Output Feedback Boundary Control of a Ginzburg–Landau Model of Vortex Shedding

An exponentially convergent observer is designed for a linearized Ginzburg-Landau model of vortex shedding in viscous flow past a bluff body. Measurements are restricted to be taken collocated with the actuation which is applied on the cylinder surface. The observer is used in conjuction with a state feedback boundary controller designed in previous work to attenuate vortex shedding. While the theoretical results apply to the linearized system under sufficiently smooth initial data that satisfy the boundary conditions, simulations demonstrate the performance of the linear output feedback scheme on the nonlinear plant model     

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.     

The normal and oblique collision of a dipole with a no-slip boundary

Benchmark results are reported of two separate sets of numerical experiments on the collision of a dipole with a no-slip boundary at several Reynolds numbers. One set of numerical simulations is performed with a finite differences code while the other set concerns simulations conducted with a Chebyshev pseudospectral code. Well-defined initial and boundary conditions are used and the accuracy and convergence of the numerical solutions have been investigated by inspection of several global quantities like the total kinetic energy, the enstrophy and the total angular momentum of the flow, and the vorticity distribution and vorticity flux at the no-slip boundaries. It is found that the collision of the dipole with the no-slip wall and the subsequent flow evolution is dramatically influenced by small-scale vorticity produced during and after the collision process. The trajectories of several coherent vortices are tracked during the simulation and show that in particular underresolved high-amplitude vorticity patches near the no-slip walls are potentially responsible for deteriorating accuracy of the computations in the course of time. Our numerical simulations clearly indicate that it is extremely difficult to obtain mode- or grid-convergence for this seemingly rather simple two-dimensional vortex-wall interaction problem.     

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.     

Numerical study of partial slip on the MHD flow of an Oldroyd 8-constant fluid

The steady flow of an Oldroyd 8-constant magnetohydrodynamic (MHD) fluid is considered for a cylindrical geometry when the no-slip condition between the cylinders and the fluid is no longer valid. The inclusion of the partial slip at boundaries modifies the governing boundary conditions, changing from a linear to a non-linear situation. The non-linear differential equation along with non-linear boundary conditions governing the flow has been solved numerically using a finite-difference scheme in combination with an iterative technique. The solution for the no-slip condition is a special case of the presented analysis. A critical assessment is made for the cases of partial slip and no-slip conditions.     

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.     

Boundary conditions for incompressible flows

A general framework is presented for the formulation and analysis of rigid no-slip boundary conditions for numerical schemes for the solution of the incompressible Navier-Stokes equations. It is shown that fractional-step (splitting) methods are prone to introduce a spurious numerical boundary layer that induces substantial time differencing errors. High-order extrapolation methods are analyzed to reduce these errors. Both improved pressure boundary condition and velocity boundary condition methods are developed that allow accurate implementation of rigid no-slip boundary conditions.     

Transitional cylindrical swirling flow in presence of a flat free surface

This article is devoted to the study of an incompressible viscous flow of a fluid partly enclosed in a cylindrical container with an open top surface and driven by the constant rotation of the bottom wall. Such type of flows belongs to a group of recirculating lid-driven cavity flows with geometrical axisymmetry and of the prescribed boundary conditions of Dirichlet type—no-slip on the cavity walls. The top surface of the cylindrical cavity is left open with an imposed stress-free boundary condition, while a no-slip condition with a prescribed rotational velocity is imposed on the bottom wall. The Reynolds regime corresponds to transitional flows with some incursions in the fully laminar regime. The approach taken here revealed new flow states that were investigated based on a fully three-dimensional solution of the Navier-Stokes equations for the free-surface cylindrical swirling flow, without resorting to any symmetry property unlike all other results available in the literature. Theses solutions are obtained through direct numerical simulations based on a Legendre spectral element method.     

