Table of Contents
An Introduction To
Computational Fluid Dynamics (CFD)
by
Dr. Sharad N. Pachpute (PhD, IIT Delhi)
Introduction to Flow Analysis Techniques
In general, science and engineering have been traditionally divided into two parts; experimental and theoretical disciplines.
Theoretical or analytical Analysis

 Theory is based on assumptions, postulations and laws of actions
 In physics, chemistry and other science subjects, many theorems are studied and mathematics helps in quantification and understanding
 These theorems are presented in terms of independent and dependent parameters
 Analytical analysis helps to provide physical dependency of parameters but it difficult to solve complex real world problem example: weather prediction, flows of rives, testing of large airships etc.
Experimental Analysis

 Using theory and mathematical formulation we can solve simple problems in our real life and nature
 Experiment is conducted for simple cases to understand the details of objectives
 Carrying out experiments is expensive, risky and time consuming
 There are many limitations to get solutions for complex problems either by theory and experiments
Modeling and Simulation

 Modeling and simulation has emerged as a third pillar in many fields like engineering, molecular dynamics and astronomy
 Modeling and simulation is a technique to prediction the science and explore new knowledge
 It is useful to solve real world problems which can not be solved by theory and may be expensive by experimental analysis
 Computational fluid dynamics (CFD) is a technique of modeling and simulation based on numerical modeling for fluid flow
 Using the CFD technique, heat and mass transfer, reactive flow, multiphase flow and combustion can be analyzed using various numerical models
 Modelings and Simulation is called as a third pillar of science and technology in the 21st century.
What is Computational Fluid Dynamics (CFD)
Definition of CFD
 Computational fluid dynamics (CFD) is a technique of flow predictions by numerical solving governing equations of fluid flows. Governing equations are conservation of mass, momentum, heat and mass transfer. Numerical methods are used to convert partial differential equations in to algebraic set of equations. After solving iteratively using computer programs, engineers visualize predicted velocity, pressure and temperature.
 Computational fluid dynamics (CFD) comprises majorly three steps:
 Preprocessing: Creation of geometry and meshing
 Simulation: Solving governing equations over a meshed model
 Postprocessing and results analysis: Predicted physical quantities need to be analyzed qualitatively and quantitatively
 Computational fluid dynamics (CFD) is an interdisciplinary subject. Hence, CFD users have to be familiar with the following subjects as per their specific requirement.
Essential Subjects for CFD Modeling
1) Mathematics:


 Partial differential equations, integration
 Numerical Methods: finite volume method (FVM), finite element method (FEM), finite difference method (FDM)

2) Flow Physics:


 Fluid mechanics: Proerties of fluid, Laminar and Turbulent Flow
 Heat and mass transfer
 Reaction: reactive or passive flow in combustion
 Phases of material: single or multiphase flow
 Other flow physics: Magnetic or electric field acting on fluid

