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This paper deals with bifurcation analysis methods based on the asymptoticnumerical method. It is used to investigate 3dimensional (3D) instabilities in a sudden expansion. To do so, highperformance computing is implemented in ELMER, ie, an opensource multiphysical software. In this work, velocitypressure mixed vectors are used with asymptoticnumerical method–based methods, remarks are made for the branchswitching method in the case of symmetrybreaking bifurcation, and new 3D instability results are presented for the sudden expansion ratio, ie, E=3. Critical Reynolds numbers for primary bifurcations are studied with the evolution of a geometric parameter. New values are computed, which reveal new trends that complete a previous work. Several kinds of bifurcation are depicted and tracked with the evolution of the spanwise aspect ratio. One of these relies on a fully 3D effect as it breaks both spanwise and topbottom symmetries. This bifurcation is found for smaller aspect ratios than expected. Furthermore, a critical Reynolds number is found for the aspect ratio, ie, A_{i}=1, which was not previously reported. Finally, primary and secondary bifurcations are efficiently detected and all postbifurcated branches are followed. This makes it possible to plot a complete bifurcation diagram for this 3D case.
This paper deals with bifurcation analysis methods based on the Asymptotic Numerical Method. Remarks are made for the branchswitching method in the case of symmetrybreaking bifurcation and new threedimensional instabilities results are presented for the sudden expansion ratio E=3. Critical Reynolds numbers for primary bifurcations are studied and several kinds of bifurcation are depicted and tracked with the evolution of the spanwise aspect ratio, moreover primary and secondary bifurcations are efficiently detected and all postbifurcated branches are followed.
No abstract is available for this article.
In this paper, the finite element method with new spherical Hankel shape functions is developed for simulating twodimensional incompressible viscous fluid problems. In order to approximate the hydrodynamic variables, the finite element method based on new shape functions is reformulated. The governing equations are the Navier–Stokes equations solved by the finite element method with the classic Lagrange and spherical Hankel shape functions. The new shape functions are derived using the first and second kind of Bessel functions. In addition, these functions have properties such as piecewise continuity, and etc. For enrichment of Hankel radial basis functions, polynomial terms are added to the functional expansion that only employs spherical Hankel radial basis functions in the approximation. Also, the participation of spherical Bessel function fields has enhanced the robustness and efficiency of the interpolation. To demonstrate the efficiency and accuracy of these shape functions, four benchmark tests in fluid mechanics are considered. Then, the present model results are compared with the classic finite element results and available analytical and numerical solutions. The results show that the proposed method, even with less number of elements, is more accurate than the classic finite element method.
A new finitevolume flow solver based on the hybrid Cartesian immersed boundary (IB) framework is developed for the solution of highspeed inviscid compressible flows. The IB method adopts a sharpinterface approach, wherein the boundary conditions are enforced on the body geometry itself. A key component of the present solver is a novel reconstruction approach, in conjunction with inverse distance weighting, to compute the solutions in the vicinity of the solidfluid interface. We show that proposed reconstruction leads to secondorder spatial accuracy while also ensuring that the discrete conservation errors diminish linearly with grid refinement. Investigations of supersonic and hypersonic inviscid flows over different geometries are carried out for an extensive validation of the proposed flow solver. Studies on cylinder liftoff and shape optimisation in supersonic flows further demonstrate the efficacy of the flow solver for computations with moving and shapechanging geometries. These studies conclusively highlight the capability of the proposed IB methodology as a promising alternative for robust and accurate computations of compressible fluid flows on nonconformal Cartesian meshes.
A new sharpinterface immersed boundary method based on inversedistance weighting reconstruction is proposed for highspeed compressible flows. The discrete conservation errors from the approach are shown to be finite but diminishing with grid refinement. The solver is applied to several flow problems in supersonic and hypersonic flows including shape optimisation and demonstrates the efficacy of the methodology.
This work explores an alternative approach to computing sensitivity derivatives of functionals, with respect to a broader range of control parameters. It builds upon the complementary character of Riemann problems that describe the Euler flow and adjoint solutions. In a previous work, we have discussed a treatment of the adjoint boundary problem, which made use of such complementarity as a means to ensure wellposedness. Here, we show that the very same adjoint solution that satisfies those boundary conditions also conveys information on other types of sensitivities. In essence, then, that formulation of the boundary problem can extend the range of applications of the adjoint method to a host of new possibilities.
This work presents a novel application of the General Theory of Sensitivity Analysis, as developed by D.G. Cacuci, to Euler compressible flows. It adopts the continuous form of the adjoint method and addresses both physical and adjoint boundary conditions in terms of complete and complementary Riemann problems. This approach enables one to compute sensitivity derivatives other than those related to geometry optimization, with respect to a large variety of objective functionals.
