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

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.     

Structure and dynamics of low Reynolds number turbulent pipe flow

Using large-scale numerical calculations, we explore the proper orthogonal decomposition of low Reynolds number turbulent pipe flow, using both the translational invariant (Fourier) method and the method of snapshots. Each method has benefits and drawbacks, making the 'best' choice dependent on the purpose of the analysis. Owing to its construction, the Fourier method includes all the flow fields that are translational invariants of the simulated flow fields. Thus, the Fourier method converges to an estimate of the dimension of the chaotic attractor in less total simulation time than the method of snapshots. The converse is that for a given simulation, the method of snapshots yields a basis set that is more optimal because it does not include all of the translational invariants that were not a part of the simulation. Using the Fourier method yields smooth structures with definable subclasses based upon Fourier wavenumber pairs, and results in a new dynamical systems insight into turbulent pipe flow. These subclasses include a set of modes that propagate with a nearly constant phase speed, act together as a wave packet and transfer energy from streamwise rolls. It is these interactions that are responsible for bursting events and Reynolds stress generation. These structures and dynamics are similar to those found in turbulent channel flow. A comparison of structures and dynamics in turbulent pipe and channel flows is reported to emphasize the similarities and differences.     

Prospects for useful research on coherent structure in turbulent shear flow

Six different flows involving coherent structures are discussed with varying amounts of detail. These are the puff in a pipe, the turbulent spot, the spiral turbulence, the vortex ring, the vortex street, and the mixing layer. One central theme is that non-steady similarity arguments and topology are of the essence of coherent structure. Another is that the Reynolds equations, which are sterile when applied to a structureless mean flow, may be quite productive when applied to a single structure. A third theme is the prospect for at least partial control of technically important flows by exploiting the concept of coherent structure.     

Asymptotic scaling in turbulent pipe flow

The streamwise velocity component in turbulent pipe flow is assessed to determine whether it exhibits asymptotic behaviour that is indicative of high Reynolds numbers. The asymptotic behaviour of both the mean velocity (in the form of the log law) and that of the second moment of the streamwise component of velocity in the outer and overlap regions is consistent with the development of spectral regions which indicate inertial scaling. It is shown that an 'inertial sublayer' in physical space may be considered as a spatial analogue of the inertial subrange in the velocity spectrum and such behaviour only appears for Reynolds numbers R+>5 x 10(3), approximately, much higher than was generally thought.     

Rotating-disk-type flow over loose boundaries

Rotating-disk-type flow of a liquid over a loose boundary, such as a layer of sand, is investigated. For this flow the formation of a new large-scale spiral pattern has been discovered. The new pattern is reminiscent of the Type-I spiral-vortex structures which characterize the laminar–turbulent transition region of boundary layers over rigid rotating disks. Flow visualizations reveal that the new pattern and the Type-I spiral vortices co-exist in the loose-boundary flow. The research investigating the origin of the new large-scale pattern is reviewed. Then photographs from flow visualizations are analysed to obtain estimates for the critical Reynolds number for which Type-I spiral vortices first appear for the loose-boundary flow and for the critical Reynolds numbers for the laminar–turbulent transition of the boundary layer. The results suggest that Type-I vortices appear at much lower Reynolds numbers over loose boundaries in comparison with flow over rigid rotating disks and that transition also appears to be advanced to much lower Reynolds numbers. The discussion of the results suggests that advanced transition arises from disturbances introduced into the flow after the loose boundary has been mobilized and not from disturbances associated with the roughness that the surfaces of the granular layer represents to the flow while grains are at rest.     

A deterministic model for the sublayer streaks in turbulent boundary layers for application to flow control

We present a relatively simple, deterministic, theoretical model for the sublayer streaks in a turbulent boundary layer based on an analogy with Klebanoff modes. Our approach is to generate the streamwise vortices found in the buffer layer by means of a vorticity source in the form of a fictitious body force. It is found that the strongest streaks correspond to a spanwise wavelength that lies within the range of the experimentally observed values for the statistical mean streak spacing. We also present results showing the effect of streamwise pressure gradient, Reynolds number and wall compliance on the sublayer streaks. The theoretical predictions for the effects of wall compliance on the streak characteristics agree well with experimental data. Our proposed theoretical model for the quasi-periodic bursting cycle is also described, which places the streak modelling in context. The proposed bursting process is as follows: (i) streamwise vortices generate sublayer streaks and other vortical elements generate propagating plane waves, (ii) when the streaks reach a sufficient amplitude, they interact nonlinearly with the plane waves to produce oblique waves that exhibit transient growth, and (iii) the oblique waves interact nonlinearly with the plane wave to generate streamwise vortices; these in turn generate the sublayer streaks and so the cycle is renewed.     

