Basics Of Fluid Dynamics And Numerics PDF 2024

BASICS OF FLUID DYNAMICS AND NUMERICS Dr. Peter Priesching 05.10.2024 Lecture CFD / FH-Joanneum / WS 2024/25 / Dr. Peter Priesching 1 BASICS OF FLUID DYNAMICS AND NUMERICS Overview ▪ Introduction...

BASICS OF FLUID DYNAMICS AND NUMERICS Dr. Peter Priesching 05.10.2024 Lecture CFD / FH-Joanneum / WS 2024/25 / Dr. Peter Priesching 1 BASICS OF FLUID DYNAMICS AND NUMERICS Overview ▪ Introduction ▪ Basics of fluid dynamics ▪ Navier – Stokes equations ▪ Turbulence ▪ Basics of the numerical solution ▪ General ▪ Discretization ▪ Numerical solution Lecture CFD / FH-Joanneum / WS 2024/25 / Dr. Peter Priesching 2 Assement ▪ No overall exam at the end of the semester ▪ Practical work in first and second half of the lecture ▪ Short Moodle test in first and second half of the lecture Practical Example 1 Moodle Test 1 Practical Example 2 Moodle Test 2 Max Points 100 100 100 100 ▪ Positive grade at ≥ 201 points Lecture CFD / FH-Joanneum / WS 2024/25 / Dr. Peter Priesching 3 BASICS OF FLUID DYNAMICS AND NUMERICS Overview ▪ Introduction ▪ Basics of fluid dynamics ▪ Navier – Stokes equations ▪ Turbulence ▪ Basics of the numerical solution ▪ General ▪ Discretization ▪ Numerical solution Lecture CFD / FH-Joanneum / WS 2024/25 / Dr. Peter Priesching 4 BASICS OF FLUID DYNAMICS AND NUMERICS Overview ▪ What is CFD? ▪ Computational Fluid Dynamics – Denomination of numerical methods for solving flow problems. ▪ Why CFD? ▪ An analytical solution of the Navier – Stokes equations is only possible for a few limited examples. Numerical methods are necessary to calculate general flow problems. Lecture CFD / FH-Joanneum / WS 2024/25 / Dr. Peter Priesching 5 BASICS OF FLUID DYNAMICS AND NUMERICS Introduction Lecture CFD / FH-Joanneum / WS 2024/25 / Dr. Peter Priesching 6 BASICS OF FLUID DYNAMICS AND NUMERICS Introduction Lecture CFD / FH-Joanneum / WS 2024/25 / Dr. Peter Priesching 7 BASICS OF FLUID DYNAMICS AND NUMERICS Introduction Example: Tractor Underhood Operating point 1 Operating point 5 Lecture CFD / FH-Joanneum / WS 2024/25 / Dr. Peter Priesching 8 BASICS OF FLUID DYNAMICS AND NUMERICS Introduction Example: Grain Drilling Machine Lecture CFD / FH-Joanneum / WS 2024/25 / Dr. Peter Priesching 9 BASICS OF FLUID DYNAMICS AND NUMERICS Introduction Example: Water Management Wading Raining Microfluidics Lecture CFD / FH-Joanneum / WS 2024/25 / Dr. Peter Priesching 10 BASICS OF FLUID DYNAMICS AND NUMERICS Introduction Example: E-Machine Cooling STATIC E-MACHINES SLIDING E-MACHINES Lecture CFD / FH-Joanneum / WS 2024/25 / Dr. Peter Priesching 11 BASICS OF FLUID DYNAMICS AND NUMERICS Introduction Example: Flow, Mixture Formation, Combustion in Engines Flow through a Diesel injector Lecture CFD / FH-Joanneum / WS 2024/25 / Dr. Peter Priesching 12 BASICS OF FLUID DYNAMICS AND NUMERICS Introduction Example: Battery Performance, Safety, Thermal Analysis Complete battery pack thermal analysis Battery pack Chevrolet BOLT Current density (vectors) Electric potential Lecture CFD / FH-Joanneum / WS 2024/25 / Dr. Peter Priesching 13 BASICS OF FLUID DYNAMICS AND NUMERICS Overview ▪ Introduction ▪ Basics of fluid dynamics ▪ Navier – Stokes equations ▪ Turbulence ▪ Basics of the numerical solution ▪ General ▪ Discretization ▪ Numerical solution Lecture CFD / FH-Joanneum / WS 2024/25 / Dr. Peter Priesching 14 BASICS OF FLUID DYNAMICS AND NUMERICS Fluid Dynamics ▪ Until the 16th century there is no prove about real calculations – except Archimedes‘(287 – 212 BC) work about ‚floating bodies‘ ▪ Only much later the work of scientists like da Vinci (1452 – 1519), Galilei (1564 – 1642), Pascal (1623 – 