In a previous paper, the authors presented an elemental enriched space to be used in a finiteelement framework (EFEM) capable of reproducing kinks and jumps in an unknown function using a fixed mesh in which the jumps and kinks do not coincide with the interelement boundaries. In this previous publication, only scalar transport problems were solved (thermal problems). In the present work, these ideas are generalized to vectorial unknowns, in particular, the incompressible NavierStokes equations for multifluid flows presenting internal moving interfaces. The advantage of the EFEM compared with global enrichment is the significant reduction in computing time when the internal interface is moving. In the EFEM, the matrix to be solved at each time step has not only the same amount of degrees of freedom (DOFs) but also the same connectivity between the DOFs. This frozen matrix graph enormously improves the efficiency of the solver. Another characteristic of the elemental enriched space presented here is that it allows a linear variation of the jump, thus improving the convergence rate, compared with other enriched spaces that have a constant variation of the jump. Furthermore, the implementation in any existing finiteelement code is extremely easy with the version presented here because the new shape functions are based on the usual finiteelement method shape functions for triangles or tetrahedrals, and once the internal DOFs are statically condensed, the resulting elements have exactly the same number of unknowns as the nonenriched finite elements.
An Elemental Enriched Space capable to reproduces kinks and jumps in the velocity and pressure field on internal interfaces which does not match with the Finite Element mesh is presented. The enriched space, which mitigates the lack of continuity required by the weak form, is evaluated for incompressible and twophase fluid mechanic problems. The condensing strategy employed does not change either: the total number of degree of freedom nor the matrix graph, leading to a significant computer time reduction, mainly when internal interfaces move though a fixed FE mesh.
SUMMARY
In this study, a depthintegrated nonhydrostatic flow model is developed using the method of weighted residuals. Using a unit weighting function depthintegrated ReynoldsAveraged NavierStokes (RANS) equations are obtained. Prescribing polynomial variations for the field variables in the vertical direction, a set of perturbation parameters remains undetermined. The model is closed generating a set of weighted averaged equations using a suitable weighting function. The resulting depthintegrated, nonhydrostatic model is solved with a semiimplicit finitevolume finitedifference scheme. The explicit part of the model is a Godunovtype finite volume scheme that uses the HLLC approximate Riemann solver to determine the nonhydrostatic depthaveraged velocity field. The implicit part of the model is solved using a Newton–Raphson algorithm to incorporate the effects of the pressure field in the solution. The model is applied with good results to a set of problems of coastal and river engineering, including steady flow over fixed bedforms, solitary wave propagation, solitary wave runup, linear frequency dispersion, propagation of sinusoidal waves over a submerged bar and dambreak flood waves.
The local smoothness indicators play an important role in the performance of a weighted essentially nonoscillatory (WENO) scheme. Due to having only two points available on each substencil, the local smoothness indicators calculated by conventional methods of Jiang and Shu [1] make the thirdorder WENO scheme too dissipative. In this paper, we propose a different method to calculate the indicators by using all the three points on the global stencil of the thirdorder WENO scheme. The numerical results demonstrate that the WENO scheme with the new indicators has less dissipation and better resolution than the ones of Jiang and Shu's for both smooth and discontinuous solutions.
Aerodynamic characteristics of various geometries are predicted using a finite element formulation coupled with several numerical techniques to ensure stability and accuracy of the method. First, an edgebased error estimator and anisotropic mesh adaptation are used to detect automatically all flow features under the constraint of a fixed number of elements, thus controlling the computational cost. A variational multiscalestabilized finite element method is used to solve the incompressible NavierStokes equations. Finally, the SpalartAllmaras turbulence model is solved using the streamline upwind PetrovGalerkin method. This paper is meant to show that the combination of anisotropic unsteady mesh adaptation with stabilized finite element methods provides an adequate framework for solving turbulent flows at high Reynolds numbers. The proposed method was validated on several test cases by confrontation with literature of both numerical and experimental results, in terms of accuracy on the prediction of the drag and lift coefficients as well as their evolution in time for unsteady cases.
In this paper, we propose an adaptive anisotropic mesh methodology for performing accurate numerical simulations of turbulent flows past complex geometries. It couples a stabilized variational multiscale NavierStokes modified solver to account for high stretched elements, a SpalartAllmaras turbulent model with a dynamic anisotropic mesh adaptation algorithm.
