ at $Re=20\,088\,000$, with an average value, $k=0.4092$. The variation of the remaining four model parameters is relatively small and, with all five parameters fixed at average values, the model profile reproduces the entire velocity data set and the wall friction reasonably well. With optimal values of the parameters used for each model profile, the fit to the PSP survey data is very good. Transforming the model velocity profile using the group, $u/u_{0}\rightarrow ku/u_{0}$, $y^{+}\rightarrow ky^{+}$ and $R_{\unicode[STIX]{x1D70F}}\rightarrow kR_{\unicode[STIX]{x1D70F}}$ where $R_{\unicode[STIX]{x1D70F}}$ is the friction Reynolds number, leads to a reduced expression for the velocity profile. When the reduced profile is cast in outer variables, the physical velocity profile is expressed in terms of $\ln (y/\unicode[STIX]{x1D6FF})$ and a new shape function $\unicode[STIX]{x1D719}(y/\unicode[STIX]{x1D6FF})$. In the limit of infinite Reynolds number, the velocity profile asymptotes to plug flow with a vanishingly thin viscous wall layer and a continuous derivative everywhere. The shape function evaluated at the pipe centreline is used to produce a new friction law with an additive constant that depends on the Kármán constant and a wall damping length scale.

scholarly journals A universal velocity profile for turbulent wall flows including adverse pressure gradient boundary layers

2021 ◽  
Vol 933 ◽  
Author(s):  
Matthew A. Subrahmanyam ◽  
Brian J. Cantwell ◽  
Juan J. Alonso
Keyword(s):  
Boundary Layer ◽  
Shear Stress ◽  
Turbulent Boundary Layer ◽  
Velocity Profile ◽  
Pressure Gradient ◽  
Channel Flow ◽  
Adverse Pressure Gradient ◽  
Turbulent Shear ◽  
Model Parameters ◽  
Mixing Length

A recently developed mixing length model of the turbulent shear stress in pipe flow is used to solve the streamwise momentum equation for fully developed channel flow. The solution for the velocity profile takes the form of an integral that is uniformly valid from the wall to the channel centreline at all Reynolds numbers from zero to infinity. The universal velocity profile accurately approximates channel flow direct numerical simulation (DNS) data taken from several sources. The universal velocity profile also provides a remarkably accurate fit to simulated and experimental flat plate turbulent boundary layer data including zero and adverse pressure gradient data. The mixing length model has five free parameters that are selected through an optimization process to provide an accurate fit to data in the range $R_\tau = 550$ to $R_\tau = 17\,207$ . Because the velocity profile is directly related to the Reynolds shear stress, certain statistical properties of the flow can be studied such as turbulent kinetic energy production. The examples presented here include numerically simulated channel flow data from $R_\tau = 550$ to $R_\tau =8016$ , zero pressure gradient (ZPG) boundary layer simulations from $R_\tau =1343$ to $R_\tau = 2571$ , zero pressure gradient turbulent boundary layer experimental data between $R_\tau = 2109$ and $R_\tau = 17\,207$ , and adverse pressure gradient boundary layer data in the range $R_\tau = 912$ to $R_\tau = 3587$ . An important finding is that the model parameters that characterize the near-wall flow do not depend on the pressure gradient. It is suggested that the new velocity profile provides a useful replacement for the classical wall-wake formulation.

The Effects of Fins on the Intermediate Wake of a Submarine Model

2010 ◽  
Vol 132 (3) ◽  
Author(s):  
Juan M. Jiménez ◽  
Ryan T. Reynolds ◽  
Alexander J. Smits
Keyword(s):  
Shear Stress ◽  
Velocity Profile ◽  
Outer Region ◽  
Reynolds Numbers ◽  
Velocity Profiles ◽  
Turbulent Wake ◽  
Self Similar ◽  
Stress Profiles

Results are presented on the behavior of the turbulent wake behind a submarine model for a range of Reynolds numbers based on the model length between 0.49×106 and 1.8×106, for test locations between 3 and 9 diameters downstream of the stern. The shape of the model emulates an idealized submarine, and tests were performed with and without stern fins. In the absence of fins, the velocity profile in planes away from the influence of the sail rapidly becomes self-similar and is well described by a function of exponentials. The fins create defects in the velocity profiles in the outer region of the wake, while yielding higher values of turbulence at locations corresponding to the tips of the fins. Measurements conducted in planes away from the midline plane show that the velocity profiles remain self-similar, while the shear stress profiles clearly show the effects of the necklace vortices trailing from the base of the fins.

