Linear instability of lid- and pressure-driven flows in channels textured with longitudinal superhydrophobic grooves

2021 ◽  
Vol 932 ◽  
Author(s):  
Samuel D. Tomlinson ◽  
Demetrios T. Papageorgiou

It is known that an increased flow rate can be achieved in channel flows when smooth walls are replaced by superhydrophobic surfaces. This reduces friction and increases the flux for a given driving force. Applications include thermal management in microelectronics, where a competition between convective and conductive resistance must be accounted for in order to evaluate any advantages of these surfaces. Of particular interest is the hydrodynamic stability of the underlying basic flows, something that has been largely overlooked in the literature, but is of key relevance to applications that typically base design on steady states or apparent-slip models that approximate them. We consider the global stability problem in the case where the longitudinal grooves are periodic in the spanwise direction. The flow is driven along the grooves by either the motion of a smooth upper lid or a constant pressure gradient. In the case of smooth walls, the former problem (plane Couette flow) is linearly stable at all Reynolds numbers whereas the latter (plane Poiseuille flow) becomes unstable above a relatively large Reynolds number. When grooves are present our work shows that additional instabilities arise in both cases, with critical Reynolds numbers small enough to be achievable in applications. Generally, for lid-driven flows one unstable mode is found that becomes neutral as the Reynolds number increases, indicating that the flows are inviscidly stable. For pressure-driven flows, two modes can coexist and exchange stability depending on the channel height and slip fraction. The first mode remains unstable as the Reynolds number increases and corresponds to an unstable mode of the two-dimensional Rayleigh equation, while the second mode becomes neutrally stable at infinite Reynolds numbers. Comparisons of critical Reynolds numbers with the experimental observations for pressure-driven flows of Daniello et al. (Phys. Fluids, vol. 21, issue 8, 2009, p. 085103) are encouraging.

1985 ◽  
Vol 160 ◽  
pp. 281-295 ◽  
Author(s):  
F. A. Milinazzo ◽  
P. G. Saffman

Computations of two-dimensional solutions of the Navier–Stokes equations are carried out for finite-amplitude waves on steady unidirectional flow. Several cases are considered. The numerical method employs pseudospectral techniques in the streamwise direction and finite differences on a stretched grid in the transverse direction, with matching to asymptotic solutions when unbounded. Earlier results for Poiseuille flow in a channel are re-obtained, except that attention is drawn to the dependence of the minimum Reynolds number on the physical constraint of constant flux or constant pressure gradient. Attempts to calculate waves in Couette flow by continuation in the velocity of a channel wall fail. The asymptotic suction boundary layer is shown to possess finite-amplitude waves at Reynolds numbers orders of magnitude less than the critical Reynolds number for linear instability. Waves in the Blasius boundary layer and unsteady Rayleigh profile are calculated by employing the artifice of adding a body force to cancel the spatial or temporal growth. The results are verified by comparison with perturbation analysis in the vicinity of the linear-instability critical Reynolds numbers.


2014 ◽  
Vol 761 ◽  
pp. 62-104 ◽  
Author(s):  
Joris C. G. Verschaeve ◽  
Geir K. Pedersen

AbstractIn the present treatise, the stability of the boundary layer under solitary waves is analysed by means of the parabolized stability equation. We investigate both surface solitary waves and internal solitary waves. The main result is that the stability of the flow is not of parametric nature as has been assumed in the literature so far. Not only does linear stability analysis highlight this misunderstanding, it also gives an explanation why Sumer et al. (J. Fluid Mech., vol. 646, 2010, pp. 207–231), Vittori & Blondeaux (Coastal Engng, vol. 58, 2011, pp. 206–213) and Ozdemir et al. (J. Fluid Mech., vol. 731, 2013, pp. 545–578) each obtained different critical Reynolds numbers in their experiments and simulations. We find that linear instability is possible in the acceleration region of the flow, leading to the question of how this relates to the observation of transition in the acceleration region in the experiments by Sumer et al. or to the conjecture of a nonlinear instability mechanism in this region by Ozdemir et al. The key concept for assessment of instabilities is the integrated amplification which has not been employed for this kind of flow before. In addition, the present analysis is not based on a uniformization of the flow but instead uses a fully nonlinear description including non-parallel effects, weakly or fully. This allows for an analysis of the sensitivity with respect to these effects. Thanks to this thorough analysis, quantitative agreement between model results and direct numerical simulation has been obtained for the problem in question. The use of a high-order accurate Navier–Stokes solver is primordial in order to obtain agreement for the accumulated amplifications of the Tollmien–Schlichting waves as revealed in this analysis. An elaborate discussion on the effects of amplitudes and water depths on the stability of the flow is presented.


