Transcritical Reynolds Numbers Research Articles

Water tunnel testing of downwind yacht sails

Downwind yacht sails, such as spinnakers, are low-aspect-ratio highly cambered wings with a sharp leading edge. They are characterised by substantial three-dimensional flow separation and are thus modelled with difficulty with numerical simulations. Furthermore, accurate full-scale validation data are not available. The first quantitative flow measurements have only recently been achieved in water tunnels. In this study, we aim to provide guidelines on this emerging sail testing methodology. We consider six model-scale rigid models at average-chord-based Reynolds numbers ranging from 5 870 to 61 870. A critical Reynolds number is identified, below which relaminarisation of the reattached boundary layer downstream of the leading edge separation bubble occurs. Both lift and drag increase monotonically at subcritical Reynolds numbers while remaining about constant at transcritical Reynolds numbers. The critical Reynolds number decreases with increasing incidence and is insensitive to the blockage ratio. Spinnakers are normally sailed in front of the mainsail, whose circulation is found to generate an approximately 3∘\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$3^{\\circ }$$\\end{document} upwash on the spinnaker and higher flow velocity on both sides of it. These findings provide guidelines for the experimental testing of spinnaker-like wings in water tunnels and provide new insights into the flow and experimental testing of highly cambered wings with massive flow separation at low Reynolds numbers.

Nucleation effects on cavitation about a sphere

The influence of nucleation on cavitation about a sphere from inception through to supercavitation at a transcritical Reynolds number of$1.5\times 10^{6}$is investigated experimentally. Two extreme free-stream nuclei populations, deplete and abundant, were investigated. Unsteady surface pressures from two sensors on opposing sides of the sphere were acquired simultaneous with high-resolution high-speed photography at cavitation numbers between 1.0 and 0.3. High-resolution spectrograms derived from these measurements reveal principally bi-modal shedding in attached and detached regimes. Correlations between unsteady pressure measurements show the high modes to be axisymmetric and low modes asymmetric. Modal topology is also discerned from the high-speed imaging. The bi-modal shedding for lower cavitation numbers is driven by coupled re-entrant jet formation and upstream shockwave propagation. The attached regime is shown to have two sub-regimes. For the abundant case, the continuous supply of activated nuclei around the sphere periphery in the first bi-modal regime has the effect of driving the high symmetric mode preferentially over the asymmetric low mode compared with the deplete case. For the first bi-modal regime, frequencies were unaffected by nucleation changes although peak responses were centred at a cavitation number of about 0.8 for the deplete and 0.825 for the abundant. For the second attached regime, where cavity lengths are of the order of the sphere size, changes in nucleation altered frequencies and amplitudes of peak unsteady pressures. For the abundant case, the continuous nuclei supply significantly reduced coherence with modal peak amplitudes reduced by an order compared with the deplete case. Continuous nuclei activation increased the probability of the high mode over the low compared with the deplete case but to a lesser extent than the first regime. Nuclei activations also significantly reduced inter-cavity and cavity durations, but not growth and collapse phases, which increased modal frequencies compared with the deplete case. The second regime asymmetric low mode topology, for both nucleation cases, is shown to be alternate shedding of oblique vortices from diametrically opposing sides of the sphere similar to low Reynolds number shedding about spheres and other axisymmetric bluff bodies in single-phase flows.

Investigation of the steady and unsteady forces acting on a pair of circular cylinders in crossflow up to ultra-high Reynolds numbers