1662), Newton (1642 – 1727), Bernoulli (1700 – 1782), Euler (1707 – 1783), d‘Alambert (1717 – 1783), Lagrange (1736 – 1813) is forming the basics for a scientific fluid mechanics. ▪ In the second half of the 18th century and in the beginning of the 19th a theory has been developed, which tries to explain fluid flow by the mechanics of solid bodies (Newton, Euler, Lagrange). ▪ In the middle of the 19th century Navier (1786 – 1836) und Stokes (1819 – 1903) have been deriving equations, which describe the flow of real (viscous) fluids. ▪ Initially the main interest has been focusing on water as a fluid. At the beginning of the 20th century aviation industry and different other industry branches start requesting air as a fluid. At that time scientists like Reynolds (1842 – 1912) and Prandtl (1865 – 1953) (boundary layer theory 1904) contribute to the research. Lecture CFD / FH-Joanneum / WS 2024/25 / Dr. Peter Priesching 15 BASICS OF FLUID DYNAMICS AND NUMERICS Fluid Dynamics Some Definitions ▪ Fluid mechanics applies the basic laws of mechanics and thermodynamics to describe the behavior of fluids. ▪ Fluids are all media, which are not solids. This means, they are mainly gases, liquids, vapors and plastic materials, … Lecture CFD / FH-Joanneum / WS 2024/25 / Dr. Peter Priesching 16 BASICS OF FLUID DYNAMICS AND NUMERICS Fluid Dynamics Concept of Continuum ▪ There are two possibilities to analyze the flow. ▪ Fluid as an assembly of molecules (laws of solid mechanics) ▪ Concept of continuum (omitting the discrete structure of the matter) ▪ The assumption of the continuum is valid as long as the mean free path length of the molecules is small compared to the characteristic length of the flow. Simplified formulation for the mean free path length of molecules: T  = 3,8 10  −5 [m] p Lecture CFD / FH-Joanneum / WS 2024/25 / Dr. Peter Priesching 17 BASICS OF FLUID DYNAMICS AND NUMERICS Fluid Dynamics Examples for Mean Free Path Length Question: From which height the continuum assumption is not valid any more, if we look at a flow in a pipe with d=40 mm? Characteristics of the atmosphere Lecture CFD / FH-Joanneum / WS 2024/25 / Dr. Peter Priesching 18 BASICS OF FLUID DYNAMICS AND NUMERICS Fluid Dynamics Physical Properties of Fluids ▪ Within macroscopic observations mean values are applied: ▪ Pressure (p), temperature (T ), density (), specific heat (cp , cv) viscosity (, ), thermal conductivity (), … m ▪ Density ( ):  = lim in [kg/m³] V → 3 V specific volume: 1 v= in [m³/kg]  Lecture CFD / FH-Joanneum / WS 2024/25 / Dr. Peter Priesching 19 BASICS OF FLUID DYNAMICS AND NUMERICS Fluid Dynamics Physical Properties of Fluids ▪ Viscosity (, ): Is a property, which results from the effect of the resistance against shear forces (Newtonian law of viscosity). du  = dy  … dynamic viscosity in [Pa.s] mainly depending on temperature  =  … kinematic viscosity in [m²/s] depending on pressure and temperature Lecture CFD / FH-Joanneum / WS 2024/25 / Dr. Peter Priesching 20 BASICS OF FLUID DYNAMICS AND NUMERICS Fluid Dynamics Physical Properties of Fluids ▪ Specific heat (cp , cv) : Is the amount of (thermal) energy which is necessary to increase the temperature of 1 kg fluid by 1 K. → Unit [J/kgK] cv … at constant volume cp … at constant pressure ▪ The ratio of both quantities is the isentropic exponent. cp = cv ▪ Within an ideal gas the difference between cp and cv is equal to the specific gas constant R. R = c p − cv Lecture CFD / FH-Joanneum / WS 2024/25 / Dr. Peter Priesching 21 BASICS OF FLUID DYNAMICS AND NUMERICS Fluid Dynamics Physical Properties of Fluids ▪ Equation of state for gases: p pv = RT or = RT  Rm  J  