1983 ◽  
Vol 133 ◽  
pp. 265-285 ◽  
Author(s):  
Günter Schewe

Force measurements were conducted in a pressurized wind tunnel from subcritical up to transcritical Reynolds numbers 2.3 × 104[les ]Re[les ] 7.1 × 106without changing the experimental arrangement. The steady and unsteady forces were measured by means of a piezobalance, which features a high natural frequency, low interferences and a large dynamic range. In the critical Reynolds-number range, two discontinuous transitions were observed, which can be interpreted as bifurcations at two critical Reynolds numbers. In both cases, these transitions are accompanied by critical fluctuations, symmetry breaking (the occurrence of a steady lift) and hysteresis. In addition, both transitions were coupled with a drop of theCDvalue and a jump of the Strouhal number. Similar phenomena were observed in the upper transitional region between the super- and the transcritical Reynolds-number ranges. The transcritical range begins at aboutRe≈ 5 × 106, where a narrow-band spectrum is formed withSr(Re= 7.1 × 106) = 0.29.


1998 ◽  
Vol 120 (3) ◽  
pp. 243-252 ◽  
Author(s):  
A. Gupta ◽  
Y. Jaluria

Experiments are performed to study forced convection water cooling of arrays of protruding heat sources with specified heat input. Each array has four rows, with three elements in each row. The arrays are mounted at the top or at the bottom of a rectangular channel. The Reynolds number, based on channel height, is varied from around 2500 to 9000. Flow visualization and temperature measurements revealed that the flow over the arrays was fully turbulent, even at the smallest Reynolds number. Different channel heights (ranging from 3 to 4 times the height of each element), different heat inputs to the modules, and different streamwise spacings between the elements are employed. The spanwise spacing between the elements is kept constant. It is found that the average Nusselt number is higher for smaller channel heights and streamwise spacing, at constant Reynolds number. The effect of buoyancy on the average heat transfer rate from the arrays is found to be small over the parametric ranges considered here. A small variation in the heat transfer coefficient is found in the spanwise direction. The observed trends are considered in terms of the underlying transport processes. The heat transfer data are also correlated in terms of algebraic equations. High correlation coefficients attest to the consistency of results. The data are compared with previous air and water cooling studies, wherever possible, and a good agreement is obtained.


1966 ◽  
Vol 24 (1) ◽  
pp. 113-151 ◽  
Author(s):  
Odus R. Burggraf

The viscous structure of a separated eddy is investigated for two cases of simplified geometry. In § 1, an analytical solution, based on a linearized model, is obtained for an eddy bounded by a circular streamline. This solution reveals the flow development from a completely viscous eddy at low Reynolds number to an inviscid rotational core at high Reynolds number, in the manner envisaged by Batchelor. Quantitatively, the solution shows that a significant inviscid core exists for a Reynolds number greater than 100. At low Reynolds number the vortex centre shifts in the direction of the boundary velocity until the inviscid core develops; at large Reynolds number, the inviscid vortex core is symmetric about the centre of the circle, except for the effect of the boundary-layer displacement-thickness. Special results are obtained for velocity profiles, skin-friction distribution, and total power dissipation in the eddy. In addition, results of the method of inner and outer expansions are compared with the complete solution, indicating that expansions of this type give valid results for separated eddies at Reynolds numbers greater than about 25 to 50. The validity of the linear analysis as a description of separated eddies is confirmed to a surprising degree by numerical solutions of the full Navier–Stokes equations for an eddy in a square cavity driven by a moving boundary at the top. These solutions were carried out by a relaxation procedure on a high-speed digital computer, and are described in § 2. Results are presented for Reynolds numbers from 0 to 400 in the form of contour plots of stream function, vorticity, and total pressure. At the higher values of Reynolds number, an inviscid core develops, but secondary eddies are present in the bottom corners of the square at all Reynolds numbers. Solutions of the energy equation were obtained also, and isotherms and wall heat-flux distributions are presented graphically.