Measurements of the steady and unsteady forces acting on a pair of circular cylinders in crossflow are performed from subcritical up to ultra-high Reynolds numbers. The two cylinders with equal diameters d are arranged inline at two centre-to-centre distances: S/d = 2.8 and 4. The trend of the drag curve for the upstream cylinder Cd_{1}(Re) at both distances is similar to that for a single circular cylinder. The development of the drag curves Cd_{2}(Re, S/d = 2.8, 4) of the downstream cylinder is inverse to that of the upstream cylinder. For both cylinder spacing values, the drag on the downstream cylinder is negative for subcritical Reynolds numbers, increases abruptly to positive values at the beginning of the supercritical regime, and shows a significant dip at transcritical Reynolds numbers. This drag inversion indicates that the critical distance Sc decreases sharply in the supercritical Reynolds number range. For S/d = 2.8 at Rerightarrow 10^{7}, the downstream cylinder experiences once more a thrust force. The curve of the Strouhal number St(Re) of the downstream cylinder for S/d = 4 is very close to that of a single cylinder. For Reynolds numbers of Reapprox 1times10^{6} - 7times10^{6}, the Strouhal numbers have nearly equal values of Stapprox 0.22 - 0.24 for both distances. This is followed by a branching. For Rerightarrow 10^{7} and the case S/d = 2.8, the Strouhal numbers dip at St = 0.17. However, for S/d = 4, they increase up to St = 0.27. In the supercritical range, two peaks occur in the power spectra for the large distance S/d = 4. Based on a wavelet analysis, we can conclude that the low-frequency mode, which does not occur for a single cylinder, is an interference effect.Graphic abstract

Mean and unsteady loading on square prisms with rounded edges: Hard marine growth, incidence, and Reynolds number effects

A set of wind tunnel experiments was performed to study the average and fluctuating wind loading on an ”infinite” 2D square prism with rounded edges of r/D = 0.16 for different heights of simulated hard marine growth at sub-to transcritical Reynolds numbers. It has already been shown, that a change in surface roughness from ks/D = 4.5×10−6 (smooth) to 1×10−3 (rough) on a similar prism at α = 0° and 45° had large effects on the widths and the Reynolds numbers marking the boundaries of the various flow states. At α = 45° it furthermore led to a large Reynolds-number independency of the lift and drag forces and of the Strouhal number. For the present wind tunnel tests two additional values of ks/D were selected to obtain smaller increments in surface roughness: 4.5×10-4 (slightly rough) and 1.4×10−3 (very rough). Mean and fluctuating lift and drag forces, the main vortex shedding frequency, distributions of the mean surface pressures in the mid-span cross-section of the prism and the mean wake profiles were recorded simultaneously. The same two static angles of incidence, α = 0° and 45°, were investigated for ReD = 60,000–12 million. For the prisms with smooth up to rough surfaces strong effects on the behaviour of all aerodynamic parameters with Reynolds number were found, whereas a further increase in ks/D induced hardly any additional effects. The absolute values of the aerodynamic parameters were found to be independent of a change in ks/D for the sub-to supercritical flow states. For the smooth and slightly rough prisms at α = 0° the supercritical flow state was present up to ReD = 12 million, whereas for the rougher prisms the upper transition and the transcritical flow states appeared at high Reynolds numbers. A change to α = 45° induced little variation with ks/D for all aerodynamic coefficients. The overall behaviour of the force coefficients and the Strouhal number with ks/D and ReD were caused by motions of the transition, separation and reattachment locations.

Experiments on the Flow around two tandem circular cylinders from sub- up to transcritical Reynolds numbers

Measurements were taken on two tandem circular cylinders in cross flow for Reynolds numbers from subcritical up to transcritical values (2 ⋅ 105≤ Re ≤ 6 ⋅ 106). The tandem arrangement with small spacing S = 1.56 d (d: diameter of the cylinders) represented a twin cable assembly of the main cable of the Messina Bridge. The tests were carried out in a special facility for high Reynolds numbers, the DNW High Pressure Wind tunnel in Göttingen. The steady forces on both cylinders were measured using a two-sided strain gauge balance. The wake profiles were taken by a pressure rake.At the angle of incidence α = 0°the appearance of the drag-curve Cd1 depending on Re for the front cylinder is similar to the behavior of a single smooth circular cylinder. The shape of the corresponding curve Cd2 of the downstream cylinder resembles the inverse of the upstream one with negative drag in the subcritical and transcritical Re-range. The critical Reynolds number regime is around Rec≈ 4⋅ 105. Here, the drag on the second cylinder changes sign from Cd2≈ -0.4 to +0.4. Further, asymmetric flow, reflected in steady lift Cl2≈ 1, was observed.The supercritical regime is followed by the upper transition, after that, the transcritical range begins at Re ≈ 3.5⋅ 106. In that case a second drag inversion occurs, that is coupled with a change of the topological structure of the flow and thus can be a source of instability.In all relevant Reynolds number ranges for both cylinders curves of the force/moment coefficients were created, i.e. the dependence of these coefficients upon α. These curves for the sub-, super-, and upper transition as well as for transcritical Reynolds numbers are strongly nonlinear in nature with a curved character, pronounced extrema, and zero-crossings. In the subcritical case there are, in addition, hysteretic jumps of lift and drag. For all Reynolds number ranges investigated, we found that for the rigidly coupled tandem cylinders, the derivative of the global lift (around α = 0°) is negative and the corresponding Den Hartog criterion indicates instability.During the experiments at transcritical Reynolds numbers there were undesired vibrations, in particular for the front cylinder. The observed vibration phenomena are likely caused by vortex resonance and they are coupled with the mentioned drag-inversion at very high Reynolds numbers.

