A Reduced-Order Model of the Mean Properties of a Turbulent Wall Boundary Layer at a Zero Pressure Gradient

2014 ◽  
Vol 136 (3) ◽  
Author(s):  
Lei Xu ◽  
Zvi Rusak ◽  
Luciano Castillo
Keyword(s):  
Boundary Layer ◽  
Shear Stress ◽  
Reynolds Shear Stress ◽  
Streamwise Velocity ◽  
Similarity Function ◽  
Mixing Length ◽  
Flow Properties ◽  
Normal Coordinate ◽  
Streamwise Location ◽  
The Mean

A novel two-equations model for computing the flow properties of a spatially-developing, incompressible, zero-pressure-gradient, turbulent boundary layer over a smooth, flat wall is developed. The mean streamwise velocity component inside the boundary layer is described by the Reynolds-averaged Navier–Stokes equation where the Reynolds shear stress is given by an extended mixing-length model. The nondimensional form of the mixing length is described by a polynomial function in terms of the nondimensional wall normal coordinate. Moreover, a stream function approach is applied with a leading-order term described by a similarity function. Two ordinary differential equations are derived for the solution of the similarity function along the wall normal coordinate and for its streamwise location. A numerical integration scheme of the model equations is developed and enables the solution of flow properties. The coefficients of the mixing-length polynomial function are modified at each streamwise location as part of solution iterations to satisfy the wall and far-field boundary conditions and adjust the local boundary layer thickness, δ99.4, to a location where streamwise speed is 99.4% of the far-field streamwise velocity. The elegance of the present approach is established through the successful solution of the various flow properties across the boundary layer (i.e., mean streamwise velocity, viscous stress, Reynolds shear stress, skin friction coefficient, and growth rate of boundary layer among others) from the laminar regime all the way to the fully turbulent regime. It is found that results agree with much available experimental data and direct numerical simulations for a wide range of Reθ based on the momentum thickness (Reθ) from 15 up to 106, except for the transition region from laminar to turbulent flow. Furthermore, results shed light on the von Kármán constant as a function of Reθ, the possible four-layer nature of the mean streamwise velocity profile, the universal profiles of the streamwise velocity and the Reynolds shear stress at high Reθ, and the scaling laws at the outer region.

scholarly journals High-resolution velocity measurement in the inner part of turbulent boundary layers over super-hydrophobic surfaces

2016 ◽  
Vol 801 ◽  
pp. 670-703 ◽  
Author(s):  
Hangjian Ling ◽  
Siddarth Srinivasan ◽  
Kevin Golovin ◽  
Gareth H. McKinley ◽  
Anish Tuteja ◽  
...  
Keyword(s):  
Boundary Layer ◽  
Shear Stress ◽  
Velocity Profile ◽  
Drag Reduction ◽  
Slip Velocity ◽  
Reynolds Shear Stress ◽  
Turbulent Boundary Layers ◽  
Mean Velocity ◽  
Roughness Elements ◽  
The Mean

Digital holographic microscopy is used for characterizing the profiles of mean velocity, viscous and Reynolds shear stresses, as well as turbulence level in the inner part of turbulent boundary layers over several super-hydrophobic surfaces (SHSs) with varying roughness/texture characteristics. The friction Reynolds numbers vary from 693 to 4496, and the normalized root mean square values of roughness $(k_{rms}^{+})$ vary from 0.43 to 3.28. The wall shear stress is estimated from the sum of the viscous and Reynolds shear stress at the top of roughness elements and the slip velocity is obtained from the mean profile at the same elevation. For flow over SHSs with $k_{rms}^{+}<1$, drag reduction and an upward shift of the mean velocity profile occur, along with a mild increase in turbulence in the inner part of the boundary layer. As the roughness increases above $k_{rms}^{+}\sim 1$, the flow over the SHSs transitions from drag reduction, where the viscous stress dominates, to drag increase where the Reynolds shear stress becomes the primary contributor. For the present maximum value of $k_{rms}^{+}=3.28$, the inner region exhibits the characteristics of a rough wall boundary layer, including elevated wall friction and turbulence as well as a downward shift in the mean velocity profile. Increasing the pressure in the test facility to a level that compresses the air layer on the SHSs and exposes the protruding roughness elements reduces the extent of drag reduction. Aligning the roughness elements in the streamwise direction increases the drag reduction. For SHSs where the roughness effect is not dominant ($k_{rms}^{+}<1$), the present measurements confirm previous theoretical predictions of the relationships between drag reduction and slip velocity, allowing for both spanwise and streamwise slip contributions.

