Natural Vortex Shedding Frequency Research Articles

Control of the von Kármán vortex street with focusing and vectoring of jet using synthetic jet array

In the present study, a novel flow control technique based on jet focusing and vectoring from a synthetic jet array (SJA) for controlling the wake of a bluff body is proposed and demonstrated. A numerical investigation into the flow past a square cylinder modified by the SJA has been carried out at a free stream Reynolds Number of 100. The SJA consists of four independently controlled synthetic jet actuators operating at a peak velocity of eight times the free stream and fifteen times the natural vortex shedding frequency of the square cylinder. The SJA is operated in two different regimes; a focusing regime involving phase delay (Δφ) with non-linear variation between the actuators and a vectoring regime with a linear phase delay without changing the geometric or operating parameters of the SJA. It has been found that jet focusing is able to reduce the coefficient of drag by as much as 43% for Δφ=90°. Focusing is also observed to reduce the fluctuations in the wake velocity with the maximum reduction in fluctuations also corresponding to Δφ=90°. Jet vectoring is able to deflect the von Kármán vortex street in a singular direction along with shifting of the front stagnation point with maximum deflection for Δφ=60°. Furthermore, vectoring leads to an asymmetry in the wake velocity field with the shifting of the velocity deficit region in the direction of the vectoring along with an asymmetry in the wake velocity fluctuations. This novel approach toward synthetic jet induced active flow control allows for greater manipulation of the flow field characteristics of bluff bodies than present methods with applications in areas of underwater and micro air vehicle maneuvering, automobile, and building aerodynamics among others.

Vortex synchronization-enabled heat-transfer enhancement in a channel with backward- and forward-facing steps

A channel with one backward-facing step and one forward-facing step is a typical configuration in engineering applications. In the channel, good heat transfer performance is often required, and the enhancement is usually achieved by employing different passive control methods, such as modification of geometric configuration or application of nanofluid. However, the other control method, i.e., active flow control (AFC), which is likely more effective, has been rarely applied in such a scenario. This study aims to bridge this gap by exploring how a rigid plate affects the heat transfer of the channel. The plate either is stationary or actively rotates, corresponding to passive flow control or AFC. The influences of the horizontal position of the plate (S) and its orientation angle (θ) on the heat transfer performance are studied when the plate is stationary to provide a baseline. Compared to the baseline, the effects of S, θ, and the rotation frequency (fr) are revealed when the plate undergoes a sinusoidal rotation. Such a thermo-fluid dynamic problem is numerically simulated by the immersed-boundary lattice Boltzmann method. The results show that the plate can improve the heat transfer performance no matter whether it rotates or not, compared to the case without a plate. The rotating plate outperforms the stationary one when θ and fr are properly chosen at each S. Substantial improvement can be achieved when vortex synchronization or resonance occurs in the channel, i.e., when the natural vortex shedding frequency is close or equal to fr.

Significant influence of fluid viscoelasticity on flow dynamics past an oscillating cylinder

This study focuses on numerically investigating the impact of fluid viscoelasticity on the flow dynamics around a transversely forced oscillating cylinder operating in the laminar vortex shedding regime at a fixed Reynolds number of $Re = 100$ . Specifically, we explore how fluid viscoelasticity affects the boundary between the lock-in and no lock-in regions and the corresponding wake characteristics compared with a simple Newtonian fluid. Our findings reveal that fluid viscoelasticity enables the synchronization of the vortex street with the cylinder motion at lower oscillation frequencies than those required for a Newtonian fluid. Consequently, the lock-in region boundary for a viscoelastic fluid differs from that of a Newtonian fluid and expands in the non-dimensional cylinder oscillation amplitude and frequency parameter space. In the primary synchronization region, the wake of a Newtonian fluid exhibits ‘2S’ (two single vortices) and ‘P+S’ (a pair of vortices and a single vortex) shedding modes. In contrast, a ‘2P’ (two pairs of vortices) vortex mode is observed for a viscoelastic fluid within the same region. To gain a deeper understanding of the differences in the coherent flow structures and their associated frequencies between the two fluids, we employ the data-driven reduced-order modelling technique, known as the dynamic mode decomposition (DMD) technique. Utilizing this technique, we successfully extract and visualize the two competing fundamental frequencies (cylinder oscillation and natural vortex shedding frequencies) and their associated flow structures in the case of the no lock-in state, whereas only the dominant cylinder oscillation frequency and associated flow structure in the case of the lock-in state. Furthermore, we propose that the presence of excess strain resulting from the stretching of polymer molecules in viscoelastic fluids leads to a distinct difference in the wake structure compared with Newtonian fluids. This observation aligns with the findings obtained from the $Q$ -criterion and vorticity transport analysis of the wake.

