Pipes used for transporting high velocity pressurized fluids often operate under timevarying

Transcription

1 Chapter 1 Introduction 1.1 Overview Pipes used for transporting high velocity pressurized fluids often operate under timevarying conditions. This can cause vibration problems. The fluid finite element based finite volume model depends on flow velocity as the variable. In turbulent flow, the relative motion of the fluid in the boundary layer generates flow disturbances in the form of vortices or eddies. As the flow rate increases so does the amount of turbulence. There is a continuous transfer of energy from the main flow into large eddies, and from the large eddies into smaller eddies, which dissipate most of the energy. This process occurs in a narrow strip inside the boundary layer, in the neighborhood of the wall. This energy dissipation produces large kinetic energy losses in the fluid. As the fluid molecules in the vortices go from locations of higher kinetic energy to regions of lower kinetic energy, i.e., from near the edge of the boundary layer to near the wall, the kinetic energy of the fluid is converted into heat and potential energy in the form of pressure.. 1

2 2 These pressure fluctuations excite vibratory oscillations in the pipe through which the fluid is flowing. The movement of the pipe also causes additional pressure fluctuations in return. This two-way interaction results in flow-induced vibration. The procedure for determining the relationship between flow rate and pipe vibration consists of solving the fluid mechanical problem. 1.2 Turbulence Turbulent flows are filled with swirling and spiraling motions. This is especially true if the object itself is spinning like a planet or star where the Coriolis effect causes winds and currents to curve and wiggle around. Turbulence consists of fluctuations in the flow field in time and space. It is a complex process, mainly because it is three dimensional, unsteady and consists of many scales. It can have a significant effect on the characteristics of the flow. Turbulence occurs when the inertia forces in the fluid become significant compared to viscous forces, and is characterized by a high Reynolds Number. In principle, the Navier-Stokes equations describe both laminar and turbulent flows without the need for additional information. However, turbulent flows at realistic Reynolds numbers span a large range of turbulent length and time scales and would generally involve length scales much smaller than the smallest finite volume mesh which can be practically used in a numerical analysis. The Direct Numerical Simulation (DNS) of these flows would require computing power which is many orders of magnitude higher than available in the foreseeable future.

3 3 To enable the effects of turbulence to be predicted, a large amount of CFD research has concentrated on methods which make use of turbulence models. Turbulence models have been specifically developed to account for the effects of turbulence without recourse to a prohibitively fine mesh and Direct Numerical Simulation. Most turbulence models are statistical turbulence model, as described below. The two exceptions to this are the Large Eddy Simulation model and the Detached Eddy Simulation model. Most widely used Turbulence Models RANS based models: Standard k-ε Model Zero Equation Model RSM- (Reynolds Stress Model) RNG - (Re-normalized Group Model) NKE - (New k-ε Model due to Shih) GIR - (Model due to Girimaji) SZL - (Shi, Zhu, Lumley Model) Standard k-ω Model SST - (Shear Stress Transport Model) 1.3 FSI (coupled analysis) and FIV (uncoupled analysis) Fluid-structure interaction (FSI) exists in liquid-carrying pipes when pressure waves in the liquid cause stresses and strains in the pipes (and vice versa). Fluid-structure

4 4 interaction in piping systems consists of the transfer of momentum and forces between piping and the contained liquid during unsteady flow. Excitation mechanisms may be caused by rapid changes in flow and pressure or may be initiated by mechanical action of the piping. The interaction is manifested in pipe vibration and perturbations in velocity and pressure of the liquid. The main sources of excitement can be summarized as in the Figure 1-1 below. Steady-flow and water hammer (transient) analyses provide information on the liquid behavior under operational conditions. Static pipe-stress analyses and structural dynamics analyses give insight into the corresponding behavior of the pipe system. Where the liquid analysis yields pressure and velocities, the structural analysis provides dynamic stresses, reaction forces and resonance frequencies. It is not unusual to perform an uncoupled calculation. Pressure histories, resulting from a transient analysis, are used as excitation loads in a structural-dynamics analysis. The calculation is called uncoupled since the predicted structural response does not influence the predicted liquid pressures. Three types of liquid-pipe coupling can be distinguished: friction, Poisson and junction coupling. Friction and Poisson coupling act along the entire pipe (distributed forces) whereas junction coupling acts at specific points (local forces) such as junctions or discontinuities in the pipe network (e.g. bends, valves). Poisson coupling relates pressures in the liquid to axial stresses in the pipe through radial contraction/expansion and leads to precursor waves (pressure changes induced by axial stress waves). Friction coupling is the interaction between the pipe wall and the fluid. It has the smallest effect on the overall response and is most often neglected.