Aspects of linear and nonlinear instabilities leading to transition in pipe and channel flows

The failure of normal-mode linear stability analysis to predict a transition Reynolds number (Retr) in pipe flow and subcritical transition in plane Poiseuille flow (PPF) has led to the search of other scenarios to explain transition to turbulence in both flows. In this work, various results associated with linear and nonlinear mechanisms of both flows are presented. The results that combine analytical and experimental approaches indicate the strong link between the mechanisms governing the transition of both flows. It is demonstrated that the linear transient growth mechanism is based on the existence of a pair of least stable nearly parallel modes (having opposite phases and almost identical amplitude distributions). The analysis that has been applied previously to pipe flow is extended here to a fully developed channel flow predicting the shape of the optimized initial disturbance (a pair of counter-rotating vortices, CVP), time for maximum energy amplification and the dependence of the latter on Re. The results agree with previous predictions based on many modes. Furthermore, the shape of the optimized initial disturbance is similar in both flows and has been visualized experimentally. The analysis reveals that in pipe flow, the transient growth is a consequence of two opposite running modes decaying with an equal decay rate whereas in PPF it is due to two stationary modes decaying with different decay rates. In the first nonlinear scenario, the breakdown of the CVPs (produced by the linear transient growth mechanism) into hairpin vortices is followed experimentally. The associated scaling laws, relating the minimal disturbance amplitude required for the initiation of hairpins and the Re, are found experimentally for both PPF and pipe flow. The scaling law associated with PPF agrees well with the previous predictions of Chapman, whereas the scaling of the pipe flow is the same as the one previously obtained by Hof et al. indicating transition to a turbulent state. In the second nonlinear scenario, the base flow of pipe when it is mildly deviated from the Poiseuille profile by an axisymmetric distortion is examined. The nonlinear features reveal a Retr of approximately 2000 associated with the bifurcation between two deviation solutions.     

Edge states intermediate between laminar and turbulent dynamics in pipe flow

We studied the dynamics near the boundary between laminar and turbulent dynamics in pipe flow. This boundary contains invariant dynamical states that are attracting when the dynamics is confined to the boundary. These states can be found by controlling a single quantity, in our case the energy content. The edge state is dominated by two downstream vortices and shows intrinsic chaotic dynamics. With increasing Reynolds number the separation between the edge state and turbulence increases. We can track it down to Re=1900, where the turbulent lifetimes are short enough that spontaneous decay can also be seen in experiments.     

Experimental study on the flow and heat transfer characteristics of a tetra-n-butyl ammonium bromide hydrate slurry (first report: Flow characteristics)

The flow characteristics of a tetra-n-butyl ammonium bromide hydrate slurry were investigated experimentally. Reynolds number, tube diameter and solid fraction were varied as the experimental parameters. For laminar flow, it was found that the ratio of the coefficients of pipe friction increases with solid fraction, and the rate of increase is high in the case of a low Reynolds number. For turbulent flow, the ratio of the coefficients of pipe friction was approximately 1 for all tube diameters at low solids fraction. It then decreased slightly at a particular solid fraction and increased again at high solid fraction. Theoretical analysis was carried out using the experimental results, and it was found that the flow characteristics of the hydrate slurry can be treated as those of a pseudoplastic fluid and clarified using the apparent Reynolds number.     

Turbulent flow in smooth and rough pipes

Recent experiments at Princeton University have revealed aspects of smooth pipe flow behaviour that suggest a more complex scaling than previously noted. In particular, the pressure gradient results yield a new friction factor relationship for smooth pipes, and the velocity profiles indicate the presence of a power-law region near the wall and, for Reynolds numbers greater than about 400x103 (R+>9x103), a logarithmic region further out. New experiments on a rough pipe with a honed surface finish with krms/D=19.4x10-6, over a Reynolds number range of 57x103-21x106, show that in the transitionally rough regime this surface follows an inflectional friction factor relationship rather than the monotonic relationship given in the Moody diagram. Outer-layer scaling of the mean velocity data and streamwise turbulence intensities for the rough pipe show excellent collapse and provide strong support for Townsend's outer-layer similarity hypothesis for rough-walled flows. The streamwise rough-wall spectra also agree well with the corresponding smooth-wall data. The pipe exhibited smooth behaviour for ks+ < or =3.5, which supports the suggestion that the original smooth pipe was indeed hydraulically smooth for ReD< or =24x106. The relationship between the velocity shift, DeltaU/utau, and the roughness Reynolds number, ks+, has been used to generalize the form of the transition from smooth to fully rough flow for an arbitrary relative roughness krms/D. These predictions apply for honed pipes when the separation of pipe diameter to roughness height is large, and they differ significantly from the traditional Moody curves.     

