On pattern selection in mixed convection in rectangular ducts

Mixed convection in a horizontal rectangular duct has the same critical Rayleigh number as natural convection in a rectangular cavity for the onset of convection. The linear stability analysis predicts either an odd or an even number of convective rolls to appear depending on the aspect ratio of the cross section. However, it has been shown both experimentally and numerically that an even number of convective rolls appears under supercritical conditions for fully developed mixed convection. The paper first presents an analytical solution for the buoyancy-induced mainstream velocity, w ^b , at the onset of buoyancy-induced motion in a forced convective flow. Then, a comparison in the initial growth rate of w ^b is made between the case of an odd and an even number of rolls; which shows the selection of an even number of rolls over an odd number in mixed convection except for low aspect ratio ducts.     

Convective stability of a nonisothermal fluid in a rotating horizontal coaxial gap

The thermal fluid convection in a coaxial horizontal gap uniformly rotating about its axis is investigated. The threshold above which convective flows are excited and the structure of these flows are studied. It is found that convection ensues irrespective of whether the inner or outer boundary temperature is higher. Convection manifests itself in the threshold development of rolls elongated in the direction of the rotation axis and is determined by two different mechanisms. If the layer is heated from outside, the centrifugal convection mechanism plays a leading part and the diameter of the convective rolls is comparable with the layer thickness. If the higher is the temperature of the inner boundary of the layer, the centrifugal inertia force has a stabilizing effect and convection development is related with the action of thermal vibrational mechanism. The latter is determined by gravity-generated oscillations of the nonisothermal fluid relative to the cavity. The wave number of the vibrational convective structures is several times smaller than under centrifugal convection. The results obtained broaden our understanding of thermal convection in systems rotating in external static force fields.     

矩形倾斜腔体中的两种对流斑图和分区

为了研究矩形倾斜腔体中普朗特数Pr=0.72的流体对流斑图和分区,本文基于流体力学方程组进行了数值模拟。在相对瑞利数r=6.0的情况下,观察了倾角θ=10°和θ=60°时对流斑图随着时间的发展,发现系统存在单圈型对流和多圈型对流两种斑图。流线随着倾角的变化说明:随着倾角增加,对流圈数逐渐减少,对流波长逐渐增加,对流波数减小;然后,随着对流圈数显著地减少,系统由多圈型对流过渡到单圈型对流。根据模拟计算结果,给出了多圈型对流过渡到单圈型对流的临界倾角θc随着相对瑞利数r变化的关系曲线。对流在θ-r平面上分为两个区域:θ<θc时系统是单圈型对流,θ>θc时系统是多圈型对流。对流最大振幅A和努塞尔数Nu随着倾角θ的变化曲线被临界倾角θc分成两段,它们有不同的变化规律。因此,临界倾角也可以由对流最大振幅A或努塞尔数Nu的变化曲线来确定。     

On Low-Dimensional Modeling of Channel Turbulence

We investigate a sequence of low-dimensional models of turbulent channel flows. These models are based on the extraction of the Karhunen–Loève (KL) eigenfunctions from a large-scale simulation in a wide channel with R _*=180 (based on the friction velocity). KL eigenfunctions provide an optimal coordinate system in which to represent the dynamics of the turbulent flow. The hierarchy of KL modes identifies the most energetic independent phenomena in the system. A series of Galerkin projections is then used to derive a dynamical approximation to flows. In order to capture essential aspects of the flow in a low-dimensional system, a careful selection of modes is carried out. The resulting models satisfy momentum and energy conservation as well as yielding the amount of viscous dissipation found in the exact system. Their dynamics includes modes which characterize the flux, rolls, and propagating waves. Unlike previous treatments the instantaneous streamwise flow is time dependent and represented by KL flux modes. The rolls correspond to the streaks observed in experiments in the viscous sublayer. Propagating waves which first appear in the model flow at low Reynolds number (R _*∼ 60) persist through the chaotic regime that appears as the Reynolds number is increased. Statistical measures of the chaotic flows which have been generated by the models compare favorably with those found in full-scale simulations. Received 13 July 1998 and accepted 8 January 1999     

Effect of a latitudinal temperature gradient on convective motions arising in a rotating spherical layer

