Computational aeroelastic simulations of self-sustained pitch oscillations of a NACA0012 at transitional Reynolds numbers

The phenomenon of low amplitude self-sustained pitch oscillations in the transitional Reynolds number regime is studied numerically through unsteady, two-dimensional aeroelastic simulations. Based on the experimental data, simulations have been limited in the Reynolds number range 5.0×10⁴<Re_c<1.5×10⁵. Both laminar and URANS calculations (using the SST k–ω model with a low-Reynolds-number correction) have been performed and found to produce reasonably accurate limit cycle pitching oscillations (LCO). This investigation confirms that the laminar separation of the boundary layer near the trailing edge plays a critical role in initiating and sustaining the pitching oscillations. For this reason, the phenomenon is being labelled as laminar separation flutter. As a corollary, it is also shown that turbulence tends to inhibit their existence. Furthermore, two regimes of LCO are observed, one where the flow is laminar and separated without re-attachment, and the second for which transition has occurred followed by turbulent re-attachment. Finally, it is established that the high-frequency, shear instabilities present in the flow which lead to von Kármán vortex shedding are not crucial, nor necessary, to the maintaining mechanism of the self-sustained oscillations.     

Large-eddy and unsteady RANS simulations of a shock-accelerated heavy gas cylinder

Two-dimensional numerical simulations of the Richtmyer–Meshkov unstable “shock-jet” problem are conducted using both large-eddy simulation (LES) and unsteady Reynolds-averaged Navier–Stokes (URANS) approaches in an arbitrary Lagrangian–Eulerian hydrodynamics code. Turbulence statistics are extracted from LES by running an ensemble of simulations with multimode perturbations to the initial conditions. Detailed grid convergence studies are conducted, and LES results are found to agree well with both experiment and high-order simulations conducted by Shankar et al. (Phys Fluids 23, 024102, 2011). URANS results using a k–L approach are found to be highly sensitive to initialization of the turbulence lengthscale L and to the time at which L becomes resolved on the computational mesh. It is observed that a gradient diffusion closure for turbulent species flux is a poor approximation at early times, and a new closure based on the mass-flux velocity is proposed for low-Reynolds-number mixing.     

Highly Resolved LES and URANS of Turbulent Buoyancy-Driven Flow Within Inclined Differentially-Heated Enclosures

In the present study, both LES and unsteady RANS computations are presented, for turbulent natural convection of air inside differentially-heated rectangular tilted cavities using a finite volume code (Code_Saturne), for an aspect ratio of H/L?=?28.6 and Rayleigh number of 0.86×10⁶. Attention is focused on two angles of inclination: 15° to the horizontal with hot lower and cold upper wall, the 15° unstable case, and the mirror image of this case where the angle is the same but with a hot upper and cold lower wall, the 15° stable case. In accordance with recent experimental data, the LES computations for both the stable and unstable tilted cavities returned three-dimensional time-averaged flow fields. In the case of the unstably stratified enclosure, the flow is highly unsteady with coherent turbulent structures in the core of the enclosure. Time-averaged temperature, velocity and resolved turbulence intensities resulting from LES computations show close agreement to measured data. Subsequent comparisons of different URANS schemes with LES are used in order to explore to what extent these models are able to reproduce the large-scale unsteady flow structures. All URANS schemes have been found to be able to reproduce the 3-D unsteady flow features present in the 15° unstable cavity. However, the low-Reynolds-number model tested, as well as requiring a high resolution near-wall grid, also needed a finer grid in the core region than the high-Reynolds-number models, thus making it computationally very expensive. Flow within the 15° stable cavity also shows some 3-D features, although it is significantly less unsteady, and the URANS models tested here have been less successful in reproducing this flow pattern. The overall heat transfer is presented here for both differentially heated enclosures.     

