Hybrid vortex method for high Reynolds number flows around three-dimensional complex boundary

The hybrid vortex method, in which the vortex particle method was combined with the vortex sheet method, was extended to flows around a three-dimensional complex geometry boundary at high Reynolds number of O(10⁶). The computing domain was decomposed into an interior domain of vortex blobs and a thin numerical boundary layer of vortex sheets. The core radii of shedding blobs were related to the size of the vortex sheets. As the result of numerical experiments on the flow over a ship propeller, the hybrid vortex method was found to be acceptable for simulations of unsteady separated flows around a solid body at high Reynolds number, since the computed results showed reasonable agreement with the measured data.     

Implicit multigrid computation of unsteady flows past cylinders of square cross-section

An implicit, multigrid scheme has been extended to treat unsteady flows using the concept of temporal subiteration (or dual-time stepping, as it sometimes is called). The efficiency and accuracy of the subiterated, multigrid approach has been discussed in a previous paper. Here, the scheme is applied to compute the unsteady flow past fixed cylinders of square cross-section at moderate Reynolds numbers. The observed pattern of periodic vortex shedding is computed and the dimensionless frequency of this phenomenon (the Strouhal number) is compared with experimentally determined values. Results of coupled aeroelastic computations also are presented that illustrate a hysteresis phenomenon as the structural frequency is varied.     

Numerical simulation of an oscillating cylinder in a cross-flow at low Reynolds number: Forced and free oscillations

A numerical simulation of the flow past a circular cylinder which is able to oscillate transversely to the incident stream is presented in this paper for a fixed Reynolds number equal to 100. The 2D Navier-Stokes equations are solved by a finite volume method with an industrial CFD code in which a coupling procedure has been implemented in order to obtain the cylinder displacement. A preliminary work is first conducted for a fixed cylinder to check the wake characteristics for Reynolds numbers smaller than 150 in the laminar regime. The Strouhal frequency f_S and the aerodynamic coefficients are thus controlled among other parameters. Simulations are then performed with forced oscillations characterized by the frequency ratio F = f₀/f_S, where f₀ is the forced oscillation frequency, and by the adimensional amplitude A. The wake characteristics are analyzed using the time series of the fluctuating aerodynamic coefficients and their power spectral densities (PSD). The frequency content is then linked to the shape of the phase portraits and to the vortex shedding mode. By choosing interesting couples (A, F), different vortex shedding modes have been observed, which are similar to those of the Williamson-Roshko map. A second batch of simulations involving free vibrations (so-called vortex-induced vibrations or VIV) is finally carried out. Oscillations of the cylinder are now directly induced by the vortex shedding process in the wake and therefore, the time integration of the motion is realized by an explicit staggered algorithm which provides the cylinder displacement according to the aerodynamic charges exerted on the cylinder wall. Amplitude and frequency response of the cylinder are thus investigated over a wide range of reduced velocities to observe the different phenomena at stake. In particular, the vortex shedding modes have also been related to the frequency response observed and our results at Re = 100 show a very good agreement with other studies using different numerical approaches.     

Numerical simulation of the flow around rows of cylinders

Two-dimensional, laminar, unsteady, water flow around cylinder arrays of unequal sizes was simulated using FLUENT™ at Reynolds numbers below 150 (based on the free-stream velocity and first row cylinder diameter). The flow pattern through two rows of inline cylinders showed incomplete vortex shedding behind the first row at a separation distance less than 2d. Karman vortices were not formed and a near-stagnant separated flow region appeared between the aligned cylinders. Cylinders in staggered arrangements shed Karman vortices regardless of the separation between the two rows. This research has shed light on the detailed flow through paper machine forming fabrics.     

Dynamic hybridization of MILES and RANS for predicting airfoil stall

Hybridization comprised of an algebraic turbulence model based on the Reynolds average Navier-Stokes (RANS) equations with a monotonically integrated large eddy simulation (MILES) is proposed to simulate transient fluid motion during separation and vortex shedding at high Reynolds numbers. The proposed hybridization utilizes the Baldwin-Lomax model with the Degani-Schiff modification as the RANS model in the near-wall region and a MILES far from the wall. Although the hybridization is assumed to be a MILES with wall modeling, the transition line between the RANS and the MILES modes is determined by the turbulent intensity that is dominated by the large eddies in the grid-scale. This hybrid model is applied to the flows past three different types of airfoils (NACA63₃-018, NACA63₁-012 and NACA64A-006) near stall, at a chord Reynolds number of Re = 5.8 × 10⁶. These airfoils are classified as trailing-edge-stall, leading-edge-stall and thin-airfoil-stall airfoils, respectively. The computed results are compared with wind tunnel experiments. The hybrid model successfully demonstrates accurate stall angle and surface pressure distribution predictions near the stall for each type of airfoil. The airfoil simulation results confirmed that the hybrid model provides a better prediction than the RANS model for unsteady turbulent flows with separation and vortex shedding simulations.     