3) Computer science:
Programming tools like C or C++ need to be learnt by CFD users initially. Familiarity with CFD software is essential for geometry, meshing, simulation and postprocessing. Parallel computing may need for complex industrial problems, large eddy simulations (LES ) and DNS.
 Hardware : Laptop, PC, HPC, Supercomputer
 Programming tools: C, C++, Python etc
 CFD software: ANSYS FLUENT, OpenFOAM, Star CCM, COMSOL
4) Experimental data for validation:
 Validation of CFD results is essential for credibility of numerical models
 Results of CFD simulations need to be reliable and consistent with experimental results
Scope of CFD Modeling
Comparison of Simulation (CFD) and Experimental Analysis
 As discussed above, simulation and experiment are two commonly used methods for design and analysis
 Simulations provide a thorough analysis at a lower cost compared to experimental analysis
 Hence CFD is used in a wide range of industrial applications where fluid flow is involved like automotive, aerospace, power generation, etc
 CFD is used for design and optimization of industrial problems
• CFD has a large potential to solve industrial problems at bigger scale due to continuous improvement in computational power and numerical models
Advantage of CFD Analysis
 The cost of analysis is low compared to experiments
 Provide detailed information
 It can applied to a variety of problems
 CFD analysis requires less time to get results compared to experimental for most of complex cases. Hence, wind tunnels are replaced with CFD simulations in many universities and industries as shown below
 CFD analysis is an effective method for design optimization in many automobile, aerospace and chemical industries
 To carry out a wind tunnel test is expensive and it requires a lot time for experimental set up
 For measurement in experiments, we can not fix the probes or sensors to measure velocity and pressure at all locations.
Applications of CFD Modeling
• Simulations provide insight into physical parameters (e.g. velocity, temperature, pressure, pollution level,.etc.) that are difficult to study using traditional (experimental) techniques.
i) To find the impact of Smoke and pollution level: Flow pattern for an oil fire
 Flow pattern for Turbomachinery is shown below
 First step of CFD is to Identify the flow physics involved in pump:turbomachinery, turbulent flow, multiphase flow (water and air)
iii) To predict the temperature distribution in boiler
 CFD analysis of boiler is shown below:Temperature for Turbulent Combustion of a coal fired boiler
 First step of CFD is to Identify the flow physics involved in boiler: turbulent flow, heat transfer (conduction, convection and radiation), multiphase flow (particle and flue gas), reactive flow (combustion)
.
Classification of Fluid Flow
 Fluid flow is classified based on type of physics involved in a particular flow. CFD user has to select correct CFD Models as per flow physics
 For each flow physics, different numerical models have been developed based on assumptions and mathematics. All models are to be consistent with experiments
 A variety of numerical models are available in most of commercial CFD solvers. Read the follwoing articles for complex flow modeling.

 Re < Recr: Laminar flow
 Re > Recr : Turbulent Flow flow
 The transition from laminar to turbulent flow depends on the surface geometry, surface roughness, freestream velocity, surface temperature, and type of fluid, among other things.
 If flow becomes turbulent for the small sectional area, then CFD user has to select an appropriate turbulence model while modeling the flows
 To simulate complex turbulent flow, turbulence models are used for numerical simulation
Regime for Laminar and turbulent Internal Flow

 Example 1: Buoyant flow from a cigarette shows that laminar and turbulent regions are formed in fluid flow

 Example 2: Water falling from a tap Water falling from a tap
 Example 3: Water channel flows show that laminar flows in the channel and turbulent around the gate due to increase in velocity
Forced Flow vs Natural (free) Flow
 Forced flow: the flow is driven by external means like fan, pump and blower etc.
 Natural (free) flow: the Flow is driven by the density difference between hot and cold fluids
Compressible Flow vs Incompressible Flow
Fluid Flow with Fixed vs Moving Boundary
 Fixed/stationary boundary :boundaries/ walls are fixed
 Example (1) of a fixed wall boundary: for flow through a circular pipe , the pipe wall is fixed and stationary (noslip)
 Example (2) of a fixed wall boundary: for flow through a control valve , the pipe wall is fixed and stationary (noslip)
 Example of a moving wall boundary: Moving/dynamic boundary : the boundaries/ wall move relative to the fluid: (a) Air flow over a moving car, (b) Air flow over a wind mill
 In flow over a windmill, air flows over blades which rotate about its axis. The lift force created due to pressure difference causes torque for rotation of blades.
Single vs Multiphase flow
 Single Phase flow : only one phase exits in the fluid system Examples: water through a pipe, air flow over a surface etc.
 Multiphase flow : more than two or more phases interact each other.
 Examples of multiphase flows: Boiling flow, condensation, bubble column etc..
 Water evaporation and boiling consists of two phases water and steam
 Condensation consists of two phases water and steam
 To model the flow when two (inter penetrating or separated)phases, multiphase models are used based on approximation and cost of simulations
Passive Flow vs Reactive Flow
 Nonreactive flows: no chemical reaction takes place between the species
 Reactive flows : chemical reaction takes place between the species
 For details of reactive flow modeling, read basics of combustion in this website. Majority of flow in combustion is turbulent for good mixing of fuel and air. It can lead to stable flames. Turbulent combustion modeling need to be studied in details before CFD modeling.
Transport Equation for CFD Modeling
Physical transport phenomenon
A caricature of physical transport phenomenon:
 Transport of water is used from source point to the location of fire extinguishing
• CFD user must know the governing equations and the physical meaning of various terms in equations.
1) Conduction (diffusion): need a medium for transfer of mass, momentum, heat and species at molecular level.
2) Advection: need a medium for transfer of mass, momentum, heat and specie at bulk level.
Convection = Advection + Diffusion
3) Radiation: no need a medium for heat (Not applicable for mass, momentum and species)
• In general, the conservation equation for any physical conserved variable is written as:
Unsteady term + Advection term = Diffusion term + Source /sink terms
Momentum and heat mechanism
General form of governing equation
General form of governing equation (in compact form)
Numerical Solution Procedure
Overview of CFD or Numerical Modeling
 Computational fluid dynamics (CFD) is the science of predicting fluid flow, heat and mass transfer, chemical reactions, and related phenomena by solving numerically the set of governing mathematical equations• Governing (PDE) equations are Conservation of fluid flow, heat and mass transfer and species, etc.
 Generally, CFD works based on the Finite volume methods (Conservative Method).
 Fluid region of pipe flow is discretized into a finite set of control volumes (mesh/cells).
 Partial differential equations (PDEs) are discretized into a system of algebraic equations using finite volume method or finite element method
Steps for CFD Modelling
 Preprocessing comprises making geometry and meshing
 Solver: selection of physical models, governing equations, solution procedures
CFD Software
 Most commonly used CFD software and tools for numerical simulations are shown below
Comprehensive CFD Software