Experimental and Theoretical Sensitivity Analysis of Turbulent Flow Past a Square Cylinder

The Siemens SGT-800 3^rd generation DLE burner fitted to an atmospheric combustion rig has been numerically investigated. Pure methane and methane enriched by 80 vol% hydrogen flames have been considered. A URANS (Unsteady Reynolds Averaged Navier-Stokes) approach was used in this study along with the k ? ω SST and the k ? ω SST-SAS models for the turbulence transport. The chemistry is coupled to the turbulent flow simulations by the use of a laminar flamelet library combined with a presumed PDF. The effect of the mesh density in the mixing and the flame region and the effect of the turbulence model and reaction rate model constant are first investigated for the methane/air flame case. The results from the k ? ω SST-SAS along with flamelet libraries are shown to be in excellent agreement with experimental data, whereas the k ? ω SST model is too dissipative and cannot capture the unsteady motion of the flame. The k ? ω SST-SAS model is used for simulation of the 80 vol% hydrogen enriched flame case without further adjusting the model constants. The global features of the hydrogen enrichment are very well captured in the simulations using the SST-SAS model. With the hydrogen enrichment the time averaged flame front location moves upstream towards the burner exit nozzle. The results are consistent with the experimental observations. The model captures the three dominant low frequency unsteady motion observed in the experiments, indicating that the URANS/LES hybrid model indeed is capable of capturing complex, time dependent, features such as an interaction between a PVC and the flame front.     

Revisiting URANS Computations of the Backward-facing Step Flow Using Second Moment Closures. Influence of the Numerics

Previous studies have shown that Unsteady Reynolds-Averaged Navier–Stokes (URANS) computations are able to reproduce the vortex shedding behind a backward-facing step. The aim of the present work is to investigate not only the quantitative predictions of the URANS methodology concerning the characteristic frequencies, but also the amplitude of the energy of the resolved eddies, by using the Elliptic Blending Reynolds Stress Model. This innovative low-Reynolds number second moment closure reproduces the non-viscous, non-local blocking effect of the wall on the Reynolds stresses, and it is compared to the standard k − ε and LRR models using wall-functions. Consistent with previous studies, in the 2D computations shown in the present article, the vortex shedding is captured with the correct Strouhal number, when second moment closures are used. To complete these previous analyses, we particularly focus here on the energy contained in the unsteady, resolved part and its dependency on the numerical method. This energy is less than 5% of the total energy and is strongly dependent on the mesh. Using a refined mesh, surprisingly, a steady solution is obtained. It is shown that this behaviour can be linked to the very small spatial oscillations at the step corner, produced by numerical dispersion, which act as perturbations that are sufficient to excite the natural mode of the shear layer, when the local Peclet number, comparing convection and diffusion effects, is high enough. This result suggests that URANS is not appropriate to quantitatively predict the amplitude of the large-scale structures developing in separated shear-layers, and that URANS results must be interpreted with care in terms of temporal variations of forces, temperatures, etc., in industrial applications using marginally fine meshes.     

Numerical simulation of the fluid–structure interaction in a tube array under cross flow at moderate and high Reynolds number

The unsteady loads in a tube bundle are studied at moderate and high Reynolds number by means of URANS and hybrid (DDES) modelling. The onset of fluid-elastic instability is analysed for different structural parameters, Scruton number and reduced velocity. The simulations have been carried out with the code NSMB (Navier–Stokes Multi Block) by using turbulence modelling methods URANS and DDES (Delayed Detached Eddy Simulation). The CEA-DIVA configuration is considered for the cylinders array for an inter-tube Reynolds number 60 000. The study is carried out for a configuration of (4×5) cylinders in static conditions as well as for the vertical free motion of one of the central cylinders in one DOF (Degree Of Freedom).The inter-tube Reynolds number is 60 000. It is found that this cylinder spontaneously displays an oscillatory motion which first corresponds to Vortex Induced Vibration (VIV), associated to a lock-in mechanism for low values of the reduced velocity and secondly develops Movement Induced Vibration, MIV, for higher values of the reduced velocity. The variation of the cylinder׳s oscillations frequency, of the unsteady loads and the structure׳s displacement are studied as a function of the reduced velocity for low and high values of the Scruton number. The increase of the phase-lag between the forces and the displacement is predicted and discussed for different Scruton number values and reduced velocities.     