Evolution of a moderately short turbulent boundary layer in a severe pressure gradient

1974 ◽  
Vol 64 (4) ◽  
pp. 763-774 ◽  
Author(s):  
R. G. Deissler
Keyword(s):  
Boundary Layer ◽  
Shear Stress ◽  
Turbulent Boundary Layer ◽  
Pressure Gradient ◽  
Reynolds Shear Stress ◽  
Pressure Gradients ◽  
Mean Velocity ◽  
Mass Injection ◽  
The Mean ◽  
Theoretical Results

The early and intermediate development of a highly accelerated (or decelerated) turbulent boundary layer is analysed. For sufficiently large accelerations (or pressure gradients) and for total normal strains which are not excessive, the equation for the Reynolds shear stress simplifies to give a stress that remains approximately constant as it is convected along streamlines. The theoretical results for the evolution of the mean velocity in favourable and adverse pressure gradients agree well with experiment for the cases considered. A calculation which includes mass injection at the wall is also given.

Characteristics of a turbulent boundary layer with an external turbulent uniform shear flow

1976 ◽  
Vol 77 (2) ◽  
pp. 369-396 ◽  
Author(s):  
Q. A. Ahmad ◽  
R. E. Luxton ◽  
R. A. Antonia
Keyword(s):  
Boundary Layer ◽  
Shear Stress ◽  
Shear Flow ◽  
Free Stream ◽  
Reynolds Shear Stress ◽  
Wall Layer ◽  
Turbulence Level ◽  
Mean Velocity ◽  
Free Stream Turbulence ◽  
The Mean

Measurements are presented of both mean and fluctuating velocity components in a turbulent boundary layer subjected to a nearly homogeneous external turbulent shear flow. The Reynolds shear stress in the external shear flow is small compared with the wall shear stress. Its transverse mean velocity gradient λ (≃ 6 s−l) is also small compared with typical gradients based on outer variables (say Uw/δ, where Uwis the value of the linear velocity profile extrapolated to the wall and δ is the boundary-layer thickness), but is of the same order as Ut/δ (Ur is the friction velocity). The influence of both positive and negative transverse velocity gradients on the turbulent wall layer is investigated over a streamwise region where the normal Reynolds stresses in the external flow are approximately equal and constant in the streamwise direction. In this region, the integral length scale of the external flow is of the same order of magnitude as that of the wall layer. Measurements in the boundary layer are also given for an un-sheared external turbulent flow (λ = 0) with a turbulence level Tu of 1.5%, approximately the same as that for h = ± 6 s−1. (Tu, is defined as the ratio of the r.m.s. longitudinal velocity fluctuation to Uw.) The measurements are in good agreement with those available in the literature for a similar free-stream turbulence level and show that the external turbulence level and length scale exert a large influence on the turbulence structure in the boundary layer. The additional effect of the external shear on the mean velocity and turbulent energy budget distributions in the inner region of the boundary layer is found to be small. In the outer region, the ‘wake’ component of the mean velocity defect is lowered by the presence of free-stream turbulence and one extra effect due to the external shear is an increase in the Reynolds shear stress when h is positive and a decrease when h is negative. Another interesting effect due to the shear is the appearance near the edge of the layer of a small but distinct region where the local mean velocity is constant and the Reynolds shear stress is negligible.