Determining boundary conditions (BCs) for incompressible flows is such a delicate matter that affects the accuracy of the results. In this research, a new characteristicbased BC for incompressible NavierStokes equations is introduced. Discretization of equations has been done via finite volume. Additionally, artificial compressibility correction has been employed to deal with equations. Ordinary extrapolation from inner cells of a domain was used as a traditional way to estimate pressure and velocities on solid wall and inlet/outlet boundaries. Here, this method was substituted by the newly proposed BCs based on the characteristics of artificial compressibility equations. To follow this purpose, a computer code has been developed to carry out series of numerical tests for a flow over a backwardfacing step and was applied to a wide range of Reynolds numbers and grid combinations. Calculation of convective and viscous fluxes was done using Jameson's averaging scheme. Employing the characteristicbased method for determining BCs has shown an improved convergence rate and reduced calculation time comparing with those of traditional ones. Furthermore, with the reduction of domain and computational cells, a similar accuracy was achieved for the results in comparison with the ones obtained from the traditional extrapolation method, and these results were in good agreement with the ones in the literature.
In this research, a new characteristics based boundary condition for incompressible NavierStokes equations is introduced. Employing the characteristicbased method for determining boundary conditions has shown an improved convergence rate and reduced calculation time comparing with those of traditional ones. Furthermore, with the reduction of domain and computational cells, a similar accuracy was achieved for the results in comparison with the ones obtained from the traditional extrapolation method and these results were in good agreement with the ones in the literature.
Whenever linear eigenmodes of open flows are computed on a numerical domain that is truncated in the streamwise direction, artificial boundary conditions may give rise to spurious pressure signals that are capable of providing unwanted perturbation feedback to upstream locations. The manifestation of such feedback in the eigenmode spectrum is analysed here for two simple configurations. First, explicitly prescribed feedback in a Ginzburg–Landau model is shown to produce a spurious eigenmode branch, named the ‘arc branch’, that strongly resembles a characteristic family of eigenmodes typically present in open shear flow calculations. Second, corresponding mode branches in the global spectrum of an incompressible parallel jet in a truncated domain are examined. It is demonstrated that these eigenmodes of the numerical model depend on the presence of spurious forcing of a local \(k^+\) instability wave at the inflow, caused by pressure signals that appear to be generated at the outflow. Multiple local \(k^+\) branches result in multiple global eigenmode branches. For the particular boundary treatment chosen here, the strength of the pressure feedback from the outflow towards the inflow boundary is found to decay with the cube of the numerical domain length. It is concluded that arc branch eigenmodes are artefacts of domain truncation, with limited value for physical analysis. It is demonstrated, for the example of a nonparallel jet, how spurious feedback may be reduced by an absorbing layer near the outflow boundary.
In the last stage of droplet growth in clouds which leads to drizzle formation, larger droplets begin to settle under gravity and collide and coalesce with smaller droplets in their path. In this article, we shall deal with the simplified problem of a large drop settling amidst a population of identical smaller droplets. We present an expression for the probability that a given large drop suffers a given number of collisions, for a general statistically homogeneous distribution of droplets. We hope that our approach will serve as a valuable tool in dealing with droplet distribution in real clouds, which has been found to deviate from the idealized Poisson distribution due to mechanisms such as inertial clustering.
We report molecular dynamics simulations designed to investigate the effective size of colloidal particles suspended in a fluid in the vicinity of a rigid wall where all interactions are defined by smooth atomic potential functions. These simulations are used to assess how the behavior of this system at the atomistic length scale compares to continuum mechanics models. In order to determine the effective size of the particles, we calculate the solvent forces on spherical particles of different radii as a function of different positions near and overlapping with the atomistically defined wall and compare them to continuum models. This procedure also then determines the effective position of the wall. Our analysis is based solely on forces that the particles sense, ensuring selfconsistency of the method. The simulations were carried out using both Weeks–Chandler–Andersen and modified LennardJones (LJ) potentials to identify the different contributions of simple repulsion and van der Waals attractive forces. Upon correction for behavior arising the discreteness of the atomic system, the underlying continuum physics analysis appeared to be correct down to much less than the particle radius. For both particle types, the effective radius was found to be \(\sim 0.75\sigma \) , where \(\sigma \) defines the length scale of the force interaction (the LJ diameter). The effective “hydrodynamic” radii determined by this means are distinct from commonly assumed values of \(0.5\sigma \) and \(1.0\sigma \) , but agree with a value developed from the atomistic analysis of the viscosity of such systems.