An Expression of Reynolds Stresses in Turbulent Lubrication Theory

1992 ◽  
Vol 114 (1) ◽  
pp. 57-60 ◽  
Author(s):  
A. K. Tieu ◽  
P. B. Kosasih
Keyword(s):  
Reynolds Number ◽  
Shear Stress ◽  
Turbulent Flow ◽  
Low Reynolds Number ◽  
Stress Gradient ◽  
Lubrication Theory ◽  
Mixing Length ◽  
Reynolds Stresses ◽  
Shear Stress Gradient ◽  
Low Reynolds

This paper proposes an alternative model of Reynolds stresses for turbulent lubrication theory. The approach relies on Prandtl’s mixing length theory which is based on a modified Van Driest mixing formula [1]. However, unlike the previous theories [2, 3] the proposed equation is capable of accounting for the effect of shear stress gradient on the mixing length. Thus it is well suited to turbulent flow analysis in bearings where the presence of shear stress gradient due to the effect of pressure gradient should be considered. A series of velocity measurements in thin channels in the low Reynolds number turbulent flow range are analysed using the theory. The data analysis shows a strong effect of shear stress gradient on the viscous sublayer in the low Reynolds number regime. As a result, a new model of mixing length applicable to the turbulent lubrication analysis in thin film at low or high Reynolds numbers or under low or high shear stress gradient is presented.

scholarly journals Featuresof velocity distribution in a turbulent flow

Vestnik MGSU ◽  
2015 ◽  
pp. 103-109
Author(s):  
Valeriy Stepanovich Borovkov ◽  
Valeriy Valentinovich Volshanik ◽  
Irina Aleksandrovna Rylova
Keyword(s):  
Reynolds Number ◽  
Shear Stress ◽  
Velocity Profile ◽  
Turbulent Flow ◽  
Velocity Distribution ◽  
Drag Coefficient ◽  
Viscous Flow ◽  
Measured Velocity ◽  
Displacement Thickness ◽  
Logarithmic Velocity Profile

In this article the questions of kinematic structure of steady turbulent flow near a solid boundary are considered. It has been established that due to friction the value of the local Reynolds number decreases and always becomes smaller than the critical value of the Reynolds number, which leads to formation of viscous flow near a wall. Velocity profiles for the area of viscous flow with constant and variable shear stress are obtained. The experimental investigations of different authors showed that in this area the flow is of unsteady character, where viscous flow occurs intermittently with turbulent flow. With increasing distance from the wall the flow becomes fully turbulent. In the area where generation and dissipation of turbulence are very intensive, there is a developed turbulent flow with increasing distance from the wall. Dissipation of turbulence is an action of viscous force. The logarithmic velocity profile was obtained by L. Prandtl disregarding the viscous component and the linear variation of the shear stress in the depth flow. The profile parameters C and k were determined from Nikuradze’s experiments. The detailed investigations of Nikuradze’s experiments established the part of the flow where the logarithmic velocity profile is correctly confirmed.This part of the flow was called “Prandtl layer”. The measured velocity distribution above this layer deviates in the direction of greater values. Processing of experimental data revealed that the thickness of the “Prandtl layer”, normalized to the radius of a pipe, depend on a drag coefficient. The formula for determining the thickness of the “Prandtl layer” with the known value of the drag coefficient is obtained. It is shown that the thickness of “Prandtl layer” almost coincides with the boundary layer displacement thickness formed on the wall of the pipe.

A Study of Fluid Velocities in Tribological Fluid Film

1994 ◽  
Vol 116 (1) ◽  
pp. 133-138 ◽  
Author(s):  
A. K. Tieu ◽  
P. B. Kosasih ◽  
M. R. Mackenzie
Keyword(s):  
Reynolds Number ◽  
Poiseuille Flow ◽  
Velocity Profiles ◽  
Fluid Film ◽  
Turbulent Regime ◽  
Mixing Length ◽  
Fluid Film Bearings ◽  
Fluid Velocities ◽  
Planar Flows ◽  
Narrow Gaps

Recently a model of Reynolds stress in turbulent lubrication theory was proposed by Tieu and Kosasih (1992) based on a modified Van Driest mixing length formula. It was developed from a study of Poiseuille flow velocities in narrow gaps. As a continuation of that study, this paper describes an investigation into fluid velocities in fluid film bearings. Experimental velocity profiles of planar flows in various film geometries are compared with the profiles calculated using the mixing length model in the transition-turbulent regime. Excellent agreements have been attained, confirming the validity of the formula in the superlaminar theory. The effects of Reynolds number and pressure gradient on nonplanar velocity profiles are also presented.