5 5 Turbo machinery vibration Seismic motion Valve Action Liquid Coupling Piping Pipe rupture Column Separation Machine Vibration Figure 1-1 Sources of excitement and interactions between liquid and piping The terms FSI (fluid-structure interaction) and FIV (flow-induced vibration) are used promiscuously in the literature. Some of the ways FSI and FIV have been defined are as follows: FSI generally involves two-way (fluid-pipe) interaction. For FIV the interaction normally is one-way. Uncoupled analysis therefore refer to FIV, and coupled analysis to FSI FSI considers vibration, noise, acoustics etc. FIV considers only vibrations. Source of excitement can be liquid or structure in FSI. FIV considers the source of excitement only to be the liquid. The term FSI is often used for unsteady flow interacting with pipe vibration, whereas the term FIV is often used for stationary flow inducing pipe vibration. In the present thesis FIV would be referring to all the first three definitions. And the uncoupled analysis would be transient. Therefore FIV would be defined as unsteady flow

6 6 inducing pipe vibration caused by one way interaction of fluid and pipe, dealing only in vibrations. 1.4 Advantages of Uncoupled Analysis The keywords that favor the partitioned or uncoupled approach are: Customization, Independent modeling, Software Reuse, and Modularity. 1. Customization: This means that each field can be treated by discretization techniques and solution algorithms that are known to perform well for the isolated system. 2. Independent Modeling: The partitioned approach facilitates the use of non-matching models. For example in a fluid-structure interaction problem the structural and fluid meshes need not coincide at their interface. This translates into project breakdown advantages in analysis of complex systems such as aircraft. Separate models can be prepared by different design teams, including subcontractors that may be geographically distributed. 3. Software Reuse: Along with customized discretization and solution algorithms, customized software (private, public or commercial) may be available. Furthermore, there is often a gamut of customized peripheral tools such as mesh generators and

7 7 visualization programs. The uncoupled approach facilitates taking advantage of existing code. This is particularly suitable to academic environments, in which software development tends to be cyclical and loosely connected from one project to another. 4. Modularity: New methods and models may be introduced in a modular fashion according to project needs. For example, it may be necessary to include local nonlinear effects in an individual field while keeping everything else the same. Implementation, testing and validation of incremental changes can be conducted in a modular fashion. 1.5 CFD code description. The following section would discuss two of the commercial CFD codes used in the process of simulating compressed methane flow through the pipe junction, FLUENT. ANSYS CFX. FLUENT is a cell centered finite volume, segregated/coupled, implicit/explicit, density based solution technique. In cell centered schemes the flow variables are stored at the centers of the mesh elements. CFX is cell-vertex finite volume, coupled implicit, pressure based solution technique (i.e., solves for pressure and velocity at the same time in the same A matrix). Pressure

8 8 and velocity are co-located, so p-v decoupling is dealt with using a Rhie-Chow approach. In vertex based schemes the flow variables are stored at the vertices of the mesh elements. CFX uses an unstructured Finite Element based Finite Volume method. The FE basis comes from the use of shape functions, common in FE techniques, to describe the way a variable changes across each element. It is also a node based code, where the solution variables are solved and stored at the centers of the finite volumes, or the vertices of the mesh. Both codes are about the same in accuracy, but hard to compare on an unstructured mesh as CFX is Cell-Vertex Fluent is Cell Centered. CFX assembles control volumes around the element vertices, resulting in polyhedral control volumes and hence there are fewer nodes than cells with a tet mesh. While this results in fewer control volumes, there are far more integration points so the resolution of gradients is more accurate per control volume. If one compares FLUENT and CFX solutions, one may find the FLUENT solution is slightly more accurate for the same mesh, but much more costly to run (since you are solving 5x the number of equations), so you can afford to run a finer mesh in CFX. The polyhedral control volumes are also much less sensitive to poor mesh quality. If your physics allows for it, you can see similar benefits in FLUENT by converting your tet mesh to polyhedral.

