An extended Reynolds analogy for excited wavy instabilities of developing streamwise vortices with applications to scalar mixing intensification

2007 ◽  
Vol 463 (2083) ◽  
pp. 1791-1813 ◽  
Author(s):  
J.T.C Liu
Keyword(s):  
Pressure Gradient ◽  
Vortex Flow ◽  
Streamwise Vortex ◽  
Streamwise Velocity ◽  
Scalar Transport ◽  
Streamwise Vortices ◽  
Reynolds Analogy ◽  
Scalar Fluxes ◽  
Transport Problems ◽  
Mixing Intensification

Studies are presented to elucidate the role of steady streamwise vortex structures, initiated upstream from weak Görtler vortices in the absence of explicit vortex generators, and their excited nonlinear wavy instabilities in the intensification of scalar mixing in a spatially developing mixing region. While steady streamwise vortex flow gives rise to significant mixing enhancement, the excited nonlinear wavy instabilities, which in turn modify the basic three-dimensional streamwise vortices, give rise to further mixing intensification which is quantitatively assessed by a mixedness parameter. Possibility of similarity between the dimensionless streamwise momentum and scalar transport problems leading to an extended Reynolds analogy is sought. This similarity is shown earlier to hold for the steady streamwise vortex flow in the absence of nonlinear wavy instabilities (Liu & Sabry 1991 Proc. R. Soc. A 432 , 1–12). In this paper, the momentum conservation equations for the nonlinear wavy or secondary instabilities together with the advected fluctuation scalar problems are examined in detail. The presence of the streamwise fluctuation pressure gradient, which prevents the similarity, is estimated in terms of the fluctuation dynamical pressure and its relative importance to advective transport. It is found from scaling that the fluctuating streamwise pressure gradient, though not completely negligible, is sufficiently unimportant so as to render similarity between fluctuation streamwise velocity and fluctuation temperature and concentration a distinct possibility. The scalar fluctuations are then inferable from the fluctuation streamwise velocity and that the Reynolds stresses of the nonlinear fluctuations and the scalar fluxes are also similar. The nonlinear instability-modified mean streamwise momentum and the modified mean heat and mass transport problems are also similar, thus providing a complete ‘Reynolds analogy’, rendering possible the interpretation of the scalar mixedness for a gaseous medium for which the Prandtl and Schmidt numbers are near unity. It is found that the nonlinearity of the wavy instability, which induces scalar fluxes modifying the mean scalar transport, further intensifies scalar mixedness over a significant streamwise region which is well above that achieved by the steady, unmodified streamwise vortices alone for the numerical example corresponding to the most amplified wavy-sinuous mode.

Nonlinear instability of developing streamwise vortices with applications to boundary layer heat transfer intensification through an extended Reynolds analogy

2008 ◽  
Vol 366 (1876) ◽  
pp. 2699-2716 ◽  
Author(s):  
J.T.C Liu
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Heat Flux ◽  
Reynolds Shear Stress ◽  
Streamwise Vortex ◽  
Secondary Instability ◽  
Streamwise Vortices ◽  
Present Contribution ◽  
Reynolds Analogy ◽  
Transport Effects

The intent of the present contribution is to explain theoretically the experimentally measured surface heat transfer rates on a slightly concave surface with a thin boundary layer in an otherwise laminar flow. As the flow develops downstream, the measured heat transfer rate deviates from the local laminar value and eventually exceeds the local turbulent value in a non-trivial manner even in the absence of turbulence. While the theory for steady strong nonlinear development of streamwise vortices can bridge the heat transfer from laminar to the local turbulent value, further intensification is attributable to the transport effects of instability of the basic steady streamwise vortex system. The problem of heat transport by steady and fluctuating nonlinear secondary instability is formulated. An extended Reynolds analogy for Prandtl number unity, Pr =1, is developed, showing the similarity between streamwise velocity and the temperature. The role played by the fluctuation-induced heat flux is similar to momentum flux by the Reynolds shear stress. Inferences from the momentum problem indicate that the intensified heat flux developing well beyond the local turbulent value is attributed to the transport effects of the nonlinear secondary instability, which leads to the formation of ‘coherent structures’ of the flow. The basic underlying pinions of the non-linear hydrodynamic stability problem are the analyses of J. T. Stuart, which uncovered physical mechanisms of nonlinearities that are crucial to the present developing boundary layers supporting streamwise vortices and their efficient scalar transporting mechanisms.

