Reynolds number dependence of the drag coefficient for laminar flow through electrically-heated photoetched screens

Experiments are performed to measure the drag coefficient of electrically-heated screens. Square-pattern 80 mesh and 100 mesh screens of 50.8 m-wide wires photoetched from 50.8 m thick Inconel sheets are examined. Ambient air is passed through these screens at upstream velocities yielding wire-width Reynolds numbers from 2 to 35, and electrical current is passed through the screens to generate heat fluxes from o to 0.17 MW/m², based on the total screen area. The dependence of the drag coefficient on Reynolds number and heat flux is determined for these two screens by measuring pressure drops across the screens for a variety of conditions in these ranges. In all cases, heating is found to increase the drag coefficient above the unheated value. A correlation relating the heated drag coefficient to the unheated drag coefficient is developed based on the idea that the main effect of heating at these levels is to modify the Reynolds number through modifying the viscosity. This correlation is seen to reproduce the experimental results closely.List of Symbols A total screen cross sectional area - C fitting coefficient, near unity - c _D heated drag coefficient - c _{D, 0} unheated drag coefficient - C _p air specific heat at constant pressure - D photoetched wire width, sheet thickness - h _s stagnation point heat-transfer coefficient - k air thermal conductivity - M distance between adjacent wires - O open area fraction - p air pressure - p air pressure drop across screen - Pr Prandtl number for air, c _p/k - Q total electrical power to screen - R radius of curvature at stagnation point - Re _D wire width Reynolds number, UD/ - T air temperature - U air speed upstream of screen - air specific heat ratio - air density - air viscosity - exponent in temperature power law for viscosity - () quantity () evaluated at heated screen temperature The authors thank John Lewin and Bob Meyer for their assistance in the design and fabrication of the heated screen test facility and Tom Grasser for his help in performing the experiments. This work was performed at Sandia National laboratories, supported by the U.S. Department of Energy under contract number DE-AC04-94AL85000.     

Reynolds number dependence of two-dimensional laminar co-rotating vortex merging

The paper reports unsteady Navier–Stokes calculations of laminar two-dimensional co-rotating vortex merging for various Reynolds numbers. The unsteady, incompressible two-dimensional Navier–Stokes equations were solved with fourth-order Runge–Kutta temporal discretization and fourth-order symmetric compact schemes for spatial discretization. Calculations of the unsteady Taylor vortex benchmark showed that fourth-order accurate solutions for all primitive variables were indeed achieved. Calculations for a pair of equal-strength co-rotating vortices show good agreement with reported direct numerical simulation and experiments for the evolution of the separation distance and core radius. It is found that the time required for merging is inversely proportional to the square root of the Reynolds number. According to previous experimental research, it was also found that complete merging in laminar regime undergoes four stages with physical meaning. The physical mechanism responsible for the merging process is investigated and it is found that the antisymmetric vorticity dynamics plays an important role until full merging.     

Studies on two-phase cross flow. Part II: Transition Reynolds number and drag coefficient

A reduction of the transition Reynolds number from laminar separation to turbulent one of cross flow around a body is anticipated in a gas—liquid bubbly flow, since there exists intensive turbulence in the main flow. A decrease in the drag coefficient of the body can also be expected. This report was aimed to classify flow patterns of the two-phase wake flow behind a cylinder and to investigate quantitatively the change of the drag coefficient corresponding to the transition of the flow patterns. From measurements of the static pressure distribution on the cylinder surface and the drag coefficient of the cylinder, it was found that the flow pattern was sure to change finally into a new one similar to the transcritical type in the single-phase flow with an increase of the mean void fraction in the main flow. It was concluded that a large reduction of the upper transition Reynolds number occurred in a two-phase flow, because the transition could be realized by a little increase of the void fraction even below the lower transition Reynolds number in the case of a large cylinder diameter.     

Reynolds number dependence of vortex patterns in accelerated flow around airfoils

Reynolds number dependence of vortical patterns visualized by smoke technique in accelerating flow behind airfoils is documented in photographic sequences at angles of attack 20° and 60°. At low Reynolds numbers the vortical pattern development is quite simple. With increasing Reynolds numbers these patterns become increasingly complex, and onset of turbulence occurs early and generates large scale turbulent vortex patterns.     