A numerical investigation is performed on the electroosmotic flow (EOF) in a surfacemodulated microchannel to induce enhanced solute mixing. The channel wall is modulated by placing surfacemounted obstacles of trigonometric shape along which the surface potential is considered to be different from the surface potential of the homogeneous part of the wall. The characteristics of the electrokinetic flow are governed by the Laplace equation for the distribution of external electric potential; the Poisson equation for the distribution of induced electric potential; the Nernst–Planck equations for the distribution of ions; and the Navier–Stokes equations for fluid flow simultaneously. These nonlinear coupled set of governing equations are solved numerically by a control volume method over the staggered system. The influence of the geometric modulation of the surface, surface potential heterogeneity and the bulk ionic concentration on the EOF is analyzed. Vortical flow develops near a surface modulation, and it becomes stronger when the surface potential of the modulated region is in opposite sign to the surface potential of the homogeneous part of the channel walls. Vortical flow also depends on the Debye length when the Debye length is in the order of the channel height. Pressure drop along the channel length is higher for a ribbed wall channel compared to the grooved wall case. The pressure drop decreases with the increase in the amplitude for a grooved channel, but increases for a ribbed channel. The mixing index is quantified through the standard deviation of the solute distribution. Our results show that mixing index is higher for the ribbed channel compared to the grooved channel with heterogeneous surface potential. The increase in potential heterogeneity in the modulated region also increases the mixing index in both grooved and ribbed channels. However, the mixing performance, which is the ratio of the mixing index to pressure drop, reduces with the rise in the surface potential heterogeneity.
The effect of local forcing on the separated, threedimensional shear layer downstream of a backwardfacing step is investigated by means of largeeddy simulation for a Reynolds number based on the step height of 10,700. The step edge is either oriented normal to the approaching turbulent boundary layer or swept at an angle of \(40^\circ \) . Oblique vortices with different orientation and spacing are generated by wavelike suction and blowing of fluid through an edge parallel slot. The vortices exhibit a complex threedimensional structure, but they can be characterized by a wavevector in a horizontal section plane. In order to determine the stepnormal component of the wavevector, a method is developed based on phase averages. The dependence of the wavevector on the forcing parameters can be described in terms of a dispersion relation, the structure of which indicates that the disturbances are mainly convected through the fluid. The introduced vortices reduce the size of the recirculation region by up to 38%. In both the planar and the swept case, the most efficient of the studied forcings consists of vortices which propagate in a direction that deviates by more than \(50^\circ \) from the step normal. These vortices exhibit a spacing in the order of 2.5 step heights. The upstream shift of the reattachment line can be explained by increased mixing and momentum transport inside the shear layer which is reflected in high levels of the Reynolds shear stress \(\rho \overline{u'v'}\) . The position of the maximum of the coherent shear stress is found to depend linearly on the wavelength, similar to twodimensional free shear layers.
The ability to manipulate and control fluid flows is of great importance in many scientific and engineering applications. The proposed closedloop control framework addresses a key issue of modelbased control: The actuation effect often results from slow dynamics of strongly nonlinear interactions which the flow reveals at timescales much longer than the prediction horizon of any model. Hence, we employ a probabilistic approach based on a clusterbased discretization of the Liouville equation for the evolution of the probability distribution. The proposed methodology frames highdimensional, nonlinear dynamics into lowdimensional, probabilistic, linear dynamics which considerably simplifies the optimal control problem while preserving nonlinear actuation mechanisms. The datadriven approach builds upon a state space discretization using a clustering algorithm which groups kinematically similar flow states into a low number of clusters. The temporal evolution of the probability distribution on this set of clusters is then described by a controldependent Markov model. This Markov model can be used as predictor for the ergodic probability distribution for a particular control law. This probability distribution approximates the longterm behavior of the original system on which basis the optimal control law is determined. We examine how the approach can be used to improve the openloop actuation in a separating flow dominated by Kelvin–Helmholtz shedding. For this purpose, the feature space, in which the model is learned, and the admissible control inputs are tailored to strongly oscillatory flows.
The stability properties of twodimensional (2D) and threedimensional (3D) compressible flows over a rectangular cavity with lengthtodepth ratio of \(L/D=6\) are analyzed at a freestream Mach number of \(M_\infty =0.6\) and depthbased Reynolds number of \(Re_D=502\) . In this study, we closely examine the influence of threedimensionality on the wake mode that has been reported to exhibit highamplitude fluctuations from the formation and ejection of largescale spanwise vortices. Direct numerical simulation (DNS) and biglobal stability analysis are utilized to study the stability characteristics of the wake mode. Using the biglobal stability analysis with the timeaveraged flow as the base state, we capture the global stability properties of the wake mode at a spanwise wavenumber of \(\beta =0\) . To uncover spanwise effects on the 2D wake mode, 3D DNS are performed with cavity widthtodepth ratio of \(W/D=1\) and 2. We find that the 2D wake mode is not present in the 3D cavity flow with \(W/D=2\) , in which spanwise structures are observed near the rear region of the cavity. These 3D instabilities are further investigated via biglobal stability analysis for spanwise wavelengths of \(\lambda /D=0.5{}2.0\) to reveal the eigenspectra of the 3D eigenmodes. Based on the findings of 2D and 3D global stability analysis, we conclude that the absence of the wake mode in 3D rectangular cavity flows is due to the release of kinetic energy from the spanwise vortices to the streamwise vortical structures that develops from the spanwise instabilities.