 These CFD solvers have been developed a wide varieties of complex flows like turbulent , multiphase and combustion etc.
 These solvers have been used in many industries and well practiced numerical models are available there
 Many industries have compared their plant (practical) data with CFD results
 Example: ANSYS FLUENT and STAR CCM
SemiComprehensive CFD Soft wares


 Some CFD solvers are developed for certain applications but not practiced on large scales of industrial problems
 These solvers need to expand their scope of applications for large scale industries
 Example: COMSOL Multiphysics, CONVEREG

Discretization of Computational Domain (Meshing)
Elements of Meshing
Computational is divided into to finite number of volumes or elements in meshing. Meshing consists of following important identitites.
 Cell : control volume into which domain is broken up
 Node : grid point
 Cell center : center of a cell
 Edge: boundary of a face
 Face : boundary of a cell
 Face defines the cell zones (solid or fluid region) for two dimensional (2D) computational domain. Hecne, CFD users has to define fluid zone for face
 For, 3D domain, face can as boundary like inlet, outlet or wall etc.
 All essential faces should be named in meshing before exporting to CFD solver
 Zone:
 It is grouping of nodes faces, and cells
 For, 3D domain, volume region defines fluid or solid regions
 All essential cell zones should be named either solid or fluid in meshing before exporting to CFD solver
 Domain : It is a group of node, face and cell zones
Structured vs nonstructured meshing
 The process of mesh generation is generally classified into two categories based on the topology of the elements that fill the domain
 After the selection of CFD domain, next step is to divide the domain into finite number of volumes or cell based on the physics to be resolved.
 Types of Meshes for Computational domain.
Assessment of Mesh Quality in CFD
 Mesh quality is essential for correct CFD simulations. Three ways are to measure of the quality of mesh
 Skewness /Orthogonal quality:
 The skewness of mesh should low as much as possible
 Higher skewness can slow down the convergence of simulation and incorrect computations fluxes near walls.
 Skewness and orthogonal both are opposite each other. Higher orthogonal quality means lower the skewness of mesh
 For complete Hexahedral elements, skewness is zero. It is nonzero value for inclined faces.
 Smoothness (change in size):
 Sudden jumping in mesh should be avoided in high gradient region
 The growth factor of 1.2 is recommended
 Smoothness in mesh is very essential near wall boundaries.
 Aspect ratio:
 Select it as per need of simulation
 In the direction of high gradient region, aspect ratio should be less ( < 20)
 In the direction of no gradient region,higher aspect ratio ( ~ 250) is enough to reduce the mesh size.
Discretization Methods for CFD
The governing equation like mass, momentum and energy are expressed in terms of partial differential equations (PDEs). These equations are space and timedependent. We can not solve PDEs like ordinary differential equations. However, partial differential equations are boundary value problems. After discrertizing with a numerical method, we can compute values based on initial and boundary conditions.
Discretization convert the transport equation (mass, momentum and energy) in partial differential form to set of algebraic equations. Three numerical methods are widely used to discretize the partial differential equations namely:
 Finite Difference Method (FDM)
 Finite Element Method (FEM)
 Finite Volume Method (FVM)
The discretization methods approximate the PDEs of fluid flows with numerical model equations, which are solved using different numerical methods.
Finite difference method (FDM)
 The computational domain is usually divided into hexahedral elements (grids) and the numerical solution is obtained at each node .
 The finite difference method (FDM) is simple to understand when the physical elements are defined in the Cartesian coordinate system, but the use of curvilinear transforms the method can be extended to domains that can not be easily represented using brickshaped elements
 The discretization results in a system of equations of the variable for grid points, and once a solution is obtained then a discrete representation of the solution is obtained.
 Note the following points for FDM:
 It is applicable only for regular grid (equal size of meshing).