Reynolds number dependence of Reynolds and dispersive stresses in turbulent channel flow past irregular near-Gaussian roughness

Direct numerical simulations of fully-developed turbulent channel flow with irregular rough walls have been performed at four friction Reynolds numbers, namely, 180, 240, 360 and 540, yielding data in both the transitionally- and fully-rough regime. The same roughness topography, which was synthesised with an irregular, isotropic and near-Gaussian height distribution, is used in each simulation. Particular attention is directed towards the wall-normal variation of flow statistics in the near-roughness region and the fluid-occupied region beneath the crests, i.e. within the roughness canopy itself. The goal of this study is twofold. (i) Provide a detailed account of first- and second-order double-averaged velocity statistics (including profiles of mean velocity, dispersive stresses, Reynolds stresses, shear stress gradients and an analysis of the mean force balance) with the overall aim of understanding the relative importance of “form-induced” and “turbulence-induced” quantities as a function of the friction Reynolds number. (ii) Investigate the possibility of predicting the levels of streamwise dispersive stress using a phenomenological closure model. Such an approach has been applied successfully in the context of idealised vegetation canopies (Moltchanov & Shavit, 2013, Water Resour. Res., vol. 49, pp. 8222-8233) and is extended here, for the first time, to an irregular rough surface. Overall, the results reveal that strong levels of dispersive stress occur beneath the roughness crests and, for the highest friction Reynolds number considered in this study, show that the magnitude (and gradient) of these “form-induced” stresses exceed their Reynolds stress counterparts. In addition, this study emphasises that the dominant source of spatial heterogeneity within the irregular roughness canopy are “wake-occupied” regions and that a suitable parameterisation of the wake-occupied area is required to obtain an accurate prediction of streamwise dispersive stress.     

Unsteady flow through screens

Summary Reflected and transmitted pressure waves take place when an incident pressure wave reaches a screen placed in a duct and crossed by a previous steady flow. The thickness of the screen is assumed to be negligible and essure losses through it in the quasi steady flow, which takes place after the passage of the incident pressure wave, are assumed be steady flow ones. The strength of reflected and transmitted pressure waves is related to the strength of the incident wave, to the porosity of the screen and to the previous flow through it. Subsonic and just supersonic quasi steady flow inside the screen are considered. Five commercial screens have been tested with incident shock waves and the experimental results have been compared with the theoretical ones: the agreement is quite satisfactory.

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Sommario L'interazione di un'onda di pressione con una rete intubata attraversata da un flusso preesistente produce un'onda riflessa ed una trasmessa. Si presenta un modello teorico basato essenzialmente sulle potesi che lo spessore della rete sia trascurabile e che le perdite carico attraverso di essa, nel flusso quasi stazionario indotto dalla suddetta interazione, coincidano con quelle proprie di un oto permanente. Si analizza il caso di flusso indotto nella rete ia subsonico che sonico. Si eseguono prove sperimentali su cinque reti di diverse porosità differenti condizioni di flusso, e si verifica l'attendibilità del odello teorico proposto.

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The present work has been supported by the “Consiglio Nazionale delle Ricerche” (C.N.R.) under Contract no. 115.4075.69.01220.

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The suffixes “u” and “d” pertain to the conditions upstream and downstream of the screen with reference to the flow through it. The suffix “j” refers to the region “j” of the wave diagrams in Fig. 1.     

Aerodynamics of laminar separation flutter at a transitional Reynolds number

Experimental observations of self-sustained pitch oscillations of a NACA 0012 airfoil at transitional Reynolds numbers were recently reported. The aeroelastic limit cycle oscillations, herein labelled as laminar separation flutter, occur in the range 5.0×10⁴≤Re_c≤1.3×10⁵. They are well behaved, have a small amplitude and oscillate about θ=0°. It has been speculated that laminar separation leading to the formation of a laminar separation bubble, occurring at these Reynolds numbers, plays an essential role in these oscillations. This paper focuses on the Re_c=7.7×10⁴ case, with the elastic axis located at 18.6% chord. Considering that the experimental rig acts as a dynamic balance, the aerodynamic moment is derived and is empirically modelled as a generalized Duffing–van-der-Pol nonlinearity. As expected, it behaves nonlinearly with pitch displacement and rate. It also indicates a dynamically unstable equilibrium point, i.e. negative aerodynamic damping. In addition, large eddy simulations of the flow around the airfoil undergoing prescribed simple harmonic motion, using the same amplitude and frequency as the aeroelastic oscillations, are performed. The comparison between the experiment and simulations is conclusive. Both approaches show that the work done by the airflow on the airfoil is positive and both have the same magnitude. The large eddy simulation (LES) computations indicate that at θ=0°, the pitching motion induces a lag in the separation point on both surfaces of the airfoil resulting in negative pitching moment when pitching down, and positive moment when pitching up, thus feeding the LCO.     