Streamwise vortex transportation and loss generation in an intermediate turbine duct

2019 ◽  
Vol 234 (6) ◽  
pp. 766-776
Author(s):  
Dong Fan ◽  
Chao Zhou
Keyword(s):  
Pressure Gradient ◽  
Streamwise Vortex ◽  
Radial Pressure ◽  
Streamwise Vortices ◽  
Loss Mechanism ◽  
Flow Physics ◽  
Tip Leakage ◽  
Turbofan Engines ◽  
Intermediate Turbine Duct ◽  
Streamwise Pressure Gradient

Annular S-shaped intermediate turbine ducts are used in modern turbofan engines with large by-pass ratios. To reduce the weight of an engine, the intermediate turbine ducts should be as short as possible, while keeping the loss at an acceptable level. Understanding the flow physics within the intermediate turbine ducts is the key to improve the intermediate turbine duct design. This paper aims to understand the transportation of the inlet streamwise vortices and loss generation in intermediate turbine ducts. First, cases with isolate incoming streamwise vortices at different spanwise locations and different axial velocities are investigated. The transportation of isolated vortex and loss generation are highly related to the interaction between vortex and boundary layer, which are mainly determined by the streamwise pressure gradient. When the axial velocity of the streamwise vortex is different to the main flow, the radial pressure gradient also has an effect. Then, the inlet condition of the intermediate turbine ducts is setup based on the flow field at the exit of a cascade, which contains the flow structures such as the tip leakage vortex, hub secondary vortex and the wake. The flow physics and the loss mechanism are analysed in detail. The formation mechanism of counter-rotating vortices pair and the influence of inlet vortex on loss generation within the intermediate turbine ducts are also presented.

scholarly journals Passive scalar transport in rotating turbulent channel flow

2018 ◽  
Vol 844 ◽  
pp. 297-322 ◽  
Author(s):  
Geert Brethouwer
Keyword(s):  
Channel Flow ◽  
Turbulent Channel Flow ◽  
Passive Scalar ◽  
Scalar Transport ◽  
Scalar Flux ◽  
Mean Velocity ◽  
Reynolds Analogy ◽  
Scalar Fluxes ◽  
Passive Scalar Transport ◽  
Gap Width

Passive scalar transport in turbulent channel flow subject to spanwise system rotation is studied by direct numerical simulations. The Reynolds number $Re=U_{b}h/\unicode[STIX]{x1D708}$ is fixed at 20 000 and the rotation number $Ro=2\unicode[STIX]{x1D6FA}h/U_{b}$ is varied from 0 to 1.2, where $U_{b}$ is the bulk mean velocity, $h$ the half channel gap width and $\unicode[STIX]{x1D6FA}$ the rotation rate. The scalar is constant but different at the two walls, leading to steady scalar transport across the channel. The rotation causes an unstable channel side with relatively strong turbulence and turbulent scalar transport, and a stable channel side with relatively weak turbulence or laminar-like flow, weak turbulent scalar transport but large scalar fluctuations and steep mean scalar gradients. The distinct turbulent–laminar patterns observed at certain $Ro$ on the stable channel side induce similar patterns in the scalar field. The main conclusions of the study are that rotation reduces the similarity between the scalar and velocity field and that the Reynolds analogy for scalar-momentum transport does not hold for rotating turbulent channel flow. This is shown by a reduced correlation between velocity and scalar fluctuations, and a strongly reduced turbulent Prandtl number of less than 0.2 on the unstable channel side away from the wall at higher $Ro$. On the unstable channel side, scalar scales become larger than turbulence scales according to spectra and the turbulent scalar flux vector becomes more aligned with the mean scalar gradient owing to rotation. Budgets in the governing equations of the scalar energy and scalar fluxes are presented and discussed as well as other statistics relevant for turbulence modelling.

scholarly journals On the formation of streamwise vortices by plasma vortex generators

2013 ◽  
Vol 733 ◽  
pp. 370-393 ◽  
Author(s):  
Timothy N. Jukes ◽  
Kwing-So Choi
Keyword(s):  
Boundary Layer ◽  
Flow Direction ◽  
Vortex Formation ◽  
Vortex Generator ◽  
Streamwise Vortex ◽  
Spanwise Direction ◽  
Formation Mechanisms ◽  
Wall Jet ◽  
Streamwise Vortices ◽  
Barrier Discharge Plasma