with the specific gas constant: R= M  kg K   J  and the universal gas constant Rm = 8,3144    mol K  1 yi and the mean molecular weight of a mixture: = M Mi Lecture CFD / FH-Joanneum / WS 2024/25 / Dr. Peter Priesching 22 BASICS OF FLUID DYNAMICS AND NUMERICS Fluid Dynamics Classification of the flow via the flow velocity ▪ Very low velocity U dh Re = →0 ▪ ‚sneaking flow‘, friction dominates, ▪ Low velocity ▪ Re low enough, laminar flow, inertia dominates. ▪ Moderate velocity ▪ Re of intermediate value, unstable condition, transition to turbulent flow ▪ Even higher velocity v Ma = ▪ Compressibility becomes important, Mach number c Lecture CFD / FH-Joanneum / WS 2024/25 / Dr. Peter Priesching 23 BASICS OF FLUID DYNAMICS AND NUMERICS Fluid Dynamics Interesting quantities in the fluid dynamics ▪ Velocity at incompressible flow ▪ Pressure ▪ Density at compressible flow ▪ Temperature ▪ Turbulence ▪ Concentration Lecture CFD / FH-Joanneum / WS 2024/25 / Dr. Peter Priesching 24 BASICS OF FLUID DYNAMICS AND NUMERICS Fluid Dynamics Some more basic definitions ▪ Continuity equation = Equation of conservation of the mass ▪ „The time rate of change of the mass in a control volume is equal to the balance of the mass flows over the boundaries of the same control volume“. ▪ 2 models: Euler Lagrange ▪ Lagrangian view: The observer moves with the CV. ▪ Eulerian view: Static CV. The observer is static. (more often used) Lecture CFD / FH-Joanneum / WS 2024/25 / Dr. Peter Priesching 25 BASICS OF FLUID DYNAMICS AND NUMERICS Fluid Dynamics Different approaches how to observe and describe the flow: finte control volume, finte control volume, fixed in space moving with the flow V V S S infinitely small control volume, infinitely small control volume, fixed in space moving with the flow dV dV  v Eulerian Approach Lagrangian Approach Lecture CFD / FH-Joanneum / WS 2024/25 / Dr. Peter Priesching 26 BASICS OF FLUID DYNAMICS AND NUMERICS Fluid Dynamics Finite control volume – fixed in space p dS  Conservation equations in integral form   V  f S  v  n Continuity equation: balance all fluxes Momentum equation: balance all forces Lecture CFD / FH-Joanneum / WS 2024/25 / Dr. Peter Priesching 27 BASICS OF FLUID DYNAMICS AND NUMERICS Fluid Dynamics Infinitely small control volume – fixed in space   ( v  v ) ( v  v ) v  v + dy v  v + dy y y Conservation equations in differential form u  v     ( u  v ) ( u )v KV u  v + dx dy x v  u v  u u  u dx  ( u  u ) u  u + dx x  ( v)v v  v u  v Continuity equation: balance all fluxes dV = dx  dy 1 Momentum equation: balance all forces Lecture CFD / FH-Joanneum / WS 2024/25 / Dr. Peter Priesching 28 BASICS OF FLUID DYNAMICS AND NUMERICS Fluid Dynamics Infinitely small control volume – fixed in space  yy dy  yy + y p p + dy  +  yx dy Conservation equations in differential form y yx y  xy  xy KV fy  xy + dx  xx dy x p fx p  xx p + dx  xx + dx dx x x y p  yx x  yy Continuity equation: balance all fluxes Momentum equation: balance all forces Lecture CFD / FH-Joanneum / WS 2024/25 / Dr. Peter Priesching 29 BASICS OF FLUID DYNAMICS AND NUMERICS Fluid Dynamics Navier-Stokes Equations Claude Louis Marie Henri Navier (1827) George Gabriel Stokes (1845) ▪ Are the basic equations of fluid mechanics. ▪ They describe the flow velocities and the pressure distribution in Newtonean fluids (liquids and gases). ▪ They form a system of non-linear partial differential equations of 2nd order. ▪ The core is the momentum equation and the continuity equation. ▪ Additionally the enthalpy equation is necessary in most real life cases.  