An Anisotropic Subgrid Model for Large Eddy Simulation of Wall Bounded Turbulent Flows

2007 ◽  
Author(s):  
Sachin S. Badarayani ◽  
Kyle D. Squires
Keyword(s):  
Reynolds Number ◽  
Large Eddy Simulation ◽  
Velocity Profile ◽  
Turbulent Flows ◽  
Model Parameters ◽  
Wall Region ◽  
Outer Flow ◽  
Eddy Simulation ◽  
Large Eddy ◽  
Near Wall

Large Eddy Simulation (LES) of high-Reynolds-number wall-bounded turbulent flows is prohibitively expensive if the energy-containing eddies in the near-wall region are resolved. This motivates the use of wall-layer models in which an approximate solution of the near wall dynamics is bridged to an LES of the outer flow. The main interest of the present work are wall-modeling strategies based on Detached Eddy Simulation (DES). In these approaches, the near-wall solution is closed using a Reynolds-averaged Navier Stokes model with a subgrid closure applied to the outer flow. As is well known, the original DES formulation applied directly as a wall model results in a shift in the velocity profile, corresponding to an under-estimation of the skin friction. A new formulation is proposed in this contribution in which the wall-parallel components of the modeled stress are reduced in order to lower the influence of the model and increase the resolved stress. The effectiveness of the new model is evaluated via comparison against DES predictions using the original and recently-proposed versions of the method. The effect of grid resolution and model parameters are also assessed using computations of turbulent channel flow at a Reynolds number based on friction velocity and channel halfwidth of 5000. The predictions show that the anisotropic form of the model stress yields an improved prediction of the mean velocity profile in better agreement with the logarithmic law and with larger resolved stress in the near-wall region.

scholarly journals Pengaruh Silinder Downstream terhadap Karakteristik Aliran Silinder Upstream Menggunakan Square Disturbance Body Tersusun Tandem

2018 ◽  
Vol 11 (2) ◽  
pp. 58
Author(s):  
Rina Rina ◽  
Sanny Ardhy
Keyword(s):  
Reynolds Number ◽  
Shear Stress ◽  
Drag Force ◽  
Turbulence Model ◽  
Bluff Body ◽  
Pressure Coefficient ◽  
Model Parameters ◽  
Square Cylinder ◽  
Shear Stress Transport ◽  
Unsteady Rans

Fluida yang mengalir di sekitar bluff body silinder sirkular, akan menimbulkan gaya-gaya aerodinamika salah satunya gaya drag. Drag sangat tidak diinginkan untuk keselamatan struktur body. Reduksi gaya drag dilakukan dengan mengontrol medan aliran seperti meningkatkan kekasaran permukaan, mengiris silinder dengan sudut iris tertentu, dan menempatkan pengganggu di sisi upstream silinder. Penelitian ini bertujuan untuk melihat pengaruh silinder downstream terhadap karakteristik aliran silinder upstream menggunakan square disturbance body yang disusun tandem pada saluran sempit. Geometri yang digunakan adalah dua silinder sirkular yang disusun tandem berdiameter (D) 25 mm dengan variasi jarak antar silinder (L/D) 1,5; 2; 2,5; 3; 3,5; 4. Square Cylinder sebagai body pengganggu ditempatkan pada sisi upstream silinder utama berdiamensi 4 mm. Posisi sudut pengganggu (?) 30°, dan jarak gap (d=0.4mm). Reynolds number berdasarkan diameter silinder, yaitu ReD 2,32x104. Penelitian iini dilakukan secara numerik 2D Unsteady-RANS menggunakan CFD software FLUENT 6.3.26 dengan model viscous Turbulence Model Shear-Stress-Transport (SST) k-?. Parameter yang diamati adalah koefisien pressure (Cp), Koefisien drag pressure (Cdp) dan visualisasi aliran berupa velocity pathline. Hasilnya menunjukkan bahwa Penambahan silinder downstream memberikan kontribusi dalam pengurangan gaya drag pada silinder upstream menggunakan square disturbance body. Pengaruh wake silinder upstream terhadap silinder downstream berkurang dengan meningkatnya rasio L/D. Interaksi wake silinder upstream terhadap silinder downstream terjadi pada konfigurasi L/D 1,5 – 3. Pengurangan gaya drag optimum terjadi pada konfigurasi L/D 3. The fluid flows around the circular cylinder bluff body will produce aerodynamic forces, one of which is the drag force. Drag is very undesirable for the safety of the body structure. Reduction of drag force is carried out by controlling the flow field such as increasing the surface roughness, slicing the cylinder with a certain iris angle, and placing the disturbance on the upstream side of the cylinder. This purpose of the study is to see the effect of downstream cylinders on the flow characteristics of upstream cylinders using a square disturbance body arranged tandem in a narrow channel. The geometry used is two circular cylinders arranged in tandem diameter (D) 25 mm with a variation of distance between cylinders (L / D) 1.5; 2; 2.5; 3; 3.5; 4. Square Cylinder as a disturbing body is placed on the side of the main cylinder upstream with a diameter of 4 mm. The position of the disturbing angle (?) is 30 °, and the gap distance (d = 0.4mm). Reynolds number is based on cylinder diameter, ie ReD 2.32x104. This research was carried out numerical 2D Unsteady-RANS using a FLUENT 6.3.26 CFD software with viscous Turbulence model Shear-Stress-Transport (SST) k-? model. Parameters observed were pressure coefficient (Cp), drag pressure coefficient (Cdp) and flow visualization in the form of velocity pathline. The results show that the addition of a downstream cylinder contributes to the reduction of the drag force on the upstream cylinder using a square disturbance body. The wake influence of upstream cylinder to downstream cylinder decreasing with increasing the ratio of L/D. The interaction of wake cylinder upstream to downstream cylinder occurs at L/D 1.5 - 3. The optimum for the drag force reduction occurs at L/D 3.

