Imran Akhtar

Proper orthogonal decomposition closure models for turbulent flows: A numerical comparison

This paper puts forth two new closure models for the proper orthogonal decomposition reduced-order modeling of structurally dominated turbulent flows: the dynamic subgrid-scale model and the variational multiscale model. These models, which are considered state-of-the-art in large eddy simulation, together with the mixing length and the Smagorinsky closure models, are tested in the numerical simulation of 3D turbulent flow past a circular cylinder at Re=1000. Five criteria are used to judge the performance of the proper orthogonal decomposition reduced-order models: the kinetic energy spectrum, the mean velocity, the Reynolds stresses, the root mean square values of the velocity fluctuations, and the time evolution of the POD coefficients. All the numerical results are benchmarked against a direct numerical simulation. Based on these numerical results, we conclude that the dynamic subgrid-scale and the variational multiscale models are the most accurate.

Two-level discretizations of nonlinear closure models for proper orthogonal decomposition

Proper orthogonal decomposition has been successfully used in the reduced-order modeling of complex systems. Its original promise of computationally efficient, yet accurate approximation of coherent structures in high Reynolds number turbulent flows, however, still remains to be fulfilled. To balance the low computational cost required by reduced-order modeling and the complexity of the targeted flows, appropriate closure modeling strategies need to be employed. Since modern closure models for turbulent flows are generally nonlinear, their efficient numerical discretization within a proper orthogonal decomposition framework is challenging. This paper proposes a two-level method for an efficient and accurate numerical discretization of general nonlinear closure models for proper orthogonal decomposition reduced-order models. The two-level method computes the nonlinear terms of the reduced-order model on a coarse mesh. Compared with a brute force computational approach in which the nonlinear terms are evaluated on the fine mesh at each time step, the two-level method attains the same level of accuracy while dramatically reducing the computational cost. We numerically illustrate these improvements in the two-level method by using it in three settings: the one-dimensional Burgers equation with a small diffusion parameter @n=10^-^3, the two-dimensional flow past a cylinder at Reynolds number Re=200, and the three-dimensional flow past a cylinder at Reynolds number Re=1000.

Modeling Steady-state and Transient Forces on a Cylinder

Numerical simulations of the flow past a stationary circular cylinder at different Reynolds numbers (Re) have been performed using a computational fluid dynamics (CFD) solver that is based on the unsteady Reynolds-averaged Navier—Stokes equations (RANS). The results obtained are used to develop reduced-order models for the lift and drag coefficients. The models not only match the numerical simulation results in the time domain, but also in the spectral domain. They capture the steady-state region with excellent accuracy. Further, the models are verified by comparing their results in the transient region with their counterparts from the CFD simulations and very good agreement is found. The work performed here is a step towards building models for vortex-induced vibrations (VIV) encountered in risers, spars, and other offshore structures.

A van der Pol–Duffing Oscillator Model of Hydrodynamic Forces on Canonical Structures

Numerical simulations of the flow past elliptic cylinders with different eccentricities have been performed using a parallel incompressible computational fluid-dynamics (CFD) solver. The pressure is integrated over the surface to compute the lift and drag forces on the cylinders. The numerical results of different cases are then used to develop reduced-order models for the lift and drag coefficients. The lift coefficient is modeled with a generalized van der Pol-Duffing oscillator and the drag coefficient is expressed in terms of the lift coefficient. The parameters in the oscillator model are computed for each elliptic cylinder. The results of the model match the CFD results not only in the time domain but also in the spectral domain.

Linear and nonlinear active feedback controls for vortex-induced vibrations of circular cylinders

We consider the problem of suppressing oscillations of an elastically mounted rigid cylinder undergoing vortex-induced vibrations by linear and nonlinear active velocity feedback controllers. Each controller relies on an actuator, which imparts an opposing force to the cylinder motion, thereby reducing its high-amplitude oscillations. A strongly coupled fluid–structure numerical model is used to solve the fluid–structure interaction equations. The results show that the choice of the active feedback controller depends on the allowable controlled amplitude of the cylinder. It is found that a cubic velocity feedback controller is more efficient than its linear velocity counterpart when very small controlled amplitudes are desired.

