Mean Velocity Defect Research Articles

Empirical Model for Low-Speed Rough-Wall Turbulent Boundary Layer Pressure Spectra

An empirical model for estimating surface pressure spectra in high Reynolds number rough-wall turbulent boundary layers in the low-speed regime is presented. The model is based on the low-frequency mean velocity defect scaling [1] and the high-frequency shear friction velocity scaling [2]. This model satisfactorily predicts surface pressure spectra measured over a wide range of roughness configurations in high Reynolds number flows free of transitional effects. These data also provide insights into the characteristics of the turbulent scales which are on the order of the roughness size and comprise the mid-frequency region. The slope of this region is highly dependent on roughness element density and element-relative measurement location. This contradicts the premise of an overlap region with slope . An evanescent pressure decay model provides evidence that the slope changes are due to a roughness-dependent pressure decay within the interstitial flow and suggests that a traditional mid-frequency similarity law may not be possible.

The low-frequency pressure fluctuations of near-equilibrium turbulent boundary layers

The low-frequency pressure fluctuations generated by a range of roughness configurations are explored for flows of roughness Reynolds numbers over 200 and blockage ratios above 73. Two-point analysis of the pressure fluctuations reveals that surfaces with high frontal solidity, $$\lambda$$ , and/or complex shear layer interactions near the elements, have structures which persist longer than structures on the smooth wall. A low-frequency pressure spectrum scaling based on the broadband convection velocity defect, $$U_{\text{e}}-{\overline{U}}_{\text{c}_{\text{b}}}$$ , is proven effective for all rough surfaces and the smooth wall. Here $$U_{\text{e}}$$ is the boundary layer edge velocity, and $${\overline{U}}_{\text{c}_{\text{b}}}$$ is the mean broadband convection velocity. A second scaling, which is more appropriate for a priori predictions, is also proposed. It uses the mean velocity defect, $$U_{\text{e}}-{\overline{U}}$$ (where $${\overline{U}}$$ is the mean boundary layer velocity), first proposed for velocity profiles by Zagarola and Smits (J Fluid Mech 373:33–79, 1998). Both scalings are effective because they are based on velocity defects proportional to the outer-layer shear stress which generates the pressure fluctuations. A re-examination of the outer-layer wall-similarity in the context of these new findings suggests that this concept is limited to a narrow range of turbulent boundary layer flows.

Generation of unsteady incoming wakes for wake/blade interaction

In this study, an effective method is proposed for the generation of unsteady incoming wakes to facilitate numerical simulation of wake/blade interaction. The wakes are specified through the inflow boundary conditions of a blade cascade, which includes the mean velocity defect as well as the velocity fluctuation induced by the vortex street in wakes. Compared to the large-eddy simulation (LES) of turbine cascades, the proposed method can be used to obtain unsteady wakes with different scales of fluctuation. Further, the analysis reveals that the velocity defect can be well fitted by the Gaussian model, and the velocity perturbation induced by wake vortex can be described by the modified Lamb-Oseen vortex or the Taylor Vortex. The vortices in wakes exhibit self-similarity behavior, which also confirms the feasibility of the proposed method. The wake/blade interaction is also simulated within T106 cascade with incoming wakes generated by the proposed method, and the results provide detailed information on the wake transportation in downstream bladerow.