Turbulent Boundary Layers on Surfaces Covered With Filamentous Algae

2000 ◽  
Vol 122 (2) ◽  
pp. 357-363 ◽  
Author(s):  
Michael P. Schultz
Keyword(s):  
Boundary Layer ◽  
Shear Stress ◽  
Turbulence Intensity ◽  
Reynolds Shear Stress ◽  
Skin Friction Coefficient ◽  
Laser Doppler Velocimeter ◽  
Mean Velocity ◽  
Rough Walls ◽  
The Mean ◽  
Stress Profiles

Turbulent boundary layer measurements have been made on surfaces covered with filamentous marine algae. These experiments were conducted in a closed return water tunnel using a two-component, laser Doppler velocimeter (LDV). The mean velocity profiles and parameters, as well as the axial and wall-normal turbulence intensities and Reynolds shear stress, are compared with flows over smooth and sandgrain rough walls. Significant increases in the skin friction coefficient for the algae-covered surfaces were measured. The boundary layer and integral thickness length scales were also increased. The results indicate that profiles of the turbulence quantities for the smooth and sandgrain rough walls collapse when friction velocity and boundary layer thickness are used as normalizing parameters. The algae-covered surfaces, however, exhibited a significant increase in the wall-normal turbulence intensity and the Reynolds shear stress, with only a modest increase in the axial turbulence intensity. The peak in the Reynolds shear stress profiles for the algae surfaces corresponded to the maximum extent of outward movement of the algae filaments. [S0098-2202(00)01902-7]

Measurements of turbulent flow downstream of a rearward-facing step

1978 ◽  
Vol 86 (3) ◽  
pp. 545-566 ◽  
Author(s):  
D. W. Etheridge ◽  
P. H. Kemp
Keyword(s):  
Boundary Layer ◽  
Shear Stress ◽  
Shear Layer ◽  
Strong Evidence ◽  
Reynolds Shear Stress ◽  
Water Channel ◽  
Local Maximum ◽  
Mixing Length ◽  
Separation Region ◽  
Made In

Measurements have been made in a water channel of the flow in and around the separation region due to a rearward-facing step. Detailed profiles of mean velocities, turbulence intensities and Reynolds shear stress are presented. The turbulence measurements reveal the development of a new shear layer, which splits at reattachment with about one-sixth of the mass flow deflected upstream. The new shear layer is associated with a region of roughly constant values of both the non-dimensional mixing lengthl/xand the shear correlation coefficientK. The mixing-length ratio is larger than that found in plane mixing layers, whereas the shear coefficient is roughly the same. There is strong evidence that near the wall the length scales increase more rapidly with distance from the wall than in an attached boundary layer, and that a local maximum occurs.

Direct numerical simulation of turbulent channel flow with spanwise rotation

2015 ◽  
Vol 788 ◽  
pp. 42-56 ◽  
Author(s):  
Zhenhua Xia ◽  
Yipeng Shi ◽  
Shiyi Chen
Keyword(s):  
Shear Stress ◽  
Friction Velocity ◽  
Reynolds Shear Stress ◽  
Turbulent Channel Flow ◽  
Streamwise Velocity ◽  
Mean Velocity ◽  
Linear Profile ◽  
Unit Slope ◽  
The Mean ◽  
Spanwise Rotation

A series of direct numerical simulations of turbulent channel flow with spanwise rotation at fixed global friction Reynolds number is performed to investigate the rotation effects on the mean velocity, streamwise velocity fluctuations, Reynolds shear stress and turbulent kinetic energy. The global friction Reynolds number is chosen to be $Re_{{\it\tau}}=u_{{\it\tau}}^{\ast }h^{\ast }/{\it\nu}^{\ast }=180$ ($u_{{\it\tau}}^{\ast }$ is the global friction velocity, $h^{\ast }$ is the channel half-width and ${\it\nu}^{\ast }$ is the kinematic viscosity), while the global-friction-velocity-based rotation number $Ro_{{\it\tau}}=2{\it\Omega}^{\ast }h^{\ast }/u_{{\it\tau}}^{\ast }$ (${\it\Omega}^{\ast }$ is the dimensional angular velocity) varies from 0 to 130. In the previously reported $2{\it\Omega}^{\ast }$-slope region for the mean velocity, a linear behaviour for the streamwise velocity fluctuations, a unit-slope linear profile for the Reynolds shear stress and a $-2Ro_{{\it\tau}}$-slope linear profile for the production term of $\langle u^{\prime }u^{\prime }\rangle$ have been identified for the first time. The critical rotation number, which corresponds to the laminar limit, is predicted to be equal to $Re_{{\it\tau}}$ according to the unit-slope linear profile of the Reynolds shear stress. Our results also show that a parabolic profile of the mean velocity can be identified around the ‘second plateau’ region of the Reynolds shear stress for $Ro_{{\it\tau}}\geqslant 22$. The parabolas at different rotation numbers have the same shape of $1/Re_{{\it\tau}}$, the radius of curvature at the vertex. Furthermore, the system rotation increases the volume-averaged turbulent kinetic energy at lower rotation rates, and then decreases it when $Ro_{{\it\tau}}\gtrsim 16$.