u j u j  p  ij  + ui  =− − + gj  t xi  x j xi   Momentum + (  ui ) = 0 t xi Continuity Lecture CFD / FH-Joanneum / WS 2024/25 / Dr. Peter Priesching 30 BASICS OF FLUID DYNAMICS AND NUMERICS Fluid Dynamics Navier-Stokes Equations Claude Louis Marie Henri Navier (1827) George Gabriel Stokes (1845) ▪ Are the basic equations of fluid mechanics. ▪ They describe the flow velocities and the pressure distribution in Newtonean fluids (liquids and gases). ▪ They form a system of non-linear partial differential equations of 2nd order. ▪ The core is the momentum equation and the continuity equation. ▪ Additionally the enthalpy equation is necessary in most real life cases.   t V  v dV = −  (  v) v  n dS −  p n dS +    n dS +  f dV S S S V Momentum   t V  dV = −   v n dS S Continuity Lecture CFD / FH-Joanneum / WS 2024/25 / Dr. Peter Priesching 31 BASICS OF FLUID DYNAMICS AND NUMERICS Fluid Dynamics General Transport Equation (  ) (  )     + ui = −    + S t xi xi  xi  with: ... arbitrary transported quantity ... diffusion coefficient S... source term Equation   S Continuity 1 0 0  Sc = Momentum - x u  -p+ u+f D Concentration c  / Sc Sc … Schmidt number Lecture CFD / FH-Joanneum / WS 2024/25 / Dr. Peter Priesching 32 BASICS OF FLUID DYNAMICS AND NUMERICS Overview ▪ Introduction ▪ Basics of fluid dynamics ▪ Navier – Stokes equations ▪ Turbulence ▪ Basics of the numerical solution ▪ General ▪ Discretization ▪ Numerical solution Lecture CFD / FH-Joanneum / WS 2024/25 / Dr. Peter Priesching 33 BASICS OF FLUID DYNAMICS AND NUMERICS Fluid Dynamics - Turbulence Turbulence is the unresolved problem of classical physics. Albert Einstein Lecture CFD / FH-Joanneum / WS 2024/25 / Dr. Peter Priesching 34 BASICS OF FLUID DYNAMICS AND NUMERICS Fluid Dynamics - Turbulence Lecture CFD / FH-Joanneum / WS 2024/25 / Dr. Peter Priesching 35 BASICS OF FLUID DYNAMICS AND NUMERICS Fluid Dynamics - Turbulence Color Experiment increasing Reynolds number A critical Re number can be defined for the transition from laminar to turbulent regime. U  L UL Re = =  for pipe flow e.g.: Recrit  2300 Lecture CFD / FH-Joanneum / WS 2024/25 / Dr. Peter Priesching 36 BASICS OF FLUID DYNAMICS AND NUMERICS Fluid Dynamics - Turbulence Transition of Boundary Layer https://news.mit.edu/ Lecture CFD / FH-Joanneum / WS 2024/25 / Dr. Peter Priesching 37 BASICS OF FLUID DYNAMICS AND NUMERICS Fluid Dynamics - Turbulence Definitions ▪ Turbulence is a property of the flow (not of the fluid) ▪ Turbulent flow is ▪ three dimensional ▪ transient (not steady) ▪ intensively mixing the fluid ▪ rotating the fluid ▪ dissipative (energy losses!) ▪ showing a large variety of length scales ▪ strongly affected by changing boundary conditions Lecture CFD / FH-Joanneum / WS 2024/25 / Dr. Peter Priesching 38 BASICS OF FLUID DYNAMICS AND NUMERICS Fluid Dynamics - Turbulence Example for large variety of length scales: Karman vortex street Landsat 7 image of clouds off the Chilean coast near the Juan Fernandez Islands, 15 September 1999, NASA Lecture CFD / FH-Joanneum / WS 2024/25 / Dr. Peter Priesching 39 BASICS OF FLUID DYNAMICS AND NUMERICS Fluid Dynamics - Turbulence Example for large variety of length scales: Karman vortex street Canary Islands, mages acquired April 24, 2000 Lecture CFD / FH-Joanneum / WS 2024/25 / Dr. Peter Priesching 40 BASICS OF FLUID DYNAMICS AND NUMERICS Fluid Dynamics - Turbulence Mean and Fluctuating Quantities The velocity, which can be measured in a turbulent flow field, shows strong fluctuations. They can be described by an addition of fluctuating velocities around a statistical mean value. u Flow velocity ui ( xi , t ) = ui ( xi ) + ui( xi , t ) Mean velocity ui( xi , t ) 1 T ui ( xi , t ) ui ( xi ) ui ( xi ) = lim  ui ( xi , t )dt