Influence of Surface Anisotropy on Turbulent Flow Over Irregular Roughness

The influence of surface anisotropy upon the near-wall region of a rough-wall turbulent channel flow is investigated using direct numerical simulation (DNS). A set of nine irregular rough surfaces with fixed mean peak-to-valley height, near-Gaussian height distributions and specified streamwise and spanwise correlation lengths were synthesised using a surface generation algorithm. By defining the surface anisotropy ratio (SAR) as the ratio of the streamwise and spanwise correlation lengths of the surface, we demonstrate that surfaces with a strong spanwise anisotropy (SAR < 1) can induce an over 200% increase in the roughness function ΔU+, compared to their streamwise anisotropic (SAR > 1) equivalent. Furthermore, we find that the relationship between the roughness function ΔU+ and the SAR parameter approximately follows an exponentially decaying function. The statistical response of the near-wall flow is studied using a “double-averaging” methodology in order to distinguish form-induced “dispersive” stresses from their turbulent counterparts. Outer-layer similarity is recovered for the mean velocity defect profile as well as the Reynolds stresses. The dispersive stresses all attain their maxima within the roughness canopy. Only the streamwise dispersive stress reaches levels that are comparable to the equivalent Reynolds stress, with surfaces of high SAR attaining the highest levels of streamwise dispersive stress. The Reynolds stress anisotropy also shows distinct differences between cases with strong streamwise anisotropy that stay close to an axisymmetric, rod-like state for all wall-normal locations, compared to cases with spanwise anisotropy where an axisymmetric, disk-like state of the Reynolds stress anisotropy tensor is observed around the roughness mean plane. Overall, the results from this study underline that the drag penalty incurred by a rough surface is strongly influenced by the surface topography and highlight its impact upon the mean momentum deficit in the outer flow as well as the Reynolds and dispersive stresses within the roughness layer.

A velocity defect chart method for estimating the friction velocity in turbulent boundary layers

A simple and new empirical method for determining the friction velocity in zero pressure gradient turbulent boundary layers is presented. The method relies on the collapse of the streamwise mean velocity defect profiles of the boundary layers over smooth and rough walls when normalised by the friction velocity. While the collapse of the velocity defect profiles is well established in the literature, this is the first time that it is being utilised for estimating the friction velocity.

Statistical properties of spherical shock waves propagating through grid turbulence, turbulent cylinder wake, and laminar flow

Wind tunnel experiments are reported for a spherical shock wave propagating through turbulent wakes of a single cylinder, double cylinders, grid-turbulence, and a laminar flow, whose influences on the shock wave are compared. Overpressure behind the shock wave is measured on a plate while streamwise velocity is measured at the flow point between the measurement plate and the location of the shock wave ejection. Average of peak-overpressure observed upon arrival of the shock wave is decreased by the mean velocity defect of the cylinder wake. Root mean squared (rms) peak-overpressure fluctuation divided by the averaged peak-overpressure is increased by turbulence, and it becomes larger with the rms velocity fluctuation. Correlation coefficients are calculated between fluctuations of peak-overpressure and low-pass filtered fluid velocity. The strong positive correlation is found for the fluid at the location where the shock ray toward the pressure measurement point passes. The length scale of velocity fluctuation with the strong correlation is related to the integral length scale of turbulence. In the double-cylinder wake experiments, the shock wave that has passed one cylinder wake interacts again with another cylinder wake before it reaches the measurement plate. The correlation coefficient for the velocity fluctuation of the first wake is weakened by the second wake, and this influence becomes more important when the rms velocity fluctuation of the second wake is larger.

Measurement of Transitional Surface Roughness Effects on Flat-Plate Boundary Layer Transition

An experimental study has been conducted to investigate the effects of transitionally rough surface on the flat-plate boundary layer transition. Transitional boundary layers with three different flat plates (ks+ = 0.07 ∼ 0.19, 2.71 ∼ 7.05, and 13.65 ∼ 41.09) have been measured with a single-sensor hot-wire probe. All of the measurements have been conducted under zero pressure gradient (ZPG) at the fixed Reynolds number (ReL) and freestream turbulence intensity (Tu) of 3.05 × 106 and 0.2%. Transitionally, rough surface does not affect the sigmoidal distribution of turbulence intermittency model; but induces earlier transition onset and shortens the transition length. For all surfaces, streamwise turbulence intensity profiles with similar values of turbulence intermittency are similar for the transition length less than 60%. Therefore, mean velocity profiles with the similar values of turbulence intermittency are similar regardless of surface conditions. However, downstream of 60% of the transition length, mean velocity defect increases as the surface roughness increases. Enhanced diffusion of turbulent kinetic energy from the near wall (y/δ &lt; 0.1) to the outer part (y/δ ≈ 0.4) of the boundary layer due to the surface roughness is responsible for the increased momentum deficit.