Settling Length for Turbulent Flow of Air in Smooth Annuli With Square-Edged or Bellmouth Entrances

1970 ◽  
Vol 37 (1) ◽  
pp. 25-28 ◽  
Author(s):  
K. Sridhar ◽  
A. A. Nicol ◽  
A. V. A. Padmanabha
Keyword(s):  
Reynolds Number ◽  
Velocity Profile ◽  
Turbulent Flow ◽  
Flow Separation ◽  
Abrupt Change ◽  
Velocity Profiles ◽  
Flat Profile

Settling or hydrodynamic entrance lengths have been determined for turbulent flow of air in three smooth concentric annuli of diameter ratios 0.306, 0.527, and 0.842. Both square-edged and bellmouth entrances were investigated for Reynolds number ranging from 7000–47,500. Flow separation caused by the abrupt change in area of the square-edged entrance resulted in skewed (distorted) velocity profiles near the entrance. This skewness was modified further downstream and finally fully developed velocity profiles were established. The settling lengths for the annuli with the square-edged entrances were between 25 and 35 dia. With the bellmouth entrance, the velocity profile developed conventionally from the nearly flat profile at the inlet. The settling lengths for the bellmouth entrances were about 10–15 equivalent diameters more than those for the square-edged entrances.

Flow and Thermal Fields in a Pendant Droplet Moving on Lyophobic Surface

2010 ◽  
Author(s):  
Basant Singh Sikarwar ◽  
K. Muralidhar ◽  
Sameer Khandekar
Keyword(s):  
Heat Transfer ◽  
Contact Angle ◽  
Reynolds Number ◽  
Shear Stress ◽  
Heat Transfer Coefficient ◽  
Wall Shear Stress ◽  
Transfer Coefficient ◽  
Maximum Shear Stress ◽  
Computational Domain ◽  
Wall Shear

Clusters of liquid drops growing and moving on physically or chemically textured lyophobic surfaces are encountered in drop-wise mode of vapor condensation. As opposed to film-wise condensation, drops permit a large heat transfer coefficient and are hence attractive. However, the temporal sustainability of drop formation on a surface is a challenging task, primarily because the sliding drops eventually leach away the lyophobicity promoter layer. Assuming that there is no chemical reaction between the promoter and the condensing liquid, the wall shear stress (viscous resistance) is the prime parameter for controlling physical leaching. The dynamic shape of individual droplets, as they form and roll/slide on such surfaces, determines the effective shear interaction at the wall. Given a shear stress distribution of an individual droplet, the net effect of droplet ensemble can be determined using the time averaged population density during condensation. In this paper, we solve the Navier-Stokes and the energy equation in three-dimensions on an unstructured tetrahedral grid representing the computational domain corresponding to an isolated pendant droplet sliding on a lyophobic substrate. We correlate the droplet Reynolds number (Re = 10–500, based on droplet hydraulic diameter), contact angle and shape of droplet with wall shear stress and heat transfer coefficient. The simulations presented here are for Prandtl Number (Pr) = 5.8. We see that, both Poiseuille number (Po) and Nusselt number (Nu), increase with increasing the droplet Reynolds number. The maximum shear stress as well as heat transfer occurs at the droplet corners. For a given droplet volume, increasing contact angle decreases the transport coefficients.

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