The hydrodynamics of planetary atmospheres and Interiors are frequently directly or indirectly connected with convective motions taking place in rotating liquid spherical layers in the field of a central force. Convective stability in a spherical layer at rest, in a central gravity field, was first discussed in [1, 2]. It was shown that the critical Rayleigh number Rao at which convective instability sets in and the wave number of the critical perturbations depend essentially on the thickness of the layer. As in the plane case, the problem of the convective stability of a spherical layer is found to be degenerate, and the form of the critical perturbations cannot be determined from the linear problem. In actuality, minimization of the Rayleigh number permits establishing only the wave numberl for the spherical harmonic Y _l ^m (θ, ?), realized at the limit of stability; the parameter m remains indeterminate and thus 2l+1 independent convective modes correspond to Rao. In [3] a study was made of the convective stability of a liquid in a slowly rotating thin spherical layer. It was shown that the presence of rotation eliminates the degeneracy; at the limit of stability there arise motions corresponding to the Y _l l (θ, ?) -harmonic with a degenerate maximum at the equator, and propagating in a wave manner toward the side opposite to the rotation. In the present work a study is made of the convective stability of a flow of liquid, arising in a rotating spherical layer due to a nonuniform distribution of the temperatures at one of the boundaries of the layer. In such a statement of the problem it is possible to model large-scale motions in the atmospheres of large planets having internal sources of heat and absorbing solar radiation near the cloud cover of the atmosphere. It is established that, depending on the relationships between the parameters imparting the rotation and the inhomogeneous distribution of the temperature, there is either stabilization or destabilization of the layer in comparison with a fixed layer of the same thickness and with the same, but uniformly distributed heat flux supplied to the layer. A study is made of the form of the corresponding critical perturbations.     

An analytic approximation of the drag coefficient for the viscous flow past a sphere

We give an analytic solution at the 10th order of approximation for the steady-state laminar viscous flows past a sphere in a uniform stream governed by the exact, fully non-linear Navier-Stokes equations. A new kind of analytic technique, namely the homotopy analysis method, is applied, by means of which Whitehead's paradox can be easily avoided and reasonably explained. Different from all previous perturbation approximations, our analytic approximations are valid in the whole field of flow, because we use the same approximations to express the flows near and far from the sphere. Our drag coefficient formula at the 10th order of approximation agrees better with experimental data in a region of Reynolds number R_d<30, which is considerably larger than that (R_d<5) of all previous theoretical ones.     

Effect of inclined temperature gradient on thermal instability in an anisotropic porous medium

Effect of anisotropy on thermal instability in a fluid saturated porous medium subjected to an inclined temperature gradient of finite magnitude is analysed using Galerkin technique. Results are compared with those of isotropic and horizontally isotropic cases. It is observed that anisotropic medium is the most stable while either isotropic situation or the horizontally isotropic situation is the most unstable one depending on the horizontal Rayleigh number (R _H ), anisotropy parametersk ₁(=k _y /k _x ), and ?₂(=?_γ/? _z ).     

Reynolds Number Dependence of Mixed Structure Functions of Temperature and Longitudinal Velocity

Reynolds number dependence of mixed structure functions of longitudinal velocity u and temperature Θ is examined over a R _λ (Taylor microscale Reynolds number) range of 180-5950 for four flows. It is found that the mixed structure functions exhibit some behaviours similar to those of individual ones of the velocity and temperature. The prediction of the bivariate lognormal model for the R _λ dependence of the mixed structure functions is approached by the present measurements only at very high R _λ. At values of the longitudinal separation r close to the integral length scale as well as Taylor microscale, the velocity and temperature increments are not statistically independent. Several methods have been used to estimate the intermittency exponents μ and μ_Θ of the velocity and temperature in the lognormal model. Each method yields a different value of μ and μ_Θ, which also depend on R _λ and the type of flows. The present measurements suggest that the best estimates for μ and μ_Θ are 0.25 ± 0.05 and 0.30 ± 0.05, respectively.     

On the Convection in a Porous Medium with Inclined Temperature Gradient and Vertical Throughflow. Part I. Normal Modes

