Paul G. A. Cizmas

Proper-Orthogonal Decomposition of Spatio-Temporal Patterns in Fluidized Beds

Abstract Numerical simulations of the hydrodynamics of a fluidized bed are carried out to investigate the complex interaction between the gas and the solid particles, and to explore the utility of a reduced-order model based on the proper orthogonal decomposition (POD). The behavior of a fluidized bed is modeled using a “two-fluid” theory, which involves conservation of mass, momentum, energy and species equations for the two interpenetrating continua. These equations are solved using a numerical algorithm that employs a conservative discretization scheme with mixed implicit and explicit formulations. We conducted simulations of gas–solid interaction in a narrow (two-dimensional) bed filled with sand particles which was uniformly fluidized at minimum fluidization but with additional air flow through a central nozzle. Aided by the proper orthogonal decomposition, spatial dominant features are identified and separated from the spatio-temporal dynamics of the simulations. The most dynamic region of the gas–solid interaction is confined to the central channel caused by the jet. The flow within this structure is successfully captured by a few POD eigenfunctions. Phase-space plots further indicate the existence of low-dimensional dynamics within the central channel. This conclusion supports the validity of a reduced-order model for fluidized beds, which can then be constructed by projecting the governing equations onto the POD modes, as it is commonly done in the Galerkin method.

Experiments and analysis for a gust generator in a wind tunnel

An experimental investigation was made of the gust field generated by a rotating slotted cylinder installed in the Duke University low-speed, closed-circuit wind tunnel. The system has a very simple configuration with low cost and can produce a controllable single or multiple harmonic gust wave in the lateral and longitudinal directions. It requires minimal power and torque input. A simplified theoretical aerodynamic model and a design estimation of the lateral and longitudinal gust flowfield is also proposed in this article. The design estimate is based on a two-dimensional dynamic lift coefficient that is given by the theoretical and experimental results. An interfering wake vortex effect is the major disadvantage of this system. Nomenclature C(k) = Theodorsens function ICleql = magnitude of equivalent lift coefficient for rotating slotted cylinder/airfoil c = airfoil chord d = cylinder diameter dLa = airfoil lift force per span length dLrsc = rotating slotted cylinder lift force per span length e, e = gap between the o.d. of the rotating slotted cylinder and trailing edge of the airfoil, elc H, H = vertical position from tunnel bottom, HIHW Hw = height of the tunnel test section

Proper Orthogonal Decomposition of Turbine Rotor-Stator Interaction

Numerical simulations of the flow in a one-stage turbine are carried out to investigate the feasibility of constructing a reduced-order model via Galerkin methods. The flow in the turbine is modeled by the unsteady Reynolds-averaged Navier-Stokes equations. The governing equations are written in the strong conservation form and solved using a fully implicit, finite difference approximation. By the use of the proper orthogonal decomposition (POD), spatial dominant features, also known as POD modes or eigenfunctions, are identified and separated from the spatiotemporal dynamics of the turbine flow. The POD reconstructed solutions indicate that a significant portion of the original dynamics is captured by a few modes. The solution reconstructed using the first 40 modes captures more than 99% of the energy spectrum, whereas the error of the energy variable is less than 0.6%, and the error of skin friction is less than 1.5%. Phase-space plots further indicate the existence of low-dimensional dynamics, which supports the validity of a reduced-order model for turbine flow.

