Author: 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
 The 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 is difficult to solve complex realworld problem example: weather prediction, flows of rivers, 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 timeconsuming
 There are many limitations to getting solutions for complex problems either by theory or experiments
Modeling and Simulation
 Modeling and simulation have emerged as a third pillar in many fields like engineering, molecular dynamics and astronomy
 Modeling and simulation is a techniques to prediction science and explore new knowledge
 It is useful to solve realworld 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
 Modeling and Simulation is called as the 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 numerically 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 into algebraic sets of equations. After solving iteratively using computer programs, engineers visualize predicted velocity, pressure, and temperature.
 This method of flow prediction has been employed in many automobile, aerospace, and power industries.
 Computational fluid dynamics (CFD) comprises 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 requirements.
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: Properties of fluid, Laminar, and Turbulent Flow
 Heat and mass transfer: Conduction, Convection, and radiation heat transfer
 Reaction in the flows: Passive flow in pipe or duct. reactive flow in the combustion
 Phases of material: One or more phases can interact with each other during the flow
 singlephase: Example – air flow through pipe
 Multiphase flow like boiling or condensation
 Other flow physics: Magnetic or electric field acting on the fluid
 Fixed or moving boundaries: Moving or stationary walls in turbomachinery
 Multiscale and multidomain: there can be multiple scales in the system like battery

3) Computer science:
Programming tools like C or C++ need to be learned 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. Most CFD solvers are developed in C or C++ which are easy to run multiple processors (HPC). Python or scripting languages are used to control the simulations on hardware.
 Hardware: Laptop, PC, HPC, Supercomputer
 Programming tools: C, C++, Python etc
 CFD software: ANSYS FLUENT, CFX, Open FOAM, Star CCM, COMSOL Multi physics
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 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
Advantages 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 of time for experimental setup
 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.
Aerospace and Aeronautics:
 CFD is used in designing aircraft, optimizing wing shapes, analyzing airflow around aircraft components, and simulating propulsion systems like jet engines.
Automotive Engineering
 CFD helps in designing cars and trucks with improved aerodynamics as presented in the post
 This numerical tool of simulation helps in optimizing engine performance and engine cooling systems
Civil Engineering:
 In civil engineering, CFD is used to study wind flow around buildings (wind load analysis), analyze ventilation systems in structures, and simulate indoor air quality.
Energy and Environmental Engineering:
 CFD is employed to analyze airflow in power plants, optimize wind turbine designs, study pollutant dispersion in the atmosphere, and simulate combustion processes in furnaces and engines.
 CFD analysis of the boiler is shown below for the temperature for Turbulent Combustion of a coalfired boiler
 The first step of CFD is to Identify the flow physics involved in any system
 turbulent flow
 heat transfer (conduction, convection, and radiation)
 Multiphase flow (particle and flue gas)
 Reactive flow (combustion)
 To find the impact of Smoke and pollution level: Flow pattern for an oil fire
Biomedical Engineering:
 CFD plays a role in studying blood flow in arteries and veins (hemodynamics), designing medical devices like stents and heart valves, and simulating drug delivery mechanisms.
Chemical and Process Engineering:
 CFD is used in designing and optimizing reactors or combustors, simulating mixing processes, analyzing heat transfer in chemical processes, and studying fluid flow in pipelines.
Ocean Engineering:
 CFD helps in designing ships with improved hydrodynamics, studying wave interactions, optimizing offshore structures, and analyzing marine propulsion systems.
Electronics Cooling
 CFD is used to analyze thermal management in electronic devices, such as optimizing heat sinks, evaluating cooling fan performance, and studying airflow in electronic enclosures.
 You can refer to the post on PCM Thermal modeling in this post
HVAC (Heating, Ventilation, and Air Conditioning):
 CFD is employed to design efficient HVAC systems, simulate airflow and temperature distribution in buildings
 CFD modeling is used to optimize ventilation for indoor comfort and air quality
Turbomachinery Design:
 CFD plays a crucial role in designing turbomachinery like pumps, turbines, and compressors, optimizing blade shapes, and analyzing fluid flow within these machines
 The flow pattern for Turbomachinery is determined based on CFD analysis
 The first step of CFD is to Identify the flow physics involved in the pump:turbomachinery, turbulent flow, multiphase flow (water and air)
Classification of Fluid Flow
 Fluid flow is classified based on the 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 following articles for complex flow modeling.
Laminar and Turbulent flow
• In general, the Reynolds number is used to determine the flow whether it is laminar or turbulent