 This method is based on Taylor series of differentiation. Finite difference methods for spatial derivatives with different order of accuracies is obtained using Taylor expansions like first or second order upwind difference scheme (UDS), central differences schemes (CDS), etc.
 The FDM is difficult to use for nonuniform grid. Hence it is rarely used for CFD solvers
 This method is not easy to implement conservation of mass, momentum and energy
 Taylor series of expansion dicretizes diferential terms in partial differential equations
 Forward, backward and central schemes are commonly used finite difference schemes to discretize partial differential equations
 Discretization is carried out over all the nodes of computational doain
Finite Volume Methods (FVM)
 This approach is suitable for both irregular or irregular meshes. is based upon an integral form of the PDE to be solved (e.g. conservation of mass, momentum, or energy).
 The governing equations are solved for a given finite volume (or cell) using finite volume methods
 The computational domain is discretized into finite volumes and then for every volume the governing equations are solved.
 The final forms equations after discretiazion involves fluxes of the conserved variable (mass, momentum and energy), and thus the calculation of fluxes is essential in this method
Note:The Finite volume method is conservative (for mass, momentum and energy etc). Hence, the FVM discretization is widely used for many CFD solvers (e.g. ANSYS FLUENT, OpenFOAM.
Finite Element Methods (FEM)
 The FDM is based upon a piece wise representation of the solution in terms of specified basis functions.
 The computational domain is divided up into smaller domains (finite elements) and the solution in each element is constructed from the basis functions.
 This can be a doubleedged situation, as the section of basis functions is essential and boundary conditions can be more difficult to formulate for complex geometries. A set of equations is obtained (for nodal values) are solved to get a solution.
 To get numerical solutions, a set of equations are obtained using the conservation equation: Field variables are written as the basis functions, the equation is multiplied with appropriate test functions, and then integrated over an element. Since the FEM solution is defined in terms of specific functions, a significant solution is obtained by solving the equations algebraically.
 The finite element method (FEM) is used to compute such approximations.Take, for example, a function u that may be the dependent variable in a PDE (i.e., temperature, electric potential, pressure, etc.) The function u can be approximated by a function u_{h} using linear combinations of basis functions according to the following expressions:
 The FEM discretization is used by some CFD solvers (e.g. COMSOL). For COMSOL CFD solver, click here: COMSOL_NUMERICAL_METHOD
Which method (FDM/FEM/FVM) is the best suited for CFD Modeling?
 The comparison of the three methods is not straightforward due to different in numerical procedures
 Finite volume method (FVM) and finite difference method (FDM) provide discrete solutions, but the finite element method (FEM) provides a continuous (up to a point) solution.
 FVM and FDM are generally considered to be easier for programming compared to FEM. This point may be vary user to user
 Most of research scholar adopt the FVM method for numerical modeling of fluid flows.
 The FVM easily provide better conservation properties for mass, momentum and energy, If you are interested to decide which method is more suitable, then go through the literatures on three methods
Interpolation Scheme in CFD Solver
 The domain of interest is divided into finite number of cells or volumes
 Based on on the spatial information of nodal point, area faces and volume cells, governing equations (mass, momentum and energy) are dicretized for both space and time
 Different numerical techniques are used for each term in governing equations like unsteady term, convection term, diffusion, viscous and body forces
1) Upwind Differentiating Scheme (UDS)
 Flow direction dependent Øe
 This approximation is of first order (O(Δx))
 Numerically diffusive
Unsteady Term discretization
 Temporal derivatives can be integrated either by the explicit method or implicit method
 Explipit Method: Euler, RungeKutta
 Implicit method : BeamWarming method
2) Implicit or backward Euler method:: accuracy in order O(Δt)
Explicit Method
Implicit Method
Solution of Linear Equations Systems
•the index l runs over the neighbor nodes involved, and the system of equations for the whole domain has the matrix form
Two types of methods for solving the system of linear algebraic equations
1) Direct methods

