Computational Fluid Dynamics F7018T. Part II: Finite volume methods

Transcription

1 Computational Fluid Dynamics F7018T Part II: Finite volume methods

2 Questions to be answered Why numerical solutions of fluid mechanical problems? What is CFD? Why is it wrong to compare CFD with FEM? Can we treat CFD-codes as black-boxes and blindly trust in the results? Which errors can be introduced when using a numerical approach?

3 Why numerical solutions of fluid mechanical problems? The equations of fluid mechanics are solvable for only a limited number of flows. Experiments can be difficult (expensive) to carry out and/or measuring equipment may disturb the flow Computer capacity increases and codes becomes more efficient

4 What is CFD?

5 What is CFD? A method that approximate partial differential equations by a system of algebraic equations. The approximations are applied to small domains in space and/or time. The results are obtained in disrete locations in space and time. Quality very much dependent on the setup of the problem and the discretization.

6 What is CFD? Carefully performed numerical simulations in combination with accurate validation experiments is a powerful tool for flow prediction and optimisation. Colourful Fluid Dynamics

7 Best Practice Guidelines Model Uncertainty There is likely to be differences between the model and reality. The most common error in this category is modelling of turbulence since all models that exist are approximations of the flow. Neglecting compressibility and/or Non-Newtonian effects are other sources of error. Discretisation or Numerical Error This error appears since the governing equations are solved in discrete points in space and time. In general, the finer grid used the better solution but also the distribution of the grid points will strongly influence on the results. Iteration or Convergence Error The governing equations are generally solved in an iterative manner starting from an initial guess and leading to a final result. An incomplete iteration procedure naturally results in an error. One reason to an iterative error is that the engineer is not patient enough, finishing the procedure too early. Another is that the iterative parameter used is not the critical one.

8 Best Practice Guidelines Round-Off Error When performing computer based numerical simulations, there is always a limit on the machine accuracy. This is of importance when the difference between two values of a parameter is in the same range as this limit. Application Uncertainties The application is often complex and all data required is usually not given regarding geometry and boundary conditions, for instance. User Error Maybe the most obvious error. This error generally decreases with the experience of the user. Code Error Typical codes usually consist of hundreds of thousands of lines and it is likely that some errors are introduced during the writing. Post processing

9 Definitions Verification: Procedure to ensure that the program solves the equations correctly Validation: Procedure to test the extent to which the model accurately represents reality

10 Components of a numerical solution Make a virtual model of reality Choose mathematical models to solve Select co-ordinate system Find appropriate boundary conditions Select a method for dicretization Define a grid Define level (type) of approximation Select solution method Set a convergence criteria Choose method to evaluate the results

11 Student diploma work at VAC 2005 The flow through cooling channel in rocket nozzle Experimental value around 750K New model with prolonged inlet Tsimulated at same point 762K, close to experimental data First model with estimated inlet boundary condition Tsimulated at a certain point = 398K

12 Methods of interest for CFD Finite Difference Methods Simple and effective Conservation not enforced Differential form Finite Volume Methods Often used in fluid mechanics Conservation of mass Integral form Finite Element Methods Often in solid mechanics Integral form Conservation of mass not enforced but it can be included in the code

13 Finite Difference Methods General form of conservation equation = typical equation to solve: ρ is the density u j is the velocity Γ is the diffusivity of φ q φ is a source or sink of φ φ is some quantity, temperature, for instance

14 Discretization of geometry

15 Approximations of first derivative

16 Truncation error

17 Finite volume methods

18 This conservation equation applies to each CV, as well as to the solution domain as a whole. If we sum equations for all CVs the global conservation equation results Thus global conservation automatically apply

19 Control Volumes and notation

20 Steps to obtain values of surface integrals 1. The integral is appr. in terms of the variable value at one or several locations on the cell wall 2. The cell wall values are appr. in terms of the nodal (CV-center) values

21 Approximation of the integrals

22 Questions to be answered Why numerical solutions of fluid mechanical problems? What is CFD? Why is it wrong to compare CFD with FEM? Can we treat CFD-codes as black-boxes and blindly trust in the results? Which errors can be introduced when using a numerical approach?

23 Vad är ett finit element? Området problemet är definierat för delas upp ett antal finita element Varje element består av ett antal noder i=1..n Lösningsvariabeln uppskattas på det finita elementet m h a n antal formfunktioner N i (shape functions) tillhörande varje nod Formfunktionen antar värdet 1 på sin egen nod och värdet 0 på övriga noder

24 Linjärt finit element i 1D Elementet har två noder x j, x j+1 Lösningsvariabeln y* approximeras efter Formfunktionerna uppfyller villkoren * ) ( ) ( ) ( + + = j j e y x N y x N x y X j+1 x x j 1 ) ( 0 ) ( 0 ) ( 1 ) ( = = = = j+ j+ j j x N x N x N x N

25 2D-case

26 Hur kan part. Diff. Ekvationer lösas med FEM? Variationsmetoder (Variational Calculus Method) Används vanligen inom strukturmekanik. Kan appliceras på vissa specialfall för strömningsproblem (potentialflöde) Viktade residualer (Method of weighted residuals)

27 The method of weighted residuals (MWR) Söker en lösning på partiell diff. ekvation på formen D[y(x),x]=0 a<x<b Approximativa lösningen y*(x) insättes i D vilket ger ett fel R (residual), enligt R=D[y*()] 0 Felet minimeras m h a en integral över hela intervallet. b a w i ( x) R( x) dx = 0 i=1,..,n

28 The method of weighted residuals (MWR) w i representerar n stycken godtyckliga viktsfunktioner. Olika variationer av MWR existerar och skillnaden är huvudsakligen val av viktsfunktion. Tre av de vanligaste är: Integral relation method (w i = 0 eller 1) Galerkin (w i = N i (x)) Minsta kvadratmetoden Resulterar i n algebraiska ekvationer för att lösa okända parametrar.

29 Hybrid I FV kan interpolationsfunktioner (formfunktioner) användas för att approximera lösningsvariabeln så att den är känd i integrationspunkterna. Ex: i CFX används formfunktioner för att approximera tryckgradienten och diffusionstermerna.