T → T 0 Velocity fluctuation ui( xi , t ) t Lecture CFD / FH-Joanneum / WS 2024/25 / Dr. Peter Priesching 41 BASICS OF FLUID DYNAMICS AND NUMERICS Fluid Dynamics - Turbulence General Description of relevant Quantities as Fluctuations plus Mean Values In 1895 O. Reynolds introduced a new general description for observation of turbulent flows. All relevant quantities should be treated as fluctuations plus mean values. Velocity: ui ( xi , t ) = ui ( xi ) + ui( xi , t ) Pressure: pi ( xi , t ) = pi ( xi ) + pi( xi , t ) Density: i ( xi , t ) = i ( xi ) + i( xi , t ) Temperature: Ti ( xi , t ) = Ti ( xi ) + Ti( xi , t ) … Reynolds - averaging Lecture CFD / FH-Joanneum / WS 2024/25 / Dr. Peter Priesching 42 BASICS OF FLUID DYNAMICS AND NUMERICS Fluid Dynamics - Turbulence Momentum equation for laminar flow:  u j u j  p  ij  + ui  =− − + gj  t xi  x j xi via Reynolds averaging Reynolds equation for turbulent flow u j p   u j   ui =− +     −  ui u j  +  g j xi x j xi  xi  … Reynolds Equation − ( ij )tot New terms can be interpreted as additional momentum transport terms. Lecture CFD / FH-Joanneum / WS 2024/25 / Dr. Peter Priesching 43 BASICS OF FLUID DYNAMICS AND NUMERICS Fluid Dynamics - Turbulence The shear stresses can be separated into a laminar and a turbulent (virtual) part: u j ( ) = − +  uiuj = ( ij )lam + ( ij )turb xi ij tot The turbulent terms are called the ‚Reynolds stress tensor‘  u12 u1u2 u1u3    uiuj =  u2 u1 u22 u2 u3         u3u1 u3u2 u32  … symmetrical These six terms in the Reynolds equation appear additionally to the laminar terms of the momentum equation. For these terms new solution methods need to be defined. (→turbulence modeling) Lecture CFD / FH-Joanneum / WS 2024/25 / Dr. Peter Priesching 44 BASICS OF FLUID DYNAMICS AND NUMERICS Fluid Dynamics - Turbulence Length Scale The smallest eddies, which appear in a turbulent flow are described by the so called Kolmogorov length scale. 1   3 4 lK =     The formulation has been derived from the assumption that the length scale in each turbulent flow is similar and only depending on the kinematic viscosity and the dissipation rate. Lecture CFD / FH-Joanneum / WS 2024/25 / Dr. Peter Priesching 45 BASICS OF FLUID DYNAMICS AND NUMERICS Fluid Dynamics - Turbulence Length Scale For small Re numbers the structures of the eddies is large and vice versa. Re small Re large Lecture CFD / FH-Joanneum / WS 2024/25 / Dr. Peter Priesching 46 BASICS OF FLUID DYNAMICS AND NUMERICS Fluid Dynamics - Turbulence Turbulence Models The goal of all the different turbulence models is to find solutions for the six unknown terms of the Reynolds stress tensor. A majority of the models tries to set up correlations between the velocity fluctuations and mean flow velocities. The definition of these kind of relations can result from theoretical hypothesis, but also from empirical observations (i.e. detailed measurements) The found models have to be seen in the context of numerical solvability. Stability of the solution and the needed calculation time are the most important parameters. Lecture CFD / FH-Joanneum / WS 2024/25 / Dr. Peter Priesching 47 BASICS OF FLUID DYNAMICS AND NUMERICS Fluid Dynamics - Turbulence Eddy Viscosity Models Boussinesq introduced an approach, where the Reynolds stresses are defined via the so called ‚eddy viscosity‘.  u j ui   uiuj = −  T  +    xi x j  eddy viscosity For the description of the eddy viscosity many approaches exist: analytical solution zero equation models (e.g. Prandtl mixing length) one equation models two equation models (k- Modell) ! … Lecture CFD / FH-Joanneum / WS 2024/25 / Dr. Peter Priesching 48 BASICS OF FLUID DYNAMICS AND NUMERICS Fluid Dynamics - Turbulence Two Equation Model – k- Model Two additional transport equations are introduced. 