1 / 66

800 likes | 1.33k Vues

AN INTRODUCTION TO COMPUTATIONAL FLUID DYNAMICS. Shuisheng He School of Engineering The Robert Gordon University. OBJECTIVES. The lecture aims to convey the following information/ message to the students: What is CFD The main issues involved in CFD, including those of Numerical methods

Télécharger la présentation
## AN INTRODUCTION TO COMPUTATIONAL FLUID DYNAMICS

**An Image/Link below is provided (as is) to download presentation**
Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.
Content is provided to you AS IS for your information and personal use only.
Download presentation by click this link.
While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.
During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

**AN INTRODUCTION TO COMPUTATIONAL FLUID DYNAMICS**Shuisheng He School of Engineering The Robert Gordon University Introduction to CFD (Pisa, 30/09/2005)**OBJECTIVES**The lecture aims to convey the following information/ message to the students: • What is CFD • The main issues involved in CFD, including those of • Numerical methods • Turbulence modelling • The limitations of CFD and the important role of validation and expertise in CFD Introduction to CFD (Pisa, 30/09/2005)**OUTLINE OF LECTURE**• Introduction • What is CFD • What can & cannot CFD do • What does CFD involve … • Issues on numerical methods • Mesh generation • Discretization of equation • Solution of discretized equations • Turbulence modelling • Why are turbulence models needed? • What are available? • What model should I use? • Demonstration • Use of Fluent Introduction to CFD (Pisa, 30/09/2005)**1. INTRODUCTION**Introduction to CFD (Pisa, 30/09/2005)**What is CFD?**• Computational fluid dynamics (CFD): • CFD is the analysis, by means of computer-based simulations, of systems involving fluid flow, heat transfer and associated phenomena such as chemical reactions. • CFD involves ... Introduction to CFD (Pisa, 30/09/2005)**What does CFD involve?**• Specification of the problem • Development of the physical model • Development of the mathematical model • Governing equations • Boundary conditions • Turbulence modelling • Mesh generation • Discretization of the governing equations • Solution of discretized equations • Post processing • Interpretation of the results Introduction to CFD (Pisa, 30/09/2005)**Depth of sea: 500m ~ 1000m**• Tidal current: 10 to 20m/s • Waves (unsteady): -5m/s to +5m/s • Diameters: 150~200mm • Gap above sea bed: 10mm An example • Initiation of the problem • DP Offshore Ltd is keen to know what (forces ) caused the damage they recently experienced with their offshore pipelines. • Development of the physical model • After a few meetings with the company, we have finally agreed a specification of the problem (For me, it defines the physical model of the problem to be solved): Introduction to CFD (Pisa, 30/09/2005)**Symmetry**Inlet: Flat inlet profiles V=25m/s Turbulence=5% 10D Outlet: fully developed zero gradient Flow Smooth wall 20D 10D An example (cont.) • Development of the mathematical model • Governing equations • Equations: momentum, thermal (x), multiphase (x), … • Phase 1: 2D, steady; Phase 2: unsteady, …, • The flow is turbulent! • Boundary conditions • Decide the computational domain • Specify boundary conditions Introduction to CFD (Pisa, 30/09/2005)**An example (cont.)**• Development of the mathematical model (cont.) • Turbulence model • Initially, a standard 2-eq k-ε turbulence model is chosen for use. • Later, to improve simulation of the transition, separation & stagnation region, I would like to consider using a RNG or a low-Re model • Mesh generation • Finer mesh near the wall but not too close to wall • Finer mesh behind the pipe Introduction to CFD (Pisa, 30/09/2005)**An example (cont.)**• Discretization of the equations • Start with 1st order upwind, for easy convergence • Consider to use QUICK for velocities, later. • There is no reason for not using the default SIMPLER for pressure. • Solver • Use Uncoupled rather than coupledmethod • Use default setup on under-relaxation, but very likely, this will need to be changed later • Convergence criterion: choose 10-5 initially: check if this is ok by checking if 10-6 makes any difference. Iteration • Start iteration Failed • Plot velocity or other variable to assist identifying the reason(s) • Potential changes in: relaxation factors, mesh, initial guess, numerical schemes, etc. Converged solution • Eventually, solution converged. Introduction to CFD (Pisa, 30/09/2005)**An example (cont.)