Large-eddy simulation of flow over a grooved cylinder up to transcritical Reynolds numbers

We report wall-resolved large-eddy simulation (LES) of flow over a grooved cylinder up to the transcritical regime. The stretched-vortex subgrid-scale model is embedded in a general fourth-order finite-difference code discretization on a curvilinear mesh. In the present study $32$ grooves are equally distributed around the circumference of the cylinder, each of sinusoidal shape with height $\unicode[STIX]{x1D716}$, invariant in the spanwise direction. Based on the two parameters, $\unicode[STIX]{x1D716}/D$ and the Reynolds number $Re_{D}=U_{\infty }D/\unicode[STIX]{x1D708}$ where $U_{\infty }$ is the free-stream velocity, $D$ the diameter of the cylinder and $\unicode[STIX]{x1D708}$ the kinematic viscosity, two main sets of simulations are described. The first set varies $\unicode[STIX]{x1D716}/D$ from $0$ to $1/32$ while fixing $Re_{D}=3.9\times 10^{3}$. We study the flow deviation from the smooth-cylinder case, with emphasis on several important statistics such as the length of the mean-flow recirculation bubble $L_{B}$, the pressure coefficient $C_{p}$, the skin-friction coefficient $C_{f\unicode[STIX]{x1D703}}$ and the non-dimensional pressure gradient parameter $\unicode[STIX]{x1D6FD}$. It is found that, with increasing $\unicode[STIX]{x1D716}/D$ at fixed $Re_{D}$, some properties of the mean flow behave somewhat similarly to changes in the smooth-cylinder flow when $Re_{D}$ is increased. This includes shrinking $L_{B}$ and nearly constant minimum pressure coefficient. In contrast, while the non-dimensional pressure gradient parameter $\unicode[STIX]{x1D6FD}$ remains nearly constant for the front part of the smooth cylinder flow, $\unicode[STIX]{x1D6FD}$ shows an oscillatory variation for the grooved-cylinder case. The second main set of LES varies $Re_{D}$ from $3.9\times 10^{3}$ to $6\times 10^{4}$ with fixed $\unicode[STIX]{x1D716}/D=1/32$. It is found that this $Re_{D}$ range spans the subcritical and supercritical regimes and reaches the beginning of the transcritical flow regime. Mean-flow properties are diagnosed and compared with available experimental data including $C_{p}$ and the drag coefficient $C_{D}$. The timewise variation of the lift and drag coefficients are also studied to elucidate the transition among three regimes. Instantaneous images of the surface, skin-friction vector field and also of the three-dimensional Q-criterion field are utilized to further understand the dynamics of the near-surface flow structures and vortex shedding. Comparison of the grooved-cylinder flow with the equivalent flow over a smooth-wall cylinder shows structural similarities but significant differences. Both flows exhibit a clear common signature, which is the formation of mean-flow secondary separation bubbles that transform to other local flow features upstream of the main separation region (prior separation bubbles) as $Re_{D}$ is increased through the respective drag crises. Based on these similarities it is hypothesized that the drag crises known to occur for flow past a cylinder with different surface topographies is the result of a change in the global flow state generated by an interaction of primary flow separation with secondary flow recirculating motions that manifest as a mean-flow secondary bubble. For the smooth-wall flow this is accompanied by local boundary-layer flow transition to turbulence and a strong drag crisis, while for the grooved-cylinder case the flow remains laminar but unsteady through its drag crisis and into the early transcritical flow range.