Numerical simulation of flow past row of square cylinders for various separation ratios

A systematic study for the flow around a row of five square cylinders placed in a side-by-side arrangement and normal to the oncoming flow at a Reynolds number of 150 is carried out through the numerical solution of the two-dimensional unsteady incompressible Navier-Stokes equations. Special attention is paid to investigate the effect of the spacing between the five cylinders on the wake structure and vortex shedding mechanism. The simulations are performed for the separation ratios (spacing to size ratio) of 1.2, 2, 3 and 4. Depending on the separation ratio the following flow patterns are observed: a flip-flopping pattern, in-phase and anti-phase synchronized pattern and non-synchronized pattern. These flow patterns are supposed to be a consequence of the interaction between two types of frequencies viz. the vortex shedding (primary) and the cylinder interaction (secondary) frequencies. At small separation ratio the flow is predominantly characterized by the jet in the gaps between successive cylinders and the secondary frequencies play a role in the resulting chaotic flow. On the contrary, at higher separation ratio the secondary frequencies almost disappear and the resulting flow becomes more synchronized dominated by the primary frequency.     

Sudden expansions in circular microchannels: flow dynamics and pressure drop

Micro particle shadow velocimetry is used to study the flow of water through microcircular sudden expansions of ratios e = 1.51 and e = 1.96 for inlet Reynolds numbers Re _d < 120. Such flows give rise to annular vortices, trapped downstream of the expansions. The dependency of the vortex length on the Reynolds number Re _d and the expansion ratio e is experimentally investigated in this study. Additionally, the shape of the axisymmetric annular vortex is quantified based on the visualization results. These measurements favorably follow the trends reported for larger scales in the literature. Redevelopment of the confined jet to the fully developed Poiseuille flow downstream of the expansion is also studied quantitatively. Furthermore, the experimentally resolved velocities are used to calculate high resolution static pressure gradient distributions along the channel walls. These measurements are then integrated into the axisymmetric momentum and energy balance equations, for the flow downstream of the expansion, to obtain the irreversible pressure drop in this geometry. As expected, the measured pressure drop coefficients for the range of Reynolds numbers studied here do not match the predictions of the available empirical correlations, which are commonly based turbulent flow studies. However, these results are in excellent agreement with previous numerical calculations. The pressure drop coefficient is found to strongly depend on the inlet Reynolds number for Re _d < 50. Although no length-scale effect is observed for the range of channel diameters studied here, for Reynolds numbers Re _d < 50, which are typical in microchannel applications, complex nonlinear trends in the flow dynamics and pressure drop measurements are discovered and discussed in this work.     

Surface effects on transient three-dimensional flows around rotating spheres at moderate Reynolds numbers

Transient wake flow patterns and dynamic forces acting on a rotating spherical particle with non-uniform surface blowing are studied numerically for Reynolds numbers up to 300 and dimensionless angular velocities up to Ω=1. This range of Reynolds numbers includes the three distinct wake regimes i.e., the steady axisymmetric, the steady non-symmetrical and the unsteady with vortex shedding. The Navier–Stokes equations for an incompressible viscous flow are solved by a finite volume method in a three-dimensional, time accurate manner. An interesting feature associated with particle rotation and surface blowing is that they can affect the near wake structure in such a way that unsteady three-dimensional wake flow with vortex shedding develops at lower Reynolds numbers as compared to flow over a solid sphere in the absence of these effects and thus, vortex shedding occurs even at Re=200. Global properties, such as the lift and drag coefficients, and the Strouhal number are also significantly affected. It is shown that the present data for the average lift and drag coefficients correlate well with:

C_L/(1+Ω)^3.6=0.11

C_D(1+20V_S)^0.2/(1+Ω)^Re/1000=24(1+Re^2/3/6)/Re

where V_S is the average surface blowing velocity normalized by the free stream velocity.     

