CFD Flashcards
Ch.1 General Introduction
Physical Experiments
Numerical Simulations
CFD -
GMMG, SAV Geometry Mathematical Model Discretization Grid Solns of Eqn. convergence Analysis Validation
Mesh Based Methods -
Mesh Free Methods -
FDM -
FEM -
FVM -
Ch.1 General Introduction
Physical Experiments
These are usually very time consuming and expensive to set up
There are limitations on extrapolation of the results obtained on scaled modelof a problem to the actual prototype.
BUT the experimentally observed data provides the closest possible approximation of the physical reality within the limits of experimental errors.
Numerical Simulations
Mathematical modelling is based on a set of assumptions with regard to thevariation of the problem variables, constitutive relations and material
properties.
Numerical simulation process introduces additional approximation errorsin the solution. Hence, results of any analytical or numerical study must be carefully validated against physical experiments to establish their practicalusefulness.
However, once validated, a numerical simulation can be easily performed onthe full scale prototype, and thereby eliminate the need of extrapolation.
CFD -CFD deals with the numerical simulation of fluid flow and heat transfer problems. CFD deals with the approximate numerical solution of governing equations based on thefundamental conservation laws of physics, namely mass, momentum and energyconservation. The CFD solution involvesConversion of the governing equations for a continuum medium into a set of discretealgebraic equations using a process called discretization. Solution of the discrete equations can using a high speed digital computer to obtainthe numerical solution to desired level of accuracy.
GMMG, SAV Geometry Mathematical Model Discretization Grid Solns of Eqn. convergence Analysis Validation
Mesh Based Methods - FDM, FEM, FVM
Mesh Free Methods - collection of nodes with no apparentconnectivity SPH, LBM, etc
Ch.2 Mathematical Modelling
Conservation Laws conservation of mass conservation of momentum conservation of energy supplemented by constitutive relations - stress and strain rate, heat diffusion law, etc
Notation
dyadic or vector (co-ordinate free form)
expanded or component form
cartesian Tensor notation form
Operators Kronecker delta Alternating tensor Gradient operator - Divergence operator -
Gauss Divergence -
RTT -
Continuity Equation
Mass Conservation -
Momentum conservation -
constitutive relation (relation between stress and strain rate), stokes hypothesis
Energy Equations
N-S eqn. -
Euler’s eqn. -
Stokes eqn. -
Mathematical Classification of PDEs -
Wave Eqn. -
Transient Heat conduction -
Steady state heat conduction or Potential Flow problems -
Unsteady Incompressible Navier-Stokes and energy equations -
Steady Incompressible Navier-Stokes equation -
Unsteady, inviscid, incompressible flows -
Steady, Inviscid, incompressible flows -
Unsteady Compressible Navier-Stokes equations -
Steady Compressible Navier-Stokes equations -
Unsteady Inviscid compressible flows -
Steady Invisicid compressible subsonic flows (Ma < 1):
Steady Inviscid compressible supersonic flows (Ma > 1):
Which type of eqns. are difficult to solve?
Boundary Conditions : In numerical simulation, we come across two types of boundary conditions
Physical & Artificial boundary conditions -
BOUNDARY CONDITIONS ON PHYSICAL BOUNDARIES -
BOUNDARY CONDITIONS ON ARTIFICIAL BOUNDARIES
Drichlet Boundary Condition - Neuman Boundary Condition - Robin Boundary Condition - Cauchy Boundary Condition - Mixed Boundary Condition -
Ch.2 Mathematical Modelling
Conservation Laws conservation of mass conservation of momentum conservation of energy supplemented by constitutive relations - stress and strain rate, heat diffusion law, etc
Notation
dyadic or vector (co-ordinate free form)
expanded or component form
cartesian Tensor notation form
Operators
Kronecker delta
Alternating tensor
Gradient operator - operated on scalar fn. gives vector fn.
Divergence operator - operated on vector fn. gives scalar qty.