AbstractThe streamwise vortices generated by dielectric-barrier-discharge plasma actuators in the laminar boundary layer were investigated using particle image velocimetry to understand the vortex-formation mechanisms. The plasma vortex generator was oriented along the primary flow direction to produce a body force in the spanwise direction. This created a spanwise-directed wall jet which interacted with the oncoming boundary layer to form a coherent streamwise vortex. It was found that the streamwise vortices were formed by the twisting and folding of the spanwise vorticity in the oncoming boundary layer into the outer shear layer of the spanwise wall jet, which added its own vorticity to increase the circulation along the actuator length. This is similar to the delta-shaped, vane-type vortex generator, except that the circulation was enhanced by the addition of the vorticity in the plasma jet. It was also observed that the plasma vortex was formed close to the wall with an enhanced wall-ward entrainment, which created strong downwash above the actuator.

The Effect of Real Turbine Roughness With Pressure Gradient on Heat Transfer

2003 ◽  
Author(s):  
Jeffrey P. Bons ◽  
Stephen T. McClain
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Pressure Gradient ◽  
Adverse Pressure Gradient ◽  
Combined Effects ◽  
Pressure Gradients ◽  
Smooth Wall ◽  
Reynolds Analogy ◽  
Integral Boundary ◽  
Favorable Pressure Gradient

Experimental measurements of heat transfer (St) are reported for low speed flow over scaled turbine roughness models at three different freestream pressure gradients: adverse, zero (nominally), and favorable. The roughness models were scaled from surface measurements taken on actual, in-service land-based turbine hardware and include samples of fuel deposits, TBC spallation, erosion, and pitting as well as a smooth control surface. All St measurements were made in a developing turbulent boundary layer at the same value of Reynolds number (Rex≅900,000). An integral boundary layer method used to estimate cf for the smooth wall cases allowed the calculation of the Reynolds analogy (2St/cf). Results indicate that for a smooth wall, Reynolds analogy varies appreciably with pressure gradient. Smooth surface heat transfer is considerably less sensitive to pressure gradients than skin friction. For the rough surfaces with adverse pressure gradient, St is less sensitive to roughness than with zero or favorable pressure gradient. Roughness-induced Stanton number increases at zero pressure gradient range from 16–44% (depending on roughness type), while increases with adverse pressure gradient are 7% less on average for the same roughness type. Hot-wire measurements show a corresponding drop in roughness-induced momentum deficit and streamwise turbulent kinetic energy generation in the adverse pressure gradient boundary layer compared with the other pressure gradient conditions. The combined effects of roughness and pressure gradient are different than their individual effects added together. Specifically, for adverse pressure gradient the combined effect on heat transfer is 9% less than that estimated by adding their separate effects. For favorable pressure gradient, the additive estimate is 6% lower than the result with combined effects. Identical measurements on a “simulated” roughness surface composed of cones in an ordered array show a behavior unlike that of the scaled “real” roughness models. St calculations made using a discrete-element roughness model show promising agreement with the experimental data. Predictions and data combine to underline the importance of accounting for pressure gradient and surface roughness effects simultaneously rather than independently for accurate performance calculations in turbines.

The anatomy of the mixing transition in homogeneous and stratified free shear layers

2000 ◽  
Vol 413 ◽  
pp. 1-47 ◽  
Author(s):  
C. P. CAULFIELD ◽  
W. R. PELTIER
Keyword(s):  
Shear Layer ◽  
Three Dimensional ◽  
Streamwise Vortex ◽  
Shear Layers ◽  
Finite Amplitude ◽  
Small Scale ◽  
Two Dimensional ◽  
Streamwise Vortices ◽  
Free Shear Layers ◽  
Nonlinear Amplification