Effect of tube spacing on the vortex shedding characteristics of laminar flow past an inline tube array: A numerical study

The effect of tube spacing on the vortex shedding characteristics and fluctuating forces in an inline cylinder array is studied numerically. The examined Reynolds number is 100 and the flow is laminar. The numerical methodology and the code employed to solve the Navier-Stokes and continuity equations in an unstructured finite volume grid are validated for the case of flow past two tandem cylinders at four spacings. Computations are then performed for a six-row inline tube bank for eight pitch-to-diameter ratios, s, ranging from 2.1 to 4. At the smallest spacing examined (s = 2.1) there are five stagnant and symmetric recirculation zones and weak vortex shedding activity occurs only behind the last cylinder. As s increases, the symmetry of the recirculation zones breaks leading to vortex shedding and this process progressively moves upstream, so that for s = 4 there is clear shedding from every row. For any given spacing, the shedding frequency behind each cylinder is the same. A critical spacing range between 3.0 and 3.6 is identified at which the mean drag as well as the rms lift and drag coefficients for the last three cylinders attain maximum values. Further increase to s = 4 leads to significant decrease in the force statistics and increase in the Strouhal number. It was found that at the critical spacing there is 180° phase difference in the shedding cycle between successive cylinders and the vortices travel a distance twice the tube spacing within one period of shedding.     

Numerical investigations on dynamic stall of low Reynolds number flow around oscillating airfoils

This paper presents a 2D computational investigation on the dynamic stall phenomenon associated with unsteady flow around the NACA0012 airfoil at low Reynolds number (Re_c ≈ 10⁵). Two sets of oscillating patterns with different frequencies, mean oscillating angles and amplitudes are numerically simulated using Computational Fluid Dynamics (CFD), and the results obtained are validated against the corresponding published experimental data. It is concluded that the CFD prediction captures well the vortex-shedding predominated flow structure which is experimentally obtained and the results quantitatively agree well with the experimental data, except when the blade is at a very high angle of attack.     

Experimental investigation on the flow transition in different pin-fin arranged microchannels

The flow characteristics of water through the in-line and staggered pin-fin microchannels with length of 25 mm, width of 2.4 mm and height of 0.11 mm were studied experimentally. The flow transition was identified as a sudden increasing slope in both pressure drop versus mass flow rate curve and friction factor versus Reynolds number curve for in-line pin-fin microchannels, but it did not occur for staggered pin-fin microchannels. The effect of pin-fin arrangements on the flow transition was not reported in the previous literature. With the aid of microparticle image velocimetry (Micro-PIV) technology, the streamlines, velocity fields and velocity fluctuation fields of flow through the pin-fin microchannels were captured to explain the flow transition, and the effect of pin-fin arrangements on the flow transition was analyzed for the first time. It was found that at the critical Reynolds number where the flow transition occurred for the in-line pin-fin microchannels, the steady double-vortex wake flow changed to the unsteady vortex-shedding wake flow. The occurrence of vortex shedding caused an obvious change in main stream from straight flow to wavy flow and further induced significant increases of transversal velocity and velocity fluctuations, which induced strong flow disturbance in transversal directions and large additional pressure drop, and finally caused the flow transition in the in-line pin-fin microchannels. For the staggered pin-fin microchannels, the main stream through the pin-fin arrays was found to be already the wavy flow before the vortex shedding. Thus, the transversal velocity and velocity fluctuations induced by the vortex shedding were relatively small, and therefore, the flow transition with an abrupt pressure drop increase was not observed in the staggered pin-fin microchannels.     

High-order accurate simulations of unsteady flow past plunging and pitching airfoils

This paper presents simulations of unsteady flow past plunging and pitching airfoils using a high-order spectral difference (SD) method. Both third-order and fourth-order SD methods are employed on unstructured quadrilateral grids for the plunging airfoil at a low Reynolds number. The vortex shedding pattern of an airfoil in an oscillating plunge motion becomes asymmetric at a sufficiently high frequency. The SD method is able to capture this effect and reveal a fine structure that closely replicates the experimental photograph. Interestingly, our simulations also predict that the degree of this asymmetry increases with Reynolds number. Unsteady flow at a higher Reynolds number past a pitching airfoil is studied using the fifth-order SD method. Our predictions show very good agreements with the available experimental data. The developed high-order accurate SD algorithms could enable high-order accurate simulations of unsteady flow past flapping Micro-Air-Vehicles (MAVs).     