LES-based Study of the Roughness Effects on the Wake of a Circular Cylinder from Subcritical to Transcritical Reynolds Numbers

This paper investigates the effects of surface roughness on the flow past a circular cylinder at subcritical to transcritical Reynolds numbers. Large eddy simulations of the flow for sand grain roughness of size k/D = 0.02 are performed (D is the cylinder diameter). Results show that surface roughness triggers the transition to turbulence in the boundary layer at all Reynolds numbers, thus leading to an early separation caused by the increased momentum deficit, especially at transcritical Reynolds numbers. Even at subcritical Reynolds numbers, boundary layer instabilities are triggered in the roughness sublayer and eventually lead to the transition to turbulence. The early separation at transcritical Reynolds numbers leads to a wake topology similar to that of the subcritical regime, resulting in an increased drag coefficient and lower Strouhal number. Turbulent statistics in the wake are also affected by roughness; the Reynolds stresses are larger due to the increased turbulent kinetic energy production in the boundary layer and separated shear layers close to the cylinder shoulders.

English

English

Field measurements on flow past a circular cylinder in transcritical Reynolds number regime

Flow around a circular cylinder is a classic scenario which invariably draws the attention of the fluid mechanics circle, because its relevant studies are of both theoretical and practical significances. However, most experiments are conducted below transcritical Reynolds number（Re) regime（Re3.5106) due to the limitations of the wind tunnel modeling technique, which makes the obtained results inapplicable to some full-scale conditions. To this end, the field measurements for wind-induced pressures on a 167-meter high large cooling tower are conducted at Re=6.59107 to enrich the experimental results of flow past a circular cylinder in transcritical Re regime. Besides, the wind effects at low Re（Re=2.1105-4.19105) are also obtained by tests on a 1:200 rigid cooling tower model in a wind tunnel with considering 4 types of wind speeds, 8 types of surface roughness, and 2 flow fields. Employing the data obtained from both field measurements and wind tunnel model tests, the variations of static/dynamic flow characteristics with Re increasing are studied. It is found that 1) with the increase of Re, the drag coefficient for the smooth-walled tower in the uniform flow field decreases dramatically in the critical Re regime and increases slowly in the supercritical regime, which accord with Roshko's and Achenbach's results; 2) for smooth-walled tower, both the base pressure coefficient and pressure coefficient increase significantly with the increase of Re in critical and supercritical regimes, which qualitatively accord with Shih's results; and 3) the finding of the Strouhal number is supportive to Shih's result（i.e., shedding from the rough cylinder persists throughout the Re range tested). More importantly, special attention is paid to the Re-independence phenomenon of fluid flow, which is a typical phenomenon occurring in transcritical Re regime. Results indicate that the Re-independence exists in an Re range from 2105 to 1108 for a circular cylinder with a relative roughness greater than 0.01, and the increased free-stream turbulence can also induce Re-independence which probably exists in a narrow low Re range. Considering the flow mechanism, a reasonable explanation can be found for the Re-independence phenomenon, i.e., the critical and supercritical regimes narrow and move to lower Re range with the increase of surface roughness or the increase of free-stream turbulence, so Re independence can occur at a very low Re.