9 9 A comparison of relative advantages of FLUENT and CFX are as discussed in the following page by Table 1-1 Comparison between FLUENT and CFX Fluent is a cell-centered code. CFX is cell-vertex finite volume. To implement boundary condition better and to avoid corner singularity better, CVFV is preferable, at the cost of slight higher CPU time. It takes a little tuning to get Fluent to converge. Their solvers have many different models. Flow solver robustness is questionable on Robustness for steady problems is better in CFX The solver is very robust. It is easy to make complex models to work. complex models (combustion, multiphase, etc..) Fluent UDF's are very well documented, archived and very flexible. A big advantage. CFX has an easy to use CFX command language to do simple things. Otherwise CFX Command Languauge is not well documented. Preprocessing (Gambit and Fluent GUI together) is fast considering unstructured meshing, yet with somewhat antiquated user interface. So, if one doesn t care about Preprocessing is maybe a bit slower than Fluent in terms of overall speed. CFX has a much more modern GUI based on QT, is fully integrated into ANSYS Workbench

10 10 a modern look and feel GUI then it will be fine. which allows direct connections with CAD, is configurable like Fluent's solver GUI. The workflow in Workbench is really nice Unstructured meshing is not strength in GAMBIT, the unstructured mesh algorithm is rather poor, and sometimes for complex geometries, it is not even possible to create an unstructured mesh. and Fluent has nothing that even comes close to this. Unstructured meshing with CFX Mesh is very good, with high quality meshes very easily produced for even complex geometries. High quality Hybrid mesh algorithm creates meshes with prisms, tetrahedrons, pyramids and hexagons. Saves a lot of time in Preprocessing and is a huge advantage. Their post processor is nice in that it is directly integrated with their solvers, but is really limited in functionality. One needs to use Ensight or Fieldview or Tecplot to do Post processing wins hands down in CFX. The CFX post processor is on par with something like Fieldview or Ensight or Tecplot anything sophisticated. FSI is much more difficult to handle with FLUENT, with little integration between other FEA and the FLUENT code. With ANSYS WORKBENCH solutions to FSI problems are very easy with direct integration for a fully coupled solution between ANSYS and CFX. The need for matching meshes at the Solid-Fluid

11 11 boundary is not there, since it can use an interpolation function. FLUENT is highly unstable with velocity and mass flow rate boundary conditions for compressible flows From the current thesis point of view, CFX has a specific advantage, since velocity boundary conditions are numerically quite stable even for incompressible fluid problems. Few other functional advantages of CFX are as follows: An automatic meshing feature is included in the current CFX software, the CFX MESH. The solver uses an unstructured tetrahedral mesh, for which the user only specifies the geometry and surface grid. The solver then automatically performs the volume meshing, using a very fast Delaunay algorithm. This greatly speeds pre-processing and produces high-quality meshes that in turn ensure faster convergence. CFX increases meshing flexibility even further by allowing mixed element types tetrahedrons, hexahedrons, pyramids and wedges to be used at the same time. Considerable controversy has revolved around the use of tetrahedral elements in CFD software. The main arguments in their favor are that the meshgeneration procedure can be automated and that high-quality meshes are easier to obtain when refining the grid locally to resolve features of interest. On the down side, tetrahedrons may be less accurate than a corresponding hexahedral mesh, especially in boundary layers where the flow is aligned with the geometry. For

12 12 these types of flow, an alternative method of meshing, developed by AEA and General Electric Corporate Research and Development in Niskayuna, N.Y., is incorporated in CFX. This uses a revolutionary technique that will grow prismatic elements normal to a triangulated surface mesh, and then switch to tetrahedrons away from the surfaces. This permits the use of meshes that are structured normal to surfaces and are thus aligned with the flow in a boundary layer. CFX coupled solver offers a radically different approach that solves all the hydrodynamic equations as a single system. After one iteration, the velocity field and pressure almost satisfy both momentum and mass conservation but not entirely, because the equations are nonlinear. Iteration is still needed, but as momentum and mass continuity are always satisfied, far fewer iterations are necessary than with the SIMPLE algorithm. Typically, CFX requires only a few dozen iterations to converge, where a segregated solver would need hundreds or thousands. Furthermore, processing times for CFX scale much better with mesh size than do those for the SIMPLE algorithm. For the large meshes often required in real engineering simulations, this approach leads to significant reductions in run times and faster project completion. Flexibility to allow combinations of multiple operating systems and hardware in a single calculation: Hardware running Windows 2000/XP, UNIX and Linux to be run at the same time using the power of your entire network. The entire ANSYS CFX feature set is supported in parallel; for example, multiple frames of reference, generalized grid interfaces, radiation, and Eulerian and Lagrangian multiphase, combustion, etc.