Helicoidal vortex model for steady and unsteady flows

A helicoidal vortex model is used to predict the flow past the blades of a wind turbine. As the tip speed ratio (TSR) varies, the environment in which the blades operate varies, and for low enough TSR, the local angle of attack α will be larger than (α)_Clmax, the incidence of maximum lift. The problem becomes highly nonlinear and it is shown that adding an artificial viscosity term to the equation allows the iterative algorithm to converge toward a smooth solution that is physically acceptable.The introduction of unsteady effects is useful to understand the cyclic forces and moments due to yaw or tower interaction, both for the design of blades to account for fatigue and for power output prediction. The 2-D impulsively plunging plate problem is solved with a semi-implicit scheme and the integral and numerical solutions compared and shown to be in excellent agreement. A 2-D test case to study the convection of a periodic shedding of vorticity downstream of a blade element is analyzed using the same semi-implicit algorithm and a stretched mesh similar to that used to model a turbine vortex sheet. The scheme captures accurately the vorticity distribution in the wake. Finally, the scheme is applied to the simulation of the NREL two-bladed rotor in yaw to assess the validity of the approach.     

Steady and unsteady flow past a rotating circular cylinder at low Reynolds numbers

Results of calculations of the steady and unsteady flows past a circular cylinder which is rotating with constant angular velocity and translating with constant linear velocity are presented. The motion is assumed to be two-dimensional and to be governed by the Navier-Stokes equations for incompressible fluids. For the unsteady flow, the cylinder is started impulsively from rest and it is found that for low Reynolds numbers the flow approaches a steady state after a large enough time. Detailed results are given for the development of the flow with time for Reynolds numbers 5 and 20 based on the diameter of the cylinder. For comparison purposes the corresponding steady flow problem has been solved. The calculated values of the steady-state lift, drag and moment coefficients from the two methods are found to be in good agreement. Notable, however, are the discrepancies between these results and other recent numerical solutions to the steady-state Navier-Stokes equations. Some unsteady results are also given for the higher Reynolds numbers of 60, 100 and 200. In these cases the flow does not tend to be a steady state but develops a periodic pattern of vortex shedding.     

Flow characteristics of two rotating side-by-side circular cylinder

This paper presents a numerical investigation of the characteristics of the two-dimensional laminar flow around two rotating circular cylinders in side-by-side arrangements. In order to consider the combined effects of the rotation and the spacing between two cylinders on the flow, numerical simulations are performed at a various range of absolute rotational speeds (|α|?2) for four different gap spacings of 3, 1.5, 0.7 and 0.2 at Reynolds number of 100 showing the typical two-dimensional vortex shedding. As |α| increases, the flow changes its condition from periodic to steady after a critical rotational speed, which depends on the gap spacing. In the cases of gap spacings of 3 and 0.2, the wake keeps the same pattern, until flow reaches the steady state. However, for the gap spacings of 1.5 and 0.7, the wake patterns change in the unsteady regimes. For the cases in which the flow is unsteady, the Strouhal number strongly depends on the gap. For a fixed gap spacing, the variation of the Strouhal number is significant when the wake pattern is changed according to the rotational speed. Regardless of the gap spacing, as |α| increases, the lift increases and the drag decreases. Quantitative information about the flow variables such as the pressure coefficient and wall vorticity distributions on the cylinders is highlighted.     

Optimal control of cylinder wakes via suction and blowing

A general method based on adjoint formulation is discussed for the optimal control of distributed parameter systems (including boundary parameter) which is especially suitable for large dimensional control problems. Strategies for efficient and robust implementation of the method are described. The method is applied to the problem of controlling vortex shedding behind a cylinder (through suction/blowing on the cylinder surface) governed by the unsteady two-dimensional incompressible Navier-Stokes equations space discretized by finite-volume approximation with time-dependent boundary conditions. Three types of objective functions are considered, with regularization to circumvent ill-posedness. These objective functions involve integration over a space-time domain. The minimization of the cost function uses a quasi-Newton DFP method.A complete control of vortex shedding is demonstrated for Reynolds numbers up to 110. The optimal values of the suction/blowing parameters are found to be insensitive to initial conditions of the model when the time window of control is larger than the vortex shedding period, the inverse of the Strouhal frequency. Although this condition is necessary for robust control, it is observed that a shorter window of control may suffice to suppress vortex shedding.     

Direct numerical simulation of flow separation around a NACA 0012 airfoil

Direct numerical simulation (DNS) for the flow separation and transition around a NACA 0012 airfoil with an attack angle of 4° and Reynolds number of 10⁵ based on free-stream velocity and chord length is presented. The details of the flow separation, detached shear layer, vortex shedding, breakdown to turbulence, and re-attachment of the boundary layer are captured in the simulation. Though no external disturbances are introduced, the self-excited vortex shedding and self-sustained turbulent flow may be related to the backward effect of the disturbed flow on the separation region. The vortex shedding from the separated free shear layer is attributed to the Kelvin-Helmholtz instability.     