**• Post processing • Interpretation of results Force vector: (1 0 0) pressure viscous total pressure viscous total zone name force force force coefficient coefficient coefficient n n n ------------------------- -------------- -------------- -------------- -------------- -------------- -------------- pipe 8.098238 0.12247093 8.2207089 13.221613 0.1999 13.421566 ------------------------- -------------- -------------- -------------- -------------- -------------- -------------- net 8.098238 0.12247093 8.2207089 13.221613 0.199 13.421566 Introduction to CFD (Pisa, 30/09/2005)**Specify the problem**Select turbulence model Generate Mesh Discretize equations Solve discretized equations Post processing CFD road map Pre-processor Solver Post-processor Introduction to CFD (Pisa, 30/09/2005)**Why CFD?**• Continuity and Navier-Stokes equations for incompressible fluids: Introduction to CFD (Pisa, 30/09/2005)**Flow in a pipe**• For laminar flow: ? • For turbulent flow: Or Why CFD? (cont.) • Analytical solutions are available for only very few problems. • Experiment combined with empirical correlations have traditionally been the main tool - an expensive one. • CFD potentially provides an unlimited power for solving any flow problems Important conclusion: There is no analytical solution even for a very simple application, such as, a turbulent flow in a pipe. Introduction to CFD (Pisa, 30/09/2005)**CFD applications**• Aerospace • Automobile industry • Engine design and performance • The energy sector • Oil and gas • Biofluids • Many other sectors Introduction to CFD (Pisa, 30/09/2005)**CFD applications (cont.)**• As a design tool, CFD can be used to perform quick evaluation of design plans and carry out parametric investigation of these designs. • As a research tool, CFD can provide detailed information about the flow and thermal field and turbulence, far beyond these provided by experiments. Introduction to CFD (Pisa, 30/09/2005)**What can CFD do?**• Flows problems in complex geometries • Heat transfer • Combustions • Chemical reactions • Multiphase flows • Non-Newtonian fluid flow • Unsteady flows • Shock waves Introduction to CFD (Pisa, 30/09/2005)**What can’t CFD do?**• CFD is still struggling to predict even the simplest flows reliably, for example, • A jet impinging on a wall • Heat transfer in a vertical pipe • Flow over a pipe • Combustion in an engine • Important conclusions: • Validation is of vital importance to CFD. • Use of CFD requires more expertise than many other areas • CFD solutions beyond validation are often sought and expertise plays an important role here. Introduction to CFD (Pisa, 30/09/2005)**Validation of CFD modelling**Errors involved in CFD results • Discretization errors • Depending on ‘schemes’ used. Use of higher order schemes will normally help to reduce such errors • Also depending on mesh size – reducing mesh size will normally help to reduce such errors. • Iteration errors • For converged solutions, such errors are relatively small. • Turbulence modelling • Some turbulence models are proved to produce good results for certain flows • Some models are better than others under certain conditions • But no turbulence model can claim to work well for all flows • Physical problem vs mathematical model • Approximation in boundary conditions • Use of a 2D model to simplify calculation • Simplification in the treatment of properties Introduction to CFD (Pisa, 30/09/2005)**Validation of CFD modelling (cont.)