1. for k (kinetic energy of turbulence) 2. for  (dissipation rate k) k2 for the eddy viscosity this results in: T = C  for length and velocity scales: 3 2 1 k lc = uc = k 2  The k- model is one of the most widely used models in numerical solutions. Lecture CFD / FH-Joanneum / WS 2024/25 / Dr. Peter Priesching 49 BASICS OF FLUID DYNAMICS AND NUMERICS Fluid Dynamics - Turbulence Example DNS (Direct Numerical Simulation) The turbulent flow is basically only calculated from the Navier – Stokes equations. This results is a very accurate solution, because no model for the description of the turbulence needs to be applied. The drawback is that the mesh resolution needs to be chosen in a way that even the smallest length scales are resolved (Kolmogorov!). → exact results possible → very long calculation times → Supercomputers necessary → the larger Re, the finer the mesh resolution Lecture CFD / FH-Joanneum / WS 2024/25 / Dr. Peter Priesching 50 BASICS OF FLUID DYNAMICS AND NUMERICS Fluid Dynamics - Turbulence Example LES (Large Eddy Simulation) LES is an intermediate solution between turbulence models (e.g. k-) and DNS. It should lead to a more accurate solution than usual TMs but with less effort than DNS. → Filtering of the turbulent flow results in a split between solution of the large eddy structures (resolved field) and the small eddy structures (sub-grid field). → SGS-models (subgrid-scale) are necessary to describe the interaction between large eddies and unresolved sub-grid turbulence. → The resolution of the filtering (mesh size) can adapt the accuracy to the available calculation resources. → The finer the mesh resolution the more accurate. Lecture CFD / FH-Joanneum / WS 2024/25 / Dr. Peter Priesching 51 BASICS OF FLUID DYNAMICS AND NUMERICS Fluid Dynamics - Turbulence What ist LES? calculation domain turbulent eddies Requirements for the CFD simulation: RANS: LES: DNS: Turbulence models Modeling only necessary No modeling of describe the influence below the mesh turbulence required of all eddies resolution → sub-grid-scale models Lecture CFD / FH-Joanneum / WS 2024/25 / Dr. Peter Priesching 52 BASICS OF FLUID DYNAMICS AND NUMERICS Fluid Dynamics - Turbulence Turbulent jet – calculated by using different approaches (source: https://www.idealsimulations.com/) Lecture CFD / FH-Joanneum / WS 2024/25 / Dr. Peter Priesching 53 BASICS OF FLUID DYNAMICS AND NUMERICS Overview ▪ Introduction ▪ Basics of fluid dynamics ▪ Navier – Stokes equations ▪ Turbulence ▪ Basics of the numerical solution ▪ General ▪ Discretization ▪ Numerical solution Lecture CFD / FH-Joanneum / WS 2024/25 / Dr. Peter Priesching 54 BASICS OF FLUID DYNAMICS AND NUMERICS Numerical Methods Method For a correct numerical solution of the different flow problems it is important, that the example is defined sufficiently. This concerns: Boundary conditions Initial conditions Type of the flow “Garbage in / garbage out” … Questions (to be answered before starting the calculation) Is friction involved? Is the flow laminar or turbulent? Is the flow compressible or incompressible? Does buoyancy play a role? Are chemical reactions involved? … Lecture CFD / FH-Joanneum / WS 2024/25 / Dr. Peter Priesching 55 BASICS OF FLUID DYNAMICS AND NUMERICS Numerical Methods Properties of Numerical Methods ▪ Consistency The finer the resolution of the mesh, the more accurate should the discretization become. ▪ Stability A solution method is stable if the appearing errors are not self amplifying. ▪ Convergence With finer mesh resolution