We investigate the detailed nature of the ‘mixing transition’ through which turbulence may develop in both homogeneous and stratified free shear layers. Our focus is upon the fundamental role in transition, and in particular the associated ‘mixing’ (i.e. small-scale motions which lead to an irreversible increase in the total potential energy of the flow) that is played by streamwise vortex streaks, which develop once the primary and typically two-dimensional Kelvin–Helmholtz (KH) billow saturates at finite amplitude.Saturated KH billows are susceptible to a family of three-dimensional secondary instabilities. In homogeneous fluid, secondary stability analyses predict that the stream-wise vortex streaks originate through a ‘hyperbolic’ instability that is localized in the vorticity braids that develop between billow cores. In sufficiently strongly stratified fluid, the secondary instability mechanism is fundamentally different, and is associated with convective destabilization of the statically unstable sublayers that are created as the KH billows roll up.We test the validity of these theoretical predictions by performing a sequence of three-dimensional direct numerical simulations of shear layer evolution, with the flow Reynolds number (defined on the basis of shear layer half-depth and half the velocity difference) Re = 750, the Prandtl number of the fluid Pr = 1, and the minimum gradient Richardson number Ri(0) varying between 0 and 0.1. These simulations quantitatively verify the predictions of our stability analysis, both as to the spanwise wavelength and the spatial localization of the streamwise vortex streaks. We track the nonlinear amplification of these secondary coherent structures, and investigate the nature of the process which actually triggers mixing. Both in stratified and unstratified shear layers, the subsequent nonlinear amplification of the initially localized streamwise vortex streaks is driven by the vertical shear in the evolving mean flow. The two-dimensional flow associated with the primary KH billow plays an essentially catalytic role. Vortex stretching causes the streamwise vortices to extend beyond their initially localized regions, and leads eventually to a streamwise-aligned collision between the streamwise vortices that are initially associated with adjacent cores.It is through this collision of neighbouring streamwise vortex streaks that a final and violent finite-amplitude subcritical transition occurs in both stratified and unstratified shear layers, which drives the mixing process. In a stratified flow with appropriate initial characteristics, the irreversible small-scale mixing of the density which is triggered by this transition leads to the development of a third layer within the flow of relatively well-mixed fluid that is of an intermediate density, bounded by narrow regions of strong density gradient.

The Evolution of Streamwise Counter-Rotating Vortices in Flat Plate Boundary Layer With Leading Edge Pattern

2015 ◽  
Author(s):  
Seyed Mohammad Hasheminejad ◽  
Hatsari Mitsudharmadi ◽  
S. H. Winoto ◽  
Kim Boon Lua ◽  
Hong Tong Low
Keyword(s):  
Boundary Layer ◽  
Flat Plate ◽  
Plate Boundary ◽  
Leading Edge ◽  
Streamwise Vortex ◽  
Streamwise Velocity ◽  
Stream Velocity ◽  
Hot Wire ◽  
Layer Flow ◽  
Different Characteristics

The evolution of streamwise counter-rotating vortices induced by different leading edge patterns is investigated quantitatively using hot-wire anemometer. A notched and triangular leading edge with the same wavelength and amplitude were designed to induce streamwise vortices over a flat plate at Reynolds number (based on the wavelength of the leading edge patterns) of 3080 corresponding to free-stream velocity of 3 m/s. The streamwise velocity at different streamwise locations collected and analyzed using a single wire probe hot-wire anemometer showed reveal different characteristics of boundary layer flow due to the presence of these two leading edge patterns. The major difference is the appearance of an additional streamwise vortex between the troughs of the notched pattern. Such vortices increase the mixing effect in the boundary layer as well as the velocity profile.

scholarly journals Analysis of Turbulent Scalar Flux Models for a Discrete Hole Film Cooling Flow

2015 ◽  
Author(s):  
Julia Ling ◽  
Kevin J. Ryan ◽  
Julien Bodart ◽  
John K. Eaton
Keyword(s):  
Temperature Distribution ◽  
Prandtl Number ◽  
Film Cooling ◽  
Turbulent Viscosity ◽  
Fluid Temperature ◽  
Turbulent Prandtl Number ◽  
Generalized Gradient ◽  
Reynolds Analogy ◽  
Scalar Fluxes ◽  
Gradient Diffusion

Algebraic closures for the turbulent scalar fluxes were evaluated for a discrete hole film cooling geometry using the results from the high-fidelity Large Eddy Simulation (LES) of Bodart et al. [1]. Several models for the turbulent scalar fluxes exist, including the widely used Gradient Diffusion Hypothesis, the Generalized Gradient Diffusion Hypothesis [2], and the Higher Order Generalized Gradient Diffusion Hypothesis [3]. By analyzing the results from the LES, it was possible to isolate the error due to these turbulent mixing models. Distributions of the turbulent diffusivity, turbulent viscosity, and turbulent Prandtl number were extracted from the LES results. It was shown that the turbulent Prandtl number varies significantly spatially, undermining the applicability of the Reynolds analogy for this flow. The LES velocity field and Reynolds stresses were fed into a RANS solver to calculate the fluid temperature distribution. This analysis revealed in which regions of the flow various modeling assumptions were invalid and what effect those assumptions had on the predicted temperature distribution.

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