**• CFD results always need validation. They can be • Compared with experiments • Compared with analytical solutions • Checked by intuition/common sense • Compared with other codes (only for coding validation!) Introduction to CFD (Pisa, 30/09/2005)**Commercial CFD packages**• Phoenix • Fluent • Star-CD • CFX (FLOW3D) • Many others • Computer design tools – integrating CFD into a design package Introduction to CFD (Pisa, 30/09/2005)**Specify the problem**Generate Mesh Select equations to solve Select turbulence models Define boundary conditions Select numerical methods Iterate – solve equations Fail – calculation does not converge or converges too slowly Improve model: Physical model Mesh Better initial guess Numerical methods (e.g., solver, convection scheme) Under-relaxations Post processing Interpretation of results – Always question the results How to use a CFD package? Introduction to CFD (Pisa, 30/09/2005)**How to use a CFD package? (cont.)**• Important issues involved in using CFD: • Mesh independence check • Selection of an appropriate turbulence model • Validation of the solution based on a simplified problem (which contains the important features similar to your problem) • Careful interpretation of your results Introduction to CFD (Pisa, 30/09/2005)**How to use a CFD package? (cont.)**• The commercial packages are so user friendly and robust, why do we still need CFD experts? Because they can provide: • Appropriate interpretation of the results and knowledge on the limitations of CFD • More accurate results (by choosing the right turbulence model & numerical methods) • Ability to obtain results (at all) for complex problems • Speed: both in terms of the time used to generate the model and the computing time Introduction to CFD (Pisa, 30/09/2005)**Basic CFD strategies**• Finite difference (FD) • Starting from the differential form of the equations • The computational domain is covered by a grid • At each grid point, the differential equations (partial derivatives) are approximated using nodal values • Only used in structured grids and normally straightforward • Disadvantage: conservation is not always guaranteed • Disadvantage: Restricted to simple geometries. • Finite Volume (FV) • Finite element (FE) Introduction to CFD (Pisa, 30/09/2005)**Basic CFD strategies (cont.)**• Finite difference (FD) • Finite Volume (FV) • Starting from the integral form of the governing equations • The solution domain is covered by control volumes (CV) • The conservation equations are applied to each CV • The FV can accommodate any type of grid and suitable for complex geometries • The method is conservative (as long as surface integrals are the same for CVs sharing the boundary) • Most widely used method in CFD • Disadvantage: more difficult to implement higher than 2nd order methods in 3D. • Finite element (FE) Introduction to CFD (Pisa, 30/09/2005)**Basic CFD strategies (cont.)**• Finite difference (FD) • Finite Volume (FV) • Finite element (FE) • The domain is broken into a set of discrete volumes: finite elements • The equations are multiplied by a weight function before they are integrated over the entire domain. • The solution is to search a set of non-linear algebraic equations for the computational domain. • Advantage: FE can easily deal with complex geometries. • Disadvantage: since unstructured in nature, the resultant matrices of linearized equations are difficult to find efficient solution methods. • Not often used in CFD Introduction to CFD (Pisa, 30/09/2005)**2. ISSUES IN NUMERICAL METHODS**Introduction to CFD (Pisa, 30/09/2005)**CFD Road Map**Specify the problem Select turbulence model Generate Mesh Discretize equations Solve discretized equations Post processing Mesh generation Why do we care? • 50% time spent on mesh generation • Convergence depends on mesh • Accuracy depends on mesh Main topics • Structured/unstructured mesh • Multi-block • body fitted • Adaptive mesh generation Introduction to CFD (Pisa, 30/09/2005)**- MESH GENERATION -Computational domain and mesh structure**• Carefully select your computational domain • The mesh needs • to be able to resolve the boundary layer • to be appropriate for the Reynolds number • to suit the turbulence models selected • to be able to model the complex geometry Introduction to CFD (Pisa, 30/09/2005)**- MESH GENERATION -Structure/unstructured mesh**• Structured grid • A structured grid means that the volume elements (quadrilateral in 2D) are well ordered and a simple scheme (e.g., i-j-k indices) can be used to label elements and identify neighbours. • Unstructured grid • In unstructured grids, volume