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.

Turbulence structure of a reattaching mixing layer

1981 ◽  
Vol 110 ◽  
pp. 171-194 ◽  
Author(s):  
C. Chandrsuda ◽  
P. Bradshaw
Keyword(s):  
Boundary Layer ◽  
Shear Stress ◽  
Reynolds Shear Stress ◽  
Mixing Layer ◽  
Three Dimensional ◽  
Flow Region ◽  
Turbulent Energy ◽  
Turbulence Structure ◽  
Hot Wire ◽  
Recirculating Flow

Hot-wire measurements of second- and third-order mean products of velocity fluctuations have been made in the flow behind a backward-facing step with a thin, laminar boundary layer at the top of the step. Measurements extend to a distance of about 12 step heights downstream of the step, and include parts of the recirculating-flow region: approximate limits of validity of hot-wire results are given. The Reynolds number based on step height is about 105, the mixing layer being fully turbulent (fully three-dimensional eddies) well before reattachment, and fairly close to self-preservation in contrast to the results of some previous workers. Rapid changes in turbulence quantities occur in the reattachment region: Reynolds shear stress and triple products decrease spectacularly, mainly because of the confinement of the large eddies by the solid surface. The terms in the turbulent energy and shear stress balances also change rapidly but are still far from the self-preserving boundary-layer state even at the end of the measurement region.

The accelerated fully rough turbulent boundary layer

1977 ◽  
Vol 82 (3) ◽  
pp. 507-528 ◽  
Author(s):  
Hugh W. Coleman ◽  
Robert J. Moffat ◽  
William M. Kays
Keyword(s):  
Boundary Layer ◽  
Shear Stress ◽  
Turbulent Boundary Layer ◽  
Pressure Gradient ◽  
Stress Tensor ◽  
Skin Friction ◽  
Shape Factor ◽  
Reynolds Shear Stress ◽  
Smooth Wall ◽  
Mean Velocity

The behaviour of a fully rough turbulent boundary layer subjected to favourable pressure gradients both with and without blowing was investigated experimentally using a porous test surface composed of densely packed spheres of uniform size. Measurements of profiles of mean velocity and the components of the Reynolds-stress tensor are reported for both unblown and blown layers. Skin-friction coefficients were determined from measurements of the Reynolds shear stress and mean velocity.An appropriate acceleration parameterKrfor fully rough layers is defined which is dependent on a characteristic roughness dimension but independent of molecular viscosity. For a constant blowing fractionFgreater than or equal to zero, the fully rough turbulent boundary layer reaches an equilibrium state whenKris held constant. Profiles of the mean velocity and the components of the Reynolds-stress tensor are then similar in the flow direction and the skin-friction coefficient, momentum thickness, boundary-layer shape factor and the Clauser shape factor and pressure-gradient parameter all become constant.Acceleration of a fully rough layer decreases the normalized turbulent kinetic energy and makes the turbulence field much less isotropic in the inner region (forFequal to zero) compared with zero-pressure-gradient fully rough layers. The values of the Reynolds-shear-stress correlation coefficients, however, are unaffected by acceleration or blowing and are identical with values previously reported for smooth-wall and zero-pressure-gradient rough-wall flows. Increasing values of the roughness Reynolds number with acceleration indicate that the fully rough layer does not tend towards the transitionally rough or smooth-wall state when accelerated.

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