Can a turbulent boundary layer become independent of the Reynolds number?

This paper examines the Reynolds number ($Re$) dependence of a zero-pressure-gradient (ZPG) turbulent boundary layer (TBL) which develops over a two-dimensional rough wall with a view to ascertaining whether this type of boundary layer can become independent of $Re$. Measurements are made using hot-wire anemometry over a rough wall that consists of a periodic arrangement of cylindrical rods with a streamwise spacing of eight times the rod diameter. The present results, together with those obtained over a sand-grain roughness at high Reynolds number, indicate that a $Re$-independent state can be achieved at a moderate $Re$. However, it is also found that the mean velocity distributions over different roughness geometries do not collapse when normalised by appropriate velocity and length scales. This lack of collapse is attributed to the difference in the drag coefficient between these geometries. We also show that the collapse of the $U_{\unicode[STIX]{x1D70F}}$-normalised mean velocity defect profiles may not necessarily reflect $Re$-independence. A better indicator of the asymptotic state of $Re$ is the mean velocity defect profile normalised by the free-stream velocity and plotted as a function of $y/\unicode[STIX]{x1D6FF}$, where $y$ is the vertical distance from the wall and $\unicode[STIX]{x1D6FF}$ is the boundary layer thickness. This is well supported by the measurements.

Outer scales and parameters of adverse-pressure-gradient turbulent boundary layers

A clear and consistent framework for the analysis of the outer region of adverse-pressure-gradient turbulent boundary layers is established in this paper based on basic principles and theory, and the help of six adverse-pressure-gradient turbulent boundary layer databases and a zero-pressure-gradient one. Outer velocity and length scales for the mean velocity defect and the Reynolds stresses are discussed first. The conditions of validity of four velocity scales are determined in terms of the shape factor, since one scale is restricted to small velocity-defect boundary layers (the friction velocity $u_{\unicode[STIX]{x1D70F}}$), one to large-defect ones (the pressure-gradient velocity $U_{po}$), while the two others are proper scales for all velocity-defect conditions (the Zagarola–Smits velocity $U_{zs}$ and the mixing-layer-type velocity $U_{m}$). The turbulent boundary layer equations are then used to bring out, in a consistent manner and without assuming any self-similar behaviour, a set of non-dimensional parameters characterizing the outer region of turbulent boundary layers with arbitrary pressure gradients. In terms of a generic outer length scale $L_{o}$ and velocity scale $U_{o}$, these non-dimensional parameters are the pressure-gradient parameter $\unicode[STIX]{x1D6FD}_{o}=L_{o}/(\unicode[STIX]{x1D70C}U_{o}^{2})\,\text{d}p_{e}/\text{d}x$, the Reynolds number $Re_{o}=U_{o}L_{o}/\unicode[STIX]{x1D708}(U_{o}/U_{e})$ and the inertial parameter $\unicode[STIX]{x1D6FC}_{o}=U_{e}V_{e}/U_{o}^{2}$, where $U_{e}$ and $V_{e}$ are respectively the streamwise and wall-normal components of mean velocity at the boundary layer edge. These parameters have a clear physical meaning: they are ratios of the order of magnitude of forces, with the Reynolds shear stress gradient (apparent turbulent force) as the reference force – inertial to apparent turbulent forces for $\unicode[STIX]{x1D6FC}_{o}$, pressure to apparent turbulent forces for $\unicode[STIX]{x1D6FD}_{o}$ and apparent turbulent to viscous forces for $Re_{o}$. We discuss at length their significance and determine under what conditions they retain their meaning depending on the outer velocity scale that is considered. The seven boundary layer databases are analysed and compared using the established framework. An astonishing and impressive result is obtained: by choosing $U_{o}=U_{zs}$, the streamwise evolution of the three ratios of forces in the outer region can be accurately followed with $\unicode[STIX]{x1D6FD}_{zs}$, $\unicode[STIX]{x1D6FC}_{zs}$ and $Re_{zs}$ in all these flows – not just the order of magnitude of these ratios. This cannot be achieved with $u_{\unicode[STIX]{x1D70F}}$ and $U_{po}$, and only imperfectly with $U_{m}$. Consequently, $\unicode[STIX]{x1D6FD}_{zs}$, $\unicode[STIX]{x1D6FC}_{zs}$ and $Re_{zs}$ can be used to follow – in a global sense – the streamwise evolution of the streamwise mean momentum balance in the outer region.