Gauss Divergence - converts surface integral to volume integral
RTT - relates rate of change in control mass system to cv system
Continuity Equation
Mass Conservation -
Momentum conservation -
constitutive relation (relation between stress and strain rate), stokes hypothesis
Energy Equations
N-S eqn. - momentum eqn for fluids
Euler’s eqn. - momentum eqn. after neglecting viscosity terms
Stokes eqn. - momentum eqn. after neglecting inertial terms
Mathematical Classification of PDEs -
governing equations of fluid flow (momentum, energy or scalar transport) are second order, coupled non-linear partial differential equations in four independent variables (time + three space coordinates), their mathematical classification is rather difficult.
Choice of the numerical solution technique andnumber of initial/boundary conditions required for their solution depends on theirmathematical character.
Nevertheless, a classification of these equations is usuallyapplied to the linearized form of Navier-Stokes equations
Elliptic - D<0 (No Real characteristics)
Parabolic - D=0 (One Real characteristics)
Hyperbolic - D>0 (Two Real characteristics)
The classification of PDE tells us about physical behaviour of the problem.
Thepropagation of information (e.g. effect of disturbance introduced in a flow field) takes placealong the characteristics.
It also sets the required number of initial/boundary conditions forthe given problem
Hyperbolic eqn. - wave eqn, 2 sets of IC
Transient Heat conduction - Parabolic in time and Elliptic in space IBV, marching type soln.
Steady state heat conduction or Potential Flow problems - Elliptic equations or BVP
Unsteady Incompressible Navier-Stokes and energy equations - parabolicin time and elliptic in space
Steady Incompressible Navier-Stokes equation - elliptic
Unsteady, inviscid, incompressible flows - Parabolic
Steady, Inviscid, incompressible flows - Elliptic
Unsteady Compressible Navier-Stokes equations are mixed hyperbolic, parabolic andelliptic equations. (How)
Steady Compressible Navier-Stokes equations - Hyperbolic & Elliptic (check)
Unsteady Inviscid compressible flows - hyperbolic
Steady Invisicd compressible subsonic flows (Ma < 1): elliptic.
Steady Inviscid compressible supersonic flows (Ma > 1): hyperbolic.
Elliptic equations are more difficult to solve than parabolic equations, which lend themselves to marching type solution procedure. Thus, in practice, steady viscous flowsare usually converted to unsteady problems, and solved using a time marching scheme
Boundary Conditions : In numerical simulation, we come across two types of boundary conditions
Physical & Artificial boundary conditions - For example, in simulation of flowover an aircraft, its solid surface represents the natural physical boundary. Flow is otherwiseunbounded, i.e. flow domain is infinite
BOUNDARY CONDITIONS ON PHYSICAL BOUNDARIES - No slip - viscous flow, Free slip - inviscid flow, No temp jump, heat flux - defined
BOUNDARY CONDITIONS ON ARTIFICIAL BOUNDARIES - inlet, outflow, symmetry, cyclic or periodic are few of the popular types of boundary conditions specified on fluid boundaries
Drichlet Boundary Condition - (V) -specifies the value of function along the boundaryof the domain
Neuman Boundary Condition - (D) - specifies the value of derivativeof function along the boundaryof thedomain
Robin Boundary Condition - (V+D) - sum of function & its derivative at the boundary
Cauchy Boundary Condition - (V&D) - both function & its derivative is specified
Mixed Boundary Condition - for one part one BC for rest part other
Ch. 3 FDM
Attractive features of FDM are:
Approach for Finite difference approximations
Truncation error -
order of accuracy -
Ch. 3 FDM
FDM -
FDM is the oldest method for numerical solution of partial differential equations. This method is also the easiest method to formulate and program for problems on simple geometries In FDM, the solution domain is discretized using a structured (usually Cartesian) grid.The main disadvantage of the finite difference method is its restriction to simplegeometries (although immersed boundary techniques do remove this restriction)
The conservation equations in differential form are approximated at each grid point byreplacing the partial derivatives by finite difference approximations in terms of nodal valuesof the unknown variables. This process results in an algebraic equation for each node. Thesealgebraic equations are collected for all the grid points and resulting system of discreteequations are solved to yield the approximate solution of the problem at the grid nodes.
Attractive features of FDM are:
- It is the easiest method to use for simple geometries, both in terms of formulation andprogramming. - It can also be adapted for problems in complex geometries using boundary fitted gridsor the concept of immersed boundaries. - Domain decomposition based solvers can be easily adapted for solution of algebraicsystems obtained from FDM, and thus, this method is uniquely suited for massivelyparallel architectures.