In this second part of our analysis of the destabilization of transverse modes in an extended horizontal layer of a saturated porous medium with inclined temperature gradient and vertical throughflow, we apply the mathematical formalism of absolute and convective instabilities to studying the nature of the transition to instability of such modes by assuming on physical grounds that the transition is triggered by growing localized wavepackets. It is revealed that in most of the parameter cases treated in the first part of the analysis (Brevdo and Ruderman 2009), at the transition point the evolving instability is convective. Only in the cases of zero horizontal thermal gradient, and in the cases of zero vertical throughflow and the horizontal Rayleigh number R _h < 49, the instability is absolute implying that, as the vertical Rayleigh number, R _v, increases passing through its critical value, R _vc, the destabilization tends to affect the base state throughout and eventually destroys it at every point in space. For the parameter values considered, for which the destabilization has the nature of convective instability, we found that, as R _v, increases beyond the critical value, while the horizontal Rayleigh number, R _h, and the Péclet number, Q _v, are kept fixed, the flow experiences a transition from convective to absolute instability. The values of the vertical Rayleigh number, R _v, at the transition from convective to absolute instability are computed. For convectively unstable, but absolutely stable cases, the spatially amplifying responses to localized oscillatory perturbations, i.e., signaling, are treated and it is found that the amplification is always in the direction of the applied horizontal thermal gradient.     

A Summary of New Predictive High Frequency Thermo-Vibrational Models in Porous Media

In this paper, we consider the effect of mechanical vibration on the onset of convection in porous media. The porous medium is saturated either by a pure fluid or by a binary mixture. The importance of a transport model on stability diagrams is presented and discussed. The stability threshold for the Darcy–Brinkman case in the Ra _Tc -R and k _c -R diagrams is presented (where Ra _Tc , k _c and R are the critical Rayleigh number, the critical wave number and the vibration parameters, respectively). It is shown that there is a significant deviation from the Darcy model. In the thermo-solutal case with the Soret effect, the influence of vibration on the reduction of multi-cellular convection is emphasized. A new analytical relation for obtaining the threshold of mono-cellular convection is derived. This relation shows how the separation factor Ψ is related to the controlling parameters of the problem, Ψ = f (R, ε*, Le), when the wave number k → 0. The importance of vibrational parameter definition is highlighted and it is shown how, by using a proper definition for vibrational parameter, we may obtain compact relationship. It is also shown how this result may be used to increase component separation.     

Direct Numerical Simulation of Turbulent Pipe Flow at Moderately High Reynolds Numbers

Fully resolved direct numerical simulations (DNSs) have been performed with a high-order spectral element method to study the flow of an incompressible viscous fluid in a smooth circular pipe of radius R and axial length 25R in the turbulent flow regime at four different friction Reynolds numbers Re _τ ?=?180, 360, 550 and $1\text{,}000$ . The new set of data is put into perspective with other simulation data sets, obtained in pipe, channel and boundary layer geometry. In particular, differences between different pipe DNS are highlighted. It turns out that the pressure is the variable which differs the most between pipes, channels and boundary layers, leading to significantly different mean and pressure fluctuations, potentially linked to a stronger wake region. In the buffer layer, the variation with Reynolds number of the inner peak of axial velocity fluctuation intensity is similar between channel and boundary layer flows, but lower for the pipe, while the inner peak of the pressure fluctuations show negligible differences between pipe and channel flows but is clearly lower than that for the boundary layer, which is the same behaviour as for the fluctuating wall shear stress. Finally, turbulent kinetic energy budgets are almost indistinguishable between the canonical flows close to the wall (up to y ^?+??≈?100), while substantial differences are observed in production and dissipation in the outer layer. A clear Reynolds number dependency is documented for the three flow configurations.     

Effect of flexure on aerodynamic propulsive efficiency of flapping flexible airfoil

The aim of present study is to investigate the effect of chord-wise flexure amplitude on unsteady aerodynamic characteristics for a flapping airfoil with various combinations of Reynolds number and reduced frequency. Unsteady, viscous flows over a single flexible airfoil in plunge motion are computed using conformal hybrid meshes. The dynamic mesh technique is applied to illustrate the deformation modes of the flexible flapping airfoil. In order to investigate the influence of the flexure amplitude on the aerodynamic performance of the flapping airfoil, the present study considers eight different flexure amplitudes (a₀) ranging from 0 to 0.7 in intervals of 0.1 under conditions of Re=10⁴, reduced frequency k=2, and dimensionless plunge amplitude h₀=0.4. The computed unsteady flow fields clearly reveal the formation and evolution of a pair of leading edge vortices along the body of the flexible airfoil as it undergoes plunge motion. Thrust-indicative wake structures are generated when the flexure amplitude of the airfoil is less than 0.5 of the chord length. An enhancement in the propulsive efficiency is observed for a flapping airfoil with flexure amplitude of 0.3 of the chord length. This study also calculates the propulsive efficiency and thrust under various Reynolds numbers and reduced frequency conditions. The results indicate that the propulsive efficiency has a strong correlation with the reduced frequency. It is found that the flow conditions which yield the highest propulsive efficiency correspond to Strouhal number St of 0.255.     