13 13 The ANSYS CFX solver is significantly less sensitive to poor mesh quality than previous releases. Automated algorithms improve solver convergence behavior in cases with poor quality meshes, and the solver can deal with cases that have negative sector volumes or flat elements. Computing resources needed for a transient calculation can be optimized through the use of Time step Adaption & Extrapolated Initial Guess for transient calculations in ANSYS CFX Time step Adaption allows the solver to automatically adjust the physical time step in a transient solution based on userspecified criteria including target number of coefficient loops or Courant Number. The Extrapolated Initial Guess extends the solution from previous time steps as the initial guess for the current time step, providing a better starting condition and minimizing the required number of coefficient loops to reach time step convergence. Additionally, key numerical transient improvements have been made, which makes it possible to achieve 2nd Order Transient with one iteration per time step, for time steps in the explicit range. 1.6 Cell Centered-Vertex Centered In the paper by Dimitri J. Mavriplis [42] an overview of current unstructured mesh discretization and solution techniques is given. Cell-centered versus vertex-centered discretizations are discussed, as well as issues of grid alignment with flow features, higher-order reconstruction, and viscous term formulation.

14 14 A cell-centered approach on a tetrahedral mesh will contain many more degrees of freedom than a vertex-centered discretization on the same mesh, and can therefore be expected to yield higher accuracy and require higher computational expense. However, the cell-centered discretization results in a relatively sparse stencil, with each tetrahedron having only four neighbors, whereas in the vertex-based discretization each vertex has on average 14 neighbors, based on the number of edges in the mesh, which can be shown to of the order of 7N, for a mesh of N vertices. The vertex-based discretization can therefore be expected to be more accurate than a cell-based discretization using equivalent numbers of unknowns, since the former approach will result in a larger number of flux calculations. Additionally, the larger stencil has the potential for more robust reconstruction techniques and limiting procedures. In the final assessment, the most effective discretization is the one which provides the highest accuracy at the lowest cost. Although numerical experiments conducted by the author [43],[44] have verified the superior accuracy of cell-centered approaches versus vertex-based approaches on identical grids, but also suggested the vertex-based approach to be the most efficient approach overall, the issue has never been decided conclusively, in large part due to the lack of fully consistent comparisons between the two approaches using identical discretizations and solvers. The definition of an equivalent grid for comparing cell-based and vertex-based discretizations at equivalent accuracy levels remains an open question. A study by Levy [45] found that matching the number of surface grid variables for both grids achieved similar accuracy for aerodynamic quantities in transonic flow cases.

15 15 Another principal difference between cell-centered and vertex-centered discretizations relates to the application of boundary conditions. A fundamental problem arises in the vertex- based discretization, since individual boundary vertices may have ill-defined boundary conditions if they are located at the intersection of two faces with different boundary conditions, as shown in Figure 1-1 this situation never occurs for the cellcentered discretization. This is due to the fact that the elemental unit of the computational boundary corresponds to a mesh face, rather than a mesh vertex. Thus it is much more appropriate to define boundary conditions based on mesh faces rather than mesh vertices. In order to implement such a boundary condition in the case of vertex-based discretizations, a weak formulation must be used, where the boundary condition is introduced into the residual through a modified boundary flux. This obviates the need to assign boundary conditions to the mesh vertices, and results in boundary condition formulations which are similar for both the cell-centered and vertex centered discretizations, as shown in figure below BC type 1 (a) BC type 1 (b) BC type 2 BC type 2 Figure 1-2 Illustration of boundary condition implementation for (a) cell-based and (b) vertex-based discretizations.