Derivatives are approximated using finite differences and written for each node resulting into set of algebraic equations.
Approach for Finite difference approximations
- Taylor series
- Polynomial fitting
- Pade approximation
Truncation error -Truncation error (TE) represents the sum of the terms in a Taylor series expansion whichwere deleted in obtaining the approximation of the derivative.
order of accuracy -If the truncation error of a finite difference approximation is O (delta x^m), then it is said tohave the accuracy of order m (or be m th order accurate).
FD approximations to first order derivatives, second order derivatives, mixed order derivatives.
General procedure to obtain 3 pt Backward difference formula, Forward difference, Central Differencing formula, etc. for uniform grid, non-uniform grid using Taylor series and Polynomial Fitting Approach.
Ch. 4. FVM
Ch. 4. FVM
FVM -
The finite volume method is based on the integral form of conservation equations. Theproblem domain is divided into a set of non-overlapping control volumes (called finitevolumes). The conservation equations are applied to each finite volume. The integralsoccurring in the conservation equations are evaluated using function values at computationalnodes (which are usually taken as centroids of finite volumes).
This process involves use ofapproximate integral formulae and interpolation methods (to obtain the values of variables atsurfaces of the CVs).
The FVM can accommodate any type of grid, and hence, it is naturally suitable forcomplex geometries. This explains its popularity for commercial CFD packages, which mustcater to problems in arbitrarily complex geometriesfinite volume method has been the most popular due to its simplicity and ease of application for problems in complex geometries. In fact, majority of commercial CFD packages (e.g. Fluent, StarCD, etc.) are based on finite volume method.
Ch. 5. FEM
Ch. 5. FEM
FEM -
The finite element method is based on the division of the problem domain into a set of finiteelements which are generally unstructured. The elements are usually triangles orquadrilaterals in two dimensions, and tetrahedra or hexahedra in three dimensions. Startingpoint of the method is conservation equation in differential form. The unknown variable isapproximated using an interpolation procedure in terms of nodal values and a set of known functions (called shape functions). This approximation is substituted into the differentialequation. The resulting residual (error) is minimized in an average sense using a weightedresidual procedure. The weighted integral statement leads to a system of discrete equations interms of unknown nodal values, which is solved to obtain the solution of the problem.
FEM is ideally suited to problems on complex geometries, and hence, this method hasbeen very popular in computational solid mechanics.
Ch. 6. Solution of Algebraic Equations
Direct Solvers:
Iterative Solvers:
Accelerated Iterative Solvers:
computational complexity
Ch. 6. Solution of Algebraic Equations
Application of FDM, FVM or FEM leads to a system of algebraic equations which may belinear or non-linear depending on the problem.
Choice of a particular solver depends on the size and nature of the system matrix.
Direct Solvers: Gauss Elimination, LU Decomposition, etc.
Iterative Solvers: Jacobi, Gauss Siedel, SOR (slow convergence)
Accelerated Iterative Solvers: Kyrolv subspace methods (conjugate Gradient, GMRES)
Number of arithmetic operations involved in numerical solution of an algebraic system iscommonly referred to as the computational complexity
Computational complexity of direct solvers grows as O(N^3 ) (specialized solvers suchas TDMA are exceptions to this rule).
In contrast, many fast iterative methods aim at a computational complexity O(N) orO(N log N).
Therefore, a direct solver is normally preferred for smaller systems, whereas iterative solversare preferred for larger systems
Ch. 7. Time Integration for First Order IVPs
Two Level Methods
Explicit or Forward Euler Method
Implicit or Backward Euler Method
Crank Nicholson(uses value at both (n & n+1) time levels)
Accuracy
Stability
Computational Aspects
Multi-point Methods -involve function values at more than two time instants
Adams-Bashforth
Adams-Moulton
Disadvantage
Predictor-corrector methods.
Predictor-Corrector -
Runge-Kutta Method -
R.K order 4
Time step constraint from convective terms and diffusive terms.
for 1D unsteady, convection, diffusion problem.
explicit scheme
c < 1
d < 1/2
CFL - courant friedrichs lewy condition
Ch. 7. Time Integration for First Order IVPs
Two Level Methods
Explicit or Forward Euler Method
Implicit or Backward Euler Method
Crank Nicholson
Accuracy
The preceding two level methods are first order accurate except for the Crank-Nicolson and
the mid-point rule which are second order accurate.