2011 ◽  
Vol 679 ◽  
pp. 77-100 ◽  
Author(s):  
QIANLONG LIU ◽  
ANDREA PROSPERETTI

The finite-Reynolds-number three-dimensional flow in a channel bounded by one and two parallel porous walls is studied numerically. The porous medium is modelled by spheres in a simple cubic arrangement. Detailed results on the flow structure and the hydrodynamic forces and couple acting on the sphere layer bounding the porous medium are reported and their dependence on the Reynolds number illustrated. It is shown that, at finite Reynolds numbers, a lift force acts on the spheres, which may be expected to contribute to the mobilization of bottom sediments. The results for the slip velocity at the surface of the porous layers are compared with the phenomenological Beavers–Joseph model. It is found that the values of the slip coefficient for pressure-driven and shear-driven flow are somewhat different, and also depend on the Reynolds number. A modification of the relation is suggested to deal with these features. The Appendix provides an alternative derivation of this modified relation.


Author(s):  
N. K. Burgess ◽  
P. M. Ligrani

Experimental results, measured on dimpled test surfaces placed on one wall of different channels, are given for a ratio of air inlet stagnation temperature to surface temperature of approximately 0.94, and Reynolds numbers based on channel height from 9,940 to 74,800. The data presented include friction factors, local Nusselt numbers, spatially-averaged Nusselt numbers, and globally-averaged Nusselt numbers. The ratios of dimple depth to dimple print diameter δ/D are 0.1, 0.2, and 0.3 to provide information on the influences of dimple depth. The ratio of channel height to dimple print diameter is 1.00. At all Reynolds numbers considered, local and spatially-resolved Nusselt number augmentations increase as dimple depth increases (and all other experimental and geometric parameters are held approximately constant). These are attributed to: (i) increases in the strengths and intensity of vortices and associated secondary flows ejected from the dimples, as well as (ii) increases in the magnitudes of three-dimensional turbulence production and turbulence transport. The effects of these phenomena are especially apparent in local Nusselt number ratio distributions measured just inside of the dimples, and just downstream of the downstream edges of the dimples. Data are also presented to illustrate the effects of Reynolds number, and streamwise development for δ/D = 0.1 dimples. Significant local Nusselt number ratio variations are observed at different streamwise locations, whereas variations with Reynolds number are mostly apparent on flat surfaces just downstream of individual dimples.


1968 ◽  
Vol 90 (1) ◽  
pp. 109-114 ◽  
Author(s):  
Ahmed R. Wazzan ◽  
T. Okamura ◽  
A. M. O. Smith