Vortex mechanism of heat transfer enhancement in a channel with spherical and oval dimples

Vortex mechanism of heat transfer enhancement in a narrow channel with dimples has been investigated numerically using LES and URANS methods. The flow separation results in a formation of vortex structures which significantly enhance heat transfer on dimpled surfaces leading to a small increase in pressure loss. The heat transfer can be significantly increased by rounding the dimple edge and use of oval dimples. To get a deep insight into flow physics LES is performed for single phase flow in a channel with a spherical dimple. The instantaneous vortex formation and separation are investigated in and around the dimple area. Considered are Reynolds numbers (based on dimple print diameter) Re_D = 20,000 and Re_D = 40,000 the depth to print diameter ratio of Δ = 0.26. Frequency analysis of LES data revealed the presence of dominating frequencies in unsteady flow oscillations. Direct analysis of the flow field revealed the presence of coherent vortex structure inclined to the mean flow. The structure changes its orientation in time causing the long period oscillations with opposite-of-phase motion. Three dimensional proper orthogonal decomposition (POD) analysis is carried out on LES pressure and velocity fields to identify spatio-temporal structures hidden in the random fluctuations. Tornado-like spatial POD structures have been determined inside dimples.     

Improved Unsteady RANS Models Applied to Jet Transverse to a Pipe Flow

An unsteady RANS model is developed in order to simulate the complex situations involving both free and bounded flows. This model tuned to catch coherent flow structures is developed both in the k-ε and k-l approaches. The full 3D geometry of a round jet exiting from a reservoir into a pipe has been computed. Periodic conditions are applied in order to compare with an experiments consisting of eight jets exiting in a cross pipe flow. Improvement has been obtained with this URANS turbulence model compared to RANS and good agreement compared with experiment has been obtained. Unsteady phenomena are reproduced by the model and provide more insight into the physical properties of the flow and of the transport of a passive scalar.     

Dynamic mode decomposition of turbulent cavity flows for self-sustained oscillations

Self-sustained oscillations in a cavity arise due to the unsteady separation of boundary layers at the leading edge. The dynamic mode decomposition method was employed to analyze the self-sustained oscillations. Two cavity flow data sets, with or without self-sustained oscillations and possessing thin or thick incoming boundary layers (Re_D = 12,000 and 3000), were analyzed. The ratios between the cavity depth and the momentum thickness (D/θ) were 40 and 4.5, respectively, and the cavity aspect ratio was L/D = 2. The dynamic modes extracted from the thick boundary layer indicated that the upcoming boundary layer structures and the shear layer structures along the cavity lip line coexisted with coincident frequency space but with different wavenumber space, whereas structures with a thin boundary layer showed complete coherence among the modes to produce self-sustained oscillations. This result suggests that the hydrodynamic resonances that gave rise to the self-sustained oscillations occurred if the upcoming boundary layer structures and the shear layer structures coincided, not only in frequencies, but also in wavenumbers. The influences of the cavity dimensions and incoming momentum thickness on the self-sustained oscillations were examined.     

Dynamic mode decomposition of turbulent cavity flows for self-sustained oscillations

Self-sustained oscillations in a cavity arise due to the unsteady separation of boundary layers at the leading edge. The dynamic mode decomposition method was employed to analyze the self-sustained oscillations. Two cavity flow data sets, with or without self-sustained oscillations and possessing thin or thick incoming boundary layers (Re_D = 12,000 and 3000), were analyzed. The ratios between the cavity depth and the momentum thickness (D/θ) were 40 and 4.5, respectively, and the cavity aspect ratio was L/D = 2. The dynamic modes extracted from the thick boundary layer indicated that the upcoming boundary layer structures and the shear layer structures along the cavity lip line coexisted with coincident frequency space but with different wavenumber space, whereas structures with a thin boundary layer showed complete coherence among the modes to produce self-sustained oscillations. This result suggests that the hydrodynamic resonances that gave rise to the self-sustained oscillations occurred if the upcoming boundary layer structures and the shear layer structures coincided, not only in frequencies, but also in wavenumbers. The influences of the cavity dimensions and incoming momentum thickness on the self-sustained oscillations were examined.     