A variational approach to the contact angle dynamics of spreading droplets

The modeling and the numerical representation of the contact line between a two-phase interface and a solid surface are still open problems from the physical, mathematical and numerical point of view. This paper deals with the numerical simulation of the spreading of a single droplet impacting over horizontal dry surfaces. A new variational approach to study the droplet spreading is presented by coupling an interface front-tracking algorithm to the single-fluid finite element formulation of the incompressible Navier-Stokes equations which are solved on a fixed mesh. Standard no-slip boundary conditions near the contact line lead to a singular behavior that in the variational approach is removed by introducing a generalized boundary condition which is the sum of a dissipation term and the dynamical contact angle law. By changing the intensity of the dissipation a large number of boundary conditions around the contact point are modeled, ranging from no-slip to free-slip. Since the impact is over horizontal surfaces, axisymmetric solutions are investigated with high mesh resolutions. A very precise implementation of the capillary force with a volumetric extension of the curvature has been adopted. We have considered a Lagrangian front-tracking method to advect the interface. The marker representing the contact point is simply advected by the computed velocity at the boundary without the need to extrapolate the vector field from the interior and to enforce locally mass-conservation. The model has been tested for the impact and the spreading of a droplet on solid substrates with a different wettability at low Reynolds numbers where the inertial, the viscous and the surface tension forces are all important. A number of droplet impacts with different outcomes, ranging from simple deposition to partial and complete rebound, have been reproduced. However, our simulations indicate that the formulas suggested in the literature for the dynamical contact angle should be modified to simulate a broad class of experiments.     

Microfluidic two-phase interactions under variable liquid to cross-flow gas momentum flux ratios

Volume of fluid (VOF) and large eddy simulations (LES) are coupled to investigate the microfluidic two-phase interactions during the liquid emergence into the cross-flow gas in a super-hydrophobic micro-channel. Spatio-temporal evolution of the gas/liquid interface is presented for nine different cases of the liquid to gas momentum flux ratios, gas/liquid Reynolds numbers and gas/liquid Weber numbers. With increased momentum of the gas flow, the liquid topology is found deflected towards the downstream. Under variable gas resistance effects, the liquid flow emerging through the square pore may or may not develop a circular cross-section governed by the axis-switching phenomenon. At strong gas inertia, vortex shedding in the downstream of the liquid generates vorticular ligaments in the wake region. Shearing effects on the liquid surface are increased at higher liquid injection velocities and/or gas densities. Depending on the competing effects of the viscous diffusion versus gas/liquid inertia, different combinations of the interactions among the three building blocks of the fluid flow problems (boundary layer, shear layer and wake) are described in microfluidics scales. The complexity of the liquid topology is found correlated with the occurrence of the phenomena such as the Kelvin–Helmholtz (KH) instability, the horseshoe vortex system, stationary/shedding vortices in the wake of the liquid topology as well as their interaction with the micro-channel wall boundary layers.     

Direct numerical simulation of the transitional boundary-layer flow induced by an isolated hemispherical roughness element

The joint application of direct numerical simulation (DNS) and a combined multiple-direct forcing and immersed boundary method (MDF/IBM) is proposed to investigate the transitional boundary-layer flow induced by three-dimensional roughness elements. The multi-direct forcing technique is used to calculate the interacting force between the solid surface of roughness elements and the fluid, and let the no-slip boundary conditions be satisfied. In order to validate the efficiency of these numerical methods, a flow past an isolated three-dimensional hemispherical roughness element mounted on a flat plate is simulated. The evolutional process of the discrete hairpin vortex and the formation of two kinds of secondary vortex structures are captured. The comparisons of profiles of streamwise mean velocity and velocity fluctuation between the simulated results and the experimental ones show great quantitative agreement. The evolution of disturbances and the growth of steady disturbance energy prove the transient growth mechanisms underlying the transitional flow induced by moderate-amplitude isolated three-dimensional roughness elements. Numerical methods used in this investigation can be extended to the simulation of the transitional boundary-layer flows induced by randomly distributed three-dimensional roughness elements.     