The effect of an acoustic field on a secondary convective flow in a layer

The effect of a traveling sonic wave on a convective flow in a horizontal layer with a fixed linear temperature distribution on the boundaries is investigated. Convective rolls with axes parallel to the basic flow (lengthwise rolls) are considered. On the basis of a weakly nonlinear analysis, it is shown that the lengthwise rolls appear smoothly and the regular flows are stable near the stability threshold. A direct numerical simulation is performed. Secondary near-critical flow regimes and regimes corresponding to finite supercriticalities are investigated.     

Convection mixte en fluide binaire avec effet Soret : étude analytique de la transition vers les rouleaux transversaux 2D

This Note deals with mixed convection in binary fluid with Soret effect in a rectangular duct heated from below. In particular, we study the transition towards transverse 2D rolls appearing at low Reynolds and Rayleigh numbers. The linear stability analysis of Poiseuille flow, with linearly stratified temperature and concentration fields, shows the influence of the separation ratio on the critical Rayleigh number at the transition towards the transversal 2D convective patterns. It highlights the presence, at low Reynolds numbers, of propagating transverse rolls in the downwards as well as in the upwards direction. Finally, we point out that, under these conditions, the propagating frequency of the rolls is the sum of two well defined frequencies: the first related to the Reynolds, the second to the separation ratio. To cite this article: E. Piquer et al., C. R. Mecanique 333 (2005).     

Axisymmetric-conical and locally conical flows without swirling

Axisymmetric steady conical and locally conical non-swirled flows of an ideal (inviscid and non-heat-conducting) gas are considered. Like two-dimensional conical flows, the examined onedimensional (axisymmetric) flows can be conically subsonic and supersonic. If the uniform flow is not considered as a conical flow, then the type of one-dimensional conical flows can change only on the shock wave, except for the junction of two one-dimensional conical flows of different types on the C ⁺ characteristic. C ^± characteristics and streamlines are constructed for a number of locally conical flows and some already known and new conical flows.     

Dependence of the aerodynamic forces on the Reynolds number with the three-dimensional flow of a vortical stream around bodies

The article discusses the dependence of the viscous stresses on the Reynolds number Re in three-dimensional flows around bodies of arbitrary form. It is shown that, with an infinite growth of a vortex with the approach to a body, singular terms appear in an asymptotic expansion in terms of ε=Re^?1/2. The infinite values of the derivatives of the velocity in flows of an incompressible liquid are due to the initial vorticity; in a supersonic flow, they can be connected with the absence of a maximum of the entropy at the critical flow line behind a leading shock wave. The singularity in the tangential stresses brings about the appearance of additional terms in the total aerodynamic forces and moments acting on the body.     

Rayleigh–Bénard convection driven by a long wavelength heating

Natural convection in a two-dimensional horizontal layer has been investigated. The layer is confined between two parallel horizontal plates. The upper plate is kept isothermal, while the lower plate has an externally imposed, long wavelength, spatially sinusoidal heating with the amplitude expressed in terms of the Rayleigh number Ra and the wavelength characterized by the wave number α. Only steady-state flow structures and their bifurcations have been considered. The detailed analysis has been carried out for two Prandtl numbers, i.e. Pr = 0.7 and Pr = 7, and only small differences in the bifurcation diagrams have been observed. When Ra < Ra _cr = 427, convection has a simple topology consisting of one pair of counter-rotating rolls per heating period. Secondary motion in the form of rolls aligned in the direction of the primary rolls and concentrated around the hot spots occurs for Ra > 427. When 427 < Ra < ～470 and α < ～0.14, the secondary motion is described by the supercritical pitchfork bifurcation. One of the branches of this bifurcation is associated with an odd number of secondary rolls per half wavelength, with rolls above the hot spots rotating in the direction opposite to the primary rolls. The other branch is associated with an even number of secondary rolls per half wavelength, with the rolls above the hot spots co-rotating with the primary rolls. The new rolls are pinched off in pairs when α decreases. When Ra > ～470 and α > ~0.14, bifurcation assumes the form of “bifurcation from infinity”. The main branch is associated with one pair of rolls per heating period for α > 0.25. Decrease in α along this branch results in the formation of secondary rolls, with the rolls at the hot spot co-rotating with the primary rolls. The lower part of the other branch is associated with one pair of rolls per heating period in the limit α → 0. Increase in α results in pinching off a single roll which counter-rotates with respect to the primary roll at the hot spot.     