Simulating air entrainment and vortex dynamics in a hydraulic jump

The air entrainment characteristics of three separate Froude number hydraulic jumps are investigated numerically using an unsteady RANS, realizable k–ε turbulence model, with a Volume of Fluid treatment for the free surface. Mean velocity profiles, average void fraction, and Sauter mean diameter compare favorably with experimental data reported in literature. In all simulations, time-averaged void fraction profiles show good agreement with experimental values in the turbulent shear layer and an accurate representation of interfacial aeration at the free surface. Sauter mean diameter is well represented in the shear layer, and free surface entrainment results indicate bubble size remains relatively unchanged throughout the depth of the jump. Several different grid resolutions are tested in the simulations. Significant improvements in void fraction and bubble size comparison are seen when the diameter to grid size ratio of the largest bubbles in the shear layer surpasses eight. A three-dimensional simulation is carried out for one Froude number jump, showing an improvement in the prediction of entrained air and bubble size compared with two-dimensional results at a substantial increase in computation time. An analysis of three-dimensional vorticity shows a complex interaction between spanwise and streamwise vortical structures and entrained air bubbles. The jump is similar to a turbulent mixing layer, constrained by the free surface, with vortex pairing and subsequent fluctuations in free surface elevation. Downstream fluctuations of the toe are associated with a roll up of the primary spanwise vortex, fluctuations of the free surface, and counter-rotating streamwise vortex pairs. The action of these flow structures is likely responsible for the improvement in three-dimensional results.     

Vorticity dynamics in turbulence growth

The mechanisms of vorticity amplification in the formation of turbulence are investigated by means of direct numerical simulations of the Navier–Stokes equations with different initial conditions and Reynolds numbers. The simulations show good universality of the enstrophy evolution, that occurs in two stages. The first stage is dominated by the effect of vortex stretching, and it finishes with a k ^?3 power-law energy spectrum. The second stage is dominated by the action of viscosity on the small scales, and it finishes with a Kolmogorov k ^?5/3 energy spectrum.     

URANS simulations for a high-speed transom stern ship with breaking waves

An exploratory study of high-speed surface ship flows is performed to identify modelling and numerical issues, to test the predictive capability of an unsteady RANS method for such flows, to explain flow features observed experimentally, and to document results obtained in conjunction with the 2005 ONR Wave Breaking Workshop. Simulations are performed for a high-speed transom stern ship (R/V Athena I) at three speeds Froude number (Fr) = 0.25, 0.43 and 0.62 with the URANS code CFDSHIP-IOWA, which utilizes a single-phase level set method for free surface modelling. The two largest Fr are considered to be high-speed cases and exhibit strong breaking plunging bow waves. Structured overset grids are used for local refinement of the unsteady transom flow at medium speed and for small scale breaking bow and transom waves at high-speeds. All simulations are performed in a time accurate manner and an examination of time histories of resistance and free surface contours is used to assess the degree to which the solutions reach a steady state. The medium speed simulation shows a classical steady Kelvin wave pattern without breaking and a wetted naturally unsteady transom flow with shedding of vortices from the transom corner. At higher speeds, the solutions reach an essentially steady state and display intense bow wave breaking with repeated reconnection of the plunging breaker with the free surface, resulting in multiple free surface scars. The high-speed simulations also show a dry transom and an inboard breaking wave, followed by outboard breaking waves downstream. In comparison to an earlier dataset, resistance is well predicted over the three speeds. The free surface predictions are compared with recent measurements at the two lowest speeds and show good agreement for both non-breaking and breaking waves.     

Slip motion and stability of a single degree of freedom elastic system with rate and state dependent friction

We consider quasistatic motion and stability of a single degree of freedom elastic system undergoing frictional slip. The system is represented by a block (slider) slipping at speed V and connected by a spring of stiffness k to a point at which motion is enforced at speed V₀ We adopt rate and state dependent frictional constitutive relations for the slider which describe approximately experimental results of Dieterich and Ruina over a range of slip speeds V. In the simplest relation the friction stress depends additively on a term A In V and a state variable θ; the state variable θ evolves, with a characteristic slip distance, to the value ? B In V, where the constants A, B are assumed to satisfy B > A > 0. Limited results are presented based on a similar friction law using two state variables.Linearized stability analysis predicts constant slip rate motion at V₀ to change from stable to unstable with a decrease in the spring stiffness k below a critical value k_cr. At neutral stability oscillations in slip rate are predicted. A nonlinear analysis of slip motions given here uses the Hopf bifurcation technique, direct determination of phase plane trajectories, Liapunov methods and numerical integration of the equations of motion. Small but finite amplitude limit cycles exist for one value of k, if one state variable is used. With two state variables oscillations exist for a small range of k which undergo period doubling and then lead to apparently chaotic motions as k is decreased.Perturbations from steady sliding are imposed by step changes in the imposed load point motion. Three cases are considered: (1) the load point speed V₀ is suddenly increased; (2) the load point is stopped for some time and then moved again at a constant rate; and (3) the load point displacement suddenly jumps and then stops. In all cases, for all values of k:, sufficiently large perturbations lead to instability. Primary conclusions are: (1) ‘stick-slip’ instability is possible in systems for which steady sliding is stable, and (2) physical manifestation of quasistatic oscillations is sensitive to material properties, stiffness, and the nature and magnitude of load perturbations.     