Vortex shedding from confined micropin arrays

The hydrodynamics in microcavities populated with cylindrical micropins was investigated using dynamic pressure measurements and fluid pathline visualization. Pressure signals were Fourier-analyzed to extract the flow fluctuation frequencies, which were in the kHz range for the tested flow Reynolds numbers (Re) of up to 435. Three different sets of flow dependent characteristic frequencies were identified, the first due to vortex shedding, the second due to lateral flow oscillation and the third due to a transition between these two flow regimes. These frequencies were measured at different locations along the chip (e.g. inlet, middle and outlet). It is established that vortex shedding initiates at the outlet and then travels upstream with increase in Re. The pathline visualization technique provided direct optical access to the flow field without any intermediate post-processing step and could be used to interpret the frequencies determined through pressure measurements. Microcavities with different micropin height-to-diameter aspect ratios and pitch-to-diameter ratios were tested. The tests confirmed an increase in the Strouhal number (associated with the vortex shedding) with increased confinement (decrease in the aspect ratio or the pitch), in agreement with macroscale measurements. The compact nature of the microscale geometry tested, and the measurement technique demonstrated, readily enabled us to investigate the flow past 4,420 pins with various degrees of confinements; this makes the measurements performed and the techniques developed here an important tool for investigating large arrays of similar objects in a flow field.     

Instability of channel flow with fluid injection and parietal vortex shedding

This paper is devoted to the computation of the stability properties and the aeroacoustic interaction in a planar channel flow driven by air injection through porous wall. The purpose is to establish a simulation method which estimates the vortex-shedding and acoustic-wave interaction. To perform these unsteady computations, turbulent motion is taken into account by a first-order model based on a nonlinear relationship between time-dependent Reynolds stresses and velocity gradients. Model coefficients are explicit functions of both strain and rotation. This model is applied to the computation of parietal vortex shedding in a simplified configuration. It is a channel flow with fluid injection. Two cases are computed here. The first one is stable and the second one is unstable. To improve the evaluation of the turbulent effects, computations of the second case are presented with and without a turbulence model. In first configuration, a good agreement with experimental data is found. In the second case both with and without turbulence model the natural unsteadiness of the flow is captured. The parietal vortex shedding is described and the turbulence effects are characterized.     

Multiple vortex formation in steady laminar axisymmetric impinging flow

Steady confined laminar axisymmetric impinging flow of a Newtonian fluid is relevant in many situations, an important application being heat and mass transfer from a solid surface to an impinging jet. This paper focuses on the evolution of the structure of the radial flow field in the channel region beyond the impingement zone. We employ an upwind scheme with an established numerical technique to solve the stream function and vorticity equations for a range of Reynolds numbers Re and geometrical aspect ratios e. Our results show the progressive complexity in the radial flow due to multiple points of flow separation and reattachment, and we provide a detailed demarcation of the Re-e plane based on flow separation behavior. In addition to the primary and secondary vortices anchored on the confining and impinging surfaces, respectively, we describe the formation and properties of a tertiary vortex which is wholly enclosed within the primary vortex. At a fixed Reynolds number, the tertiary vortex is observed only for a specific range of the aspect ratio, and we catalog its birth, growth and demise as the aspect ratio is varied. The range of aspect ratios over which the tertiary vortex exists is seen to increase with the Reynolds number. These results show that the fine structure of the radial flow at high Reynolds number continues to be dependent on the aspect ratio in a complex manner. At a given aspect ratio, the sizes of the vortices increases with Reynolds number, scaling as ∼Re^1/3, and for sufficiently large Re, the length of the tertiary vortex can exceed that of the secondary vortex. The primary and secondary vortex lengths satisfy an asymptotic relationship independent of Re and e, the numerically computed value of α being ∼2. Similarly, the locations of these vortices bear simple linear relationships independent of Re and e. Furthermore, despite the complex fine structure of the flow field, macroscopic flow properties such as vortex circulation and excess pressure loss continue to exhibit relatively simple dependence on Re and e, in accordance with previous results at much lower Reynolds numbers. Finally, some comments are made regarding the possibility of additional cascaded or isolated vortices occurring at even higher Reynolds numbers and aspect ratios.