elements (triangular or quadrilateral in 2D) can be joined in any manner, and special lists must be kept to identify neighbouring elements Introduction to CFD (Pisa, 30/09/2005)**- MESH GENERATION -Structure/unstructured mesh**• Structured grid Advantages: • Economical in terms of both memory & computing time • Easy to code/more efficient solvers available • The user has full control in grid generation • Easy in post processing Disadvantages • Difficult to deal with complex geometries • Unstructured grid • Advantages/disadvantages: opposite to above points! Introduction to CFD (Pisa, 30/09/2005)**- MESH GENERATION -Multi-Block and Overset Mesh**Introduction to CFD (Pisa, 30/09/2005)**- MESH GENERATION -Body fitted mesh - transformation**Regular mesh Body fitted mesh Introduction to CFD (Pisa, 30/09/2005)**- MESH GENERATION -Adaptive mesh generation**• Adaptive mesh generation • The mesh is modified according to the solution of the flow • Two types of adaptive methods • Local mesh refinement • Mesh re-distribution • Dynamic adaptive method • Mesh refinement/redistribution are automatically carried out during iterations • Demonstration – flow past a cylinder Introduction to CFD (Pisa, 30/09/2005)**CFD Road Map**Specify problem Select turbulence model Generate Mesh Discretize equations Solve discretized equations Post processing Equation discretization Relevant issues • Convergence strongly depends on numerical methods used. • Accuracy – discretization errors Main topics • Staggered/collocated variable arrangement • Convection schemes • Accuracy • Artificial diffusion • Boundedness • Choice of many schemes • Pressure-velocity link • Linearization of source terms • Boundary conditions Introduction to CFD (Pisa, 30/09/2005)**V**U,V,P,T U P,T - EQUATION DISCRETIZATION -Staggered/collocated variable arrangement • Collocated variable arrangement • All variables are defined at nodes • Staggered variable arrangement • Velocities are defined at the faces and scalars are defined as the nodes Collocated Arrangement Staggered Arrangement Introduction to CFD (Pisa, 30/09/2005)**The problem:**Unless special measures are taken, the collocated arrangement often results in oscillations The reason is the weak coupling between velocity & pressure Staggered variable arrangement Almost always been used between 60’s and early 80’s Still most often used method for Cartesian grids Disadvantage: difficult to treat complex geometry Collocated variable arrangement Methods have been developed to over-come oscillations in the 80’s and such methods are often being used since. Used for non-orthogonal, unstructured grids, or, for multigrid solution methods - EQUATION DISCRETIZATION -Staggered/collocated variable arrangement Introduction to CFD (Pisa, 30/09/2005)**- EQUATION DISCRETIZATION -Convection schemes**The problem • To discretize the equations, convections on CV faces need to be calculated from variables stored on nodal locations • When the 2nd order-accurate linear interpolation is used to calculate the convection on the CV faces, undesirable oscillation may occur. • Development/use of appropriate convection schemes have been a very important issue in CFD • There are no best schemes. A choice of schemes is normally available in commercial CFD packages to be chosen by the user. Introduction to CFD (Pisa, 30/09/2005)**- EQUATION DISCRETIZATION -Convection schemes (cont.)**The requirements for convection schemes: • Accuracy: Schemes can be 1st, 2nd, 3rd...-order accurate. • Conservativeness: Schemes should preserve conservativeness on the CV faces • Boundedness: Schemes should not produce over-/under-shootings • Transportiveness: Schemes should recognize the flow direction Introduction to CFD (Pisa, 30/09/2005)**- EQUATION DISCRETIZATION -Convection schemes (cont.)