 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 the flow becomes turbulent for the small sectional area, then the 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
 CFD solver can not decide the flow either laminar or turbulent. It solely depends on CFD users to decide the flow type and relevant numerical models.
The 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
Regions of fluid flows: Laminar Transition, and Turbulent 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 a fan, pump, 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.
 For flow through the pump, the impeller rotates and pushes the fluid through the outlet pipe. Due to a change in boundaries, the flow physics and CFD models are also changed compared to simple pipe flow problems
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 with each other.
 Examples of multiphase flows are boiling flow, condensation, bubble column, etc.
 Water evaporation and boiling consist of two phases water and steam
 Condensation consists of two phases water and steam
 To model the flow when two (interpenetrating 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. Combustion is considered a flow with fast reactions. Air, fuel, and ignition source are major elements of combustion.
 In the combustion chamber, the reaction takes place between fuel species and oxygen in air. Products of combustion are released during the reactions.
 Heat is released during combustion. The liquid phase changes to the gas phase by absorbing heat from the combustion. The evaporated fuel species react with oxygen in the surrounding air.
 Solid combustion comprises the pyrolysis stage and devolatilization, a reaction with oxygen
 For details of reactive flow modeling, read the basics of combustion in this website. The majority of flow in combustion is turbulent for good mixing of fuel and air. It can lead to stable flames. Turbulent combustion modeling needs to be studied in detail before CFD modeling.
Transport Equation for CFD Modeling
Physical transport phenomenon
A caricature of physical transport phenomenon:
 Transport of water is used from the source point to the location of fire extinguishing
• CFD users must know the governing equations and the physical meaning of various terms in equations.
1) Conduction (diffusion): need a medium for the transfer of mass, momentum, heat, and species at the molecular level.
2) Advection: need a medium for transfer of mass, momentum, heat, and species at the bulk level.
Convection = Advection + Diffusion
3) Radiation: no need for 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
The general form of governing equation
 The general form of a governing equation in the context of physics and engineering often takes the form of a differential equation.
 The specific form of the equation depends on the physical system and the laws that govern its behavior
The 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).
 The 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 Modelings
 Preprocessing comprises making geometry and meshing
 Solver selection of physical models, governing equations, solution procedures
 Postprocessing of CFD results
 The steps for CFD modelings are given as below:
 Define the objectives of modeling
 Create the geometry or computational domain using the CAD tools like Space claim, solid works
 Create the meshing using meshing software like ANSYS Meshing, Hyper mesh
 Set up the solver and physical models
 Compute and monitor the solution
 Examine and save the results in plots, contours, and flux reports
 Consider revisions to the numerical or physical model parameters, if necessary
CFD Software for Commerical and OpenSource
 The most commonly used CFD software and tools for numerical simulations are shown below
Comprehensive CFD Software

 In the last four decades, CFD solvers have been developed for a wide variety of complex flows like turbulent, multiphase and combustion etc.
 These solvers have been used in many industries and wellpracticed numerical models are available there
 Many industries have compared their plant (practical) data with CFD results
 Example: ANSYS FLUENT, STAR CCM, and OpenFOAM
 Click here : ANSYS_FLUENT_THEORY_GUIDE
Semicomprehensive CFD Softwares


 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 largescale industries
 Example: COMSOL Multiphysics, CONVERGE