Transition from Convective to Absolute Instability in a Porous Layer with Either Horizontal or Vertical Solutal and Inclined Thermal Gradients,and Horizontal Throughflow

Recently, in Diaz and Brevdo (J Fluid Mech 681: 567–596, 2011), further in the text referred to as D&B, we found an absolute/convective instability dichotomy at the onset of convection in a flow in a saturated porous layer with either horizontal or vertical solutal and inclined thermal gradients, and horizontal throughflow. The control parameter in D&B triggering the destabilization is the vertical thermal Rayleigh number, R _v. In this article, we treat the parameter cases considered in D&B in which the onset of convection has the character of convective instability and occurs through longitudinal modes. By increasing the vertical thermal Rayleigh number starting from its critical value, R _vc, we determine the value R _vt of R _v at which the transition from convective to absolute instability takes place and compute the physical characteristics of the emerging absolutely unstable wave packet. In some cases, the value of the transitional vertical thermal Rayleigh number, R _vt, is only slightly greater than the critical value, R _vc, meaning that at the onset of convection the base convectively unstable state can be viewed as marginally absolutely unstable. However, in several cases considered, the value of R _vt is significantly greater than the critical value, R _vc, implying that the base state is not marginally but essentially absolutely stable at the point of destabilization.     

Numerical investigations of the vortex interactions for a flow over a pitching foil at different stages

The fluid–structure interaction is investigated numerically for a two-dimensional flow (Re=2.5·10⁶) over a sinusoid-pitching foil by the SST (Shear Stress Transport) k–ω model. Although discrepancies in the downstroke phase, which are also documented in other numerical studies, are observed by comparing with experimental results, our current numerical results are sufficient to predict the mean features and qualitative tendencies of the dynamic stall phenomenon. These discrepancies are evaluated carefully from the numerical and experimental viewpoints.In this study, we have utilized Λ, which is the normalized second invariant of the velocity gradient tensor, to present the evolution of the Leading Edge Vortex (LEV) and Trailing Edge Vortex (TEV). The convective, pressure, and diffusion terms during the dynamic stall process are discussed based on the transport equation of Λ. It is found that the pressure term dominates the rate of the change of the rotation strength inside the LEV. This trend can hardly be observed directly by using the vorticity transport equation due to the zero baroclinic term for the incompressible flow.The mechanisms to delay the stall are categorized based on the formation of the LEV. At the first stage before the formation of the LEV in the upper surface, the pitching foil provides extra momentum into the fluid flows to resist the flow separation, and hence the stall is delayed. At the second stage, a low-pressure area travels with the evolution of the LEV such that the lift still can be maintained. Three short periods at the second stage corresponds to different flow patterns during the dynamic stall, and these short periods can be distinguished according to the trend of the pressure variation inside the LEV. The lift stall occurs when a reverse flow from the lower surface is triggered during the shedding of the LEV. For a reduced frequency k_f=0.15, the formation of the TEV happens right after the lift stall, and the lift can drop dramatically. With a faster reduced frequency k_f=0.25, the shedding of the LEV is postponed into the downstroke, and the interaction between the LEV and TEV becomes weaker correspondingly. Thus, the lift drops more gently after the stall. In order to acquire more reliable numerical results within the downstroke phase, the Large Eddy Simulation (LES), which is capable of better predictions for the laminar-to-turbulent transition and flow reattachment process, will be considered as the future work.     

Combined radiation and convection heat transfer in simultaneously developing flow in ducts with semi-circular and right triangular cross sections

In this paper, the interaction of radiation and forced convection in simultaneously developing laminar flow through semicircular and right triangular ducts with isothermal non-black wall is investigated. The three dimensional momentum and energy equations are discretized by the method of lines and solved numerically by the marching method. The method of momentum is employed to consider the radiation contribution which models the radiation in the partial differential equation, instead of the partial integrodifferential equation. The effects of three major parameters, radiation-conduction parameter,N, optical thickness, τ _b , and wall emissivity, ε _w , in the entry region of these irregular geometry ducts are discussed. The numerical results in terms of the variation of the bulk temperature and the mean Nusselt number indicate that the thermal radiation not only enhances the heat transfer rate, but also changes the characteristics of the convective heat transfer. Furthermore, the results compare very well with the available data published in the open literature for the pure convective case.