Across-wind aerodynamic parameters of tall chimneys with circular cross section—a new empirical model

The prediction of across-wind response of circular cylinders remains a challenging task, despite extensive research efforts. An attempt has been made to correlate the rms lift coefficient due only to vortex shedding, C′ L,VS both in wind tunnel and full-scale conditions by separating the local rms lift coefficient, C′ L, into two components, one due to the lateral turbulence and the other due to the vortex shedding. Based on the literature, and also using test results measured by the authors, it is found that the final value of C′ L,V, as discussed in the paper, show a mean value of 0.089 with a coefficient of variation of 18%, independent of Reynolds number regime. The above value plus 1.66 times the standard deviation gives a value of 0.115 which is in excellent agreement with the value of C′ L,VS=0.12 recommended for design of chimneys under open terrain conditions, as per the Indian Standard Code of Practice: IS: 4998 Part-2, 1992. Further it is shown that the Griffin universal Strouhal number, G attains a mean value of 0.065 with a coefficient of variation (cov) of 8%, independent of sub-critical and trans-critical Reynolds number regime.

Computation of Vortex Shedding and Radiated Sound for a Circular Cylinder: Subcritical to Transcritical Reynolds Numbers

The Lighthill acoustic analogy combined with Reynolds-averaged Navier–Stokes flow computations are used to investigate the ability of existing technology to predict the tonal noise generated by vortex shedding from a circular cylinder for a range of Reynolds numbers (100 < Re < 5 million). Computed mean drag, mean coefficient of pressure, Strouhal number, and fluctuating lift are compared with experiment. Two-dimensional calculations produce a Reynolds number trend similar to experiment but incorrectly predict many of the flow quantities. Different turbulence models give inconsistent results in the critical Reynolds number range (Re≈ 100000). The computed flow field is used as input for noise prediction. Two-dimensional inputs overpredict both noise amplitude and frequency; however, if an appropriate correlation length is used, predicted noise amplitudes agree with experiment. Noise levels and frequency content agree much better with experiment when three-dimensional flow computations are used as input data.

The Surface Flow Structure of Large Scale Vortices at the rear of a Cylindrical Pier for the Flow of Transcritical Reynolds Number

The surface flow structure of large scale vorticies at the rear of a cylindrical pier is studied for the flow of transcritical Reynolds number is over 108. The field observation around AKASHIKAIKYOU OHASHI bridge pier was carried out. Water surface images are taken from top of the main tower using a video camera. And an image analysis method using a correlation method for the calculation of two dimensional velocity distribution of surface is examined. In the rear side of the pier, large scale vortices like Karman vortices could be seen by video and MEM spectrum of surface velocity fluctuation.

The effect of a wake splitter plate on the flow amound a circular cylinder.

Experiments were carried out using a circular model with a wake splitter plate having length ratio (L/d)=1/6, 1 or 3 in the Reynolds number range of 5×104≤Re≤107 using a cryogenic wind tunnel. The resulting variations of pressure distributions, drag coefficients, back pressure coefficients, the minimum values of pressure coefficients and Strouhal numbers with the length of wake splitter plate are presented. The effect of the wake splitter plate on the drag and Strouhal number are shown to change depending on the Reynolds number range and on the length of the wake splitter plate. Effects of the wake splitter plate on the reduction of the drag coefficient is most remarkable in the subcritical Reynolds number range. In the supercritical Reynolds number range the wake splitter plate serves to suppress the separation bubble. In the transcritical Reynolds number range the effects of the wake splitter plate can hardly be recognized.

Simulation of viscous flow around a circular cylinder in the boundary layer near a wall

A numerical method for solving the problem of an arbitrary shear flow around one or more circular cylinders is presented. This extends the applicability of the vortex-in-cell method to compute the viscous flow around a cylinder in the shear layer close to a wall at transcritical Reynolds numbers. The effect of varying the gap between the wall and the cylinder and the thickness of the boundary layer is investigated. The computations show that, if the cylinder is partly within the boundary layer, the effect due to the shear of the onset flow is important. Comparison with experimental data is encouraging.