Numerical experiments on feedback EMHD control of large scale coherent structures in channel turbulence

Summary The current work focusses on the spatio-temporal evolution of large scale coherent structures in the turbulent boundary layer of a plane channel both with and without microtile-based EMHD control. The heuristic concept behind the microtile designs that we have simulated apparently does not yield a successful drag reduction strategy (for the open-loop case [1]). In this work we investigate the flow response when the Lorentz force is applied with feedback conditioned on the advection of large-scale flow structures (e.g. hairpin vortices). We performed a long-time simulation conditioned on the passage of strong ejection events but obtained no reduction in skin friction. Based on short-time simulations we found that the near-wall flow structures undergo merely a spatial phase-shift when advecting above a single Lorentz force actuator: the structures are simply decelerated or accelerated with little change in their appearance, based on flow visualization. During this interaction of the applied Lorentz force with the flow, the Reynolds stress is unchanged.     

Turbulent fluid flows in a circular pipe and plane channel and models of mesoscale turbulence

Based on the model of anisotropic wall turbulence in the near-wall layer and the momentum model in the flow core, velocity profiles in the entire region of the turbulent flow of an incompressible fluid in a circular pipe and plane channel have been obtained. The differences in the profiles among the layers are due to the change in the structure of turbulent vortices.     

Large eddy simulation of turbulent heat transfer in pipe flows with respect to Reynolds and Prandtl number effects

Heat transfer in a fully developed turbulent pipe flow is investigated by the use of the large eddy simulation technique. Isoflux condition is imposed at the wall. Four Prandtl numbers are considered (0.71, 3, 5, and 7) and three Reynolds numbers (5,500, 10,000, and 20,000). The effects of Reynolds and Prandtl numbers on turbulent heat transfer in pipe flow are investigated in order to obtain a more detailed knowledge of the thermal field in circular pipe flow. The objective of this study is also to examine the effectiveness of the large eddy simulation approach for predicting the turbulent heat transfer at different Prandtl numbers larger than 0.71, for various Reynolds numbers up to 20,000. Validation is achieved by comparing the present predictions to the available results of the literature. The effects of Prandtl and Reynolds numbers on many statistical quantities, such as mean temperature profiles, RMS of fluctuating temperature, turbulent heat fluxes, higher-order statistics, and heat transfer coefficient, are examined. Visualizations of instantaneous filtered temperature fields are analyzed.     

Gas-particle turbulent flow of foursquare tangential jets in a simplified pulverized-coal firing boiler

An experimental cold-model of a simplified tangential firing boiler was established to investigate the mesoscale turbulent flow behaviors, including gas vortex structures, particle motions and interactions between two phases. A modified PIV technology, employing two pairs of lasers and cameras, was applied to measure the velocity and velocity gradient of turbulent flow in foursquare tangential jets alternatively. At a given initial gas velocity and particle mass loading, the interaction between gas and particles was studied at three different particle sizes. It was found that two main coherent vortex structures, circular eddy and hairpin eddy, distributed mainly in low speed area and heavy impingement area, respectively. The characteristics of particle motion in foursquare tangential jets correlated with gas turbulence dissipation, particle size, particle concentration and particle density. Small particles were easily entrained by gas vortex, so that they consumed more turbulence energy and attenuated the gas turbulence intensity. On the contrary, large particles had more inertia and led to heavier impingement in the chamber center, resulting in particle random distribution and complex momentum transfer between gas and particles. Moreover, large particles stretched the coherent vortex to be narrow and long, while small particles pulled down the vortices rotation intensity.     

Direct resonance analysis and modeling for a turbulent boundary layer over a corrugated surface

Summary Effects of surface corrugation on turbulent flow in a boundary layer are studied using a model based on the direct resonance theory. The induced mean flow due to weakly nonlinear waves, superimposed on the mean and fluctuating components of turbulence, is determined. The mean turbulent flow is affected by the surface corrugation throughout the boundary layer. The corrugated surface generates higher harmonics and affects the streamwise vortices generated by the waves superimposed on turbulence whose mean flow includes secondary induced mean flow components due to the corrugation.     

Large- and very-large-scale motions in channel and boundary-layer flows

Large-scale motions (LSMs; having wavelengths up to 2-3 pipe radii) and very-LSMs (having wavelengths more than 3 pipe radii) have been shown to carry more than half of the kinetic energy and Reynolds shear stress in a fully developed pipe flow. Studies using essentially the same methods of measurement and analysis have been extended to channel and zero-pressure-gradient boundary-layer flows to determine whether large structures appear in these canonical wall flows and how their properties compare with that of the pipe flow. The very large scales, especially those of the boundary layer, are shorter than the corresponding scales in the pipe flow, but otherwise share a common behaviour, suggesting that they arise from similar mechanism(s) aside from the modifying influences of the outer geometries. Spectra of the net force due to the Reynolds shear stress in the channel and boundary layer flows are similar to those in the pipe flow. They show that the very-large-scale and main turbulent motions act to decelerate the flow in the region above the maximum of the Reynolds shear stress.     