the calculation result should become closer to the exact solution. ▪ Conservation (of mass, momentum, energy, …) The basic conservation laws have to be fulfilled locally and globally. Lecture CFD / FH-Joanneum / WS 2024/25 / Dr. Peter Priesching 56 BASICS OF FLUID DYNAMICS AND NUMERICS Numerical Methods Properties of Numerical Methods ▪ Boundedness The results of the numerical solution have to be within the physical boundaries (e.g. concentration > 0). ▪ Realizibility Applied models shall be able to describe the physical behavior. ▪ Accuracy Numerical solutions are only approximations of the reality. There are three main errors, which appear: ▪ Modeling errors ▪ Discretization errors ▪ Convergence errors Lecture CFD / FH-Joanneum / WS 2024/25 / Dr. Peter Priesching 57 BASICS OF FLUID DYNAMICS AND NUMERICS Overview ▪ Introduction ▪ Basics of fluid dynamics ▪ Navier – Stokes equations ▪ Turbulence ▪ Basics of the numerical solution ▪ General ▪ Discretization ▪ Numerical solution Lecture CFD / FH-Joanneum / WS 2024/25 / Dr. Peter Priesching 58 BASICS OF FLUID DYNAMICS AND NUMERICS Numerical Methods Discretization Methods ▪ Finite Difference Method – FD ▪ Starting from the conservation equations in differential form ▪ Solution domain is represented by a mesh ▪ Differential equations are solved in each node. Partial derivaties are replaced by relations between the nodes. ▪ Taylor expansion for first and second derivatives ▪ Advantage: for a structured mesh quite simple and effective ▪ Disadvantage: Conservation is not guaranteed. Problematic especially with complex geometries Lecture CFD / FH-Joanneum / WS 2024/25 / Dr. Peter Priesching 59 BASICS OF FLUID DYNAMICS AND NUMERICS Numerical Methods ▪ Finite Volume Method – FV ▪ Starting from integral conservation equations ▪ The flow domain is split into finite volume elements ▪ Conservation equations are applied to the volume elements ▪ Values of the quantities are calculated for the centers of the volume elements ▪ Values on the bounding faces are calculated via interpolation ▪ Only the volume elements on the boundary are defined by the geometry surface ▪ Advantage: Works for all geometries. Conservation laws fulfilled. Simple to program. ▪ Disadvantage: Interpolation methods of higher order are difficult to be applied ▪ Finite Element Method – FE ▪ Similar to FV – flow domain split into volume elements ▪ Weighting factors for each equation ▪ Advantage: Simply applicable for unstructured meshes ▪ Disadvantage: Numerical solution might be difficult Lecture CFD / FH-Joanneum / WS 2024/25 / Dr. Peter Priesching 60 BASICS OF FLUID DYNAMICS AND NUMERICS Numerical Methods Courant Number The Courant number is the ratio between time and space resolution. It basically defines over how many cell layers a physical quantity is moving during one time step. t C =u 1 x Due to stability reasons especially in explicit methods Courant should be: C1. Additionally C should be quite close to one due to accuracy reasons. Lecture CFD / FH-Joanneum / WS 2024/25 / Dr. Peter Priesching 61 BASICS OF FLUID DYNAMICS AND NUMERICS Numerical Methods Spatial Discretization left: nodes in the cell center, right: faces in the middle between nodes Lecture CFD / FH-Joanneum / WS 2024/25 / Dr. Peter Priesching 62 BASICS OF FLUID DYNAMICS AND NUMERICS Numerical Methods Interpolation Methods Figure: Naming of nodes and faces in a discretized volume. For ‘structured’ meshes there is a clear relation of each cell to its 6 neighbors (in 3D). Lecture CFD / FH-Joanneum / WS 2024/25 / Dr. Peter Priesching 63 BASICS OF FLUID DYNAMICS AND NUMERICS Numerical Methods Interpolation Methods sb k. nc2 if