Behaviour of Mean Velocity in the Turbulent Axisymmetric Outer Near Wake

An asymptotic study of the outer near wake of a long slender body of revolution is carried out. The long slender body is a cylinder which is kept parallel to the flow and takes the shape of a simplified geometry such as that of an underwater vehicle, a rocket or hull form of a ship model. The wake flow is axisymmetric and the analysis has been carried out without any assumption on the eddy viscosity but utilizing the general behaviour of turbulent shear stress in the near wake. The governing equations are solved with appropriate boundary conditions. Similarity analysis for the mean velocity characteristics is carried out which shows the existence of a logarithmic region in the normal direction in the overlap region between the outer near wake and the inner near wake. Also shown is the exponential decay of the mean velocity defect as freestream velocity is reached. Validation of the results of the analysis is done using available experimental data.

Structural differences between small and large momentum-defect turbulent boundary layers

We analyse how coherent structures in turbulent boundary layers (TBL) change in response to a strong adverse pressure gradient by using two direct numerical simulation databases. One zone of a zero-pressure-gradient TBL and three zones of a strongly decelerated TBL, corresponding to three ranges of mean velocity defect, with the shape factor varying from 1.54 to 3.75, are considered. We investigate the properties of three-dimensional sweeps, ejections and streamwise-velocity structures. The identified sweeps and ejections contribute everywhere more than 30% to the Reynolds shear stress in both flows. The effect of increasing mean velocity defect in the adverse-pressure-gradient TBL is significant. Streamwise-velocity structures and all wall-attached sweeps and ejections, that is near-wall ones and taller ones that reach the wall region (important in the logarithmic layer of the zero-pressure-gradient TBL), lose their streamwise elongation and become less organised. Wall-attached sweeps and ejections become progressively less numerous and spanwise elongated structures become more frequent. Their strength diminishes in comparison to wall-detached ones and they lose their role as the main contributors to the Reynolds shear stress. In terms of spatial organisation, a pair of side-by-side sweep and ejection is still the dominant configuration, but the probability of such an event has decreased. This fact, together with other results of spatial organisation of structures, does not point toward the presence of a Kelvin-Helmholtz-type instability or a varicose instability of low-speed structures in the outer region, two instabilities that have been suggested in the literature to explain the turbulence regeneration process in large velocity defect turbulent boundary layers.

Comparison of turbulent boundary layers over smooth and rough surfaces up to high Reynolds numbers