Self-excited vortex-acoustic lock-in in a bluff body combustor

We investigate self-excited vortex-acoustic lock-in in a bluff body combustor. The bluff body not only anchors the flame but also sheds vortices. Mutual interaction between vortex shedding and the combustor acoustic field leads to lock-in. A lower-order model for vortex shedding is coupled to the acoustic equations to obtain a set of discrete dynamical maps, which are then solved numerically. Lock-in is identified when a definite phase relationship between vortex shedding and acoustic field remains unaltered with time. The common frequency during (phase) lock-in is either close to the natural acoustic or vortex shedding frequencies, accordingly termed A- or V-lock-in, respectively. Instability and amplitude suppression occur with most parts of the A- and V-lock-in regions, respectively. Furthermore, we relate the lock-in and pre-lock-in regimes observed in our previous experiments (Singh & Mariappan, Combust. Sci. Technol., 2019, pp. 1–29) to A- and V-lock-in phenomena, respectively. Among the two lock-in phenomena, combustion instability is favoured by $1:1$ A-lock-in, whose boundaries in the parameter space can be obtained from a forced response of the vortex shedding process. Finally, we discuss the bifurcations leading to lock-in.

Modification of the wake of a wall-mounted bathymetry obstacle submitted to waves opposing a tidal current

In this paper, the effects of regular waves propagating against the current on the wake of a wide bathymetric obstacle are experimentally investigated. For that purpose, a square cylinder of height representing 1/8th of the water column, mounted perpendicular to the flow, is used. Both Reynolds and Froude numbers are close to the ones encountered at sea and swell properties (amplitude, frequency and wavelength) which are typical conditions in the English Channel. To study this complex hydrodynamic interaction, PIV measurements are used to capture the velocity field around and in the wake of the cylinder. The wake of this wide cylinder has been extensively studied and is characterised by its direction towards the water surface and by energetic and wide vortices arising from the sheared layer at a Strouhal number of approximatively 0.06, after a merging of smaller eddies. In this work, the focus is on how surface gravity waves, propagating against the current, modify the generation of these turbulent structures. The results show that the cylinder wake is strongly modified in the presence of waves, even if, on average, the main wake structure is globally conserved. The waves act both on the vortex generation and the wake development with modifications in terms of turbulent energy distribution and level, wake recovery and ascent rate. The vortex shedding becomes highly energetic and well-organised from the cylinder position. Its return period is extremely regular and corresponds to half of the wave frequency which is really close to the natural vortex shedding frequency. That results in a doubling of the statistical length scale of the vortices.

Wind energy generation by forced vortex shedding

Oscillating wind energy conversion systems that rely on vortex-induced vibrations (VIV) suffer from low energetic efficiency, but active flow control can produce significant improvements. This research investigates wind energy generation resulting from alternating slot blowing (ASB) on a circular cylinder, experimentally and theoretically. On the basis of measured peak lift coefficients, a low momentum coefficient (<0.04) excitation regime and a high momentum coefficient (>0.04) forcing regime were identified. In the excitation regime, a clear amplitude peak was observed close to the nominal natural vortex shedding frequency. In the forcing regime, separation or circulation control time-scales resulted in mild peaks at approximately half of the natural shedding frequency. In all regimes, the natural vortex shedding amplitudes were exceeded; in particular, by up to a factor of three in the forcing regime. Using a mathematical simulation model, it was shown that an ASB system produces 3.8 times higher net peak power coeffcients − comparable to small wind turbines − with a 7.3 times greater bandwidth than a conventional VIV energy harvester. Economic benefits can potentially be realized by significantly upscaling the system and adding a second degree-of-freedom. Future research will focus on improved slot design, supercritical flow excitation and forcing, and non-linear mathematical modelling of the dynamical system.