Mesh Generation and Deformation Algorithm for Aeroelasticity Simulations

H IGH-FIDELITY flow solvers for aeroelastic applications require the use of computational meshes that deform as the structure is being displaced. High-aspect-ratio wings increase the demands on the robustness of the mesh-deforming algorithm, because these wings are extremely flexible and attain deformations that are a significant fraction of the span of the wing. The mesh deformation algorithm must be not only robust but also computationally inexpensive to avoid penalizing the turnaround time of the aeroelastic computations. Different approaches have been developed to solve the movingmesh problem. For meshes generated by using overlapping grids, a natural way to allow for gridmotion is to slide the overlapping region of the grids [1,2]. The advantage of thismethod is that the body-fitted meshes do not deform during the body motion. A disadvantage of this approach is that the interpolation algorithm that communicates the solution between grids has to be updated for each overlapping position of the grids. The tension spring analogy [3] is one of themostwidely usedmesh deformation strategies. In this approach, each edge of the mesh is represented by a spring for which the stiffness is proportional to the reciprocal of the length of the edge. By replacing the edges with springs, a deformation of the boundary translates into a deformation of the spring network, which adjusts its shape to the equilibrium position of the network. The displacements in each direction are decoupled and the equation that updates the position of the nodes is relatively easy to solve. A disadvantage of this method is that for highly distorted meshes, collapsed or negative volume cells may appear, especially on high-aspect-ratio cells such as those used for viscous flows. An improvement over the tension spring analogy is the torsion spring analogy [4,5]. The torsion spring analogy consists of adding a torsional spring to the tension-spring-analogy technique. The stiffness of the torsional spring is related to the angle between the edges. As the angle tends to zero, the stiffness tends to infinity, thus preventing vertices from crossing over edges and avoiding negativevolume cells. The disadvantage of this method is the higher complexity and computational cost than with the tension spring analogy. The transfinite-interpolation mesh deformation technique is based on the linear interpolation of the boundary motion [6]. The motion of a node located between amoving and a fixed boundary is equal to the motion of themoving boundary times a scale factor. This scale factor, assigned to each node of themesh, depends on the distances from the node to the moving and the fixed surfaces. The scale factor is 1.0 for nodes on the moving boundary and 0.0 for nodes on the fixed boundary. The method guarantees a smooth transition between the moving boundaries and the fixed boundaries. One disadvantage of this method is that it cannot guarantee the mesh orthogonality at deforming surfaces, a condition that is important for viscous flows. Another approach to simulatemesh deformation is to use the linear elasticity equations [7]. The deformed grid is obtained by solving the equilibrium equations for the stress field. Themodulus of elasticity is chosen to be inversely proportional to the cell volume or to the distance from the deforming boundaries. Therefore, the cells close to the moving boundaries have small deformations, and the majority of the mesh deformation is relegated to the regions farther away from the moving boundary. This Note presents a grid generation and deformation algorithm for wings with large deformations. The computational domain was discretized using a hybrid grid that consisted of structured hexahedra around the wing and unstructured triangular prisms elsewhere. The mesh was divided in layers that were topologically identical in the spanwise direction. The mesh deformation algorithm was applied in two steps. First, the spring analogy technique was applied to deform the nodes within a mesh layer. Second, the layers were deformed to be perpendicular to the boundaries of the domain and to the surface of the wing. The Note describes the mesh generation algorithm and the mesh deformation algorithm and shows results for a wing with large tip deformation.

A reduced-order model for a bubbling fluidized bed based on proper orthogonal decomposition

This paper presents the development of a reduced-order model (ROM) for dynamics of non-reactive, isothermal fluidized beds, based on the proper orthogonal decomposition (POD) method. Several implementations of this ROM were developed for a two-dimensional bubbling fluidized bed using numerical results from a full computational fluid dynamics (CFD) simulation of the bed. The solutions of the ROM were used to investigate the influence of the size of the particles on the motion of the bed. The solutions of the ROM were compared with the full model solutions for different particle diameters. The differences between the ROM and the full order solutions (the CFD results) were less than 3% within the range of diameters used for POD generation. The computational time of the ROM varied between 25 and 33% of the computational time of the full CFD solution. The computational speed-up depended on the complexity of the transport phenomena, the ROM methodology and the reconstruction error.

PARALLEL COMPUTATION OF TURBINE BLADE CLOCKING

This paper presents a numerical study of airfoil clocking of a six-row test turbine configuration with equal pitches. Since the rotor-stator interaction flow is highly unsteady, the numerical simulation of airfoil clocking requires the use of time marching methods, which can be computationally expensive. The large turnaround time and the associated cost for such simulations makes it unacceptable for the turbomachinery design process. To reduce the turnaround time and cost/MFLOP, a parallel code based on Message-Passing Interface libraries was developed. The relative circumferential positions of the three stator and three rotor rows in an industrial steam turbine were varied to increase turbine efficiency. A grid density study was performed to verify the grid independence of the computed solutions. The clocking of the second-stage airfoils gave approximately a 50% greater efficiency variation than the clocking of the third-stage airfoils. This was true for clocking both rotor and stator airfoils. Rotor clocking produces an efficiency variation which is approximately twice the efficiency variation produced by stator clocking. For both stator and rotor clocking, the maximum efficiency is obtained when the wake impinges on the leading edge of the clocked airfoil. NOMENCLATURE p Pressure T Temperature η Efficiency γ Ratio of specific heats of a gas