Turbulent boundary layer measurements above a smooth wall and sandpaper roughness are presented across a wide range of friction Reynolds numbers, ${\it\delta}_{99}^{+}$, and equivalent sand grain roughness Reynolds numbers, $k_{s}^{+}$ (smooth wall: $2020\leqslant {\it\delta}_{99}^{+}\leqslant 21\,430$, rough wall: $2890\leqslant {\it\delta}_{99}^{+}\leqslant 29\,900$; $22\leqslant k_{s}^{+}\leqslant 155$; and $28\leqslant {\it\delta}_{99}^{+}/k_{s}^{+}\leqslant 199$). For the rough-wall measurements, the mean wall shear stress is determined using a floating element drag balance. All smooth- and rough-wall data exhibit, over an inertial sublayer, regions of logarithmic dependence in the mean velocity and streamwise velocity variance. These logarithmic slopes are apparently the same between smooth and rough walls, indicating similar dynamics are present in this region. The streamwise mean velocity defect and skewness profiles each show convincing collapse in the outer region of the flow, suggesting that Townsend’s (The Structure of Turbulent Shear Flow, vol. 1, 1956, Cambridge University Press.) wall-similarity hypothesis is a good approximation for these statistics even at these finite friction Reynolds numbers. Outer-layer collapse is also observed in the rough-wall streamwise velocity variance, but only for flows with ${\it\delta}_{99}^{+}\gtrsim 14\,000$. At Reynolds numbers lower than this, profile invariance is only apparent when the flow is fully rough. In transitionally rough flows at low ${\it\delta}_{99}^{+}$, the outer region of the inner-normalised streamwise velocity variance indicates a dependence on $k_{s}^{+}$ for the present rough surface.

Scaling and statistics of large-defect adverse pressure gradient turbulent boundary layers

The purpose of this article is to test similarity laws and scaling ideas, as well as characterize turbulence behaviour of large-defect adverse-pressure gradient turbulent boundary layers using six experimental and numerical databases including a new direct numerical simulation of a strongly decelerated non-equilibrium turbulent boundary layer. In the latter flow, at a moderate Reynolds number, the mean velocity profiles depart from the classical law of the wall throughout the inner region including in the viscous sublayer and they do not follow the log law. However, the agreement is excellent with the extended law of the wall that accounts for the pressure gradient for the viscous sublayer. The Reynolds stress components are not self-similar in the viscous sublayer when the velocity defect is important, but they scale reasonably well with the pressure-viscous scales.Detailed comparisons of the six different flows are made in the outer region. In order to do such comparisons, an outer region velocity scale analogous to the commonly defined free shear layer velocity scales is introduced. It is found that the investigated one-point velocity statistics in the upper half of large-defect boundary layers resemble those of a mixing layer: mean velocity defect, Reynolds stresses, turbulent kinetic energy budgets, uv correlation factor and structure parameter −〈uv〉/2k. The dominant peaks of turbulence production and Reynolds stresses are located roughly in the middle of the boundary layer. The profiles of the uv correlation factor reveal that u and v become less correlated throughout the boundary layer as the mean velocity defect increases, especially near the wall. The structure parameter is low in the large-defect disequilibrium boundary layers, similar to large-defect equilibrium flows and mixing layers and decreases as the mean velocity defect increases. All large-velocity-defect boundary layers analysed are found to be less efficient in extracting turbulent energy from the mean flow than zero-pressure-gradient turbulent boundary layers, even throughout the outer region.

Optimization and analysis of shock wave/boundary layer interaction for drag reduction by Shock Control Bump

Shock Control Bump (SCB) is a flow control method which reduces the wave drag through introducing a small bump over the wing surface. The present paper is mainly devoted to numerical investigation of shock wave/boundary layer interaction (SWBLI) as the main factor influencing the aerodynamic performance of transonic bumped airfoils. The survey is conducted for three airfoils through detailed SCB shape optimization processes employing differential evolution algorithm (DE). SWBLI is analyzed thoroughly for clean and bumped airfoils and it is shown how the modified wave structure originating from upstream of SCB reduces the wave drag while simultaneously improving the boundary layer velocity profiles downstream of the shock wave. The present work extends the conventional approach for SCB design via detailed interpretation of the decay of mean velocity defect downstream of the bumped airfoils. The numerical analysis of bumped airfoils in case of high transonic Mach number shows that taller SCB required for weakening the strong shock wave results in large separated flow zone and its performance improvement is deteriorated.