A two-dimensional lattice Boltzmann model for uniform channel flows

In this paper a novel two-dimensional lattice Boltzmann model (LBM) is developed for uniform channel flows. The axial velocity is solved from a momentum diffusion equation over the cross-sectional plane. An extrapolation boundary condition is also introduced to enhance the no-slip boundary in the momentum equation. This boundary treatment can also be applied to LBM simulations of other diffusion processes. The algorithm and boundary treatment are validated by simulations of steady Poiseuille and pulsatile Womersley flows in circular pipes. The numerical convergence and accuracy are comparable to those of existing models. Moreover, comparison with general three-dimensional lattice Boltzmann simulations demonstrates the advantages of our two-dimensional model, including lower computational resource requirements (memory and time), easier boundary treatment for arbitrary cross-sectional shapes, and no velocity constraint. These features are attractive for practical applications with uniform channel flows.     

Implementation of a transient adaptive sub-cell module for the parallel-DSMC code using unstructured grids

A method of transient adaptive sub-cells (TAS) suitable for unstructured grids that is modified from the existing one for the structured grids of DSMC is introduced. The TAS algorithm is implemented within the framework of a parallelized DSMC code (PDSC). Benchmarking tests are conducted for steady driven cavity flow, steady hypersonic flow over a two-dimensional cylinder, steady hypersonic flow over a cylinder/flare and the unsteady vortex shedding behind a two-dimensional cylinder. The use of TAS enables a reduction in the computational expense of the simulation since larger sampling cells and less simulation particles can be employed. Furthermore, the collision quality of the simulation is maintained or improved and the preservation of property gradients and vorticity at the scale of the sub-cells enables correct unsteady vortex shedding frequencies to be predicted. The use of TAS in a parallel-DSMC code allows simulations of unsteady processes at a level to be carried out efficiently, accurately and with acceptable computational time.     

A new wall boundary condition in particle methods

We present a new formulation of the boundary condition for solid walls in particulate fluid flows. It integrates traditional treatments of bounce-back and Maxwellian reflections, and hence ensures both no-slip condition and effective thermostat on the wall. The formulation was validated in a series of simulations with pseudo-particle modeling [W. Ge, J. Li, Chem. Eng. Sci. 58 (2003) 1565-1585].     

Modeling of delta-wing type vortex generators

A numerical model of delta-wing type vortex generator was developed in two steps.The first step was to obtain a parameterized model of the shedding vortex based on delta-wing theory,which relates the geometry parameters and flow field parameters to the strength of shedding vortex which directly decides the source term.In the second step,a method was proposed to add source terms into the flow control equations so that the shedding vortex could be simulated numerically.As soon as the numerical model was compl...     

Spectral simulations of an unsteady compressible flow past a circular cylinder

An unsteady compressible viscous wake flow past a circular cylinder has been successfully simulated using spectral methods. A new approach in using the Chebyshev collocation method for a periodic problem is introduced. We have further proved that the eigenvalues associated with the differentiation matrix are purely imaginary, reflecting the periodicity of the problem. It has been shown that the solution of a model problem has exponential growth in time if an ‘improper’ boundary conditions procedure is used. A characteristic boundary conditions, which is based on the characteristics of the Euler equations of gas dynamics, has been derived for the spectral code. The primary vortex shedding frequency computed agrees well with the results in the literature for Mach = 0.4, Re = 80. No secondary frequency is observed in the power spectrum analysis of the pressure data.     

Weak imposition of Dirichlet boundary conditions in fluid mechanics

Weakly enforced Dirichlet boundary conditions are compared with strongly enforced conditions for boundary layer solutions of the advection-diffusion equation and incompressible Navier-Stokes equations. It is found that weakly enforced conditions are effective and superior to strongly enforced conditions. The numerical tests involve low-order finite elements and a quadratic NURBS basis utilized in the Isogeometric Analysis approach. The convergence of the mean velocity profile for a turbulent channel flow suggests that weak no-slip conditions behave very much like a wall function model, although the design of the boundary condition is based purely on numerical, rather than physical or empirical, conditions.     

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.     

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).     

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.