Pressure and force fluctuations on isolated circular cylinders of finite height in boundary layer flows

Abstract This paper presents spatio-temporal measurements of a fluctuating pressure field acting on isolated roughened cylinders of finite height in simulated turbulent boundary layer flows at subcritical Reynolds numbers. The cylinder surface is roughened by placing discrete roughness elements along the height at four circumferential locations. The resulting surface roughness configuration helps to modify the flow over the cylinder, which leads to the pressure field that exhibits characteristics similar to the transcritical Reynolds numbers. The space-time measurements of the random pressure field allowed computations of the mean and r.m.s. pressure coefficients, power spectral density, spanwise and circumferential correlations, and their eigenfunction expansions. A manifold-pneumatic averaging technique was utilized to evaluate time varying area-averaged loads at various levels along the model height. The spatially averaged measurements of the pressure field over the roughened cylinder facilitated the estimation of local and mode-generalized alongwind and acrosswind aerodynamic force coefficients, their spectral descriptions and multi-level force correlations. An increase in the turbulence level influences the fluctuating pressure field through modifications which take place in the structure of the boundary layer over the cylinder and the structure of the separated shear layers.

Drag and vortex shedding from circular cylinder in the high reynolds numbers range. Effect of surface roughness.

Discussions are made on the effect of surface roughness on the drag coefficient and Strouhal numbers 5×105≤Re≤107. There is the maximum value of surface roughness which does not effect the drag coefficient and Strouhal number in the subcritical and transitional range and at the critical point. This maximum value is determined as (Ks/d)crit=5×10-4. For a larger relative surface roughness value, there are some functional relations between the relative surface roughness (Ks/d) and the minimum drag coefficient and the Reynolds number which corresponds to CD min. In the transcritical Reynolds number range, the drag coefficient depends on the relative surface roughness. It increases with an increase in the relative surface roughness. There arise vortex streets in the wake of a cylinder which satisfy the Bearman's condition, i. e. St*=0.181 in the subcritical and transcritical Reynolds number ranges.

Effect of end plates on the flow around rough cylinders

Measurements of mean and fluctuating pressures on a rough circular cylinder in uniform flow are presented. The experiments were carried out with and without end plates, at critical and transcritical Reynolds numbers. The effect of the end plates was found to be very significant, of the order of 15 to 20 percent. The characteristics of the flow transitions around rough cylinders are examined in the light of these and other results.

On the force and vortex shedding on a circular cylinder from subcritical up to transcritical Reynolds numbers.

Pressure distribution and vortex shedding were measured in a cryogenic wind tunnel from subcritical up to transcritical Reynolds numbers(105 ≤ Re ≤ 107 ) and Mach numbers up to 0.3 without changing experimental arrangement. Drag coefficients were calculated using pressure distributions. Pressure distributions and drag coefficients show characteristic changes for subcritical, lower transition, critical, under transition and transcritical Reynolds number ranges respectively. Strouhal number takes constant value with an increase in Reynolds number for 105 ≤ Re < 3.2 × 105. Narrowband vortex shedding could not be measured in the critical Reynolds number range ( 3.2 × 105 ≤ Re ≤ 6.9 × 105). It was also measured again in the upper transition and transcritical Reynolds number ranges. Spectra of the pressure distributions and velocity fluctuations measured are presented for several characteristic values of Reynolds numbers. Results are also compared with those of the other authors.

On the Force and Vortex Shedding on a Circular Cylinder from Subcritical up to Transcritical Reynolds Numbers

Pressure distributions and vortex shedding were measured in a cryogenic wind tunnel from subcritical up to transcritical Reynolds numbers (105 ≤ Re ≤ 107) and Mach numbers up to 0.3 without changing experimental arrangement. Drag coefficients were calculated using pressure distributions. Pressure distributions and drag coefficients show characteristic changes for subcritical, lower transition, critical, upper transition and transcritical Reynolds numbers respectively. Strouhal number is constant in the range of Reynolds numbers 105 ≤ Re ≤ 3.2 ×105. Narrow band vortex shedding could not be found in the critical Reynolds numbers range (3.2×105 ≤ Re ≤6.9×105), but it was found again in the upper transition and transcritical Reynolds number ranges. The pressure distributions and spectra of velocity fluctuations are presented for several characteristic values of Reynolds numbers. Results are also compared with those of other authors.

On the force fluctuations acting on a circular cylinder in crossflow from subcritical up to transcritical Reynolds numbers

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.