Discretization of Computational Domain (Meshing)
Elements of Meshing
Computational is divided into to finite number of volumes or elements in meshing. Meshing consists of the following important identities.
 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. Hence, CFD users has to define the fluid zone for the face
 For, the 3D domain, the face can be a boundary like an inlet, outlet or wall etc.
 All essential faces should be named in meshing before exporting to the CFD solver
 Zone:
 It is a grouping of nodes’ faces and cells
 For, 3D domain, the 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
Element of mesh in CFD model
Methods of Meshing in CFD
 In Computational Fluid Dynamics (CFD), meshing is a crucial step that involves dividing the geometric domain into a finite number of elements to facilitate numerical simulations.
 The quality of the mesh significantly affects the accuracy and efficiency of the CFD simulation.
 Choosing the appropriate meshing technique depends on the specific requirements of the CFD simulation, including the geometry, flow characteristics, and computational resources available.
 A combination of these techniques depends on the desired balance between accuracy and computational efficiency.
 Various meshing techniques are employed to generate meshes suitable for different types of simulations. Here are some common meshing techniques in CFD:
Structured Meshing:

 In structured meshing, the computational domain is divided into logically structured grids, such as quadrilaterals in 2D or hexahedral cells in 3D.
 Regular grid structures simplify the application of numerical methods and are wellsuited for simpler geometries.
Unstructured Meshing:
 Unstructured meshing allows for more flexibility in handling complex geometries by using arbitrary shapes like triangles (2D) or tetrahedra (3D).
 It is particularly useful for simulations involving irregular geometries and complex boundaries.
Hybrid Meshing:

 Hybrid meshing combines elements of both structured and unstructured meshes to take advantage of their respective benefits.
 This approach is often used to address specific challenges in the geometry, such as refining the mesh in certain regions while maintaining a structured grid elsewhere.
Delaunay Triangulation:
 Delaunay triangulation is a technique primarily used for generating unstructured triangular meshes in 2D.
 It ensures that no points in the domain are inside the circumcircle of any triangle, promoting a more uniform distribution of elements.
Advancing Front Method:

 The Advancing Front Method is commonly employed for unstructured mesh generation, particularly in 3D.
 It involves growing the mesh from an initial seed and advancing towards the unmeshed domain while maintaining mesh quality.
QuadDominant Meshing:

 Quaddominant meshing aims to generate meshes with a majority of quadrilateral elements, which can be computationally advantageous for certain simulations.
 It is often used in combination with triangular elements to improve mesh quality.
Anisotropic Meshing:

 Anisotropic meshing involves adapting the mesh resolution based on the flow features and gradients, allowing for higher accuracy in regions of interest.
 It is particularly useful in capturing boundary layer phenomena and other flow gradients efficiently.
Automatic Mesh Refinement:
 Automatic mesh refinement techniques dynamically adjust the mesh resolution based on flow gradients, geometric complexity, or userdefined criteria.
 This allows for efficient allocation of computational resources and accurate capture of flow features in regions of interest.
 Some commercial CFD solvers like CONVERGE and Star CCM generate meshing based on gradient in the domain for optimum computational cost
Boundary Layer Meshing:
 Boundary layer meshing techniques focus on refining the mesh near solid boundaries to accurately capture boundary layer phenomena.
 These techniques typically involve layering cells along the boundary surface and adjusting cell sizes to resolve boundary layer gradients.
Mesh Refinement Techniques:

 Adaptive mesh refinement involves dynamically adjusting the mesh density based on the solution or gradient, focusing computational resources where they are needed most.
 Local refinement can be applied to resolve features like shocks or boundary layers with higher accuracy.
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 the CFD domain, the next step is to divide the domain into a finite number of volumes or cells 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 the mesh should be below 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
Meshing Thumb rules