Numerical investigations of the vortex interactions for a flow over a pitching foil at different stages

The fluid–structure interaction is investigated numerically for a two-dimensional flow (Re=2.5·10⁶) over a sinusoid-pitching foil by the SST (Shear Stress Transport) k–ω model. Although discrepancies in the downstroke phase, which are also documented in other numerical studies, are observed by comparing with experimental results, our current numerical results are sufficient to predict the mean features and qualitative tendencies of the dynamic stall phenomenon. These discrepancies are evaluated carefully from the numerical and experimental viewpoints.In this study, we have utilized Λ, which is the normalized second invariant of the velocity gradient tensor, to present the evolution of the Leading Edge Vortex (LEV) and Trailing Edge Vortex (TEV). The convective, pressure, and diffusion terms during the dynamic stall process are discussed based on the transport equation of Λ. It is found that the pressure term dominates the rate of the change of the rotation strength inside the LEV. This trend can hardly be observed directly by using the vorticity transport equation due to the zero baroclinic term for the incompressible flow.The mechanisms to delay the stall are categorized based on the formation of the LEV. At the first stage before the formation of the LEV in the upper surface, the pitching foil provides extra momentum into the fluid flows to resist the flow separation, and hence the stall is delayed. At the second stage, a low-pressure area travels with the evolution of the LEV such that the lift still can be maintained. Three short periods at the second stage corresponds to different flow patterns during the dynamic stall, and these short periods can be distinguished according to the trend of the pressure variation inside the LEV. The lift stall occurs when a reverse flow from the lower surface is triggered during the shedding of the LEV. For a reduced frequency k_f=0.15, the formation of the TEV happens right after the lift stall, and the lift can drop dramatically. With a faster reduced frequency k_f=0.25, the shedding of the LEV is postponed into the downstroke, and the interaction between the LEV and TEV becomes weaker correspondingly. Thus, the lift drops more gently after the stall. In order to acquire more reliable numerical results within the downstroke phase, the Large Eddy Simulation (LES), which is capable of better predictions for the laminar-to-turbulent transition and flow reattachment process, will be considered as the future work.     

CFD modelling of the cross-flow through normal triangular tube arrays with one tube undergoing forced vibrations or fluidelastic instability

A CFD methodology involving structure motion and dynamic re-meshing has been optimized and applied to simulate the unsteady flow through normal triangular cylinder arrays with one single tube undergoing either forced oscillations or self-excited oscillations due to damping-controlled fluidelastic instability. The procedure is based on 2D URANS computations with a commercial CFD code, complemented with user defined functions to incorporate the motion of the vibrating tube. The simulation procedure was applied to several configurations with experimental data available in the literature in order to contrast predictions at different calculation levels. This included static conditions (pressure distribution), forced vibrations (lift delay relative to tube motion) and self-excited vibrations (critical velocity for fluidelastic instability). Besides, the simulation methodology was used to analyze the propagation of perturbations along the cross-flow and, finally, to explore the effect on the critical velocity of the Reynolds number, the pitch-to-diameter ratio and the degrees of freedom of the vibrating cylinder.     