The theory of two-dimensional instability of laminar flow of water over solid surfaces is extended to include the effects of heat transfer. The equation that governs the stability of these flows to Tollmien-Schlichting disturbances is the Orr-Sommerfeld equation “modified” to include the effect of viscosity variation with temperature. Numerical solutions to this equation at high Reynolds numbers are obtained using a new method of integration. The method makes use of the Gram-Schmidt orthogonalization technique to obtain linearly independent solutions upon numerically integrating the “modified Orr-Sommerfeld” equation using single precision arithmetic. The method leads to satisfactory answers for Reynolds numbers as high as Rδ* = 100,000. The analysis is applied to the case of flow over both heated and cooled flat plates. The results indicate that heating and cooling of the wall have a large influence on the stability of boundary-layer flow in water. At a free-stream temperature of 60 deg F and wall temperatures of 60, 90, 120, 135, 150, 200, and 300deg F, the critical Reynolds numbers Rδ* are 520, 7200, 15200, 15600, 14800, 10250, and 4600, respectively. At a free-stream temperature of 200F and wall temperature of 60 deg F (cooled case), the critical Reynolds number is 151. Therefore, it is evident that a heated wall has a stabilizing effect, whereas a cooled wall has a destabilizing effect. These stability calculations show that heating increases the critical Reynolds number to a maximum value (Rδ* max = 15,700 at a temperature of TW = 130 deg F) but that further heating decreases the critical Reynolds number. In order to determine the influence of the viscosity derivatives upon the results, the critical Reynolds number for the heated case of T∞ = 40 and TW = 130 deg F was determined using (a) the Orr-Sommerfeld equation and (b) the present governing equation. The resulting critical Reynolds numbers are Rδ* = 140,000 and 16,200, respectively. Therefore, it is concluded that the terms pertaining to the first and second derivatives of the viscosity have a considerable destabilizing influence.


2019 ◽  
Vol 20 (1) ◽  
pp. 85-95
Author(s):  
O. Ya. Maslikova ◽  
I. I. Gritsuk ◽  
D. N. Ionov ◽  
V. K. Debolskiy

One of the most important issues of river hydraulics is the movement of water and the formation of a channel in a stream that has a non-straight-line outline in the plan. Under natural conditions for rivers characteristic winding shape in the plan. The curvature of the jet occurs when the flow is divided into sleeves, at the inflow into the river, the confluence of flows, etc. Therefore, the study of channel processes in rivers is impossible without knowledge of the flow patterns at the curve of the channel. When designing hydraulic structures, including bridge crossings on the meandering sections of rivers, one should know the features of the dynamics of the channel in the sections of the flow turning. In winter, such areas may be narrowed due to the freezing of the channel, and during the period of ice thawing they are clogged with ice fragments. The narrowing of the canal causes an increase in the Reynolds number and a redistribution of velocity diagrams in the area under consideration, which causes a change in the erosion pattern. In laboratory conditions, the nature of the distribution of velocities and the formation of vortices on the installation, creating a rounded flow. It is shown that, at critical Reynolds numbers, a vortex countercurrent occurs in the rounded flow at the inner shore. The impact of this velocity distribution on the erosion pattern of the various slopes of the rounded flow was analyzed.


Author(s):  
Sebastian Ruck ◽  
Frederik Arbeiter

Abstract The velocity field of the fully developed turbulent flow in a one-sided ribbed square channel (rib-height-to-channel-height ratio of k/h = 0.0667, rib-pitch-to-rib-height ratio of p/k = 9) were measured at Reynolds numbers (based on the channel height h and the mean bulk velocity uB) of Reh = 50 000 and 100 000 by means of Laser-Doppler-Anemometry (LDA). Triple velocity correlations differed slightly between both Reynolds numbers when normalized by the bulk velocity and the channel height, similarly to the first- and second-order statistical moments of the velocity. Their near-wall behavior reflected the crucial role of turbulent transport near the rib crest and within the separated shear layer. Sweep events occurred with the elongated flow structures of the flapping shear layer and gained in importance towards the channel bottom wall, while strong ejection events near the rib leading and trailing edges coincided with flow structures bursting away from the wall. Despite the predominant occurrence of sweep events close to the ribbed wall within the inter-rib spacing, ejection events contributed with higher intensity to the Reynolds shear stress. Ejection and sweep events and their underlying transport phenomena contributing to the Reynolds shear stress were almost Reynolds number-insensitive in the resolved flow range. The invariance to the Reynolds number can be of benefit for the use of scale-resolving simulation methods in the design process of rib structures for heat exchange applications.


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