Stability
Explicit Euler conditionally stable
Implicit Euler produce smooth solutions even for larger delta t
Computational Aspects
Explicit methods are easy to program, use little memory and computational time per step; but
are unstable for large delta t .
Implicit methods are much more stable, but require iterative
solution (at least, solution of a linear system for a linear problem) at each time step.
Multi-point Methods -involve function values at more than two time instants
Adams-Bashforth
Adams-Moulton
Disadvantage
These methods require initial data at many points. Hence, these are not self-starting. Thus, at
the first time step, we have to use a lower order Adams method or a Runge-Kutta method.
Explicit methods for time integrationdiscussed in previous lecture are easy to program and use, but are conditionally stable.
Implicit methods (such as backward Euler method) offer better stability but are
computationally expensive. Predictor-corrector methods offer a compromise between these
choices.There exists a wide-variety of such methods based on the choice of the base methods
and time instants used in predictor and corrector steps
Predictor-Corrector -A multi-level predictor-corrector method can be constructed based on an Adams-Bashforth
method as predictor and an Adams-Moulton method as corrector
Runge-Kutta Method -These methods are two level multi-point methods which are easy to use and self-starting, but
require more computational effort per time step as compared to multi-point methods.These methods are generally more expensive but are more accurate and stablethan multi-point methods of the same order.
R.K order 4 step 1. Explicit Euler predictor step 2. Implicit Euler corrector step 3. Mid-point rule predictor step 4. Simpson's rule corrector
The choice of an explicit orimplicit method depends on the objectives of the numerical simulation and nature of the
problem (which dictates the stability requirements).
when Primary Objective is steady state solution - implict methods, which allow greater delta t, are preferred.
If accurate time history is required, small time step may meet stability condition for explicit method. Explicit methods are preferred as they are computationally more efficient.
Time step constraint from convective terms and diffusive terms.
for 1D, explicit scheme
c < 1
d < 1/2
CFL - courant friedrichs lewy condition
courant number, c - ratio of time step to convection time (i.e time required by adisturbance to be convected a distance dx)
d - ratio of time stepto characteristic diffusion time (i.e. the time required for transmissionof a disturbance by diffusion)
Ch.8. Grid Generation
Mesh or Grid -
FDM -
FEM & FVM -
Types of Grid -
structured grid -
Block structured grid
Grid generation techniques
Unstructured grid -
Ch.8. Grid Generation
Mesh or Grid - A mesh or grid is a set of points distributed over the problem domain for a numerical solutionof a set of partial differential equations (PDEs). In CFD analysis, type of grid/mesh woulddepend on the discretization technique (FDM/FVM/FEM), geometry of the problem domainand underlying physics
FDM - structured grid
FEM & FVM - structured or unstructured grid
grid/mesh strongly affects the accuracy of the numerical solution, and hence, significanteffort is usually invested in generation of a suitable grid. In industrial CFD analyses whichinvolve numerical solution of problems in complex geometries, mesh generation process mayconsume nearly 50% of overall simulation time
Types of Grid - structured or unstructured grid
structured grid - In structured grids, grid points follow a fixed structure, which can be mapped to a rectangle (in 2-D) or parallelepiped (in 3-D).
Grid points are located at the intersections of the grid lines & Interior grid points have a fixed number of neighbouring points.
structured grid
cartesian - simple rectangular geometries
body-fitted - curved bodies
orthogonal (grid lines are perpendicular)
oblique (grid lines intersect obliquely)
Block structured grid
Decompose the body in different blocks and generate structured grid in each block separately. Block-structured grids allow different mesh densities in different regions (blocks), and thusfine grids can be easily put in regions likely to show steeper gradients of problem variables.
Grid generation techniques
Algebraic methods - for 2d & 3d both
Conformal mapping - for 2d only
PDE -for 2d & 3d both
Algebraic techniques are usually fast and simple to program, but may not provide a very good quality mesh.Thus, these are often used to provide initial guess to mesh generation methods based on PDEs.