16 16 Finally it is concluded that for transonic flow simulations, unstructured mesh methods can be equal or even superior to most block-structured and overset mesh methods in terms of delivered accuracy at fixed computational cost. On the one hand, equivalent accuracy on equivalent unstructured and structured grids can be expected for subsonic and transonic flows, and solver technology for unstructured mesh methods is in no way inferior to that used for structured mesh methods. Furthermore, unstructured mesh approaches generally scale better on massively parallel computer hardware than their structured counterparts. 1.7 CFX Meshing Application The Meshing Application CFX MESH in the ANSYS Workbench is an easy-to-use CFD mesh generation tool which produces high quality hybrid meshes for both simple and complex geometries. It is an integrated component within the Meshing Application of ANSYS Workbench. This highly-automated meshing tool is closely coupled to ANSYS DesignModeler for geometry creation and meshing. 3-D Proximity: 3-D proximity detection automatically refines the mesh in areas where geometric features are in close proximity and result in small spaces within the volume. This improves robustness and reduces user input. Prismatic inflation: Flow adjacent to walls is characterized by high velocity and turbulence gradients in the normal direction. While tetrahedral cells can be used in such

17 17 boundary layers, greater accuracy can be achieved using prism elements. ANSYS meshing software automatically creates prisms by extruding the triangular surface mesh, thereby ensuring the highest quality of solutions. This meshing tool provides a number of different prismatic inflation schemes. 1.8 Motivation and Importance of Present Work Finite-element methods are widely used in the analysis of solids and structures and they provide great benefits in product design. Today it is nearly impossible to ignore the advances that have been made in the computer analysis of structures without losing an edge in innovation and productivity. In recent decades some significant advances have been made also in the use of finite element methods for the analysis of fluids. Computational finite element modeling has been used in the past for fluid flow simulations but mostly in the aeronautics. However, the application of these methods for simulating fluid flows in mechanical engineering analysis is more recent. Both of these applications, the flow simulations and the structural analysis of mechanical engineering systems, have been rapidly advancing in the past decade largely due to the analysis capabilities that are becoming widely available for many practical applications. The evolution of these methods and in particular their implications on mechanical engineering design initiated the development of a new field of analysis. This new emerging field of analysis requires a either uncoupled solution or even better, a fully coupled solution of fluid flows with structural interactions and is commonly referred to as fluid-structure interaction.

18 18 Symptoms of FSI include vibrations, noise and fatigue damage to piping, supports and machinery. Other disruptions include leaking flanges, burst rupture disks, relief valve discharges and pipes jumping off their supports. FSI is not a widely recognised phenomenon and it is quite feasible that it is responsible for a significant number of unexplained piping failures and other unacceptable behaviour. Failures due to FSI are often attributed to other factors. For example, failure due to fatigue could in fact be FSI induced; failure due to corrosion could again be partially attributed to FSI, i.e. a weakened area together with an FSI event resulting in rupture. The aim of this thesis is twin pronged. One, it tries to identify how Flow Induced Vibrations affect the structure of the pipes and pipe junctions carrying fluids. Second, it tries to identify the regions of the pipe system where the effect of these vibrations are most evident, hence providing for a means to reduce the number of sensors needed in the field and their optimum locations, to pick up adverse vibrations in the structure. Most industries use these kinds of pipes to transport different fluids across their facilities; it becomes very imperative to study what kind of pipe geometries, supports etc...are needed to avoid fatal causalities due to induced vibrations. It is imperative to identify if there is a possibility of Flow Induced Vibrations in the system and its probable adverse affects. In course of this investigation if it seems like

19 19 Flow Induced Vibrations are not the cause then focus on other modes of possible failures need to be studied. 1.9 Thesis Structure The thesis is divided importantly into 7 major Chapters. This particular section is part of Chapter 1, which was an Introduction to Turbulence and FSI in Pipes carrying fluids. Chapter 2, corresponds to literature review. The Literature review is sorted according to the relevance of the papers to the thesis and according to CFD first and FSI later, rather than alphabetically or chronological order. With the most pertinent papers being mentioned at the start and the proceeding towards the less applicable ones. Chapter 3 extensively discusses about the Theory behind the Turbulence models used for Fluid Analysis, models for Structural modal and transient analysis, and expected errors in CFD and FEA. Chapter 4 deals with Methodology, procedures and objectives. Chapter 5 discusses about modeling the Fluid problem, steps involved in determining its convergence and discussion about wall functions and fully developed profile boundary conditions. Chapter 6 does the same for the Structural analysis, outlining the steps involved, modeling, meshing and convergence of the structural mesh.

20 20 Chapter 7 gives a detailed discussion of the results, their validity, and comparison between CFD analyses with two different simulation codes. It also discusses structural modal and transient results. The chapter then ends with the conclusion of the research.