A New Closure Strategy for Proper Orthogonal Decomposition Reduced-Order Models

Proper orthogonal decomposition (POD) is one of the most significant reduced-order modeling (ROM) techniques in fluid mechanics. However, the application of POD based reduced-order models (POD-ROMs) is primarily limited to laminar flows due to the decay of physical accuracy. A few nonlinear closure models have been developed for improving the accuracy and stability of the POD-ROMs, which are generally computationally expensive. In this paper we propose a new closure strategy for POD-ROMs that is both accurate and effective. In the new closure model, the Frobenius norm of the Jacobian of the POD-ROM is introduced as the eddy viscosity coefficient. As a first step, the new method has been tested on a one-dimensional Burgers equation with a small dissipation coefficient 1⁄4 10 . Numerical results show that the Jacobian based closure model greatly improves the physical accuracy of the POD-ROM, while maintaining a low computational cost. [DOI: 10.1115/1.4005928]

Model Based Control of Laminar Wake Using Fluidic Actuation

Control of fluid-structure interaction is of practical importance from the perspective of wake modification and reduction of vortex-induced vibrations (VIVs). The aim of this study is to design a control to suppress vortex shedding. We perform a two-dimensional simulation of the flow past a circular cylinder using a parallel Computational Fluid Dynamics (CFD) solver. We record the velocity and pressure fields over a shedding cycle and compute the proper orthogonal decomposition (POD) modes of the divergence-free velocity and pressure, respectively. The Navier―Stokes equations are projected onto these POD modes to reduce the dynamical system to a set of ordinary-differential equations (ODEs). This dynamical system exhibits a limit cycle with negative linear damping and positive nonlinear damping. The reduced-order model is then modified by placing a pair of suction actuators and applying a control strategy using a control function method. We use the pressure POD mode distribution on the cylinder surface to optimally locate the actuators. We design a controller based on the linearized system and make it positively damped using pole-placement technique. The control-input settles to a constant value, suggesting constant suction through the actuators. We validate the results using CFD simulations in an open-loop setting and observe suppression of the hydrodynamic forces acting on the cylinder.

Shape Sensitivity Analysis in Flow Models Using a Finite-Difference Approach

Reduced-order models have a number of practical engineering applications for unsteady flows that require either low-dimensional approximations for analysis and control or repeated simulation over a range of parameter values. The standard method for building reduced-order models uses the proper orthogonal decomposition (POD) and Galerkin projection. However, this standard method may be inaccurate when used “off-design” (at parameter values not used to generate the POD). This phenomena is exaggerated when parameter values describe the shape of the flow domain since slight changes in shape can have a significant influence on the flow field. In this paper, we investigate the use of POD sensitivity vectors to improve the accuracy and dynamical system properties of the reduced-order models to problems with shape parameters. To carry out this study, we consider flows past an elliptic cylinder with varying thickness ratios. Shape sensitivities (derivatives of flow variables with respect to thickness ratio) computed by finite-difference approximations are used to compute the POD sensitivity vectors. Numerical studies test the accuracy of the new bases to represent flow solutions over a range of parameter values.

MODELING THE TRANSIENT AND STEADY-STATE FLOW OVER A STATIONARY CYLINDER

A reduced-order model for the two-dimensional flow over a stationary circular cylinder is examined. The lift is modeled with the van der Pol equation with three parameters; it models self-excited self-limiting systems. The drag is modeled as the sum of a mean term and a time-varying term proportional to the product of the lift and its time derivative. The transient and steady-state flows are calculated using a CFD code based on the unsteady Reynolds-averaged Navier-Stokes equations. The steady-state lift and drag CFD results are used to identify the three parameters in the lift model using a combination of higher-order spectral techniques and perturbation methods. The model is validated using steady-state numerical simulations for three cases describing low, moderate, and high Reynolds number flows. Then, the model is shown to reproduce the transient lift and drag calculated with the CFD code.Copyright

A Biologically Inspired Computational Study of Flow Past Tandem Flapping Foils

using Particle Image Velocimetry (PIV). The simulations have been carried out using a Cartesian grid method that allows us to simulate flows with complex moving boundaries on stationary Cartesian grids. The simulations indicate that vortex shedding from the upstream dorsal fin is indeed capable of increasing the thrust of the tail fin significantly. However, this thrust augmentation is found to be quite sensitive to the phase relationship between the two flapping fins. The numerical simulations allows us to examine the underlying physical mechanism for this thrust augmentation. I. Introduction