Investigations on the effects of structural damping on vortex-induced vibration response of an airfoil at a high angle of attack via the aero-damping map

Large-scale modern wind turbines at standstill are prone to vortex-induced vibrations. In this study, we propose the use of the aero-damping map to investigate the complex vibration responses of the wind turbine airfoil at 90° of attack angle with different levels of structural dampings. The vibration amplitude and response frequency in the lock-in condition and soft lock-in conditions agree well with the contour line on which the sum of aerodynamic damping and structural damping is equal to zero. The mechanism of frequency soft lock-in is explored from the aspect of energy transfer that when the equilibrium state cannot be maintained at the natural frequency due to high structural damping, the system locks to a frequency between the natural frequency and vortex shedding frequency of the stationary airfoil to achieve lower aerodynamic damping and more energy absorption from the air. The transient response of the beat vibration is also investigated with the aero-damping map combined with the dynamic mode decomposition method. It is found that the lock-in mode and von Kármán mode coexist in the unsteady flow field during beat vibration. The competition between the two modes causes the system to be in an intermittent state of alternating frequency lock-in stage with lower aerodynamic damping and unlock-in stage with higher aerodynamic damping, hence resulting in the amplitude amplification and attenuation alternately.

Interaction between coherent and turbulent oscillations in non-reacting and reacting wake flows

The presence of large-scale coherent structures can significantly impact the dynamics of a turbulent flow field and the behaviour of a flame stabilized in that flow. The goal of this study is to analyse how increasing free-stream turbulence can change the response of the flow to longitudinal acoustic excitation of varying amplitudes. We study the flow in the wake of a cylindrical bluff body at both non-reacting and reacting conditions, as the presence of a flame can significantly alter the global stability of the flow. The frequency of longitudinal acoustic excitation is set to match the natural frequency of anti-symmetric vortex shedding for this configuration and we vary the free-stream turbulence using perforated plates upstream of the bluff body. The results show that varying the level of free-stream turbulence can influence not only the amplitude of the coherent flow response, but also the symmetry of vortex shedding in the presence of longitudinal acoustic excitation. Increasing the turbulence intensity can fundamentally change the structure of the time-averaged flow and can directly impact the coherent flow response in two ways. First, increasing turbulence intensity can enhance the amplitude of the natural anti-symmetric vortex shedding mode in the wake. Second, increasing turbulence intensity weakens the symmetric response of the flow to the longitudinal acoustic excitation. In the non-reacting and reacting conditions, both symmetric and anti-symmetric modes are present and are characterized using a spectral proper orthogonal decomposition. We see evidence of interaction between the symmetric and anti-symmetric modes, which leads to an interference pattern in the coherent vorticity response in the shear layers. We conclude by presenting a conceptual model for the influence that turbulence has on these flows.

Impact of turbulence on flame brush development of acoustically excited rod-stabilized flames

Coherent structures, such as those arising from hydrodynamic instabilities or excited by thermoacoustic oscillations, can significantly impact flame structure and, consequently, the nature of heat release. The focus of this work is to study how coherent oscillations of varying amplitudes can impact the growth of the flame brush in a bluff-body stabilized flame and how this impact is influenced by the free stream turbulence intensity of the flow approaching the bluff body. We do this by providing external acoustic excitation at the natural frequency of vortex shedding to simulate a highly-coupled thermoacoustic instability, and we vary the in-flow turbulence intensity using perforated plates upstream of the flame. We use high-speed stereoscopic particle image velocimetry to obtain the three-component velocity field and we use the Mie-scattering images to quantify the behavior of the flame edge. Our results show that in the low-turbulence conditions, presence of high-amplitude acoustic excitation can cause the flame brush to exhibit a step-function growth, indicating that the presence of strong vortical structures close to the flame can suppress flame brush growth. This impact is strongly dependent on the in-flow turbulence intensity and the flame brush development in conditions with higher levels of in-flow turbulence are minimally impacted by increasing amplitudes of acoustic excitation. These findings suggest that the sensitivity of the flow and flame to high-amplitude coherent oscillations is a strong function of the in-flow turbulence intensity.

On the wake response of a square cylinder subjected to an inline nonharmonic forcing

In this work, we study the effect of two spectrally simple nonharmonic inline forcing waveforms on the wake response of a square cylinder. The characteristics of wake response under nonharmonic forcing were studied and compared to wake response under harmonic forcing for a constant Reynolds number 100. By varying the relative strength of the spectral components and the secondary frequency of the nonharmonic forcing of the inflow, an asymmetric modulated wake and an asymmetric 2P + 2S wake were observed for a two-tone nonharmonic forcing. These modes combine the features of the wake response of individual spectral components. In some regions of the parameter space, the wake response did not correlate well with the spectral components of harmonic forcing. An amplitude-modulated forcing was also studied. For amplitude-modulated forcing, the additional frequency component did not affect the wake structure. This observation indicates that the wake may act as a low pass filter for inline flow frequency components above the natural vortex shedding frequency.