Linear global instability analysis has been performed in the wake of a low aspect ratio threedimensional wing of elliptic cross section, constructed with appropriately scaled Eppler E387 airfoils. The flow field over the airfoil and in its wake has been computed by full threedimensional direct numerical simulation at a chord Reynolds number of \(Re_{c}=1750\) and two angles of attack, \(\mathrm{{AoA}}=0^\circ \) and \(5^\circ \) . Pointvortex methods have been employed to predict the inviscid counterpart of this flow. The spatial BiGlobal eigenvalue problem governing linear smallamplitude perturbations superposed upon the viscous threedimensional wake has been solved at several axial locations, and results were used to initialize linear PSE3D analyses without any simplifying assumptions regarding the form of the trailing vortex system, other than weak dependence of all flow quantities on the axial spatial direction. Two classes of linearly unstable perturbations were identified, namely strongeramplified symmetric modes and weakeramplified antisymmetric disturbances, both peaking at the vortex sheet which connects the trailing vortices. The amplitude functions of both classes of modes were documented, and their characteristics were compared with those delivered by local linear stability analysis in the wake near the symmetry plane and in the vicinity of the vortex core. While all linear instability analysis approaches employed have delivered qualitatively consistent predictions, only PSE3D is free from assumptions regarding the underlying base flow and should thus be employed to obtain quantitative information on amplification rates and amplitude functions in this class of configurations.
Reference solutions are important in several applications. They are used as base states in linear stability analyses as well as initial conditions and reference states for sponge zones in numerical simulations, just to name a few examples. Their accuracy is also paramount in both fields, leading to more reliable analyses and efficient simulations, respectively. Hence, steadystates usually make the best reference solutions. Unfortunately, standard marching schemes utilized for accurate unsteady simulations almost never reach steadystates of unstable flows. Steady governing equations could be solved instead, by employing Newtontype methods often coupled with continuation techniques. However, such iterative approaches do require large computational resources and very good initial guesses to converge. These difficulties motivated the development of a technique known as selective frequency damping (SFD) (Åkervik et al. in Phys Fluids 18(6):068102, 2006). It adds a source term to the unsteady governing equations that filters out the unstable frequencies, allowing a steadystate to be reached. This approach does not require a good initial condition and works well for selfexcited flows, where a single nonzero excitation frequency is selected by either absolute or global instability mechanisms. On the other hand, it seems unable to damp stationary disturbances. Furthermore, flows with a broad unstable frequency spectrum might require the use of multiple filters, which delays convergence significantly. Both scenarios appear in convectively, absolutely or globally unstable flows. An alternative approach is proposed in the present paper. It modifies the coefficients of a marching scheme in such a way that makes the absolute value of its linear gain smaller than one within the required unstable frequency spectra, allowing the respective disturbance amplitudes to decay given enough time. These ideas are applied here to implicit multistep schemes. A few chosen test cases shows that they enable convergence toward solutions that are unstable to stationary and oscillatory disturbances, with either a single or multiple frequency content. Finally, comparisons with SFD are also performed, showing significant reduction in computer cost for complex flows by using the implicit multistep MGM schemes.
The pressure drag of blunt bluff bodies is highly relevant in many practical applications, including to the aerodynamic drag of road vehicles. This paper presents theory revealing that a mean drag reduction can be achieved by manipulating wake flow fluctuations. A linear feedback control strategy then exploits this idea, targeting attenuation of the spatially integrated base (back face) pressure fluctuations. Largeeddy simulations of the flow over a Dshaped blunt bluff body are used as a testbed for this control strategy. The flow response to synthetic jet actuation is characterised using system identification, and controller design is via shaping of the frequency response to achieve fluctuation attenuation. The designed controller successfully attenuates integrated base pressure fluctuations, increasing the timeaveraged pressure on the body base by 38%. The effect on the flow field is to push the rollup of vortices further downstream and increase the extent of the recirculation bubble. This control approach uses only bodymounted sensing/actuation and input–output model identification, meaning that it could be applied experimentally.