Pressure Gradient Effects on Smooth and Rough Surface Turbulent Boundary Layers—Part I: Favorable Pressure Gradient

The effects of the pressure gradient and surface roughness on turbulent boundary layers have been experimentally investigated. In Part I, smooth- and rough-surface turbulent boundary layers with and without favorable pressure gradients (FPG) are presented. All of the tests have been conducted at the same Reynolds number (based on the length of the flat plate) of 900,000. Streamwise time-mean and fluctuating velocities have been measured using a single-sensor hot-wire probe. For smooth surfaces, the FPG decreases the mean velocity defect and increases the wall shear stress; however, the friction coefficient hardly changes due to the increased freestream velocity. The FPG effect on the streamwise normal Reynolds stress has been examined. The FPG increases the streamwise normal Reynolds stress for y less than 0.6 times the boundary layer thickness. With the zero pressure gradient (ZPG), the roughness increases the mean velocity defect throughout the boundary layer and increases the normal Reynolds stress for y greater than twice the average roughness height. The roughness effect on the mean velocity defect is stronger under the FPG than under the ZPG for y less than 30 times the average roughness height. For y less than 25 times the average roughness height, the roughness effect of increasing normal Reynolds stress is also stronger under the FPG than under the ZPG. Consequently, for a rough surface, the FPG increases the integrated streamwise turbulent kinetic energy, friction coefficient, roughness Reynolds number, and roughness shift. Furthermore, the FPG increases the roughness effects on the integral boundary layer parameters—the boundary layer thickness, momentum thickness, ratio of the displacement thickness to the boundary layer thickness, and shape factor. Thus, the FPG augments the roughness effects on turbulent boundary layers.

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

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.

Analysis of a Turbulent Boundary Layer Subjected to a Strong Adverse Pressure Gradient

A strongly decelerated turbulent boundary layer is investigated by direct numerical simulation. Transition to turbulence is triggered by a trip wire which is modelled using the immersed boundary method. The Reynolds number close to the exit of the numerical domain is Reθ = 2175 and the shape-factor is H = 2.5. The analysis focuses on the latter portion of the flow with large velocity defect, at higher Reynolds numbers and further from the transition region. Mean velocity profiles do not reveal a logarithmic law. Departure from the law of the wall occurs throughout the inner region. The production and Reynolds stress peaks move to roughly the middle of the boundary layer. The profiles of the uv correlation factor reveal that u and v become less correlated throughout the boundary layer as the mean velocity defect increases, especially near the wall. The structure parameter is low in the present flow, similar to equilibrium APG flows and mixing layers, and decreases as the mean velocity defect increases. The statistics of the upper half of the boundary layer resemble those of a mixing layer. Furthermore, various two-dimensional two-point correlation maps are obtained. The Cvv and Cww correlations obtained far from the transition region at Reθ = 2175 and at y/δ = 0.4 coincide with results obtained for a ZPG boundary layer, implying that the structure of the v,w fluctuations is the same as in ZPG. However, Cuu indicates that the structure of the u fluctuation in this APG boundary layer is almost twice as short as the ZPG one. The APG structures are also less correlated with the flow at the wall. The near-wall structures are different from ZPG flow ones in that streaks are much shorter or absent.

Attenuation of the wake of a sphere in an intense incident turbulence with large length scales

We report an investigation of the wake of a sphere immersed in a uniform turbulent flow for sphere Reynolds numbers ranging from 100 to 1000. An original experimental setup has been designed to generate a uniform flow convecting an isotropic turbulence. At variance with previous works, the integral length scale of the turbulence is of the same order as the sphere diameter and the turbulence intensity is large. In consequence, the most intense turbulent eddies are capable of influencing the flow in the close vicinity of the sphere. Except in the attached region downstream of the sphere where the perturbation of the mean velocity is larger than the standard deviation of the incident turbulence, the flow is controlled by the incident turbulence. The distortion of the turbulence while the flow goes round the sphere leads to an increase in the longitudinal fluctuation and a decrease in the transversal one. The attenuation of the transversal fluctuations is still significant at 30 radii downstream of the sphere whereas the longitudinal fluctuations relax more rapidly toward the incident value. The more striking result however concerns the evolution of the mean velocity defect with the distance x from the sphere. It decays as x−2 and scales with the standard deviation of the incident turbulence instead of scaling with the mean incident velocity.