Stochastic dynamics of vortex-acoustic lock-in: stochastic bifurcation and resonance

We perform a theoretical study to understand vortex-acoustic lock-in in the presence of an additive Gaussian white noise using a lower-order model. The acoustic field is realized as an external harmonic excitation. Frequency, harmonic excitation and noise amplitudes are varied over ranges encountered in practical and lab-scale combustors. Probability density functions (PDFs) for shedding time periods and phase instants are obtained in the Fokker–Planck framework. Unlike the general scenario, where stochastic bifurcation is identified by a qualitative change in the stationary PDFs, in this case, stochastic lock-in is identified by a qualitative change in the spectrum of the transition probability matrix. The effect of the noise is to reduce the extent of lock-in while preserving the underlying deterministic dynamical features. Although various orders of lock-in are identified, 1 : 1 stochastic lock-in with the excitation frequency close to the natural vortex shedding frequency is found to be the most favourable condition for combustion instability to occur. We show that lock-in can be accompanied by both instability and amplitude suppression. Stochastic resonance is also observed; however, its contribution to instability is marginal.

Numerical investigation on synthetic jet control for flow over two-dimensional square cylinder

Flow around a square cylinder with synthetic jet located at the cylinder's rear surface is investigated numerically. The numerical analysis is carried out at low Reynolds number of 100 with the excitation frequency normalized by the natural vortex-shedding frequency fe/f0 varying from 1 to 6 and the peak velocity normalized by the free-stream velocity Ue/U∞ varying from 1 to 5. It is found that drag reduction can be achieved by applying synthetic jet control, and a maximum drag reduction rate of about 29.2% is obtained for the case of Ue/U∞= 5 and fe/f0 = 3. For the wake vortex structures under present synthetic jet control, four typical patterns, named as no-effect mode, alternative shedding mode with half frequency lock-in, asymmetric shedding mode of unilateral deflection and symmetric shedding mode, are observed.

Transition to Periodic Behaviour of Flow Past a Circular Cylinder under the Action of Fluidic Actuation in the Transitional Regime

This study focuses on the numerical investigation of the underlying mechanism of transition from chaotic to periodic dynamics of circular cylinder wake under the action of time-dependent fluidic actuation at the Reynolds number = 2000. The forcing is realized by blowing and suction from the slits located at ±90∘ on the top and bottom surfaces of the cylinder. The inverse period-doubling cascade is the underlying physical mechanism underpinning the wake transition from mild chaos to perfectly periodic dynamics in the spanwise-independent, time-dependent forcing at twice the natural vortex-shedding frequency.

Acoustic analysis of a forced-oscillating cylinder in flow using a hybrid method

An acoustic analysis based on a hybrid methodology is performed for a transversely forced-oscillating circular cylinder with the aim of understanding the sound generation mechanism. The sound is predicted both in moving-observer and moving-medium systems using the unsteady near-field flow solutions as input. Different forms of integration formulation, the effect of placement of the data surface, the impact of surface closure and averaging among various closed surfaces are investigated over a range of F=0.2 to 1.4 where F is the frequency ratio of cylinder oscillation to the natural vortex shedding frequency. The results show that the sound is mainly radiated from the cylinder surface, induced by the unsteady aerodynamic loading and cylinder movement. The impact of the hydrodynamic structures of the cylinder wake on the predicted sound signal varies in different regimes of F. For 0.2≤F≤0.7, the hydrodynamic perturbation introduces spurious noise sources on the intersecting portion of the permeable data surface. For 0.8≤F≤1.05, the mean flow of the wake affects sound propagation through a convective effect, which must be taken into account in hybrid prediction. For 1.1≤F≤1.4, the effect of the wake flow on the sound generation, propagation and prediction is small. The accuracy of hybrid prediction can be significantly improved by both the open permeable surface and surface-averaging.