 Gauss elimination
 LU decomposition
 Tridiagonal matrix algorithm (TDMA)
2) Indirect or iterative methods:

 Jacobi method
 GaussSeidel method
 Successive overrelaxation (SOR)
 Conjugate gradient method (CG)
 Multigrid methods
Grid based Coupling of Pressure and Velocity
Colocated grid
 Node for pressure and velocity components at the control volume (CV) center.
 Same CV for all variables.
 Possible oscillations of pressure
Staggered grid
 The different unknown variables are located at different grid nodes.
 Pressure located in the cell centers, velocities at cell faces.
 Strong coupling between the velocities and pressure helps to avoid oscillations
 Other staggering method is the Arbitrary LagrangianEulerian (ALE).
Pressure Velocity Coupling – Algorithm
After the discretised form of the governing equation (NavierStokes system), we get a set of equations with linear dependence of pressure and velocity, or viceversa. This interequation coupling is called a pressure velocity coupling. A special numerical treatment is necessary in order to the pressurevelocity coupling. This is achieved by following algorithm such as SIMPLE, SIMPLEC, PISO, Coupled solver.
SIMPLE Algorithm
 Semiimplicit method for Pressurelinked Equations

 1.Advance to the next iteration t(n)=t(n+1)
 Initialize u(n+1) and p(n+1) using latest available values of u and p
 Construct the momentum equations
 Underrelax the momentum matrix
 Solve the momentum equations to obtain a prediction for u (n+1)
 Construct the Pressure equation
 Solve the pressure equation for p(n+1)
 Underrelax p(n+1)
 Correct the velocity for u(n+1)
 If not converged, go back to step 2
 Correct the fluxes for ϕ(n+1)
b) SIMPLEC:SemiImplicit Method for Pressure Linked EquationsConsistent
 The pressurecorrection underrelaxation factor is generally set to 1.0, which helps for faster convergence
 This method is suitable for turbulent flow with pressure varying fields
C) PISO: Pressure–Implicit with Splitting of Operators (PISO)
 PISO consist of one predictor step and two corrector steps
 This methods helps to check the mass conservation in each iteration using predictorcorrector steps
d) Fractional step method


 This method solves the incompressible NavierStokes equations with primitive variables by block LU decomposition.
 This method helps to solve issues of boundary conditions for the intermediate flow variables and the pressure