Detached eddy simulation of flow around two wall-mounted cubes in tandem

We investigate the performance of unsteady Reynolds-averaged Navier–Stokes (URANS) computation and various versions of detached eddy simulation (DES) in resolving coherent structures in turbulent flow around two cubes mounted in tandem on a flat plate at Reynolds number (Re) of 22,000 and for a thin incoming boundary layer. Calculations are carried out using four different coherent structure resolving turbulence models: (1) URANS with the Spalart–Allmaras model; (2) the standard DES [Spalart, P.R., Jou, W.H., Strelets, M., Allmaras, S.R., 1997. Comments on the feasibility of LES for wings, and on a hybrid RANS/LES approach. In: Liu, C., Liu, Z., (Eds.), Advances in DNS/LES. Greyden Press, Columbus, OH]; (3) the Delayed DES (DDES); and (4) the DES with a low-Re modification (DES-LR) [Spalart, P., Deck, S., Shur, M., Squires, K., Strelets, M., Travin, A., 2006. A new version of detached eddy simulation, resistant to ambiguous grid densities. Theor. Comput. Fluid Dyn. 20 (3), 181–195]. The grid sensitivity of the computed solutions is examined by carrying out simulations on two successively refined grids. The computed results for all cases are compared with the experimental measurements of Martinuzzi and Havel [Martinuzzi, R., Havel, B., 2000. Turbulent flow around two interfering surface-mounted cubic obstacles in tandem arrangement. ASME J. Fluids Eng. 122, 24–31] for two different cube spacings. All turbulence models reproduce essentially identical separation of the approach thin boundary layer and yield an unsteady horseshoe vortex system consisting of multiple vortices in the leading edge region of the upstream cube. Significant discrepancies between the URANS and all DES solutions are observed, however, in other regions of interest such as the shear layers emanating from the cubes, the inter-cube gap and the downstream wake. Regardless of the grid refinement, URANS fails to capture key features of the mean flow, including the second horseshoe vortex in the upstream junction and recirculating flow on the top surface of the downstream cube for the large cube spacing, and underestimates significantly turbulence statistics in most regions of the flow for both cases. On the coarse mesh, all three DES approaches appear to yield very similar results and fail to reproduce the second horseshoe vortex. The standard DES and DDES solutions obtained on the fine meshes are essentially identical and both suffer from premature switching to unresolved DNS, due to the mis-interpretation of grid refinement as wall proximity, which leads to spurious vortices in the inter-cube region. Numerical solutions show that the low-Re modification (DES-LR) is critical prerequisite in DES on the ambiguously fine – not fine enough for full LES – mesh to prevent excessive nonlinear drop of the subgrid eddy viscosity in low cell-Re regions like in the inter-obstacle gap. Mean flow quantities and turbulence statistics obtained with DES-LR on the fine mesh are in good overall agreement with the measurements in most regions of interest for both cases.     

Scrutinizing URANS in Shedding Flows: The Case of Cylinder in Cross-Flow in the Subcritical Regime

To unravel the widespread perception that the RANS (Reynolds-averaged Navier-Stokes) concept is unreliable in predicting the dynamics of separated flows, we assessed the performance of two RANS closure levels, the linear eddy-viscosity (LEVM) and the second-moment (Reynolds stress, RSM) approaches in a massively separated generic flow over a bluff body. Considered is the canonical, zero-turbulence, cross-flow over an infinite cylinder with reference to our LES and the available DNS and experiments at two Reynolds numbers, Re = 3.9 × 10³ and 1.4 × 10⁵, both within the sub-critical regime with laminar separation. Both models capture successfully the vortex shedding frequency, but the low frequency modulations are detected only by the RSM. At high Reynolds numbers the RSM is markedly superior to the LEVM showing very good agreement with the LES and experimental data. The RSM, accounting naturally for the stress anisotropy and phase lag between the stress and strain eigenvectors, is especially successful in reproducing the growth rate of the turbulent kinetic energy in the initial shear layer which proved to be crucial for accurate prediction of the separation-induced transition. A scrutiny of the unsteady RANS (URANS) stress terms based on the conditional phase-averaged LES data shows a remarkable similarity of the normalized coherent and stochastic (modeled) stress components for the two Reynolds numbers considered. The mixed (cross) correlations, while non-negligible at the low Re number, diminish fast relative to the stochastic ones with increasing Reynolds number and, in the whole, are not significant to undermine the URANS concept and its applicability to high Re flows of industrial relevance.