**Examples of convection schemes • 1st order schemes: • Upwind scheme (UW): most often used scheme! • Power law scheme • Skewed upwind • Higher order schemes • Central differencing scheme (CDS) – 2nd order • Quadratic Upwind Interpolation for Convective Kinematics (QUICK) – 3rd order and very often used scheme • Bounded higher-order schemes • Total Variation Diminishing (TVD) – a group of schemes • SMART Introduction to CFD (Pisa, 30/09/2005)**- EQUATION DISCRETIZATION -Pressure-velocity link**• The problem • The pressure appears in the momentum equation as the driving force for the flow. But for incompressible flows, there is no transport equation for the pressure. • In stead, the continuity equation will be satisfied if the appropriate pressure field is used in the momentum equations • The non-linear nature of and the coupling between, the various equations also pose problems that need care. • The remedy • Iterative guess-and-correct methods have been proposed – see next slide. Introduction to CFD (Pisa, 30/09/2005)**- EQUATION DISCRETIZATION -Pressure-velocity link (cont.)**Most widely used methods • SIMPLE (Semi-implicit method for pressure-linked equations) • A basic guess-and-correct procedure • SIMPLER (SIMPLE-Revised): used as default in many commercial codes • Solve an extra equation for pressure correction (30% more effort than SIMPLE). This is normally better than SIMPLE. • SIMPLEC (SIMPLE-Consistent): Generally better than SIMPLE. • PISO (Pressure Implicit with Splitting of Operators) • Initially developed for unsteady flow • Involves two correction stages Introduction to CFD (Pisa, 30/09/2005)**- EQUATION DISCRETIZATION -Linearization of source terms**• This slide is only relevant to those who develops CFD codes. • The treatment of source terms requires skills which can significantly increase the stability and convergence speed of the iteration. • The basic rule is that the source term should be linearizated and the linear part can the be solved directly. • An important rule is that only those of linearization which result in a negative gradient can be solved directly Introduction to CFD (Pisa, 30/09/2005)**- EQUATION DISCRETIZATION -Boundary conditions**• Specification of boundary conditions (BC) is a very important part of CFD modelling • In most cases, this is straightforward but, in some cases, it can be very difficult ..., • Typical boundary conditions: • Inlet boundary conditions • Outlet boundary conditions • Wall boundary conditions • Symmetry boundary conditions • Periodic boundary conditions Introduction to CFD (Pisa, 30/09/2005)**Relevant issues**• Cost/speed • Stability/Convergence Main topics • Solver – solution of the discretized equation system • Convergence criteria • Under-relaxation • Solution of coupled equations • Unsteady flow solvers CFD Road Map Specify problem Select turbulence model Generate Mesh Discretize equations Solve discretized equations Post processing Solution of discretized equations Introduction to CFD (Pisa, 30/09/2005)**- SOLUTION OF DISCRETIZED EQUATIONS -Solvers**• Discretized Equations – large linearized sparse matrix = * Introduction to CFD (Pisa, 30/09/2005)**- SOLUTION OF DISCRETIZED EQUATIONS -Solvers (cont.)**• The discretized governing equations are always sparse,non-linear but linearizated, algebraic equation systems • The ‘matrix’ from structured mesh is regular and easier to solve. • A non-structured mesh results in an irregular matrix. • Number of equations = number of nodes • Number of molecules in each line: • Upwind, CDS for 1D results in a tridiagonal matrix • QUICK for 1D results in a penta-diagonal matrix • 2D problems involves 5 & more molecules • 3D problems involves 7 & more molecules Introduction to CFD (Pisa, 30/09/2005)**Very expensive!**Very effective method used for tridiagonal matrix Simple and probably most often used method Used for more ‘complex’ problems Effective method for more ‘complex’ problems - SOLUTION OF DISCRETIZED EQUATIONS -Solvers (cont.) • Direct methods • Gauss elimination: • Tridiagonal Matrix Algorithm (TDMA): • Indirect methods • Basic methods: • Jacobi • Gauss-Seidel • Successive over-relaxation (SOR) • ADI-TDMA • Strongly implicit procedure (SIP) • Conjugate Gradient Methods (CGM) • Multigrid Methods Introduction to CFD (Pisa, 30/09/2005)**- SOLUTION OF DISCRETIZED EQUATIONS -Convergence criteria**• Two basic methods: • Changes between any two iterations are less than a given level • Residuals in the transport equations are less than a given value • Criteria can be specified using absolute or relative values Introduction to CFD (Pisa, 30/09/2005)

More Related