Critical transition Reynolds number for plane channel flow

The determination of the critical transition Reynolds number is of practical importance for some engineering problems. However, it is not available with the current theoretical method, and has to rely on experiments. For supersonic/hypersonic boundary layer flows, the experimental method for determination is not feasible either. Therefore,in this paper, a numerical method for the determination of the critical transition Reynolds number for an incompressible plane channel flow is proposed. It is basically aimed to test the feasibility of the method. The proposed method is extended to determine the critical Reynolds number of the supersonic/hypersonic boundary layer flow in the subsequent papers.     

The numerical solution of the laminar flow in a constricted channel at moderately high Reynolds number using Newton iteration

The numerical solution of the flow in a stepped channel constricted to half its width has been obtained for Reynolds numbers up to 2000 using Newton's iteration to solve the ensuing algebraic system. In order to avoid high-frequency errors, a locally fine grid is effected near the corner by transformation of the independent variables. The results predict a downstream recirculation region, observed in experiments but not found in earlier numerical calculations. The inclusion of the Dennis–Hudson upwinding, added for stability in SOR methods, whilst giving the same characteristics of the flow, is less accurate by at least an order of magnitude.     

Analysis of a meshless solver for high Reynolds number flow

In this paper, the extension of an upwind least‐square based meshless solver to high Reynolds number flow is explored, and the properties of the meshless solver are analyzed both theoretically and numerically. Existing works have verified the meshless solver mostly with inviscid flows and low Reynolds number flows, and in this work, we are interested in the behavior of the meshless solver for high Reynolds number flow, especially in the near‐wall region. With both theoretical and numerical analysis, the effects of two parameters on the meshless solver are identified. The first one is the misalignment effect caused by the significantly skewed supporting points, and it is found that the meshless solver still yields accurate prediction. It is a very interesting property and is opposite to the median‐dual control volume based vertex‐centered finite volume method, which is known to give degraded result with stretched triangular/tetrahedral cells in the near‐wall region. The second parameter is the curvature, and according to theoretical analysis, it is found in the region with both large aspect ratio and curvature, and the streamwise residual is less affected; however, the wall‐normal counterpart suffers from accuracy degradation. In this paper, an improved method that uses a meshless solver for the streamwise residual and finite difference for wall‐normal residual is developed. This method is proved to be less sensitive to the curvature and provides improved accuracy. This work presents an understanding of the meshless solver for high Reynolds number flow computation, and the analysis in this paper is verified with a series of numerical experiments. Copyright © 2012 John Wiley & Sons, Ltd.     

Nonmonotonic relationship between the Reynolds number and the lift coefficient of a plate in a hypersonic rarefied gas flow

The aerodynamic coefficients of a plate in a hypersonic stream at low Reynolds numbers are studied over a wide range of similarity parameters. The dependence of the lift coefficientC _Y on the tangential force coefficient, the finite Mach number at the outer edge of the boundary layer and the velocity-slip and temperature-jump boundary conditions is taken into consideration. The nonmonotonic character of the relationship betweenC _Y and the Reynolds number, revealed previously in experiments, is explained within the framework of the viscous hypersonic interaction model.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 186–189, January–February, 1996.     

Dependence of flow classification on the Reynolds number for a two-cylinder wake

This work aims to investigate the dependence of flow classification on the Reynolds number (Re) for the wake of two staggered cylinders. The Re examined ranges from 1.5×10³ to 2.0×10⁴. The pitch ratio, P^⁎=P/d examined is 1.2–6.0 (d is the cylinder diameter), and angle (α) is 0–90°, where P is the center-to-center spacing between two cylinders and α is the angle between the incident flow and the line through the cylinder centers. Two single hotwires were used to measure simultaneously the fluctuating streamwise velocities (u) in the vortex streets generated by the two cylinders. The power spectral density functions and the Strouhal numbers were then obtained from the u signals, based on which the flow structure pattern or mode could be determined. Over two hundred configurations of two staggered cylinders have been examined for each Re. It is found that Re has an appreciable effect on the dependence of the flow mode on P^⁎ and α. The observation is connected to the Re effect on the generic features of a two-cylinder wake such as flow separation, boundary layer thickness, gap flow deflection and vortex formation length.     