d.. db ib nc1 j i Figure: Unstructured mesh. For ‘unstructured’ meshes the interpolation methods are similar, but the clear relation of the cell orientation and the cartesian coordinate system doesn’t exist. Lecture CFD / FH-Joanneum / WS 2024/25 / Dr. Peter Priesching 64 BASICS OF FLUID DYNAMICS AND NUMERICS Numerical Methods Interpolation Methods For the calculation the values on the surfaces of each control volume are needed. They are not resulting directly from the solution of the equations. Therefore they need to be found on another way. ▪ Upwind Interpolation (UDS) ▪ Value on the face is the same as the one in the upstream cell. ▪ + : ‚boundedness‘ always fulfilled, no numerical oscilations ▪ - : numerical diffusive P if u0 e = E if u0 Lecture CFD / FH-Joanneum / WS 2024/25 / Dr. Peter Priesching 65 BASICS OF FLUID DYNAMICS AND NUMERICS Numerical Methods Interpolation Methods ▪ Linear Interpolation (CDS) ▪ linear interpolation of two cell values ▪ + : simplest scheme of second order (two neighbouring values used) ▪ - : can lead to numerical oscilations ▪ Higher Order Schemes ▪ Interpolation with functions of higher order ▪ + : more accurate than other schemes ▪ - : Information about more neighbours necessary. Difficult in unstructured meshes. Time consuming Lecture CFD / FH-Joanneum / WS 2024/25 / Dr. Peter Priesching 66 BASICS OF FLUID DYNAMICS AND NUMERICS Overview ▪ Introduction ▪ Basics of fluid dynamics ▪ Navier – Stokes equations ▪ Turbulence ▪ Basics of the numerical solution ▪ General ▪ Discretization ▪ Numerical solution Lecture CFD / FH-Joanneum / WS 2024/25 / Dr. Peter Priesching 67 BASICS OF FLUID DYNAMICS AND NUMERICS Numerical Methods Definition of Boundary Conditions There are different methods how to define boundary conditions: ▪ Values of quantities defined at the boundary (Diriclet BC) ▪ open BC: Velocity, Temperature, … ▪ no-slip at walls (velocity = 0) ▪ Gradient of quantities defined at the boundary (Neumann BC) ▪ pressure gradient ▪ temperature gradient (heat flux) Lecture CFD / FH-Joanneum / WS 2024/25 / Dr. Peter Priesching 68 BASICS OF FLUID DYNAMICS AND NUMERICS Numerical Methods Solution of the Navier – Stokes Equations The system of the NS equations is coupled over the pressure. During the solution this coupling has to be taken into account by a special iterative solution. A widely used and easy to understand method is the so called SIMPLE (Semi Implicit Method for Pressure Linked Equations, Caretto et al., 1972) algorithm. Initialization Structure of the assumption of the pressure field iterate until converged SIMPLE algorithm: solution of momentum equations time loop solution of pressure correction equation calculation of the corrected pressure solution of other transport equations output Lecture CFD / FH-Joanneum / WS 2024/25 / Dr. Peter Priesching 69 BASICS OF FLUID DYNAMICS AND NUMERICS Numerical Methods Solution of the Navier – Stokes Equations The PISO (Pressure Implicit with Splitting of Operators) algorithm is an extension of the SIMPLE. It can reduce calculation time and increase stability. Initialization assumption of the pressure field iterate until converged Structure of the solution of momentum equations PISO algorithm: time loop solution of pressure correction equation calculation of the corrected pressure and velocities 2nd solution of pressure correction equation calculation of the corrected pressure and velocities solution of other transport equations output Lecture CFD / FH-Joanneum / WS 2024/25 / Dr. Peter Priesching 70 BASICS OF FLUID DYNAMICS AND NUMERICS Dr. Peter Priesching Lecture CFD / FH-Joanneum / WS 2024/25 / Dr. Peter Priesching 72