 Highquality mesh and correct numerical schemes mean good convergence of residuals
 For ANSYS FLUENT: Orthogonal quality = 1 – cell skewness
 For complete Hexahedral elements, skewness is zero. It is a nonzero value for inclined faces
 Try to keep skewness below 0.98 so that the CFD simulation will not diverge due to unbounded solutions
 Mesh quality can be improved by converting the tetrahedral cells to polyhedral cells in CFD solvers (ANSYS FLUENT, Star CCM)
 For poorquality mesh, the solutions can be obtained by controlling the relaxation factor or selecting nearwall gradient schemes
 Try to keep at least threeelement in narrow gaps which is a shadow or wake region.
 Add more elements (at least 10 cells) in highvelocity fluid zones to better numerical convergence
 Smoothness (change in size):
 Sudden jumping in the mesh should be avoided in the high gradient region (like inlet and nearwall)
 A growth factor of 1.2 is recommended between adjacent cells
 Smoothness in the mesh is very essential near wall boundaries.
 Aspect ratio:
 Select it as per the need for simulation
 In the direction of the high gradient region, the aspect ratio should be less ( < 20) for laminar and turbulent flow (RANS Modeling).
 A very low aspect ratio of 23 is needed for multiphaseseparated flows and large eddy simulations (LES)
 In the direction of no gradient region, a higher aspect ratio ( ~ 250) is enough to reduce the mesh size.
How to Check Mesh Quality for CFD Simulations
 Checking mesh quality is an essential step in Computational Fluid Dynamics (CFD) simulations, as it directly affects the accuracy and reliability of the results.
 Remember that mesh quality checks should be performed before starting the simulation, and if you find issues, it’s best to correct them and regenerate the mesh as necessary. A goodquality mesh is a fundamental prerequisite for obtaining reliable and accurate CFD results
 Poor mesh quality can lead to inaccurate solutions, convergence issues, and increased computational costs. There are several metrics and tools available to assess mesh quality.
 Here’s a general guide on how to check mesh quality in CFD simulations:
Mesh Visualization

 Start by visually inspecting the mesh to get a rough idea of its quality.
 Use your CFD software’s mesh visualization tools to display the mesh in 2D and 3D.
 Look for irregularities, poorly shaped elements, skewed faces, abrupt changes in element size, etc.
Mesh Statistics:

 Most CFD software provides mesh statistics that give you quantitative information about the mesh quality.
 Look for metrics such as minimum and maximum element volumes, aspect ratios, skewness, orthogonality, etc.
Aspect Ratio

 The aspect ratio is the ratio of the longest edge to the shortest edge of an element.
 High aspect ratios can cause accuracy issues, especially in boundary layers.
 A wellshaped element should have an aspect ratio close to 1.
Skewness

 Skewness measures the deformation of an element from its ideal shape (usually an equilateral triangle or cube).
 High skewness can adversely affect solution accuracy and convergence. Aim for low skewness values.
 Some CFD meshing tools create polyhedral cells to reduce skewness and overall mesh count.
Orthogonality

 Orthogonality measures the angle between the face normal and the cell face.
 Low orthogonality can lead to interpolation errors and reduced solution accuracy, particularly in finitevolume methods. Higher orthogonality is desired.
Mesh Resolution:

 Ensure that the mesh is adequately refined in areas where significant flow gradients or complex flow features are expected.
 Use mesh refinement techniques such as adaptive meshing or multizone meshing to improve resolution where necessary.
Y+ Value for Turbulent flow

 For turbulence modeling using the wall functions approach, it’s crucial to check the y+ value.
 The y+ value represents the distance of the first cell center from the wall in wall units.
 Maintaining y+ in an appropriate range (typically < 5 for boundary layer flows) ensures accurate wall modeling. It is between 30 and 200 for highspeed flows.
 High mesh density in the normal direction of the wall need to given for resolution of turbulent boundary layers
 To save computational cost for resolution of viscous sublayer, salable wall functions are used during turbulent simulation
Mesh Sensitive Study:

 Perform a mesh convergence study to determine the sensitivity of your results to mesh refinement.
 This involves running simulations with progressively finer meshes and comparing the results.
 Convergence is achieved when further mesh refinement does not significantly change the results.
Boundary Layer Resolution

 Ensure that the boundary layer is adequately resolved with a sufficient number of cells in the region near the walls.
 This is crucial for capturing accurate boundary layer behavior.
Mesh Quality Improvement Tools

 Based on the identified mesh quality issues, you may need to clean up or remesh certain regions of your domain to improve mesh quality.
 Many CFD software packages like ANSYS FLUENT offer mesh improvement tools to help with this process by removing highly skewed cell
 Maximum skewness should be 0.97. You can simulate but you need to reduce the underrelaxation factors for pressure, momentum, and turbulent quantities.
Discretization Methods for CFD
 The governing equations 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 discrediting with a numerical method, we can compute values based on initial and boundary conditions.
 Discretization converts the transport equation (mass, momentum, and energy) in the partial differential form to a 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 with 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 grids (equal size of meshing).
 This method is based on Taylor’s series of differentiation. Finite difference methods for spatial derivatives with different order of accuracies is obtained using Taylor expansions like first or secondorder upwind difference scheme (UDS), central differences schemes (CDS), etc.
 The FDM is difficult to use for nonuniform grids. 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 discretized differential 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 the computational domain
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 discretization involve 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, Open FOAM)
 For Refere ANSYS FLUENT User Guide: FLUENT_NUMERICAL_ METHODS
 For OpenFOAM, click here: OpenFOAM_Numerical_Scheme
Finite Element Methods (FEM)
 This numerical method is based upon a piecewise 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 the COMSOL CFD solver, click here: COMSOL_NUMERICAL_METHOD
 Finite Difference Method (FDM) is a numerical technique for solving partial differential equations (PDEs) that arise in various fields of science and engineering.
 COMSOL Multiphysics is a commercial software platform used for modeling and simulating physical systems, which includes tools for solving PDEs through various numerical methods, including the Finite Element Method (FEM), Finite Volume Method (FVM), and Finite Difference Method (FDM)
Which method (FDM/FEM/FVM) is the best suited for CFD Modeling?
 The comparison of the three methods is not straightforward due to differences 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 research scholars adopt the FVM method for numerical modeling of fluid flows.
 The FVM easily provides better conservation properties for mass, momentum, and energy, If you are interested in deciding which method is more suitable, then go through the literature 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 directiondependent Øe
 This approximation is of firstorder (O(Δx))
 Numerically diffusive
• This interpolation is of second order, O(Δx2)
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
Gridbased 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 is located in the cell centers, velocities at cell faces.
 Strong coupling between the velocities and pressure helps to avoid oscillations
 Another 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 pressurevarying fields
C) PISO: Pressure–Implicit with Splitting of Operators (PISO)
 PISO consists of one predictor step and two corrector steps
 This method 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 algorithms refer the post.
 simpleFoam: based on SIMPLE algorithm
 pisoFoam: based on PISO formulation
 pimpleFoam: a 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 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 conditions: noslip (fluid is at rest at the wall) or freeslip (no frictional losses at the boundary).
 Wall can be slip or noslip type
 A wall can be stationary or moving wall (like a 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 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. The 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: Laminar Flow Through the Circular Pipe
Performing a Computational Fluid Dynamics (CFD) analysis of laminar flow through a circular pipe involves using specialized software like ANSYS Fluent, COMSOL, OpenFOAM, or other CFD tools to model and simulate the fluid flow in the pipe.
Here’s a stepbystep guide on how to set up and conduct a CFD analysis for laminar flow in a circular pipe:
Geometry and Meshing
 Create a 3D model of a circular pipe with appropriate dimensions.
 Generate a mesh that discretizes the pipe geometry. For laminar flow, a structured mesh is often used, but unstructured meshes can also be employed. Ensure a finer mesh near the pipe walls to capture boundary layer effects
 Meshing of Geometry:
 Determine the mesh parameters, such as element size, growth rate, and the number of divisions for different regions of the pipe.
 Choose the meshing method appropriate for your simulation. There are two primary methods: structured and unstructured meshing.
 Structured Meshing is used in simulation
 This method is suitable for simple geometries and is often used for laminar flows
 It involves creating a gridlike structure with uniform or structured element sizes within the pipe.
 Meshing a pipe in a Computational Fluid Dynamics (CFD) simulation is a crucial step to accurately capture the flow characteristics. The goal is to create a mesh that adequately represents the geometry and fluid domain while being efficient in terms of computation. Here are the steps to mesh a pipe:
CFD Solver Set Up
Define the Physics:
 Specify the type of fluid you are simulating and its properties (density, viscosity).
 Set up the laminar flow physics.
 In most CFD software, you can specify laminar flow using the appropriate turbulence model.
 For laminar flow, this typically means using the “laminar” option.
 Specify the governing equations for the flow. In the case of laminar flow, you are solving the steadystate incompressible NavierStokes equations. These equations describe the conservation of mass and momentum in the fluid.
Boundary Conditions
 Define the boundary conditions for the problem, including the inlet and outlet conditions.
 For laminar flow, you usually specify a known velocity at the inlet and either a fixed pressure or outflow condition at the outlet.
 Physical Model: Viscous Laminar
 Boundary Conditions
 Inlet: fixed mass flow rate or velocity
 Outlet: outflow (zero gradient for velocity)
Solver Settings:
 Choose a solver appropriate for your problem. Laminar flow problems are typically wellbehaved and can often be solved using pressurebased solvers. Configure the solver settings, such as convergence criteria, time step (if transient), and numerical schemes.
 Pressure Velocity Coupling: SIMPLE
 Numerical Discretization Schemes
 Pressure: Standard
 Momentum: QUICK
 Convergence criterion: 10^06
Initialization:
 Provide an initial guess for the flow field. For laminar flows, a uniform velocity field is often a good starting point.
Run the Simulation:
 Start the simulation and let the CFD software iterate to reach a converged solution.
 Monitor the convergence and adjust settings as necessary.
PostProcessing:
 Once the simulation is complete, you can visualize and analyze the results.
 Common postprocessing tasks include velocity and pressure profiles, shear stress distribution, and streamlines within the pipe.
CFD Results and Analysis
 Compare your simulation results with experimental data or analytical solutions if available to validate your CFD model.
 It’s important to note that the accuracy of the results depends on the quality of the mesh, appropriate boundary conditions, and accurate modeling of the fluid properties.
 Additionally, always ensure that the software you are using is properly validated and that you have a good understanding of the underlying physics to set up the simulation correctly
 Qualitative results: Contours and velocity vectors
Validation of CFD Results
 Comparison of CFD results with theory/ experiment
 To get the credibility of CFD models, results obtained for numerical simulations need to be validated against theoretical or experimental data
 In the following figure, the pressure drop and fully developed velocity profiles are the same with those of CFD simulations and theoretical data laminar flow through pipe
Conclusion
 Computational Fluid Dynamics (CFD) is a flow prediction technique
 It is useful for design and optimization of problems in fluid mechanics, heat transfer, multiphase flows
 Due to the 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.
Respected sir,
I have used the CFD modelling study materials in ” https://cfdflowengineering.com/basicsofcfdmodelingforbeginners/” published by you. They were of great help in my preparation for the final year UG project. Thank you for the great resource. Also, I have seen a comment by you on the page with your mail ID to get in touch with you for more study resources on the same. It would be very kind of you if you could share the resources with me.
Thanking you in anticipation.
–
Priyanka Rajeev
UG Student
Dept. of Mechanical Engineering
National Institute of Technology Puducherry
it is interesting ppt for beginners to have some hints about CFD techniques to solve their complex problems which cannot be addressed /touch with practical labs so far now.