Unstructured grid - sides of a cell/element have no relation to the co-ordinate directions in unstructured grids. The un-structured grid consists of triangles or quadrilateral in 2D and tetrahedra, wedge, hexahedra or polyhedra in 3D. Unstructured grids are much easier to generate in complex domains, and hence, majority of commercial CFD solvers have adopted unstructured grid based solvers. In finite volume applications, quadrilateral (hexahedral) elements are preferred for better accuracy in interpolation and integrations.
Delaunay - Voronoi methods
Advancing Front Method
Quad Tree / Octree Methods -
Ch.9. Soln of N-S eqn
collocated grid -
staggered grid -
modified SMAC scheme was used which is 2 step predictor-corrector scheme
u provisional - Gauss Siedel
pressure poisson eqn - GMRES
u corrected
Schemes for Compressible Flows-
wave based -
flux based -
Ch.9. Soln of N-S eqn
collocated grid - computing V and P grid/node points
staggered grid - computing V at mid of grid points P at grid point
Incompressible - No pressure eqn., so PPE formed by taking div. of momentum eq.
modified SMAC scheme was used which is 2 step predictor-corrector scheme
u provisional - Gauss Siedel
pressure poisson eqn - GMRES
u corrected
Schemes for Compressible Flows- separate methods for shock and shock free simulations wave-based and flux based methods wave based - AUSM, Roe, HLL flux based - ENO/WENO, PVUM
Ch. 10. Numerical Simulation of Turbulent Flows
IMPORTANT ASPECTS OF REAL LIFE CFD ANALYSIS
VERIFICATION AND VALIDATION
Verification -
Validation -
Ch. 10. Numerical Simulation of Turbulent Flows
IMPORTANT ASPECTS OF REAL LIFE CFD ANALYSIS
VERIFICATION AND VALIDATION
Verification - Verification stands for quantitative estimation of the closeness to the numerical simulation results to the exact solution of the mathematical model. The verification process requires comparison of the computational solution with known analytical solutions OR high-accuracy benchmark solutions.
Grid Independence Test & Benchmark Comparison
Validation - Validation is the process of determining the closeness of the approximate numerical simulation to actual real world problem. CFD simulations should ideally be validated with experimental measurements performed on the real system (or its physical model).
Non-dimensionalization Reasons
- Reduction in no. of parameters
- Permits a relative comparison of magnitudes of various terms. (we can get simplified and reasonably accurate model).
- Permits normalization of magnitude of various terms helps in achieving and sustaining accurate solutions.
Consistency
Stability
Convergence
Stability Analysis for ?
Monotonicity
Consistent- Exact soln of algebraic eqn approaching the analytical soln of Governing eqn. for limit delta x, delta y towards 0. This property is called consistency.
A - Analytical soln of Governing Equations
D - Exact soln of Discrete Equations
N - Actual Numerical Soln of Discrete Equationns
Truncation error = A - D
Round off Error = D - N
Stability - Round off errors as soln is marched from one time level to the next time level.
Stability Analysis for linear FDE done using Fourier Analysis/ Von-Neuman Method
Even if the stability criteria is satisfied spurious oscillations/ wiggles that can lead to non-physical values
property that prevents spurious oscillations is termed as Monotonicity.
Numerical viscosity
Dissipation and Dispersion
Examining The leading order term of T.E helps in Identifying the behavior of the numerical solution.
Even order terms - cause dissipation
Odd order terms - cause dispersion
Chapter 5 - Incompressible Viscous Flows via Finite Difference Methods
In dealing with incompressible flows, there are two approaches:
the transition between incompressible and compressible flows involves a complex process of interactions between inviscid and viscous properties, it is reasonable to seek a unified approach in which both incompressible and compressible flows can be accommodated.
ARTIFICIAL COMPRESSIBILITY METHOD
SIMPLE - SEMI-IMPLICIT METHOD FOR PRESSURE-LINKED EQUATIONS
SIMPLER - SIMPLE Revised
SIMPLEC - SIMPLE Consistent
PISO - PRESSURE IMPLICIT WITH SPLITTING OF OPERATORS
MARKER-AND-CELL (MAC) METHOD
Vortex methods
Chapter 5 - Incompressible Viscous Flows via Finite Difference Methods
In dealing with incompressible flows, there are two approaches: primitive variable methods and vortex methods.