Asymptotic and numerical analysis of momentumless turbulent wakes

The asymptotic behavior of turbulent axisymmetric and plane momentumless wakes was studied using the Reynolds-averaged momentum equations and the second-order model of turbulence. The similarity solutions were obtained analytically and the process of transition to self-similarity was studied numerically. It was found that a single-point spectrum of the solutions of the corresponding eigenvalue problem for turbulent energy and dissipation rate existed. However, the spectra of solutions for the normal components of the anisotropy tensor and for the mean velocity defect were discrete. The numerical solution of a non-self-similar problem shows, in accordance with experiments and analytic solutions, that mean and fluctuating velocities decay with different rates, shear stresses decay faster than normal stresses and the anisotropic component of normal stresses decays faster than the isotropic component. The analysis of solutions for a full system of the Reynolds stress equations showed the presence of ‘partial similarity’ of the turbulent momentumless wake when some variables (velocity defect and shear stress) remain non-similar in flow that is self-similar as a whole.

A direct numerical simulation study on the mean velocity characteristics in turbulent pipe flow

Fully developed incompressible turbulent pipe flow at bulk-velocity- and pipe-diameter-based Reynolds number ReD=44000 was simulated with second-order finite-difference methods on 630 million grid points. The corresponding Kármán number R+, based on pipe radius R, is 1142, and the computational domain length is 15R. The computed mean flow statistics agree well with Princeton Superpipe data at ReD=41727 and at ReD=74000. Second-order turbulence statistics show good agreement with experimental data at ReD=38000. Near the wall the gradient of $\mbox{ln}\overline{u}_{z}^{+}$ with respect to ln(1−r)+ varies with radius except for a narrow region, 70 &lt; (1−r)+ &lt; 120, within which the gradient is approximately 0.149. The gradient of $\overline{u}_{z}^{+}$ with respect to ln{(1−r)++a+} at the present relatively low Reynolds number of ReD=44000 is not consistent with the proposition that the mean axial velocity $\overline{u}_{z}^{+}$ is logarithmic with respect to the sum of the wall distance (1−r)+ and an additive constant a+ within a mesolayer below 300 wall units. For the standard case of a+=0 within the narrow region from (1−r)+=50 to 90, the gradient of $\overline{u}_{z}^{+}$ with respect to ln{(1−r)++a+} is approximately 2.35. Computational results at the lower Reynolds number ReD=5300 also agree well with existing data. The gradient of $\overline{u}_{z}$ with respect to 1−r at ReD=44000 is approximately equal to that at ReD=5300 for the region of 1−r &gt; 0.4. For 5300 &lt; ReD &lt; 44000, bulk-velocity-normalized mean velocity defect profiles from the present DNS and from previous experiments collapse within the same radial range of 1−r &gt; 0.4. A rationale based on the curvature of mean velocity gradient profile is proposed to understand the perplexing existence of logarithmic mean velocity profile in very-low-Reynolds-number pipe flows. Beyond ReD=44000, axial turbulence intensity varies linearly with radius within the range of 0.15 &lt; 1−r &lt; 0.7. Flow visualizations and two-point correlations reveal large-scale structures with comparable near-wall azimuthal dimensions at ReD=44000 and 5300 when measured in wall units. When normalized in outer units, streamwise coherence and azimuthal dimension of the large-scale structures in the pipe core away from the wall are also comparable at these two Reynolds numbers.