Pressure Velocity Coupling in OpenFOAM
For details of OpenFOAM schemes and algorithm refer the post.
 simpleFoam: based on SIMPLE algorithm
 pisoFoam: based on PISO formulation
 pimpleFoam: combination of SIMPLE and PISO
Flow solvers available in ANSYS FLUENT
Properties of Fluid
• A CFD user must be familiar with what fluid/solid properties are given to the CFD solver (e.g. FLUENT, OpenFOAM and COMSOL etc.) as input data and what properties will be calculated by the CFD solver
Boundary Conditions
The following boundary conditions which are commonly used in a variety of CFD domains:
1) Inflow (inlet)condition:
 Convective flux is prescribed like velocity or mass flow rate inlet
 Input to be specified at he inlet: Momentum (velocity, mass flow rate, pressure, turbulent conditions), Thermal (temperature, emissivity and shell conduction), Species ( mole/mass fraction), Multiphase (volume fraction of phase, particle size or distribution.
2) Wall:
 No fluid penetrates the boundary, i.e. convective flux is zero.
 Two types of wall condition: noslip (fluid is at rest at the wall) or freeslip (no frictional loses at the boundary).
 Wall can be slip or noslip type
 Wall can be stationary or moving wall (like rotor for turbomachinery)
 Input to be specified at the wall: Momentum (velocity, roughness), Thermal (temperature, convective HTC, emissivity and shell conduction), Species, Multiphase
3) Outflow (outlet) condition:
 Convective flux independent of the coordinate
 Normal to the boundary. zero gradient BC for all variables except pressure
 It is more suitable for a longer pipe flow with a fully developed profiles of velocity or temperature
 It is also called zerogradient BC for (U,T,Y except P) in OpenFOAM
4) Symmetry condition:
 This boundary is used when flow does not cross the boundary
 Across the boundary all gradient are zero
5) Periodic (cyclic) condition:
 Repetitive boundary conditions
 It can be transnational or angular
 Boundary conditions used for pipe flow or free stream flow over a circular cylinder
 Boundary conditions for flow through a pipe or duct is shown below. Inlet is specified with a fixed velocity
 Initial Boundary conditions for flow over a cylinder are defined as below
 Most of boundary conditions are defined based on fixed value and fixed gradients of velocity, pressure and temperature. That understanding will help decoding open FOAM CFD solver: Adiabatic wall ( dT/dx =0) and Outlet (dT/dx=0)
Example of CFD Modeling : Pipe Flow
CFD Analysis of Laminar Flow Through the Circular Pipe
ANSYS FLUENT Set up
 Solver setting : Pressure based incompressible solver
 Physical Model: Viscous Laminar
 Boundary Conditions
 Inlet: fixed mass flow rate or velocity
 Outlet: outflow (zero gradient for velocity)
 Pressure Velocity Coupling: SIMPLE
 Numerical Discretization Schemes
 Pressure : Standard
 Momentum: QUICK
 Convergence criterion: 10^06
CFD Results and Analysis
Qualitative results: Contours and velocity vectors
Validation of CFD Results
 Comparison of CFD results with theory/ experiment
 To get credibility of CFD models, results obtained for numerical simulations need to be validated against theoretical or experimental data
 In the follwing figure, the pressure drop and fully developed velocity profiles are same with that of CFD simulations and theoretical datafor lamiar flow through pipe
Conclusion
 Computational fluid Dyanmic (CFD) is a flow prediction technqiues
 It is useful for design and optimization of problems in fluid mechanics, heat transfer, multiphase flows
 Due to low cost of flow analysis, CFD modeling is popular in many industries
References
For CFD Beginners:
 Jiyuan Tu Guan Heng Yeoh Chaoqun Liu, Computational Fluid Dynamics: A Practical Approach,ButterworthHeinemann, Elsevier Publication,. 2018
For Intermediate CFD users:
 Henk Kaarle Versteeg, WeeratungeMalalasekera, An Introduction to Computational Fluid Dynamics: The Finite Volume Method,Pearson Education Limited, 2007
 Petrila, Titus, Trif, Damian, Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics, Springer Publication, 2012
Excellent
Thanks for your comment
This is nice.please could i get resource materials for 3D meshes and solvers?
Thank for your feedback, You can email to: info@cfdflowengineering.com to get learning materials
Very well written with nice figures for CFD beginners.
It’s really nice and very good presentation for better understanding. Hats off for your great work.
CFD is the acronym for computational fluid dynamics and, as the name suggests, is the branch of fluid mechanics that makes use of computers to analyze the behavior of fluids and physical systems. CFD modeling and analysis became a popular online simulation solution as the difficulty grew in applying the laws of physics directly to reallife scenarios in order to make analytical predictions. This fact became especially prevalent for fluid flow and heat transfer engineering problems.