The primitive variable approach includes the
artificial compressibility method (ACM), and the
pressure correction methods (PCM) including the
marker and cell (MAC) method, the
semi-implicit method for pressure linked equations (SIMPLE), and the
pressure implicit with splitting of operators
the transition between incompressible and compressible flows involves a complex process of interactions between inviscid and viscous properties, it is reasonable to seek a unified approach in which both incompressible and compressible flows can be accommodated.
Preconditioning Process for Compressible Flows and Viscous Flows,
Flowfield-dependent variation (FDV) methods
ARTIFICIAL COMPRESSIBILITY METHOD
In the artificial compressibility method (ACM), the continuity equation is modified to include an artificial compressibility term which vanishes when the steady state is reached.
SIMPLE - SEMI-IMPLICIT METHOD FOR PRESSURE-LINKED EQUATIONS
- Pressure field is estimated in the domain
- Intermediate Velocity value is found using momentum equation
- Solve the pressure correction Poissons equation to find correction pressure. since corner grid points avoided, the scheme is semi-implicit
- Pressure and velocity are corrected using correction values.
- Now these are our next estimates and this process is iterated untill convergence
SIMPLER - SIMPLE Revised
- convergence of the SIMPLE is not satisfactory because of the tendency for overestimation of p . A remedy is use of under-relaxation parameter
- complete Poisson equation is used for pressure corrections
SIMPLEC - SIMPLE Consistent
PISO - PRESSURE IMPLICIT WITH SPLITTING OF OPERATORS
No iterations. In this scheme, the conservation of mass is designed to be satisfied within the predictor-corrector steps.
MARKER-AND-CELL (MAC) METHOD
The solution is advanced in time by solving the momentum equations for velocity components using the current estimates of the pressure distributions.
The pressure is improved by numerically solving the Poisson equation.
The improved pressure may then be used in the momentum equations for a better solution at the present time step.
Vortex methods
in which pressure terms are absent are preferred in dealing with rotational incompressible flows as they are computationally efficient.
CHAPTER SIX
Compressible Flows via Finite Difference Methods
Compressible inviscid flows are analyzed using the potential or Euler equations, whereas compressible viscous flows are solved from the Navier-Stokes system of equations.
Shock waves may occur in compressible flows and require special attention as to the solution methods.
EULER EQUATIONS
The most basic requirement for the solution of the Euler equations is to assure that solution schemes provide an adequate amount of artificial viscosity required for rapid convergence toward an exact solution.
Solution schemes for the Euler equations may be grouped into three major categories:
(1) central schemes,
(2) first order upwind schemes, and
(3) second order upwind schemes and essentially nonoscillatory schemes.
Central Schemes - Lax-Friendrichs Lax-Wendroff MacCormack Beam and Warming Runge-Kutta
First Order Upwind Schemes
Flux Vector Splitting (VanLeer)
Godunov Methods-Riemann solver (Exact Riemann solver, Approximate Riemann solver)
Second Order Upwind Schemes
MUSCL monotone upstream centered schemes for conservation laws
TVD Total variation diminishing
ENO Essentially Non Oscillatory
CHAPTER SIX
Compressible Flows via Finite Difference Methods
Compressible inviscid flows are analyzed using the potential or Euler equations, whereas compressible viscous flows are solved from the Navier-Stokes system of equations.
Shock waves may occur in compressible flows and require special attention as to the solution methods.
EULER EQUATIONS
The most basic requirement for the solution of the Euler equations is to assure that solution schemes provide an adequate amount of artificial viscosity required for rapid convergence toward an exact solution.
Solution schemes for the Euler equations may be grouped into three major categories:
(1) central schemes,
(2) first order upwind schemes, and
(3) second order upwind schemes and essentially nonoscillatory schemes.
Central Schemes - Lax-Friendrichs Lax-Wendroff MacCormack Beam and Warming Runge-Kutta
First Order Upwind Schemes
Flux Vector Splitting (VanLeer)
Godunov Methods-Riemann solver (Exact Riemann solver, Approximate Riemann solver)
Second Order Upwind Schemes
MUSCL monotone upstream centered schemes for conservation laws
TVD Total variation diminishing
ENO Essentially Non Oscillatory