微管道内湍流转捩的实验研究

研究旨在确定微管道内流动从层流到湍流转捩的临界雷诺数。利用微观粒子图像测速技术(Micro-PIV)研究了去离子水在内径为230μm的圆形截面玻璃微管道内的流场结构,得到了从层流到充分发展湍流各流动状态下的轴向平均速度分布和湍流度分布,实验雷诺数为1020~3145,同时研究了微管道内的流动阻力特性。平均速度场和脉动速度场的实验结果表明微管道内从层流到湍流的转捩发生在Re=1800~1900左右,与流动阻力的测量结果一致,与宏观流动比较,并未发现微管道内的流动转捩有明显提前。实验结果还显示,当Re>2700时,微管道内的平均流速分布和相对湍流度分布呈现典型的充分发展湍流状态特征。     

Comparison of measured and computed Strehl ratios for light propagated through a channel flow of a He-N(2) mixing layer at high Reynolds numbers

A lateral shearing interferometer was used to measure the slope of perturbed wave fronts after they propagated through a He-N(2) mixing layer in a rectangular channel. Slope measurements were used to reconstruct the phase of the turbulence-corrupted wave front. The random phase fluctuations induced by the mixing layer were captured in a large ensemble of wave-front measurements. Phase structure functions, computed from the reconstructed phase surfaces, were stationary in first increments. A five-thirds power law is shown to fit streamwise and cross-stream slices of the structure function, analogous to the Kolmogorov model for isotropic turbulence, which describes the structure function with a single parameter. Strehl ratios were computed from the phase structure functions and compared with a measured experiment obtained from simultaneous point-spread function measurements. Two additional Strehl ratios were calculated by using classical estimates that assume statistical isotropy throughout the flow. The isotropic models are a reasonable estimate of the optical degradation only within a few centimeters of the initial mixing, where the Reynolds number is low. At higher Reynolds numbers, Strehl ratios calculated from the structure functions match the experiment much better than Strehl ratio calculations that assume isotropic flow.     

Investigation of plane mixing layer using large eddy simulation

Spatially evolving turbulent mixing layers are investigated in this paper using large eddy simulation. The evolutions of large vortex structures are obtained, in which the processes of vortex roll up, growth, pairing and breaking up are shown in details. The simulated flow patterns agree well with experimental visualization results. Predicted results of mean properties of streamwise velocity, fluctuating velocity and Reynolds stress at different sections show good self-similarity and agree well with experimental measurements. Linear growth of the momentum thickness along the streamwise direction is then obtained, indicating that pairing and amalgamating of large vortex structures in plane mixing layers occur randomly in space and time. The effects of convection velocity and the rate of shear on the evolution of vortex structures, self-similarity as well as momentum thickness are also investigated. The rate of shear has a significant effect on the evolution of coherent structure and the slope of momentum thickness growth, while the convection velocity only affects the space interval of the adjoining vortices. The rate of shear and convection velocity have no significant effect on the non-dimensional distributions of turbulence statistics. Self-similarities are therefore obtained for different rates of shear and convection velocities.This work was partially supported by the National Natural Science Foundation of China (Grant No.10002009) and The Research Committee of The Hong Kong Polytechnic University (Project Account Codes G-T609 and G-T429).     

Vortex motion in the early stages of unsteady flow around a circular cylinder

In this paper, processes in the early stages of vortex motion and the development of flow structure behind an impulsively-started circular cylinder at high Reynolds number are investigated by combining the discrete vortex model with boundary layer theory, considering the separation of incoming flow boundary layer and rear shear layer in the recirculating flow region. The development of flow structure and vortex motion, particularly the formation and development of secondary vortex and a pair of secondary vortices and their effect on the flow field are calculated. The results clearly show that the flow structure and vortices motion went through a series of complicated processes before the symmetric main vortices change into asymmetric: development of main vortices induces secondary vortices; growth of the secondary vortices causes the main vortex sheets to break off and causes the symmetric main vortices to become “free” vortices, while a pair of secondary vortices is formed; then the vortex sheets, after breaking off, gradually extend downstream and the structure of a pair of secondary vortices becomes relaxed. These features of vortex motion look very much like the observed features in some available flow field visualizations. The action of the secondary vortices causes the main vortex sheets to break off and converts the main vortices into free vortices. This should be the immediate cause leading to the instability of the motion of the symmetric main vortices. The flow field structure such as the separation position of boundary layer and rear shear layer, the unsteady pressure distributions and the drag coefficient are calculated. Comparison with other results or experiments is also made. This work was presented at the First Asian Congress of Fluid Mechanics, Bangalore in December 1980.