Anomalous heat transfer and drag in laminar flow of viscoelastic fluids

A boundary-layer approach is used to derive an expression for the heat transfer coefficient when a viscoelastic fluid flows past a cylinder. The heat-transfer coefficient becomes independent of velocity at large values of the latter, thus explaining the experimental results of James and Acosta. The agreement with the observations of these authors is, however, only qualitative, since their experiments were carried out at relatively low Reynolds numbers. By changing the exponent of the Reynolds number from the boundary-layer approximation value to one which is valid in the range used by James and Acosta, a correlation is suggested which is in satisfactory agreement with their data. Similar considerations are used to explain the additional experimental observation of James and Acosta, i.e. that the drag coefficient for the flow past a cylinder also becomes independent of velocity at large values of the latter quantity.     

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.     

Small Reynolds number instabilities in two-layer Couette flow

Instabilities in two-layer Couette flow are investigated from a small Reynolds number expansion of the eigenvalue problem governing linear stability. The wave velocity and growth rate are given explicitly, and previous results for long waves and short waves are retrieved as special cases. In addition to the inertial instability due to viscous stratification, the flow may be subject to the Rayleigh–Taylor instability. As a result of the competition of these two instabilities, inertia may completely stabilise a gravity-unstable flow above some finite critical Froude number, or conversely, for a gravity-stable flow, inertia may give rise to finite wavenumber instability above some finite critical Weber number. General conditions for these phenomena are given, as well as exact or approximate values of the critical numbers. The validity domain of the many asymptotic expansions is then investigated from comparison with the numerical solution. It appears that the small-Re expansion gives good results beyond Re = 1, with an error less that 1%. For Reynolds numbers of a few hundred, we show from the eigenfunctions and the energy equation that the nature of the instability changes: instability still arises from the interfacial mode (there is no mode crossing), but this mode takes all the features of a shear mode. The other modes correspond to the stable eigenmodes of the single-layer Couette flow, which are recovered when one fluid is rigidified by increasing its viscosity or surface tension.     

Analytical solution for laminar flow through leaky tube

The laminar flow through a leaky tube is investigated, and the momentum and conservation of energy equations are solved analytically. By using the Hagen-Poiseuille velocity profile and defining unknown functions for the axial and radial velocity components, the pressure and mass transfer equations are obtained, and their profiles are plotted according to different parameters. The results indicate that the axial velocity, the radial velocity, the mass transfer parameter, and the pressure in the tube decrease as the fluid moves along the tube.     

The development of a plane laminar flow at low Reynolds numbers

Summary The development of a plane Poiseuille flow at low Reynolds numbers is studied; given any velocity distribution in the section x=0, it is possible to evaluate its evolution along the direction of motion, by means of quite simple calculation. A numerical example is also given.

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Sommario Si esamina l'evoluzione del profilo di velocità in moto piano laminare a bassi numeri di Reynolds. Assegnata una qualsiasi distribuzione di velocità nella sezione x=0, è possibile calcolare come questa si modifichi lungo il percorso. Si fornisce un esempio numerico.

     

Stability of low Reynolds number flow with viscous heating

Summary The stability of fully developed plane Couette flow and pipe flow with viscous heating is studied at low Reynolds number for a Newtonian liquid with a temperature-dependent viscosity. The solution is obtained by a direct integration method of the eigenfunction equations, with eigenvalues located in the complex plane by means of the argument principle of complex variable theory. An instability will occur in plane Couette flow, but outside the parameter range which will be encountered in practice. There is no comparable instability in pipe flow. It can be concluded that a thermal mechanism does not cause the low Reynolds number instabilities observed in polymer processing operations.

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Zusammenfassung Für eine newtonsche Flüssigkeit mit temperaturabhängiger Viskosität wird die Stabilität der voll entwickelten ebenen Couette-Strömung und Rohrströmung bei niedrigen Reynolds-Zahlen untersucht. Die Lösung wird durch direkte Integration der Eigenwert-Gleichungen gewonnen, wobei das Argument-Prinzip der Funktionentheorie auf die in der komplexen Ebene gelegenen Eigenwerte angewandt wird. In der ebenen Couette-Strömung wird eine Instabilität gefunden, jedoch außerhalb des in der Praxis realisierten Parameter-Bereichs. In der Rohrströmung gibt es dagegen keine vergleichbare Instabilität. Man kommt zu dem Schluß, daß thermische Mechanismen nicht für die in Polymer-Verarbeitungsprozessen beobachteten bei niedrigen Reynolds-Zahlen auftretenden Instabilitäten verantwortlich gemacht werden können.

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With 6 figures and 1 table