A reduced-order model for heat transfer in multiphase flow and practical aspects of the proper orthogonal decomposition

Abstract This paper discusses two practical aspects of reduced-order models (ROMs) based on proper orthogonal decomposition (POD) and presents the derivation and implementation of a ROM for non-isothermal multiphase flow. The POD method calculates basis functions for a reduced-order representation of two-phase flow by calculating the eigenvectors of an autocorrelation matrix composed of snapshots of the flow. The flow is divided into transient and quasi-steady regions and two methods are shown for clustering snapshots in the transient region. Both methods reduce error as compared to the constant sampling case. The ROM for non-isothermal flow was developed using numerical results from a full-order computational fluid dynamics model for a two-dimensional non-isothermal fluidized bed. Excellent agreement is shown between the reduced- and full-order models. The composition of the autocorrelation matrix is also considered for an isothermal case. An approach treating field variables separately is shown to produce less error than a coupled approach.

Acceleration techniques for reduced-order models based on proper orthogonal decomposition

This paper presents several acceleration techniques for reduced-order models based on the proper orthogonal decomposition (POD) method. The techniques proposed herein are: (i) an algorithm for splitting the database of snapshots generated by the full-order model; (ii) a method for solving quasi-symmetrical matrices; (iii) a strategy for reducing the frequency of the projection. The acceleration techniques were applied to a POD-based reduced-order model of the two-phase flows in fluidized beds. This reduced-order model was developed using numerical results from a full-order computational fluid dynamics model of a two-dimensional fluidized bed. Using these acceleration techniques the computational time of the POD model was two orders of magnitude shorter than the full-order model.

Aeroelastic Analysis for Future Air Vehicle Concepts Using a Fully Nonlinear Methodology

Abstract : Future air vehicles will be highly flexible and will include deformable sub-systems resulting in new physical interactions between a vehicles structure, the surrounding flowfleld, and the dynamics of the vehicle that are fundamentally nonlinear. Although existing aeroelastic methodologies might be considered reliable for traditional applications, they fail (as evidenced by current experiences) to properly capture the complex physics expected for these vehicles. Challenges include non-traditional and time-varying geometries, separated flows, nonlinear dynamic vehicle states, and high-fidelity modeling requirements for highly integrated vehicles. In short, there are no means for understanding the basic interactions that occur in systems dominated by nonlinearities in all three disciplines -structure, flow, and dynamics, nor are the computational interfaces adequate to handle the nonlinear interdisciplinary interactions.

Parallel Multigrid Algorithm for Aeroelasticity Simulations

This paper presents the development of a multigrid parallel algorithm for a nonlinear aeroelastic analysis. The aeroelastic model consists of 1) a nonlinear structural model that captures in-plane, out-of-plane, and torsional couplings; 2) an unsteady viscous aerodynamic model that captures compressible flow effects for transonic flows with shock/boundary-layer interaction; and 3) a solution methodology that assures a tightly coupled solution of the nonlinear structure and the fluid flow, including a consistent geometric interface between the highly deforming structure and the flowfield. A domain-decomposition parallel computation algorithm based on a message-passing interface was developed for the flow solver. A three-level multigrid algorithm was implemented in the flow solver to further reduce the computational time. A grid generation and deformation algorithm was developed concurrently with the flow solver in order to improve the efficiency of the computation. The grid deformation methodology kept the mesh topology unchanged as the structure deformed. Consequently, it was not necessary for either the parallel computation or the multigrid algorithm to update their communication pointers while the structure deformed. The validation of the numerical solver was done using experimental results of the F-5 wing. The aeroelastic solver was then used to assess the effect of structural nonlinearities on the aeroelastic response of the heavy Goland wing.