Numerical investigation of the separated and reattaching flow over a 5:1 rectangular cylinder in streamwise sinusoidal flow

Three-dimensional Large-Eddy Simulations (3D LES) are performed to study the separated and reattaching flow over a 5:1 rectangular cylinder in streamwise sinusoidal flow. The ratio of inflow frequency to natural vortex-shedding frequency is varied from 0 to 6 ​at five amplitudes up to 60% of the time-mean velocity. The effect on the two-dimensional (2D) and three-dimensional (3D) characteristics of the separation bubble and surface pressures are reported. The relationship between the time-averaged flow field and the statistics of the chordwise pressure distribution are obtained firstly. The flow-pressure relationships are qualitatively similar in uniform and sinusoidal flows. Then, the amplitude-frequency domain of the sinusoidal flow is divided into three regimes to describe the effects on the mean reattachment length. The simulations capture the interferences between leading-edge vortices and trailing edge vortices for longer separation bubbles. However, the interactions are not observed in shorter separation bubbles due to the diffusions of leading-edge vortices. Finally, the flow three-dimensionality is studied by performing proper orthogonal decomposition (POD) and spanwise correlation analysis on the surface pressures. The oncoming flow oscillations correspond to the most energetic structures dominating the pressure field, which are constant in the spanwise direction. While the local vortical structures mainly influence the 3D features of the flow dynamics.

Numerical Simulation of Wake Deflection Control around NACA0012 Airfoil Using Active Morphing Flaps

This study demonstrates an active flow control for deflecting a direction of wake vortex structures behind a NACA0012 airfoil using an active morphing flap. Two-dimensional direct numerical simulations are performed for flows at the chord Reynolds number of 10,000, and the vortex pattern in the controlled and noncontrolled wakes as well as the effect of an actuation frequency on the control ability are rigorously investigated. It is found that there is an optimum actuation-frequency regime at around F + = 2.00 which is normalized by the chord length and freestream velocity. The wake vortex pattern of the well-controlled case is classified as the 2P wake pattern according to the Williamson’s categorization [1] [2], where the forced oscillation frequency corresponds to the natural vortex shedding frequency without control. The present classification of wake vortex patterns and finding of the optimum frequency regime in the wake deflection control can lead to a more robust design suitable for vortex-induced-vibration (VIV) related engineering systems.

Heat transfer in flow around a rotary oscillating cylinder at a high subcritical Reynolds number: A computational study

We studied numerically the heat transfer in flow over a rotationally oscillating cylinder at a subcritical Reynolds number (Re=1.4×105) that is an order of magnitude higher than previously reported in the literature. This paper is a follow-up of the earlier study of hydrodynamics and drag force in a range of forcing frequencies and amplitudes (Palkin et al., 2018). This time we focus on heat transfer and its correlation with the observed flow field and vortical patterns. Four forcing frequencies f=fe/f0=0,1,2.5,4 for two forcing amplitudes Ω=ΩeD/2U∞=1 and 2 are considered, where f0 is the natural vortex-shedding frequency, U∞ the free-stream velocity and D the cylinder diameter. The parametric study was performed by solving three-dimensional unsteady Reynolds-averaged Navier–Stokes (URANS) equations closed by a wall-integrated second-moment (Re-stress) model, verified earlier by Large-eddy simulations and experiments in several reference cases including flows over a stagnant, as well as rotary oscillating cylinders at the same Re number. The thermal field, treated as a passive scalar, was obtained from the simultaneous solution of the energy equation, closed by the standard (GGDH) anisotropic eddy-diffusivity model. The computations showed that for the unforced cylinder heat transfer is characterized by very high local rates due to a strong thinning of the thermal boundary layer as a result of the impact and interactions of large coherent structures with the wall. The overall average Nusselt number does not change much for the forced cylinder but its time-averaged, phase-averaged and instantaneous circumferential profiles show some profound differences compared to the stationary cylinder. The distribution of Nu on the back surface becomes more uniform with less frequent occurrence of high values, especially for the higher frequencies f=2.5 and f=4. This is attributed to diminishing of the mean-recirculation zone as well as to the overall suppression of turbulent fluctuations. The rotary oscillation of the cylinder appears potentially efficient in achieving a more uniform circumferential distribution of Nu and avoiding local overheats and hot spots.

Lock-in phenomenon of vortex shedding in oscillatory flows: an analytical investigation pertaining to combustors

An analytical investigation is performed to understand the lock-in phenomenon, observed in vortex shedding combustors. Several aeroengine afterburners and ramjets use a bluff body to stabilize the flame. The bluff body sheds vortices. During the occurrence of high-amplitude combustion instability, the frequency of vortex shedding locks in to the frequency of the chamber acoustic field. This phenomenon is termed vortex-acoustic lock-in. In general, there is a two-way coupling between the vortex shedding process and the acoustic field, making analytical investigation difficult. Since the frequency of the latter remains largely unaltered, performing an investigation to study the response of vortex shedding to external excitation not only allows one to gain insights, but also make the problem analytically tractable. We begin with a lower-order model available in the literature to describe the vortex shedding process in non-reacting flows, arising from sharp corners in the presence of upstream velocity excitation. The continuous time domain model is transformed to a discrete map, which connects the time instances of two successive vortex shedding events. The frequency and amplitude of excitation are varied to study the instantaneous vortex shedding time period, as the response of the system. In the absence of forcing, the iterates of the map form a period-1 solution with the frequency equalling the natural vortex shedding frequency. On increasing the amplitude of excitation, quasi-periodic behaviour of the iterates is observed, followed by a period-1 lock-in solution, where vortex shedding occurs at the excitation frequency. On further increasing the amplitude, de-lock-in occurs. From the map, an analytical solution is extracted, which represents the lock-in state. The condition and thereby the region in the frequency–amplitude parameter space where a general$p:1$lock-in occurs is then identified. Several important analytical expressions, such as for (1) critical threshold frequency above which lock-in occurs, (2) boundary of lock-in region in the parameter space, that are of direct importance to the design of quieter combustors are obtained. The study also identifies the transition of higher-order$p:1$to$1:1$lock-in state, through a series of lock-in and de-lock-in steps, whose occurrence could be verified from future experiments.

Transient cross flow and heat transfer over a rotationally oscillating cylinder subjected to gust impulse

Numerical analysis is carried out to investigate the effect of an upstream gust-impulse on the transient fluid flow and associated convective heat transfer from a two dimensional rotationally oscillating circular cylinder. Three distinct oscillation scenarios based on the oscillation amplitude are investigated. Forcing frequency for each rotational amplitude is varied such that the system remains within the lock-on state. Further, three different gust scenarios are chosen by relating the gust frequency, StG, to the natural vortex shedding frequency in the streamlined flow for a fixed Reynolds number of 110 at Prandtl number of 7. Balances for mass, momentum, and energy are solved by considering incompressible, viscous fluid subjected to no-slip CWT boundary at the cylinder wall. Phase diagrams, the spatiotemporal mean of the pressure coefficient and Nusselt number i.e. CP‾‾ and Nu‾‾, and vorticity and temperature contours are presented and discussed in detail. Temporal evolution of lift coefficient (CL) and Lissajous curves of lift coefficient against drag coefficient depict the perturbing effect of the gust-impulse. Time averaged drag coefficient (CD‾) and RMS values of lift coefficient (CL-RMS) show contrasting shifts in their peak values as the rotational-oscillation amplitude is increased. In addition to causing a transient phase difference between the cylinder angular velocity and the lift coefficient, introduction of the gust-impulse in the flow domain induces secondary frequencies causing transient amplification of the lift force which tends to make the system temporarily unstable by shifting it out of the lock-on state. The induced phase difference causes short term shear layer stretching and early vortex detachment around the cylinder surface thereby affecting the overall flow and the resultant convective heat transfer physics in the system. Moreover, a regression correlation is presented relating the oscillation amplitude, oscillation forcing Strouhal frequency and gust Strouhal frequency to the spatiotemporal averaged Nusselt number.

Fully Implicit Chebyshev Time-Spectral Method for General Unsteady Flows

An efficient fully implicit Chebyshev time-spectral method (TSM) is developed to solve general unsteady flows with or without periodicity. High accuracy of the Chebyshev time-spectral operator is demonstrated by simulating the one-dimensional shock motion at constant speed. A space-time lower–upper symmetric Gauss–Seidel (LU-SGS) fully implicit scheme with multigrid acceleration is used to efficiently solve the equations resulting from the Chebyshev TSM. The implicit temporal coupling term is properly split so that the LU-SGS sweeps are extended to the time domain. For cases in which the physical time step is small, a modification to the implicit scheme is suggested to enhance numerical stability. Computational results of nonperiodic flows over a pitching airfoil are presented for different physical time steps and various numbers of Chebyshev collocation points. The fully implicit Chebyshev TSM is also tested to predict the natural vortex shedding frequency for a stationary circular cylinder. This efficient approach can be easily applied to the time-accurate computation of general nonperiodic unsteady flows with a given initial condition.